MASARYK UNIVERSITY
Faculty of Science
Department of Condensed Matter Physics
Exotic Magnetism
in Relativistic Transition Metal Compounds
Habilitation thesis
Brno, 2020 Jiˇr´ı Chaloupka
Overview of the thesis
The thesis reviews a decade of my activities in the field of quantum magnetism of transition metal
compounds containing heavier transition metal ions where a strong spin-orbit coupling leads to
an on-site entanglement of spin and orbital degrees of freedom. Under certain circumstances, this
may give rise to an exotic magnetic behavior, very different from the conventional Heisenberg
magnets. The main focus is put on two kinds of materials – iridium Ir4+
and ruthenium Ru3+
compounds described by effective spin-1
2
models with bond-selective anisotropic interactions and
on ruthenium Ru4+
oxides realizing soft-spin physics due to the spin length fluctuating between
spinless and spinful states.
The study of such materials is at present not motivated by immediate technological applications
but more fundamentally, aiming at the understanding of the basic processes driving the
magnetism of these compounds and at the explanation of the experimental observations which
presents a challenge per se. The potential utilization of the new unusual mechanisms of spin
behavior in condensed matter encountered here is envisioned in a more distant future. Here,
“topological quantum computing” may serve as an example of a particularly attractive keyword
that is often used in this context.
When thinking about the structure of the thesis, I took into consideration various (sometimes
conflicting) requests that appeared, as well as my limited enthusiasm to work on extensive texts.
The latter was finally compensated by the convenience of having a text that may deliver the
necessary knowledge to the students I supervise. The resulting thesis is composed of several
parts with varying level of difficulty and the amount of details:
(i) The first part (Sec. 1) contains a “popular” introduction – a description of the emergence
of magnetic models in insulating transition metals oxides (TMO) and a brief discussion of their
properties targeted to a broader audience and characterized by a rather colloquial style. In the
very end of this part I have tried to give a few hints on the features that make the magnetism of
above iridates and ruthenates so special.
(ii) The following part including Secs. 2.1-2.4 is supposed to serve as a supplementary material for
a graduate-level course on strongly correlated electron systems to be started at our faculty next
year and hence may be easily skipped by expert audience. Here a lot of technical details are given
and I do not avoid some illustrations of concepts from basic solid-state courses (such as the tightbinding
approximation) as it often happens that the students are lacking a solid understanding of
those and yet another exposition may be quite helpful. Being conceived as a study material, this
part systematically guides the reader through a derivation of an effective model with localized
degrees of freedom. It starts with a description of single-electron orbitals in a TMO crystal
environment (Sec. 2.1), followed by a discussion of intraionic interactions and multiplet structure
stemming from Coulomb repulsion of the valence electrons and spin-orbit coupling (Sec. 2.2).
After the preparatory sections concerning the ionic states, in Sec. 2.3 the ions will get connected
by electronic hopping captured in tight-binding approximation. The second part culminates by
introducing the Mott transition and illustrating the emergence of an effective model along with
a discussion of the virtual processes leading to superexchange interactions (Sec. 2.4).
(iii) The third part focuses on the two particular areas of my research – the pseudospin-1
2
materials
with dominant Kitaev interactions (Sec. 3) and the soft-spin singlet-triplet systems (Sec. 4).
Rather than a detailed review of the already substantial body of literature on the subject (in
particular in the first area where many reviews appeared during the last few years), the goal is
to show the route toward the relevant microscopic models for these materials and to give a brief
introduction on their physical properties emphasizing the peculiar features not present in the
i
conventional magnets. Here I have also used the opportunity to partially collect the results that
sometimes appear rather scattered in the attached papers and give them a coherent presentation.
(iv) Finally, the closing and in fact the largest part contains reprints of the relevant papers and
a brief summary of their content including remarks on the “historical” context (Secs. 5 and 6).
In conformance with the Masaryk University habilitation rules, in the following list of the
attached papers I specify my contributions in terms of both quantity and content. The papers
are listed in the same order as appearing in Secs. 5 and 6.
Publication Share Content contribution
1. J. Chaloupka, G. Jackeli, and G. Khaliullin,
Kitaev-Heisenberg Model on a Honeycomb
Lattice: Possible Exotic Phases in Iridium
Oxides A2IrO3, Physical Review Letters
105, 027204 (2010)
33% construction of the phase diagram
and exploration of its phases via
exact diagonalization (ED) ap-
proach
2. J. Chaloupka, G. Jackeli, and G. Khaliullin,
Zigzag Magnetic Order in the Iridium Oxide
Na2IrO3, Physical Review Letters 110,
097204 (2013)
33% construction of the phase diagram
via ED, calculations of the magnetic
susceptibility using finitetemperature
Lanczos method, fitting
the experimental data
3. S. H. Chun, J. W. Kim, J. Kim,
H. Zheng, C. C. Stoumpos, C. D. Malliakas,
J. F. Mitchell, K. Mehlawat, Y. Singh,
Y. Choi, T. Gog, A. Al-Zein, M. M. Sala,
M. Krisch, J. Chaloupka, G. Jackeli,
G. Khaliullin, and B. J. Kim, Direct evidence
for dominant bond-directional interactions in
a honeycomb lattice iridate Na2IrO3, Nature
Physics 11, 462 (2015)
6% simulations of the diffuse magnetic
X-ray scattering using ED,
contributions to the theoretical
model development
4. J. Chaloupka and G. Khaliullin, Hidden symmetries
of the extended Kitaev-Heisenberg
model: Implications for the honeycomblattice
iridates A2IrO3, Physical Review B
92, 024413 (2015)
75% development of the general
method to identify hidden symmetries
and its application to
the particular case studied, other
calculations used to interpret the
experimental data, writing most
of the manuscript
5. J. Chaloupka and G. Khaliullin, Magnetic
anisotropy in the Kitaev model systems
Na2IrO3 and RuCl3, Physical Review B 94,
064435 (2016)
75% calculations via the spin-coherent
state method (including its development),
part of the smaller supportive
calculations, model analysis
in the context of experimental
data, writing most of the
manuscript
ii
6. D. Gotfryd, J. Rusnaˇcko, K. Wohlfeld,
G. Jackeli, J. Chaloupka, and A. M. Ole´s,
Phase diagram and spin correlations of
the Kitaev-Heisenberg model: Importance
of quantum effects, Physical Review B 95,
024426 (2017)
25% methodological developments in
the CMFT part, calculations
of dynamic spin susceptibility,
preparation of parts of the
manuscript
7. J. Rusnaˇcko, D. Gotfryd, and J. Chaloupka,
Kitaev-like honeycomb magnets: Global
phase behavior and emergent effective models,
Physical Review B 99, 064425 (2019)
40% symmetry analysis and derivation
of emergent effective models,
part of the phase-diagram calculations,
writing about half of the
manuscript
8. J. Kim, J. Chaloupka, Y. Singh, J. W. Kim,
B. J. Kim, D. Casa, A. Said, X. Huang,
and T. Gog, Dynamic Spin Correlations in
the Honeycomb Lattice Na2IrO3 Measured by
Resonant Inelastic x-Ray Scattering, Physical
Review X 10, 021034 (2020)
40% all the theoretical parts that constitute
the indicated share of the
manuscript – extensive numerical
simulations of the RIXS spectra
to identify the experimentally relevant
parameter window in the
global phase diagram, model interpretation
of the pseudospin dynamics
in the relevant regime
9. H. Liu, J. Chaloupka, and G. Khaliullin,
Kitaev Spin Liquid in 3d Transition Metal
Compounds Physical Review Letters 125,
047201 (2020)
25% ED calculations of the phase diagrams
and spin excitations for
the microscopically obtained parameters
of the extended KitaevHeisenberg
model
10. J. Chaloupka and G. Khaliullin, Spin-State
Crossover Model for the Magnetism of Iron
Pnictides, Physical Review Letters 110,
207205 (2013)
75% development of the theoretical
model and its extensive analysis
by a large number of methods,
major part in the manuscript
preparation
11. J. Chaloupka and G. Khaliullin, DopingInduced
Ferromagnetism and Possible
Triplet Pairing in d4
Mott Insulators,
Physical Review Letters 116, 017203 (2016)
67% analytical and numerical exploration
of the developed theoretical
model – magnetic phase diagram,
calculations of the dynamic
spin response and superconducting
pairing, major part in the
manuscript preparation
12. A. Jain, M. Krautloher, J. Porras,
G. H. Ryu, D. P. Chen, D. L. Abernathy,
J. T. Park, A. Ivanov, J. Chaloupka,
G. Khaliullin, B. Keimer, and B. J. Kim,
Higgs mode and its decay in a twodimensional
antiferromagnet, Nature
Physics 13, 633 (2017)
15% calculations of the dynamic magnetic
spectra including the interplay
of the Higgs mode and twomagnon
continuum, part of the
experimental data analysis
iii
13. S.-M. Souliou, J. Chaloupka, G. Khaliullin,
G. Ryu, A. Jain, B. J. Kim,
M. Le Tacon, and B. Keimer, Raman
Scattering from Higgs Mode Oscillations
in the Two-Dimensional Antiferromagnet
Ca2RuO4, Physical Review Letters 119,
067201 (2017)
40% fitting the temperature dependent
Raman spectra to extract the
magnetic response, model calculations
and interpretation of the
data, writing a large part of the
manuscript
14. J. Chaloupka and G. Khaliullin, Highly frustrated
magnetism in relativistic d4
Mott insulators:
Bosonic analog of the Kitaev honeycomb
model, Physical Review B 100, 224413
(2019)
75% all the theoretical developments
and model calculations apart from
the derivation of the model itself,
preparation of most of the
manuscript
iv
Contents
1 Introduction 1
1.1 How the transition-metal compounds are built up . . . . . . . . . . . . . . . . . . . . 1
1.2 Emergence of an effective spin system . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Spin systems at a glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Kitaev systems – a quick introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Soft-spin systems – a quick introduction . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Effective models with localized degrees of freedom 13
2.1 Orbital splitting in a crystal environment . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Local correlations and multiplet structure of transition metal ions . . . . . . . . . . . 20
2.3 Electronic hopping and tight-binding approximation . . . . . . . . . . . . . . . . . . . 37
2.4 Mott limit and interactions emerging from residual hopping . . . . . . . . . . . . . . 44
3 Kitaev materials 55
3.1 Microscopic origin of the Kitaev interaction . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Extended Kitaev–Heisenberg model and its phase diagram . . . . . . . . . . . . . . . 60
3.3 Specific features of the zigzag phase in the regime of the dominant Kitaev interaction 67
4 Soft-spin systems 71
4.1 Singlet-triplet model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Revisions of the model to reflect tetragonal splitting . . . . . . . . . . . . . . . . . . 74
4.3 Magnetic order due to triplon condensation . . . . . . . . . . . . . . . . . . . . . . . 77
4.4 Excitation spectra probed by neutron and Raman scattering . . . . . . . . . . . . . . 80
5 Papers addressing the Kitaev–Heisenberg model 85
6 Papers addressing the singlet–triplet model 207
7 Conclusions and outlook 257
Bibliography 259
v
1 Introduction
Among the various classes of solid state materials, the transition metal compounds are probably
the richest one in terms of diverse physical phenomena as well as their complexity [1–3]. To
name a few prominent examples, we mention high-Tc cuprate superconductors [4, 5], colossally
magnetoresistant manganites [6], multiferroics with strong magnetoelectric coupling [7], or the
more recently studied exotic properties of materials with large spin-orbit coupling [8] including
quantum spin liquid behavior [9,10]. The complexity of the physics of transition metal compounds
stems from several key aspects:
ˆ Geometry – many possible crystal lattices including those enforcing reduced dimensionality
ˆ Localized versus itinerant behavior – competition of delocalization of the electrons preferring
metallic state and strong Coulomb repulsion among electrons at the individual ions that
supports insulating behavior
ˆ Multiplet structure – we can experience many-body physics at the level of the individual
transition metal ions – electrons residing in various atomic orbitals of the valence shell are
subject to local correlations generated by Coulomb repulsion and Pauli principle
ˆ Inter-ionic interactions – the electronic connection between the relevant ions may attain
many forms due to the various bonding geometries and several valence orbitals involved.
Combined with local correlations, non-trivial interactions emerge, giving rise to the multifaceted
behavior of transition metal oxides.
ˆ Feedback of the lattice – the lattice does not merely provide a rigid playground for the electrons
but may actively participate via electron-phonon coupling/crystal field effects. The
interplay of the electronic degrees of freedom with the lattice may generate new phenomena
such as Jahn-Teller effect.
In this and the following part of the thesis, we try to illustrate most the above points. Here in
Sec. 1, we stay at a basic level and focus on spin systems as a prototype example of systems
with localized degrees of freedom. After this initial exposition, Sec. 2 brings up the omitted
“details”, e.g. those related to the orbital structure. To this end, we will follow the standard
scheme used when deriving an effective model for a strongly correlated electronic system with
localized degrees of freedom and discuss the ingredients one mixes in.
1.1 How the transition-metal compounds are built up
The key information about a transition metal compound is provided by its chemical composition
and the type of the crystal lattice formed by the ions. Apart from the transition metal elements
(d-elements of the periodic table), the chemical formulas include electronegative p-elements (typically
oxygen, but also other chalcogens, as well as halogens and pnictogens) and electropositive
elements from the left part of the periodic table (alkali metals, alkaline earth metals, and rareearth
metals). The contrast in electronegativities has consequences for the valence/electronic
occupation of the individual ions but for now let us focus solely on the structure.
Several examples of crystal lattices demonstrating the structural features of transition metal
compounds are shown in Fig. 1. A common element of the selected sample lattices is a MO6
octahedron where the central transition metal ion M is surrounded by an octahedral cage of
oxygen anions O, often called ligands. Though most frequently met, this is not the only possibility,
the ligands surrounding the transition metal can be found also to form a tetrahedral cage, a
trigonal bipyramid etc.
1
2 1.1 How the transition-metal compounds are built up
b a
c
O72IrR2IrA O32
La Sr2−x xCuO4 Sr2CuO3
O6Ir2A
A
OIrA
R
O Ir
1D chain
2D plane
(d)(c)
(a) (b)
OCu
La/Sr
Fig. 1: Sample crystal lattices of transition metals oxides: (a) Unit cell of a high-Tc superconductor
La2−xSrxCuO4 showing the octahedral cages of oxygen ions around the Cu ions (left) and the CuO2
planes (right) as the basic constituents of the crystal lattice. Geometrically, the individual layers of
corner-shared CuO6 octahedra correspond to square lattices of copper ions with 180◦ Cu–O–Cu bonding.
(b) Unit cell of another cuprate material Sr2CuO3 that contains not only CuO2 planes in its structure
but also quasi-1D chains formed by copper and oxygen ions. (c) Layered crystal structure of honeycomb
iridates A2IrO3 where the honeycomb lattice of the iridium atoms results from placing the IrO6 octahedra
into edge-shared configurations. The side view on the left shows weakly bound AIr2O6 layers separated
by A ions. In the top view, the honeycomb lattice of Ir ions as well as the 90◦ M–O–M bonds are
clearly visible. (d) Pyrochlore-lattice iridate R2Ir2O7 (adapted from [8]).
The relevant transition metal ions may form various kinds of lattices depending on the way
their cages are attached to each other. Figure 1(a) and (b) illustrate the case of corner-shared
octahedra using two cuprate materials. Due to their ligand cages, the transition metals are
connected indirectly via oxygens in 180◦
M–O–M bonding geometry. The other ions, here La
and Sr, fill in the voids and hold the structure together. The octahedra are not equally stacked
1.2 Emergence of an effective spin system 3
in all directions. Compared to M–O–M bonds, the other links are much weaker so that one
can observe effectively isolated structures with reduced dimensionality which are composed of
strongly bound Cu and O ions. One can thus talk about weakly coupled square-lattice CuO2
planes [Fig. 1(a)] or even about almost decoupled one-dimensional chains [Fig. 1(b)].
A different situation occurs when the octahedra share their edges [see Fig. 1(c)]. The bonds
are now due to 90◦
M–O–M bridges but the direct M–M bonding becomes also important. Edgesharing
of the octahedra implies a possibility of M–M bonds taking non-rectangular 60◦
or 120◦
angles so that for example triangular or honeycomb two-dimensional networks of transition metal
ions can be generated. Figure 1(c) shows an example of a honeycomb iridate A2IrO3 where A is
an alkali metal Na or Li. Here the octahedra form honeycomb layers intercalated by A ions that
appear also in the voids of the honeycomb layers. Interestingly, a similar honeycomb structure
is displayed by the ruthenium halide α-RuCl3 but it is completely missing the A ions. These
honeycomb compounds will be discussed in Sec. 3 as examples of so-called Kitaev materials.
Finally, as an example of a truly three-dimensional structure, in Fig. 1(d) we present pyrochlore-lattice
iridates R2Ir2O7 with R being a rare-earth element such as Nd, Sm, or Yb. The
Ir ions in these compounds create a three-dimensional network of tetrahedra connected by their
corners. A closer inspection reveals Ir–O–Ir bonds at an angle of approximately 120◦
in this case.
1.2 Emergence of an effective spin system
The different levels of electronegativity cause an electron transfer among the elements constituting
the transition metal compound. The formal valence can be easily counted when assuming filled
valence shells by all the elements other than transition metals. Each oxygen ion is expected to
attract two electrons making it O2−
whereas the electrons are donated by ions such as Na+
, Sr2+
or La3+
. Adopting these rules, we find for the compounds shown in Fig. 1: La3+
2−xSr2+
x Cu2+x
O2−
4 ,
Sr2+
2 Cu2+
O2−
3 , A+
2 Ir4+
O2−
3 , and R3+
2 Ir4+
2 O2−
7 , respectively. Even though such a formal valence
should be taken as indicative only, often it provides a reasonable starting estimate.
Being stripped of their outer s electrons and partly d electrons by the more electronegative
p-element ions, the transition metal ions expose their partially filled d shells. The extent of
the wave functions of the valence d orbitals turns out to be quite suitable for interesting things
to happen. Staying on a qualitative level of discussion, we may evaluate this extent from two
viewpoints: On one hand, d orbitals are small enough so that the Coulomb repulsion among the
valence electrons is sufficiently strong, leading to strongly correlated behavior of those electrons.
On the other hand, their extent still allows for a large electronic contact between the neighboring
ions which gives rise to significant interactions.
We thus have two opposing mechanisms simultaneously at play – a tendency of the electrons
to delocalize (travel through the crystal) and their Coulomb repulsion which tries to keep them
apart. Depending on the balance between the two, we may end up with a correlated metallic
state [Fig. 2(a)] or with an insulating state [Fig. 2(b)]. Note that the latter so-called Mott
insulator arises due to electron-electron interactions. This is very different to conventional band
insulator, where the insulating state appears when properly filling up the bands generated by
periodic crystal potential. In fact, simple band theory would easily predict a Mott insulator to
be metallic instead.
Both mechanisms are captured in their simplest form by a prototype model – single-band
Hubbard model. It considers a single orbital per site that may be, according to Pauli principle,
occupied by up to two electrons. However, a simultaneous presence of two electrons activates the
Coulomb repulsion denoted as Hubbard U. The “electronic contact” is described by an amplitude
t of the hopping process where an electron moves to a neighboring site. In formal terms, the
4 1.2 Emergence of an effective spin system
d1
d1
t J
d1
d1
d2
d0
U
t Heff = J S1 · S2
J =
4t2
U
(b)
Mott insulator
(c)
Hubbard model
(d)
effective spin model
(a)
correlated metal
Fig. 2: Schematic pictures of (a) correlated metallic state and (b) Mott-insulating state for a system
with one active orbital per site and half-filled case (on average we have one electron per site out of
two possible). The arrows indicate the spin of the electrons. (c) Virtual process in the single-orbital
Hubbard model that generates the exchange interaction on a bond. (d) An effective spin model that
reproduces the low-energy behavior of the Hubbard model at the particular bond.
corresponding Hamiltonian reads as
H = −t
ij σ
(c†
iσcjσ + c†
jσciσ) + U
i
ni↑ni↓ , (1.1)
where ciσ is the usual fermionic operator annihilating an electron of spin σ at site i, the first
sum runs through all nearest-neighbor bonds ij , and the second one in effect counts the doubly
occupied sites that are penalized by the energy U. In the limit of strong repulsion compared to
the electronic hopping amplitude, U t, and for the total number of electrons being equal to
the number of sites, double occupations of the sites are avoided and the system becomes a Mott
insulator depicted in Fig. 2(b). In the context of the single-band Hubbard model, we talk about
half-filled situation (one half of the maximum number of electrons are present), the real materials
with an integer formal valence of d ions that realize Mott insulating state are usually termed
“undoped”. By introducing extra electrons or holes (missing electrons), the material becomes
“doped” and typically turns into a metal at a sufficient doping level.
In the Mott insulating limit of a half-filled system, one does not need to give the positions
of electrons, the state is sufficiently described by specifying their spins only. We arrive at an
example of a model with localized degrees of freedom, here a spin model. Even though the
positions are frozen, there is still some “life” left. The spins are interacting and have a particular
ground-state configuration and a specific low-energy dynamics. The interactions are generated
by the virtual processes of the type shown in Fig. 2(c). An electron for a short while visits its
neighbor, creating a virtual excited state with an energy U, and by the second hopping in the
opposite direction, the one-electron-per-site rule is restored. This process and the resulting small
kinetic energy gain are possible only when the two electrons can actually meet in the same orbital
– they need to be in a singlet state. In a way, the process can be understood as a formation of
a bonding orbital occupied by the two electrons in a singlet state, which is caused by a weak
hybridization of the localized ionic orbitals. The preference of singlets of neighboring spins is
the desired spin-spin interaction. More quantitatively, the virtual processes can be accounted for
1.3 Spin systems at a glance 5
within a second-order perturbation theory giving rise to the Heisenberg Hamiltonian
Heff = J
ij
Si · Sj (1.2)
with the exchange constant J = 4t2
/U and S = (Sx
, Sy
, Sz
) being the spin operators describing
spin-1
2
of the localized electrons.1
The label “exchange” is explained in a pictorial way by Fig. 2(c)
where the result of the virtual process is a configuration with the two spins exchanged. Formally,
the operator Si·Sj can be also written as 1
4
(2Pij −1) where Pij is Dirac permutation operator that
exchanges the quantum states of two particles [11]. For the effective model to adequately capture
the low energy physics of the original Hubbard model [H of Eq. (1.1)], the energy scale of J needs
to be well below U which is satisfied in the assumed U t limit. More detailed illustration of the
emergence of the effective spin model will be given in Sec. 2.4 based on numerical simulations of
the Hubbard model. In the next section we will briefly review the basic features of spin models.
1.3 Spin systems at a glance
As introduced in the previous section, a spin system is a system of spins residing at sites of a given
lattice that are subject to spin-spin interactions. Despite the apparent simplicity compared to
the underlying real material, there is still a large variability of the spin systems. One can consider
various lattice geometries, interactions beyond nearest neighbors, anisotropic interactions of the
general matrix form α,β=x,y,z JαβSα
i Sβ
j replacing the isotropic scalar product in Eq. (1.2), or
more than two-site interactions. For simplicity, here we limit ourselves to the isotropic Heisenberg
Hamiltonian as appearing in Eq. (1.2). The exchange constant J came out positive for the
particular mechanism discussed above but in principle it may also take a negative value favoring
a triplet on the bond.
The general tendency of physical systems is an evolution towards equilibrium driven by minimization
of the free energy F = U −TS. This contains two competing contributions – the internal
energy U and the entropy term −TS. The balance between the two is determined by the temperature.
In the low-temperature regime where the entropy does not matter, the bond interactions
captured by the Hamiltonian in Eq. (1.2) and contributing to U will be optimized. The coupled
spins will thus try to become aligned in the negative J case or contra-aligned for positive J. On
a square lattice this leads to long-range ferromagnetic (FM) or antiferromagnetic (AF) order presented
as examples in Fig. 3(a) and (b), respectively.2
When measuring the spins or the magnetic
moments, we will find a position-dependent average of the type SR ∼ mR ∼ eiQ·R
where Q is
the characteristic wavevector of the ordering – the ordering vector. For a ferromagnet it is equal
to zero since all the spins point in the same direction. For an antiferromagnet, the spin direction
alternates with the site index n as eiπn
in both x and y directions and the ordering vector is
Q = (π/a, π/a) where a denotes the lattice parameter.3
The corresponding Fourier component
MQ of the magnetization may serve as an order parameter determining the “strength” of the
magnetic order. At elevated temperatures the order becomes gradually weakened by thermal
fluctuations [see Fig. 3(c)]. This may be understood as a consequence of the term −TS that
now prefers a state with higher entropy, i.e. less ordered. At certain critical temperature, the
magnetic order ceases to exist and the system becomes paramagnetic with zero order parameter.
1
It is customary to work with dimensionless spin operators. We will follow this convention and use dimensionless
variants (i.e. divided by ) of all angular momentum operators through the whole text.
2
To be precise, a strictly two-dimensional Heisenberg system considered here would not order at finite temperatures,
however, in reality one encounters quasi-2D systems.
3
The reciprocal space is often measured in units of 1/a, in that case Q = (π, π).
6 1.3 Spin systems at a glance
0
1
2
3
4
(0,0) (π,π)
ω/|J|
q
0.0
0.5
1.0
1.5
2.0
(0,0) (π,π)
ω/J
q
Q = (0,0) Q = (π, π)
MR = MQ eiQ·R
MQ
M
E
x yM
(a) (b) (c)
0
0
TTN
(d) (e) (f)
Fig. 3: (a) Ferromagnetic and (b) antiferromagnetic ordering of spins on a square lattice. The ordering
vector Q is expressed in units of 1/(lattice parameter). (c) Typical temperature dependence of the order
parameter (Fourier component of the magnetization at the ordering vector Q). TN stands for the N´eel
temperature – the critical temperature of an antiferromagnet. (d) Mexican-hat profile of the energy
depending on the magnetization. Blue/red arrows indicate the direction of magnetization changes that
are “for free” or cost energy, respectively. (e) Schematic spin-wave dispersion (energy ω as function of
the wavevector q) in the case of a ferromagnet on a square lattice. (f) The same for a square-lattice
antiferromagnet.
With the order established, the behavior of the system becomes more collective and specific
modes – wave-like excitations of the spins – can be observed in its low-energy dynamics. For
an isotropic system, the magnetization can take any direction with the same resulting energy
[see Fig. 3(d)], only its length is fixed by energy minimization. Therefore, a global simultaneous
rotation of all the magnetic moments does not cost any energy and the excitations of the system
that are close in nature to such a rotation will be the lowest-energy modes with the dispersion
ω ∼ qα
. More generally, these modes – called Goldstone modes – appear whenever the system
spontaneously breaks a continuous symmetry as our system did when developing long-range
magnetic order with certain magnetization direction. The determination of the rules for the
number and type of the Goldstone modes is a deep theoretical problem [12, 13]. As a result
for our particular model, the Heisenberg ferromagnet shows one Goldstone mode with α = 2 at
Q = (0, 0) while in the Heisenberg antiferromagnet we find two modes with α = 1 at q = (0, 0)
and the ordering Q = (π, π). The overall dispersion of the spin waves for these two cases
is depicted in Figs. 3(e) and (f), respectively. When such a dispersion of spin waves for a
real spin system is resolved experimentally, for example by inelastic scattering of neutrons, it
brings relatively rich information enabling to narrow down the type and strength of the spin-spin
interactions constituting the corresponding spin model.
Nevertheless, such a luxury is often not accessible, neutron experiments frequently suffer
from low intensity, have to be performed on powders instead of monocrystals providing only
direction-averaged information etc., so that alternative ways to get at least some insights into
the spin-spin interactions are needed. A standard tool used to estimate the magnitude of the
1.3 Spin systems at a glance 7
cT
(a) (b)
TN
χ
T0
B
J
χ
χ
χ
||
⊥
T0
χ
−1
Θ
T0 TN
Fig. 4: (a) Schematic temperature dependence of the static spin susceptibility χ = dM/dB for a
ferromagnet. The inset tries to suggest the physical picture behind the curve that is described in the
text. (b) The same for an antiferromagnet where the susceptibility becomes anisotropic below the N´eel
temperature and differs for the field polarized perpendicular or parallel to the magnetization. The inset
shows an extrapolation of χ−1 to get the Curie-Weiss scale Θ.
spin-spin interactions and one of the first experimental probes applied to freshly baked samples
is the temperature dependence of the static spin susceptibility illustrated in Fig. 4. Here one uses
the high-temperature behavior of the spins in homogeneous magnetic field that tries to align them
in parallel fashion. Without the spin-spin interactions the high-temperature spin susceptibility
would approximately follow the Curie law χ ∼ 1/T for isolated moments. This is modified by
the interactions that try to imprint their characteristic correlations onto the partially polarized
spins. The result above the critical temperature is the Curie-Weiss law χ ∼ 1/(T − Θ) where
the Curie-Weiss scale Θ combines the spin-spin interaction parameters in some way. Roughly
speaking, FM J < 0 supports the parallel alignment induced by the magnetic field and leads to
positive Θ (enhances the susceptibility), AF J > 0 works in the opposite way (suppresses the
susceptibility). Going down in temperature, an anomaly at the critical temperature signals a
phase transition into a magnetic ordered state. Unless a more sophisticated fitting is involved,
one typically plots χ−1
(T) and extrapolates the linear part to get Θ as shown in Fig. 4(b).
So far we were naively discussing the spin systems as a set of arrows that want to align and
this is only prevented at high temperatures by thermal fluctuations. However, the situation is
more complex and the magnetic order has to fight with additional enemies. An intrinsic enemy is
the very nature of spin as a quantum object. To see its consequences, we rewrite the Heisenberg
Hamiltonian (1.2) in terms of the spin raising/lowering operators S±
= Sx
± iSy
:
Heff = J
ij
Si · Sj = J
ij
1
2
(S+
i S−
j + S−
i S+
j ) + Sz
i Sz
j . (1.3)
Assuming the AF ordering with the moments along z direction, the last Sz
Sz
part of the interaction
would be completely happy. But more energy can be gained from the resonance of
configurations with flipped spins as sketched in Fig. 5(a). The degree of the ordering is thus
sacrificed for a gain of “kinetic” energy. The processes bringing disorder by misaligning spins
are now of quantum origin and may be thus termed as quantum fluctuations in analogy with the
thermal ones. The severity of quantum fluctuations strongly depends on the spin length (with
larger spins behaving more classically and spin-1
2
being the most “quantum” one), the nature of
the spin-spin interactions and the dimensionality of the system (with lower-dimensional systems
being more susceptible to quantum fluctuations). As an example of the level of quantum fluctuations
found in quantum antiferromagnet, in Fig. 5(b) we sketch an exact ground state of a
8 1.3 Spin systems at a glance
i j |GS = 1√
3
|↑↓↑↓ − 1
2 |↑↑↓↓ − 1
2 |↓↑↑↓ − 1
2 |↓↓↑↑ − 1
2 |↑↓↓↑ + |↓↑↓↑
S+
i S−
jS−
i S+
j
|GS =
(a)
+J
(b)
Fig. 5: (a) Resonant spin-flip processes driven by the part of the Heisenberg interaction that is perpendicular
to the ordered moment direction. Spin-1
2 is considered here so that only two Sz
i eigenstates
exist, those are indicated by up/down arrows. (b) Exact ground state of a Heisenberg square consisting
of four spin-1
2 sites. It can be represented either as a superposition of configurations of properly aligned
or misaligned spins (bottom) or via a resonance of two coverings of the square by singlets of neighboring
spins (right). A singlet state of two spins is marked by a gray oval. An analogous resonance of several
possible coverings is familiar from quantum chemistry of benzene and other aromatic molecules. In that
case coverings by double bonds between carbon atoms are resonating.
square of four Heisenberg-interacting spins. Here the configurations with properly contra-aligned
spins (N´eel configurations) represent only two thirds of the whole state in terms of probability.
Focusing on the topology of the bonds, the square is in fact a 1D system with periodic boundary
conditions. Even though the amount of quantum fluctuations (measured by the energy gain with
respect to the N´eel configuration) is reduced when the 1D system grows in length, the quantum
fluctuations are still strong enough to completely melt the long-range order in the case of a 1D
Heisenberg chain at T = 0.
Sometimes the enemy is more visible – in case of frustrated spin systems one can explicitly see
that the simple orderings of e.g. N´eel type cannot satisfy the spin-spin interactions. An example
of a geometric frustration is presented in Fig. 6(a) where we take a piece of so-called kagome lattice
and try to populate it with AF-interacting spins. This attempt is soon over as we inevitably fail
to make three interacting spins on a triangle mutually antiparallel. The nature of the ground
state of the Heisenberg model on kagome lattice is still a subject of intense studies and it seems
that the long-range order is indeed destroyed by the geometric frustration [14]. This is in contrast
to the triangular lattice which, despite being composed from frustrated triangles as well, shows
?
columnarNeel
Tc
m
0
0
1/2 J J2 1
/
1J 2J
(a) (b)
Fig. 6: (a) Frustration of spins that are subject of AF interaction when placed on a kagome lattice.
(b) Competition of nearest-neighbor and next-nearest-neighbor interaction in J1-J2 model that,
depending on the J2/J1 ratio shows two distinct types of spin ordering separated by a disordered phase.
1.4 Kitaev systems – a quick introduction 9
a long-range order with a “compromise” angle of 120◦
between neighboring spins [15]. Another
case of frustration is demonstrated in Fig. 6(b) for a geometrically non-frustrated square lattice.
Once we activate the next-nearest-neighbor Heisenberg interaction J2 in addition to the nearestneighbor
J1, those two interactions start to compete as they support different spin arrangements.
Depending on the ratio of J1 and J2, this competition leads to a reduction of the magnetic order
strength or – around the balanced ratio J2 ≈ J1/2 – even to a complete absence of any long-range
order [16,17].
1.4 Kitaev systems – a quick introduction
In the end of this section we briefly introduce the spin systems that are the main subject of
the thesis – Kitaev-like and soft-spin systems. Both of them go in some sense far beyond the
concepts of a Heisenberg magnet. Leaving a detailed discussion for the respective sections 3 and
4, here we stay at the “popular” level and address only the essential features that make those
systems special.
The Kitaev-like magnets are strongly frustrated systems of spins residing on honeycomb
lattice that interact predominantly via a particular bond-selective interaction introduced by
Alexei Kitaev in his famous model [18]. In contrast to the isotropic Heisenberg interaction, the
Kitaev interaction picks only one component of the two spins on a bond, i.e. it replaces the
scalar product Si · Sj by for example Sz
i Sz
j . This is not that unique yet, the same kind of
anisotropy can be found in the famous Ising model which is one of the prototype models for
magnetism introduced in early 1920’s [19]. The key point of the Kitaev model, however, is that
the interaction axis is bond-dependent and follows the pattern depicted in Fig. 7(a). Formally,
the Hamiltonian of the Kitaev model can be written as
HKitaev = −Kx
ij x
Sx
i Sx
j − Ky
ij y
Sy
i Sy
j − Kz
ij z
Sz
i Sz
j , (1.4)
where each of the three sums run through the bonds of a particular direction. Often an “isotropic”
version of the model with Kx = Ky = Kz = K is considered. The interaction constant may
take both signs, in fact there is a one-to-one mapping between ferromagnetic (K > 0) and
antiferromagnetic (K < 0) cases. The model can be extended to arbitrary spins but let us limit
our discussion to the spin-1
2
variant proposed by Kitaev.
The honeycomb lattice is a geometrically non-frustrated lattice, the antiferromagnetic Heisenberg
interaction discussed earlier would therefore establish long-range AF order similarly to the
case of the square lattice depicted in Fig. 3(b). The Kitaev interaction, either ferromagnetic or
antiferromagnetic, fails to do so. The reason is its intrinsic frustration stemming from a competition
of the interactions at the three bond directions. As observed in Fig. 7, each site of the lattice
is a member of three bonds exhausting all the possible bond directions. To optimize the energy
of the attached z-bond, both spins it connects should be parallel (for positive K) or anti-parallel
(for negative K) and pointing along the z direction. On the other hand, the y-bond wants them
to point in the y direction, and the x-bond makes yet another request. The three options for
an optimum spin direction are mutually orthogonal. An intuitive compromise might be to align
spins along the (x+y+z)/
√
3 direction. This would bring a classical energy 3× 1
3
1
4
K = 1
4
K which
is the same as if we fully satisfied one bond direction and therefore does not seem promising.
The spins choose a different way how to cope with the inherent frustration of their interactions
and form so-called Kitaev quantum spin liquid. It is a exotic state where nearest-neighbor spins
show well-defined correlations while further neighbors have zero correlations. Remarkably, this
ground state can be obtained by an exact calculation making the Kitaev model one of the few
examples of exactly solvable models in the field of quantum magnetism. Related to the exact
10 1.5 Soft-spin systems – a quick introduction
z
y
x
x
z
y
Ir
O
Na
Sz
S z
Sy
S y
Sx
S x
z
x
y
IrO
Na
(a) (c)(b)
Fig. 7: (a) Honeycomb lattice with three distinct directions of bonds indicated by colors. The Kitaev
interaction between nearest-neighbor spins is of the form Sα
i Sα
j with the active component α = x, y, z
determined by the bond direction. (b) Top view of the honeycomb lattice of edge-shared octahedra in
the iridate Na2IrO3. More details on the structure of the compound can be found in Fig. 1(c). The
main axes of the octahedra labeled by x, y, and z coincide with the spin axes of the Kitaev interaction.
The active one for a particular bond is perpendicular to the bond direction. (c) A closer view on the
Ir-O-Ir bonds with marked square Ir2O2 plaquettes where the main exchange processes happen. The
Kitaev axis for the given bond is perpendicular to the corresponding plaquette.
solution are other fancy features of the model – an extensive number of conserved quantities
(their number grows linearly with the system size), topological quantum order, or fractionalized
elementary excitations that cannot be understood as simple wave-like rotations of the spins like
it is in the case of conventional magnets.
While the original Kitaev model was introduced on purely theoretical grounds in the context
of topological quantum computing, it was the proposal by George Jackeli and Giniyat Khaliullin
[20] of its possible realization in Mott insulators with large spin-orbit coupling – i.e. actual
materials – that attracted the attention of a broader solid state community. The possible realization
of a quantum spin liquid accessible to an exact solution triggered a lot of interest and
made the “Kitaev materials” one of the recent hot topics in condensed matter physics. The
intense research on Na2IrO3 and other candidate compounds revealed that the dominant Kitaev
interaction is supplemented by several other interactions that drive those materials away from
the desired quantum spin liquid into a long-range ordered state, though some Kitaev-like features
are preserved. In Sec. 3, after explaining the microscopic origin of the strongly anisotropic
and bond-selective interactions, we will provide details on the theoretical investigations of this
situation as well as the most important experimental results for the proper context.
1.5 Soft-spin systems – a quick introduction
In the soft-spin systems, the elementary building object itself is redefined. Instead of a rigid spin
such as spin-1
2
appearing in the Heisenberg model of Sec. 1.2, each site carries now a superposition
of various spinless and spinfull states. The balance between them brings a new degree of freedom.
In the situations when this balance and thus the fraction of the spinfull states is easily changed,
the average spin moment becomes “soft” and prone to fluctuations. It should be emphasized
that this effect is different to spin rotations, i.e. fluctuations of the spin direction of rigid spins.
Having a basic idea of what the soft-spin systems are, we may address the question how they
actually emerge in nature. The key elements are the quasidegeneracy of the above spinless and
1.5 Soft-spin systems – a quick introduction 11
(π,π)
ρ
√
1−ρ
√
ρ
T0
M1
M2
J > λ>J < λ<
magnetic LROsinglets
4
d 4
d
(b) (d)
J / λ
Q =
MyMx
(a)
T+1 T−1
s
singlet
triplet
+
(c)
Higgs
magnon
Q0 q
J
λ
singlet
triplet
condensation
E
0 QCP
ωq
triplon
Q0 q
3× 2×
1×
Fig. 8: (a) Schematic energy-level structure of a d4 ion with large spin-orbit coupling. The ground
state is a nonmagnetic singlet. The magnetic moment is contributed primarily by the transition between
singlet and triplets (M1), only a small part of it is carried by the triplet (M2). (b) The two competing
energy scales of the model – the cost of the triplet λ and the inter-ionic exchange J. (c) Quantum
critical point separating the state full of singlets and a condensate of triplets established at sufficiently
large J/λ. In the condensate state, singlet a triplets are mixed on-site in a quantum superposition with
the relative fractions given by the condensate density ρ. The condensate carries staggered magnetic
moments corresponding to long-range antiferromagnetic order. (d) Change of the excitation spectrum
upon condensation. Dispersing triplet excitations touch zero energy at QCP and transform into a spinwave
like rotational mode (magnon) and an oscillation of the spin length (“Higgs” mode). Numbers
indicate the degeneracy of the modes.
spinfull states and the presence of inter-ionic exchange interactions that are capable of mixing
them. When these two are well balanced, the soft-spin scenario may occur. The soft-spin
system that will be extensively discussed in Sec. 4 is based on transition metal ions containing
four valence electrons that are subject of moderate spin-orbit coupling. The spin-orbit coupling
arranges the low-energy spectrum of valence states according to the level scheme in Fig. 8(a).
The lowest state is a nonmagnetic singlet, above it, separated by the spin-orbit coupling strength
λ, are three excited states forming a triplet. The situation with the magnetic moment is more
complicated compared to the above introductory description. The largest part of the moment
sits primarily on the transition between the singlet and triplet states, while the triplet states
carry only a minor contribution. Nevertheless, the essence of the soft-spin scenario remains the
same. The exchange interactions between the d4
ions under consideration are most naturally
formulated as processes altering the configurations of singlets and triplets on a particular bond.
When the microscopic details are considered, one finds that there are two dominant processes
of similar strength – an exchange of a triplet and a singlet at the two sites and a creation or
annihilation of triplet pairs on the bonds. Both bring down triplets in energy and give them
dispersion. The fate of the system now depends on the ratio of the two energy scales – the triplet
cost λ and the strength J of the exchange interactions [see Fig. 8(b)–(d) for a vague depiction].
Starting in the limit λ J, triplets are relatively costly and the system is full of singlets
12 1.5 Soft-spin systems – a quick introduction
occupying most of the lattice sites. Dispersing triplets serve as elementary excitations moving in
this singlet background, their dispersion is still relatively weak and the bandwidth determined
by J by far does not exceed the basic energy scale λ. When increasing J, the triplet dispersion
becomes more pronounced [Fig. 8(d) left] and at some critical J/λ it will touch zero level at the
AF ordering vector Q. Being zero energy excitations, the triplets will get incorporated into the
ground state changing its nature. We just experienced a quantum phase transition – the system
passed a quantum-critical point (QCP) separating the state full of singlets from the phase that
could be regarded as a condensate of triplets (objects of a bosonic character). In the condensed
state, each site can be imagined as a superposition of both singlet and triplet whose proportions
are related to the condensate density ρ. This superposition carries a magnetic moment of M1
type [c.f. Fig. 8(a)] with the value proportional to ρ (1 − ρ). The structure of the condensate
imposes AF staggering of those magnetic moments so the condensation in fact created a longrange
AF order. Near the QCP, the condensate is still not very rigid and shows pronounced
fluctuations of its amplitude, giving rise to an unusual spectrum of magnetic excitations. This
and other problems related to the peculiar AF order established by a condensation of triplets
will be addressed in detail in Sec. 4.
2 Effective models with localized degrees of freedom
The purpose of this part is to expose in detail the way to arrive at an effective model for a
transition metal compound where the localization tendencies discussed in Sec. 1.2 take over. The
system is then effectively described as consisting of localized degrees of freedom that are subject
to mutual interactions.
The derivation of the corresponding model may be a rather complex task involving several
stages. First, one has to identify the localized degrees of freedom themselves. This typically
consist in an inspection of the multiplet structure of the active ions that depends on their valence,
the crystal environment determining single-electron orbital levels, and the intra-ionic interactions
that generate the many-electron states forming the multiplet structure. The low-energy states of
the multiplet structure are then used as a local basis of the effective model. We will cover these
topics in the following two sections, focusing in particular on the cases of interest: d4
and d5
transition metal ions with strong spin-orbit coupling. Next, one has to open the communication
channels between the individual ions. They are provided by the electronic hopping to be described
within the framework of the tight-binding approximation in the third section of this part. Finally,
as will be discussed in the last section, one combines the above pieces of information and obtains
the interactions among the localized degrees of freedom by considering perturbatively the virtual
processes generated by the electronic hopping.
2.1 Orbital splitting in a crystal environment
The valence electrons in transition metal ions occupy d-type orbitals. In contrast to the case of
a free-standing ion, they are exposed to a crystal environment which necessarily modifies their
wavefunctions and energy levels. The Coulomb interaction with the charges of the surrounding
ions as well as the electronic coupling to their orbitals both imprint the symmetry of the crystal
environment to the new orbitals of the ion under consideration. Leaving the intra-ionic interactions
aside for a moment, we are going to inspect the restructuring of the single-electron levels
by the additional crystal field. The general prediction for the symmetry of the new eigenstates
depending on the symmetry of the environment can be made based on group theory. However,
a more intuitive approach is to explicitly calculate the new levels when including the crystal
field potential of a proper symmetry. It is handled by using first-order perturbation theory for
the originally five-fold degenerate d-orbitals which leads to split energy levels corresponding to
certain combinations of those orbitals. Such an approach is acceptable in particular in our case of
interest where the valence electrons of the transition metal ions retain their localized character.
The advantage of an explicit, though crude calculation, is that one gets an idea about the relative
values of the splittings.
2.1.1 Crystal field within point-charge model
A simple picture of the crystal field is provided by so-called point-charge model where we imagine
the ligands as point charges acting electrostatically on the valence electrons of the transition
metal ion. In the case of the most frequent structural unit – the MO6 octahedron, the oxygen
ions surrounding the transition metal M are supposed to carry the nominal charge Q = −2e
corresponding to the formal valence O2−
. The valence electrons with the charge q = −e are then
perturbed by the potential
VCF(r) =
qQ
4π 0 n
1
|r − Rn|
(2.1)
13
14 2.1 Orbital splitting in a crystal environment
with n going through the six oxygens at positions Rn. These positions may form an ideal
octahedron or incorporate its distortion. Both situations will be inspected later in this section.
Note that the prefactor containing qQ is positive due to the repulsion of the valence electrons and
the negatively charged oxygen ions. Such a treatment of the crystal environment is of course much
simplified and in reality its prediction can be substantially modified, e.g. by covalency effects
(electronic coupling). Nevertheless, the symmetry of the environment is properly embedded in
VCF(r). The next step is the multipole expansion of the crystal field based on the general formula
1
|r − r |
=
∞
l=0
rl
<
rl+1
>
Pl(cos α) =
∞
l=0
rl
<
rl+1
>
4π
2l + 1
+l
m=−l
Y ∗
lm(ϑ , ϕ )Ylm(ϑ, ϕ) . (2.2)
Here r< (r>) is the smaller (larger) number from the pair |r| and |r | and α is the angle between
r and r . The angles ϑ, ϕ and ϑ , ϕ are the conventional spherical angles specifying the direction
of r and r , respectively. By inserting (2.2) into Eq. (2.1), we obtain the final multipole expansion
VCF(r) =
qQ
4π 0
∞
l=0
+l
m=−l
Alm rl
Ylm(ϑ, ϕ) (2.3)
with the multipole coefficients Alm given by
Alm =
4π
2l + 1 n
1
Rl+1
n
Y ∗
lm(ϑn, ϕn) . (2.4)
This expansion may be also understood as a power series in r/Rn. Since the typical distance
of the electron from the octahedron center is visibly less than the distance Rn of the ligands,
the potential terms will weaken with increasing l. Moreover, as we will see below, only terms
with l ≤ 4 will actually contribute to the VCF matrix elements between d-type orbitals due to
symmetry reasons.
2.1.2 Matrix elements of the crystal field
In the absence of the crystal field, the orbitals of the d shell with the angular momentum l = 2
are five-fold degenerate and their wavefunctions are of the form
dm(r, ϑ, ϕ) = f(r) Y2m(ϑ, ϕ) (m = −2, −1, 0, +1, +2) . (2.5)
The radial part of the wavefunction f(r) is common to all the orbitals while they differ in the angular
dependence captured by the spherical harmonics. These wavefunctions will be the starting
point of the perturbation theory to incorporate the crystal field. For our purposes it is sufficient
to consider the first order, the resulting energy shifts and the corresponding combinations of
the dm orbitals are therefore obtained simply by diagonalizing the 5 × 5 matrix of the perturbation
VCF expressed in the unperturbed dm basis. Employing the multipole expansion (2.3), the
necessary matrix elements read as
dm1 |VCF|dm2 =
qQ
4π 0
∞
l=0
+l
m=−l
Alm
∞
0
rl+2
f2
(r) dr Y ∗
2m1
YlmY2m2 dΩ . (2.6)
The middle integral is just the average rl
of some power of the radial distance. The second
integral is more tricky but can be evaluated using Clebsch-Gordan coefficients4
following the
4
Some of the formulas involving Clebsch-Gordan coefficients would be more elegantly formulated using Wigner
3j symbols. However, CG coefficients will be frequently utilized in 2.2 when adding various angular momenta so
we keep using them everywhere for simplicity.
2.1 Orbital splitting in a crystal environment 15
formula (c.f. Eq. 3.8.73 from Ref. [21])
Y ∗
lmYl1m1 Yl2m2 dΩ =
(2l1 + 1)(2l2 + 1)
4π(2l + 1)
l1l200 | l0 l1l2m1m2 | lm . (2.7)
In our case, some of the quantum numbers are fixed and we arrive at
dm1 |VCF|dm2 =
qQ
4π 0
∞
l=0
+l
m=−l
Alm rl 2l + 1
4π
l200 | 20 l2mm2 | 2m1 . (2.8)
Looking at the structure of the Clebsch-Gordan coefficients or the original integral containing
spherical harmonics, we notice that we are in fact adding angular momenta with quantum numbers
l and 2 and evaluating the overlap with an eigenstate of total angular momentum having
the quantum number 2. To have nonzero matrix elements, this implies that the multipolar order
l cannot exceed l = 4 (hexadecapole) as mentioned earlier.
Since the angular momentum conservation is broken by the non-spherical environment, it is
not very helpful to strictly keep the corresponding eigenstates |dm of Eq. (2.5) as the working
basis. Instead, it is more convenient to utilize their real combinations that are be better adjusted
to the octahedral symmetry. We follow the convention by Tanabe and Sugano [22] and introduce
them as
|ξ = i√
2
(|d+1 + |d−1 ) ∼ yz (2.9)
|η = − 1√
2
(|d+1 − |d−1 ) ∼ zx (2.10)
|ζ = − i√
2
(|d+2 − |d−2 ) ∼ xy (2.11)
|u = |d0 ∼ 3z2
− r2
(2.12)
|v = 1√
2
(|d+2 + |d−2 ) ∼ x2
− y2
(2.13)
Here the polynomials on the right indicate the symmetry resulting from the particular combination
of spherical harmonics. The matrix elements of VCF obtained via Eq. (2.8) need to be
converted to the basis of real harmonics leading to a new 5 × 5 matrix for the diagonalization.
When showing the corresponding matrices in the following paragraphs, we will be always using
the above order of the basis states.
2.1.3 Cubic case – t2g and eg orbitals
Let us now apply the above general results to the specific case of an ideal octahedron of oxygen
ions surrounding the transition metal ions. The crystal field for this situation expressed using
Eq. (2.3) and expanded up to l = 4 takes the form (assuming qQ = 2e2
)
VCF(r) =
e2
2π 0R
6 +
7
√
π
3
r4
R4
Y40 +
5
14
(Y4,−4 + Y4,+4) (2.14)
or, in a more familiar Cartesian representation involving the main octahedron axes [see Fig. 9(a)]
VCF(r) =
e2
2π 0R
6 +
35
4R4
x4
+ y4
+ z4
−
3
5
r4
. (2.15)
From the symmetry point of view, the above potential is characterized by the point group Oh
capturing the symmetries of a cube or an octahedron. The first contribution to VCF is a monopole
16 2.1 Orbital splitting in a crystal environment
term from the six charges at distance R and can be ignored since it shifts all the orbitals equally.
The second term with l = 4 is a hexadecapole potential and leads to the actual splitting of the
orbitals. To see the splitting, we express the second term in the orbital basis. First using the
original states |d−2 , |d−1 , |d0 , |d+1 , |d+2 described by spherical harmonics, this gives us a
non-diagonal matrix
V
(l=4)
CF =
e2
2π 0R
r4
R4
1
6











1 0 0 0 5
0 −4 0 0 0
0 0 6 0 0
0 0 0 −4 0
5 0 0 0 1











(2.16)
that is, however easy to diagonalize. In the Tanabe-Sugano basis (2.9)-(2.13), the matrix representing
VCF comes already in a diagonal form
V
(l=4)
CF =
e2
2π 0R
r4
R4
diag −2
3
, −2
3
, −2
3
, +1, +1 (2.17)
separating in energy two sets of states as shown in Fig. 9(a). At the lower level we find the
three states |ξ , |η , |ζ called t2g orbitals, since they form a basis of the three-dimensional T2g
representation of the group Oh [23]. The upper level contains so-called eg orbitals (forming a
basis of the Eg representation of this group). The t2g-eg splitting
∆CF = E(eg) − E(t2g) =
5e2
6π 0R
r4
R4
(2.18)
is for historical reasons denoted as 10Dq with D being the prefactor in the x4
+ y4
+ z4
part
of the electrostatic potential and q being proportional to e r4
. The typical value of ∆CF for
TMO is a few electronvolts. The shapes of the t2g and eg orbitals depicted in Fig. 9(a) enable
us to intuitively understand the origin of the splitting. The eg orbitals have their lobes oriented
directly toward the negatively charged oxygen ions. Therefore they experience a larger Coulomb
repulsion compared to the t2g orbitals and move higher in energy.5
2.1.4 Further splitting due to tetragonal and trigonal distortion of the octahedra
The MO6 octahedra in transition metal oxides are subject to various forms of distortions. Therefore
their symmetry is not ideal and further splitting of orbitals occurs. We will discuss two
important cases: (i) tetragonal distortion i.e. compression or elongation of the octahedron along
one of its main axes, (ii) trigonal distortion where the compression or elongation happens in a
direction perpendicular to one pair of faces of the octahedron.
In both cases we will consider a volume-conserving deformation – contracting or elongating
the octahedron geometry by factor (1 − ε) in one direction and compensating this change in the
remaining two directions:
R = (1 − ε) R +
1
√
1 − ε
R⊥ . (2.19)
Here R stands for the component of the ligand position R in the selected contraction (ε > 0)
or elongation (ε < 0) direction and R⊥ is perpendicular to it.
5
Thanks to this direct contact between the pair of eg orbitals and the surrounding octahedral cage, strong
Jahn-Teller effects may happen in the eg orbital systems [3]. The system spontaneously distorts the octahedra in
a systematic way to gain energy by eg orbital splitting.
2.1 Orbital splitting in a crystal environment 17
x −y2 2
z3 2
−r2
t2g
eg
t2g
eg
x −y2 2
z3 2
−r2
t2g
eg
eg
π
+ eg
π
−,
Y
X
x
y
z
Z
(a)
d
ideal octahedron
xy
yz zx
z
x
y
∆CF
xy
zx
(b)
yz ,
(c)
a1g
compression
tetragonal
compression
trigonal
Fig. 9: (a) Splitting of d-type orbitals in a cubic field of an ideal octahedron. The crystal field of the
negatively charged oxygen ions makes a distinction between the t2g orbitals xy, yz, and zx whose lobes
point inbetween the oxygens and eg orbitals x2 − y2 and 3z2 − r2 with their lobes oriented toward the
oxygens. (b) Further splitting of the orbitals under the tetragonal compression of the octahedron. The
planar orbitals xy and x2 − y2 shift down in energy. (c) Splitting of the orbitals by the crystal field
of a trigonally compressed octahedron. Upper right corner: Relation between the cubic xyz axes and
trigonal XY Z axes with Z being the axis of the compression. The cubic axes and the Z axis point
above the paper plane.
Let us start with the simpler case of the tetragonal distortion. Following Fig. 9(b) we select
the z axis. Evaluating the matrix elements of the distorted crystal field in the Tanabe-Sugano
basis, we arrive at the correction to the original matrix shown in Eq. (2.17). To linear order in
the relative compression factor ε it reads as
δV
(l≤4)
CF = ε
e2
2π 0R
9
7
r2
R2
diag(1, 1, −2, 2, −2) +
25
42
r4
R4
diag(−2, −2, 4, 3, −3) . (2.20)
The first contribution stems from the leading quadrupolar correction to the potential which is
proportional to Y20 ∼ 3z2
− r2
. In the case of a compression, it supports orbitals lying in the xy
plane, i.e. xy and x2
− y2
, and lifts up those that are oriented out of the xy plane. This result is
natural when one considers the relative changes of the positions of the negative oxygen ions as
suggested in Fig. 9(b). For completeness, we have presented also a correction to the hexadecapole
which can be neglected. It merely adjusts a bit the splittings obtained from the quadrupolar
term, bringing no qualitative change.
Somewhat more complicated is the case of a trigonal distortion illustrated in Fig. 9(c). We
focus only on the quadrupolar contribution to the crystal field. To the linear order in ε the
corresponding matrix in Tanabe-Sugano basis takes the form
δV
(l=2)
CF = ε
e2
2π 0R
r2
R2


A C
CT
B

 (2.21)
18 2.1 Orbital splitting in a crystal environment
with the matrix blocks A, B, C given by
A = −
6
7





0 1 1
1 0 1
1 1 0





, B =


0 0
0 0

 , and C =
2
7





−
√
3 3
−
√
3 −3
2
√
3 0





. (2.22)
As we can see, there is no direct trigonal splitting of the eg orbitals, it only happens indirectly by
t2g-eg mixing via the matrix elements in the C block. Such an effect is negligible if the relevant
matrix elements are small compared to the t2g-eg splitting (2.18). The main effect is the splitting
among the t2g orbitals ξ, η, ζ. To determine it, we have to diagonalize the 3 × 3 matrix of the
block A. A convenient choice of its eigenvectors (the convenience will become clear in the next
paragraph) is as follows:
|a1g = 1√
3
(|ξ + |η + |ζ ) , (2.23)
|eπ
g+ = + 1√
3
e+2πi/3
|ξ + e−2πi/3
|η + |ζ , (2.24)
|eπ
g− = − 1√
3
e−2πi/3
|ξ + e+2πi/3
|η + |ζ . (2.25)
The eigenvalue of A for the a1g state is −12
7
, the eπ
g pair of states is degenerate with the A
eigenvalue 6
7
. The value of the splitting is thus a bit different compared to the tetragonal case
but there are again a singlet and a doublet. The a1g singlet has a pronounced elongated shape
with the lobes pointing along the [111] direction in cubic coordinates x, y, z [see Fig. 9(c)]. For
a trigonal compression (ε > 0) it goes down in energy since the oxygen ions move away from the
lobes, relieving a bit the Coulomb repulsion. In contrast, the eπ
g doublet has the electron density
localized closer to the plane perpendicular to [111] and gets shifted up in energy.
2.1.5 Quenched angular momentum
The particular combinations of orbitals that got split in energy by the crystal field do not form
anymore the set of eigenstates of the Lz operator like it was the case of |dm . By separating
the t2g and eg subspaces, the original angular momentum with l = 2 got quenched as one can
see by considering for example the eg orbitals. One of them is |dm with m = 0, another one
combines m = ±2. Since the magnetic quantum numbers differ by 2, all the matrix elements of
Lx and Ly are zero. Lz matrix elements come zero as well so that L projected to the eg subspace
is strictly zero, i.e. the angular momentum is fully quenched. Still, remnants of the original
angular momentum are present in the set of t2g orbitals that are combinations of m = ±1 and
m = ±2, so that it is possible to have nonzero matrix elements of Lx,y. In fact, as we explicitly
show below, they host an effective angular momentum with the quantum number leff
= 1. To
this end, we rearrange the basis a bit by combining
|a = 1√
2
(−i|η − |ξ ) , |b = |ζ , |c = 1√
2
(−i|η + |ξ ) . (2.26)
In the new basis consisting of a, b, c, u, v, the angular momentum operators Lx, Ly, Lz are
represented by the matrices
Lx =
1
√
2










0 −1 0 +
√
3 i +i
−1 0 −1 0 0
0 −1 0 −
√
3 i −i
−
√
3 i 0 +
√
3 i 0 0
−i 0 +i 0 0










, Ly =
1
√
2










0 +i 0 −
√
3 +1
−i 0 +i 0 0
0 −i 0 −
√
3 +1
−
√
3 0 −
√
3 0 0
+1 0 +1 0 0










,
2.1 Orbital splitting in a crystal environment 19
Lz =











−1 0 0 0 0
0 0 0 0 +2i
0 0 +1 0 0
0 0 0 0 0
0 −2i 0 0 0











. (2.27)
The t2g subspace operators found in the indicated upper left corner show a structure precisely
corresponding to L operator expressed in an l = 1 basis up to an overall sign change. The
effective angular momentum operator may be thus defined as Leff
= −L projected to the t2g
subspace. The states |a , |b , |c then correspond to the Leff
z eigenstates with m = +1, 0, and −1,
respectively. At the same time, these states are (Leff
)2
eigenstates corresponding to the quantum
number leff
= 1. In this basis Leff
= [Leff
x , Leff
y , Leff
z ] is represented by the proper set of angular
momentum matrices:
Leff
=





1
√
2





0 1 0
1 0 1
0 1 0





,
1
√
2





0 −i 0
+i 0 −i
0 +i 0





,





+1 0 0
0 0 0
0 0 −1










. (2.28)
The effective angular momentum Leff
just defined may be used to express the tetragonal
splitting in a compact way. The same shifts of the t2g levels as observed in Fig. 9(b) are obtained
using the Hamiltonian ∆tetra[(Leff
z )2
− 2
3
] with ∆tetra denoting the splitting value. This is not
surprising, since the splitting ∆tetra stems from a quadrupolar correction to the crystal field
(proportional to ε) and is thus naturally captured by a quadrupolar angular momentum operator.
Based on this symmetry argument, one can generalize the form of the t2g splitting to other uniaxial
compression/elongation directions by taking
Hsplit = ∆ (n · Leff
)2
− 2
3
, (2.29)
where n is a unit vector along the selected direction. For example, the trigonal splitting of
Fig. 9(c) corresponds to n = [1, 1, 1]/
√
3. This brings us to a connection between the eigenstates
in Eqs. (2.23)–(2.25) and Leff
. Namely, the states |a1g and |eπ
g± that split under trigonal
distortion into a singlet and doublet are eigenstates of n · Leff
with m = 0 and m = ±1,
respectively. Even more explicitly, we can rotate the effective angular momentum operators from
the cubic coordinates xyz to the new coordinates XY Z [depicted in Fig. 9(c)] more appropriate
to the trigonal case:





LX
LY
LZ





eff
=






1√
6
1√
6
− 2
3
− 1√
2
1√
2
0
1√
3
1√
3
1√
3











Lx
Ly
Lz





eff
. (2.30)
Using the three states |eπ
g+ , |a1g , and |eπ
g− as a basis, we find that the components LX, LY ,
and LZ are represented by matrices identical to those in Eq. (2.28). These states therefore play
the same role as |a , |b , |c of the tetragonal case.
Finally, let us comment on the off-diagonal blocks in (2.27) connecting the t2g and eg states.
These are not frequently used as the two subspaces are separated by the largest crystal-field
20 2.2 Local correlations and multiplet structure of transition metal ions
splitting. However, in some situations they may become important. For example when considering
a t2g system with large spin orbit coupling, the above matrix elements may bring a sizable
admixture of eg states to the predominantly t2g ones.
2.2 Local correlations and multiplet structure of transition metal ions
In the previous section we have investigated the valence shell of a d ion from the single-electron
point of view, studying the orbitals and their splitting due to the crystal field. When the valence
shell is occupied by more than one electron, the Coulomb repulsion among them makes their
motion strongly correlated and organizes the electrons into many-body eigenstates that form the
ionic multiplet structure. In the following, we first analyze the structure of the Hamiltonian
capturing the Coulomb repulsion among valence electrons in d orbitals and then discuss some
examples of its diagonalization arriving at the multiplet structure. Apart from the Coulomb interaction,
the situation is further complicated by spin-orbit coupling that appears as a relativistic
quantum mechanical effect and tries to contra-align the spins of the individual electrons and their
orbital angular momenta. Such effects are of a crucial importance in iridates and ruthenates to
be discussed in the next parts of the thesis, we therefore devote a substantial part of this section
to a detailed exploration of the relevant ionic states restructured by the spin-orbit coupling.
2.2.1 Coulomb interactions among valence electrons
The starting point for a discussion of the many-body effects of the Coulomb repulsion among
electrons in a d shell is the second-quantized form of the corresponding Hamiltonian:
HCoul =
1
2 αβγδ σσ
Vαβγδ α†
σβ†
σ γσ δσ . (2.31)
Here the indices α, β, γ, δ run through the orbitals and the spin summation got restricted
because we are dealing with a spin-conserving charge-charge interaction. The matrix elements
of the Coulomb interaction take the form
Vαβγδ =
e2
4π 0
d3
r d3
r ψ∗
α(r) ψ∗
β(r )
1
|r − r |
ψγ(r ) ψδ(r) . (2.32)
We will again use the orbitals of the Tanabe-Sugano basis ξ, η, ζ, u, v that are described by
real wavefunctions, making the above matrix elements real. They are also subject to obvious
symmetry relations
Vαβγδ = Vαγβδ = Vδβγα = Vβαδγ , (2.33)
reducing the number of the independent values from 54
= 625 to 120. This number will be
further (and drastically) reduced once the symmetry of the orbitals themselves is employed.
While the use of the Cartesian orbitals is convenient to easily incorporate the crystal field
splittings, the evaluation of the matrix elements is simpler when working with the spherical
orbitals dm of Eq. (2.5). In that case one considers the matrix elements
Vm1m2m3m4 =
e2
4π 0
d3
r d3
r d∗
m1
(r) d∗
m2
(r )
1
|r − r |
dm3 (r ) dm4 (r) (2.34)
and constructs Vαβγδ as linear combinations of Vm1m2m3m4 . The key tool is the multipole expansion
(2.2) of 1/|r − r | with the subsequent evaluation of the angular integrals via Eq. (2.7). The
radial integration is reduced to an evaluation of a set of so-called Slater-Condon parameters
F(l)
=
e2
4π 0
∞
0
dr r2
∞
0
dr r 2 rl
<
rl+1
>
f(r)f(r ) . (2.35)
2.2 Local correlations and multiplet structure of transition metal ions 21
After some manipulations involving also the symmetry property of the spherical harmonics Y ∗
lm =
(−1)m
Yl,−m, the matrix element (2.34) can be rearranged into
Vm1m2m3m4 =
l=0,2,4
25 F(l)
(2l + 1)2
2200|l0 2
+l
m=−l
(−1)m+m1+m2
22, −m1m4|l, −m 22, −m2m3|lm .
(2.36)
An important feature is the Clebsch-Gordan coefficient 2200|l0 entering the above expression.
It limits the necessary Slater-Condon parameters to three numbers F(0)
, F(2)
, F(4)
. These are
most conveniently expressed via Racah parameters [24]
A = F(0)
− 1
9
F(4)
B = 1
49
F(2)
− 5
441
F(4)
C = 5
63
F(4)
(2.37)
that lead to compact formulas for Vαβγδ containing round coefficients.
As we found out, all the Coulomb matrix elements Vαβγδ can be expressed as linear combinations
of just three parameters A, B, C given by the radial part of the orbital wavefunctions.
Together with the crystal field splittings, they determine the multiplet structure which therefore
depends on just very few parameters. Such a huge reduction is a result of the assumed
spherical symmetry i.e. it happens if the orbitals share the radial part of the wavefunction and
their angular dependence is given by spherical harmonics or their combinations. While this is in
general not true, in the case of localized orbitals in transition metal oxides it is still a reasonable
approximation and the corresponding description of the multiplet structure is often sufficient.
Going beyond this approximation, the number of independent parameters necessarily increases.
For example, when assuming the cubic symmetry, we have to deal with ten of them instead of
the Racah A, B, C [25].
When considering the Coulomb repulsion in a valence shell, the sum in the Hamiltonian (2.31)
is usually systematically truncated. We will now discuss the set of matrix elements to be included
and rearrange the selected subset of terms into the conventional form of intra-ionic interactions.
The dominant matrix elements correspond to a Coulomb repulsion of two electrons sharing the
same orbital. The corresponding matrix element Vαααα is the same for all the orbitals of the
Tanabe-Sugano basis and will be termed as the intra-orbital Hubbard U. In terms of the Racah
parameters, it is expressed as
U = A + 4B + 3C = Vαααα . (2.38)
The relevant contributions extracted from Eq. (2.31) may be cast to the familiar form of the
intra-orbital Hubbard interaction
H1 = U
α
nα↑nα↓ . (2.39)
The second kind of contributions to be included are two-orbital interactions with two distinct
pairs of identical indices in αβγδ. The corresponding matrix elements are of two types – Coulomb
integral for two different orbitals
Uαβ =
e2
4π 0
d3
r d3
r ψ2
α(r)
1
|r − r |
ψ2
β(r ) = Vαββα = Vβααβ (2.40)
and the exchange integral
Jαβ =
e2
4π 0
d3
r d3
r ψα(r)ψβ(r)
1
|r − r |
ψα(r )ψβ(r ) = Vααββ = Vββαα = Vαβαβ = Vβαβα .
(2.41)
Both can be evaluated via Eq. (2.36) with a subsequent conversion to Tanabe-Sugano orbital
22 2.2 Local correlations and multiplet structure of transition metal ions
Jαβ ξ η ζ u v
ξ − 3B + C 3B + C B + C 3B + C
η 3B + C − 3B + C B + C 3B + C
ζ 3B + C 3B + C − 4B + C C
u B + C B + C 4B + C − 4B + C
v 3B + C 3B + C C 4B + C −
Table 2: Exchange integrals Jαβ in terms of Racah parameters
basis. Table 2 gives the values of the exchange integrals in terms of the Racah parameters, the
Coulomb integrals may be obtained via the relation
U = Uαβ + 2Jαβ (2.42)
valid for our case of orbitals derived from the spherical harmonics. After collecting all the
contributions involving either Uαβ or Jαβ, we arrive at the following two-orbital Hamiltonian:
H2 =
α<β σ
(Uαβ − Jαβ) nασnβσ + Uαβnασnβ,−σ +
α=β
Jαβ α†
↑β†
↓α↓β↑ + α†
↑α†
↓β↓β↑ . (2.43)
To make it more transparent, it is possible to combine the spin-dependent part of the interorbital
density-density interaction σ(nασnβσ − nασnβ,−σ) and the second term from the right
(α†
↑β†
↓α↓β↑) into an inter-orbital spin-spin interaction. It is clear that these terms may only be
active between two singly-occupied orbitals. For the singly-occupied orbitals we can introduce
the corresponding spin operator Sα (here associated with the orbital α) as
Sα = 1
2
(S+
α +S−
α ), 1
2i
(S+
α −S−
α ), Sz
α with S+
α = α†
↑α↓ , S−
α = α†
↓α↑ , Sz
α = 1
2
(nα↑−nα↓) (2.44)
that enables us to bring the intra-ionic Coulomb interaction into the final form
HCoul = U
α
nα↑nα↓ +
α<β
(Uαβ − 1
2
Jαβ) nαnβ
− 2
α<β
Jαβ Sα · Sβ +
α=β
Jαβ α†
↑α†
↓β↓β↑ , (+ neglected terms) (2.45)
where the individual terms represent the intra-orbital Hubbard interaction, inter-orbital Hubbard
interaction, Hund’s exchange, and inter-orbital pair-hopping term, respectively.
By inspecting Table 2 we notice, that the parametrization of the interactions in HCoul simplifies
when one considers a strictly t2g-only or eg-only system. In that case all relevant exchange
integrals Jαβ are equal to JH = 3B + C or JH = 4B + C, respectively, and consequently
Uαβ = U = U − 2JH. The resulting two-parameter Hamiltonian HCoul(U, JH) is the HubbardKanamori
Hamiltonian [26].
There are still many non-zero matrix elements in Eq. (2.31) that were omitted when constructing
(2.45). These are summarized in Table 3. Note that they always involve both t2g and
eg orbitals at the same time so that they only lead to perturbative corrections for purely t2g
or eg systems. However, they may become important when dealing with mixed t2g-eg situation.
An important example – spin-state crossover of d6
configuration – will be discussed in the next
paragraph.
2.2 Local correlations and multiplet structure of transition metal ions 23
three-orbital four-orbital
ξ2 1
r12
u v = +2
√
3B ξζ 1
r12
η v = +3B
ξξ 1
r12
uv = −
√
3B ηζ 1
r12
ξ v = −3B
η2 1
r12
u v = −2
√
3B ξη 1
r12
ζ u = −2
√
3B
ηη 1
r12
uv = +
√
3B ηζ 1
r12
ξ u = +
√
3B
ξζ 1
r12
η u = +
√
3B
Table 3: Nonzero matrix elements involving three or four different orbitals. The integrals are written in
an abbreviated form with the prime indicating the argument r of the corresponding orbital wavefunction
while the absence of prime indicates the argument r. Further, r12 stands for |r − r |. Together with
the symmetry property (2.33), the left and right columns of the table generate 24 and 40 Vαβγδ matrix
elements, respectively.
2.2.2 Ionic Hubbard model and multiplet structure
Together with the energy levels of the individual orbitals discussed in the Sec. 2.1, the above
Coulomb interaction Hamiltonian forms the ionic Hubbard model
Hion =
ασ
εαα†
σασ + HCoul . (2.46)
Before extending it further with the spin-orbit interaction in the next paragraph, we will inspect
the resulting multiplet structure (i.e. the spectrum of eigenstates for a fixed number of electrons)
in a few interesting cases.
Let us first comment on the general tendencies that can be intuitively inferred from the
structure of Hion. For an isolated ion, all the orbital energies Eα are equal and the eigenstates
are decided solely by the Coulomb interaction. The leading role takes the intra-orbital Hubbard
repulsion (given by the dominant parameter U) which tries to place electrons to different orbitals
whenever possible. The unpaired spins in singly-occupied orbitals are then organized by the
ferromagnetic Hund’s coupling to form the largest possible total spin. These observations are in
fact the content of the first Hund’s rule as discussed in standard textbooks [27]. The situation is
changed by the crystal field which splits the orbital energies Eα. The primary splitting due to the
octahedral crystal field is the t2g-eg one. Its value is substantial so that it may be energetically
favorable to keep electrons in the lower t2g levels, even though more doubly-occupied orbitals
appear and this scenario thus leads to a larger Hubbard repulsion and an energy loss in Hund’s
coupling. On this occasion, it should be also noted that the crystal values of the model parameters
are strongly affected by screening. This does not significantly modify the Hund’s exchange but
the Hubbard repulsion is quite reduced compared to free ions, increasing the relative importance
of the crystal field splitting.
From the formal point of view, the first three terms in Eq. (2.45) do not change the distribution
of electrons among the orbitals. This type of dynamics comes only due to the last term and is
constrained by the necessity to transfer a complete electron pair from a doubly occupied orbital
24 2.2 Local correlations and multiplet structure of transition metal ions
η
ζ
ξ
t2g
6
t2g
6
↓ξ t2g
6
t2g
6(a) (b)
η
ζ
ξ
η
ζ
ξ
η
ζ
ξ
↓ ↓ζη
Fig. 10: (a) Schematic representation of the six-electron many-body state corresponding to the t6
2g
electron configuration. The orbitals ξ, η, ζ are considered to be degenerate; in this and the following
figure they are vertically separated for clarity only. In second quantization, the state depicted in panel
(a) corresponds to ξ†
↑ ξ†
↓ η†
↑ η†
↓ ζ†
↑ ζ†
↓ | , where | is an empty state (no electrons in the valence shell). At
the bottom a visual representation of the six-electron cloud is shown. The angular distribution of the
total electron density (integrated in the radial direction) is captured as a surface plot where the distance
of the surface points to the origin is proportional to the integral density in the corresponding direction.
(b) States of the t5
2g configuration obtained by removing a single electron from the t6
2g configuration
shown in the panel (a). The electron clouds at the bottom bear some similarity to the px, py, and pz
orbitals which is related to the effective angular momentum leff = 1 carried by the t2g orbitals.
into an empty one. Thanks to the limited mixing of electron configurations, the identification of
the ionic eigenstates may be relatively simple, in spite of dealing with a correlated many-body
problem. The real troubles only start once we let the ions interact.
Our first illustrative example will be the t5
2g electron configuration which is relevant for the
physics of Kitaev materials Na2IrO3 or α-RuCl3. We assume that the crystal field splitting is
large enough so that it practically eliminates the eg orbitals.6
For simplicity, we further assume
that the t2g orbitals are not split. The starting point is the t6
2g configuration with fully populated
t2g orbitals shown in Fig. 10(a). The t5
2g configurations are obtained by removing one electron
from it. There are three possible choices of the corresponding orbital and two options for the
spin projection, leading altogether to six-fold degeneracy. The resulting layout of electrons among
orbitals allows only the Hubbard repulsion to be active so that the Coulomb energy evaluates
simply to 2U +8(U − 1
2
JH) = 10(U −2JH). By observing the angular distribution of the electron
clouds as presented in Fig. 10, we notice that the t6
2g configuration has a full cubic symmetry
whereas the t5
2g configurations vaguely resemble the shapes of the Cartesian px, py, pz orbitals.
This visual similarity is related to the fact that by extracting a t2g electron as an leff
= 1 object
from the fully symmetric t6
2g configuration with total Leff
= 0, we created a configuration with
Leff
= 1 again. A more precise symmetry classification is embedded in the usual notation of the
members of the multiplet structure. Being eigenstates of some symmetric Hamiltonian, each set of
degenerate eigenstates constitutes a basis for a certain representation of the respective symmetry
group. In our case the rotation group O of a cube/octahedron is the relevant one (extending it by
the spatial inversion, we get the group Oh). It has two one-dimensional representations A1, A2,
one two-dimensional representation E, and two three-dimensional representations T1, T2. The
three eigenstates presented in Fig. 10(b) are a basis for the T2 representation of O [25]. When
6
This low-spin situation is the case in the relevant Ir4+
(5d) and Ru3+
(4d) ions; the 3d ions usually prefer to
employ the eg orbitals to form a larger total spin and hence optimize Hund’s exchange.
2.2 Local correlations and multiplet structure of transition metal ions 25
ξ2
η2
ζ2
η
ζ
ξ
t2g
4
ζη ζξ ξη
T
3
1
T
1
2
E
1
A
1
1
3JH
2JH
η
ζ
ξ
ζξζη ξη
(b)
(a) (c)
15×
1×
9×
5×
EE
1 1
xz
Fig. 11: (a), (b) Schematic representation of the many-body states obtained by removing two electrons
from the t6
2g configuration, either from two different orbitals (a), or from the same orbital (b). In the case
(a), only one of the four possible spin configurations for each electron distribution is shown. (c) Level
splitting due to the Coulomb interaction. The lowest nine states are spin-triplets 3T1 based on the states
of the type shown in panel (a). The same three orbital populations may form also singlets 1T2 that
merge with the two 1E spin-singlets to form a five-fold degenerate level. Well above the others is the
fully symmetric 1A1 singlet.
referring to a multiplet member, the label of the group representation comes together with the
multiplicity in front. By multiplicity one means the number of possible spin projection values
for the total spin of the state. Here we have three spin-1
2
doublets so the full conventional label
for the six-fold degenerate t5
2g states is 2
T2.
The case of the t5
2g configuration discussed so far is fairly trivial. A bit more involved is
the analysis of the t4
2g configuration. Here one can consider two independent classes of states
presented in Fig. 11(a) and (b). The former one consists of two singly-occupied orbitals and one
doubly occupied. Again there are three possibilities how to arrange them. All three configurations
have the same Hubbard repulsion energy U + 5(U − 1
2
JH) = 6(U − 2JH) − 1
2
JH. Due to the two
singly-occupied orbitals, the Hund’s coupling comes now into play while the pair hopping term
is clearly inactive. The two unpaired spins-1
2
can form either a triplet or a singlet state. The
respective full Coulomb energies of the 3
T1 triplets and 1
T2 singlets are then 6U − 13JH and
6U − 11JH differing by 2JH. The corresponding states can be easily expressed in the second
quantization formalism, for example the four states based on |η ζ of Fig. 11(a) read as:
|3
T1(η ζ, Sz =+1) = ξ†
↑ ξ†
↓ η†
↑ ζ†
↑ | , (2.47)
|3
T1(η ζ, Sz =0) =
1
√
2
ξ†
↑ ξ†
↓ (η†
↑ ζ†
↓ + η†
↓ ζ†
↑) | , (2.48)
|3
T1(η ζ, Sz =−1) = ξ†
↑ ξ†
↓ η†
↓ ζ†
↓ | , (2.49)
|1
T2(ηζ) =
1
√
2
ξ†
↑ ξ†
↓ (η†
↑ ζ†
↓ − η†
↓ ζ†
↑) | . (2.50)
As a consequence of having one t2g orbital populated more than the others, the electron-density
clouds of the twelve eigenstates 3
T1, 1
T2 become asymmetric. As shown in Fig. 11(c), they
26 2.2 Local correlations and multiplet structure of transition metal ions
resemble those of the dyz, dzx, dxy orbitals. The second class of states that do not interact with
the former ones is shown in Fig. 11(b). In each of these states, one of the three t2g orbitals is
empty while the other two are doubly occupied which makes the states singlets in terms of total
spin. This configuration precludes Hund’s coupling but activates the pair hopping proportional
to JH as well. The eigenstates are obtained by diagonalizing the pair hopping term expressed in
the basis states |ξ2
, |η2
, and |ζ2
of Fig. 11(b):
JH





0 1 1
1 0 1
1 1 0





. (2.51)
Doing so, we get a two-fold degenerate eigenvalue −JH and the associated 1
E states
|1
Ez =
1
√
6
|ξ2
+ |η2
− 2|ζ2
, (2.52)
|1
Ex =
1
√
2
|η2
− |ξ2
, (2.53)
and a non-degenerate eigenvalue +2JH associated with a fully symmetric 1
A1 state
|1
A1 =
1
√
3
|ξ2
+ |η2
+ |ζ2
. (2.54)
The Hubbard repulsion energy amounts to 2U + 4(U − 1
2
JH) = 6U − 10JH, by adding the
eigenvalues of the pair hopping we find that 1
E merge with the 1
T2 level while 1
A1 is singled out
at the top of the multiplet level scheme in Fig. 11(c).
The Cartesian formulation of the multiplet structure as demonstrated above may be convenient
when considering e.g. further splitting of the orbitals in a tetragonal crystal field. However,
we need to prepare ground for the inclusion of the spin-orbit coupling in the next paragraph. In
this context the role of Leff
is essential so that the multiplet structure of both t5
2g and t4
2g should
be reformulated as Leff
eigenstates. The Hamiltonian including the Coulomb interaction commutes
with Leff
and S, the eigenstates will be therefore classified by the corresponding quantum
numbers and denoted by |Leff
, Leff
z , S, Sz .
To obtain compact expressions in the following, we adopt a hole picture and think about
one-hole or two-hole configurations on top of the t6
2g “vacuum”. The one-hole states will carry
Leff
= 1 and S = 1
2
. It is convenient to introduce the hole operators in such a way that they
create states with the selected spin and Leff
quantum numbers and give them suitable phases.
For our purpose, the best choice of connecting the hole and electron operators is
h†
mσ = (−1)m
(−σ) c−m,−σ (2.55)
with m denoting the magnetic quantum number associated with Leff
and σ = ±1 indicating the
up/down configuration of spin. The logic behind the above construction is as follows: (i) To create
a configuration with mσ based on the fully symmetric (Leff
= 0) and spinless (S = 0) t6
2g state,
we need to remove an electron with the opposite quantum numbers. (ii) The phase factor (−1)m
is related to a conversion between Ylm and Y ∗
l,−m. (iii) The factor (−σ) is generated by reordering
the fermionic operators to follow the sequence ξ†
↑ ξ†
↓ η†
↑ η†
↓ ζ†
↑ ζ†
↓ in the final t5
2g configuration. In
analogy with the notation of the a, b, c electron states in Eq. (2.26), the hole operators a, b, c will
2.2 Local correlations and multiplet structure of transition metal ions 27
be associated with m = +1, 0, and −1, respectively. Put explicitly in terms of the annihilation
operators ξσ, ησ, ζσ, they read as
a†
↑ = + 1√
2
(iη↓ + ξ↓) , b†
↑ = −ζ↓ , c†
↑ = + 1√
2
(iη↓ − ξ↓) ,
a†
↓ = − 1√
2
(iη↑ + ξ↑) , b†
↓ = +ζ↑ , c†
↓ = − 1√
2
(iη↑ − ξ↑) . (2.56)
Defined this way, a†
σ (b†
σ, c†
σ) acting on the state with fully occupied t2g orbitals
|t6
2g = ξ†
↑ ξ†
↓ η†
↑ η†
↓ ζ†
↑ ζ†
↓ | (2.57)
creates the t5
2g state with spin σ and the Leff
z quantum number equal to +1 (0, −1). Using the
systematic labeling of the eigenstates by |Leff
, Leff
z , S, Sz , we have for example
1, +1, 1
2
, +1
2
= a†
↑ |t6
2g . (2.58)
The phases of all the six generated states follow Condon-Shortley convention so that they give
canonical matrix elements of both total spin and total effective orbital momentum operators
S and Leff
. This feature will be useful later when combining them following the rules for the
addition of angular momenta.
The selection of Leff
quantization axis proceeds in accordance with the potential further
reduction of the octahedral symmetry. Here we have used Leff
z for the tetragonal case, for the
trigonal case one has to implement the a1g and eπ
g± states of Eqs. (2.23)–(2.25) into the definition
of the hole operators to reflect the quantization in the [111] direction. Moreover, when including
the spin-orbit coupling, the spin quantization axis should be consistently changed to [111] as
well. The derivation of an explicit linear relation between the electron operators ξσ, ησ, ζσ and
the trigonal hole operators is a useful exercise though the relation itself has a limited use. It may
be formulated as follows:
a†
↑ = A↑ eiπ/4
sin θ − A↓ cos θ a†
↓ = A↑ eiπ/4
cos θ + A↓ sin θ with tan 2θ =
√
2 (2.59)
and similarly for b, c. The operators A, B, C are defined as
Aσ = 1√
3
(e+2πi/3
ξσ + e−2πi/3
ησ + ζσ) ,
Bσ = 1√
3
(ξσ + ησ + ζσ) ,
Cσ = − 1√
3
(e−2πi/3
ξσ + e+2πi/3
ησ + ζσ) . (2.60)
The hole operators are labeled according to [111] projections of the effective orbital momentum
and spin, the electron operators use the z spin quantization axis. Equations (2.59) and (2.60)
constitute a trigonal analog of Eq. (2.56). When expressing the matrices for the X, Y , and Z
components of S and Leff
operators [c.f. Fig. 9(c)], the result is identical to the matrices for the
x, y, and z components in the basis given by Eq. (2.56).
Now we are going to assemble the eigenstates with two holes that will correspond to the t4
2g
multiplet structure. This can be achieved simply by forming Leff
and S eigenstates of the two
holes. The reason for this simplification is that the Coulomb Hamiltonian for a t2g-only valence
shell can be cast to a transparent form
Ht2g = 1
2
(U − 3JH)N(N − 1) + 5
2
JHN − 2JHS2
− 1
2
JH(Leff
)2
, (2.61)
where N stands for number of electrons and S, Leff
are the total spin and total effective orbital
momentum as before. For a fixed number of electrons, this form of the Hamiltonian demonstrates
the tendency to maximize total spin in the first place and Leff
in the second.
28 2.2 Local correlations and multiplet structure of transition metal ions
By combining the two Leff
= 1 holes, we can obtain the possible total Leff
= 0, 1, 2 states with
the help of the tables of Clebsch-Gordan coefficients. The total spin will automatically follow by
observing the symmetry of the state. Symmetric or antisymmetric states in terms of orbitals have
to be of a complementary symmetry in terms of spins. This means that the symmetric orbital
combinations will be spin singlets while the two holes in an antisymmetric orbital combination
have to form a spin triplet. The nondegenerate topmost energy state 1
A1 shown in Fig. 11(b) is
a singlet in both orbital and spin sector, i.e. it has Leff
= 0 and S = 0:
|Leff
, Leff
z , S, Sz = |0, 0, 0, 0 = 1√
3
a†
↑c†
↓ − b†
↑b†
↓ + c†
↑a†
↓ |t6
2g . (2.62)
The five-fold degeneracy of the level encompassing 1
T2 and 1
E in Fig. 11(b) is also explained, as
it corresponds to Leff
= 2 and S = 0 with the five Leff
z eigenstates given explicitly by
|2, +2, 0, 0 = a†
↑a†
↓ |t6
2g ,
|2, +1, 0, 0 = 1√
2
a†
↑b†
↓ + b†
↑a†
↓ |t6
2g ,
|2, 0, 0, 0 = 1√
6
a†
↑c†
↓ + 2 b†
↑b†
↓ + c†
↑a†
↓ |t6
2g ,
|2, −1, 0, 0 = 1√
2
b†
↑c†
↓ + c†
↑b†
↓ |t6
2g ,
|2, −2, 0, 0 = c†
↑c†
↓ |t6
2g . (2.63)
Finally, the low-energy triplets 3
T1 have Leff
= 1 and S = 1. We write down the corresponding
states using an abbreviated |Leff
z , Sz notation:
|+1, +1 = a†
↑b†
↑ |t6
2g , |+1, 0 = 1√
2
a†
↑b†
↓ + a†
↓b†
↑ |t6
2g , |+1, −1 = a†
↓b†
↓ |t6
2g ,
| 0, +1 = a†
↑c†
↑ |t6
2g , | 0, 0 = 1√
2
a†
↑c†
↓ + a†
↓c†
↑ |t6
2g , | 0, −1 = a†
↓c†
↓ |t6
2g ,
|−1, +1 = b†
↑c†
↑ |t6
2g , |−1, 0 = 1√
2
b†
↑c†
↓ + b†
↓c†
↑ |t6
2g , |−1, −1 = b†
↓c†
↓ |t6
2g .
(2.64)
The phases of the t4
2g states given in Eqs. (2.62), (2.63), (2.64) are again compatible with the
Condon-Shortley convention so that they generate canonical matrices of Leff
and S operators
when used as a basis.
Having explored in detail the t4
2g and t5
2g configurations, we will now have a brief look at
a combined t2g-eg situation. As an illustration, we will use a particularly rich case of d6
ions
such as Co3+
or Fe2+
well known for spin-state crossover phenomena [28]. The basic physics
can be understood based on the sketches in Fig. 12(a). These show so-called low-spin (LS),
intermediate-spin (IS), and high-spin (HS) configurations of a d6
ion differing in the distribution
of the six electrons among the t2g and eg levels. Placing an electron to the eg levels costs the
extra crystal-field energy ∆ but doing so, we can create two unpaired spins that are subject
to Hund’s coupling and may therefore bring an energy gain proportional to JH. For simplicity,
we first ignore the differences in JH for t2g and eg orbitals. More precisely, we set the Racah
parameter B to zero leading to Jαβ = JH = C and vanishing matrix elements in Table 3. We
will also not consider the perturbative corrections by pair hopping of t2g electrons into empty eg
levels. By evaluating the Hubbard repulsion in the t6
2g configuration, we get the reference energy
E(t6
2g) = E0 = 15(U − 2JH) . (2.65)
2.2 Local correlations and multiplet structure of transition metal ions 29
S=0
LS IS HS
ge
t2g
∆
S=2S=1
JH JH
8B
3C∆−
T
3
2
T
3
1
t2g
6
T
1
2
T
1
1
16B
2C
(a)
3∆−0energy
relative
∆−82
3z2 −r2 electron
2x −y2 electron
xy hole
xy hole
(b)
Fig. 12: (a) Schematic representation of the low-spin, intermediate-spin, and high-spin states of d6 ion
such as Co3+. The relative energies of the states are determined by the balance between the crystal
field splitting ∆ supporting configurations with less eg electrons and Hund’s coupling that gains energy
by forming the maximum total spin. (b) Multiplet structure of the t5
2ge1
g configuration. The lower two
triplets 3T1,2 correspond to the intermediate-spin states. The electron clouds for T1 and T2 slightly
differ. Their internal structure can be understood in terms of a hole in t6
2g configuration combined with
the eg electrons, leading to either a triplet or a singlet total spin state. T1 states involve eg orbital lying
in the plane defined by the t2g hole, the out-of-plane eg orbital is employed in the T2 states.
The configuration with one eg electron contains two unpaired spins-1
2
so that the state may be
total singlet or triplet. The respective energies are E0 + ∆ − 3JH and E0 + ∆ − JH with the
triplet state (intermediate spin S = 1) being supported by Hund’s coupling. In case of the
configurations with two eg electrons in different eg orbitals, we have to sort out the interaction of
four unpaired spins which leads to the energy E = E0 + 2∆ − [S(S + 1) + 2]JH as function of the
total spin S. The lowest state is here the high-spin S = 2 with the energy 2∆ − 8JH above E0.
Depending on the balance between the crystal field splitting and Hund’s coupling strength, the
LS, IS, and HS states may be close in energy so that the total spin of the ions may be changed
by affecting slightly their crystal/ligand environment, e.g. by pressure. The low-lying excited
states may also be thermally activated or brought into play by inter-ionic interactions. Later in
Section 4 we will be dealing with an analogous situation created by spin-orbit coupling.
We conclude this paragraph by discussing the detailed multiplet structure of the t5
2ge1
g configuration
for nonzero Racah B whose presence further splits the triplet and singlet levels. As
presented in Fig. 12(b), the non-uniform interaction of the t2g orbitals with the eg ones (see
Table. 2) makes an energy distinction between the various combinations of the orbitals. When
one interprets the missing electron in t2g orbitals as a positively charged hole, it is intuitively
expected that the negatively charged eg electron will be preferably put into the eg orbital better
matching this hole in shape. This is indeed observed in Fig. 12(b), the configurations where the
eg orbital x2
− y2
matches the plane defined by the xy hole are lower than those involving the
out-of-plane 3z2
− r2
orbital. Of course, the hole can be put to the other t2g orbitals as well.
In that case one selects other in-plane/out-of-plane combinations of the eg orbitals. Each of the
levels shown in Fig. 12(b) is thus three-fold orbitally degenerate, giving in total 24 states of the
multiplet structure as it has to be for the t5
2ge1
g configuration. At this point it is important to
30 2.2 Local correlations and multiplet structure of transition metal ions
fully employ the matrix elements from Table 3 that we neglected in Eq. (2.45). By omitting
them, we would artificially split the levels further and break the octahedral symmetry.
2.2.3 Spin-orbit coupling
Apart from the Coulomb interaction discussed in detail in the previous paragraphs, additional
interactions originating in relativistic quantum mechanics come into play. In the hydrogen atom
these just give the fine structure of the energy levels and can be neglected in the first approximation.
This will not be the case in the heavier transition metal ions under our consideration.
As we progress down through the periodic table of elements, one of the relativistic corrections –
the spin-orbit coupling – becomes increasingly important and essentially rearranges the multiplet
structures. Intuitively it may be understood by considering the electron in its rest frame where
it is encircled by the positively charged nucleus. The magnetic field generated by the “current
loop” provided by the nucleus then acts on the spin of the electron. More generally and on a
quantitative level, we may estimate this effect by taking a static electrostatic field E the electron
moves in (e.g. that of the nucleus), performing Lorentz transformation to the rest frame of
the electron which gives the magnetic field B ≈ −(v × E)/c2
, and letting this magnetic field
act on the magnetic moment associated with the electronic spin via the usual 2µB s · B . This
simple estimate leads to a correct result up to a factor of 2. A proper derivation is based on an
expansion of the solution of the Dirac equation up to the order v2
/c2
and gives the interaction
of the form [29]
HSOC =
e
2m2c2
s · (E × p) . (2.66)
Here the electron spin operator s is again dimensionless (i.e. it does not include ). For the
centrally-symmetric potential of an atom, the electric field is radial and related to the radial
derivative of the potential energy V for the electron via
E = − Φ = −
r
r
dΦ
dr
=
1
e
r
r
dV
dr
. (2.67)
The use of the above radial field in Eq. (2.66) brings the dimensionless electron angular momentum
operator l = (r × p)/ and the interaction Hamiltonian becomes
HSOC =
2
2m2c2
1
r
dV
dr
s · l (2.68)
giving the interaction its name: spin-orbit coupling. Since the binding potential for electrons is an
increasing function of distance, the prefactor in (2.68) is positive so that the spin of an electron
is preferably antiparallel to its orbital momentum. To make HSOC practical, the prefactor is
approximated by its average with the main contribution apparently coming from the vicinity of
the nucleus. Often it is referred to a result obtained from the scaling of the Schr¨odinger equation
for hydrogen-like atoms with the potential V ∝ −Ze2
/r which gives
1
r
dV
dr
∝
Z
r3
= Z4 1
r3
hydrogen
(2.69)
suggesting the spin-orbit coupling strength proportional to Z4
. However, as argued by Landau
and Lifshitz [30], one has to combine the above scaling of unscreened estimate (∝ Z4
) with the
scaling of the probability for a valence electron being close to the nucleus (∝ 1/Z2
), leading to
an alternative estimate of the strength proportional to Z2
. Even when considering this reduced
2.2 Local correlations and multiplet structure of transition metal ions 31
scaling, the spin-orbit coupling quickly increases with Z and becomes well visible in transition
metal ions. We therefore have to extend our Hion with an extra term
HSOC = ζ
i
si · li , (2.70)
where the sum runs through our valence electrons and ζ is a positive constant. In 3d elements,
ζ takes the values of tens of meV, in the heavier 5d elements it reaches hundreds of meV, for
example ζ ≈ 0.4 eV in Ir4+
[31].
There are two standard schemes how to incorporate the above spin-orbit coupling into the
multiplet structure. Within the framework of so-called j-j coupling scheme one first adds the
spin and orbital momenta of the individual electrons to form the total angular momenta j = s+l,
diagonalizing thereby the spin-orbit interaction, and then takes care of the Coulomb interaction.
The multiplet structure is then described in terms of the total angular momentum J = ji
and its constituting elements ji. This approach is more adequate for the case of a very strong
spin-orbit coupling such as that encountered in lanthanides or actinides. We will use the LS
coupling scheme that is appropriate for 3d, 4d, and with some reservations also for 5d transition
metal ions, where the spin orbit-coupling is weaker and Hund’s coupling is decisive. Here one first
forms the total angular momentum L and total spin S states as we did in the previous paragraph
and later mixes them via spin-orbit coupling. The formal tool to perform this operation is the
Wigner-Eckart theorem. As its consequence, when working in the fixed L and fixed S subspaces,
the spin orbit coupling (2.70) turns out to be equivalent to properly scaled S · L [32]. We will
therefore make a replacement
HSOC = λ S · L (within a subspace with fixed L and S) (2.71)
making the diagonalization of the spin-orbit coupling particularly simple. One can invoke also
an intuitive argument to support this step: Hund’s coupling tends to align the electrons to form
states with the maximum S. In that case we can utilize the relation si = S/2S with 2S being
the number of electrons, which leads to
HSOC ≈
ζ
2S
S ·
i
li =
ζ
2S
S · L . (2.72)
This way we also obtained a connection between ζ and λ in a form λ = ζ/2S which is correct
for less than half-filled t2g shell. In the case of more than half-filled t2g shell, the sign is opposite,
λ = −ζ/2S. We will make one more sign twist by utilizing the more convenient effective orbital
momentum Leff
which is −L projected to t2g orbitals.
As we have learned from Eq. (2.61), the JH part of the Coulomb interaction for a t2g shell
separates the subspaces with fixed S and Leff
, which is the necessary prerequisite for the above
replacement in HSOC. Taking all the relevant factors and signs into account, the spin-orbit
coupling term to be diagonalized reads as
HSOC ≈ λ S · Leff
=
λ
2
(Leff
+ S)2
− (Leff
)2
− S2
=
λ
2
J2
− (Leff
)2
− S2
(2.73)
with λ = ζ for the t5
2g configuration and λ = ζ/2 for t4
2g. We have already incorporated the total
angular momentum J = Leff
+S that enables an elegant solution of the problem by constructing
the eigenstates of J. The Condon-Shortley phase convention that we have consistently kept in
the previous paragraph makes this task straightforward – it is sufficient to combine the Leff
z and
Sz eigenstates using standard tables of Clebsch-Gordan coefficients.
32 2.2 Local correlations and multiplet structure of transition metal ions
1
√
3
2
3
1
√
3
2
3
J = 1
2
Jz = +1
2 Leff
z = +1, Sz = −1
2Leff
z = 0, Sz = +1
2
J = 3
2
J = 1
2
J = 3
2
Jz = −1
2 Leff
z = 0, Sz = −1
2 Leff
z = −1, Sz = +1
2
3
2
λ
doublet
3
2
+
1
2
+
3
2
−
1
2
−J =z
(a)
2g
t
5
(b)
quartet
↓
↓
spin polarization
= + −
= − +
Fig. 13: (a) Energy levels of a t5
2g ion created by spin-orbit coupling. The “shapes” of the states
corresponding to the upper quartet and lower doublet are depicted in a way similar to the previous orbital
figures. For better clarity, we show only the part corresponding to the hole in the t6
2g configuration.
The spin polarization of the hole density obtained as normalized ρ↑ − ρ↓ is indicated by color. (b) Spin
and orbital decomposition of the Jz = ±1
2 holes. The spin polarization is again indicated by color as in
panel (a), the states with nonzero effective orbital momentum have a circular arrow attached.
Let us first focus on the simpler case of the t5
2g configuration. Here we combine Leff
= 1 and
S = 1
2
states which results in a multiplet structure consisting of a J = 1
2
doublet and a J = 3
2
quartet [see Fig. 13(a)]. The level splitting due to the term 1
2
λJ2
in HSOC amounts to 3
2
λ. The
explicit wavefunctions can be written using the |Leff
z , Sz states or the hole operators introduced
e.g. in Eq. (2.56). For the lower doublet we get7
|J = 1
2
, Jz =+1
2
= + 2
3
|+1, ↓ − 1
3
|0, ↑ = + 2
3
a†
↓ − 1
3
b†
↑ |t6
2g = D†
+1
2
|t6
2g ,
|J = 1
2
, Jz =−1
2
= + 1
3
|0, ↓ − 2
3
|−1, ↑ = + 1
3
b†
↓ − 2
3
c†
↑ |t6
2g = D†
−1
2
|t6
2g . (2.74)
The “shapes” of the holes corresponding to these states depicted in Fig. 13(a) and in a decomposed
form in Fig. 13(b) enable to appreciate one particular aspect of the wavefunctions
influenced by spin-orbit coupling. Whenever in some part of its wavefunction the hole shows an
orbital motion, the spin-orbit coupling gets activated and tries to contra-align the hole spin and
the corresponding orbital momentum. This effect generates a nonuniform spatial distribution of
the spin polarization of the hole (or electron) density indicated by color in Fig. 13. As we will see
in Sec. 3, the resulting entanglement of spin and orbital degrees of freedom may have a crucial impact
on the inter-ionic exchange interactions. For completeness, we also give explicit expressions
for the states of the upper quartet, again following the Condon-Shortley phase convention:
|J = 3
2
, Jz =+3
2
= a†
↑ |t6
2g = Q†
+3
2
|t6
2g ,
|J = 3
2
, Jz =+1
2
= 1
3
a†
↓ + 2
3
b†
↑ |t6
2g = Q†
+1
2
|t6
2g ,
7
Let us note, that the double degeneracy of the ionic ground state for t5
2g configuration is guaranteed by
Kramers theorem. Since the ionic Hamiltonian is invariant with respect to time-reversal symmetry, its ground
state manifold for an odd number of electrons will be spanned by two degenerate partners related by the timereversal
operation which is well visible in Eq. (2.74) and its pictorial representation in Fig. 13(b).
2.2 Local correlations and multiplet structure of transition metal ions 33
|J = 3
2
, Jz =−1
2
= 2
3
b†
↓ + 1
3
c†
↑ |t6
2g = Q†
−1
2
|t6
2g ,
|J = 3
2
, Jz =−3
2
= c†
↓ |t6
2g = Q†
−3
2
|t6
2g . (2.75)
In a similar way, one can include the spin-orbit interaction into the multiplet structure of the
t4
2g configuration. The Leff
= 0 and Leff
= 2 spin-singlets of Eqs. (2.62) and (2.63), respectively,
are not affected, the spin-orbit interaction in the LS coupling scheme only reorganizes the Leff
=
1, S = 1 states of Eq. (2.64). Three energy levels corresponding to J = 0, 1, and 2 are generated
within this sector as depicted in Fig. 14(a). The lowest state is a nonmagnetic singlet of total
angular momentum J with fully compensated spin and orbital momentum. Written explicitly
using |Leff
z , Sz notation for the constituting parts or using the hole operators, it reads as
|J =0, Jz =0 =
1
√
3
(|+1, −1 − |0, 0 + |−1, +1 ) =
1
√
3
a†
↓b†
↓ − 1√
2
a†
↑c†
↓ + a†
↓c†
↑ + b†
↑c†
↑ |t6
2g .
(2.76)
The three components mutually compensating their spin and orbital momentum are presented
in Fig. 14(b) along with the final shape of the two-hole cloud corresponding to this state. It has
cubic symmetry and shows no spin polarization. The next three states separated by excitation
energy λ = 1
2
ζ form a triplet with total angular momentum J = 1. They carry certain magnetic
moment but as it was mentioned in the introductory Sec. 1.5 and as we will see in detail in the
next paragraph, the magnetism of the t4
2g configuration is primarily of Van Vleck type, residing
“on the transition” between the above J = 0 state and J = 1 triplet states. The states of the
triplet can be written in a standard way using the |Leff
z , Sz eigenstates of Eq. (2.64)
|J =1, Jz =+1 = 1√
2
(|+1, 0 − |0, +1 ) ,
|J =1, Jz =0 = 1√
2
(|+1, −1 − |−1, +1 ) ,
1
√
3
Leff
z = 0
Sz = 0
Leff
z = +1
Sz = −1
Leff
z = −1
Sz = +1
Leff
= 2
S = 0
Leff
= 0
S = 0
Leff
= 1
S = 1
↓
↓
spin polarization
+2 +1 0 −1 −2J =z
− +=
1×
9×
5×
3JH
2JH
J = 2
J = 1
2g
t4
(a) (b)
λ
2λ
J = 0
Fig. 14: (a) Final energy level scheme of t4
2g configuration (LS coupling). Hund’s coupling first separates
the low-energy triplet sector with Leff = 1 from the two singlet sectors. Spin-orbit coupling rearranges
the triplet sector into three sets of J eigenstates with total angular momentum J = 0, 1, and 2.
(b) Shapes of the two-hole states corresponding to J = 0, 1, and 2 (from bottom to top). The lowest
state with J = 0 is decomposed into its spin and orbital components.
34 2.2 Local correlations and multiplet structure of transition metal ions
|J =1, Jz =−1 = 1√
2
(|0, −1 − |−1, 0 ) . (2.77)
Finally, the topmost states generated within the Leff
= 1, S = 1 sector are the J = 2 states at
the energy 3λ relative to the J = 0 ionic ground state. They are given by
|J =2, Jz =+2 = |+1, +1 ,
|J =2, Jz =+1 = 1√
2
(|+1, 0 + |0, +1 ) ,
|J =2, Jz =0 = 1√
6
(|+1, −1 + 2|0, 0 + |−1, +1 ) ,
|J =2, Jz =−1 = 1√
2
(|0, −1 + |−1, 0 ) ,
|J =2, Jz =−2 = |−1, −1 . (2.78)
At this point we are ready to briefly discuss the differences between the LS and j-j coupling
schemes. It should be first noted, that the spin-orbit coupling was not fully diagonalized in the
above procedure. The problem is the separate consideration of each of the Leff
, S sectors. In
fact, the spin-orbit coupling includes contributions beyond λS ·Leff
that bring additional mixing
between the sectors. The conservation of the total angular momentum is respected but this still
allows the mixing of the two J = 0 singlets of Eq. (2.62) (Leff
= 0, S = 0) and Eq. (2.76)
(Leff
= 1, S = 1) or the mixing of J = 2 states with the same Jz projection (the J = 1 triplet
states have no partners to mix with). To put things explicitly, by using |Leff
, S; J, Jz = |1, 1; 0, 0
and |0, 0; 0, 0 as the basis of the J = 0 subspace, the ionic Hamiltonian takes the form
6U − 13JH +


−ζ −
√
2ζ
−
√
2ζ 5JH

 (2.79)
with the diagonal part coinciding with the J = 0 levels shown in Fig. 14(a). As one can see, the
mixing due to the spin-orbit coupling is not significant provided that ζ is much smaller than the
separation of the two levels ≈ 5JH. Now we can address the same subspace using the j-j coupling
scheme. To this end we take the hole operators producing the J = 1
2
and J = 3
2
eigenstates of
a single hole (i.e. t5
2g configuration) and combine them to form two-hole states of given total
angular momentum. In Eqs. (2.74) and (2.75) we have denoted the respective hole operators as
D†
Jz
(Jz = ±1
2
) and Q†
Jz
(Jz = ±1
2
, ±3
2
). The J = 0 states can be obtained by combining either
two j = 1
2
holes
|j1, j2; J, Jz = 1
2
, 1
2
; 0, 0 = D†
+1
2
D†
−1
2
|t6
2g (2.80)
or two j = 3
2
holes
|j1, j2; J, Jz = 3
2
, 3
2
; 0, 0 =
1
√
2
Q†
+3
2
Q†
−3
2
− Q†
+1
2
Q†
−1
2
|t6
2g . (2.81)
Using these two states as the basis, we get for the matrix of the ionic Hamiltonian
6U − 12JH +


2
3
JH − 2ζ 5
√
2
3
JH
5
√
2
3
JH
7
3
JH + ζ

 . (2.82)
The roles of Hund’s coupling and spin-orbit coupling are interchanged now. Spin-orbit coupling
is diagonalized while Hund’s coupling brings the mixing of the two configurations that would
be neglected in the j-j coupling scheme. Nevertheless, when including the off-diagonal matrix
2.2 Local correlations and multiplet structure of transition metal ions 35
J= 2
1
2
3
2
3
J= 1
J= 2
J= 0
2
3
J=
J= 2
J= 1
J= 0 2
1
J=
J= 2
3
2
3
Leff
Leff
Leff
Leff
ζ
ζ2
1
ζ
ζ
ζ2
1
ζ
J=
SS S S S
d5
d4
dd2
d1 3
Fig. 15: Lowest energy levels for various dn configurations composed of t2g electrons only. The singleelectron
spin-orbit coupling constant ζ is used in all cases to indicate the splitting. The schematics at
the bottom illustrate the addition of the spin and effective orbital momentum in the ground state.
elements both in Eq. (2.79) (going beyond the LS coupling scheme) and Eq. (2.82) (going beyond
the j-j coupling scheme), we arrive at the same exact eigenvalues of the ionic Hamiltonian.
To conclude this paragraph, we show in Fig. 15 the low-energy level structure established
by the spin-orbit coupling for all the non-trivial tn
2g electron configurations with degenerate t2g
orbitals. The figure illustrates the reciprocity of the pairs of complementary configurations d1
–d5
,
d2
–d4
that is caused by the sign change of the spin orbit coupling λ as we increase the number of
electrons. While the effective orbital momentum and spin support each other in the case of the
less than half-filled configurations, they try to compensate each other in the more than half-filled
case. The middle configuration with three electrons maximizes the spin but does not carry the
effective orbital momentum.
2.2.4 Magnetic moment
Magnetic materials are most naturally investigated by probes that couple to magnetic moments.
In the short final paragraph of this section, we will study the connection between the ionic
magnetic moment and the various angular momenta used in the previous text. The magnetic
moment is contributed by both the orbital momenta and spins of the individual electrons of the
open valence shell that sum up to total orbital momentum and total spin
M = −µB
i
(li + g0si) = −µB(L + g0S) (2.83)
with µB ≈ 0.05788 meV T−1
being the Bohr magneton. The prefactors for the orbital part and
spin part differ by the electron g-factor g0 = 2.0023 . . . which is well approximated by 2 for our
purposes. The magnetic moments couple to magnetic fields via the Hamiltonian
Hfield = −B · M = µBB · (L + g0S) (2.84)
analogous to the interaction of a classical magnetic dipole with magnetic field. For convenience,
the magnetic moment is sometimes redefined as M = L + g0S removing the minus sign and
measuring in Bohr magnetons. The interaction with the field in the form µBB ·M should be
used in such a case.
Magnetic moment is a vector operator and by virtue of the Wigner-Eckart theorem, its matrix
elements between the states within a subspace with fixed L, S, J are proportional to the matrix
36 2.2 Local correlations and multiplet structure of transition metal ions
elements of the other vector operators L, S, J. In the case of free ions, the relation between the
magnetic moment and the total angular momentum is given by the well-known Land´e g-factor
g(LSJ) =
3
2
+
S(S + 1) − L(L + 1)
2J(J + 1)
. (2.85)
In the crystalline environment the above formula does not apply since the orbital part of
the magnetic moment is fully or partially suppressed. As a first example we may start with
the situation of undoped high-Tc cuprates where the Cu2+
ions have d9
electron configuration
with one hole in the eg orbital of x2
− y2
symmetry. As evident from Eq. (2.27), the orbital
moment of eg orbitals is fully quenched so that the magnetic moment appears only due to the
spin-1
2
of the hole made in the fully populated and thus fully symmetric d10
shell. A more
complicated situation occurs if the t2g orbitals carrying reduced Leff
participate in the formation
of the magnetic moment M = 2S − κLeff
(expressed within the “positive” convention). The
second contribution comes with a reduction factor κ due to covalency effects between transition
metal ion and oxygens, but let us ignore this in the following and set κ = 1.
For the t5
2g configuration, the spin-orbit coupling led to the J = 1
2
doublet ground state.
When comparing the corresponding operator matrices, we find, that within the two-dimensional
J = 1
2
subspace the following relations hold: S = −1
3
J and Leff
= 4
3
J. The magnetic moment
thus reads as
M = 2S − Leff
= −2J (2.86)
giving the g factor of −2 for the J = 1
2
doublet. For the upper J = 3
2
quartet of the same
configuration we find S = 1
3
J and Leff
= 2
3
J, so that the spin and orbital contributions cancel
in 2S − Leff
= 0 and the upper quartet is thus nonmagnetic. The entanglement of spin and
orbital angular momentum generates also nonzero matrix elements of the magnetic moment that
connect the J = 1
2
and J = 3
2
states, making the quartet visible in magnetic excitation spectra.
Such Van Vleck type of magnetic moment is even more interesting in the case of the t4
2g
configuration. Having the future applications in Sec. 4 in mind, we will focus on the lowest
energy sector consisting of the nonmagnetic J = 0 singlet serving as the ionic ground state and
the J = 1 triplet. The triplet states themselves carry the magnetic moment M = 2S−Leff
= 1
2
J
corresponding to a relatively small g-factor 1
2
. The main part of the magnetic moment is available
in the transitions between the J = 0 and J = 1 states. To capture this Van Vleck moment in
a transparent way we introduce the set of four operators s, T+1, T0, T−1 via the creation of the
J = 0, 1 states given in Eqs. (2.76) and (2.77) 8
s†
|t6
2g = |J =0, Jz =0 T†
m |t6
2g = −|J =1, Jz =m (2.87)
and arrange the triplet ones to a Cartesian form T = (Tx, Ty, Tz) with
Tx =
1
i
√
2
(T+1 − T−1) , Ty =
1
√
2
(T+1 + T−1) , Tz = iT0 . (2.88)
By observing the nonzero matrix elements of the magnetic moment operator, we find two cate-
gories:
s| (2S − Leff
)α |Tβ = −
√
6 i δαβ and Tβ| (2S − Leff
)α |Tβ = −
i
2
αββ . (2.89)
8
Note the negative sign in the triplet states in Eq. (2.87), this was introduced to have the notation compatible
with the papers cited later in Sec. 4.
2.3 Electronic hopping and tight-binding approximation 37
The magnetic moment operator reproducing these matrix elements in the selected four-dimensional
J = 0, 1 subspace can be written down in an elegant way
M = 2S − Leff
= −
√
6 i(s†
T − T †
s) − 1
2
i(T †
× T ) . (2.90)
In this form it is clear that the major potential to generate a magnetic moment have the transitions
between s and T states, the second part is the already mentioned contribution of the J
moment of the triplet that is equivalent to −i(T †
× T ) within the J = 0, 1 subspace.
2.3 Electronic hopping and tight-binding approximation
So far we have been dealing with the (rather complex) physics of correlated valence shells of
the individual ions. In this section we are going to activate connections between the ions in the
form of electronic hopping. There will not be any many-body aspects discussed here as our main
goal is just to get the matrix elements enabling a single electron to move from site to site – socalled
tight-binding parameters entering a single-electron hopping Hamiltonian. As a motivating
example we start by considering independent electrons moving in a crystal consisting of identical
atoms arranged in a simple lattice. Their wavefunctions obey the Schr¨odinger equation
−
2
2m
2
+
R
Vat(r − R) Ψ = EΨ , (2.91)
where Vat(r − R) is the atomic potential for an atom placed at site R. Summed through the
lattices sites, the atomic potentials generate a periodic crystal potential. In the tight-binding
approximation to the problem (2.91), one assumes that the relevant states are well localized so
that the electron wavefunctions can be constructed as linear combinations of atomic orbitals.
This concept is illustrated by Fig. 16 where we construct a virtual two-dimensional crystal made
out of potential wells of circular symmetry and study the evolution of its energy levels when
reducing the lattice spacing, i.e. bringing the initially isolated atoms closer to each other. At
very large lattice spacing, the spectrum of energy levels has a discrete structure below the top of
the crystal potential, corresponding to the individual bound states of the isolated wells. Above
that threshold energy, delocalized states forming a continuum are found. As we bring the “atoms”
closer and closer, the localized states start to overlap and their interaction produces energy bands
of increasing bandwidth. The higher-energy bound states are forming bands sooner because they
have a larger spatial extent and overlap more easily. This is an analogy of the atomic orbitals in a
crystal - the valence ones form bands while the deep electron levels retain their atomic character.
It is intuitively clear that in the situation with rather well localized states (the electrons are
“tightly bound” to their atoms), the appropriate model Hamiltonian should be of the form
HTB =
nR
εn c†
nRcnR −
n ∆R
tnn (∆R) c†
n ,R+∆R cnR , (2.92)
where the operators c†
nR and cnR create/annihilate an electron in the state |φnR corresponding
to orbital n at site R. The first part of this tight-binding Hamiltonian HTB just counts the
energies of the occupied orbitals [c.f. the energies εα in (2.46)], the second part captures the
hopping of electrons between the orbitals located at R and R + ∆R. The amplitudes of the
hopping processes are the matrix elements of the original crystal Hamiltonian such as that of
Eq. (2.91): tnn (∆R) = − φn ,R+∆R| H |φnR . The signs are introduced in such a way that the
hopping parameters t will be mostly positive. For the sake of brevity, we ignore spin that is
38 2.3 Electronic hopping and tight-binding approximation
1/a [nm-1
]
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
10-3
10-2
10-1
100
E − Emin [eV] DOS
p2x
s1
p1yp1x
s2
d1xyd1x2
−y2
p2y
Fig. 16: (left) Wavefunctions of the lowest eigenstates in the potential well described by the 2D potential
Vat(r) = V0 exp(−κr) r0/(r + r0) with V0 = 5 eV, κ = 0.5 nm−1, r0 = 0.5 nm. The indicated levels are
either non-degenerate or two-fold degenerate and they are labeled in analogy with atomic orbitals.
(right) Density of states for a square lattice of the above wells as function of the inverse lattice spacing
1/a. The energy is measured from the lowest eigenstate. For a large spacing (small 1/a) the wells are
practically isolated and the density of states shows discrete peaks at the energies of bound states. Blue
dotted line indicates the average potential level, the red dashed line the top of the potential. The data
to construct this figure were obtained by solving Eq. (2.91) by plane-wave expansion method.
conserved during the hopping and would come as an extra index σ together with σ. While the
values of hopping amplitudes are not known yet, one can expect that the nearest-neighbor and
possibly second nearest-neighbor ones will be most important and – in the case of more orbitals
involved – also anticipate their symmetry structure [see Fig. 17(a) and (b) for two examples].
Owing to the periodicity of the lattice, the Hamiltonian can be easily diagonalized by employing
Bloch waves assembled as linear combinations of the atomic orbitals:
|nk =
1
√
N R
eik·R
|φnR . (2.93)
Here N denotes the total number of sites in the crystal and normalizes |nk to unity when the
overlaps of orbitals at different sites are negligible. By inserting the consistently transformed
electron operators cnR = N−1/2
k eik·R
cnk into HTB, it acquires the form with separated contributions
of the individual Bloch vectors k
HTB =
k nn
εnδnn −
∆R
tnn (∆R) e−ik·∆R
c†
n kcnk . (2.94)
For each k, it remains to diagonalize a matrix whose dimension is equal to the number of orbitals
involved (no diagonalization is thus needed in case of one relevant orbital). For the two examples
2.3 Electronic hopping and tight-binding approximation 39
-6.5
-6.0
-5.5
-5.0
-4.5
-4.0
-3.5
Γ X M Γ
Bloch vector
-21
-20
-19
-18
-17
-16
-15
Γ X M Γ
Bloch vector
-6.1135
-6.1130
-6.1125
-6.1120
Γ X M Γ
-5.16
-5.14
-5.12
-5.10
-5.08
Γ X M Γ
1a=
tπ
tσ
t
t’
(d)
(b)
(a) 2a=
a
(c)
Fig. 17: (a) Hopping processes included in the simplest tight-binding approximation for the bands
derived from s “orbitals” of Fig. 16 in a square lattice. Nearest-neighbor and next nearest-neighbor
hopping amplitudes t and t are indicated. (b) Hopping processes involving the p orbitals on a square
lattice. The symmetry of these states makes certain hopping amplitudes to vanish, the non-zero ones
depend on the relative orientation of the orbitals (tσ and tπ). (c) Band structure obtained for the
setup of Fig. 16 and the value a = 2 nm of the lattice parameter. The weakly dispersing bands derived
from the s1 and p1 levels are shown in detail on the left. The green dashed lines are fits by the
corresponding nearest-neighbor tight-binding dispersion relations. The band structure is plotted along
the conventional path involving Γ = (0, 0), X = (π/a, 0) and M = (π/a, π/a) points in the Brillouin
zone. (d) Band structure for a = 1nm where even the lowest level already shows a significant dispersion.
Its profile seems to be just a scaled version of that from panel (c), demonstrating the applicability of
the tight-binding scheme.
in Fig. 17(a),(b) we get
HTB =
k
εs − 2t(cos kxa + cos kya) − 4t cos kxa cos kya c†
kck (2.95)
and
HTB =
k
c†
pxk c†
pyk


εp − 2tσ cos kxa − 2tπ cos kya 0
0 εp − 2tπ cos kxa − 2tσ cos kya




cpxk
cpyk


(2.96)
giving directly the dispersion relations of electrons. In the latter case, nonzero off-diagonal
elements would be generated by next nearest-neighbor hopping, nearest-neighbor pairs of px
and py orbitals are not connected due to symmetry reasons. The band structures obtained
numerically by solving the full problem (2.91) are presented in Fig. 17 and contrasted to those
resulting in nearest-neighbor tight-binding approximation. A remarkable agreement is obtained
when choosing the proper values of the few parameters (εs and t or εp and tσ, tπ), in particular
for the s band derived from the most localized bound state.
As we have just seen, the tight-binding approximation is a useful tool well capturing the
dispersion of the bands derived from localized states. Its success relies on a limited range of
40 2.3 Electronic hopping and tight-binding approximation
significant overlaps (in the sense of the matrix element of H) of those localized states, reaching
only few nearest neighbors. Our motivational example was based on a collection of weakly coupled
atoms. However, single-electron problems similar to Eq. (2.91) also arise as auxiliary problems
in ab-initio calculations within the framework of density functional theory (DFT). There exist
sophisticated approaches how to construct the local orbital bases such as maximally localized
Wannier orbitals and to extract the values of the corresponding hopping matrix elements making
the tight-binding scheme applicable in a broader context. In a way, by considering the limit
of weakly coupled atoms, we obtain hints about the symmetry/structure of the corresponding
tight-binding model, a realistic DFT calculation then fills in the actual values of the parameters.
After the initial exposition of the tight-binding approach, we will now focus in more detail on
the symmetry properties of the tight-binding matrix elements tnn (∆R) = − φn ,R+∆R| H |φnR .
for transition metal compounds. The relevant ones are those connecting an oxygen ion and
a transition metal ion (i.e. p and d orbitals), and two transition metal ions (only d orbitals
involved). A general approach of their symmetry reduction to as few parameters as possible
under the assumption of spherically symmetric atomic wavefunctions9
was developed by Slater
and Koster [33]. Let us write down the matrix element tnn (∆R) explicitly
tnn (∆R) = − φ∗
n (r − ∆R) −
2
2m
2
+
R
Vat(r − R ) φn(r) d3
r . (2.97)
We can ignore the on-site elements (∆R = 0), these can be incorporated into the local level
structure by renormalizing the energies εn. The basic trick is to use the fact that φn(r) and
φ∗
n (r−∆R) are eigenstates of the atomic Hamiltonian −
2
2m
2
+Vat(r) or −
2
2m
2
+Vat(r−∆R),
respectively. This enables a decomposition of the integral in (2.97) into three contributions
tnn (∆R) = − φ∗
n (r − ∆R) 1
2
Vat(r) + 1
2
Vat(r − ∆R) φn(r) d3
r (2.98)
−
εn + εn
2
φ∗
n (r − ∆R) φn(r) d3
r − φ∗
n (r − ∆R)
R = 0, ∆R
Vat(r − R ) φn(r) d3
r .
We will study just the contribution on the first line and analyze its symmetry for p-d and d-d
orbital pairs and spherically symmetric Vat. The contributions on the second line – orbital
overlaps and a sum of so-called three-center integrals – are usually neglected. In principle, they
can be assumed to renormalize the tight-binding parameters.
To evaluate the two-center integrals 1
2
φ∗
n (r − ∆R) [Vat(r) + Vat(r − ∆R)] φn(r) d3
r one
observes that the term in the bracket has a rotational symmetry with ∆R being the rotational
axis. It is therefore convenient to take the decomposition of the orbitals into spherical harmonics
φn(r) = f(r)
+l
m=−l
cmYlm(ϑ, ϕ) (2.99)
and rotate the angular part to the new set of spherical coordinates ϑrot, ϕrot, where the polar
angle ϑrot is measured from ∆R. This operation amounts to a linear transformation of the set
of coefficients cm. After the transformation, the expression for the two-center integrals contains
azimuthal integrals of the type Y ∗
l m (ϑrot, ϕrot)Ylm(ϑrot, ϕrot) dϕrot where we have to distinguish
the polar angles ϑrot and ϑrot since the origin of the spherical coordinate system differs for the
9
This means wavefunctions of the form (common radial part) × (linear combinations of spherical harmonics)
such as the case explored in Sec. 2.1.
2.3 Electronic hopping and tight-binding approximation 41
dzx
px
dxy
dxyY22
22Y
ϕi
e+ 2
ϕi
e+ 2
dzx
dzx
21Y
Y11
e ϕi
−
e ϕi
+
e ϕi
−
Y21
21Y
e ϕi
+
e ϕi
−
e ϕi
+
e ϕi
−
20Y
Y20
pdσ ddδddπddσpdπ
Y10
20Y
(b) (c)(a) (d) (e)
Fig. 18: Illustration of the basic Slater-Koster integrals for p-d and d-d situation. The main rotational
axis is z pointing upwards. (a) (pdσ) bonding of the in-bond oriented orbitals is the strongest link
between p and d orbitals. (b) (pdπ) bonding is weaker and due to the phases of the spherical harmonics,
it comes with an opposite sign to (pdσ). A real-orbital example is shown on the right – the integral
between p orbital perpendicular to the bond and a matching d orbital is just equal to Slater-Koster
(pdπ), here including the sign. (c) Strongest (ddσ) bonding between two d orbitals. (d) Weaker (ddπ)
bonding with the same sign issues as (pdπ). The integral between the two dzx orbitals shown in the
right example is equal to (ddπ). (e) (ddδ) bonding and a real-orbital example with the same value of
the integral.
two orbitals. The azimuthal angle ϕrot can be chosen as common. The above integrals vanish
for m = m due to the rotational symmetry of the spherical harmonics. The result can thus be
expressed as a linear combination of Slater-Koster integrals (l1l2m) that are defined, following
Eq. (2.98), as the two-center integrals (l1l2m) = 1
2
α∗
(r − ∆R) [Vat(r) + Vat(r − ∆R)] β(r) d3
r
with β(r) = f(r)Yl1m(ϑ1rot, ϕrot) and α(r) = f(r)Yl2m(ϑ2rot, ϕrot). The values of l1,2 = 0, 1, 2, . . .
are specified by the conventional letters for atomic orbitals s, p, d, . . . and those of m = 0, 1, 2
by σ, π, δ, following the chemical bonding nomenclature. Note that the values for +m and −m
are identical.
The basic set of Slater-Koster integrals needed for the analysis of hoppings in transition
metal oxides is presented in Fig. 18. The main contribution to the Slater-Koster integrals will be
presumably collected near the central area of the bond. Taking into consideration the angular
distribution of the spherical harmonics, it may be expected that σ bonding is in general stronger
than π bonding and that is stronger than δ bonding. One can also anticipate the signs: Since
Vat is negative, the Slater-Koster integral typically has a negative sign when the closest lobes
of the two orbitals have equal signs (or same complex phase). Accordingly, the indicated σ
and δ Slater-Koster integrals will be probably negative while the π ones positive. Following
our definition, the hopping parameters tnn will be of opposite signs, for example the hopping t
between s orbitals is positive which is indeed observed in Fig. 17. The above “rules of thumb”
are useful when inspecting the actual hopping channels between ions as we will do soon. To have
a more intuitive notation, we incorporate the anticipated signs into the newly introduced labels
for hopping parameters:
tpdσ = −(pdσ) , tpdπ = +(pdπ) , tddσ = −(ddσ) , tddπ = +(ddπ) , tddδ = −(ddδ) (2.100)
that will be used below and that are not always following Eq. (2.98). In this convention the
hopping parameter will be taken positively if orbital lobes of the same sign “meet” on the bond.
42 2.3 Electronic hopping and tight-binding approximation
−tpdπ
tpdπ −
√
3
2 tpdσ √
3
2 tpdσ
1
2tpdσ √
3
2 tpdσ
tpdπ
1
2tpdσ
−1
2tpdσ−tpdπ
3
4tddσ + 1
4tddδ
−tpdπ
−tpdπ
dx y2− 2
dx y2− 2
dx y2− 2
px
px
d z2−3 r2
dxy
dxy
py
pz
py
px
dxy
(d) (e) (f)
M1
M2
M1
M2
x y
z
O
O
dxy
d z2−3 r2
dxy
(b) (c)
y
z
x
z
y
x
(a)
z
x y yx
z
O
x
y
dzx
dyz
Fig. 19: Examples of hopping channels contributing to nearest-neighbor hopping between two transition
metal ions in the case of 180◦ metal-oxygen-metal bonds (a,b,c) and 90◦ bonds (d,e,f). The indicated
hopping amplitudes use the quantities defined by Eq. (2.100). The insets show the geometry of the
connected octahedra and label the transition metal ions by M1,2. The setup of the axes for the 90◦ case
is identical to Fig. 7(b),(c) (z-bond). There are a few more contributing options that were not shown:
(a) Connection of two dzx orbitals via pz, (b) connection of two d3z2−r2 orbitals via px, (c) situation
with interchanged d orbitals, (d) hopping between dzx and dyz mediated by pz of the front oxygen of
the metal2-oxygen2 plaquette.
When dealing with hoppings in transition metal compounds, we most frequently encounter
either 180◦
metal-oxygen-metal bonds or 90◦
ones (see e.g. Fig. 1). The various options that we
need to consider when connecting two transition metal ions are summarized in Fig. 19. A crucial
observation is that eg orbitals are even when mirrored by any plane containing the bond so that
they can only couple to oxygen p orbitals via σ-bonding, while t2g orbitals have an odd-parity
mirror plane so that only the π-bonding to oxygen orbitals is possible. As a consequence, the
180◦
bond geometry enables only separate t2g–t2g and eg–eg hoppings, t2g–eg mixing is absent.
For the 90◦
bond geometry, on the other hand, the t2g–eg mixing channel is the dominant one and
eg–eg hopping is forbidden by symmetry. Most of the channels depicted in Fig. 19 correspond
to second-order processes involving two successive p-d hoppings. We will now go through the
individual cases and inspect the resulting d-d hoppings.
We start with the 180◦
bond and t2g hopping channel. According to Fig. 19(a) showing an
x-bond situation, a pair of dxy orbitals becomes connected through the mediating py orbital. The
same type of connection, now via pz orbital, can be found for two dzx orbitals. The remaining
t2g orbital dyz is inactive on an x-bond. To derive the effective d-d hopping, we consider the
M1–O–M2 bond in the initial configuration dm
–p6
–dn
corresponding to a completely filled va-
2.3 Electronic hopping and tight-binding approximation 43
lence shell of O2−
. The process of hopping from the transition metal ion M1 to M2 needs to start
with a p-d hopping from the oxygen to M2. This creates a virtual configuration dm
–p5
–dn+1
with
the excitation energy ∆pd (to be discussed in detail later in Sec. 2.4) that can be relaxed when
another electron moves from M1 to the oxygen and fills the hole in p orbitals. The final configuration
is dm−1
–p6
–dn+1
with one electron moved from M1 to M2. Thinking within the framework
of the second-order perturbation theory, the amplitude of the whole process can be estimated as
−tO→2t1→O/∆pd which incorporates the amplitudes of the two successive hoppings. The corresponding
term in the effective d-d hopping Hamiltonian then reads as (+tO→2t1→O/∆pd)c†
2c1. The
extra sign originates in reordering the electron operators that appear in the sequence (p†
c1)(c†
2p)
when following the partial hoppings in the perturbation term. Here p, p†
are electron operators
corresponding to the participating p orbital. A similar process can be constructed when working
out the 2 → 1 direction of the d-d hopping. Taking into account also the opposite signs of the
partial hopping amplitudes observed in Fig. 19(a), we get for the t2g hopping Hamiltonian on the
x-bond
H
(x)
ij = −(t2
pdπ/∆pd)(d†
xydxy + d†
zxdzx)ij + H.c. (2.101)
Hopping Hamiltonians for the other bond directions can be obtained by cyclic permutation.
The situation is more complex for eg orbitals on a 180◦
bond, where all possible combinations
of eg orbitals are connected. Shown in Fig. 19(b), (c) are two options, there is in addition a
connection between two d3z2−r2 orbitals. Evaluating the effective d-d hopping in this case, we
arrive at the bond Hamiltonian
H
(x)
ij = −
t2
pdσ
∆pd
d†
3z2−r2 d†
x2−y2
i


+1
4
−
√
3
4
−
√
3
4
+3
4




d3z2−r2
dx2−y2


j
+ H.c. (2.102)
For a y-bond the hopping matrix is almost the same, the only change is an opposite sign of the
off-diagonal elements as a consequence of the opposite signs of the dx2−y2 orbital lobes pointing
in x and y directions. For a z-bond we have essentially the situation from Fig. 18(a) giving
H
(z)
ij = −(t2
pdσ/∆pd) (d†
3z2−r2 d3z2−r2 )ij + H.c. The dx2−y2 orbital is completely disconnected in that
case. The partial hopping amplitude tpdσ is typically two times stronger than tpdπ which makes
the eg hopping more powerful. A well known example of this type of hopping is the motion of
holes in the CuO2 planes of high-Tc cuprates. Residing in the planar dx2−y2 orbital, they can
fully utilize the geometry of the square lattice with Cu–O–Cu bonds.
Moving on to the 90◦
bond geometry and t2g orbitals, we find that the hopping channel via
oxygen is quite similar to the 180◦
case, with two orbitals out of three being active. However,
the path is bent now which results in an interchange of t2g orbitals. Specifically, for the bond
along 1√
2
(y − x) direction presented in Fig. 19(d) [the geometry coincides with the z-bonds of
Fig. 7(b),(c)], we get
Hij = +(t2
pdπ/∆pd)(d†
zxdyz + d†
yzdzx)ij + H.c. (2.103)
with the two contributions being mediated separately by the two oxygen ions in the M2O2
plaquette. In this geometry there might also be a significant direct overlap of dxy orbitals as
shown in Fig. 19(e). For the other two t2g orbitals, a direct hopping is also possible but weak
because its matrix element contains 1
2
(ddπ) while the dxy orbital uses stronger 3
4
(ddσ). The
orbitals active in oxygen-mediated hopping and direct hopping are thus basically complementary.
As before, we can get the t2g hoppings for the other bond directions by cyclic permutation.
Finally, let us consider the t2g–eg hopping on a 90◦
bond. As demonstrated in Fig. 19(f), the
necessary orientation of the p orbitals is only compatible with the dxy orbital. When connecting it
to the eg orbital, an interesting quantum interference effect occurs. For the d3z2−r2 orbital the two
44 2.4 Mott limit and interactions emerging from residual hopping
hopping channels via front and rear oxygen ions come with the same total phases accumulated
from the partial p-d hoppings so that they add up to
Hij = +(tpdσtpdπ/∆pd)(d†
3z2−r2 dxy + d†
xyd3z2−r2 )ij + H.c. (2.104)
However, for the dx2−y2 orbital with alternating signs of its lobes, the amplitudes of the two
channels add to zero and dx2−y2 is thus disconnected. The Hamiltonians for the other bond
directions may be obtained by cyclic permutation but in this case a subsequent decomposition
of the resulting d3x2−r2 and d3y2−r2 eg orbitals into the conventional dx2−y2 and d3z2−r2 pair is
needed.
2.4 Mott limit and interactions emerging from residual hopping
Having explored both the physics of the individual ions as well as the way how to connect them
via electronic hopping, we are now in position to assemble all together in a form of so-called
multiorbital Hubbard model
H =
i
Hion(i) +
ij
Hhopping(ij) . (2.105)
The first sum goes through the lattice sites and collects the intra-ionic contributions Hion =
ασ εαnασ + HCoul + HSOC that we have thoroughly analyzed in Secs. 2.1 and 2.2. The second
sum runs through the bonds (quite often nearest-neighbor ones but further neighbors can be
included if needed) and activates the various hopping channels as introduced in Sec. 2.3.
At a closer inspection the problem defined by Eq. (2.105) looks intricate and it indeed is.
Without HCoul we would be just facing a band-structure calculation on a single-electron level,
readily performed by an application of the Bloch theorem. However, electron correlations due
to two-body interactions contained in HCoul, that we assume to be strong, make it a genuine
many-body problem.10
We have already successfully handled the electron correlations when
diagonalizing the individual Hion which was a relatively simple task due to a limited Hilbert space
of an individual ion with given number of electrons. This is no more true for a lattice of connected
ions since the Hilbert space dimension grows in a terrifying way – essentially exponentially with
the number of lattice sites. Moreover, the base for this exponential is not small due to several
orbitals involved and combined with spin-1
2
. When resorting to a fully numerical diagonalization,
even the huge computational power easily accessible nowadays enables to exactly treat clusters
with a few transition metal ions only.
One way out is to simplify the model by identifying the relevant ionic states – typically the
low-energy multiplet states – and formulate an effective model in terms of those. The actual model
may be obtained, for example, by getting rid of the high-energy states in a perturbative manner.
A proper choice of the elementary objects for the model and processes to be included can make
the physics behind the particular material more transparent and guide further approximations.
Even though the results may be qualitative only, the insights gained are sometimes more valuable
than a quantitative treatment of the original Hubbard model by some complex numerical method.
Our focus is on models with localized degrees of freedom appearing as effective models for
undoped Mott insulators, the canonical example being a spin model. In the introductory section
1.2 we took a very simplistic approach to the problem of its emergence. The aim of the
present section is to put it on a bit more solid ground to get ready for a derivation of “realistic”
models in Secs. 3 and 4.
10
In principle, there are also inter-ionic interactions of two-body character that could be included in the model,
such as Hubbard repulsion of the electrons residing at neighboring ions, but these are only needed in special
situations and we do not need to address them in our cases of interest. Consequently, the only source of correlated
behavior of electrons in our models will be the intra-ionic electron-electron interactions in HCoul.
2.4 Mott limit and interactions emerging from residual hopping 45
rA+rO
f =
rA + rO
√
2(rB + rO)
tolerance factor
n 5 n+1
n6n
n+15n−1
d p d
d p d
dpd
∆
Ueff
pd
rB+rO
A
B
O
(c)(a)
2
1
(b)
Fig. 20: (a) Hierarchy of energies of charge excitations for a undoped Mott-Hubbard insulator. The
lowest elementary state is the balanced dn-p6-dn configuration of metal-oxygen-metal bond with a full
p shell of the O2− ion. Process denoted as 1 is a charge-transfer process leading to a p-d excitation
– a virtual state with the excitation energy ∆pd. It may get relaxed by a backward process 1 or by
process 2 producing a d-d excitation. (b) Phase diagram of the RNiO3 family of compounds as function
of the tolerance factor and temperature (from Torrance et al. [34]) (c) Definition of the tolerance factor
quantifying the deformations of the unit cell in ABO3 perovskites. The labels rA, rB, and rO should
bring associations with the respective ionic radii. For an ideal cube f = 1.
2.4.1 Metal-insulator transition
As it was emphasized in Sec. 1.2, a key element in transition metal oxides is the competition
between the tendency of electrons to delocalize and form bands and Coulomb repulsion that
wants to keep the electrons apart. Mott insulating state sketched in Fig. 2(b) and forming a
basis for our model description appears as a consequence of the latter mechanism taking over.
Considering the details, one has to be careful because the role of the oxygen bridges between
the transition metal ions may be more complex than mere mediators of d-d hopping. According
to the scheme developed by Zaanen, Sawatzky, and Allen [35, 36], the insulating states may
be classified as Mott-Hubbard insulators or charge-transfer insulators. The idea behind this
classification is that the elementary hopping which moves an electron between p orbital of oxygen
ion and d orbital of the transition metal ion may create a virtual state that is more convenient
than the one generated by full d-d hopping. When “deriving” the effective d-d hopping via oxygen
in Sec. 2.3 we have handled it in a bit handwaving way having the Mott-Hubbard regime in mind.
Let us now analyze this issue in more detail following the scheme in Fig. 20(a) that applies to
the Mott-Hubbard situation. The starting point is the most probable dn
–p6
–dn
configuration
of a metal-oxygen-metal bond. Since the valence shell of O2−
is full in this configuration, the
action may only start by moving one of its electrons to a neighboring transition metal ion via
p-d hopping. This creates p-d charge excitation dn
–p5
–dn+1
whose energy ∆pd is defined as the
difference of the energies of the two states [see Fig. 20(a)]. It is contributed by the energy
difference between the d and p orbitals εd − εp and the Coulomb energy associated with the
change dn
→ dn+1
.11
In the next step, we can either restore the initial situation by backward
p-d hopping or continue to reach d-d excitation with the configuration dn−1
–p6
–dn+1
that has an
energy Ueff. This is now contributed by the Coulomb energy associated with the simultaneous
11
The changes of the Coulomb energy in the p shell of oxygen related to the p6
→ p5
process may be absorbed
into the definition of εd − εp.
46 2.4 Mott limit and interactions emerging from residual hopping
5
10
15
20
0
2
4
6
8
10
12
14
16
0.0
0.5
1.0
1.5
2.0
site U / t
〈n〉
0.0
0.5
1.0
1.5
2.0
0 5 10 15 20 25
〈n〉
site
U/t=12
0.0
0.5
1.0
1.5
2.0
〈n〉
U/t=1
insulator< UW <
metalU W<~
V( )x
(a) (c)(b)
Fig. 21: (a) Two states of a Hubbard chain in a parabolic trap that can be distinguished by observing
the electron density. They appear depending on the ratio between the “bandwidth” W ∝ t and Hubbard
repulsion U. (b) Average site-dependent occupation in the ground state of the chain with 24 sites and
12 fermions (N↑ = N↓ = 6) as function of the U/t ratio. The parabolic potential reaches 30t at the
chain ends. Mott transition can be observed around U/t ≈ 8 in this case. (c) Density profiles well below
and well above the critical U/t corresponding to the Mott transition.
changes dn
→ dn+1
and dn
→ dn−1
. In our schematics in Fig. 20(a) it is the Hubbard U but often
one encounters Hund’s coupling effects in this virtual state as well. In a charge-transfer insulator,
∆pd < Ueff so that the bonds excited by p-d hopping are not eager to continue towards forming
the d-d excitation and prefer to spend more time with a hole on oxygen. The resulting state of
the entire system is still an insulator that looks like in Fig. 2(b) but the picture behind clearly
differs from Fig. 2(c). This is the case of late 3d transition metal oxides like high-Tc cuprates. In
a Mott-Hubbard insulator the lower excitation is the d-d one (i.e. Ueff < ∆pd), so that the states
with p5
configuration of oxygen are indeed by a larger part just the mediators of the effective
d-d hopping as we have implicitly assumed before. Of course, the distinction is not strict, both
p-d and d-d charge excitations are in the game, but their proportion depends on the ∆pd to Ueff
ratio and we label the particular system according to the prevailing one.
Both insulating scenarios above require the hopping amplitudes to be substantially smaller
than the the lower of the two charge-excitation gaps ∆pd and Ueff, otherwise the insulating state
is not kept and we end up in a metallic state. Sometimes we are able to tune the balance
between the competing kinetic energy and strong correlations and drive the system through the
metal–insulator transition. We will now illustrate it in two examples. The first one is a set
of experimental data on RNiO3 family of perovskite nickelates that was assembled into a very
instructive phase diagram by Torrance et al. [34]. The phase diagram is reproduced in Fig. 20(b).
Its horizontal axis needs some explanation. The quantity shown there is so-called tolerance factor
which measures how much the unit cell got deformed from an ideal cube. By changing the R
cation, one can significantly influence this deformation. Thinking intuitively, the closer we are
to the cubic situation with straight bonds, the larger is the overlap of the orbitals and hence
the hopping. Going away from the cubic case, we may therefore experience a metal to insulator
transition as indeed visible in the the phase diagram by Torrance et al. who collected data for
various R elements and also mixed solid solutions. At low-enough temperatures the insulator
develops antiferromagnetic order, consistently with our intuitive picture.
Another example that we will analyze more thoroughly is the metal-insulator transition observed
in a one-dimensional Hubbard chain that is actually sketched in Fig. 2(b),(c) as a prototype
system. We have simulated this system numerically using exact diagonalization for short
2.4 Mott limit and interactions emerging from residual hopping 47
chains so that we can freely tune the ratio of the main parameters – hopping t and Hubbard
repulsion U. The first results presented in Fig. 21 concern a Hubbard chain with an extra potential
of a parabolic trap.12
This problem was studied in the context of atoms trapped in an
optical lattice [37, 38] so we should be talking about fermionic spin-1
2
atoms instead of electrons.
Nevertheless, the physics of the Hubbard model works the same way. When looking at
the metal-insulator transition, one should compare the bandwidth W to the Coulomb repulsion
represented by Hubbard U. Here the concept of bandwidth is not well defined but we can still
understand it as “some quantity proportional to the hopping t.” The inhomogeneous situation
due to the parabolic trap brings some specifics. We can distinguish two phases just by observing
the distribution of the density of fermions n as sketched in Fig. 21(a) – a bell-shaped fermionic
cloud with notable double occupancy replaces the metallic phase of a homogeneous system, the
insulating phase is represented by a state where a Mott insulating core develops, having a flat
density profile with n ≈ 1. The transition around U/t ≈ 8 is well visible when studying the
evolution of the density profile with U/t [see Fig. 21(b),(c)].
Another way of looking at the Mott transition in 1D Hubbard chain, somewhat less exotic
compared to the above one, is an inspection of the spin excitations as evolving with U/t. In
Fig. 22(a) we perform this inspection for a periodic chain of 16 sites, again simulated numerically
by exact diagonalization. The finite-sized chain has a discrete set of wavevectors q but the shape
and intensity of the spin excitation spectrum is sufficiently densely covered for our purposes.
The bandwidth W = 4t stemming from the dispersion εk = −2t cos ka is well defined in this
case and appears in the metallic limit t U also as the bandwidth of the spin excitations. In
12
Here our calculation reproduces the results presented in Ref. [37] that were obtained using Earth Simulator –
the most powerful supercomputer in the world from 2002 to 2004. Not even two decades later, we were able to get
them using a refurbished server which well demonstrates the remarkable steady progress in computer technology.
ω/t
qa
U = t/10
0
2
4
6
8
10
12
0 π 2π
qa
U = t
0 π 2π
qa
U = 2t
0 π 2π
qa
U = 5t
0 π 2π
5×
qa
U = 10t
χ″(q,ω)
0 π 2π
20×
ω/(4t2
/U)
0
1
2
3
0 π/2 π 3π/2 2π
0
1
2
3
4
5
ω/J
0
1
2
3
0 π/2 π 3π/2 2π
0
1
2
3
4
5
∝ J =4t2
/U
(b)
(c)
(a)
W
U
qa
Fig. 22: (a) U/t dependence of the spin susceptibility of the one-dimensional Hubbard chain with 16
sites and periodic boundary conditions applied. Plotted is the imaginary part χ (q, ω) as function of
one-dimensional momentum q ∈ [0, 2π] and energy ω. The intensity of the upper part of the spectrum
is magnified in the last two panels to make it better visible. (b) Detailed view on the low-energy part
of the spin susceptibility for U/t = 10 with ω scaled by the anticipated exchange constant J = 4t2/U.
(c) Spin susceptibility obtained for the Heisenberg model on the same chain. The dotted lines indicate
the lower and upper bound of the spin excitations known from the exact solution of an infinite chain by
Bethe Ansatz.
48 2.4 Mott limit and interactions emerging from residual hopping
this regime Fig. 22(a) essentially shows the Lindhard function for a 1D band. With increasing U
there is more and more spectral weight appearing at low energies at the AF wavevector q = π/a,
this is a sign of incipient AF correlations. The overall shape of the excitation spectrum remains
the same, however. A dramatic change happens around U/t ≈ 5 where the spectrum reveals a
clear separation of two energy scales and a birth of an effective model. Tracing the two parts
of the spectra to even higher U/t regime where the Mott insulator is well developed, we can
attribute the low-energy part to an effective Heisenberg model with the excitation bandwidth
proportional to the effective exchange parameter J = 4t2
/U. The high-energy excitations start
around the energy U signifying the presence of doubly occupied sites in the participating states.
With increasing U/t, their intensity gradually weakens. A conclusive test of the emergence of the
Heisenberg model is provided by a comparison of the low-energy part of the spin susceptibility
obtained within Hubbard model [Fig. 22(b)] and that of the Heisenberg model on the same chain
[Fig. 22(c)]. Due to its one-dimensionality, the Heisenberg chain does not order magnetically and
the spin excitations form a continuum instead of narrow magnon branch(es) but this difference to
higher-dimensional antiferromagnets does not matter here, since we are only interested in finding
signatures of the emergent model. We scale the energy axis by the exchange constant in both
cases and observe an excellent agreement. There are some tiny shifts visible near the top of the
two spectra in Fig. 22(b) and Fig. 22(c) but otherwise they display no noticeable differences.
Therefore, the description of the Hubbard chain by an effective spin model seems to be perfectly
justified in this large U/t limit with roughly U 2W.
2.4.2 Superexchange interactions
Having prepared the foundations for the effective model in a form of a Mott insulator, we now
come to the question of the effective interactions. When mediated by oxygen ions or some other
bridging ions, they are commonly called superexchange interactions to make a distinction between
direct exchange interactions of two neighboring ions and the more complex mechanism. The first
theory of superexchange interactions was proposed in 1934 by H. A. Kramers [39], later in 1950’s
it was refined by P. W. Anderson in his seminal works on the topic [40, 41]. In this paragraph
we are going to expose the general formalism how to obtain exchange interactions and illustrate
it on a relatively simple but nontrivial example of superexchange interactions in cuprates.
The exchange interactions stem from the short excursions of electrons to the neighboring ions,
the simplest example being sketched in Fig. 2(c). These charge excitations are rather costly. As
we have discussed in the previous paragraph, they are separated by the gap Ueff or ∆pd that
is assumed to be significantly larger than the hopping parameters tpd. The problem apparently
calls for a perturbative treatment of the hoppings and we will indeed proceed that way, treating
the ionic part of the Hamiltonian (2.105) as the unperturbed Hamiltonian and the hopping part
as a perturbation. The situation is complicated by two facts: First, we have to perform the
perturbation theory to a relatively high order. Following the scheme in Fig. 20(a) for MottHubbard
case, we need in total four hopping processes to get to the virtual excited state and
back. Second, the states we are perturbing (i.e. products of ionic multiplet states) are degenerate/quasidegenerate.
The usual Rayleigh-Schr¨odinger perturbation theory found in textbooks
is not very convenient here since it leads to relatively complex expressions. In the following
derivation, we will employ the more suitable Brillouin-Wigner variant of the perturbation theory
adapted for quasidegenerate levels.
We will first consider a general problem with the full Hamiltonian H = H0 + λV consisting of
the unperturbed one H0 and the perturbation V accompanied by the usual expansion parameter
λ to be varied from 0 to 1. The level structure for a quasidegenerate case of our interest is
schematically represented in Fig. 23(a). The key concept, introduced independently by Feshbach
2.4 Mott limit and interactions emerging from residual hopping 49
t
t
















































Heff
HA
HB
λV
HA
HBλV
λV
(c)(a)
∆
(b)
Fig. 23: (a) Level scheme of the problem to be treated by the perturbation theory. By virtue of the
unperturbed Hamiltonian H0, the Hilbert space splits into low-energy states contained in the subspace
HA and high-energy states in HB. The states of these subspaces are separated by the energy gap
∆ and connected by a weak perturbation λV. (b) Block structure of the corresponding Hamiltonian
matrix. The unperturbed Hamiltonian H0 does not connect the two Hilbert subspaces HA and HB.
This connection is provided by the off-diagonal blocks of λV which may also contribute to the diagonal
blocks. The matrix elements of the off-diagonal blocks λV are assumed to be significantly smaller than
the energy gap ∆ enabling a perturbative treatment. (c) The desired structure of the Hamiltonian.
After performing a proper basis rotation, the off-diagonal blocks connecting low-energy sector to highenergy
virtual states are eliminated and the effective Hamiltonian for the low-energy sector is found in
the corresponding diagonal block.
[42, 43] and L¨owdin [44, 45], is the partitioning of the Hilbert space. Based on unperturbed
energies (i.e. the spectrum of H0), we single out the low-energy subspace HA and let its states
interact by a weak perturbation λV with the states of the high-energy subspace HB. The ultimate
goal is to find an effective Hamiltonian acting on states in the lower subspace but leading to the
same results as the full one. To be more concrete, we introduce a decomposition of a state from
the full Hilbert space utilizing the projection operators onto HA and HB subspaces:
|ψ = |ψA + |ψB = P|ψ + Q|ψ , Q = 1 − P . (2.106)
We now want to construct the effective Hamiltonian Heff in such a way, that a projection of the
true eigenstate of H onto HA is an eigenstate of Heff with the same eigenvalue:
(H − E)|ψ = 0 → (Heff − E) P|ψ = 0 . (2.107)
Vaguely speaking, the effective Hamiltonian has to take the low-energy component P|ψ , complete
it with the corresponding high-energy part, act with the full Hamiltonian, and return a
low-energy component of the result. This “hidden” admixture of the high-energy sector that effectively
eliminates λV roughly corresponds to the basis rotation that is suggested in Fig. 23(b),(c).
The above task is quite demanding and may be precisely fulfilled only by an energy-dependent
Heff(E). We are, in an indirect way, facing the inherent problem of the Brillouin-Wigner perturbation
theory where the final perturbed energies themselves enter the expressions for the
perturbation corrections. It is not a big issue, however, since the target energies E of interest
form a relatively narrow band compared to the intermediate excitation energies so that we can
make approximations easily. An explicit formula for Heff(E) is obtained by inserting (2.106)
into (H − E)|ψ = 0 and decomposing the results by applying the projectors P and Q. For the
high-energy component we get
QHP|ψ + QHQ|ψ − EQ|ψ = 0 → Q|ψ =
1
E − QHQ
QHP|ψ (2.108)
50 2.4 Mott limit and interactions emerging from residual hopping
with the operator/matrix inversion happening within the HB subspace. The above expression
can be used in the low-energy component of the original equation to yield (Heff − E)P|ψ = 0
with
Heff(E) = PHP + PHQ
1
E − QHQ
QHP . (2.109)
This result is still exact and general, though not very useful without the level separation of
Fig. 23(a). By observing the structure of the full Hamiltonian matrix depicted in Fig. 23(b), we
can see the correspondence of its blocks to the individual terms appearing in Eq. (2.109). The
second-order effective Hamiltonian can be obtained very quickly by approximating the denominator
by −∆, assuming that the relevant states from HA, HB form narrow bands as compared
to the gap ∆. To reach higher orders, we need to systematically expand the resolvent of the
projected Hamiltonian:
G(E) =
1
E − QHQ
(2.110)
entering Eq. (2.109) and operating in HB. To this end we introduce the unperturbed resolvent
G0(E) =
1
E − QH0Q
(2.111)
that is related to the full one by the operator/matrix relation
G−1
0 (E) − G−1
(E) = λQVQ . (2.112)
By applying G0(E) from the left and G(E) from the right, we get Dyson’s equation
G(E) = G0(E) + λ G0(E) QVQ G(E) (2.113)
that can be solved by repeated iteration, adding one order in λ in each step. When inserting the
resulting series into Eq. (2.109), we obtain the perturbation series for the effective Hamiltonian
in the form
Heff(E) = PH0P + λ PVP + λ2
PVQ
1
E − QH0Q
QVP+
+ λ3
PVQ
1
E − QH0Q
QVQ
1
E − QH0Q
QVP + . . . , (2.114)
where we have used the fact that H0 does not connect HA and HB. This concludes the derivation
of the general Brillouin-Wigner perturbation theory for quasidegenerate levels.
The perturbation expansion (2.114) may look intimidating at the first sight, but in the examples
below we will see that it actually leads to a relatively simple and intuitive scheme for the
derivation of the exchange interactions. A convenient choice of the basis are product states where
each ion is in some multiplet state, which means that the ionic part of Eq. (2.105) (our H0) is already
diagonalized. To get the total energy of the configuration, one just sums up the energies of
the participating multiplet states. This makes the evaluation of (E − QH0Q)−1
entering (2.114)
trivial. The perturbation V is represented by the hopping Hamiltonian. When evaluating the
second-order contribution to Heff via (2.114), we start with the initial low-energy configuration,
jump into the high-energy subspace by the first hopping and leave it by the second one, reaching
the final low-energy configuration. The corresponding matrix element of Heff is the product of
the used hopping matrix elements divided by negatively taken excitation energy. To get the
higher-order contributions, we are additionally traveling through the high-energy subspace via
further hoppings, collecting the hopping matrix elements used on the way and dividing by the
2.4 Mott limit and interactions emerging from residual hopping 51
excitation energies of the visited states. Here one often ignores the finer structure of the excited
multiplet levels and counts only U contributions, neglecting the smaller shifts due to JH, for
example.
Let us now inspect the half-filled one-band Hubbard model as the simplest example. The
second order process from Fig. 2(c) concerns only two sites, we can therefore consider an isolated
bond as well. Due to the spin conservation in the hopping processes, it pays off to start with total
spin eigenstates as the initial configurations. In our case, the two electrons on a bond may form
a singlet and three triplet states. Due to the Pauli principle, we find that only a singlet state
|s = 1√
2
(c†
1↑c†
2↓ − c†
1↓c†
2↑)| on the bond is connected to the high-energy sector, here the relevant
state is the doubly occupied combination |d = 1√
2
(c†
1↑c†
1↓ + c†
2↑c†
2↓)| with the excitation energy
U. The corresponding matrix element is d|(−t) σ(c†
1σc2σ + c†
2σc1σ)|s = −2t, the second-order
effective Hamiltonian thus reads as
Heff(E) =
4t2
E − U
|s s| ≈ −
4t2
U
|s s| . (2.115)
The above preference of singlet state of the two electron spins at every bond can be encoded into
the final Heisenberg Hamiltonian
H = J
ij
Si · Sj − 1
4
with J =
4t2
U
. (2.116)
A bit more involved is the evaluation of the superexchange interaction in cuprates [46] that
serves as a good introductory example of a fourth-order calculation. Additional examples with
more complex orbital structure will be presented in Secs. 3 and Sec. 4. Because the d shell is
almost filled in this case, it is better to work with a hole representation using the configuration
d10
as a “vacuum”. In the undoped CuO2 plane, each Cu2+
site with d9
configuration is occupied
by a hole that goes to the planar dx2−y2 orbital. The lowest-order superexchange processes appear
in fourth order in the p–d hopping and are depicted in Fig. 24. Again we need to consider one
Cu–O–Cu bond only and this choice determines the active p orbital bridging the dx2−y2 orbitals.
To derive the superexchange, we start with singlet of the two holes |s = 1√
2
(h†
1↑h†
2↓ − h†
1↓h†
2↑)| .
Here the empty state | corresponds to the d10
–p6
–d10
bond configuration. Triplet configurations
2+ 2− 2+
Cu O Cu
2+ 2− 2+
Cu O Cu
U
2∆+U
d
p
1 2(b)
1 2(a)
43
43
Fig. 24: (a) Virtual process of charge transfer type on a CuO2 bond. Apart from two p-d excitations
(each of the energy cost ∆pd), the holes meeting in the oxygen p orbital experience Hubbard repulsion
Up. (b) Virtual process involving d-d transition. The energy cost of the intermediate state is the usual
Hubbard repulsion U in the d orbital.
52 2.4 Mott limit and interactions emerging from residual hopping
are again disqualified because the two holes have to meet at some point in a single orbital (either
p or d). As shown in Fig. 24, after two p-d hoppings we can reach two kinds of intermediate
states. In so-called charge-transfer process, the two holes meet at the oxygen ion. This state
can be reached in two ways – the one shown in Fig. 24(a) and by a hopping sequence with
interchanged hoppings 1 and 2. The other possibility is shown in Fig. 24(b) and corresponds to
a d-d charge excitation. Here we can end up with two holes at the right Cu site (as depicted) or
the left one. Collecting the possible hopping routes to get to the middle intermediate states, we
obtain for the corresponding piece of the full fourth order formula
QVQ
1
E − QH0Q
QVP =
2
√
2 t2
pd
E − ∆pd
|p↑p↓ s| +
√
2 t2
pd
E − ∆pd
|h1↑h1↓ + |h2↑h2↓ s| . (2.117)
The intermediate state |p↑p↓ has energy 2∆pd + Up due to two p-d excitations and Hubbard
repulsion of the two holes that meet at the bridging oxygen. The d-d excitations |h1↑h1↓ and
|h2↑h2↓ have energy equal to Hubbard Ud for d orbitals. The final expression is obtained by using
those energies in the denominator of the middle resolvent 1
E−QH0Q
of the fourth order formula
and applying conjugate operator to that of (2.117) to get back to the initial state. Neglecting E
in the denominators, we get for the two types of processes:
H
(CT)
eff ≈ −
8 t4
pd
∆2
pd(2∆pd + Up)
|s s| and H
(dd)
eff ≈ −
4 t4
pd
∆2
pdUd
|s s| , (2.118)
which gives the superexchange constant
J =
4t4
pd
∆2
pd
1
∆pd + 1
2
Up
+
1
Ud
. (2.119)
Cuprates are charge-transfer insulators with ∆pd < Ud (roughly by a factor of two), the first
contribution to J is thus the larger one. As we can see the subdominant d-d contribution has
the same form like the second-order result (2.116) with the effective d-d hopping t = t2
pd/∆pd.
As an illustration of the superexchange mechanism between two d9
Cu2+
ions in a real material,
we will discuss the interesting case of copper fluoride KCuF3. Despite having a threedimensional
perovskite structure, this compound surprisingly hosts quasi one-dimensional magnetism.
The key to the quasi-1D situation is the particular orbital arrangement due to Jahn-Teller
qa/2πwavevector
(a) (b)
energy(meV)
= 34meV = −1.6meVJ J
J
J
Fig. 25: (a) Two types of orbital ordering in KCuF3 and the superexchange constants. The small
balls represent fluorine ions, the larger ones potassium ions. Copper d-type orbitals form quasi onedimensional
chains (vertical) that are almost decoupled due to the orthogonality of orbitals. (b) Spin
excitations in KCuF3 measured by inelastic neutron scattering [47]. Dashed lines have the same meaning
as in Fig. 22(b),(c).
2.4 Mott limit and interactions emerging from residual hopping 53
effect [48]. The orbital ordering comes in two polytypes presented in Fig. 25(a). From the point
of view of the superexchange mechanism, fluorine ions act similarly to oxygen ions. They have
higher electronegativity and the relative energy levels change, but the physical picture shown
in Fig. 24 is preserved. According to Fig. 25(a), the superexchange mechanism discussed above
works only along one-dimensional chains formed by the properly oriented d orbitals. In the
perpendicular direction the p-d hopping is broken due to orthogonality of the orbitals. As a
consequence, the effective J between the chains is only about 5% of the intrachain one. The
quasi-one-dimensionality is very clearly confirmed by the spin excitation spectrum in Fig. 25(b)
obtained by inelastic neutron scattering [47]. It can be directly compared to the one calculated
for a Heisenberg chain and presented in Fig. 22(c).
As a concluding remark let us note, that even though we tried to develop the perturbation
expansion in a formally clean way, one should keep in mind that it frequently gives only indicative
results. For example it may have problematic convergence due to insufficiently large ratio of the
charge-excitation gap to the hopping amplitudes in the realistic parameter setup, the approach
to the ionic states and hopping matrix elements may be too simplistic etc.
54 2.4 Mott limit and interactions emerging from residual hopping
3 Kitaev materials
This part of the thesis is devoted to a more in-depth discussion of the so-called Kitaev materials
that were briefly introduced in Sec. 1.4. They feature highly anisotropic and bond-selective spin
interactions which result in a very peculiar magnetic behavior. Since these materials have been
intensely studied in the last ten years, there already exists a large volume of literature which is
pointless to review here since this task has been performed many times already and there are
several review papers taking the subject from various perspectives [8–10,49–55]. Instead, we keep
the “pedagogical” character of the text and focus on a few subjectively selected aspects that we
consider essential.
Being armed with the tools developed in Sec. 2, we will first study the origin of the bondselective
interactions, highlighting the crucial role of the entanglement of spin and orbital degrees
of freedom created by spin-orbit coupling and combined with the bond-selectivity of the hopping
processes. The effective spin model derived for a simplified case will be later further extended
and we will discuss its very rich phase diagram that is not fully explored up to now. In the last
section of this part, we will consider the experimentally observed zigzag phase and illustrate,
in the context of measured data, the specific features that appear in the regime with dominant
Kitaev interaction.
3.1 Microscopic origin of the Kitaev interaction
The most heavily studied Kitaev materials Na2IrO3 and α-RuCl3 are essentially a stack of honeycomb
lattices made of edge-sharing octahedra as shown in Fig. 1(c) and Fig. 7. This is not
the only option, the transition metal ions may form also three-dimensional structures such as
the hyperhoneycomb [56] or harmonic honeycomb lattice [57] and other types of two-dimensional
lattices, for instance the triangular [58] or kagome lattice [59], are considered theoretically as
well. Two elements are common among the Mott insulating Kitaev magnets: (i) approximately
90◦
metal-oxygen-metal bonding, (ii) ionic ground state being a doublet generated by spin-orbit
coupling. The latter is realized by the t5
2g configuration of 5d Ir4+
(see Refs. [60,61] for an interesting
experimental verification of the complex spin-orbit entangled wavefunctions) or 4d Ru3+
ions with relatively large spin-orbit coupling and we will focus on this particular situation in the
following, but even the 3d elements with smaller spin-orbit coupling may be relevant according
to the recent proposals [62–64] for t5
2ge2
g Co2+
systems.
The above two elements generate highly anisotropic spin interactions as we will show by
explicitly deriving the corresponding spin model. The local basis for the model is the J = 1
2
doublet of t5
2g configuration that was thoroughly analyzed in Sec. 2.2.3. Choosing the z axis for
J quantization, the two basis states read as
|˜↑ = 2
3
a†
↓ − 1
3
b†
↑ |t6
, |˜↓ = 1
3
b†
↓ − 2
3
c†
↑ |t6
. (3.1)
Here we have used tilded arrows to emphasize that our local degree of freedom is J = 1
2
pseudospin
instead of a regular spin-1
2
. The states are formulated as in Eq. (2.74) in terms of holes in t6
2g
configuration created by the operators a, b, c of Eq. (2.56), corresponding to the Leff
z = +1, 0, −1
eigenstates. The superposition of various spin and orbital labels in the pseudospin states leads
to rather complex superexchange processes because the two orbital components have different
fate when considering hopping on the honeycomb lattice as summarized in Fig. 26.
In the honeycomb lattice one encounters three types of bonds, each characterized by a different
orientation of the square M2O2 plaquette with two 90◦
nearest-neighbor M–O–M bonds. The
focus in Fig. 26(a),(b) is on a z-bond that is perpendicular to the cubic axis z. The bond direction
55
56 3.1 Microscopic origin of the Kitaev interaction
ηξ
ζ
η
ξ
a†
σ = (+σ)(iη-σ +ξ-σ )/
√
2
b†
σ = (−σ)ζ-σ
c†
σ = (+σ)(iη-σ −ξ-σ )/
√
2
η
ξ
y
z
x
z
x y
(a)
(b)
y
yx
x
c
b
+it
−it
t’
z
z
a
b
(c)
Fig. 26: (a) Oxygen-mediated hopping in the square M2O2 plaquette perpendicular to the cubic axis z.
The two elementary bonding routes connecting ξ and η orbitals combine into imaginary hopping between
a and c orbitals which changes Leff
z by the maximum ∆Leff
z = ±2. (b) Direct hopping connects primarily
only b orbitals (i.e. ζ) and preserves therefore their Leff
z = 0. There is in addition a weaker ddπ and
ddδ coupling between the other orbitals that we neglect. (c) Orientation of the two relevant octahedra
for the z-bond used in panels (a) and (b) and cyclic exchange rules in the honeycomb lattice. When
switching between the three bond directions, the definitions of a, b, c hole operators shown in the middle
change by permuting the Cartesian orbitals ξ, η, ζ. This is a consequence of the corresponding rotation
among the cubic axes x, y, z that can be used when the C3 symmetry of the system is not broken.
selects the two orbitals active in the effective hopping mediated by oxygen ions, in this case they
are ξ ∼ yz and η ∼ zx [see Fig. 26(a)]. The 90◦
bond geometry also allows for a significant direct
overlap of the orbitals. Considering just the strongest ddσ bonding, the direct hopping concerns
only the orbital complementary to the above two [Fig. 26(b)]. In total, the electron hopping
Hamiltonian on a z-bond takes the form
Htt = t
σ
ξ†
iσ ηjσ + η†
iσ ξjσ − t
σ
ζ†
iσ ζjσ + H.c. , (3.2)
where t = t2
pdπ/∆pd and H.c. adds the opposite hopping direction i → j. With the pseudospin
basis being expressed using hole operators a, b, c, it is more convenient to switch to the hole
picture and convert the above Htt to
Htt = −it
σ
a†
iσ cjσ − c†
iσ ajσ + t
σ
b†
iσ bjσ + H.c. (3.3)
For the other two bond directions, we can construct Htt based on Eq. (3.2) by a cyclic permutation
as suggested in Fig. 26(c) or we can keep Eq. (3.3) and modify the a, b, c operators following
the change of the Leff
quantization axis from z to x or y.
Having clarified the hopping geometry, we can now proceed with the derivation of the effective
pseudospin Hamiltonian following the scheme of the second-order perturbation theory as outlined
in Sec. 2.4.2. Doing so we will adopt j-j coupling scheme to describe the virtual states and extend
it with off-diagonal Hund’s coupling. This approach is not the most practical one for the actual
calculation but best exposes the internal structure of the superexchange processes. We start by
considering the action of the perturbation Htt on a z-bond ij with two pseudospins up. When
3.1 Microscopic origin of the Kitaev interaction 57
including only the t contribution to the hole hopping in the direction j → i, we generate d4
-d6
intermediate state
|˜↑ i ⊗ |˜↑ j = D†
+1
2
|t6
i ⊗ D†
+1
2
|t6
j
hole hopping j →i via t channel
−−−−−−−−−−−−−−−−−−→ −it 2
3
Q†
−3
2
D†
+1
2
|t6
i ⊗ |t6
j (3.4)
that we have written down using the doublet and quartet hole operators introduced in Eqs. (2.74)
and (2.75) [see also Fig. 27(a) for a schematic picture]. The t4
2g state with two holes at site i
can be decomposed into |j1, j2; J, Jz states of j-j coupling scheme discussed near the end of
Sec. 2.2.3. Explicitly they read as
3
2
, 1
2
; 1, −1 = 1
2
Q†
−1
2
D†
−1
2
−
√
3
2
Q†
−3
2
D†
+1
2
|t6
,
3
2
, 1
2
; 2, −1 =
√
3
2
Q†
−1
2
D†
−1
2
+ 1
2
Q†
−3
2
D†
+1
2
|t6
. (3.5)
We can adopt a shorthand notation QDJ,Jz for d4
-d6
bond configurations where the d4
site
is in the |3
2
, 1
2
; J, Jz state. The state at the right-hand side of Eq. (3.4) then decomposes as
it√
2
QD1,−1 − it√
6
QD2,−1. Considering now all four possible initial pseudospin configurations and
including both t and t hoppings in Htt in the direction j → i, we arrive at the following set of
d4
-d6
intermediate states
|˜↑ i ⊗ |˜↑ j
Htt (j→i)
−−−−−−−−→ +
it
√
2
QD1,−1 −
it
√
6
QD2,−1 −
t
3
√
2
QD1,+1 +
t
√
6
QD2,+1 ,
|˜↓ i ⊗ |˜↓ j
Htt (j→i)
−−−−−−−−→ −
it
√
2
QD1,+1 −
it
√
6
QD2,+1 −
t
3
√
2
QD1,−1 −
t
√
6
QD2,−1 ,
|˜↑ i ⊗ |˜↓ j
Htt (j→i)
−−−−−−−−→ −it
2
3
QD2,+2 +
t
3
QD1,0 +
t
3
(DD0 − QD2,0) ,
|˜↓ i ⊗ |˜↑ j
Htt (j→i)
−−−−−−−−→ −it
2
3
QD2,−2 +
t
3
QD1,0 −
t
3
(DD0 − QD2,0) . (3.6)
Here DD0 is a shorthand notation for d4
-d6
configuration with the site i populated by |1
2
, 1
2
; 0, 0
of Eq. (2.80). These relations represent one key element of the derivation of the effective Hamiltonian
using Eq. (2.114), the other being the excitation energies entering through the (E − H)
operator that acts in the subspace of high-energy virtual states. Before analyzing the latter ones,
let us inspect the implications of Eqs. (3.6) themselves. In the case of purely t hopping, the virtual
states are completely disconnected so that when leaving the virtual state by the second hopping,
the initial pseudospin configuration will be restored. In such a case, the exchange interaction
can only be of Ising-like form. This way we already arrive at the Kitaev interaction KSz
i Sz
j that
is enabled only by differing excitation energies for ˜↑˜↑, ˜↓˜↓ and ˜↑˜↓, ˜↓˜↑ configurations because the
integral matrix elements for the second order processes are equal: (t/
√
2)2
+(t/
√
6)2
= (t 2/3)2
.
The direct hopping t opens new possibilities and enables a cross-talk of the pseudospin configurations.
The shared virtual states for the parallel pseudospin configurations ˜↑˜↑ and ˜↓˜↓ activate
the effective process flipping both pseudospins at the same time: ˜↑i
˜↑j ↔ ˜↓i
˜↓j. In the case of
the antiparallel configurations ˜↑˜↓ and ˜↓˜↑, we now have the possibility of the effective process
˜↑i
˜↓j ↔ ˜↓i
˜↑j which is just a part of the usual Heisenberg exchange among the pseudospins. The
parallel and antiparallel configurations remain mutually disconnected even with t included.
When dealing with the excitation energies, we will include only U and JH interactions, neglecting
the effects of spin-orbit coupling in the virtual states. The situation is simple for the
58 3.1 Microscopic origin of the Kitaev interaction
3JH
2JH
QQ0
DD0
D−1
2
D+1
2
, Q+1
2
D+1
2
Q−3
2
↑ b ↑ a ↓ ↑ b ↑ a ↓ QQ2m
QD1m
U −3JH
JH
JH
QD2m
D+1
2
↓ b ↓ c ↑
b ↑
JH
c ↓ 2g
t4
& &
~ ~
=0
= ± 2
(a) (b)
9×
1×
5×
z
t’
t
~
L∆ z∆eff L
eff
Fig. 27: (a) Virtual processes starting with a nearest-neighbor pair of J = 1
2 pseudospins on a z-bond.
The t hopping operating on the b component of the pseudospins can generate usual Heisenberg exchange.
The hopping t mediated by oxygens takes the a component of the pseudospin up and flips its Leff
z ,
producing a fully polarized quartet hole. To get back to two J = 1
2 pseudospins this hole needs to be
removed in the next step. If this happens via a backward t hopping, the initial pseudospin configuration
is restored, enabling only Ising-type of exchange. The exchange is FM due to the Hund’s coupling in the
virtual state. (b) Multiplet structure of a t4
2g ion constructed by combining two j = 1
2, 3
2 holes into total
J eigenstates |j1, j2, J, Jz and handling off-diagonal Hund’s coupling afterwards. The lowest state is a
threefold degenerate level QD1m (m = 0, ±1) made of a quartet hole and a doublet hole that combine
into total J = 1 state. The pair of a quartet hole and a doublet hole may form also the total J = 2
state QD2m (m = 0, ±1, ±2). Further possibilities include a pair of quartet holes combining into total
J = 2 state QQ2m or total J = 0 pairs DD0 and QQ0. Hund’s coupling couples pairwise the QD2m and
QQ2m states with the same m, as well as the pair of DD0 and QQ0 states (both options are marked
by shading) and produces the energy levels of LS coupling scheme shown on the right. The indicated
energies are excitation energies of the particular d4-d6 bond configuration with respect to the initial
d5-d5 configuration of two J = 1
2 pseudospins.
QD1m excited states that are already composed of eigenstates of the respective ionic Hamiltonians.
To evaluate the excitation energy, we need to consider the energy 6U −13JH of the t4
2g state
and the energy 15(U − 2JH) of t6
2g configuration and compare them with the initial energy which
is twice the energy 10(U − 2JH) of t5
2g configuration. We thus obtain
E1 = U − 3JH (3.7)
as the excitation energy to the QD1m d4
-d6
state. The case of QD2m and DD0 excitations is more
complicated since these states are coupled via JH to the combinations QQ2m and QQ0, respectively,
that are not accessible by the hopping directly. As indicated in Fig. 27(b), this coupling
restores the proper energy levels that would be simply described in LS coupling scheme. To get
the effective excitation energies, we need to consider the two-dimensional subspaces spanned by
the pairs QD2m, QQ2m and DD0 [Eq. (2.80)], QQ0 [Eq. (2.81)]. The latter case was already
studied in Sec. 2.2.3 as an example. The corresponding Hamiltonians shifted by the energy of
the d5
-d5
configuration to give the excitation energies read as
Hexc = E1 +
JH
3


4 2
√
2
2
√
2 2

 and Hexc = E1 +
JH
3


5 5
√
2
5
√
2 10

 . (3.8)
3.1 Microscopic origin of the Kitaev interaction 59
Since only the first member of the pair is accessed by the hopping, the matrix element (H−1
exc)11 of
the inverse matrix to Hexc serves as the inverse excitation energy in the perturbation expansion
(2.114). This gives us the effective excitation energies E2 for QD2m and E0 for DD0 state that
can be inserted into the second-order perturbation term in a straightforward way. Consistently
with Fig. 27(b), they are higher than E1 and are explicitly given by E−1
2 = E−1
1 − ∆2 and
E−1
0 = E−1
1 − ∆0 with the reduction factors
∆2 =
4
3
JH
(U − 3JH)(U − JH)
and ∆0 =
5
3
JH
(U − 3JH)(U + 2JH)
. (3.9)
We are now ready to utilize Eqs. (3.6) and combine them together with the three excitation
energies to obtain the effective Hamiltonian. It will have a form of a 4 × 4 matrix that is
constructed by considering all the possible virtual connections of the four bond-basis states ˜↑˜↑,
˜↓˜↓, ˜↑˜↓, ˜↓˜↑. When doing so, we additionally multiply by two to include d6
-d4
virtual states as
well. In the final matrix, one can remove a constant shift
H0 = −
4
3
t2
E1
−
5
9
t 2
E1
(3.10)
found on the diagonal, since this overall kinetic energy gain is irrelevant for the pseudospin
interactions. The resulting matrix contains the following nonzero matrix elements:
H11 = H22 = +
t 2
9E1
+
t2
+ t 3
3
∆2 , H21 = H∗
12 =
2i tt
3
∆2 , (3.11)
H33 = H44 = −
t 2
9E1
+
4t2
3
∆2 +
2t 2
9
(∆2 + ∆0) , H34 = H43 =
2t 2
9
1
E1
− ∆2 − ∆0 .
To create a spin model, the effective interaction should be formulated using the pseudospin-1
2
operators S acting on the ˜↑, ˜↓ basis states in the usual way. The structure of the effective
Hamiltonian obtained perturbatively is reproduced if we consider the pseudospin-1
2
model
JSi · Sj + KSz
i Sz
j + Γ(Sx
i Sy
j + Sy
i Sx
j ) . (3.12)
Expressing this Hamiltonian in the same bond basis ˜↑˜↑, ˜↓˜↓, ˜↑˜↓, ˜↓˜↑ as used above, we find
Heff =









J+K
4
−iΓ
2
+ iΓ
2
J+K
4
−J+K
4
J
2
J
2
−J+K
4









(3.13)
which enables to quantify the values of the exchange parameters J, K, Γ by a comparison with
the matrix elements in Eq. (3.11):
J =
4t 2
9
1
E1
− ∆2 − ∆0 , K = −2t2
+
2
3
t 2
∆2 , Γ =
4tt
3
∆2 . (3.14)
Let us note that the derivation was performed for the particular case of the z-bond. Using
the C3 symmetry of the problem as illustrated in Fig. 26(c), the effective interaction for the
other bond directions is obtained simply by cyclic permutation, for example on the level of the
60 3.2 Extended Kitaev–Heisenberg model and its phase diagram
pseudospin Hamiltonian (3.12) by replacing the x, y, z components of the pseudospin operators
by the cyclically permuted ones.
The structure of the final expressions for the three exchange parameters highlights several
aspects. First of all, the usual isotropic exchange of Heisenberg type originated only from the
supposedly smaller t hopping13
and compared to the canonical cases (2.116) or (2.119) it is
further reduced by factor of 1
9
due to the relatively small participation of the relevant orbital
in the pseudospin wavefunctions. There is no other contribution of the usual type t2
/U, the
remaining terms are solely due to the Hund’s coupling in the virtual states which brings a
smallness factor of roughly JH/U. This is an important observation, suggesting that the overall
energy scale of the interactions will be rather low. Among all those “small” interactions, the FM
Kitaev interaction can be expected to be the dominant one, since it is generated by the larger
t hopping. The off-diagonal exchange Γ resulting from a combined action of oxygen-mediated
and direct hopping could be the second most important one, following our naive estimates. The
interactions are thus expected to be highly anisotropic as it is indeed found experimentally.
Of course, our analysis considered only one class of virtual processes, namely the d-d transitions
within the t2g sector. There is a number of other contributing channels – one can additionally
consider charge transfer processes and cyclic exchange (see [62] for a thorough discussion of those
in the d7
case), and involve also the eg orbitals that are higher in energy than the t2g ones but
are coupled by stronger hopping amplitudes [c.f. the discussion related to Fig. 19(f)]. Moreover,
the pseudospin wavefunctions become modified when including the trigonal splitting among t2g
orbitals that is present in the actual materials. Nevertheless, even our limited approach provided
a number of insights concerning the emergence of strongly anisotropic and bond-selective spin
interactions characteristic for Kitaev materials.
3.2 Extended Kitaev–Heisenberg model and its phase diagram
3.2.1 Effects of the trigonal splitting
In the previous section, we have considered an ideal case of two connected octahedra of undistorted
shape and arrived at the JKΓ model of Eq. (3.12) describing the effective interaction of
J = 1
2
pseudospins. A more realistic model should include also trigonal splitting of the t2g orbitals
that significantly modifies the pseudospin wavefunctions and brings a new interaction term that
was not symmetry-allowed in the previous example. Below we will incorporate the effects of the
trigonal distortion and later include also selected further-neighbor coupling to arrive at the final
interaction Hamiltonian that is often used as a basic model for the magnetism of honeycomb
Kitaev materials.
The trigonal splitting of orbitals arises in a twofold way. First, the structure of edge shared
octahedra may be compressed (or elongated) in the direction perpendicular to the honeycomb
plane, i.e. in the [111] direction in the cubic coordinate system associated with the octahedral
axes. There are of course other types of possible distortion, breaking the C3 symmetry of the
lattice, but the trigonal one may be expected to have the largest impact. Second, the hopping of
electrons supports the extension of their wavefunctions in the direction of the honeycomb plane,
this effect thus acts similarly to the trigonal distortion. Formally, we need to amend the ionic
Hamiltonian with the trigonal field that splits the a1g orbital of Eq. (2.23) from the eπ
g pair of
Eqs. (2.24) and (2.25). Following Eq. (2.29), this is achieved by the term ∆[(Leff
Z )2
− 2
3
] where
Z is the out-of-plane direction as introduced in Fig. 9(c). On the level of point-charge model,
the a1g orbital should sink down under a trigonal compression which corresponds to positive ∆.
13
As an example, Winter et al. [65] give ab-initio values t ≈ 0.25eV and about ten times smaller t for Na2IrO3,
while for α-RuCl3 the estimated t, t are approximately the same 0.15 eV.
3.2 Extended Kitaev–Heisenberg model and its phase diagram 61
0
15
30
45
60
75
90
−∞ -5 -2 -1 -0.5 0 0.5 1 2 5 +∞
ϑ[deg]
∆ / λ
-50
-40
-30
-20
-10
0
10
20
30
40
50
−∞ -5 -2 -1 -0.5 0 0.5 1 2 5 +∞
exchangeparameter[meV]
∆ / λ
J
K
Γ
Γ′
-15
-10
-5
0
5
10
15
20
−∞ -5 -2 -1 -0.5 0 0.5 1 2 5 +∞
∆ / λ
∆ ≫ λ∆ = 0∆ ≪ −λ
|˜↓
|˜↑
(a) (c)
(b)
(d)
Fig. 28: (a) Pseudospin wavefunctions in the limit of large negative trigonal field ∆, cubic case with
∆ = 0, and the limit of large positive trigonal field ∆. The graphical representation shows the hole
distribution in the same way as in Fig. 13. The shapes in the ∆ = 0 limit are again cubic but the spin
polarization distribution is modified because of the Z quantization axis that applies to both Leff
and
S. (b) Mixing angle ϑ entering Eq. (3.15) as it depends on the trigonal field. (c) Exchange parameters
resulting from second order perturbation theory with U = 1.7 eV, JH = 0.3 eV, λ = 0.4 eV, t = 0.26 eV,
t = 0.03 eV. Solid lines are calculated neglecting spin-orbit coupling in the virtual states, dotted lines
include this effect. (d) The same for the values U = 3.0 eV, JH = 0.6 eV, λ = 0.15 eV, t = 0.15 eV,
t = 0.15 eV. The parameter sets are inspired by Ref. [65] and are supposed to imitate the situation in
Na2IrO3 and α-RuCl3, respectively. To have continuous evolution of the exchange parameters, we had
to change the relative phase of the two pseudospin wavefunctions below ∆ = −2.25λ.
When adopting the hole picture to get the t5
2g pseudospin states, we need to revert the sign and
diagonalize H = λS · Leff
− ∆(Leff
Z )2
. The new Hamiltonian still conserves one component of
J, namely JZ, but does not conserve J2
as it did before. The extra term mixes members of the
J = 1
2
doublet and J = 3
2
quartet with matching JZ. The wavefunctions of the lower doublet are
then modified to
|˜↑ = cos ϑ a†
↓ − sin ϑ b†
↑ |t6
, |˜↓ = sin ϑ b†
↓ − cos ϑ c†
↑ |t6
, tan 2ϑ =
2
√
2
1 + 2∆
λ
. (3.15)
The shapes and spin polarization of the hole densities corresponding to the above pseudospin
wavefunctions are presented in Fig. 28(a) and the orbital mixing angle ϑ in Fig. 28(b), covering the
whole range of ratios of the trigonal field ∆ to the spin-orbit coupling λ. For the cubic case with
∆ = 0, the mixing angle takes the “magic” value of ϑ = 1
2
arctan(2
√
2) = arctan(1/
√
2) ≈ 35.26◦
that is at the same time the angle of the cubic axes to the honeycomb plane. With this value,
the cubic pseudospin wavefunctions of Eq. (3.1) are reproduced. In contrast to the JZ = ±1
2
quartet state, the JZ = ±3
2
ones are unaffected by the mixing so that the quartet splits into two
doublets. Such a splitting may be accessible experimentally e.g. by a resonant inelastic x-ray
scattering, enabling a quantification of the trigonal field ∆ [66].
Performing the same kind of perturbation calculation as in the previous section, but starting
with the low-energy doublet (3.15), we can derive the effective pseudospin Hamiltonian including
62 3.2 Extended Kitaev–Heisenberg model and its phase diagram
now an extra anisotropic interaction Γ that scales linearly with the trigonal splitting near the
cubic limit. At a nearest-neighbor z-bond, the pseudospin interaction reads as
H ij z = JSi · Sj + KSz
i Sz
j + Γ(Sx
i Sy
j + Sy
i Sx
j ) + Γ (Sx
i Sz
j + Sz
i Sx
j + Sy
i Sz
j + Sz
i Sy
j ) . (3.16)
To get the interactions at other bond directions, we again apply the cyclic permutation as shown
in Fig. 26(c). This form of the Hamiltonian already includes all the symmetry-allowed nearestneighbor
terms compatible with the C3 symmetry of the lattice and is often called as the extended
Kitaev-Heisenberg model. Further often employed extension consists in including further neighbor
couplings that are likely to be significant due to the extended nature of 4d and 5d orbitals.
Among these, Heisenberg third-neighbor coupling J3 seems to be most important.
Before inspecting the phase diagram of the model, it is instructive to have a look at the
interaction parameters as functions of the trigonal splitting presented in Fig. 28(c),(d). We use
two sets of microscopic parameters that roughly correspond to the values suggested by Winter
et al. [65] for Na2IrO3 and α-RuCl3, respectively. Being 5d and 4d materials, they differ in the
relative strength of the spin-orbit coupling and Hund’s coupling. In the case of Fig. 28(c), these
two energy scales are comparable, JH λ. This cast some doubts at the approximation made
in the previous section, i.e. neglecting spin-orbit coupling in the virtual states. Indeed, if both
spin-orbit coupling and Hund’s coupling are incorporated in the virtual states and excitation
energies, some of the newly obtained interaction parameters show a significant reduction. Most
importantly, this concerns the Kitaev interaction K, which still remains dominant, however. The
situation is much different in Fig. 28(d) that illustrates the JH λ regime relevant for 4d Ru3+
ions. Here the difference caused by the neglection of spin-orbit coupling in the virtual states is
rather minor. The set of parameters used in this case also assumes equal hopping amplitudes
t = t . As can be expected based on Eq. (3.14), this choice makes the anisotropic interactions K
and Γ of the same order of magnitude.
Regarding the dependence on the trigonal field, we note that the interaction is most anisotropic
near the cubic limit. The reason is that the trigonal field, either positive or negative, quenches
further the orbital angular momentum and makes the interactions more Heisenberg-like. For a
large trigonal compression, the pseudospin is reduced to the eπ
g doublet states having entangled
spin and the only unquenched Leff
Z orbital component. This leaves space for Ising-like anisotropy
with differing interactions of the in-plane and out-of-plane components. When worked out, such a
constraint implies K = 0 and Γ = Γ as can be indeed observed in Fig. 28(c),(d). In the opposite
limit – large negative trigonal splitting – the orbital component is completely suppressed and
we are left with spin-1
2
residing in the a1g orbital. The interaction is then necessarily isotropic
and only the parameter J takes a nonzero value. Note, that in this limit the wavefunctions as
introduced in Eq. (3.15) bring an extra minus sign to the relative phase of |˜↑ and |˜↓ , so that
at some point we have to redefine the pseudospin to match the spin-1
2
carried by the a1g orbital.
3.2.2 Phase diagram of the extended Kitaev-Heisenberg model
The main attraction of the Kitaev materials is of course the connection to the Kitaev honeycomb
model for spins-1
2
[18] that appears as a special case of the pseudospin model in Eq. (3.16) when
J, Γ and Γ are zero. In this limit the model is exactly solvable and at the same time it features
an exotic ground state – quantum spin liquid touched upon in the introductory Sec. 1.4. An
essential element of the exact solution [18] is the exploitation of an extensive set of conserved
quantities. These are associated with the individual hexagonal plaquettes of the honeycomb
lattice and are constructed as a product Wp of six spin operators indicated in Fig. 29(a). The
conserved quantities have eigenvalues ±1 and by considering all the possible distributions of +1’s
and −1’s at the plaquettes of the honeycomb lattice we exhaust the Hilbert space of the model
3.2 Extended Kitaev–Heisenberg model and its phase diagram 63
0.0
0.2
0.4
0.6
0.8
1.0
10
-4
10
-3
10
-2
10
-1
10
0
10
1
T
η=0.5
0.0
0.2
0.4
0.6
0.8
1.0
10
-2
10
-1
10
0
10
1
η=1.0
1
2 ln2
S
z
S
z
S
z
S
z
+1 +1
−1 −1
or
or
Wp
Wp
CV
CV
(a)
yy
zz zz
yyxx
xx
S
S
Wp = 26
Sx
1Sy
2Sz
3Sx
4Sy
5Sz
6
j i
i j
i
j
i
j
(b)
6
1
2 4
5
3
(c)
Fig. 29: (a) Conserved plaquette quantity Wp of the Kitaev model. (b) Origin of the local nature of spin
correlations Sα
i Sβ
j that are limited to nearest-neighbor bonds. When acting by a spin operator on the
flux-free ground state of the Kitaev model, two fluxes are generated at the adjacent plaquettes and need
to be removed again when applying the second spin operator. Moreover, there is a link between the spin
component and the orientation of the pair of the concerned plaquettes, making the correlations highly
anisotropic – only the component Sα
i Sα
j is nonzero at α-bond. (c) Temperature-dependent specific
heat CV , entropy S, and thermal average of the plaquette quantities Wp for the Kitaev honeycomb
model. The upper panel shows results for the isotropic version with equal Kitaev interaction strength
for all bond directions, Kx = Ky = Kz [c.f. Eq. (1.4)], the lower panel is for the anisotropic situation
Kx = Ky = η/3, Kz = 1 − 2η/3 with η = 0.5. Shading indicates the two crossover temperature
ranges. At lowest temperatures the ground-state sector without fluxes dominates, hence Wp ≈ 1. In
the middle regime, the plaquette quantities Wp fluctuate and Wp ≈ 0. The entropy per site staying
flat around 1
2 ln 2 in certain interval of temperatures suggests the fractionalization of spins-1
2 into two
parts (these are the localized and running Majorana fermions) that breaks down at higher temperatures
where S ≈ ln 2 corresponding to spin-1
2. Panel (c) is adapted from Ref. [67].
by disconnected subspaces. When utilizing the Majorana spin representation (here for Pauli spin
operators σα = 2Sα)
σx = ibxc, σy = ibyc, σz = ibzc (3.17)
that employs the “gauge” Majorana fermions bα and “matter” fermions c, we find that the
bα fermions are static and become constituents of Wp quantities while c fermions are freely
running on the lattice. In this representation the diagonalization of the spin Hamiltonian is thus
equivalent to a problem of noninteracting Majorana fermions hopping in a magnetic background
which can be easily solved. The magnetic background differs in each of the symmetry sectors,
i.e. subspaces of states with different configurations of Wp’s. The spin liquid ground state is
found in the homogeneous sector with Wp = +1 at every plaquette. The above diagonalization
procedure applied to this sector shows that the spectrum of excitations that inherit unusual
fermionic character is gapless. However, this does not imply that we are dealing here with
gapless magnetic excitations. In fact, by applying the spin operator at a selected site, we flip
two plaquette quantities Wp from +1 to −1 as shown in Fig. 29(b). From the viewpoint of the
Majorana fermion representation, we have introduced two magnetic fluxes at those plaquettes.
This brings us out of the ground-state Wp = +1 sector and costs certain energy that appears
as a spin gap in the magnetic excitation spectra [68,69]. This feature also leads to short-range
spin correlations limited to nearest neighbors. When evaluating the static correlator Sα
i Sβ
j ,
the first application of the Sβ
j flips Wp at two hexagons attached to site j (the particular pair
is decided by β) and this defect has to be restored by the application of Sα
i which requires i
to be either equal to j (trivial) or forming a nearest-neighbor bond. As a consequence, there
64 3.2 Extended Kitaev–Heisenberg model and its phase diagram
cannot be nonzero spin correlations beyond nearest neighbors. Despite the relatively simple exact
solution, the evaluation of thermodynamic quantities for the Kitaev model is challenging. As a
reward we can observe an interesting hierarchy of regimes that appear due to the combination of
fermionic excitations and fluxes with finite cost [67,70]. There are two characteristic temperature
scales, that can be clearly identified in the temperature-dependent specific heat or entropy as
illustrated by Fig. 29(c). Below the lower one the thermodynamics is given by Majorana fermionic
excitations within the flux-free ground-state sector (all Wp = +1). Above this temperature scale,
thermally excited fluxes start to appear and disturb running Majorana fermions. These can
still maintain their coherent motion, however. Finally, above the higher temperature scale the
fractionalization of spins into Majorana fermions is lost and the system behaves as a thermally
excited paramagnet.
While the fancy features such as spin liquid with short-range correlations or fermionic character
of excitations are appealing theoretically, the actual materials do not realize the pure Kitaev
limit and are less frustrated anisotropic magnets that support long-range order. Their magnetic
behavior can be explained by invoking the other terms of the extended Kitaev-Heisenberg model
(3.16) that “spoil” the Kitaev limit but on the other hand make the systems more approachable
by methods developed for conventional spin systems.
A central issue related to the Kitaev materials, for instance Na2IrO3 or α-RuCl3, is the identification
of the relevant parameter regime. To estimate it from first principles is not a simple
task since, as we have learned in Sec. 3.1, the typical leading superexchange contribution ∝ t2
/U
is missing and the balance of the existing subleading interactions largely depends on microscopic
details. Accepting the form of the extended Kitaev-Heisenberg model, we can get to the values
of the model parameters by evaluating various measurable quantities within the model and comparing
them to experimental data. The first experimental constraint is the presence of long-range
magnetic order of zigzag type [71–74] in both abovementioned compounds.14
This observation
calls for a determination of the phase diagram of the model. Due to the high anisotropy and
several participating interactions, it is quite complex in its entirety and some of its parts are still
being debated. In general this concerns the more frustrated regions with potential spin liquid
ground states [76,77], in most of its parameter space the extended Kitaev-Heisenberg model shows
long-range orderings that are well understood. In Fig. 30 we try to give an idea of the above
complexity, presenting selected sections through the phase diagram obtained by two methods.
In the first one we stay on the classical level, treating spins as regular vectors of fixed length
1
2
in Eq. (3.16) and trying to optimize their orientations to minimize the resulting classical
energy. One of the possible approaches, that is also used in Fig. 30(a)-(c), is the LuttingerTisza
method first introduced in Ref. [78] when studying crystals with dipolar interactions and
later conveniently reformulated to the Fourier domain [79]. Here one represents the spins on the
two sublattices A, B of the honeycomb lattice by Fourier expansions SAR = q eiqR
SAq and
SBR = q eiqR
SBq with R denoting the unit cell position. Due to the translational symmetry,
the spin Hamiltonian then assumes the form
H =
q
Ψ†
qHqΨq , where Ψq =


SAq
SBq

 . (3.18)
The explicit forms of the 6×6 matrices Hq can be found e.g. in Ref. [80]. Treating spins classically
in Eq. (3.18), we would have to minimize the energy under the constraint |SAR|2
= |SBR|2
= 1
4
that should be valid for each R. This is complicated to achieve in the Fourier representation
14
One can also find spiral orders such as that observed in α-Li2IrO3 [75], these can be still captured on the
level of highly anisotropic models of extended Kitaev-Heisenberg type.
3.2 Extended Kitaev–Heisenberg model and its phase diagram 65
but in Luttinger-Tisza method one makes an approximation that the above constraints should
be satisfied on average only, i.e. R(|SAR|2
+ |SBR|2
) = 1
4
N with N being the total number of
spins. Minimization under this relaxed constraint which translates to the requirement of constant
q Ψ†
qΨq is simple, one just evaluates the eigenvalues of the matrices Hq and finds the lowest
one among them. The solution found this way has to satisfy the original spin-length constraints
otherwise the Luttinger-Tisza methods is said to fail for that parameter point. In such cases an
alternative method is needed, for example Monte Carlo minimization of the classical energy for
a large piece of the honeycomb lattice.
The second method is fully quantum and consists in finding the ground state of a small
honeycomb cluster via exact diagonalization. For this clearly finite-size system, the symmetry
cannot be spontaneously broken and instead of a single ordered state, we find a fluctuating
superposition of all the degenerate possibilities. To disentangle the information about potential
orderings, the ground state can be analyzed by the method of spin coherent states introduced in
Ref. [81]. Here one works with the coherent spin-1
2
states that “point” in certain directions and
based on them assembles a product state for the cluster that best fits its exact ground state. To
put things explicitly, let us capture the direction by a unit vector n and denote by |n spin-1
2
state that is an eigenstate of n · S operator (i.e. the component of S in the direction of n) with
the eigenvalue +1
2
. Using spherical coordinates, we get n = (cos φ sin θ, sin φ sin θ, cos θ) and
|n = e−iφ/2
cos θ
2
| ↑ + e−iφ/2
sin θ
2
| ↓ . Based on a set of spin coherent states for the individual
sites we define a product state of the N-site cluster as |Ψ = |n1 ⊗|n2 ⊗. . .⊗|nN and measure
its overlap with the ground state |GS . The ordering pattern is then detected by varying the
directions ni and trying to maximize the probability P(n1 . . . nN ) = | Ψ|GS |2
. This approach
provides a relatively good overview of the phase diagram including details of the ordered phases
and is consistent, apart from a few problematic regions, with more advanced but less transparent
numerical approaches such as DMRG [82] or tensor networks [83,84] used to investigate certain
parameter regions of the model.
The rich phase diagram of the extended Kitaev-Heisenberg model that is partially revealed
by Fig. 30 contains a few phases that can be anticipated in the various limits of the model. These
include the ferromagnetic (FM) and antiferromagnetic (AF) phases linked to J < 0 and J > 0
Heisenberg limit, respectively, which are very extended in contrast to the Kitaev spin liquid
phases found in the Kitaev-limit areas of FM or AF character [see Fig. 30(d)]. The Kitaev spin
liquid state is able to withstand only a rather limited amount of non-Kitaev perturbations and
gives up soon enabling a development of a long-range order. Apart from the expected FM and
AF order, there is in addition a stripy one, two types of zigzag order differing in the direction of
the ordered moments, and two non-collinear orders of vortex type (only one of them appears in
the sections shown in Fig. 30). The respective ordering patterns are presented in Fig. 30(g)-(k).
In fact, by using nontrivial sublattice-dependent spin rotations, the model can be shown [80]
to be equivalent to Heisenberg model for certain parameter combinations – points of hidden
SU(2) symmetry. These hidden relations give somewhat deeper insight into the location and
characteristics of the individual phases [80,85]. The stripy and zigzag phases were found to be
related to the FM and AF Heisenberg points by a four-sublattice transformation and the vortex
phase seen in Fig. 30 was linked to the FM Heisenberg point by a six-sublattice transformation.
A problematic part of the phase diagram for the exact diagonalization method limited to small
clusters is the area indicated by the IC label. It is partially covered by incommensurate spiral
phase but it may also host further spin liquid phases not directly linked to the closest Kitaev
point [76,77].
We are not going to discuss the details of the phase diagram and the (hidden) symmetry
properties of the model that were thoroughly analyzed e.g. in Ref. [85]. Instead, we briefly
inspect the experimentally relevant zigzag phases. As already mentioned, there are two distinct
66 3.2 Extended Kitaev–Heisenberg model and its phase diagram
Γ′=0 J3=0
fail
FM
stripy
vortex
zigzag
AF
Γ′= −0.1 J3=0
fail
FM
stripy
vortex
zigzag
AF
Γ′=0 J3=0.1
fail
FM
stripy
vortex
zigzag
AF
P [%]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
α
0°
10°
20°
30°
40°
50°
α
0°
10°
20°
30°
40°
50°
AF K
AF J
AF
FM
IC
zz2
zz1
FM J
FM K
vortex KSL
KSL
stripy
(g) (h) (i) (j) (k)
(a) (b) (c)
(d) (e) (f)
AFFM stripy zigzag vortex
Fig. 30: (a) Classical phase diagram of the extended Kitaev-Heisenberg model as obtained by the
Luttinger-Tisza method. The interactions are parametrized as J = sin θ cos φ, K = sin θ sin φ, Γ = cos θ
with θ being the radial coordinate in the circle (distance from the center varying from 0 up to π/2
at the outer rim) and φ being the conventional polar angle measured from the horizontal axis. Four
points of purely Heisenberg or Kitaev character at the outer rim are indicated by dots. In the black
region the Luttinger-Tisza method suggests an incommensurate order. (b) The same with fixed nonzero
Γ = −0.1. (c) The same as in (a) with fixed nonzero third-neighbor Heisenberg exchange J3 = 0.1.
(d) Phase diagram obtained by coherent-state analysis of the exact ground states for 24-site cluster.
Shown are the added probabilities of coherent states with zigzag (unscaled), ferromagnetic (unscaled),
antiferromagnetic (reduced 10 times), stripy (reduced 5 times), and vortex (reduced 10 times) patterns.
The values above 3.2% are shown using the topmost color of the scale. The AF phase extends up to
the outer rim, its probability gets reduced there due to increased quantum fluctuations compared to
the inner part of the circle. Tiny patches of Kitaev spin liquid (KSL) phase are encircled. The large
area showing almost zero probability hosts an incommensurate (IC) or large-unit-cell order. (e) Zigzag
region in the phase diagram with fixed Γ = −0.1 obtained by the same method as in panel (d). The
area with P(zigzag) ≥ 0.5% is highlighted. The color indicates the angle of the ordered moments to the
honeycomb plane. (f) The same as in panel (e) for fixed J3 = 0.1. (g–k) Pseudospin patterns of the
commensurate ordered phases.
zigzag phases. They are not completely unrelated, however, according to Ref. [80] one can
actually find a dual transformation of the model that converts one into another. The ordered
3.3 Specific features of the zigzag phase in the regime of the dominant Kitaev interaction 67
moments in the upper phase zz1 of Fig. 30(d) prefer one of the cubic axes x, y, z, depending
on the direction of the zigzag chains. For the pattern shown in Fig. 30(j) with zigzag chains
running along the x and y bonds, the ordered moments are close to (or coinciding with) the z
axis. This spin arrangement is stabilized in the regime of large AF K > 0 that profits from the
AF correlations of z pseudospin components at the vertical bonds and FM J < 0 interaction
that, being component-insensitive, gains energy due to FM correlations at the majority of bonds
(FM x and y bonds versus AF z bonds). The lower zigzag phase zz2 has the ordered moments
pointing along an almost orthogonal direction, for the pattern in Fig. 30(j) they are found roughly
inbetween the cubic x and y axes. In this case, the zigzag order is supported by large FM K < 0
that exploits the FM correlated x and y pseudospin components on the zigzag chains formed by
x- and y-bonds, and large Γ > 0 interaction that contributes on the interchain z-bonds where it
simultaneously utilizes the conveniently oriented x and y components of the pseudospins. The
latter situation is observed experimentally, in agreement with the microscopic expectations of
the ferromagnetic Kitaev interaction and positive Γ interaction (see [53] and references therein)
that we have found also in Sec. 3.1 based on a simplified calculation. As illustrated in Fig. 30(e),
the zigzag phase zz2 is supported by negative Γ interaction (i.e. positive crystal field ∆). The
phase zz1 does not feature the properly correlated pseudospin components compatible with the
negative Γ interaction and is supported by small positive Γ instead. Both zigzag phases benefit
from the anticipated third-nearest-neighbor Heisenberg interaction J3 that accepts any moment
direction in the zigzag arrangement [see Fig. 30(f)]. The precise ordered moment direction is
decided by the balance of the anisotropic interactions K, Γ, Γ and can be used as a guide to
narrow down the relevant parameter region [81]. It is automatically resolved by the method of
coherent states and indicated in Figs. 30(e),(f) to give an example of this effect.
Finally, let us note that while the qualitative appearance of the phase diagrams obtained
classically [Fig. 30(a)-(c)] and by exact diagonalization of a cluster Hamiltonian [Fig. 30(d)-(f)]
is quite similar, the precise positions of the phase boundaries may differ significantly. Generally
speaking, the phases involving stronger quantum fluctuations (AF, zigzag) become more extended
when going from the classical to the quantum calculation since the properly included quantum
fluctuations bring them an energy advantage in the competition with the less fluctuating phases
(FM, stripy, vortex). These trends are clearly visible in Fig. 30.
3.3 Specific features of the zigzag phase in the regime of the dominant
Kitaev interaction
Focusing now on the zigzag phase, we will study a few signatures of the highly anisotropic and
bond-selective pseudospin interactions that appear in static and dynamic pseudospin correlations.
The component- and bond-selective Kitaev interaction induces anisotropic and bond-dependent
correlations of the pseudospins. This is true not only for the ordered states, as it has been
discussed in the previous section, but can be observed also in the short-range correlated state
above the ordering temperature.
In the case when the model supports zigzag ordering, the state above TN contains – in a shortrange
correlated form – all three possible zigzag patterns depicted in Fig. 31(a). Each of them is
characterized by different preferred direction of the pseudospin moments and at the same time
corresponds to different pair of characteristic wavevectors [Bragg spots shown in Fig. 31(a)]. This
link enables a direct experimental proof of anisotropic bond-selective interactions in a real Kitaev
material by measuring momentum-dependent correlations of the pseudospins resolved into the
individual components. For example, assuming the above FM K < 0 situation, the zigzag pattern
with FM chains along x and y bonds places pseudospins into the direction roughly inbetween the
x and y cubic axes. Considering only the diagonal components of the pseudospin correlations,
68 3.3 Specific features of the zigzag phase in the regime of the dominant Kitaev interaction
Sx
-qSx
q S
y
-qS
y
q
ξ=0.0 ξ=0.2 ξ=0.4 ξ=0.6 ξ=0.8 ξ=1.0
0
1
0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.0 0.2 0.4 0.0 0.1 0.2 0.3 0.0 0.1 0.2
T=1K T=5K T=10K T=20K T=50K T=100K
0.0 0.1 0.2
T=500K
-0.2 0.0 0.2 0.4 -0.2 0.0 0.2 0.4 -0.2 0.0 0.2 0.4 -0.2 0.0 0.2 0.4 -0.2 0.0 0.2 0.4 -0.2 0.0 0.2 0.4 -0.2 0.0 0.2 0.4
Sz
-qSz
q
zigzag 1 zigzag 2 zigzag 3
(a) (b)
(c)
(d)
Fig. 31: (a) Three possible zigzag patterns differing in the direction of the FM zigzag chains and the
corresponding Bragg spots in the reciprocal space. The inner dashed hexagons in the bottom part
are Brillouin zones of the honeycomb lattice, the blue filled hexagons mark the Brillouin zone of the
triangular lattice generated by filling the voids in the honeycomb lattice. (b) Component-resolved
pseudospin correlations measured by diffuse magnetic x-ray scattering on Na2IrO3 [86]. The experiment
was performed at T = 17 K, i.e. above TN ≈ 12 − 15 K (sample dependent). The dashed hexagons have
the same meaning as in panel (a). (c) Simulated pseudospin correlations Sz
−qSz
q obtained by exact
diagonalization on 24-site cluster when interpolating between Heisenberg and Kitaev limits of the model.
With the selected parametrization J = J2 = J3 = 1 − ξ and K = −ξ for ξ ∈ [0, 1], the ground state
is of zigzag type up to ξ very close to 1 where it switches to the Kitaev spin liquid. (d) Temperature
dependent pseudospin correlations uncovering the hierarchy of the energy scales. The upper row shows
Sz
−qSz
q correlations, the bottom row Sx
−qSy
q . Model parameters J = −0.5, K = −5.0, Γ = 2.5,
J3 = 0.5 in units of meV were adopted from Refs. [87,88]. The correlations are calculated for 24-cluster
using finite-temperature Lanczos method [89,90] in LTLM variant [91].
the characteristic momenta of the above zigzag pattern should be thus visible mainly in Sx
−qSx
q
and Sy
−qSy
q correlations. Alternatively, the Sx
−qSx
q correlations should display characteristic
momenta of zigzag patterns 2 and 3, but a negligible contribution from zigzag 1. Such a correspondence
is indeed observed in the diffuse magnetic x-ray maps presented in Fig. 31(b). When
3.3 Specific features of the zigzag phase in the regime of the dominant Kitaev interaction 69
added together, the maps of the three components Sα
−qSα
q (α = x, y, z) reveal a very symmetric
picture with all the zigzag momenta showing similar intensities.
To simulate these effects and estimate the amount of interaction anisotropy needed to explain
the experimental observations, in Fig. 31(c) we perform exact diagonalization on a small 24-site
cluster interpolating between Heisenberg and Kitaev limits of a model that supports zigzag
ordering most of the time. This is achieved by using AF Heisenberg interactions of the same
strength up to third nearest neighbors and complementing them by FM Kitaev interaction.
Although our calculation is in principle for T = 0 only, the inability of the cluster ground state
to spontaneously break symmetry comes as an advantage here and we get all three zigzag patterns
mixed in, imitating therefore the situation just above TN. As seen in Fig. 31(c), the Heisenberg
limit of the model retains fully isotropic correlations so that all the zigzag points in a selected
correlation component Sα
−qSα
q are equally intense. With a relatively modest relative strength
of the Kitaev interaction as compared to the Heisenberg ones, the symmetry is lost and two
of the characteristic momenta start to vanish. However, to make the “unwanted” Bragg spots
completely invisible (in the given color scale), a relatively large Kitaev interaction is needed,
suggesting its dominance in the Na2IrO3 compound [86].
Another interesting fictitious experiment is performed in Fig. 31(d). Here we take a parameter
set with dominant FM K and substantial Γ > 0, accompanied by small J < 0 and J3 > 0 to
0
10
20
30
40
50
60
Γ M Γ’ X K Γ YK’ Γ’
energy(meV)
wavevector q
Γ M Γ’ X K Γ YK’ Γ’
wavevector q
0
10
20
30
40
50
60
Γ M Γ’ X K Γ Y K’ Γ’
energy(meV)
Γ M Γ’ X K Γ Y K’ Γ’
0
1
2
3
4
5
6
Γ M Γ’ X K Γ Y K’ Γ’
spectralweight(a.u.)
wavevector q
24a
32a
32b1
32b2
32b3
exp
Γ K X
Γ ’K’Y
M
(a) (b) (c)
(d) (e) (f)
Fig. 32: (a) RIXS response calculated by exact diagonalization of the extended Kitaev-Heisenberg
model on 24-site and various 32-site clusters (see Ref. [92] for details). The model parameters used are:
J ≈ 10 meV, K ≈ −15 meV, Γ ≈ 16 meV, Γ = −2.4 meV, J2 = 1.2 meV, J3 = 2.4 meV. (b) The
same dynamic response function evaluated within linear spin-wave approximation (LSW), averaging
over all three zigzag directions. (c) Brillouin zone of the honeycomb lattice (dashed hexagon), extended
Brillouin zone (solid hexagon) and positions of the high-symmetry points used in panels (a) and (b).
(d) Spectral weight of the response function from panels (a), (b) as obtained by exact diagonalization
for the various clusters (points), within LSW approximation (dashed line), and compared to (arbitrarily
scaled) experimental data. (e) Measured RIXS response on Na2IrO3 at T = 7 K [92] compared to
(f) the broadened data from panel (a).
70 3.3 Specific features of the zigzag phase in the regime of the dominant Kitaev interaction
stabilize the zigzag order, and calculate the pseudospin correlations by a finite-temperature exact
diagonalization approach. By elevating the temperature, the correlations gradually reveal the
hierarchy of energy scales. Requiring the support of the smallest J and J3 interactions, the sharp
zigzag correlations give up first, their degradation is clearly visible in the 10 K ≈ 1 meV map.
The Γ interaction correlating the Sx
and Sy
components is defeated by the thermal fluctuations
in the interval T ≈ 20 − 50 K where the corresponding correlations almost vanish. The largest
energy scale K is able to withstand higher temperatures and gives the correlations a wave-like
profile in momentum space, characteristic for the nearest-neighbor Kitaev correlations.
As a final illustration of the features brought about by the dominance of the Kitaev interaction,
in Fig. 32 we present dynamical correlations in a form of resonant-inelastic x-ray scattering
(RIXS) spectrum that was supposed to simulate the experimental data on Na2IrO3 based on
the extended Kitaev-Heisenberg model [92]. In contrast to gapless magnons observed for the
isotropic Heisenberg model that conserves total spin, here we get a gapped spin excitation spectrum,
though the gap can be quite small as it happens to be in our example. The violation
of total pseudospin conservation is also manifested by large q = 0 intensity at finite energies,
that could not be present if Sq=0 – which is proportional to the total pseudospin – commuted
with the Hamiltonian. A comparison of exact diagonalization, here rather limited by momentum
resolution, and linear spin-wave approximation reveals large decay rates due to highly anisotropic
interactions. The dynamics is characterized by two energy scales with the smaller Heisenberg
interactions influencing the low-energy magnons that keep to be well defined, while the large
anisotropic interactions determine the overall shape of the spectrum up to the high energies that
correspond to the fast dynamics of the pseudospins. This is subject to strong decay processes
leading to a large broadening of the high-energy spectral features. On the other hand, being generated
by the large K and Γ interactions, the high-energy spectral features survive even above TN
where the low-energy magnon features are lost [92]. As discussed before, in this regime one can
imagine short-range correlated zigzag fragments, still showing their specific high-energy dynamics
due to K and Γ. This effect is a dynamic analog of the gradual suppression of characteristic static
correlations by elevated temperature that is captured in Fig. 31(d). Finally, the bond-selectivity
of the interactions that creates links between the momentum space and pseudospin components
as observed on the correlations in Fig. 31(a),(b), also enters the game here. For example, even
though all the three zigzag directions are equally employed in both Fig. 32(a),(b), the particular
selection of the pseudospin components entering the RIXS response makes a distinction between
otherwise symmetry-equivalent points M and Y – the former hosts an intense magnon cone while
the latter does not.
4 Soft-spin systems
In this part of the thesis, we will move beyond the concept of rigid spin systems, where the
local moments are represented by spins or pseudospins of a given spin length, and consider
soft-spin systems that were briefly introduced in Sec. 1.5. Here the magnetic ions may host
several quasidegenerate spin states that are dynamically mixed by superexchange interactions.
This dynamic mixing makes the magnetic moments soft, which brings specific features to e.g.
spin excitation spectra. On the material side, our motivation is the ruthenate Ca2RuO4 that
is Mott insulating below approximately 360 K [93] and shows antiferromagnetic order below
TN ≈ 110 K [94–96]. Its crystal structure depicted in Fig. 33(a) is similar to that of highTc
cuprates and consists of stacked RuO2 planes where the Ru ions connected by O ions are
arranged into a square lattice. The ordered magnetic moments lie within the RuO2 planes and
are oriented diagonally with respect to the square lattice.
Below we will formulate the magnetic model for this compound based on the low-energy
multiplet states of Ru4+
ions with t4
2g configuration. As observed in Ref. [97], the orbital angular
momentum of t2g orbitals is largely unquenched, supporting the low-energy structure as in Fig. 14
that is formed by spin-orbit coupling. It features nonmagnetic J = 0 singlet ionic ground state
and J = 1 triplet states that are separated by the energy λ. Based on this local basis, we will first
formulate a singlet-triplet superexchange model serving as an introductory example. The model
will be later refined to account for the tetragonal and small orthorhombic splitting in Ca2RuO4
and become in fact a singlet-doublet model. Finally, we will solve the model, demonstrate the
peculiar antiferromagnetic order due to a condensation of J = 1 triplet levels and obtain spin
response and Raman response that can be successfully compared to experiments on Ca2RuO4.
In the following, we will limit our discussion to the square lattice case applicable to Ca2RuO4.
However, another interesting situation occurs when applying the same ideas to the honeycomb
lattice case where Kitaev-like frustration of interactions may appear [98–100]. The details of the
resulting frustrated singlet-triplet model are discussed in Ref. [100] attached in Sec. 6.
4.1 Singlet-triplet model
As a first step toward a microscopic model for the magnetism of Ca2RuO4, we will consider
the multiplet structure of t4
2g configuration discussed in Sec. 2.2.3. The moderate spin-orbit
coupling constant λ ≈ 70 − 80 meV for Ru4+
ions [31,97] leads to the hierarchy of energy scales
U JH λ that fits the successive level splitting suggested by Fig. 14. A natural selection of
basis states for the minimal model includes the nonmagnetic J = 0 ionic ground state and – to
have some magnetic moment available – low-lying excited J = 1 states. As we will see below,
these states are easily accessible by second-order superexchange processes on a bond. The J = 2
states at three times higher excitation energy will be ignored. It was already noted in Sec. 2.2.4
that we encounter here a special situation of predominantly Van Vleck type of magnetic moment
– the largest part of the magnetic moment contained within the J = 0, 1 subspace is obtained
from the transition between J = 0 and J = 1 states.
The local part of the singlet-triplet model embodies the energy cost of the triplet state as
compared to a singlet. This is captured by the simple Hamiltonian λ i nTi = ET i nTi counting
the number of triplet excitations with the excitation energy ET = λ. The derivation of the bond
part will follow the general scheme utilized in the previous sections. With the singlet-triplet basis
selected, we could proceed similarly to Sec. 3.1 by forming all possible combinations of singlets
and triplets on a bond and obtain their energy gains and mutual connections by considering
the virtual processes within second-order perturbation theory. However, here we will illustrate
another route to arrive at the same result. We will first obtain more general model of Kugel-
71
72 4.1 Singlet-triplet model
T s s
TTss
singlet
triplet
α Tα
tα
κα
E =T λ
α α
(a) (b)
y
x
Fig. 33: (a) Crystal structure and magnetic ordering of Ca2RuO4. Ru4+ ions are depicted as gray
balls surrounded by oxygen octahedra. Magnetic moments lying in the RuO2 planes and pointing along
crystallographic b axis are indicated by yellow arrows. In-plane bond directions are labeled as x and y.
Adapted from Ref. [101]. (b) Schematic representation of singlet-triplet processes corresponding to the
Hamiltonian terms in Eq. (4.5). Each site may host a singlet t4
2g configuration s or a triplon excitation
Tα (α = x, y, z) of the cost ET = λ. The upper exchange process is a triplon hopping (exchange of s
and T), the lower process is a creation/annihilation of a singlet pair of triplons.
Khomskii type15
operating in the subspace involving all nine Leff
= 1, S = 1 ionic states selected
by Hund’s coupling. This model will be later projected on the J = 0, 1 states to get the singlettriplet
model. The most natural basis for the Kugel-Khomskii model are the states of the type
(2.64) that enable to easily capture the action of the two hoppings involved in the second-order
virtual process by properly assembling the spin operators Si, Sj and orbital operators Leff
i , Leff
j
associated with the two sites of the bond ij . Due to the spin conservation during the hopping,
the spin operators may only appear in a form of the isotropic product Si · Sj. The orbital part
of the interaction is not limited this way and may take various forms depending on the hopping
geometry and the particular bond direction. In the case of 180◦
metal–oxygen–metal bonding,
we have a bond-selected pair of active orbitals that are subject to oxygen-mediated hopping
preserving the orbital label (i.e. it is diagonal in orbitals). For concreteness, we will consider a
z-bond16
where the hopping Hamiltonian reads as [c.f. Eq. (2.101) for an x-bond]
Ht = −t
σ
ξ†
iσ ξjσ + η†
iσ ηjσ + H.c. , (4.1)
with t = t2
pdπ/∆pd. In contrast to Eqs. (3.2) and (3.3) for the 90◦
bond geometry encountered
in Sec. 3.1, here we only get diagonal hopping of a and c holes associated with Leff
z eigenstates.
Combining this information with the a, b, c composition of the basis states (2.64), we can infer
that their individual Leff
z will be mostly preserved in the superexchange process, but there is also
a possibility to exchange Leff
z = +1 and Leff
z = −1 on the bond. All these observations are indeed
consistent with the actual form of the effective Hamiltonian derived for the z-bond:
t2
2U
(Si·Sj+1)
(L+
i )2
(L−
j )2
+ (L−
i )2
(L+
j )2
2
+ (Lz
i )2
−2 (Lz
j )2
−2 + Lz
i Lz
j +
t2
U
(Lz
i )2
+ (Lz
j )2
.
(4.2)
15
This type of spin-orbital models has been introduced by K. I. Kugel and D. I. Khomskii in Ref. [102].
16
Note, that in the planar structure of Ca2RuO4 only the x- and y-bonds of M–O–M type are present, as seen
in Figs. 33(a) and 34(a). The z-bond is considered here just to have a familiar notation used before.
4.1 Singlet-triplet model 73
For brevity, we denoted Leff
by L in the above equation. Now we need to project the Hamiltonian
(4.2) onto the singlet-triplet basis. This is achieved by considering the site operators such as
Sx
i (Lz
i )2
contained in the fragments of (4.2) and expressing them in the multiplet basis with spinorbit
coupling included. By throwing away the terms involving the J = 2 quintuplet and keeping
only s and Tα operators associated with J = 0, 1 states, we arrive at the required projection. For
example, the above site operator can be written as
Sx
(Lz
)2
= −
i
√
6
(s†
Tx −T†
xs)−
i
2
(T†
y Tz −T†
z Ty ) + neglected terms involving J = 2 states. (4.3)
When converted into the singlet-triplet basis, the bond Hamiltonian (4.2) generates a large
number of terms. The diagonal ones can be written as a renormalization of the triplet energy and
mutual density-density interactions of triplets. To this end we first make a constant energy shift so
that the energy gain of |ss bond configuration becomes zero. The other energy gains can be then
distributed among the local ET shifts ∆ETx = ∆ETy = −11
6
t2
U
, ∆ETz = −4
3
t2
U
and the repulsion
terms Vαα nTαinTα j with Vxx =Vyy =Vzz = 23
12
t2
U
, Vxy = 5
3
t2
U
, and Vxz =Vzx =Vyz =Vzy = 17
12
t2
U
. Note
that the z-bond naturally makes a distinction between the Tz excitation and the Tx, Ty pair. For
the other bond directions, all the Hamiltonian terms are obtained by a cyclic permutation. The
above diagonal contributions will be later ignored, we may imagine them as being partly absorbed
into renormalized ET and partly neglected under an assumption of low density of triplet states.
More interesting are the off-diagonal terms of the effective Hamiltonian that can be visualized
as bond processes involving triplet excitations. These are of hardcore boson nature and will be
called triplons in the following. To make the notation compact, we introduce Hubbard operators
switching between s and Tα states of the t4
2g configuration:
T †
α = |Tα s| , Tα = |s Tα| . (4.4)
The off-diagonal contributions quadratic in triplon operators may be summarized as (again for
a z-bond)
txy T †
xiTxj + T †
yiTyj + tzT †
ziTzj − κxy T †
xiT †
xj + T †
yiT †
yj − κzT †
ziT †
zj + H.c. (4.5)
They can be understood as hopping of triplons and their pairwise creation and annihilation
depicted in Fig. 33(b). There are also terms involving three or four triplon operators – conversion
of a single triplon on a bond to a pair of complementary ones, exchange of triplons on the bond,
and pair-conversion terms ∝ |TαiTαj Tα iTα j|. All these may be ignored when focusing on the
cases with small enough density of triplons. Including first order corrections in η = JH/U, the
interaction parameters in Eq. (4.5) read as
txy ≈
t2
U
1 −
5
6
η , tz ≈
2
3
t2
U
, κxy ≈
5
6
t2
U
1 −
8
5
η , κz ≈
2
3
t2
U
. (4.6)
When assuming the typical η ≈ 0.2 [see e.g. the Ru parameters used in Fig. 28(d)], the values
of the above parameters are relatively close to each other, so it is reasonable to simplify the
singlet-triplet Hamiltonian to the isotropic form
H = ET
i
nTi + J
ij
T †
i ·Tj − T †
i ·T †
j + H.c. (4.7)
with J ≈ 2t2
/3U and the vector operator T = (Tx, Ty, Tz). Based on Eq. (4.7) applied to a square
lattice as appropriate for Ca2RuO4, we could already study the competition of the triplon cost ET
and the superexchange J, the emergence of the magnetic order due to triplon condensation, and
the implications for magnetic excitation spectra. However, to make the model more realistic, we
will still include tetragonal splitting of the orbitals that essentially eliminates one of the triplon
flavors, a small splitting due to orthorhombicity, and also consider some of the higher-order terms.
74 4.2 Revisions of the model to reflect tetragonal splitting
4.2 Revisions of the model to reflect tetragonal splitting
Apart from the major t2g-eg splitting, the orbitals in Ca2RuO4 are subject to further splittings
due to tetragonal and small orthorhombic components of crystal field illustrated by Fig. 34(a),(b).
The main effect is due to the tetragonal crystal field ∆(nyz +nzx−2nxy)/3 that brings down the
planar xy orbital for positive ∆ as shown Fig. 9(b). As we will see below, the tetragonal splitting
pushes the magnetic moment to the RuO2 plane by essentially deactivating one member of the
triplet states. The small orthorhombic crystal field ∆ generates in-plane anisotropy selecting
the b crystallographic axis as preferential for the ordered moments.
The tetragonal splitting acts on the two-hole states |Leff
z , Sz of Eq. (2.64) via an extra term
∆[(Leff
z )2
− 2
3
], changing the proportions of these states in the multiplet eigenstates. This leads to
wavefunction modifications similar to what we already encountered in Eq. (3.15) for pseudospin-1
2
states of the t5
2g configuration. The wavefunction of the singlet ground state gets adjusted to the
tetragonal splitting as
|s = cos ϑ0
1√
2
(|+1, −1 + |−1, +1 ) − sin ϑ0 |0, 0 , tan 2ϑ0 =
2
√
2
1 − 2δ
(4.8)
with the crystal field being quantified by δ = ∆/2λ = ∆/ζ. For nonzero ∆, the orbital-mixing
angle ϑ0 deviates from its cubic-limit value arctan(1/
√
2) ≈ 35.26◦
that reproduces the original
J = 0 singlet of Eq. (2.76). The triplet states split into a degenerate pair
|T±1 = cos ϑ1 |±1, 0 ± sin ϑ1|0, ±1 , tan ϑ1 =
1
√
1 + δ2 − δ
(4.9)
that is linked to |Tx = i√
2
(|T+1 − |T−1 ) and |Ty = 1√
2
(|T+1 + |T−1 ) used in Sec. 2.2.4 to
handle the magnetic moment of the t4
2g configuration, and the non-degenerate state
|T0 = − 1√
2
(|+1, −1 − |−1, +1 ) (4.10)
that is linked to the remaining |Tz = −i|T0 and only shifts in energy. The above states are
still eigenstates of Jz but J is not a good quantum number anymore due to the mixing of the
original J = 0, 1 states with the J = 2 quintuplet states caused by the tetragonal crystal field.
The energy level splitting is plotted in full in Fig. 34(c) and the two angles entering Eqs. (4.8)
and (4.9) in Fig. 34(d).
The newly introduced splitting among the triplet levels should be reflected in the singlettriplet
model. As seen in Fig. 34(e) showing the ionic excitation spectrum, for positive ∆ case that
applies to Ca2RuO4, the T0 level goes quickly up and can be omitted in the model basis, leading
to a singlet-doublet model. In contrast, the excitation energy of the doublet T±1 levels becomes
significantly reduced compared to the original λ = ζ/2, making them even better accessible by
the superexchange processes. Ref. [101] estimates δ ≈ 1.5 and the corresponding reduction of the
T±1 excitation energy from λ ≈ 70 − 80 meV to about 25 meV. When leaving out the T0 ∼ Tz
state, the three active basis states are conveniently described by pseudospin-1. Using the above
states, we can form the basis of pseudospin-1 according to
|+1 = 1√
2
(i|Tx − |Ty ) = −|T+1 , |0 = |s , |−1 = 1√
2
(i|Tx + |Ty ) = |T−1 (4.11)
defining the eigenstates of the z-projection of the pseudospin-1. The corresponding pseudospin
operators that fulfill the spin-1 algebra are then expressed as
Sx = −i(s†
Tx − T†
xs) , Sy = −i(s†
Ty − T†
y s) , Sz = −i(T†
xTy − T†
y Tx) . (4.12)
4.2 Revisions of the model to reflect tetragonal splitting 75
0°
15°
30°
45°
60°
75°
90°
−∞ -5 -2 -1 -0.5 0 0.5 1 2 5 +∞
∆ / ζ
ϑ0
ϑ1
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
−∞ -5 -2 -1 -0.5 0 0.5 1 2 5 +∞
E/ζ
∆ / ζ
Jz = 0
Jz = ±1
Jz = ±2
0.0
0.5
1.0
1.5
2.0
2.5
3.0
-4 -3 -2 -1 0 1 2 3 4
(E−Es)/ζ
∆ / ζ
Jz = 0
Jz = ±1
Jz = ±2
∆′
∼∆
J =0z J=0
J =0z J=1
J =0z
J =z ±1
1
2
pseudospin
aT
bT
s
yTxT ,
s
Tz
TE
a
c
b
x
y
singlet−triplet pseudospin 1
(a)
(b)
(e)
(f)
(c)
(d)
Fig. 34: (a) Coordinate systems for the RuO2 plane. The crystallographic axes are labeled by a and b.
(b) Splitting of triplet levels under crystal field of distorted octahedra. Tetragonal deformation giving
rise to the ∆ field lifts up the Tz triplet state, small additional orthorhombic distortion represented by
the field ∆ further splits the Tx and Ty states into the combinations Ta,b = 1√
2
(Tx Ty). (c) Energy
levels within the Leff = 1 and S = 1 sector depending on the ratio of the tetragonal crystal field ∆ and
spin-orbit coupling strength ζ. (d) Auxiliary angles ϑ0 and ϑ1 entering the eigenstates in Eqs. (4.8) and
(4.9). (e) Excitation energies measured from the singlet ionic ground-state level. The insets illustrate
the tetragonal deformation of the octahedron as connected to the sign of ∆ within point-charge model.
(f) Electron densities of low-energy t4
2g states including their spin polarization. The low-energy states
form a basis of a singlet-triplet model (cubic limit ∆ = 0), effective spin-1 model (large positive ∆), or
effective spin-1
2 model (large negative ∆).
Pseudospin-1 introduced this way also perfectly captures the surviving components of the magnetic
moment. The x and y components of S are proportional to the in-plane Van Vleck moment
[see Eq. (2.90)] while the z component describes the magnetic moment hosted by the triplons
themselves. The g-factors that connect magnetic moment 2S − Leff
and pseudospin-1 operators
within the pseudospin-1 subspace are plotted in Fig. 35(a). For the cubic limit, we notice the
coincidence with the factors entering Eq. (2.90). At large tetragonal compression, the orbital
component is suppressed as evident from Eqs. (4.8) and (4.9) and the usual spin g-factor 2 is
recovered.
Adopting the pseudospin-1 notation, we will first implement the level splitting depicted in
Fig. 34(b) into the local part of the new singlet-doublet model. Combining both the reduced T±1
excitation cost ET and the in-plane anisotropy attracting the moments to the b axis, we have
Hloc = ET nT + 1
2
∆ (nTa − nTb) = ET S2
z + 1
2
∆ (S2
a − S2
b ) = ET S2
z − ∆ (SxSy + SySx) , (4.13)
where ET is now much smaller than λ. Still, when the experimental data on Ca2RuO4 are fit by
the pseudospin-1 model, ET turns out to be the dominant parameter, constraining the magnetic
moments to the RuO2 plane.
76 4.2 Revisions of the model to reflect tetragonal splitting
0.0
0.5
1.0
1.5
2.0
2.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0
g-factor
∆ / ζ
gab
gc
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
exchangeparameter[t
2
/U]
∆ / ζ
bilinear
η=0.0
η=0.1
η=0.2
J
A
α
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
∆ / ζ
biquadratic
η=0.0
η=0.1
η=0.2
Q2
Qxy
Qxz
other
(a) (b) (c)
Fig. 35: (a) In-plane (gab) and out-of-plane (gc) g-factors connecting the magnetic moment and
pseudospin-1 S. The scaling of the in-plane and out-of-plane components is highly anisotropic near
the cubic limit but the pseudospin-1 picture only starts to apply around ∆ ≈ ζ where the g-factors are
already rather close to their ∆ → ∞ value of 2. (b),(c) Exchange parameters of the effective spin-1
model depending on the crystal field and Hund’s coupling strength η = JH/U. Panel (b) concerns the
dominant bilinear part of the interaction showing the parameters J, A, α defined by Eq. (4.14), the
smaller biquadratic part is addressed in panel (c). Here we have singled out three quadrupolar interaction
channels Q2,iQ2,j, Qxy,iQxy,j, and Qxz,iQxz,j, the other nonzero ones are indicated by thin gray
lines only (shown for all three η values).
Much more demanding task is the recalculation of the superexchange part of the singlet-triplet
model with the modified basis wavefunctions and its subsequent conversion to the pseudospin-1
operators. The resulting contribution of the x-bonds in RuO2 plane can be written as
Hx =
ij x
(J + A)Sx
i Sx
j + (J − A)Sy
i Sy
j + J(1 − α)Sz
i Sz
j +
αβ
q
(x)
αβ Qα
i Qβ
j . (4.14)
A similar form with interchanged x and y components is obtained for the y-bonds. Together
with Hloc we get the full Hamiltonian of the pseudospin-1 model, H = Hloc + Hx + Hy. Let
us now analyze the superexchange interaction in Eq. (4.14) in detail. The first part contains
diagonal spin-spin interactions SαiSαj, each component coming with its own interaction constant.
By inserting the definitions of Sx = −i(Tx − T †
x ) and Sy = −i(Ty − T †
y ) in terms of Hubbard
operators, one finds, that the J(Sx
i Sx
j +Sy
i Sy
j ) part of the pseudospin-1 interaction just reproduces
the Tx,y part of the quadratic interactions in the singlet-triplet model of Eq. (4.7). On top of
this, we also get a bond-selective contribution associated with the parameter A [this effect was
already neglected at the level of Eq. (4.7)] and Sz
i Sz
j interaction parametrized using Jz = J(1−α)
that was among the ignored four-triplon terms in Sec. 4.1. The respective interaction constants
in units of t2
/U and with c0,1 = cos 2ϑ0,1, s0,1 = sin 2ϑ0,1, and η = JH/U are given explicitly by
J =
7−c0+c1−7c0c1−2
√
2s0s1
48(1 − 3η)
+
13−c0−5c1+29c0c1+13
√
2s0s1
96
+
9−c0−c1+5c0c1+3
√
2s0s1
32(1 + 2η)
,
A =
5−3c0+3c1−5c0c1+2
√
2s0s1
48(1 − 3η)
+
−10−6c1+4c0c1−7
√
2s0s1
96
−
2c0+2c0c1+
√
2s0s1
32(1 + 2η)
,
Jz =
1 − cos 4ϑ1
24(1 − 3η)
+
17 − 24c1 + 7 cos 4ϑ1
96
+
7 − 8c1 + cos 4ϑ1
32(1 + 2η)
. (4.15)
4.3 Magnetic order due to triplon condensation 77
Ignoring the corrections due to Hund’s coupling in virtual states, they reduce to
J =
9−c0−c1+5c0c1+3
√
2s0s1
16
, A = −
2c0+2c0c1+
√
2s0s1
16
, Jz =
7−8c1+cos 4ϑ1
16
. (4.16)
As can be seen in Fig. 35(b) where we plot J, A, α as functions of the tetragonal field ∆, the
anisotropy is quickly getting marginal with increasing ∆ and in the large ∆ limit we approach
the isotropic Heisenberg situation with Jx = Jy = Jz = t2
/U. This effect is a consequence of
the quenched orbital component of the pseudospin-1 which then coincides with the original spin
S = 1 of t4
2g configurations and hence may only be subject to isotropic interactions. Vanishing
anisotropy can be also readily verified (for any JH/U ratio) using the above explicit expressions
when ϑ0 = ϑ1 = π/2 giving c0 = c1 = −1, s0 = s1 = 0, and cos 4ϑ1 = 1.
Since we deal with an effective spin-1 situation, the bilinear terms of the general form SαiSβj
may not be sufficient to fully describe the spin-spin interactions, in contrast to spin-1
2
models,
and this is indeed the case here. In Eq. (4.14), we have included also biquadratic terms that are
expressed using products of the five components of spin-1 quadrupolar moments:
Q0 = 1√
3
(2S2
z −S2
x −S2
y ) , Q2 = S2
x −S2
y , Qxy = SxSy +SySx , Qyz, Qzx (analogous). (4.17)
They can be in general arbitrarily combined for the two sites of the bond leading to the superposition
αβ q
(x)
αβ Qα
i Qβ
j but in our case the coefficients q
(x)
αβ are diagonal with the exception of
Q0 to Q2 coupling. All the nonzero biquadratic interactions are presented in Fig. 35(c) which
demonstrates that they are minor in the ∆/ζ interval of interest. In the following section we will
thus consider just the bilinear part of the superexchange.
4.3 Magnetic order due to triplon condensation
Before attempting a detailed comparison of the magnetic model introduced in the previous sections
and the experimental data on Ca2RuO4, we will demonstrate the basic features of the
model phase diagram. In particular, we will focus on the long-range magnetic order supported
by the model, that can be interpreted as resulting from triplon condensation. The introductory
singlet-triplet model (4.7) and the refined pseudospin-1 model of Sec. 4.2 show similar overall
behavior in this respect. Here we will consider the main part of the pseudospin-1 model that is
contained also in Eq. (4.7) and can be written explicitly as
H = ET
i
nTi + J
ij α=x,y
(T†
αs)i(s†
Tα)j − (T†
αs)i(T†
αs)j + H.c. . (4.18)
The model can be easily handled in the limit ET J where the intuitive picture of its ground
state is a dilute gas of triplons moving on the square lattice. To obtain the dispersion of moving
triplons we treat the local constraint ns + nTx + nTy = 1 by adopting dynamical Gutzwiller
approximation. Namely, we replace both s and s†
operators in Eq. (4.18) by 1 − nTx − nTy
and expand the square root assuming small density of triplons. We keep terms up to quadratic
order in the triplon operators Tx, Ty and regard T as regular bosonic operators afterwards. In
momentum space, we get the Hamiltonian of the canonical form
Hharm =
q α=x,y
AqT†
αqTαq − 1
2
Bq(TαqTα,−q + T†
αqT†
α,−q) (4.19)
with
Aq = ET + 4Jγq , Bq = 4Jγq , γq = 1
2
(cos qx + cos qy) (4.20)
78 4.3 Magnetic order due to triplon condensation
that is readily solved by Bogoliubov transformation giving elementary excitations with the dis-
persion
ωq = A2
q − B2
q = ET (ET + 8Jγq) . (4.21)
According to this result, the triplon excitations that start at the energy ET in the J → 0 limit
gradually soften around the q = (π, π) point when J is increased [see also Fig. 36(d)]. We can
continue with this solution up to a relatively modest Jcrit = 1
8
ET . At this point the excitation
dispersion touches zero energy, signaling a quantum phase transition to a phase with condensed
triplons as it has been briefly introduced in Sec. 1.5. Let us note here, that the position of the
quantum critical point is approximate only and Jcrit is revised to a higher value when going beyond
the harmonic approximation, i.e. including interaction effects between triplons (the density of
which is no longer negligible near the critical point).
To cover the entire parameter range of the model at the same level of approximation, we
utilize a trial wavefunction that allows for the condensation of triplons (as hardcore bosons) by
locally mixing the s and T states:
|Ψ =
R
cos θ s†
R + sin θ
α=x,y
d∗
αRT†
αR |vac . (4.22)
Having a condensate of vector bosons, we need to deal with its internal structure which is represented
here by a position-dependent complex vector dR. The local constraint ns + nT = 1
is maintained when |d| = 1. By minimizing the Hamiltonian average Ψ|H|Ψ with respect to
the variational parameter θ and optimizing simultaneously the dR structure, we get the phase
diagram shown in Fig. 36(a). The average can be conveniently expressed by introducing condensate
density ρ = Ψ|nT |Ψ = sin2
θ and separating the real and imaginary parts of the d
vectors as dR = uR + ivR with the constraint u2
+ v2
= 1. With this notation we obtain for the
Hamiltonian average, termed also the “classical” energy of the condensate
Eclass = Ψ|H|Ψ = ET
i
ρ + J
ij
4ρ(1 − ρ) vi · vj . (4.23)
Up to J = 1
8
ET the minimization gives θ = 0 and hence zero condensate density, recovering the
above result of the harmonic approximation. Above this critical J/ET strength, nonzero ρ appears
and the structure of Eq. (4.23) forces us to maximize the v component and make it antiparallel
at nearest-neighbor sites to gain energy. To uncover the magnetic nature of this type of triplon
condensate, we evaluate Ψ|Mα|Ψ = gab Ψ|(−i)(s†
Tα − T†
αs )|Ψ = −2gabvαR ρ(1 − ρ) (α =
x, y) and Ψ|Mz|Ψ = gc Ψ|(−i)(T†
xTy − T†
y Tx)|Ψ = 0, finding that the staggered vR of the
condensate translates to in-plane ordering of magnetic moments with the ordering vector Q =
(π, π) [see Fig. 36(b)]. Any direction of v and therefore any normalized combination of Tx and Ty
is equally good in terms of energy, this corresponds to the freedom to choose the magnetization
direction in the lattice plane. For later convenience, we set vxR = −eiQ·R
, vy = 0 giving Mx =
2gab ρ(1 − ρ)eiQ·R
, My = 0 and bringing Eq. (4.22) to its final form
|Ψ =
R
1 − ρ s†
R +
√
ρ i eiQ·R
T†
xR |vac (4.24)
with the optimum condensate density and θ parameter given by
ρ = sin2
θ =
1
2
1 −
ET
8J
(J > Jcrit = 1
8
ET ) . (4.25)
4.3 Magnetic order due to triplon condensation 79
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5J / ET
=
0
1
2
Γ X M Γ
ω/ET
J = 0.04 ET
0
1
2
Γ X M Γ
J = 0.11 ET
0
1
2
Γ X M Γ
J = 0.14 ET
0
1
2
Γ X M Γ
J = 0.20 ET
0
1
2
3
4
Γ X M Γ
J = 0.45 ET
Q = (π, π)
M = 2 ρ(1−ρ)
ρ
MyMx
Eamplitude
mode
XΓ
(a)
(d)
(c)
magnon
(b)
M
−π +π0
−π
0
+π
QCP
triplon gas
AF triplon condensate
Fig. 36: (a) Condensate density and ordered moment obtained from the variational Ansatz (4.22) as
functions of the J to ET ratio. Quantum critical point at J/ET = 1
8 separating the triplon gas phase and
condensed phase is indicated by the black square and the dashed line. (b) Magnetic ordering pattern
in the square lattice having Q = (π, π) ordering vector. (c) Brillouin zone of the square lattice with
highlighted conventional path around the irreducible Brillouin zone. (d) Dispersions of the elementary
excitations at selected J/ET points plotted along the conventional path of panel (c). Initially, we
observe twofold degenerate branch of triplon excitations. After the condensation occurs, the excitation
dispersion splits giving rise to a gapless magnon mode and gapful mode that can be interpreted as an
amplitude mode of the triplon condensate.
Let us now inspect the evolution of the elementary excitations. To calculate their spectra
we need to perform a harmonic expansion around |Ψ of Eq. (4.24), similar to that used to
arrive at Eqs. (4.19)-(4.21). The situation is complicated by the more complex structure of |Ψ
for the condensed case with nonzero ρ. This can be gauged away by performing the bosonic
transformation into new bosons a, b, c according to17





s
Tx
Ty





R
=





cos θ i eiQ·R
sin θ 0
i eiQ·R
sin θ cos θ 0
0 0 1










c
a
b





R
(4.26)
which converts |Ψ into a suitable form |Ψ = R c†
R|vac . Now we are in a position to employ
the dynamical Gutzwiller approximation for the condensed boson c via c, c†
→
√
1 − na − nb
followed by harmonic expansion in a and b operators. On the harmonic level, both a and b
bosons obey the Hamiltonian of the form (4.19) with
Aq = ET cos 2θ + 8J sin2
2θ + 4J cos2
2θ γq , Bq = 4J cos2
2θ γq , (a-bosons) (4.27)
Aq = ET cos2
θ + 4J sin2
2θ + 4J cos2
θ γq , Bq = 4J cos2
θ γq , (b-bosons) (4.28)
17
The form of the transformation matrix is chosen based on two requirements - to preserve bosonic commutation
relations in general and to maintain continuity with Eqs. (4.19)-(4.21) in particular.
80 4.4 Excitation spectra probed by neutron and Raman scattering
which leads to elementary excitations independently carried by the a and b bosons and having
the dispersions ωq = A2
q − B2
q shown in Fig. 36(d). For the uncondensed case θ = 0, the
dispersions are degenerate and coincide with Eq. (4.21) for the moving triplons, in the condensed
case they split and the corresponding excitations become two distinct species. The excitation
carried by b bosons is always gapless at the wavevector Q [symmetry point M in Fig. 36(c),(d)]
and corresponds to a magnon. Not surprisingly, it is linked to the boson Ty associated with
the direction perpendicular to the ordered moment. At q = Q, the mode merely rotates the
ordered moment among energetically equivalent positions, hence its gapless nature. A rather
specific feature of the magnon dispersion is that it reaches its maximum at q = 0, making it very
distinct from the conventional magnon dispersion obtained within antiferromagnetic Heisenberg
model. The excitation carried by a bosons can be interpreted as an amplitude fluctuation of
the condensate (sometimes dubbed as the condensed-matter Higgs mode [103]). Through the
oscillations in the angle θ, it effectively shakes the ratio of s and Tx in the local superposition and
hence modulates the condensate density. Such a mode necessarily costs some energy and is thus
gapful as seen in Fig. 36(d). Close to the quantum critical point the amplitude fluctuations are
still cheap and the gap is relatively small but when increasing J in the J > Jcrit interval, making
the condensate more and more robust, the amplitude mode gradually shifts to higher energies.
Fig. 36(e) schematically represents the two distinct modes as oscillations in a “Mexican hat”
depiction of the condensate potential. Both of them enter various dynamic response functions
and can be probed experimentally as we will discuss in the next paragraph.
4.4 Excitation spectra probed by neutron and Raman scattering
The peculiar magnetic state formed by the condensation of triplon excitations should most naturally
manifest itself in the dynamical magnetic susceptibility. The traditional experimental way
to access this quantity is the inelastic neutron scattering whose results will be discussed in the
following. Apart from the unusual magnon dispersion having a maximum at q = 0, we will be
mainly interested in the signatures of the amplitude mode detected in the experiments.
To be able to appreciate the inelastic neutron scattering data presented in Fig. 37, we first
inspect the corresponding susceptibilities on a simplified level of the previous Sec. 4.3. We start
by transforming the pseudospin S or magnetic moment operators via Eq. (4.26) followed by the
replacement of the condensed c boson operator. This gives us the connection of the magnetic
moment components and the elementary excitations carried by a, b bosons with already known
dispersions. Keeping only terms up to the quadratic order as before, we have
Sx = Mx/gab ≈ eiQ·R
sin 2θ + cos 2θ (a − a†
) , (4.29)
Sy = My/gab ≈ cos θ (b − b†
) , (4.30)
Sz = Mz/gc ≈ −eiQ·R
sin θ (b + b†
) . (4.31)
Equations (4.29)-(4.31) enable us to separately assess three distinct polarizations of the magnetic
modes as depicted in Fig. 37(a): the longitudinal polarization parallel to the ordered moment direction
(represented here by Sx) is complemented by in-plane and out-of-plane transverse options
(Sy and Sz, respectively). Note that Fig. 37(a) shows the actual situation in Ca2RuO4 where the
ordered moments point along the b axis while we are assuming the x direction. For the moment
we can ignore this inconsistency as it is irrelevant in the initial inspection of the in-plane isotropic
model. What matters here is just the relative orientation of the particular magnetic component
with respect to the ordered moment direction and lattice plane.
Based on Eq. (4.29) it is clear that the oscillating part of longitudinal component of the
magnetic moment is directly connected to the a bosons. The corresponding excitations – the
4.4 Excitation spectra probed by neutron and Raman scattering 81
T′
L
T
ω[meV]
wavevector q
0
10
20
30
40
50
60
70
(π/2,π/2)(π,0) (π,π) (0,0) (π,0)
L
T T′
ω [meV]
q=(π,π)
0
2
4
6
0 10 20 30 40 50
T
L
T′
q=(0,0)
ab
c
0
2
4
6 T L
T′
Tk+q
Lq Lq
T−k+Q
(b)
(a) polarized INS, spin−flip channel(d)(c)
Fig. 37: (a) Polarization of the magnetic modes with respect to the ordered moment direction. Apart
from the two transverse modes, in-plane T and out-of-plane T , the longitudinal mode L is visible
in the experimental data. (b) Renormalization of the longitudinal mode via decay into two-magnon
continuum. (c) Dynamical susceptibility as measured by unpolarized inelastic neutron scattering on
Ca2RuO4. The lines show the theoretical dispersions of the magnetic modes including anharmonic
effects. Model parameters used to fit the data: ET = 25 meV, J = 5.8 meV, α = 0.15, A = 2.3 meV,
and ∆ = 4 meV. The insets provide an intuitive picture of the nature of the longitudinal (top) and
transverse modes (bottom). (d) Results of polarized neutron measurements (squares and dots) for
two wavevectors q = (0, 0) (upper panel) and q = (π, π) (lower panel) compared to the theoretical
spectral profiles (shaded areas) of the modes calculated including anharmonic effects. The experimental
conditions enabled to distinguish the in-plane (ab plane) response containing T and L modes and the
out-of-plane (c axis) response containing the low-intensity T mode. Details on the experiments and
theoretical fits can be found in Ref. [101].
amplitude oscillations of the condensate – should therefore get imprinted into the susceptibility in
a form of a longitudinal mode (L). In the harmonic approximation it will share the dispersion with
the a bosons corresponding to the amplitude mode in Fig. 36(d). Similarly, the transverse in-plane
component of the magnetic moment in Eq. (4.30) is linked to the b bosons that were interpreted
as magnons. Finally, the transverse out-of-plane component in Eq. (4.31) is again linked to the b
bosons, but it includes an extra factor eiQ·R
which translates to a shift in momentum space. Both
transverse components of the dynamic susceptibility will therefore contain magnon-like modes
T, T but the dispersion of the out-of-plane T mode will be shifted by Q = (π, π) compared to
the original magnons.
This rough picture inferred from Eqs. (4.29)-(4.31) is consistent with the experimental observations
represented by the map of unpolarized neutron scattering intensity shown in Fig. 37(c)
and complemented by polarized neutron scattering spectra in Fig. 37(d). While the overall interpretation
of the data in terms of the L, T, T modes discussed above seems acceptable, the
simplified approach of Sec. 4.3 apparently does not capture certain features of the experimental
data. The deficiencies that are easily addressed on the theory side is the lack of the opening
of the magnon gap and discrepancies in the details of the dispersions. These are remedied by
the inclusion of the terms of the full pseudospin-1 model that were missing in Eq. (4.18). The
low-energy magnon gap opens due to the orthorhombic field represented by ∆ term in the local
82 4.4 Excitation spectra probed by neutron and Raman scattering
part (4.13) of the full model. It prefers the b-axis direction in the RuO2 plane so that in-plane
rotations of the ordered moments are not for free anymore.18
The dispersions are “cured” by
considering the bond-selective interaction A and the interaction of the out-of-plane pseudospin-1
components Sz. Much more challenging is the proper treatment of the longitudinal response.
The longitudinal mode is somewhat poorly visible in the unpolarized data but the polarized ones
clearly suggest that its effective lifetime drastically changes through the Brillouin zone. The
mode is observed as quite sharp at q = (0, 0), however, in the region around the ordering vector
Q = (π, π) it is subject to a strong decay. The explanation of this observation lies in the simultaneous
excitation of two-magnon continuum in the longitudinal channel and its interplay with
the amplitude mode. While being minor in three-dimensional cases [104,105], the interplay with
Goldstone modes is of a crucial importance in our two-dimensional setting. Without going into
details that can be found in the Supplementary Information to Ref. [101], we just note that the
most important correction when analyzing the spin excitations beyond harmonic approximation
is the coupling of the form
kq
Γkq aq bk b−k−q−Q with Γkq ∝ J sin 2θ cos 2θ γq + 1
2
(γk + γ−k−q−Q) (4.32)
which generates a selfenergy for the bare L mode obtained within harmonic approximation. In
the above equation, a and b should be understood as a shorthand notation for the imaginary
parts of the corresponding fields associated with the operators such as a−a†
entering Eq. (4.29).
The coupling activates a decay of the L mode into two-magnon continuum composed of pairs of
T modes, the simplest example of such process being illustrated in Fig. 37(b). The selfenergy
turns out to be indeed strongest for q around Q = (π, π) due to the common action of a large
matrix element combined with the employment of low-energy magnons with momentum ≈ Q.
The magnon gap due to ∆ is actually an advantage here and enables us to stay with RPAlike
approximation for the two-magnon decay. Having gapless magnons, we would have to face
delicate issues of infrared divergences. When renormalizing the L mode (and to a much smaller
extent also the T mode) by incorporating the anharmonic corrections, we obtain a nice detailed
agreement with the experimental data, including the extreme broadening of the L mode near
q = Q as demonstrated by the spectra in Fig. 37(d).
With the amplitude mode being obscured by the two-magnon continuum in the longitudinal
magnetic susceptibility, we would like to find another probe allowing for a more direct access.
Perhaps a bit surprising answer to this request is that the amplitude mode can be found in
the typically magnetically silent channel of Raman scattering. As it has been shown in several
theoretical studies, the appearance of the amplitude mode strongly depends on the symmetry of
the probe used [103,106]. We have observed its large decay in the longitudinal response (that was
still reduced thanks to the magnon gap) while in so-called scalar susceptibility it can maintain
its coherence [107–109]. The latter situation is the case of the Raman scattering.
In formal terms, the Raman scattering on a superexchange-driven magnetic system probes
dynamic correlation function of the Raman operator that is expressed within Fleury-Loudon
approximation [110] as
R ∝
ij
( in · rij)( out · rij)Hij . (4.33)
Here Hij is the superexchange Hamiltonian for the bond ij connecting two sites with the
relative position rij and in ( out) is the polarization vector of the incoming (outgoing) photon.
18
Lifting the degeneracy in the subspace spanned by Tx, Ty as shown in Fig. 34(b), this field also selects a
particular triplon combination Tb = (Tx + Ty)/
√
2 to condense. The calculation in Sec. 4.3 where we have used
Tx condensation for simplicity has to be revised accordingly.
4.4 Excitation spectra probed by neutron and Raman scattering 83
0 20 40 60 80 100
Ramanresponseχ″(ω)[a.u.]
ω [meV]
230K
175K
110K
95K
70K
50K
30K
T=10K
B1g − channel
0 20 40 60 80 100
ω [meV]
230K
175K
110K
95K
70K
50K
30K
T=10K
Ag − channel
χ″(ω)[a.u.]
ω [meV]
Ag − channel N=16
N=18
N=20
0 20 40 60 80 100 120χ″(ω)[a.u.]
B1g − channel N=16
N=18
N=20
composite
excitationsamplitude
mode
B1g Ag
δnT=Eδ ETSx Sxδ( SySy− )A=Eδ
B
B’ A’
in
out
x
y
(c)
(b)
magnon
(a) amplitude modemagnon E
S Sx y
~ ~~~~~
A
x
y
in
out
Fig. 38: (a) Temperature dependent Raman spectra of Ca2RuO4 sample obtained using the B1g and Ag
polarization setups [111]. The insets indicate the relative orientation of the in-plane polarization vectors
in, out with respect to the lattice. The magnetic part of the signal developing below TN ≈ 110 K is
highlighted for the lowest T = 10K by gray shading. Sharp peaks corresponding to the phonons that are
present at all temperatures were subtracted. (b) Schematic representation of the Raman excitation of
the magnetic condensate via modulation of the bond energy given by Eq. (4.33). In the B1g channel, the
A-term deforms the bottom of the potential driving a magnon rotation. In the Ag channel, modulations
via nT lead to an isotropic oscillation of the condensate amplitude – a direct excitation of the amplitude
mode occurs in this channel. (c) Raman spectra obtained by exact diagonalization on clusters with
N = 16, 18, and 20 sites compared to experimental data. The numerical spectra for B1g and Ag
channels are presented in identical scales and overlayed by the magnetic signal from panel (a) (dashed
lines). The calculation was performed using ET = 31 meV, J = 7.5 meV, α = 0.15, A = 2.3 meV, and
∆ = 4 meV.
By playing around with various combinations of the polarizations, we can test the system in
rather different ways.
For Heisenberg-like magnets the usual channel of interest is the B1g one that is probed when
using cross-polarized setup as shown in the inset of Fig. 38(a). It corresponds to the Raman
operator of the form R ∝ Hx − Hy, where Hx and Hy are sums of Hij over x- and y-bonds,
respectively. In the Heisenberg case this term excites a two-magnon continuum. The same
happens in our case with mostly Heisenberg-like bond interactions of pseudospins-1 as given for
the x-bonds by Hx in Eq. (4.14). However, thanks to the bond-selective anisotropic component
of the interaction A ij (±1)(Sx
i Sx
j − Sy
i Sy
j ), we get an additional feature that is unusual in the
magnetic Raman scattering – a direct excitation of a single magnon. To see this we consider the A
part of the Raman operator in more convenient axes, A ij (Sa
i Sb
j + Sb
i Sa
j ), and approximate Sb
which is the component along the ordered moment direction by Sb
R = S eiQ·R
. This way the A
part of the B1g Raman operator becomes ∝ A S Sa
Q and probes therefore the magnon at q = Q.
Looking at the experimental data in Fig. 38(a), we can see the magnetic signal developing below
TN. Consistently with the above expectations, it is composed of the two contributions – twomagnon
continuum (feature B ) encompassing energies around two times the magnon bandwidth
84 4.4 Excitation spectra probed by neutron and Raman scattering
≈ 40meV observed in Fig. 37(c) and a very sharp peak (feature B) at the energy coinciding with
the bottom of the magnon dispersion at Q = (π, π).
Let us now focus on the Ag channel that is normally not very interesting in magnetic Raman
scattering. In this case the Raman operator can be expressed as R ∝ Hx + Hy and is typically
proportional to the magnetic Hamiltonian of the system itself, making this channel silent at
finite frequencies. However, our Hamiltonian is equipped with a large local term ET nT that is
a source of highly interesting spectra. Finite-energy dynamics is equally well described by the
Raman operator equal to the difference of scaled R and the Hamiltonian, R = Hx + Hy − H =
−ET nT ∝ nT , which means that in the A1g channel we are in fact probing the nT susceptibility
and hence the amplitude mode. To have a more precise connection, we can take the nT operator
approximated in the same way as the longitudinal Sx in Eq. (4.29). It reads as
nT ≈ ρ − 1
2
i eiQ·R
sin 2θ (a − a†
) (4.34)
which implies that the magnetic signal in A1g Raman channel (essentially the nT susceptibility
for q → 0) will contain the q → Q amplitude mode of Fig. 36(d) that failed to show up clearly
in the longitudinal magnetic response. The experimental data for the Ag channel in Fig. 38(a)
indeed show a spectral profile consisting of a peak (feature A) followed by a long tail (feature
A ) that resembles the unspoiled “Higgs” spectral functions found in the literature [107–109]. To
have a one-to-one comparison for our particular model, Fig. 38(c) presents the theoretical Raman
spectra obtained numerically for small clusters and contrasts them to the magnetic part of the
signal plotted in Fig. 38(a). Overall, the magnetic Raman features in both channels are well
reproduced by the model calculations including their relative intensities, successfully concluding
our discussion on the singlet-triplet magnetism of Ca2RuO4, where the magnetic order is due to
Bose-Einstein condensation of initially gapped magnetic excitations.
5 Papers addressing the Kitaev–Heisenberg model
ˆ Kitaev-Heisenberg Model on a Honeycomb Lattice:
Possible Exotic Phases in Iridium Oxides A2IrO3
J. Chaloupka, G. Jackeli, and G. Khaliullin
Physical Review Letters 105, 027204 (2010) DOI: 10.1103/PhysRevLett.105.027204
In this paper we made an initial exploration of the phase diagram of the Kitaev-Heisenberg
model for honeycomb iridates that was proposed in Ref. [20]. Assuming FM Kitaev interaction
and AF Heisenberg interaction, we have demonstrated that the FM Kitaev spin liquid survives
the perturbation by the Heisenberg interaction up to a sizable strength, leading to a finite
window of the liquid phase in the phase diagram.
ˆ Zigzag Magnetic Order in the Iridium Oxide Na2IrO3
J. Chaloupka, G. Jackeli, and G. Khaliullin
Physical Review Letters 110, 097204 (2013) DOI: 10.1103/PhysRevLett.110.097204
This paper is a continuation of the study of the Kitaev-Heisenberg model reflecting the experimental
data that appeared since the publication of the first paper in 2010. Providing
microscopic arguments, we have extended the parameter regime to include further combinations
of signs of the Kitaev and Heisenberg interactions and determined the parameter values
that were consistent with the experimental data on Na2IrO3 available that time – zigzag type
of the magnetic ordering, temperature-dependent static magnetic susceptibility, and powder
inelastic neutron scattering.
ˆ Direct evidence for dominant bond-directional interactions
in a honeycomb lattice iridate Na2IrO3
S. H. Chun, J. W. Kim, J. Kim, H. Zheng, C. C. Stoumpos, C. D. Malliakas, J. F. Mitchell,
K. Mehlawat, Y. Singh, Y. Choi, T. Gog, A. Al-Zein, M. M. Sala, M. Krisch, J. Chaloupka,
G. Jackeli, G. Khaliullin, and B. J. Kim
Nature Physics 11, 462 (2015) DOI: 10.1038/NPHYS3322
By studying spin correlations above the N´eel temperature using magnetic diffuse x-ray scattering,
this paper brought an experimental evidence, that the spin interactions in Na2IrO3 are
highly anisotropic and this anisotropy is bond selective. Based on a comparison to numerical
simulations, we have quantified the dominance of the anisotropic interactions. Another important
result of this paper is the determination of the ordered moment direction by resonant
x-ray diffraction that fixed the FM sign of the Kitaev interaction in Na2IrO3.
85
86
ˆ Hidden symmetries of the extended Kitaev-Heisenberg model:
Implications for the honeycomb-lattice iridates A2IrO3
J. Chaloupka and G. Khaliullin
Physical Review B 92, 024413 (2015) DOI: 10.1103/PhysRevB.92.024413
This paper explored the rich hidden symmetries of the extended Kitaev-Heisenberg model
(including Γ and Γ interactions) through the method of dual transformations that map the
model to an equivalent form but with different parameters. We have identified several points
of hidden SU(2) symmetry where the model reduces (in a nontrivial way) to either FM or AF
Heisenberg model. The complete exploration of these symmetries provided deeper insights
into the global phase diagram of the model. The results were also used in the discussion of
the relevant parameter regime of honeycomb iridates Na2IrO3 and Li2IrO3.
ˆ Magnetic anisotropy in the Kitaev model systems Na2IrO3 and RuCl3
J. Chaloupka and G. Khaliullin
Physical Review B 94, 064435 (2016) DOI: 10.1103/PhysRevB.94.064435
This paper is devoted to a detailed study of the ordered moment direction as a very sensitive
probe of the anisotropic interactions. We have developed a method based on spin-coherent
states that enabled to analyze cluster ground states obtained by exact diagonalization and
precisely determine the moment direction. Utilizing the above methodology we have attempted
to narrow down the parameter regime in Na2IrO3 and the newly discovered (at that time)
α-RuCl3. Doing so, we have emphasized the role of the trigonal field that brings a distinction
between the pseudospin direction and those of the magnetic moment and a special vector
entering the polarization dependence of the resonant x-ray scattering.
ˆ Phase diagram and spin correlations of the Kitaev-Heisenberg model:
Importance of quantum effects
D. Gotfryd, J. Rusnaˇcko, K. Wohlfeld, G. Jackeli, J. Chaloupka, and A. M. Ole´s
Physical Review B 95, 024426 (2017) DOI: 10.1103/PhysRevB.95.024426
In this paper we tried to quantify the role of quantum fluctuations through the phase diagram
of the (non-extended) Kitaev-Heisenberg model by contrasting several methods that handle
quantum fluctuations in different ways. We have illustrated and provided insights into several
interesting effects, such as the striking difference of the robustness of the FM and AF Kitaev
spin liquid phases against Heisenberg perturbations.
87
ˆ Kitaev-like honeycomb magnets: Global phase behavior and emergent effective models
J. Rusnaˇcko, D. Gotfryd, and J. Chaloupka
Physical Review B 99, 064425 (2019) DOI: 10.1103/PhysRevB.99.064425
This paper presents a comprehensive discussion of the global phase diagram of the extended
Kitaev-Heisenberg model. By combining numerical simulations on small clusters and analytical
techniques based on symmetry analysis of this spin Hamiltonian, we were able to thoroughly
interpret the global trends in the phase diagram and uncover peculiar links to well-known simpler
models on the honeycomb lattice. The paper can also serve as a methodological example
how to deal with a complex spin model with bond-selective anisotropic interactions based on
symmetry grounds.
ˆ Dynamic Spin Correlations in the Honeycomb Lattice Na2IrO3 Measured
by Resonant Inelastic x-Ray Scattering
J. Kim, J. Chaloupka, Y. Singh, J. W. Kim, B. J. Kim, D. Casa,
A. Said, X. Huang, and T. Gog
Physical Review X 10, 021034 (2020) DOI: 10.1103/PhysRevX.10.021034
In this paper we present results of resonant inelastic x-ray (RIXS) measurements on Na2IrO3
utilizing a state-of-the-art setup at APS Argonne that are interpreted through an extensive set
of numerical simulations. The experiment provided unique momentum and energy resolved
spectra of spin excitations in this compound. These are compared to exact diagonalization
calculations of the RIXS response based on the extended Kitaev-Heisenberg model. This way,
important conclusions about the interactions in Na2IrO3 were obtained.
ˆ Kitaev Spin Liquid in 3d Transition Metal Compounds
H. Liu, J. Chaloupka, and G. Khaliullin
Physical Review Letters 125, 047201 (2020) DOI: 10.1103/PhysRevLett.125.047201
This paper extends the idea of Ref. [62] that suggested a potential realization of the Kitaev
honeycomb model in cobalt compounds with 3d7
valence configuration. Via a microscopic
derivation of the exchange interactions when including the effects of the trigonal crystal field,
we found that the trigonal distortion should be an efficient tool to drive the material toward
the Kitaev spin liquid phase. Apart from a general discussion we tried to apply our theory
to the candidate compound Na3Co2SbO6 and estimated the parameters of the corresponding
model.
88
Kitaev-Heisenberg Model on a Honeycomb Lattice:
Possible Exotic Phases in Iridium Oxides A2IrO3
Jirˇı´ Chaloupka,1,2
George Jackeli,2,* and Giniyat Khaliullin2
1
Department of Condensed Matter Physics, Masaryk University, Kotla´rˇska´ 2, 61137 Brno, Czech Republic
2
Max-Planck-Institut fu¨r Festko¨rperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
(Received 16 April 2010; published 9 July 2010)
We derive and study a spin one-half Hamiltonian on a honeycomb lattice describing the exchange
interactions between Ir4þ
ions in a family of layered iridates A2IrO3 (A ¼ Li; Na). Depending on the
microscopic parameters, the Hamiltonian interpolates between the Heisenberg and exactly solvable Kitaev
models. Exact diagonalization and a complementary spin-wave analysis reveal the presence of an
extended spin-liquid phase near the Kitaev limit and a conventional Ne´el state close to the Heisenberg
limit. The two phases are separated by an unusual stripy antiferromagnetic state, which is the exact ground
state of the model at the midpoint between two limits.
DOI: 10.1103/PhysRevLett.105.027204 PACS numbers: 75.10.Jm, 75.25.Dk, 75.30.Et
Magnetic systems exhibit, most commonly, long-range
classical order at sufficiently low temperatures. An exception
are frustrated magnets, in which the topology of the
underlying lattice and/or competing interactions lead to an
extensively degenerate manifold of classical states. In such
systems, exotic quantum phases of Mott insulators (spin
liquids, valence bond solids, etc.) can emerge as the true
ground states (for reviews, see Refs. [1,2]). In quantum
spin liquids, strong zero-point fluctuations of correlated
spins prevent them to ‘‘freeze’’ into magnetic or statically
dimerized patterns, and conventional phase transitions that
break time-reversal and lattice symmetries are avoided.
Spin liquids have attracted particular attention since
Anderson proposed their possible connection to superconductivity
of cuprates [3].
Recently, spin-liquid states of matter have been exemplified,
on a quantitative level, by an exactly solvable
model by Kitaev [4]. His model deals with spins one-half
that live on a honeycomb lattice. The nearest-neighbor
(NN) spins interact in a simple Ising-like fashion but,
because different bonds use different spin components
[see Fig. 1(a)], the model is highly frustrated. Its ground
state is spin-disordered and supports the emergent gapless
excitations represented by Majorana fermions [4]. Spinspin
correlations are, however, short-ranged and confined
to NN pairs [5,6]. This may suggest the robustness of the
disordered state to spin perturbations. Indeed, Tsvelik has
shown [7] that there is a window of stability for the spinliquid
state in the Kitaev model perturbed by isotropic
Heisenberg exchange.
Finding a physical realization of this remarkable model
is a great challenge, also because of its special properties
attractive for quantum computation [4]. As the key element
of the model is a bond-selective spin anisotropy, one
possible idea [8] is to explore Mott insulators of late
transition metal ions with orbital degeneracy, in which
the bond directional nature of electron orbitals can be
translated into a desired anisotropy of magnetic interactions
through strong spin-orbit coupling.
In this Letter, we examine the iridium oxides A2IrO3
from this perspective. In these compounds, the Ir4þ ions
have an effective spin one-half moment and form weakly
coupled honeycomb-lattice planes. Our analysis of the
underlying exchange mechanisms shows that the spin
Hamiltonian comprises two terms, ferromagnetic (FM)
and antiferromagnetic (AF), in the form of Kitaev and
Heisenberg models, respectively. The model has an interx
x
S S z z
S S
y y
S S
z( )
y( )
(c)
stripy AF
)b()a(
Heisenberglimit
Kitaevlimit
spin liquidNeel AF
α=0 α=1α=1/2
x( )
FIG. 1. (a) Three types of bonds in the honeycomb lattice and
Kitaev part of the interaction. (b) The supercell of the foursublattice
system enabling the transformation of the model (1)
into the Hamiltonian of a simple ferromagnet at  ¼ 1
2 . This
supercell with periodic boundary conditions applied was used as
a cluster for the exact diagonalization. (c) Schematic phase
diagram: With increasing , the ground state changes from the
Ne´el AF order to the stripy AF state (being a fluctuation-free
exact solution at  ¼ 1
2 ) and to the Kitaev spin liquid. See the
text for the critical values of .
PRL 105, 027204 (2010) P H Y S I C A L R E V I E W L E T T E R S
week ending
9 JULY 2010
0031-9007=10=105(2)=027204(4) 027204-1 Ó 2010 The American Physical Society
89
esting phase behavior and hosts, in addition to the spinliquid
state, an unusual AF order that is also an exact
solution at a certain point in phase space.
Experimental studies of iridium compounds are rather
scarce, and the nature of their insulating behavior is not yet
fully understood. In fact, Na2IrO3 was suggested as an
interesting candidate for a topological band insulator [9].
Given that high temperature magnetic susceptibilities of
Na2IrO3 and Li2IrO3 obey the Curie-Weiss law with an
effective moment corresponding S ¼ 1=2 per Ir ion [10–
13], we start here with the Mott insulator picture.
The Hamiltonian.—We recall that the Ir4þ ion in the
octahedral field has a single hole in the threefold degenerate
t2g level hosting an orbital angular momentum l ¼ 1.
Strong spin-orbit coupling lifts this degeneracy, and the
resulting ground state is a Kramers doublet with total
angular momentum one-half [14], referred to as ‘‘spin’’
hereafter. In fact, it is predominantly of orbital origin, and
this is what makes the magnetic interactions highly anisotropic
due to the spin-orbit entanglement of magnetic and
real spaces. In A2IrO3 compounds, the IrO6 octahedra
share the edges, and Ir ions can communicate through
two 90 Ir-O-Ir exchange paths [8] or via direct overlap
of their orbitals. Collecting the possible exchange processes
(discussed below) and projecting them onto the
lowest Kramers doublet with S ¼ 1=2, we obtain the following
spin Hamiltonian on a given NN ij bond:
H ð
Þ
ij ¼ ÀJ1S
i S
j þ J2Si Á Sj: (1)
Here spin quantization axes are taken along the cubic axes
of IrO6 octahedra. In a honeycomb lattice formed by Ir
ions, there are three distinct types of NN bonds referred
to as 
 (¼x; y; z) bonds because they host the Ising-like
J1 coupling between the 
 components of spins [see
Fig. 1(a)]. The first part of Eq. (1) is thus nothing but the
FM Kitaev model, and the J2 term is a conventional AF
Heisenberg model. The exchange constants J1 and J2 are
derived from a multiorbital Hubbard Hamiltonian consisting
of the local interactions and the hopping term. The
latter describes tpd hopping between Ir 5d and O 2p
orbitals via the charge-transfer gap Ápd, and a direct dd
overlap t0
between NN Ir t2g orbitals [15]. We find J1 ¼
ð1 þ 22Þ and J2 ¼ ð2 þ 3Þ. Hereafter, we use
4t2
=9Ud as our energy unit, where t ¼ t2
pd=Ápd, and Ud
stands for the Coulomb repulsion on the same d orbitals.
There are three physically distinct virtual processes that
determine the set of  parameters and thus the ratio J2=J1.
The 1 ¼ 6JH
UdÀ3JH
Ud
UdÀJH
term appears due to the multiplet
structure of the excited levels induced by Hund’s coupling
JH [8]. The processes when two holes meet at the same
oxygen site (and experience Up repulsion) and when they
are cyclically exchanged around a Ir2O2 square plaquette
bring together a 2 ¼
Up
ÁpdþUp=2
Ud
Ápd
contribution. Further, a
direct dd-hopping t0
between NN Ir t2g orbitals contributes
to the Heisenberg term with exchange coupling 3 ¼
ðt0
=tÞ2
. It is difficult to estimate the values of all the
parameters involved; however, we expect 1 to be the
largest, of the order of 1, and 2;3 < 1.
We parametrize the exchange couplings as J1 ¼ 2 and
J2 ¼ 1 À  and study the properties of Kitaev-Heisenberg
model (1) in the whole parameter space 0  1.
Phase diagram.—At  ¼ 0, we are left with the
Heisenberg model exhibiting the Ne´el order with a staggered
moment reduced to hSz
i ’ 0:24 [16]. The opposite
limit,  ¼ 1, corresponds to the exactly solvable Kitaev
model with a short-range spin-liquid state [4], where spin
correlation functions are identically zero beyond the NN
distance and, on a given NN bond, only the components of
spins matching the bond type are correlated [5].
Interestingly, the model is exactly solvable at  ¼ 1
2 ,
too. At this point Eq. (1) reads, e.g., on a z-type bond,
as H ðzÞ
ij ¼ 1
2 ðSx
i Sx
j þ Sy
i Sy
j À Sz
i Sz
jÞ. This anisotropic
Hamiltonian can be mapped to that of a simple
Heisenberg model on all bonds simultaneously [17].
Specifically, we divide the honeycomb lattice into four
sublattices [see Fig. 1(b)] and introduce the rotated operators
~S: While ~S ¼ S in one of the sublattices, ~S on the
remaining three sublattices differs from the original S by
the sign of two appropriate components, depending on the
sublattice they belong to. In the new basis, Eq. (1) takes the
form
H ð
Þ
ij ¼ À2ð2 À 1Þ~S
i
~S
j À ð1 À Þ~Si Á ~Sj: (2)
At  ¼ 1
2 , the first term vanishes and we obtain the isotropic,
both in spin and real spaces, Heisenberg model
H ð
Þ
ij ¼ À 1
2
~Si Á ~Sj with FM coupling. Thus, at  ¼ 1
2 ,
i.e., at J1 ¼ 2J2, the exact ground state of model (1) is a
fully polarized FM state in the rotated basis. Now consider
the FM array of spins with, e.g., h~Sz
i ¼ 1=2, and map it
back to the original spin basis. The resulting order corresponds
to a stripy AF pattern of the original magnetic
moments depicted in Fig. 1(c). Note that such a stripy
order, despite being of AF type, is fluctuation-free at
 ¼ 1
2 and would thus show a fully saturated AF order
parameter.
The above discussion suggests three possible ground
state phases of the model (1) as shown in Fig. 1(c):
(i) Ne´el order near  ¼ 0, (ii) stripy AF order around  ¼
1
2 , and (iii) a spin-liquid phase close to  ¼ 1.
We first consider the ordered phases. Except special
cases of  ¼ 0 and  ¼ 1
2 just discussed, the Hamiltonian
(1) does not have any spin-rotational symmetry. However,
a spurious SUð2Þ continuous symmetry and associated
pseudo-Goldstone mode appear in a linear spin-wave
(SW) description. As in the case of a similar model on a
cubic lattice [18], we find that quantum fluctuations restore
the underlying discrete (hexagonal) symmetry of the
model, selecting thereby the direction of ordered moments
along one of the cubic axes (of IrO6 octahedra), and also
open a gap in SW spectra. Considering the quantum energy
PRL 105, 027204 (2010) P H Y S I C A L R E V I E W L E T T E R S
week ending
9 JULY 2010
027204-2
90
cost for rotating the order parameter by a small angle
away from a cubic axis, we find a quantum SW gap Á ’
2
 ð À 1
2Þ2
for  $ 1
2 .
The classical phase boundary between Ne´el and stripy
AF orderings is at  ¼ 1
3 , where linear SW spectra of both
states develop zero-energy lines [19], reflecting the infinite
degeneracy of classical states. At  ¼ 1
3 , Eq. (1) reads,
e.g., on z-type bonds, as H ðzÞ
ij ¼ 2
3 ðSx
i Sx
j þ Sy
i Sy
jÞ; i.e., only
two spin components are coupled on a given bond.
Considering Ne´el or stripy AF with ordered spins parallel
to the z axis, one finds that flipping all the spins along a
zigzag chain, formed by x- and y-type bonds, does not
change classical energy. This degeneracy is again accidental
(an artifact of classical treatment) and can thus be lifted
by quantum fluctuations. They favor the Ne´el state and
shift the classical phase boundary to a larger value  ’ 0:4.
This estimate is obtained by comparing the energies
of the Ne´el [e1 ’ À 3
16 ð3 À 5Þ] and the stripy [e2 ’
À 1
8 ð5 À 3 þ 1
Þ] states including quantum corrections
via second-order perturbation theory and matches well
the numerical result found below.
Now we discuss the phase behavior at 1
2 <  < 1, i.e., in
between two exact solutions (stripy AF at  ¼ 1
2 and a
Kitaev spin liquid at  ¼ 1). In terms of rotated spins, all
the couplings are of FM nature in this region [see Eq. (2)].
Thus, the FM order (read stripy AF of the original spins)
is the only possible magnetic phase here to compete
with the spin-liquid state. Since the latter is stable against
a weak Heisenberg-type perturbation [7], a critical value
of  for the spin order/disorder transition must be located
at some point less than 1. We give its naive estimate based
on the energetics of these two phases. The energy of the
stripy AF state is given above. The upper boundary for the
energy of spin-liquid state is given by the expectation value
of Eq. (2) using the exact result hS
i S
j i ¼ 0:13 at  ¼ 1
[5]. As a result, we find the transition from stripy AF order
to a spin liquid at  ’ 0:86 (close to the numerical result
below).
Single-magnon excitations fail to detect this transition
(since, as said above, there is not any other competing
magnetic state). As  increases, the lower branch of the
linear SW spectrum just gradually softens, to become
completely flat in the limit of  ¼ 1 where the classical
ground state is extensively degenerate [20]. We therefore
suspect that the instability responsible for the collapse of
magnetic order resides in the two-magnon sector [21].
Leaving this subtle issue for a future work, we now turn
to our numerical results, which describe the evolution of
spin correlations across the entire phase diagram.
Numerical study.—We use the Lanczos exact diagonalization
method to study a 24-site cluster [see Fig. 1(b)]
with periodic boundary conditions. The cluster is compatible
with the above discussed four-sublattice transformation
of Eq. (1) into Eq. (2). This provides an exact reference
point  ¼ 1=2, which is useful for the interpretation of
numerical data shown in Figs. 2 and 3 in terms of the
original as well as transformed spins.
Figure 2 clearly locates the two phase transitions. In
particular, a pronounced maximum in the second derivative
of the ground state energy [Fig. 2(c)] indicates a first-order
transition from Ne´el to stripy AF phase at  ’ 0:4. The
much weaker (note the log scale) and wider second peak at
 ’ 0:8 suggests a second- (or a weakly first-) order transition
from stripy AF to a spin-liquid state.
Figure 2(a) shows the squared total spins ~S2
tot and S2
tot
normalized to ~Sð~S þ 1Þ with ~S ¼ N=2 that can be reached
in the FM state. Although these are not conserved quantities
in the model, they characterize the phase map quite
well. In particular, a long tail of ~S2
tot above  ¼ 0:8 indicates
a ‘‘leakage’’ of stripy AF correlations into a
spin-liquid phase. This is also evidenced by the behavior
of longer range, beyond NN, spin correlations that are
still visible in a spin-liquid regime, except close to the
10
1
102
10
3
0.0 0.2 0.4 0.6 0.8 1.0
−d
2
EGS/dα
2
α
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
NNspincorrelations
〈 ˜S ˜S 〉
〈 ˜Sγ
˜Sγ
〉
〈 S S 〉
0.0
0.2
0.4
0.6
0.8
1.0
squaredtotalspin
(a)
(b)
(c)
〈 ˜S
2
tot 〉 〈 S
2
tot 〉
spin
liquid
FAlee´N
stripy AF
0
1/4
1/2 1
〈˜S˜S〉
α
FIG. 2. (a) Squared total spin of the 24-site cluster, normalized
to its value in the fully polarized FM state, as a function of .
The solid (dot-dashed) line corresponds to the rotated (original)
spin basis. (b) The NN spin correlations: The solid (dot-dashed)
line corresponds to a scalar product of the rotated (original)
spins. The component of the correlation function matching the
bond direction is indicated by a dotted line. This quantity is the
same in both bases. The inset compares NN spin correlations
(solid line) above  ¼ 0:5 with longer range spin correlations up
to third-nearest neighbors (dotted lines). (c) Negatively taken
second derivative of the ground state energy with respect to . Its
maxima indicate the phase transitions at  ’ 0:4 and 0.8.
PRL 105, 027204 (2010) P H Y S I C A L R E V I E W L E T T E R S
week ending
9 JULY 2010
027204-3
91
Kitaev limit where they vanish completely [see the inset in
Fig. 2(b)].
Figure 2(b) highlights how the NN spin correlations
evolve as their interactions change from one type to another.
In the Ne´el state, where the model is more
Heisenberg-like for the original spins, we reproduce hSi Á
Sji ’ À0:37 [16]. At the ‘‘hidden’’ FM Heisenberg point
 ¼ 1=2, one finds h~Si Á ~Sji ¼ 1
4 , equally contributed by
all three components of the rotated spin ~S. Things change
dramatically in the spin-liquid phase: Here, a particular
component of spin correlations h~S
i
~S
j i, dictated by the
Kitaev model, dominates. Its value of 0.132 that we find
at  ¼ 1 agrees well with the exact result 0.131 for an
infinite lattice [5].
Finally, we discuss the response to a weak magnetic field
~Bz
which, in terms of original spins, linearly couples to the
stripy AF order parameter. Figure 3 shows that even a very
weak field induces a nearly saturated moment in the entire
region of the stripy AF phase. As the system switches to the
Ne´el phase, a response to the ‘‘stripy field’’ ~Bz
drops
abruptly to zero, as expected. The induced moment sharply
reduces near  ¼ 0:8, too, but remains finite in a spinliquid
phase. Here the magnetization curve shows a linear
dependence on ~Bz
, and we may extract from its slope the
susceptibility  ¼ h~Sz
toti=N ~Bz. Shown in Fig. 3 is the inverse
value of  as a function of . This quantity scales
with the energy gap between the ground state and the
excited states that are accessible by the magnetic field.
According to Kitaev’s solution [4], these states must belong
to the flux sectors located at energies of the order of 1.
The observed À1 / ð À 0:8Þ behavior shows that this
characteristic (spin) gap gradually softens towards the  ’
0:8 critical point, as the spin correlations beyond the NN
distances start to grow [see Fig. 2(b), inset].
Experimental data [10–13] are rather insufficient to
conclusively locate the position of A2IrO3 compounds in
our phase diagram. Also, Na=Ir site disorder [13] has to be
kept in mind: Often, nonmagnetic impurities induce local
moments [22], and this has been shown to happen in the
Kitaev model as well [23].
In conclusion, we have examined the interactions and
possible magnetic states in iridates A2IrO3. The obtained
Kitaev-Heisenberg model shows rich behavior including a
spin liquid and unusual stripy AF phases. We hope that
these results will motivate experimental studies of layered
iridates and similar compounds of late transition metal
ions, where the physics of the Kitaev model might be
within reach.
We thank B. Keimer, A. Schnyder, S. Trebst, and M.
Zhitomirsky for discussions. We are grateful to H. Takagi
and A. M. Tsvelik for valuable discussions and for communicating
the unpublished results. Support from
MSM0021622410 (J. C.) and GNSF/ST09-447 (G. J.) is
acknowledged.
*Also at Andronikashvili Institute of Physics, 0177 Tbilisi,
Georgia.
[1] L. Balents, Nature (London) 464, 199 (2010).
[2] G. Misguich and C. Lhuillier, in Frustrated Spin Systems,
edited by H. T. Diep (World Scientific, Singapore, 2005).
[3] P. W. Anderson, Science 235, 1196 (1987).
[4] A. Kitaev, Ann. Phys. (N.Y.) 321, 2 (2006).
[5] G. Baskaran, S. Mandal, and R. Shankar, Phys. Rev. Lett.
98, 247201 (2007).
[6] H.-D. Chen and Z. Nussinov, J. Phys. A 41, 075001
(2008).
[7] A. M. Tsvelik (unpublished).
[8] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205
(2009).
[9] A. Shitade et al., Phys. Rev. Lett. 102, 256403 (2009).
[10] I. Felner and I. M. Bradaric´, Physica (Amsterdam) 311B,
195 (2002).
[11] H. Kobayashi et al., J. Mater. Chem. 13, 957 (2003).
[12] H. Takagi (private communication).
[13] Y. Singh and P. Gegenwart, arXiv:1003.0973.
[14] B. J. Kim et al., Science 323, 1329 (2009).
[15] On a Ir2O2 plaquette, e.g., in the xy plane, the hopping
term reads as Àtpdpy
1ð2Þ;zðdi;xzðyzÞ þ dj;yzðxzÞÞ À
t0dy
i;xydj;xy þ H:c:, where p1ð2Þ;z refers to a 2pz orbital of
oxygen 1(2) shared by NN Ir ions i and j.
[16] J. B. Fouet, P. Sindzingre, and C. Lhuillier, Eur. Phys. J. B
20, 241 (2001).
[17] G. Khaliullin, Prog. Theor. Phys. Suppl. 160, 155 (2005).
[18] G. Khaliullin, Phys. Rev. B 64, 212405 (2001).
[19] The details of SW analysis will be given elsewhere.
[20] G. Baskaran, D. Sen, and R. Shankar, Phys. Rev. B 78,
115116 (2008).
[21] Preliminary calculations indicate that two-magnon bound
states form at large . In case of the Kitaev toric code
model, the relevance of multimagnon bound states to a
quantum phase transition has been discussed by J. Vidal,
R. Thomale, K. P. Schmidt, and S. Dusuel, Phys. Rev. B
80, 081104(R) (2009).
[22] G. Khaliullin, R. Kilian, S. Krivenko, and P. Fulde, Phys.
Rev. B 56, 11 882 (1997).
[23] A. J. Willans, J. T. Chalker, and R. Moessner, Phys. Rev.
Lett. 104, 237203 (2010).
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
magneticmoment
inversesusceptibility
α
χ
-1
˜Bz = 10-3
˜Bz = 10-4
FIG. 3. Magnetic moment 2h~Sz
toti=N induced by field ~Bz
(Zeeman coupled to the rotated spins). The circles at  > 0:8
show the inverse spin susceptibility to this field.
PRL 105, 027204 (2010) P H Y S I C A L R E V I E W L E T T E R S
week ending
9 JULY 2010
027204-4
92
Zigzag Magnetic Order in the Iridium Oxide Na2IrO3
Jirˇı´ Chaloupka,1,2
George Jackeli,1,* and Giniyat Khaliullin1
1
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
2
Central European Institute of Technology, Masaryk University, Kotla´rˇska´ 2, 61137 Brno, Czech Republic
(Received 23 September 2012; published 28 February 2013)
We explore the phase diagram of spin-orbit Mott insulators on a honeycomb lattice, within the KitaevHeisenberg
model extended to its full parameter space. Zigzag-type magnetic order is found to occupy a
large part of the phase diagram of the model, and its physical origin is explained as due to interorbital
t2g À eg hopping. The magnetic susceptibility, spin wave spectra, and zigzag order parameter are
calculated and compared to the experimental data, obtaining thereby the spin coupling constants in
Na2IrO3 and Li2IrO3.
DOI: 10.1103/PhysRevLett.110.097204 PACS numbers: 75.10.Jm, 75.25.Dk, 75.30.Et
In the quest for materials with novel electronic phases,
iridium oxide Na2IrO3 came into focus recently [1–7] due
to theoretical predictions [8,9] that this system may host
Kitaev model physics and the quantum spin Hall effect.
Na2IrO3 is an insulator with a sizable and temperature
independent optical gap ’ 0:35 eV [7], and shows
Curie-Weiss type susceptibility [1,6] with moments corresponding
to an effective spin one-half Ir4þ
ion with a t5
2g
configuration [10]. These facts imply that Na2IrO3 is a
Mott insulator with well-localized Ir moments.
Collective behavior of local moments in Mott insulators
is governed by three distinct and often competing forces:
(i) orbital-lattice [Jahn-Teller (JT)] coupling, (ii) virtual
hopping of electrons across the Mott gap resulting in
exchange interactions, and (iii) relativistic spin-orbit
coupling (see Ref. [11] for extensive discussions). The
corresponding energy scales EJT, J, and  vary broadly
depending on the type of magnetic ions and chemical
bonding [12]. When  > ðEJT; JÞ, as often realized for
Co, Rh, and Ir ions in an octahedral environment, local
moments acquire a large orbital component which may
result in a strong departure from spin-only Heisenberg
models [8,11]. The direct observation of large spin-orbit
splitting 3=2 $ 0:6–0:7 eV in insulating iridates Sr2IrO4
[13], Sr3Ir2O7 [14], and Na2IrO3 [15] made it certain that
 > ðEJT; JÞ. Thus, the low-energy physics of Na2IrO3 is
governed by interactions among the spin-orbit entangled
Kramers doublets of Ir ions.
It is also established now [3–5] that Ir moments in
Na2IrO3 undergo antiferromagnetic (AF) order at TN ’
15 K. The fact that TN is much smaller than the paramagnetic
Curie temperature (À125 K) [6] and spin-wave
energies [4] implies that the underlying interactions are
strongly frustrated. This is natural in the so-called KitaevHeisenberg
(KH) model [16] where long range order is
suppressed by the proximity to the Kitaev spin-liquid (SL)
state. However, the observed ‘‘zigzag’’ magnetic pattern
[ferromagnetic (FM) zigzag chains, AF coupled to each
other] came as a surprising challenge to this simple and
attractive model. To resolve the ‘‘zigzag puzzle’’, a number
of proposals, ranging from various modifications of the KH
model [4,6,17–19] to a complete denial [20] of a local
moment picture in Na2IrO3, have been put forward.
In this Letter, we show that the zigzag order is in fact a
natural ground state (GS) of the KH model, in a previously
overlooked parameter range. Next, we identify the exchange
process that supports a zigzag-phase regime.
Further, we calculate spin-wave spectra, the ordered
moment, and magnetic susceptibility of the model in the
zigzag phase, and find a nice agreement with experiment.
This lends strong support to the KH model as a dominant
interaction in Na2IrO3 and related oxides.
The model.—Nearest-neighbor (NN) interaction
between isospin one-half Kramers doublets of Ir4þ
ions,
coupled via 90
-exchange bonds, reads as follows (the
exchange processes are described later):
H ð
Þ
ij ¼ 2KS
i S
j þ JSi Á Sj: (1)
Here, 
ð¼ x; y; zÞ labels 3 distinct types of NN bonds of a
honeycomb lattice [16] of Ir ions in Na2IrO3, and spin axes
oriented along the Ir-O bonds of IrO6 octahedron. The
bond-dependent Ising coupling between the 
 components
of spins is nothing but the Kitaev model [21], and the
second term stands for the Heisenberg exchange.
Let us introduce the energy scale A ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K2
þ J2
p
and the
angle ’ via K ¼ A sin’ and J ¼ A cos’; the model (1)
takes then the following form:
H ð
Þ
ij ¼ Að2 sin’S
i S
j þ cos’Si Á SjÞ: (2)
We let the ‘‘phase’’ angle ’ vary from 0 to 2, uncovering,
thereby, additional phases of the model that escaped attention
previously [16], including its zigzag ordered state
which is of a particular interest here.
It is instructive to introduce, following Refs. [11,16], 4
sublattices with the fictitious spins ~S, which are obtained
from S by changing the sign of its two appropriate
PRL 110, 097204 (2013) P H Y S I C A L R E V I E W L E T T E R S
week ending
1 MARCH 2013
0031-9007=13=110(9)=097204(5) 097204-1 Ó 2013 American Physical Society
93
components depending on the sublattice index. This transformation
results in the ~S Hamiltonian of the same form as
(1), but with effective couplings ~K ¼ K þ J and ~J ¼ ÀJ,
revealing a hidden SUð2Þ symmetry of the model at
K ¼ ÀJ (where the Kitaev term ~K vanishes). For the
angles, the mapping reads as tan~’ ¼ À tan’ À 1.
Phase diagram.—In its full parameter space, the KH
model accommodates 6 different phases, best visualized
using the phase-angle ’ as in Fig. 1(a). In addition to the
previously discussed [16,22,23] Ne´el-AF, stripy-AF, and
SL states near ’ ¼ 0, À 
4 , and À 
2 , respectively, we
observe 3 more states. First one is ‘‘AF’’ (K > 0) Kitaev
spin-liquid near ’ ¼ 
2 . Second, the FM phase broadly
extending over the third quadrant of the ’ circle. The
FM and stripy-AF states are connected [see Fig. 1(a)] by
the 4-sublattice transformation, which implies their identical
dynamics. Finally, near ’ ¼ 3
4 , the most wanted
phase, zigzag AF, appears occupying almost a quarter of
the phase space. Thanks to the above mapping, it is understood
that the zigzag and Ne´el states are isomorphic, too.
In particular, the ’ ¼ 3
4  zigzag state is identical to the
Heisenberg-AF state of the fictitious spins [24].
To obtain the phase boundaries, we have diagonalized
the model numerically, using a hexagonal 24-site cluster
with periodic boundary conditions. The cluster is compatible
with the above 4-sublattice transformation and ’ $ ~’
mapping. As seen in Fig. 1(b), the second derivative of the
GS energy EGS with respect to the ’ well detects the phase
transitions. Three pairs of linked transition points are
found: ’ ð88; 92Þ and (À 76, À108) for the spin
liquid-order transitions around Æ 
2 , and (162
, À34
) or
the transitions between ordered phases.
The transitions from zigzag-AF to FM, and from stripyAF
to Ne´el-AF are expected to be of first order by symmetry;
the corresponding peaks in Fig. 1(b) are indeed very
sharp. The spin liquid-order transitions near ’ ¼ À 
2 lead
to wider and much less pronounced peaks, suggesting a
second- (or weakly first-) order transition [16]. On the
contrary, liquid-order transitions around ’ ¼ 
2 show up
as very narrow peaks; on the finite cluster studied, they
correspond to real level crossings. The nature of these
phase transitions remains to be clarified [25].
While at J ¼ 0 (i.e., ’ ¼ Æ 
2 ) the sign of K is irrelevant
[21], the stability of the AF- and FM-type Kitaev spin
liquids against J perturbation is very different: the SL
phase near 
2 (À 
2 ) is less (more) robust. This phase
behavior is related to a different nature of the competing
ordered phases: for the 
2 SL, these are highly quantum
zigzag and Ne´el states, while the SL near À 
2 is sandwiched
by more classical (FM and ‘‘fluctuation free’’stripy
[16]) states which are energetically less favorable than the
quantum SL state.
Exchange interactions in Na2IrO3.—Having fixed the
parameter space (K > 0, J < 0) for the zigzag phase, we
turn now to the physical processes behind the model (1).
Exchange interactions in Mott insulators arise due to virtual
hoppings of electrons. This may happen in many
different ways, depending sensitively on chemical bonding,
intra-ionic electron structure, etc. The case of present
interest (i.e., strong spin-orbit coupling, t5
2g configuration,
and 90-bonding geometry) has been addressed in several
papers [8,11,16,26]. There are the following four physical
processes that contribute to K and J couplings.
Process 1: Direct hopping t0 between NN t2g orbitals.
Since no oxygen orbital is involved, 90
bonding is irrelevant;
the resulting Hamiltonian is H1 ¼ I1Si Á Sj with I1 ’
ð2
3 t0
Þ2
=U [16]. Here, U is the Coulomb repulsion between
t2g electrons. Typically, one has t0
=t < 1, when compared
to the indirect hopping t of t2g orbitals via oxygen ions.
Process 2: Interorbital NN t2g À eg hopping ~t. This is
the dominant pathway in 90 bonding geometry since
it involves strong tpd overlap between oxygen-2p
and eg orbitals; typically, ~t=t $ 2. The corresponding
Hamiltonian is [11]
-18
-16
-14
-12
-10
-8
-6
0 π/2 π 3π/2 2π
0
100
200
300
400
500
EGS/A
−d
2
EGS/Adϕ2
ϕ
lee´NypirtsdiuqilMFgazgizdiuqillee´N
Neel
stripy
ϕ
liquid
liquid
zigzag
FM
(b)
(a)
FIG. 1 (color online). (a) Phase diagram of the KitaevHeisenberg
model containing 2 spin-liquid and 4 spin-ordered
phases. The transition points (open dots on the ’ circle) are
obtained by an exact diagonalization. The gray lines inside the
circle connect the points related by the exact mapping (see text).
Open and solid circles in the insets indicate up and down spins.
The rectangular box in the zigzag pattern (top-left) shows the
magnetic unit cell. (b) Ground-state energy EGS and its second
derivative Àd2EGS=d’2 revealing the phase transitions.
PRL 110, 097204 (2013) P H Y S I C A L R E V I E W L E T T E R S
week ending
1 MARCH 2013
097204-2
94
Hð
Þ
2 ¼ I2ð2S
i S
j À Si Á SjÞ: (3)
This is nothing but the model (1) with K ¼ ÀJ ¼ I2 > 0,
i.e., at its SUð2Þ symmetric point ’ ¼ 3
4  inside the zigzag
phase; see Fig. 2. For the Mott-insulating iridates (as
opposed to charge-transfer cobaltates [11]), we estimate
I2 ’ 4
9 ð~t= ~UÞ2 ~JH, where ~U is the (optically active) excitation
energy associated with t2g À eg hopping, and ~JH is
Hund’s interaction between the t2g and eg orbitals. The
physics behind this expression is clear: ð~t= ~UÞ2 measures
the amount of t2g spin which is transferred to the NN eg
orbital; once arrived, it encounters the ‘‘host’’ t2g spin and
has to obey the Hund’s rule.
For its remarkable properties, the Hamiltonian H2 (3)
deserves a few more words. On a triangular lattice, it shows
a nontrivial spin vortex ground state [11,27]; however, the
elementary excitations are simple SUð2Þ magnons of a
conventional Heisenberg-AF state. When regarded as the
‘‘J’’ part of a doped t À J model, it leads to an exotic
pairing [11,28].
Process 3: Indirect hopping t between NN t2g orbitals
via oxygen ions. This gives rise to the Kitaev model Hð
Þ
3 ¼
ÀI3S
i S
j , with I3 ’ 8
3 ðt2=UÞðJH=UÞ [8], where JH is
Hund’s coupling between t2g electrons. This process supports
’ ¼ À 
2 SL state; see Fig. 2.
Process 4: Mechanisms involving pd charge-transfer
excitations with energy Ápd. Two holes may meet at an
oxygen and experience Coulomb Up and Hund’s Jp
H interactions,
or cycle around a Ir2O2 plaquette (Fig. 2). The
resulting Hamiltonian H4 has the form of H2 (3). The
coupling constant I4 ’ 8
9 t2
ð 2
2ÁpdþUpÀJp
H
À 1
Ápd
Þ is negative
[29], supporting the stripy-AF state not observed in
Na2IrO3.
Putting things together, we observe that it is the interorbital
t2g À eg hopping H2 process that uniquely supports
zigzag order in Na2IrO3. This implies in general that multiorbital
Hubbard-type models, when applied to iridates with
90-bonding geometry, must include eg states as well, even
though the moments reside predominantly in the t2g shell.
Up to this point, we neglected trigonal field splitting Á
of the t2g level due to the c axis compression present in
Na2IrO3. This approximation is valid as long as Á is much
smaller than the spin-orbit coupling  ’ 0:4 eV [13,15,30]
and seems to be justified, since the recent ab initio calculations
[20] suggest that Á ’ 75 meV only [31].
We have also examined the longer-range couplings,
using the hopping matrix of Ref. [20], and found that the
second-NN interaction has the form of (3) (as previously
noticed Refs. [32,33]), while the third-NN coupling is of
the AF-Heisenberg type [the corresponding coupling constants
are 4
9 ðt2
2;3=UÞ]. The second (third)-NN interaction
would oppose (support) zigzag order; however, we believe
that these couplings are not significant in Na2IrO3 because
the hoppings t2 and t3 are small [34].
We do not attempt here to evaluate the parameters
involved in H1 À H4; ab-initio calculations as in
Ref. [35] might be more useful in this regard. Instead,
having obtained a zigzag order in our model (1) and
identified the physical process driving this order, we turn
now to the experimental data. The J and K values in
Na2IrO3 and Li2IrO3 will be extracted below from analysis
of the neutron scattering and magnetic susceptibility data.
Spin waves in the zigzag phase.—Consider a single
domain zigzag state, e.g., with FM chains running perpendicular
to z-type bonds. Following Ref. [4], we introduce a
rectangular a  b magnetic unit cell [
ffiffiffi
3
p
a0 Â 3a0 in terms
of hexagon-edge a0; see Fig. 1(a)], and define the ab-plane
wave vector q in units of (h, k) as q ¼ ð2
a h; 2
b kÞ. Standard
spin-wave theory gives four dispersive branches:
!2
1;2ðh; kÞ ¼ ½K2
þ ðK þ JÞ2
Šc2
h À KJð1 À shskÞ
Æ jðK þ JÞchj
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð2K À JÞ2 À ð2Ksh À JskÞ2
q
;
(4)
and !3;4ðh; kÞ ¼ !1;2ðÀh; kÞ, with ch ¼ cosh, sh ¼
sinh, and sk ¼ sink. If K ¼ ÀJ, i.e., at the ’ ¼ 3
4 
point of hidden SUð2Þ symmetry, two branches are degenerate
(!1 ¼ !2) and become true Goldstone modes. Away
from this special point, the small magnon gap is expected
to open by quantum effects not considered here. For q
with h ¼ k, the dispersions (4) simplify to !1ðh; hÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2Kð2K þ JÞ
p
jchj and !2ðh; hÞ ¼
ffiffiffi
2
p
jJchj, revealing two
different energy scales in the magnon spectra set by K and
J couplings.
t~
xz yz xzyz
H2 H1
3H
H4
xy xyxy2 2
3z −r
ϕ
t’
3 4
1
4
2
3
yz xz
xz yz
21
t
FM Heisenberg AF Heisenberg
FMKitaevAFKitaev
FIG. 2 (color online). Schematics of four different exchange
processes (see text for details), arranged around the ’ phase
diagram of Fig. 1(a). Taken separately, the Hamiltonians H1, H2,
H3, and H4 would favor ‘‘pure’’ Ne´el-AF, zigzag-AF, Kitaev-SL,
and stripy-AF states, respectively, as indicated by arrows connecting
Hi with the dots on the ’ circle. The circle is divided
into the phase sectors by gray lines; SL phases are shaded.
PRL 110, 097204 (2013) P H Y S I C A L R E V I E W L E T T E R S
week ending
1 MARCH 2013
097204-3
95
While the bandwidth of the lowest dispersive mode (set
by J) is already known to be about 5–6 meV [4], we are not
aware of the high energy magnon data to estimate K in
Na2IrO3. We have therefore examined (see below) the
magnetic susceptibility data [1,6], and obtained ðJ; KÞ ’
ðÀ4:0; 10:5Þ meV that well fit the susceptibility as well as
the neutron scattering data [4]. With this, we predict the
magnon spectra for Na2IrO3 shown in Fig. 3. The lowest
dispersive (J) mode is as observed [4], indeed. However,
mapping out entire magnon spectra is highly desirable to
quantify the Kitaev term K directly.
Magnetic susceptibility.—We have calculated the uniform
magnetic susceptibility ðTÞ of the model (1) on
8- and 14-site clusters by exact diagonalization, and on
24-site cluster using the finite-temperature Lanczos
method [36,37]. The parameters are varied such that J ¼
A cos’ is consistent with the neutron data [4] while ’ stays
within the zigzag sector of Fig. 1(a); this strongly narrows
the possible K window. For the data fits, we let the g factor
of the Ir4þ
ion deviate from 2 (due to the covalency effects
[10]), and include the T-independent Van Vleck term 0.
The result for J ¼ À4:0 meV, K ¼ 10:5 meV, g ¼ 1:78,
0 ¼ 0:16 Â 10À3
cm3
=mol fits the Na2IrO3 data nicely
(Fig. 4); deviations occur at low temperatures only, when
correlation length exceeds the size of the cluster used. The
fit is quite robust: similar results can be found for small
only variations, locating Na2IrO3 near ’ ¼ 111 Æ 2 of
the model phase diagram Fig. 1(a). The spin couplings
obtained are reasonable for the 90-exchange bonds (as
expected [8,11], they are much smaller than in 180-bond
perovskites [13,14]). The magnitude of the Van Vleck
term also agrees with our estimate 0 ’ 8
3 2
BNA ’ 0:2 Â
10À3 cm3=mol for the Ir4þ ion, considering spin-orbit
coupling  ’ 0:4 eV [13,15,30].
Dominance of the Kitaev term (2K=J $ 5 in Na2IrO3)
implies strong frustration hence enhanced quantum
fluctuations; this explains the reduced ordered moment
m ’ 0:22B [5]. With the J, K, and g values above, we
calculated the leading order spin-wave correction to m and
obtained m ’ 0:33B [38].
For the sake of curiosity, we have also fitted the ðTÞ
data of Li2IrO3 [6], a sister compound of Na2IrO3.
Acceptable results have been found for the angle window
’ ¼ 124 Æ 6
; a representative plot for J ¼ À5:3 meV,
K ¼ 7:9 meV, g ¼ 1:94, 0 ¼ 0:14 Â 10À3 cm3=mol is
shown in Fig. 4. It is worth noticing that the value of J,
which controls the bandwidth of the softest spin-wave
mode (see Fig. 3), appears to be similar in both compounds.
This may explain why they undergo magnetic
transition at similar TN ’ 15 K, despite very different
high temperature susceptibilities.
To conclude, we have clarified the origin of zigzag
magnetic order in Na2IrO3 in terms of nearest-neighbor
Kitaev-Heisenberg model for localized Ir moments. The
model well agrees with the low-energy magnon and
high temperature magnetic susceptibility data. A general
implication of this work is that the interactions considered
here should hold a key for understanding the magnetism
of a broad class of spin-orbit Mott insulators with
90-exchange bonding geometry, including triangular,
honeycomb, and hyperkagome lattice iridates.
We thank R. Coldea, Y. Singh, H. Takagi, and I. I. Mazin
for discussions. J. C. acknowledges support by the
Alexander von Humboldt Foundation, ERDF under
Project No. CEITEC (CZ.1.05/1.1.00/02.0068), and the
EC 7th Framework Programme (286154/SYLICA). G. J.
is supported by Grant No. GNSF/ST09-447 and in part by
the NSF under Grant No. NSF PHY11-25915.
0
5
10
15
20
(-1,0) (0,0) (0,1) (1,1) (½,½) (0,0)
X Γ Y Γ’ M Γ
ωq[meV]
’Γ
M
Γ
Y
X
FIG. 3. Magnon spectra in the zigzag phase calculated using
Eq. (4) with ðJ; KÞ ¼ ðÀ4:0; 10:5Þ meV. The inset shows the
path along the symmetry directions in the reciprocal space; the
notation of Ref. [4] is used.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0 50 100 150 200 250 300 350 400
χ[10
-3
cm
3
/mol]
T [K]
theory
Na2IrO3
Li2IrO3
0
3
6
9
12
0 100 200 300 400
χ-1
[102
mol/cm3
]
T [K]
FIG. 4. Experimental magnetic susceptibilities ðTÞ for
Na2IrO3 [1,6] (squares) and Li2IrO3 [6] (circles) fitted by the
theoretical calculations. Exact  of the 8-site (14-site) cluster
is shown as solid (dashed) lines. Lanczos results for the
24-site cluster are indicated by shading [37]. Their comparison
suggests that the calculated  gives the thermodynamic
limit down to T % 100 K where the finite-size effects become
significant.
PRL 110, 097204 (2013) P H Y S I C A L R E V I E W L E T T E R S
week ending
1 MARCH 2013
097204-4
96
*Also at Andronikashvili Institute of Physics, 0177 Tbilisi,
Georgia.
[1] Y. Singh and P. Gegenwart, Phys. Rev. B 82, 064412
(2010).
[2] H. Takagi (unpublished).
[3] X. Liu, T. Berlijn, W.-G. Yin, W. Ku, A. Tsvelik, Y.-J.
Kim, H. Gretarsson, Y. Singh, P. Gegenwart, and J. P. Hill,
Phys. Rev. B 83, 220403 (2011).
[4] S. K. Choi, R. Coldea, A. N. Kolmogorov, T. Lancaster,
I. I. Mazin, S. J. Blundell, P. G. Radaelli, Y. Singh, P.
Gegenwart, K. R. Choi, S.-W. Cheong, P. J. Baker, C.
Stock, and J. Taylor, Phys. Rev. Lett. 108, 127204 (2012).
[5] F. Ye, S. Chi, H. Cao, B. C. Chakoumakos, J. A.
Fernandez-Baca, R. Custelcean, T. F. Qi, O. B. Korneta,
and G. Cao, Phys. Rev. B 85, 180403 (2012).
[6] Y. Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale, W.
Ku, S. Trebst, and P. Gegenwart, Phys. Rev. Lett. 108,
127203 (2012).
[7] R. Comin, G. Levy, B. Ludbrook, Z.-H. Zhu, C. N.
Veenstra, J. A. Rosen, Y. Singh, P. Gegenwart, D.
Stricker, J. N. Hancock, D. van der Marel, I. S. Elfimov,
and A. Damascelli, Phys. Rev. Lett. 109, 266406 (2012).
[8] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205
(2009).
[9] A. Shitade, H. Katsura, J. Kunesˇ, X.-L. Qi, S.-C. Zhang,
and N. Nagaosa, Phys. Rev. Lett. 102, 256403 (2009).
[10] A. Abragam and B. Bleaney, Electron Paramagnetic
Resonance of Transition Ions (Clarendon Press, Oxford,
1970).
[11] G. Khaliullin, Prog. Theor. Phys. Suppl. 160, 155 (2005).
[12] J. B. Goodenough, Magnetism and the Chemical Bond
(Interscience, New York, 1963).
[13] J. Kim, D. Casa, M. H. Upton, T. Gog, Y.-J. Kim, J. F.
Mitchell, M. van Veenendaal, M. Daghofer, J. van den
Brink, G. Khaliullin, and B. J. Kim, Phys. Rev. Lett. 108,
177003 (2012).
[14] J. Kim, A. H. Said, D. Casa, M. H. Upton, T. Gog, M.
Daghofer, G. Jackeli, J. van den Brink, G. Khaliullin, and
B. J. Kim, Phys. Rev. Lett. 109, 157402 (2012).
[15] H. Gretarsson, J. P. Clancy, X. Liu, J. P. Hill, E. Bozin, Y.
Singh, S. Manni, P. Gegenwart, J. Kim, A. H. Said, D.
Casa, T. Gog, M. H. Upton, H.-S. Kim, J. Yu, V. M.
Katukuri, L. Hozoi, J. van den Brink, Y.-J. Kim,
arXiv:1209.5424.
[16] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev.
Lett. 105, 027204 (2010).
[17] I. Kimchi and Y.-Z. You, Phys. Rev. B 84, 180407
(2011).
[18] S. Bhattacharjee, S.-S. Lee, and Y. B. Kim, New J. Phys.
14, 073015 (2012).
[19] C. H. Kim, H. S. Kim, H. Jeong, H. Jin, and J. Yu, Phys.
Rev. Lett. 108, 106401 (2012).
[20] I. I. Mazin, H. O. Jeschke, K. Foyevtsova, R. Valenti, and
D. I. Khomskii, Phys. Rev. Lett. 109, 197201 (2012).
[21] A. Kitaev, Ann. Phys. (Amsterdam) 321, 2 (2006).
[22] H.-C. Jiang, Z.-C. Gu, X.-L. Qi, and S. Trebst, Phys. Rev.
B 83, 245104 (2011).
[23] J. Reuther, R. Thomale, and S. Trebst, Phys. Rev. B 84,
100406 (2011).
[24] After initial submission of this work, we became aware of
the recent derivation [Y. Yu, L. Liang, Q. Niu, and S. Qin,
Phys. Rev. B 87, 041107(R) (2013)] of the KH model and
zigzag phase from a single band Hubbard model with the
spin- and link-dependent NN-hoppings. However, such
hoppings are not present in the hexagonal iridates
A2IrO3 [8,9].
[25] C. C. Price and N. B. Perkins, [Phys. Rev. Lett. 109,
187201 (2012)] demonstrated the highly nontrivial phase
behavior of the KH model also at finite temperatures.
[26] G. Chen and L. Balents, Phys. Rev. B 78, 094403 (2008).
[27] I. Rousochatzakis, U. K. Ro¨ssler, J. van den Brink, and M.
Daghofer, arXiv:1209.5895.
[28] G. Khaliullin, W. Koshibae, and S. Maekawa, Phys. Rev.
Lett. 93, 176401 (2004).
[29] The results for I4 of Refs. [8,11,26] were incomplete.
[30] O. F. Schirmer, A. Forster, H. Hesse, M. Wohlecke, and S.
Kapphan, J. Phys. C 17, 1321 (1984).
[31] Typically, ‘‘noncubic’’ corrections to the interactions between
Kramers doublets scale as ð3 cos2À1
2 Þ2 [28], which is
about 0.01 if Á ¼ 75 meV. [ is given by tan2 ¼
2
ffiffiffi
2
p
=ð þ 2ÁÞ]. The case of Á >  can be excluded
also on the grounds that, in this limit, the interactions
become bond-independent and support either Ising-FM or
xy-AF states [11], instead of zigzag order observed.
[32] J. Reuther, R. Thomale, and S. Rachel, Phys. Rev. B 86,
155127 (2012).
[33] M. Kargarian, A. Langari, and G. A. Fiete, Phys. Rev. B
86, 205124 (2012).
[34] The second-NN hopping t2 ’ 75 meV is about 4 times
less than NN hopping t via oxygen, and t3 ’ 30 meV [20].
[35] V. M. Katukuri, H. Stoll, J. van den Brink, and L. Hozoi,
Phys. Rev. B 85, 220402(R) (2012).
[36] J. Jaklicˇ and P. Prelovsˇek, Adv. Phys. 49, 1 (2000).
[37] We used M ¼ 100 Lanczos steps and Nst ¼ 1024 random
sampling vectors. The values of  and their statistical error
are presented in Fig. 4 in the form of 3 intervals
estimated by taking many sets of the sampling vectors.
[38] Hybridization of the Ir-5d and O-2p orbitals on antiferromagnetic
bonds [5], as well as the higher order quantum
corrections may further reduce m.
PRL 110, 097204 (2013) P H Y S I C A L R E V I E W L E T T E R S
week ending
1 MARCH 2013
097204-5
97
98
LETTERS
PUBLISHED ONLINE: 11 MAY 2015 | DOI: 10.1038/NPHYS3322
Direct evidence for dominant bond-directional
interactions in a honeycomb lattice iridate Na2IrO3
Sae Hwan Chun1
, Jong-Woo Kim2
, Jungho Kim2
, H. Zheng1
, Constantinos C. Stoumpos1
,
C. D. Malliakas1
, J. F. Mitchell1
, Kavita Mehlawat3
, Yogesh Singh3
, Y. Choi2
, T. Gog2
, A. Al-Zein4
,
M. Moretti Sala4
, M. Krisch4
, J. Chaloupka5
, G. Jackeli6,7
, G. Khaliullin6
and B. J. Kim6
*
Heisenberg interactions are ubiquitous in magnetic materials
and play a central role in modelling and designing quantum
magnets. Bond-directional interactions1–3
offer a novel
alternative to Heisenberg exchange and provide the building
blocks of the Kitaev model4
, which has a quantum spin
liquid as its exact ground state. Honeycomb iridates, A2IrO3
(A = Na, Li), offer potential realizations of the Kitaev magnetic
exchange coupling, and their reported magnetic behaviour
may be interpreted within the Kitaev framework. However,
the extent of their relevance to the Kitaev model remains
unclear, as evidence for bond-directional interactions has so
far been indirect. Here we present direct evidence for dominant
bond-directional interactions in antiferromagnetic Na2IrO3 and
show that they lead to strong magnetic frustration. Diffuse
magnetic X-ray scattering reveals broken spin-rotational
symmetry even above the Néel temperature, with the three
spin components exhibiting short-range correlations along
distinct crystallographic directions. This spin- and real-space
entanglement directly uncovers the bond-directional nature of
these interactions, thus providing a direct connection between
honeycomb iridates and Kitaev physics.
Iridium (IV) ions with pseudospin-1/2 moments form in
Na2IrO3, a quasi-two-dimensional (2D) honeycomb network, which
is sandwiched between two layers of oxygen ions that frame
edge-shared octahedra around the magnetic ions and mediate
superexchange interactions between neighbouring pseudospins
(Fig. 1a). Owing to the particular spin–orbital structure of the
pseudospin5,6
, the isotropic part of the magnetic interaction is
strongly suppressed in the 90◦
bonding geometry of the edgeshared
octahedra2,3
, thereby allowing otherwise subdominant
bond-dependent anisotropic interactions to play the main role
and manifest themselves at the forefront of magnetism. This
bonding geometry, common to many transition-metal oxides, in
combination with the pseudospin that arises from strong spin–orbit
coupling gives rise to an entirely new class of magnetism beyond
the traditional paradigm of Heisenberg magnets. On a honeycomb
lattice, for instance, the leading anisotropic interactions take the
form of the Kitaev model3
, which is a rare example of exactly
solvable models with non-trivial properties such as Majorana
fermions and non-Abelian statistics, and with potential links to
quantum computing4
.
Realization of the Kitaev model is now being intensively
sought out in a growing number of materials7–13
, including 3D
extensions of the honeycomb Li2IrO3, dubbed ‘hyper-honeycomb’7
and ‘harmonic-honeycomb’8
, and 4d transition-metal analogues
such as RuCl3 (ref. 12) and Li2RhO3 (ref. 13). Although most of
these are known to magnetically order at low temperature, they
exhibit a rich array of magnetic structures, including zigzag14–16
,
spiral17
and other more complex non-coplanar structures18,19
that
are predicted to occur in the vicinity of the Kitaev quantum spin
liquid (QSL) phase20–23
, which hosts many degenerate ground states
frustrated by three bond-directional Ising-type anisotropies. All of
these magnetic orders are captured in an extended version of the
Kitaev model written as
H =
ij γ
KSγ
i Sγ
j +JSi ·Sj +Γ (Sα
i Sβ
j +Sβ
i Sα
j )
which includes, in addition to the Kitaev term K, the Heisenberg
exchange J, which may be incompletely suppressed in the
superexchange process and/or arise from a direct exchange
process21
, and the symmetric off-diagonal exchange term Γ , which
is symmetry-allowed even in the absence of lattice distortions23–25
.
This ‘minimal’ Hamiltonian couples pseudospins S (hereafter
referred to as ‘spin’) only on nearest-neighbour bonds ij , neglecting
further-neighbour couplings, which may be non-negligible. The
bond-directional nature of the K and Γ terms is reflected in the spin
components [α = β = γ ∈ (x, y, z)] which they couple for a given
bond (γ ∈x-, y-, z-bonds; Fig. 1a). For example, the K term couples
only the spin component normal to the Ir2O6 plaquette containing
the particular bond. Despite these extra terms that may account for
finite-temperature magnetic orders in the candidate materials, the
fact that the Kitaev QSL phase has a finite window of stability against
these perturbations20,25
calls for investigation of competing phases
and a vigorous search for the Kitaev QSL phase.
Although the notion of magnetic frustration induced by
competing bond-directional interactions is compelling, it remains
a theoretical construct without an existence proof for such
interactions in a real-world material. Moreover, theories for
iridium compounds based on itinerant electrons suggest alternative
pictures26–28
. In principle, measurement of the dynamical structure
factor through inelastic neutron scattering (INS) or resonant
1Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA. 2Advanced Photon Source, Argonne National Laboratory,
Argonne, Illinois 60439, USA. 3Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, Sector 81, Mohali 140306, India.
4European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble Cedex, France. 5Central European Institute of Technology, Masaryk University,
Kotlářská 2, 61137 Brno, Czech Republic. 6Max Planck Institute for Solid State Research, Heisenbergstraße 1, D-70569 Stuttgart, Germany. 7Institute for
Functional Matter and Quantum Technologies, University of Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany. *e-mail: bjkim@fkf.mpg.de
462 NATURE PHYSICS | VOL 11 | JUNE 2015 | www.nature.com/naturephysics
© 2015 Macmillan Publishers Limited. All rights reserved
99
NATURE PHYSICS DOI: 10.1038/NPHYS3322 LETTERS
Na2lrO3
a
b
c
z
x
y
Ir
O
z-bond
x-bond
y-bond
c
0 −50 −100 −150
Q = (0, 1, 3.5)
T = 6.4 K
40°
44.3 ± 1.24°
−200
0
5
10
Integratedintensity(a.u.)
104
103
102
101
100
Intensity(a.u.)
H (r.l.u.) K (r.l.u.) L (r.l.u.)
−0.3 0.0 1.0 6.50.3 0.7 1.3 6.2 6.8
5 K
11 K
14 K
13 K
a
c
b
10
100
1,000
T (K)
TN
6 8 10 12 14 16
Correlationlength(Å)
a axis
S
Q
kf
ki
y
x
z
f
g
e
a
b
d
Ψ
eσ
eπ
Θ
= 50°Θ
Ψ (°)
e ′σ
e ′π
Figure 1 | Magnetic easy axis and temperature dependence of the zigzag order. a, Honeycomb layers of Ir4+ in the monoclinic Bravais lattice. Green,
yellow and blue planes show Ir2O6 plaquettes normal to the local x-, y- and z-axes (black arrows), respectively, which point along three nearest-neighbour
Ir–O bonds in an octahedron. Ir–Ir bonds are labelled following the plaquettes they belong to. Na+ is not shown for clarity. Blue arrows show the spins in the
static zigzag order propagating along the b direction. Spins are antiparallel between the layers (not shown). b, Illustration of the scattering geometry. Shown
in blue is the scattering plane defined by the incident (ki) and outgoing (kf) wavevector (red arrows). Green arrows show the X-ray polarizations. The
azimuth, Ψ , is defined as the angle between the a-axis and the scattering plane. c, Ψ -dependence of the magnetic Bragg peak (blue filled circle) intensity at
(0, 1, 3.5) measured in the σ–π channel. The black hexagon is the Brillouin zone of the honeycomb net. The red solid line shows the best fitting result to the
data with Θ =44.3◦, with a standard error of ±1.24◦. We note that the actual error may be larger owing to systematic errors arising from factors such as
changes in the beam footprint on the sample. Green and blue lines shows the calculated Ψ dependence for Θ =40◦ and 50◦, respectively. d–f, H, K and L
scans, respectively, of the magnetic Bragg peak at (0, 1, 6.5) for selected temperatures. g, Temperature dependence of the correlation lengths along the a-,
b- and c-axes from Gaussian fitting to the scans. Error bars represent the standard deviation in the fitting procedure. The solid lines are guides to the eye.
inelastic X-ray scattering (RIXS) provides the most direct access to
the Hamiltonian describing the magnetic interactions. However, a
fully momentum- and energy-resolved dynamical structure factor
thus far remains elusive for any of the candidate materials;
RIXS suffers from insufficient energy resolution29
and INS is at
present limited by unavailability of large-volume single crystals15
.
In this Letter, we take a new approach using diffuse magnetic
X-ray scattering to provide direct evidence for predominant bonddirectional
interactions in Na2IrO3 through the measurement of
equal-time correlations of spin components above the ordering
temperature (TN =12–15 K, see Supplementary Fig. 1).
We start by establishing the spin orientation in the static
zigzag order14–16
below TN, as shown in Fig. 1a, using standard
resonant magnetic X-ray diffraction. In this measurement, the
X-ray polarization projects out a certain spin component; the
intensity depends on the spin orientation through the relation I ∝
|kf ·S|2
for the σ–π channel measured, where kf is the scattered Xray
wavevector (Fig. 1b). Figure 1c shows the intensity variation as
the sample is rotated about the ordering wavevector Q = (0, 1, 3.5)
by an azimuthal angle Ψ , which causes S to precess around Q.
Earlier studies14,16
have established that S is constrained to lie in
the ac-plane, so this measurement of I(Ψ ) determines the spin
orientation by resolving the tilting angle Θ of S with respect
to the a-axis. The best fitting result with Θ = 44.3◦
indicates
that the magnetic easy axis is approximately half way between
the cubic x- and y-axes (Fig. 1a). This static spin orientation is
a compromise among all anisotropic interactions present in the
system, and is strongly tied to the magnetic structure because
of their bond-directional nature. To see this point, consider, for
example, the K term: in the zigzag structure propagating along
the b direction, where the spins are antiferromagnetically aligned
on the z-bond and ferromagnetically aligned on the x-bond
and y-bond, a ferromagnetic (antiferromagnetic) K favours spins
pointing perpendicular to (along) the z-axis for a pair of spins on
the z-bond, and along (perpendicular to) the x-axis and y-axis for
the pairs on the x-bond and y-bond, respectively.
The zigzag order is one of the many magnetic states (including
the aforementioned spiral and non-coplanar structures) that are
NATURE PHYSICS | VOL 11 | JUNE 2015 | www.nature.com/naturephysics 463
© 2015 Macmillan Publishers Limited. All rights reserved
100
LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS3322
0 0.5−0.5
H (r.l.u.)
0 0.5−0.5
0 0.5−0.5
H (r.l.u.)
0 0.5−0.5
H (r.l.u.)
0 0.5−0.50 0.5−0.5 0 0.5−0.5
0 0.5−0.5
H (r.l.u.)
0 0.5−0.5
H (r.l.u.)
1
−1
0
K(r.l.u.)
30° 60° 90° 120°
1
−1
0
K(r.l.u.)
Sxx Sxx + Syy + Szz
Syy Szz
1
−1
0
K(r.l.u.)
0.0
0.5
1.0
Intensity(a.u.)
0 120 240 360
cb
d e
a
±(0.5, 0.5)±(0.5, −0.5)
±(0, 1)
y
x
z
Ψ = 0°
Ψ (°)
H (r.l.u.) H (r.l.u.)H (r.l.u.) H (r.l.u.)
Intensity(a.u.)
Figure 2 | Diffuse magnetic X-ray scattering intensities above TN. a, Intensity plots in the HK-plane (L varying between 6.5 and 7) measured at T =17 K for
selected azimuth angles summing π–σ and π–π channels, sensitive to spin components parallel to ki and perpendicular to the scattering plane,
respectively. For example, Ψ =0◦ measures the sum of correlations Sxx and Syy. The dashed hexagon indicates the first Brillouin zone of the honeycomb
net. b, Spin-component-resolved equal-time correlations extracted from a. c, Spin-component-integrated equal-time correlations extracted from a. Peaks
are located at Q=±(0,1), ±(0.5,0.5), and ±(0.5,−0.5). d, Ψ -dependence of the diffuse peak intensities for Samples 1 (open symbol) and 2 (closed
symbol). Solid lines show the calculated Ψ -dependence for x-, y- and z-zigzag states shown in e for the π–σ and π–π polarization channels summed.
e, Zigzag orders propagating along three equivalent directions. Blue zigzag is the static structure, and green and yellow zigzags are generated by 120◦
rotation of the blue zigzag.
classically degenerate in the pure Kitaev limit30
and comprise the
micro-states in the QSL phase. Away from the pure Kitaev limit,
depending on their energy separations, signatures of other magnetic
states and their associated magnetic anisotropies may become
observable in the paramagnetic phase through diffuse magnetic
scattering. In particular, zigzag orders propagating in two other
directions, ±120◦
rotated from the static one, are expected for a
honeycomb net with C3 symmetry. (The actual 3D crystal structure
has an only approximate C3 symmetry because of a monoclinic
distortion, which singles out one propagation direction for the longrange
ordered state (along b direction) out of the three possible
under the ideal C3 symmetry15
.)
With other magnetic correlations possibly emerging at high
temperature in mind, we follow the temperature evolution of the
zigzag order. Figure 1d–f shows H, K and L scans, respectively, of
the magnetic Bragg peak at Q = (0, 1, 6.5) for selected temperatures.
Figure 1g shows the correlation lengths along the a-, b- and
c-axes as a function of temperature. As the temperature increases
above TN, the zigzag correlations diminish rather isotropically,
despite dominant 2D couplings in the honeycomb net. This
3D characteristic of the magnetic correlations contrasts with
that of the quasi-2D Heisenberg antiferromagnet Sr2IrO4, which
exhibits 2D long-range correlations well above TN (ref. 31), and
implies that the critical temperature in Na2IrO3 is limited by the
anisotropic interactions rather than the interlayer coupling; the
Mermin–Wagner theorem requires either the symmetry to be lower
than SU(2) or the dimension to be higher than 2D for a finitetemperature
phase transition. The zigzag correlations survive on a
length scale of several nanometres (approximately three unit cells
wide) above TN, but the peak intensities drop by two orders of
magnitude. To isolate such small signals from the background, we
used an experimental set-up that maximizes the signal-to-noise
ratio, as described in the Methods.
Figure 2a maps the diffuse scattering intensity over a region
in momentum space encompassing a full Brillouin zone of the
honeycomb net, at several different Ψ angles to resolve the spin
components (see Supplementary Fig. 2). These maps integrate
the dynamic structure factor over the range 0 ≤ ω ≤ 100 meV,
464 NATURE PHYSICS | VOL 11 | JUNE 2015 | www.nature.com/naturephysics
© 2015 Macmillan Publishers Limited. All rights reserved
101
NATURE PHYSICS DOI: 10.1038/NPHYS3322 LETTERS
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
Spectralweightratio
ξHeisenberg J1 + J2 + J3 FM Kitaev
Zig-zag
Liquid
0 1
Intensity (a.u.)
Figure 3 | Simulation of diffuse scattering using exact diagonalization.
The Kitaev–Heisenberg model including up to third nearest-neighbour
Heisenberg interactions was considered. ξ interpolates between the pure
Heisenberg model and the pure Kitaev model via J1 =J2 =J3 =1−ξ and
K=−ξ. A ferromagnetic K with finite J1, J2 and J3 stabilizes the zigzag state
for most values of ξ. The black curve shows the spectral asymmetry,
defined as the ratio of spectral weight at Q=(0.5,0.5) to that at Q=(0,1).
Images show equal-time correlations Sx
QSx
−Q obtained by exact
diagonalization using a 24-site cluster and plotted in the extended Brillouin
zone for selected ξ. The correlations for y and z components (not shown)
can be generated by ±120◦ rotations of the images shown.
covering the entire range of magnetic excitations, and serve
as an excellent approximation for the equal-time correlation
Sαα ≡ Sα
QSα
−Q (α =x,y,z). When averaged over the three spin
components, the intensity map (Fig. 2c) indeed shows three zigzag
correlations above TN, with peaks at Q = ±(0, 1), ±(0.5, 0.5) and
±(0.5, −0.5) of equal intensities, confirming the near-ideal C3
symmetry. However, the spin-component-resolved maps, shown
in Fig. 2b, manifestly break the C3 symmetry. The system is left
invariant only when C3 rotation is performed simultaneously in the
real space and in the spin space—that is, cyclic permutation of spin
indices. This ‘global’ C3 symmetry implies a strong entanglement
between the real space and the spin space. Specifically, the full
azimuthal dependence of each zigzag state, shown in Fig. 2d, closely
follows the curves simulated for spin orientation fixed relative to
the propagation direction, as depicted in Fig. 2e. In other words,
specifying a spin component amounts to fixing the momentum
direction and vice versa. This one-to-one correspondence between
the spin space and the real space is a direct consequence of the bonddependent
nature of the anisotropic exchange terms.
Qualitatively, it is immediately seen that the anisotropic
interactions dominate over the isotropic interactions and the
system is very far away from the pure Heisenberg limit, in
which case the spatial correlations must be spin-componentindependent
with three zigzag peaks having equal intensities by
symmetry (as in the spin-averaged correlation shown in Fig. 2c
preserving C3 symmetry). A measure of how close the system is
to either the Heisenberg or the Kitaev limits is provided by the
intensity ratio of the weakest peak to the two bright peaks in the
spin-component-resolved correlations (Fig. 2b). To quantify this
measure, represented by a variable linearly interpolating between
these two limits, ξ, requires specifying the Hamiltonian, which
is not precisely known. For an estimation at a semi-quantitative
level, we adopt a simple Hamiltonian that neglects all anisotropic
terms beyond the K term. (This in turn requires including furtherneighbour
Heisenberg couplings J2 and J3 to stabilize the zigzag
order32
, which we take to be equal to J1 for simplicity.) Figure 3 shows
the simulated patterns for selected ξ. It is clear that the observed
diffuse pattern is consistent with the simulated pattern for the large
ξ limit. In fact, the observed intensity ratio of ≈0.2 is even smaller
Intensity(a.u.)
T = 9 K T = 17 K
Energy loss (meV)
−100 0 100 200−200
0
100
200
300
400
Ψ = 180° Ψ = 180°
Figure 4 | Resonant inelastic X-ray scattering spectra below TN.
RIXS spectra recorded at T =9 K and Ψ =180◦. Q=(0,1), (0.5,0.5) and
(0.5,−0.5), shown as blue, green and yellow filled symbols, respectively,
marked on the Brillouin zone of the honeycomb net and colour-coded with
the spectra. At this Ψ angle, S lies approximately along kf and π–σ and
π–π channels measure the two spin components transverse to S. For
comparison, the inset shows the diffuse map at T =17 K for the same Ψ
angle, generated by rotating the Ψ =60◦ data shown in Fig. 2a clockwise
by 120◦.
than calculated (Fig. 3) for the largest ξ in the zigzag phase, which
confirms the predominant anisotropic interactions.
Interpreted within this model, our calculations would imply that
the system is very close to the Kitaev limit. However, it is becoming
increasingly evident that other anisotropic terms beyond the Kitaev
interaction do play a role22,23
. This is, in fact, evident from the static
spin not pointing along one of the cubic axes favoured by the K term;
all other anisotropic terms conspire to rotate the spin away from
the principal axes. This in turn suggests that the zigzag structure is
further stabilized by other anisotropic terms. The zigzag correlations
survive at least up to ∼70 K (see Supplementary Fig. 3), which is in
accord with the observation that coherent spin waves15
disperse up
to ≈5 meV. This energy scale coincides with the temperature scale
(≈100 K) below which the magnetic susceptibility deviates from the
Curie–Weiss behaviour33
. This energy scale is, however, still far too
small in comparison with the energy (≈100 meV) spanned by the
magnetic excitations (Fig. 4), suggesting that the zigzag order is an
emergent phenomenon. Despite the macroscopic degeneracy in the
Kitaev QSL phase being reduced down to three zigzags, the highenergy
Kitaev interactions leave their signature in the low-energy
sector: the three spin components, each carrying its own zigzag,
compete and melt the long-range order at a temperature much lower
than that suggested by the Weiss temperature (ΘW), leading to a
large frustration parameter33
(≡ΘW/TN) approximately equal to 8.
The fluctuations among three zigzag states remain even below
TN, albeit with subtle spectral changes (Supplementary Fig. 3d),
implying that they are primarily quantum rather than thermal
fluctuations. At Ψ = 180◦
(Fig. 4), the intensity remains highest
at Q = ±(0.5, 0.5) and Q = ±(0.5, −0.5), away from the Bragg
peaks at Q=±(0,1), and peaked at zero energy within the energy
resolution of 24 meV. Note that this scattering geometry probes two
spin components transverse to the static component. A profound
consequence of the unusual nature of the fluctuations is that the soft
excitations are located away from the Bragg peak15
. This is a notable
exception to the universality held in conventional magnets that
spin waves emanate from Bragg peaks by virtue of the Goldstone
theorem, and magnetic anisotropy is manifested as a spin-wave
gap, even in systems with extremely large magnetic anisotropy34
.
By contrast, the spin gap in our system is small (unresolved in our
NATURE PHYSICS | VOL 11 | JUNE 2015 | www.nature.com/naturephysics 465
© 2015 Macmillan Publishers Limited. All rights reserved
102
LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS3322
spectra and estimated to be smaller than 2 meV from INS data15
)
in comparison with the overall energy scale of the system, despite
the fact that the magnetism is dominated by the anisotropic terms.
Rather, the anisotropy is manifested as the separation of the longwavelength
spin waves from the Bragg peaks, which is a natural
consequence of each spin component exhibiting its own real-space
correlations. Our results directly reveal the key building blocks of
the Kitaev model in Na2IrO3, and establish a new design strategy
for the long-sought quantum spin liquids via the bond-directional
magnetic coupling.
Methods
Methods and any associated references are available in the online
version of the paper.
Received 11 January 2015; accepted 1 April 2015;
published online 11 May 2015
References
1. van Vleck, J. H. On the anisotropy of cubic ferromagnetic crystals. Phys. Rev.
52, 1178–1198 (1937).
2. Khaliullin, G. Orbital order and fluctuations in Mott insulators. Prog. Theor.
Phys. Suppl. 160, 155–202 (2005).
3. Jackeli, G. & Khaliullin, G. Mott insulators in the strong spin–orbit coupling
limit: From Heisenberg to a quantum compass and Kitaev models. Phys. Rev.
Lett. 102, 017205 (2009).
4. Kitaev, A. Anyons in an exactly solved model and beyond. Ann. Phys. 321,
2–111 (2006).
5. Kim, B. J. et al. Novel Jeff =1/2 Mott state induced by relativistic spin–orbit
coupling in Sr2IrO4. Phys. Rev. Lett. 101, 076402 (2008).
6. Kim, B. J. et al. Phase-sensitive observation of a spin–orbital Mott state in
Sr2IrO4. Science 323, 1329–1332 (2009).
7. Takayama, T. et al. Hyper-honeycomb iridate β-Li2IrO3 as a platform for Kitaev
magnetism. Phys. Rev. Lett. 114, 077202 (2015).
8. Modic, K. A. et al. Realization of a three-dimensional spin-anisotropic
harmonic honeycomb iridate. Nature Commun. 5, 4203 (2014).
9. Kimchi, I. & Vishwanath, A. Kitaev–Heisenberg models for iridates on the
triangular, hyperkagome, kagome, fcc, and pyrochlore lattices. Phys. Rev. B 89,
014414 (2014).
10. Hermanns, M. & Trebst, S. Quantum spin liquid with a Majorana Fermi surface
on the three-dimensional hyperoctagon lattice. Phys. Rev. B 89, 235102 (2014).
11. Lee, S. B., Jeong, J-S., Hwang, K. & Kim, Y. B. Emergent quantum phases in a
frustrated J1–J2 Heisenberg model on the hyperhoneycomb lattice. Phys. Rev. B
90, 134425 (2014).
12. Plumb, K. W. et al. α-RuCl3: A spin–orbit assisted Mott insulator on a
honeycomb lattice. Phys. Rev. B 90, 041112 (2014).
13. Luo, Y. et al. Li2RhO3: A spin-glassy relativistic Mott insulator. Phys. Rev. B 87,
161121 (2013).
14. Liu, X. et al. Long-range magnetic ordering in Na2IrO3. Phys. Rev. B 83,
220403 (2011).
15. Choi, S. K. et al. Spin waves and revised crystal structure of honeycomb iridate
Na2IrO3. Phys. Rev. Lett. 108, 127204 (2012).
16. Ye, F. et al. Direct evidence of a zigzag spin-chain structure in the honeycomb
lattice: A neutron and X-ray diffraction investigation of single-crystal Na2IrO3.
Phys. Rev. B 85, 180403 (2012).
17. Reuther, J., Thomale, R. & Rachel, S. Spiral order in the honeycomb iridate
Li2IrO3. Phys. Rev. B 90, 100405 (2014).
18. Biffin, A. et al. Noncoplanar and counterrotating incommensurate magnetic
order stabilized by Kitaev interactions in γ-Li2IrO3. Phys. Rev. Lett. 113,
197201 (2014).
19. Biffin, A. et al. Unconventional magnetic order on the hyperhoneycomb Kitaev
lattice in β-Li2IrO3: Full solution via magnetic resonant X-ray diffraction. Phys.
Rev. B 90, 205116 (2014).
20. Chaloupka, J., Jackeli, G. & Khaliullin, G. Kitaev–Heisenberg model on a
honeycomb lattice: Possible exotic phases in iridium oxides A2IrO3. Phys.
Rev. Lett. 105, 027204 (2010).
21. Chaloupka, J., Jackeli, G. & Khaliullin, G. Zigzag magnetic order in the iridium
oxide Na2IrO3. Phys. Rev. Lett. 110, 097204 (2013).
22. Lee, E. K-H. & Kim, Y. B. Theory of magnetic phase diagrams in
hyperhoneycomb and harmonic-honeycomb iridates. Phys. Rev. B 91,
064407 (2015).
23. Rau, J. G., Lee, E. K-H. & Kee, H-Y. Generic spin model for the honeycomb
iridates beyond the Kitaev limit. Phys. Rev. Lett. 112, 077204 (2014).
24. Katukuri, V. M. et al. Kitaev interactions between j=1/2 moments in
honeycomb Na2IrO3 are large and ferromagnetic: Insights from ab initio
quantum chemistry calculations. New J. Phys. 16, 013056 (2014).
25. Yamaji, Y., Nomura, Y., Kurita, M., Arita, R. & Imada, M. First-principles study
of the honeycomb-lattice iridates Na2IrO3 in the presence of strong spin–orbit
interaction and electron correlations. Phys. Rev. Lett. 113, 107201 (2014).
26. Shitade, A. et al. Quantum spin Hall effect in a transition metal oxide Na2IrO3.
Phys. Rev. Lett. 102, 256403 (2009).
27. Mazin, I. I., Jeschke, H. O., Foyevtsova, K., Valenti, R. & Khomskii, D. I.
Na2IrO3 as a molecular orbital crystal. Phys. Rev. Lett. 109, 197201 (2012).
28. Kim, C. H., Kim, H. S., Jeong, H., Jin, H. & Yu, J. Topological quantum phase
transition in 5d transition metal oxide Na2IrO3. Phys. Rev. Lett. 108,
106401 (2012).
29. Gretarsson, H. et al. Magnetic excitation spectrum of Na2IrO3 probed with
resonant inelastic X-ray scattering. Phys. Rev. B 87, 220407 (2013).
30. Price, C. & Perkins, N. B. Finite-temperature phase diagram of the classical
Kitaev–Heisenberg model. Phys. Rev. B 88, 024410 (2013).
31. Fujiyama, S. et al. Two-dimensional Heisenberg behavior of Jeff =1/2 isospins
in the paramagnetic state of the spin–orbital Mott insulator Sr2IrO4. Phys. Rev.
Lett. 108, 247212 (2012).
32. Kimchi, I. & You, Y-Z. Kitaev–Heisenberg-J2-J3 model for the iridates A2IrO3.
Phys. Rev. B 84, 180407 (2011).
33. Singh, Y. & Gegenwart, P. Antiferromagnetic Mott insulating state in single
crystals of the honeycomb lattice material Na2IrO3. Phys. Rev. B 82,
064412 (2010).
34. Kim, J. et al. Large spin-wave energy gap in the bilayer iridate Sr3Ir2O7:
Evidence for enhanced dipolar interactions near the Mott metal–insulator
transition. Phys. Rev. Lett. 109, 157402 (2012).
Acknowledgements
Work in the Materials Science Division of Argonne National Laboratory (sample
preparation, characterization, and contributions to data analysis) was supported by the
US Department of Energy, Office of Science, Basic Energy Sciences, Materials Science
and Engineering Division. Use of the Advanced Photon Source, an Office of Science User
Facility operated for the US Department of Energy (DOE) Office of Science by Argonne
National Laboratory, was supported by the US DOE under Contract No.
DE-AC02-06CH11357. K.M. acknowledges support from UGC-CSIR, India. Y.S.
acknowledges DST, India for support through Ramanujan Grant #SR/S2/RJN-76/2010
and through DST grant #SB/S2/CMP-001/2013. J.C. was supported by ERDF under
project CEITEC (CZ.1.05/1.1.00/02.0068) and EC 7th Framework Programme
(286154/SYLICA).
Author contributions
B.J.K. conceived the project. S.H.C., J-W.K., J.K. and B.J.K. performed the experiment
with support from Y.C., T.G., A.A-Z., M.M.S. and M.K. H.Z. and K.M. grew the single
crystals; C.C.S., C.D.M. and K.M. characterized the samples under the supervision of
J.F.M. and Y.S. S.H.C., J-W.K. and B.J.K. analysed the data. J.C. performed the numerical
calculations. J.C., G.J. and G.K. developed the theoretical model. All authors discussed
the results. B.J.K. led the manuscript preparation with contributions from all authors.
Additional information
Supplementary information is available in the online version of the paper. Reprints and
permissions information is available online at www.nature.com/reprints.
Correspondence and requests for materials should be addressed to B.J.K.
Competing financial interests
The authors declare no competing financial interests.
466 NATURE PHYSICS | VOL 11 | JUNE 2015 | www.nature.com/naturephysics
© 2015 Macmillan Publishers Limited. All rights reserved
103
NATURE PHYSICS DOI: 10.1038/NPHYS3322 LETTERS
Methods
Single-crystal growth. Single crystals of Na2IrO3 were grown following two
different recipes using Na2CO3 flux (Sample 1) and self-flux (Sample 2). For
Sample 1, a mixture of Na2CO3 and IrO2 with a molar ratio of 50:1 was melted at
1,050 ◦
C for 6 h followed by fast cooling at a rate of 100 ◦
C h−1
down to 1,000 ◦
C,
slow cooling at a rate of 1 ◦
C h−1
down to 800 ◦
C and furnace cooling to room
temperature in sequence. Hexagonal pillar-shaped crystals with typical dimensions
of 0.2 mm × 0.2 mm × 0.4 mm were obtained after dissolving Na2CO3 flux in
acetone and water. For Sample 2, powders of Na2CO3 were mixed with 10–20%
excess IrO2 and were calcined at 700 ◦
C for 24 h. Single crystals were grown on top
of a powder matrix in a subsequent heating at 1,050 ◦
C. Plate-like crystals with
typical dimensions of 5 mm × 5 mm × 0.1 mm were physically extracted.
Resonant X-ray scattering. Incident X-rays were tuned to the Ir L3 edge
(11.2145 keV). The resonant X-ray diffraction experiments were carried out at the
6 ID-B beamline of the Advanced Photon Source. The polarization analysis was
performed in the vertical scattering geometry using a pyrolytic graphite analyser
probing the σ–π channel. The RIXS was performed at ID20 of the European
Synchrotron Radiation Facility. The total instrumental energy resolution of 24 meV
was achieved with a monochromator and a diced spherical analyser made from Si
(844) and a position-sensitive area detector placed on a Rowland circle with a 2 m
radius. The diffuse magnetic scattering was performed using the RIXS
spectrometers at the 9 ID, 27 ID and 30 ID (MERIX) beamlines of the Advanced
Photon Source, where a monochromator of 90 meV bandwidth was used for an
order-of-magnitude higher incident photon flux than that from the Si (844)
monochromator. In these experiments, a horizontal scattering geometry was used
with the π-incident X-ray polarization measuring the sum of π–σ and π–π
channels. The 2θ angle was fixed at 90◦
to minimize the contribution from
Thompson elastic scattering. As a result, L values in the HK maps shown in Fig. 2a
vary in the range between 6.5 and 7. The in-plane momentum resolution of the
RIXS spectrometer was ±0.048 Å−1
. The use of RIXS spectrometers rejecting all
inelastically scattered X-rays outside of the 100 meV energy window centred at the
elastic line led to a significant improvement in the signal-to-noise ratio. A typical
counting time of 2 h was required for a map shown in Fig. 2.
NATURE PHYSICS | www.nature.com/naturephysics
© 2015 Macmillan Publishers Limited. All rights reserved
104
1
Supplementary Information
A. Sample characterization
The powder x-ray diffraction patterns of both Sample #1 and #2 were consistent with the crystal structure in the
C2/m space group as previous reported15
. Sample #1 had a slightly lower TN=12 K compared to Sample #2 with
TN=15 K, as measured by SQUID magnetometry (Fig. S1a) and by resonant x-ray diffraction through the magnetic
Bragg peaks at Q = (0 1 n+1
2 ) (n: integer) (Fig. S1b). Sample #1 was found to be of multi domains but had a
superior crystallinity with 0.1◦
mosaicity (as compared to 0.5◦
mosaicity of Sample #2) (Fig. S1c), and thus was used
for the resonant diffraction experiment (polarization analysis and measurement of the magnetic correlation length.)
Sample #2 was found to be of a single domain and was used for the RIXS measurement. Both Sample #1 and #2
were used for the diffuse scattering measurement and gave identical results.
46.0 46.5 47.0 47.5 48.0
theta (degree)
Intensity(A.U.)
46.0 46.5 47.0 47.5
theta (degree)
Intensity(A.U.)
Sample#1
(0 0 7)
Sample#2
(0 0 7)
(0 1 3.5)
6 K
7
8
9
10
11
12
13
Sample#1
(0 1 6.5)
8.5 K
10
12
14
15
16
Sample#2
theta (degree) theta (degree)
16.0 16.5 17.0 17.5 40 41 42 43 44 45
Intensity(A.U.)
Intensity(A.U.)
χ(10-3
cm3
/mol)
H = 3 T c axis
Sample#2
χ(10-3
cm3
/mol)
b
C
1.4
1.6
1.8
2.0
T (K)
0 10 20 30 40 50 60
T (K)
0 10 20 30 40 50 60
4
5
H = 2 T
Sample#1
arbitrary orientation
a
3
TN
= 12 K TN
= 15 K
Supplementary Figure 1. Characterization of Sample #1 and Sample #2. (a) Temperature dependence
of magnetic susceptibility. Dotted lines indicate TN. Black arrows indicate that the data were measured while
warming after zero-field cooling. (b) Temperature dependence of the magnetic Bragg peaks. (c) Sample mosaicity.
B. Extraction of the spin-component resolved equal-time correlators
The x-ray scattering intensity measured without using a polarization analyzer contains contributions from both π-σ
and π-π channels, probing spin components along ki and perpendicular to the horizontal scattering plane, respectively.
In other words, two spin components perpendicular to kf are measured in the 90◦
horizontal scattering geometry used.
For example, when Ψ=0◦
, the z local cubic axis points approximately along kf (Fig. S2), and thus the scattering
Direct evidence for dominant bond-directional
interactions in a honeycomb lattice iridate Na2IrO3
SUPPLEMENTARY INFORMATION
DOI: 10.1038/NPHYS3322
NATURE PHYSICS | www.nature.com/naturephysics	 1
© 2015 Macmillan Publishers Limited. All rights reserved
105
2
eπ
Q
eπ'
eσ'
kf
ki
y
x
z
Ψ = 0 °
Q
x
z
y
Ψ = 120 °
Q
z
y
x
Ψ = 240 °
a axis
a axis
a axis
Supplementary Figure 2. Scattering configurations. (a) Ψ= 0◦
, (b) Ψ = 120◦
, and (c) Ψ = 240◦
. The
azimuth Ψ is the angle between the a axis and the scattering plane. Thick black arrows denote the local x, y, and z
axes in the IrO6 octahedron. a axis is indicated by a thin black arrow. Cyan arrows are the incident and scattered
x-rays with the wave vectors ki and kf , respectively. Green arrows indicate the x-ray polarizations. Q (dark blue
arrow) is the momentum transfer.
intensity measures the correlation Sxx+Syy. Likewise, Ψ=120◦
(Ψ=240◦
) measures Sxx+Szz(Syy+Szz). Then, Sxx,
Syy, and Szz can be extracted by solving a set of linear equations.
C. Temperature dependence of the diffuse magnetic peak and RIXS spectra
The short-range zig-zag order is observable at least up to T≈70 K (Fig. S3a). The magnetic correlation length (1.6-1.8
nm) along the a axis does not vary significantly in the measured temperature region. Figs. S3b and S3c plot the
diffuse map at T=50 K for Ψ=0◦
and Ψ=30◦
, respectively, which is similar to the diffuse map recorded at T=17 K
(shown in Fig. 2a) apart from thermal broadenings. Fig. S3d depicts the RIXS spectra at Q=(0.5 0.5) and (0.5 -0.5)
for Ψ=180◦
, which show subtle spectral change in the temperature range, 9 K-17 K, below and above TN.
15 K
20 K
30 K
50 K
70 K
(H 1 6.5)
a
d
0.0 0.5-0.5
H (r. l. u.)
Intensity(A.U.)
1
-1
0
0 0.5-0.5
H (r. l. u.)
K(r.l.u.)
Y = 0
50 K
b
T (K)
0 50
Correlation
length(A)
15
20
Y = 30
50 K
c
Ψ = 0
0
100
200
300
400
Intensity(A.U.)
Energy loss (meV)
Q = (0.5 0.5)
9 K
13 K
15 K
17 K
9 K
13 K
15 K
17 K
2001000-100-200 2001000-100-200
Energy loss (meV)
Q = (0.5 -0.5)
Sample #2 Sample #2
Ψ = 180 Ψ = 180
Supplementary Figure 3. Diffuse magnetic x-ray scattering intensities above TN and RIXS spectra
below and above TN. (a) H profiles of the diffuse peak at Q=(0 1). Shown in the inset is the magnetic correlation
length along the a axis derived from Gaussian fitting (solid curves)of the data for Sample #1. (b,c) Diffuse maps at
T=50 K for Ψ=0◦
and Ψ=30◦
. The dashed hexagon indicates the first Brillouin zone of the honeycomb net. (d)
Temperature evolution (9 K-17 K) of RIXS spectra at Q=(0.5 0.5) and Q=(0.5 -0.5) for Ψ=180◦
.
2	 NATURE PHYSICS | www.nature.com/naturephysics
SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHYS3322
© 2015 Macmillan Publishers Limited. All rights reserved
106
3
D. Calculated azimuthal angle dependence for all polarization channels
In order to facilitate comparison between the two azimuth angle dependence curves shown in Fig. 1c and Fig. 2d measured
with different analyzer settings, we provide in Fig. S4 calculated azimuth angle dependence for all polarization
channels for the two measured Q positions.
0.0
0.5
1.0
Intensity(A.U)
0 120
Ψ (degree)
240 360
π-π’ channel π-σ’ channel
π-π’ channel + π-σ’ channel
σ-π’ channelQ = (0 1 6.74)
0.0
0.5
1.0
Intensity(A.U)
0 120
Ψ (degree)
240 360
π-π’ channel π-σ’ channel
π-π’ channel + π-σ’ channel
σ-π’ channelQ = (0 1 3.5)
ba
Supplementary Figure 4. Azimuthal angle dependence for all polarization channels. (a) For Q=(0 1
6.74) (b) For Q=(0 1 3.5).
NATURE PHYSICS | www.nature.com/naturephysics	 3
SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS3322
© 2015 Macmillan Publishers Limited. All rights reserved
107
108
PHYSICAL REVIEW B 92, 024413 (2015)
Hidden symmetries of the extended Kitaev-Heisenberg model:
Implications for the honeycomb-lattice iridates A2IrO3
Jiˇr´ı Chaloupka1
and Giniyat Khaliullin2
1
Central European Institute of Technology, Masaryk University, Kotl´aˇrsk´a 2, 61137 Brno, Czech Republic
2
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
(Received 19 February 2015; revised manuscript received 5 May 2015; published 13 July 2015)
We have explored the hidden symmetries of a generic four-parameter nearest-neighbor spin model, allowed
in honeycomb-lattice compounds under trigonal compression. Our method utilizes a systematic algorithm to
identify all dual transformations of the model that map the Hamiltonian on itself, changing the parameters and
providing exact links between different points in its parameter space. We have found the complete set of points of
hidden SU(2) symmetry at which a seemingly highly anisotropic model can be mapped back on the Heisenberg
model and inherits therefore its properties such as the presence of gapless Goldstone modes. The procedure
used to search for the hidden symmetries is quite general and may be extended to other bond-anisotropic spin
models and other lattices, such as the triangular, kagome, hyperhoneycomb, or harmonic-honeycomb lattices.
We apply our findings to the honeycomb-lattice iridates Na2IrO3 and Li2IrO3, and illustrate how they help to
identify plausible values of the model parameters that are compatible with the available experimental data.
DOI: 10.1103/PhysRevB.92.024413 PACS number(s): 75.10.Jm, 75.25.Dk, 75.30.Et
I. INTRODUCTION
When relativistic spin-orbit coupling dominates over the
exchange and orbital-lattice interactions, the orbital moment
L of an ion remains unquenched and a total angular momentum
J = S + L is formed. This was known to happen
in compounds of late transition metal ions such as of cobalt
(see, e.g., Ref. [1]); however, the “cleanest” examples of
spin-orbit coupled magnets emerged more recently: these are
the iridium oxides Sr2IrO4 and Na2IrO3 with perovskite and
honeycomb-lattice structures, correspondingly.
By construction, magnetic ordering in these systems necessarily
involves interactions between orbital moments L,
in addition to a conventional Heisenberg exchange among
the spin part of total angular momentum J [2]. Since the
L moment, hosted by t2g orbital in a crystal, is only an
“effective” one [3], it need not be conserved during the
electron hoppings; thus the L-moment exchange interactions
are generally not SU(2) invariant [4]. Moreover, the orbital
moments have a “shape” and hence the L interactions are
anisotropic in real space, too, and thus strongly frustrated even
on simple cubic lattices. Altogether, this results in nontrivial
L Hamiltonians and orderings, including, e.g., noncoplanar
(multi-Q) states, “hidden” Goldstone modes, etc. [5,6]. Via
the spin-orbit coupling, these peculiar features of orbital
physics are inherited by the “pseudospin-J” wave functions
and interactions [6–14]. In essence, the frustrated nature
and quantum behavior of t2g-orbital moments [15,16] are
transferred to those of low-energy pseudospins J.
Depending on the electron configuration of ions, the ground
state pseudospin may take different values J = 0,1/2,1, . . . ,
and a variety of magnetic Hamiltonians with different symmetries
and diverse behavior emerge in each case, because of
different admixture of non-Heisenberg L interactions. Perhaps
the most radical departure from a conventional magnetism is
realized in compounds with apparently “nonmagnetic” J = 0
ions, where a competition between spin-orbit and exchange
interactions results in a nonmagnetic-magnetic quantum phase
transition [17–20].
The case of pseudospin J = 1/2 iridates is of special
interest. This is because Sr2IrO4 perovskite was found
[22–24] to host cuprate-like magnetism, and honeycomblattice
iridates A2IrO3 (A = Na, Li) have been suggested [9]
as a candidate material where the Kitaev model [25] physics
might be realized. Following this proposal, a subsequent work
[11] has introduced the minimal magnetic Hamiltonian for
iridates A2IrO3: the Kitaev-Heisenberg model (KH model)—a
frustrated spin model with many attractive properties. Most
importantly, its phase diagram contains a finite window of
a quantum spin-liquid phase which emanates from the pure
Kitaev point of the model with a known exact solution [25].
To reflect the later experimental findings in iridates, such as
the zigzag (Na2IrO3 [26–28]) and spiral (Li2IrO3 [29]) type
magnetic orderings, the initially proposed model was modified
by including longer-range Heisenberg [28,30] or anisotropic
[31] interactions, extending the parameter range [32–34],
by considering further anisotropic terms in the Hamiltonian
[35–42], or by including spatial anisotropy of the model
parameters [43]. An alternative picture based on an itinerant
approach has been also suggested [44].
Despite the extensive efforts, no consensus concerning
the minimal model for the honeycomb-lattice iridates has
thus far been reached. A reliable microscopic derivation of
the exchange interactions is difficult and does not lead to a
conclusive suggestion for the minimal Hamiltonian and its
parameters. On the experimental side, the richest information
about the underlying spin model would be provided by mapping
momentum-resolved spin excitation spectrum. However,
due to the lack of large enough monocrystals, the inelastic
neutron scattering (INS) has been performed on powders
only [28]. Another possible probe—resonant inelastic x-ray
scattering (RIXS)—suffers from a small resolution at present.
While it could be successfully applied in the case of perovskite
iridates [23], here the limitation comes from the much smaller
energy scale of the excitations to be studied in detail by
RIXS; however, the overall strength of magnetic interactions
in Na2IrO3 has been quantified [45,46].
1098-0121/2015/92(2)/024413(14) 024413-1 ©2015 American Physical Society
109
JI ˇR´I CHALOUPKA AND GINIYAT KHALIULLIN PHYSICAL REVIEW B 92, 024413 (2015)
Nevertheless, the experimental data collected to date puts
rather strong constraints on the possible models. First, the
RIXS-derived magnetic energies [45,46] (of the order of 40
meV) are much higher than the ordering temperature (∼15 K),
suggesting strong frustration. Second, the magnetic scattering
intensity, measured by RIXS at zero momentum, Q = 0, is
as strong as elsewhere in the Brillouin zone, which implies a
dominance of anisotropic, non-Heisenberg spin interactions.
Third, the recent resonant x-ray scattering data [46] have
revealed nearly ideal C3 symmetry of the spin correlations
in momentum space. Moreover, inelastic neutron scattering
data [28] have indicated that a spin gap, if present, would be
relatively small (less than 2 meV). All these observations taken
together imply that the dominant pseudospin interactions in iridates
are strongly frustrated, highly anisotropic in spin space,
and yet highly symmetric in real space. By very construction,
all these features are in fact the intrinsic properties of the KH
model and its extended versions.
The KH model, supplemented by other C3 symmetry
allowed terms (see below), is therefore physically sound and
plausible. However, there is a problem of its large parameter
space (four parameters even within the nearest-neighbor
model) resulting in complex phase diagrams, which makes the
analysis of experimental data and the extraction of the model
parameters a difficult task. In such cases, clarification of the
underlying symmetry properties of the model is often of a great
help. In general, the spin-orbital models in Mott insulators
possess peculiar symmetries [6,14] which are rooted in the
bond-directional nature of orbitals. In this context, a special
four-sublattice rotation [6] within spin space has proved itself
as an extremely useful tool in the case of the original twoparameter
KH model [11,32,47–50]. It maps the Hamiltonian
on itself but changes the Hamiltonian parameters, connecting
thereby different points in the parameter space. Being an exact
transformation, it transfers the complete knowledge about
some point in the phase diagram, including the ground state,
excitation spectrum, response functions, etc., to its partner.
Based solely on this self-duality of the model, the entire phase
diagram could be sketched and the deep relations between the
phases understood. In addition, it also reveals points of hidden
SU(2) symmetry, where the system is exactly equivalent to a
Heisenberg model for the rotated spins. Given its usefulness, it
is highly desirable to find and analyze similar transformations
for the extended versions of the KH model.
In this paper, we introduce a systematic method to derive
dual transformations of bond-anisotropic spin Hamiltonians
and demonstrate its results and their physical implications
in the case of honeycomb iridates adopting the full nearestneighbor
model [36,37,39]. We find all the hidden SU(2)symmetry
points of the model, the most peculiar one being
characterized by a “vortex”-like pattern with a six-site unit
cell, and demonstrate how the characteristics of the hidden
Heisenberg magnet manifest themselves in the anisotropic
situations. By identifying the SU(2) points we characterize all
the possible gapless Goldstone modes that may be encountered
within the model. This is relevant in the context of real
materials as the spin gap was found to be well below 2 meV
[28,29], suggesting a connection to some of the SU(2) points.
Finally, using a self-duality of the model, we will provide a link
between our fits of the earlier Na2IrO3 data [32] and the recent
experimental observation of the ordered moment direction
[46]. We argue that this observation provides a direct access
to the strength of the additional terms “extending” the KH
model, and quantify the spin easy axis direction in terms of this
“departure” from the pure KH model. This allows us to suggest
plausible values of the model parameters that are compatible
with the current data. While we focus here on the case of a
honeycomb lattice as realized in Na2IrO3 and more recently
in RuCl3 [51], the method is general and expected to produce
interesting results also in the context of the new structural
families of iridates—recently synthesized hyperhoneycomb
[52,53] and harmonic-honeycomb lattices [54,55], or the
theoretically proposed hyperoctagon lattice [56].
The paper is organized as follows. Section II introduces the
Hamiltonian and discusses its parameters. Sections III and IV
introduce the method and derive and discuss the main results
of the paper—the hidden symmetries of the model. Section V
and Appendix B discuss the implications of the results for
honeycomb iridates.
II. EXTENDED KITAEV-HEISENBERG MODEL
We start by specifying the model Hamiltonian including
all symmetry-allowed spin interactions on nearest-neighbor
bonds. An ideal, undistorted structure of the honeycomb
NaIr2O6 plane is shown in Fig. 1(a). We will utilize its
rotational C3 symmetry and the three sets of parallel mirror
z = z~
S
S
S
z
y
x
c
ba
(b)
O
Ir
(a)
Ir
Na O
z
y
Y
x
Z X
x
y~
~
(c)
B
A
c
a b
FIG. 1. (Color online) (a) Top view of the honeycomb NaIr2O6
plane, the definition of global X,Y,Z axes, and the xyz reference
frame for the spin components. The X and Y directions coincide with
the crystallographic a and b axes. The three bond directions of the
honeycomb lattice are labeled as a, b, and c; its two sublattices are
labeled by A and B. (b) Two edge-shared IrO6 octahedra of a c bond
and the definition of the local spin axes ˜x, ˜y, ˜z [used in Eq. (1)]. (c)
Simultaneous cyclic permutation of the Ir-Ir bond directions a, b, c
and the spin components x, y, z when applying a C3 rotation to the
model.
024413-2
110
HIDDEN SYMMETRIES OF THE EXTENDED KITAEV- . . . PHYSICAL REVIEW B 92, 024413 (2015)
planes containing the shared edges of the IrO6 octahedra and
cutting the Ir-Ir bonds into halves. The C3 symmetry links the
interactions for different bond directions while the presence of
the mirror planes restricts the possible interactions for a given
bond direction. A trigonal distortion (compression or elongation
along the Z axis) fully preserves these symmetries so
that our Hamiltonian applies in that case as well. Furthermore,
recent experiments [46] indicate a nearly ideal C3 symmetry
of the spin properties and hence suggest that additional
terms, possibly induced by a monoclinic distortion present in
Na2IrO3, can be neglected. Physically, this observation implies
the robustness of the pseudospin wave functions against weak
monoclinic distortions.
The bond Hamiltonian is most compactly expressed in
a local, bond-dependent ˜x ˜y˜z reference frame for spins,
presented in Fig. 1(b) for a c bond. Due to the mirror symmetry,
the in-bond S ˜x
component is forbidden to interact with the S ˜y
and S˜z
components [36]. Following the notation of Ref. [36]
we arrange the allowed terms into the form
H ij c = J Si · Sj + K S˜z
i S˜z
j
+D S ˜x
i S ˜x
j − S
˜y
i S
˜y
j + C S
˜y
i S˜z
j + S˜z
i S
˜y
j . (1)
This four-parameter Hamiltonian extends the KH model (J and
K terms) by the D term bringing further anisotropy among
the diagonal components of the interaction, and the C term
determining the only symmetry-allowed nondiagonal element
in the exchange interaction tensor. Parameter C would vanish
for an isolated pair of undistorted octahedra; it becomes finite
due to a trigonal distortion and/or due to the extended nature of
orbitals in a crystal (“recognizing” the fact that the octahedra
are canted relative to the crystal axis Z).
To capture the C3 symmetry, it is convenient to switch to
cubic axes xyz, introduced in Fig. 1(a) and pointing from an
Ir ion to neighboring O ion positions in an ideal structure. The
c-bond Hamiltonian in the cubic reference frame, as derived
in Ref. [37], reads then as
H ij c = J Si · Sj + K Sz
i Sz
j + Sx
i S
y
j +S
y
i Sx
j
+ Sx
i Sz
j +Sz
i Sx
j +S
y
i Sz
j +Sz
i S
y
j , (2)
with the correspondence = −D and = 1√
2
C, often used
below. For the other bond directions, the Hamiltonian is
obtained by a cyclic permutation [see Fig. 1(c)], resulting in
one-to-one correspondence between the three types of bonds
and interactions, as required by C3 symmetry. Physically, each
type of bond favors its own distinct “orbital setup” to optimize
the hopping energy, and this is fingerprinted in pseudospin
interactions via spin-orbit coupling. For completeness, Appendix
A shows the Hamiltonian in the global axes XYZ; it
has certain advantages moving the bond dependence from the
operator forms to the coupling constants.
A few comments are in order concerning the model parameters.
In general, calculation of exchange integrals in transition
metal compounds with 90◦
d − p − d bonding geometry is
an intricate task, because more hopping pathways are allowed
as compared to a simpler case of 180◦
d − p − d bonding
in perovskites (where theory [9] has correctly predicted the
strength of dominant exchange constants). For instance, t2g
orbitals may also overlap directly, in addition to oxygenmediated
hoppings; there is a large overlap between orbitals of
t2g and eg symmetries (forbidden in perovskites), etc., resulting
in a number of competing ferromagnetic and antiferromagnetic
contributions which are difficult to evaluate, in particular in
compounds with small Mott and/or charge-transfer excitation
gaps. The uncertainties in interaction parameters U and JH
further affect the theoretical estimates.
Initial consideration [6] of the pseudospin one-half exchange
interactions in 90◦
-bonding geometry resulted in K =
−2J [hitting a “hidden” SU(2) point by chance] in the cubic
limit; later work [8,9,11] using different approximations has
changed this estimate both in terms of the signs and values of
J and K, illustrating the difficulties described above. It was
also found that the nondiagonal element allowed in cubic
symmetry may take sizable values [36,37,39,40]. Further,
is expected to become as large as the other parameters
if trigonal splitting of the t2g orbital level, caused by
a compression along the Z axis, becomes comparable to
spin-orbit coupling λ; also, the trigonal field suppresses the
parameter K. These trends are easy to understand: large
trigonal field suppresses the in-plane components of orbital
moment LX and LY , leaving the axial LZ component the only
unquenched one; thus the pseudospin one-half Hamiltonian,
written most conveniently in global axes in this limit, may
not contain anything but XX + YY and ZZ type terms:
JXY (SX
i SX
j +SY
i SY
j ) + JZSZ
i SZ
j , identical for all bonds. This
is what has indeed been found by explicit calculations [6,35]
in the limit of λ. This implies K = 0 and =
in this limit (see also Appendix A), while JXY = (J − )
and JZ = (J + 2 ) may take any values depending on the
microscopic details. Although this limit is not very realistic for
the Ir4+
ion with large spin-orbit constant λ ∼ 0.4 eV [3,57],
we may expect sizable values of both and in Na2IrO3
where seems to exceed 0.1 eV [58,59]. The role of and
terms should further increase in other compounds based on
pseudospin-1/2 Co4+
, Ru3+
, and Rh4+
ions with smaller λ.
In general, the high-energy behavior of spins and orbitals
in transition metal compounds is well captured by the KugelKhomskii
models [4] and their descendants [6]. However, the
low-energy physics and ultimate magnetic “fixed-point” are
heavily influenced by many “unpleasant” details originating
from orbital-lattice coupling and distortions, unavoidable in
real materials. In perovskites, the Kugel-Khomskii energy
scale is given by 4t2
/U independent of spin-orbit coupling;
however, this leading term drops out for pseudospins-1/2 in
the edge-shared, 90◦
-bonding geometry [6,9], so the “highenergy”
scale is set up by the subleading terms. In iridates,
the hope [9] is that the Kitaev-type coupling is the leading one
among these subleading terms. Since this coupling is itself a
correction to 4t2
/U, this expectation may or may not hold in
reality.
To summarize up to now: in real materials even with an ideal
C3 symmetry, all four of the exchange parameters may play a
significant role. This motivates us to regard the Hamiltonian
(1) and (2) as an effective model with arbitrary parameters,
and look for some general symmetry arguments that may help
to identify plausible parameter windows in the analysis of
experimental data.
024413-3
111
JI ˇR´I CHALOUPKA AND GINIYAT KHALIULLIN PHYSICAL REVIEW B 92, 024413 (2015)
III. SYSTEMATIC CONSTRUCTION OF DUAL
TRANSFORMATIONS
Having fixed the model Hamiltonian, we are ready to
explore its dual transformations. By a dual transformation we
mean a prescription for site-dependent rotations in the spin
space, Si = Ri Si, which transforms a spin Hamiltonian H(S)
into a formally new Hamiltonian H (S ). We are interested in
self-dual transformations of H that map the model onto itself,
preserving all its symmetry properties. That is, the rotated
partner H (i) has the same four terms albeit with different
parameters J K D C , and (ii) it respects the C3 rotation rules
encoded in Fig. 1(c), hence preserving the original distribution
of the three types of bond-dependent interactions on a lattice.
Starting with the JKDC Hamiltonian expressed as H(S) =
ij ST
i Hij Sj where Hij are 3 × 3 matrices, we obtain
H(S) = H (S ) = ij Si
T
Hij Sj with Hij = RiHij RT
j . For
a self-dual transformation, the matrices Hij are identical to Hij ,
but the parameters JKDC are replaced by J K D C , and the
one-to-one correspondence between the bond directions and
interactions remains intact. These two points in the parameter
space are linked by the transformation and knowing the
solution at one of the points, we may “rotate” it to the other one.
In this section we give an algorithm to find the self-dual
transformations for the extended KH model that map it
onto itself. We have found a single self-dual transformation
JKDC ↔ J K D C operating in full parameter space of the
model; we will show it shortly below and return to it later when
discussing experimental data.
However, studying the hidden symmetries of the model, we
have identified a number of restricted self-dual transformations
that operate only in some regions of the parameter space, where
constants J,K,D,C are all finite but obey certain relations, or
some of them are simply zero. Our primary interest is in the
special class of such transformations of the type J0 ↔ JKDC,
which convert the Heisenberg model into the full JKDC
model and vice versa. These transformations, to be discussed
in the next section, reveal points of hidden SU(2) symmetry;
by inverting the transformation the anisotropic model with
the parameters JKDC can be exactly mapped back to the
Heisenberg model with the exchange constant J0.
A. Algorithm
A systematic search for the dual transformations seems to
be an intricate task. Fortunately, it can be easily performed by
computer on a finite cluster of the lattice using the following
simple algorithm. We give it specifically for the case of a
self-dual transformation:
(A) As a first step, we choose two rotation matrices Ri,
Rj on a selected bond ij . They have to preserve the JKDC
form given by (1), which leaves us with only a few choices,
each having only one free angular parameter.
(B) Next, we randomly choose nonzero values of the initial
parameters JKDC and use the relation Hij = RiHij RT
j together
with the C3 symmetry to determine the new Hamiltonian
matrices for the three bond directions.
(C) Knowing all the bond Hamiltonians, we may now
determine further rotation matrices by utilizing relations of
the type Rj = (Hij )−1
RiHij and proceeding neighbor-byneighbor.
To fully determine the rotation matrices, about two
thirds of the bonds need to be used.
(D) The bonds of the remaining third are used to check
consistency; the Hamiltonian matrix determined by using the
rotation matrices belonging to the bond has to be identical to
that determined in step B. If the total difference on all the
remaining bonds equals zero, we have just constructed a selfdual
transformation. By scanning through the entire interval
of the free parameter introduced in step A, we find all the
self-dual transformations.
The above procedure may be easily adapted to find the dual
transformations such as J0 ↔ JKDC. In this case, in step
A of the algorithm, we use the symmetry of the Heisenberg
model and choose Ri as an identity matrix. The choice of
the second matrix Rj is restricted by the requirement that
Hij = RiHij RT
j = JRT
j be of the JKDC form.
By inspecting the rotation matrices of the cluster, we can
identify the particular unit cell of the transformation. Note
that even if our cluster is smaller than this unit cell, we do not
miss the corresponding transformation, so that the method is
completely systematic [60].
B. Self-duality of the extended Kitaev-Heisenberg model
The systematic procedure described above has identified
only a single self-dual transformation JKDC ↔ J K D C .
This is not surprising given the complexity of the model. The
corresponding parameter transformation may be written in a
matrix form
⎛
⎜
⎝
J
K
D
C
⎞
⎟
⎠ =
⎛
⎜
⎜
⎜
⎜
⎜
⎝
1 +4
9
+4
9
+2
√
2
9
0 −1
3
−4
3
−2
√
2
3
0 −4
9
+5
9
−2
√
2
9
0 −2
√
2
9
−2
√
2
9
+7
9
⎞
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎝
J
K
D
C
⎞
⎟
⎠. (3)
For convenience, we also give the transformation of the
parameters JK entering the Hamiltonian (2):
⎛
⎜
⎝
J
K
⎞
⎟
⎠ =
⎛
⎜
⎜
⎜
⎜
⎝
1 +4
9
−4
9
+4
9
0 −1
3
+4
3
−4
3
0 +4
9
+5
9
+4
9
0 −2
9
+2
9
+7
9
⎞
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎝
J
K
⎞
⎟
⎠. (4)
In terms of the spins, the transformation, labeled for future
reference as T1, is simply a global π rotation about the Z axis
defined in Fig. 1(a). The individual SX
, SY
, and SZ
components
transform according to
T1 : (X ,Y ,Z ) = (−X, − Y,Z) (5)
at every site. By applying the transformation twice, we get an
identity and the matrices in (3) and (4) are thus self-inverse.
Despite its apparent triviality, this transformation will play an
essential role when discussing the real materials; see Sec. V
below.
IV. POINTS OF HIDDEN SU(2) SYMMETRY
In this paragraph we find and characterize all the points of
hidden SU(2) symmetry present in the extended KH model.
At these special points in the parameter space, the anisotropic
024413-4
112
HIDDEN SYMMETRIES OF THE EXTENDED KITAEV- . . . PHYSICAL REVIEW B 92, 024413 (2015)
TABLE I. Parameter values for the SU(2) points in units of the
exchange constant J0 of the hidden Heisenberg model.
J/J0 K/J0 ( ≡−D)/J0 ( ≡ 1√
2
C)/J0
T2 −1/3 0 2/3 2/3
T4 −1 2 0 0
T6 0 −1 −1 0
T1T4 −1/9 −2/3 8/9 −4/9
T2T6 −2/3 1 1/3 −2/3
model can be mapped back to a Heisenberg ferromagnet or
antiferromagnet. The SU(2) points of the original KH model
have been identified [11] by virtue of the four-sublattice
transformation introduced in Ref. [6]. The corresponding
ordering patterns on the honeycomb lattice are of stripy and
zigzag type. A similar symmetry analysis of the KH model
was performed for other relevant lattices [47].
The extended KH model of course inherits the SU(2)
points of the KH model and contains several new ones in
addition. They are identified by dual transformations of the
type J0 ↔ JKDC which is less general than JKDC ↔
J K D C . Because of this, we obtain a relatively rich set
of dual transformations characterized by two-, four-, and
six-sublattice structure of the rotations. In terms of parameters,
all the nontrivial SU(2) points are listed in Table I. We now
proceed with the detailed description of the corresponding
transformations.
A. Summary of the SU(2) points and the corresponding
rotations on the sublattices
We first give a summary of the transformations as represented
by rotations in the real space. Each of them generates
an infinite number of orderings, since the ordered moment
direction in the underlying Heisenberg model can be chosen
arbitrarily. Figure 2 shows a few important examples.
The simplest transformation T2 is π rotation about the Z
axis at one of the two sublattices of the honeycomb lattice:
T2 : (X ,Y ,Z ) = (X,Y,Z) (sublattice A) ,
(X ,Y ,Z ) = (−X, − Y,Z) (sublattice B). (6)
Its physical relevance is small due to the dominance of (≡
1√
2
C) and the complete absence of K (corresponding to the
case of strong trigonal field splitting, as explained above). As
a curiosity, if we choose the spins to lie in the honeycomb
plane, T2 converts the FM pattern to AF and vice versa. We
may thus have an AF/FM ordered pattern, but the hidden nature
revealing itself, e.g., in the spin dynamics is that of Heisenberg
FM/AF, respectively.
The next transformation T4 has a four-sublattice structure
depicted in Fig. 2(a) with π rotations about cubic x, y, and
z axes applied at sublattices 1, 2, and 3, respectively, and no
rotation involved at sublattice 4. Written explicitly:
T4 : (x ,y ,z ) = (x, − y, − z) (sublattice 1),
(x ,y ,z ) = (−x,y, − z) (sublattice 2),
(x ,y ,z ) = (−x, − y,z) (sublattice 3),
(x ,y ,z ) = (x,y,z) (sublattice 4). (7)
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Qb
Q3
QcQ1
2Q
Qa
X
M
Γ
Y
Γ″
Γ′
5
1
2
4
6
3
2
1
1
3
4 3
2
4
FIG. 2. (Color online) (a), (b) Unit cells for the four- and sixsublattice
transformations. (c), (d) Stripy and zigzag patterns related
to the FM and AF order of a hidden Heisenberg magnet via the
four-sublattice transformation T4. The spins take the z-axis direction.
(e), (f) “Vortex”-like patterns generated by the six-sublattice transformation
T6. The spins are lying in the lattice plane in the case presented.
The colors of the arrows in panels (d) and (f) indicate the sublattices
of the hidden AF order. (g) Brillouin zones of the honeycomb (inner
hexagon) and the completed triangular lattice (outer hexagon). The
characteristic vectors Qa,b,c of the four-sublattice transformation and
Q1,2,3 of the six-sublattice transformation are shown in red and blue,
respectively. (h) Bragg spots of the patterns in panels (c)–(f). The dot
size is proportional to |SQ|.
This transformation, introduced earlier in Ref. [6], is a selfdual
transformation of the original two-parameter KH model
and has been already heavily used in this context. Applying
the transformation to an ordered Heisenberg FM/AF with the
moments pointing along the z axis, we get the stripy/zigzag
order shown in Figs. 2(c) and 2(d).
Perhaps the most surprising SU(2) point of the model is
linked to the six-sublattice transformation T6. Its rotations are
024413-5
113
JI ˇR´I CHALOUPKA AND GINIYAT KHALIULLIN PHYSICAL REVIEW B 92, 024413 (2015)
most conveniently described in the cubic coordinates. On the
lattice sites 1, 3, and 5 [see Fig. 2(b)] they correspond to cyclic
permutations among the spin components. On the lattice sites
2, 4, and 6 the rotations correspond to anticyclic permutations
which have to be followed by a spin inversion. Altogether the
transformation can be written as
T6 : (x ,y ,z ) = (x,y,z) (sublattice 1),
(x ,y ,z ) = (−y, − x, − z) (sublattice 2),
(x ,y ,z ) = (y,z,x) (sublattice 3),
(x ,y ,z ) = (−x, − z, − y) (sublattice 4),
(x ,y ,z ) = (z,x,y) (sublattice 5),
(x ,y ,z ) = (−z, − y, − x) (sublattice 6). (8)
It is easy to see that for K = (≡ −D) and J = = 0, these
rotations lead to the isotropic Heisenberg Hamiltonian. As
an example, we consider the c bond 1 − 2 of Fig. 2(b). By
exchanging x and y at site 2, the nondiagonal term in (2)
becomes diagonal and the inversion ensures its proper sign.
Sample patterns generated by T6 and showing a “vortex”-like
structure are presented in Figs. 2(e) and 2(f). The peculiarity
of the SU(2) points is now best demonstrated: the Hamiltonian
is completely anisotropic containing K and (≡ −D) terms
only, the ordered spins form a very unusual pattern, yet the
hidden nature of the system is exactly that of the Heisenberg
FM or AF, including, e.g., the presence of gapless Goldstone
modes.
Apart from revealing a hidden SU(2) point of the present
model, the T6 transformation has a remarkable property that
deserves special attention. Namely, applying T6 to the Kitaev
Hamiltonian, we notice that it redistributes three types of Isinginteractions
on a honeycomb lattice such that at each hexagon
a Kekul´e-type pattern is formed [61]. We thus arrive at the
so-called Kekul´e-Kitaev model [62]. In other words, the Kitaev
and Kekul´e-Kitaev models are exact dual partners linked via
the T6 transformation. This observation should be helpful in
studying both models, in particular of their extended versions
including a Heisenberg term [62,63].
Two more transformations providing SU(2) points are
obtained as the combinations T1T4 and T2T6. They share the
sublattice structure with T4 and T6, respectively. Adopting
the extended KH model, the former one is probably the
SU(2) point closest to the real situation in Na2IrO3 as will
be discussed in Sec. V.
B. Implications for the phase diagram
After examining the nature of the individual SU(2) points,
we want to visualize now their positions in the parameter space,
get a sketch of the phase diagram, and infer the relations
between the individual phases. The result can be compared
with the published phase diagrams of Refs. [37] and [38], obtained
by classical analysis and partly complemented by exact
diagonalization. For this reason, we adopt the representation
of the parameter space introduced in Ref. [37]. The overall
energy scale irrelevant for the phase diagram is removed and
J, K, are parametrized using “spherical” angles θ and φ
via J = sin θ cos φ, K = sin θ sin φ, and = −D = ± cos θ,
FM AF
stripy
zigzag
"vortex"
A
B
C
zigzag
B: Γ’= -0.398 T1T4
A: Γ’=+0.894 T2
C: Γ’= -0.535 T2T6
Γ’= -0.365 Na2IrO3
Γ’= -0.395 Li2IrO3
FM AF
stripy
zigzag
"vortex"
B
A
C
stripy
B: Γ’=+0.398 T1T4
(a)
(b)
Γ > 0
Γ < 0
A: Γ’= -0.894 T2
C: Γ’=+0.535 T2T6
FIG. 3. (Color online) (a) Depiction of the SU(2) points using
the parametrization of Ref. [37], J = sin θ cos φ, K = sin θ sin φ,
= cos θ. The distance from the center of the circle corresponds
to θ going from 0 (center) through π/4 (dashed circle) to π/2 (solid
circle). The polar angle is φ. Filled squares show the SU(2) points with
= 0, open squares those with nonzero values given on the right
along with the transformation label. The color of the points indicates
their hidden FM (blue) or AF (red) nature. The green square (circle)
shows the parameter values specified in Sec. V when discussing
Na2IrO3 (Li2IrO3). (b) The same as in panel (a) but with = − cos θ.
keeping (≡ 1√
2
C) as a separate parameter of the phase
portrait.
Shown in Fig. 3 is the complete set of SU(2) points of
the extended KH model. The outer rings correspond to the
original KH model and contain the trivial SU(2) points and
the two well-known T4 hidden SU(2) points of the KH model
characterized by a stripy and zigzag pattern. Still within the
JK plane is the “vortex” T6 point associated with a “vortex”like
pattern. The corresponding phases determined by these
SU(2) points can be observed in Figs. 2 and 3 of Ref. [37],
with the T6 point lying in their 120◦
phase.
Three more SU(2) points A, B, and C characterized by a
nonzero value of are shown as projected onto the JK
plane. For the > 0 case presented in Fig. 3(a), they are
of AF character; one of them appears for positive (point A)
and two for negative (points B and C) values of . The
point A (given by T2) of hidden AF nature can possess FM
pattern as discussed in the previous paragraph. The region
between the true FM Heisenberg point and the point A in the
phase diagram obtained classically is therefore filled by the
024413-6
114
HIDDEN SYMMETRIES OF THE EXTENDED KITAEV- . . . PHYSICAL REVIEW B 92, 024413 (2015)
FM phase extending as increases (see panels (c) and (e)
of Fig. 2 of Ref. [38]). However, the (hidden) nature of this
phase changes from FM to AF which should manifest itself,
e.g., on the character of the magnon dispersion. Similarly, the
presence of the points B (T1T4) and C (T2T6) of zigzag and
“vortex” character, respectively, explains the enlarged region
of the corresponding phases in the classical phase diagram
for < 0 (see panels (a) and (d) of Fig. 2 of Ref. [38]). We
also observe an intimate relation between the zigzag phase
emanating from the B (T1T4) point and that connected to the
zigzag SU(2) point of the original KH model (given by T4). Due
to the additional T1 rotation, their ordered moment directions
are related by π rotation about the global Z axis. This point
will be further discussed in Sec. V. Finally, similar conclusion
as for the > 0 case presented in Fig. 3(a) can be drawn
for the < 0 case shown in Fig. 3(b). The SU(2) points are
related by inversion with respect to the center of the circle and
the opposite FM/AF nature.
In summary, we have illustrated that the gross features of
the phase diagram of the extended, four-parameter KH model
can be deduced solely by inspecting the nature of the points
of hidden SU(2) symmetry and their location in the parameter
space.
C. Spin excitation spectra
We proceed further by inspecting the spin excitation spectra
at the SU(2) points associated with T4 and T6 transformations,
and see how they are related to those of the simple Heisenberg
model. To this end, the dual transformations have to be
expressed in Fourier space and relations between the Fourier
components Sq of the dual partners have to be established.
The situation is somewhat complicated by the two-sublattice
structure of the honeycomb lattice, requiring us to introduce
an additional index [see the labels A and B in Fig. 1(a) for the
convention used below].
In both cases, it is convenient to use the cubic axes xyz. The
four-sublattice transformation has three characteristic vectors
Qa/b = (∓π/
√
3, − π/3) and Qc = (0,2π/3) touching the
Brillouin zone boundary in the middle of its edges [see
Fig. 2(g)]. The rotation matrices have a simple diagonal form,
reflecting only the sign changes of the respective components
RA/B = diag (±ei Qa·R
, ± ei Qb·R
,ei Qc·R
). (9)
The six-sublattice transformation written in Fourier representation
has a full matrix structure
RA/B = ±1
3
(I + MA/Bγ + M∗
A/Bγ ∗
) (10)
with the factor γ = 1
3
(ei Q1·R
+ ei Q2·R
+ ei Q3·R
) and the ma-
trices
I =
⎛
⎝
1 1 1
1 1 1
1 1 1
⎞
⎠, MA =
⎛
⎝
1 c∗
c
c 1 c∗
c∗
c 1
⎞
⎠,
MB =
⎛
⎝
c 1 c∗
1 c∗
c
c∗
c 1
⎞
⎠, (11)
where c = e2πi/3
. The characteristic vectors Q1,2 =
(−2π/3
√
3, ± 2π/3) and Q3 = (4π/3
√
3,0) shown in
Fig. 2(g) again touch the boundary of the Brillouin zone,
now in its corners. The dual transformation takes a general
form SAR = Q ei Q·R
RA Q SAR (here for sublattice A) which
translates into
SAq =
Q
RA Q SA,q− Q; (12)
i.e., the Fourier components get shifted by the characteristic
vectors. As a side result, the above relation gives the Bragg
spots derived from the Bragg spots of Heisenberg FM/AF
(SA,q=0 = ±SB,q=0 = 1) and presented in Fig. 2(h).
To study the spin excitations, we employ the spin susceptibility
tensor defined as
χαβ(q,ω) = i
∞
0
Sα
q (t),S
β
−q(0) ei(ω+iδ)t
dt. (13)
It is evaluated at the SU(2) points by first decomposing Sq into
the A- and B-sublattice contributions via
Sq =
1
√
2
ei
√
3qx /2
(eiqy /2
SAq + e−iqy /2
SBq), (14)
applying the dual transformation in the Fourier form of Eq. (12)
to get back to the underlying Heisenberg model, and using the
spin susceptibility for the Heisenberg model obtained within
linear-spin-wave (LSW) approximation. In the case of T4, this
brings simple q shifts by Qa, Qb, and Qc for the individual
components. For T6, the corresponding expressions are
somewhat more involved containing a nonshifted contribution
and shifted contributions combining pairs of the characteristic
vectors Q1, Q2, and Q3. Without going into details, the
presence of both shifted and nonshifted parts can be easily
understood based on Eq. (10).
Presented in Fig. 4 are the traces of the spin susceptibility
tensor of the Heisenberg model and the extended KH model
at the two hidden SU(2) points under consideration. For
completeness, we demonstrate both hidden FM and AF cases
characterized by quadratic and linearly dispersing Goldstone
modes, respectively. The situation is more transparent for
the four-sublattice patterns—stripy (hidden FM) and zigzag
(hidden AF)—since the spin wave dispersions are just shifted
with the q = M points replacing the Goldstone points and
of the Heisenberg case. In our example, we have chosen
the z axis as the ordered moment direction. For the magnons,
which are in fact deviations of the ordered moment in x and y
directions, only Qa and Qb shifts are active, selecting four out
of the six M points in total. The remaining two are the Bragg
spots reached from and by Qc shifts active for the ordered
z spin component. The Bragg spots and the Goldstone points
are thus complementary in this case. The spin excitations
associated with the six-sublattice patterns are significantly
more complicated. They contain both shifted Goldstone modes
[in Fig. 4(c) such a mode appears at q = K point coinciding
with Q3] and Goldstone modes at the characteristic momenta
q = and q = of the underlying Heisenberg model. In the
latter case just the intensity of the modes has been transferred
by the dual transformation, making, e.g., the linear Goldstone
mode at q = the most intense one in the hidden AF case.
A similar analysis of the spin excitations as presented here
for T4 and T6 SU(2) points can be performed for the remaining
SU(2) points. Due to the nature of the relevant transformations,
024413-7
115
JI ˇR´I CHALOUPKA AND GINIYAT KHALIULLIN PHYSICAL REVIEW B 92, 024413 (2015)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Γ″ Y Γ M Γ′ X K Γ
ωq/|J0|
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Γ″ Y Γ M Γ′ X K Γ
ωq/|J0|
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Γ″ Y Γ M Γ′ X K Γ
ωq/|J0|
(a)
(b)
(c)
FM
AF
XΓ
Γ″
Γ′
FIG. 4. (Color online) (a) LSW dispersion of the Heisenberg FM
(blue) and AF (red) on the honeycomb lattice. The width of the lines
indicates the trace of the spin susceptibility tensor, α χαα(q,ω),
calculated in the LSW approximation. (b) The same for the stripy and
zigzag state presented in Figs. 2(c) and 2(d). Energy is scaled by J0
of the hidden Heisenberg magnet. (c) The same for the “vortex”-like
patterns presented in Figs. 2(e) and 2(f).
no other characteristic vectors appear. Therefore, q = , q =
and their counterparts shifted by the vectors Qa,b,c and
Q1,2,3 entering the transformations T4 and T6 constitute the
entire set of the wave vectors of the Goldstone modes that can
be observed within the extended KH model.
V. APPLICATION TO THE REAL MATERIALS
The aim of this work was to study the basic symmetry
properties of the extended KH model—a promising spin
Hamiltonian for the magnetism of honeycomb iridates. Below
we illustrate how this knowledge, taken together with the
experimental data, helps to locate the plausible windows in
otherwise very large parameter space even for this nearestneighbor
(NN) model. We will show that, despite having only
a single result, the search for full self-dual transformations
JKDC ↔ J K D C of the extended KH model provides
us with a surprisingly useful tool in the context of Na2IrO3.
This utility of T1 emerges due to the recent observation of
the magnetic moment direction [46] which, as we see shortly,
imposes an important constraint on the model parameters. This
is because, in general, the data on magnetic easy axes in a
crystal, along with the magnon gaps and torque magnetometry
data, provide direct information on the symmetry and strength
of the anisotropy terms in spin Hamiltonians, and the case of
Na2IrO3 is of course not at all special in this sense.
To begin with, we recall that Na2IrO3 shows so-called
zigzag order, where the spins on a and b bonds are parallel and
form ferromagnetic chains that run along the X direction and
couple antiferromagnetically along the Y axis. This relatively
simple collinear magnetic structure has been first explained
[28,30] as due to 2nd-NN J2 and 3rd-NN J3 Heisenberg
couplings (which are often relevant in compounds with
90◦
-bonding geometry; a well-known example is quasi-onedimensional
cuprates). This model emphasizes a geometrical
frustration which is realized at large values of J2,3 and resolved
by the C3 symmetry breaking zigzag formation.
However, as argued in the Introduction, more recent data
[45,46] suggest that the origin of frustrations is largely
related to the non-Heisenberg-type interactions which are
bond-dependent and hence highly frustrated even on the level
of NN models. A minimal NN model of this sort is the KH
model, which has been shown [32] to host zigzag order in its
phase diagram indeed. We follow this way of reasoning and
explore below the extended version of the KH model as the
basic NN model for iridates. On the way, we will also see the
point where the data may require the presence of additional
terms J2,3 too, suggesting that the both “zigzag theories” above
are the part of a full story.
In Ref. [32], the available experimental data on Na2IrO3
have been fitted using the two-parameter KH model, regarding
it as a phenomenological spin Hamiltonian with arbitrary
parameters. For K = 21 meV and J = −4 meV, the model
was found consistent with experiments in terms of the type
of magnetic ordering, the temperature dependence of static
magnetic susceptibility, and the low-energy spin-excitation
spectrum being compared to powder INS. Later, RIXS experiments
[45,46] confirmed the presence of a high-energy branch
of spin excitations, with an even better agreement obtained if
the LSW calculation of Ref. [32] is replaced by a more suitable
exact diagonalization [65].
However, the recent data [46] on the moment direction
came about as an unexpected surprise, challenging at first
glance the above coherent description of Na2IrO3. The point is
that within the original two-parameter KH model, the zigzag
order is characterized by the spins pointing towards one of the
oxygen ions [see Fig. 1(a)]. This expectation is generic and
guaranteed by the “order-from-disorder” physics [66] which
typically selects one of the high-symmetry cubic axes as the
easy one, when a spin Hamiltonian contains the compasstype
or Kitaev-type bond-dependent anisotropy [11,16,40,67],
independently of parameter values. The resonant magnetic xray
diffraction data [46] show instead that the magnetic easy
axis is in fact far away from any of the Ir-O bond directions: it
is oriented “nowhere” slightly below a midpoint between the
two, x and y, oxygen ions in Figs. 1(a) and 5(a). This is a clear
indication of the significance of the D and C terms in the spin
Hamiltonian [68].
To reconcile all the data at hand using now the full
four-parameter model, we first notice that the above two easy
axis directions—the one observed in Na2IrO3 and the one
expected from the KH model as used in Ref. [32]—are roughly
024413-8
116
HIDDEN SYMMETRIES OF THE EXTENDED KITAEV- . . . PHYSICAL REVIEW B 92, 024413 (2015)
30°
35°
40°
45°
50°
55°
0.0 0.2 0.4 0.6 0.8 1.0
α
r = D / (K+C/ 2) = −Γ / (K+Γ′)
K~D
|K|>>|D|
α0⎯
α0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.2 0.4 0.6 0.8 1.0
J2=J3[meV]
r = D / (K+C/ 2) = −Γ / (K+Γ′)
(a)
(b)
zigzag
incommensurate
stripy
J2
J3
0α
Z
XY
α0
Sz x,y
α
FIG. 5. (a) Pseudospin angle α relative to the XY plane (see inset)
as a function of the parameter r = D/(K + C/
√
2) = − /(K + ).
Dashed lines show the “magic” angle α0 35◦
and its complement
¯α0 55◦
, determined by the z axis and xy plane, respectively,
as sketched in the inset. (b) The phase diagram as a function of
long-range couplings J2 = J3 and anisotropy parameter r. Starting
with the “bare,” T1-derived values of J = 5.3 meV, K = −7.0 meV,
D = −9.3 meV, and C = −6.6 meV, we have scaled D and C
simultaneously to vary r. To stay within the zigzag phase at the smaller
values of r 0.59, one needs to have finite J2,3 couplings. Otherwise,
the NN-only extended KH model with negative K < 0 switches to
the incommensurate and “stripy” [11,32,69] ground states. The inset
shows the exchange bonds J2 and J3.
related to each other simply by a π rotation about the Z axis.
This observation gives an immediate hint of how to obtain a
starting parameter point when fitting the current data set for
Na2IrO3 within the extended, four-parameter KH model in an
appealingly easy way, and resolve the above apparent problem
with the moment direction.
As discussed in Sec. IV B, the T4-associated zigzag phase
of the KH model with K > 0 is related to the zigzag phase
connected to the SU(2) point B (T1T4) of Fig. 3(a) via T1.
Remarkably, due to the nature of T1—a global π rotation of the
magnetic moments about the Z axis—all the aforementioned
consistent results [32] of the two-parameter KH model are
fully preserved if we apply Eq. (3) to the parameters K
and J of Ref. [32] given above; the only change is the
spin easy axis being rotated to the proper direction as in
experiment. The corresponding set of parameters obtained via
(3) is J = 5.3 meV, K = −7.0 meV, D ≡ − = −9.3 meV,
C ≡
√
2 = −6.6 meV. We would like to emphasize that
these numbers should not be taken literally; rather, they fix the
signs of the parameters involved and put an upper limit for D
and C, as we explain below.
For the representative parameters given above, the pseudospin
makes a “magic” angle of α0 35◦
from the XY plane,
as follows from T1 construction. This is slightly lower than
observed [46,70]. Now, we inspect how the angle α varies as a
function of the anisotropy parameters. The result is illustrated
in Fig. 5(a) and shows that the exact value of α heavily
influences the “departure” from the KH model quantified
by the parameter r = D/(K + C/
√
2) = − /(K + ). [The
corresponding Eqs. (B13), (B14), and (B16) for the spin
direction are derived in the Appendix B by minimizing the
classical energy]. The “magic” angle α = α0 appears at r =
0.8 – as obtained for the above parameters. Yet, as observed
in Fig. 5(a), by increasing α for example by 10◦
only, we
already find r 0.3 and get closer to the |K| |D| regime.
Therefore, more detailed measurements and fits of the ordered
spin direction are highly desirable to get the actual values of
the parameters D and C relative to the Kitaev term K. Doing
so, it is crucial to take into account the fact that the pseudospin
direction and magnetic moment direction are not the same
in general; while they coincide in the cubic limit, a sizable
trigonal-field splitting might be present in Na2IrO3 [45,59].
It is thus important to quantify this splitting by independent
measurements.
At this point, longer-range couplings J2,3 become a part
of the full spin model for iridates, for the following reason.
As a T1 partner of the zigzag phase of Ref. [32], the present
NN-model with large D is well in its zigzag ordered state.
But this is not so at smaller values of D (e.g., for r ∼ 0.5),
which are required to get the spin angles α ∼ 40◦
or above;
see Fig. 5(a). Incorporating moderate J2 and J3 couplings into
the model, we can however stabilize the zigzag phase, see
Fig. 5(b), and hence obtain the ordered spin angles above the
“magic” one. The values of J2,3 of the order of 1–2 meV
are indeed suggested by ab initio calculations [39]. This
shows again the key importance of the experimental data on
moment directions for quantifying the balance between the
two zigzag-supporting mechanisms discussed above: based on
J2,3 geometrical frustration, and on frustration driven by the
non-Heisenberg nature of interactions in spin-orbit coupled
magnets. Recent observations [46] of a pronounced spin-space
anisotropy on one hand, and an “intermediate” spin direction
that requires finite J2,3 values on the other hand, suggest that
both mechanisms are at play in Na2IrO3.
Altogether, the present analysis using the symmetry properties
of the model, taking into account the recent data on
moment direction [46], as well as considering the role of the
J2,3 couplings, suggests a plausible window in the parameter
space of an effective spin model for Na2IrO3: J2,3 < J ∼
|C| < |D| < |K|, with positive (AF) Heisenberg couplings
J2,3 and J. The leading anisotropy terms K < 0 and D < 0
are both negative, while a smaller term C may in principle take
any sign. This parameter window is globally consistent with
experimental observations on Na2IrO3 we are aware of to date,
and may be used as a guide in future analysis, in particular
once q-resolved spin response becomes available, and the
ordered pseudospin and magnetic moment directions (they
differ in general) are obtained and confirmed by independent
measurements.
Even though this general result still leaves quite a freedom,
it is of great help by fixing the signs of most relevant couplings
and their hierarchy. This is the main outcome of the present
theory in the context of real materials. Further, we note that
the Kitaev coupling K can be deduced from overall magnetic
024413-9
117
JI ˇR´I CHALOUPKA AND GINIYAT KHALIULLIN PHYSICAL REVIEW B 92, 024413 (2015)
energy scale, and the spin and magnetic moment directions
should determine the parameter r hence D. From a careful
analysis of the zigzag stability condition, magnon gaps and
dispersions, paramagnetic susceptibility data, etc., one should
be able to quantify all the model parameters including C, J,
and J2,3.
Considering this result in the context of microscopic theories,
we notice first that the signs of J > 0 and K < 0 above are
consistent with the original calculations of these parameters
for honeycomb iridates [9,11] as well as with the later studies
[36–40]. Next, we may conclude that a contribution from t2g-eg
hopping that favors pseudospin interaction with K > 0 [6] is
not significant in iridates; this is also consistent with the recent
calculations [36,73]. Further, the present symmetry analysis
resolves an apparent conflict with the theoretical K < 0 [9,11]
and the positive K > 0 that follows from the best data fit using
the KH model [32]: in fact, the pure KH model with K > 0
and the extended one with K < 0 and sizable D,C terms are
T1-dual partners (the latter one being physical).
More surprisingly, a relatively large D (≡ − ) anisotropy
term is required to “turn” the moment direction well away
from the pure KH model position. A positive implication
of this observation is that this term makes it much easier to
stabilize the zigzag order (the pure KH model with large K < 0
would require large long-range J2,3 couplings otherwise). In a
view of the discussion in Sec. II, this suggests a presence
of sizable trigonal field effects in Na2IrO3. Eventually, an
unusual—out of any crystal symmetry axis—orientation of
pseudospins [46] should originate from a competition among
theseveral anisotropyterms K, D, andC of different symmetry
and physical origin.
To conclude our discussion of Na2IrO3: it seems that the
extended KH model, likely further “extended” by moderate
longer-range couplings, is indeed a good candidate model
for this compound. Even though these extensions (to be still
quantified by future experiments) reduce the chances for
“pure” Kitaev-model physics in iridates, the model itself is
highly interesting due to its rich internal structure and hidden
symmetries that we have uncovered in this work.
Motivated by the above, we further consider the case of
Li2IrO3. Since the data are limited here, the discussion will
be brief and suggestive only. Due to the smaller Curie-Weiss
temperature and more “ferromagnetic” behavior of its spin
susceptibility [74,75], this compound was located closer
to the SU(2) point of the KH model [32]. Even though
the parameters K = 15.8 meV and J = −5.3 meV given in
Ref. [32] correspond to the zigzag phase while Li2IrO3 shows
a spiral magnetic ordering [29], these parameters can be used to
get a hint of the direction in the parameter space to consider. We
therefore transform the above parameters using (3) to obtain
J = 1.7 meV, K = −5.3 meV, D ≡ − = −7.0 meV, C ≡√
2 = −5.0 meV. Representing the parameters for Na and
Li compounds obtained via the T1 transformation (3) in Fig. 3,
we see that both are close to the SU(2) point B (T1T4), with Li
being closer, as expected. To approach the spiral state observed
in Li2IrO3, we first note that, in first approximation, K ∼ D <
0 in both cases and that Li compound is characterized by a
much smaller J. For simplicity, we set J = 0 meV, assume
K = D = −10 meV to roughly preserve the overall energy
scale, and reduce the parameter C associated with the trigonal
0.0
0.5
1.0
1.5
2.0
-6 -4 -2 0 2 4
Q/(2π/3)
C [meV]
(a) (b)
(c)
(d)
zigzag spiral
spiral AF
a
b
-3.0
-2.5
-2.0
-1.5
-3.2
-3.0
-2.8
-2.6
0
5
10
15
20
Γ’’ Y Γ M Γ’ X K Γ
ωq[meV]
0
5
10
15
20
Γ’’ Y Γ M Γ’ X K Γ
FIG. 6. (a) Map of the q-dependent classical energy (per site, in
units of meV) obtained by Luttinger-Tisza method [71,72] for the
“bare,” i.e., T1-derived parameters J = 5.3 meV, K = −7.0 meV,
D = −9.3 meV, C = −6.6 meV relevant for Na2IrO3. The hexagon
indicates the first Brillouin zone. (b) The same for the parameters
K = D = −10 meV, J = C = 0 relevant to Li2IrO3. (c) Length
of the ordering vector for varying C, keeping the other parameter
values unchanged. The dashed (solid) line was calculated using the
above parameters JKD relevant to Na2IrO3 (Li2IrO3). Points a and
b show the C values used in panels (a) and (b), respectively. (d) LSW
dispersions for the parameters used in panel (a) (left) and panel (b)
(right). In the latter case, we have taken C −1.2 meV instead of
C = 0 meV to stay in the zigzag phase at the border to the spiral
phase.
distortion, which is expected to be much smaller in Li2IrO3
with the bond angles being closer to 90◦
. The Luttinger-Tisza
[71,72] maps of the classical energy for the Na and Li case
presented in Figs. 6(a) and 6(b) confirm the zigzag and
incommensurate magnetic ordering, respectively. For C = 0
the incommensurate ordering wave vector is obtained as Q
2
3
Qc [see Figs. 6(b) and 6(c)]; this would predict a magnetic
Bragg peak in powder neutron diffraction experiments at
a | Q| value that could be consistent with experiments on
powder Li2IrO3 [29]. Finally, Fig. 6(d) compares the spin
excitations obtained using the LSW approximation. In the case
of Na2IrO3, the dispersion is identical to that presented in Fig. 3
of Ref. [32], possessing low- and high-energy branches. As the
parameter J is reduced, these two branches gradually merge,
leading to a steeper dispersion compared to Na2IrO3, which
might be consistent with powder inelastic neutron scattering
experiments on powder Li2IrO3 [29]. The predicted dispersion
024413-10
118
HIDDEN SYMMETRIES OF THE EXTENDED KITAEV- . . . PHYSICAL REVIEW B 92, 024413 (2015)
is illustrated in Fig. 6(d) for a point on the boundary between
the zigzag and the spiral phase. A further minor reduction
of C to enter the spiral phase and get the proper ordering
vector should not affect this result dramatically, apart from
the changes at low energies forming an “hour-glass” shape
characteristic of spiral magnets (see, e.g., Ref. [76]).
VI. CONCLUSIONS
To summarize, we have analyzed nontrivial symmetries
of the extended Kitaev-Heisenberg model on the honeycomb
lattice. As a main result, we have identified the complete set
of points in the parameter space where this bond-anisotropic
model can be transformed to a simple Heisenberg model and
is therefore characterized by hidden SU(2) symmetry. Such a
dual transformation can be performed using a particular choice
of sublattice rotations of the spins, specific for each of the
SU(2) points. The sublattice structure of the transformations
creates a number of ordering patterns which together with the
location of the hidden SU(2) points in the parameter space
give a good overview of the global phase diagram of the
model. In terms of the spin excitations, the hidden SU(2)
symmetry manifests itself by the presence of Goldstone modes
inherited from the SU(2) symmetric Heisenberg FM/AF on the
honeycomb lattice. Their characteristic vectors and even the
full spin excitation spectra are easily obtained by an explicit
transformation of the FM/AF case.
One of the special transformations linked to the hidden
SU(2) points reveals at the same time an exact duality between
the Kitaev and Kekul´e-Kitaev models; this result should be
useful in theoretical studies of these and related models.
We emphasize that, adopting the extended KH model, all
the above results are necessary consequences of its symmetry
which is in turn dictated by the underlying C3 symmetry of the
lattice.
Having the results of the general symmetry analysis at
hand, we were able to find the region of the parameter
space that is consistent with the observed properties of the
honeycomb lattice iridates Na2IrO3 and Li2IrO3. Further, a
relation between the ordered moment direction and the model
parameters is derived, which may help to quantify these
parameters from future experiments.
Finally, our method to systematically explore the hidden
symmetries is general and can be applied to other bondanisotropic
models as well. In the context of the iridate
materials, the symmetry analysis of the extended KH model
on hyperhoneycomb and harmonic-honeycomb lattices is of a
great interest.
ACKNOWLEDGMENTS
We would like to thank B. J. Kim for sharing with us
the experimental data [46] which motivated this study, and
R. Coldea, G. Jackeli, I. Kimchi, N. B. Perkins, S. Trebst,
and S. E. Sebastian for helpful discussions and comments.
J.C. acknowledges support by ERDF under project CEITEC
(CZ.1.05/1.1.00/02.0068), EC 7th Framework Programme
(286154/SYLICA), and Czech Science Foundation (GA ˇCR)
under Project No. 15-14523Y.
APPENDIX A: XYZ FORM OF THE HAMILTONIAN
The Hamiltonian expressed in terms of the spin components
SX
, SY
, and SZ
, corresponding to the XYZ reference frame in
Fig. 1(a), takes the form
H ij γ = JXY SX
i SX
j + SY
i SY
j + JZSZ
i SZ
j
+ A cγ SX
i SX
j −SY
i SY
j − sγ SX
i SY
j +SY
i SX
j
− B
√
2 cγ SX
i SZ
j +SZ
i SX
j + sγ SY
i SZ
j +SZ
i SY
j .
(A1)
Here the C3 symmetry of the model is embodied in the
factors cγ ≡ cos φγ and sγ ≡ sin φγ , where the angles φγ are
determined by the bond directions: φγ = 0,2π
3
,4π
3
for the c, a,
and b bonds, respectively. In terms of the original parameters
JK , the exchange constants entering (A1) read as
A = 1
3
K + 2
3
( − ), (A2)
B = 1
3
K − 1
3
( − ), (A3)
JXY = J + B − , (A4)
JZ = J + A + 2 . (A5)
Note that it is the A and B terms which bring about the
bond directionality of the interactions, and hence they naturally
support C3 symmetry breaking orderings such as zigzag in the
present model. Physically, these terms arise from the exchange
processes that involve the in-plane components of orbital
momentum LX and LY which “know” the bond directions,
like the orbitals do in the Kugel-Khomskii models.
It is also noticed that the A and B terms change the
Z component of total angular momentum by ±2 and ±1,
correspondingly. This is because the t2g-orbital angular
momentum L is not a conserved quantity in a crystal, and this
commonly shows up in effective spin Hamiltonians due to the
spin-orbit coupling.
A strong trigonal field splits the t2g level such that the lowest
Kramers doublet (pseudospin) wave functions |˜↑ ,|˜↓ become
simple products of LZ = ±1 and spin |↓ , |↑ states, correspondingly;
i.e., there will be a one-to-one correspondence
between the real spin and pseudospin directions. Since the
total spin is conserved during the hoppings, pseudospin is then
conserved, too. Thus, the spin-nonconserving terms A and B
must vanish in this limit, which implies K → 0 and →
simultaneously. Physically, a strong compression along the
trigonal axis dictates that this axis becomes the “easy” (or
“hard”) one for moments. Since this limit is not realized in
iridates, we will not use the XYZ form of the Hamiltonian
in this paper; however, it might be useful for pseudospin-1/2
Co4+
, Rh4+
, and Ru3+
compounds where the spin-orbit and
crystal field effects may strongly compete.
APPENDIX B: ANALYSIS OF THE CLASSICAL ENERGY
AND MOMENT DIRECTION
In this Appendix we show the expressions used in the
classical energy analysis. We first give the Hamiltonian in
its momentum space form utilized within the Luttinger-Tisza
024413-11
119
JI ˇR´I CHALOUPKA AND GINIYAT KHALIULLIN PHYSICAL REVIEW B 92, 024413 (2015)
method. By minimizing the classical energy in the zigzag phase
we then find the ordered moment direction.
Transforming the spin operators via SAR = q eiq·R
SAq
and similarly for the B sublattice, we cast the Hamiltonian
into the form
H =
q
†
q Hq q with q =
SAq
SBq
, (B1)
where the q vectors cover the first Brillouin zone of the
triangular lattice of R. The simplest expressions for the 6 × 6
matrices Hq of the momentum-space Hamiltonian (B1) are
obtained using the cubic axes x, y, z. Complementing the
interactions in Eq. (2) by long-range J2 and J3, we arrive at
Hq = Nsite
Fq Gq
G
†
q Fq
with Fq = 1
2
J2
⎛
⎝
1 0 0
0 1 0
0 0 1
⎞
⎠ η2q
(B2)
and
Gq = 1
4
(J1η1q + J3η3q)
⎛
⎝
1 0 0
0 1 0
0 0 1
⎞
⎠
+1
4
K
⎛
⎝
e1 0 0
0 e2 0
0 0 1
⎞
⎠ + 1
4
⎛
⎝
0 1 e2
1 0 e1
e2 e1 0
⎞
⎠
+1
4
⎛
⎝
0 e1 + e2 e1 + 1
e1 + e2 0 e2 + 1
e1 + 1 e2 + 1 0
⎞
⎠. (B3)
Here the momentum-dependent factors read as
e1,2 = e−i 1
2 (±
√
3qx +3qy )
, (B4)
η1q = 1 + 2 cos
√
3qx
2
e−i 3
2 qy
, (B5)
η2q = cos
√
3qx + 2 cos
√
3qx
2
cos
3qy
2
, (B6)
η3q = e−i3qy
+ 2 cos
√
3qx. (B7)
In the Luttinger-Tisza method [71,72], the matrices
Hq/2Nsite for q running through the Brillouin zone are
diagonalized. The q vector and the eigenvector corresponding
to the minimum eigenvalue then determine the ordering
resulting on a classical level and the minimum eigenvalue
itself gives the classical energy per site. This approach relaxes
the spin-length constraint which should be checked afterward.
Next, we evaluate the classical energy for the zigzag
state with the ordering vector Q = Qc = (0,2π/3) and an
arbitrary ordered moment direction given by a unit vector
u. The corresponding zigzag pattern is captured by Q =
(+1
2
u, − 1
2
u)T
. Using (B1), we get for the classical energy
per site
Eclass = 1
8
(J1 − K − 2J2 − 3J3) + 1
8
uT
Mu (B8)
with the matrix
M =
⎛
⎝
2K − + 2
− + 2 2K
0
⎞
⎠ (B9)
or equivalently
M =
⎛
⎝
2K D +
√
2C −D
D +
√
2C 2K −D
−D −D 0
⎞
⎠. (B10)
The ordered moment direction can now be obtained as
the eigenvector of M corresponding to its lowest eigenvalue.
However, as will be clear in a moment, it is more convenient
to switch to the reference frame which coincides with the local
˜x, ˜y, ˜z axes for c bonds [see Fig. 1(b)]. The matrix M is then
transformed to
˜M =
⎛
⎝
2K−D−
√
2C 0 0
0 2K+D+
√
2C −
√
2D
0 −
√
2D 0
⎞
⎠,
(B11)
which can be readily diagonalized and the angle α of the
ordered pseudospin to the XY plane can be found.
As discussed in the main text, if we rotate the spins by 180◦
around the global Z axis, the observed moment would come
close to the ˜z axis. Since the latter is an attractive point for
the two-parameter KH model [11], we guess that this rotation
will transform the actual J,K,D,C Hamiltonian (K < 0, large
D) for Na2IrO3 into an effective J ,K ,D ,C one, with K >
0 and small only D and C values, i.e., into a nearly twoparameter
KH model (which guarantees that the corresponding
effective easy axis is close to ˜z). We therefore first apply the
T1 transformation via Eq. (3), calculate the moment direction
for effective J ,K ,D ,C , and later make use of the expected
smallness of the transformed D . The first two steps yield an
analytical expression for the angle α:
α = α0 +
1
2
arctan
2
√
2D
2K + D +
√
2C
(B12)
with the first contribution being the “magic” angle α0 =
arcsin(1/
√
3) 35.3◦
of the ˜z axis to the XY plane and the
second contribution supposed to be small. Now, we return to
the original spin axes by applying the T1 transformation again.
This does not alter the angle α but rotates the moment into its
physical position: below a midpoint between two oxygen ions
[46]. In terms of the original parameters we have
α = α0 +
1
2
arctan 2
√
2
4K − 5D + 2
√
2C
14K + 23D + 7
√
2C
. (B13)
For D = C = 0, this equation gives the moment direction
towards a midpoint of two oxygens, as expected for negative
K values of the Kitaev coupling [46] on a classical level (but
it will turn to either the x or y oxygen direction once the
order-by-disorder mechanism is switched on [11,40]). The
moment moves down from this position once the model is
extended by D and C terms of a proper sign. At the parameter
set given in the main text, the moment takes the “magic” angle.
By expanding the arctangent near this point, we arrive at the
following formula for the deviation from α0:
δα =
1
2
arctan
4
√
2
7
1 − 5
4
r
1 + 23
14
r
≈
2
√
2
7
1 − 5
4
r
1 + 23
14
r
(B14)
024413-12
120
HIDDEN SYMMETRIES OF THE EXTENDED KITAEV- . . . PHYSICAL REVIEW B 92, 024413 (2015)
with the single parameter
r =
D
K + 1√
2
C
= −
K +
. (B15)
This parameter quantifies the “departure” from the KH model
and can be measured by resonant x-ray [46] or neutron
diffraction experiments; as mentioned in the main text, care
has to be taken in the fits by considering the crystal field effects
on pseudospin wave functions.
In terms of the parameter r, Eq. (B13) can be rewritten as
tan 2α = 4
√
2
1 + r
7r − 2
. (B16)
Note that this and the previous equations for α hold at finite
J2,3 Heisenberg corrections as well, since the easy axis is
determined solely by the anisotropy terms.
[1] W. J. L. Buyers, T. M. Holden, E. C. Svensson, R. A. Cowley,
and M. T. Hutchings, J. Phys. C: Solid State Phys. 4, 2139
(1971).
[2] R. J. Elliott and M. F. Thorpe, J. Appl. Phys. 39, 802 (1968).
[3] A. Abragam and B. Bleaney, Electron Paramagnetic Resonance
of Transition Ions (Clarendon Press, Oxford, 1970).
[4] K. I. Kugel and D. I. Khomskii, Sov. Phys. Usp. 25, 231 (1982).
[5] G. Khaliullin and S. Okamoto, Phys. Rev. Lett. 89, 167201
(2002); ,Phys. Rev. B 68, 205109 (2003).
[6] G. Khaliullin, Prog. Theor. Phys. Suppl. 160, 155 (2005).
[7] G. Khaliullin, W. Koshibae, and S. Maekawa, Phys. Rev. Lett.
93, 176401 (2004).
[8] G. Chen and L. Balents, Phys. Rev. B 78, 094403 (2008).
[9] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205
(2009).
[10] A. Shitade, H. Katsura, J. Kuneˇs, X.-L. Qi, S.-C. Zhang, and
N. Nagaosa, Phys. Rev. Lett. 102, 256403 (2009).
[11] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett.
105, 027204 (2010).
[12] S. Okamoto, Phys. Rev. Lett. 110, 066403 (2013).
[13] W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, Annu.
Rev. Condens. Matter Phys. 5, 57 (2014).
[14] Z. Nussinov and J. van den Brink, Rev. Mod. Phys. 87, 1 (2015).
[15] G. Khaliullin and S. Maekawa, Phys. Rev. Lett. 85, 3950
(2000).
[16] G. Khaliullin, Phys. Rev. B 64, 212405 (2001).
[17] G. Chen, L. Balents, and A. P. Schnyder, Phys. Rev. Lett. 102,
096406 (2009).
[18] G. Khaliullin, Phys. Rev. Lett. 111, 197201 (2013).
[19] O. N. Meetei, W. S. Cole, M. Randeria, and N. Trivedi, Phys.
Rev. B 91, 054412 (2015).
[20] More specifically, some ions with an even number of electrons
(e.g., d4
-Ru in Ca2RuO4 [18] or d6
-Fe in FeSc2S4 [17]) adopt
a singlet J = 0 ground state. When the exchange interactions
become comparable to the spin-orbit induced singlet-triplet
splitting, magnetic order emerges as a condensation of the
excited levels with J = 1. At the transition, the spin-length
fluctuations diverge; the ordered state is characterized by “soft”
moments and the associated amplitude (Higgs) mode, in addition
to the usual transverse magnons [18]. In essence, this is similar to
the spin-state crossover physics and “soft” magnetism realized
in some compounds of Co3+
and Fe2+
ions, where the ground
state has S = 0 (due to large covalency and crystal field effects)
while the S = 0 levels are not too high; for details, see the recent
work [21] and references therein.
[21] J. Chaloupka and G. Khaliullin, Phys. Rev. Lett. 110, 207205
(2013); ,Prog. Theor. Phys. Suppl. 176, 50 (2008).
[22] B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita, H. Takagi,
and T. Arima, Science 323, 1329 (2009).
[23] J. Kim, D. Casa, M. H. Upton, T. Gog, Y.-J. Kim, J. F.
Mitchell, M. van Veenendaal, M. Daghofer, J. van den Brink, G.
Khaliullin, and B. J. Kim, Phys. Rev. Lett. 108, 177003 (2012).
[24] S. Fujiyama, H. Ohsumi, T. Komesu, J. Matsuno, B. J. Kim, M.
Takata, T. Arima, and H. Takagi, Phys. Rev. Lett. 108, 247212
(2012).
[25] A. Kitaev, Ann. Phys. 321, 2 (2006).
[26] X. Liu, T. Berlijn, W.-G. Yin, W. Ku, A. Tsvelik, Y.-J. Kim, H.
Gretarsson, Y. Singh, P. Gegenwart, and J. P. Hill, Phys. Rev. B
83, 220403(R) (2011).
[27] F. Ye, S. Chi, H. Cao, B. C. Chakoumakos, J. A. Fernandez-Baca,
R. Custelcean, T. F. Qi, O. B. Korneta, and G. Cao, Phys. Rev.
B 85, 180403(R) (2012).
[28] S. K. Choi, R. Coldea, A. N. Kolmogorov, T. Lancaster, I. I.
Mazin, S. J. Blundell, P. G. Radaelli, Y. Singh, P. Gegenwart,
K. R. Choi, S.-W. Cheong, P. J. Baker, C. Stock, and J. Taylor,
Phys. Rev. Lett. 108, 127204 (2012).
[29] R. Coldea, presentation at SPORE13, Dresden, 2013 (unpublished);
S. Choi, R. Coldea et al. (unpublished).
[30] I. Kimchi and Y.-Z. You, Phys. Rev. B 84, 180407(R) (2011).
[31] J. Reuther, R. Thomale, and S. Rachel, Phys. Rev. B 90,
100405(R) (2014).
[32] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett.
110, 097204 (2013).
[33] Y. Yu, L. Liang, Q. Niu, and S. Qin, Phys. Rev. B 87, 041107(R)
(2013).
[34] S. Okamoto, Phys. Rev. B 87, 064508 (2013).
[35] S. Bhattacharjee, S.-S. Lee, and Y. B. Kim, New J. Phys. 14,
073015 (2012).
[36] V. M. Katukuri, S. Nishimoto, V. Yushankhai, A. Stoyanova, H.
Kandpal, S. Choi, R. Coldea, I. Rousochatzakis, L. Hozoi, and
J. van den Brink, New J. Phys. 16, 013056 (2014).
[37] J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Phys. Rev. Lett. 112,
077204 (2014).
[38] J. G. Rau and H.-Y. Kee, arXiv:1408.4811.
[39] Y. Yamaji, Y. Nomura, M. Kurita, R. Arita, and M. Imada, Phys.
Rev. Lett. 113, 107201 (2014).
[40] Y. Sizyuk, C. Price, P. W¨olfle, and N. B. Perkins, Phys. Rev. B
90, 155126 (2014).
[41] K. Shinjo, S. Sota, and T. Tohyama, Phys. Rev. B 91, 054401
(2015).
[42] I. Kimchi, R. Coldea, and A. Vishwanath, Phys. Rev. B 91,
245134 (2015).
[43] E. Sela, H.-C. Jiang, M. H. Gerlach, and S. Trebst, Phys. Rev. B
90, 035113 (2014).
024413-13
121
JI ˇR´I CHALOUPKA AND GINIYAT KHALIULLIN PHYSICAL REVIEW B 92, 024413 (2015)
[44] I. I. Mazin, H. O. Jeschke, K. Foyevtsova, R. Valent´ı, and D. I.
Khomskii, Phys. Rev. Lett. 109, 197201 (2012).
[45] H. Gretarsson, J. P. Clancy, Y. Singh, P. Gegenwart, J. P. Hill, J.
Kim, M. H. Upton, A. H. Said, D. Casa, T. Gog, and Y.-J. Kim,
Phys. Rev. B 87, 220407(R) (2013).
[46] S. H. Chun, J.-W. Kim, Jungho Kim, H. Zheng, C. Stoumpos,
C. Malliakas, J. F. Mitchell, K. Mehlawat, Y. Singh, Y. Choi, T.
Gog, A. Al-Zein, M. Moretti Sala, M. Krisch, J. Chaloupka, G.
Jackeli, G. Khaliullin, and B. J. Kim, Nat. Phys. 11, 462 (2015).
[47] I. Kimchi and A. Vishwanath, Phys. Rev. B 89, 014414 (2014).
[48] I. Rousochatzakis, U. K. R¨ossler, J. van den Brink, and M.
Daghofer, arXiv:1209.5895.
[49] M. Becker, M. Hermanns, B. Bauer, M. Garst, and S. Trebst,
Phys. Rev. B 91, 155135 (2015).
[50] K. Li, S.-L. Yu, and J.-X. Li, New J. Phys. 17, 043032
(2015).
[51] K. W. Plumb, J. P. Clancy, L. J. Sandilands, V. V. Shankar,
Y. F. Hu, K. S. Burch, H.-Y. Kee, and Y.-J. Kim, Phys. Rev. B
90, 041112(R) (2014).
[52] T. Takayama, A. Kato, R. Dinnebier, J. Nuss, H. Kono, L. S. I.
Veiga, G. Fabbris, D. Haskel, and H. Takagi, Phys. Rev. Lett.
114, 077202 (2015).
[53] A. Biffin, R. D. Johnson, S. Choi, F. Freund, S. Manni, A.
Bombardi, P. Manuel, P. Gegenwart, and R. Coldea, Phys. Rev.
B 90, 205116 (2014).
[54] K. A. Modic, T. E. Smidt, I. Kimchi, N. P. Breznay, A. Biffin,
S. Choi, R. D. Johnson, R. Coldea, P. Watkins-Curry, G. T.
McCandless, J. Y. Chan, F. Gandara, Z. Islam, A. Vishwanath,
A. Shekhter, R. D. McDonald, and J. G. Analytis, Nat. Commun.
5, 4203 (2014).
[55] A. Biffin, R. D. Johnson, I. Kimchi, R. Morris, A. Bombardi,
J. G. Analytis, A. Vishwanath, and R. Coldea, Phys. Rev. Lett.
113, 197201 (2014).
[56] M. Hermanns and S. Trebst, Phys. Rev. B 89, 235102 (2014).
[57] B. N. Figgis and M. A. Hitchman, Ligand Field Theory and Its
Applications (Wiley-VCH, New York, 2000).
[58] H. Gretarsson, J. P. Clancy, X. Liu, J. P. Hill, E. Bozin, Y. Singh,
S. Manni, P. Gegenwart, J. Kim, A. H. Said, D. Casa, T. Gog,
M. H. Upton, H.-S. Kim, J. Yu, V. M. Katukuri, L. Hozoi, J.
van den Brink, and Y.-J. Kim, Phys. Rev. Lett. 110, 076402
(2013).
[59] A relative strength of the trigonal field splitting /λ in Na2IrO3
can be deduced from a splitting BC ∼ 0.1 eV [58] of the excited
J = 3/2 quartet, using a relation BC/λ = 1√
2
tan−1
θ −
1, where the angle θ is given by tan 2θ = 2
√
2/(1 + 2
λ
). This
gives /λ 3/8. With this ratio, we obtain the pseudospin g
factors (see Ref. [3] for details) gc −2.6 and gab 1.7 for
out-of-plane and in-plane directions, correspondingly, in fair
agreement with the observed values |gc| 2.7 and |gab| 1.9
[74].
[60] Unfortunately, the requirement of the Hamiltonian being represented
by nonsingular matrices prevents us from finding
all the hidden Kitaev points by using the transformations
K0 ↔ JKDC. We can only reveal a part of them by using
J0K0 ↔ JKDC and setting J0 = 0.
[61] M. Kamfor, S. Dusuel, J. Vidal, and K. P. Schmidt, J. Stat. Mech:
Theory Exp. (2010) P08010.
[62] E. Quinn, S. Bhattacharjee, and R. Moessner, Phys. Rev. B 91,
134419 (2015).
[63] Recently we came across a preprint [64] which also introduces
the six-sublattice transformation, and claims that it maps the
extended KH model onto itself. We emphasize again that this
transformation is not self-dual in general; is becomes such only
at the special parameter setup for this model (linking J0 ↔ K =
). In most of the phase space, it breaks the bond distribution
pattern dictated by the C3 rotation rules in Fig. 1(c).
[64] J. Lou, L. Liang, Y. Yu, and Y. Chen, arXiv:1501.06990.
[65] J. Chaloupka, G. Jackeli, and G. Khaliullin, presentation at
SPORE13, Dresden, 2013 (unpublished).
[66] For a discussion of the order-from-disorder phenomena in
frustrated spin systems, see A. M. Tsvelik, Quantum Field
Theory in Condensed Matter Physics (Cambridge University
Press, Cambridge, 1995), Chap. 17, and references therein.
[67] Regarding the terminology used here, we have in mind that the
compass-type and Kitaev-type interactions are very distinct in
their physical appearance and origin. While the compass-spins
(pseudodipoles) as introduced in Ref. [4] align themselves
along the bond direction as real compasses do (hence the
name), the spins in the Kitaev honeycomb model tend rather
to avoid the bond directions. Placed on a honeycomb lattice,
the bond-directional compasses would feel much less frustrated
than the Kitaev spins, and show “order-from-disorder” behavior
instead. As to the physical origin, the compass-type and Kitaevtype
spin anisotropy terms arise in compounds with distinct
chemical bonding geometries—corner-shared [16,9] and edgeshared
[6,9] structures, correspondingly.
[68] In general, obtaining the right moment orientation is a crucial
test for the completeness of the model Hamiltonians used.
For instance, the compass-type anisotropy (only allowed for
spin-1/2 in an ideal perovskite) favors the easy spin axis
along a bond direction (say [100]), but in real perovskites
it often takes a “wrong” direction (e.g., [110] in Sr2IrO4),
due to (orthorhombic) distortions and/or other factors not
included in idealized “Heisenberg plus compass” type spin
models. Similarly, an unexpected easy axis direction in Na2IrO3
is a manifestation of the other anisotropy terms beyond the
“Heisenberg plus Kitaev” spin model.
[69] H.-C. Jiang, Z.-C. Gu, X.-L. Qi, and S. Trebst, Phys. Rev. B 83,
245104 (2011).
[70] Ref. [46] has estimated the angle α 44◦
based on the fits that
did not account for a trigonal field splitting. Positive (negative)
values may decrease (increase) the actual value of α from this
estimate.
[71] J. M. Luttinger and L. Tisza, Phys. Rev. 70, 954 (1946).
[72] D. B. Litvin, Physica 77, 205 (1974).
[73] K. Foyevtsova, H. O. Jeschke, I. I. Mazin, D. I. Khomskii, and
R. Valent´ı, Phys. Rev. B 88, 035107 (2013).
[74] Y. Singh and P. Gegenwart, Phys. Rev. B 82, 064412 (2010).
[75] Y. Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale, W. Ku, S.
Trebst, and P. Gegenwart, Phys. Rev. Lett. 108, 127203 (2012).
[76] J.-H. Kim, A. Jain, M. Reehuis, G. Khaliullin, D. C. Peets, C.
Ulrich, J. T. Park, E. Faulhaber, A. Hoser, H. C. Walker, D. T.
Adroja, A. C. Walters, D. S. Inosov, A. Maljuk, and B. Keimer,
Phys. Rev. Lett. 113, 147206 (2014).
024413-14
122
PHYSICAL REVIEW B 94, 064435 (2016)
Magnetic anisotropy in the Kitaev model systems Na2IrO3 and RuCl3
Jiˇr´ı Chaloupka1,2
and Giniyat Khaliullin3
1
Central European Institute of Technology, Masaryk University, Kamenice 753/5, 62500 Brno, Czech Republic
2
Department of Condensed Matter Physics, Faculty of Science, Masaryk University, Kotl´aˇrsk´a 2, 61137 Brno, Czech Republic
3
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
(Received 19 July 2016; revised manuscript received 10 August 2016; published 31 August 2016)
We study the ordered moment direction in the extended Kitaev-Heisenberg model relevant to honeycomb
lattice magnets with strong spin-orbit coupling. We utilize numerical diagonalization and analyze the exact
cluster ground states using a particular set of spin-coherent states, obtaining thereby quantum corrections to the
magnetic anisotropy beyond conventional perturbative methods. It is found that the quantum fluctuations strongly
modify the moment direction obtained at a classical level and are thus crucial for a precise quantification of the
interactions. The results show that the moment direction is a sensitive probe of the model parameters in real
materials. Focusing on the experimentally relevant zigzag phases of the model, we analyze the currently available
neutron-diffraction and resonant x-ray-diffraction data on Na2IrO3 and RuCl3 and discuss the parameter regimes
plausible in these Kitaev-Heisenberg model systems.
DOI: 10.1103/PhysRevB.94.064435
I. INTRODUCTION
Due to their intermediate spatial extension, d electrons
in transition-metal compounds comprise both the localized
and itinerant features. This duality is manifested in a rich
variety of metal-insulator transitions [1,2]. Even deep in the
Mott-insulating phase, the d electrons partially retain their
kinetic energy, by making virtual hoppings to the neighboring
sites and forming the covalent bonds. The internal structure
of these bonds is dictated by the orbital shape of d electrons
as well as by Pauli principle and Hund’s interactions among
spins. This results in an intimate link between the nature of
chemical bonds (“orbital order”) and magnetism [3], which can
be cast in terms of phenomenological Goodenough-Kanamori
rules.
The Kugel-Khomskii models [4] form a theoretical framework
where the “spin physics” and “orbital chemistry” are
treated on equal footing. A special feature of these models is
that the d orbital is spatially anisotropic and hence cannot satisfy
all the bonds simultaneously. In high-symmetry crystals,
this results in a picture of fluctuating orbitals [5,6], where
the frustration among different covalent bonds is resolved
by virtue of their quantum superposition, lifting the orbital
degeneracy without a static order.
It might seem that a relativistic spin-orbit coupling, which
lifts the orbital degeneracy already on a single ion level [3,4],
will readily eliminate the orbital frustration problem. This
coupling does indeed greatly reduce the initially large spinorbital
Hilbert space of d ions, leaving often just a twofold
degenerate Kramers level with an effective (“pseudo”) spin
one-half [7]. It turns out, however, that the pseudospins
still well “remember” the orbital frustration, by inheriting
the bond-directional nature of orbital interactions via the
spin-orbit entanglement [6].
The bond-directional nature of pseudospin interactions
has profound consequences for magnetism (as well as for
the properties of doped systems [8]). The most remarkable
example, pointed out in Ref. [9], is a possible realization of
Kitaev’s honeycomb model [10] in materials with the d5
(t2g)
electronic configuration such as Na2IrO3. This theoretical
proposal has sparked a broad interest in honeycomb lattice
pseudospin systems (see the recent review paper [11] and
references therein).
There is a direct experimental evidence [12] that the
Kitaev-type interactions are indeed dominant in Na2IrO3.
Unusual features pointing towards the Kitaev model have been
observed [13] also in spin excitation spectra of RuCl3 (this
compound was suggested [14] to host pseudospin physics,
too). On the other hand, it is also clear that there are terms in
the pseudospin Hamiltonian that take these systems away from
the Kitaev spin-liquid phase window [15]. The identification
of these “undesired” interactions and clarification of their
dependence on material parameters is an important issue that
has been in the focus of many recent studies.
Experimentally, the strength of a dominant Kitaev coupling
|K| can readily be evaluated from an overall bandwidth of
spin excitations; however, the determination of its sign and
quantification of the subdominant terms is not straightforward
and needs a theory support. The aim of this paper is to show
that the direction of the ordered moments, which can be
extracted from the neutron-diffraction and x-ray-diffraction
data, contains valuable information on the model parameters,
including the sign of K. Considering a symmetry dictated
form of the model Hamiltonian, we calculate the pseudospin
direction fully including quantum fluctuations which are
expected to be crucial in frustrated spin models. We will point
out that the pseudospin itself is not directly probed by neutrons;
rather, they detect the direction of the magnetic moment which
is not the same as that of the pseudospin. Similarly, we will
describe how to extract the pseudospin direction from resonant
x-ray-scattering (RXS) data.
The paper is organized as follows. Section II introduces
the model Hamiltonian. Section III briefly discusses the
pseudospin easy axis direction on a classical level. Section IV
introduces the method of deriving the moment direction from
exact diagonalization (ED) data. Section V presents the ED
results on moment direction as a function of model parameters.
Section VI considers a relation between the pseudospins and
magnetic moments probed by neutron-diffraction and RXS
experiments, and discusses implications of the theory for
2469-9950/2016/94(6)/064435(11) 064435-1 ©2016 American Physical Society
123
JI ˇR´I CHALOUPKA AND GINIYAT KHALIULLIN PHYSICAL REVIEW B 94, 064435 (2016)
Na2IrO3 and RuCl3. Appendix A compares the method of
Sec. IV with the standard approach. Appendix B derives
the equations used in the analysis of RXS data. Finally,
Appendix C discusses how the trigonal field can be extracted
from J = 3/2 magnetic excitation spectra.
II. EXTENDED KITAEV-HEISENBERG MODEL
To describe the interactions among the pseudospins (referred
to as “spins” below), we adopt a model containing
all symmetry allowed nearest-neighbor (NN) terms and the
longer-range Heisenberg interactions:
H =
ij ∈NN
H
(γ )
ij +
ij /∈NN
Jij Si · Sj . (1)
The nearest-neighbor contribution is the extended KitaevHeisenberg
model [16–18] that, apart from the Heisenberg
interaction, includes all the bond-anisotropic interactions
compatible with the symmetries of a trigonally distorted
honeycomb lattice. Its z-bond contribution (see Fig. 1 for the
definitions of the bonds and spin axes) takes the following
form:
H(z)
ij = K Sz
i Sz
j + J Si · Sj + Sx
i S
y
j + S
y
i Sx
j
+ Sx
i Sz
j + Sz
i Sx
j + S
y
i Sz
j + Sz
i S
y
j . (2)
The Hamiltonian contributions for the other bonds (x and y)
are obtained by a cyclic permutation among Sx,Sy,Sz. The
resulting alternation of the local easy axis directions from
bond to bond, imposed by the Ising-like term K, brings about
a strong frustration which, as discussed above, can be traced
back to the orbital frustration problem in Kugel-Khomskii type
models. An extensive discussion of the above Hamiltonian
and its nontrivial symmetry properties can be found in
Ref. [19].
With the Kitaev-coupling K alone, the model has a spinliquid
ground state. Both Na2IrO3 and RuCl3 show spin order
where the zigzag-type ferromagnetic (FM) chains, running
along the a direction, are coupled to each other antiferromagnetically
[see Fig. 1(b)]. This order becomes a ground state of
the Kitaev model with K > 0 [antiferromagnetic (AF) sign],
when a small FM J < 0 Heisenberg coupling is added [20]. If
the Kitaev coupling is negative, K < 0 (FM sign), then zigzag
order emerges due to longer-range AF couplings [21,22] and/or
, terms [17–19]. Given that the stability of the Kitaev-liquid
phase against perturbations strongly depends on the sign of
K [20], which scenario is realized in a given compound
becomes an important issue.
Leaving aside the “orbital chemistry” aspects that decide
the sign of K as well as the other model parameters, we just
mention that various ab initio estimates (see, e.g., [16,23,24])
generally support the FM K < 0 regime, most likely reflecting
the decisive role of Hund’s coupling effect on K emphasized
earlier [9,15]. However, we take here a phenomenological
approach, considering the model with free parameter values
including both signs of K. The J, , and values are varied
such that the ground state stays within the zigzag phase. Based
on a recent result [24] that third-NN Heisenberg coupling J3
is more significant than second-NN J2 in both Na2IrO3 and
ac−plane
z
x
y
xy−plane
AF Kitaev
FM Kitaev
ac
b
O Ir
Na
z
y
x
ϕ
x
z
y
(c)
( )b()a
Γ < 0
Γ = 0
Γ > 0
Γ < 0
Γ > 0
FIG. 1. (a) Top view of the honeycomb lattice of the edge-shared
IrO6 octahedra in Na2IrO3. (b) Three types of bonds and zigzag-AF
state where x and y bonds connecting similar dots are FM, while the
z bonds are AF (top), and the orientation of the cubic axes x, y, z with
respect to the octahedra (bottom). (c) The possible directions of the
ordered moment in the above zigzag state. In the AF Kitaev case the
moment is tied to the cubic z axis and deviates from it only slightly
with nonzero . In the FM Kitaev case with = 0, it is constrained
to the xy plane classically, and pinned to a cubic x or y axis when
quantum fluctuations are included. Nonzero < 0 gradually pushes
the moment direction towards the b axis in the honeycomb plane,
while positive drives it first towards the ac plane [which is reached
at ≈ 0.05|K|, see Fig. 3(a)], and then rotates the moment within
the ac plane towards the a axis.
RuCl3, we replace Jij in Eq. (1) by J3, reducing thereby the
parameter space.
The magnetic anisotropy in the present model is a nontrivial
problem, since the leading term K is anisotropic by itself, and,
on top of this highly frustrated interaction, the other terms
which eventually drive a magnetic order in real compounds
have a strong impact on magnetic energy profile. As illustrated
in Fig. 1(c) and discussed in detail below, the ordered moment
direction is very sensitive to the model parameters, and it shows
a qualitatively different behavior in case of FM and AF Kitaev
couplings. We note that the “moment direction” in this figure
refers to that of pseudospin; Sec. VI explains how it is related
to the magnetic moments probed by neutron-diffraction and
x-ray-diffraction experiments.
064435-2
124
MAGNETIC ANISOTROPY IN THE KITAEV MODEL . . . PHYSICAL REVIEW B 94, 064435 (2016)
III. CLASSICAL MOMENT DIRECTION
Let us briefly mention the results of a classical analysis
(for details see Appendix B of Ref. [19]) assuming the zigzag
order with antiferromagnetic z bonds as shown in Fig. 1(b).
On this level, the moment direction is determined solely by
the anisotropy parameters K, , and and corresponds to the
eigenvector of the matrix
M =
⎛
⎝
2K − + 2
− + 2 2K
0
⎞
⎠ (3)
that has the lowest eigenvalue. This minimizes the anisotropic
contribution in the classical energy per site of the zigzag
phase, Eclass = 1
8
(J − K − 3J3) + 1
8
mT
Mm, where m is a
unit vector. The dominant Kitaev interaction contributing by
the diagonal terms makes the main choice—it prefers either the
xy plane (FM K < 0) or the z axis (AF K > 0). The smaller
and terms lead to a finer selection of the ordered moment
direction.
In the case of the zigzag order stabilized by AF K > 0 and
FM J < 0, the ordered moment direction is close to the z axis
being slightly tilted in the ac plane mainly by virtue of [see
Fig. 1(c)].
The FM K < 0 case, where the zigzag order is stabilized
by and J3 terms, is more complex. With = = 0, the
entire xy plane is degenerate on a classical level. Further
selection depends on the sign of − 2 , with the positive
and negative sign making the moment to jump into the ac
plane or the b axis in the honeycomb plane, respectively.
In the former case, an increasing further pushes the
moment closer to the honeycomb plane. As it has been found
earlier [15,25] and discussed below, the Kitaev term generates
an additional magnetic anisotropy due to quantum and/or
thermal fluctuations, pinning the moment direction to the cubic
axes. This will turn the above jumps into a gradual rotation
of the easy axis with changing , along the path shown in
Fig. 1(c).
IV. EXTRACTION OF THE MOMENT DIRECTION FROM
A CLUSTER GROUND STATE
To determine the ground state of the Hamiltonian (1) and
obtain the moment direction as a function of model parameters
more rigorously than in the previous perturbative methods, we
have performed an exact diagonalization using a hexagonshaped
24-site supercell covering the honeycomb lattice. This
cluster is highly symmetric and compatible with all the hidden
symmetries of the model [19] so that no bias induced by the
cluster geometry is expected.
Since the cluster ground state does not spontaneously break
the symmetry and corresponds to a superposition of all possible
degenerate orderings, the identification of the ordered moment
direction is not straightforward. One possibility is to evaluate
the 3 × 3 correlation matrix Sα
− QS
β
Q (α,β = x,y,z) at the
ordering vector Q and to take the direction of the eigenvector
corresponding to its largest eigenvalue. Because of specific
problems of this standard approach in the present context
(see Appendix A for details), we have developed here another
method that brings a more intuitive picture of the exact ground
state by “measuring” the presence of the classical states with
a varying moment direction. As a basic building block, we
utilize the spin-1
2
coherent state
|θ,φ = Rz(φ)Ry(θ)|↑ = e−iφSz
e−iθSy
|↑ (4)
that is fully polarized along the (θ,φ) direction [26]. Here the
cubic axes are used as a convenient reference frame and θ and
φ are the conventional spherical angles. A spin-coherent state
on the cluster is constructed as a direct product
| =
N
j=1
|θj ,φj (5)
with the unit vectors mj = (cos φ sin θ, sin φ sin θ, cos θ)j
forming the desired pattern. In this fully polarized classical
state |Sα
i S
β
j | = 1
4
mα
i m
β
j and the energy |H| is thus
equal to the classical energy. We consider only collinear states
of FM, AF, and zigzag type. For example, a FM state with the
moment direction (θ,φ) is explicitly expressed as
| =
N
j=1
e−iφ/2
cos θ
2
|↑ j + e+iφ/2
sin θ
2
|↓ j . (6)
By varying θ and φ and evaluating the overlap with the exact
cluster ground state |GS , we obtain the probability map
P(θ,φ) = | |GS |2
. The ordered moment direction is then
identified by locating the maxima of P(θ,φ).
There is an intrinsic width of the peaks in P(θ,φ) due
to the nonzero overlap of the spin-coherent states, namely,
| | |2
= cos2N
(1
2
), where is the angle between the
directions (θ,φ) and (θ ,φ ). This gives an approximate half
width at half maximum of
√
2/N (in terms of the angular distance
from the maximum), evaluating to about 17◦
for N = 24.
Despite this sizable intrinsic width, the ordered moment
direction can be detected with a high accuracy (limited only
by the accuracy of the ground-state vector), as we see below.
V. MOMENT DIRECTION—EXACT DIAGONALIZATION
RESULTS
A. Testing the method: Nearly Heisenberg limit
Before discussing in detail the ordered moment direction in
the zigzag phases, relevant for actual compounds Na2IrO3 and
RuCl3, let us demonstrate the above method by considering
the Kitaev-Heisenberg model close to the Heisenberg limit,
|J| |K|, with both signs of J. In such a situation, the FM or
AF order is established by the dominant isotropic interaction,
while the anisotropic Kitaev interaction merely selects the easy
axis direction via an order-from-disorder mechanism [27].
We start with the FM case J < 0. Figure 2(a) is the
corresponding probability map obtained by the method of
Sec. IV for K/J = 0.2. The probability is clearly peaked at
the directions of the cubic axes attaining there the maximum
value Pmax slightly less than 1
6
. This is due to the cluster ground
state being a superposition of six possible classical states and
a small contribution of quantum fluctuations. The width of the
peaks matches well the intrinsic width estimated in Sec. IV.
That the K term favors cubic axes for the ordered moment
follows also from simple analytical calculations. By treating
the quantum fluctuations within second-order perturbation
064435-3
125
JI ˇR´I CHALOUPKA AND GINIYAT KHALIULLIN PHYSICAL REVIEW B 94, 064435 (2016)
0
2
4
6
8
10
12
14
16
0.0
0.5
1.0
1.5
2.0
0.0
0.5
1.0
1.5
2.0
2.5
0.0
0.5
1.0
1.5
2.0
2.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
(a) (b)
(c) (d)
(e) (f)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
P[%] ΔP[%]
P[%] P[%]
P[%] P[%]
×10
-3
J=-1 K=-0.2 J=+1 K=-0.2
K=+1 J=-0.2 K=-1 J=J3=0.2
panel d & Γ=+0.1 panel d & Γ=-0.1
x
y
z
α=90°
α≈35°
α=0°
ϕ
FIG. 2. (a) Map of the probability of the spin-coherent state given
by Eq. (6) in the FM ground state of the KH model near the Heisenberg
limit. The radial coordinate gives the angle α to the honeycomb plane;
the polar angle ϕ matches that defined in Fig. 1(b). (b) Probability
map for the AF ground state obtained using small K and dominant
J > 0. Only the variation P on top of P0 = 2.923% is shown.
(c) Probability map for the zigzag phase of the KH model with K > 0,
J < 0 reveals a strong pinning to the z axis. The coherent state
corresponding to the zigzag pattern in Fig. 1(b) was used. Directions
lying in the xy plane are indicated by the dashed line. (d) Soft xy
plane for FM K < 0 zigzag stabilized by J3. Cubic axes x and y are
selected but the moment strongly fluctuates in the plane. (e,f) The
same as in panel (d) but extended by a sizable term forcing the
moment into the ac plane (left) or the b axis (right).
expansion (see Ref. [28] for details), we obtain the magnetic
anisotropy energy
δE(2)
FM ≈
K2
64|J|
1 − m4
x − m4
y − m4
z , (7)
depending on the moment direction given by a unit vector m =
(mx,my,mz). This quantum correction on top of the isotropic
classical energy is minimized for m pointing along the cubic
axes x,y, and z that become the easy axes, consistent with the
ED result.
The case of the AF J > 0 is rather different due to the
presence of large quantum fluctuations already in the Heisenberg
limit. This is manifested in an almost flat probability
profile with P of about 3% [see Fig. 2(b)]. Nevertheless, the
probability maxima again precisely locate the x,y, and z
directions for the ordered moments, consistent with the “orderfrom-disorder”
calculations [15,25,28–30] in the models containing
compass- or Kitaev-type bond-directional anisotropy.
B. Moment direction in the zigzag phases
Having verified the method, we now move to the zigzag
phases observed in Na2IrO3 and RuCl3. We first inspect the
case of , = 0 when the anisotropy is due to the Kitaev
term alone. Shown in Fig. 2(c) is the probability map for AF
K > 0 and FM J < 0, where the z axis is selected already on
the classical level as discussed in Sec. III [31]. The probability
is indeed strongly peaked at the direction of the z axis. The
small Pmax of about 3% is again a signature of large quantum
fluctuations in the ground state. Note that this number contains
an overall reduction factor of 1
6
due to the six possible zigzag
states being superposed in the cluster ground state.
The probability map Fig. 2(d) for the FM K < 0 zigzag
case reveals the moment being constrained to the vicinity
of the xy plane, as expected from classical considerations.
Within this plane, the order-from-disorder mechanism selects
the cubic axes x and y where the probability reaches its
maxima. Concluding the survey of the probability maps, we
show P calculated including a large enough that leads to the
selection of a direction within the ac plane [ > 0, Fig. 2(e)]
or the b axis [ < 0, Fig. 2(f)].
The above three examples for the FM K zigzag indicate
a rather complex behavior of the moments in this case, as
already suggested in Fig. 1(c). In the following, we therefore
focus on the full dependence presented in Fig. 3(a) in the
form of the angles α( ) (the angle to the honeycomb plane)
and ϕ( ) (polar angle of the projection into the honeycomb
plane). Instead of the jump in α( ) obtained on a classical
level, we find a finite window | | 0.05|K| of an orderfrom-disorder
stabilized phase, where the moment direction
gradually moves from the cubic axis ( = 0) to either the b
axis ( < 0) or the ac plane ( > 0). Once the critical value
of is reached, the moment either stays along the b axis or
is pushed down within the ac plane closer to the honeycomb
plane. Figure 3(b) illustrates the evolution of α( ) for different
values of J3 stabilizing the zigzag order. For small J3, the
dominant directional Kitaev term makes the moment more
pinned to the cubic axes, which is manifested by a significantly
reduced slope of α( ) near = 0 compared to the large-J3
case. On the other hand, the critical values of are only slightly
affected by J3.
The above crossover behavior near = 0 may be easily
understood and even semiquantitatively reproduced by
considering a competition of the classical energy and the
order-from-disorder potential as follows. Keeping the moment
m = (cos φ, sin φ,0) within the xy plane preferred by K < 0,
we can evaluate the classical energy per site:
Eclass = 1
8
(K − 3J3 + J) − 1
8
( − 2 ) sin 2φ. (8)
In this contribution, the anisotropy is due to the and
terms only. Eclass is complemented by an order-from-disorder
potential Efluct(φ) that should contain four equivalent minima
at φ = 0,1
2
π,π,3
2
π corresponding to the cubic axes (supported
by the K term). Such a potential can be represented by the
064435-4
126
MAGNETIC ANISOTROPY IN THE KITAEV MODEL . . . PHYSICAL REVIEW B 94, 064435 (2016)
0°
10°
20°
30°
40°
50°
60°
70°
80°
90°
-0.10 -0.05 0.00 0.05 0.10
angle
Γ
b-axis xy-plane ac-plane
α
ϕ
0°
10°
20°
30°
40°
50°
0° 30° 60° 90°
ϕ
J3=0.1
0.2
0.5
0°
10°
20°
30°
40°
50°
-0.10 -0.05 0.00 0.05 0.10
α
Γ
J3=0.1
0.2
0.5
0°
10°
20°
30°
40°
50°
60°
70°
-0.3 -0.2 -0.1 0.0 0.1 0.2
α
Γ
Γ′=0
-0.1
-0.2
0°
10°
20°
30°
40°
50°
60°
-1.0 -0.5 0.0 0.5 1.0
Γ
classical α
J3=0
0.1
0.2
30°
35°
40°
45°
50°
55°
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
α
Γ
(a)
(b)
(c)
(d) (e)
classical α
J3=J=0.2
J3=0.1
J3=0.5
J=0.5
FIG. 3. (a) -dependent angles α, ϕ specifying the moment
direction reveal three regimes for FM K zigzag supported by small
J3. The values K = −1 and J = J3 = 0.2 were used. At = 0,
the angles give the direction towards an oxygen ion. A crossover in
the interval | | 0.05 corresponds to the path shown in Fig. 1(c).
(b) Left panel shows the angle α for K = −1, J = 0.2 and several
J3 values manifesting a stronger pinning to the cubic axis at smaller
J3. The same data are presented as α(ϕ) in the right panel together
with α(ϕ) corresponding to the xy plane (dashed). The black dot
indicates the cubic axis direction. (c) The angle α for larger values
of > 0 compared to the classical result of Ref. [19] (dotted). The
blue solid curve is a continuation of that of panel (a), red and green
curves are calculated using different J3 values used in panel (b), and
the blue dashed curve denotes a larger J value. (d) The angle α
for the parameters K = −1, J = J3 = 0.2, and several values.
(e) -dependent α in the AF K = +1 case with J = −0.2 and
several J3 values compared to the classical result of Ref. [19] (dotted).
The endpoints of the curves are determined by a sharp drop of the
probability of the classical zigzag state indicating a phase boundary.
following form:
Efluct = V sin2
2φ, (9)
approximating Efluct(φ) by its lowest harmonic. This function
is characterized by a single unknown parameter—the barrier
height V , determined mainly by the dominant K. Assuming
= 0, the minimization of the total energy Eclass + Efluct
gives φ( ) = 1
2
arcsin 16V
and the critical value crit = 16V .
This enables us to extract effective V from our numerical data.
By taking crit ≈ 0.05|K| observed in Figs. 3(a) and 3(b)
we get V ≈ 0.003|K|. Furthermore, converting φ in the xy
plane to the angle α to the honeycomb plane, we obtain
“phenomenological” α( ) = arcsin 1
3
(1 + 16V
) that roughly
approximates the numerical α( ) data. The agreement between
these two α( ) profiles improves with increasing J3, when the
order-from-disorder potential becomes more harmonic and the
deviation of the moment direction from the xy plane for > 0
reduces [see Fig. 3(b)]. In fact, Eqs. (8) and (9), together with
the value of V ≈ 0.003|K| extracted from the ED data, may
be used for a semiquantitative determination of the easy axis
direction within the xy plane.
For curiosity, we have evaluated the potential barrier V
also analytically, by two slightly different methods. First, as
in Sec. V A, we estimated quantum corrections for the zigzag
phase along the lines of Ref. [28]. This reproduced the above
form (9) of the anisotropy potential, and provided a consistent
estimate of V ≈ 0.005|K|. An alternative evaluation of the
anisotropy potential within the linear spin-wave framework
resulted in zero-point energy of the same form as Eq. (9)
again, but with an overestimated value of V ≈ 0.014|K|.
In Na2IrO3 the moment direction was found [12] in the ac
plane suggesting that > crit for this material. We thus focus
on this particular case and investigate how the precise value of
α is affected by the model parameters in more detail. Already
on a classical level, finite > 0 rotates the moment within
the ac plane from α ≈ 54.7◦
(corresponding to the xy plane)
toward the honeycomb plane (α = 0). Such an effect is well
visible also in Figs. 3(a) and 3(b). Presented in Fig. 3(c) are
a few representative α( ) curves for larger values of up to
|K| that serve as a test of the classical prediction
tan 2α = 4
√
2
1 + r
7r − 2
with r = −
K +
(10)
derived in Ref. [19]. As we find, the quantum fluctuations
included in the exact ground state push the ordered moments
much closer to the honeycomb plane. The difference is
substantial and needs to be considered when trying to quantify
the model parameters based on the experimental data.
So far, we have considered = 0 only, while a small
negative is expected to be generated by a trigonal compression
[18,19,32]. Based on Eq. (8), is expected to effectively
shift the value of in the first approximation. Indeed, as shown
in Fig. 3(d), the rough three-phase picture as in Fig. 3(a) is
preserved and the negative shifts the α( ) curve in the
negative direction. This enables α to reach higher values, even
above the xy-plane angle 54.7◦
.
Finally, in Fig. 3(e) we briefly analyze the AF K situation
with the moment near the z axis. In contrast to the FM K case,
small has a relatively little effect here, because the z axis
064435-5
127
JI ˇR´I CHALOUPKA AND GINIYAT KHALIULLIN PHYSICAL REVIEW B 94, 064435 (2016)
is classically selected by the dominant K > 0 itself. Quantum
fluctuations are found to generate an even stronger pinning to
the z axis, compared to the classical solution of Ref. [19]. Only
a very large coupling is able to take the spin away from the
z axis.
VI. COMPARISON TO EXPERIMENT
A. Extracting pseudospin direction from resonant x-ray and
neutron-scattering data
Having quantified the pseudospin easy axis direction as
a function of the Hamiltonian parameters, we consider now
how this “pseudomoment” direction is related to that of
real magnetic moments measured by neutron-scattering and
x-ray scattering experiments. To this end, we first define the
pseudospin one-half wave functions including crystal field of
trigonal symmetry. The latter splits the t2g manifold into an
orbital singlet a1g = 1√
3
(xy + yz + zx) and the eg doublet
{ 1√
6
(yz + zx − 2xy) ; 1√
2
(zx − yz)}. Denoting this splitting
by and using the hole representation, we have
H = 1
3
[2n(a1g) − n(eg)]. (11)
Within a point-charge model, positive (negative) would
correspond to a compression (elongation) of octahedra along
the trigonal c axis. The actual value of in real material is
decided by various factors, but this issue is not relevant in the
present context.
In terms of the effective angular momentum l = 1 of the t2g
shell, the a1g state corresponds to the lc = 0 state, while the eg
doublet hosts the lc = ±1 states, using the quantization axis c
suggested by the trigonal crystal field. Explicitly,
|0 =
1
√
3
(|yz + |zx + |xy ), (12)
|±1 = ±
1
√
3
(e±2πi/3
|yz + e∓2πi/3
|zx + |xy ). (13)
Via these lc states, pseudospin-1
2
wave functions are defined
as
+1
2
= + sin ϑ |0,↑ − cos ϑ |+1,↓ , (14)
−1
2
= − sin ϑ |0,↓ + cos ϑ |−1,↑ , (15)
where ↑ and ↓ refer to the projections of the hole spin on the
trigonal c axis. The spin-orbit “mixing” angle 0 ϑ π/2
is given by tan 2ϑ = 2
√
2/(1 + δ), where δ = 2 /λ.
Using the wave functions (14) and (15), we may express the
spin s and orbital l moments of a hole via the pseudospin S. In
a cubic limit, i.e., = 0, one has s = −1
3
S, l = 4
3
S, and total
magnetic moment M = (2s − l) = −2S (note a negative g
factor g = −2). These relations imply that the pseudospin easy
axis direction is identical to that of spin, orbital, and magnetic
moments when the trigonal field is zero. However, this is
no longer valid at finite . For instance, strong compression
(ϑ = 0) would completely suppress the ab-plane components
of magnetic moments, so the pseudospin and magnetic
moment will not be parallel anymore (unless pseudospin is
ordered along the c axis).
The x rays and neutrons couple initially to the spin and
orbital moments, and the scattering operator has to be projected
onto the pseudospin basis. We first consider an effective RXS
operator. For pseudospin one-half in a trigonal field, it has to
have a form ˆR ∝ ifab(PaSa + PbSb) + ifcPcSc, where P =
ε × ε and ε (ε ) is the polarization of the incoming (outgoing)
photon. This can be written as ˆR ∝ i P · N, introducing a
vector N = (faSa,fbSb,fcSc) with fa = fb ≡ fab. The RXS
data determine a direction of this auxiliary vector N; in
Na2IrO3, it was found to make an angle αN ≈ 44.3◦
to the ab
plane [12]. However, this is not yet the pseudospin direction,
since fab = fc and hence αS = αN , unless the trigonal field
is exactly zero (unlikely in real materials). To access the
pseudospin angle αS and quantify the model parameters, one
has to know the “RXS factors” fab and fc.
We have derived the f factors (see Appendix B for details).
For the L3 edge, they read as
fab =
1
2
+
5
6
√
2
s2ϑ −
1
6
c2ϑ , (16)
fc = 1 +
2
3
c2ϑ −
1
3
√
2
s2ϑ . (17)
Here, s2ϑ = 2
√
2/r, c2ϑ = (1 + δ)/r, and r = 8 + (1 + δ)2.
Figure 4(a) shows the f factors as a function of trigonal field
parameter δ. In the cubic limit, one has fab = fc and hence N
is parallel to S, as expected.
For completeness, we show also the f factors for the L2
edge:
fab = 2fc = −3
2
+ 1
2
c2ϑ +
√
2 s2ϑ , (18)
which vanish at the δ = 0 limit, as a consequence of the spinorbit
entangled nature of pseudospins [33].
In neutron-diffraction experiments, the magnetic moment
M = (gaSa,gbSb,gcSc) is probed. For the pseudospins as defined
above, the g factors are (neglecting covalency effects [7])
gab = −(1 +
√
2 s2ϑ − c2ϑ ), (19)
gc = −(1 + 3 c2ϑ ). (20)
The g-factor anisotropy can quantify the strength of the trigonal
field, as illustrated in Fig. 4(b). Again, magnetic moment
direction is in general different from that of pseudospin, and
to access the latter one needs to know the g factors.
These considerations imply that the orientations of the
(x-ray) N vector and magnetic moment M differ from each
other, and also from that of pseudospin S which enters the
model Hamiltonian. As we show in Fig. 4(c), their relative
angles come in the order αM > αN > αS for positive , and
in reversed order αS > αN > αM for negative . Ideally,
having measured both N and M directions in the same
compound, one could extract the crystal-field parameter δ
using the above equations, and uniquely fix the pseudospin
easy axis angle αS. In principle, the g-factor anisotropy
provides the same information on δ, but obtaining g factors in
magnetically concentrated systems is a somewhat nontrivial
task. Alternatively, one could extract the value and sign of
directly from the splitting and anisotropy of the high-energy
J = 3/2 quartet in single crystals (see Appendix C for details).
064435-6
128
MAGNETIC ANISOTROPY IN THE KITAEV MODEL . . . PHYSICAL REVIEW B 94, 064435 (2016)
-4
-3
-2
-1
0
1
2
3
4
-∞ ∞-3 -2 -1 0 1 2 3
δ = 2Δ / λ
gab
gc
gc /gabg-factors
RuCl3 Na2IrO3
0.0
0.5
1.0
1.5
2.0
-∞ ∞-3 -2 -1 0 1 2 3
δ=-1 δ=+0.75 δ=+2
αS=38°
a
c
a
c
a
c
(a)
(b)
(c)
(d) (e)
fab
fc
fc /fabL3 RXS factors
20°
30°
40°
50°
60°
-1 0 1
δ = 2Δ / λ
αM
αN
αS
αN=44.3°
20°
30°
40°
50°
60°
70°
-1 0 1
δ = 2Δ / λ
αM
αN
αS
αM=35°
S
N
M
S
M
N
FIG. 4. (a) Factors f entering the relation between the pseudospin
S and L3 RXS vector N presented as functions of the trigonal field.
(b) g factors as functions of the trigonal field. Intervals of δ consistent
with the g factors suggested by the experimental data on RuCl3
[34,35] and Na2IrO3 [36,37] are indicated by shading. (c) Directions
of the S, N, and M vectors for sample values of the trigonal field
parameter δ and a fixed pseudospin angle αS = 38◦
. The case with
the negative δ = −1 could be relevant for RuCl3, while positive
δ = +0.75 with the reverse order of the vectors M, N, and S for
Na2IrO3. (d),(e) Angles αS, αN , and αM of the vectors S, N, and M
to the honeycomb plane as functions of δ keeping fixed αN = 44.3◦
(d) or αM = 35◦
(e). The shaded δ intervals are the same as in
panel (b).
B. Implications for Na2IrO3 and RuCl3
Armed with the above relations between different moments,
and using the results of Sec. V B, let us now analyze the
available experimental data on Na2IrO3 and RuCl3.
Starting with the case of Na2IrO3, we utilize the value
αN ≈ 44.3◦
determined recently by RXS [12]. Keeping this
experimental constraint, in Fig. 4(d) we plot the remaining
angles αM and αS as functions of the relative strength of the
trigonal crystal field δ. In Ref. [19], the value /λ ≈ 3/8 was
deduced based on the splitting BC ≈ 0.1 eV of the J = 3/2
quartet [37]. As seen in Fig. 4(b), the corresponding δ ≈ 0.75
is also roughly consistent with the anisotropy of the g factors,
gc/gab ≈ 1.4, obtained by fitting the temperature-dependent
magnetic susceptibilities χc > χab [36]. The data in Fig. 4(d)
then suggest that the magnetic moment takes an angle
of about αM ≈ 50◦
to the honeycomb plane, while the
pseudospin angle αS is roughly 38–40◦
. Such a deviation
of the pseudospin from the xy plane (α ≈ 54.7◦
) implies a
sizable value. Based on Fig. 3(c) we may naively expect
the /|K| ratio in the range 0.3–0.5. We emphasize, however,
that this conclusion relies on the above estimate of the trigonal
field, that should be verified by measuring the “magnetic”
angle αM directly by neutron scattering.
Compared to Na2IrO3, RuCl3 shows an opposite
magnetic anisotropy behavior with χc χab [34]. The
magnetic structure has been recently investigated by neutron
scattering [38], with the result αM ≈ 35◦
and ϕ being equal
to either 0 or 180◦
. Similarly to Fig. 4(d), in Fig. 4(e) we
keep the measured angle, now αM , fixed at its experimental
value, and plot αS and αN for varying δ = 2 /λ. This
parameter could be obtained from the anisotropy of J = 3/2
transitions in single crystals (see Appendix C). We are not
aware of such a direct measurement in RuCl3, so the trigonal
field is best assessed by considering the anisotropy of the g
factors. References [34,35] reported in-plane and out-of-plane
magnetization curves measured for high fields up to 60 T.
Even though the saturation was not reached, the data indicate
the value gc/gab ≈ 0.4–0.5. A similar ratio was also found by
Yadav et al. [39] using quantum chemistry methods and by
fitting the high-field data of Ref. [35]. The corresponding δ
puts the pseudospin angle αS at relatively high values of about
αS 50◦
[see Fig. 4(e)]. Adopting this estimate, we will try
to identify a consistent parameter window.
Unfortunately, the present neutron experiment [38] could
not directly resolve the orientation of the moments with respect
to the a axis, i.e., whether ϕ = 0 or 180◦
. The absence of this
most conclusive evidence for the sign of the Kitaev interaction
requires us to consider both possibilities.
We assume first FM K < 0 as obtained in two recent ab initio
calculations of the exchange interactions in RuCl3 [24,39].
Figure 3(c) gives a hint that the estimated αS 50◦
can be
reached for small only. As seen in Fig. 3(d), by including
small negative that shifts the crossover towards negative
, the pseudospin direction may rotate even far above the
xy plane. Interestingly, the corresponding parameter regime
J ∼ − ∼ − ∼ 0.2|K| matches well the prediction by
quantum chemistry calculations [39].
Now we analyze the AF K > 0 case, proposed for RuCl3 in
Refs. [13,38,40]. In this case, the zigzag order is obtained on
the level of the two-parameter Kitaev-Heisenberg model [20]
alone, and this simplicity makes the AF K scenario particularly
attractive. In the zigzag phase of the two-parameter model, the
pseudospins point along the cubic z axis leading to αS ≈ 35◦
.
This can be reconciled with the experimental value αM ≈ 35◦
only in a nearly cubic situation with a small trigonal distortion.
Considering, however, the large anisotropy of the g factors
064435-7
129
JI ˇR´I CHALOUPKA AND GINIYAT KHALIULLIN PHYSICAL REVIEW B 94, 064435 (2016)
discussed above and the resulting αS 50◦
, it seems that
the AF Kitaev interaction needs to be supplemented by other
anisotropic interactions lifting the pseudospin considerably
up. This scenario is addressed in Fig. 3(e). We have found
that does not influence αS much so that we focus on the
dependence. Since the AF K zigzag phase becomes fragile if
the other anisotropy terms are included, the model has to be
additionally extended by J3. Based on the data of Fig. 3(e),
we may conclude that large negative comparable to K is
needed to obtain αS 50◦
. It should be carefully checked if
such a substantially extended model is still consistent with
other experimental data, in particular with the spin excitation
spectrum with only small gaps [13].
We would like to stress again that our analysis of RuCl3 for
both K < 0 and K > 0 heavily relied on the relative trigonal
field strength /λ inferred solely from the magnetization
anisotropy in high magnetic fields. It is thus highly desirable
to measure the complementary angle αN by RXS and quantify
/λ more precisely, as suggested in the previous subsection.
As discussed in Appendix C, measuring the anisotropy of
J = 3/2 states by inelastic neutron scattering in single crystals
would be also very helpful.
To summarize this section, in Na2IrO3, the measured
moment direction [12] with ϕ = 0◦
well fixes the FM sign
of the Kitaev interaction, and our analysis of its angle from the
ab plane suggests that ∼ 0.3 − 0.5|K| coupling is present.
Concerning RuCl3, the current ambiguity in the angle ϕ (0
or 180◦
) leaves open the issue of the sign of K. There is
also an uncertainty in the trigonal field value ; based so
far on the g-factor anisotropy, we found that FM K < 0 with
relatively small and values would be consistent with
the data, while the AF K > 0 situation requires large < 0
couplings comparable to K.
VII. CONCLUSIONS
We have investigated the ordered moment direction in the
zigzag phases of the extended Kitaev-Heisenberg model for
honeycomb lattice magnets. Our method analyzes the exact
cluster ground states using a particular set of spin coherent
states and as such fully accounts for the quantum fluctuations.
The interplay among the various anisotropic interactions leads
to a complex behavior of the ordered moment direction as a
function of the model parameters. We have found substantial
corrections to the results of a classical analysis that are
important when quantifying the exchange interactions based
on the experimental data.
We have pointed out that, away from the ideal cubic
situation, the notion of the “ordered moment direction” has
to be precisely specified. Assuming a trigonal field relevant to
the layered honeycomb systems, we have derived relations
among the directions of (i) the pseudospins entering the
model Hamiltonian, (ii) the magnetic moments measured by
neutron diffraction, and (iii) the moment direction as probed
by resonant magnetic x-ray scattering. These relations and a
combination of neutron and x-ray data should enable a reliable
quantification of the trigonal field as well as the pseudospin
direction in future experiments.
Using the above results, we have analyzed the currently
available experimental data on Na2IrO3 and RuCl3 and
identified plausible parameter regimes in these compounds.
ACKNOWLEDGMENTS
We would like to thank G. Jackeli, B. J. Kim, S. E. Nagler,
and J. Rusnaˇcko for helpful discussions. J.C. acknowledges
support by Czech Science Foundation (GA ˇCR) under Project
No. GJ15-14523Y and MˇSMT ˇCR under NPU II project
CEITEC 2020 (Project No. LQ1601).
APPENDIX A: COMPARISON OF NUMERICAL METHODS
As mentioned in the main text, the standard method to
obtain the ordered moment direction using the ED ground state
is to evaluate the spin-spin correlation matrix Sα
− QS
β
Q (α,β =
x,y,z) at the ordering vector Q and to find its eigenvector
corresponding to the largest eigenvalue. However, there are
two main problems associated with this simple method, both
emerging since the cluster ground state is a linear superposition
of degenerate orderings where the individual orderings have
equal weights.
(i) If there are several equivalent easy axis directions
associated with the selected ordering vector Q, they will be
characterized by the same eigenvalue. This leads to a degenerate
eigenspace and prevents us from resolving such directions.
The most severe cases are those with a dominant Heisenberg
interaction presented in Figs. 2(a) and 2(b). Here we have three
degenerate easy axes x, y, and z which makes the correlation
matrix proportional to a unit matrix and thus isotropic. In the
FM K < 0 zigzag situation shown in Fig. 2(d) and the entire
middle phase in Fig. 3(a), two degenerate moment directions
for a particular zigzag pattern (selected by Q) are possible and
the correlation matrix therefore just uncovers the softness of
the xy plane. Only after these two directions merge for a large
enough | |, the moment direction can be identified.
(ii) The zigzag pattern to be probed is selected by choosing
the ordering vector Q. In contrast to an infinite lattice, at a
finite cluster this separation of the three zigzag directions is
0°
10°
20°
30°
40°
50°
60°
-0.2 0.0 0.2 0.4 0.6 0.8 1.0
α
Γ
classical
ED & overlaps
ED & correlation matrix
FIG. 5. Comparison of the angle α of the pseudospin direction
to the ab plane obtained using various methods. The parameters
K = −1 and J = J3 = 0.2 were used. The blue curve is identical to
the one shown in Figs. 3(a)–3(d).
064435-8
130
MAGNETIC ANISOTROPY IN THE KITAEV MODEL . . . PHYSICAL REVIEW B 94, 064435 (2016)
not perfect. The range of spin correlations is limited by the
size of the cluster and the corresponding momentum space
peaks become broad. The correlation matrix at given Q is thus
“polluted” by small contributions of the two other zigzags
in the ground state, that are associated with the remaining
ordering vectors.
Our method introduced in Sec. IV does not suffer from
the above problems and is able to handle all the situations
encountered. This is due to the full resolution of the various
degenerate orderings present in the cluster ground state by
using a prescribed ordering pattern and by a construction of a
full directional map.
If applicable, the standard method gives results very similar
to our method. We demonstrate this in Fig. 5 that compares
the two methods for the parameters K = −1, J = J3 = 0.2
and varying used in Fig. 3. The slight deviations observed
for > 0 can be interpreted as a manifestation of the second
problem discussed above.
APPENDIX B: DERIVATION OF THE L-EDGE
RXS OPERATOR
Resonant x-ray scattering is conceptually similar to the
Raman light scattering, in a sense that both processes involve
the intermediate states created and subsequently eliminated
by incoming and outgoing photons. However, the nature
of the intermediate states in these two cases is radically
different: while the Raman light scattering involves intersite
d-d transitions, the x rays create the high-energy on-site
p-d transitions. As a result, the Raman light scattering
probes intersite (two-magnon) spin flips, while the presence
of a strong spin-orbit coupled 2p-core hole in the RXS
intermediate states makes single-ion spin flips a dominant
magnetic scattering channel (see the recent review [41] and
references therein for details).
A complex time dynamics of the intermediate states makes
the x-ray-scattering process hard to analyze microscopically.
However, as far as one is concerned with the low-energy
excitations in Mott insulators, the problem of the intermediate
states can be disentangled and cast in the form of frequencyindependent
phenomenological constants [42–44]. This results
is an effective RXS operator formulated in terms of low-energy
(orbital, spin, etc.) degrees of freedom alone. The form of this
operator is dictated by symmetry. In essence, this approach is
similar to that of Fleury and Loudon [45] widely used in the
theories of Raman light scattering in quantum magnets.
While the RXS operator used in the main text follows from
an underlying trigonal symmetry, the ratio between fab and
fc constants requires specific calculations. This can be easily
done, with some routine modifications of the previous work
for the case of tetragonal symmetry [46,47], as outlined below.
In cubic axes x,y, and z (see Fig. 1), a dipolar 2p to 5d
transition operator reads as
D = εxTx + εyTy + εzTz, (B1)
where εx,y,z are the polarization factors, and Tx = d
†
zxpz +
d
†
xypy, Ty = d
†
xypx + d
†
yzpz, Tz = d
†
yzpy + d
†
zxpx. Here and
below, it is implied that d and p operators carry also the spin
quantum numbers (↑, ↓) over which summation is taken.
In the quantization axes a,b, and c, suggested by the
trigonal crystal field, this operator takes the following form:
D =
1
√
6
(εaTa + εbTb + εcTc), (B2)
where
Ta = (d
†
0 + 2d
†
−1)p1 + (d
†
1 − d
†
−1)p0 + (2d
†
1 − d
†
0)p−1,
iTb = (−d
†
0 + 2d
†
−1)p1 + (d
†
1 + d
†
−1)p0 − (2d
†
1 + d
†
0)p−1,
Tc =
√
2 (2d
†
0p0 − d
†
1p1 − d
†
−1p−1). (B3)
Here, the indices 0 and ±1 stand for the lc orbital quantum
numbers of d and p electrons.
Within the above Fleury-Loudon-like approach to the xray-scattering
problem, the effective RXS operator is given by
D†
(ε )D(ε), and its part responsible for the magnetic scattering
reads as ˆR ∝ i(ε × ε ) · (T†
× T).
Next, the core-hole operators p in Eq. (B3) are expressed
in terms of spin-orbit split j = 1/2 and 3/2 eigenstates of the
2p level, resulting in two sets of T operators active in L2 and
L3 edges, correspondingly. After “integrating out” these 2p1
2
and 2p3
2
operators, the product (T†
× T) becomes a simple
quadratic form of d operators. Finally, projecting this form
onto a pseudospin doublet [given by Eqs. (14) and (15) of the
main text], we arrive at the RXS operator ˆR ∝ ifab(PaSa +
PbSb) + ifcPcSc, with the f factors shown in the main text.
Via the pseudospin wave functions, the RXS f factors are
sensitive to a trigonal field strength.
APPENDIX C: DETERMINATION OF THE TRIGONAL
FIELD FROM J= 3/2 MAGNETIC EXCITATION SPECTRA
Under spin-orbit coupling λ and trigonal crystal field , t2ghole
states split into three levels A, B, and C [see Fig. 6(a)]. The
A level hosts a Kramers pseudospin one-half (corresponding
to J = 1/2 in the cubic limit), with the wave functions
|A+ = + sin ϑ |0,↑ − cos ϑ |+1,↓ , (C1)
|A− = − sin ϑ |0,↓ + cos ϑ |−1,↑ , (C2)
as were given by Eqs. (14) and (15) of the main text. The upper
Kramers doublets B and C are derived from the spin-orbit
J = 3/2 quartet. The former correspond to pure Jc = ±3/2
states of J = 3/2 moment:
|B+ = |+1,↑ , (C3)
|B− = |−1,↓ , (C4)
while the C level wave functions are given by
|C+ = cos ϑ |0,↑ + sin ϑ |+1,↓ , (C5)
|C− = cos ϑ |0,↓ + sin ϑ |−1,↑ , (C6)
corresponding to Jc = ±1/2 states of the J = 3/2 quartet
in the cubic limit, and containing some admixture of
064435-9
131
JI ˇR´I CHALOUPKA AND GINIYAT KHALIULLIN PHYSICAL REVIEW B 94, 064435 (2016)
1.0
1.1
1.2
1.3
1.4
1.5
1.6
-3 -2 -1 0 1 2 3
〈IB〉/〈IC〉
δ = 2Δ / λ
0.0
0.5
1.0
1.5
2.0
intensity
IB
(ab)
IB
(c)
IC
(ab)
IC
(c)
-1.5
-1.0
-0.5
0.0
0.5
1.0
energy/λ
(a)
(b)
(c)
A
B
C
ΔBC
3λ/2
J=3/2 quartet
J=1/2 doublet
IB IC
1.0 1.5 2.0 2.5λ 1.0 1.5 2.0 2.5λ
BC B C
FIG. 6. (a) Level structure of a d5
(t2g) ion upon trigonal field
splitting characterized by δ = 2 /λ (hole picture). (b) Intensities
of the magnetic transitions A → B and A → C for the ab-plane
and c-axis components of the dynamical spin structure factor as
given by Eqs. (C11) and (C13). (c) Ratio of the powder-averaged
intensities. The insets show the broadened (HWHM = 1
4
λ) peak
structure assuming δ = −1 (left) and δ = +1 (right), respectively.
the original J = 1/2 doublet at finite . The energies of
these states are EA,C/λ = 1
4
[∓ 8 + (1 + δ)2 − 1] + 1
12
δ and
EB/λ = 1
2
− 1
6
δ.
Transitions from the ground-state A level to B and C states
are magnetically active; their spectral weights in the dynamical
spin structure factor are determined by matrix elements of
the magnetic moment M = 2s − l:
∓ B±|Ma|A± =
1
i
B±|Mb|A± = cos ϑ +
1
√
2
sin ϑ,
(C7)
± C∓|Ma|A± =
1
i
C∓|Mb|A± =
1
2
(s2ϑ +
√
2c2ϑ ) . (C8)
Out-of-plane moment Mc matrix elements between A and B
vanish (independent of the spin-orbit mixing angle ϑ), while
C±|Mc|A± = 3
2
s2ϑ . (C9)
In the magnetic excitation spectra, a transition A → B gives a
peak at the energy
EB − EA =
λ
4
[ 8 + (1 + δ)2 + 3 − δ], (C10)
with the following intensities for different components of the
dynamical spin structure factor:
IB =
1
4
(3 + c2ϑ + 2
√
2s2ϑ ) (ab plane)
0 (c axis)
. (C11)
The second transition A → C is peaked at the energy
EC − EA =
λ
2
8 + (1 + δ)2 (C12)
and has the intensity
IC =
1
4
(s2ϑ +
√
2c2ϑ )2
(ab plane)
9
4
s2
2ϑ (c axis)
. (C13)
The B and C peaks are separated by BC/λ =
1
4
[ 8 + (1 + δ)2 − 3 + δ]; at small trigonal splitting λ,
this can be approximated as BC ≈ 2
3
. At positive (negative)
, the B peak position is lower (higher) than that of the C peak
[see Fig. 6(a)].
Figure 6(b) shows that the intensities of both transitions
are highly anisotropic with respect to ab-plane and c-axis
polarizations, with the opposite behavior of B and C contributions.
The out-of-plane response is due to the C transition
exclusively, while the B peak dominates the ab-plane intensity.
This should enable one to distinguish them and determine
thereby both the sign and value of trigonal field parameter δ
from single-crystal spin-polarized neutron-scattering data.
On the other hand, the powder averaged intensities of B
and C peaks are nearly the same for realistic values of δ [see
Fig. 6(c)]. Even at |δ| = 1, the two peaks may overlap to
give a single broad line, leaving an ambiguity in the sign of
parameter δ.
[1] N. F. Mott, Metal-Insulator Transitions (Taylor and Francis,
London, 1974).
[2] M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 70, 1039
(1998).
[3] J. B. Goodenough, Magnetism and the Chemical Bond (Interscience,
New York, 1963).
[4] K. I. Kugel and D. I. Khomskii, Sov. Phys. Usp. 25, 231 (1982).
[5] G. Khaliullin and S. Maekawa, Phys. Rev. Lett. 85, 3950
(2000).
[6] G. Khaliullin, Prog. Theor. Phys. Suppl. 160, 155 (2005).
[7] A. Abragam and B. Bleaney, Electron Paramagnetic Resonance
of Transition Ions (Clarendon, Oxford, 1970).
[8] G. Khaliullin, W. Koshibae, and S. Maekawa, Phys. Rev. Lett.
93, 176401 (2004).
064435-10
132
MAGNETIC ANISOTROPY IN THE KITAEV MODEL . . . PHYSICAL REVIEW B 94, 064435 (2016)
[9] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205
(2009).
[10] A. Kitaev, Ann. Phys. 321, 2 (2006).
[11] J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Annu. Rev. Condens.
Matter Phys. 7, 195 (2016).
[12] S. H. Chun, J.-W. Kim, Jungho Kim, H. Zheng, C. C. Stoumpos,
C. D. Malliakas, J. F. Mitchell, K. Mehlawat, Y. Singh, Y. Choi,
T. Gog, A. Al-Zein, M. Moretti Sala, M. Krisch, J. Chaloupka,
G. Jackeli, G. Khaliullin, and B. J. Kim, Nat. Phys. 11, 462
(2015).
[13] A. Banerjee, C. A. Bridges, J.-Q. Yan, A. A. Aczel, L. Li,
M. B. Stone, G. E. Granroth, M. D. Lumsden, Y. Yiu, J. Knolle,
S. Bhattacharjee, D. L. Kovrizhin, R. Moessner, D. A. Tennant,
D. G. Mandrus, and S. E. Nagler, Nat. Mater. 15, 733 (2016).
[14] K. W. Plumb, J. P. Clancy, L. J. Sandilands, V. V. Shankar,
Y. F. Hu, K. S. Burch, H.-Y. Kee, and Y.-J. Kim, Phys. Rev. B
90, 041112(R) (2014).
[15] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett.
105, 027204 (2010).
[16] V. M. Katukuri, S. Nishimoto, V. Yushankhai, A. Stoyanova, H.
Kandpal, S. Choi, R. Coldea, I. Rousochatzakis, L. Hozoi, and
J. van den Brink, New J. Phys. 16, 013056 (2014).
[17] J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Phys. Rev. Lett. 112,
077204 (2014).
[18] J. G. Rau and H.-Y. Kee, arXiv:1408.4811.
[19] J. Chaloupka and G. Khaliullin, Phys. Rev. B 92, 024413 (2015).
[20] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett.
110, 097204 (2013).
[21] I. Kimchi and Y.-Z. You, Phys. Rev. B 84, 180407(R) (2011).
[22] S. K. Choi, R. Coldea, A. N. Kolmogorov, T. Lancaster, I. I.
Mazin, S. J. Blundell, P. G. Radaelli, Y. Singh, P. Gegenwart,
K. R. Choi, S.-W. Cheong, P. J. Baker, C. Stock, and J. Taylor,
Phys. Rev. Lett. 108, 127204 (2012).
[23] Y. Yamaji, Y. Nomura, M. Kurita, R. Arita, and M. Imada,
Phys. Rev. Lett. 113, 107201 (2014).
[24] S. M. Winter, Y. Li, H. O. Jeschke, and R. Valent´ı, Phys. Rev. B
93, 214431 (2016).
[25] Y. Sizyuk, P. W¨olfle, and N. B. Perkins, Phys. Rev. B 94, 085109
(2016).
[26] A. Auerbach, Interacting Electrons and Quantum Magnetism
(Springer, New York, 1994).
[27] For a discussion of the order-from-disorder phenomena in
frustrated spin systems, see A. M. Tsvelik, Quantum Field
Theory in Condensed Matter Physics (Cambridge University,
Cambridge, England, 1995), Chap. 17, and references therein.
[28] G. Jackeli and A. Avella, Phys. Rev. B 92, 184416 (2015).
[29] G. Khaliullin, Phys. Rev. B 64, 212405 (2001).
[30] Z. Nussinov and J. van den Brink, Rev. Mod. Phys. 87, 1 (2015).
[31] However, the separation of the individual zigzag chain directions
is an order-from-disorder effect, since a linear combination of
different zigzag patterns is a classical ground state as well.
[32] S. Bhattacharjee, S.-S. Lee, and Y. B. Kim, New J. Phys. 14,
073015 (2012).
[33] B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita, H. Takagi,
and T. Arima, Science 323, 1329 (2009).
[34] Y. Kubota, H. Tanaka, T. Ono, Y. Narumi, and K. Kindo,
Phys. Rev. B 91, 094422 (2015).
[35] R. D. Johnson, S. C. Williams, A. A. Haghighirad, J. Singleton,
V. Zapf, P. Manuel, I. I. Mazin, Y. Li, H. O. Jeschke, R. Valent´ı,
and R. Coldea, Phys. Rev. B 92, 235119 (2015).
[36] Y. Singh and P. Gegenwart, Phys. Rev. B 82, 064412 (2010).
[37] H. Gretarsson, J. P. Clancy, X. Liu, J. P. Hill, E. Bozin, Y. Singh,
S. Manni, P. Gegenwart, J. Kim, A. H. Said, D. Casa, T. Gog,
M. H. Upton, H.-S. Kim, J. Yu, V. M. Katukuri, L. Hozoi, J.
van den Brink, and Y.-J. Kim, Phys. Rev. Lett. 110, 076402
(2013).
[38] H. B. Cao, A. Banerjee, J.-Q. Yan, C. A. Bridges, M. D.
Lumsden, D. G. Mandrus, D. A. Tennant, B. C. Chakoumakos,
and S. E. Nagler, Phys. Rev. B 93, 134423 (2016).
[39] R. Yadav, N. A. Bogdanov, V. M. Katukuri, S. Nishimoto, J. van
den Brink, and L. Hozoi, arXiv:1604.04755.
[40] H.-S. Kim, Vijay Shankar V., A. Catuneanu, and H.-Y. Kee,
Phys. Rev. B 91, 241110(R) (2015).
[41] L. J. P. Ament, M. van Veenendaal, T. P. Deveraux, J. P. Hill,
and J. van den Brink, Rev. Mod. Phys. 83, 705 (2011).
[42] L. J. P. Ament and G. Khaliullin, Phys. Rev. B 81, 125118
(2010).
[43] M. W. Haverkort, Phys. Rev. Lett. 105, 167404 (2010).
[44] L. Savary and T. Senthil, arXiv:1506.04752.
[45] P. A. Fleury and R. Loudon, Phys. Rev. 166, 514 (1968).
[46] L. J. P. Ament, G. Khaliullin, and J. van den Brink, Phys. Rev.
B 84, 020403(R) (2011).
[47] J. Kim, D. Casa, M. H. Upton, T. Gog, Y.-J. Kim, J. F.
Mitchell, M. van Veenendaal, M. Daghofer, J. van den Brink,
G. Khaliullin, and B. J. Kim, Phys. Rev. Lett. 108, 177003
(2012).
064435-11
133
134
PHYSICAL REVIEW B 95, 024426 (2017)
Phase diagram and spin correlations of the Kitaev-Heisenberg model:
Importance of quantum effects
Dorota Gotfryd,1,2
Juraj Rusnaˇcko,3,4
Krzysztof Wohlfeld,1
George Jackeli,5,6
Jiˇr´ı Chaloupka,3,4
and Andrzej M. Ole´s2,6
1
Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, PL-02093 Warsaw, Poland
2
Marian Smoluchowski Institute of Physics, Jagiellonian University, prof. S. Łojasiewicza 11, PL-30348 Krak´ow, Poland
3
Central European Institute of Technology, Masaryk University, Kamenice 753/5, CZ-62500 Brno, Czech Republic
4
Department of Condensed Matter Physics, Faculty of Science, Masaryk University, Kotl´aˇrsk´a 2, CZ-61137 Brno, Czech Republic
5
Institute for Functional Matter and Quantum Technologies, University of Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany
6
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
(Received 16 August 2016; revised manuscript received 28 November 2016; published 23 January 2017)
We explore the phase diagram of the Kitaev-Heisenberg model with nearest neighbor interactions on the
honeycomb lattice using the exact diagonalization of finite systems combined with the cluster mean field
approximation, and supplemented by the insights from analytic approaches: the linear spin-wave and second-order
perturbation theories. This study confirms that by varying the balance between the Heisenberg and Kitaev term,
frustrated exchange interactions stabilize in this model either one of four phases with magnetic long range order:
N´eel phase, ferromagnetic phase, and two other phases with coexisting antiferromagnetic and ferromagnetic
bonds, zigzag and stripy phase, or one of two distinct spin-liquid phases. Out of these latter disordered phases,
the one with ferromagnetic Kitaev interactions has a substantially broader range of stability as the neighboring
competing ordered phases, ferromagnetic and stripy, have very weak quantum fluctuations. Focusing on the
quantum spin-liquid phases, we study spatial spin correlations and dynamic spin structure factor of the model
by the exact diagonalization technique, and discuss the evolution of gapped low-energy spin response across the
quantum phase transitions between the disordered spin liquid and phases with long range magnetic order.
DOI: 10.1103/PhysRevB.95.024426
I. INTRODUCTION
Frustration in magnetic systems occurs by competing
exchange interactions and leads frequently to disordered spinliquid
states [1–3]. Recent progress in understanding transition
metal oxides with orbital degrees of freedom demonstrated
many unusual properties of systems with active t2g degrees of
freedom—they are characterized by anisotropic hopping [4–8]
which generates Ising-like orbital interactions [9–17], similar
to the orbital superexchange in eg systems [18,19]. Particularly
challenging are 4d and 5d transition metal oxides, where the
interplay between strong electron correlations and spin-orbit
interaction leads to several novel phases [20,21]. In iridates
the spin-orbit interaction is so strong that spins and orbital
operators combine to new S = 1/2 pseudospins at each site
[22], and interactions between these pseudospins decide about
the magnetic order in the ground state.
The A2IrO3 (A = Na, Li) family of honeycomb iridates
has attracted a lot of attention as these compounds have t2g
orbital degree of freedom and lie close to the exactly solvable
S = 1/2 Kitaev model [23]. This model has a number of
remarkable features, including the absence of any symmetry
breaking in its quantum Kitaev spin-liquid (KSL) ground
state, with gapless Majorana fermions [23] and extremely
short-ranged spin correlations confined to nearest neighbors
[24]. We emphasize that below we call a KSL also disordered
spin-liquid states which arise near the Kitaev points in presence
of perturbing Heisenberg interactions ∝ J.
By analyzing possible couplings between the Kramers
doublets it was proposed that the microscopic model
adequate to describe the honeycomb iridates includes Kitaev
interactions accompanied by Heisenberg exchange in the
form of the Kitaev-Heisenberg (KH) model [25]. Soon after
the experimental evidence was presented that several features
of the observed zigzag order are indeed captured by the
KH model [26–34]. Its parameters for A2IrO3 compounds
are still under debate at present [35,36]. One finds also
a rather unique crossover from the quasiparticle states
to a non-Fermi-liquid behavior by varying the frustrated
interactions [37]. Unfortunately, however, it was recently
realized that this model is not sufficient to explain the observed
direction of magnetic moments in Na2IrO3, and its extension
is indeed necessary to describe the magnetic order in real
materials [38,39]. For example, bond-anisotropic interactions
associated with the trigonal distortions have to play a role to
explain the differences between Na2IrO3 and Li2IrO3 [40], the
two compounds with quite different behavior reminiscent of
the unsolved problem of NaNiO2 and LiNiO2 in spin-orbital
physics [19]. On the other hand, the KH model might be
applicable in another honeycomb magnet α-RuCl3, see, e.g.,
a recent study of its spin excitation spectrum [41].
Understanding the consequences of frustrated Heisenberg
interactions on the honeycomb lattice is very challenging and
has stimulated several studies [42–45]. The KH model itself is
highly nontrivial and poses an even more interesting problem
in the theory [25,34,46,47]: The Kitaev term alone has intrinsic
frustration due to directional Ising-like interactions between
the spin components selected by the bond direction [23]. In
addition, these interactions are disturbed by nearest neighbor
Heisenberg exchange which triggers long-range order (LRO)
sufficiently far from the Kitaev points [25,34,46,47]. In
general, ferromagnetic (FM) and antiferromagnetic (AF) interactions
coexist and the phase diagram of the KH model is quite
rich as shown in several previous studies [25,34,46–49]. Finally,
the KH model has also a very interesting phase diagram
on the triangular lattice [50–53]. These studies motivate better
2469-9950/2017/95(2)/024426(11) 024426-1 ©2017 American Physical Society
135
DOROTA GOTFRYD et al. PHYSICAL REVIEW B 95, 024426 (2017)
understanding of quantum effects in the KH model on the honeycomb
lattice in the full range of its competing interactions.
The first purpose of this paper is to revisit the phase diagram
of the KH model and to investigate it further by comparing the
exact diagonalization (ED) result [34] with the self-consistent
cluster mean field theory (CMFT), supplemented by the
insights from the linear spin-wave theory (LSWT) and the
second-order perturbation theory (SOPT). The main advantage
of CMFT is that it goes beyond a single site mean field classical
theory and gives not only the symmetry-broken states with
LRO, but partly includes quantum fluctuations as well, namely
the ones within the considered clusters [43,54,55]. In this way
the treatment is more balanced and may allow for disordered
states in cases when frustration of interactions dominates.
We present below a complete CMFT treatment of the phase
diagram which includes also the Kitaev term in MF part of the
Hamiltonian and covers the entire parameter space (in contrast
to the earlier prototype version of CMFT calculation on a single
hexagon for the KH model [56]). Note that the CMFT complements
the ED which is unable to get symmetry breaking for a
finite system, but nevertheless can be employed to investigate
the phase transitions in the present KH model by evaluating
the second derivative of the ground state energy to identify
phase transitions by its characteristic maxima [25,34]. The
ED result can be also used to recognize the type of magnetic
order by transforming to reciprocal space and computing spinstructure
factor. The second purpose is to investigate further the
difference between quantum KSL regions around both Kitaev
points mentioned in Ref. [34] and LRO/KSL boundaries.
The paper is organized as follows: In Sec. II we introduce
the KH model and define its parameters. In Sec. III we
present three methods of choice: (i) the exact diagonalization
in Sec. III A, (ii) the self-consistent CMFT in Sec. III B, and
(iii) linear spin wave theory in Sec. III D. An efficient method
of solving the self-consistence problem obtained within the
CMFT is introduced in Sec. III C. The numerical results are
presented and discussed in Sec. IV: (i) the phase transitions
and the phase diagram are introduced in Sec. IV A, and
(ii) the phase boundaries, the values of the ground state energies,
and the magnetic moments obtained by different methods
are presented and discussed in Secs. IV B and IV C, and
(iii) we discuss the impact of the Kitaev interaction on different
spin ordered states in Sec. IV C. Spin correlations obtained
for various phases are presented in Sec. V. The dynamical
spin susceptibility and spin structure factor are introduced and
analyzed for different phases in Sec. VI. Finally, in Sec. VII
we present the main conclusions and short summary. The
paper is supplemented with the Appendix where we explain
the advantages of the linearization procedure implemented on
the CMFT on the example of a single hexagon.
II. KITAEV-HEISENBERG MODEL
We start from the KH Hamiltonian with nearest neighbor
interactions on the honeycomb lattice in a form,
H ≡ K
ij γ
S
γ
i S
γ
j + J
ij
Si · Sj , (2.1)
where γ = x,y,z labels the bond direction. The Kitaev term
∝ K favors local bond correlations of the spin component
interacting on the particular bond. The superexchange J is
of Heisenberg form and alone would generate a LRO state,
antiferromagnetic or ferromagnetic, depending on whether
J > 0 or J < 0. We fix the overall energy scale,
J2
+ K2
= 1, (2.2)
and choose angular parametrization:
K = sin ϕ, J = cos ϕ, (2.3)
varying ϕ within the interval ϕ ∈ [0,2π]. This parametrization
exhausts all the possibilities for nearest neighbor interactions
in the KH model.
While zigzag AF order was observed in Na2IrO3 [28–32],
its microscopic explanation has been under debate for a
long time. The ab initio studies [35,57] give motivation
to investigate a broad regime of parameters K and J, see
Eqs. (2.3). Further motivation comes from the honeycomb
magnet α-RuCl3 [41]. Note that we do not intend to identify the
parameter sets representative for each individual experimental
system, but shall concentrate instead on the phase diagram of
the model Eq. (2.1) with nearest neighbor interactions only.
III. CALCULATION METHODS
A. Exact diagonalization
We perform Lanczos diagonalization for an N = 24-site
cluster with periodic boundary conditions (PBC). This cluster
respects all the symmetries of the model, including hidden
ones. Among the accessible clusters it is expected to have the
minimal finite-size effects.
B. Cluster mean field theory
A method which combines ED with an explicit breaking
of Hamiltonian’s symmetries is the so-called self-consistent
CMFT. It has been applied to several models with frustrated
interactions, including the Kugel-Khomskii model [54,55].
The method was also extensively used by Albuquerque et al.
[43] as one of the means to establish the full phase diagram of
the Heisenberg-J2-J3 model on the honeycomb lattice.
Within CMFT the internal bonds of the cluster [connecting
the circles in Fig. 1(a)] are treated exactly. The corresponding
part HIN of the Hamiltonian is the nearest neighbor KH
Hamiltonian, Eq. (2.1). The external bonds that connect the
boundary sites (•) with the corresponding boundary sites of
periodic copies of the cluster ( ) are described by the MF part
of the Hamiltonian,
HMF ≡ K
[ij] z
Sz
i Sz
j + J
[ij]
Sz
i Sz
j , (3.1)
where [ij] marks the external bonds. Since the ordered
moments in the KH model align always along one of the cubic
axes x, y, z (see, e.g., Ref. [25]), we have put
Si · Sj ≡ Sz
i Sz
j (3.2)
in HMF to simplify the calculations.
The averages Sz
i generate effective magnetic fields acting
on the boundary sites of the cluster. The total Hamiltonian
H ≡ HIN + HMF (3.3)
024426-2
136
PHASE DIAGRAM AND SPIN CORRELATIONS OF THE . . . PHYSICAL REVIEW B 95, 024426 (2017)
Γ’
(a)
(b) (c)
4π/3
q
z
x y
1 y
qx
Γ
X
x
y
z
x
z
y
M
FIG. 1. (a) 24-site cluster and the introduction of the mean fields.
Gray (black) circles indicate internal (boundary) sites. In CMFT the
internal bonds of the cluster are treated exactly while the external
bonds crossing the cluster boundary (dashed) are treated on the MF
level. The sites marked by generate effective magnetic fields on
the boundary sites •. Labels x, y, and z stand for three inequivalent
bond directions determining the active products S
γ
i S
γ
j in the Kitaev
part of the Hamiltonian (2.1), e.g., bonds of x direction contribute
with the Sx
i Sx
j product to the Hamiltonian, etc. The pseudospin axes
used here are parallel to the cubic axes indicated in the top view of a
single octahedron. (b) Unit cells: for honeycomb lattice (coinciding
with a single hexagon of that lattice), for triangular lattice (inner
dotted hexagon), and zigzag magnetic unit cell (dashed rectangle).
Black and white circles stand for up/down spin and indicate one of
three equivalent zigzag patterns. (c) Corresponding Brillouin zones
and special q points for the lattice constant a = 1. The q vectors
compatible with the 24-site cluster in (a) are also shown.
is diagonalized in a self-consistent manner, taking a slightly
different approach than the one presented in Ref. [43]: Instead
of starting with a random wave function our algorithm begins
with expectation values Sz
i in on each boundary site i of the
cluster. These can represent a certain pattern (zigzag, stripy,
N´eel, FM) or be set randomly to have a “neutral” starting
point. After diagonalizing the Hamiltonian (3.3) (again by
the ED Lanczos method) the ground state of the system is
obtained and we recalculate the expectation values Sz
i to be
used in the second iteration. The procedure is repeated until
self-consistency is reached.
C. Linearized cluster mean field theory
A single iteration of the self-consistent MF calculation may
be viewed as a nonlinear mapping of the set of initial averages
0
2
4
6
8
10
12
14
0 π/2 π 3π/2 2π
−d2
EGS/dϕ2 ϕ
lee´N zigzag FM stripy lee´N
10-1
10
0
10
1
102
0 π/2 π 3π/2 2π
λ
(a)
(b)
FIG. 2. (a) The values of λ obtained by the linearization of CMFT
for an embedded cluster of N = 24 sites with fixed magnetic order
patterns: FM, AF, stripy, and zigzag. Leading λ > 1 indicates the
order that sets in. The disordered KSL phases near ϕ = π/2 and
3π/2 are indicated by red. (b) Second derivative of the ground state
energy, −d2
E0(ϕ)/dϕ2
, obtained by ED. Adopted from Ref. [34].
{ Sz
i in} to the resulting averages { Sz
i fin}. The self-consistent
solution is then a stable stationary point of such a mapping.
To find the leading instability, we may consider the case of
small initial averages in the CMFT calculation and identify
the pattern characterized by the fastest growth during the
iterations. To this end we linearize the above mapping.
In the lowest order the mapping corresponds to the
multiplication of the vector of the averages { Sz
i in} by the
matrix,
Fij =
∂ Sz
i fin
∂ Sz
j in
, (3.4)
where i and j run through the cluster boundary sites.
During iterations, the patterns corresponding to the individual
eigenvectors of the matrix F grow as λn
after n iterations for
a particular eigenvalue λ. The ordering pattern obtained by
CMFT is then given by the eigenvector with largest λmax > 1.
In the quantum KSL regimes, all the eigenvalues are less than
1 and no magnetic order emerges. An example of linearized
CMFT applied to a single hexagon with PBC can be found in
the Appendix.
A modified version of this method, used to obtain Fig. 2(a),
assumes a particular ordered pattern (N´eel, zigzag, FM, or
stripy phase) and uses a single spin average Sz
in distributed
along the boundary sites outside the cluster, with the signs
consistent with this pattern. The resulting values, Sz
i fin, are
then averaged correspondingly. In this case the matrix F is
reduced to a single value λ plotted in Fig. 2(a). We observe that
the largest eigenvalue either drops below 1 when the disordered
024426-3
137
DOROTA GOTFRYD et al. PHYSICAL REVIEW B 95, 024426 (2017)
KSL state takes over, or interchanges with another eigenvalue
at a quantum phase transition to a different ordered phase.
D. Linear spin-wave theory
The LSWT is a basic tool to determine spin excitations and
quantum corrections in systems with LRO [58]. For systems
with coexisting AF and FM bonds quantum corrections are
smaller than for the N´eel phase on the same lattice but are
still substantial for S = 1/2 spins [59]. For the KH model the
LSWT [25,29,34] has to be implemented separately for each of
the four ordered ground states: N´eel (N), zigzag (ZZ), FM, or
stripy (ST). Then for a particular ground state the Hamiltonian
is rewritten in terms of the Holstein-Primakoff bosons [29,60]
and only quadratic terms in bosonic operators are kept. The
spectrum of such a quadratic Hamiltonian is finally obtained
using the successive Fourier and Bogoliubov transformations.
While the spin wave dispersion relations are usually of
prime interest [25,29,34,60], there are also two other quantities
which can easily be calculated using LSWT and which will
be important in the discussion that follows: (i) the value of the
total ordered moment M per site, and (ii) the total energy
per site E . These observables are calculated in a standard
way [58,59] and expressed in terms of the eigenvalues, i.e.,
spin-wave energies ωkα, and the eigenvector components
{vkαλ} of the bosonic Hamiltonian before the Bogoliubov
transformation:
M = S −
1
LV α,λ=1,...,L k∈BZ
|vkα,λ|2
d2
k, (3.5)
and
E =Ecl [S2
→ S(S + 1)]
+
S
2LV α=1,...,L k∈BZ
ωkα d2
k, (3.6)
where the choice of the sign of the eigenvalues and the
normalization of their eigenvectors is described in Ref. [58].
Here Ecl is the classical ground state energy per site, e.g.,
Ecl = −JzS2
/2, (3.7)
with z = 3 for the N´eel phase at K = 0 and S = 1/2 is the
value of spin quantum number. L in Eqs. (3.5)–(3.6) is the
number of the eigenvalues of the problem (spin-wave modes)
and α enumerates these modes. For all cases except for the
zigzag order [25], the integrals go over the two-sublattice
(L = 2) rectangular Brillouin zone (BZ) [61] with its volume
V = 8π2
/3
√
3 and −π/
√
3 kx π/
√
3, −2π/3 ky
2π/3 (as already mentioned we assume the lattice constant
a = 1). For the zigzag state L = 4 and the rectangular BZ can
be chosen as: −π/
√
3 kx π/
√
3 and −π/3 ky π/3
and its volume is V = 4π2
/3
√
3.
IV. QUANTUM PHASE TRANSITIONS
A. Phase diagram
Here we supplement the ED-based phase diagram for the
KH model Eq. (2.1) established in Ref. [34] with the one
obtained within CMFT. Figure 3 displays the phase boundaries
obtained with ED [34], within CMFT, as well as classical
ϕ
FM
leeNgazgiz
stripy
FIG. 3. T = 0 phase diagram for KH model. The outer ring is
composed from ED data for the 24-site cluster, reproducing the
result from Ref. [34] in the new parametrization, the middle ring
shows CMFT results also for 24-site cluster and the inner black circle
represents the classical result. The convention used for the angular
parameter ϕ which determines coupling constants [see Eqs. (2.3)] is
shown in the center of the inner circle. The colors represent particular
phases, shown also as mini drawings next to suitable regions of the
phase diagram. Starting from ϕ = 0 green colored region corresponds
to N´eel order, red—KSL, yellow—zigzag order, dark blue—FM,
red—KSL, light blue—stripy phase, and again green—N´eel phase.
(Luttinger-Tisza) phase boundaries. The latter are included
for completeness and to highlight the fact that the quantum
fluctuations stabilize the KSL phases beyond single points,
see below. To examine them in more detail it is instructive to
analyze the data in Fig. 2(a) for the boundaries obtained from
linearized CMFT and Fig. 2(b) for the peaks in the second
derivative of energy, −d2
E0(ϕ)/dϕ2
, giving phase boundaries
in ED [34].
It is clearly visible that all the methods that include quantum
fluctuations give quantum versions of the four classically
established magnetic phases: N´eel, zigzag, FM, and stripy.
As the most important effect we note that when quantum
fluctuations are included within a classical phase, the energy
is generally lowered and that the emerging phase is expected
to expand beyond the classical boundaries, but only in cases
when a phase which competes with it has weaker quantum
fluctuations. This implies that phases of AF nature will expand
at the expense of the FM ones as the latter phases have lower
energy gains by quantum fluctuations (which even vanish
exactly for the FM order at K = 0 and J < 0).
We summarize the phase boundaries obtained within
different methods in Table I. One finds substantial corrections
to the quantum phase transitions which follow from quantum
fluctuations. These corrections are quite substantial in both
KSLs at the Kitaev points (K = +1, ϕ = 1
2
π and K = −1,
024426-4
138
PHASE DIAGRAM AND SPIN CORRELATIONS OF THE . . . PHYSICAL REVIEW B 95, 024426 (2017)
TABLE I. Phase boundaries for the KH model, parameterized
by the angle ϕ (in units of π), see Eqs. (2.3). Columns: classical
Luttinger-Tisza approximation, second-order perturbation theory
(SOPT), exact diagonalization (ED), and self-consistent cluster mean
field theory (CMFT).
Boundary Classical SOPT ED CMFT
N´eel/KSL 0.5 0.492 0.494 0.496
KSL/zigzag 0.5 0.507 0.506 0.505
zigzag/FM 0.75 0.813 0.814 0.825
FM/KSL 1.5 1.463 1.448 1.478
KSL/stripy 1.5 1.530 1.539 1.519
stripy/N´eel 1.75 1.705 1.704 1.699
ϕ = 3
2
π, first column of Table I). Indeed, in the classical
approach massively degenerate ground states exist just at
isolated points, but they are replaced by disordered spin-liquid
states that extend to finite intervals of ϕ when quantum
fluctuations are included, see the second, third, and fourth
column in Table I. The expansion of N´eel and zigzag phases
beyond classical boundaries is given by particularly large
corrections and is well visible.
The most prominent feature in the phase diagram described
above is however the difference in size between two KSL
regions, already addressed before using ED [34] and also
visible now in the CMFT data. Therefore, the CMFT result
supports the claim from Ref. [34] that the stability of KSL
perturbed by relatively small Heisenberg interaction depends
on the nature of the phases surrounding the spin liquid
and the amount of quantum fluctuations that they carry. In
the following we discuss the above issues more thoroughly,
examining: (i) ground state energy curves emerging from ED,
CMFT, SOPT within the linked cluster expansion and LSWT,
(ii) the ordered moment given by various methods, (iii) the
spin-spin correlation functions, and (iv) the spin structure
factor as well as the dynamical spin susceptibility in the
vicinity of the Kitaev points.
B. Quantum corrections: Energetics
We start the discussion of quantum corrections to the energy
of the ordered phases by noting that, even though it properly
captures finite order parameters, the CMFT looses quantum
energy on the external bonds and would therefore not provide
a reliable estimate of the ground-state energy. However, if
one calculates instead the energy based on the correlations on
the bonds of the central hexagon, the estimate is significantly
improved. Here we choose the energy obtained using the ED
calculations [see Fig. 4(a)] as a reference value because of
all the bonds treated in an exact manner. This observation
is also supported by the fact that the ED phase boundaries
were roughly confirmed by tensor networks (iPEPS) [49]
and density matrix renormalization group (DMRG) results
[48]: The iPEPS phase boundaries agree with ED for AF
KSL/LRO transitions and the boundaries between different
LRO phases differ only slightly from those found in ED
(iPEPS: zigzag/FM—0.808π, stripy/N´eel—1.708π). For FM
KSL/LRO transition however the iPEPS result deviates more,
i.e., KSL/stripy—1.528π. On the other hand, DMRG bound-
0.0
0.1
0.2
0.3
0.4
0.5
0 π/2 π 3π/2 2π
orderedmoment
ϕ
lee´NypirtsMFgazgizlee´N
LSWT
CMFT
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0 π/2 π 3π/2 2π
energy/site
ϕ
(a)
(b)
ED
classical
LSWT
SOPT
CMFT
FIG. 4. (a) Comparison between ground state energies per site
obtained using various methods: classical Luttinger-Tisza approximation
(dashed black), SOPT (solid red), LSWT (dashed red), ED
for 24-site cluster (solid blue, see Ref. [34] for this result in a different
parametrization), and CMFT (energy given by the central hexagon,
solid green). (b) Ordered moment obtained from CMFT (solid green
line for the central hexagon, dashed green line for the value for
intermediate and boundary sites) and LSWT (dashed red line).
aries agree perfectly with ED and due to four-sublattice dual
transformation [10,25] one can reproduce the FM/zigzag as
well as FM/KSL boundaries. Only the extent of the AF
spin-liquid phase cannot be extracted from this result, but that
is already confirmed by iPEPS.
Figure 4(a) shows a quite remarkable agreement between
the energy values and critical values of ϕ obtained by the
simplest SOPT [25] and our reference ED results. This
suggests that this analytical method can be utilized to get better
insight to the quantum contributions to the ground state energy.
For a phase X with LRO, the energy per site EX, written as a
sum of the classical energy Ecl and the quantum fluctuation
contribution EX, is obtained as:
EN = −
1
8
(K + 3J) −
1
16
(K + 3J), (4.1)
EZZ = −
1
8
(K − J) −
1
16
(K − J), (4.2)
EFM = +
1
8
(K + 3J) +
1
16
K2
K + 2J
, (4.3)
024426-5
139
DOROTA GOTFRYD et al. PHYSICAL REVIEW B 95, 024426 (2017)
EST = +
1
8
(K − J) +
1
16
(K + 2J)2
K
. (4.4)
In addition, to get the LRO/KSL phase boundary points in
Table I, we estimate the energy of the KSL phase as
EKSL
3
2
(K + J) Sγ
Sγ
Kitaev, (4.5)
using the analytical result for the Kitaev points [24],
Sγ
Sγ
Kitaev ≈ ±0.131.
The two spin-liquid phases in the phase diagram of the
KH model differ strongly in their extent, despite the formal
equivalence of the FM (K = −1) and AF (K = 1) Kitaev
points provided by an exact mapping of the Kitaev Hamiltonian
[23]. As mentioned earlier, this is due to the fact that the
two KSLs compete with LRO phases of a distinct nature.
Here we give a simple interpretation based on the strength
of the quantum corrections of the LRO phases estimated using
Eqs. (4.1)–(4.4). Later, in Secs. V and VI we illustrate the
different nature of the transitions between FM and AF KSL
and the surrounding it LRO phases in terms of spin correlations
and spin dynamics.
Let us now compare the quantum fluctuation contribution
and the classical one. For the LRO phases surrounding the AF
spin liquid—N´eel and zigzag—we always have E/Ecl = 1
2
as deduced from Eqs. (4.1) and (4.2), i.e., only 2
3
EN and 2
3
EZZ
are found in the classical approach. This guarantees that the
quantum phase transition between these two types of order
occurs at the same value of ϕ = π/2 in SOPT and in the
classical approach that do not capture the spin-liquid phase
in between these ordered states, see Fig. 4(a). In contrast, the
phases neighboring to the FM spin liquid—FM and stripy—
would reach the value of E/Ecl = 1
2
only at the FM Kitaev
point with J = 0 and away from this point the contribution
of quantum fluctuations decreases rapidly allowing for a large
extent of the FM spin-liquid phase. Note that both these latter
phases contain a point which is exactly fluctuation free—for
the FM phase when frustration is absent (K = 0), and for the
stripy phase it is related to the FM one by the interaction
transformation [39] at K = −2J.
Moving to the CMFT energy analysis one should also keep
in mind that within the CMFT method the external bonds
between Sz
i and Sz
j do not include quantum fluctuations fully.
This implies a worse estimate of the energy (of the whole
cluster) for regions of the phase space that allow quantum
fluctuations. As a consequence the region of stability of FM
spin-liquid phase is smaller than that obtained in the ED.
Significantly better energy estimate is given by the central
hexagon, for which all the bonds experience exact interactions.
As a result, this CMFT energy curve [green line in Fig. 4(a)]
lies almost as close to ED energy as the SOPT one. Finally, the
estimates obtained from LSWT, which represents a harmonic
approximation to the quantum fluctuations, are not as good as
those from central hexagon via CMFT and SOPT, see dashed
red lines in Fig. 4(a). As expected, the energy obtained from the
LSWT agrees well with ED curve for phases with less quantum
fluctuations, FM and stripy phase, and starts to diverge when
these phases are unstable beyond quantum phase transitions
within N´eel and zigzag phases.
C. Quantum corrections: Ordered moment
As usual, getting the correct value of the ordered moment
turns out to be a more difficult task than estimating the
ground state energy. This is primarily due to the fact that
the ED does not capture the symmetry-broken states and the
ordered moment can only be indirectly extracted from the
m2
; moreover, the SOPT may not be reliable here. Hence,
we are mostly left with the results obtained with CMFT
and LSWT. We discuss the corresponding data [shown in
Fig. 4(b)] together with the several values given already in the
literature.
Let us begin with the Heisenberg AF point ϕ = 0: Here
it is expected that the ordered moment should be strongly
reduced by quantum fluctuations. LSWT estimates the ordered
moment value at 0.248 [61]. Similar values were extracted
from m2
in quantum Monte Carlo (0.268 [62–64]) and ED
(0.270 [43]) calculations. In the last case however the authors
admit that the set of clusters for finite size scaling was chosen
so as to make the best agreement with quantum Monte Carlo.
Another method—series expansion (high order perturbation
theory) [47] sets ordered moment value at a somewhat higher
value of 0.307. While all the above results seem roughly
consistent, CMFT value obtained from the boundary sites
seems to stand out (0.374 for ϕ = 0). Nevertheless, the
central-hexagon value (0.330 for ϕ = 0) lies much closer to
the results from the methods mentioned above. Moreover, one
should note that the ordered moment estimated from m2
for
24-site cluster ED equals 0.45 [43] which is above the CMFT
value. This suggests that at this point the finite size scaling is
important.
Before moving to the frustrated regime we briefly mention
that the trivial ordered moment value at ϕ = π is here
correctly reproduced by both CMFT and LSWT. Besides,
for the regions around the fluctuation-free FM (and stripy)
point the ordered moments predicted by CMFT and LSWT
also match. Following the ground state energy analysis,
LSWT gives the correct result because quantum fluctuations
contribution is small compared to the classical state. The
further one moves towards the Kitaev points, however, the
more incorrect the LSWT approximation should be because of
the strong reduction of the ordered moment due to increasing
frustration.
In contrast, the lack of quantum fluctuations on the external
bonds generates systematic errors within CMFT except for
FM and stripy phases. The ordered moment obtained from
the boundary sites experiences the errors discussed above.
However, the ordered moment values for intermediate sites
and the central hexagon become largely reduced in the whole
N´eel and zigzag regions due to the fact that for the internal
part of the cluster the fluctuations are fully included. Still,
the best estimate comes from the central hexagon where
quantum fluctuations on the bonds are included and CMFT
gives more realistic results than LSWT in frustrated parts of
the phase diagram. Here it is also important to stress that the
series expansion captures correctly the fluctuation-free point
at ϕ = π (FM) and ϕ = − arctan 2 (stripy) and predicts a
broader region of the FM KSL phase [47]. The order parameter
is also qualitatively correctly estimated and is reduced more
to m 0.3 for both N´eel and zigzag phases [47]. However,
while the ordered moment values obtained by CMFT are
024426-6
140
PHASE DIAGRAM AND SPIN CORRELATIONS OF THE . . . PHYSICAL REVIEW B 95, 024426 (2017)
consistent with the four-sublattice dual transformation, the
ordered moment data from the high-order perturbation theory
[47] are not as the values of ordered moment differ at the points
connected by the mapping. Unfortunately the largest difference
appears near the FM LRO/KSL boundaries. This observation
uncovers certain shortcomings of the high-order perturbation
theory.
D. Quantum corrections: Naive interpretation
Let us conclude the discussion of the quantum corrections
with the following more general observation: Developing the
argumentation presented by Iregui, Corboz, and Troyer [49],
the dependence of the quantum corrections to the energy and
to the ordered moment on the angle ϕ suggests that the Kitaev
interaction is less “compatible” with the FM/stripy ground
states than with the N´eel/zigzag ones. This can be understood
in the simple picture of the KH model on a four-site segment
of the honeycomb lattice consisting of three bonds attached to
a selected lattice site, as presented below.
Starting with ϕ = π (FM ground state, e.g. along the z
quantization axis), increasing ϕ leads to gradual increase of
the FM Kitaev term which favors ferromagnetically aligned
spins along the x, y, and z quantization axes for the x, y, and
z directional bonds, respectively. It can easily be seen that,
e.g., for the x bond, the eigenstate of the FM Kitaev-only
Hamiltonian on that bond (|↑x↑x ) has a 25% overlap with the
FM ground state, | ↑z↑z|↑x↑x |2
= 1
4
. While again a similar
situation happens for the y bond, the overlap between such
states for the z bond is maximal, i.e., these states are identical
(we assume the same phase factors 1).
Next, we perform a similar analysis for ϕ = 0 and firstly
assume that we deal with a classical N´eel ground state, |↑z↓z .
In this case for the “unsatisfied” bonds from the point of view
of the increasing AF Kitaev interaction we also obtain that the
eigenstate of the AF Kitaev-only Hamiltonian (|↑x↓x ) on that
bond has a 25% overlap with the classical N´eel ground state—
e.g., | ↑z↓z|↑x↓x |2
= 1
4
. However, this situation changes
once we consider that the spin quantum fluctuations dress the
classical N´eel ground state. This can be best understood if we
assumed the unrealistic but insightful case of very strong quantum
fluctuations destroying the classical N´eel ground state:
then for the x bond a singlet could be stabilized and the overlap
between such a state and the state “favored” by the Kitaev
term increases to 50%: | 0|↑x↓x |2
= 1
2
. This suggests that the
N´eel ground state, which contains quantum spin fluctuations,
is more “compatible” with the states “favored” by the Kitaev
terms than the FM ground state, resulting in more stable values
of ordered moment for N´eel phase. It seems that the above difference
is visible in CMFT data but not in LSWT ones. We shall
discuss this issue further by analyzing spin correlations below.
V. SPIN CORRELATIONS
Additional information about the ground state is given by
spin-spin correlation functions. In Fig. 5(a) one can observe
isotropic stable S
γ
i S
γ
j correlations in almost the entire AF
phase (with Si · Sj ≈ −0.36 for ϕ = 0), while for the FM
phase the anisotropy quickly develops when moving away
from the FM Heisenberg point ϕ = π (here Si · Sj reaches
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0 π/2 ϕ = π 3π/2 2π
bondcorrelation
total
Kitaev
complementary
0.00
0.05
0.10
0.15
0.20
88° 89° 90° 91° 92°
|bondcorrelation|
2
3
4
5
0.00
0.05
0.10
0.15
0.20
88° 89° 90° 91° 92°
|bondcorrelation|
ϕ
2
3
4
5
260° 270° 280°
2
3
4
5
260° 270° 280°
ϕ
(a)
)c()b(
)e()d(
2
3
4
5
FIG. 5. (a) Spin correlations Si · Sj obtained within ED for the
bonds between nearest neighbors (black line), spin correlations of the
components active in the Kitaev interaction, S
γ
i S
γ
j (blue line), and
complementary spin components, S
¯γ
i S
¯γ
j (red line). Below further
neighbor spin correlations | Si · Sj | are shown (colors correspond
here to different neighbors). ED: (b) near the AF spin-liquid phase,
and (c) for the angle ϕ interval including the FM spin-liquid phase.
CMFT—the neighborhood of the: (d) AF spin-liquid, and (e) FM
spin-liquid region.
the classical value 0.25). This again demonstrates that the AF
(and zigzag) phase is more robust and uniform than the FM
(and stripy) phase.
Moreover, spin-spin correlations allow us to confirm the
disordered regions around the Kitaev points as critical cases
of quantum spin liquid [65]. At the Kitaev points (J = 0)
we observe the expected undisturbed KSL pattern: nonzero
values of nearest neighbor correlations between spin
components active in the Kitaev interaction [blue curve in
Fig. 5(a)] and vanishing correlations between complementary
components (red curve). In contrast, the next nearest and
further neighbor correlations disappear, see Figs. 5(b) and 5(c).
While moving away from the Kitaev points the absolute values
of the correlations enter the regions of slow growth—these
are signatures of the critical spin-liquid phases and they look
024426-7
141
DOROTA GOTFRYD et al. PHYSICAL REVIEW B 95, 024426 (2017)
similar in AF and FM spin liquid cases. At some point however
proceeding further results in rapidly growing absolute values
which mark KSL/LRO boundaries.
Furthermore, Figs. 5(b) and 5(c) prove that there is a
qualitative difference between the two spin-liquid regimes.
This is observed in the rapid growth of spin correlations at the
onset of LRO: The steplike jump visible in Fig. 5(b) contrasts
with the smoother crossover seen in Fig. 5(c). Below we
investigate this distinct behavior by analyzing the dynamical
spin susceptibility for various available phases. After Fourier
transformation of the z-component correlations, we obtain the
spin structure factor to be discussed in the context of the spin
susceptibility also in Sec. VI.
As a supplement we present the further neighbor spin
correlations obtained via CMFT [Figs. 5(d) and 5(e)]. One
should remark that within KSL the averages Sz
i are 0 and
CMFT is thus equivalent to ED for an isolated cluster (open
boundary conditions). This leads to stronger finite size effects
and larger inhomogeneity of the correlations. Nevertheless,
considering the central part of the cluster, the emergence
of the longer-range correlations away from the Kitaev point
presented in Figs. 5(d) and 5(e) is almost identical to that
calculated by ED, see Figs. 5(b) and 5(c).
VI. SPIN SUSCEPTIBILITY AND EXCITATIONS IN THE
VICINITY OF THE KITAEV POINTS
Below we study the spin dynamics within the KH model
by analyzing the dynamical spin susceptibility at T = 0,
χαα(q,ω) = i
∞
0
0| Sα
q (t),Sα
−q(0) | 0 eiωt
dt, (5.1)
with the Fourier-transformed spin operator defined via
Sα
q =
1
√
N R
e−iq·R
Sα
R , (5.2)
and | 0 denoting the cluster ground state. For ω > 0, the
imaginary part of χ(q,ω)αα reads as
χαα(q,ω) = −Im 0| Sα
q
1
ω + EGS − H + iδ
Sα
−q | 0 ,
(5.3)
which can be conveniently expressed as a sum over the excited
states {|ν },
χαα(q,ω) = π
|ν
ν|Sα
−q| 0
2
δ(ω − Eν), (5.4)
where the excitation energy Eν is measured relative to the
ground state energy EGS. We have evaluated χαα(q,ω) by ED
on a hexagonal cluster of N = 24 sites. In the ED approach,
the exact ground state of the cluster | 0 is found by Lanczos
diagonalization, the operator Sα
−q is applied, and the average
of the resolvent 1/(z − H) is determined by Lanczos method
using normalized Sα
−q| 0 as a starting vector [66].
In our case of the KH model, the calculation generally
requires a relatively large number of Lanczos steps (up to one
thousand) to achieve convergence of the dense high-energy
part of the spectrum. Having the advantage of being exact, the
method is limited by the q vectors accessible for a finite cluster
0.0
0.1
0.2
0.3
0.4
88° ϕ=90° 92°
ω
0
50
100
150
200
250
300
0.0
0.1
0.2
0.3
0.4
88° ϕ=90° 92°
ω
0
20
40
60
80
100
0.0
0.1
0.2
0.3
0.4
260° ϕ=270° 280°
ω
0
100
200
300
400
500
0.0
0.1
0.2
0.3
0.4
260° ϕ=270° 280°
ω
0
20
40
60
80
100
120
140
160
0.0
0.2
0.4
0.6
0.8
1.0
)b()a(
(c)
)e()d(
(f)
0.0
0.2
0.4
0.6
0.8
1.0
q = Γ’ q = M
q = Γ q = X
FIG. 6. (a) Dynamical spin susceptibility χ (q,ω) obtained by
ED near the AF KSL phase at the characteristic wave vector of the
AF order, q = . (b) The same for the zigzag wave vector q = M.
(c) Brillouin zone portraits of the spin-structure factor Sz
−qSz
q at
ϕ = 87.5◦
, 90◦
, and 92.5◦
(interpolated from the ED data). The inner
hexagon is the Brillouin zone of the honeycomb lattice; the outer one
corresponds to the triangular lattice with the missing sites filled in.
(d), (e) The same as in panels (a), (b) but for the interval containing
the FM (q = ) and stripy (q = X) phase. (f) Brillouin zone portraits
of the spin-structure factor obtained at ϕ = 255◦
, 270◦
, and 285◦
.
and compatible with the PBC, and by finite-size effects due
to small N. These concern mainly the low-energy part of χ
and lead, e.g., to an enlarged gap of spin excitations in LRO
phases of AF nature. Nevertheless, a qualitative understanding
can still be obtained.
The evolution of numerically obtained χ (q,ω)αα (5.4) with
varying ϕ is presented in Figs. 6(a) and 6(b) for the region
including AF spin-liquid phase, as well as in Figs. 6(d) and
6(e) for the region including the FM spin-liquid phase. The
transitions are well visible at the characteristic q vectors of
the individual LRO phases. The structure factor pattern, see
Figs. 6(c) and 6(f), changes accordingly between the sharply
peaked one in LRO phases and a wavelike form characteristic
for nearest neighbor correlations in the spin-liquid phases.
After entering the spin-liquid phase, further changes of the
spin response are very different for the AF and FM case. In the
024426-8
142
PHASE DIAGRAM AND SPIN CORRELATIONS OF THE . . . PHYSICAL REVIEW B 95, 024426 (2017)
AF case, there is a sharp transition—a level crossing for our
cluster, so that the ground state changes abruptly. The original
intense pseudo-Goldstone mode as well as many other excited
states become inactive in the spin-liquid phase. The observed
low-energy gap in χ (q,ω)αα varies only slightly with ϕ.
In contrast, when entering the FM spin-liquid phase the
excitation that used to be the gapless magnon mode is
characterized by a gradually increasing gap which culminates
at the Kitaev point. Starting from the Kitaev point, the gradual
reduction of the low-energy gap in χ (q,ω)αα due to the
Heisenberg perturbation manifests itself by a development
of finite spin correlations beyond nearest neighbors (already
reported in Fig. 2 of Ref. [25]) and an increase of the
static susceptibility to the magnetic field Zeeman coupled
to the order parameter of the neighboring LRO phase. This
susceptibility then diverges at the transition point (see also
Fig. 3 of Ref. [25]).
VII. SUMMARY AND CONCLUSIONS
In the present paper we report a study of the phase
diagram of the Kitaev-Heisenberg model by a combination of
exact diagonalization and cluster mean field theory (CMFT),
supplemented by the insights from linear spin-wave theory and
the second-order perturbation theory. Both methods allowed
to stabilize previously known phases with long range order:
N´eel, zigzag, FM, and stripy. Moreover, the ordered moment
analysis provided by cluster mean field approach demonstrates
N´eel-zigzag and FM-stripy connections described before [34].
Compared to the previous CMFT studies utilizing N = 6
site cluster (see Ref. [56] or the Appendix), we have used
a sufficiently large cluster of N = 24 sites preserving the
lattice symmetries and improving the ratio between internal
and boundary bonds. This led to a balanced approach which
allowed us to treat both ordered and disordered (spin-liquid)
states on equal footing.
As the main result, the present study uncovers a fundamental
difference between the onset of broken symmetry phases
in the vicinity of Kitaev points with antiferromagnetic or
ferromagnetic interactions. While the spin liquids obtained
at K = +1 and K = −1 are strictly equivalent and can be
transformed one into the other in the absence of Heisenberg
interactions (at J = 0), spin excitations and quantum phase
transitions emerging at finite J are very different in both cases.
For the antiferromagnetic Kitaev spin liquid phase (K 1)
one finds that a gap opens abruptly in χ (q,ω) at q =
and q = M when the ground state changes to the critical
Kitaev quantum spin liquid. This phase transition is abrupt
and occurs by level crossing. In contrast, for ferromagnetic
spin liquid K −1 the gaps in χ (q,ω) at q = and q = X
open gradually from the points of quantum phase transition
from ordered to disordered phase. With much weaker quantum
corrections for ordered phases in the regime of ferromagnetic
Kitaev interactions, the spin liquid is more robust near K = −1
as a phase that contains quantum fluctuations and survives in
a broader regime than near K = 1 when antiferromagnetic
Kitaev interactions are disturbed by increasing (antiferromagnetic
or ferromagnetic) Heisenberg interactions. This behavior
is reminiscent of the ferromagnetic Kitaev model in a weak
magnetic field [65].
ACKNOWLEDGMENTS
We thank Giniyat Khaliullin for insightful discussions.
We kindly acknowledge support by Narodowe Centrum
Nauki (NCN, National Science Center) under Project No.
2012/04/A/ST3/00331. The CMFT spin-spin correlations
were calculated at the Interdisciplinary Centre for Mathematical
and Computational Modelling (ICM) of the University of
Warsaw under Grant No. G66-22. J.R. and J.C. were supported
by Czech Science Foundation (GA ˇCR) under Project No.
GJ15-14523Y and by the project CEITEC 2020 (LQ1601) with
financial support from the Ministry of Education, Youth and
Sports of the Czech Republic under the National Sustainability
Programme II. Access to computing and storage facilities
owned by parties and projects contributing to the National
Grid Infrastructure MetaCentrum, provided under the program
“Projects of Large Research, Development, and Innovations
Infrastructures” (CESNET LM2015042), is acknowledged.
G.J. is supported in part by the National Science Foundation
under Grant No. NSF PHY11-25915.
APPENDIX: COMPARISON BETWEEN CMFT AND
LINEARIZED CMFT FOR A SINGLE HEXAGON
Here we compare linearization results for a single hexagon
with full CMFT to see how well linearized CMFT performs as a
shortcut method. It is important to realize that this cluster is not
compatible with stripy or zigzag order because of their four-site
magnetic unit cell, see Fig. 1(b), and they are suppressed within
vast regions of ϕ compared to the 24-site case. The size of
the system allows for quick CMFT computations and enables
detailed comparison between the two approaches. Moreover,
specific problems linked to the above incompatibility make the
N = 6-site cluster a good test case to illustrate the linearized
CMFT.
Following the procedure described in Sec. III C, 6 eigenvalues
λi are produced for each value of ϕ parameter. The
corresponding spin patterns are inferred by inspecting the
eigenvectors. Only the patterns associated with λi > 1 are
able to grow during iterations and eventually stabilize as a
10-2
10-1
100
10
1
10
2
103
0 π/2 π 3π/2 2π
λ
ϕ
λmax > 1
FIG. 7. Full linearized CMFT result for a single hexagon. Blue
lines represent all emerged positive eigenvalues λ, while maximal λ
larger than 1 is indicated in red.
024426-9
143
DOROTA GOTFRYD et al. PHYSICAL REVIEW B 95, 024426 (2017)
0.0
0.1
0.2
0.3
0.4
0.5
0 π/2 π 3π/2 2π
|〈S
z
i〉|
ϕ
(b)
(a)
FIG. 8. (a) Spin patterns obtained for a single hexagon by CMFT.
From the left: N´eel, zigzag, FM, and stripy. (b) Phase diagram for a
single hexagon determined by | Sz
i |. Red and blue sites (see inset)
are nonequivalent in the present CMFT due to the approximation
given by Eq. (3.2) which generates the terms ∝ J that add to the
Kitaev term only on the vertical bonds ij z in the MF part of the
Hamiltonian (2.1).
self-consistent solution of full CMFT. Comparison of both
methods presented in Figs. 7 and 8 provides the phase diagram
for a single hexagon: N´eel phase for ϕ ∈ [0,0.5)π, KSL for
ϕ = π
2
, zigzag phase for ϕ ∈ (0.5,0.555)π, disordered region
I for ϕ ∈ (0.555,0.864)π, FM phase for ϕ ∈ (0.864,1.5)π,
KSL for ϕ = 3
2
π, stripy phase for ϕ ∈ (1.5,1.62)π (linearization),
ϕ ∈ (1.5,1.64)π (CMFT), disordered region II
for ϕ ∈ (1.62,1.684)π (linearization) and ϕ ∈ (1.64,1.684)π
(CMFT), and N´eel phase for ϕ ∈ (1.684,2]π. In contrast to
N = 24 cluster the two spin-liquid regions are replaced by
single points ϕ = π
2
and ϕ = 3
2
π.
Striking difference between phase diagrams for 24-site and
6-site clusters is the reduction of the zigzag and stripy phases
and the emergence of two regions of disorder indicated by
two gray-shaded regions. Here all λi < 1 and no spin pattern
is strong enough to stabilize. Zigzag pattern emerges from
CMFT with random initial values of Sz
i without additional
help. Stripy pattern however is more difficult to catch. As
one can see in Fig. 7, two different λi corresponding to two
stripy patterns exchange at ϕ = 1.568π. Unfortunately, huge
parasitic oscillations make these patterns extremely difficult
to stabilize within CMFT. These stem from a large negative
λi that previously corresponded to FM pattern and decreased
rapidly for ϕ > 1.5π. If one recalls that the equivalent of
one iteration in the linearized version of CMFT is in fact
multiplication by λi, one can easily see that large negative
λi would cause oscillations with an exponentially growing
amplitude when performing the iterations of the self-consistent
loop. To overcome this issue we introduce a damping into
a self-consistent loop by taking (1 − d) Sz
i fin + d Sz
i in as
the new averages. Here d < 1 is a suitably chosen damping
factor. With this modification CMFT produces one finite stripy
order suggested by linearization. However since the parasitic
negative λi grows enormously in magnitude as we approach
the phase boundary an extreme damping has to be included
making the phase boundary hard to determine by using
CMFT.
In conclusion, it is evident that the ordered patterns suggested
by linearization were reproduced by CMFT within regions
dictated by the maximal λi > 1. Moreover, the linearized
procedure indicated possible difficulties with stabilizing stripy
phases that had to be cured by a strong damping introduced
into the self-consistent loop.
[1] Bruce Normand, Cont. Phys. 50, 533 (2009).
[2] Leon Balents, Nature (London) 464, 199 (2010).
[3] L. Savary and L. Balents, Rep. Prog. Phys. 80, 016502 (2017).
[4] G. Khaliullin and S. Maekawa, Phys. Rev. Lett. 85, 3950 (2000);
G. Khaliullin, P. Horsch, and A. M. Ole´s, ibid. 86, 3879 (2001).
[5] A. B. Harris, T. Yildirim, A. Aharony, O. Entin-Wohlman, and
I. Ya. Korenblit, Phys. Rev. Lett. 91, 087206 (2003).
[6] M. Daghofer, K. Wohlfeld, A. M. Ole´s, E. Arrigoni, and
P. Horsch, Phys. Rev. Lett. 100, 066403 (2008); K. Wohlfeld,
M. Daghofer, A. M. Ole´s, and P. Horsch, Phys. Rev. B 78,
214423 (2008).
[7] M. Daghofer, A. Nicholson, A. Moreo, and E. Dagotto, Phys.
Rev. B 81, 014511 (2010); A. Nicholson, W. Ge, X. Zhang,
J. Riera, M. Daghofer, A. M. Ole´s, G. B. Martins, A. Moreo,
and E. Dagotto, Phys. Rev. Lett. 106, 217002 (2011).
[8] P. Wr´obel and A. M. Ole´s, Phys. Rev. Lett. 104, 206401 (2010);
P. Wr´obel, R. Eder, and A. M. Ole´s, Phys. Rev. B 86, 064415
(2012).
[9] S. Di Matteo, G. Jackeli, C. Lacroix, and N. B. Perkins,
Phys. Rev. Lett. 93, 077208 (2004); S. Di Matteo, G. Jackeli,
and N. B. Perkins, Phys. Rev. B 72, 024431 (2005).
[10] G. Khaliullin, Prog. Theor. Phys. Suppl. 160, 155 (2005).
[11] G. Jackeli and D. A. Ivanov, Phys. Rev. B 76, 132407
(2007).
[12] G. Jackeli and D. I. Khomskii, Phys. Rev. Lett. 100, 147203
(2008).
[13] F. Kr¨uger, S. Kumar, J. Zaanen, and J. van den Brink, Phys. Rev.
B 79, 054504 (2009).
[14] Gia-Wei Chern and N. Perkins, Phys. Rev. B 80, 220405(R)
(2009).
[15] A. van Rynbach, S. Todo, and S. Trebst, Phys. Rev. Lett. 105,
146402 (2010).
[16] F. Trousselet, A. Ralko, and A. M. Ole´s, Phys. Rev. B 86, 014432
(2012).
[17] G. Chen and L. Balents, Phys. Rev. Lett. 110, 206401
(2013).
[18] M. Daghofer, A. M. Ole´s, and W. von der Linden, Phys. Rev. B
70, 184430 (2004).
[19] A. Reitsma, L. F. Feiner, and A. M. Ole´s, New J. Phys. 7, 121
(2005).
[20] W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, Annu.
Rev. Condens. Matter Phys. 5, 57 (2014).
024426-10
144
PHASE DIAGRAM AND SPIN CORRELATIONS OF THE . . . PHYSICAL REVIEW B 95, 024426 (2017)
[21] W. Brzezicki, A. M. Ole´s, and M. Cuoco, Phys. Rev. X 5, 011037
(2015); W. Brzezicki, M. Cuoco, and A. M. Ole´s, J. Supercond.
Novel Magn. 29, 563 (2016); 30, 129 (2017).
[22] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205
(2009).
[23] A. Kitaev, Ann. Phys. (NY) 321, 2 (2006).
[24] G. Baskaran, S. Mandal, and R. Shankar, Phys. Rev. Lett. 98,
247201 (2007).
[25] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett.
105, 027204 (2010).
[26] Y. Singh and P. Gegenwart, Phys. Rev. B 82, 064412 (2010);
F. Trousselet, G. Khaliullin, and P. Horsch, ibid. 84, 054409
(2011).
[27] X. Liu, T. Berlijn, W.-G. Yin, W. Ku, A. Tsvelik, Y.-J. Kim,
H. Gretarsson, Y. Singh, P. Gegenwart, and J. P. Hill, Phys. Rev.
B 83, 220403(R) (2011).
[28] Y. Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale, W. Ku, S.
Trebst, and P. Gegenwart, Phys. Rev. Lett. 108, 127203 (2012).
[29] S. K. Choi, R. Coldea, A. N. Kolmogorov, T. Lancaster, I. I.
Mazin, S. J. Blundell, P. G. Radaelli, Y. Singh, P. Gegenwart,
K. R. Choi, S.-W. Cheong, P. J. Baker, C. Stock, and J. Taylor,
Phys. Rev. Lett. 108, 127204 (2012).
[30] F. Ye, S. Chi, H. Cao, B. C. Chakoumakos, J. A. Fernandez-Baca,
R. Custelcean, T. F. Qi, O. B. Korneta, and G. Cao, Phys. Rev.
B 85, 180403 (2012).
[31] R. Comin, G. Levy, B. Ludbrook, Z.-H. Zhu, C. N. Veenstra,
J. A. Rosen, Y. Singh, P. Gegenwart, D. Stricker, J. N. Hancock,
D. van der Marel, I. S. Elfimov, and A. Damascelli, Phys. Rev.
Lett. 109, 266406 (2012).
[32] H. Gretarsson, J. P. Clancy, X. Liu, J. P. Hill, E. Bozin, Y. Singh,
S. Manni, P. Gegenwart, J. Kim, A. H. Said, D. Casa, T. Gog,
M. H. Upton, H.-S. Kim, J. Yu, V. M. Katukuri, L. Hozoi, J. van
den Brink, and Y.-J. Kim, Phys. Rev. Lett. 110, 076402 (2013).
[33] F. Trousselet, M. Berciu, A. M. Ole´s, and P. Horsch, Phys. Rev.
Lett. 111, 037205 (2013).
[34] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett.
110, 097204 (2013).
[35] V. M. Katukuri, S. Nishimoto, V. Yushankhai, A. Stoyanova,
H. Kandpal, S. Choi, R. Coldea, I. Rousochatzakis, L. Hozoi,
and J. van den Brink, New. J. Phys. 16, 013056 (2014).
[36] S. M. Winter, Y. Li, H. O. Jeschke, and R. Valent´ı, Phys. Rev. B
93, 214431 (2016).
[37] F. Trousselet, P. Horsch, A. M. Ole´s, and W.-L. You, Phys. Rev.
B 90, 024404 (2014).
[38] S. H. Chun, J.-W. Kim, J. Kim, H. Zheng, C. C. Stoumpos,
C. D. Malliakas, J. F. Mitchell, K. Mehlawat, Y. Singh, Y. Choi,
T. Gog, A. Al-Zein, M. Moretti Sala, M. Krisch, J. Chaloupka,
G. Jackeli, G. Khaliullin, and B. J. Kim, Nat. Phys. 11, 462
(2015).
[39] J. Chaloupka and G. Khaliullin, Phys. Rev. B 92, 024413 (2015).
[40] J. G. Rau and H. Y. Kee, arXiv:1408.4811.
[41] A. Banerjee, C. A. Bridges, J.-Q. Yan, A. A. Aczel, L. Li, M. B.
Stone, G. E. Granroth, M. D. Lumsden, Y. Yiu, J. Knolle, S.
Bhattacharjee, D. L. Kovrizhin, R. Moessner, D. A. Tennant, D.
G. Mandrus, and S. E. Nagler, Nat. Mater. 15, 733 (2016).
[42] J. Reuther, D. A. Abanin, and R. Thomale, Phys. Rev. B 84,
014417 (2011).
[43] A. F. Albuquerque, D. Schwandt, B. Het´enyi, S. Capponi, M.
Mambrini, and A. M. La¨uchli, Phys. Rev. B 84, 024406 (2011).
[44] D. C. Cabra, C. A. Lamas, and H. D. Rosales, Phys. Rev. B 83,
094506 (2011); A. Kalz, M. Arlego, D. Cabra, A. Honecker,
and G. Rossini, ibid. 85, 104505 (2012); H. D. Rosales, D. C.
Cabra, C. A. Lamas, P. Pujol, and M. E. Zhitomirsky, ibid. 87,
104402 (2013).
[45] X. Y. Song, Y. Z. You, and L. Balents, Phys. Rev. Lett. 117,
037209 (2016).
[46] J. G. Rau, Eric Kin-Ho Lee, and H. Y. Kee, Phys. Rev. Lett. 112,
077204 (2014).
[47] J. Oitmaa, Phys. Rev. B 92, 020405(R) (2015).
[48] H.-C. Jiang, Z.-C. Gu, X.-L. Qi, and S. Trebst, Phys. Rev. B
83, 245104 (2011); J. Reuther, R. Thomale, and S. Trebst, ibid.
84, 100406 (2011); I. Kimchi and Y. Z. You, ibid. 84, 180407
(2011); R. Schaffer, S. Bhattacharjee, and Y. B. Kim, ibid. 86,
224417 (2012); Y. Yu, L. Liang, Q. Niu, and S. Qin, ibid.
87, 041107 (2013); E. Sela, H.-C. Jiang, M. H. Gerlach, and
S. Trebst, ibid. 90, 035113 (2014).
[49] J. Osorio Iregui, P. Corboz, and M. Troyer, Phys. Rev. B 90,
195102 (2014).
[50] K. Li, S.-L. Yu, and J.-X. Li, New J. Phys. 17, 043032 (2015).
[51] M. Becker, M. Hermanns, B. Bauer, M. Garst, and S. Trebst,
Phys. Rev. B 91, 155135 (2015).
[52] G. Jackeli and A. Avella, Phys. Rev. B 92, 184416 (2015).
[53] I. Rousochatzakis, U. K. R¨ossler, J. van den Brink, and
M. Daghofer, Phys. Rev. B 93, 104417 (2016).
[54] W. Brzezicki, J. Dziarmaga, and A. M. Ole´s, Phys. Rev. Lett.
109, 237201 (2012); Phys. Rev. B 87, 064407 (2013).
[55] W. Brzezicki and A. M. Ole´s, Phys. Rev. B 83, 214408 (2011).
[56] D. Gotfryd and A. M. Ole´s, Acta Phys. Polon. A 127, 318
(2015).
[57] K. Foyevtsova, H. O. Jeschke, I. I. Mazin, D. I. Khomskii, and
R. Valent´ı, Phys. Rev. B 88, 035107 (2013).
[58] L. R. Walker, Spin Waves and Other Magnetic Modes
in Magnetism (Academic Press, New York and London,
1963).
[59] M. Raczkowski and A. M. Ole´s, Phys. Rev. B 66, 094431
(2002).
[60] P. A. Maksimov and A. L. Chernyshev, Phys. Rev. B 93, 014418
(2016).
[61] Zheng Weihong, J. Oitmaa, and C. J. Hamer, Phys. Rev. B 44,
11869 (1991).
[62] J. D. Reger, J. A. Riera, and A. P. Young, J. Phys.: Condens.
Matter 1, 1855 (1989).
[63] E. V. Castro, N. M. R. Peres, K. S. D. Beach, and A. W. Sandvik,
Phys. Rev. B 73, 054422 (2006).
[64] U. L¨ow, Condens. Matter Phys. 12, 497 (2009).
[65] K. S. Tikhonov, M. V. Feigel’man, and A. Yu. Kitaev, Phys.
Rev. Lett. 106, 067203 (2011).
[66] P. Fulde, Electron Correlations in Molecules and Solids,
Springer Series in Solid-State Sciences, Vol. 100 (SpringerVerlag,
Berlin/Heidelberg/New York, 1995).
024426-11
145
146
PHYSICAL REVIEW B 99, 064425 (2019)
Kitaev-like honeycomb magnets: Global phase behavior and emergent effective models
Juraj Rusnaˇcko,1,2
Dorota Gotfryd,3,4
and Jiˇrí Chaloupka1,2
1
Central European Institute of Technology, Masaryk University, Kamenice 753/5, CZ-62500 Brno, Czech Republic
2
Department of Condensed Matter Physics, Faculty of Science, Masaryk University, Kotláˇrská 2, CZ-61137 Brno, Czech Republic
3
Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, PL-02093 Warsaw, Poland
4
Marian Smoluchowski Institute of Physics, Jagiellonian University, Łojasiewicza 11, PL-30348 Kraków, Poland
(Received 14 October 2018; revised manuscript received 28 January 2019; published 19 February 2019)
Compounds of transition metal ions with strong spin-orbit coupling recently attracted attention due to the
possibility to host frustrated bond-dependent anisotropic magnetic interactions. In general, such interactions
lead to complex phase diagrams that may include exotic phases, e.g., the Kitaev spin liquid. Here we report on
our comprehensive analysis of the global phase diagram of the extended Kitaev-Heisenberg model relevant to
honeycomb lattice compounds Na2IrO3 and α-RuCl3. We have utilized recently developed method based on spin
coherent states that enabled us to resolve arbitrary spin patterns in the cluster ground states obtained by exact
diagonalization. Global trends in the phase diagram are understood in combination with the analytical mappings
of the Hamiltonian that uncover peculiar links to known models—Heisenberg, Ising, Kitaev, or compass models
on the honeycomb lattice—or reveal entire manifolds of exact fluctuation-free ground states. Finally, our study
can serve as a methodological example that can be applied to other spin models with complex bond-dependent
non-Heisenberg interactions.
DOI: 10.1103/PhysRevB.99.064425
I. INTRODUCTION
In contrast to simple examples of Heisenberg magnets discussed
in standard textbooks, frustrated spin systems [1] offer
much wider range of phenomena, including the exotic spinliquid
behavior [2,3] or the emergence of effective monopoles
in spin-ice pyrochlores [4,5]. The usual sources of frustration
are frustrated geometry of the lattice (e.g., kagome [6]) or
the presence of longer-range interactions competing with the
nearest-neighbor ones (as, e.g., in J1-J2 model [7–9]) and possibly
among themselves. Within the last decade, pseudospin-
1
2
systems with frustrated bond-dependent non-Heisenberg
interactions emerging in Mott insulators as a consequence
of spin-orbit coupling (SOC) became a subject of intense
research [10–16]. While one of the main motivations has
been a possible realization of the Kitaev honeycomb model
[17], the presence of additional interactions leads to very rich
magnetic behavior that is particularly attractive as well as
challenging to study.
The basic element enabling the realization of the above
models possessing bond-dependent anisotropic interactions
has been well known for a long time. It relies on a d5
valence
configuration of heavy transition-metal ions with large SOC,
which combines the spin s = 1
2
and effective orbital angular
momentum leff = 1 of the hole in the t5
2g configuration into
Jeff = 1
2
Kramers doublet ground state [18,19]. A direct experimental
evidence for the spin-orbital entangled multiplet
structure [20] was obtained, e.g., by resonant x-ray scattering
on Sr2IrO4 [21] containing d5
Ir4+
ions. It was the seminal
theoretical proposal by Jackeli and Khaliullin [22] that suggested
how to exploit the Jeff = 1
2
pseudospins in Mott insulators
with large SOC. Two lines of intense research followed.
The first one focuses on the square lattice case with the resulting
Heisenberg interactions among the pseudospins—a situation
appealingly analogous to undoped cuprates. Cupratelike
magnetism was indeed found in perovskite iridates [23] and
certain observations support the idea to extend the analogy
to the doped case [24,25]. Yet bigger excitement was initiated
by a proposal that the honeycomb Jeff = 1
2
compounds
may be close to the Kitaev limit where Ising-like bonddependent
interactions lead to a spin-liquid ground state. Such
an exotic effective spin system may naturally arise when
translating the bond-anisotropic interactions of the d orbitals
appearing in Kugel-Khomskii models [26] into the pseudospin
space via the SOC-induced spin-orbital entanglement
[19,27].
In the search of materials close to the Kitaev limit, much
attention has been paid to the honeycomb iridates Na2IrO3,
α-Li2IrO3, and the ruthenate α-RuCl3 [28] that is claimed
to show signatures of Kitaev physics in the excitation spectra
[29–32]. However, these compounds were found to host
long-range magnetic order instead—zigzag type in Na2IrO3
[33–35] and α-RuCl3 [29,36,37] and spiral type in α-Li2IrO3
[38]. Only very recently, an evidence for a liquid state was
found in a related compound H3LiIr2O6 [39]. Even though the
zigzag phase is present in the phase diagram of the originally
proposed Kitaev-Heisenberg model [40], later experiments
on Na2IrO3 showed that it gives an inconsistent ordered
moment direction [41] and additional bond-anisotropic and/or
further-neighbor interactions have to be invoked [42–47]. In
the resulting extended Kitaev-Heisenberg models, the highly
anisotropic interactions lead to complex phase behavior (see
Refs. [15,42,43] for examples) or unusual spin excitation
spectra showing, e.g., a breakdown of the magnon picture
even in the long-range ordered phase away from the Kitaev
limit [48] or topological features [49,50].
2469-9950/2019/99(6)/064425(18) 064425-1 ©2019 American Physical Society
147
RUSNA ˇCKO, GOTFRYD, AND CHALOUPKA PHYSICAL REVIEW B 99, 064425 (2019)
The (extended) Kitaev-Heisenberg models are not limited
to the honeycomb lattice. A large number of other situations
have been discussed, including triangular [19,51–54] and
kagome [55] lattices and suitable types of three-dimensional
structures such as experimentally realized hyperhoneycomb
[56–60], harmonic honeycomb [57,59,61–63], hyperkagome
[27,64], fcc [65–67], and pyrochlore lattices [68,69], or hypothetical
hyperoctagon lattice [70]. Finally, the concept of
Kitaev interactions in pseudospin Jeff = 1
2
systems has been
recently extended to d7
compounds such as those containing
Co2+
[71,72].
In general, a thorough inspection of an extended KitaevHeisenberg
model in terms of spin structures, excitations,
etc. through the parameter space is desired. Apart from theoretical
interest, this is mostly in order to establish it as
an effective model for a concrete material and to narrow
down the parameter regime. Methodologically, the inspection
is complicated by the new kind of frustration stemming
from the bond dependence of the interactions. Since exotic
features such as spin-liquid ground states and fractionalized
excitations are “around,” simple approaches—for instance,
the Luttinger-Tisza method [73] or linear spin waves—often
have a limited success and one has to resort to unbiased
numerical methods fully incorporating quantum effects. Of
a great value are also exact symmetry properties, such as
dual mappings of the Hamiltonian utilizing sublattice spin
rotations [11,19,69,74,75] that proved surprisingly powerful
when establishing and interpreting the phase diagram.
The aim of this paper is to perform a detailed analysis of
the phase diagram of the extended Kitaev-Heisenberg model
(EKH) relevant for honeycomb materials. Portions of the
phase diagram have been reported before by several studies,
both on the classical level [42,46,76] as well as including the
quantum effects [15,47,77–79]. Here we take a global view of
the phase diagram, trying to understand its trends based on
the competition/cooperation of the interactions and general
symmetry properties. We also analyze the internal structure
of the phases including the ordered moment direction that is
useful when fixing the model parameters based on experimental
data [41,80]. To this end, we build on previous work [80]
and use exact diagonalization combined with ground-state
analysis based on spin-1/2 coherent states and complemented
by cluster mean-field theory. This allows us to determine
the spin structures through the phase diagram, including the
noncollinear ones and estimate the amount of quantum fluctuations.
The global analysis revealed two surprising features
that underline the richness of the EKH model and enable a
deeper understanding of its phase behavior: (i) sets of exact
fluctuation-free ground states forming entire manifolds in
the parameter space and (ii) possibility to map part of the
phase space of the EKH model to a model characterized by
a single bond-dependent interaction axis. This way several
models of separate interest “emerge” from the EKH model:
Ising, Kitaev, and compass [11,26] models as well as their
combinations.
The paper is organized as follows: The model and numerical
methods are introduced in Secs. II and III, respectively.
Section IV contains the phase diagram of the model along
with a discussion of its phases. Section V analyzes the manifolds
of fluctuation-free ground states. Finally, Sec. VI is
devoted to the study of the Ising-Kitaev-compass case and its
links to EKH model.
II. EXTENDED KITAEV-HEISENBERG MODEL
A. Model Hamiltonian
According to the currently available prevailing evidence
for honeycomb materials [15] and following Ref. [80],
we choose to study the nearest-neighbor extended KitaevHeisenberg
model [42–44] complemented by third-nearest
neighbor Heisenberg exchange. The nearest-neighbor (NN)
part of the model contains—in addition to the usual Heisenberg
exchange—all possible anisotropic terms that are allowed
by symmetry of the trigonally distorted honeycomb
lattice [44,75]. It is most conveniently expressed in cubic
coordinates x, y, z introduced in Fig. 1(a) that allow to easily
incorporate the discrete rotational C3 symmetry. For a bond of
c direction, the Hamiltonian contribution reads as
H(c)
i j = J Si · Sj + K Sz
i Sz
j + Sx
i Sy
j +Sy
i Sx
j
+ Sx
i Sz
j +Sz
i Sx
j +Sy
i Sz
j +Sz
i Sy
j , (1)
whereas the contributions for the other bond directions are
obtained by a cyclic permutation of the spin components
Sx
, Sy
, and Sz
. The J and K terms alone constitute the KitaevHeisenberg
model [22,74] that has been subject to extensive
studies [40,74,81–88] and still serves as a prototype model
to capture a departure from the Kitaev physics. In light of
experimental data [41], it has been generally recognized that
further anisotropic terms are needed, leading to the addition
of the and terms introduced in Refs. [42–44]. When
studying the phase diagram we keep signs of J and K flexible
and fix the signs > 0 and < 0 following the ab initio
calculations as well as the perturbative evaluation of the effective
interactions [43,47]. According to the latter one, small
negative should correspond to a trigonal compression of the
y
x
z
Na
Ir
O
a b
c
(a) (b)
(c)
(d)
FIG. 1. (a) Honeycomb NaIr2O6 layer. Iridium ions form a honeycomb
lattice with a sodium atom in the middle of each hexagon.
Each iridium atom is surrounded by an octahedron of oxygens; the
neighboring octahedra share an edge. The figure shows also the
definition of cubic x, y, z axes and the bond directions a, b, c. [(b)–
(d)] Sublattices of the two-, four- and six-sublattice transformations
T2, T4, T6 that reveal the points of hidden SU(2) symmetry.
064425-2
148
KITAEV-LIKE HONEYCOMB MAGNETS: GLOBAL PHASE … PHYSICAL REVIEW B 99, 064425 (2019)
lattice [43], observed in Na2IrO3 [34,35]. Moreover, several
ab initio studies have evaluated the importance of furtherneighbor
couplings (see, e.g., Refs. [45,47]). In Ref. [47],
the effective spin Hamiltonians for Na2IrO3, α-RuCl3, and
α-Li2IrO3 were constructed using a combination of DFT and
cluster exact diagonalization that equally treated interactions
up to third nearest neighbors. Among the further-neighbor
interactions, a significant value of J3 > 0 was found for all
three compounds, which leads us to the complete model
considered here
H =
i j ∈NN
H(γ )
i j +
i j ∈3rd NN
J3Si · Sj . (2)
B. Hidden symmetries of the model
The NN part of the above model (J3 = 0) has rich symmetry
properties explored in detail in the previous work
[75]. First, it supports a self-dual transformation T1 that
corresponds to a global π rotation of the spins around the
axis perpendicular to the honeycomb plane. Such a transformation
fully preserves the form of the Hamiltonian but
replaces the values of the parameters JK by another set
of values. Second, Ref. [75] has also identified a number
of special parameter combinations—the points of “hidden”
SU(2) symmetry in the parameter space—for which the
NN model maps to ferromagnetic (FM) or antiferromagnetic
(AF) Heisenberg model on the honeycomb lattice. This is
achieved by employing either two-, four-, or six-sublattice
coverings of the honeycomb lattice as depicted in Figs. 1(b)–
1(d) and performing selected sublattice-dependent rotations
of the spins. The neighboring spins that belong to different
sublattices are therefore rotated in a different fashion and
the interaction among those spins takes a modified form, in
certain cases, the simple Heisenberg one. For these particular
cases, the seemingly anisotropic model is thus exactly
equivalent to the Heisenberg model on the honeycomb lattice.
By using the same transformation backwards, we can
exploit the known properties of Heisenberg model obtaining
thereby, e.g., the ordering pattern or excitation spectra at
the points of “hidden” SU(2) symmetry. Due to the sublattice
structure of the transformation, the simple ordering
patterns of Heisenberg FM or AF transform to more complex
ones such as stripy, zigzag, or even noncollinear vortex
pattern.
As a well-known example, we can consider the KitaevHeisenberg
model with the parameters satisfying the relation
K = −2J and the four-sublattice covering shown in Fig. 1(c).
Keeping the spins at the sites marked by unrotated, and applying
π rotations around the x, y, or z axes to the spins at the
sites attached to the sites by the a, b, or c bond, respectively,
we obtain the Heisenberg Hamiltonian H = −J i j S i · S j
in the rotated spin variables S . In the notation of Ref. [75],
this transformation is called T4. The other possibilities include
two-sublattice transformation T2, the six-sublattice T6, and the
combinations T1T4 and T2T6. All these points of “hidden”
SU(2) symmetry summarized in Table I and Fig. 3 of Ref. [75]
provide exact reference points in the parameter space and will
be extensively utilized in the present study.
III. METHODS
To solve the model, we use the standard Lanczos exact
diagonalization (ED) technique employing a finite cluster
[89]. The calculated cluster ground state is subsequently
analyzed utilizing spin-1/2 coherent states [80] as detailed
below. The ED technique is complemented by the cluster
mean-field theory (CMFT). This combination is useful for a
global characterization of the phase diagram—ED gives the
ground-state characteristics such as energies and spin correlations,
the analysis based on spin-1/2 coherent states enables
to better assess the ordering patterns and the direction of
magnetic moments, and CMFT supplements this information
by the length of the ordered moments, which is not directly
accessible by ED.
In both cases, we use a hexagonal 24-site cluster with
periodic boundary conditions applied. This cluster has a fully
symmetric shape and supports all the phases with hidden
SU(2) symmetry [75]. It is therefore expected to provide a
fair environment for the competition of the phases, with the
exception of the possible spiral phases that are forced to fit
the periodic boundary conditions and may be thus slightly
suppressed. In this specific case, we have extended our ED
analysis to 32-site clusters.
A. Spin-1/2 coherent states for noncollinear phases
The analysis of the exact ground state of the cluster obtained
by ED presents a challenge—the cluster ground state
does not spontaneously break symmetry but instead contains
a linear combination of all the degenerate spin configurations.
To resolve the dominant configuration and obtain the direction
of the pseudospins from the ED ground state, we follow
Ref. [80] and employ spin-1/2 coherent states. Such a state,
polarized in a direction given by spherical angles θ and φ is
given by
|θ, φ = e−iφSz
e−iθSy
|↑ , (3)
where we make a standard choice of cubic z direction as the
quantization axis. The cluster spin-coherent state is then a
direct product of coherent states on each site j:
| =
N
j=1
|θj, φj . (4)
This state can be understood as a classical (fluctuation-free)
spin pattern with the individual spins pointing in the directions
determined by the angles θj and φj. By calculating the overlap
|GS and maximizing its absolute value by varying the
angles, we can identify the classical pattern that best fits the
exact ground state |GS .
For collinear phases (in the case of EKH model, these are
FM, AF, zigzag, and stripy) the cluster spin-coherent state
is captured by a single pair (θ, φ), which makes it easy to
find the moment direction by inspecting the probability map
P(θ, φ) = | |GS |2
and finding the maximum. However,
already the analysis of hidden SU(2) points revealed the
existence of several noncollinear “vortex” phases in the phase
diagram of the EKH model [75]. In the general case, the
probability P = | |GS |2
has to be maximized with respect
to all 2N angles. For our cluster with N = 24 sites, this poses
064425-3
149
RUSNA ˇCKO, GOTFRYD, AND CHALOUPKA PHYSICAL REVIEW B 99, 064425 (2019)
a nontrivial computational problem of global optimization in
48-dimensional space. To this end, we use the particle swarm
method for global optimization, which yields a result further
refined by a local optimization algorithm.
The demanding task can be partly avoided by estimating
in advance the parameter windows where noncollinear phases
can be found. This can be achieved by first finding the optimal
spin configuration among the collinear ones and calculating
the full Hessian matrix of second derivatives (with respect
to all 48 angular parameters) for such a configuration. The
potential instability of the collinear phase can be identified by
analyzing the eigenvalues of this Hessian matrix.
B. Cluster mean-field theory
Similarly to ED, within CMFT we periodically cover the
lattice by copies of a given cluster. The bonds connecting
the cluster copies (external bonds) are treated in a meanfield
approximation, replacing the contributions to the bond
Hamiltonian according to the recipe
Sα
i Sβ
j ≈ Sα
i Sβ
j + Sα
i Sβ
j − Sα
i Sβ
j , (5)
while the internal bonds of the cluster are kept fully quantum
[81]. The mean-field approximation generates effective
magnetic fields acting at the outer sites of the cluster and polarizing
the cluster ground state to be determined by ED. The
polarizing fields depend on the averages Sα
i measured on the
polarized ground state, which leads to a selfconsistent problem
with much higher computational demands than the pure
ED. On the other hand, by explicitly breaking the ground-state
symmetry, the CMFT method allows to directly determine the
ordering pattern and estimate the ordered moment length.
The introduction of the mean-field boundary makes the
sites of the cluster nonequivalent. In combination with
the highly anisotropic bond-dependent interactions, the spin
structures show a tendency towards various forms of artificial
canting. To prevent this, we limit ourselves to the case of
collinear spin structures and follow the approach described
in Ref. [81], where an averaged ordered moment through the
cluster is taken and distributed on the boundary sites following
a particular ordering pattern.
IV. GLOBAL PHASE DIAGRAM
By optimizing the spin configurations using the methods
described in the previous section and evaluating the corresponding
probabilities, we are able to construct a detailed
phase diagram of the model. We present the slices through
the phase diagram using a common parametrization for the
main interactions [42], that is J = cos ϕ sin θ, K = sin ϕ sin θ,
= cos θ with ϕ ∈ [0, 2π] and θ ∈ [0, π/2]. This way all
the J, K sign combinations and interaction strength ratios for
positive 0 are explored. The remaining model parameters
and J3 are kept fixed for a given slice. Figure 2(a) shows
the phase diagram for = J3 = 0, which is the special case
of the JK model, first analyzed in Ref. [42]. We shall now
use this diagram to survey the main properties of the phases
and move on to their evolution with and J3 afterwards. The
reader may also consult Appendix C containing an extensive
set of phase diagram slices for selected values.
vortex-a vortex-b
y
x
z
y
z
x
b
a
FM-b
FM-a
(b)
(a) vortex
FM AF
stripy
zigzag
a
b
20°
24°
28°
32°
36°
40°
AF
FIG. 2. (a) Phase diagram of the extended Kitaev-Heisenberg
model with = J3 = 0 using the parametrization J = cos ϕ sin θ,
K = sin ϕ sin θ, and = cos θ with ϕ ∈ [0, 2π] being the azimuthal
angle and θ ∈ [0, π/2] the radial coordinate measured as a distance
from the center of the circle. Color intensity and contours show
the probabilities of classical spin patterns for the respective phases.
Dashed lines separate distinct regions within one phase. White areas
represent regions where no clear signatures of a long-range ordered
phase were detected using the 24-site cluster. The gray dots indicate
points of (hidden) SU(2) symmetry. The hatched part of the central
region with a large probability of the zigzag pattern shows a tendency
to form a noncollinear spin arrangement. (b) The angle of the ordered
moments to the honeycomb plane for the zigzag phase—in the upper
region, the moment points near the cubic z direction (assuming
zigzags running along a and b bonds), whereas in the central region
it is located between x and y axes. The panel also depicts the in-plane
spin patterns for distinct regions (labeled as a and b) within vortex
and FM phases and the out-of-plane pattern of the AF phase. Further
details can be found in Fig. 9(e) of Appendix C.
064425-4
150
KITAEV-LIKE HONEYCOMB MAGNETS: GLOBAL PHASE … PHYSICAL REVIEW B 99, 064425 (2019)
A. Collinear phases of the JK model
We first focus on the simpler collinear phases which occupy
most of the phase diagram. Two phases, FM and AF,
dominating Fig. 2(a) are directly linked to Heisenberg points.
Though they may seem trivial at the first sight, our inspection
revealed their interesting internal structure due to the complex
interplay of the bond-anisotropic interactions. We start with
the FM phase, which is expected to be most accessible due to
the small amount of quantum fluctuations. In the JK limit (the
outer circle of the diagram), the ordered moments point along
one of the three cubic directions x, y, z selected by virtue
of the “order-from-disorder” mechanism on top of isotropic
classical energy [80,90]. Since the cluster ground state is a
superposition of the six degenerate possibilities, the probability
approaching the value 1/6 near the FM point indicates
vanishing quantum fluctuations. With the presence of the
term, the magnetic moment is quickly pushed into the honeycomb
plane, lying either directly within the plane or close to
it [with 10◦
deviation, see also Fig. 9(e) in Appendix C].
This can be understood by evaluating the classical energy
for the FM phase: Eclass ∝ 3J + K − + (nx + ny + nz)2
,
where the unit vector n = (nx, ny, nz ) represents the moment
direction. The honeycomb plane is thus preferred by interaction
on a classical level which makes it easy to outweigh
the fluctuation-selected cubic axis. A small value of the
order 10−2
to 10−1
of the dominant JK is typically sufficient
to achieve this with the value dropping even lower near the
FM Heisenberg point. Within the honeycomb plane, moments
point either in the bond direction, or perpendicular to the bond
in two separate regions of the FM phase [see Fig. 2(b)]. In
accord with the intuition, departing from the FM Heisenberg
point, quantum fluctuations intensify, lowering thus the plotted
probability.
Linked to the FM phase by means of the four-sublattice
transformation (T4 in the notation of Ref. [75]) is the stripy
phase. Its hidden FM nature is manifested by a large probability,
reaching 1/6 at the hidden SU(2) point K = −2J < 0 that
is an image of the FM Heisenberg point in the T4 mapping.
In contrast to the FM phase, the magnetic moment direction
is tied to the vicinity of the cubic axes throughout the stripy
phase, lifting a bit with increasing instead of moving to
the honeycomb plane. This is because the interaction is
not compatible with the T4 transformation and acts differently
here.
In the AF phase, the moment direction is classically degenerate
in the JK limit, and the cubic directions are chosen again
by the “order-from-disorder” mechanism. The addition of the
anisotropy fixes now the moments in the (111) direction—
perpendicular to the honeycomb plane. This state minimizes
the classical energy including contribution: Eclass ∝ −3J −
K + − (nx + ny + nz )2
. Similarly to the FM phase, the
fluctuation energy selecting the cubic directions is small and
the change to the (111) direction occurs already at a minute
of the order 10−4
to 10−2
of the dominant JK with the
critical value of decreasing to zero at the AF Heisenberg
point. Going deeper into the AF phase, the probability of the
classical Néel configuration increases with steadily, peaking
at 1/2 on a line near the circle center that starts at the K =
hidden SU(2) symmetry point. For the (111) AF state, there
are two equivalent configurations of the moments, meaning
that the peaking probability of 50% represents a classical
state without any quantum fluctuations. Indeed, as we later
explicitly demonstrate in Sec. V, terms that would lead to
quantum fluctuations are present but their remarkable cancellation
for the particular order causes the highly anisotropic
model to support a fluctuation-free AF state on an entire
manifold of its parameter space. The same AF phase may
thus be represented by fluctuation-free ground states as well
as those with significant quantum fluctuations, depending on
the location in the parameter space.
Analogous to the FM/stripy case, T4 maps the AF Heisenberg
point to the hidden SU(2) point K = −2J > 0. The top
zigzag region of the phase diagram extends around this point;
in the JK limit, the moment direction coincides again with
one of the cubic directions. Adding further anisotropy with
increasing , the moments are pushed continuously towards
the honeycomb plane, as shown in Fig. 2(b).
Of a greater experimental relevance is the second zigzag
phase near the center of the phase diagram. It is also linked
to a hidden SU(2) point which, however, occurs at finite
< 0 [75]. In this phase, the moment direction is located
roughly between the cubic x and y axes, near the direction
found experimentally [41]. We will show later, that it is this
zigzag region that largely expands and dominates the phase
diagram after the inclusion of and/or J3 coupling terms.
A comprehensive discussion of the moment direction in both
zigzag phases in the context of the experimental data can be
found in Ref. [80].
B. Vortex phase
The vortex phase is a noncollinear phase “emanating” from
the most peculiar hidden SU(2) symmetry point of the model
that is revealed by a six-sublattice spin rotation T6 of Ref. [75].
T6 maps the ferromagnetic J < 0 Heisenberg model to EKH
model at the parameter point J = 0, K = > 0 indicated in
Fig. 2(a). Owing to its hidden FM nature and six degenerate
spin configurations, the optimized probabilities reach 1/6
in the vicinity of this exact vortex point, and continuously
decrease with the departure away from it.
The phase comprises regions with two different most probable
classical configurations of moments labeled as vortex-a
and vortex-b in Fig. 2(b). Let us note, however, that these two
patterns have very close probabilities and are continuously
connected, implying a presence of a soft mode oscillating between
them. In partial agreement with the classical treatment
[42], spins are found to lie within or close to the honeycomb
plane. The vortex-b pattern is always planar while in the
vortex-a regions near the boundary with AF or zigzag phase,
the spins start to tilt away from the honeycomb plane in a
staggered AF fashion. The tilt is largest in the right part of
the vortex phase [see Fig. 9(e)] which we interpret as the
proximity effect of the robust AF order with the moments
perpendicular to the honeycomb plane.
A deeper understanding of the internal structure of the vortex
phase is possible by utilizing four reference points where
the EKH maps to simpler models. One of them is the vortex
SU(2) point in = 0 slice. The freedom associated with
the selection of the ordered moment direction in the hidden
FM at this point creates a continuous family of degenerate
064425-5
151
RUSNA ˇCKO, GOTFRYD, AND CHALOUPKA PHYSICAL REVIEW B 99, 064425 (2019)
patterns including vortex-a and vortex-b. Another hidden
SU(2) symmetry point but of AF nature is found for ≈
−0.5 at the opposite edge of the vortex phase [see Fig. 9(j)
of Appendix C]. It is associated with T2T6 transformation of
Ref. [75]. For a planar structure, the staggering of the hidden
AF order is compensated by the two-sublattice π rotation T2
such that this point supports the same vortex-a and vortex-b
patterns as for the = 0 hidden FM point associated with
just T6. However, in contrast to the latter point, the corresponding
state has pronounced quantum fluctuations because
of the hidden AF nature. Departing away from the hidden FM
point or the hidden AF point, the degeneracy is lifted and
one of the configurations is chosen as the energetically most
favorable. Here the proximity to the remaining two reference
points decides. As we find in Sec. VI, the point K = =
−J > 0 (the “meeting” point of four phases) corresponds to a
FM compass model on the honeycomb lattice while at another
nearby point with K > 0, J = > 0, and < 0, the model
maps to AF compasslike model with the interaction direction
perpendicular to the bond. These two compass(like) models
prefer patterns vortex-b and vortex-a, respectively, which
qualitatively explains the location of vortex-a,b subphases.
C. Remaining phases of the JK model
The remaining parts of the phase diagram slice for = 0
and J3 = 0 [kept white in Fig. 2(a)] are to a small extent occupied
by the two known Kitaev spin liquids associated with the
FM and AF Kitaev points. Here the optimization of spin-1/2
coherent states described in Sec. III A finds a large number
of configurations consisting of aligned/contra-aligned pairs
of the nearest-neighbor spins, as appearing in classical S →
∞ limit of the Kitaev model [91,92] (see also Appendix A
for several details concerning the behavior of the method in
the presence of Kitaev spin liquids).
However, much bigger portion of the phase diagram is
taken by the white region in the lower central part which
shows a particularly puzzling behavior. Parts of it were suggested
earlier to host incommensurate phases [42,78]. The
vertical J = 0 line seems to play a special role as it clearly
separates the middle zigzag as well as the vortex region from
the other phases on the right [see Fig. 2(a) and the detail in
Fig. 3(a)]. The K- model corresponding to the J = 0 line
has been recently studied separately and its ground state for
ferromagnetic K was found to bear signatures of a spin liquid
[93,94].
Using the method of Sec. III A for the above region, we
find tendencies to form complex spin structures, though the
probability of such configurations is quite small, hinting towards
a possibility of phase(s) without a long-range order. Interestingly,
the region with a large probability of the collinear
zigzag structure is also partially unstable towards a formation
of a noncollinear spin arrangement—see the hatched pattern
in Fig. 2(a) or 3(a). Although the clusters accessible to ED
are not in general large enough to properly capture potential
spin orderings with large unit cells, we still try to provide a
further analysis based on momentum-space correlations. Here
we utilize two more clusters in ED, a 32-site cluster of a
FIG. 3. (a) Position of the four selected points 1–4 in the phase
diagram. In addition, the FM Kitaev point is taken as a reference.
[(b)–(h)] Sz
−qSz
q correlations at the selected parameter points calculated
for the 24-site cluster [(b), (c), (f), and (g)] and 32-site clusters
of hexagonal [(d) and (h)] and rectangular shape (e). The nearestneighbor
correlations in the liquid state are manifested by a wavelike
pattern (b)—such a pattern seems to be present as a “background”
in the other maps [(c)–(h)] as well. At point 1, the larger 32-site
clusters already support incommensurate correlations [(d) and (e)],
while the 24-site cluster shows zigzaglike correlations (c) though
collinear zigzag is not the most probable configuration anymore.
Incommensurate correlations are visible at the 24-site cluster for
point 3 (g) and merge with the zigzag ones on the K- line (f).
Deeper in the white region, the incommensurate wave vector moves
out of the first Brillouin zone (h). All the panels (b)–(h) show the
available q resolution for the given cluster. In (c), the high-symmetry
points in the Brillouin zone are labeled.
hexagonal shape and a rectangular one (4
√
3 × 6 in lattice
spacings), in addition to our default 24-site cluster.
Figure 3(a) shows the positions of four parameter points
selected for a comparison: 1 in the unstable zigzag region,
2 on the K- boundary, 3 in the expected spiral phase close
to J = 0 line, and point 4 deeper in the expected spiral
phase. The FM Kitaev point is added for reference. Plotted
in Figs. 3(b)–3(h) are the maps of the equal-time spin-spin
correlation function Sz
−qSz
q . It should be emphasized, that the
cluster ground states do not spontaneously break symmetry
and contain, e.g., a linear combination of several ordering
patterns that differ by the direction of the ordering wave vector
and hence the ordered moment direction. The selection of
the spin component of the correlation function then provides
access to various components of this combination. For the
hexagonal clusters, where a rotation by 2π/3 is in effect just
a cyclic permutation among the Sx
, Sy
, and Sz
components,
064425-6
152
KITAEV-LIKE HONEYCOMB MAGNETS: GLOBAL PHASE … PHYSICAL REVIEW B 99, 064425 (2019)
the other correlation functions Sx
−qSx
q and Sy
−qSy
q are merely
2π/3-rotated copies of the maps shown in Fig. 3.
By combining various sets of maps from Fig. 3, several
trends can be illustrated. (i) The wavelike background identical
to momentum-represented nearest-neighbor correlations
of the Kitaev liquid [Fig. 3(b)] is universally present at all
points, less apparently in the case of peaked structures on top
of the background because of an extended color scale range.
(ii) Panels (c)–(e) show the influence of the cluster size and
shape at the parameter point 1 that we demonstrate now to be
in the incommensurate region. For the smallest 24-site cluster,
the correlation map in Fig. 3(c) still includes peaks located
at the M momenta, which corresponds to a zigzag arrangement.
However, using the method of Sec. III A, the zigzag
pattern is found unstable which already hints towards another
type of ordering. This is fully revealed by the larger 32-site
clusters. By providing a denser momentum-space coverage,
they enable the preferred incommensurate state to develop
[Figs. 3(d) and 3(e)]. The difference between Figs. 3(d) and
3(e) is an effect of the cluster shape. The symmetric hexagonal
32-site cluster [panel (d)] supports three degenerate directions
for the ordering wave vector that coexist in the ground state
(two of them visible aside the main maxima near the Brillouin
zone center), while the rectangular shape of the second 32-site
cluster selects only one of those directions [panel (e)].
(iii) Panels (c), (f), and (g) demonstrate, for the 24-site
cluster, the evolution from commensurate correlations [point
1, Fig. 3(c)] to incommensurate ones [point 3, Fig. 3(g)] found
in the white region. At the boundary point 2 with J = 0,
the corresponding states show a level crossing and we obtain
the average spin-correlation pattern displayed in Fig. 3(f)
resembling that of the Kitaev point.
(iv) Panels (d) and (h) illustrate, for the symmetric 32-site
cluster, the transfer of the incommensurate wave vector from
the inside of the first Brillouin zone [point 1, Fig. 3(d)] to the
outside [point 4, Fig. 3(h)] when moving in the direction of
positive J. This trend was also obtained by classical Monte
Carlo simulations [43].
In conclusion, the studied region of the phase diagram
shows a complex behavior with the spin correlations indicating
tendencies towards various incommensurate orders.
However, the common wavelike background to the spin correlations
suggests a presence of strong liquidlike features.
D. Effect of nonzero and J3 parameters
We shall now investigate the evolution of the phases found
in the = J3 = 0 slice of the phase diagram when the
parameters and J3 are varied. As argued in Sec. II, we
limit ourselves to the experimentally most relevant case of
small < 0 and J3 > 0. Additional data to establish a fuller
picture are presented in Appendix C. The observed trends can
be successfully explained either simply by considering the
classical energy or, more fundamentally, correlated with the
positions of the points of special symmetry in the parameter
space, as inspected in Ref. [75] (points of hidden SU(2)
symmetry) and the following Secs. V and VI.
Figures 4(a) and 4(b) shows phase diagrams for two moderate
values of < 0. Most notable effect of negative is
the large expansion of the vortex phase and mainly of the
in-plane
in-plane
0.50
0.15
FIG. 4. Phase diagrams for nonzero values of and J3 represented
by probabilities of optimized collinear and vortex spin
patterns in the ED ground state (left) and, focusing on zigzag
phases, by the angle of the moments to the honeycomb plane (right).
Hatched/white areas in the zigzag phase indicate the instability of the
collinear pattern. Bottom part of (a) also shows the ordered moment
length calculated by CMFT (left) and the angle to the honeycomb
plane in zigzag phases (right). Shown in (b) are projected positions
of a hidden SU(2) point (gray •) and a compasslike point ( ).
064425-7
153
RUSNA ˇCKO, GOTFRYD, AND CHALOUPKA PHYSICAL REVIEW B 99, 064425 (2019)
central zigzag phase. The former trend can be understood as
a proximity effect of the point where the EKH model maps
to AF compasslike model (to be analyzed in Sec. VI). Its
position is at ≈ −0.6 in the chosen parametrization and the
projection to JK plane is indicated in Fig. 4(b). Though the
difference in is still quite large, this special point efficiently
enforces the vortexlike correlations of type a so that the vortex
phase not only grows but also becomes dominated by vortex-a
pattern (cf. Appendix C). The expansion of the central zigzag
phase is linked to approaching the hidden SU(2) point that is
an image of the AF Heisenberg point in T1T4 transformation.
This point, having ≈ −0.4 and the projection onto JK
plane as indicated in Fig. 4(b), enforces the zigzag order with
the moment direction consistent with experiments and may be
actually regarded as the source of the central zigzag phase.
On the other hand, the top zigzag region related to the SU(2)
point in the = 0 plane is suppressed with < 0, to the
extent that it is not even discernible already for = −0.2.
The FM and AF phases develop more complex internal
structure when < 0 is added. This is due to the competition
of the energy contributed by and that are decisive for
the moment direction at a classical level. The anisotropic
part of these contributions is proportional to ±( + 2 )(nx +
ny + nz )2
for FM and AF, respectively. In the FM phase,
< 0 creates a new subphase where the moments pushed
originally to the honeycomb plane due to > 0 [Fig. 2(b)]
take the direction perpendicular to the honeycomb plane. This
subphase extends near the outer rim of the FM phase where
is sufficiently weak. An opposite effect is observed in
the AF phase. Here, in addition, the absence of the moment
confinement by anisotropic classical energy in the case of
+ 2 = 0 leads to an enhancement of quantum fluctuations
and the probability plotted in e.g., Fig. 4(b) therefore drops at
the corresponding circle.
Based on the data presented so far, the probabilities of the
best-fitting classical configurations represent a good measure
of quantum fluctuations in the ground state. To have an
independent quantification and to cross-check our results, we
compare them to a complementary approach, namely CMFT
described in Sec. III B. Its advantage is the ability to estimate
the ordered moment length that we plot in Fig. 4(a). The phase
boundaries of the collinear phases are in a good agreement
with the method based on ED and the moment length reveals
the less fluctuating FM and stripy phases, and the gradual
decrease of quantum fluctuations when going deeper into the
AF phase. The data on the moment angle to the honeycomb
plane show a somewhat larger spread but the trend is identical.
The evolution of the phases with increasing third nearestneighbor
coupling J3 is illustrated in Figs. 4(c) and 4(d).
As expected already at the level of the classical energy, the
antiferromagnetic J3 > 0 coupling further favors zigzag and
AF phases. The stripy and vortex phases of hidden FM nature
as well as the FM phase get quickly suppressed and the two
zigzag regions merge filling the entire left half of the phase
diagram. In both zigzag and AF phases, the third nearestneighbor
bonds have contra-aligned spins favorable for AF
J3 interaction. The energy gain brought by J3 therefore does
not visibly shift the zigzag/AF boundary. The two zigzag
regions have incompatible moment directions. When merging
them, the system makes a compromise by pushing the moment
direction to the honeycomb plane so that it can easily flip between
z and (x + y)/
√
2 directions projected onto honeycomb
plane [cf. Fig. 2(b)]. Near the boundary between the zigzag
subphases where the moment lies in the honeycomb plane,
the quantum fluctuations are significantly suppressed.
We reach the conclusion that both < 0 and J3 > 0—
expected to be present in real materials—strongly stabilize
the central zigzag phase that is consistent with experimental
observations in Na2IrO3 in both the magnetic ordering and
direction of magnetic moment. As for the precise moment
direction (figures in the right column of Fig. 4), the evolution
seems to be dictated by K, , while , J3 influence mostly
the extent of the phase. We note that one has to distinguish
the real pseudospin direction and the moment direction as
probed by various techniques such as neutron or resonant xray
scattering [80]. Based solely on the moment direction with
the experimental data [41] translating to the pseudospin angle
of about 38◦
–40◦
[80], it seems that the FM K < 0 should be
the largest interaction, followed by possibly still large > 0.
Being in accord with the conclusions of Ref. [80], this also
falls in line with ab initio estimates of dominant ferromagnetic
K and comparable J > 0, > 0, < 0, and J3 > 0 [15].
Finally, small hatched/white areas in the zigzag phase
shown in Fig. 4 again indicate the instability of the collinear
zigzag pattern that may be interpreted as a protrusion of
the possible incommensurate phase. They appear at small
and J3 which together with the link between and trigonal
distortion suggests an explanation for the spiral order in less
distorted α-Li2IrO3 compared to Na2IrO3 with zigzag order.
This point was analyzed at a basic classical level in Ref. [75].
V. FLUCTUATION-FREE MANIFOLDS
As noticed in Sec. IV A when inspecting the phase diagram
of the JK model [Fig. 2(a)], the AF phase contains an
unusual line of fluctuation-free ground states located near
the center of the phase diagram. The distance to this line
seems to determine the magnitude of quantum fluctuations
throughout the entire AF phase—the probability of the Néel
state in the ground state increases more or less monotonously
starting from the outer rim and approaching the line radially
inwards. In fact, similar lines are present also for nonzero
slices in a certain range and form thus an entire surface in
the parameter space. This is quite unexpected since at those
parameter points, all the interactions are active and there is
no apparent cancellation leading to the absence of quantum
fluctuations. What is more, a manifold of fluctuation-free
ground states is found also in part of the FM phase away
from the trivially fluctuation-free FM Heisenberg point. This
is demonstrated in Fig. 5(a) for two values of < 0. Below
we address both cases, starting with the simpler FM one.
A. FM phase
A common feature of the fluctuation-free ground states is
the moment direction being perpendicular to the honeycomb
plane, suggesting to rewrite the Hamiltonian into the XY Z
reference frame [Fig. 5(b)] where this perpendicular direction
is singled out. The Hamiltonian contributions for all the bond
064425-8
154
KITAEV-LIKE HONEYCOMB MAGNETS: GLOBAL PHASE … PHYSICAL REVIEW B 99, 064425 (2019)
(d)
(b)
Y
X
Z
x y
zKitaev Kekulé
6
x y
z
z
y
zy xx
x
x
x
1
2
3
4
5
6
(c)
y
x
z
(f)
FIG. 5. (a) Lower left quadrant of the phase diagram for =
−0.1 and −0.2 showing the FM phase. The color indicates the
difference Pmax − P between the probability P of the classical state
in the ED ground state and its maximum value of Pmax = 1/2. The
maximum probability, corresponding to a fluctuation-free ground
state, is reached in the FM subphase with the moments perpendicular
to the honeycomb plane (darker color) at the line given by K +
2 − 2 = 0 (dashed). (b) Coordinate frames used to express spin
interactions. (c) Schematic representation of the T6 transformation
on the honeycomb lattice. At each of the six sublattices, a different
rotation of spin components is applied. (d) Correspondence
between the bonds and interaction Hamiltonians H(x)
, H(y)
, and H(z)
for the EKH model and extended Kekulé-Kitaev-Heisenberg model
obtained when performing the T6 transformation. (e) The fluctuationfree
line in the AF phase of the JK model. The line is determined
by 3J + K − − 2 = 0 and crosses the hidden SU(2) symmetric
vortex point (gray •). (f) Shifted line for a case of nonzero : the
line no longer enters the AF phase.
directions can be cast to a common form [75]:
H(γ )
i j = JXY SX
i SX
j + SY
i SY
j + JZ SZ
i SZ
j
+ A SX
i SX
j −SY
i SY
j cos φγ − SX
i SY
j +SY
i SX
j sin φγ
− B SX
i SZ
j +SZ
i SX
j cos φγ + SY
i SZ
j +SZ
i SY
j sin φγ .
(6)
The bond-dependence of the interactions is expressed via
the trigonometric factors containing the angles of the bonds
measured from the Y axis, i.e., φγ = 0, 2π
3
, 4π
3
for the c, a, and
b bonds, respectively. Equation (6) is obtained by inserting
into Eq. (1) the transformation relations
⎛
⎜
⎝
Sx
Sy
Sz
⎞
⎟
⎠ =
⎛
⎜
⎜
⎝
1√
6
− 1√
2
1√
3
1√
6
1√
2
1√
3
− 2
3
0 1√
3
⎞
⎟
⎟
⎠
⎛
⎜
⎝
SX
cos φγ + SY
sin φγ
−SX
sin φγ + SY
cos φγ
SZ
⎞
⎟
⎠,
(7)
which represent a conversion from the cubic xyz to XY Z
reference frame for a c bond as well as the necessary cyclic
permutation among xyz (rotation around Z axis), and using the
fact that cos 2φγ = cos φγ , sin 2φγ = − sin φγ for the allowed
values of φγ .
The interaction parameters in (6) expressed in terms of the
original J, K, , and read as
JXY = J + 1
3
(K − − 2 ), (8)
JZ = J + 1
3
(K + 2 + 4 ), (9)
A = 1
3
(K + 2 − 2 ), (10)
B =
√
2
3
(K − + ). (11)
Let us now consider a FM state polarized in the Z direction
and inspect the terms that could lead to quantum fluctuations.
As in usual Heisenberg magnets, the JXY interaction containing
S+
i S−
j and S−
i S+
j does not act on the polarized state. The
above state is an eigenstate of the SZ
i operators, the action of
B terms in the Hamiltonian therefore sums up to
−B
sites
⎡
⎣SX
i
γ =a,b,c
cos φγ + SY
i
γ =a,b,c
sin φγ
⎤
⎦, (12)
which drops out since both γ cos φγ and γ sin φγ are
zero. Only the remaining A terms containing S+
i S+
j and S−
i S−
j
are active. Setting A = 0, all the S−
i or S−
i S−
j terms that could
lead to quantum fluctuations are cut off by zero prefactors
and we are left with an exact eigenstate. The two conditions
for a fluctuation-free FM ground state, i.e., moments being
perpendicular to the honeycomb plane and A = 0 translating
to
K + 2 − 2 = 0, (13)
are checked in Fig. 5(a). Approaching the line given by
Eq. (13) within the (111) polarized FM phase, the probability
indeed reaches the maximum value of 1/2, reflecting the two
degenerate configurations (moments along Z or −Z) being
064425-9
155
RUSNA ˇCKO, GOTFRYD, AND CHALOUPKA PHYSICAL REVIEW B 99, 064425 (2019)
superposed in the cluster ground state. Exactly at this line,
we find a doubly degenerate ground state. Finally, let us
note that the ground state remains fluctuation-free even in the
presence of J3 provided that the (111) polarized FM pattern is
preserved.
B. AF phase
A more complex situation is encountered in the case of
(111) polarized AF phase. Here the fluctuation-free manifold
is attached to the vortex point of hidden SU(2) symmetry
hosting an infinite number of fluctuation-free states. The (111)
polarized AF state is one of them, the others being e.g., the
vortex-a and vortex-b configurations shown in Fig. 2(b). The
connection to the SU(2) vortex point suggests a special role of
the T6 transformation which we discuss in more detail here.
The T6 transformation is a six-sublattice mapping that
rotates the spins according to the recipe:
sublattice 1: (Sx
, Sy
, Sz
) = (Sx
, Sy
, Sz
),
sublattice 2: (Sx
, Sy
, Sz
) = (−Sy
, −Sx
, −Sz
),
sublattice 3: (Sx
, Sy
, Sz
) = (Sy
, Sz
, Sx
),
(14)
sublattice 4: (Sx
, Sy
, Sz
) = (−Sx
, −Sz
, −Sy
),
sublattice 5: (Sx
, Sy
, Sz
) = (Sz
, Sx
, Sy
),
sublattice 6: (Sx
, Sy
, Sz
) = (−Sz
, −Sy
, −Sx
).
For a better understanding, the transformation is depicted in
Fig. 5(c). On sites 1, 3, and 5 marked by a square symbol,
the spins are rotated around the (111) axis, on sites 2, 4, and
6 marked by a circle, the mapping consists of π rotations
around axes lying in the honeycomb plane. This in effect
changes the (111) polarized AF pattern into (111) polarized
FM one, making a first step towards the understanding of the
AF fluctuation-free line.
The second step involves the transformation of the Hamiltonian.
Performing the T6 spin rotations, we find that the resulting
model is similar to EKH in the sense that three types of
bond interactions of the form of Eq. (1) appear, H(x)
≡ H(a)
,
H(y)
≡ H(b)
, and H(z)
≡ H(c)
, with the parameters modified
according to
(J, K, , )Kekulé = (− , −J − K + , −J, − ). (15)
However, the assignment of H(x,y,z)
to the bonds is not simply
by the bond direction anymore. Instead, as shown in Fig. 5(d),
a network of benzene-like rings governed by alternating
H(y)
and H(z)
is formed. They are interconnected by bonds
possessing the H(x)
type of interactions. This way, the T6
transformation maps the EKH model to an extended variant
of Kekulé-Kitaev model [95].
We are now in position to combine the result of T6 transformation
with the argumentation of Sec. V A. Since the transformation
led to (111) polarized FM pattern and in the new model
each site is a member of three bonds governed by H(x)
, H(y)
,
and H(z)
, the cancellation of the terms leading to quantum
fluctuations proceeds exactly the same way. Substituting the
parameters in Eq. (13) according to (15), we thus arrive at the
condition for the fluctuation-free AF state:
3J + K − − 2 = 0. (16)
As demonstrated in Fig. 5(e), this line coincides with the
region where the probability of Néel state peaks at 1/2. For
a negative , the line quickly gets out of the AF phase. However,
going in the positive direction, the fluctuation-free
line gets even deeper into the AF phase (cf. Appendix C). At
the special point J = = > 0, K = 0 on the fluctuationfree
manifold, the model even reduces to AF Ising model
with the (111) Ising axis, as can be seen from Eqs. (6)–(11).
Unlike in the previous FM case, the addition of J3 spoils the
fluctuation-free nature of the ground state since the J3 interaction
generates terms of A type under the T6 transformation.
VI. ISING-KITAEV-COMPASS MODEL
In this section, we address yet another feature of the model
that enables further insights into its phase behavior. Namely,
we find points in the parameter space where the four interactions
JK can be combined into a single one, characterized
by a single interacting spin component (interaction axis) that
depends on the bond direction. This way, the model in Eq. (6)
may realize combinations of Ising, Kitaev, or compass model
on the honeycomb lattice.
A. Compass point in the phase diagram
Inspecting the JK phase diagram, we find a degenerate
point, in which several phases seem to meet: vortex, ferromagnet,
and both zigzag phases. Writing the interaction as
H = i j ST
i Hi jSj, we find that the Hamiltonian matrices in
this parameter point K = = −J > 0, = 0 have a symmetrical
block shape:
Ha =
⎛
⎜
⎝
J + K
J
J
⎞
⎟
⎠ =
⎛
⎜
⎝
0 0 0
0 −K K
0 K −K
⎞
⎟
⎠, (17)
Hb =
⎛
⎜
⎝
J
J + K
J
⎞
⎟
⎠ =
⎛
⎜
⎝
−K 0 K
0 0 0
K 0 −K
⎞
⎟
⎠, (18)
Hc =
⎛
⎜
⎝
J
J
J + K
⎞
⎟
⎠ =
⎛
⎜
⎝
−K K 0
K −K 0
0 0 0
⎞
⎟
⎠. (19)
The matrices can be diagonalized by a change of basis to the
rotating coordinate frame ˜xγ , ˜yγ , ˜zγ , γ ∈ {a, b, c}, where ˜xγ
axis points in the bond direction, ˜yγ is perpendicular to the
bond direction and lies in the honeycomb plane, and ˜zγ points
out of the honeycomb plane—see Fig. 6(a) for a sketch of
this coordinate system. After the change of basis, all three
interaction matrices have the same form for all bond directions
a, b, c:
H =
⎛
⎜
⎝
−2K 0 0
0 0 0
0 0 0
⎞
⎟
⎠, (20)
064425-10
156
KITAEV-LIKE HONEYCOMB MAGNETS: GLOBAL PHASE … PHYSICAL REVIEW B 99, 064425 (2019)
FIG. 6. (a) Rotating coordinate system ˜xγ , ˜yγ , ˜zγ on a honeycomb
lattice: color distinguishes three bond types a, b, c. (b) The
direction of the interaction axis for each bond. All of them are at
an angle ϑ with the Ising (111) direction shown in black. [(c) and
(d)] Ising-Kitaev-compass parameter line in the phase diagram and
the corresponding values of the parameter. The Ising point ( )
emerges for positive , while the π-rotated Kitaev point (◦) and
the perpendicular “compass” point ( ) are found for negative
values. The true compass point ( ) appears for = 0. (e) The angle
ϑ of the interaction axis to the (111) direction depending on the
position on the parameter line. ϕ = π/2 is assumed. The Kitaev
point (•) is connected to the π-rotated Kitaev point (◦) by the T1
dual transformation [75]—a π rotation around the (111) axis.
which represents FM interaction in the bond direction, concisely
written as
H =
i j ∈NN
−2K(Si · ri j )(Sj · ri j ), (21)
where the unit vector ri j points from site i to site j. This form
of interaction is known in the literature as the 120◦
honeycomb
compass model [11,96–98]. Similar to the Kitaev model, it
features frustration due to competing interactions for the three
bond directions. However, the exact ground state is not known
in this case and its nature is in fact not clear, as several past
works came to inconsistent conclusions. One study found a
Néel state [96], others suggested a stabilization of a dimer
pattern [97], a superposition of dimer coverings [98], or a
quantum spin liquid state [99]. With the link to the compass
model, the apparent special role of the −J = K = point
marked by a competition of four long-range ordered phases
in its vicinity is confirmed. This competition was noticed also
in the tensor-network analysis of the JK model [77], which
claimed a small region surrounding this point to harbor a
valence bond solid phase.
B. Ising-Kitaev-compass line in the phase diagram
Motivated by the previous example, we now demonstrate
that the EKH model provides also a more general case of a single
interaction axis lying anywhere between the honeycomb
plane and the perpendicular (111) direction. Dictated by the
C3 symmetry of the EKH model, the interaction axis has to
rotate together with the bond direction as shown in Fig. 6(b).
Representing the bond-dependent interaction axis by a unit
vector nγ , the Hamiltonian of the single-axis model hidden in
the EKH model has to be of the form
HIKc =
i j ∈NN
K(nγ · Si)(nγ · Sj ). (22)
We specify the interaction-axis direction by the deviation ϑ
from the (111) direction and the azimuthal angle ϕ measured
from the bond. In the rotating reference frame ˜xγ ˜yγ ˜zγ , we
have
nγ = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ), (23)
while in the XY Z reference frame of Fig. 5(b)
nγ = (− sin ϑ sin(ϕ+φγ ), sin ϑ cos(ϕ+φγ ), cos ϑ), (24)
with φγ being the bond angles defined in Sec. V. In the
direction of increasing ϑ, the Hamiltonian (22) encompasses
Ising model (ϑ = 0), Kitaev model [ϑ = arccos(1/
√
3), ϕ =
π/2], and compass model (ϑ = π/2, ϕ = 0). Note that the
physical properties do not depend on ϕ so that it is sufficient
to focus on ϑ as the relevant parameter. For example, true
compass model has ϕ = 0 (the interaction axis coincides with
the bond direction) but ϕ = π/2 (the in-plane interaction
axis is perpendicular to the bonds) leads to the same ground
state, apart from a trivial rotation. We thus call the latter one
“compass” to suggest a small only distinction.
We now establish the link to EKH model by converting
its interactions into main axes. This is most conveniently performed
in the rotating reference frame where the Hamiltonian
is represented by a matrix common to all bond directions. If
two of its eigenvalues are zero, we are left with the single
interaction axis corresponding to the model of Eq. (22). In
Sec. VI A, we have already encountered a situation in which
the Hamiltonian was readily diagonalized merely by casting
it into the rotating frame and the only nonzero eigenvalue
for the in-bond direction generated the compass interaction.
The general inspection is left for Appendix B, here we only
summarize the results presented in Figs. 6(c)–6(e). It turns
out that the compass case with nγ = ˜xγ (ϕ = 0) is singular for
0 and in the other cases the interaction axis is found in
064425-11
157
RUSNA ˇCKO, GOTFRYD, AND CHALOUPKA PHYSICAL REVIEW B 99, 064425 (2019)
the perpendicular ˜yγ ˜zγ plane (ϕ = π/2)—Fig. 6(b) contains a
sketch of such a bond-dependent interaction axis.
Assuming > 0, the diagonalized interaction has only one
nonzero component on the line determined by
J = , J(J + K) = 2
, J > 0, J + K > 0, (25)
which is indicated in Figs. 6(c) and 6(d). In our parametrization,
it covers the range | | 1
2
1 +
√
3 ≈ 0.83.
Figure 6(e) shows how the deviation ϑ of the interaction
axis from (111) direction evolves on the parameter line (25);
the colors differentiate the two branches for > 0 and
< 0, respectively, and correspond to colors in Figs. 6(c)
and 6(d). The interaction constant ˜K = 3J + K is always
positive, hence the interaction is antiferromagnetic. Several
distinct points are labeled in the figure: the limit ϑ = 0
achieved for J = = = 1
3
˜K, K = 0 corresponds to an
antiferromagnetic Ising point that lies at the same time
at the fluctuation-free manifold discussed in Sec. V. The
“compass” limit ϑ = ±π/2 is reached for J = = 1
6
˜K,
K = 1
2
˜K, and = −1
3
˜K. By varying the model parameters,
one can arbitrarily interpolate between these two limits. A
special role is played by the Kitaev case ϑ ≈ ±54.7◦
that is
characterized by a spin-liquid ground state. It can be found
either at AF Kitaev point K = ˜K, J = = = 0 with
ϑ = arccos(1/
√
3) or, less trivially, in a form rotated by π
about the (111) axis: J = = 4
9
˜K, K = −1
3
˜K, = −2
9
˜K
with ϑ = − arccos(1/
√
3). In general, the parameters for ϑ
and −ϑ are related by the T1 transformation of Ref. [75] that
corresponds to a π rotation about the (111) axis (for details
see Appendix B).
C. Phases of Ising-Kitaev-compass model
Having identified the line in the phase diagram where the
EKH model effectively interpolates between Ising, Kitaev,
and “compass” model captured by Eq. (22) (all AF for >
0), we now briefly address the phase diagram on this line
for varying ϑ. Let us note, that the corresponding type of
model (dubbed “tripod”) has been studied before using tensor
networks [100], although with the axis in the ˜xγ ˜zγ plane and
not in connection with the EKH model. As before, we apply
the method of Sec. III A combined with an analysis of the spin
correlations.
The resulting phase diagram is shown in Fig. 7. We formally
plot it as a function of |ϑ|. For negative values of ϑ
it covers the range Ising—π-rotated Kitaev—“compass” that
is continuously visible in Fig. 6(e). The phase diagram for
positive ϑ is identical, even though corresponding to different
line in the parameter space of the EKH model.
More than half of the phase diagram is occupied by the
AF Ising phase with the moments pointing in the (111)
direction. It starts as fluctuation free at the Ising point ϑ = 0◦
and gradually acquires more quantum fluctuating nature until
|ϑ| ≈ 49.5◦
where a transition to a spin-liquid associated
with the (π-rotated) Kitaev point at |ϑ| ≈ 54.7◦
happens.
The increasing content of quantum fluctuations is manifested
by decreasing spin correlations [Fig. 7(a)] or the probability
of Néel state in the ground state, which follows a similar
curve. At approximately 62◦
the spin liquid state breaks
FIG. 7. (a) Trace of the spin structure factor S(q) = α Sα
−qSα
q
obtained by ED using hexagonal 24-site cluster. The calculation
reveals an Ising antiferromagnet and a Kitaev liquid phase; for
high values of |ϑ| (interaction axis near the honeycomb plane), the
correlations show a mixture of contributions mainly at , X, and
K points. [(b)–(d)] Most probable spin configurations determined by
the method of spin-1/2 coherent states (using ϕ = π/2). The Néel
state is out of plane in (b) and only slightly tilted from the plane in (c).
The spin arrangement (d) is the same as the vortex-a configuration
shown in Fig. 2(b). [(e)–(h)] Spin-spin correlations Sz
−qSz
q for a few
selected values of ϑ and ϕ = π/2. In (f), the typical cosine wave
pattern characterizing a spin liquid state appears.
down. Compared to the tensor network study [100], we find
a significantly wider spin liquid window: (49.5◦
, 62◦
) versus
(52.7◦
, 57.6◦
). The reason for this discrepancy is not clear, an
earlier tensor-network study of the Kitaev-Heisenberg model
[85] was in a good agreement with ED. One possibility is
that a different procedure of finding the level crossings or
extrapolating the bond dimension has been used for the two
tensor networks studies.
For |ϑ| 62◦
, the ground state is characterized by peaks
in the structure factor at the and X point [see Fig. 7(g)],
while around |ϑ| ≈ 77◦
the correlations at point drop
and the K point becomes dominant [Fig. 7(h)]. In the first
part of this interval, the method of spin-1/2 coherent states
finds a Néel state with the moment direction slightly tilted
( 5◦
) from the honeycomb plane [Fig. 7(c)] and the amount
064425-12
158
KITAEV-LIKE HONEYCOMB MAGNETS: GLOBAL PHASE … PHYSICAL REVIEW B 99, 064425 (2019)
of quantum fluctuations comparable to the ground state of
the AF Heisenberg model. After a transition at about 77◦
, a
vortex-a pattern is found [Fig. 7(d)], but with a probability
P ≈ 3% significantly smaller than observed earlier inside the
vortex phase. This suggests that the second phase might be
possibly disordered, as claimed previously by Refs. [98,99]
for the compass model. The absence of visible changes in the
correlations in the region of the second phase indicates that
the features of “compass” limit |ϑ| = 90◦
are kept throughout
this phase. Similarly to the vortex phase, the most probable
pattern (vortex-a) is accompanied by the complementary pattern
(vortex-b) having a very close probability (P ≈ 3.16%
versus 3.05% in the AF “compass” limit). The same pair but
with the swapped probabilities is found for FM compass point
discussed in Sec. VI A. This is natural since the two models
as well as the two patterns are linked by a simple 90◦
rotation
of the spins (interchanging the in-bond and perpendicular
components), followed by a 180◦
rotation at every second site
(converting AF to FM and vice versa).
Finally, we note that the presence of two distinct phases
between the spin liquid phase and the compass limit is at odds
with the conclusion of Ref. [100] that the whole interval is
occupied by a dimer phase. To check the reliability of our
phase diagram, we have performed an additional ED for 32site
clusters of two different shapes, confirming the existence
of the Néel phase for those clusters as well.
VII. CONCLUSIONS
We have performed a detailed numerical investigation of
the global phase diagram of the extended Kitaev-Heisenberg
model including the analysis of the internal structure of the
individual phases. To this end, we have used mainly exact
diagonalization combined with a recently developed groundstate
analysis based on spin-1/2 coherent states.
In the context of real materials such as Na2IrO3 or αRuCl3,
our results are useful when judging the extent of
the experimentally observed zigzag phase and comparing the
direction of the ordered moments, fixing thereby the relevant
window in the parameter space.
In more general terms, we have interpreted the trends
in the phase diagram based on several types of symmetry
features found in the extended Kitaev-Heisenberg model,
providing a number of reference points of expected behavior.
They include points of hidden SU(2) symmetry, manifolds of
fluctuation-free ground states, and mappings to other models:
Ising, compass, and hidden Kitaev. We have demonstrated that
while being in principle simple results of linear algebra, these
symmetry features have far-reaching consequences and well
fix the overall structure of the global phase diagram. As we
believe, our symmetry-guided study can serve as a methodological
template that can be applied to other spin models with
bond-dependent non-Heisenberg interactions emerging in the
field of Mott insulators with strong spin-orbit coupling.
Among the unusual symmetry properties brought about
by the bond-dependent non-Heisenberg interactions, we have
highlighted two interesting features that, to the best of our
knowledge, escaped attention so far:
(i) Fluctuation-free ground states on entire manifolds of parameter
points, possessing both FM and AF ordering patterns.
These are enabled due to a partial cancellation of interactions
for the particular spin structure. However, above the ground
state, these interactions are fully active and shall lead to
excitation spectra quite distinct from those of e.g., Heisenberg
FM.
(ii) Models with bond-dependent non-Heisenberg interactions
may realize not only the sought-after Kitaev model
but also other models with a single bond-dependent interacting
spin component (“interaction axis”). In the case of
the extended Kitaev-Heisenberg model, they range from the
simple Ising model, through Kitaev, to the 120◦
compass
model on honeycomb lattice, whose ground-state nature is still
debated in the literature. The above models are continuously
connected in the parameter space of the extended KitaevHeisenberg
model and the corresponding line in the phase
diagram contains both trivial (Ising limit) as well as highly
nontrivial phases—perturbed Kitaev spin liquid and the phase
associated with the perturbed compass model. Such links to
the extended Kitaev-Heisenberg model may motivate a search
for candidate materials realizing, e.g., a compass model.
ACKNOWLEDGMENTS
We would like to thank Giniyat Khaliullin, Andrzej M.
Ole´s, Krzysztof Wohlfeld, and Kurt Hingerl for helpful discussions.
J.C. and J.R. acknowledge support from the Ministry
of Education, Youth and Sports of the Czech Republic
within the project CEITEC 2020 (LQ1601) under the National
Sustainability Programme II, European Research Council
via project TWINFUSYON (No. 692034), Czech Science
Foundation (GA ˇCR) under project No. GJ15-14523Y, and
Masaryk University internal projects MUNI/A/1310/2016 and
MUNI/A/1291/2017. J.R. is Brno Ph.D Talent Scholarship
Holder—Funded by the Brno City Municipality. D.G. was
supported by Polish National Science Center (NCN) under
projects 2012/04/A/ST3/00331 and 2016/23/B/ST3/00839.
Computational resources were provided by the CESNET
LM2015042 and the CERIT Scientific Cloud LM2015085,
provided under the programme “Projects of Large Research,
Development, and Innovations Infrastructures”. The CMFT
calculations were performed at the Interdisciplinary Centre
for Mathematical and Computational Modelling (ICM) of the
University of Warsaw under Grant Nos. G66-22 and G72-9.
APPENDIX A: OPTIMIZED SPIN-1/2 COHERENT STATES
IN THE PRESENCE OF SPIN LIQUID PHASES
In contrast to the ordered phases, the ground states of
Kitaev spin liquid (KSL) phases are highly entangled and
cannot be well described by a spin-1/2 coherent state (4) that
is a simple product of spin-1/2 states. This is indeed observed
when applying the method of Sec. III A to the spin-liquid
ground state. Nevertheless, the spin-liquid ground state still
has a significant overlap with the configurations found in
the classical S → ∞ limit of the Kitaev model. Shown in
Figs. 8(a) and 8(b) are the most probable configurations found
in the exact ground state of the 24-site cluster near the AF or
FM Kitaev point, respectively. They are characterized by spins
pointing along the cubic axes x, y, z and forming aligned (FM
case) or contra-aligned (AF case) pairs on nearest-neighbor
064425-13
159
RUSNA ˇCKO, GOTFRYD, AND CHALOUPKA PHYSICAL REVIEW B 99, 064425 (2019)
z
y
x
Pmax[%]
ϕ
KSL
zigzag
AF
0.0
0.5
1.0
1.5
2.0
88°89°90°91°92°
ϕ
KSL
FM
stripy
0
2
4
6
8
10
260° 270° 280°
Pmax [%]
0.1
1
10
)b()a(
)d()c(
(e)
AF
zigzag
vortex
KSL
FIG. 8. Most probable configurations near the (a) AF and (b) FM
Kitaev points. Spins pointing along the x, y, and z cubic axes are
marked by red, green, and blue color, respectively. (c) Probability
Pmax of the optimized configuration near the AF Kitaev point in the
Kitaev-Heisenberg model parametrized as J = cos ϕ and K = sin ϕ.
Phase transitions to the neighboring ordered phases are indicated by
dashed lines. (d) The same for the region around the FM Kitaev point.
(e) Map of Pmax in a small area near the AF Kitaev spin liquid phase
in the extended Kitaev-Heisenberg model with = J3 = 0 (belongs
to the same slice as shown in Fig. 2).
bonds. Their orientation is determined by the bond direction
which is linked to the active spin component in the Kitaev
interaction.
It is instructive to inspect the evolution of probability Pmax
of the optimized spin-1/2 coherent state when going from
the KSL phase through a quantum phase transition to the
neighboring ordered phases. This is done in Fig. 8(c) for the
case of AF Kitaev model perturbed by Heisenberg interaction.
While the KSL is characterized by small Pmax ≈ 0.1%, at the
phase boundary either to zigzag or to AF ordered phase Pmax
jumps by about one order of magnitude to values above 1%
typically encountered in the main text for the ordered phases
with significant quantum fluctuations. A different behavior
is found in the FM case. Here the probability rises near the
phase boundaries to FM and stripy phases and continuously
connects to Pmax of their ground states. However, a discontinuity
still appears in the derivative with respect to model parameters,
with Pmax shooting up after crossing the phase boundary.
This is yet another manifestation of the different nature of
the phase transitions involving AF and FM KSL phases that
shows up, e.g., in the behavior of the spin-excitation gap [81].
Finally, in Fig. 8(e), we consider a larger portion of the
phase diagram near the AF KSL phase which also involves
the nonzero > 0 parameter case. The AF KSL phase can be
easily distinguished again by a drop of Pmax. The case of FM
KSL phase at the bottom of the corresponding phase diagram
slice is more complicated by the less pronounced transition
and more complex phase behavior in the surrounding region
as discussed in Sec. IV C. Fig. 8(e) also illustrates the difficulties
of the global optimization in the complete 48-dimensional
space of θ and φ parameters corresponding to the 24-site
cluster. As seen in Fig. 8(e), the danger of getting trapped in a
local maximum increases e.g., near the zigzag/vortex phase
boundary or in the KSL phase characterized by competing
configurations.
APPENDIX B: ISING-KITAEV-COMPASS MODEL:
DERIVATION
The EKH Hamiltonian has a single matrix form for all
bond directions when written in the rotating reference frame
˜xγ , ˜yγ , ˜zγ :
H =
⎛
⎜
⎝
J − 0 0
0 J + 2K
3
+ 3
− 4
3
√
2
3
(K − + )
0
√
2
3
(K − + ) J + K
3
+ 2
3
+ 4
3
⎞
⎟
⎠.
(B1)
The matrix consists of two blocks, therefore one principal axis
is ˜xγ and the other two lie in the ˜yγ ˜zγ plane. The eigenvalues
of this matrix are
λ1 = J − ,
λ2 = J + 1
2
(K + + (K − )2 + 8 2), (B2)
λ3 = J + 1
2
(K + − (K − )2 + 8 2).
If only the first eigenvalue λ1 = J = is nonzero, we obtain
the 120◦
honeycomb compass model, the situation discussed
in Sec. VI A. For > 0, one can find a parameter region
where only λ2 is nonzero:
J = , J(J + K) = 2
, J > 0, J + K > 0. (B3)
Analogous to this case, for < 0 only λ3 is nonzero in the
region given by
J = , J(J + K) = 2
, J < 0, J + K < 0. (B4)
The sole nonzero eigenvalue in both of these cases equals
3J + K and its sign is the same as the sign of , hence in
the case > 0 used in the main text 3J + K > 0 and the
interaction is antiferromagnetic. The parameter region (B3)
[or (B4)] can be parametrized by a single variable, which
we choose as the angle ϑ between the ˜zγ = Z axis and the
064425-14
160
KITAEV-LIKE HONEYCOMB MAGNETS: GLOBAL PHASE … PHYSICAL REVIEW B 99, 064425 (2019)Γ′=+0.894
FM:9.5% AF:50.0% zigzag:0.0% stripy:0.0% vortex:0.0%(a)
Γ′=+0.707
FM:10.7% AF:50.0% zigzag:0.0% stripy:0.0% vortex:0.0%(b)
Γ′=+0.200
FM:14.9% AF:50.0% zigzag:2.3% stripy:31.5% vortex:0.0%(c)
Γ′=+0.100
FM:15.7% AF:50.0% zigzag:2.8% stripy:27.1% vortex:8.1%(d)
Γ′=+0.000
FM:16.7% AF:50.0% zigzag:2.9% stripy:17.8% vortex:16.1%(e)
Γ′=-0.100
FM:50.0% AF:47.2% zigzag:2.7% stripy:28.5% vortex:16.7%(f)
Γ′=-0.200
FM:50.0% AF:30.3% zigzag:2.8% stripy:32.9% vortex:15.3%(g)
Γ′=-0.312
FM:50.0% AF:10.6% zigzag:2.9% stripy:0.9% vortex:13.8%(h)
Γ′=-0.398
FM:50.0% AF:12.5% zigzag:2.9% stripy:0.3% vortex:12.7%(i)
Γ′=-0.535
FM:49.9% AF:13.7% zigzag:2.9% stripy:1.5% vortex:11.1%(j)
Γ′=-0.603
FM:49.6% AF:14.1% zigzag:2.8% stripy:0.9% vortex:10.4%(k)
Γ′=-1.000
FM:46.4% AF:14.7% zigzag:2.3% stripy:0.3% vortex:7.3%(l)
0° 15° 30° 45° 60° 75° 90°
angle to the honeycomb plane α
P=0
P=Pmax
β=0°
β=30°
β=60°
β=180°
FIG. 9. Phase diagram slices for selected values and J3 = 0. Left circle in each panel shows a color map of the probabilities of the
patterns: FM (blue), AF (green), zigzag (red), stripy (blue), and vortex (violet). The ranges of the corresponding scales (bottom left) are
determined by the maximum probability Pmax as given at the top of the individual panels. Middle circles show the angle to the honeycomb
plane (common scale at bottom center). Right circles show the azimuthal angle ϕ within the honeycomb plane. Fluctuation-free lines given by
Eq. (13) (FM phase, light blue) and (16) (AF phase, light green) are indicated along with the special points: (hidden) SU(2) symmetry points
[• in (a), (e), (i), and (j)], Kitaev points [• in (e), ◦ in (h)], Ising point [ in (b)], compass point [ in (e)] and compass-like point [ in (k)].
064425-15
161
RUSNA ˇCKO, GOTFRYD, AND CHALOUPKA PHYSICAL REVIEW B 99, 064425 (2019)
interaction direction. In terms of the model parameters, ϑ is
given by
tan 2ϑ = 2
√
2
9
J − K + 8
− 1 . (B5)
It is also easy to obtain the parameters J, K, , and
realizing the model (22) with arbitrary ϑ and ϕ = π/2:
J = = ˜K 1
6
[1 + cos ϑ(cos ϑ −
√
8 sin ϑ)], (B6)
K = ˜K 1
2
[1 − cos ϑ(cos ϑ −
√
8 sin ϑ)], (B7)
= ˜K 1
3
cos 2ϑ + 1√
8
sin 2ϑ) . (B8)
Finally, as demonstrated in Ref. [75], the nearest-neighbor
EKH model preserves its form under global rotation of the
spin axes by π around the (111) direction. This transformation,
labeled as T1 in Ref. [75], maps the model parameters
JK onto another set according to Eq. (4) of Ref. [75]. In
the context of the Ising-Kitaev-compass model, the rotation
by π around (111) axis corresponds to a sign change ϑ →
−ϑ. Indeed, inserting the transformed set of parameters into
Eq. (B5), one observes the sign change. Obviously, the T1
transformation does not have any effect in the Ising (ϑ = 0)
and compass/“compass” (ϑ = ±π/2) case. Apart from these
cases, it connects various pairs of points on the curves in
Figs. 6(c)–6(e), most importantly, the Kitaev model and its
π-rotated variant.
APPENDIX C: DETAILED EVOLUTION OF THE PHASE
DIAGRAM FOR NONZERO
Figure 9 presents a detailed evolution of the phases of
the nearest neighbor model (J3 = 0) with the parameter
attaining both positive and negative values. It was obtained
using the method of Sec. III A for the hexagon-shaped 24-site
cluster. The values were selected to include all the special
symmetry points discussed in Ref. [75] and in the present
paper. The lines of fluctuation-free ground states discussed in
Sec. V are also indicated.
The plots use the same parametrization of the interactions
as that of Fig. 2 and show also the moment direction in the
form of an angle to the honeycomb plane α and an azimuthal
angle in the honeycomb plane β. Using the XY Z reference
frame of Fig. 5(b), the moment direction in the collinear
bvortex−avortex−
(b) (c)(a)
1
6
5
4
3
2
β = 30°β = 0°
β
FIG. 10. (a) Moment directions defined by Eq. (C2) capturing
vortex in-plane structure and out-of-plane AF staggering. For
the in-plane deviation angle β = 0◦
and 30◦
, we get the patterns
(b) vortex-a and (c) vortex-b of Fig. 2(b), respectively.
phases is given by
n = (cos α cos β, cos α sin β, sin α). (C1)
For the moments lying in the honeycomb plane, β = 0◦
corresponds
to the direction perpendicular to a bond while β = 30◦
is in-bond direction. The values β = 60◦
and β = 180◦
are
used in the out-of-plane cases where a further distinction is
necessary. In the vortex phase, the directions of the moments
at the six sublattices marked in Fig. 5(c) are
nk = (cos α cos βk, cos α sin βk, (−1)k−1
sin α), (C2)
where βk = (−1)k−1
β − 60◦
k and k = 1, 2, . . . , 6 labels the
sublattice. This ansatz, depicted in Fig. 10, captures both patterns
vortex-a (β = 0◦
or 60◦
) and vortex-b (β = 30◦
) shown
in Fig. 2(b) and AF staggering in the direction perpendicular
to the honeycomb plane (nonzero α).
Not indicated in Fig. 9 are the regions of the instability
of the zigzag phase observed in Figs. 2(a), 4(a), and 4(b).
Apart from the ones shown earlier, the regions with noncollinear
tendencies appear for large negative −0.5 near
the meeting point of the zigzag phase with FM and vortex
phases and, to a smaller extent, also at the bottom near the
white region in Figs. 9(j)–9(l). Since the smaller zigzag phase
visible in Figs. 9(c)–9(f) is a copy of the larger zigzag phase
linked by the exact T1 transformation, it also contains such
patches of instability. Similarly, the top and bottom white
areas hosting incommensurate orderings [visible in Figs. 9(a)–
9(c) and Figs. 9(c)–9(f), respectively] are linked by the T1
transformation. The white area of Figs. 9(j)–9(l) maps to the
< 0 case.
[1] Introduction to Frustrated Magnetism, edited by C. Lacroix, P.
Mendels, and F. Mila (Springer, Berlin, 2011).
[2] L. Balents, Nature (London) 464, 199208 (2010).
[3] L. Savary and L. Balents, Rep. Prog. Phys. 80, 016502
(2017).
[4] S. T. Bramwell and M. J. P. Gingras, Science 294, 14951501
(2001).
[5] C. Castelnovo, R. Moessner, and S. L. Sondhi, Annu. Rev.
Condens. Matter Phys. 3, 35 (2012).
[6] M. R. Norman, Rev. Mod. Phys. 88, 041002 (2016).
[7] P. Chandra and B. Doucot, Phys. Rev. B 38, 9335 (1988).
[8] S. Chakravarty, B. I. Halperin, and D. R. Nelson, Phys. Rev. B
39, 2344 (1989).
[9] B. Schmidt and P. Thalmeier, Phys. Rep. 703, 1 (2017).
[10] W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents,
Annu. Rev. Condens. Matter Phys. 5, 57 (2014).
[11] Z. Nussinov and J. van den Brink, Rev. Mod. Phys. 87, 1
(2015).
[12] J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Annu. Rev. Condens.
Matter Phys. 7, 195 (2016).
[13] R. Schaffer, E. K.-H. Lee, B.-J. Yang, and Y. B. Kim, Rep.
Prog. Phys. 79, 094504 (2016).
064425-16
162
KITAEV-LIKE HONEYCOMB MAGNETS: GLOBAL PHASE … PHYSICAL REVIEW B 99, 064425 (2019)
[14] S. Trebst, arXiv:1701.07056.
[15] S. M. Winter, A. A. Tsirlin, M. Daghofer, J. van den Brink, Y.
Singh, P. Gegenwart, and R. Valentí, J. Phys.: Condens. Matter
29, 493002 (2017).
[16] M. Hermanns, I. Kimchi, and J. Knolle, Annu. Rev. Condens.
Matter Phys. 9, 17 (2018).
[17] A. Kitaev, Ann. Phys. 321, 2 (2006).
[18] A. Abragam and B. Bleaney, Electron Paramagnetic Resonance
of Transition Ions (Clarendon Press, Oxford, 1970).
[19] G. Khaliullin, Prog. Theor. Phys. Suppl. 160, 155 (2005).
[20] B. J. Kim, H. Jin, S. J. Moon, J.-Y. Kim, B.-G. Park, C. S.
Leem, J. Yu, T. W. Noh, C. Kim, S.-J. Oh, J.-H. Park, V.
Durairaj, G. Cao, and E. Rotenberg, Phys. Rev. Lett. 101,
076402 (2008).
[21] B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita, H.
Takagi, and T. Arima, Science 323, 1329 (2009).
[22] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205
(2009).
[23] J. Kim, D. Casa, M. H. Upton, T. Gog, Y.-J. Kim, J. F.
Mitchell, M. van Veenendaal, M. Daghofer, J. van den Brink,
G. Khaliullin, and B. J. Kim, Phys. Rev. Lett. 108, 177003
(2012).
[24] Y. K. Kim, O. Krupin, J. D. Denlinger, A. Bostwick, E.
Rotenberg, Q. Zhao, J. F. Mitchell, J. W. Allen, and B. J. Kim,
Science 345, 187 (2014).
[25] Y. K. Kim, N. H. Sung, J. D. Denlinger, and B. J. Kim, Nat.
Phys. 12, 37 (2016).
[26] K. I. Kugel and D. I. Khomskii, Sov. Phys. Usp. 25, 231
(1982).
[27] G. Chen and L. Balents, Phys. Rev. B 78, 094403 (2008).
[28] K. W. Plumb, J. P. Clancy, L. J. Sandilands, V. V. Shankar,
Y. F. Hu, K. S. Burch, H.-Y. Kee, and Y.-J. Kim, Phys. Rev. B
90, 041112(R) (2014).
[29] A. Banerjee, C. A. Bridges, J.-Q. Yan, A. A. Aczel, L. Li, M.
B. Stone, G. E. Granroth, M. D. Lumsden, Y. Yiu, J. Knolle, S.
Bhattacharjee, D. L. Kovrizhin, R. Moessner, D. A. Tennant,
D. G. Mandrus, and S. E. Nagler, Nat. Mater. 15, 733 (2016).
[30] K. Ran, J. Wang, W. Wang, Z.-Y. Dong, X. Ren, S. Bao, S.
Li, Z. Ma, Y. Gan, Y. Zhang, J. T. Park, G. Deng, S. Danilkin,
S.-L. Yu, J.-X. Li, and J. Wen, Phys. Rev. Lett. 118, 107203
(2017).
[31] A. Banerjee, J. Yan, J. Knolle, C. A. Bridges, M. B. Stone, M.
D. Lumsden, D. G. Mandrus, D. A. Tennant, R. Moessner, and
S. E. Nagler, Science 356, 1055 (2017).
[32] S.-H. Do, S.-Y. Park, J. Yoshitake, J. Nasu, Y. Motome, Y. S.
Kwon, D. T. Adroja, D. J. Voneshen, K. Kim, T.-H. Jang, J.-H.
Park, K.-Y. Choi, and S. Ji, Nat. Phys. 13, 1079 (2017).
[33] X. Liu, T. Berlijn, W.-G. Yin, W. Ku, A. Tsvelik, Y.-J. Kim, H.
Gretarsson, Y. Singh, P. Gegenwart, and J. P. Hill, Phys. Rev.
B 83, 220403(R) (2011).
[34] F. Ye, S. Chi, H. Cao, B. C. Chakoumakos, J. A. FernandezBaca,
R. Custelcean, T. F. Qi, O. B. Korneta, and G. Cao, Phys.
Rev. B 85, 180403(R) (2012).
[35] S. K. Choi, R. Coldea, A. N. Kolmogorov, T. Lancaster, I. I.
Mazin, S. J. Blundell, P. G. Radaelli, Y. Singh, P. Gegenwart,
K. R. Choi, S.-W. Cheong, P. J. Baker, C. Stock, and J. Taylor,
Phys. Rev. Lett. 108, 127204 (2012).
[36] R. D. Johnson, S. C. Williams, A. A. Haghighirad, J.
Singleton, V. Zapf, P. Manuel, I. I. Mazin, Y. Li, H. O. Jeschke,
R. Valentí, and R. Coldea, Phys. Rev. B 92, 235119 (2015).
[37] J. A. Sears, M. Songvilay, K. W. Plumb, J. P. Clancy, Y. Qiu,
Y. Zhao, D. Parshall, and Y.-J. Kim, Phys. Rev. B 91, 144420
(2015).
[38] S. C. Williams, R. D. Johnson, F. Freund, S. Choi, A. Jesche,
I. Kimchi, S. Manni, A. Bombardi, P. Manuel, P. Gegenwart,
and R. Coldea, Phys. Rev. B 93, 195158 (2016).
[39] K. Kitagawa, T. Takayama, Y. Matsumoto, A. Kato, R. Takano,
Y. Kishimoto, S. Bette, R. Dinnebier, G. Jackeli, and H.
Takagi, Nature (London) 554, 341345 (2018).
[40] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett.
110, 097204 (2013).
[41] S. H. Chun, J.-W. Kim, Jungho Kim, H. Zheng, C. C.
Stoumpos, C. D. Malliakas, J. F. Mitchell, K. Mehlawat, Y.
Singh, Y. Choi, T. Gog, A. Al-Zein, M. Moretti Sala, M.
Krisch, J. Chaloupka, G. Jackeli, G. Khaliullin, and B. J. Kim,
Nat. Phys. 11, 462 (2015).
[42] J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Phys. Rev. Lett. 112,
077204 (2014).
[43] J. G. Rau and H.-Y. Kee, arXiv:1408.4811.
[44] V. M. Katukuri, S. Nishimoto, V. Yushankhai, A. Stoyanova,
H. Kandpal, S. Choi, R. Coldea, I. Rousochatzakis, L. Hozoi,
and J. van den Brink, New J. Phys. 16, 013056 (2014).
[45] Y. Yamaji, Y. Nomura, M. Kurita, R. Arita, and M. Imada,
Phys. Rev. Lett. 113, 107201 (2014).
[46] Y. Sizyuk, C. Price, P. Wölfle, and N. B. Perkins, Phys. Rev. B
90, 155126 (2014).
[47] S. M. Winter, Y. Li, H. O. Jeschke, and R. Valentí, Phys. Rev.
B 93, 214431 (2016).
[48] S. M. Winter, K. Riedl, P. A. Maksimov, A. L. Chernyshev, A.
Honecker, and R. Valentí, Nat. Commun. 8, 1152 (2017).
[49] P. A. McClarty, X.-Y. Dong, M. Gohlke, J. G. Rau, F.
Pollmann, R. Moessner, and K. Penc, Phys. Rev. B 98,
060404(R) (2018).
[50] D. G. Joshi, Phys. Rev. B 98, 060405(R) (2018).
[51] M. Becker, M. Hermanns, B. Bauer, M. Garst, and S. Trebst,
Phys. Rev. B 91, 155135 (2015).
[52] I. Rousochatzakis, U. K. Rössler, J. van den Brink, and M.
Daghofer, Phys. Rev. B 93, 104417 (2016).
[53] K. Li, S.-L. Yu, and J.-X. Li, New J. Phys. 17, 043032
(2015).
[54] A. Catuneanu, J. G. Rau, H.-S. Kim, and H.-Y. Kee, Phys. Rev.
B 92, 165108 (2015).
[55] K. Morita, M. Kishimoto, and T. Tohyama, Phys. Rev. B 98,
134437 (2018).
[56] T. Takayama, A. Kato, R. Dinnebier, J. Nuss, H. Kono, L. S. I.
Veiga, G. Fabbris, D. Haskel, and H. Takagi, Phys. Rev. Lett.
114, 077202 (2015).
[57] E. K.-H. Lee and Y. B. Kim, Phys. Rev. B 91, 064407
(2015).
[58] H.-S. Kim, E. K.-H. Lee, and Y. B. Kim, Europhys. Lett. 112,
67004 (2015).
[59] E. K.-H. Lee, J. G. Rau and, Y. B. Kim, Phys. Rev. B 93,
184420 (2016).
[60] V. M. Katukuri, R. Yadav, L. Hozoi, S. Nishimoto, and J. van
den Brink, Sci. Rep. 6, 29585 (2016).
[61] K. A. Modic, T. E. Smidt, I. Kimchi, N. P. Breznay, A.
Biffin, S. Choi, R. D. Johnson, R. Coldea, P. Watkins-Curry,
G. T. McCandless, J. Y. Chan, F. Gandara, Z. Islam, A.
Vishwanath, A. Shekhter, R. D. McDonald, and J. G. Analytis,
Nat. Commun. 5, 4203 (2014).
064425-17
163
RUSNA ˇCKO, GOTFRYD, AND CHALOUPKA PHYSICAL REVIEW B 99, 064425 (2019)
[62] A. Biffin, R. D. Johnson, I. Kimchi, R. Morris, A. Bombardi,
J. G. Analytis, A. Vishwanath, and R. Coldea, Phys. Rev. Lett.
113, 197201 (2014).
[63] I. Kimchi, J. G. Analytis, and A. Vishwanath, Phys. Rev. B 90,
205126 (2014).
[64] Y. Okamoto, M. Nohara, H. Aruga-Katori, and H. Takagi,
Phys. Rev. Lett. 99, 137207 (2007).
[65] G. Cao, A. Subedi, S. Calder, J.-Q. Yan, J. Yi, Z. Gai, L.
Poudel, D. J. Singh, M. D. Lumsden, A. D. Christianson, B. C.
Sales, and D. Mandrus, Phys. Rev. B 87, 155136 (2013).
[66] A. M. Cook, S. Matern, C. Hickey, A. A. Aczel, and A.
Paramekanti, Phys. Rev. B 92, 020417(R) (2015).
[67] A. A. Aczel, A. M. Cook, T. J. Williams, S. Calder, A. D.
Christianson, G.-X. Cao, D. Mandrus, Y.-B. Kim, and A.
Paramekanti, Phys. Rev. B 93, 214426 (2016).
[68] H. Kuriyama, J. Matsuno, S. Niitaka, M. Uchida, D.
Hashizume, A. Nakao, K. Sugimoto, H. Ohsumi, M. Takata,
and H. Takagi, Appl. Phys. Lett. 96, 182103 (2010).
[69] I. Kimchi and A. Vishwanath, Phys. Rev. B 89, 014414
(2014).
[70] M. Hermanns and S. Trebst, Phys. Rev. B 89, 235102
(2014).
[71] H. Liu and G. Khaliullin, Phys. Rev. B 97, 014407 (2018).
[72] R. Sano, Y. Kato, and Y. Motome, Phys. Rev. B 97, 014408
(2018).
[73] J. M. Luttinger and L. Tisza, Phys. Rev. 70, 954 (1946).
[74] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett.
105, 027204 (2010).
[75] J. Chaloupka and G. Khaliullin, Phys. Rev. B 92, 024413
(2015).
[76] L. Janssen, E. C. Andrade, and M. Vojta, Phys. Rev. B 96,
064430 (2017).
[77] J. Lou, L. Liang, Y. Yu, and Y. Chen, arXiv:1501.06990.
[78] K. Shinjo, S. Sota, and T. Tohyama, Phys. Rev. B 91, 054401
(2015).
[79] S. Nishimoto, V. M. Katukuri, V. Yushankhai, H. Stoll, U. K.
Rössler, L. Hozoi, I. Rousochatzakis, and J. van den Brink,
Nat. Commun. 7, 10273 (2016).
[80] J. Chaloupka and G. Khaliullin, Phys. Rev. B 94, 064435
(2016).
[81] D. Gotfryd, J. Rusnaˇcko, K. Wohlfeld, G. Jackeli, J.
Chaloupka, and A. M. Ole´s, Phys. Rev. B 95, 024426 (2017).
[82] C. C. Price and N. B. Perkins, Phys. Rev. Lett. 109, 187201
(2012).
[83] R. Schaffer, S. Bhattacharjee, and Y. B. Kim, Phys. Rev. B 86,
224417 (2012).
[84] J. Reuther, R. Thomale, and S. Trebst, Phys. Rev. B 84,
100406(R) (2011).
[85] J. Osorio Iregui, P. Corboz, and M. Troyer, Phys. Rev. B 90,
195102 (2014).
[86] M. Gohlke, R. Verresen, R. Moessner, and F. Pollmann, Phys.
Rev. Lett. 119, 157203 (2017).
[87] F. Trousselet, M. Berciu, A. M. Ole´s, and P. Horsch, Phys. Rev.
Lett. 111, 037205 (2013).
[88] F. Trousselet, P. Horsch, A. M. Ole´s, and W.-L. You, Phys.
Rev. B 90, 024404 (2014).
[89] Strongly Correlated Systems: Numerical Methods, edited by
A. Avella and F. Mancini (Springer, Berlin, 2013).
[90] For a discussion of the order-from-disorder phenomena in
frustrated spin systems, see A. M. Tsvelik, Quantum Field
Theory in Condensed Matter Physics (Cambridge University
Press, Cambridge, 1995), Chap. 17, and references therein.
[91] G. Baskaran, D. Sen, and R. Shankar, Phys. Rev. B 78, 115116
(2008).
[92] I. Rousochatzakis, Y. Sizyuk, and N. B. Perkins, Nat.
Commun. 9, 1575 (2018).
[93] M. Gohlke, G. Wachtel, Y. Yamaji, F. Pollmann, and Y. B.
Kim, Phys. Rev. B 97, 075126 (2018).
[94] P. P. Stavropoulos, A. Catuneanu, and H.-Y. Kee, Phys. Rev. B
98, 104401 (2018).
[95] E. Quinn, S. Bhattacharjee, and R. Moessner, Phys. Rev. B 91,
134419 (2015).
[96] E. Zhao and W. V. Liu, Phys. Rev. Lett. 100, 160403 (2008).
[97] C. Wu, Phys. Rev. Lett. 100, 200406 (2008).
[98] J. Nasu, A. Nagano, M. Naka, and S. Ishihara, Phys. Rev. B
78, 024416 (2008).
[99] C.-C. Chen, L. Muechler, R. Car, T. Neupert, and J. Maciejko,
Phys. Rev. Lett. 117, 096405 (2016).
[100] H. Zou, B. Liu, E. Zhao, and W. V. Liu, New J. Phys. 18,
053040 (2016).
064425-18
164
Dynamic Spin Correlations in the Honeycomb Lattice Na2IrO3
Measured by Resonant Inelastic x-Ray Scattering
Jungho Kim ,1,*
Jiˇrí Chaloupka ,2,3,†
Yogesh Singh,4
J. W. Kim,1
B. J. Kim,5,6
D. Casa,1
A. Said,1
X. Huang,1
and T. Gog1
1
Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA
2
Central European Institute of Technology, Masaryk University,
Kamenice 753/5, CZ-62500 Brno, Czech Republic
3
Department of Condensed Matter Physics, Faculty of Science, Masaryk University,
Kotláˇrská 2, CZ-61137 Brno, Czech Republic
4
Department of Physical Sciences, Indian Institute of Science Education and Research Mohali,
Knowledge City, Sector 81, Mohali 140306, India
5
Department of Physics, Pohang University of Science and Technology, Pohang 790-784,
Republic of Korea
6
Center for Artificial Low Dimensional Electronic Systems, Institute for Basic Science (IBS),
77 Cheongam-Ro, Pohang 790-784, Republic of Korea
(Received 16 April 2019; revised manuscript received 14 March 2020; accepted 23 March 2020; published 13 May 2020)
A Kitaev quantum spin liquid is a prime example of novel quantum magnetism of spin-orbit entangled
pseudospin-1=2 moments in a honeycomb lattice. Most candidate materials such as Na2IrO3 have many
competing exchange interactions beyond the minimal Kitaev-Heisenberg model whose small variations in
the strength of the interactions produce huge differences in low-energy dynamics. Our incomplete
knowledge of dynamic spin correlations hampers identification of a minimal model and quantification of
the proximity to the Kitaev quantum spin-liquid phase. Here, we report momentum- and energy-resolved
magnetic excitation spectra in a honeycomb lattice Na2IrO3 measured using a resonant inelastic x-ray
scattering spectrometer capable of 12 meV resolution. Measured spectra at a low temperature show that the
dynamic response lacks resolution-limited coherent spin waves in most parts of the Brillouin zone but has a
discernible dispersion and spectral weight distribution within the energy window of 60 meV. A systematic
investigation using the exact diagonalization method and direct comparison of high-resolution experimental
spectra and theoretical simulations allow us to confine a parameter regime in which the extended
Kitaev-Heisenberg model reasonably reproduces the main feature of the observed magnetic excitations.
Hidden Kitaev quantum spin-liquid and Heisenberg phases found in the complex parameter space are used
as references to propose the picture of renormalized magnons as explaining the incoherent nature of
magnetic excitations. Magnetic excitation spectra are taken at elevated temperatures to follow the
temperature evolution of the resonant inelastic x-ray scattering dynamic response in the paramagnetic
state. Whereas the low-energy excitation progressively diminishes as the zigzag order disappears, the
broad high-energy excitation maintains its spectral weight up to a much higher temperature of 160 K. We
suggest that the dominant nearest-neighbor interactions keep short-range correlations up to quite high
temperatures with a specific short-range dynamics which has a possible connection to a proximate spinliquid
phase.
DOI: 10.1103/PhysRevX.10.021034 Subject Areas: Condensed Matter Physics, Magnetism,
Strongly Correlated Materials
I. INTRODUCTION
A Kitaev quantum spin liquid (KQSL) is a topological
phase of matter resulting from an exactly solvable
Hamiltonian of nearest-neighbor bond-directional interactions
between half-integer spins in a honeycomb lattice
[1,2]. Their long-range quantum entanglement and topologically
protected fractional excitations are of particular
interest for potential quantum computing platforms [3].
It has been pointed out that the bond-directional Kitaev
*
jhkim@aps.anl.gov
†
chaloupka@physics.muni.cz
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
and DOI.
PHYSICAL REVIEW X 10, 021034 (2020)
2160-3308=20=10(2)=021034(20) 021034-1 Published by the American Physical Society
165
interactions arise naturally in a honeycomb magnet with
strong spin-orbit coupling, which triggered a wave of
searching for a material realization of the KQSL [4].
The Kitaev-Heisenberg model and related spin models
indicate extended stability of the spin-liquid phase away
from the pure Kitaev limit [5–10], which widens the scope
of candidate materials.
Na2IrO3 is one of the first and most extensively studied
candidate materials despite having a conventional symmetry-breaking
magnetic order as in other honeycomb
magnets [11–14]. A zigzag antiferromagnetic (AFM) order
was found by resonant x-ray magnetic scattering and
inelastic neutron scattering [12–14]. It has been shown
that the inclusion of small Heisenberg terms, omnipresent
in all materials, can explain the zigzag order [13,14].
Subsequently, many other models were proposed which
may be more realistic but are of increased complexity
[6–10]. Resonant magnetic x-ray scattering measurements
provide important constraints on the minimal model and the
sign of the Kitaev term [15]. It was found that the ordered
magnetic moment direction approximately bisects the angle
between the cubic x and y axes and that two other
dynamically fluctuating zigzag orders related to the static
one by the approximate C3 symmetry of the lattice have
their corresponding moment directions [15]. These results
imply a dominant ferromagnetic Kitaev term (K) with a
non-negligible off-diagonal exchange (Γ), which is supported
by quantum chemistry [16] and other ab initio
calculations [17,18]. The off-diagonal exchange which is
symmetry allowed can be sizable when direct exchange is
effective [7]. At a classical level, a spin-liquid phase is
theoretically found in models with a large off-diagonal
exchange [19]. The infinite density matrix renormalization
group study on the K-Γ model found strong numerical
evidence for the existence of a quantum spin liquid for
ferrolike Kitaev interactions [20].
The materials search for the KQSL led to the discovery
of many other honeycomb materials [21–26]. For example,
Cu2IrO3 and hydrogen-intercalated H3LiIr2O6 are found to
bear no sign of a magnetic order down to the lowest
temperatures [24,25]. The 4d compound α-RuCl3 is found
to be a Jeff ¼ 1=2 Mott insulator despite having a much
smaller magnitude of spin-orbit coupling and has received
much attention recently [26]. Dynamic spin correlations of
α-RuCl3 have been extensively studied by inelastic neutron
scattering (INS) [27–30]. The dynamical structure factor
through INS reveals a highly unusual intensity distribution
over a large energy interval around the zone center [29,30].
Although incoherent excitations originating from strong
magnetic anharmonicity can naturally occur in a highly
anisotropic frustrated magnet [31,32], several theoretical
works support that the broad feature is a dynamic response
of Majorana fermions—a salient nonlocal feature of the
Kitaev quantum spin-liquid phase [33–37].
Phenomenologically, Na2IrO3 and α-RuCl3 share several
common features: an unusual broad continuum in their
Raman scattering spectra [38,39]; magnetic entropy recovered
or released in two widely separated temperature scales
in heat capacity measurements [30,40]; and high-field
evolution of the zigzag-ordered phase to a nonmagnetic
phase in magnetic torque measurements [41,42]. However,
measurement of the dynamical structure factor providing
the most direct information thus far remains elusive for
Na2IrO3, because resonant inelastic x-ray scattering (RIXS)
suffers from insufficient energy resolution and INS is
difficult for Ir compounds due to their high neutron absorption.
The INS measurement on polycrystalline Na2IrO3
samples at a low temperature observes spin-wave excitations
with a sinusoidal-like low-momentum dispersion, which can
be best understood by including substantial further-neighbor
exchanges that stabilize the zigzag magnetic order [14].
Previous RIXS measurement on single-crystal Na2IrO3
observed only a broad low-energy excitation interpreted as
containing signals of magnetic and phonon origins [43],
which limits a detailed comparison with theoretical calculations.
A more recent RIXS measurement rules out the
phonon interpretation of the low-energy excitation by showing
different peak energies of isostructural Na2IrO3 and
Li2IrO3 and suggests a magnetic origin of the broad
excitation [44], which is phenomenologically similar to
the unusual broad scattering in α-RuCl3 [29,30]. Both
RIXS works can observe only the broad excitation and
do not provide detailed information on the magnetic
ground state.
Here, we report magnetic excitation spectra in a honeycomb
lattice Na2IrO3 measured by the state-of-the-art
RIXS spectrometer providing an unprecedented energy
resolution of 12 meV. The measurements are carried out
along all high-symmetry paths including the second
Brillouin zone (BZ) center. The 25 meV resolution spectra
at T ¼ 7 K reveal a discernible dispersion and spectral
distribution within the energy window of 60 meV. The
12 meV resolution allows us to characterize the spectral
width of the excitation peak and better define the lowenergy
excitation feature [45,46]. The zone center spectrum
lacks a resolution-limited peak typical of a coherent
collective excitation and has a broad feature only at high
energy. The incoherent feature persists over the whole BZ.
As approaching the zone boundary, a spectral weight shifts
toward lower energy. Remarkably, a sharp collective
excitation peak whose width is comparable to the spectrometer
resolution is resolved at the K point of the
honeycomb BZ. An exact diagonalization method on
finite-size clusters is used to find the relevant parameter
regime of the extended Kitaev-Heisenberg model which
captures the main features of the measured magnetic
excitation spectra at a low temperature in terms of spectral
dispersion and intensity. Hidden KQSL and Heisenberg
phases found in the complex parameter space of the model
JUNGHO KIM et al. PHYS. REV. X 10, 021034 (2020)
021034-2
166
provide useful references for the nature of the magnetic
excitations appearing in the measured spectra. We present
the temperature evolution of the RIXS dynamic response
up to 280 K, which reveals an anomalous behavior of the
broad high-energy excitation.
The paper is organized as follows: Section II presents
experimental RIXS spectra at low temperatures. Section III
describes the simulations of the low-temperature RIXS data
based on the extended Kitaev-Heisenberg model and
discusses the nature of the observed magnetic excitations.
The zigzag phase of the model is systematically explored,
and several kinds of differing pseudospin dynamics are
observed. Comparing directly the experiment and theoretical
RIXS spectra, this observation is used to identify the
parameter regime consistent with the experimental data. A
picture of renormalized magnons is proposed to explain the
main spectral features in this regime. Section IV presents
experimental RIXS spectra at high temperatures and discusses
the anomalous higher-energy spectral intensities in
the context of Kitaev systems at a finite temperature.
Section V concludes the paper and presents a perspective
on a RIXS probe for higher-order correlations that detects
the full continuum of the Majorana fermions of the KQSL.
The Appendixes provide details of our experimental setup,
the description of the numerical computations, and a
discussion of the hidden-symmetry points of the extended
Kitaev-Heisenberg model that are utilized in Sec. III.
II. MAGNETIC EXCITATION SPECTRA
AT A LOW TEMPERATURE
A. RIXS scatterings over the entire Brillouin zone
Figure 1(d) shows a RIXS intensity map recorded at T ¼
7 K along the Γ-M-Γ0-X-K-Γ-Y-K0-Γ0 path of the in-plane
FIG. 1. Magnetic excitation spectra in Na2IrO3 along high-symmetry Brillouin zone directions taken at T ¼ 7 K. (a) Scattering
geometry. Yellow arrows indicate incident and scattered x rays, which define the scattering plane (gray). Brown arrows indicate x-ray
polarizations. Green arrows indicate the cubic axes ðx; y; zÞ with respect to the octahedra (all of them point above the paper plane).
(b) One of the collinear zigzag patterns and the corresponding direction of the ordered moments. The left-facing arrows have an out-ofplane
component pointing above the paper plane; i.e., the corresponding moment direction lies approximately between the x and y axes.
(c) Two-dimensional reciprocal space diagram showing the measured path along the symmetry directions. The inner hexagon (blue
dashed line) indicates the first Brillouin zone of the honeycomb lattice. (d) RIXS intensity map of magnetic excitations in Na2IrO3 as
functions of the wave vector and energy loss. (e) The intensity profiles integrated over [60, 105] (open squares) and [105, 135] meV
(filled triangles) show that the high-energy spectral intensities are broadly peaked at the Γ point, extending up to 105 meV. (f) The
intensity profile integrated over ½−30; 60Š meV (filled squares) shows a distinctive distribution of the spectral weight along the
K-Γ-Y-K0
-Γ0
path. Passing through the K point, the excitation intensity rapidly increases and then decreases, which is followed by a
near-constant intensity along the Γ-Y-K0
path. Large intensities at the Γ and Γ0
wave vectors correspond to elastic scatterings. The zeroenergy
loss intensity at the M point is the diffuse magnetic Bragg peak. In the used scattering geometry, the magnetic Bragg peak at the Y
point is suppressed.
DYNAMIC SPIN CORRELATIONS IN THE HONEYCOMB … PHYS. REV. X 10, 021034 (2020)
021034-3
167
momentum transfer as shown in Fig. 1(c). These RIXS
spectra are obtained from a standard setup with conventional
25 meV resolution using a Si(844) diced spherical
analyzer [47,48]. The out-of-plane momentum transfer is
varied to keep the scattering angle close to 90° to minimize
elastic scatterings. Singular intensities at zero energy loss at
Γ (Γ0
) and M originate from a specular elastic scattering and
a static zigzag magnetic order [11–14], respectively.
Polarization factors in the current scattering geometry
depicted in Fig. 1(a) lead to a vanishing magnetic Bragg
peak at the Y point and a weak scattering intensity at the K0
point relative to the K point [15].
The main dispersing feature is observed in the first BZ
along the K-Γ-Y-K0
path below approximately 60 meV. An
intense low-energy excitation around 10 meV is clearly
seen near the K point. The spectral intensity moves to
higher energy along the K-Γ path, reaching its highest
energy at the Γ point, and disperses toward lower energy
along the Γ-Y-K0
path.
For each wave vector, the intensity is integrated over an
energy window of interest to obtain the distribution of the
spectral weight in the BZ. Figure 1(e) shows the intensity
profiles on the high-energy loss and the energy gain sides.
The intensity over the far energy loss region ([105, 135] meV,
filled triangles) shows a nearly constant value which is
comparable to that of the far energy gain region
(½−70; −30Š meV, open circles). The intensity profile integrated
over [60, 105] meV(open squares) in Fig. 1(f) shows a
distribution, peaked broadly around the BZ center Γ. These
indicate that the spectral intensity of the observed excitation
extends roughly to 105 meV.
Figure 1(f) shows the intensity distribution of the main
feature (½−30; 60Š meV, filled squares) which reveals a
distinctive distribution of the spectral weight as a function
of the wave vector. Weak intensities are seen along the Γ0
-X
line (outside the first Brillouin zone). Passing through the K
point, the excitation intensity rapidly increases and then
decreases, which is followed by a nearly constant intensity
along the Γ-Y-K0
path. The intensity is weakened as it
approaches the Γ0
point.
B. High-energy resolution RIXS spectra
Recently, a higher energy resolution has been achieved
for the Ir L3 RIXS by using the quartz(309) crystal [45,46].
In this work, the high-resolution quartz analyzer is used to
better examine spectral widths of magnetic excitations at
the Γ, near M, Y, and K wave vectors, where prominent
low-energy spectral weights below 20 meV are seen from
the 25 meV RIXS spectra in Fig. 1(d). The measured
energy resolution function of the quartz analyzer is plotted
at the bottom in Fig. 2(a), which can be described with the
pseudo-Voigt function with a 12 meV full width at half
maximum (FWHM).
Strong elastic scatterings at Q ¼ ð0 0 6.75Þ (Γ) and (0.45
0.45 6.5) (near M) are due to a specular scattering and a
quasielastic scattering due to the diffuse magnetic peak,
respectively. At both wave vectors, resolution-limited peaks
characteristic of coherent spin waves are not found, but
broad incoherent scatterings are seen at high energies. At
the Γ, the broad incoherent scattering has a clear peak
structure at around 40 meV. At Q ¼ ð0 1 6.5Þ (Y), a glimpse
of a low-energy peak is detected, and a broad peak is
centered around 50 meV. On the other hand, a narrow width
peak below 20 meV is clearly revealed at the Q ¼
ð0.67 0 6.6Þ (K) point, which is also followed by a broad
feature.
Figure 2(b) shows the Γ point spectrum with a fit in
which the elastic peak is fitted to the pseudo-Voigt
resolution function and the broad incoherent peak is fitted
by a damped harmonic oscillator (DHO) function convoluted
by the pseudo-Voigt resolution function. Note that
the high-resolution data have rather a large ratio of background
to signal and poor statistics. The background level
is determined in a way that the energy gain data below
−30 meV are distributed around the zero, and such a
background is subtracted from the raw data. The DHO
function is expressed as AnðTÞγf1=½ðE − E0Þ2
þ γ2
Š−
1=½ðE þ E0Þ2
þ γ2
Šg, where A is the amplitude, nðTÞ is
the Bose factor, E0 is the peak energy, and γ is the peak
width. The fitted curves are overlaid with the Γ point data in
Fig. 2(b). A damped peak of ðE0; γÞ ¼ ð36; 22Þ meV is used
to describe the broad incoherent peak. Figure 2(c) shows the
K point spectrum, where the background level estimated for
the Γ spectrum is assumed. The low-energy peak is described
by a narrow width peak of ðE0; γÞ ¼ ð3; 7Þ meV. The broad
feature is fit by the ðE0; γÞ ¼ ð38; 20Þ meV DHO.
Figure 2(d) shows the temperature dependence of the
high-resolution RIXS spectrum at the K point. Na2IrO3 is a
Mott-like correlated insulator with a 340 meV energy gap
[49]. Within the Mott gap of 340 meV, the lattice and spin
degrees of freedom could be associated with the appearance
of excitations. The temperature dependence data in Fig. 2(d),
for example, provide a means to distinguish excitations of
distinct origins. If the lattice degree of freedom is involved,
the Bose population factor leads to an increasing intensity
with an increasing temperature. On the other hand, at a
temperature above a characteristic spin exchange energy, the
contribution of the spin degree of freedom vanishes.
The low-energy peak in Fig. 2(d) clearly decreases at
T ¼ 70 K and becomes featureless at T ¼ 150 K. The
long-range zigzag order disappears above 15 K, but the
diffuse magnetic scattering study finds that the short-range
zigzag correlations survive at least up to 70 K [15]. Thus,
this temperature dependence data establish that the lowenergy
peak is a magnon peak of the zigzag magnetic order.
On the other hand, the broad feature intensity barely
changes between 7 and 150 K. This observation is in
contrast to the previous RIXS study [43], which finds a
temperature-dependent broad scattering and interprets it as
a resonant phonon contribution. However, a more recent
JUNGHO KIM et al. PHYS. REV. X 10, 021034 (2020)
021034-4
168
RIXS [44] reports that the broad scattering intensity hardly
changes between 5 and 90 K and persists up to 300 K,
which is consistent with the current observation. This
recent RIXS work rules out the phonon interpretation of
the low-energy excitation and claims the magnetic origin of
the broad scattering intensity by showing different peak
energies of isostructural Na2IrO3 and Li2IrO3 and the same
resonance behavior of all low-energy signals [44]. Here, we
assign the broad feature as having a magnetic origin and
discuss in more detail in the next two sections.
III. MODEL DESCRIPTION OF
THE LOW-T RIXS SPECTRA
A. Spin Hamiltonian
To perform a quantitative model analysis of the magnetic
excitations as observed by RIXS, we adopt the extended
Kitaev-Heisenberg model for Jeff ¼ 1=2 pseudospins
[7,16,50]. Compared to the originally proposed KitaevHeisenberg
model for Na2IrO3 [4] comprising a dominant
Kitaev interaction supplemented by a smaller Heisenberg
interaction, the model is extended by two kinds of
off-diagonal exchange interactions. The nearest-neighbor
Hamiltonian for the pseudospins S then takes the form
H
ðzÞ
ij ¼ KSz
i Sz
j þ JSi · Sj þ ΓðSx
i Sy
j þ Sy
i Sx
jÞ
þ Γ0
ðSx
i Sz
j þ Sz
i Sx
j þ Sy
i Sz
j þ Sz
i Sy
jÞ ð1Þ
shown here for a z bond [vertical bond in Fig. 1(b); the
bond direction is perpendicular to the z axis]. In the case of
the other bond directions, a cyclic permutation of the
pseudospin components is applied. Various ab initio estimates
of the interaction parameters (e.g., Refs. [8,16,17])
generally suggest a dominant ferromagnetic (FM) Kitaev
interaction (K < 0) and a positive off-diagonal Γ interaction.
This parameter setup, combined with suitable values
of the smaller interaction parameters J and Γ0
, favors the
zigzag magnetic order with the magnetic moments pointing
approximately in between the x and y axes (assuming the
zigzag order with FM x and y bonds) which corresponds to
the experimental situation [15]. In addition, motivated by
sizable further-neighbor interactions found by the ab initio
estimates (see, e.g., Ref. [17]), we also include isotropic
FIG. 2. High-resolution RIXS spectra recorded at T ¼ 7 K. (a) The measured energy resolution function of the quartz analyzer is
plotted at the bottom. At Q ¼ ð0 0 6.75Þ (Γ), an elastic scattering is followed by a broad excitation without any indication of a narrow
excitation. At Q ¼ ð0.45 0.45 6.5Þ (near M), a soft low-energy excitation is unresolved with 12 meV energy resolution, and a
quasielastic scattering is followed by a broad shoulder excitation. At Q ¼ ð0 1 6.5Þ (Y), a low-energy excitation is seen with a broad
high-energy feature. At Q ¼ ð0.67 0 6.6Þ (K), a narrow excitation is discovered at a low energy. (b),(c) The measured RIXS spectra at
the Γ and K wave vectors, respectively, are fit by the pseudo-Voigt function (elastic scattering) and damped harmonic oscillator (DHO)
function convoluted by the 12 meV resolution function. (d) Temperature dependence of the RIXS excitation spectrum at the K point.
The low-energy peak shows a clear decrease at T ¼ 70 K and becomes featureless at T ¼ 150 K, indicating that it is an excitation peak
of the zigzag magnetic order.
DYNAMIC SPIN CORRELATIONS IN THE HONEYCOMB … PHYS. REV. X 10, 021034 (2020)
021034-5
169
Heisenberg interactions among second and third nearest
neighbors and arrive at the full model Hamiltonian
H ¼
X
hiji∈NN
H
ðγÞ
ij þ
X
hiji∈2ndNN
J2Si · Sj þ
X
hiji∈3rdNN
J3Si · Sj;
ð2Þ
where γ labels the nearest-neighbor (NN) bond direction.
Following the prevailing expectations, in our analysis we
assume that the main interactions are K, Γ, and J, while Γ0
,
J2, and J3 are significantly smaller in magnitude.
B. Simulations of the low-temperature RIXS data
We simulate the RIXS spectra by calculating the
dynamical pseudospin structure factor and combining its
components according to the recipes given in Refs. [51,52].
Namely, we utilize the effective RXS operator expressed
within the Kramers doublet manifold via
R ∝ iðε × ε0
Þ · ðfabSab þ f⊥S⊥Þ: ð3Þ
Here, ε and ε0
are the polarization vectors of the incident
and scattered x rays, respectively, and Sab and S⊥ denote
the component of the pseudospin lying within the honeycomb
plane (crystallographic ab) and being perpendicular
to it, respectively. For the L3 edge resonant process, the
factors connecting the pseudospin and the RXS operators
read as fab ¼ 1
2 þ ð5=6
ffiffiffi
2
p
Þ sin 2ϑ − 1
6 cos 2ϑ and f⊥ ¼
1 þ 2
3 cos 2ϑ − ð1=3
ffiffiffi
2
p
Þ sin 2ϑ with the angle tan 2ϑ ¼
2
ffiffiffi
2
p
=ð1 þ 2Δ=λÞ being determined by the ratio of the
trigonal field Δ and the spin-orbit coupling constant λ
[51,52]. The values of ϑ for Na2IrO3 can be estimated from
the splitting ΔBC ≈ 0.1 eV of the Jeff ¼ 3=2 quartet [53],
leading to slightly anisotropic fab ≈ 0.91 and f⊥ ≈ 1.15.
The RIXS intensity is then calculated as the dynamical
correlation function of the R operator: Iðq; ωÞ ∝ χ00
Rðq; ωÞ
with χRðq; ωÞ ¼ ih½RqðtÞ; R−qð0ފiω. This quantity can be
conveniently expressed via the pseudospin susceptibility
tensor χαβðq; ωÞ ¼ ih½Sα
qðtÞ; Sβ
−qð0ފiω calculated either by
exact diagonalization (ED) on small clusters (see
Appendix B for details) or within the linear spinwave
(LSW) approximation. For the geometry shown in
Fig. 1(b), the RIXS intensity is roughly proportional to
1
2 ðχ00
xx þ χ00
yyÞ þ χ00
zz (neglecting the small off-diagonal contribution
and summing up the π − π0
and π − σ0
scattering
channels to account for the unpolarized detection).
C. Identification of relevant parameter regime
To narrow down the parameter regime consistent with
the experimental data, we perform a systematic scan
through the parameter space of the model, inspecting the
type of the magnetic order and the excitation spectra
obtained by ED. As the six parameters present in the
model make this scan a challenging task, we use fixed small
values of the subsidiary interactions (Γ0
and J2;3) and vary
only the main ones (J, K, and Γ). This limitation is not
severe, since the overall dispersion and intensity distribution
in the calculated RIXS spectra is determined by the
dominant interactions, while the smaller ones affect only
finer details of the spin dynamics not resolved in the
experiment. Despite that, the Γ0
and J2;3 interactions still
play important roles in stabilizing the zigzag phase and
extending it to the regime where the dominant interactions
can reproduce the RIXS data.
The phase diagram focusing on the zigzag phase is
presented in Fig. 3. To construct it, we assume small Γ0
< 0
associated with the trigonal compression [50] that is fixed
at Γ0
¼ −0.1A with A being the overall energy scale of the
dominant interactions defined as A ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
J2
þ K2
þ Γ2
p
.
Similarly, the further-neighbor interactions J2 and J3 are
fixed at J2 ¼ 0.05A and J3 ¼ 0.1A, respectively, putting
thus more emphasis on the third-neighbor interaction as
suggested by ab initio estimates [17]. The main interactions
are parametrized using two angles θ, ϕ as ðJ; K; ΓÞ ¼
ðA sin θ cos ϕ; A sin θ sin ϕ; A cos θÞ with the ranges θ ∈
½0; π=2Š and ϕ ∈ ½0; 2πŠ covering the entire JKΓ parameter
space with Γ > 0. The resulting phase diagram in Fig. 3
shows two zigzag regions connected by a narrow “neck”
which substantially differ in the direction of the ordered
moments. Let us consider for concreteness one of the
degenerate zigzag patterns with zigzag chains running
along the x and y bonds shown in Fig. 1(b). The ordered
moments in the upper zigzag phase are then found close to
the z axis, while the lower zigzag phase is characterized by
the moment direction pointing roughly in between the x
and y axes. These observations can be understood using
simple energy-based arguments when comparing Fig. 1(b)
and Eq. (1). The zigzag order of the upper phase found for
K > 0 and J < 0 fully satisfies the AFM K interaction on
the z bonds, where it picks up the dominant z component of
the ordered moments as seen in Eq. (1). The energy gain
from the remaining x and y bonds is due to FM J
interaction. In the bottom zigzag phase covering mainly
the K < 0 and Γ > 0 case, the FM Kitaev interaction
profits from the FM bonds within the chains by using the x
and y components of the pseudospins separately. The
positive Γ interaction brings energy gain on the AFM
interchain z bonds by utilizing the simultaneous presence
of the x and y components. This phase is also significantly
supported by the negative Γ0
interaction (see Ref. [54] for a
more detailed discussion). The longer-range AFM interactions
J2 and particularly J3 further stabilize the zigzag
state—now independently on the moment direction due to
their isotropic character—and expand significantly the
zigzag phases in the parameter space.
The experimentally determined direction of the ordered
moments as given by Ref. [15] is consistent with the bottom
zigzag region. Using the precise ordered moment direction
JUNGHO KIM et al. PHYS. REV. X 10, 021034 (2020)
021034-6
170
as a very sensitive probe of the anisotropic exchange
interactions, a further refinement of the relevant parameter
regime is possible. Namely, we can utilize the angle of the
ordered moments to the honeycomb plane which is
estimated in a subsequent analysis of the experimental
data in Ref. [15] to be around 38°–40° [51]. The theoretical
values of this angle obtained by ED through the zigzag
phase are indicated in Fig. 3. Considering solely the
experimental moment direction, the matching parameter
sets form a strip near the A2 and B1–B4 points in Fig. 3.
Let us now focus on the phase diagram in Fig. 3 from the
point of view of our RIXS data. We calculate RIXS maps via
ED for several selected parameter points assuming the same
scattering geometry as in the experiment. Because of the
finite size of the clusters used in ED, their ground states
entering the calculation of the dynamic response comprise all
possible zigzag patterns in an equal-weight superposition.
This mixing actually imitates the experimental conditions,
since below TN the long-range zigzag order in the sample is,
in fact, accompanied by short-range zigzag orders with the
complementary directions of the zigzag chains [15]. The
maps, presented in Fig. 3, can thus be directly compared to
the data in Fig. 1(d) up to the energy scale that is flexible by
tuning A. Within the bottom zigzag phase, which covers a
broad range of parameters, several kinds of spin-excitation
behavior can be observed. The differences concern the wave
vectors away from the M point that universally hosts an
intense low-energy magnon through the whole zigzag phase.
In the left part of the phase diagram with significant
negative J (representative B4 and C3 points), the overall
0
10
20
30
40
50
60
70
80
M ’ X K YK’ ’
Energy(meV)
0
10
20
30
40
50
60
70
80
M ’ X K YK’ ’
Energy(meV)
0
10
20
30
40
50
60
70
80
M ’ X K YK’ ’
Energy(meV)
0
10
20
30
40
50
60
70
80
M ’ X K YK’ ’
Energy(meV)
0
10
20
30
40
50
60
70
80
M ’ X K YK’ ’
0
10
20
30
40
50
60
70
80
M ’ X K YK’ ’
0
10
20
30
40
50
60
70
80
M ’ X K YK’ ’
0
10
20
30
40
50
60
70
80
M ’ X K YK’ ’
0
10
20
30
40
50
60
70
80
M ’ X K YK’ ’
0
10
20
30
40
50
60
70
80
M ’ X K YK’ ’
Moment angle to the honeycomb plane
hH
hK
A1
B1B2
B3
B4
C1
C2
C3
A3
A2
0 10 20 30 40 50
FIG. 3. Zigzag phase in the phase diagram of the extended Kitaev-Heisenberg model with the main interactions parametrized using the
radial coordinate θ and the azimuth ϕ as J ¼ A sin θ cos ϕ, K ¼ A sin θ sin ϕ, Γ ¼ A cos θ, and the smaller interactions kept constant:
Γ0
¼ −0.1A, J2 ¼ 0.05A, and J3 ¼ 0.1A. Here, A is the overall energy scale, which is irrelevant for the phase diagram but determines
the characteristic energies of the magnetic excitations. The color indicates the angle of the zigzag-ordered moments to the honeycomb
plane. The data are obtained using the exact-diagonalization-based method in Ref. [51]. Small panels around the phase diagram show,
for selected parameter points, the theoretical RIXS intensity maps calculated by exact diagonalization of the extended KitaevHeisenberg
model on 24- and 32-site clusters (see Appendix B for details). We assume the scattering geometry depicted in Fig. 1(a) and
present a sum of π − π0
and π − σ0
scattering channels (imitating an unpolarized detection) plotted along the same path through the
Brillouin zone as in Fig. 1. The energies are determined by taking the value A ¼ 29 meV giving the best match between the A2
parameter point and the experimental data. Gaussian broadening with a FWHM of 25 meV is applied to the spectra. Finally, the violet
lines show the pathways in the parameter space connecting the selected points A2 and A3 with the points of hidden symmetry utilized in
Sec. III D 2. They should be understood as projections only, since the hidden Heisenberg point (hH) has Γ0
≈ −0.4A, whereas the hidden
Kitaev point (hK) has Γ0
≈ −0.3A, and J2;3 ¼ 0 for both points.
DYNAMIC SPIN CORRELATIONS IN THE HONEYCOMB … PHYS. REV. X 10, 021034 (2020)
021034-7
171
shape of the intensity cloud hints to the proximity of the
FM phase. There are intense low-energy excitations around
the Γ point whose dispersion goes up toward the AFM
wave vector Γ0
—such features are shared with FM magnons.
In the bottom part, one can notice a shift to the
magnon breakdown regime reported for α-RuCl3 by
Ref. [32] (representative B3, B2, and B1 points). It is
most apparent near the J ¼ 0 line (B1 and B2 points),
which shows a rather flat dispersion of excitations with a
pronounced high-energy tail. Yet more incoherent scattering
is seen at Kitaev-dominant point A1. The most robust
features of the experimental RIXS spectra in Fig. 1(d) are
the dispersing excitations in the K-Γ-Y-K0 part, reaching
the maximum energy at the Γ point and maximum intensity
at the K point. This feature can be clearly observed in the
theoretical maps on the right corresponding to the parameter
points with a sizable AFM J > 0 (C1, A3, and A2
points). A hint of this feature is displayed also at the A1 and
C2 points of the highly dominant Γ interaction and FM
J < 0. In the C2 case, the intensity resides at relatively high
energies, being connected to the M-point magnons by a
steep excitation branch.
Taking into account also the experimental constraint on
the ordered moment angle to the honeycomb plane, the
suggested parameter range for Na2IrO3 lies around points
A2 and A3 in Fig. 3, i.e., in the regime with Γ comparable to
or somewhat less than jKj and quite sizable AFM J > 0.
For the A2 point, the moment angle to the honeycomb plane
fits well the experimental value; the A3 point may seem to
match slightly better the experimental RIXS map. In the
following, we thus discuss both these points in parallel to
give a broader picture of the relevant parameter range.
To enable a one-to-one comparison, Figs. 4(b) and 4(c)
present the theoretical RIXS spectra for the A2 and A3
points, respectively, with the resolution matching the
experimental one and with the values of the energy scale
A tuned to fit the position of the main peak at the q ¼ 0
wave vector (Γ point). Figure 4(a) shows the experimental
RIXS intensity map for T ¼ 7 K with the positions of the
intensity maxima for the individual wave vectors indicated
as open circles, where the specular elastic scattering
peaks at Γ and Γ0
are subtracted. The broad maps in
Figs. 4(a)–4(c) show a good overall agreement of the
theoretical maps with the RIXS data in both the dispersion
0
10
20
30
40
50
60
M ’ X K Y K’ ’
Energy(meV)
0
10
20
30
40
50
M ’ X K Y K’ ’
A2
0
10
20
30
40
50
M ’ X K Y K’ ’
A3
0
1
2
3
4
5
M ’ X K Y K’ ’
(a) )c((b)
(d) (e)
Intensity(a.u.)
Totalspectralweight(a.u.)
Wave vector q
A2
A3
exp
-50 0 50 100 150
q = K
Energy (meV)
-50 0 50 100 150
q = Y
Energy (meV)
q =A2
A3
q = M
FIG. 4. (a) Experimental RIXS intensity map for T ¼ 7 K with the positions of the intensity maxima for the individual wave vectors
indicated as open circles. (b) Theoretical RIXS intensity map calculated for the parameter values corresponding to the A2 point in Fig. 3
and the energy scale A ¼ 29 meV, which gives J ≈ 12 meV, K ≈ −24 meV, Γ ≈ 11 meV, Γ0 ≈ −3 meV, J2 ≈ 1.5 meV, and
J3 ≈ 3 meV. The scattering geometry depicted in Fig. 1(a) is assumed, and a sum of π − π0 and π − σ0 scattering channels is taken,
corresponding to an unpolarized detection. The spectra are broadened in energy by Gaussians with a FWHM of 25 meV to imitate the
experimental resolution as in (a). (c) The same as in (b) but for the parameter values corresponding to the A3 point in Fig. 3 and the
energy scale of A ¼ 24 meV: J ≈ 10 meV, K ≈ −15 meV, Γ ≈ 16 meV, Γ0 ¼ −2.4 meV, J2 ¼ 1.2 meV, and J3 ¼ 2.4 meV.
(d) Energy-integrated RIXS response (total spectral weight) corresponding to (a)–(c). The experimental data, obtained by integrating
in the ½−30; 60Š meV range, are arbitrarily scaled with respect to theoretical ones. (e) RIXS intensity profiles at selected wave vectors.
High-resolution experimental data (gray dots) are compared to theoretical spectra for the above parameters (red and blue lines). The
theoretical spectra are broadened by the experimental resolution function of the high-resolution setup.
JUNGHO KIM et al. PHYS. REV. X 10, 021034 (2020)
021034-8
172
and intensity distribution in the Brillouin zone. The latter is
supported by the total spectral weight (energy-integrated
spectra) in Fig. 4(d), which seems to match quite well to the
corresponding experimental data in Fig. 1(f). Here, essentially
no difference between A2 and A3 points is observed.
Figure 4(e) compares the high-resolution experimental
RIXS spectra in Fig. 2(a) to the theoretical spectra convoluted
with the experimental resolution function. The
main incoherent and coherent features of the Γ and K
spectra are well captured by the theoretical spectra. A
discrepancy is that the theoretical spectra show relatively
less spectral weight in high-energy parts. At the M point,
the incoherent high-energy part is reasonably reproduced
by the model. At the Y point, the discrepancy in the
incoherent high-energy intensity seems bigger. Overall, the
theoretical spectra clearly capture the main features of the
spectra at the individual wave vectors including, to some
extent, also the shape of intensity profiles following the
central peaks.
Since the data for the various cluster sizes and shapes
seem to be quite consistent in the energy ranges and profiles
of the modes, we cannot attribute the discrepancy in the
incoherent high-energy parts solely to finite-size effects.
Even though the extended Kitaev-Heisenberg model captures
the essential dynamics of the Jeff ¼ 1=2 pseudospins,
it may lack certain scattering processes that lead to the
enhancement of the high-energy tails. One such mechanism
may be a magnetoelastic coupling that comes into play due to
the large orbital component of the Jeff ¼ 1=2 pseudospins
and may act as an additional decay channel. Pronounced
effects of coupled lattice and pseudospin dynamics are
observed, for example, in phonon line shapes in Raman
spectra of perovskite iridates [55,56]. Reference [43] introduces
the phonon contributions as producing a series of
harmonic peaks to describe the incoherent high-energy
scatterings. Few facts, however, suggest that there are no
big direct phonon contributions to the incoherent highenergy
parts. It is known that the resonant phonon contribution
becomes visiblewhen the inverse core-hole lifetime is
in the range of a few hundred meV, and a 1 eV detuning of the
lifetime leads to a near-zero cross section [57]. At the Ir L3,
the inverse core-hole lifetime is more than 5 eV [58], and this
shorter-lived core hole state is not expected to provide
sufficient time for the lattice to respond. A majority of
optical phonon modes of Na2IrO3 and Li2IrO3 reside around
60 meV [59]. Recent O K-edge RIXS on Li2IrO3 report the
resonant phonon spectra where the fundamental phonon is at
70 meV [60]. The Ir L3-edge RIXS on Li2IrO3 [44], on the
other hand, shows that the low-energy excitation feature is
centered at 20 meV, which is notably lower than 70 meVand
so of a different origin from phonon. Our high-resolution
spectra in Fig. 4(e) do not show maximum intensities at this
energy range but show decreasing intensities, for example, at
Γ, K, and M. As mentioned earlier, Ref. [44] argues against
the phonon interpretation by showing that the energy scale of
the incoherent feature is very different in two isostructural
Na2IrO3 and Li2IrO3. Hence, we assign the incoherent highenergy
scattering as of magnetic origin.
The x-ray and neutron diffraction studies [11,13,14]
show that there are inherent imperfections in crystal
structure such as stacking faults and site disorders, which
result in a structural diffuse scattering and a remaining
short-range order below the ordering temperature [13,15].
Small variations in the structure can alter various exchange
interactions in Na2IrO3. So it is possible that the current
single parameter set should be supplemented with nearby
parameter sets among many in Fig. 3 to explain the
discrepancy in the incoherent high-energy parts and fully
reproduce the whole observed spectra.
D. Nature of the spin excitations
Having the main magnetic intensity in the RIXS spectra
below approximately 60 meV reasonably well captured by
the excitations of the extended Kitaev-Heisenberg model, it
is natural to ask what the character is of those excitations. In
the following, we address this issue by comparing ED
results to the linear spin-wave approximation and by
inspecting the evolution of the spectra when moving in
the parameter space toward points of hidden symmetry. In
the latter case, we use a nearby hidden Heisenberg point
where a magnon picture is valid and a nearby hidden Kitaev
point with the excitations carried by Majorana fermions.
1. Comparison to linear spin-wave approximation
Each of Figs. 5(a) and 5(b) presents fine energy-resolved
RIXS spectra obtained by ED and LSW calculation for
the A2 and A3 points, respectively, used also in Figs. 4(b)
and 4(c). In the case of the A3 point with Γ ≈ jKj shown in
Fig. 5(b), the LSW approximation seems quite successful in
capturing the overall dispersion and intensity of the spin
excitations, the main difference being the high-energy
magnon branches that get significantly renormalized in
terms of both the energy shift and broadening. Figure 5(d)
shows that the distributions of the total spectral weight are in
excellent agreement between ED and LSW. In the case of the
A2 point with Γ ≈ 1
2 jKj shown in Fig. 5(a), the agreement is
spoiled by the fact that LSW places this parameter point
closer to a competing order with the characteristic wave
vector K. The reason is that the minimization of purely
classical energy in the LSW approximation completely
neglects quantum fluctuations supporting the zigzag phase.
As a consequence, LSW brings down the excitations at both
K and K0
and lifts the M-point magnon. There is also an
associated shift of the spectral weight from the M point to the
K and K0
points as shown in Fig. 5(c). This result is in
contrast to the ED calculations, which give similar lowenergy
spectra for both A2 and A3 points. However, apart
from these trends related to a phase boundary shift, the LSW
description of the A2 point successfully gives a rough picture
DYNAMIC SPIN CORRELATIONS IN THE HONEYCOMB … PHYS. REV. X 10, 021034 (2020)
021034-9
173
of the excitation branches. The high-energy branches at the
Γ point are now visibly split. According to the LSW
calculation, the lower branch corresponds to oscillations
of the moment angle to the honeycomb plane and the upper
one to the oscillations in the in-plane direction perpendicular
to the zigzag chains.
Based on the above results, it seems quite likely that the
spin-excitation spectra could be reproduced when going
beyond LSW and include the anharmonic effects, e.g., by
correcting the dispersions via the self-consistent spin-wave
theory and by implementing finite magnon lifetimes due to
magnon decay processes [31]. An attempt to evaluate the
magnon decay in a Kitaev system, utilizing the so-called
imaginary self-consistent Dyson equation approach, was
recently performed by Winter et al. [32] when interpreting
the neutron data on α-RuCl3. Note that in their case the
multimagnon contribution seems stronger (Fig. 6 in
Ref. [32]) due to a different parameter regime within the
zigzag phase.
2. Proximate hidden Kitaev and Heisenberg points
The applicability of the magnon picture can be further
checked by inspecting the evolution of the calculated RIXS
spectra when going through the parameter space toward
suitable reference points. The parameter region identified
as relevant for Na2IrO3 lies close to two of the points of
special symmetry [61], where the nearest-neighbor
extended Kitaev-Heisenberg model can be exactly mapped
to either the Kitaev or Heisenberg model. Since in both the
pure Kitaev and Heisenberg cases the spin excitations are
well known, the special-symmetry points provide convenient
references for us. In the following, we keep the
discussion of both special-symmetry points very brief;
further details can be found in Appendix C or the original
Ref. [61].
Working in the representation utilizing the energy scale
A, our point A2 corresponds to
ðJ; K; Γ; Γ0
ÞA2 ≈ ð0.4; −0.8; 0.4; −0.1ÞA ð4Þ
complemented by the small J2;3. Similarly, for the A3 point,
we have
ðJ; K; Γ; Γ0ÞA3 ≈ ð0.4; −0.6; 0.7; −0.1ÞA: ð5Þ
The hidden Kitaev point is obtained for the parameter set
ðJ; K; Γ; Γ0
ÞhK ≈ ð0.6; −0.5; 0.6; −0.3ÞA; ð6Þ
K X
’K’Y
M
0
10
20
30
40
50
60
M X K Y K’
Energy(meV)
Point A2 - ED
M X K Y K’
Point A2 - LSW
0
10
20
30
40
50
60
M X K Y K’
Energy(meV)
Point A3 - ED
M X K Y K’
Point A3 - LSW
0
1
2
3
4
5
6
M X K Y K’
Totalspectralweight(a.u.)
Wave vector q
LSW
24a
32a
32b1
32b2
32b3
M X K Y K’
Wave vector q
0
1
2
3
4
5
6
M X K Y K’
Totalspectralweight(a.u.)
Wave vector q
LSW
24a
32a
32b1
32b2
32b3
M X K Y K’
Wave vector q
(a) (b)
(c) (d)
FIG. 5. (a) RIXS intensity maps for the parameter point A2 and the same energy scale as in Fig. 4(b). The left and right are obtained by
exact diagonalization and the linear spin-wave approach, respectively. To better resolve the finer features, Gaussian broadening with a
small FWHM of only 2 meV is used in both cases. In the case of LSW, the spectra are averaged by equally employing all three possible
zigzag pattern directions. This procedure leads to a map that could be directly compared to the ED one, since the cluster ground state used
in ED contains all zigzag pattern in an equal-weight superposition. (b) The same for the parameter point A3 and the energy scale as in
Fig. 4(c). (c) Energy-integrated RIXS response (total spectral weight) corresponding to (a). Data for all the clusters are presented,
showing negligible finite-size effects on the spectral weight. The curve obtained by the linear spin-wave approach (right) is shown also
on the left by a dashed line. (d) Total spectral weight for the data in (b).
JUNGHO KIM et al. PHYS. REV. X 10, 021034 (2020)
021034-10
174
where J2 and J3 are zeros as the longer-range interactions
are absent at this special-symmetry point. It may be derived
by taking the canonical Kitaev model and rotating the
interaction axes x, y, and z by 180° around the axis
perpendicular to the honeycomb plane (see Appendix C
for a detailed discussion). This way, we arrive at the
extended Kitaev-Heisenberg model with the above parameter
set. Since the transformation is exact, all the features
including, e.g., the fermionic excitation spectrum, are
exactly preserved. At the hidden Kitaev point, we thus
find the behavior of the extended Kitaev-Heisenberg model
to be identical to the Kitaev model with interaction
parameter K0 ≈ 1.4A, up to a simple global rotation. At
the second considered hidden-symmetry point with the
parameter set
ðJ; K; Γ; Γ0
ÞhH ≈ ð−0.1; −0.6; 0.8; −0.4ÞA; ð7Þ
the extended Kitaev-Heisenberg model exactly maps to the
Heisenberg model with the interaction constant J0 ≈ 0.9A.
In this case, the derivation is more complicated and
includes a four-sublattice transformation connecting zigzag
and N´eel order (cf. Refs. [61,62] and Appendix C). The
spin-excitation spectra are then directly linked to those of
the simple Heisenberg antiferromagnet on the honeycomb
lattice, but a momentum shift by q ¼ M and equivalent
wave vectors is involved.
Figure 6 presents the parameter evolution of the RIXS
response, focusing on the Γ-point (q ¼ 0) intensity containing
the prominent high-energy feature that is a signature
of the lack of the global rotational symmetry of the model.
The model parameter sets are linearly interpolated between
the A2 and A3 points and the hidden-symmetry points, all
given by Eqs. (4)–(7). Note that the latter parameter points
differ essentially in J only while having roughly the same
Γ=jKj ratio as our A3 point and an enhanced negative value
of Γ0
compared to A2 and A3. An explicit plot of the
parameters used in Fig. 6 can be found in Appendix C,
and the corresponding pathways are indicated in the
overall phase diagram in Fig. 3 in a projected form (note
that Γ0 and J2;3 change once we depart from the A2 and A3
points).
0
1
2
3
Energy/A
0
1
2
3
Energy/A
0
1
2
3
M XK YK’
Energy/A
M XK YK’ M XK YK’ M XK YK’ M XK YK’
0.0
0.2
0.4
0.6
0.8
1.0
A3
Point
A2
Point
HiddenKitaevpoint
HiddenHeisenbergpoint(a)
(c)
(b)
(d)
24a
32a
avg
(e)
FIG. 6. (a) Evolution of the theoretical Γ-point (q ¼ 0) RIXS intensity on a line through the parameter space going from the hidden
Heisenberg point ðJ; K; Γ; Γ0Þ ≈ ð−0.1; −0.6; 0.8; −0.4ÞA to the point A2 of Fig. 3. The line lies completely in the zigzag phase. Finitesize
effects can be estimated by comparing data for various clusters shown in Appendix B. The average is taken over all of them (i.e.,
24a, 32a, and 32b1 to 32b3). The spectra are presented in units of the energy scale A and only a tiny broadening is used to resolve the
structure of the continuum. (b) The same for a line connecting the point A2 and the hidden Kitaev point
ðJ; K; Γ; Γ0Þ ≈ ð0.6; −0.5; 0.6; −0.3ÞA. Apart from the last point, the ground state remains zigzag ordered. The dotted line indicates
scaled exact solution in the AFM Kitaev limit [63]. (c,d) Same as panels (a,b) but for the A3 point of Fig. 3. (e) Two-magnon density of
states for selected points of panels (c) and (d) calculated using the linear spin-wave dispersions (gray lines). Single zigzag pattern of
Fig. 1(b) was assumed, its Bragg spots correspond to the Y points.
DYNAMIC SPIN CORRELATIONS IN THE HONEYCOMB … PHYS. REV. X 10, 021034 (2020)
021034-11
175
The evolution from the hidden Heisenberg point toward
the A2 and A3 points shown in Figs. 6(a) and 6(c),
respectively, starts with a simple response profile containing
a sharp magnon peak and a two-magnon continuum
characteristic for the Heisenberg model. The magnon at the
Γ point calculated for the zigzag phase is, in fact, a copy of
the N´eel AFM magnon at the M point shifted by means of
the hidden-symmetry transformation. Therefore, it appears
at a high energy (
ffiffiffi
2
p
J0 in the LSW approximation). Going
away from the hidden Heisenberg point, the high-energy
magnon branches get broadened via magnon scattering
and slowly merge with the two-magnon continuum forming
the broad high-energy features observed at our points.
The gradual onset of the magnon scattering is further
illustrated in Fig. 6(e) by comparing the LSW dispersions
and the two-magnon density of states (DOS): D2ðωÞ ¼P
nq;n0q0 δðω − ωnq − ωn0q0 Þ, where ωnq stands for the
dispersion of the nth magnon branch. Neglecting the
magnon-magnon interaction vertex, this quantity indicates
the strength of the scattering continuum accessible when
keeping the kinematic constraint [31]. It becomes gradually
activated as we depart from the hidden Heisenberg point,
where the two-magnon decay does not occur and the basic
decay channel involves three-magnon processes [31,32].
Continuing further from our points to the hidden Kitaev
point, Figs. 6(b) and 6(d) show that the zigzag order is
maintained almost up to the hidden Kitaev point. This result
is natural, since the hidden KQSL comes with an antiferromagnetic
effective Kitaev interaction. As revealed by
the study of the Kitaev-Heisenberg model [62], the AFM
KQSL phase has a limited extent compared to the FM
KQSL phase because of a stronger competition with the
surrounding phases that gain energy due to quantum
fluctuations [64]. The high-energy continuum in the Γpoint
response is present up to the zigzag to KQSL
boundary with moderate changes in its shape that seem
to be correlated with the two-magnon DOS. Upon entering
the KQSL, there is an abrupt change in the character of
the spectrum that later only negligibly evolves when
approaching the exact hidden AFM Kitaev point as shown
in Figs. 6(b) and 6(d). The continuum spreads over a larger
spectral range and becomes composed of sharp peaks
reflecting the excitations being carried by (almost) noninteracting
Majorana fermions with a set of possible
momenta strongly limited by the cluster. Nevertheless,
the overall distribution of the spectral weight can be
successfully compared to the exact result for an infinite
lattice [63].
The above observations suggest that the picture of
renormalized magnons adequately captures the spin excitations
within the extended Kitaev-Heisenberg model in the
parameter regime matching the low-temperature RIXS data
below ≲60 meV. This suggestion is supported by the good
overall agreement of LSW dispersions, intensity, and
spectral weight distribution in the Brillouin zone with
the ED results. Moreover, the magnon broadening seems
to correlate well with the two-magnon DOS that gives hints
about the decay rates of the individual magnon branches.
Compared to the recent analysis of spin excitations in
α-RuCl3 [32] that placed α-RuCl3 to the regime of magnon
“breakdown” within the same spin model as used here, we
encounter better-defined magnons with a less extended
background of the multimagnon character.
Despite the relative proximity of the KQSL associated
with the hidden Kitaev point, in particular, for the A3 point,
the model spectra for zero temperature do not clearly bear
the features characteristic for the Kitaev limit, most
importantly, the flat dispersions of the spin excitations that
are seen in the exact results for the Kitaev model [63,65] or
the study of its perturbed variant [66]. An interesting result
in this context is the recent finding by Gohlke et al. [20]
that the K-Γ-only model shows signatures of a spin-liquid
ground state in a wide parameter range. A later study by
Wang, Normand, and Liu [67] using the variational
quantum Monte Carlo method revisits those results and
finds a proximate Kitaev spin liquid with a different
structure than KQSL for negative K and Γ=jKj up to
roughly 0.6. In the parameter area that we identify as
relevant for Na2IrO3, the large K and Γ are complemented
by sizable J and a number of smaller interactions; nevertheless,
the spin dynamics within the K-Γ-only model and a
possible connection to our case is a highly relevant
problem.
IV. MAGNETIC EXCITATION SPECTRA
AT A HIGH TEMPERATURE
A. Temperature evolution of
magnetic excitation spectra
A defining feature of quantum spin liquids is an
emergent magnetic excitation carrying fractional quantum
numbers. In the case of KQSL, the fractionalized magnetic
excitations are represented by Majorana fermions [1–3].
When a conventional magnetic order is thermally melted, a
signature of QSL may be found in the spin dynamics over a
wide temperature range. A recent INS study of α-RuCl3
reports a highly unusual temperature-stable signal around
the zone center, which is interpreted as a dynamical
response of the Majorana fermions of the KQSL due to
thermal fluctuations of fluxes [27–30,35,68]. The recent
RIXS work [44] on Na2IrO3 suggests that the dynamical
spin-spin correlation of the broad scattering is restricted to
nearest neighbors and phenomenologically similar to the
unusual broad INS scattering of α-RuCl3. In Fig. 2(d), the
broad scattering at the K point is presented to survive up to
150 K, showing that Na2IrO3 also has a highly unusual
temperature-stable signal. In this section, we present the
temperature evolution of the RIXS spectra over the full
Brillouin zone up to 280 K to show the thermal characteristics
of dynamic spin correlations in a paramagnetic phase
of Na2IrO3.
JUNGHO KIM et al. PHYS. REV. X 10, 021034 (2020)
021034-12
176
Figure 7(b) shows an intensity color map of RIXS
spectra at T ¼ 70 K. The specular elastic peaks at Γ and
Γ0 are seen as in the T ¼ 7 K map in Fig. 7(a). Because the
short-range zigzag order disappears at this temperature, the
magnetic Bragg peak at the M point is largely suppressed,
exposing an underneath low-energy excitation [15]. The
low-energy excitation at the K, Y, and K0
and the highenergy
excitation at the Γ stay more or less the same at
T ¼ 70 K. On the other hand, it is seen that the spectral
intensity near the Γ point unusually grows up at T ¼ 70 K,
connecting the high-energy excitation at the Γ to the lowenergy
excitations at the M, K, Y, and K0
points. This
temperature evolution of the spectral intensity distribution
becomes more pronounced up to T ¼ 160 K as shown in
Figs. 7(c) and 7(d).
Noticeable changes are observed at T ¼ 200 K. The
overall RIXS scattering intensity becomes weakened and
moves toward lower energy, confining its significant weight
within 60 meVas shown in Fig. 7(e). The scattering intensity
displays a triangular shape along the K-Γ-Y-K0 path. At
T ¼ 280 K, the RIXS scattering intensity over the whole
Brillouin zone substantially diminishes, resulting in a featureless
spectrum as in a paramagnet as shown in Fig. 7(f).
The weakening RIXS intensity at a high temperature is at
odds with the Ref. [43] RIXS work but consistent with
Ref. [44]. As discussed in Secs. II B and III C, it is not
necessary to invoke the lattice degrees offreedom to describe
our spectra, and the observed excitations are assigned as of
magnetic origin.
In a two-dimensional system, short-ranged spin correlations
above the long-range order temperature are visible
as a diffusive scattering at the vicinity of characteristic
points in the Brillouin zone. As the temperature increases
further, the short-range order dies and the corresponding
spectral weight disappears. Seemingly, the temperature
dependencies of the low-energy spectral weights around
the M and K points follow this general tendency. However,
the observed intensity modulations in other areas than the
K and M are unusual. In particular, this temperaturedependent
intensity modulation is seen at a high energy,
whose energy is beyond the thermal energy of room
temperature. Magnetic excitations at intermediate temperatures
are broad in energy and momentum and remind us of
unusual scatterings over a large energy interval revealed in
α-RuCl3 through the INS [27,29,30].
B. Dynamic spin correlations in a paramagnetic phase
In this section, we discuss the observation that the
spectral intensity in a large region surrounding the Γ point
unusually grows up to 160 K, connecting the high-energy
feature at the Γ point to the low-energy feature at the M, K,
and Y points and becomes a diffusive low-energy scattering
FIG. 7. Temperature evolution of magnetic excitation spectra. (a) T ¼ 7 K. (b) T ¼ 70 K. The magnetic Bragg peak at the M
disappears, indicating disappearing zigzag correlations. Spectral intensities fill in between Γ and all first Brillouin zone symmetric points
(M, K, K0
, and Y), connecting the high-energy feature at the Γ to the low-energy features at the M, K, and Y. This trend becomes more
pronounced at (c) T ¼ 120 K and (d) T ¼ 160 K. (e) T ¼ 200 K. Spectral intensities move toward lower energy. (f) T ¼ 280 K. The
RIXS intensity over the whole Brillouin zone substantially diminishes.
DYNAMIC SPIN CORRELATIONS IN THE HONEYCOMB … PHYS. REV. X 10, 021034 (2020)
021034-13
177
with negligible momentum dependence at 280 K (Fig. 7).
As the ED calculations in Fig. 6 show, the broad feature
below the ordering temperature contains the spontaneous
magnon decay into a two-magnon continuum. Twomagnon
excitations include pairs of very short wavelength
spin waves and can remain at higher temperatures than
single-magnon excitations. However, two-magnon scattering
cannot explain the increasing spectral weights over the
large Brillouin region, because the zigzag correlations
decrease as the temperature increases as evidenced by
decreasing intensities near the K and M points.
In a frustrated spin system above the ordering temperature,
the larger interactions can keep short-range correlations
up to quite high temperatures comparable to the
energy scale of these interactions. The corresponding
fragments of a few specifically correlated spins have a
specific dynamics determined by the dominant interactions.
In our case, the dynamics at elevated temperatures may be
influenced by the proximity of hidden KQSL in the
parameter space as well as the large scales of K and Γ
that are suggestive of a possible connection to spin-liquid
ground states in the K-Γ model that are currently being
investigated [20,67,69]. Reliable finite-temperature calculations
of the dynamic response are challenging and out of
the scope of the present ED scheme. Fortunately, a number
of results exist for the integrable pure Kitaev model. An
exact solution for spin dynamics at zero temperature is
known [63,65], and detailed finite-temperature behaviors
of the FM and AFM Kitaev systems are available from
studies combining the cluster dynamical mean-field theory
and the continuous-time quantum Monte Carlo method
(CDMFT þ CTQMC) [35,70]. Given the connections mentioned
above, here we discuss our observations in the
context of Kitaev systems at a finite temperature.
The thermal characteristics of Kitaev systems are understood
in terms of fractionalization of spins 1
2 into itinerant
Majorana fermions coupled to Z2 fluxes represented by
localized Majorana fermions [1]. Two characteristic crossover
temperatures appear [68]. At the lowest temperatures
below TL ≈ 0.012 K related to the Z2 flux gap, an almost
flux-free state is found, with only low-energy itinerant
Majorana fermions being thermally excited. Intermediate
temperatures are characterized by thermally activated Z2
fluxes, but the itinerant Majorana fermions still retain their
coherence. Finally, around TH ≈ 0.375 K, the fluxes and
Majorana fermions recombine into spins, the nearestneighbor
spin correlations decay, and the system is adiabatically
connected to a conventional spin-1
2 paramagnet.
Much of this physics is discussed in the context of
α-RuCl3. The successive thermal fractionalization finds its
thermodynamic signatures in magnetic specific heat data in
α-RuCl3 where two separated broad peaks exist and a
plateau in between two peaks is pinned at half of the ideal
R ln 2 magnetic entropy [30,35,70]. Raman scattering
observes a polarization-independent broad continuum [33]
successfully interpreted as due to pairs of itinerant
Majorana fermions [34]. Banerjee et al. report a highly
unusual scattering in α-RuCl3 through the INS which is
broad in energy and momentum and remains at a high
temperature, stimulating much research directed at identifying
unique dynamic correlations of emergent Majorana
fermions in systems close to KQSL [27,29]. Theoretical
works indicate that the characteristic broad scattering of the
KQSL is preserved in a proximate phase with long-range
ordered spins (proximate KQSL) [37]. The proximate
KQSL picture has been further elaborated experimentally
and theoretically to understand the finite-temperature
behavior [30,35,65,70].
Similarly to α-RuCl3, Na2IrO3 shows properties in the
paramagnetic phase that can be interpreted as arising from
the fractionalization to Majorana fermions. The two separated
broad maxima in magnetic specific heat are also
present in Na2IrO3 with one around 20 K and the other
around 110 K [40]. Half of the ideal R ln 2 entropy is gained
at around 60 K, and the full R ln 2 entropy is recovered at
more than 150 K, whose behavior is in good agreement
with theoretical predictions [68,71]. Signatures of Kitaevlike
correlations are seen in Raman scattering [38]. In this
work, we measure the low-energy RIXS response, which,
adopting the fast-collision approximation, is closely related
to the dynamic structure factor by the INS [52,72]. The
momentum- and energy-resolved magnetic excitation spectra
show that magnetic excitations in Na2IrO3 are broad in
energy and momentum at intermediate temperatures,
reminding of the unusual scatterings over a large energy
interval in INS on α-RuCl3 [27,29].
An apparent difference between Na2IrO3 and α-RuCl3 is
in the energy scales. Magnetic excitations in Na2IrO3 are
observed at a much higher energy than those in α-RuCl3
which are confined within the 15 meVenergy window [27–
30,43]. The broad excitation at the Γ point [Fig. 2(b)]
locates at 36 meV, while the one in α-RuCl3 can beviewed as
a diffusive quasielastic scattering. The short-ranged correlations
of three spin domains carrying their own zigzags
persist at a much higher temperature than in α-RuCl3 [15].
The attempts to quantify the interactions based on the INS
data in α-RuCl3 produce ðJ; KÞ ¼ ð−4.6; 7Þ meV in the
J-K model, ðK; ΓÞ ¼ ð−6.8; 9.5Þ meV in the K-Γ model,
and ðJ;K;ΓÞ¼ð−0.5;−5;2.5ÞmeV in the nearest-neighbor
model Hamiltonian [27,28,32], while our analysis of the
RIXS data in Na2IrO3 gives larger values; for example, the
A2 and A3 points correspond to ðJ; K; ΓÞ ¼ ð12; −24; 11Þ
and ð10; −15; 16Þ meV, respectively. Related to the discussion
in terms of the Kitaev model, the hidden KQSL found
near our fit point is driven by the effective Kitaev interaction
of AFM type with the strength of about K0 ≈ 35–40 meV
(see Appendix C), giving, e.g., the crossover temperature
of TH ≈ 140–170 K.
In studies of α-RuCl3, the temperature evolution of the
broad scattering features is described by the isotropic
JUNGHO KIM et al. PHYS. REV. X 10, 021034 (2020)
021034-14
178
Kitaev model with an FM Kitaev interaction, K ¼
−16.5 meV [30]. The CDMFT þ CTQMC calculation
on the FM Kitaev system finds that a quasielastic response
at zero energy is large around the Γ point at a low
temperature and becomes diffusively broadened in energy
at a high temperature, ending up in a conventional paramagnetic
phase, which is consistent with the experimental
observations by the INS [30,35,70]. On the other hand, the
CDMFT þ CTQMC calculation on the AFM Kitaev system
shows that an incoherent flat feature at ω ∼ K0 is seen
around the Γ point at a low temperature, while a quasielastic
response is distributed on the Brillouin zone boundary
[35,70]. As the temperature increases, the incoherent
feature becomes diffusively broadened in energy and is
connected to the quasielastic response on the Brillouin zone
boundary, losing its flat dispersion. The incoherent broad
feature at ω ∼ K0 merges to a diffusive response at zero
energy when the AFM Kitaev system adiabatically enters
into a conventional paramagnetic phase. These finitetemperature
behaviors of the AFM Kitaev system bear a
similarity to the observed temperature evolution of our
magnetic excitation spectra. The spectral intensity in a large
region surrounding the Γ unusually grows up to 160 K,
connecting the high-energy feature at the Γ to the lowenergy
feature at the M, K, and Y points as shown in Fig. 7.
The overall spectral weight moves toward lower energy at
200 K and shows a diffusive response with negligible
momentum dependence at 280 K.
V. CONCLUSION AND OUTLOOK
In this study, magnetic excitation spectra in a honeycomb
lattice Na2IrO3 were obtained for the wide-range reciprocal
space up to the second Brillouin zone using the RIXS
spectrometer. The state-of-the-art 12 meV measurements
could identify the low-energy sharp magnon peak below
the AFM order temperature and verify the broad widths of
magnetic excitations. These sets of data allow a detailed
comparison with theoretical calculations. The dispersion
and spectral intensity distribution in the reciprocal space of
RIXS spectra are well reproduced by the simulation using
the exact diagonalization method on finite-size clusters.
The parameter regime is characterized by large K < 0 and
Γ > 0 complemented by sizable J > 0 with small J2 and
J3, and Γ0
< 0. We examine two of the points of special
symmetry, i.e., AFM Heisenberg and AFM Kitaev, close to
the parameter region of Na2IrO3 and investigate the
evolution of the spin excitations along paths connecting
the special-symmetry points and the parameter points of
Na2IrO3. This inspection suggests that the main magnetic
intensity below the ordering temperature can be reasonably
explained by the picture of renormalized magnons.
Magnetic excitation spectra in Na2IrO3 show unusual
spectral intensity modulations in a large region surrounding
the Γ point at elevated temperatures. Finite-temperature
calculations of the complex spin Hamiltonian are
challenging and not available at the moment. We conjecture
that the dominant nearest-neighbor interactions keep shortrange
correlations up to quite high temperatures with a
specific short-range dynamics which has a possible connection
to a proximate spin-liquid phase. An interesting
experimental direction is given by a theoretical suggestion
that the full continuum of the Majorana fermions of the
KQSL can be mapped without interference with flux
excitations using the spin-conserving scattering of the
RIXS [72]. The spin-conserving measurements require
two instrumental capabilities which cannot be achieved
using a standard (spherical-analyzer-based) RIXS spectrometer:
a high-energy resolution and an efficient scattered
x-ray polarization analysis [73,74]. Recently, a new flat
crystal RIXS analyzer system was developed, which
provides a polarization analysis without compromising
the energy resolution and with high efficiency [46]. If
successful, the spin-conserving RIXS measurements will
give a transparent description of the existence of a
proximate KQSL in Na2IrO3.
ACKNOWLEDGMENTS
The use of the Advanced Photon Source at the Argonne
National Laboratory was supported by the U.S. Department
of Energy under Contract No. DE-AC02-06CH11357. J. C.
acknowledges support by Czech Science Foundation
(GAČR) under Project No. 19-16937S and MŠMT ČR
under NPU II Project No. CEITEC 2020(LQ1601).
Computational resources were supplied by the Ministry of
Education,YouthandSports oftheCzechRepublic under the
Project CERIT-Scientific Cloud (Project No. LM2015085)
provided within the program Projects of Large Research,
Development and Innovations Infrastructures.
APPENDIX A: SAMPLE
AND RIXS MEASUREMENT
Single-domain single crystals of Na2IrO3 are grown by
the self-flux. Powders of Na2CO3 are mixed with 10%–
20% excess IrO2 and are calcined at 700 °C for 24 h. Single
crystals are grown on top of a powder matrix in subsequent
heating at 1050 °C. Platelike crystals with typical dimensions
of 5 mm × 5 mm × 0.1 mm are physically extracted.
The sample is mounted in a Displex closed-cycle cryostat.
The RIXS measurements are performed using the RIXS
spectrometer at the 27-ID beam line of the Advanced
Photon Source where the sample, analyzer, and detector are
positioned in the Rowland geometry. The diamond(111)
high-heat-load monochromator reflects x rays from two inline
undulators into a high-resolution monochromator. The
two-bounce monochromator of single monolithic Si(844)
channel-cut crystal produces an energy bandpass of
14.8 meV. The four-bounce monochromator of two monolithic
Si(844) channel-cut crystals results in an energy
bandpass of 8.9 meV. The beam is then focused by a set of
DYNAMIC SPIN CORRELATIONS IN THE HONEYCOMB … PHYS. REV. X 10, 021034 (2020)
021034-15
179
Kirkpatrick-Baez mirrors, yielding a typical spot size of
10 × 40 μm2
FWHM (v × h) at the sample. A horizontal
scattering geometry is used with the incident photon
polarization in the scattering plane. Mapping of the full
Brillouin zone is carried out within only a few degrees of
90° scattering geometry to minimize the contribution from
the Thompson elastic scattering. For the 25 meV RIXS
measurement, a Si(844) diced spherical analyzer with 1-in
diameter and a position-sensitive silicon microstrip detector
are used with the 14.8 meV incident bandpass. For the
12 meV RIXS measurement, a quartz(309) diced spherical
analyzer with 1-in diameter is used with the 8.9 meV
incident bandpass.
APPENDIX B: EXACT DIAGONALIZATION
ON FINITE CLUSTERS
The theoretical RIXS intensity is obtained by combining
the components of the pseudospin susceptibility tensor
calculated for zero temperature:
χαβðq; ωÞ ¼ i
Z ∞
0
hGSj½Sα
qðtÞ; Sβ
−qð0ފjGSieiωtdt; ðB1Þ
where
Sα
q ¼
1
ffiffiffiffiffiffiffiffiffi
Nsite
p
X
R
Sα
Re−iq·R
: ðB2Þ
The ground state jGSi and, subsequently, the dynamic
response embodied in χαβðq; ωÞ are evaluated by the
standard Lanczos exact diagonalization method [75] based
on periodic tiling of the honeycomb lattice with small
clusters. Since the intensity profiles contain broad features
corresponding to continua of densely spaced levels, to
achieve convergence, we use a large number of Lanczos
steps in the calculation—500 to get the data in Fig. 4 and
1200 to get the fine-resolved data presented in Fig. 6. A
combination of symmetric hexagonal clusters 24a and 32a
and rectangular clusters 32b1–32b3 shown in Fig. 8 enable
us to access a number of wave vectors along the
Γ-M-Γ0
-X-K-Γ-Y-K0
-Γ0
path used to plot the maps. The
maps are constructed by nearest-point interpolation with an
additional averaging if the given wave vector is compatible
with several clusters.
To account for the simultaneous presence of the three
zigzag patterns in the sample—one long-range and two
short-range correlated [15]—the response for the three
possibilities with different directions of zigzag chains needs
to be averaged. This average is, in fact, automatically
included in the exact diagonalization calculation, because
the symmetry is not spontaneously broken and the cluster
ground state is a superposition of the zigzag patterns
(equal-weight superposition in the case of 24a and 32a
and approximately equal-weight for 32b). The explicit
averaging is performed in the case of the linear spin-wave
calculation only.
A side note on the selection of the clusters is in order:
Even though there are other 24-site clusters (of elongated or
asymmetric shape) that could bring better q resolution, their
dynamic response contains artifacts, e.g., due to the
creation of very short zigzag chain loops of just a few
bonds when periodic boundary conditions are applied.
When compared to the dataset for 24a, 32a, and 32b,
the corresponding intensity profiles clearly stand out and
are thus not included as unreliable.
APPENDIX C: HIDDEN HEISENBERG
AND KITAEV POINTS
In this Appendix, we briefly elaborate on the points of
hiddensymmetryandthelinksbetweentheexcitationspectra
of the extended Kitaev-Heisenberg model and the “hidden”
models. The hidden-symmetry points are revealed by rotating
spin axes, either globally or in a sublattice-dependent
fashion, to convert the extended Kitaev-Heisenberg model
K X
M
’Y
24a
32a
32b1
32b2
32b3
(a)
(b)
32b1
32b3
32b2
24a
32a
FIG. 8. (a) Clusters used in the exact diagonalization calculations:
fully symmetric hexagonal 24- and 32-site clusters and a
rectangular 32-site cluster in three possible orientations with
respect to the honeycomb lattice. (b) Wave vectors compatible
with periodic boundary conditions applied to the individual
clusters. They are shown in one quadrant of the first Brillouin
zone of the honeycomb lattice (dashed black line) and that of the
completed triangular lattice (solid black line). High-symmetry
points and the path along which the measured or simulated data
are presented (gray line) are indicated [cf. Fig. 1(a)].
JUNGHO KIM et al. PHYS. REV. X 10, 021034 (2020)
021034-16
180
with a particular set of parameters to a simpler model [61].
This one-to-one correspondence enables us to transfer the
known features such as the excitation spectra of the simpler
model to the extended Kitaev-Heisenberg one.
We start with the hidden Kitaev point. To reveal its
presence, one has to utilize the self-dual transformation
[61] of the pseudospin Hamiltonian between the original
axes xyz used in Eq. (1) and the new axes x0
y0
z0
that are
180° rotated around the axis perpendicular to the honeycomb
plane (see the insets in Fig. 9). The two reference
frames for the spins are linked by the relation
0
B
@
Sx0
Sy0
Sz0
1
C
A ¼
1
3
0
B
@
−1 2 2
2 −1 2
2 2 −1
1
C
A
0
B
@
Sx
Sy
Sz
1
C
A: ðC1Þ
Applying the transformation to the case of Ising-type
interaction in the new coordinate system x0
y0
z0
, we find
the correspondence
Sz0
i Sz0
j ¼ −
1
3
Sz
i Sz
j þ
4
9
Si · Sj þ
4
9
ðSx
i Sy
j þ Sy
i Sx
jÞ
−
2
9
ðSx
i Sz
j þ Sz
i Sx
j þ Sy
i Sz
j þ Sz
i Sy
jÞ: ðC2Þ
The 180° rotation is compatible with the cyclic permutation
among xyz and x0y0z0 axes so that similar relations can be
found for Sx0
i Sx0
j and Sy0
i Sy0
j interactions. Altogether, the
Ising-type interaction distributed on x, y, and z bonds
constitutes the Kitaev model, while the right-hand side of
the relation (C2) (and the two other ones) is just the extended
Kitaev-Heisenberg model with a particular combination of
parameters. Therefore, at the hidden Kitaev point given by
the parameters J ¼ 4
9 K0, K ¼ − 1
3 K0, Γ ¼ 4
9 K0, and
Γ0
¼ − 2
9 K0, the extended Kitaev-Heisenberg model exactly
maps to a Kitaev model with the interaction constant K0.
Since the transformation is just a global rotation of the spin
axes, all the features of the Kitaev model are exactly
reproduced at the hidden Kitaev point. In particular, the
excitation spectra are identical, and the spin susceptibility is
obtained from that of the Kitaev model by a simple linear
combination of the components.
The hidden Heisenberg point is given by the parameters
J ¼ − 1
9 J0, K ¼ − 2
3 J0, Γ ¼ 8
9 J0, and Γ0
¼ − 4
9 J0, where J0
is the effective Heisenberg interaction constant. Here, the
connection is less apparent, since, by applying the above
global rotation (C1), we get to the Kitaev-Heisenberg
model only. Its parameters read as K0
¼ 2J0 and J0
¼
−J0 (Γ and Γ0
are zero). To establish the relation to the final
Heisenberg model, we have to invoke the four-sublattice
transformation connecting zigzag and N´eel order [5,61,76].
It is depicted in Fig. 9(a) and consists of 180° rotations
around one of the cubic axes or identity applied on the
respective sublattices. This transformation preserves the
form of the Kitaev-Heisenberg model but changes the
balance between the Kitaev and Heisenberg term and, for
the above parameters K0
and J0
, leads to a pure Heisenberg
model with the exchange parameter J0. Again, all the
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
Parameter(xyz-frame)Parameter(xyz-frame)
Heisenberg Point A2/A3 Kitaev
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
y’
x’
z’
y
z
x
Sx Sy
Sz
SxSy
Sz
K
J
K
J
(b)
(c)
(a)
FIG. 9. (a) Four-sublattice transformation used in the text. At the
individual sublattices, the spin axes are rotated by 180° around the
cubic axes x, y, and z or left intact. In the momentum space (right),
the spin components are shifted by the zigzag ordering vectors.
(b) Model parameter values along the lines used in Fig. 6. The
values are expressed using the original xyz-coordinate system for
the spins. The full lines and symbols correspond to Figs. 6(a)
and 6(b), and dashed lines and open symbols to Figs. 6(c) and 6(d).
The figure is organized like Fig. 6 with A2 or A3 parameters being
in the middle and the evolution toward the hidden Heisenberg
(Kitaev) point corresponding to the left (right) direction from the
middle. Only nearest-neighbor interaction parameters JKΓΓ0 are
presented; J2;3 linearly vanish when approaching the points of
hidden symmetry. The marked parameter points correspond to the
spectra shown in Fig. 6. (c) The same lines through the parameter
space but expressed using the x0
y0
z0
-coordinate system rotated by
180° aroundtheaxisperpendicular tothehoneycomb plane.In both
xyz and x0
y0
z0
frames depicted by the insets, the axes point above
the paper plane.
DYNAMIC SPIN CORRELATIONS IN THE HONEYCOMB … PHYS. REV. X 10, 021034 (2020)
021034-17
181
features of the Heisenberg model such as the magnon
spectra can be transferred to the hidden Heisenberg point.
However, due to the four-sublattice transformation,
momentum shifts depicted in Fig. 9(a) are involved. The
Fourier components of Sx, Sy, and Sz are shifted by wave
vectors with the directions identical to the corresponding
bond directions of the honeycomb lattice [61]. The intense
AFM Heisenberg magnons residing at the Γ0
points in the
corners of the extended Brillouin zone are then translated to
the zigzag M points.
In the main text, we study the parameter evolution of the
q ¼ 0 spectra when going toward the hidden Heisenberg
and Kitaev points. The corresponding parameter values
are shown in Fig. 9(b) in the xyz reference frame and in
Fig. 9(c) also rotated via Eq. (C1) to the x0
y0
z0
reference
frame. Note that, in the x0
y0
z0
frame, the dominant interaction
at our points A2 and A3 is K > 0, accompanied by
small J < 0 and Γ and Γ0 interactions. The calculated
response is thus similar to the one obtained for the upper
zigzag phase in Fig. 3 that is stabilized in the KitaevHeisenberg
model itself.
[1] A. Kitaev, Anyons in an Exactly Solved Model and Beyond,
Ann. Phys. (Amsterdam) 321, 2 (2006).
[2] S. R. Elliott and M. Franz, Colloquium: Majorana Fermions
in Nuclear, Particle, and Solid-State Physics, Rev. Mod.
Phys. 87, 137 (2015).
[3] A. Yu. Kitaev, Fault-Tolerant Quantum Computation by
Anyons, Ann. Phys. (Amsterdam) 303, 2 (2003).
[4] G. Jackeli and G. Khaliullin, Mott Insulators in the Strong
Spin-Orbit Coupling Limit: From Heisenberg to a Quantum
Compass and Kitaev Models, Phys. Rev. Lett. 102, 017205
(2009).
[5] J. Chaloupka, G. Jackeli, and G. Khaliullin, KitaevHeisenberg
Model on a Honeycomb Lattice: Possible
Exotic Phases in Iridium Oxides A2IrO3, Phys. Rev. Lett.
105, 027204 (2010).
[6] I. Kimchi and Y.-Z. You, Kitaev-Heisenberg-J2-J3 Model
for the Iridates A2IrO3, Phys. Rev. B 84, 180407(R) (2011).
[7] J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Generic Spin Model
for the Honeycomb Iridates beyond the Kitaev Limit, Phys.
Rev. Lett. 112, 077204 (2014).
[8] Y. Yamaji, Y. Nomura, M. Kurita, R. Arita, and M. Imada,
First-Principles Study of the Honeycomb-Lattice Iridates
Na2IrO3 in the Presence of Strong Spin-Orbit Interaction
and Electron CorrelationsPhys. Rev. Lett. 113, 107201
(2014).
[9] K. Shinjo, S. Sota, and T. Tohyama, Density-Matrix
Renormalization Group Study of the Extended KitaevHeisenberg
Model, Phys. Rev. B 91, 054401 (2015).
[10] E. Sela, H.-C. Jiang, M. H. Gerlach, and S. Trebst, Orderby-Disorder
and Spin-Orbital Liquids in a Distorted
Heisenberg-Kitaev Model, Phys. Rev. B 90, 035113 (2014).
[11] Y. Singh and P. Gegenwart, Antiferromagnetic Mott Insulating
State in Single Crystals of the Honeycomb Lattice
Material Na2IrO3, Phys. Rev. B 82, 064412 (2010).
[12] X. Liu, T. Berlijn, W.-G. Yin, W. Ku, A. Tsvelik, Y.-J. Kim,
H. Gretarsson, Y. Singh, P. Gegenwart, and J. P. Hill, LongRange
Magnetic Ordering in Na2IrO3, Phys. Rev. B 83,
220403(R) (2011).
[13] F. Ye, S. Chi, H. Cao, B. C. Chakoumakos, J. A. FernandezBaca,
R. Custelcean, T. F. Qi, O. B. Korneta, and G. Cao,
Direct Evidence of a Zigzag Spin-Chain Structure in the
Honeycomb Lattice: A Neutron and x-ray Diffraction
Investigation of Single-Crystal Na2IrO3, Phys. Rev. B 85,
180403(R) (2012).
[14] S. K. Choi, R. Coldea, A. N. Kolmogorov, T. Lancaster,
I. I. Mazin, S. J. Blundell, P. G. Radaelli, Y. Singh, P.
Gegenwart, K. R. Choi, S.-W. Cheong, P. J. Baker, C. Stock,
and J. Taylor, Spin Waves and Revised Crystal Structure of
Honeycomb Iridate Na2IrO3, Phys. Rev. Lett. 108, 127204
(2012).
[15] S. H. Chun, J.-W. Kim, J. Kim, H. Zheng, C. Stoumpos, C.
Malliakas, J. F. Mitchell, K. Mehlawat, Y. Singh, Y. Choi, T.
Gog, A. Al-Zein, M. M. Sala, M. Krisch, J. Chaloupka, G.
Jackeli, G. Khaliullin, and B. J. Kim, Direct Evidence for
Dominant Bond-Directional Interactions in a Honeycomb
Lattice Iridate Na2IrO3, Nat. Phys. 11, 462 (2015).
[16] V. M. Katukuri, S. Nishimoto, V. Yushankhai, A. Stoyanova,
H. Kandpal, S. Choi, R. Coldea, I. Rousochatzakis, L. Hozoi,
and J. van den Brink, Kitaev Interactions between J ¼ 1=2
Moments in Honeycomb Na2IrO3 are Large and Ferromagnetic:
Insights from ab initio Quantum Chemistry Calculations,
New J. Phys. 16, 013056 (2014).
[17] S. M. Winter, Y. Li, H. O. Jeschke, and R. Valentí, Challenges
in Design of Kitaev Materials: Magnetic Interactions
from Competing Energy Scales, Phys. Rev. B 93, 214431
(2016).
[18] S. M. Winter, A. A. Tsirlin, M. Daghofer, J. van den Brink,
Y. Singh, P. Gegenwart, and R. Valentí, Models and
Materials for Generalized Kitaev Magnetism, J. Phys.
Condens. Matter 29, 493002 (2017).
[19] I. Rousochatzakis and N. B. Perkins, Classical Spin
Liquid Instability Driven by Off-Diagonal Exchange in
Strong Spin-Orbit Magnets, Phys. Rev. Lett. 118, 147204
(2017).
[20] M. Gohlke, G. Wachtel, Y. Yamaji, F. Pollmann, and Y. B.
Kim, Quantum Spin Liquid Signatures in Kitaev-Like
Frustrated Magnets, Phys. Rev. B 97, 075126 (2018).
[21] Y. Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale, W. Ku,
S. Trebst, and P. Gegenwart, Relevance of the HeisenbergKitaev
Model for the Honeycomb Lattice Iridates A2IrO3,
Phys. Rev. Lett. 108, 127203 (2012).
[22] T. Takayama, A. Kato, R. Dinnebier, J. Nuss, H. Kono,
L. S. I. Veiga, G. Fabbris, D. Haskel, and H. Takagi,
Hyperhoneycomb Iridate β-Li2IrO3 as a Platform for Kitaev
Magnetism, Phys. Rev. Lett. 114, 077202 (2015).
[23] K. A. Modic, T. E. Smidt, I. Kimchi, N. P. Breznay,
A. Biffin, S. Choi, R. D. Johnson, R. Coldea, P. WatkinsCurry,
G. T. McCandless, J. Y. Chan, F. Gandara, Z. Islam,
A. Vishwanath, A. Shekhter, R. D. McDonald, and J. G.
Analytis, Realization of a Three-Dimensional Spin–
Anisotropic Harmonic Honeycomb Iridate, Nat. Commun.
5, 4203 (2014).
[24] M. Abramchuk, C. Ozsoy-Keskinbora, J. W. Krizan, K. R.
Metz, D. C. Bell, and F. Tafti, Cu2IrO3: A New Magnetically
JUNGHO KIM et al. PHYS. REV. X 10, 021034 (2020)
021034-18
182
Frustrated Honeycomb Iridate, J. Am. Chem. Soc. 139,
15371 (2017).
[25] K. Kitagawa, T. Takayama, Y. Matsumoto, A. Kato, R.
Takano, Y. Kishimoto, S. Bette, R. Dinnebier, G. Jackeli,
and H. Takagi, A Spin–Orbital-Entangled Quantum Liquid
on a Honeycomb Lattice, Nature (London) 554, 341 (2018).
[26] K. W. Plumb, J. P. Clancy, L. J. Sandilands, V. V. Shankar,
Y. F. Hu, K. S. Burch, H.-Y. Kee, and Y.-J. Kim, α-RuCl3: A
Spin-Orbit Assisted Mott Insulator on a Honeycomb, Phys.
Rev. B 90, 041112(R) (2014).
[27] A. Banerjee, C. A. Bridges, J.-Q. Yan, A. A. Aczel, L. Li,
M. B. Stone, G. E. Granroth, M. D. Lumsden, Y. Yiu, J.
Knolle, S. Bhattacharjee, D. L. Kovrizhin, R. Moessner,
D. A. Tennant, D. G. Mandrus, and S. E. Nagler, Proximate
Kitaev Quantum Spin Liquid Behaviour in a Honeycomb
Magnet, Nat. Mater. 15, 733 (2016).
[28] K. Ran, J. Wang, W. Wang, Z.-Y. Dong, X. Ren, S. Bao, S.
Li, Z. Ma, Y. Gan, Y. Zhang, J. T. Park, G. Deng, S.
Danilkin, S.-L. Yu, J.-X. Li, and J. Wen, Spin-Wave
Excitations Evidencing the Kitaev Interaction in Single
Crystalline α-RuCl3, Phys. Rev. Lett. 118, 107203 (2017).
[29] A. Banerjee, J. Yan, J. Knolle, C. A. Bridges, M. B. Stone,
M. D. Lumsden, D. G. Mandrus, D. A. Tennant, R. Moessner,
and S. E. Nagler, Neutron Scattering in the Proximate
Quantum Spin Liquid α-RuCl3, Science 356, 1055 (2017).
[30] S.-H. Do, S.-Y. Park, J. Yoshitake, J. Nasu, Y. Motome, Y. S.
Kwon, D. T. Adroja, D. J. Voneshen, K. Kim, T.-H. Jang,
J.-H. Park, K.-Y. Choi, and S. Ji, Majorana Fermions in the
Kitaev Quantum Spin System α-RuCl3, Nat. Phys. 13, 1079
(2017).
[31] M. E. Zhitomirsky and A. L. Chernyshev, Colloquium:
Spontaneous Magnon Decays, Rev. Mod. Phys. 85, 219
(2013).
[32] S. M. Winter, K. Riedl, P. A. Maksimov, A. L. Chernyshev,
A. Honecker, and R. Valentí, Breakdown of Magnons in a
Strongly Spin-Orbital Coupled Magnet, Nat. Commun. 8,
1152 (2017).
[33] L. J. Sandilands, Y. Tian, K. W. Plumb, Y.-J. Kim, and K. S.
Burch, Scattering Continuum and Possible Fractionalized
Excitations in α-RuCl3, Phys. Rev. Lett. 114, 147201
(2015).
[34] J. Nasu, J. Knolle, D. L. Kovrizhin, Y. Motome, and R.
Moessner, Fermionic Response from Fractionalization in
an Insulating Two-Dimensional Magnet, Nat. Phys. 12, 912
(2016).
[35] J. Yoshitake, J. Nasu, and Y. Motome, Fractional Spin
Fluctuations as a Precursor of Quantum Spin Liquids:
Majorana Dynamical Mean-Field Study for the Kitaev
Model, Phys. Rev. Lett. 117, 157203 (2016).
[36] I. A. Leahy, C. A. Pocs, P. E. Siegfried, D. Graf, S.-H. Do,
K.-Y. Choi, B. Normand, and M. Lee, Anomalous Thermal
Conductivity and Magnetic Torque Response in the Honeycomb
Magnet α-RuCl3, Phys. Rev. Lett. 118, 187203
(2017).
[37] M. Gohlke, R. Verresen, R. Moessner, and F. Pollmann,
Dynamics of the Kitaev-Heisenberg Model, Phys. Rev. Lett.
119, 157203 (2017).
[38] S. N. Gupta, P. V. Sriluckshmy, K. Mehlawat, A. Balodhi,
D. K. Mishra, S. R. Hassan, T. V. Ramakrishnan, D. V. S.
Muthu, Y. Singh, and A. K. Sood, Raman signatures of
strong Kitaev exchange correlations in ðNa1−xLixÞ2IrO3:
Experiments and theory, Europhys. Lett. 114, 47004 (2016).
[39] J. Knolle, G.-W. Chern, D. L. Kovrizhin, R. Moessner, and
N. B. Perkins, Raman Scattering Signatures of Kitaev Spin
Liquids in A2IrO3 Iridates with A ¼ Na or Li, Phys. Rev.
Lett. 113, 187201 (2014).
[40] K. Mehlawat, A. Thamizhavel, and Y. Singh, Heat Capacity
Evidence for Proximity to the Kitaev Quantum Spin Liquid
in A2IrO3 (A ¼ Na, Li), Phys. Rev. B 95, 144406 (2017).
[41] I. A. Leahy, C. A. Pocs, P. E. Siegfried, D. Graf, S.-H. Do,
K.-Y. Choi, B. Normand, and M. Lee, Anomalous Thermal
Conductivity and Magnetic Torque Response in the Honeycomb
Magnet α-RuCl3, Phys. Rev. Lett. 118, 187203
(2017).
[42] S. D. Das, S. Kundu, Z. Zhu, E. Mun, R. D. McDonald, G.
Li, L. Balicas, A. McCollam, G. Cao, J. G. Rau, H.-Y. Kee,
V. Tripathi, and S. E. Sebastian, Magnetic Anisotropy of the
Alkali Iridate Na2IrO3 at High Magnetic Fields: Evidence
for Strong Ferromagnetic Kitaev Correlations, Phys. Rev.
B 99, 081101(R) (2019).
[43] H. Gretarsson, J. P. Clancy, Y. Singh, P. Gegenwart, J. P.
Hill, J. Kim, M. H. Upton, A. H. Said, D. Casa, T. Gog, and
Y.-J. Kim, Magnetic Excitation Spectrum of Na2IrO3
Probed with Resonant Inelastic x-ray Scattering, Phys.
Rev. B 87, 220407(R) (2013).
[44] A. Revelli, M. M. Sala, G. Monaco, C. Hickey, P. Becker, F.
Freund, A. Jesche, P. Gegenwart, T. Eschmann, F. L.
Buessen, S. Trebst, P. H. M. van Loosdrecht, J. van den
Brink, and M. Gruninger, Fingerprints of Kitaev Physics in
the Magnetic Excitations of Honeycomb Iridates
arXiv:1905.13590.
[45] A. H. Said, T. Gog, M. Wieczorek, X. Huang, D. Casa, E.
Kasman, R. Divan, and J. H. Kim, High-Energy-Resolution
Diced Spherical Quartz Analyzers for Resonant Inelastic
X-ray Scattering, J. Synchrotron Radiat. 25, 373 (2018).
[46] J. Kim, D. Casa, A. Said, R. Krakora, B. J. Kim, E. Kasman,
X. Huang, and T. Gog, Quartz-Based Flat-Crystal Resonant
Inelastic x-ray Scattering Spectrometer with sub-10 meV
Energy Resolution, Sci. Rep. 8, 1958 (2018).
[47] T. Gog, D. M. Casa, A. H. Said, M. H. Upton, J. Kim, I.
Kuzmenko, X. Huang, and R. Khachatryan, Spherical
Analyzers and Monochromators for Resonant Inelastic
Hard X-ray Scattering: A Compilation of Crystals and
Reflections, J. Synchrotron Radiat. 20, 74 (2013).
[48] Yu. V. Shvyd’ko, J. P. Hill, C. A. Burns, D. S. Coburn, B.
Brajuskovic, D. Casa, K. Goetze, T. Gog, R. Khachatryan,
J.-H. Kim, C. N. Kodituwakku, M. Ramanathan, T. Roberts,
A. Said, H. Sinn, D. Shu, S. Stoupin, M. Upton, M.
Wieczorek, and H. Yavas, Merix—Next Generation Medium
Energy Resolution Inelastic X-Ray Scattering Instrument at
the APS, J. Electron Spectrosc. Relat. Phenom. 188, 140
(2013).
[49] R. Comin, G. Levy, B. Ludbrook, Z.-H. Zhu, C. N. Veenstra,
J. A. Rosen, Y. Singh, P. Gegenwart, D. Stricker, J. N.
Hancock, D. van der Marel, I. S. Elfimov, and A.
Damascelli, Na2IrO3 as a Novel Relativistic Mott Insulator
with a 340-meV Gap, Phys. Rev. Lett. 109, 266406 (2012).
[50] J. G. Rau and H.-Y. Kee, Trigonal Distortion in the
Honeycomb Iridates: Proximity of Zigzag And Spiral
Phases in Na2IrO3, arXiv:1408.4811.
DYNAMIC SPIN CORRELATIONS IN THE HONEYCOMB … PHYS. REV. X 10, 021034 (2020)
021034-19
183
[51] J. Chaloupka and G. Khaliullin, Magnetic Anisotropy in the
Kitaev Model Systems Na2IrO3 and RuCl3, Phys. Rev. B 94,
064435 (2016).
[52] B. J. Kim and G. Khaliullin, Resonant Inelastic x-ray
Scattering Operators for t2g Orbital Systems, Phys. Rev.
B 96, 085108 (2017).
[53] H. Gretarsson, J. P. Clancy, X. Liu, J. P. Hill, E. Bozin, Y.
Singh, S. Manni, P. Gegenwart, J. Kim, A. H. Said, D. Casa,
T. Gog, M. H. Upton, H.-S. Kim, J. Yu, V. M. Katukuri, L.
Hozoi, J. van den Brink, and Y.-J. Kim, Crystal-Field
Splitting and Correlation Effect on the Electronic Structure
of A2IrO3, Phys. Rev. Lett. 110, 076402 (2013).
[54] J. Rusnačko, D. Gotfryd, and J. Chaloupka, Structural and
Magnetic Studies of Sr2IrO4, Phys. Rev. B 99, 064425
(2019).
[55] H. Gretarsson, N. H. Sung, M. Höppner, B. J. Kim, B.
Keimer, and M. Le Tacon, Two-Magnon Raman Scattering
and Pseudospin-Lattice Interactions in Sr2IrO4 and
Sr3Ir2O7, Phys. Rev. Lett. 116, 136401 (2016).
[56] H. Gretarsson, J. Sauceda, N. H. Sung, M. Höppner, M.
Minola, B. J. Kim, B. Keimer, and M. Le Tacon, Raman
Scattering Study of Vibrational and Magnetic Excitations in
Sr2−xLaxIrO4, Phys. Rev. B 96, 115138 (2017).
[57] L. Braicovich, M. Rossi, R. Fumagalli, Y. Peng, Y. Wang, R.
Arpaia, D. Betto, G. M. D. Luca, D. D. Castro, K. Kummer,
M. M. Sala, M. Pagetti, G. Balestrino, N. B. Brookes, M.
Salluzzo, S. Johnston, J. van den Brink, and G. Ghiringhelli,
Determining the Electron-Phonon Coupling in Superconducting
Cuprates by Resonant Inelastic X-ray Scattering:
Methods and Results on Nd1þxBa2−xCu3O7−δ arXiv:
1906.01270.
[58] M. O. Krause and J. H. Oliver, Natural Widths of Atomic K
and L Levels,Kα X‐ray Lines and Several KLL Auger Lines,
J. Phys. Chem. Ref. Data 8, 329 (1979).
[59] V. Hermann, J. Ebad-Allah, F. Freund, I. M. Pietsch, A.
Jesche, A. A. Tsirlin, J. Deisenhofer, M. Hanfland, P.
Gegenwart, and C. A. Kuntscher, High-Pressure versus
Isoelectronic Doping Effect on the Honeycomb Iridate
Na2IrO3, Phys. Rev. B 96, 195137 (2017).
[60] J. G. Vale, C. D. Dashwood, E. Paris, L. S. I. Veiga, M.
Garcia-Fernandez, A. Nag, A. Walters, K. J. Zhou, I. M.
Pietsch, A. Jesche, P. Gegenwart, R. Coldea, T. Schmitt, and
D. F. McMorrow, High-Resolution Resonant Inelastic x-ray
Scattering study of the Electron-Phonon Coupling in
Honeycomb α-Li2IrO3, Phys. Rev. B 100, 224303 (2019).
[61] J. Chaloupka and G. Khaliullin, Hidden Symmetries of the
Extended Kitaev-Heisenberg Model: Implications for the
Honeycomb-Lattice Iridates A2IrO3, Phys. Rev. B 92,
024413 (2015).
[62] J. Chaloupka, G. Jackeli, and G. Khaliullin, Zigzag Magnetic
Order in the Iridium Oxide Na2IrO3, Phys. Rev. Lett.
110, 097204 (2013).
[63] J. Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moessner,
Dynamics of Fractionalization in Quantum Spin Liquids,
Phys. Rev. B 92, 115127 (2015).
[64] D. Gotfryd, J. Rusnačko, K. Wohlfeld, G. Jackeli, J.
Chaloupka, and A. M. Oleś, Phase Diagram and Spin
Correlations of the Kitaev-Heisenberg Model: Importance
of Quantum Effects, Phys. Rev. B 95, 024426 (2017).
[65] J. Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moessner,
Dynamics of a Two-Dimensional Quantum Spin Liquid:
Signatures of Emergent Majorana Fermions and Fluxes,
Phys. Rev. Lett. 112, 207203 (2014).
[66] J. Knolle, S. Bhattacharjee, and R. Moessner, Dynamics of a
Quantum Spin Liquid Beyond Integrability: The KitaevHeisenberg-Γ
Model in an Augmented Parton Mean-Field
Theory, Phys. Rev. B 97, 134432 (2018).
[67] J. Wang, B. Normand, and Z.-X. Liu, One Proximate Kitaev
Spin Liquid in the K-J-Γ Model on the Honeycomb Lattice,
Phys. Rev. Lett. 123, 197201 (2019).
[68] J. Nasu, M. Udagawa, and Y. Motome, Thermal Fractionalization
Of Quantum Spins in a Kitaev Model: TemperatureLinear
Specific Heat and Coherent Transport of Majorana
Fermions, Phys. Rev. B 92, 115122 (2015).
[69] A. Catuneanu, Y. Yamaji, G. Wachtel, Y. B. Kim, and H.-Y.
Kee, Path to Stable Quantum Spin Liquids in Spin-Orbit
Coupled Correlated Materials, npj Quantum Mater. 3, 23
(2018).
[70] J. Yoshitake, J. Nasu, Y. Kato, and Y. Motome, Majorana
Dynamical Mean-Field Study of Spin Dynamics at Finite
Temperatures in the Honeycomb Kitaev Model, Phys. Rev.
B 96, 024438 (2017).
[71] Y. Yamaji, T. Suzuki, T. Yamada, S.-I. Suga, N. Kawashima,
and M, Imada, Clues and Criteria for Designing a Kitaev
Spin Liquid Revealed by Thermal and Spin Excitations of
the Honeycomb Iridate Na2IrO3, Phys. Rev. B 93, 174425
(2016).
[72] G. B. Halász, N. B. Perkins, and J. van den Brink, Resonant
Inelastic X-Ray Scattering Response of the Kitaev Honeycomb
Model, Phys. Rev. Lett. 117, 127203 (2016).
[73] K. Ishii, S. Ishihara, Y. Murakami, K. Ikeuchi, K. Kuzushita,
T. Inami, K. Ohwada, M. Yoshida, I. Jarrige, N. Tatami, S.
Niioka, D. Bizen, Y. Ando, J. Mizuki, S. Maekawa, and Y.
Endoh, Polarization-Analyzed Resonant Inelastic x-ray
Scattering of the Orbital Excitations in KCuF3, Phys.
Rev. B 83, 241101(R) (2011).
[74] X. Gao, D. Casa, J. Kim, T. Gog, C. Li, and C. Burns,
Toroidal Silicon Polarization Analyzer for Resonant Inelastic
x-ray Scattering, Rev. Sci. Instrum. 87, 083107
(2016).
[75] A. Avella and F. Mancini, Strongly Correlated Systems:
Numerical Methods (Springer, Berlin, 2013).
[76] G. Khaliullin, Orbital Order and Fluctuations in Mott
Insulators, Prog. Theor. Phys. Suppl. 160, 155 (2005).
JUNGHO KIM et al. PHYS. REV. X 10, 021034 (2020)
021034-20
184
Kitaev Spin Liquid in 3d Transition Metal Compounds
Huimei Liu,1
Jiˇrí Chaloupka ,2,3
and Giniyat Khaliullin 1
1
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
2
Department of Condensed Matter Physics, Faculty of Science, Masaryk University, Kotláˇrská 2, Brno 61137, Czech Republic
3
Central European Institute of Technology, Masaryk University, Kamenice 753/5, Brno 62500, Czech Republic
(Received 10 February 2020; accepted 2 July 2020; published 21 July 2020)
We study the exchange interactions and resulting magnetic phases in the honeycomb cobaltates. For a
broad range of trigonal crystal fields acting on Co2þ ions, the low-energy pseudospin-1=2 Hamiltonian is
dominated by bond-dependent Ising couplings that constitute the Kitaev model. The non-Kitaev terms
nearly vanish at small values of trigonal field Δ, resulting in spin liquid ground state. Considering
Na3Co2SbO6 as an example, we find that this compound is proximate to a Kitaev spin liquid phase, and can
be driven into it by slightly reducing Δ by ∼20 meV, e.g., via strain or pressure control. We argue that, due
to the more localized nature of the magnetic electrons in 3d compounds, cobaltates offer the most
promising search area for Kitaev model physics.
DOI: 10.1103/PhysRevLett.125.047201
The Kitaev honeycomb model [1], demonstrating the key
concepts of quantum spin liquids [2] via an elegant exact
solution, has attracted much attention (see the recent
reviews [3–7]). In this model, the nearest-neighbor (NN)
spins S ¼ 1=2 interact via a simple Ising-type coupling
Sγ
i Sγ
j. However, the Ising axis γ is not global but bond
dependent, taking the mutually orthogonal directions
(x, y, z) on the three adjacent NN bonds on the honeycomb
lattice. Having no unique easy axis and being frustrated, the
Ising spins fail to order and form instead a highly entangled
quantum many-body state, supporting fractional excitations
described by Majorana fermions [1].
Much effort has been made to realize the Kitaev spin
liquid (SL) experimentally. From a materials perspective,
the Ising-type anisotropy is a hallmark of unquenched
orbital magnetism. As the orbitals are spatially anisotropic
and bond directional, they naturally lead to the desired
bond-dependent exchange anisotropy via spin-orbit coupling
[8]. Along these lines, 5d iridates have been suggested
[9] to host Kitaev model; later, 4d RuCl3 was added
[10] to the list of candidates. To date, however, the Kitaev
SL remains elusive, as this state is fragile and destroyed by
various perturbations, such as small admixture of a conventional
Heisenberg coupling [11] caused by direct overlap
of the d orbitals. Even more detrimental to Kitaev SL are
the longer range couplings [12], unavoidable in weakly
localized 5d- and 4d-electron systems with the spatially
extended d wave functions. We thus turn to 3d systems
with more compact d orbitals [13].
While the idea of extending the search area to 3d materials
is appealing, and plausible theoretically [15,16], it raises an
immediate question crucial for experiment: is spin-orbit
coupling (SOC) in 3d ions strong enough to support the
orbital magnetism prerequisite for the Kitaev model design?
This is a serious concern, since noncubic crystal fields
present in real materials tend to quench orbital moments and
suppress the bond dependence of the exchange couplings
[8]. In this Letter, we give a positive answer to this question.
Our quantitative analysis of the crystal field effects on the
magnetism of 3d cobaltates shows that the orbital moments
remain active and generate a Kitaev model as the leading
term in the Hamiltonian. In fact, we identify the trigonal
crystal field as the key and experimentally tunable parameter,
which decides the strength of the non-Kitaev terms in
3d compounds.
Our main results are summarized in Fig. 1, displaying
various magnetic phases of spin-orbit entangled pseudospin-
1=2 Co2þ
ions on a honeycomb lattice. The phase diagram is
shown as a function of trigonal field Δ, in a window relevant
for honeycomb cobaltates, and a ratio of Coulomb repulsion
U and the charge-transfer gap Δpd [17]. From the analysis of
experimental data, we find that Na3Co2SbO6 [18–20] is
located atjust∼20 meV“distance” fromtheKitaevSLphase
(see Fig. 1), and could be driven there by a c-axis compression
that reduces Δ. This seems feasible, given that Δ
variations within a window of ∼70 meV were achieved
by strain control in a cobalt oxide [21].
We now describe our calculations resulting in Fig. 1.
In short, we first derive the pseudospin exchange interactions
from a microscopic theory, as a function of various
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
and DOI. Open access publication funded by the Max Planck
Society.
PHYSICAL REVIEW LETTERS 125, 047201 (2020)
0031-9007=20=125(4)=047201(6) 047201-1 Published by the American Physical Society
185
parameters, and then obtain the corresponding ground
states numerically by exact diagonalization.
Exchange interactions.—In an octahedral environment,
Co2þ
ion with t5
2ge2
g configuration possesses spin S ¼ 3=2
and effective orbital moment L ¼ 1, which form, via spinorbit
coupling, a pseudospin ˜S ¼ 1=2 [14]. Over decades,
cobaltates served as a paradigm for quantum magnetism,
providing a variety of pseudospin-1=2 models ranging from
the Heisenberg model in perovskites with corner-sharing
octahedra [22,23] to the Ising model when the CoO6
octahedra share their edges [24].
A microscopic theory of Co2þ
interactions in the edgesharing
geometry has been developed just recently [15,16],
assuming an ideal cubic symmetry. Here we consider a
realistic case of trigonally distorted lattices, where t2g
orbitals split as shown in Fig. 2(a). Our goal is to see if
such distortions leave enough room for the Kitaev model
physics in real compounds. This is decided by the spinorbital
structure of the pseudospin ˜S ¼ 1=2 wave functions;
in terms of jSZ; LZi states (the trigonal axis Zkc is
perpendicular to the honeycomb plane), they read as:



 Æ
˜1
2

¼ C1



 Æ
3
2
; ∓ 1

þ C2



 Æ
1
2
; 0

þ C3



 ∓
1
2
; Æ1

:
ð1Þ
The coefficients C1;2;3 depend on a relative strength Δ=λ of
the trigonal field ΔðL2
Z − 2
3Þ and SOC λL · S [25,26]. At
Δ ¼ 0, one has ðC1; C2; C3Þ ¼ ð1=
ffiffiffi
2
p
; −1=
ffiffiffi
3
p
; 1=
ffiffiffi
6
p
Þ, and
all the three components of L are equally active. A positive
(negative) Δ field tends to quench LZ (LX=Y).
The next step is to project various spin-orbital exchange
interactions in cobaltates [15] onto the above pseudospin-
1=2 subspace. The calculations are standard but very
lengthy; the readers interested in details are referred to
the Supplemental Material [26]. At the end, we obtain the
˜S ¼ 1=2 Kitaev model K ˜Sγ
i
˜Sγ
j, supplemented by Heisenberg
J and off-diagonal anisotropy Γ, Γ0
terms; for γ ¼ z type NN
bonds, they read as:
H
ðzÞ
ij ¼ K˜Sz
i
˜Sz
j þ J˜Si · ˜Sj þ Γð˜Sx
i
˜Sy
j þ ˜Sy
i
˜Sx
jÞ
þ Γ0
ð˜Sx
i
˜Sz
j þ ˜Sz
i
˜Sx
j þ ˜Sy
i
˜Sz
j þ ˜Sz
i
˜Sy
jÞ: ð2Þ
InteractionsH
ðγÞ
ij for γ ¼ x,ytype bondsfollow froma cyclic
permutation among ˜Sx
j, ˜Sy
j, and ˜Sz
j.
While the Hamiltonian (2) is of the same form as in d5
Ir=Ru systems [5,34], the microscopic origin of its parameters
K; J; Γ; Γ0
is completely different in d7
Co compounds.
This is due to the spin-active eg electrons of
Coðt5
2ge2
gÞ ions, which generate new spin-orbital exchange
U / Δpd
Δ(meV)
FIG. 1. The calculated magnetic phase diagram of honeycomb
cobaltates. The Kitaev SL phase is surrounded by ferromagnetic
(FM) states with moments in the honeycomb ab plane and along
the c axis, zigzag-type states with moments in the ab plane (zz1),
along Co-O bonds (zz2), and in the ac plane (zz3). Vortex- and
stripy-type phases take over at smaller U=Δpd. The color map
shows the second-NN spin correlation strength (leading eigenvalue
of the correlation matrix h˜Sα
i
˜Sβ
j i normalized by ˜S2
¼ 1=4),
which drops sharply in the SL phase. The star indicates the rough
position of Na3Co2SbO6.
t2U
(c) K (d) J
(e) Г (f) Г
t2g-eg
eg-eg
t2g-t2g
Δ / Δ /
eg-eg
t2g-eg
t2g-t2g
eg-eg
t2g-eg
t2g-t2g
t2g-t2g
t2g-t2g
eg-eg
total
t2g-eg
Δ
cubic trigonal
t2g
(a)
t2g t2g
eg eg
(b)
FIG. 2. (a) Splitting of t2g-electron level under trigonal crystal
field. (b) Schematic of the spin-orbital exchange channels for d7
ions. (c)–(f) Exchange parameters K, J, Γ, and Γ0
(red solid lines)
as a function of Δ=λ, calculated at U=Δpd ¼ 2.5 and Hund’s
coupling JH ¼ 0.15U. On each panel, dashed lines show individual
contributions of t2g-t2g (black), t2g-eg (blue), and eg-eg
(green) exchange channels. The couplings J, Γ, and Γ0
nearly
vanish in the cubic limit Δ ¼ 0.
PHYSICAL REVIEW LETTERS 125, 047201 (2020)
047201-2
186
channels t2g-eg and eg-eg, shown in Fig. 2(b), in addition to
the t2g-t2g ones operating in d5 systems with t2g-only
electrons. In fact, the new terms make a major contribution
to the exchange parameters, as illustrated in Figs. 2(c)–2(f).
In particular, Kitaev coupling K comes almost entirely
from the t2g-eg process. It is also noticed that t2g-eg and
eg-eg contributions to J, Γ, and Γ0
are of opposite signs and
largely cancel each other, resulting in only small overall
values of these couplings.
Figure 2 shows that the trigonal field Δ, which acts via
modification of the pseudospin wave function (1), has an
especially strong impact on the non-Kitaev couplings J, Γ,
Γ0
. As a result, the relative strength (J=K, etc.) of these
“undesired” terms is very sensitive to Δ variations. This
suggests the orbital splitting Δ as an efficient (and
experimentally accessible) parameter that controls the
proximity of cobaltates to the Kitaev-model regime.
Another important parameter in the theory is the U=Δpd
ratio. In contrast to Ir=Ru-based Mott insulators with small
U=Δpd ∼ 0.5, cobaltates are charge-transfer insulators [17],
with typical values of U=Δpd ∼ 2–3 depending on the
material chemistry. Including both Mott-Hubbard U and
charge-transfer Δpd excitations, we have calculated [26]
the exchange couplings as a function of U=Δpd and Δ=λ.
Figure 3(a) shows that Kitaev coupling K is not much
sensitive to U=Δpd variations. On the other hand, the nonKitaev
terms, especially Heisenberg coupling J, are quite
sensitive to U=Δpd, see Figs. 3(b)–3(d). However, their
values relative to K remain small over a broad range of
parameters.
Phase diagram.—Having quantified the exchange
parameters in Hamiltonian (2), we are now ready to address
the corresponding ground states. As Kitaev coupling is the
leading term, the model is highly frustrated. We therefore
employ exact diagonalization (ED) which has been widely
used to study phase behavior of the extended KitaevHeisenberg
models (see, e.g., Refs. [11,35–39]). In particular,
by utilizing the method of coherent spin states
[38,39], we can detect and identify the magnetically
ordered phases (including easy-axis directions for the
ordered moments). When non-Kitaev couplings are small
(roughly below 10% of the FM K value), a quantum spinliquid
state is expected. Reflecting the unique feature of the
Kitaev model [1], this state is characterized by short-range
spin correlations that are vanishingly small beyond nearestneighbors
[11].
The resulting phase diagram, along with the data
quantifying spin correlations beyond NN distances, is
presented in Figs. 3(e) and 3(f). The main trends in the
phase map are easy to understand considering the variations
of non-Kitaev couplings with Δ=λ and U=Δpd. As we see
in Figs. 3(c) and 3(d), Γ0
exactly vanishes at the Δ ¼ 0 line,
and Γ is very small too. Thus, in the cubic limit, the model
(2) essentially becomes the well-studied K − J model, with
large FM Kitaev K term, and J correction changing
from AF J > 0 to FM J < 0 as a function of U=Δpd.
Consequently, the ground state changes from stripy AF (at
small U=Δpd) to FM order at large U=Δpd, through the
Kitaev SL phase in between [35]. In the SL phase, spin
correlations are indeed short-ranged and bond-selective: for
z-type NN bonds, we find h˜Sz ˜Sz
i=˜S2
≃ 0.52 (as in the
Kitaev model), while they nearly vanish at farther distances,
see Figs. 3(e) and 3(f).
As we switch on the trigonal field Δ, the Γ0
term comes
into play confining the SL phase to the window of jΔj=λ <
1 (where jΓ0=Kj < 0.1). In the FM phases, the sign of Γ0
decides the direction of the FM moments. On the left-top
(left-bottom) part of the phase map, where Heisenberg
coupling J is AF, the stripy state gives way to a vortex-type
[34] (zigzag-type) ordering, stabilized by the combined
effect of Γ and Γ0
terms.
0
0.1
0.2
-0.1
-0.2
0
0
-0.05
-0.05
0.05
/
(c) Г/|K| (d) Г /|K|
-3.6
U / Δpd
/ (e)
U / Δpd
(f)
SL
0
0.2 -0.2
(b) J/|K|
-4
-4
-3.6
-3.6
/
(a) K
SL
2nd-NN 3rd-NN
FIG. 3. (a) Kitaev coupling K (in units of t2
=U), and (b)–(d) the
relative values of J=jKj, Γ=jKj, and Γ0
=jKj as a function of Δ=λ
and U=Δpd. For convenience, specific values of parameters are
indicated by contour lines. (e)–(f) The corresponding phase
diagram obtained by ED of the model on a hexagon-shaped
24-site cluster. As in Fig. 1, the color maps quantify the strength
of (e) second-NN and (f) third-NN spin correlations, which drop
sharply in the SL phase (small but finite values are due to
deviations from the pure Kitaev model [11]).
PHYSICAL REVIEW LETTERS 125, 047201 (2020)
047201-3
187
To summarize up to now, the nearest-neighbor pseudospin
Hamiltonian is dominated by the FM Kitaev model,
which appears to be robust against trigonal splitting of
orbitals. Subleading terms, represented mostly by J and Γ0
couplings, shape the phase diagram, which includes a
sizeable SL area. While these observations are encouraging,
it is crucial to inspect how the picture is modified by
longer range interactions, especially by the third-NN
Heisenberg coupling J3
˜Si · ˜Sj, which appears to be one
of the major obstacles on the way to a Kitaev SL in 5d and
4d compounds [5,12]. We have no reliable estimate for J3,
since long-range interactions involve multiple exchange
channels and are thus sensitive to material chemistry
details. As such, they have to be determined experimentally.
We note that jJ3=Kj ≃ 0.1 was estimated [40,41] in
the 4d compound RuCl3; in cobaltates with more localized
3d orbitals [13], this ratio is expected to be smaller.
Adding a J3 term to the model (2), we have reexamined
the ground states and found that the Kitaev SL phase is
stable up to jJ3=Kj ∼ 0.06 [26]. The modified phase diagram,
obtained for a representative value of J3 ¼ 0.15t2
=
U ≃ 0.04jKj, is shown in Fig. 1 [42]. Its comparison with
Fig. 3 tells that the main effect of J3 is to support the
zigzag-type states (with different orientation of moments)
at the expense of other phases. Note also that the SL area is
shifted to the right, where FM J and AF J3 tend to frustrate
each other. The phase diagram in Fig. 1 should be generic
to Co2þ
honeycomb systems, and will be used in the
following discussion.
Honeycomb lattice cobaltates.—A number of such
compounds are known: A3Co2SbO6 (A ¼ Na; Ag; Li)
[18–20,43,44], Na2Co2TeO6 [18,45–47], BaCo2ðXO4Þ2
(X ¼ As, P) [48–51], CoTiO3 [52–54], CoPS3 [55,56].
They are quasi-two-dimensional magnets; within the ab
planes, zigzag or FM order is most common.
Traditionally, experimental data in Co2þ
compounds is
analyzed in terms of an effective ˜S ¼ 1=2 models of XXZ
type [48,50,54,57–59]. As ˜S ¼ 1=2 magnons (∼10 meV)
are well separated from higher lying spin-orbit excitations
(∼30 meV), the pseudospin picture itself is well justified;
however, a conventional XXZ model neglects the bonddirectional
nature of pseudospin ˜S ¼ 1=2 interactions. A
general message of our work is that a proper description of
magnetism in cobaltates should be based on the model of
Eq. (2), supplemented by longer-range interactions. We
note in passing that the XXZ model also follows from
Eq. (2) when the Kitaev-type anisotropy is suppressed [34];
however, such an extreme limit is unlikely for realistic
trigonal fields, given the robustness of the K coupling,
see Fig. 3.
As an example, we consider Na3Co2SbO6 which has low
N´eel temperature and a reduced ordered moment [20].
Analyzing the magnetic susceptibility data [20] including
all spin-orbit levels [26], we obtain a positive trigonal field
Δ ≃ 38 meV and λ ≃ 28 meV; these values are typical for
Co2þ ions in an octahedral environment (see, e.g.,
Ref. [54]). With Δ=λ ≃ 1.36, we evaluate ˜S ¼ 1=2 doublet
g factors gab ≃ 4.6 and gc ≃ 3, from which a saturated
moment of 2.3μB, consistent with the magnetization data
[20], follows.
Zigzag-ordered moments in Na3Co2SbO6 are confined
to the ab plane [20]; this corresponds to the zz1 phase in
Fig. 1. The easy-plane anisotropy is due to the Γ0 term,
which is positive for Δ > 0, see Fig. 3(d). Regarding the
location of Na3Co2SbO6 on the U=Δpd axis of Fig. 1, we
believe it is close to the FM==ab phase, based on the
following observations. First, a sister compound
Li3Co2SbO6 has ab-plane FM order [44] (most likely
due to smaller Co-O-Co bond angle, 91° versus 93°,
slightly enhancing the FM J value). Second, zigzag order
gives way to fully polarized state at small magnetic fields
[18,20]. These facts imply that zz1 and FM==ab states are
closely competing in Na3Co2SbO6.
Based on the above considerations, we roughly locate
Na3Co2SbO6 in the phase diagram as shown in Fig. 1. In
this parameter area, the exchange couplings are K≃
−3.6t2
=U, J=jKj ∼ −0.14, Γ=jKj ∼ −0.03, and Γ0
=jKj∼
0.16, see Figs. 3(a)–3(d). The small values of J; Γ; Γ0 imply
the proximity to the Kitaev model, explaining a strong
reduction of the ordered moments from their saturated values
[20]. As a crucial test for our theory, we show in Fig. 4 the
expected spin excitations. The large FM Kitaev interaction
enhances magnon spectral weight near q ¼ 0 and leads to
its anisotropy in momentum space, see Figs. 4(a) and 4(b).
Zq(t2
/U)
0
1
2
3
4
5
Y * K X
Z(t2
/U)
0
2
4
6
Y * K X
(a)
spectral weight
(q,Z)
LSW
ED
XK*
Y
0
max(b)
(c)
4223
FIG. 4. Spin excitation spectrum expected in Na3Co2SbO6. The
parameters K ¼ −3.6, J ¼ −0.5, Γ ¼ −0.1, Γ0
¼ 0.6 (in units of
t2
=U) follow from our theory, while J3 ¼ 0.15 is added “by
hand” [63] to stabilize the zigzag order. (a) Magnon dispersions
and intensities from linear spin wave (LSW) theory. (b) The
energy-integrated magnon intensity over the Brillouin zone. The
intensity is largest around Γ, i.e., away from the Bragg point Y.
(c) Exact diagonalization results for hexagonal 24- and 32-site
clusters. Plotted is the trace χ00
ðq; ωÞ of the spin susceptibility
tensor [26], which comprises the low-energy magnon peak and a
broad continuum.
PHYSICAL REVIEW LETTERS 125, 047201 (2020)
047201-4
188
The ED results in Fig. 4(c) show that, as a consequence of the
dominant Kitaev coupling, magnons are strongly renormalized
and only survive at low energies, and a broad continuum
of excitations [41,60] as in RuCl3 [61,62] emerges. Neutron
scattering experiments on Na3Co2SbO6 are desired to verify
these predictions.
If the above picture is confirmed by experiments, the
next step should be to drive Na3Co2SbO6 into the Kitaev
SL state. As suggested by Fig. 1, this requires a reduction of
the trigonal field by ∼20 meV, e.g., by means of strain or
pressure control. At this point, the relative smallness of
SOC for 3d Co ions comes as a great advantage: while
strong enough to form the pseudospin moments, it makes
the lattice manipulation of the ˜S ¼ 1=2 wave functions (and
hence magnetism) far easier than in iridates [64].
Monitoring the magnetic behavior of Na3Co2SbO6 and
other honeycomb cobaltates under uniaxial pressure would
be thus very interesting.
To conclude, we have presented a comprehensive theory
of exchange interactions in honeycomb cobaltates, and
studied their magnetic phase behavior. The analysis of
Na3Co2SbO6 data suggests that this compound is proximate
to a Kitaev SL phase and could be driven there by a caxis
compression. A broader message is that as one goes
from 5d Ir to 4d Ru and further to 3d Co, magnetic d
orbitals become more localized, and this should improve
the conditions for realization of the nearest-neighbor-only
interaction model designed by Kitaev.
We thank A. Yaresko, T. Takayama, and A. Smerald for
discussions, and M. Songvilay for sharing unpublished
data. G. Kh. acknowledges support by the European
Research Council under Advanced Grant No. 669550
(Com4Com). J. Ch. acknowledges support by Czech
Science Foundation (GAČR) under Project No. GA19-
16937S and MŠMT ČR under NPU II Project No. CEITEC
2020 (LQ1601). Computational resources were supplied
by the Project “e-Infrastruktura CZ” (No. e-INFRA
LM2018140) provided within the program Projects of
Large Research, Development and Innovations
Infrastructures.
[1] A. Kitaev, Ann. Phys. (N.Y.) 321, 2 (2006).
[2] L. Savary and L. Balents, Rep. Prog. Phys. 80, 016502
(2017).
[3] M. Hermanns, I. Kimchi, and J. Knolle, Annu. Rev.
Condens. Matter Phys. 9, 17 (2018).
[4] S. Trebst, arXiv:1701.07056.
[5] S. M. Winter, A. A. Tsirlin, M. Daghofer, J. van den Brink,
Y. Singh, P. Gegenwart, and R. Valentí, J. Phys. Condens.
Matter 29, 493002 (2017).
[6] H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E.
Nagler, Nat. Rev. Phys. 1, 264 (2019).
[7] Y. Motome and J. Nasu, J. Phys. Soc. Jpn. 89, 012002
(2020).
[8] G. Khaliullin, Prog. Theor. Phys. Suppl. 160, 155 (2005).
[9] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205
(2009).
[10] K. W. Plumb, J. P. Clancy, L. J. Sandilands, V. V. Shankar,
Y. F. Hu, K. S. Burch, H.-Y. Kee, and Y.-J. Kim, Phys. Rev.
B 90, 041112(R) (2014).
[11] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett.
105, 027204 (2010).
[12] S. M. Winter, Y. Li, H. O. Jeschke, and R. Valentí, Phys.
Rev. B 93, 214431 (2016).
[13] Cf. hr2
i3d ¼ 1.25 and hr2
i4d ¼ 2.31 for Co2þ
and Ru3þ
ions
(in a.u.), respectively [14].
[14] A. Abragam and B. Bleaney, Electron Paramagnetic
Resonance of Transition Ions (Clarendon Press, Oxford,
1970).
[15] H. Liu and G. Khaliullin, Phys. Rev. B 97, 014407 (2018).
[16] R. Sano, Y. Kato, and Y. Motome, Phys. Rev. B 97, 014408
(2018).
[17] J. Zaanen, G. A. Sawatzky, and J. W. Allen, Phys. Rev. Lett.
55, 418 (1985).
[18] L. Viciu, Q. Huang, E. Morosan, H. W. Zandbergen, N. I.
Greenbaum, T. McQueen, and R. J. Cava, J. Solid State
Chem. 180, 1060 (2007).
[19] C. Wong, M. Avdeev, and C. D. Ling, J. Solid State Chem.
243, 18 (2016).
[20] J.-Q. Yan, S. Okamoto, Y. Wu, Q. Zheng, H. D. Zhou, H. B.
Cao, and M. A. McGuire, Phys. Rev. Mater. 3, 074405
(2019).
[21] S. I. Csiszar, M. W. Haverkort, Z. Hu, A. Tanaka, H. H.
Hsieh, H.-J. Lin, C. T. Chen, T. Hibma, and L. H. Tjeng,
Phys. Rev. Lett. 95, 187205 (2005).
[22] T. M. Holden, W. J. L. Buyers, E. C. Svensson, R. A.
Cowley, M. T. Hutchings, D. Hukin, and R. W. H.
Stevenson, J. Phys. C 4, 2127 (1971).
[23] W. J. L. Buyers, T. M. Holden, E. C. Svensson, R. A.
Cowley, and M. T. Hutchings, J. Phys. C 4, 2139 (1971).
[24] R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska,
D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, and
K. Kiefer, Science 327, 177 (2010).
[25] M. E. Lines, Phys. Rev. 131, 546 (1963).
[26] See the Supplemental Material at http://link.aps.org/
supplemental/10.1103/PhysRevLett.125.047201 for Co2þ
ionic wave functions, computational details of the exchange
couplings and phase diagrams, spin excitation spectra, and
the analysis of experimental data in Na3Co2SbO6, which
includes Refs. [27–33].
[27] G. W. Pratt, Jr. and R. Coelho, Phys. Rev. 116, 281 (1959).
[28] V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev. B
44, 943 (1991).
[29] W. E. Pickett, S. C. Erwin, and E. C. Ethridge, Phys. Rev. B
58, 1201 (1998).
[30] H. Jiang, R. I. Gomez-Abal, P. Rinke, and M. Scheffler,
Phys. Rev. B 82, 045108 (2010).
[31] K. Foyevtsova, H. O. Jeschke, I. I. Mazin, D. I. Khomskii,
and R. Valentí, Phys. Rev. B 88, 035107 (2013).
[32] J. Chaloupka and G. Khaliullin, Prog. Theor. Phys. Suppl.
176, 50 (2008).
[33] S. H. Chun, J.-W. Kim, Jungho Kim, H. Zheng, C. C.
Stoumpos, C. D. Malliakas, J. F. Mitchell, K. Mehlawat,
Y. Singh, Y. Choi, T. Gog, A. Al-Zein, M. Moretti Sala,
PHYSICAL REVIEW LETTERS 125, 047201 (2020)
047201-5
189
M. Krisch, J. Chaloupka, G. Jackeli, G. Khaliullin, and B. J.
Kim, Nat. Phys. 11, 462 (2015).
[34] J. Chaloupka and G. Khaliullin, Phys. Rev. B 92, 024413
(2015).
[35] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett.
110, 097204 (2013).
[36] S. Okamoto, Phys. Rev. Lett. 110, 066403 (2013).
[37] J. G. Rau, Eric Kin-Ho Lee, and H.-Y. Kee, Phys. Rev. Lett.
112, 077204 (2014).
[38] J. Chaloupka and G. Khaliullin, Phys. Rev. B 94, 064435
(2016).
[39] J. Rusnačko, D. Gotfryd, and J. Chaloupka, Phys. Rev. B 99,
064425 (2019).
[40] S. M. Winter, K. Riedl, D. Kaib, R. Coldea, and R. R.
Valentí, Phys. Rev. Lett. 120, 077203 (2018).
[41] S. M. Winter, K. Riedl, P. A. Maksimov, A. L. Chernyshev,
A. Honecker, and R. R. Valentí, Nat. Commun. 8, 1152
(2017).
[42] In Fig. 1, the Δ=λ axis of Fig. 3 is replaced by Δ, using
λ ¼ 28 meV for Co2þ ion.
[43] E. A. Zvereva, M. I. Stratan, A. V. Ushakov, V. B.
Nalbandyan, I. L. Shukaev, A. V. Silhanek, M. AbdelHafiez,
S. V. Streltsov, and A. N. Vasiliev, Dalton Trans.
45, 7373 (2016).
[44] M. I. Stratan, I. L. Shukaev, T. M. Vasilchikova, A. N.
Vasiliev, A. N. Korshunov, A. I. Kurbakov, V. B.
Nalbandyan, and E. A. Zvereva, New J. Chem. 43, 13545
(2019).
[45] E. Lefrançois, M. Songvilay, J. Robert, G. Nataf, E.
Jordan, L. Chaix, C. V. Colin, P. Lejay, A. Hadj-Azzem,
R. Ballou, and V. Simonet, Phys. Rev. B 94, 214416
(2016).
[46] A. K. Bera, S. M. Yusuf, A. Kumar, and C. Ritter, Phys. Rev.
B 95, 094424 (2017).
[47] W. Yao and Y. Li, Phys. Rev. B 101, 085120 (2020).
[48] L. P. Regnault, C. Boullier, and J. Y. Henry, Physica
(Amsterdam) 385B, 425 (2006).
[49] R. Zhong, T. Gao, N. P. Ong, and R. J. Cava, Sci. Adv. 6,
eaay6953 (2020).
[50] H. S. Nair, J. M. Brown, E. Coldren, G. Hester, M. P.
Gelfand, A. Podlesnyak, Q. Huang, and K. A. Ross, Phys.
Rev. B 97, 134409 (2018).
[51] R. Zhong, M. Chung, T. Kong, L. T. Nguyen, S. Lei, and
R. J. Cava, Phys. Rev. B 98, 220407(R) (2018).
[52] R. E. Newnham, J. H. Fang, and R. P. Santoro, Acta Crystallogr.
17, 240 (1964).
[53] A. M. Balbashov, A. A. Mukhin, V. Y. Ivanov, L. D.
Iskhakova, and M. E. Voronchikhina, Low Temp. Phys.
43, 965 (2017).
[54] B. Yuan, I. Khait, G.-J. Shu, F. C. Chou, M. B. Stone, J. P.
Clancy, A. Paramekanti, and Y.-J. Kim, Phys. Rev. X 10,
011062 (2020).
[55] R. Brec, Solid State Ionics 22, 3 (1986).
[56] A. R. Wildes, V. Simonet, E. Ressouche, R. Ballou, and
G. J. McIntyre, J. Phys. Condens. Matter 29, 455801 (2017).
[57] M. T. Hutchings, J. Phys. C 6, 3143 (1973).
[58] K. Tomiyasu, M. K. Crawford, D. T. Adroja, P. Manuel,
A. Tominaga, S. Hara, H. Sato, T. Watanabe, S. I. Ikeda,
J. W. Lynn, K. Iwasa, and K. Yamada, Phys. Rev. B 84,
054405 (2011).
[59] K. A. Ross, J. M. Brown, R. J. Cava, J. W. Krizan, S. E.
Nagler, J. A. Rodriguez-Rivera, and M. B. Stone, Phys. Rev.
B 95, 144414 (2017).
[60] M. Gohlke, R. Verresen, R. Moessner, and F. Pollmann,
Phys. Rev. Lett. 119, 157203 (2017).
[61] A. Banerjee, P. Lampen-Kelley, J. Knolle, C. Balz, A. A.
Aczel, B. Winn, Y. Liu, D. Pajerowski, J. Yan, C. A.
Bridges, A. T. Savici, B. C. Chakoumakos, M. D. Lumsden,
D. A. Tennant, R. Moessner, D. G. Mandrus, and S. E.
Nagler, npj Quantum Mater. 3, 8 (2018).
[62] L. J. Sandilands, Y. Tian, K. W. Plumb, Y.-J. Kim, and K. S.
Burch, Phys. Rev. Lett. 114, 147201 (2015).
[63] In fact, the choice of J3 is dictated by the close proximity of
zz1 and FM==ab states in Na3Co2SbO6. Classically, they
differ by J − Γ þ 3J3; this gives a rough idea of J3 ∼ −J=3
(as Γ is very small).
[64] H. Liu and G. Khaliullin, Phys. Rev. Lett. 122, 057203
(2019).
PHYSICAL REVIEW LETTERS 125, 047201 (2020)
047201-6
190
Supplemental Material for
Kitaev Spin Liquid in 3d Transition Metal Compounds
Huimei Liu,1
Jiˇr´ı Chaloupka,2, 3
and Giniyat Khaliullin1
1
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
2
Department of Condensed Matter Physics, Faculty of Science,
Masaryk University, Kotl´aˇrsk´a 2, 61137 Brno, Czech Republic
3
Central European Institute of Technology, Masaryk University, Kamenice 753/5, 62500 Brno, Czech Republic
(Dated: May 6, 2020)
I. Single-ion wavefunctions
The d7
Co2+
ions in an octahedral crystal field have predominantly t5
2ge2
g configuration with a high spin S = 3/2 [1].
A trigonal distortion along Z-axis splits the t2g manifold into an orbital singlet a1g and a doublet eg by energy ∆, see
Fig. S1(a,b). In the electron representation, it is captured by the Hamiltonian H∆ = 1
3 ∆(2na1g
− neg
). In terms of
the effective angular momentum L = 1 of the Co2+
ions, the a1g-hole configuration corresponds to LZ = 0, while the
eg doublet hosts the LZ = ±1 states. Consequently, the trigonal field Hamiltonian translates into H∆ = ∆(L2
Z − 2
3 ).
The following relations between the L-states and orbitals hold:
|LZ = 0 =
1
√
3
(|a + |b + |c ) ,
|LZ = ±1 = ±
1
√
3
e±i
2π
3 |a + e i
2π
3 |b + |c , (S1)
where shorthand notations a = dyz, b = dzx, and c = dxy are used.
Diagonalization of H∆ = ∆(L2
Z − 2
3 ) and Hλ = λL · S results in a level structure shown in Fig. S1(c). The states
are labeled according to the total angular momentum Jeff = 1
2 , 3
2 , and 5
2 . The ground state Kramers doublet hosts a
pseudospin S = 1/2; its wavefunctions, written in the basis of |SZ, LZ , read as:
1
2
, ±
1
2
= C1 ±
3
2
, 1 + C2 ±
1
2
, 0 + C3
1
2
, ±1 . (S2)
The coefficients obey a relation C1 : C2 : C3 =
√
6
r1
: −1 :
√
8
r1+2 , where the parameter r1 > 0 is determined by the
equation ∆
λ = r1+3
2 − 3
r1
− 4
r1+2 [2]. The ground state energy is
EGS =
∆
3
−
λ
2
(r1 + 3). (S3)
The exchange Hamiltonian between the pseudospins S = 1/2 is obtained by projecting the Kugel-Khomskii type
spin-orbital Hamiltonians onto the ground state doublet (S2).
We also specify the excited states, which will be needed in Sec. IV to calculate the magnetic susceptibility. The
wavefunctions and energies for 3
2 , ±1
2 and 5
2 , ±1
2 states share the same form as of Eq. S2 and Eq. S3, but with
different r1. Namely, the above equation ∆
λ = r1+3
2 − 3
r1
− 4
r1+2 has three roots. The root r1 > 0 corresponds to
the ground state. The other two roots with −2 < r1 < 0 and r1 < −2 correspond to 3
2 , ±1
2 and 5
2 , ±1
2 states,
respectively. The wavefunctions and energies of the remaining states are:
3
2
, ±
3
2
= cϕ ±
3
2
, 0 − sϕ ±
1
2
, ±1 , E(3
2 , ±3
2 ) = −1
2 ∆ + 1
2 λ
2
+ 6λ2 + 1
4 λ − 1
6 ∆ ,
5
2
, ±
3
2
= sϕ ±
3
2
, 0 + cϕ ±
1
2
, ±1 , E(5
2 , ±3
2 ) = 1
2 ∆ + 1
2 λ
2
+ 6λ2 + 1
4 λ − 1
6 ∆ ,
5
2
, ±
5
2
= ±
3
2
, ±1 , E(5
2 , ±5
2 ) = 3
2 λ + 2
3 ∆ . (S4)
Here, cϕ = cos ϕ, sϕ = sin ϕ, and tan 2ϕ = 2
√
6λ/(2∆ + λ).
191
2
Co
(a)
X
Y
Z
x
y
z
eg(b)
d7
e'g
a1g
Δ
cubic trigonal
t2g
4
Δ/λ
0 1 2-3 -2
3
0
1
2
(c)
-1 3
5
6
(z)
(x) (y)
Δ/λ=1.36
FIG. S1: (a) Top view of the honeycomb cobaltate plane, x, y, and z type NN-bonds are shown in blue, green, and red colors,
respectively. The definition of global X, Y , Z and the local cubic x, y, z axes are shown in insets. (b) High-spin d7
(t5
2ge2
g)
configuration in the trigonal crystal field ∆. (c) Splitting of S = 3/2, L = 1 manifold of Co2+
ion under spin-orbit coupling λ
and trigonal field ∆. At ∆/λ = 1.36 (appropriate for Na3Co2SbO6), the first excited state energy is about λ ∼ 30 meV.
II. Pseudospin S = 1/2 Hamiltonian and calculation of its parameters
In the cubic reference frame, pseudospin-1/2 interactions on z-type bonds have a general form
H
(z)
ij = KSz
i Sz
j + JSi · Sj + Γ(Sx
i Sy
j + Sy
i Sx
j ) + Γ (Sx
i Sz
j + Sz
i Sx
j + Sy
i Sz
j + Sz
i Sy
j ) . (S5)
The interactions on x and y type bonds are obtained by cyclic permutations among Sx
j , Sy
j , and Sz
j .
The Hamiltonian in Eq. S5 can also be written in global XY Z reference frame [3]:
H
(γ)
ij =JXY SX
i SX
j + SY
i SY
j + JZSZ
i SZ
j
+A cγ SX
i SX
j − SY
i SY
j − sγ SX
i SY
j + SY
i SX
j
−B
√
2 cγ SX
i SZ
j + SZ
i SX
j + sγ SY
i SZ
j + SZ
i SY
j , (S6)
with cγ ≡ cos φγ and sγ ≡ sin φγ. The angles φγ = 0, 2π
3 , 4π
3 refer to the z, x, and y type bonds, respectively. The
transformations between the two sets of parameters entering Eq. S5 and Eq. S6 are:
JXY = J + 1
3 K − 1
3 (Γ + 2Γ ) , K = A + 2B ,
JZ = J + 1
3 K + 2
3 (Γ + 2Γ ) , J = 1
3 (2JXY + JZ − A − 2B) ,
A = 1
3 K + 2
3 (Γ − Γ ) , Γ = 2
3 (A − B) + 1
3 (JZ − JXY ) ,
B = 1
3 K − 1
3 (Γ − Γ ) , Γ = 1
3 (JZ − JXY + B − A) . (S7)
Since the pseudospin wavefunctions (S2) are defined in the trigonal XY Z basis, it is technically simpler to derive
S = 1/2 Hamiltonian in a form of (S6), and then convert the results onto a cubic xyz reference frame via Eqs. S7.
As discussed in the main text, there are three basic exchange channels in d7
systems, which we consider now in
detail. General form of the Kugel-Khomskii type spin-orbital Hamiltonians were obtained earlier [4]; for completeness,
they will be reproduced below. Here, the major task is to derive the corresponding pseudospin-1/2 Hamiltonians in a
realistic case of finite trigonal splitting of t2g orbitals. As the S = 1/2 wavefunctions (S2) are somewhat complicated,
the calculations are tedious but can still be done analytically.
192
3
1. t2g-t2g exchange contributions
1.1 Intersite U processes
The spin-orbital Hamiltonian for these exchange processes is given by equations (A2) and (3) of Ref. [4]:
H
(z)
11 =
4t2
9
1
E1
(Si · Sj + S2
)(a†
i bia†
jbj + b†
i aib†
jaj)
+
4t2
27
1
E3
+
2
E2
(Si · Sj + S2
)(nianjb + nibnja)
−
t2
6
1
E1
−
1
E2
(Si · Sj +S2
) (nia −njb)2
+(nib −nja)2
−
4t2
27
1
E2
−
1
E3
(Si · Sj − S2
)(a†
i bib†
jaj + b†
i aia†
jbj)
+
t2
6
3
E1
+
1
E2
−
4
E3
(nianjb + nibnja)
−
4
9
tt
U
(Si · Sj + S2
) (a†
i cic†
jbj + c†
i aib†
jcj) + (a ↔ b)
+
4
9
t 2
U
(Si · Sj − S2
) nicnjc. (S8)
Here na = a†
a, etc. denote the orbital occupations, t is the hopping between a and b orbitals via ligand ions, t
is the direct overlap of c orbitals. The Mott-Hubbard excitation energies are E1 = U − 3JH, E2 = U + JH, and
E3 = U + 4JH, where U and JH are Coulomb repulsion and Hund’s coupling on Co2+
ions.
Now, we need to express various combinations of the spin and orbital operators above in terms of the pseudospins
S = 1/2 defined by Eq. S2. To this end, we have derived a general projection table, presented in subsection 4 below.
Using this table, we obtain the pseudospin Hamiltonian in the form of Eq. S6, with the following exchange constants:
JXY
11 =
4t2
27
3
E1
−
1
E2
+
1
E3
2u2
4 + 2u2
6 − 13
2 u2
5 +
t2
27
9
E1
−
1
E2
+
4
E3
2
9 u2
1 − u2
4 − 1
2 u2
5
−
2t2
9
1
E1
−
1
E2
u2
1 −
4
9
tt
U
4u2
6 − 2u2
4 + 13
2 u2
5 +
4
9
t 2
U
1
9 u2
1 + u2
4 + 1
2 u2
5 , (S9)
JZ
11 =
4t2
27
3
E1
−
1
E2
+
1
E3
2u2
7 + u2
3 − 3
8 (u2 − 1)2
+
t2
27
9
E1
−
1
E2
+
4
E3
2
9 u2
2 − 2u2
3
−
2t2
9
1
E1
−
1
E2
u2
2 −
4
9
tt
U
4u2
7 − u2
3 − 3
4 (u2 − 1)2
+
4
9
t 2
U
1
9 u2
2 + 2u2
3 ,
A11 = −
4t2
27
3
E1
−
1
E2
+
1
E3
4u4u6 + 13
2 u2
5 +
t2
27
9
E1
−
1
E2
+
4
E3
2
3 u1u4 + u2
5
−
t2
3
1
E1
−
1
E2
u1u4 −
4
9
tt
U
4u4u6 − 13u2
5 +
4
9
t 2
U
1
2 u2
5 − 2
3 u1u4 ,
B11 =
4t2
27
3
E1
−
1
E2
+
1
E3
u3(u6 − u4) −
u5
√
2
u7 − 9
4 u2 + 9
4 +
t2
27
9
E1
−
1
E2
+
4
E3
u3
1
3 u1 + 2u4 − 1
3
√
2
u2u5
−
t2
6
1
E1
−
1
E2
u1u3 − 1√
2
u2u5 +
4
9
tt
U
u3(2u4 + u6) −
u5
√
2
u7 − 9
4 u2 + 9
4 −
4
9
t 2
U
u3
1
3 u1 − u4 − 1
3
√
2
u2u5 .
Coefficients ui (i = 1, 2, . . . , 7) are given by Eqs. S33 below; they depend on the spatial shape of the pseudospin
wavefunctions (S2), and thus decide how the relative values of the pseudospin interactions vary as a function of
trigonal field ∆.
193
4
1.2 Charge-transfer processes
The spin-orbital Hamiltonian is (Eq. 9 in Ref. [4]):
H
(z)
12 =
4
9
t2
∆pd +
Up
2
(Si · Sj − S2
)(nianjb + nibnja) −
2
9
t2
Jp
H
(∆pd +
Up
2 )2
Si · Sj(nic + njc), (S10)
where ∆pd is charge-transfer gap. Up and Up = Up − 2Jp
H are the intra- and inter-orbital Coulomb repulsion of the
ligand p orbitals, respectively, and Jp
H is the Hund’s coupling.
Using the projection table of subsection 4, we find the exchange constants in the form of Eq. S6:
JXY
12 =
4
9
t2
∆pd +
Up
2
2
9 u2
1 − u2
4 − 1
2 u2
5 −
4
27
t2
Jp
H
(∆pd +
Up
2 )2
u2
1 ,
JZ
12 =
4
9
t2
∆pd +
Up
2
2
9 u2
2 − 2u2
3 −
4
27
t2
Jp
H
(∆pd +
Up
2 )2
u2
2 ,
A12 =
4
9
t2
∆pd +
Up
2
2
3 u1u4 + u2
5 +
4
9
t2
Jp
H
(∆pd +
Up
2 )2
u1u4 ,
B12 =
4
9
t2
∆pd +
Up
2
u3
1
3 u1 + 2u4 − 1
3
√
2
u2u5 +
2
9
t2
Jp
H
(∆pd +
Up
2 )2
u1u3 − 1√
2
u2u5 . (S11)
1.3 Cyclic exchange processes
The spin-orbital Hamiltonian is (Eq. 11 in Ref. [4]):
H
(z)
13 =
4
9
t2
∆pd
(Si · Sj + S2
)(a†
i bia†
jbj + b†
i aib†
jaj). (S12)
After projection, we obtain the exchange constants as:
JXY
13 =
4
9
t2
∆pd
2u2
4 + 2u2
6 − 13
2 u2
5 , JZ
13 =
4
9
t2
∆pd
2u2
7 + u2
3 − 3
8 (u2 − 1)2
,
A13 = −
4
9
t2
∆pd
4u4u6 + 13
2 u2
5 , B13 =
4
9
t2
∆pd
u3(u6 − u4) −
u5
√
2
u7 − 9
4 u2 + 9
4 . (S13)
The total contribution from t2g-t2g hopping channel to Eq. S6 is given by
JXY
1 = JXY
11 + JXY
12 + JXY
13 , JZ
1 = JZ
11 + JZ
12 + JZ
13 ,
A1 = A11 + A12 + A13 , B1 = B11 + B12 + B13 . (S14)
The corresponding K, J, Γ, and Γ values can be obtained using Eqs. S7.
2. t2g-eg exchange contributions
2.1 Intersite U processes
The corresponding spin-orbital exchange Hamiltonian is (Eq. A5 in Ref. [4]):
H
(z)
21 =
4α1
9
tte
U
(Si · Sj − S2
)(nic + njc) −
tte
6
∆e
∆pd
1
E1 + D
−
1
E2 + D
Si · Sj (2 − nic − njc). (S15)
Here, te = t2
pdσ/∆e, with tpdσ representing hopping between p and eg orbitals via the charge-transfer gap ∆e = ∆pd+D.
Parameter D is the splitting between t2g and eg levels. The constants α1 and 1/U are:
α1 = 1 −
D2
2∆pd∆e
∆pd + ∆e
U + 2JH
− 1 ,
1
U
=
1
6
2
E2 + D
+
1
E3 + D
+
3
U + 2JH − D
. (S16)
194
5
After projection onto pseudospin-1/2 doublet (S2), we get the exchange constants in the form of Eq. S6:
JXY
21 =
8α1
27
tte
U
−
2tte
9
∆e
∆pd
1
E1 + D
−
1
E2 + D
u2
1 ,
JZ
21 =
8α1
27
tte
U
−
2tte
9
∆e
∆pd
1
E1 + D
−
1
E2 + D
u2
2 ,
A21 = −
8α1
9
tte
U
+
tte
3
∆e
∆pd
1
E1 + D
−
1
E2 + D
u1u4 ,
B21 =
4α1
9
tte
U
+
tte
6
∆e
∆pd
1
E1 + D
−
1
E2 + D
u2u5
√
2
− u1u3 . (S17)
2.2 Charge-transfer processes
The spin-orbital Hamiltonian describing these processes is (Eq. 19 in Ref. [4]):
H
(z)
22 =
8α2
9
tte
∆pd +
Up
2
(Si · Sj − S2
)(nic + njc) −
2α3
9
tte Jp
H
(∆pd +
D+Up
2 )2
Si · Sj (2 − nic − njc), (S18)
where
α2 = 1 −
D
4(∆e +
Up
2 )
+
D Up
8∆pd(∆e +
Up
2 )
−
D
4∆e
, α3 =
(∆pd + ∆e)2
4∆pd∆e
. (S19)
The corresponding pseudospin exchange constants are:
JXY
22 =
16α2
27
tte
∆pd +
Up
2
−
8α3
27
tte Jp
H
(∆pd +
D+Up
2 )2
u2
1 ,
JZ
22 =
16α2
27
tte
∆pd +
Up
2
−
8α3
27
tte Jp
H
(∆pd +
D+Up
2 )2
u2
2 ,
A22 = −
16α2
9
tte
∆pd +
Up
2
+
4α3
9
tte Jp
H
(∆pd +
D+Up
2 )2
u1u4 ,
B22 =
8α2
9
tte
∆pd +
Up
2
+
2α3
9
tte Jp
H
(∆pd +
D+Up
2 )2
u2u5
√
2
− u1u3 . (S20)
2.3 Cyclic exchange processes
The corresponding spin-orbital Hamiltonian is (Eq. 22 in Ref. [4]):
H
(z)
23 = −
2α4
9
tte
∆pd
(Si · Sj + S2
)(nic + nic), (S21)
with α4 = 1 − 1
2
D
∆pd+D .
After projection onto pseudospin-1/2 doublet, we obtain:
JXY
23 = −
4α4
27
tte
∆pd
u2
1 , JZ
23 = −
4α4
27
tte
∆pd
u2
2 ,
A23 =
4α4
9
tte
∆pd
u1u4 , B23 = −
2α4
9
tte
∆pd
u2u5
√
2
− u1u3 . (S22)
The total contribution from t2g-eg exchange channel to Eq. S6 is given by
JXY
2 = JXY
21 + JXY
22 + JXY
23 , JZ
2 = JZ
21 + JZ
22 + JZ
23 ,
A2 = A21 + A22 + A23 , B2 = B21 + B22 + B23 . (S23)
195
6
3. eg-eg exchange contribution
The corresponding Hamiltonian is very simple (see Eq. 27 in Ref. [4]):
H
(z)
3 = −
4
9
t2
e Jp
H
(∆e +
Up
2 )2
Si · Sj. (S24)
Note that no orbital operators are involved in this interaction and thus it has no bond-dependence. This is because
eg doublet hosts two electrons with parallel spins, leaving no eg-orbital degeneracy. After projecting Eq. S24 onto
pseudospin subspace, we find
JXY
3 = −
4
9
t2
e Jp
H
(∆e +
Up
2 )2
u2
1 , JZ
3 = −
4
9
t2
e Jp
H
(∆e +
Up
2 )2
u2
2 , (S25)
while the bond-dependent terms A3 = B3 = 0. The latter implies that eg-eg interaction channel supports the
XXZ-type model. In the cubic reference frame, Eq. S5, this translates into K = 0 and Γ = Γ .
Total values of the exchange constants are obtained by summing up t2g-t2g, t2g-eg, and eg-eg contributions [Eqs. S14,
S23, and S25, respectively], and converted into K, J, Γ, and Γ using Eqs. S7.
4. Projection table
Calculating the matrix elements of spin-orbital operators within the pseudospin S = 1/2 doublet (S2), we obtain
the correspondence:
S+ = u1S+ , S− = u1S− , SZ = u2SZ , (S26)
S+na =
√
2u3ei
2π
3 SZ +
u1
3
S+ − u4ei
4π
3 S− ,
S−na =
√
2u3e−i
2π
3 SZ +
u1
3
S− − u4e−i
4π
3 S+ ,
SZna =
u2
3
SZ +
u5
2
(SX −
√
3SY ) , (S27)
S+nb =
√
2u3e−i
2π
3 SZ +
u1
3
S+ − u4e−i
4π
3 S− ,
S−nb =
√
2u3ei
2π
3 SZ +
u1
3
S− − u4ei
4π
3 S+ ,
SZnb =
u2
3
SZ +
u5
2
(SX +
√
3SY ) , (S28)
S+nc =
√
2u3SZ +
u1
3
S+ − u4S− ,
S−nc =
√
2u3SZ +
u1
3
S− − u4S+ ,
SZnc =
u2
3
SZ − u5SX , (S29)
a†
b = i
2
√
3
[(1 − u2)SZ − 6u5SX] , S+a†
b = −
u3
√
2
SZ + u6S+ − u4S− ,
S−a†
b = −
u3
√
2
SZ + u6S− − u4S+ , SZa†
b = u7SZ +
u5
2
SX , (S30)
b†
c = i
2
√
3
[(1 − u2)SZ − 3u5(
√
3SY − SX)] , S+b†
c =
u3
√
2
e−i
π
3 SZ + u6S+ − u4e−i
2π
3 S− ,
S−b†
c =
u3
√
2
ei
π
3 SZ + u6S− − u4ei
2π
3 S+ , SZb†
c = u7SZ −
u5
4
(SX −
√
3SY ) , (S31)
196
7
c†
a = i
2
√
3
[(1 − u2)SZ + 3u5(
√
3SY + SX)] , S+c†
a =
u3
√
2
ei
π
3 SZ + u6S+ − u4ei
2π
3 S− ,
S−c†
a =
u3
√
2
e−i
π
3 SZ + u6S− − u4e−i
2π
3 S+ , SZc†
a = u7SZ −
u5
4
(SX +
√
3SY ) , (S32)
The parameters ui (i = 1, 2, . . . , 7) are determined by the pseudospin wavefunction (S2) parameters C1,2,3 as:
u1 = 2
√
3C1C3 + 2C2
2 , u2 = 1 + 2(C2
1 − C2
3 ) , u3 = 2
√
2
3 C2C3 − 2
3 C1C2 , u4 = 2
3 C2
3 ,
u5 = 2
3 C2C3 , u6 = 2
3 C2
2 − 1√
3
C1C3 , u7 = 1
3 C2
2 + 1
6 C2
3 − 1
2 C2
1 . (S33)
In the cubic limit, where (C1, C2, C3) = ( 1√
2
, −1√
3
, 1√
6
), they are
u1 = u2 =
5
3
, u3 = u4 =
1
9
, u5 = −
√
2
9
, u6 =
1
18
, u7 = −
1
9
. (S34)
5. Microscopic parameters used in the calculations
Apart from an overall energy scale t2
/U, a number of microscopic parameters appeared in the above expressions
for exchange constants. Hund’s coupling JH ∼ 0.8 eV follows from optical data in CoO [5]; cubic splitting D for 3d
ions is of the order of 1.0 − 1.5 eV. With the ab initio estimates of U ∼ 5.0 − 7.8 eV [6–8], this gives JH/U ∼ 0.1 − 0.2
and D/U ∼ 0.13 − 0.30. Specifically, we set JH/U = 0.15 and D/U = 0.20. Hund’s coupling on oxygen is large,
Jp
H ∼ 1.2 − 1.6 eV [9], while Up is about ∼ 4 eV, so we use the representative values of Jp
H/Up = 0.3 and Up/U = 0.7.
We set a direct hopping t = 0.2t (i.e. smaller than in 5d/4d compounds [10]), but this value is nearly irrelevant
here since t2g-t2g exchange is of minor importance anyway, see Fig. 2 of the main text. A ratio tpdσ/tpdπ = 2 [11] is
used. Regarding ∆/λ and U/∆pd values, we vary them rather broadly, as they most sensitively control the exchange
interactions. With the above input parameters, we arrive at K, J, Γ, and Γ values presented in the main text. We
have verified that while variations of the input parameters result in some changes of the exchange constants, they do
not affect the overall picture and conclusions.
III. Exact diagonalization: Phase diagrams based on static correlations and coherent-state analysis
We consider the nearest-neighbor (NN) interaction model (Eq. 2 of the main text or S5 in the previous section),
supplemented by the third-NN Heisenberg exchange J3 that appears as the major one among the long-range interactions
in ab-initio studies [10]. In this section we show the full evolution of the phase diagram with the parameter J3
and also demonstrate the robustness of our picture with respect to variations of the Hund’s exchange JH. The data
presented here complements Fig. 1 and Fig. 3(e,f) of the main manuscript.
To determine the magnetic state, we have performed exact diagonalization using the values of exchange parameters
derived in Sec. II. Utilizing the Lanczos method, we have obtained exact ground states of the exchange Hamiltonian for
a symmetric, hexagon-shaped cluster containing 24 sites. Periodic boundary conditions were applied, corresponding
to a periodic tiling of an infinite lattice. Since the small cluster does not allow for spontaneous symmetry breaking,
we inspect its magnetic state by analyzing the static spin correlations and by employing the method of coherent spin
states introduced in Ref. [12].
We focus on real-space correlations that enable us to judge the extent of the Kitaev spin liquid phase which should
be characterized by vanishing correlations beyond nearest neighbors. By evaluating the static spin correlations in
momentum space, we would be able to detect the magnetically ordered states that show peaks at the characteristic
momenta of the particular ordering pattern. Here, however, it is favorable to utilize the method of coherent spin
states that provides a better access to the magnetic order encoded in the complex cluster wavefunction. In essence, it
constructs “classical” states (coherent spin states) with spins pointing in prescribed directions and identifies a “classical”
state having maximum overlap with the exact cluster ground state. Thanks to its full flexibility in the individual
spin directions, the method can precisely determine both collinear patterns as well as non-collinear ones. The “classical”
trial state is a product state of spins pointing in prescribed directions (in the sense of finding spin up with 100%
probability when measuring in that particular direction) and as such it excludes quantum fluctuations. The maximum
overlap is therefore a useful indicator of the amount of quantum fluctuations. For a fluctuation-free state and nondegenerate
cluster ground state, the corresponding probability reaches the value 1/(number of degenerate patterns).
197
8
(a) FM (c) stripy(b) zigzag (d) vortex
FIG. S2: Sketch of the magnetic structures for (a) FM, (b) zigzag, (c) stripy, and (d) vortex orders. Open and closed circles
represent opposite spin directions.
In contrast, Kitaev spin liquid is highly fluctuating and does not contain a pronounced “classical” state which leads
to a tiny maximum overlap (see [12, 13] for details).
Figures S3 and S4 show phase diagram data as functions of U/∆pd and ∆/λ for several values of J3. The static
correlations up to fourth NN presented in upper three rows of panels clearly localize the Kitaev spin liquid phase
spreading in the area with dominant K. It is surrounded by several phases with long-range correlations that are
identified by the method of coherent spin states. For J3 = 0, these include two types of FM orders with the magnetic
moments lying in the honeycomb plane and perpendicular to it, respectively, stripy phase, zigzag phase zz3, and
finally a vortex phase of the type depicted in Fig. S2.
The effect of nonzero antiferromagnetic J3 may be estimated by considering the correlations of third NN in the
individual phases. Strongly supported by J3 is the zigzag phase that is characterized by AF oriented spins on all
third-neighbor bonds. Similarly, a large suppression may be expected for FM and stripy phases that have FM aligned
third NN spins. The effect on the vortex phase is weak as each spin has one FM aligned third neighbor and two
third neighbors at an angle of 120◦
, leading to a cancellation of J3 in energy on classical level. Finally, in the Kitaev
spin liquid phase the third neighbors are not correlated at all, so that small J3 has a moderate negative impact when
trying to align them in AF fashion. The consequences of the above energetics are well visible in Figs. S3 and S4.
Once including nonzero J3, the Kitaev spin liquid phase slightly grows first, at the expense of FM and stripy phases.
At the same time, the Kitaev spin liquid phase is also being expelled from the bottom left corner by the expanding
zz3 phase. With increasing J3 between J3 = 0.05 and 0.15 in t2
/U units, two new zigzag phases zz1 and zz2 around
Kitaev SL are successively formed. Once J3 reaches 0.25t2
/U, the zigzag order quickly takes over, suppressing the
Kitaev SL phase completely.
In the large area covered by the zigzag order, various ratios and combinations of signs of the nearest-neighbor
interactions are realized. This is the origin of three distinct zigzag phases zz1, zz2, and zz3, differing in their moment
directions as seen in bottom panels of Figs. S3 and S4. Negative Γ and positive Γ found in zz1 phase space [see
Fig. 3(c,d) of the main text] lead to the ab-plane moment direction. The zz3 phase is characterized by opposite
signs of Γ and Γ interactions which stabilizes the zigzag order as in Na2IrO3 [12, 14]. Finally, in the zz2 phase, Γ
and Γ terms maintain only small values and moment directions pointing along cubic axes x, y, z are selected by
order-from-disorder mechanism [12].
To check the robustness of our picture, we have also performed the exact diagonalization for a different JH value.
The trends discussed above remain quite similar as demonstrated in Figs. S5 and S6 calculated for JH/U = 0.2.
Roughly speaking, when we increase the JH/U value, the whole scenario merely shifts to smaller U/∆pd region.
198
9
∆/λ
J3=0.00 J3=0.05 J3=0.10
2nd
NNcorrs.3rd
NNcorrs.4th
NNcorrs.P(FM)[%]P(stripy)[%]P(zigzag)[%]zigzagangletoh.p.
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
10-1
10
0
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
10-1
10
0
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
10-1
10
0∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
10
-1
100
∆/λ -1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0
10
20
30
40FM // ab
FM // c
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0
10
20
30
40FM // ab
FM // c
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0
10
20
30
40FM // ab
FM // c
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0
5
10
15
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0
5
10
15
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0
5
10
15
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0
1
2
3
4
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0
1
2
3
4
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0
1
2
3
4
∆/λ
U / ∆pd
zz3
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0°
10°
20°
30°
40°
50°
∆/λ
U / ∆pd
zz3
zz1
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0°
10°
20°
30°
40°
50°
∆/λ
U / ∆pd
zz3
zz1
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0°
10°
20°
30°
40°
50°
JH/U=0.15
FIG. S3: The first three rows present second-NN, third-NN and fourth-NN spin correlations. The color indicates the largest
absolute value among the eigenvalues of the 3 × 3 spin correlation matrix for the respective bond. It is normalized by the
maximum possible value of S2
= 0.25. The next three rows are the probability of FM, stripy, and zigzag classical states
contained in the cluster ground state as determined by the method of coherent spin states. The last row shows the angle
between the honeycomb plane (h.p.) and the magnetic moments for the zigzag order. JH /U = 0.15 is fixed and three columns
correspond to J3 = 0, J3 = 0.05, and J3 = 0.1 in units of t2
/U.
199
10
∆/λ
J3=0.15 J3=0.20 J3=0.25
2nd
NNcorrs.3rd
NNcorrs.4th
NNcorrs.P(FM)[%]P(stripy)[%]P(zigzag)[%]zigzagangletoh.p.
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
10-1
10
0
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
10-1
10
0
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
10-1
10
0
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0
10
20
30
40FM // ab
FM // c
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0
10
20
30
40FM // ab
FM // c
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0
10
20
30
40FM // ab
FM//c
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0
5
10
15
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0
5
10
15
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0
5
10
15
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0
2
4
6
8
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0
2
4
6
8
∆/λ
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0
2
4
6
8
∆/λ
U / ∆pd
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0°
10°
20°
30°
40°
50°
zz3
zz2
zz1
∆/λ
U / ∆pd
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0°
10°
20°
30°
40°
50°
zz3
zz2
zz1
∆/λ
U / ∆pd
-1.0
0.0
1.0
2.0
1.5 2.0 2.5 3.0 3.5
0°
10°
20°
30°
40°
50°
zz3
zz2
zz1
JH/U=0.15
FIG. S4: The same as in Fig. S3 for larger J3 values. The three columns correspond to J3 = 0.15, 0.20, and 0.25 (t2
/U).
200
11
∆/λ
J3=0.00 J3=0.05 J3=0.10
2nd
NNcorrs.3rd
NNcorrs.4th
NNcorrs.P(FM)[%]P(stripy)[%]P(zigzag)[%]zigzagangletoh.p.
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
10-1
10
0
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
10-1
10
0
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
10-1
10
0∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
10
-1
100
∆/λ -1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0
10
20
30
40FM // ab
FM // c
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0
10
20
30
40FM // ab
FM // c
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0
10
20
30
40FM // ab
FM // c
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0
2
4
6
8
10
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0
2
4
6
8
10
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0
2
4
6
8
10
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0
1
2
3
4
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0
1
2
3
4
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0
1
2
3
4
∆/λ
U / ∆pd
zz3
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0°
10°
20°
30°
40°
50°
∆/λ
U / ∆pd
zz3
zz1
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0°
10°
20°
30°
40°
50°
∆/λ
U / ∆pd
zz3
zz1
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0°
10°
20°
30°
40°
50°
JH/U=0.20
FIG. S5: The same as in Fig. S3 for a larger value JH /U = 0.2. The three columns correspond to J3 = 0, J3 = 0.05, and
J3 = 0.1 (t2
/U).
201
12
∆/λ
J3=0.15 J3=0.20 J3=0.25
2nd
NNcorrs.3rd
NNcorrs.4th
NNcorrs.P(FM)[%]P(stripy)[%]P(zigzag)[%]zigzagangletoh.p.
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
10-1
10
0
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
10-1
10
0
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
10-1
10
0
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
10
-1
100
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0
10
20
30
40FM // ab
FM // c
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0
10
20
30
40FM // ab
FM // c
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0
10
20
30
40FM // ab
FM // c
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0
2
4
6
8
10
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0
2
4
6
8
10
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0
2
4
6
8
10
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0
2
4
6
8
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0
2
4
6
8
∆/λ
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0
2
4
6
8
∆/λ
U / ∆pd
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0°
10°
20°
30°
40°
50°
zz3
zz2
zz1
∆/λ
U / ∆pd
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0°
10°
20°
30°
40°
50°
zz3
zz2
zz1
∆/λ
U / ∆pd
-1.0
0.0
1.0
2.0
1.0 1.5 2.0 2.5 3.0
0°
10°
20°
30°
40°
50°
zz3
zz2
zz1
JH/U=0.20
FIG. S6: The same as in Fig. S5 for larger J3 values. The three columns correspond to J3 = 0.15, 0.20, and 0.25 (t2
/U).
202
13
IV. Trigonal crystal field ∆ in Na3Co2SbO6
The parameter ∆ determines the effective magnetic moment values µα
eff (α = ab or c), and thus can be obtained
from paramagnetic susceptibility χα
(T). One has to keep in mind that extracting the moments from a standard
Curie-Weiss fit χ(T) = C/(T − Θ) + χ0 assumes that the excited levels are high in energy (as compared to kBT) and
hence thermally unpopulated. The Curie constant C is then indeed temperature independent, providing the ground
state g-factors and moments. For Co2+
ions, where the excited level at ∼ 30 meV is thermally activated already at
the room temperature, we have to use instead a general expression for a single-ion susceptibility:
χα
ion =
1
Z(T) n,m
e−βEn
− e−βEm
Em − En
(Mα
nm)2
. (S35)
Here, n and m run over all the 12 states (6 doublets in Fig. S1), with the wavefunctions and energies calculated
in Sec. I. The partition function Z(T) = n e−βEn
, and β = 1/kBT. Mα
nm = n|Mα|m is matrix element of the
magnetic moment operator M = (2S − 3
2 κL) (in units of Bohr magneton µB). We use the covalency reduction factor
κ = 0.8 typical for Co2+
ion [1]. χα
ion includes both the Curie and Van-Vleck contributions and depends on two
parameters, ∆ and λ.
We have fitted the data of Ref. [15] with χα
(T) = χα
ion + χα
0 , and obtained a fair agreement with experiment for
both χab
and χc
, using ∆ = 38 meV and λ = 28 meV, see Fig. S7(a,b). In particular, the characteristic changes in
the slopes of both 1/χab
and 1/χc
data are well reproduced by the calculations. In fact, this behavior is common for
layered cobaltates and deserves some discussion.
It is instructive to divide Eq. S35 into two parts, χα
ion = χα
1 +χα
2 , where χα
1 term accounts for the transitions within
S = 1/2 doublet. Using the wavefunctions (S2), we obtain
χα
1 = p1/2
(µα
eff )2
3kBT
. (S36)
The effective moments µα
eff = gα S(S + 1), with the S = 1/2 doublet g-factors given by
gab = 4
√
3C1C3 + 4C2
2 − 3
√
2κC2C3 ,
gc = (6 + 3κ)C2
1 + 2C2
2 − (2 + 3κ)C2
3 . (S37)
In Eq. S36, p1/2 = 2/Z(T) measures the occupation of the ground state. As the excited levels of Co2+
are relatively low,
the weight p1/2 of the Curie term, as well as Van-Vleck contribution χα
2 of the excited states depend on temperature.
The characteristic changes in the slopes of 1/χab
(1/χc
) around 200 K (100 K) originate from the interplay between
χ1(T) and χ2(T) which become of a similar order at these temperatures, see Fig. S7(c,d).
The g-factors (S37) are plotted in Fig. S7(g); with ∆ and λ values obtained above, we get gab 4.6 and gc 3.
This gives the in-plane saturated magnetic moment Mab = gabS = 2.3µB consistent with experiment [15].
Apparent deviations at low temperatures are due to short-range correlations between the pseudospins, which can
partially be accounted for in a molecular field approximation, i.e. replacing the Curie term χα
1 by χα
1 · T/(T − Θα).
The result is shown in Fig. S7(e,f). The paramagnetic Curie temperatures Θab = 17K and Θc = 6K are rather small
and anisotropic. We can evaluate Θ values using our theoretical exchange constants given in Fig. 4 caption of the
main text; the result is:
Θab = −3
4 J + J3 + 1
3 K − 1
3 (Γ + 2Γ ) 1.4 (t2
/U),
Θc = −3
4 J + J3 + 1
3 K + 2
3 (Γ + 2Γ ) 0.6 (t2
/U). (S38)
Curiously enough, this gives the Θ-anisotropy close to what we get from the susceptibility fits. This comparison also
suggests the energy scale of t2
/U ∼ 1 meV, setting thereby the magnon bandwidth of the order of 10 meV. The
relative smallness of t2
/U is due to large U and more localized nature of 3d orbitals.
It is worth to comment on a positive sign of ∆ > 0 in Na3Co2SbO6. Within a simple model only considering
contribution from O6 octahedron, which is slightly compressed along the c-axis [15], one would find a negative ∆ < 0
instead. However, this approximation is too crude in layered structures, where the non-cubic Madelung potential of
more distant ions has to be considered. In Na3Co2SbO6, we think that ∆ > 0 is due to a positive contribution of the
high-valence Sb5+
ions residing within the ab-plane. A c-axis compression would enhance a negative contribution of
the oxygen octahedra, reducing thereby a total value of the trigonal field ∆.
203
14
λ=28 meV
Δ=38 meV
χ(10-2emu/mol)
T (K)
c
ab
1/χ(mol/emu)
T (K)
c
ab
susceptibility(10-2emu/mol)
T (K)T (K)
(c) (d)(b)(a)
χ1 χ1
χ2 χ2
0
5
10
15
20
0 100 200 300 0 100 200 300 0 100 200 300 0 100 200 300
0
2
4
6
8
10
0
2
4
6
8
10
0
10
20
30
40
50
60
70
λ=28 meV
Δ=38 meV
Θab = 17 K
Θc = 6 K
χ(10-2emu/mol)
T (K)
c
ab
1/χ(mol/emu)
c
ab
T (K)
(f)(e)
0
5
10
15
20
0 100 200 300 0 100 200 300
0
10
20
30
40
50
60
70
Δ / λ
Δ / λ =1.36
gab
gc
210-1
0
1
2
3
4
5
6
(g)
H//ab H//c
FIG. S7: (a),(b) Temperature dependence of magnetic susceptibility χ and its inverse 1/χ in Na3Co2SbO6. Open circles
represent the experimental data extracted from Ref. [15], and solid lines are the fits using single-ion approximation χα
=
χα
ion + χα
0 , with χab
0 = −10−3
emu/mol and χc
0 = 1.5 × 10−3
emu/mol. (c),(d) Decomposition of single-ion susceptibility χα
ion
into pseudospin-1/2 χ1 and Van-Vleck χ2 contributions. (e),(f) The fitting results including the pseudospin interactions within
a molecular field approximation. Here, χab
0 = −1.5 × 10−3
emu/mol and χc
0 = 1.5 × 10−3
emu/mol. (g) The g-factors gab (red)
and gc (blue) as a function of ∆/λ. ∆/λ = 1.36 corresponds to Na3Co2SbO6.
V. Dynamical spin susceptibility
1. Linear spin wave theory
The dispersions and intensities of magnons presented in Fig. 4(a,b) of the main text were determined by standard
linear spin wave (LSW) theory. Zigzag pattern with FM x and y bonds was assumed, i.e. the zigzags are running along
the X direction in Fig. S1(a). By applying Holstein-Primakoff transformation, harmonic expansion, and Bogoliubov
transformation numerically, we have calculated diagonal components of the spin susceptibility tensor and evaluated
its trace that is plotted in Fig. 4(a,b), including artificial lorentzian broadening with FWHM of 0.4 in units of t2
/U.
2. Exact diagonalization
The dynamical spin susceptibility profiles presented in Fig. 4(c) of the main text were determined by exact diagonalization
(ED) using the hexagonal clusters with N = 24 and N = 32 sites shown in Fig. S8(a) and (b), respectively.
Utilizing Lanczos algorithm, we have obtained the exact cluster ground state |GS and calculated the dynamical spin
susceptibility tensor χαβ(q, ω) = i GS|[Sα
q (t), Sβ
−q(0)]|GS exp(iωt)θ(t)dt. Here Sα
q = R Sα
R exp(−iqR)/
√
N is the
Fourier component combining spin operators at cluster sites R. The accessible wavevectors q that are compatible with
periodic tiling of the honeycomb lattice by the clusters are depicted in Fig. S8(c). As in the case of the LSW theory, in
Fig. 4(c) we have plotted the imaginary part of the trace of the spin susceptibility tensor: χ (q, ω) = Im α χαα(q, ω).
204
15
The spectra were broadened by lorentzians with FWHM of 0.1 in units of t2
/U and the quasielastic peaks at momenta
corresponding to the zigzag Bragg points were removed.
χ″(q=Γ,ω)
ω (t2
/ U)
32
24
0
1
2
3
4
5
0 1 2 3 4 5
(a) (d)(b) (c)
continuum
N=24 N=32
24
32
Γ
Γ X
magnon peak
′′
FIG. S8: (a) 24-site cluster used in ED to obtain phase diagrams and spin susceptibility. (b) 32-site cluster used in ED
calculations of the spin susceptibility. (c) Wavevectors compatible with the periodic tiling of the honeycomb lattice by 24- and
32-site clusters. Inner dotted hexagon indicates the Brillouin zone of the honeycomb lattice, outer hexagon corresponds to the
Brillouin zone of the triangular lattice formed when adding sites at hexagon centers to the honeycomb lattice. (d) Imaginary
part of the trace of the spin susceptibility tensor at q = Γ = 0 calculated by ED for 24- and 32-site clusters. The values of model
parameters are the same as in Fig. 4 of the main text. The thick black bars show the positions and relative spectral weights of
the magnon peaks obtained within LSW theory. Note that the ED results for 24- and 32-site clusters are qualitatively similar
to each other.
Compared to the LSW approximation result, the ED profiles show highly renormalized magnons that only survive
at low energies, and broad continua of excitations that emerge as a consequence of the dominant Kitaev interactions.
In fact, the most spectral weight is taken by the continuum. This is illustrated in detail for the FM wavevector
q = Γ = 0 in Fig. S8(d) and can be seen in Fig. 4(c) of the main text for other wavevectors q as well. To properly
capture such broad continua, we have used 1000 Lanczos steps in the dynamical susceptibility evaluation.
Finally, we want to notice an important aspect that one has to keep in mind while comparing the above results
with the experimental data. Namely, the cluster ground state is fully symmetric and contains all degenerate ordering
patterns. In our case these correspond to the three possible zigzag directions that are represented with equal weights
for the hexagonal shape clusters. As a result, the dynamical spin susceptibility obtained via ED contains contributions
from all these zigzag patterns. In practice, this would correspond to the dynamical spin structure factor measured
on the twinned samples with three types of zigzag domains. On the other hand, the intensities calculated using the
LSW theory correspond to a single-domain crystal with one particular zigzag pattern.
[1] A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Ions (Clarendon Press, Oxford, 1970).
[2] M. E. Lines, Phys. Rev. 131, 546 (1963).
[3] J. Chaloupka and G. Khaliullin, Phys. Rev. B 92, 024413 (2015).
[4] H. Liu and G. Khaliullin, Phys. Rev. B 97, 014407 (2018).
[5] G. W. Pratt Jr. and R. Coelho, Phys. Rev. 116, 281 (1959).
[6] V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev. B 44, 943 (1991).
[7] W. E. Pickett, S. C. Erwin, and E. C. Ethridge, Phys. Rev. B 58, 1201 (1998).
[8] H. Jiang, R. I. Gomez-Abal, P. Rinke, and M. Scheffler, Phys. Rev. B 82, 045108 (2010).
[9] K. Foyevtsova, H. O. Jeschke, I. I. Mazin, D. I. Khomskii, and R. Valent´ı, Phys. Rev. B 88, 035107 (2013).
[10] S. M. Winter, Y. Li, H. O. Jeschke, and R. Valent´ı, Phys. Rev. B 93, 214431 (2016).
[11] J. Chaloupka and G. Khaliullin, Prog. Theor. Phys. Suppl. 176, 50 (2008).
[12] J. Chaloupka and G. Khaliullin, Phys. Rev. B 94, 064435 (2016).
[13] J. Rusnaˇcko, D. Gotfryd, and J. Chaloupka, Phys. Rev. B 99, 064425 (2019).
[14] S. H. Chun, J.-W. Kim, Jungho Kim, H. Zheng, C. C. Stoumpos, C. D. Malliakas, J. F. Mitchell, K. Mehlawat, Y. Singh,
Y. Choi, T. Gog, A. Al-Zein, M. Moretti Sala, M. Krisch, J. Chaloupka, G. Jackeli, G. Khaliullin, and B. J. Kim, Nature
Phys. 11, 462 (2015).
[15] J.-Q. Yan, S. Okamoto, Y. Wu, Q. Zheng, H. D. Zhou, H. B. Cao, and M. A. McGuire, Phys. Rev. Materials 3, 074405
(2019).
205
206
6 Papers addressing the singlet–triplet model
ˆ Spin-State Crossover Model for the Magnetism of Iron Pnictides
J. Chaloupka and G. Khaliullin
Physical Review Letters 110, 207205 (2013) DOI: 10.1103/PhysRevLett.110.207205
This paper represents our first attempt to employ the ideas of triplon condensation within
singlet-triplet model, here in the context of iron pnictides. The singlet and triplet states
correspond to low-spin and intermediate-spin states of 3d6
configuration of Fe2+
ions that
get dynamically mixed by superexchange processes. We have demonstrated that the unusual
features of the iron-pnictide magnetism can be explained by AF condensate with a large
quadrupolar component. Such a situation can be effectively described by a spin-1 system with
a significant biquadratic interaction.
ˆ Doping-Induced Ferromagnetism and Possible Triplet Pairing in d4
Mott Insulators
J. Chaloupka and G. Khaliullin
Physical Review Letters 116, 017203 (2016) DOI: 10.1103/PhysRevLett.116.017203
In this paper we study a doped variant of the singlet-triplet model for d4
systems where
large-enough spin orbit coupling forms J = 0 ionic ground state. We find that the doped
electrons strongly affect the triplon condensation supporting the FM order. Already a modest
amount of doping may convert the AF condensate into FM one or induce FM condensation
in the nonmagnetic phase. In the vicinity of the FM phase, the low-energy spin response
is dominated by intense paramagnons. We have explored a scenario where these excitations
serve as mediators of triplet pairing.
ˆ Higgs mode and its decay in a two-dimensional antiferromagnet
A. Jain, M. Krautloher, J. Porras, G. H. Ryu, D. P. Chen, D. L. Abernathy, J. T. Park,
A. Ivanov, J. Chaloupka, G. Khaliullin, B. Keimer, and B. J. Kim
Nature Physics 13, 633 (2017) DOI: 10.1038/NPHYS4077
This paper presents results of inelastic neutron scattering on Ca2RuO4 that are interpreted
along the lines of singlet-triplet model (more precisely, we use the refined model of Sec. 4.2). We
demonstrate the peculiar spectrum of magnetic excitations including the amplitude (“Higgs”)
mode associated with triplon condensate that appears in the longitudinal spin response, as
well as magnons with unusual dispersion having a maximum at q = 0. The interplay of the
amplitude mode with the two-magnon continuum is observed, leading to a broadening of the
longitudinal mode that largely depends on the position in the Brillouin zone.
207
208
ˆ Raman Scattering from Higgs Mode Oscillations in the Two-Dimensional
Antiferromagnet Ca2RuO4
S.-M. Souliou, J. Chaloupka, G. Khaliullin, G. Ryu, A. Jain, B. J. Kim, M. Le Tacon, and
B. Keimer
Physical Review Letters 119, 067201 (2017) DOI: 10.1103/PhysRevLett.119.067201
In this paper we demonstrate that the Raman scattering on Ca2RuO4 can be successfully
interpreted in terms of the magnetic model introduced in the above Ref. [101] to fit the
neutron data. We present the temperature-dependent Raman spectra in B1g and Ag channels,
identify their magnetic parts, and interpret the magnetic features in terms of magnons and the
amplitude mode. Most importantly, the Ag spectrum is shown to directly reveal the amplitude
mode without a parasitic excitation of the two-magnon continuum as in the longitudinal spin
response.
ˆ Highly frustrated magnetism in relativistic d4
Mott insulators:
Bosonic analog of the Kitaev honeycomb model
J. Chaloupka and G. Khaliullin
Physical Review B 100, 224413 (2019) DOI: 10.1103/PhysRevB.100.224413
The paper analyzes the singlet-triplet model for d4
compounds with the honeycomb lattice.
We focus on the limit of dominant direct hopping between t2g orbitals which leads to Kitaevlike
frustration of exchange interactions. The symmetry of the model is explored in detail, the
links to the Kitaev honeycomb model are discussed, and effective models emerging in various
parameter regimes are identified. We construct the overall phase diagram of the model based
on numerical simulations complemented by the above symmetry analysis and discuss possible
implications for the magnetism of honeycomb ruthenium oxides.
Spin-State Crossover Model for the Magnetism of Iron Pnictides
Jirˇı´ Chaloupka1,2
and Giniyat Khaliullin1
1
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
2
Central European Institute of Technology, Masaryk University, Kotla´rˇska´ 2, 61137 Brno, Czech Republic
(Received 18 March 2013; published 14 May 2013)
We propose a minimal model describing magnetic behavior of Fe-based superconductors. The key
ingredient of the model is a dynamical mixing of quasidegenerate spin states of Fe2þ
ion by intersite
electron hoppings, resulting in an effective local spin Seff. The moments Seff tend to form singlet pairs and
may condense into a spin nematic phase due to the emergent biquadratic exchange couplings. The longrange
ordered part m of Seff varies widely, 0 m Seff, but magnon spectra are universal and scale with
Seff, resolving the puzzle of large but fluctuating Fe moments. Unusual temperature dependences of a local
moment and spin susceptibility are also explained.
DOI: 10.1103/PhysRevLett.110.207205 PACS numbers: 75.10.Jm, 71.27.+a, 74.70.Xa
Since the discovery of superconductivity (SC) in doped
LaFeAsO [1], a number of Fe-based SCs have been found
and studied [2]. Evidence is mounting that quantum magnetism
is an essential part of the physics of Fe-based SCs.
However, the origin of magnetic moments and the mechanisms
that suppress their long-range order (LRO) in favor
of SC are still not well understood.
The magnetic behavior of Fe-based SCs is unusual. The
ordered moments range from 0:1–0:4B, as in spin-density
wave (SDW) metals like Cr, to 1–2B typical for Mott
insulators, causing debates whether the spin-Heisenberg
[3–8] or fermionic-SDW pictures [9–13] are more adequate.
At the same time, irrespective of the strength or
very presence of LRO, the Fe-ions possess the fluctuating
moments $1–2B [14,15], even in apparently ‘‘nonmagnetic’’
LiFeAs and FeSe. In fact, it was noticed early on
that the Fe-moments, ‘‘formed independently on fermiology’’
[16] and ‘‘present all the time,’’ [3] are instrumental
to reproduce the measured bond lengths and phonon spectra
[3,16–18]. Recent experiments [19–21] observe intense
high-energy spin waves that are almost independent of
doping, further supporting a notion of local moments
induced by Hund’s coupling [22] and coexisting [23–25]
with metallic bands.
While the formation of the local moments in multiorbital
systems is natural, it is puzzling that these moments (residing
on a simple square lattice) may remain quantum disordered
in a broad phase space despite a sizable interlayer
coupling; moreover, the Fe-pnictides are semimetals with
strong tendency of the electron-hole pairs to form SDW
state, further supporting classical LRO of the underlying
moments. A fragile nature of the magnetic-LRO in Fepnictides
thus implies the presence of a strong quantum
disorder effects, not captured by ab initio calculations that
invariably lead to magnetic order over an entire phase
diagram. The ideas of domain wall motion [17] and local
spin fluctuations [22] were proposed as a source of spin
disorder, but no clear and tractable model of quantum
magnetism in Fe-based SCs has emerged to date. Here
we propose such a model.
Since Fe pnictides are distinct among the other (Mn, Co,
Ni) families, their unique physics should be rooted in
specific features of the Fe ion itself. In fact, Fe2þ is famous
for its spin crossover [26]; it may adopt either of S ¼ 0, 1,
2 states depending on orbital splitting, covalency, and
Hund’s coupling. As the ionic radius of Fe is sensitive to
its spin, Fe-X bond length (X is a ligand) is also crucial. In
oxides, S ¼ 2 is typical and S ¼ 0,1 occur at high pressures
only [27]. In compounds with more covalent Fe-X
bonds (X ¼ S, As, Se), S ¼ 0 is more common while
S ¼ 1, 2 levels are higher. The basic idea of this Letter is
that when the covalency and Hund’s coupling effects compete,
the many-body ground state (GS) is a coherent
superposition of different spin states intermixed by electron
hoppings, resulting in an average effective spin Seff
whose length depends on pressure, etc. We explore this
dynamical spin-crossover idea, and find that (i) local
moment Seff may increase with temperature explaining
recent data [28], (ii) interactions between Seff contain large
biquadratic exchange, and resulting spin-nematic correlations
compete with magnetic LRO, (iii) the ordered
moment m varies widely, but magnon spectra are universal
and scale with Seff as observed [19,20,29], and (iv) singlet
correlations among Seff lead to the increase of the spin
susceptibility with temperature [30].
The Fe ions in pnictides have a formal valence state
Fe2þðd6Þ. Among its possible spin states [Fig. 1(a)], lowspin
ones are expected to be favored; otherwise, the
ordered moment would be too large and robust. The S ¼
0, 1 states, ‘‘zoomed-in’’ further in Fig. 1(b), are most
important since they can overlap in the many-body GS
by an exchange of just two electrons between ions;
see Fig. 1(c). The corresponding  process converts
FeðS ¼ 0Þ-FeðS ¼ 0Þ pair into FeðS ¼ 1Þ-FeðS ¼ 1Þ singlet
pair and vice versa; this requires the interorbital hopping
which is perfectly allowed for $109 Fe-As-Fe
PRL 110, 207205 (2013) P H Y S I C A L R E V I E W L E T T E R S
week ending
17 MAY 2013
0031-9007=13=110(20)=207205(5) 207205-1 Ó 2013 American Physical Society
209
bonding. Basically,  is a part of usual exchange process
when local Hilbert space includes different spin states
S ¼ 0,1; hence,  $ J. Coupling J between S ¼ 1 triplets
is contributed also by their indirect interaction via
the electron-hole Stoner continuum and, as expected, it
reduces with doping [31] as the electron-hole balance of
a parent semimetal becomes no longer perfect.
The Hamiltonian describing the above physics comprises
the following three terms: on-site energy ET of
S ¼ 1 triplet T relative to S ¼ 0 singlet s, and the bond
interactions , J,
H ¼ ET
X
i
nTi
þ
X
hiji
½ÀijðDy
ijsisj þ H:c:Þ þ JijSi Á SjŠ:
(1)
The operator Dy
ij creates a singlet pair of spin-full T
particles on bond hiji. For a general spin S of T particles,
Dij ¼
P
MðÀ1ÞMþSTi;þMTj;ÀM with M ¼ ÀS; . . . ; S
denoting the N ¼ 2S þ 1 projections; physically, N ¼ 3.
The constraint nsi þ nTi ¼ 1 is implied [32,33].
The above model rests on three specific features of Fe
pnictides or chalcogenides: (i) spin-state flexibility of Fe2þ
that can be tuned by pressure increasing ET, (ii) edgesharing
FeX4 tetrahedral structure allowing ‘‘spin-mixing’’
 term, and (iii) semimetallic nature which makes J values
to decrease upon doping [31].
Figures 1(d)–1(f) demonstrate the behavior of spin-1 T
particles (N ¼ 3) on a single bond. The GS wave function
jc GSi ¼ cosjAi þ sinjBi is a superposition of two singlets
A ¼ sy
1 sy
2 and B ¼ Àð1=
ffiffiffi
3
p
Þ
P
MðÀ1ÞM
Ty
1;MTy
2;ÀM,
with the ‘‘spin-mixing’’ angle tan2 ¼
ffiffiffi
3
p
=ðET À JÞ.
The GS energy EGS ¼ ðET À JÞ À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðET À JÞ2
þ 32
p
. At
 ¼ 0, there is a sudden jump [Fig. 1(e)] from S ¼ 0 state
to S ¼ 1 once the J-energy compensates the cost of having
two T particles. At finite , the dynamical mixing of spin
states converts this transition into a spin-crossover, where
the effective spin-length Seff ¼ nT ¼ sin2
 increases
gradually. Figure 1(f) shows that  term strongly stabilizes
the singlet pair of T particles; this leads (see later) to a large
biquadratic coupling ðS1 Á S2Þ2
which is essential in Fe
pnictides [31,34,35].
We are ready to show the model in action, explaining
recent observation of an unusual increase of the local
moment upon warming [28]. This fact is at odds with
Heisenberg and SDW pictures but easy to understand
within the spin-crossover model. Indeed, the spin-length
Seff may vary as a function of ET which, in turn, is sensitive
to lattice expansion; in fact, Gretarsson et al. found that the
moment value follows c-axis thermal expansion  ¼ c=c.
We add (magnetoelastic) coupling ÀAnT in Eq. (1),
affecting ET value, and evaluate  and hnTi selfconsistently.
This is done by minimizing the elastic energy
ð1=2ÞK2
À K0T þ ð1=4ÞQ4
(0 is the usual thermal
expansion coefficient), together with the GS energy EGS
given above. This results in a linear relation  ’ 0T þ
ðA=KÞhnTi between the magnetic moment (¼ 2nT)
and lattice expansion. They both strongly increase with
temperature if lattice is ‘‘soft’’ enough (i.e., small K), as
demonstrated in Figs. 1(g) and 1(h) by employing
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 100 200 300
effectivemoment[µB]
T [K]
0
1
2
3
4
0 100 200 300
ε(T)ε(0)[%]
T [K]
α0T
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0
effectivespin
J / ET
κ=0
κ=J/4
κ=J
T
(e)
(g) (h)
Jκ
1 2
κ
(c)
S=0 S=1 S=2 Fe (S=0)
2+ 2+
Fe (S=1)
(d)
(a) (b)
cosα sinαψGS
j ,−1Ti ,+1Tis js
−
-3
-2
-1
0
1
2
3
4
0.0 0.5 1.0 1.5 2.0
energy/ET
κ / ET
J=0.5ET
0
0
1
2
s T
)f(
GS
FIG. 1. (a) Schematic view of low (S ¼ 0), intermediate
(S ¼ 1), and high (S ¼ 2) spin states of Fe2þ
ð3d6
Þ. (b) S ¼ 0
and S ¼ 1 states differ in two electrons (out of six) occupying
either the same or two different t2g orbitals. The S ¼ 1 state has
a larger ionic radius. (c) The  process generating a singlet pair
of S ¼ 1 triplets T of two Fe2þ
ions, both originally in the S ¼ 0
state (denoted by s). (d) The GS wave function of a Fe2þ-Fe2þ
pair is a coherent superposition of two total-singlet states.
(e) Effective spin (average occupation of S ¼ 1 state) depending
on the ratio of the coupling J between S ¼ 1 states and their
energy ET. (f) Energy levels labeled by the total spin value of the
Fe2þ
-Fe2þ
pair. Only singlet pairs are affected by . With
increasing , the S ¼ 1 states are gradually mixed into the
GS. (g) Temperature dependence of the local magnetic moment
2nT, and (h) the c axis thermal expansion. Squares in (g,h)
represent experimental data on Ca0:78La0:22Fe2As2 [28].
Dashed line in (h) is a thermal expansion excluding magnetoelastic
term.
PRL 110, 207205 (2013) P H Y S I C A L R E V I E W L E T T E R S
week ending
17 MAY 2013
207205-2
210
the parameters ET À J ¼ 160 meV,  ¼ 60 meV,
A ¼ 1:5 eV, K ¼ 4:55 eV, Q ¼ 250 eV, and 0 ¼ 0:2 Â
10À4
KÀ1
, providing a good fit to the experimental data
of Ref. [28].
Turning to collective behavior of the model, we notice
first that for N ! 1 and large , the GS is dominated by
tightly bound singlet dimers derived from the single-bond
solution. The resonance of dimers on square-lattice plaquettes
then supports a columnar state [36] breaking lattice
symmetry without magnetic LRO [33]. In the opposite
limit of N ¼ 1, the model shows a condensation of T
bosons. We found that the N ¼ 3 model relevant here is
also unstable towards a condensation of T particles with
S ¼ 1. This condensate hosts interesting properties not
present in a conventional Heisenberg model. We discuss
them based on the following wave function describing
Gutzwiller-projected condensate of spin-1 T bosons:
jÉi ¼
Y
i
 ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À 
p
sy
i þ
ffiffiffiffi

p X
¼x;y;z
dÃ
iTy
i

jvaci; (2)
where  2 ½0; 1Š is the condensate density to be understood
as the effective spin-length Seff. The complex unit
vectors di ¼ ui þ ivi (u2
i þ v2
i ¼ 1) determine the spin
structure of the condensate in terms of the coherent states
of spin-1 [37,38] corresponding to Tx ¼ðTþ1 ÀTÀ1Þ=
ffiffiffi
2
p
i,
Ty ¼ ðTþ1 þ TÀ1Þ=
ffiffiffi
2
p
, Tz ¼ iT0. The GS phase diagram
obtained by minimizing hÉjH jÉi and cross-checked by
an exact diagonalization on a small cluster is presented in
Fig. 2. We have included nearest-neighbor (NN) and nextNN
interactions and fixed their ratio at J2=J1 ¼ 2=1 ¼
0:7, reflecting large next-NN overlap via As ions. Like in
J1 À J2 model, this ratio decides between (, ) and (, 0)
order. Figures 2(a) and 2(b) contain, apart from a disordered
(uncondensed) phase ( ¼ 0) at small , J, three
distinct phases depending on =ET and J=ET values.
(i) Ferroquadrupolar (FQ) phase with ui ¼ u and vi ¼ 0.
This phase has zero magnetization and is characterized by
the quadrupolar order parameter hS
S
À ð1=3ÞS2
i ¼
ðð1=3Þ À uuÞ with u playing the role of the director
[38]. This state, often referred to as ‘‘spin nematic,’’
appears in biquadratic exchange [37–40] and optical lattice
models [41–44]. (ii) Nonsaturated antiferromagnetic
(ns-AF) phase with stripy magnetic order, specified by
ui ¼ ð0; 0; uÞ and vi ¼ ð0; v; 0ÞeiQÁRi with Q ¼ ð; 0Þ.
The LRO moment hSi given by m ¼ 2uv can take values
from 0 to Seff ¼ . (iii) Saturated antiferromagnet
(AF) with the same Q vector, but now with u ¼ v ¼
1=
ffiffiffi
2
p
and m ¼ Seff ¼ 1.
The part of the phase diagram relevant to pnictides is
shown in Fig. 2(d). The decrease of J is associated with
doping that changes the nesting conditions [31], while
the increase of ET is related to external or chemical pressure.
Figures 2(e) and 2(f), shows that the LRO moment m
quickly vanishes as J (ET) values decrease (increase);
however, the spin-length Seff ¼  remains almost constant
($1=2), corresponding to a fluctuating magnetic
moment $1B. This quantum state is driven by the 
process, which generates the spin-1 states in a form of
singlet pairs.
We consider now the excitation spectrum. It is convenient
to separate fast (density) and slow (spin) fluctuations.
We introduce pseudospin  ¼ 1=2 indicating the presence
of a T particle, and a vector field d defining the spin-1
operator as S ¼ Àiðdy
 dÞ. The resulting Hamiltonian,
H ¼ET
X
i

1
2
Àz
i

À
X
hiji
ijðþ
i þ
j di Ádj þH:c:Þ
À
X
hiji
Jij

1
2
Àz
i

1
2
Àz
j

ðdy
i ÂdiÞÁðdy
j ÂdjÞ (3)
is decoupled on a mean-field level. The condensate spin
dynamics is then given by Oð3Þ-symmetric Hamiltonian
0
π/4
π/2
0 π/4 π/2
ϕ
ϑ
(a)
0 π/4 π/2
ϑ
(b)
0.0
0.2
0.4
0.6
0.8
1.0
FQ
ns-AF
AF
0
π/4
π/2
0 π/4 π/2
ϕ
ϑ
(c)
0
200
0.00.20.40.60.81.0
ET[meV]
J / J
(0)
(d)
A B C
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
FQ
ns-AF
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.00.20.40.60.81.0
J / J (0)
(e)
ρ
m
A
B C
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 50 100 150 200 250
ET [meV]
(f)
ρ
m
B
erusserpgnipod
FIG. 2 (color online). (a) Condensate density ð SeffÞ obtained
from Eq. (2) as a function of angles #, ’ which parametrize
the model (1) via ET ¼ cos#, 1 ¼ sin# cos’, and
J1 ¼ sin# sin’. We set 2=1 ¼ J2=J1 ¼ 0:7. (b) The ordered
spin moment value m. (c) T occupation per site nT obtained by
an exact diagonalization of 12-site cluster, to be compared with
 of panel (a). (d) The ordered moment m as a function of ET
and relative J-strength for fixed 1 ¼ 100 meV, 2 ¼ 0:71,
Jð0Þ
1 ¼ 140 meV, Jð0Þ
2 ¼ 0:7Jð0Þ
1 . (e,f) Effective spin length  ¼
Seff and ordered moment m at the (e) ET ¼ 100 meV and (f)
J=Jð0Þ
¼ 0:75 lines through the phase diagram in (d).
PRL 110, 207205 (2013) P H Y S I C A L R E V I E W L E T T E R S
week ending
17 MAY 2013
207205-3
211
H d ¼ À
X
hiji
~ijðdi Ádj þH:c:ÞÀ
X
hiji
~Jijðdy
i ÂdiÞÁðdy
j ÂdjÞ
(4)
with the renormalized ~ij ¼ ijhþ
i þ
j i % ijð1 À Þ
and ~Jij % Jij2
. The excitations are found by introducing
a, b, c bosons according to d ¼ ðdx; dy; dzÞ ¼ ða; ub À
iveiQÁRc; ÀiveiQÁRb þ ucÞ and replacing the condensed
one as c, cy !
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À na À nb
p
. The resulting (a, b)
Hamiltonian is solved by the Bogoliubov transformation.
A similar approach is used for the  sector describing the
condensate density fluctuations  ¼ Seff.
Shown in Fig. 3 is the excitation spectra for several
points of the phase diagram. The spin-length fluctuations
Seff are high in energy. Figure 3(b) focuses on the magnetic
excitations. In the FQ phase, quadrupole and magnetic
modes are degenerate and gapless at q ¼ 0. As the
AF phase is approached, the gap at Q decreases and closes
upon entering the magnetic phase. However, the higher
energy magnons (which scale with Seff) are not much
affected by transition, apart from getting (softer) harder
in a (dis)ordered phase; this explains the persistence of
well-defined high-energy magnons into nonmagnetic
phases [19,20].
The magnetic modes in Fig. 3(b) resemble excitations of
bilinear-biquadratic spin model [38]. In fact, the dispersion
in FQ phase can be exactly reproduced [45] from an
effective spin-1 model
P
hiji
~JijSi Á Sj À ~ijðSi Á SjÞ2, with
~J and ~ given above. A large biquadratic coupling was
indeed found to account for many observations in Fe
pnictides [8,31,34]. We note, however, that this model
possesses FQ and AF phases only and misses the ns-AF
phase, where the ordered moment is reduced already at the
classical level; also, it does not contain the key notion of
the original model, i.e., formation of the effective spin Seff
and its fluctuations.
Singlet correlations inherent to the model may also lead
to an increase of the paramagnetic susceptibility ðTÞ with
temperature [30]. Considering the nonmagnetic phase, we
find that for the field parallel to the director u,  is
temperature dependent, k ¼ 1
2T
R
d!N ð!ÞsinhÀ2 !
2T ,
where N ð!Þ ¼
P
qð! À !qÞ is the density of states
(DOS) of magnetic excitations, while ? is constant. The
average  ¼ ðk þ 2?Þ=3 (with additional factor of
42
2
BNA) gives the measured ðTÞ, assuming slow rotations
of the director. The DOS shown in Fig. 4(a) is contributed
mainly by the regions around (, 0) and (0, )
hosting AF correlations. The corresponding thermal excitations
lead to the increase of  [Fig. 4(b)].
To conclude, we proposed the model describing quantum
magnetism of Fe pnictides. Their universal magnetic
spectra, wide-range variations of the LRO moments, and
emergent biquadratic-spin couplings are explained. The
model stands also on its own: extending the Heisenberg
models to the case of ‘‘mixed-spin’’ ions, it represents a
novel many-body problem. Of particular interest is the
effect of band fermions, which should have a strong impact
on low-energy dynamics of the model, e.g., converting the
q ¼ 0 Goldstone modes into overdamped spin-nematic
fluctuations. Understanding the effects of coupling
between local moments and band fermions, including
0
50
100
150
200
250
300
350
(0,0) (π,0) (π,π) (0,0)
ωq[meV]
(b)
A
B
C
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(0,0) (π,0) (π,π) (0,0)
ωq[eV]
(a)
| |
ρδ
rotationm
mδ
FIG. 3 (color online). (a) Dispersion of the condensate density
(, solid-black) and the ordered moment-length (jmj, dottedblue)
fluctuations, and the magnon dispersion (solid-blue), at the
point A in the phase diagram of Fig. 2(d). All three modes are
active in resonant x-ray scattering, and the latter two in neutron
scattering. (b) Evolution of the magnetic excitations going
from FQ to the ns-AF phase [C ! B ! A in Fig. 2(d)].
Twofold degenerate quadrupole waves (C) split into the magnon
(solid lines) and the jmj mode (dotted lines). The latter
represents oscillations between the nematic and magnetic orderings
and is gapful.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 100 200 300 400
χ[10-4
emu/mol]
T [K]
χ
χ⊥
χ||
(b)
0
2
4
6
8
0 50 100 150 200
DOS[eV
-1
]
ω [meV]
Γ=0
(a)
FIG. 4. (a) Density of states of the magnetic excitations calculated
for the point C of Fig. 2(d). We included the damping (e.g.,
due to coupling to the Stoner continuum) in a form Àð!Þ ¼
minð!; ÀÞ with À ¼ !Q=2. The result with À ¼ 0 is shown for
comparison. (b) Temperature dependence of the uniform susceptibility
. The components k (?) parallel (perpendicular)
to the local director u are also shown.
PRL 110, 207205 (2013) P H Y S I C A L R E V I E W L E T T E R S
week ending
17 MAY 2013
207205-4
212
implications for SC, should be the next step towards a
complete theory of Fe pnictides.
J. C. acknowledges support by the Alexander von
Humboldt Foundation, ERDF under project CEITEC
(CZ.1.05/1.1.00/02.0068) and EC 7th Framework
Programme (286154/SYLICA).
[1] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono,
J. Am. Chem. Soc. 130, 3296 (2008).
[2] For a review of the experimental data, see, e.g., D. C.
Johnston, Adv. Phys. 59, 803 (2010).
[3] T. Yildirim, Physica (Amsterdam) 469C, 425 (2009).
[4] C. Xu, M. Mu¨ller, and S. Sachdev, Phys. Rev. B 78,
020501(R) (2008).
[5] Q. Si and E. Abrahams, Phys. Rev. Lett. 101, 076401
(2008).
[6] C. Fang, H. Yao, W. F. Tsai, J. P. Hu, and S. A. Kivelson,
Phys. Rev. B 77, 224509 (2008).
[7] G. Uhrig, M. Holt, J. Oitmaa, O. P. Sushkov, and R. R. P.
Singh, Phys. Rev. B 79, 092416 (2009).
[8] D. Stanek, O. P. Sushkov, and G. S. Uhrig, Phys. Rev. B
84, 064505 (2011).
[9] I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du,
Phys. Rev. Lett. 101, 057003 (2008).
[10] K. Kuroki, S. Onari, R. Arita, H. Usui, Y. Tanaka, H.
Kontani, and H. Aoki, Phys. Rev. Lett. 101, 087004 (2008).
[11] A. V. Chubukov, D. V. Efremov, and I. Eremin, Phys.
Rev. B 78, 134512 (2008).
[12] S. Graser, T. A. Maier, P. J. Hirschfeld, and D. J. Scalapino,
New J. Phys. 11, 025016 (2009).
[13] E. Kaneshita, T. Morinari, and T. Tohyama, Phys. Rev.
Lett. 103, 247202 (2009).
[14] H. Gretarsson et al., Phys. Rev. B 84, 100509(R) (2011).
[15] P. Vilmercati et al., Phys. Rev. B 85, 220503(R) (2012).
[16] M. D. Johannes, I. I. Mazin, and D. S. Parker, Phys. Rev. B
82, 024527 (2010).
[17] I. I. Mazin and M. D. Johannes, Nat. Phys. 5, 141 (2009).
[18] D. Reznik et al., Phys. Rev. B 80, 214534 (2009).
[19] M. Liu et al., Nat. Phys. 8, 376 (2012).
[20] K.-J. Zhou et al., Nat. Commun. 4, 1470 (2013).
[21] M. Wang et al., arXiv:1303.7339.
[22] Z. P. Yin, K. Haule, and G. Kotliar, Nat. Mater. 10, 932
(2011).
[23] J. Dai, Q. Si, J.-X. Zhu, and E. Abrahams, Proc. Natl.
Acad. Sci. U.S.A. 106, 4118 (2009).
[24] S.-P. Kou, T. Li, and Z.-Y. Weng, Europhys. Lett. 88,
17010 (2009).
[25] W. Lv, F. Kru¨ger, and P. Phillips, Phys. Rev. B 82, 045125
(2010).
[26] Spin Crossover in Transition Metal Compounds I, edited
by P. Gu¨tlich and H. A. Goodwin (Springer, Berlin, 2004).
[27] S. Stackhouse, Nat. Geosci. 1, 648 (2008).
[28] H. Gretarsson, S. R. Saha, T. Drye, J. Paglione, J. Kim, D.
Casa, T. Gog, W. Wu, S. R. Julian, and Y.-J. Kim, Phys.
Rev. Lett. 110, 047003 (2013).
[29] J. T. Park et al., Phys. Rev. B 86, 024437 (2012).
[30] R. Klingeler et al., Phys. Rev. B 81, 024506 (2010).
[31] A. N. Yaresko, G.-Q. Liu, V. N. Antonov, and O. K.
Andersen, Phys. Rev. B 79, 144421 (2009).
[32] See Supplemental Material at http://link.aps.org/
supplemental/10.1103/PhysRevLett.110.207205 for
derivation of the Hamiltonian (1) from a two-orbital
Hubbard model.
[33] To address a tetra/ortho structural (‘‘orbital order’’) transition,
we may include also xz=yz orbital degeneracy of
S ¼ 1 triplets; this is left for future work.
[34] A. L. Wysocki, K. D. Belashchenko, and V. P. Antropov,
Nat. Phys. 7, 485 (2011).
[35] R. Yu, Z. Wang, P. Goswami, A. H. Nevidomskyy, Q. Si,
and E. Abrahams, Phys. Rev. B 86, 085148 (2012).
[36] N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694
(1989).
[37] B. A. Ivanov and A. K. Kolezhuk, Phys. Rev. B 68, 052401
(2003).
[38] A. La¨uchli, F. Mila, and K. Penc, Phys. Rev. Lett. 97,
087205 (2006).
[39] H. Tsunetsugu and M. Arikawa, J. Phys. Soc. Jpn. 75,
083701 (2006).
[40] K. Harada and N. Kawashima, Phys. Rev. B 65, 052403
(2002).
[41] E. Demler and F. Zhou, Phys. Rev. Lett. 88, 163001
(2002); A. Imambekov, M. Lukin, and E. Demler, Phys.
Rev. A 68, 063602 (2003).
[42] S. K. Yip, Phys. Rev. Lett. 90, 250402 (2003).
[43] C. M. Puetter, M. J. Lawler, and H.-Y. Kee, Phys. Rev. B
78, 165121 (2008).
[44] M. Serbyn, T. Senthil, and P. A. Lee, Phys. Rev. B 84,
180403(R) (2011).
[45] This can be understood using the identity ðSi Á SjÞ2 ¼ jdi Á
djj2 þ 1. If v ( u % 1, like in the FQ phase or close
to it in the ns-AF phase, we recover the  term of Eq. (4),
ðSi Á SjÞ2
% di Á dj þ dy
i Á dy
j .
PRL 110, 207205 (2013) P H Y S I C A L R E V I E W L E T T E R S
week ending
17 MAY 2013
207205-5
213
214
Supplemental Material for
Spin-state crossover model for the magnetism of iron pnictides
Jiˇr´ı Chaloupka1,2 and Giniyat Khaliullin1
1
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
2
Central European Institute of Technology, Masaryk University, Kotl´aˇrsk´a 2, 61137 Brno, Czech Republic
Here we analyze two-orbital Hubbard model in the regime of large Hund’s coupling and large interorbital
hopping, and explicitly demonstrate the emergence of the effective model proposed in the main paper.
We also provide estimates of the model parameters in terms of the microscopic parameters of the Hubbard
model.
Based on the ”orbital-differentiation” mechanism – which is particularly pronounced in multiorbital
systems with large Hund’s coupling (see Ref. [1] for recent discussion) – we assume a coexistence of
strongly correlated orbitals (hosting magnetic moments) and more itinerant bands (responsible for the
charge transport and Fermi-surface related physics). For the Fe-pnictide/chalcogenide families, a minimal
model for the ”magnetic” sector is a two-orbital Hubbard model which may accommodate magnetic
moments ranging from zero to 2µB per Fe-ion, depending on the parameter regime. This possible momentwindow
is what observed in Fe-pnictide/chalcogenides [2] (and also consistent with the model of the main
text). We assume that these two orbitals (labeled a and b below) are populated by two electrons per site on
average, while the remaining four electrons out of Fe-d6 configuration form a semimetallic band structure.
The itinerant bands are not a prime source of magnetic moments but, as noticed in the main text, we keep
in mind that they may mediate the interactions between local moments and hence support their long-range
order [3, 4].
Let us focus now on the ”magnetic” sector, i.e. two-orbital Hubbard Hamiltonian. As usual, it comprises
two parts, local interactions and intersite hoppings: H = Hloc + Hkin. Its local part includes the
crystal field splitting ∆ between a and b orbitals (their precise structure in terms of original d states is not
essential here) and local correlations:
Hloc =
∆
2 i
(nib − nia) + U
i,γ=a,b
niγ↑niγ↓ +
i
U′
− JH 2Sia · Sib + 1
2 nianib . (S1)
The local pair-hopping term is neglected, and the relation U′ = U − 2JH between inter- and intra-orbital
Coulomb interactions will be used. The kinetic term Hkin of the Hamiltonian contains the intersite hopping
of both intra- and inter-orbital character
Hkin = −t
ij ,σ
a†
iσajσ + b†
iσbjσ + h.c. − ˜t
ij ,σ
a†
iσbjσ + b†
iσajσ + h.c. . (S2)
Similar model was recently considered in Ref. [5] to address the spin-transition physics in cobaltates.
The key difference of our model is the presence of interorbital hopping ˜t, which converts the transitions
found in Ref. [5] into a smooth spin-crossover such that the ground state magnetic moment length (not
long-range order parameter!) may acquire any value from zero to 2 µB.
Our aim is to obtain the model Hamiltonian of the main paper as an effective low-energy Hamiltonian
resulting from H = Hloc + Hkin in the appropriate regime of parameters ∆, JH, etc. This is achieved by
a standard procedure – we select the relevant d2
i − d2
j bond states from the eigenbasis of Hloc and obtain
effective interactions on the bonds by eliminating Hkin perturbatively, employing the low-energy d3
i − d1
j
and d1
i − d3
j configurations as the intermediate states. To check the validity of this approach, the exact
eigenstates of H on a single bond are calculated and the results compared with those of the effective
Hamiltonian Heff we have derived.
In the spin-crossover regime discussed in the main paper, large Hund’s coupling nearly compensates
the crystal field splitting (i.e., ∆ ∼ 3JH) and makes the on-site singlet |s = a†
↑
a†
↓
| and the three triplet
1
215
states |T+1 = a†
↑
b†
↑
| , |T0 = 1√
2
(a†
↑
b†
↓
+ a†
↓
b†
↑
) | , and |T−1 = a†
↓
b†
↓
| quasidegenerate. These states thus
form the relevant low-energy sector while the other states (such as a†
↑
b†
↓
| ) are much higher in energy and
can be ignored.
To be able to extract the effective Hamiltonian on a bond, it is convenient to consider the subspaces
with total spin Stot = 0, 1, 2 separately. In Stot = 0 sector, the relevant bond states are the two d2
i − d2
j
configurations depicted in Fig. S1(a, b): |ss = |s i|s j with the local energy Ess = 2U − 2∆, and |TT =
1√
3
|T+1 i|T−1 j − |T0 i|T0 j + |T−1 i|T+1 j with the local energy ETT = Ess + 2(∆ − 3JH). The bond
interaction originates from virtual processes employing as the intermediate states mainly the low-lying
d3
i − d1
j and d1
i − d3
j configurations presented in Fig. S1(c, d). They are denoted as |A and |B and their
local energy amounts to EA = EB = Ess+U′+∆−3JH. The other bond states have a negligible contribution
to the groundstate, due to their high energy or due to kinematic (no hopping) reasons. The lowest state
in the Stot = 1 sector is composed of a pair of on-site triplets |T and states analogous to |A and |B but
having total spin one. Finally, the only states in the Stot = 2 are the combinations of two on-site triplets.
These states are unaffected by hopping.
The validity of the above classification of low-energy levels of Hubbard model is demonstrated in
Fig. S2 showing the results of an exact diagonalization of full two-orbital model H on a single bond.
We consider a representative set of parameters ∆ and JH such that a spin-crossover regime, where the
on-site singlet and triplet states are quasidegenerate, is realized: ∆ − 3JH = 0.1 eV. Focusing on Stot = 0
sector in Fig. S2(b), we can observe that with increasing ˜t, the state |TT gets gradually involved into the
groundstate, which becomes a mixture of |ss , |TT and the higher energy states |A , |B serving as the
intermediate states for the κ-processes. The contribution of the other states, which are neglected in our
derivation of the effective Hamiltonian below, is indeed negligible.
Having selected our basis states and evaluated their local energy, we proceed now by incorporating the
intersite hopping within this basis. First, the initial Hamiltonian H is projected to the selected subspace
of total spin Stot and denoted accordingly as H(S ) (where S = 0, 1, 2). In the next step, the intermediate
states are eliminated from H(S )-matrix by perturbation theory. After these steps, we will obtain an effective
Hamiltonian H(S )
eff
that operates within d2
i − d2
j configuration alone, and compare it with the model
Hamiltonian Hmodel used in the main paper.
In the most interesting Stot = 0 subspace, after the elimination of intermediate states, the Hamiltonian
H projected to the subspace spanned by |ss , |TT , |A , |B states
H(0)
=


Ess 0 −
√
2 ˜t −
√
2 ˜t
0 ETT − 3
2
˜t − 3
2
˜t
−
√
2 ˜t − 3
2
˜t EA 0
−
√
2 ˜t − 3
2
˜t 0 EB


becomes H(0)
eff
=


Ess −
4 ˜t2
ε
−
2
√
3 ˜t2
ε
−
2
√
3 ˜t2
ε
ETT −
3 ˜t2
ε


(S3)
t~t~
(a) (b) (c) (d)
ss BA
∆
U HJ ,U’
TT
Figure S1: Basis states dominating the groundstate of the Hubbard model in the discussed spin-crossover regime.
All the states have zero total spin. The two d2
i − d2
j configurations |ss and |TT are shown in panels (a) and (b),
respectively, together with a schematic depiction of the microscopic parameters. The d3
i − d1
j configuration |A and
d1
i − d3
j configuration |B , shown in panels (c) and (d), respectively, are connected to the d2
i − d2
j configurations by
virtue of the interorbital hoppings ˜tab or ˜tba indicated by arrows.
2
216
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
0.0 0.2 0.4 0.6 0.8 1.0
energy[eV]
˜t [eV]
Stot=0
1
2
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
probability
˜t [eV]
(a) (b)
P(ss)
P(TT)
P(A)=P(B)
P(other)
Figure S2: (a) Exact energy levels on the bond as a function of interorbital hopping ˜t for U = 3 eV, JH = 1 eV,
∆ − 3JH = 0.1 eV, and t = 0 (solid lines) and t = 0.5 ˜t (dotted lines). States with different values of the total spin are
distinguished by color. (b) Probabilities of selected basis states |ss , |TT , |A , and |B in the groundstate. The other
states ignored in the effective model derivation have a negligible total weight [see corresponding P(other) curve].
For a moderate value of t, the energy and the composition of the groundstate remains practically unaffected.
operating now within the |ss and |TT singlet states of d2
i − d2
j configuration. Here, ε = EA − E = EB − E
denotes the excitation energy. In the second order perturbation theory E = Ess, but by diagonalizing the
energy dependent H(0)
eff
self-consistently, one can exactly reproduce the groundstate energy and the ratio
of |ss and |TT coefficients obtained by diagonalizing the original matrix H(0). In the following, we
therefore take E = EGS with EGS being the groundstate energy of H(0)
eff
.
Using the same procedure, the pairs of local triplets T of total spin Stot = 1 obtain an energy H(1)
eff
=
E1 = ETT − 2˜t2/ε′ with ε′ = EA − E1 being the excitation energy, and those of total spin Stot = 2 remain
at an energy H(2)
eff = ETT .
The effective Hamiltonian Heff can now be exactly mapped to the model Hamiltonian Hmodel proposed
in the main paper. For a single bond, using the same notations, the corresponding matrix elements of
Hmodel read as
H(0)
model =
0 −
√
3 κ
−
√
3 κ 2ET − 2J − 4K
, H(1)
model = 2ET − J − K , H(2)
model = 2ET + J − K . (S4)
To make the correspondence between Heff and Hmodel matrices complete, we had to include small biquadratic
exchange −K(S1 · S2)2. The term-by-term comparison of the matrix elements of H(S )
eff
and
H(S )
model
yields the following values of the model parameters
κ =
2˜t2
ε
, J =
˜t2
ε′
, K = ˜t2 1
ε
−
1
ε′
, ET = (∆ − 3JH) + ˜t2 5
2ε
−
1
ε′
. (S5)
As evidenced by Fig. S3(a), the effective model gives an adequate description of the lowest states of the
Hubbard model. The obtained model parameters entering Eqs. (S4) and (S5) are presented in Fig. S3(b)
as functions of the interorbital hopping amplitude ˜t. The realistic range of ET ≈ 0.1 − 0.2 eV and κ, J ≈
0.05−0.20eV is obtained by taking ˜t ≈ 0.2−0.4eV. The small biquadratic exchange contained in Heff can
be neglected at this point because the much larger effective biquadratic contribution is in fact generated
by the κ-processes dynamically (see main text).
3
217
It is worth noticing that the strength κ of the key process of the model is finite due to interorbital
hopping ˜t. This process is thus ineffective in perovskite lattices, but it is perfectly allowed for the Fe(As/Te)-Fe
bonding geometry of Fe-pnictides/chalcogenides and leads to the spin-crossover mechanism
(”soft” magnetism) in these compounds (see main text). Concerning the role of intra-orbital t-hopping in
the mapping, it did not enter the above formulas, since t does not connect any pair of the selected low
energy states. The intermediate states that can be reached by t have an energy higher by ∆ than those
involved by ˜t, so that the t-effect on κ and ET values is relatively weak. It is only found to increase J by
about 2t2/(∆ + EA − ETT ).
To conclude, we have shown that the model Hamiltonian proposed in the paper naturally emerges from
the two-orbital Hubbard model with strong Hund’s coupling, when a regime of spin-state quasidegeneracy
is realized. The model parameters that follow from this derivation are well within the ranges that we have
explored in our study.
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
0.0 0.2 0.4 0.6 0.8 1.0
energy[eV]
˜t [eV]
exact
effective, Stot=0
1
2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.0 0.2 0.4 0.6 0.8 1.0
parameter[eV]
˜t [eV]
(a) (b)
ET
κ
J
−K
Figure S3: (a) Energy levels resulting from the diagonalization of Heff compared to the exact levels of the original
Hubbard Hamiltonian. The same parameters as in Fig. S2 are used and t = 0. (b) Values of the effective model
parameters as functions of interorbital hopping ˜t.
References
[1] A. Georges, L. de’ Medici, and J. Mravlje, arXiv:1207.3033.
[2] H. Gretarsson et al., Phys. Rev. B 84, 100509(R) (2011).
[3] M.D. Johannes, I.I. Mazin, and D.S. Parker, Phys. Rev. B 82, 024527 (2010).
[4] M.D. Johannes and I.I. Mazin, Phys. Rev. B 79, 220510(R) (2009).
[5] J. Kuneˇs and V. Kˇr´apek, Phys. Rev. Lett. 106, 256401 (2011).
4
218
Doping-Induced Ferromagnetism and Possible Triplet Pairing in d4 Mott Insulators
Jiří Chaloupka1
and Giniyat Khaliullin2
1
Central European Institute of Technology, Masaryk University, Kotlářská 2, 61137 Brno, Czech Republic
2
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
(Received 26 October 2015; published 8 January 2016)
We study the effects of electron doping in Mott insulators containing d4
ions such as Ru4þ
, Os4þ
, Rh5þ
,
and Ir5þ with J ¼ 0 singlet ground state. Depending on the strength of the spin-orbit coupling, the undoped
systems are either nonmagnetic or host an unusual, excitonic magnetism arising from a condensation of the
excited J ¼ 1 triplet states of t4
2g. We find that the interaction between J excitons and doped carriers
strongly supports ferromagnetism, converting both the nonmagnetic and antiferromagnetic phases of the
parent insulator into a ferromagnetic metal, and further to a nonmagnetic metal. Close to the ferromagnetic
phase, the low-energy spin response is dominated by intense paramagnon excitations that may act as
mediators of a triplet pairing.
DOI: 10.1103/PhysRevLett.116.017203
A distinct feature of Mott insulators is the presence of
low-energy magnetic degrees of freedom, and their coupling
to doped charge carriers plays the central role in transition
metal compounds [1]. In large spin systems like manganites,
this coupling converts parent antiferromagnet (AF) into a
ferromagnetic (FM) metal and gives rise to large magnetoresistivity
effects. The doping of spin one-half compounds
like cuprates and titanites, on the other hand, suppresses
magnetic order and a paramagnetic (PM) metal emerges. In
general, the fate of magnetism upon charge doping is
dictated by the spin-orbital structure of parent insulators.
In compounds with an even number of electrons on the d
shell, one may encounter a curious situation when the ionic
ground state has no magnetic moment at all, yet they may
order magnetically by virtue of low-lying magnetic levels
with finite spin, if the exchange interactions are strong
enough to overcome the single-ion magnetic gap. The d4
ions such as Ru4þ
, Os4þ
, Rh5þ
, Ir5þ
possess exactly this
type of level structure [2] due to spin-orbit coupling
λðS · LÞ: the spin S ¼ 1 and orbital L ¼ 1 moments form
a nonmagnetic ground state with total J ¼ 0 moment,
separated from the excited level J ¼ 1 by λ. A competition
of the exchange and spin-orbit couplings results then in a
quantum critical point (QCP) between the nonmagnetic
Mott insulator and the magnetic order [3,4]. Since the
magnetic order is due to the condensation of the virtual
J ¼ 1 levels and hence “soft,” the amplitude (Higgs) mode
is expected. The corollary of the “d4
excitonic magnetism”
[3] is the presence of the magnetic QCP that does not
require any special lattice geometry, and the energy scales
involved are large. The recent neutron scattering data [5] in
d4 Ca2RuO4 seem to support the theoretical expectations.
As we show in this Letter, the unusual magnetism of d4
insulators, where the soft J spins fluctuate between 0 and 1,
results also in anomalous doping effects that differ drastically
from conventional cases as manganites and cuprates.
Indeed, while common wisdom suggests that the PM phase
with yet uncondensed J moments near QCP would get even
“more PM” upon doping, we find that mobile carriers
induce long-range order instead. The order is of the FM
type and is promoted by the carrier-driven condensation
of J moments. By the same mechanism, the exchange
dominated AF phase also readily switches to the FM metal,
as observed in La-doped Ca2RuO4 [6,7]. The theory might
be relevant also to the electric-field-induced FM of
Ca2RuO4 [8] and the FM state of the RuO2 planes in
oxide superlattices [9]. Further doping suppresses any
magnetic order, and we suggest that residual FM correlations
may lead to a triplet superconductivity (SC).
Model.—There are a number of d4 compounds, magnetic
as well nonmagnetic, with various lattice structures [10–17].
To be specific, we consider a square lattice d4
insulator
lightly doped by electrons. Assuming relatively large spinorbit
coupling (SOC), the relevant states are pseudospin
J ¼ 0, 1 states of t4
2g and J ¼ 1=2 states of t5
2g [see Fig. 1(a)].
The d4
singlet s (J ¼ 0) and triplon T0;Æ1 (J ¼ 1) states
obey the Hamiltonian derived in Ref. [3]. Adopting the
Cartesian basis Tx ¼ ðT1 − T−1Þ=
ffiffiffi
2
p
i, Ty ¼ ðT1 þ T−1Þ=ffiffiffi
2
p
, and Tz ¼ iT0, it can be written as
Hd4 ¼ λ
X
i
T†
i · Ti þ
1
4
K
X
hiji

sis†
j

T†
i · Tj −
1
3
T†
iγTjγ

−s†
i s†
j

5
6
Ti · Tj −
1
6
TiγTjγ

þ H:c:

; ð1Þ
where γ is determined by the bond direction. The model
shows the AF transition due to a condensation of T at a
critical value Kc ¼ 6
11 λ of the interaction parameter
K ¼ 4t2
0=U. The degenerate Tx;y;z levels split upon
material-dependent lattice distortion, affecting the details
of the model behavior [18]. We will consider the cubic
PRL 116, 017203 (2016) P H Y S I C A L R E V I E W L E T T E R S
week ending
8 JANUARY 2016
0031-9007=16=116(1)=017203(5) 017203-1 © 2016 American Physical Society
219
symmetry case and make a few comments on the possible
effects of the tetragonal splitting.
The d4
system is doped by introducing a small number of
d5
objects—fermions fσ carrying the pseudospin J ¼ 1=2
of t5
2g. The on-site constraint ns þ nT þ nf ¼ 1 is implied.
The Hamiltonian describing the correlated motion of f is
derived by calculating matrix elements of the nearestneighbor
hopping ˆTij ¼ −t0ða†
iσajσ þ b†
iσbjσÞ between multielectron
configurations hd5
i d4
jj ˆTijjd4
i d5
ji. Here, a and b are
the t2g orbitals active on a given bond, e.g., xy and zx for x
bonds. The resulting hopping Hamiltonian comprises three
contributions, Hd4−d5 ¼
P
ijðh1 þh2 þh3Þ
ðγÞ
ij . The first one,
depicted schematically in Figs. 1(b) and 1(c), is a spinindependent
motion of f, accompanied by a backflow of s
and T:
h
ðγÞ
1 ¼ −tf†
iσfjσ

s†
jsi þ
15
16

T†
j · Ti −
3
5
T†
jγTiγ

: ð2Þ
The second contribution is a spin-dependent motion of f
generating J ¼ 0 ↔ J ¼ 1 magnetic excitation in the d4
background [see Fig. 1(d)]:
h
ðγÞ
2 ¼ i~t

σγ
ijðs†
jTiγ − T†
jγsiÞ −
1
3
σij · ðs†
j Ti − T†
j siÞ

: ð3Þ
Here, σij ¼ f†
iαταβfjβ with Pauli matrices τ denotes the
bond-spin operator. The derivation for the cubic symmetry
gives t ¼ 4
9 t0 and ~t ¼ ð1=
ffiffiffi
6
p
Þt0 with the ratio ~t=t ≈ 1.
However, these values are affected by the lattice distortions
(via the pseudospin wave functions) and f-band renormalization
reducing the effective t. We thus consider ~t=t as a free
parameter and set ~t ¼ 1.5t below. The last contribution to
Hd4−d5 reads as coupling between the bond spins residing
in f and T sectors: h
ðγÞ
3 ¼ 9
16 tðσγ
ijJγ
ji þ 1
3 σij · JjiÞ, where
Jji ¼ −iðT†
j × TiÞ. At small doping and near QCP where
the density of T excitons is small, the scattering term h3 can
be neglected.
Phase diagram.—We first inspect the phase behavior of
the model as a function of doping x and interaction
parameters K and ~t. The magnetic order is linked to the
condensation of triplons induced by their mutual interactions
and the interaction with the doped fermions f. In
contrast to the cubic lattice where all the T flavors are
equivalent, on the two-dimensional square lattice the Tz
flavor experiences the strongest interactions and is selected
to condense, provided that it is not suppressed by a large
tetragonal distortion. We thus focus on Tz and omit the
index z.
Following the standard notation for spin-1 condensates,
we express complex T ¼ u þ iv using two real fields u, v.
The ordered dipolar moment residing on Van Vleck
transition s↔T is then m ¼ 2
ffiffiffi
6
p
v [3]. Assuming either
FM order (condensation prescribed by T → iv) or AF order
(T → Æiv in a Néel pattern), we evaluate the classical
energy of the T condensate and add the energy of the f
bands polarized due to the condensed T. Doing so, we
replace si by
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 − x − v2
p
to incorporate the constraint, on
average. The resulting total energy EðvÞ ¼ ET þ Eband is
minimized with respect to the condensate strength v
and compared for the individual phases: FM, AF, and
PM (v ¼ 0). The condensate energy amounts to ET ¼
½λ Æ 11
6 Kð1 − x − v2
ފv2
, with the þ (−) sign for FM (AF)
phase, respectively. The band energy Eband ¼
P
kσεkσnkσ is
calculated for a particular doping level x ¼
P
kσnkσ using
the band dispersion εkσ ¼ −4ðt1 − σt2Þγk, where γk ¼
1
2 ðcos kx þ cos kyÞ. The hopping parameter t1 stemming
from h1 reads as t1 ≃ tð1 − xÞ and t1 ≃ tð1 − x − 2v2
Þ for
FM and AF, respectively. This captures the doubleexchange
nature of h1—only FM-aligned T allow for a
free motion of f, while the AF order of T blocks it.
The parameter t2 quantifies the polarization of the bands by
virtue of h2 and is nonzero in the FM case
only: t2 ¼ 2
3
~tv
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 − x − v2
p
.
Shown in Fig. 2 are the resulting phase diagrams along
with the total ordered moment m½μBŠ ¼ 2
ffiffiffi
6
p
v þ n↑ − n↓.
d
4
i d
5
j d
5
i d
4
j
5
2gt
fσ
4
2gt(a)
(c)
(b)
(d)
2λ
J=3/2
J=1/2s
TM
J=0
J=1
J=2
t
t
t~
λ
fj sjsi fi
fj fi
Ti Tj
fjsi fi
Tj
FIG. 1. (a) Spin-orbital level structure of t4
2g and t5
2g configurations.
The lowest states including singlet s and triplet TM states
of d4
, and pseudospin 1=2 fσ states of d5
configurations form a
basis for the effective low-energy Hamiltonian. (b)–(d) Schematics
of electron hoppings that lead to Eqs. (2) and (3): (b) Free
motion of a doped fermion fσ in a singlet background. (c) The
fermion hopping is accompanied by a triplon backflow supporting
the double-exchange type ferromagnetism. (d) Fermionic
hopping generates a singlet-triplet excitation. This process leads
to a coupling between the Stoner continuum and T moments
promoting magnetic condensation.
PRL 116, 017203 (2016) P H Y S I C A L R E V I E W L E T T E R S
week ending
8 JANUARY 2016
017203-2
220
In both phase diagrams for constant ~t=λ [Figs. 2(a) and 2(b)]
at x ¼ 0 we recover the QCP of the d4
model. Nonzero
doping causes a suppression of the AF phase via the doubleexchange
mechanism in h1, and an appearance of the FM
phase strongly supported by h2 that directly couples the
moment m ∼ v of T exciton to the fermionic spin σij,
promoting magnetic condensation. With an increasing ~t the
FM phase quickly extends as seen also in Figs. 2(c) and 2(d)
containing the phase diagrams for constant K=λ ¼ 0.65
(selected to roughly reproduce the experimental value 1.3 μB
for Ca2RuO4 [19]) and K=λ ¼ 0.30. The constant ~t=λ cut in
Fig. 2(c) is strongly reminiscent of the phase diagram of
La-doped Ca2RuO4 [6,7,20], where the AF phase is almost
immediately replaced by the FM phase present up to a
certain doping level. To estimate realistic values of ~t=λ, we
assume t0 ∼ 300 meV. The large SOC in d4
Ir5þ
with λ ∼
200 meV [22–24] leads to ~t=λ ∼ 1 and places it strictly to the
AF/PM (c) or PM/PM (d) regime. In contrast to this, the
moderate λ ∼ 70–80 meV in Ru4þ
[2,25] makes the FM
phase easily accessible.
Spin susceptibility, emergence of paramagnons.—The
tendency toward FM ordering naturally manifests itself
in the dynamic spin response of the coupled T exciton and
f-band system. Here we study it in detail for the PM phase,
focusing again on Tz being the closest to condense. The
magnetic moment m is carried mainly by the dipolar
component v ¼ ðT − T†Þ=2i of triplons so that the dominant
contribution to the spin susceptibility is given by the v
susceptibility χðq; ωÞ. To evaluate it, we replace si →ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 − x − nTi
p
, and decouple h1 (3) into f and T parts on
a mean-field level. This yields a fermionic Hamiltonian
Hf ¼
P
kσεkf†
kσfkσ with εk ¼ −4tð1 − xÞγk, and a quadratic
form for Tz boson: HT ¼
P
q½AqT†
qTq − 1
2 Bq
ðTqT−q þ T†
qT†
−qފ. Here, Aq ¼ λ þ 4thnijið1 − γqÞþ
Kð1 − xÞγq, Bq ¼ 5
6 Kð1 − xÞγq, and hniji ¼
P
kσγknkσ.
Bogoliubov diagonalization provides the bare triplon
dispersion ωq ¼ ðA2
q − B2
qÞ1=2
and the bare v susceptibility
χ0ðq; ωÞ ¼ 1
2 ðAq − BqÞ=½ω2
q − ðω þ iδÞ2
Š. The susceptibility
is further renormalized by the coupling h2 (3), which can
be viewed as an interaction between a dipolar component v
of the triplons and the Stoner continuum of f fermions:
Hint ¼ g
X
q
vq ~σ−q; ~σ−q ¼
X
k
Γkqf†
kþq;ατz
αβfk;β: ð4Þ
The coupling constant g ¼ 8
3
~t
ffiffiffiffiffiffiffiffiffiffiffi
1 − x
p
, and the vertex Γkq ¼
1
2 ðγk þ γkþqÞ is close to 1 in the limit of small k, q.
By treating this coupling on a RPA level, we arrive at the
full v susceptibility χ ¼ χ0=ð1 − χ0ΠÞ with the v self-energy
Πðq; ωÞ ¼ g2
X
kσ
Γ2
kq
nkσ − nkþqσ
εkþq − εk − ω − iδ
: ð5Þ
The interplay of the coupled excitonic and band spin
responses is demonstrated in Fig. 3. The high-energy
component of χ linked to χ0 follows the bare triplon
dispersion ωq. In an undoped system, due to the AF K
interaction, ωq has a minimum at q ¼ ðπ; πÞ and χ0 would
be most intense there. By doping, the double exchange
mechanism in h1 disfavoring AF correlations pushes ωq up
near ðπ; πÞ. Further, due to a dynamical mixing [Eqs. (3)
and (4)] of triplons with the fermionic continuum, the lowenergy
component of χ gains spectral weight as ~t=λ
approaches the critical value, and a gradually softening
FM paramagnon is formed [see Fig. 3(b)]. The emergence
of the paramagnon and the increase of its spectral weight is
shown in detail in Fig. 3(e). Finally, once the critical ~t=λ is
reached, triplons, whose spectral weight was pulled down
by the coupling to the Stoner continuum, condense and
the FM order sets in, signaled by the divergence of
χðq ¼ 0; ω ¼ 0Þ [cf. Figs. 3(c) and 3(d)].
Triplet pairing.—Intense paramagnons emerging in the
proximity to the FM phase may serve as mediators of a
triplet pairing interaction [26]. In the following, we perform
semiquantitative estimates for this triplet SC.
While the dominant contribution to the pairing
strength is due to the vz fluctuations, in order to assess
the structure of the triplet order parameter, the full coupling
Hint ¼ g
P
qvq · ~σ−q leading to the effective interaction
− 1
2 g2
P
qαχαðq; ω ¼ 0Þ~σα
q ~σα
−q has to be considered. The
K/λ
x
~t / λ = 1.7
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3
PM
FM
AF
x
~t / λ = 2.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3
0.0
0.5
1.0
1.5
2.0
PM
FM
AF
m[μB]
x
0.0
0.5
1.0
0.0 0.1 0.2 0.3
FM
AF
x
0.0
0.5
1.0
0.0 0.1 0.2 0.3
FM
~t/λ
K / λ = 0.65
0
1
2
3
4
PM
FM
AF
K / λ = 0.30
(b)(a)
(d)(c) m/μB
m/μB
0
1
2
3
4
0.0
0.5
1.0
1.5
2.0
PMFM
FIG. 2. (a),(b) Phase diagrams and the ordered magnetic
moment value for varying doping x and K=λ keeping fixed
~t=λ of 1.7 and 2.5. (c) Phase diagram for varying doping and ~t=λ
and fixed K ¼ 0.65λ above the critical Kc ¼ 6
11 λ of the d4
system. The bottom panel shows mðxÞ along the cut at
~t=λ ¼ 3.5. (d) The same for K ¼ 0.3λ and the cut at ~t=λ ¼ 3.
PRL 116, 017203 (2016) P H Y S I C A L R E V I E W L E T T E R S
week ending
8 JANUARY 2016
017203-3
221
vα susceptibility χα for α ¼ x, y may be calculated the same
way as χz above, using now Aα
q ¼ Az
q þ ½6
5 thniji − 1
6 K
ð1 − xފ cos qα and Bα
q ¼ Bz
q − 1
12 Kð1 − xÞ cos qα. Thecoupling
vertex for vx and vy obtains an additional contribution,
Γα
kq ¼ Γz
kq − 3
4 ½cos kα þ cosðkα þ qαފ. The resulting BCS
interaction in terms of tþ1k ¼ fk↑f−k↑, t0k ¼ 1ffiffi
2
p ðfk↓f−k↑
þfk↑f−k↓Þ, and t−1k ¼ fk↓f−k↓ takes the form
HBCS ¼ −
1
2
X
kk0
½Vzðt†
1t1 þ t†
−1t−1Þkk0
þ ðVx − VyÞðt†
1t−1 þ t†
−1t1Þkk0
þ ðVx þ Vy − VzÞt†
0kt0k0 Š; ð6Þ
where Vα denotes the properly symmetrized Vαkk0 ¼
g2
ðΓα
k;k0−kÞ2 1
2 ½χαðk − k0
Þ − χαðk þ k0
ފ. Decomposed into
the Fermi surface harmonics, the BCS interaction is
well approximated by Vzkk0 ≈ 2V0 cosðϕk − ϕk0 Þ and
ðVx − VyÞkk0 ≈ 2V1 cosðϕk þ ϕk0 Þ with V0;1 > 0 [see
Figs. 3(d) and 4(a)]. The relatively small V1 ≪ V0 fixes
the relative phase of the tþ1 and t−1 pairs so that the SC order
parameter becomes ΔÆ1k ¼ ΔeÆiϕk . This ordering type is
capturedby thedvectord ¼ −iΔðsin ϕk; cos ϕk; 0Þ ∼ ˆxky þ
ˆykx shown in Fig. 4(b). In the classification of Ref. [28], it
forms the Γ−
4 irreducible representation of tetragonal group
D4h.However,thisresultappliestothecubicsymmetrycase.
Lattice distortions that cause splitting among Tx;y;z and
modify the pseudospin wave functions may in fact offer a
possibility to “tune” the symmetry of the order parameter.
If distortions favor Tx;y, the potentials Vx;y are expected to
dominate in Eq. (6), supporting the chiral t0 pairing represented
by the last term in (6).
Data in Figs. 4(c) and 4(d) serve as a basis for a rough Tc
estimate. Figure 4(c) shows the BCS parameter λBCS ≈ V0N
(N is DOS per spin component of the f band) which attains
sizable values near the FM phase boundary, where the
paramagnons are intense. To avoid complex physics near the
very vicinity of the FM QCP [29–31], we take a conservative
upper limit λBCS ≈ 0.5. Extending V0 by the ω dependence
of the underlying χzðq; ωÞ, we define λBCSðωÞ. Its imaginary
part to be understood as the conventional α2
F is plotted in
Fig. 4(d) yielding an estimate of the BCS cutoff Ω ≲ 0.1λ.
With λ ∼ 100 meV, this gives Tc ≈ Ωe−1=λBCS of about 10 K.
In conclusion, we have explored the doping effects in
spin-orbit d4
Mott insulators. The results show that the
doped electrons moving in the d4
background firmly favor
ferromagnetism, explaining, e.g., the observed behavior of
~t/λ
x
λBCS=1
0.50
0.25
0
1
2
3
0.0 0.1 0.2 0.3
FM
ImλBCS(ω)
ω / λ
λBCS=1
0.50
0.25
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.1 0.2 0.3
-0.4 0.4
−π
0
+π
−π 0 +π
χx − χy
k
φk
(a)
(b)
(c)
(d)
FIG. 4. (a) Combination ðχx − χyÞω¼0 that determines the
symmetry of the pairing potential Vx − Vy. The parameters are
the same as in Figs. 3(b) and 3(d). The dashed circle indicates the
Fermi surface. (b) Representation of ΔÆ1k ¼ ΔeÆiϕk using the d
vector along the Fermi surface. (c) Contours of λBCS ¼ V0N in
the phase diagram of Fig. 2(d). (d) Imaginary part of ω-dependent
λBCSðωÞ for x ¼ 0.1 and the values of ~t=λ corresponding to
λBCS ¼ 1, 0.5, and 0.25.
ω/λ
x=0.1
~t / λ = 1.5
0.0
0.5
1.0
1.5
2.0
(0,0) (π/2,π/2) (π,π)
Im χ(q,ω)
x=0.1
~t / λ = 2.1
0.0
0.5
1.0
1.5
2.0
(0,0) (π/2,π/2) (π,π)
0
1
2
3
Im χ(q,ω)
Reχ(q,ω=0)
q
bare
full
0.0
0.5
1.0
1.5
(0,0) (π/2,π/2) (π,π)
q
bare
full
0
1
2
3
4
5
(0,0) (π/2,π/2) (π,π)
Imχ(qx=qy=π/10,ω)
ω / λ
(e)
(c) (d)
(a) (b)
paramagnon
"bare" Stoner
Van-Vleck exciton
~t / λ = 1.2
1.6
1.9
2.1
2.2
0
1
2
3
4
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
FIG. 3. (a) Imaginary part of the vz susceptibility χðq; ωÞ in the
ðπ; πÞ direction calculated for x ¼ 0.1, ~t=λ ¼ 1.5, K=λ ¼ 0.3.
χðq; ωÞ is shown in units of λ−1
. The black (gray) dashed line
shows the bare triplon dispersion for x ¼ 0.1 (x ¼ 0). (b) The
same for ~t=λ ¼ 2.1 closer to the FM transition point ~t=λ ≈ 2.25.
(c),(d) The static susceptibility corresponding to panels (a) and
(b). (e) Imaginary part of χðq; ωÞ at q ¼ ðπ=10; π=10Þ for several
values of ~t=λ gradually approaching the FM transition point.
PRL 116, 017203 (2016) P H Y S I C A L R E V I E W L E T T E R S
week ending
8 JANUARY 2016
017203-4
222
La-doped Ca2RuO4. In the paramagnetic phase near the
FM QCP, the incipient FM correlations are manifested by
intense paramagnons that may provide a triplet pairing.
We thank G. Jackeli for useful comments. J. C. acknowledges
support by the Czech Science Foundation (GA ČR)
under Project No. 15-14523Y and ERDF under Project
CEITEC (CZ.1.05/1.1.00/02.0068).
[1] M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 70,
1039 (1998).
[2] A. Abragam and B. Bleaney, Electron Paramagnetic
Resonance of Transition Ions (Clarendon, Oxford, 1970).
[3] G. Khaliullin, Phys. Rev. Lett. 111, 197201 (2013).
[4] O. N. Meetei, W. S. Cole, M. Randeria, and N. Trivedi,
Phys. Rev. B 91, 054412 (2015).
[5] A. Jain, M. Krautloher, J. Porras, G. H. Ryu, D. P. Chen,
D. L. Abernathy, J. T. Park, A. Ivanov, J. Chaloupka, G.
Khaliullin, B. Keimer, and B. J. Kim, arXiv:1510.07011.
[6] G. Cao, S. McCall, V. Dobrosavljevic, C. S. Alexander, J. E.
Crow, and R. P. Guertin, Phys. Rev. B 61, R5053 (2000).
[7] G. Cao, C. S. Alexander, S. McCall, J. E. Crow, and R. P.
Guertin, J. Magn. Magn. Mater. 226–230, 235 (2001).
[8] F. Nakamura, M. Sakaki, Y. Yamanaka, S. Tamaru, T.
Suzuki, and Y. Maeno, Sci. Rep. 3, 2536 (2013).
[9] C. R. Hughes, T. Harada, R. Ashoori, A. V. Boris, H.
Hilgenkamp, M. E. Holtz, L. Li, J. Mannhart, D. A. Muller,
D. G. Schlom, A. Soukiassian, X. Renshaw Wang, and H.
Boschker (unpublished).
[10] S. Nakatsuji, S. Ikeda, and Y. Maeno, J. Phys. Soc. Jpn. 66,
1868 (1997).
[11] Y. Miura, Y. Yasui, M. Sato, N. Igawa, and K. Kakurai,
J. Phys. Soc. Jpn. 76, 033705 (2007).
[12] M. Bremholm, S. E. Dutton, P. W. Stephens, and R. J. Cava,
J. Solid State Chem. 184, 601 (2011).
[13] G. Cao, T. F. Qi, L. Li, J. Terzic, S. J. Yuan, L. E. DeLong, G.
Murthy, and R. K. Kaul,Phys. Rev. Lett. 112, 056402 (2014).
[14] Y. Shi, Y. Guo, Y. Shirako, W. Yi, X. Wang, A. A. Belik, Y.
Matsushita, H. L. Feng, Y. Tsujimoto, M. Arai, N. Wang, M.
Akaogi, and K. Yamaura, J. Am. Chem. Soc. 135, 16507
(2013).
[15] P. Khalifah, R. Osborn, Q. Huang, H. W. Zandbergen, R.
Jin, Y. Liu, D. Mandrus, and R. J. Cava, Science 297, 2237
(2002).
[16] S. Lee, J.-G. Park, D. T. Adroja, D. Khomskii, S. Streltsov,
K. A. McEwen, H. Sakai, K. Yoshimura, V. I. Anisimov,
D. Mori, R. Kanno, and R. Ibberson, Nat. Mater. 5, 471
(2006).
[17] Hua Wu, Z. Hu, T. Burnus, J. D. Denlinger, P. G. Khalifah,
D. G. Mandrus, L.-Y. Jang, H. H. Hsieh, A. Tanaka, K. S.
Liang, J. W. Allen, R. J. Cava, D. I. Khomskii, and L. H.
Tjeng, Phys. Rev. Lett. 96, 256402 (2006).
[18] A. Akbari and G. Khaliullin, Phys. Rev. B 90, 035137
(2014).
[19] M. Braden, G. André, S. Nakatsuji, and Y. Maeno, Phys.
Rev. B 58, 847 (1998).
[20] For a detailed comparison, one has to keep in mind polaronic
and phase coexistence effects [21] generic to weakly doped
Mott insulators, and deviations from two dimensionality.
[21] W. Brzezicki, A. M. Oleś, and M. Cuoco, Phys. Rev. X 5,
011037 (2015).
[22] B. N. Figgis and M. A. Hitchman, Ligand Field Theory and
Its Applications (Wiley-VCH, New York, 2000).
[23] J. Kim, D. Casa, M. H. Upton, T. Gog, Y.-J. Kim, J. F.
Mitchell, M. van Veenendaal, M. Daghofer, J. van den
Brink, G. Khaliullin, and B. J. Kim, Phys. Rev. Lett. 108,
177003 (2012).
[24] Note a relation λ ¼ ξ=2S between spin-orbit coupling
constant λ for spin S ¼ 1 and the single-electron
one ξ [2].
[25] T. Mizokawa, L. H. Tjeng, G. A. Sawatzky, G. Ghiringhelli,
O. Tjernberg, N. B. Brookes, H. Fukazawa, S. Nakatsuji,
and Y. Maeno, Phys. Rev. Lett. 87, 077202 (2001).
[26] We recall that the pseudospins are spin-orbit entangled
objects [27], and hence a pseudospin triplet is in fact a
mixture of real-spin singlets and triplets.
[27] G. Khaliullin, Prog. Theor. Phys. Suppl. 160, 155 (2005).
[28] M. Sigrist and K. Ueda, Rev. Mod. Phys. 63, 239
(1991).
[29] A. V. Chubukov, A. M. Finkeĺstein, R. Haslinger, and D. K.
Morr, Phys. Rev. Lett. 90, 077002 (2003).
[30] A. V. Chubukov, C. Pépin, and J. Rech, Phys. Rev. Lett. 92,
147003 (2004).
[31] A. V. Chubukov and D. L. Maslov, Phys. Rev. Lett. 103,
216401 (2009).
PRL 116, 017203 (2016) P H Y S I C A L R E V I E W L E T T E R S
week ending
8 JANUARY 2016
017203-5
223
224
LETTERS
PUBLISHED ONLINE: 27 MARCH 2017 | DOI: 10.1038/NPHYS4077
Higgs mode and its decay in a two-dimensional
antiferromagnet
A. Jain1,2†
, M. Krautloher1†
, J. Porras1†
, G. H. Ryu1‡
, D. P. Chen1
, D. L. Abernathy3
, J. T. Park4
, A. Ivanov5
,
J. Chaloupka6
, G. Khaliullin1
, B. Keimer1
* and B. J. Kim1,7
*
Condensed-matter analogues of the Higgs boson in particle
physics allow insights into its behaviour in different symmetries
and dimensionalities1
. Evidence for the Higgs mode has
been reported in a number of different settings, including
ultracold atomic gases2
, disordered superconductors3
, and
dimerizedquantummagnets4
.However,decayprocessesofthe
Higgsmode(whichareeminentlyimportantinparticlephysics)
have not yet been studied in condensed matter due to the lack
of a suitable material system coupled to a direct experimental
probe. A quantitative understanding of these processes is
particularly important for low-dimensional systems, where
the Higgs mode decays rapidly and has remained elusive
to most experimental probes. Here, we discover and study
the Higgs mode in a two-dimensional antiferromagnet using
spin-polarized inelastic neutron scattering. Our spin-wave
spectra of Ca2RuO4 directly reveal a well-defined, dispersive
Higgs mode, which quickly decays into transverse Goldstone
modes at the antiferromagnetic ordering wavevector. Through
a complete mapping of the transverse modes in the reciprocal
space, we uniquely specify the minimal model Hamiltonian
and describe the decay process. We thus establish a novel
condensed-matterplatformforresearchonthedynamicsofthe
Higgs mode.
For a system of interacting spins, amplitude fluctuations of the
local magnetization—the Higgs mode—can exist as well-defined
collective excitations near a quantum critical point (QCP). We
consider here a magnetic instability driven by the intra-ionic spin–
orbit coupling, which tends towards a non-magnetic state through
complete cancellation of orbital (L) and spin (S) moments when
they are antiparallel and of equal magnitude5,6
. This mechanism
should be broadly relevant for d4
compounds of such ions as
Ir(V), Ru(IV), Os(IV) and Re(III) with sizable spin–orbit coupling
but remains little explored. We investigate the magnetic
insulator Ca2RuO4, a quasi-two-dimensional antiferromagnet7
with
nominally L = 1 and S = 1 (Fig. 1). Because the local symmetry
around the Ru(IV) ion is very low8,9
(having only inversion
symmetry), it is widely believed that the orbital moment is
completely quenched by the crystalline electric field10–13
, which is
dominated by the compressive distortion of the RuO6 octahedra
along the c-axis (Fig. 1). In the absence of an orbital moment,
the nearest-neighbour magnetic exchange interaction is necessarily
isotropic. Deviations from this behaviour are a sensitive
indicator of an unquenched orbital moment. If this moment is
sufficiently strong, it can drive Ca2RuO4 close to a QCP with novel
Higgs physics.
Our comprehensive set of time-of-flight (TOF) inelastic neutron
scattering (INS) data over the full Brillouin zone (Fig. 2a)
indeed reveal qualitative deviations of the transverse spin-wave
dispersion from those of a Heisenberg antiferromagnet. In particular,
the global maximum of the dispersion is found at q=(0,0),
in sharp contrast to a Heisenberg antiferromagnet, which has a
minimum there (Fig. 1). This striking manifestation of orbital
magnetism in Ca2RuO4 leads us to consider the limit of strong
spin–orbit coupling described in terms of a singlet and a triplet
separated in energy by λ (Fig. 1), which was estimated in earlier
experiments14–16
to be in the range ∼75–100 meV. In this limit, the
ground state is non-magnetic with zero total angular momentum,
and therefore a QCP separating it from a magnetically ordered
phase is expected as a matter of principle. Although this QCP
can be pre-empted by an insulator–metal transition17,18
or rendered
first-order by coupling to the lattice or other extraneous
factors, it is sufficient that the system is reasonably close to the
hypothetical QCP.
To assess the proximity to the QCP and the possibility of
finding the Higgs mode, we first reproduce the observed transverse
spin-wave modes by applying the spin-wave theory19,20
to the
following phenomenological Hamiltonian dictated by general
symmetry considerations:
H =J
ij
( ˜Si · ˜Sj −α ˜Szi
˜Szj)+E
i
˜S2
zi
+
i
˜S2
xi
A
ij
( ˜Sxi
˜Syj + ˜Syi
˜Sxj) (1)
Here, ˜S denotes a pseudospin-1 operator describing the entangled
spin and orbital degrees of freedom. This model includes single-ion
terms (E and ) of tetragonal (z c) and orthorhombic (x a)
symmetries, correspondingly, as well as an XY-type exchange
anisotropy (α > 0) and the bond-directional pseudodipolar
interaction (A); note that its sign depends on the bond. Also
symmetry allowed—but neglected here—are the Dzyaloshinskii–
Moriya interaction (which can be gauged out by a suitable local
coordinate transformation) and further-neighbour interactions.
1
Max Planck Institute for Solid State Research, Heisenbergstraße 1, D-70569 Stuttgart, Germany. 2
Solid State Physics Division, Bhabha Atomic Research
Centre, Mumbai 400085, India. 3
Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA. 4
Heinz
Maier-Leibnitz Zentrum, TU München, Lichtenbergstraße 1, D-85747 Garching, Germany. 5
Institut Laue-Langevin 6, rue Jules Horowitz, BP 156, 38042
Grenoble Cedex 9, France. 6
Central European Institute of Technology, Masaryk University, Kotlářská 2, 61137 Brno, Czech Republic. 7
Department of Physics,
Pohang University of Science and Technology, Pohang 790-784, Republic of Korea. †
These authors contributed equally to this work. ‡
Present address:
Max-Planck-Institute for Chemical Physics of Solids, Nöthnitzerstraße 40, D-01187 Dresden, Germany. *e-mail: B.Keimer@fkf.mpg.de; bjkim@fkf.mpg.de
NATURE PHYSICS | VOL 13 | JULY 2017 | www.nature.com/naturephysics 633
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
225
LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS4077
c
ba
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
λ
|L + S| = 0
|L + S| = 1
(π,π) (0,0) (π,π) (0,0)
xy
L = 0, S =1
Heisenberg
magnet
L = 1, S = 1
Spin−orbit
singlet
Δ
λ >> ΔΔ >> λ
yz/zx
Ca2RuO4
Figure 1 | Crystal, magnetic, and electronic structures of Ca2RuO4.
Ca2RuO4 crystallizes in the orthorhombic Pbca space group, a distorted
variant of the layered perovskite structure with a quasi-two-dimensional
square lattice. For clarity, Ca ions are shown as small, light grey balls and
oxygen ions, at all corners of the octahedra, are not shown. The distortion
involves 2% compression of the RuO6 octahedra along the c-axis, and their
rotation about the c-axis and tilting about an axis that lies in the
a–b plane8,9
. The unit cell for the undistorted square lattice is shown as the
blue square. (π,π) magnetic order develops below TN ≈110 K with the
moment (orange arrow) aligned approximately along the b-axis. The
compressive distortion of the RuO6 octahedra leads to the splitting ∆
between the orbitals of xy and yz/zx symmetry. If ∆ is much larger than the
spin–orbit splitting (λ), the orbital degrees of freedom are completely
quenched and a S=1 Heisenberg magnet is obtained. In the other limit
λ ∆, a non-magnetic singlet ground state is stabilized. These two distinct
phases exhibit qualitatively different magnetic excitation spectra. See
Supplementary Figs 1 and 2 for the evolution of the electronic structure and
the spin-wave dispersions between these two limiting cases. Error bars
denote one standard deviation.
The coupling constants resulting from fits of the model to the
measured spectra are provided in the caption of Fig. 2. We stress
that this model gives the unique minimal description of the system,
which we also derive explicitly starting from the microscopic
electronic structure (see Supplementary Information).
We find that the single-ion term E overwhelms all other coupling
constants, particularly the nearest-neighbour exchange coupling J,
and thus confines the pseudospins to the a–b basal plane. This
accounts for the XY-like dispersion, which has a maximum at
q=(0,0). This important aspect was missed in a recent INS study of
Ca2RuO4, because the dispersion along the path (π
2
,π
2
)–(0,0) was not
measured21
. The large E implies unquenched spin–orbit coupling;
in cubic symmetry E is λ itself (see Supplementary Fig. 1). This
term also acts towards suppressing the magnetic order by favouring
the ˜Sz = 0 singlet ground state—known in the literature as ‘spin
nematic’22
—and is responsible for the significant reduction of the
moment size from the nominal 2µB for pure S = 1 moments to
approximately 1.3µB (ref. 8). Other terms allow fine tuning of the
spin-wave dispersions; the pseudodipolar term accounts for the
weak dispersion along the magnetic zone boundary (π
2
,π
2
)-(π,0),
and is responsible for gapping the transverse mode. The latter
means that the Goldstone bosons acquire a finite mass through
the explicit breaking of the continuous symmetry, the significance
of which for the Higgs mode decay will be discussed later on.
Our calculation (Fig. 2b) predicts an intense Higgs mode in
this parameter regime, visible as a longitudinal spin wave, which
heralds a proximate QCP. Although not evident in the image
plot in Fig. 2a, a peak that matches well the predicted position
of the Higgs mode is clearly seen in the energy spectra plotted
in Fig. 2c.
We further pursue the Higgs mode using spin-polarized INS,
using the scattering geometry that maximizes its neutron crosssection.
We use the standard XYZ-difference method to filter out
all non-magnetic and incoherent scattering signals and to resolve all
three spin-wave polarizations: the longitudinal mode (L) oscillates
along the crystallographic b-axis, and the transverse Goldstone
modes (T and T ) along the a- and c-axes. Because our sample
mosaic consisting of ∼100 crystals is ‘twinned’—that is, approximately
half of them are rotated 90◦
about the c-axis with respect
to the other half—we can distinguish only between in-plane (a–b)
and out-of-plane (c) polarized modes. However, this is sufficient to
identify the Higgs mode (see Supplementary Information).
Figure 3a shows the measured (symbols with error bars) and
calculated (solid lines) dynamical susceptibility at q = (0,0). We
observe three peaks in total as expected, but not all of them
were clearly seen in the TOF data (see also Supplementary Fig. 3)
because their intensities are maximized in different scattering
geometries. The highest-energy peak at approximately 52 meV is
unambiguously identified as the Higgs mode by its magnetic and
in-plane-polarized character, because the second in-plane-polarized
mode at approximately 45 meV has already been identified as the
T mode (Fig. 2). Further, the data are in excellent accord with the
model calculation, which has no adjustable parameter after fitting
the dispersion of the T modes. The intensity ratio between the L
and T modes is 0.55 ± 0.11, which is a quantitative measure of
the proximity to the QCP (Supplementary Fig. 7), at which the
distinction between the L and T modes vanishes and their intensities
become identical.
Having established the existence of the Higgs amplitude mode,
we now look at its long-wavelength behaviour. It is at the ordering
wavevector where the stability of the Higgs mode critically depends
on the dimensionality of the system. In three dimensions, earlier
INS studies on a dimerized quantum magnet have established
a well-defined Higgs mode4
, which was then used to study its
critical behaviour across a QCP23,24
. In sharp contrast, our in-planepolarized
spectrum measured at q = (π,π) shows only one clear
peak for the T mode at approximately 14 meV, followed by a broad
magnetic intensity distribution in the energy range 20–50 meV,
which is, however, well above the detection limit (Fig. 3b). The
Higgs mode has decayed to the extent that a high-flux spin-polarized
neutron spectrometer is required to detect its trace—although in
retrospect a hint of this feature could be found in the unpolarized
data shown in Fig. 3c.
However, it is also known that the response of the Higgs
mode depends strongly on the symmetry of the probe being
used. Therefore, its rapid decay in the longitudinal susceptibility
measured by INS does not necessarily imply its instability in two
dimensions. In fact, it has been shown in other two-dimensional
systems, such as disordered superconductors3
and superfluids of
cold atoms2
, that the Higgs mode is clearly visible in the scalar
susceptibility with its characteristic ∼ω3
onset in the energy
spectrum. Indeed, theory predicts a contrasting behaviour of the
Higgs mode in the scalar and longitudinal susceptibilities; in the
latter, the Higgs mode quickly loses its coherence by decaying into a
pair of Goldstone modes25,26
. This results in an infrared divergence
in two dimensions and renders the Higgs mode elusive.
Conversely, the INS spectrum at q = (π,π) encodes detailed
information on the decay process of the Higgs mode that is not
available from other measurements. To model the decay process,
we go beyond the harmonic approximation used in the spin-wave
634
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
NATURE PHYSICS | VOL 13 | JULY 2017 | www.nature.com/naturephysics
226
NATURE PHYSICS DOI: 10.1038/NPHYS4077 LETTERS
Energy(meV)
0
10
20
30
40
50
60
a (π,0) (π,π) (0,0) (π,0)
(π,0) (π,π) (0,0) (π,0)
0
5
10
15
20
Energy(meV)
Momentum (H, K)
Momentum (H, K)
0
10
20
30
40
50
60
0
5
10
15
20
T
L
T′
(1,0) (0,0)
Energy (meV)
Intensity(a.u.)
10 20 30 40 50 60
(0,0)
(π,0)
b
c
( ,0)
1
2 ( , )
1
2
1
2
( , )
π
2
π
2
( ,0)π
4
( ,0)π
2
( ,0)
3π
4
( , )
π
2
π
2
( , )
1
2
1
2
(1,0) (0,0)( ,0)
1
2 ( , )
1
2
1
2 ( , )
1
2
1
2
Figure 2 | Spin-wave dispersions strongly deviating from the Heisenberg model. a, TOF INS spectra along high-symmetry directions measured at T =5 K
(see Supplementary Fig. 3 for more details). The momenta indicated on the top axis refer to the undistorted square lattice unit cell (see Fig. 1), which is
doubled for the magnetic unit cell. The dotted line is from b for direct comparison between theory and experiment. b, The excitation spectra of the model in
equation (1) calculated with the parameters E 25 meV, J 5.8 meV, α =0.15, 4.0 meV and A 2.3 meV. The spectra were convolved with the
instrumental resolution (4.2 meV full-width at half-maximum). Transverse and longitudinal modes are labelled as ‘T’ and ‘L’, respectively, and their motions
are depicted. The T mode arises from back-folding of the T mode by the magnetic (π,π) scattering and thus its intensity vanishes when approaching the
QCP. The L mode carries the Higgs amplitude oscillation. The latter is not subject to the back-folding because it is insensitive to the staggered vector field.
The black arrows show the momentum- and energy-conserving decay process of the L mode into a pair of T modes. The total momentum of the pair is that
of the L mode shifted by (π,π) momentum provided by the condensate (see Supplementary Information). c, Energy spectra for several q points along
(0,0)–(π,0) obtained using an integration window of ±0.0475 in H and K after symmetrization. The integration ranges were doubled for q=(0,0) to
obtain similar statistics. The red marks indicate the position of the longitudinal mode as calculated by theory. The black solid curves are a guide to the eye.
Error bars denote one standard deviation.
theory to include the coupling of the longitudinal mode to the twomagnon
continuum (see Supplementary Information). The solid
lines in Fig. 3 show the result of the final calculation, which give an
excellent description of the data both at q = (0,0) and q = (π,π);
the decay process (Fig. 2b) is kinematically restricted away from
the ordering wavevector, and the Higgs mode is well identified
at q = (0,0). Although continuum structures are generic to lowdimensional
quantum magnets (for example, Haldane chains27,28
),
the present case is distinct because the continuum structure near
q=(π,π) is derived from a Higgs mode that exists as a well-defined
quasiparticle away from the ordering wavevector.
As mentioned earlier, we have a rather unusual situation where
all the transverse modes are massive (gapped), as a result of the
orthorhombic symmetry of the crystal structure parameterized by
, which significantly modifies the Higgs mode decay process at low
energies. The transverse gap cuts off the infrared singularity and the
spectral weight piles up at non-zero energy. We illustrate this point
in Fig. 4 by simulating the change in the longitudinal spectrum as the
system approaches the QCP. At q = (π,π), the decay of the Higgs
mode into a pair of minimum-energy transverse modes is still the
dominant channel, which generates a ‘resonance’ at twice the energy
of the gap. This resonance steals much of the spectral weight from
the bare longitudinal mode, thus obscuring its spectral signature,
especially near the QCP. As the system moves away from the QCP,
the longitudinal mode progressively hardens and becomes weaker,
and its spectral weight spans a larger energy range. The spectral
evolution at q = (0,0) shows this trend with the decay process
suppressed; the Higgs mode remains a well-defined excitation even
away from the QCP, although its intensity quickly diminishes.
Now that we have established a two-dimensional material system,
future studies can reveal further aspects of the Higgs mode. In
particular, it is uncertain at this point whether the decay process
considered above fully describes its dynamics. In addition to
the multimagnon continuum27–30
, other channels such as decays
into vortex-like excitations are conceivable in two-dimensional
planar magnets and require further investigation. It would be
also interesting to compare the results presented herein with the
spectra from resonant inelastic X-ray scattering, which can in
principle access both the scalar and longitudinal susceptibilities.
Finally, it is interesting to note that the Higgs boson in particle
physics is detected through its decay products, such as pairs of
photons, W and Z bosons, or leptons. The Higgs potential can be
determined through the decay rates and branching ratios of these
processes, which have been calculated to very high precision. Our
NATURE PHYSICS | VOL 13 | JULY 2017 | www.nature.com/naturephysics
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
635
227
LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS4077
Energy (meV)
q = (π,π)
T
L
L
T′
0
0
2
4
6
8
10
0 10 20 30 40 50 60
Energy (meV)
0 10 20 30 40 50 60
χ″(a.u.)χ″(a.u.)
q = (0,0)
T
L
T′
a
ab
c
ab
c
0
2
4
6
8
10
Intensity(a.u.)
0.4 0.6 0.8 1.0 1.2 1.4 1.6
(π, π)
(H,0) (r.l.u.)
T T
Gaussian fit
Continuum
TOF data
b
c ( , )
π
2
π
2 ( , )
3π
2
3π
2
Figure 3 | Identification of the magnetic modes with polarized INS and
their comparison to model calculation. a,b, Imaginary part of the dynamic
spin susceptibility obtained by normalizing the INS spectra measured at
T =2 K at q=(0,0) (a)(Supplementary Fig. 5) and q=(π,π) (b)
(Supplementary Fig. 6) with respect to the orientation factor and the
isotropic form factor of the Ru ion (Supplementary Fig. 4). Blue (red)
symbols indicate in-plane (out-of-plane) polarized magnetic intensities.
Solid symbols show data with the background removed by taking the
difference between two spin-flip channels, and open symbols show data
from a single spin-flip channel (see Supplementary Information). Solid lines
show the calculated spectra, which were convolved using Gaussian
functions with 0.19π and 2.5 meV full-width at half-maximum to account
for the instrumental momentum and energy resolutions, respectively. The
decay process of the L mode into T modes is described in the
Supplementary Information. The shaded area indicates the spectral weight
of the L mode. The intensities in a,b are in the same arbitrary units. The T
mode arises from back-folding of the T mode by the magnetic (π,π)
scattering. c, A feature consistent with the L mode in the unpolarized data;
an H-cut obtained by integrating the data in the range 25≤E≤31 meV,
−0.05≤K ≤0.05, and 0≤L≤6. A quadratic background has been fitted
and subtracted. The solid line is a Gaussian fit to single magnon peaks (T),
and the shaded region indicates the intensity not accounted for by the T
magnons but instead consistent with the longitudinal mode. Error bars
denote one standard deviation.
study represents the first step towards a parallel development in
condensed-matter physics.
Methods
Methods, including statements of data availability and any
associated accession codes and references, are available in the
online version of this paper.
QCPCa2RuO4Heisenberg
χ″(a.u.)
Energy (meV) Energy (meV)
0
2
0
2
4
0
2
4
6
0
4
8
12
0 10 20 30 40 50 60 70
q = (π,π) q = (0,0)
ρ = 0.33
ρ = 0.28
ρ = 0.23
ρ = 0.18
×4
×4
×4
×4
0
2
0
2
0
2
4
0
2
4
6
50 60 70
Figure 4 | Evolution of the Higgs mode towards the QCP. Imaginary part of
the longitudinal susceptibility calculated for several values of J/E, which can
be expressed in terms of the condensate density ρ via the relation
ρ =1/2(1−E/8J). The ρ values of 0.18, 0.23, 0.28, 0.33 approximately
correspond to J/E=0.20, 0.23, 0.28, 0.37, respectively. The dotted lines
show the bare susceptibility before taking the decay process into account.
Received 19 February 2016; accepted 22 February 2017;
published online 27 March 2017
References
1. Pekker, D. & Varma, C. M. Amplitude/Higgs modes in condensed matter
physics. Annu. Rev. Condens. Matter Phys. 6, 269–297 (2014).
2. Endres, M. et al. The ‘Higgs’ amplitude mode at the two-dimensional
superfluid/Mott insulator transition. Nature 487, 454–458 (2012).
3. Sherman, D. et al. The Higgs mode in disordered superconductors close to a
quantum phase transition. Nat. Phys. 11, 188–192 (2015).
4. Rüegg, C. et al. Quantum magnets under pressure: controlling elementary
excitations in TlCuCl3. Phys. Rev. Lett. 100, 205701 (2008).
5. Khaliullin, G. Excitonic magnetism in Van Vleck-type d4
Mott insulators.
Phys. Rev. Lett. 111, 197201 (2013).
6. Meetei, O. N., Cole, W. S., Randeria, M. & Trivedi, N. Novel magnetic state in
d4
Mott insulators. Phys. Rev. B 91, 054412 (2015).
7. Nakatsuji, S., Ikeda, S. I. & Maeno, Y. Ca2RuO4: new Mott insulators of layered
ruthenate. J. Phys. Soc. Jpn 66, 1868–1871 (1997).
8. Braden, M., André, G., Nakatsuji, S. & Maeno, Y. Crystal and magnetic
structure of Ca2RuO4: magnetoelastic coupling and the metal-insulator
transition. Phys. Rev. B 58, 847–861 (1998).
9. Friedt, O. et al. Structural and magnetic aspects of the metal-insulator
transition in Ca2−x Srx RuO4. Phys. Rev. B 63, 174432 (2001).
10. Anisimov, V. I., Nekrasov, I. A., Kondakov, D. E., Rice, T. M. & Sigrist, M.
Orbital-selective Mott-insulator transition in Ca2−x Srx RuO4. Eur. Phys. J. B 25,
191–201 (2002).
11. Fang, Z., Nagaosa, N. & Terakura, K. Orbital-dependent phase control in
Ca2−x Srx RuO4. Phys. Rev. B 69, 045116 (2004).
12. Liebsch, A. & Ishida, H. Subband filling and Mott transition in Ca2−x Srx RuO4.
Phys. Rev. Lett. 98, 216403 (2007).
13. Gorelov, E. et al. Nature of the Mott transition in Ca2RuO4. Phys. Rev. Lett. 104,
226401 (2010).
14. Mizokawa, T. et al. Spin–orbit coupling in the Mott insulator Ca2RuO4.
Phys. Rev. Lett. 87, 077202 (2001).
15. Haverkort, M. W., Elfimov, I. S., Tjeng, L. H., Sawatzky, G. A. & Damascelli, A.
Strong spin–orbit coupling effects on the Fermi surface of Sr2RuO4 and
Sr2RhO4. Phys. Rev. Lett. 101, 026406 (2008).
16. Fatuzzo, C. G. et al. Spin–orbit-induced orbital excitations in Sr2RuO4 and
Ca2RuO4: a resonant inelastic X-ray scattering study. Phys. Rev. B 91,
155104 (2015).
17. Taniguchi, H. et al. Anisotropic uniaxial pressure response of the Mott
insulator Ca2RuO4. Phys. Rev. B 88, 205111 (2013).
18. Nakamura, F. et al. From Mott insulator to ferromagnetic metal: a pressure
study of Ca2RuO4. Phys. Rev. B 65, 220402(R) (2002).
636
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
NATURE PHYSICS | VOL 13 | JULY 2017 | www.nature.com/naturephysics
228
NATURE PHYSICS DOI: 10.1038/NPHYS4077 LETTERS
19. Matsumoto, M., Normand, B., Rice, T. M. & Sigrist, M. Field- and
pressure-induced magnetic quantum phase transitions in TlCuCl3. Phys. Rev. B
69, 054423 (2004).
20. Sommer, T., Vojta, M. & Becker, K. Magnetic properties and spin waves of
bilayer magnets in a uniform field. Eur. Phys. J. B 23, 329–339 (2001).
21. Kunkemöller, S. et al. Highly anisotropic magnon dispersion in Ca2RuO4:
evidence for strong spin orbit coupling. Phys. Rev. Lett. 115, 247201 (2015).
22. Podolsky, D. & Demler, E. Properties and detection of spin nematic order in
strongly correlated electron systems. New J. Phys. 7, 59 (2005).
23. Giamarchi, T., Rüegg, C. & Tchernyshyov, O. Bose–Einstein condensation in
magnetic insulators. Nature 4, 198–204 (2008).
24. Merchant, P. et al. Quantum and classical criticality in a dimerized quantum
antiferromagnet. Nat. Phys. 10, 373–379 (2014).
25. Podolsky, D., Auerbach, A. & Arovas, D. P. Visibility of the amplitude (Higgs)
mode in condensed matter. Phys. Rev. B 84, 174522 (2011).
26. Gazit, S., Podolsky, D. & Auerbach, A. Fate of the Higgs mode near quantum
criticality. Phys. Rev. Lett. 110, 140401 (2013).
27. Kenzelmann, M. et al. Multiparticle states in the S=1 chain system CsNiCl3.
Phys. Rev. Lett. 87, 017201 (2001).
28. Zaliznyak, I. A., Lee, S.-H. & Petrov, S. V. Continuum in the spin-excitation
spectrum of a Haldane chain observed by neutron scattering in CsNiCl3.
Phys. Rev. Lett. 87, 017202 (2001).
29. Stone, M. B., Zaliznyak, I. A., Hong, T., Broholm, C. L. & Reich, D. H.
Quasiparticle breakdown in a quantum spin liquid. Nature 440,
187–190 (2006).
30. Masuda, T. et al. Dynamics of composite Haldane spin chains in IPA-CuCL3.
Phys. Rev. Lett. 96, 047210 (2006).
Acknowledgements
We acknowledge financial support from the German Science Foundation (DFG) via
the coordinated research programme SFB-TRR80, and from the European Research
Council via Advanced Grant 669550 (Com4Com). The experiments at Oak Ridge
National Laboratory’s Spallation Neutron Source were sponsored by the Division of
Scientific User Facilities, US DOE Office of Basic Energy Sciences. J.C. was supported
by GACR (project no. GJ15-14523Y) and by MSMT CR under NPU II project CEITEC
2020 (LQ1601).
Author contributions
M.K., G.H.R. and D.P.C. grew the single crystals. A.J., M.K. and J.P. characterized and
co-aligned the crystals. A.J., M.K., J.P. and B.J.K. performed INS experiments and
analysed the data. D.L.A., J.T.P. and A.I. supported the INS experiments. G.K. developed
the theoretical model. J.C. and B.J.K. performed the numerical calculations. B.J.K. wrote
the manuscript with contributions from G.K., J.C., B.K., J.P., A.J. and M.K. and
discussions with all authors. B.J.K. and B.K. managed the project.
Additional information
Supplementary information is available in the online version of the paper. Reprints and
permissions information is available online at www.nature.com/reprints. Publisher’s note:
Springer Nature remains neutral with regard to jurisdictional claims in published maps
and institutional affiliations. Correspondence and requests for materials should be
addressed to B.K. or B.J.K.
Competing financial interests
The authors declare no competing financial interests.
NATURE PHYSICS | VOL 13 | JULY 2017 | www.nature.com/naturephysics
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
637
229
LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS4077
Methods
Sample synthesis and characterization. Single crystals of Ca2RuO4 were grown by
the floating zone method with RuO2 self-flux31
. The lattice parameters a=5.409 Å,
b=5.505 Å, and c =11.9312 Å were determined by X-ray powder diffraction, in
good agreement with the parameters reported in the literature8
for the ‘S’ phase
with short c-axis lattice parameter. The magnetic ordering temperature TN =110 K
was determined using magnetization measurements in a Quantum Design
SQUID-VSM device. Polarized neutron diffraction measurements indicate that
most of the array orders in the ‘A-centred’ magnetic structure with magnetic
ordering vector Q=(1,0,0)8
. The fraction of the sample with ordering vector
Q=(0,1,0), that is, ‘B-centred’, is estimated to be less than 5%.
Time-of-flight inelastic neutron scattering. For the TOF measurements,
we co-aligned about 100 single crystals with a total mass of ∼1.5 g into a mosaic on
Al plates. Approximately half of the crystals were rotated 90◦
about the c-axis from
the other half (Supplementary Fig. 2). The in-plane and c-axis mosaicities of the
aligned crystal assembly were 3.2◦
and 2.7◦
, respectively. The measurements
were performed on the ARCS time-of-flight chopper spectrometer at the Spallation
Neutron Source, Oak Ridge National Laboratory, Tennessee, USA. The incident
neutron energy was 100 meV. The Fermi chopper and T0 chopper frequencies were
set to 600 and 90 Hz, respectively, to optimize the neutron flux and energy resolution.
The measurements were carried out at T =5 K. The sample was mounted with the
(H,0,L) plane horizontal. The sample was rotated over 90◦
about the vertical c-axis
with a step size of 1◦
. At each step data were recorded over a deposited proton
charge of 3 Coulombs (∼45 minutes) and then converted into 4D S(Q,ω) using
the HORACE software package32
and normalized using a vanadium calibration.
Polarized inelastic neutron scattering. Preliminary triple-axis measurements, to
reproduce the TOF results and determine the feasibility of the polarized
experiment, were done on the thermal triple-axis spectrometer PUMA at the
FRM-II, Garching, Germany. The measurements were done on crystals from the
same batch as the ones used for the TOF experiment. To optimize the flux and
energy resolution, double-focused PG (002) and Cu (220) monochromators, for
measurements below and above 30 meV, respectively, and a double-focused PG
(002) analyser were used, keeping kf =2.662 Å−1
constant. For the polarized
triple-axis measurement we remounted the crystals from the TOF experiment on Si
plates and increased the number of crystals to obtain a total sample mass of ∼3 g.
The mosaicity of this sample was 3.2◦
and 2.6◦
for in-plane and c-axis,
respectively. The experiment was performed on the IN20 three-axis spectrometer
at the Institute Laue-Langevin, Grenoble, France. For the XYZ polarization
analysis, we used a Heusler (111) monochromator and analyser in combination
with Helmholtz coils at the sample position. Throughout the experiment we used a
fixed kf =2.662 Å−1
and performed polarization analysis in energy and H scans at
(π,π) and (0,0), keeping L as small as permitted by kinematic constraints. The
measurements were carried out at T =2 K.
Data availability. The data that support the plots within this paper and other
findings of this study are available from the corresponding author upon reasonable
request. The polarized inelastic neutron scattering data are available at
http://doi.ill.fr/10.5291/ILL-DATA.4-01-1431.
References
31. Nakatsuji, S. & Maeno, Y. Synthesis and single-crystal growth of Ca2−x Srx RuO4.
J. Solid State Chem. 156, 26–31 (2001).
32. Ewings, R. A. et al. HORACE: software for the analysis of data from single
crystal spectroscopy experiments at time-of-flight neutron instruments.
Nucl. Instrum. Methods Phys. Res. A 884, 132–142 (2016).
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
NATURE PHYSICS | www.nature.com/naturephysics
230
In the format provided by the authors and unedited.
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
Supplementary Information:
Higgs mode and its decay in a two-dimensional antiferromagnet
A. Jain, M. Krautloher, J. Porras, G. H. Ryu, D. P. Chen, D. L. Abernathy,
J. T. Park, A. Ivanov, J. Chaloupka, G. Khaliullin, B. Keimer, and B. J. Kim
A. Microscopic model
We derive here the phenomenological model in eq. (1)
starting from the microscopic electronic structure. The
compressive tetragonal distortion ∆, is the key parameter
that determines the proximity of Ca2RuO4 to the QCP,
because the spin-orbit splitting λ is known for Ru(IV)
ion, and the nearest-neighbor exchange coupling J is to
a large extent fixed by the measured bandwidth W 2zJ
of the spin-wave (z is the coordination number). In the
absence of ∆, the low energy physics is described in terms
of a singlet-triplet model7
, formally similar to that used
for dimerized quantum magnets23
, such as TlCuCl3 and
BaCuSi2O6. The magnetic transitions in these systems
have been extensively studied as a Bose-Einstein condensation
of triplons, where the magnetic field H plays the
role of the chemical potential µ. In our case, ∆ plays the
role of µ (see Fig. S1); it splits the triplets into a doublet
and a singlet and thereby lowers the energy cost E to
create an exciton (Tx or Ty). The quantum phase transition
occurs when E W; the equality holds for classical
consideration. With the free-ion value λ ≈ 75 meV and
W ≈ 45 meV, we estimate that QCP is at δ(≡∆/2λ) ≈ 1.
Because Ca2RuO4 is on the right hand side of the QCP
where the Tz singlet is in very high energy and hence can
be neglected, the low-energy physics of Ca2RuO4 can be
λ=
ξ
Tx
,Ty
Tz
s S=1
E W
Tx,y,z
δ(=Δ/2λ,)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
1
2
3
Energy(λ)
spin-orbit singlet
QCP
Heisenberg AF
δ ∞
2
FIG. S1. Microscopic mechanism of the QCP driven
by tetragonal lattice distortion. Crystal-field splitting of
the triplet effectively lowers the energy scale of SOC from λ
to E. λ is equal to one-half of the single-electron SOC ξ for
d4
low-spin electron configuration. The QCP occurs when E
becomes equal to the strength of the exchange field W ≈ 2zJ.
The blue shading indicates the region where the effective S = 1
model is valid.
described by the three levels {s, Tx, Ty}. These three
levels constitute the effective S = 1 degrees of freedom in
the phenomenological model in eq. (1). We note that the
large energy scale of E microscopically originates from λ
and depends on δ.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.2
0.4
0.6
0.8
1.0
J A
α
η=0
η=0.2
η=0
η=0.2
δ
0.0
0.2
0.1
0.0
-0.2
-0.2
0.5 1.0 1.5 2.0 2.5 3.0
δ
a
b
0
20
40
60
80
y
X
Z
T
T’L
0
20
40
60
80
0
20
40
60
80
0
20
40
60
80
δ=0, ρ=0
δ=0.4, ρ=0
δ=0.5, ρ=0.02
δ=1.0, ρ=0.14
δ=1.5, ρ=0.23
δ=3.0, ρ=0.36
δ=10, ρ=0.46
δ=104
, ρ=0.5
0
20
40
60
80
80
0
20
40
60
0
20
40
60
80
0
20
40
60
80
(π,π) (0,0)(π,0) (π,0) (π,π) (0,0)(π,0) (π,0)
Energy(meV)
Momentum
π
2
π
2( , ) π
2
π
2( , )
FIG. S2. Coupling constants and spin-wave dispersion
from the microscopic model. a, J and A (see eq. 1) are
in units of t2
/U. b, Evolution of the spin-wave dispersions
from the non-magnetic spin-orbit singlet to the Heisenberg
limit as a function of δ. η = 0.25 was used. A non-zero ρ, the
condensate density, indicates the magnetic order. At δ = 0.5,
the magnetic order has set in; now the Tx becomes the transverse
mode and Ty the longitudinal mode. At δ = 1.5, the
simulation is very similar to the experimental data. Further
increase of δ leads to a development of a local minimum at
q = (0,0), and eventually to a Heisenberg-like dispersion.
SUPPLEMENTARY INFORMATION
DOI: 10.1038/NPHYS4077
NATURE PHYSICS | www.nature.com/naturephysics  1
231
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
2
-1.0
0.0
1.0
-1.0
0.0
1.0
-1.0
0.0
1.0
-1.0
0.0
1.0
-1.0
0.0
1.0
-1.0
0.0
1.0
-1.0
0.0
1.0
-1.0
0.0
1.0
-1.0 0.0 1.0 -1.0 0.0 1.0 -1.0 0.0 1.0 -1.0 0.0 1.0
-1.0 0.0 1.0 -1.0 0.0 1.0 -1.0 0.0 1.0 -1.0 0.0 1.0
806040200
806040200
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
806040200
806040200
H (r.l.u.)H (r.l.u.)H (r.l.u.)
(π,π) (π,π)
(π,π) (π,π)(π/2,π/2)
(H,0) (H,0.5) (H,-H ) (H,-H-1)
K(r.l.u.)Energy(mEV)K(r.l.u.)
H (r.l.u.)
cut 1 cut 2 cut 3 cut 4
2
1
3
4
a b c d
e f g h
i j k l
(0,0) (0,0) (0,0) (0,0)
(π,0) (π,0) (π,0) (π,0)(π,0)
15 meV 20 meV 25 meV 30 meV
35 meV 40 meV 45 meV 50 meV
20
15
10
5
0
FIG. S3. Time-of-flight INS spectra of Ca2RuO4. a-h, Constant energy maps of INS intensity in the (H,K) plane with
L integrated over the range 0 ≤ L ≤ 6. The wave-vectors are expressed in reciprocal lattice units (r.l.u.) of a tetragonal cell
with a∗
= b∗
= 1.16 ˚A−1
and c∗
= 0.53 ˚A−1
. At E = 14 meV, intense spin waves are observed at the reduced wave-vectors (±1, 0)
and (0, ±1), which correspond to the antiferromagnetic ordering wave-vector (π, π) for the 2D square lattice Brillouin-zone. i-l,
Energy spectra along high-symmetry directions as shown in blue lines in panel a. The intensity is in arbitrary units.
In terms of the z projections of the S=1 and L=1 moments
|SM , LM , the wave functions for the basis states
{s, Tx, Ty} with Tx = 1
i
√
2
(T1 − T−1), Ty = 1√
2
(T1 + T−1)
are given by
|s = sin θ0
1√
2
(|1, −1 + | − 1, 1 ) − cos θ0 |0, 0 , (1)
|T+1 = cos θ1|1, 0 − sin θ1|0, 1 , (2)
|T−1 = sin θ1|0, −1 − cos θ1| − 1, 0 . (3)
Here the angles θ0, θ1 are defined through
tan θ1 =
1
δ +
√
1 + δ2
, tan θ0 = 1 + β2 − β, (4)
where β = 1√
2
(δ− 1
2 ). The energy E in eq. (1) and Fig. S1
is given by
E =
ξ
2
√
2
β + 1 + β2
−
1
δ +
√
1 + δ2
. (5)
Using the above wavefunctions in the standard secondorder
perturbation theory, we calculate the coupling constants
incorporating the Hund’s coupling η = JH/U measured
in units of the Coulomb interaction U. Figure
S2a shows the exchange constant J, two-ion XY-type
anisotropy α, and pseudodipolar interaction A as functions
of δ, for η = 0 and η = 0.2; the latter would be more
realistic. The calculation shows that α 1 in the entire
range of δ where the model is relevant to Ca2RuO4, insensitive
to the value of η, confirming that the XY-type
anisotropy due to two-ion exchange is small. Thus, the
single-ion term E is mostly responsible for the XY-type
anisotropy.
Using these coupling constants, we simulate in Fig. S2b
the evolution of the spin-wave spectra over the entire
phase diagram from the non-magnetic singlet to the
Heisenberg limit using δ as the only tuning parameter.
In the simulation, we used λ = 50 meV, η = 0.25, and
t2
/U = 5.75 meV. Additionally the term in eq. (1) was
added to reproduce the transverse mode gap of ≈14 meV.
Note that at δ = 1.5, the simulated spectra becomes very
similar to the experimental spectra. The above parameters
translate to J 5.2 meV, α = 0.1, E = 21.5 meV and
NATURE PHYSICS | www.nature.com/naturephysics	 2
SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS4077
232
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
3
A 1.0 meV, which are in excellent agreement with those
found from the fitting, considering that the model is minimal
and the coupling constants absorb various renormalization
effects in the solid not taken into account
in the microscopic model. In particular, A absorbs the
effect of further neighboring couplings, which also contribute
to the dispersion along the magnetic zone boundary
(π/2,π/2)-(π,0).
B. Time-of-flight inelastic neutron scattering
Figures S3a-h and S3i-l exhibit constant-energy maps
and energy spectra along high-symmetry directions, respectively,
measured by TOF INS comprising spin-wave
dispersions in the energy range 14 <
∼ ¯hω <
∼ 45 meV and
phonon modes above ∼50 meV. The magnetic nature of
the former is explicitly confirmed by using spin-polarized
neutrons (Fig. 3), and the non-magnetic nature of the latter
is inferred from exhaustion of all magnetic modes and
also through comparison with the known phonon modes.
The data has been integrated along L because the magnetic
excitations are close to the 2D limit, which can also
be seen from the narrow linewidth of the excitations after
the integration. Indeed, a recent INS study shows that
the excitations at (π, π) are almost dispersionless (less
than 1 meV), with no significant change in amplitude21
.
Figures S3m and S3n show energy spectra and an H-cut,
respectively, indicated in which are intensities consistent
with the longitudinal mode.
C. Polarization analysis
In the standard reference frame for the neutron polarization
with ˆx Q, ˆy ⊥ Q in the scattering plane of the
spectrometer and ˆz = ˆx × ˆy, the magnetic intensity in
the spin flip channels is extracted from the differences:
My = Ix − Iy, (6)
Mz = Ix − Iz,
where Ix, Iy, Iz are the raw intensities of the respective
polarizations. Note that any contribution from the
background is suppressed in the difference. For conversion
from INS intensity to dynamic spin susceptibility,
we used the isotropic form factor for Ru+
, which gave a
good description of the data at 15 meV (Fig. S4).
C.1 Flipping ratio
The flipping ratio is estimated using the strong magnetic
signal at 15 meV to be around F = 10; minor differences
in the flipping ratio for different channels Fx = 14.8,
Fy = 14.4 and Fz = 14.1 measured using several nuclear
Bragg reflections, do not alter the analysis significantly.
All data were taken in regions where the signal in the
non-spin flip channel ˆx was at the background level, and
therefore any leakage from non-spin flip processes can be
neglected in the differences.
C.2 Twinning ratio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
qc
2
My / f 2
(Q)
Mz / f 2
(Q)
NormalizedIntensity
FIG. S4. Determination of the twinning ratio. Magnetic
intensities My(blue squares) and Mz(red circles) normalized
by the squared magnetic form factor as a function of q2
c at
Q = (1,0,L) with L = 2, 4, and 6 at energy transfer 15 meV.
For one type of domain the intensity is constant, and for the
other type of domain the intensity increases linearly with q2
c .
A one-parameter fit (red and blue solid lines) to the data
points determines the twinning ratio p.
In this study the a and b orientation of the crystals in
the array are not distinguished. In other words, for the
volume fraction p of the sample the scattering plane is
(H,0,L), and for the fraction (1 − p) the scattering plane
is (0,H,L). Taking into account the polarization factor,
the intensities in each channel are related to excitations
Ma, Mb and Mc along the crystallographic directions by:
My = q2
c [p Ma + (1 − p) Mb] + 1 − q2
c Mc (7)
Mz = (1 − p) Ma + p Mb
where q2
c = (Qc/|Q|)
2
.
The twinning ratio p can be estimated from rocking
scans through the Bragg reflections (4,0,0) and (0,4,0)
where the separation in the scattering angle is large
enough to distinguish the two peaks (not shown). Alternatively,
p can be estimated from the inelastic measurements
by considering the L-dependence of the 15 meV
feature at (π,π) as shown in Fig. S4. Since this is an inplane
transverse mode, Mb and Mc vanish and eq. (14)
greatly simplifies. From the one-parameter fit to the
data, a twinning ratio p = 0.498 ± 0.014 is determined,
consistent with the first method. For the analysis we
used p = 0.5.
From polarization analysis on this “twinned” array,
only in-plane (ab) or out-of-plane (c) polarization can
be distinguished, as Ma and Mb give equal contributions
in each channel. Nevertheless, two in-plane-polarized
modes T and L and one out-of-plane-polarized mode
T are expected which are non-degenerate. We can indeed
distinguish them from the energy scans at (0,0) and
(π, π).
C.3 Energy scans at q = (0,0)
For the energy scans at q = (0,0), it is useful to use two
NATURE PHYSICS | www.nature.com/naturephysics	 3
SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS4077
233
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
4
0 10 20 30 40 50 60
-5
0
5
10
15
20
bQ =(2,0,0.4)
M y
M z
Q =(0,0,L)
M y
M z
a
Energy (meV)
Intensity(arb.unit)
FIG. S5. Energy scans at q = (0,0). Magnetic intensities
a, at Q = (2, 0, 0.4), and b, at Q = (0, 0, L). The value of
L was varied along the scan to minimize the magnitude of
Q. Blue squares denote My, red circles Mz, and the lines are
guides to the eye.
different Brillouin zones to maximize the intensity for the
different modes. The measurements at Q = (2,0,0.4) at
15 meV give conclusive evidence for the folded mode T
(Fig. S5a), as the out-of-plane polarization gives rise to
a signal in My but not in Mz. We confirmed that the
signal is peaked at (2,0,0.4) by scanning along the H direction
(not shown). To avoid a sharp spurion a small L
component was used. For the energy scan at Q = (0,0,L),
shown in Fig. S5b, the signal exclusively originates from
in-plane polarized modes. We observe two magnetic excitations
clearly separated in energy, both with equal contributions
from the My and Mz channels. Given the dispersion
obtained from TOF, the peak at 45 meV is unambiguously
assigned to the transverse mode T; thus the
peak at 52 meV must be associated with the longitudinal
mode L.
C.4 Energy scans at q = (π, π)
Energy scans at Q = (1,0,L) shown in Fig. S6, corresponding
to q = (π, π) of the tetragonal unit cell, reveal
three magnetic excitations above a gap of 14 meV. For
the two features lowest in energy we observe a signal in
both My and Mz channels, characterizing them as inplane
polarized magnetic excitations. The third mode is
unambiguously identified as the folded mode as the polarization
factor suppresses the intensity in the Mz channel
completely for out-of-plane excitations. The My and Mz
signals allow separation of the in-plane and out-of-plane
responses, because the My signal exclusively originates
from in-plane polarized modes, whereas the Mz signal
has contributions from both in-plane and out-of-plane
polarized modes.
D. Proximity to the QCP
To quantify the proximity to the QCP, we introduce
τ = J/Jcr ≈ 8J/E. At the QCP (τ = 1), where the distinction
between transverse and amplitude modes vanEnergy
(meV)
c
0 10 20 30 40 50 60
0
50
100
150
200
Ix
-BG
My
+Mz
My
Mz
0
5
10
20 30 40 50 60
0
5
10
Energy (meV)
Intensity(arb.units)
Intensity(arb.unit)
FIG. S6. Energy scan at q = (π,π). The value of L was varied
along the scan to minimize the magnitude of Q. The data
denoted Ix-BG (empty black diamonds) is obtained from the
raw data in the Mx channel after subtraction of a small background;
this method is only reliable when the signal is much
larger than the background. The intensities My+Mz(filled
black triangles), My (blue squares), and Mz(red circles) are
obtained using eq. (6) and the lines are guides to the eye. The
inset shows in detail the region above 20 meV for My (top)
and Mz(bottom). Dashed lines represent the tail of the main
transverse mode.
ishes, their intensity ratio at q = (0,0) is 1 (equality
holds when the gap is zero), and approaches zero as the
moment saturates (Fig. S7). The measured intensity ratio
of 0.55 ± 0.11 translates to τ ≈ 1.8. In principle, the
size of the static moment contains the same information,
but only after corrections due to g-factors, covalency, and
quantum fluctuations, have been properly taken into account,
which are model-dependent and fraught with systematic
uncertainties.
m/g
L/T
τ
τ ∞
0.0
0.2
0.4
0.6
0.8
1.0
1.0 1.5 2.0 2.5 3.0
FIG. S7. Quantification of the proximity to the QCP.
Evolution of the intensity ratio between the amplitude (L) and
the transverse (T) modes at q = (0,0), and the static magnetic
moment normalized by the in-plane g-factor as a function of
τ = J/Jcr. τ = 1 at the QCP.
NATURE PHYSICS | www.nature.com/naturephysics	 4
SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS4077
234
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
5
E. Mode dispersions and intensities
The excitation spectra for the model in eq. (1) formulated
in the basis {s, Tx, Ty} were calculated using the modified
spin-wave theory for the models where the QCP is associated
with triplet condensation (see refs. 19,20). The
energy and magnetic intensity of the longitudinal mode
obtained within the harmonic approximation reads as
ωLq = W 1 +
γq
τ2
, ILq ∝
1
τ
1
τ2 + γq
, (8)
where W = 8J is the energy scale and γq = 1
2 (cos qx +
cos qy). This mode is most dispersive and intense near
the QCP (τ ∼ 1) while in the rigid-spin limit (τ 1),
it flattens and vanishes. To describe the main (T) and
folded (T ) transverse modes, we introduce two auxiliary
quantities
aq = 1
2 W(1 + 1
τ )(1 + γq) + ,
bq = 1
2 W(1 + 1
τ ) 1 − τ−1
τ+1 (1 − α)γq + . (9)
Then the energy and intensity of the T mode may be
expressed as
ωT q = aqbq , IT q ∝
1
τ
τ + 1
2
bq
aq
(10)
and for the T mode we have
ωT q = ωT ˜q , IT q ∝
1
τ
τ − 1
2
a˜q
b˜q
, (11)
where ˜q = q + (π, π). The intensity contrast between
the T and T modes is most pronounced in the softspin
situation where τ ∼ 1. At the crossing point of their
dispersions, q = (π
2 , π
2 ), γq is zero and IT /IT becomes
(τ − 1)/(τ + 1), vanishing as τ → 1. Note that in the
standard Heisenberg or XY models the intensity ratio is
1 (consider τ → ∞). For the relative intensity of the L
and T mode at q = (0, 0), used to quantify the proximity
to the QCP, we get IL/IT = 2/ (τ + 1)(τ2 + 1) (α = 0,
= 0) corrected by a multiplicative factor 1− τ2
2(τ+1) W for
small nonzero . Non-zero pseudodipolar term in eq. (1)
mixes the L mode and T mode leading to two modes with
the modified dispersions
ω2
1,2q =
ω2
Lq + ω2
T q
2
±
ω2
Lq − ω2
T q
2
2
+ c2
q , (12)
where c2
q = W3
bq(A/2Jτ)2
(cos qx − cos qy)2
. Due to the
d-wave type form-factor (cos qx − cos qy), this correction
is only relevant near the (π, 0) area.
The two-dimensional situation requires us to go beyond
the harmonic approximation for the amplitude mode. Its
coupling to the two-magnon continuum modifies the bare
susceptibility
χL0(q, ω) =
W
2(ω2
Lq − ω2)
(13)
associated with the amplitude mode as χ−1
L = χ−1
L0 − ΠL.
Collecting the leading terms, the self-energy Π is obtained
as
ΠL(q, ω) =
k
M2
Lkk bkbk (ω−1
T k + ω−1
T k )
(ωT k + ωT k )2 − (ω + iΓ)2
. (14)
Here k = −k + q + (π, π) and the matrix element
M2
Lkk =
W2
4
1 −
1
τ2
γq
τ
+
γk + γk
2
2
. (15)
Note that the (π, π) shift in the momentum conservation
relation above arises due to the condensate acting as a
source and sink of the (π, π) momentum. In the calculations,
we have used the broadening parameter Γ = 6meV.
The self-energy is largest for q ≈ (π, π), where the dominant
contribution comes from k ≈ −k ≈ (π, π) (supported
by both small ωT and large M2
kk ), and turns
the amplitude mode into a broad feature. A sizable gap
ωT (π,π) of the spin-wave dispersion prevents the infrared
singularity of Π, whose imaginary part would diverge like
1/ω in the gapless case. In our case it is zero below
the cutoff energy 2ωT (π,π) comparable to W making the
above perturbative approach well controlled.
The T mode is the subject of a similar, albeit much
smaller renormalization due to a coupling to the L mode.
The relevant bare susceptibility χT 0(q, ω) = 1
2 bq/(ω2
T q −
ω2
) is modified according to χ−1
T = χ−1
T 0 − ΠT by employing
the self-energy
ΠT (q, ω) =
k
M2
T kk Wbk (ω−1
Lk + ω−1
T k )
(ωLk + ωT k )2 − (ω + iΓ)2
(16)
containing the matrix element
M2
T kk =
W2
8
τ + 1
τ2(τ − 1)
1+γq −γk −γq−k +2τ
W
2
.
(17)
Note that, as a consequence of the rotational symmetry,
this coupling vanishes for the q = (π,π) magnons in the
gapless situation ( = 0).
NATURE PHYSICS | www.nature.com/naturephysics	 5
SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS4077
235
236
Raman Scattering from Higgs Mode Oscillations in the Two-Dimensional
Antiferromagnet Ca2RuO4
Sofia-Michaela Souliou,1,2
Jiˇrí Chaloupka,3,4
Giniyat Khaliullin,1
Gihun Ryu,1
Anil Jain,1,6
B. J. Kim,1,7,8
Matthieu Le Tacon,1,5
and Bernhard Keimer1
1
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
2
European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble Cedex, France
3
Central European Institute of Technology, Masaryk University, Kamenice 753/5, 62500 Brno, Czech Republic
4
Department of Condensed Matter Physics, Faculty of Science, Masaryk University, Kotláˇrská 2, 61137 Brno, Czech Republic
5
Karlsruhe Institute of Technology, Institut für Festkörperphysik, D-76021 Karlsruhe, Germany
6
Solid State Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India
7
Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Republic of Korea
8
Center for Artificial Low Dimensional Electronic Systems, Institute for Basic Science (IBS),
77 Cheongam-Ro, Pohang 790-784, Republic of Korea
(Received 12 April 2017; revised manuscript received 30 May 2017; published 8 August 2017)
We present and analyze Raman spectra of the Mott insulator Ca2RuO4, whose quasi-two-dimensional
antiferromagnetic order has been described as a condensate of low-lying spin-orbit excitons with
angular momentum Jeff ¼ 1. In the Ag polarization geometry, the amplitude (Higgs) mode of the spinorbit
condensate is directly probed in the scalar channel, thus avoiding infrared-singular magnon
contributions. In the B1g geometry, we observe a single-magnon peak as well as two-magnon and
two-Higgs excitations. Model calculations using exact diagonalization quantitatively agree with the
observations. Together with recent neutron scattering data, our study provides strong evidence for excitonic
magnetism in Ca2RuO4 and points out new perspectives for research on the Higgs mode in two dimensions.
DOI: 10.1103/PhysRevLett.119.067201
The notion of Goldstone and Higgs modes, corresponding
to phase and amplitude oscillations of a condensate
of quantum particles, appears in many areas of physics
including magnetism [1]. In quantum magnets, especially
near quantum criticality [2], the magnetization density is far
from being saturated and, hence, allowed to oscillate near
its mean value, forming a collective amplitude mode.
The “magnetic” Higgs mode has been observed [3] in
quantum dimer systems, where the magnetic order is due to
Bose-Einstein condensation of spin-triplet excitations [4]. A
conceptually similar, but physically distinct case is expected
in Van Vleck–type Mott insulators, where the “soft”
moments result from condensation of spin-orbit excitons
[5], that is, magnetic transitions between spin-orbit Jeff ¼ 0
and Jeff ¼ 1 levels propagating via exchange interactions.
Recent inelastic neutron scattering (INS) experiments [6]
on Ca2RuO4 have indeed revealed Higgs oscillations of
the magnetization in this material, which is based on
nominally nonmagnetic, spin-orbit singlet Ru4þ
ions. A
detailed analysis of the dispersion relations of the Higgs
mode and magnons determined by INS showed that
Ca2RuO4 is close to a quantum critical point associated
with the condensation of Jeff ¼ 1 excitons [6].
The unique aspect of Ca2RuO4 is that it hosts Higgs
physics in a two-dimensional setting, which has been a focus
of many theoretical studies [7–15]. As the magnetization
density is not a conserved quantity, the Higgs mode is not
symmetry protected, and various decay processes convert it
into a many-body resonance with ∼ω3
onset. It was also
emphasized [7] that the actual appearance of this resonance
strongly depends on the symmetry of the probe. In INS
experiments, which probe the longitudinal magnetic susceptibility,
the low-energy behavior of the Higgs resonance
is masked by the infrared-singular two-magnon contribution.
To avoid contamination by the Goldstone modes, the probe
should couple to the condensate in the scalar channel (i.e.,
insensitively to the phase or direction). Precisely this type of
experiment has been done in ultracold atomic systems [16].
In this Letter, we demonstrate that Raman light scattering
in the fully symmetric, i.e., Ag channel can serve as a scalar
probe in magnetic systems, thus providing direct access to
Higgs oscillations of soft moments. While in conventional
Heisenberg magnets with rigid spins (such as La2CuO4 or
Sr2IrO4) the Ag channel is magnetically silent, the size of
the local moments, and hence the magnetization density in
excitonic systems is determined by a balance between the
spin-orbit λ and exchange J interactions [5,6], and the Ag
modulation of the latter directly shakes the condensate
density.
The Raman scattering data in Ca2RuO4 presented below
indeed reveal a pronounced magnetic contribution in the Ag
channel, which we identify and describe using the same
excitonic model that has already been parametrized in the
INS study [6]. In the B1g channel, we observe the expected
PRL 119, 067201 (2017) P H Y S I C A L R E V I E W L E T T E R S
week ending
11 AUGUST 2017
0031-9007=17=119(6)=067201(5) 067201-1 © 2017 American Physical Society
237
two-magnon scattering and an additional two-Higgs scattering
contribution, as well as a single-magnon peak. All the
observations are coherently explained by model calculations.
Experiment.—Single crystals of Ca2RuO4 with TN ¼
110 K were grown by a floating zone method, as described
elsewhere [17]. The Raman data were recorded on a Labram
(Horiba Jobin-Yvon) single-grating Raman spectrometer,
using the 632.817 nm line of a Heþ
=Neþ
mixed gas laser.
The experiments were performed in backscattering geometry
along the crystallographic c axis. Ca2RuO4 crystallizes in
the orthorhombic Pbca-D15
2h space group. Excitations in the
B1g and Ag representations of the point group D2h were
probed in crossed and parallel configurations, respectively,
with the polarization of the incident light at 45° to the Ru-Ru
bonds [see Figs. 1(c) and 2(c)]. The spectra were corrected
for the Bose thermal factor to obtain the Raman response
functions χ00
ðωÞ.
Temperature-dependent χ00
ðωÞ spectra in the range of 5 to
110 meVare plotted in Figs. 1 and 2. The frequencies of the
observed phonon modes are in good agreement with previous
Raman studies [18]. The phonon modes are superimposed on
top of a broad continuum. As the temperature is lowered, the
continuumevolves into distinct spectral features B, B0
(Fig.1)
and A, A0
(Fig. 2). The temperature dependence of the new
features follows closely that of the magnetic order parameter
and strongly suggests their magnetic origin. The fact that
these excitations are well inside the optical gap exceeding
0.5 eV [19] further supports this interpretation.
More specifically, in the B1g channel, the feature B
appears around 12 meVand gradually sharpens [Fig. 1(b)].
Earlier Raman studies attributed it either to two-magnon
scattering [20,21] or to a zone-boundary folded phonon in
the magnetically ordered state [18]. However, we find
below that the two-magnon scattering is represented by the
B0
structure around 80 meV, while the B peak is identified
as a single-magnon excitation.
In the Ag channel, the A structure in the range of 25
to 50 meV develops in the magnetically ordered state
[Fig. 2(b)]. The phonon modes in this spectral region
exhibit pronounced Fano-type asymmetric line shapes—a
clear signature of the presence of a continuum of excitations
coupled to the phonons. As noticed above, the large
optical gap implies a magnetic origin of the continuum.
Extraction of the magnetic response.—We adopt the
Green’s function approach [22–24] to the Raman response
of the coupled system of phonons and a continuum. We
describe the system by a matrix propagator whose inverse
G−1
ðωÞ contains the response functions of the magnetic
½G−1
ðωފ00 ¼ RðωÞ þ iSðωÞ and phonon ½G−1
ðωފnn ¼
ωn − ω − iΓn (n ¼ 1…N) subsystems as the diagonal
elements. The coupling between phonon n and the
continuum is provided by nondiagonal matrix elements
0 20 40 60 80 100
0 200 400 600 800 cm
-1
Ramanresponseχ″(ω)[a.u.]
ω [meV]
295K
230K
175K
130K
110K
95K
70K
50K
30K
20K
T=10K
B1g − crossed geometry )b()a(
(c)
10 15 20meV
50 100 150 cm
-1
95K
70K
50K
30K
10K
J
J
y
in
out
x
B
B’
B +δ
−δ
y
−∝ x
FIG. 1. (a) Raman spectra in B1g scattering geometry with two
magnetic features B, B0 appearing below TN. The background
(dashed line) is subtracted in further analysis. (b) Detailed view
on the feature B. (c) Polarization vectors of incoming and
outgoing photons with respect to the Ru lattice. In the Raman
process, the exchange is modulated with opposite signs on x and
y bonds leading to the Raman operator R ∝ Hx − Hy.
Ramanresponseχ″(ω)[a.u.]
ω [meV]
295K
230K
175K
130K
110K
95K
70K
50K
30K
20K
T=10K
Ag − parallel geometry(a) (b)
(c)
0 20 40 60 80 100
0 200 400 600 800 cm
-1
95K
50K
10K
10 20 30 40 50meV
100 200 300 400 cm
-1
J
J
A
A A’
y
xin
out
+δ
+δ
y
+∝ x
FIG. 2. (a) Raman spectra in Ag scattering geometry with two
magnetic features A, A0
appearing below TN. (b) Detailed view on
the feature A. (c) The polarization vectors in the Ag setup. In this
case, the exchange is modulated equally on x and y bonds and the
Raman operator R ∝ Hx þ Hy.
PRL 119, 067201 (2017) P H Y S I C A L R E V I E W L E T T E R S
week ending
11 AUGUST 2017
067201-2
238
½G−1
ðωފn0 ¼ ½G−1
ðωފ0n ¼ Vn. After invertingG−1
ðωÞ, the
Raman response is obtained as χ00ðωÞ ¼
PN
j¼0 Wj
½ImGðωފjj, where Wj are spectral weights of the normal
modes of the coupled spin-phonon system.
The magnetic response functions SðωÞ, determined by
fitting χ00
ðωÞ to the low-temperature spectra, are presented
in Fig. 3. While in the B1g case the above procedure just
confirms the expected result, in the Ag case it proved
essential to obtain the actual SðωÞ profile. The feature A is
found to be peaked at about 40 meVand has a long tail that
merges with the high-energy continuum (A0
), much flatter
than the B1g one (B0).
Magnetic model.—In the following, we give a quantitative
interpretation of the magnetic features using the excitonic
model of Ref. [5], refined further by a comparison to INS
data [6]. The model utilizes the local basis depicted in
Fig. 4(b) stabilized by intraionic spin-orbit coupling. The
dominant energy scale corresponds to the energy cost ET of a
triplon T (derived from Jeff ¼ 1 states) relative to that of the
singletgroundstate s (Jeff ¼ 0).Itscompetitionwiththespinorbital
exchange interaction results in a quantum critical point
separating the paramagnetic phase (dilute “gas” of T on top of
an s background) and the antiferromagnetic phase (condensate
with coherently mixed T and s). In terms of hardcore
bosons s and Tx=y associated with the relevant low-energy
levels and obeying the local constraint ns þ nT ¼ 1, these
main constituents of the model are expressed as
H¼ET
X
i
nTi
þJ
X
hiji;γ¼x;y
ðT†
γisis†
j Tγj −T†
γisiT†
γjsj þH:c:Þ:
ð1Þ
χ″(ω)[a.u.]
ω [meV]
Ag
0 20 40 60 80 100
χ″(ω)[a.u.]
B1g
(a)
(b)
0 20 40 60 80 100
0 200 400 600 cm
-1
’AA
B B’
FIG. 3. Fits of T ¼ 10 K Raman spectra in B1g (a) and Ag
(b) channels using a model of phonons interacting with a
magnetic continuum. The model response (blue) is compared
to the experimental points (black). The obtained magnetic signal
SðωÞ is indicated by shading. The Ag phonons marked by blue
triangles are most strongly affected by the spin-phonon interaction
which changes their line shape dramatically compared to
the noninteracting case (red dashed line in the inset). The
associated spectral-weight transfer is moderate only.
χ″(ω)[a.u.]
ω [meV]
Ag − channel
N=16
N=18
N=20
0 20 40 60 80 100 120
χ″(ω)[a.u.]
(d)
(e)
B1g − channel N=16
N=18
N=20
0 20 40 60 80 100 120
0 200 400 600 800 cm
-1
yTxT ,
s
aT
bT
(b)
Tz
TE
s
amplitude
mode
composite
excitations
δnT=Eδ ET Sx Sxδ( SySy− )A=Eδ
magnon
B1gAg
E
S Sx y
(a)
x
y
a
c
b
(c)
magnon
amplitude mode
FIG. 4. (a) Coordinate frames for the Ru lattice. The ordered
moments point along the b axis. (b) Multiplet structure of Ru4þ
ions with a ground-state singlet s with Jeff ¼ 0 and higher-lying
magnetic states derived from Jeff ¼ 1 triplet. A tetragonal crystal
field removes the degeneracy of the triplet states T by lifting up
Tz [6]. An orthorhombic distortion further splits Tx=y into Ta=b ¼
ðTx ∓ TyÞ=
ffiffiffi
2
p
levels forming together with s the basis for the
low-energy model (shaded). (c) Modulation of the condensate
energy (Mexican-hat potential) in the Raman process leading to
an excitation of the amplitude mode (Ag channel). Note that
δnT ≡ δðS2
x þ S2
yÞ, i.e., the Ag coupling is rotationally invariant.
In contrast, the B1g coupling leads to the shape deformations,
leaving the condensate density intact. A single magnon is excited
instead of the amplitude mode. (d),(e) Raman spectra obtained
by exact diagonalization on clusters with N ¼ 16, 18, and 20
sites [25] using ET ¼ 31 meV, J ¼ 7.5 meV, A ¼ 2.3 meV,
α ¼ 0.15, and Δ0
¼ 4 meV. The ED data for B1g (a) and Ag
(b) channels are presented in identical scales and overlayed by the
magnetic SðωÞ from Figs. 3(a), 3(b) (dashed line).
PRL 119, 067201 (2017) P H Y S I C A L R E V I E W L E T T E R S
week ending
11 AUGUST 2017
067201-3
239
The exchange interaction J comprises triplon hopping and
pair creation or annihilation which act together to form AFaligned
pairs of Van Vleck moments.
The full model is most conveniently expressed using a
pseudospin S¼1 formed by the three levels fs; Tx; Tyg [6].
The corresponding in-plane operators Sγ ¼ −iðs†
Tγ − T†
γ sÞ
for γ ¼ x, y are directly linked to the dominating
Van Vleck part of the magnetic moment, while Sz ¼
−iðT†
xTy − T†
yTxÞ is related to the moment residing in
the excited T levels. In this basis, the J term in Eq. (1) takes
a form of the XY model JðSx
i Sx
j þ Sy
i Sy
jÞ. Supplemented by
the bond-directional interaction A and coupling between
the out-of-plane Sz components, the exchange Hamiltonian
for the x bonds reads as
Hx ¼
X
hiji∥x
½ðJ þ AÞSx
i Sx
j þ ðJ − AÞSy
i Sy
j þ Jð1 − αÞSz
i Sz
jŠ:
ð2Þ
The signs of the A terms are opposite for y bonds. The Tx=ylevel
orthorhombic splitting [see Figs. 4(a), 4(b)] orienting
the moments along the b axis translates into a singleion
anisotropy HΔ0 ¼ −Δ0
ðSxSy þ SySxÞ ¼ 1
2 Δ0
ðS2
a − S2
bÞ.
The full Hamiltonian used below is then H ¼ ETnT þ
Hx þ Hy þ HΔ0 , with nT ¼ S2
z.
Model calculations and interpretation of the data.—
We employ the Loudon-Fleury [26] Raman scattering
operator R ∝
P
hijiðϵin · rijÞðϵout · rijÞHij, which modulates
the exchange interactions Hij in a way determined
by the incoming ϵin and outgoing ϵout polarization vectors
[27]. Specifying ϵin (ϵout) by its angle φ (φ0
) to the a axis, R
becomes
R∝cosðφ−φ0
ÞðHx þHyÞþsinðφþφ0
ÞðHx −HyÞ: ð3Þ
For B1g (φ ¼ 0, φ0 ¼ π=2) and Ag (φ ¼ φ0 ¼ π=2) symmetries,
only the Hx − Hy or Hx þ Hy term above is
active, respectively.
We first discuss the implications of Eq. (3) on a
qualitative level. Consider the Ag scattering channel with
R ∝ Hx þ Hy. While in the usual rigid spin systems (e.g.,
cuprates) this operator is proportional to the Hamiltonian
itself and does not bring any dynamics, here we may
replace it by its complement in the Hamiltonian, i.e.,
R ∝ ETnT (and a small Δ0
term), and obtain a nontrivial
spectrum. Most importantly, ETnT globally changes the
balance between the s and Tx=y components coherently
mixed in the condensate, exciting thus directly the amplitude
mode of the condensate. This Ag Raman process
may be intuitively understood as a forced expansion and
contraction of the Mexican-hat potential in Fig. 4(c). In
contrast to INS, the amplitude mode is probed here in a
rotationally invariant way, using a scalar coupling to the
condensate density. We thus avoid the contamination by the
two-magnon response that leads to a drastic broadening of
the longitudinal mode in the dynamical spin susceptibility.
In the B1g channel, the modulation of the exchange J
contained in R ∝ Hx − Hy produces a high-energy twomagnon
continuum, as in usual Heisenberg magnets. Here
it is additionally supported by other composite excitations
such as a two-Higgs continuum (similar to what was found
inasoft-spinmodel[28]).Aspecialroleisplayedbythebond-
anisotropicAtermcontributingtoRasA
P
hijiðSx
i Sx
j − Sy
i Sy
jÞ.
The resulting quadrupolar modulation of the condensate
energy [see Fig. 4(c)] drives the ordered moment toward
the x or y directions, hence exciting a magnon.
To confirm the above expectations and make a quantitative
comparison to the experiment, in Figs. 4(d), 4(e) we
show Raman spectra calculated by exact diagonalization
(ED). The best fit to the magnetic intensity extracted in
Fig. 3 is obtained for the parameters ET ¼ 31 meV,
J ¼ 7.5 meV, A ¼ 2.3 meV, α ¼ 0.15, and Δ0
¼ 4 meV,
well matching those from the INS data [6]. The small
differences in ET and J are due to the different methods—
the spin-wave approach [6] versus ED used here.
In accord with the above discussion, the B1g model
spectrum in Fig. 4(d) contains a high-energy continuum
and a single-magnon peak due to the bond-directional
A part of R that sums up to A
P
hijiðSa
i Sb
j þ Sa
i Sb
j Þ.
Approximating S along the ordered moment direction
by Sb
R ≈ hS∥ieiQ·R
with Q ¼ ðπ; πÞ, this part becomes
AhS∥iSa
Q, thus probing the magnon at the ordering vector.
The energy of the experimental feature B of about
12.5 meV indeed agrees with that of the INS ðπ; πÞ-magnon
peak [6,29]. The spectral weight (SW) of the peak B is
roughly proportional to A2
, enabling us to estimate A by
comparing the SW of B and that of the B0
continuum. The
experimental SW ratio obtained from Fig. 3(a) amounts
to 0.27. In the model calculations, the average over the
three clusters gives a consistent value of 0.30, confirming
A ≃ 2.3 meV taken from INS fits.
In the Ag channel, the model spectrum in Fig. 4(e) is
dominated by the amplitude mode appearing at 40 meV
in agreement with the expected position of the bare
amplitude mode based on INS (see Fig. 4 of Ref. [6]).
The amplitude mode peak is accompanied by a high-energy
continuum [Fig. 4(e)]. Since it is a part of the nT
susceptibility, its profile is rather different from that of
the (mainly) two-magnon continuum in the B1g channel.
The limited scattering possibilities on the small clusters do
not allow us to access the mode profile by ED in detail. The
available results for the relativistic quantum OðNÞ model in
2 þ 1 dimensions [11–14] suggest a Higgs peak with ∼ω3
onset and an extended tail, which is in qualitative agreement
with SðωÞ extracted in Fig. 3(b).
Finally, we comment on the notable interplay of phonons
with the amplitude mode observed in Fig. 3(b). First, Ag
PRL 119, 067201 (2017) P H Y S I C A L R E V I E W L E T T E R S
week ending
11 AUGUST 2017
067201-4
240
phonons involving rotations and tiltings of RuO6 octahedra
modify the Ru-O-Ru bond angle, thus modulating the
exchange J in a symmetric fashion. Second, deformations
of the octahedra affect the splitting among t2g orbitals, thus
modulating ET owing to the different orbital composition
of the s and Tx=y states. Both mechanisms provide a natural
coupling of phonons to oscillations of the condensate
density that is determined by the ratio ET=J.
In conclusion, we have presented Raman light scattering
data on Ca2RuO4 and fully interpreted its magnetic features
in terms of the excitonic model [5,6]. As demonstrated, the
Ag scattering channel enables direct access to the amplitude
(Higgs) mode of the spin-orbit condensate. In contrast to
INS, the Higgs mode is probed here via a scalar coupling
and is not obscured by the two-magnon continuum. The
overall agreement with both the neutron and Raman
experiments strongly supports the excitonic picture as
the basis for magnetism of Ca2RuO4. More generally,
our results encourage future experimental efforts to explore
other compounds based on Van Vleck-type ions such as
Ru4þ
, Os4þ
, and Ir5þ
.
J. C. acknowledges support by the Czech Science
Foundation (GAČR) under Project No. GJ15-14523Y and
MŠMT ČR under NPU II project CEITEC 2020 (LQ1601).
B. K. acknowledges support by the European Research
Council under Advanced Grant No. 669550 (Com4Com)
and by the German Science Foundation (DFG) under the
coordinated research program SFB-TRR80.
[1] D. Pekker and C. M. Varma, Annu. Rev. Condens. Matter
Phys. 6, 269 (2015).
[2] S. Sachdev and B. Keimer, Phys. Today 64, 29 (2011).
[3] Ch. Rüegg, B. Normand, M. Matsumoto, A. Furrer, D. F.
McMorrow, K. W. Krämer, H.-U. Güdel, S. N. Gvasaliya, H.
Mutka, and M. Boehm, Phys. Rev. Lett. 100, 205701 (2008).
[4] T. Giamarchi, Ch. Rüegg, and O. Tchernyshyov, Nat. Phys.
4, 198 (2008).
[5] G. Khaliullin, Phys. Rev. Lett. 111, 197201 (2013).
[6] A. Jain, M. Krautloher, J. Porras, G. H. Ryu, D. P. Chen,
D. L. Abernathy, J. T. Park, A. Ivanov, J. Chaloupka, G.
Khaliullin, B. Keimer, and B. J. Kim, Nat. Phys. 13, 633
(2017).
[7] D. Podolsky, A. Auerbach, and D. P. Arovas, Phys. Rev. B
84, 174522 (2011).
[8] D. Podolsky and S. Sachdev, Phys. Rev. B 86, 054508
(2012).
[9] L. Pollet and N. Prokof’ev, Phys. Rev. Lett. 109, 010401
(2012).
[10] K. Chen, L. Liu, Y. Deng, L. Pollet, and N. Prokof’ev, Phys.
Rev. Lett. 110, 170403 (2013).
[11] S. Gazit, D. Podolsky, and A. Auerbach, Phys. Rev. Lett.
110, 140401 (2013).
[12] S. Gazit, D. Podolsky, A. Auerbach, and D. P. Arovas, Phys.
Rev. B 88, 235108 (2013).
[13] A.RançonandN.Dupuis,Phys.Rev.B89,180501(R)(2014).
[14] F. Rose, F. Léonard, and N. Dupuis, Phys. Rev. B 91,
224501 (2015).
[15] Y. T. Katan and D. Podolsky, Phys. Rev. B 91, 075132
(2015).
[16] M. Endres, T. Fukuhara, D. Pekker, M. Cheneau, P. Schauß,
Ch. Gross, E. Demler, S. Kuhr, and I. Bloch, Nature
(London) 487, 454 (2012).
[17] S. Nakatsuji and Y. Maeno, J. Solid State Chem. 156, 26
(2001).
[18] H. Rho, S. L. Cooper, S. Nakatsuji, H. Fukazawa, and Y.
Maeno, Phys. Rev. B 71, 245121 (2005).
[19] J. H. Jung, Z. Fang, J. P. He, Y. Kaneko, Y. Okimoto, and Y.
Tokura, Phys. Rev. Lett. 91, 056403 (2003).
[20] C. S. Snow, S. L. Cooper, G. Cao, J. E. Crow, H. Fukazawa,
S. Nakatsuji, and Y. Maeno, Phys. Rev. Lett. 89, 226401
(2002).
[21] H. Rho, S. L. Cooper, S. Nakatsuji, H. Fukazawa, and Y.
Maeno, Phys. Rev. B 68, 100404(R) (2003).
[22] A. Nitzan, Mol. Phys. 27, 65 (1974).
[23] M. V. Klein, in Light Scattering in Solids, edited by M.
Cardona (Springer-Verlag, Heidelberg, 1975).
[24] X. K. Chen, E. Altendorf, J. C. Irwin, R. Liang, and W. N.
Hardy, Phys. Rev. B 48, 10530 (1993).
[25] Clusters 16A, 18A, and 20A from D. D. Betts, H. Q. Lin,
and J. S. Flynn, Can. J. Phys. 77, 353 (1999) were used. The
spectra were broadened by Gaussians (FWHM ¼ 8 meV),
apart from the low-energy B1g peak whose line shape is
taken from the B feature in Fig. 3(a).
[26] P. A. Fleury and R. Loudon, Phys. Rev. 166, 514 (1968).
[27] Note that a direct excitation of the magnetic continuum
requires ϵin and ϵout to be in the xy plane. We have verified
that the feature A and the associated Fano asymmetries are
indeed absent in the zz-polarized Raman spectra (not shown).
[28] S. A. Weidinger and W. Zwerger, Eur. Phys. J. B 88, 237
(2015).
[29] S. Kunkemöller, D. Khomskii, P. Steffens, A. Piovano, A.
A. Nugroho, and M. Braden, Phys. Rev. Lett. 115, 247201
(2015).
PRL 119, 067201 (2017) P H Y S I C A L R E V I E W L E T T E R S
week ending
11 AUGUST 2017
067201-5
241
242
PHYSICAL REVIEW B 100, 224413 (2019)
Highly frustrated magnetism in relativistic d4
Mott insulators: Bosonic analog
of the Kitaev honeycomb model
Jiˇrí Chaloupka 1,2
and Giniyat Khaliullin 3
1
Department of Condensed Matter Physics, Faculty of Science, Masaryk University, Kotláˇrská 2, 61137 Brno, Czech Republic
2
Central European Institute of Technology, Masaryk University, Kamenice 753/5, 62500 Brno, Czech Republic
3
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
(Received 24 September 2019; published 16 December 2019)
We study the orbitally frustrated singlet-triplet models that emerge in the context of spin-orbit coupled Mott
insulators with t4
2g electronic configuration. In these compounds, low-energy magnetic degrees of freedom can
be cast in terms of three-flavor “triplon” operators describing the transitions between spin-orbit entangled J = 0
ionic ground state and excited J = 1 levels. In contrast to a conventional, flavor-isotropic O(3) singlet-triplet
models, spin-orbit entangled triplon interactions are flavor-and-bond selective and thus highly frustrated. In
a honeycomb lattice, we find close analogies with the Kitaev spin model—an infinite number of conserved
quantities, no magnetic condensation, and spin correlations being strictly short-ranged. However, due to the
bosonic nature of triplons, there are no emergent gapless excitations within the spin gap, and the ground
state is a strongly correlated paramagnet of dense triplon pairs with no long-range entanglement. Using exact
diagonalization, we study the bosonic Kitaev model and its various extensions, which break exact symmetries
of the model and allow magnetic condensation of triplons. Possible implications for magnetism of ruthenium
oxides are discussed.
DOI: 10.1103/PhysRevB.100.224413
I. INTRODUCTION
Frustrated magnets where competing exchange interactions
result in exotic orderings and spin-liquid phases [1–3]
has been a subject of active research over the years. In general,
the magnetic moments in solids are composed of spin and
orbital angular momentum, with rather different symmetry
properties of interactions in spin and orbital sectors. While the
spin-exchange processes are described by isotropic Heisenberg
model, the orbital moment interactions are far more
complex—they are strongly anisotropic in real and magnetic
spaces [4–6] and frustrated even on simple cubic lattices.
The physical origin of this frustration is that the orbitals are
spatially anisotropic and hence cannot simultaneously satisfy
all the interacting bond directions in a crystal.
In late transition metal ion compounds, the bond directionality
and frustration of the orbital interactions are inherited
by the total angular momentum J = L + S [5]. Consequently,
the low-energy “pseudospin” J models may obtain nontrivial
symmetries and host exotic ground states. The best example of
this sort is the emergence of the Kitaev honeycomb model [7]
in spin-orbit coupled Mott insulators of transition metal ions
with low-spin d5
(S = 1/2, L = 1) [8,9] and high-spin d7
(S =
3/2, L = 1) [10,11] electronic configurations, both possessing
pseudospin J = 1/2 Kramers doublet in the ground state.
The present paper studies the consequences of orbital frustration
in another class of spin-orbit Mott insulators, which
are based on transition metal ions with low-spin d4
(S = 1,
L = 1) electronic configuration such as 4d Ru4+
and 5d Ir5+
.
In these compounds, spin-orbit coupling λL · S favors nonmagnetic
J = 0 ionic ground state, and magnetic order—if
any—is realized via the condensation of excited J = 1 triplet
states [12,13]. Near a magnetic quantum critical point, where
spin-orbit coupling and exchange interactions are of a similar
strength and compete, magnetic condensate can strongly fluctuate
both in phase and amplitude, as it has been observed in
d4
Mott insulator Ca2RuO4 [14,15].
A minimal low-energy model describing the J = 0 Mott
insulators is a singlet-triplet model, which can be written
down in terms of three-flavor “triplon” operators Tα with α =
x, y, z [12]. Distinct from a conventional triplet excitations
in spin-only models, the spin-orbit entangled triplons keep
track of the spatial shape of the t2g orbitals. Therefore their
dynamics is expected to be flavor-and-bond selective and frustrate
the triplon condensation process. In broader terms, J = 0
Mott insulators provide a natural route to a phenomenon of
frustrated magnetic criticality [3].
The bond-directional nature of triplon dynamics is most
pronounced in compounds with 90◦
-exchange geometry as,
e.g., in honeycomb lattice Li2RuO3 [16] with RuO6 octahedra
sharing the edges. Previous work [12,17] has already
addressed singlet-triplet honeycomb models, and found that
the frustration effects can strongly delay triplon condensation,
or suppress it completely in the limit when only one particular
triplon flavor out-of-three Tα is active on a given bond. Here,
we perform a comprehensive symmetry analysis and exact
diagonalization of the model in this limit, where it features
a number of properties of Kitaev model. For instance, we
observe that the model has an extensive number of conserved
quantities, magnetic correlations are highly anisotropic
and confined to nearest-neighbor sites. We also find that
the model is closely related to the bilayer spin-1/2 Kitaev
model [18–20]. However, unlike the Kitaev spin-liquid with
emergent nonlocal excitations, the ground state of the model
2469-9950/2019/100(22)/224413(14) 224413-1 ©2019 American Physical Society
243
JI ˇRÍ CHALOUPKA AND GINIYAT KHALIULLIN PHYSICAL REVIEW B 100, 224413 (2019)
is a strongly correlated paramagnet smoothly connected to
the noninteracting triplon gas, and the lowest excitations are
of a single-triplon character at any strength of the exchange
interactions.
We analyze the model behavior also in the parameter
regime where singlet-triplet level is reversed (formally corresponding
to the sign-change of spin-orbit coupling), and find
that triplon pairs condense into a valence-bond-solid (VBS).
This state is identical to the plaquette-VBS phase of hard-core
bosons on kagome lattice [21]; this is not accidental, since
the symmetry properties of the model allow a mapping of
triplon-pair configurations on honeycomb lattice to a system
of spinless bosons on dual kagome lattice. Further, adding the
terms that relax the model symmetries, we find a rich phase
diagram including the magnetic and quadrupolar orderings.
The paper is organized as follows. Section II introduces
the model and sketches its derivation. In Sec. III, we analyze
the model symmetries and find analogies to the Kitaev honeycomb
model. The phase diagrams and spin excitations are
studied in Secs. IV and V—for the simpler one-dimensional
(1D) analog of the model providing useful insights, and the
full model on the honeycomb lattice, respectively. Section VI
summarizes the main results.
II. HONEYCOMB SINGLET-TRIPLET MODEL
We consider a transition metal ion with four electrons on
t2g level, e.g., Ru4+
. Spin-orbit coupling results in a multiplet
structure shown in Fig. 1(a). A minimal model for magnetism
of such van Vleck-type ions includes the lowest excited
Jeff = 1 states |T±1 and |T0 , in addition to the ground state
Jeff = 0 singlet |s . It is convenient to work with three triplon
operators Tα of Cartesian flavors (“colors”) α = x, y, z. Using
the above Jz eigenstates, they are defined as
Tx =
1
i
√
2
(T1 − T−1),
Ty =
1
√
2
(T1 + T−1), (1)
Tz = iT0,
and together form a Cartesian vector T. A constraint nx +
ny + nz + ns = 1 with nα = T †
α Tα and ns = s†
s is implied.
Spin-orbit splitting reads then as a chemical potential for Tα
bosons: λ nT = λ (nx + ny + nz).
As illustrated in Fig. 1(b), local magnetic moment is composed
of two terms, M = M1 + M2, where M1 originates
from dipolar-active transitions between s and Tα states [12]:
M1 = 2
√
6 v = −
√
6 i (s†
T − T†
s), (2)
while M2 is derived from triplon-spin J = −i(T†
× T ) with
g factor 1/2:
M2 = 1
2
J = −1
2
i (T†
× T ). (3)
In Eq. (2), v = −1
2
i(s†
T − T†
s) keeps track of the imaginary
part of T (real part of T carries a quadrupolar moment).
Triplon interactions are derived from Kugel-Khomskiitype
exchange Hamiltonian, projected onto singlet-triplet basis
[12,17]. In honeycomb lattice of the edge-shared RuO6
octahedra, see Fig. 1(c), there are two types of electron
dxydxy
x
y
z
t
~
t2g
4
SOC
J
J
eff
eff
=1
=0
eff =2
3λ
λ
J
s
TαJeff =1
Jeff =0
M2
M1
(a)
(c)
(d)
z dyzp
o
90
KK ~
yx
z
t
dzx
z
yx
(b)
)f()e(
O Ru
A
FIG. 1. (a) Energy levels of t4
2g configuration in LS coupling
scheme. Low-energy Jeff = 0 singlet and Jeff = 1 triplet form the
local basis of our effective model. (b) The magnetic moment consists
of dominant van Vleck-type contribution M1 residing on the Jeff =
0 ↔ 1 transition, and a smaller contribution M2 carried by Jeff = 1
triplet. (c) Top view of the honeycomb lattice of edge-shared RuO6
octahedra in A2RuO3. Cubic axes x, y, z pointing from Ru towards O
ions, as well as the three types (red, green, blue) of nearest-neighbor
bonds in the honeycomb lattice are indicated. (d) Hopping within
Ru2O2 plaquette of a z bond proceeds through direct overlap of d
orbitals (left) or indirectly via oxygen ions (right). (e) Kitaev-like
pattern of active bond colors for the interaction K in Eq. (10) in
the direct-hopping case. On z bonds, the blue-color triplons Tz are
active, etc. (f) Complementary xy-type pattern for the interaction ˜K
emerging in the case of indirect hopping; on z bonds, the red (Tx) and
green (Ty) color triplons are active.
exchange processes, generated by (i) a direct hopping t of
d electrons between nearest-neighbor (NN) Ru ions, and
(ii) indirect hopping ˜t via oxygen ions, as depicted in Fig. 1(d).
224413-2
244
HIGHLY FRUSTRATED MAGNETISM IN RELATIVISTIC … PHYSICAL REVIEW B 100, 224413 (2019)
Consider, for example, direct t hopping; for a z-type bond,
it reads as −t(d†
xy,idxy,j + H.c.). Second-order perturbation
theory gives the exchange Hamiltonian, written in terms of
spin S = 1 and orbital L = 1 operators of d4
configuration:
H(z)
i j =
t2
U
(Si · Sj + 1)L2
ziL2
z j −L2
zi −L2
z j . (4)
Next, one has to calculate the matrix elements of operators
in Eq. (4) between Jeff = 0 and Jeff = 1 wave functions [12],
and represent them in terms Tα and s. For example, Sz
L2
z =
8
3
vz, while Sx/y
L2
z = 2
3
vx/y + 1
2
Jx/y, with “van-Vleck”
moments v and triplon-spin J already defined above. The
projected Hamiltonian (4) takes the form of H(z)
i j = 8
3
t2
U
(h2 +
h3 + h4)(z)
i j . It contains two-, three-, and four-triplon operator
terms:
h(z)
2 = vzivz j +
1
4
(vxivx j + vyivy j ), (5)
h(z)
3 =
1
8
3
2
(vxiJx j + vyiJy j ) + (i ↔ j), (6)
h(z)
4 =
3
32
(JxiJx j + JyiJy j ) +
1
32
Q3iQ3j, (7)
where Q3 = (nx + ny − 2nz )/
√
3 is a quadrupole operator of
Eg symmetry. Interactions H(x)
i j and H(y)
i j for x- and y-type
bonds follow from symmetry. The largest term in the above
Hamiltonian is represented by vzivz j coupling in h2; physically,
this is Ising-type coupling between van-Vleck moments.
Indirect hopping via ligands −˜t(d†
yz,idzx,j + H.c.) generates
triplon Hamiltonian of the same form, ˜H(z)
i j = 3 ˜t2
U
(˜h2 + ˜h3 +
˜h4)(z)
i j . In contrast to the above case, however, a dominant term
here is represented by xy-type coupling (vxivx j + vyivy j ) in ˜h2
(explicit forms of the other terms can be found in Ref. [12]).
The full models H and ˜H are clearly rich but complicated;
considering their dominant terms represented by Ising- and
xy-type couplings between v-moments should provide some
useful insights. Even though these couplings look as simple
quadratic forms, the hard-core nature of triplons and their
bond-directional anisotropy lead to nontrivial consequences
[12,17].
We introduce the bond operator Oα
i j = 4vαivα j, which in
terms of singlet s and triplon Tα operators reads as
Oα
i j = T †
αisi s†
j Tα j − T †
αisi T †
α jsj + H.c. (8)
We recall that s and Tα are subject to local constraint ns +
nT = 1. Alternatively,
Oα
i j = T †
αiTα j − T †
αiT †
α j + H.c., (9)
where T †
= T †
s is a hard-core boson with nT 1. In terms
of Oα
i j, a minimal singlet-triplet model Hamiltonian can concisely
be written as
H =
i
ET nTi
+
i j α
KOα
i j + ˜K O¯α
i j + O
¯¯α
i j . (10)
Here, the color α ∈ {x, y, z} is given by the direction of the
bond i j , and ¯α, ¯¯α are the two complementary colors; e.g.,
for z-type bond i j z one has α = z, while ¯α = x and ¯¯α = y.
As derived, the model parameters are ET = λ, K = 2
3
t2
/U,
and ˜K = 1
4
K + 3
4
˜t 2
/U.
The K ( ˜K) term in Eq. (10) features Kitaev-like (xy-type)
symmetry, with one (two) active components of T vector on
a given bond. The resulting color-and-bond selective interaction
patterns K and ˜K are shown in Figs. 1(e) and 1(f),
correspondingly. At K = ˜K, the model is equivalent to a
conventional O(3) singlet-triplet models [22] that appear, e.g.,
in low-energy description of a bilayer Heisenberg system.
In this isotropic limit, the model is free of frustration and
would undergo a magnetic transition at large enough coupling
strength K ∼ ET . In this paper, the Kitaev-like model with
K term, where triplon dynamics is most frustrated, is of
primary interest. In particular, we are interested in the nature
of magnetically disordered ground state at strong coupling
limit of K ET . In real materials, an admixture of the complementary
interaction ˜K is expected, and we will consider its
impact on the phase behavior of the model.
III. SYMMETRY PROPERTIES AND LINKS TO KITAEV
HONEYCOMB MODEL
The color-bond correspondence of the above model in
the K = 0, ˜K = 0 case is strongly reminiscent of the Kitaev
honeycomb model. In this section, we focus exclusively on
this limit, draw the corresponding analogies, and find an exact
link between our model and a particular variant of bilayer
Kitaev model.
A. Extensive number of conserved quantities
In the Kitaev-like limit of the model in Eq. (10), i.e., ˜K = 0,
the number of Tα stays either even (0 and 2) or odd (1) on
a bond of direction α. The parity of this number is thus a
conserved Z2 quantity that can be formally written as
Pi j = (1 − 2nαi)(1 − 2nα j ) (11)
with nαi counting Tα-triplon number on site i. Being associated
with the individual bonds, the parities form an extensive
set of conserved Z2 quantities that decompose the Hilbert
space into subspaces with fixed bond-parity configurations.
The total Hilbert space dimension equals 4N
for a system with
N sites. With one Z2 conserved quantity per bond (amounting
to 3/2 per site), the average subspace dimension is reduced to
4N
/2
3N
2 = (
√
2)N
. This is actually the same scaling as in the
case of the Kitaev honeycomb model [7], where the conserved
Z2 quantities are associated with hexagonal plaquettes (giving
1/2 of Z2 quantity per site) and the average subspace dimension
thus becomes 2N
/2
N
2 = (
√
2)N
. In accord with intuitive
expectation, our numerical calculations found the ground state
to have all-even bond-parity configuration.
B. Mapping to hardcore bosons on a dual lattice
When working in the subspaces with fixed bond parities,
most of the 4N
configurations of triplons on the honeycomb
lattice of size N are irrelevant. To remove this redundancy in
the description, here we develop an auxiliary particle representation
by mapping to a system of spinless hardcore bosons
on dual (kagome) lattice.
224413-3
245
JI ˇRÍ CHALOUPKA AND GINIYAT KHALIULLIN PHYSICAL REVIEW B 100, 224413 (2019)
(a)
(b)
T
T
T
x
y
z
b
2’
1’ 5’
4’
3’
6’
p
p’
5
4
6
3
2
1
TE
21
K
K
1’ 2’
K
FIG. 2. (a) Sample triplon configuration on the honeycomb lattice
(left) and the corresponding configuration of spinless hardcore
bosons b (black dots) on a dual kagome lattice (right). In the case
of all-even bond parities depicted here, the presence of a T dimer
with a proper color (i.e., same as that of the bond) is represented by b
boson on the dual lattice. No nearest-neighbor pairs of b on a kagome
lattice is allowed. (b) Equivalent spin-1/2 bilayer model realizing
the singlet-triplet basis on interlayer vertical bonds that are subject
to Heisenberg interaction J = ET . In the model, Kitaev interaction K
active on intralayer bonds (such as 1-2 and 1 -2 ) is complemented by
a Kitaev interaction −K of opposite sign, acting on interlayer cross
links (such as 1-2 and 1 -2). Vertically adjacent hexagonal plaquettes
p, p are used to construct the conserved quantities.
For simplicity, we limit ourselves to the case of all-even
bond parities, similar one-to-one mappings can be found also
in the other cases. The mapping is illustrated in Fig. 2(a). A
given bond of the honeycomb lattice can either be occupied by
a pair of triplons of the proper color or be empty. These two
states are represented by the presence/absence of a hardcore
boson b on the corresponding dual lattice site. Starting from
a b configuration, the state of a given honeycomb site can be
uniquely reconstructed by checking the surrounding kagome
sites for b bosons. Either (i) one of them is found, selecting
one of the Tα states with α depending on the bond occupied
by the b boson, or (ii) none is present, corresponding to the
“empty” states s. The constraint for the b bosons is now
evident—a nearest-neighbor pair of b at the dual lattice is
forbidden.
Altogether, we can formulate the Hamiltonian for the b
bosons on the dual lattice as
Hdual =
i
[2ET nb − K(b + b†
)]i + U
i j
nbinbj, (12)
where i runs through the sites of the kagome lattice, and the
repulsive interaction with U → ∞ enforces the constraint of
“no nearest-neighbor occupation” for b bosons. Without this
constraint, the sum of local Hamiltonians in Eq. (12) would
be easy to diagonalize leading to bond eigenstates that involve
|ss and |TαTα pairs. In the form of Eq. (12), the peculiarity
of the model is fully exposed—the K interaction forms bond
dimers that communicate via constraint only. Adding intersite
hopping terms b†
i bj in the model would generate boson dispersion
and phase relations between them on different sites,
leading to a superfluid condensate; however, we have so far
no clear microscopic mechanism that would result in such a
triplon-pair hopping process.
Finally, let us recall that the above Hdual is valid for the
all-even sector only; the formulation of the constraint in the
other cases is more complicated.
C. Local nature of the spin correlations
Similarly to the Kitaev model, the presence of the local
conserved quantities has consequences for the spin correlations,
both of van Vleck moments of Eq. (2) as well as triplon
spins entering Eq. (3). Let us consider static correlations of the
type MαMα = Z−1
n n|MαMα|n e−βEn
or the corresponding
dynamic correlations. The eigenstates |n of Eq. (10) with
˜K = 0 have fixed bond-parity configurations. When acting by
the α-component of the van Vleck moment operator M1α ∝
(s†
Tα − T †
α s) on a given site, the bond parity of the attached
α bond is switched. Bond parities are conserved by the
Hamiltonian, the introduced parity defect is thus immobile,
and to get back to original parity configuration, one has to act
with M1α either at the same site or on the second site of the
affected α bond. Therefore MαMα correlator is strictly zero
beyond a nearest-neighbor distance. Similarly, the triplon spin
operator −i(T†
× T ) flips parities of two attached bonds, the
original bond-parity configuration therefore has to be restored
by acting at the same site. As a result, the Kitaev-like limit of
the model is characterized by nearest-neighbor only correlations
of the magnetic moments (stemming from the van Vleck
component matching the bond color), and a localized nature
of the dynamic spin response. This mechanism is completely
analogous to the Kitaev model, where a spin flip introduces
two localized fluxes [7].
D. Links to the Kitaev honeycomb model
In the previous Secs. III A and III C, we have noticed
several striking similarities between the bosonic K model
and Kitaev’s model for spins-1/2 residing on the honeycomb
lattice. A deeper connection of the two models can be thus
anticipated, motivating the search for a spin-1/2 equivalent of
our model that could reveal such a link. A natural search direction
is the class of bilayer spin-1/2 systems with Heisenberg
interlayer interaction forming a local singlet-triplet basis on
the interlayer rungs.
Indeed, the Hamiltonian in Eq. (10) can be exactly mapped
onto spin-1/2 bilayer honeycomb system with the interactions
K and ˜K transforming into nearest-neighbor intralayer links
and second nearest-neighbor interlayer links as depicted in
Fig. 2(b). For ˜K = 0, the Hamiltonian involving the nearestneighbor
bond 1-2 and the adjacent one 1 -2 in the other layer
reads as
H121 2 = ET (S1S1 + S2S2 )
+ K Sα
1 Sα
2 + Sα
1 Sα
2 − K Sα
1 Sα
2 + Sα
1 Sα
2 . (13)
224413-4
246
HIGHLY FRUSTRATED MAGNETISM IN RELATIVISTIC … PHYSICAL REVIEW B 100, 224413 (2019)
The first two terms form nothing but a pair of Kitaev models
linked by vertical Heisenberg bonds. This so-called bilayer
Kitaev model was recently studied in Refs. [18–20]. The last
term in Eq. (13) provides additional Kitaev-like cross-links of
the sign opposite to the intra-layer Kitaev interaction and, as
we find later, drastically changes the behavior of the system
from that of standard bilayer Kitaev model.
With the above mapping, we are ready to consider the relations
between various local conserved quantities. The single
layer Kitaev model conserves the product of spin operators at
a hexagonal plaquette [see Fig. 2(b)]
Wp = 26
Sx
1Sy
2Sz
3Sx
4Sy
5Sz
6. (14)
In a bilayer Kitaev model, one has to construct productsWpWp
of Kitaev’s Wp for vertically adjacent plaquettes [18]. These
conserved Z2 quantities bring about certain features of Kitaev
physics to the bilayer Kitaev model. In the case of our model
(13), the extra Kitaev-like crosslinks are present. However, the
products WpWp are still conserved, as can be verified by a
direct calculation. Surprisingly, this does not make them yet
another set of conserved quantities. In fact, it turns out that
WpWp are merely products of our bond parities Pi j
WpWp = P12P23P34P45P56P61 (15)
in the original formulation, and appear as a simple consequence
of the bond-parity conservation in the model. The
above connection also translates the all-even parity configuration
of the ground state into the absence of fluxes in the
ground state, i.e., WpWp = +1 for all plaquettes. The local
symmetries of our model are thus more powerful than in the
Kitaev model or its simple bilayer extension. Intuitively it
may be expected that this denser covering by local conserved
quantities will lead to less entangled (more factorized) ground
states, as we indeed find below.
As a side remark, we note that while the Hamiltonian
(10) contains a balanced combination of hopping and pair
terms, differing only by the sign [see Eq. (9)], it is possible to
generalize the above mapping to the case A T †
α Tα − B T †
α T †
α +
H.c. with A = B. The resulting spin-1/2 interactions consist
of H121 2 of Eq. (13) with K = 1
2
(A + B) and an additional
four-spin interaction
H121 2 = 2(A − B)(S1 × S1 )α
(S2 × S2 )α
. (16)
By introducing symmetric off-diagonal exchange i j =
Sx
i Sy
j + Sy
i Sx
j (for a z bond, x and y-bond expressions are
obtained by cyclic permutation), it can be brought to a
form 2(B − A)( 12 1 2 − 12 1 2) resembling somewhat the
structure in Eq. (13). All the arguments concerning conserved
quantities remain valid also in the A = B case, because the
original interactions in expressed using the T particles manifestly
conserve bond parities. For example, despite the complicated
structure of Eq. (16), it commutes with the plaquette
products WpWp keeping them still conserved.
IV. KITAEV-LIKE SINGLET-TRIPLET ZIGZAG CHAIN
Before studying the full model on the honeycomb lattice,
we first focus on its one-dimensional analog. The 1D system
is more accessible to numerics and enables easier insights.
As an example of such an approach in the context of the
Kitaev-Heisenberg model, Ref. [23] studies the corresponding
1D chains and subsequently makes an interpretation of the 2D
honeycomb model behavior in terms of coupled 1D chains.
To form a 1D model analogous to the honeycomb one,
we remove Tz triplon and keep only one zigzag chain of the
honeycomb lattice, consisting of x and y bonds. In the Kitaevlike
limit ˜K = 0, these two changes are equivalent as only Tz
is active on the z bonds. Going away from the Kitaev-like
limit, as an alternative to the complementary ˜K interaction,
it is more transparent here to add bond-independent JXY
interaction. Instead of the model in Eq. (10), we therefore deal
with
H =
i
ET nTi
+
i j α
KOα
i j + JXY Ox
i j + Oy
i j (17)
formulated for a zigzag chain of alternating x and y bonds with
the bond direction determining again the color α. Note that
for JXY = 0, a slight change to the original parametrization
occurs: K = (K − ˜K)orig and JXY = ˜Korig.
Having now only three levels s, Tx, and Ty in the 1D
model, it is possible to convert it to a spin-1 chain using the
transformation
Sx
= −i(s†
Tx − T †
x s), (18)
Sy
= −i(s†
Ty − T †
y s), (19)
Sz
= −i(T †
x Ty − T †
y Tx ), (20)
where the first two components of the effective spin-1 correspond
to van Vleck moments v while the last one is linked
to the triplon spin J [see Eqs. (2) and (3)]. The resulting
equivalent spin-1 model is a Kitaev-XY spin-1 chain with
single-ion anisotropy ET :
H =
i
ET Sz
i
2
+
i j α
KSα
i Sα
j + JXY Sx
i Sx
j + Sy
i Sy
j . (21)
The phase diagram of this model for the K = 0 case (no
bond-alternation) was thoroughly explored in the context
of spin-1 XXZ chain with single-ion anisotropy (see, e.g.,
Refs. [24–27]). Later in Sec. IV B, we will use these corresponding
studies as a reference.
A. Chain with pure Kitaev-like interaction
As the first step, we consider the Kitaev-like limit of the
model (17), i.e., neglecting JXY term. In general, the behavior
of all our models is determined by a competition of the triplon
energy cost ET with the energy gain due to the interactions.
One can thus expect a quantum critical point (QCP) separating
a triplon gas phase with small triplon densities (dominant ET
regime) from a phase characterized by strongly interacting
triplons at larger densities (dominant K regime).
Such a competition is captured by Fig. 3 presenting an
evolution for varying K to ET ratio. For a better understanding
and to actually reach the QCP in this case, we have extended
the parameter range to the (nonrealistic in the present physical
context but interesting theoretically) regime of ET < 0 with
reversed s and T levels. The data obtained by exact diagonalization
(ED) are shown for two chain lengths to assess
the finite-size effects that are quite small here. As seen in
224413-5
247
JI ˇRÍ CHALOUPKA AND GINIYAT KHALIULLIN PHYSICAL REVIEW B 100, 224413 (2019)
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
spingap
ϑ / π
N=32
N=16
0.4
0.5
0.6
0.7
0.8
0.9
1.0
EGS/Ebond
1 / 2
- d
2
EGS/dϑ
2
N=32
N=16
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
nTpersite
ϑ / π
(a)
(b)
(c)
(d)
N=32
N=16
-N/2 δ=0 +N/2 -N/2 δ=0 +N/2 -N/2 δ=0 +N/2
〈D1D1+δ〉
N=32
1
0
gas
VBS
FIG. 3. (a) Occupation of the triplon states Tx,y within the chain
model parametrized as ET = cos ϑ, K = sin ϑ obtained by exact
diagonalization. Results for the chains of the length N = 16 and
N = 32 are nearly identical. The insets show a cartoon picture of the
ground state. At positive ET K, each bond can be predominantly
in the bonding state as it is mostly composed of s. At negative ET with
|ET | K, the bonding states are incompatible and are realized only
on every second bond creating thus a valence bond solid. (b) Groundstate
energy per bond EGS measured by bonding state energy Ebond
(red) and the second derivative of EGS with respect to ϑ revealing the
quantum critical point (blue). (c) Spin gap obtained as the difference
of ground-state energies within all-even sector and the sector with
a single odd-parity bond. (d) Examples of bond-dimer correlations
D1D1+δ with the dimer operator defined as a projector to the
bonding state of Eq. (22): D = |B B|. The value of the correlator at
δ = 0 gives the probability of the bonding state PB, at large distances
it approaches P2
B in the gas phase. Oscillations near δ = 0 growing
with ϑ are due to the increasing incompatibility of bonding states
at adjacent bonds. The correlation length diverges approaching the
QCP and long-range correlations are seen in the VBS phase.
Fig. 3(a), for an increasing interaction strength K, the triplon
density nT gradually increases with just a single hint of a
change of the regime located already at negative ET . This QCP
is clearly revealed by the second derivative of the ground-state
energy EGS with respect to model parameters as presented in
Fig. 3(b).
To understand the energetics of the evolution together with
the nature of the two phases, it is convenient to measure the
chain EGS per bond by a ground-state energy of an isolated
bond, as it is done in Fig. 3(b). The ground state of a
single bond—the bonding state—mixes a pair of proper-color
triplons and a pair of s in the wave function
|B = cos φ |ss + sin φ |TαTα (22)
with φ given by tan 2φ = K/ET , and has the energy Ebond =
ET −
√
E2
T + K2
. This approximately evaluates to −K2
/2ET
for small K, capturing the perturbative incorporation of a
triplon pair (energy 2ET ) by a process with an amplitude
K. The orthogonal combination to |B is the antibonding
state |A = cos φ |TαTα − sin φ |ss whose energy starts at
2ET in K → 0 limit. Similarly to nT , the ground-state energy
measured by Ebond shows a gradual evolution with the model
parameters for most of the parameter range apart from a
change at QCP [see Fig. 3(b)]. In the ET K regime, EGS
reveals the dominance of the bonding states that seem to fill up
the system. This is possible since the bonding states are composed
mostly of s states that can be shared by the neighboring
bonds. For increasing K and thus an increasing admixture
of T pairs with bond-dependent color, the bonding states at
neighboring bonds have less overlap and the energy gain from
K is reduced compared to that of isolated bonds. At the QCP,
EGS/Ebond approaches 1/2 and stays flat indicating a valence
bond solid (VBS) phase with a rigid structure where every
second bond hosts a bonding state. A more detailed inspection
shows that in the VBS phase, EGS/Ebond positively deviates
from 1/2 with the difference scaling as K6
. This energy gain
can be understood within second order perturbation theory as
an effect of virtual processes where two neighboring T pairs
disappear to make space for an emerging middle T pair (total
amplitude is K3
) being an intermediate state.
Interestingly, the smooth evolution observed in Fig. 3
suggests a picture of the dilute triplon gas at ET K being
continuously connected with the dense triplon state close to
the QCP. It is further supported by a gradual reduction of the
spin gap closing at QCP Fig. 3(c) and an exponential decay of
dimer correlations [Fig. 3(d)] with the decay length diverging
at QCP when the VBS is formed.
In the Kitaev model, the spin gap separates the flux-free
ground state from the topological sector with two fluxes.
Within this gap, excitations from the flux-free sector carried
by itinerant Majorana fermions can be found. In our model,
the spin gap shown in Fig. 3(c) separates the ground state
with all-even bond configuration and the sector with one odd
bond that is being flipped by the van Vleck moment operator.
However, in contrast to the Kitaev model, here the excitation
to one-odd sector is lower than excitations within all-even
sector all the way up to QCP. In other words, no modes (e.g.,
Majorana bands) are present within the spin gap. At the QCP,
the lowest excitations merge, including also partly the lowest
excitations to the other sectors with more odd bonds. The
special role of the QCP will be further demonstrated in the
next section—an antiferromagnetic condensate will be found
224413-6
248
HIGHLY FRUSTRATED MAGNETISM IN RELATIVISTIC … PHYSICAL REVIEW B 100, 224413 (2019)
to emanate from it and an intuitive picture of the critical
excitations near QCP will be inferred.
B. Extension towards XY chain
Having explored the Kitaev-like limit of the model on
the chain, we now consider its extension by JXY interaction
introduced in Eqs. (17) and (21). As our main interest is
the qualitative illustration of the concepts that appear in the
honeycomb case as well, we do not focus on the specifics
that are related to one-dimensionality but rather on the generic
features that will be inherited by the 2D lattice case.
Let us start the discussion with the pure XY-limit where
our model can be related to spin-1 XXZ chain with single-ion
anisotropy for which extensive studies are available [24–28],
mainly in the connection with the Haldane gap problem. Its
phase diagram is quite complex containing a number of phases
depending on the JZ/JXY ratio and single-ion anisotropy D
[corresponding to our ET in Eq. (21)]: large-D phase, Haldane
phase, two XY phases, the ferromagnetic phase, and the Néel
phase [27]. For the relevant JZ = 0 case that matches to
our model, it shows two quantum critical points. The first
one at ET ≈ 0.34JXY corresponds to the transition between
the large-D phase and the Haldane phase and its precise
determination requires the detection of topological features of
the Haldane phase such as the edge spin-1/2 pair [27]. The
second transition to the Néel phase occurs at ET ≈ −2JXY and
in contrast to the first one is easy to capture precisely [28].
The XY-limit of our model is numerically studied in
Fig. 4(a) by means of spin correlations obtained by ED.
Since the local conserved quantities are lost when introducing
the JXY interaction, we are now limited to a shorter chain
length (at most N = 20 sites) compared to the previous
paragraph. We employ both van Vleck moments [Sx
and Sy
components of the effective spin-1 defined by Eqs. (18) and
(19)] and triplon spin [Sz
component defined in Eq. (20)].
The static correlators Sα
q Sα
−q of their Fourier components
Sα
q = N
l=1 Sα
l exp(−iql)/
√
N at the characteristic momentum
q = π are plotted for two different lengths of the chain
and subtracted. The difference uncovers a correlation contribution
scaling with the system size (on top of a sizeindependent
contribution) that we regard as a signature of a
particular phase. Though oversimplified compared to a full
finite-size scaling analysis, this approach will later provide a
rough sketch of the phase diagram of the model in its entire
parameter space.
Figure 4(a) shows three regimes of the correlations for
the XY limit. The first one for ET JXY corresponds to
the triplon gas with the correlations generated exclusively
by triplon excitations. It is quickly replaced at about ET ≈
1.5JXY with a triplon condensate characterized by antiferromagnetic
(AF) correlations of van Vleck moments M1α ∝
(s†
Tα − T †
α s). An intuitive picture of the condensate can be
based on a trial ground-state wave function that explicitly
mixes the condensed T states into a “pool” of s states
| =
N
l=1
( 1 − ρ |s +
√
ρ i eiπl
|T )l , (23)
creating thus van Vleck moments. Here |T stands for any
normalized combination of |Tx and |Ty and ρ is the conden-
ϕ/π
ϑ / π
c e
gd
f
h
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
〈S
α
πS
α
π〉
α=x α=z
0
2
4
6
8
ω
0
1
2
3
4
χxx
ω
0
1
2
3
4
χxx
ω
0
1
2
3
4
0 π 2π
χxx
0
1
2
3
χxx
0
1
2
3
χxx
(a)
(b)
(c)
(e)
(g)
(d)
(f)
(h)
0.00
0.02
0.04
0.06
0.08
0 π 2π
χzz
16
8
gas
VBS
Bethe
AF condensate
FIG. 4. (a) Correlations Sα
q Sα
−q with q = π for the chain
model with K = 0 parametrized as ET = cos ϑ, JXY = sin ϑ. Data
for chains of the length N = 16 and 8 are shown. Their difference
(shaded areas) represents the part of the correlations scaling
with the system size and could be used to estimate the extent
of the corresponding phases. (b) Approximate phase diagram
of ET -K-JXY model parametrized as ET = cos ϑ, K = sin ϑ cos ϕ,
JXY = sin ϑ sin ϕ. The contours capture the difference of the correlations
Sα
π Sα
π between N = 16 and 8 chains. Red and blue lines
are based on Sx
and Sz
correlations, respectively. Black squares are
reference points from Langari et al. [28] (ϕ = π/2 line) and from
Fig. 3 (ϕ = 0 line). The top line with ϕ = π/2 matches panel (a).
[(c)–(h)] Imaginary part of Sx
or Sz
susceptibilities calculated for a
chain with N = 20 sites at the selected parameter points marked in
(b). (h) shows also the boundaries (dashed lines) of the excitation
continuum for the effective spin-1/2 Heisenberg chain.
sate density, 0 ρ 1. At each site l the hardcore condition
nT 1 is obeyed. While the wave function (23) is more
appropriate for the 2D case with static long-range AF order, it
224413-7
249
JI ˇRÍ CHALOUPKA AND GINIYAT KHALIULLIN PHYSICAL REVIEW B 100, 224413 (2019)
still captures the transition between the regime of a triplon gas
(ρ = 0) and the condensate with pronounced AF correlations
(ρ > 0) and gives a very crude estimate ET = 4JXY of the
transition point. This is based on minimizing the energy
per site ET ρ − 4JXYρ(1 − ρ) with respect to the variational
parameter ρ. Later in Sec. V B, we will develop a quantitative
mean-field treatment of the honeycomb model based on the
same type of condensation.
The second quantum phase transition (QPT) appearing
at ET ≈ −2JXY is to a phase associated with the limit of
large negative ET . At this transition, the energy gain from
negative ET overcomes the energy gain from correlated van
Vleck moments and the system collapses to a state full of
Tx and Ty triplons leaving the triplon color as the only active
degree of freedom. The costly s state can be integrated out
leading to an effective interaction among pseudospins 1/2
describing the Tx, Ty doublets. In the isotropic XY case under
consideration, the resulting effective model valid for JXY
−ET is simply a spin-1/2 Heisenberg chain with the exchange
parameter Jeff = J2
XY/|ET |. It can be obtained by removing
s via second order perturbation theory and introducing the
sublattice dependent mapping
|Tx → |↑ , |Ty → |↓ (even sites), (24)
|Tx → |↓ , |Tx → −|↑ (odd sites). (25)
Because of the connection to the exactly solvable spin-1/2
Heisenberg chain, hereafter we call the corresponding phase
the Bethe phase (Néel phase in the context of spin-1 XXZ
chains).
It has to be noted that while our second QPT for negative
ET well corresponds to the reference data by Chen et al. [27],
the first change of the regime occurs for much smaller JXY in
our data than in Ref. [27]. On the other hand, we obtain a good
agreement with the quantum renormalization group (QRG)
and ED study [28] of the ground-state fidelity. This may be
interpreted within the triplon condensation picture as follows.
The true Haldane phase appears around ET ≈ 0.34JXY but this
QPT is preceded much earlier by our “transition” associated
with the onset of van Vleck correlations and corresponding to
the emergence of a triplon (quasi)condensate. Such a change
in the ground-state structure is also reflected in the groundstate
fidelity inspected in Ref. [28]. While probably not a real
QPT, it is a crossover determined by the energy balance of
the triplon cost and the energy gain due to a formation of
correlated van Vleck moments. In nonfrustrated situations,
this energy balance shall lead to a crossover/transition at
similar J/ET ratios, depending mainly on the connectivity of
the particular lattice. Therefore the apparent discrepancy is not
essential because in the 2D honeycomb case, the emergence
of the condensate will correspond to establishing a real longrange
AF order of van Vleck moments.
After discussing both the Kitaev-like limit explored in
Sec. IV A and the XY limit in the above, we now extend
the correlation-based approach to the full ET -K-JXY model to
obtain a sketch of the phase diagram presented in Fig. 4(b).
The topology of the phase diagram follows from the features
already met above when inspecting the limiting cases.
Most of the phase diagram, in particular all of its physically
sensible part (ET > 0), is taken up by the competition of the
triplon gas and the triplon condensate with AF correlations
of van Vleck moments—components Sx
, Sy
of the effective
spin-1. The crossover is more and more delayed when going
from the XY limit (ϕ/π = 0.5) to the Kitaev-like one (ϕ = 0).
This is easily understood by an increasing frustration in this
direction and thus a smaller gain from creating correlated
van Vleck moments. A larger interaction strength is thus
needed to overcome the ET cost. The remaining phases are
restricted to the area of large negative ET . Depending on the
balance between K and JXY, the system selects either the
VBS phase with every second bond essentially inactivated,
or the Bethe phase linked to the hidden effective spin-1/2
model—a Heisenberg chain—and revealed by AF correlations
of Sz
components of the effective spin-1.
To complement the phase diagram, Figs. 4(c)–4(h) present
dynamical correlations χαα(q, ω) = Sα
q Sα
−q ω, i.e., spin susceptibility
associated with the effective spin-1, calculated
for several points in the phase diagram. The gapped van
Vleck susceptibility χxx = χyy in Figs. 4(c) and 4(d) for the
triplon gas phase shows the difference between the local-like
response consisting of two flat parts in the Kitaev-like limit
[Fig. 4(c)] and (almost) continuous dispersion at large JXY
[Fig. 4(d)]. Similarly, Figs. 4(e)–4(g) capture the evolution
from the Kitaev-like to the XY-limit response of the AF
condensate. Here the low-energy part is dominated by an
intense linear mode centered at the AF wave vector q = π.
Extrapolation of data up to N = 20 suggests gapless response
inside the AF condensate region, within the precision limited
by finite-size effects that are pronounced mainly in the
transition region. Finally, inspecting the susceptibility χzz for
a point deep inside the Bethe phase, we notice that the dynamical
response clearly reveals the hidden spin-1/2 Heisenberg
chain. For example, its excitation continuum perfectly
matches the expected analytical boundaries obtained using
Jeff = J2
XY/|ET |, see the dashed lines in Fig. 4(h).
An interesting feature is the “emanation” of the AF condensate
from the QCP of the Kitaev-like model. Around that
point, indicated by a black square on ϕ = 0 line of Fig. 4(b),
the triplon gas consists of bonding states that are about to form
VBS, while for the AF condensate above QCP we expect the
state of the type (23). A naive picture of the link between the
two states that is related to low-energy van Vleck excitations
observed in Fig. 3(c) can be constructed as follows. For
simplicity, let us consider mixing of s and T states in 1:1 ratio
and ignore the triplon color. Adopting the condensate wave
function (23) with ρ = 1
2
, at two neighboring sites l, l + 1 we
have a state proportional to
|cond ∝ (|s + i|T )l ⊗ (|s − i|T )l+1
= [|ss + |T T − i(|sT − |T s )]l,l+1. (26)
On the other hand, the bonding state with 1:1 mixing is
|B ∝ |ss + |T T . Applying the van Vleck operator Mq ∝
−i N
l=1 e−iql
(s†
l Tl − T †
l sl ) with the critical q = π on |B ,
we obtain −2i(|sT − |T s ) which is exactly the missing part
to get bond state |cond of (26). The presence of low-energy
van Vleck excitations that become gapless at the QCP of the
Kitaev-like limit therefore makes the “pool” of bonding states
224413-8
250
HIGHLY FRUSTRATED MAGNETISM IN RELATIVISTIC … PHYSICAL REVIEW B 100, 224413 (2019)
susceptible to the formation of the AF condensate of the type
(23) and this condensate is indeed formed once JXY is added.
V. HONEYCOMB LATTICE CASE
With the basic physical features of our model being illustrated
by the 1D simplified case, we now focus on its original
version on the honeycomb lattice. While the 2D lattice and
one more degree of freedom bring an increased complexity
compared to what discussed in Sec. IV, the overall behavior
will turn out to be rather similar.
A. The case of pure Kitaev-like interaction
This similarity is seen already in Figs. 5(a) and 5(b) which
is a direct analogy of Figs. 3(a) and 5(b) capturing the competition
between the gas and VBS phase in the Kitaev-like limit
of the model. Again a single QCP is detected, now with a position
shifted to a smaller K/|ET | ratio in the negative ET region.
The reduction of the VBS phase is a consequence of the lattice
connectivity—the VBS state can only host a bonding state on
one third of the bonds compared to one half in the chain case,
leading to a less competitive energy gain. The formation of
VBS consisting of maximum geometrically possible number
of dimers (bonding states), i.e., Nbond/3, is seen also in the GS
energy per bond measured by Ebond. This quantity stays close
to 1/3 and there is again a small positive deviation scaling
as K6
that indicates residual interactions among the dimers.
They establish a specific dimer arrangement that we detect
in Fig. 5(c) using dual b-boson representation on the kagome
lattice as described in Sec. III B (note that all the bond parities
are even in the case inspected). The VBS phase is marked by
size-dependent reciprocal-space correlations of the b-boson
density nb at the characteristic momenta q = K lying in the
corners of the extended Brillouin zone of the kagome lattice.
This suggests the real-space pattern shown in Fig. 5(d) that
is also the most probable b configuration in the ground state.
Translating it into the T -dimer picture, we obtain the pattern
in Fig. 5(e) which maximizes the number of plaquettes carrying
three dimers. However, the true structure of the ground
state is more complicated and requires the following deeper
analysis.
In fact, working in the basis of maximum dimer coverings,
an effective quantum dimer model (QDM) with O(K6
)
interactions can be formulated. This QDM “lives” in the alleven
parity sector and captures both the ground state and the
lowest excitations. Leaving the details aside, we note that the
dimer model involves two kinds of interactions. First, there
is an energy gain ∝K6
from dimer-dimer bonds which is
constant for all the coverings and which was already noticed
in the 1D case with just two trivial dimer coverings. The
second contribution, being the actual driving force stabilizing
the VBS pattern, is enabled by the geometry of the honeycomb
lattice and corresponds to hexagonal plaquette flips
with an amplitude ∝K6
. They are captured by the QDM
Hamiltonian
(27)
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
nTpersite
ϑ / π
(a)
(b)
(c)
N=30
N=24
0.0
0.2
0.4
0.6
0.8
1.0
EGS/Ebond
1 / 3
- d
2
EGS/dϑ
2
N=36
N=24
0
1
2
3
4
5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
〈nbnb〉q
ϑ / π
q=Γ
q=K′
N=30
N=24
K′
Γ Γ
(e) )f()d(
VBSgas
1
√
2
FIG. 5. (a) Occupation of the triplet state within the honeycomb
model parametrized as ET = cos ϑ, K = sin ϑ. Presented for a
hexagonal 24-site cluster and rectangular 30-site cluster. (b) Groundstate
energy per bond EGS measured by bonding-state energy Ebond
(red) and the second derivative of EGS with respect to ϑ revealing
the quantum critical point (blue). (c) Correlations nbnb q of the
bosons b on the dual kagome lattice. Size-dependent correlations at
the characteristic vector q = K detect the VBS state. The insets show
the correlations at ϑ = 0.95π plotted in the extended Brillouin zone
of the kagome lattice at all momenta accessible when using the 24and
30-site clusters, respectively. (d) Static pattern of the b bosons
on the dual lattice suggested by their reciprocal-space correlations.
(e) Corresponding arrangement of T dimers in the original representation.
Red, green, and blue ellipses represent bonding states
involving Tx, Ty, and Tz, respectively. The shaded hexagons indicate
flippable plaquettes. (f) The actual plaquette order in VBS phase. The
shaded hexagons indicate resonating plaquettes.
with t = 3K6
/16|ET |5
, i.e., by the kinetic term of the
Rokhsar–Kivelson (RK) model for the honeycomb lattice
[29,30]. Using the connection to this well-studied model we
224413-9
251
JI ˇRÍ CHALOUPKA AND GINIYAT KHALIULLIN PHYSICAL REVIEW B 100, 224413 (2019)
can fix the type of order in the VBS phase now. The phase
diagram of the honeycomb RK model, depending on the
ratio of the flippable plaquette energy cost V and the flipping
amplitude t, was precisely determined by Monte Carlo
simulations [30–32]. Our V/t = 0 case falls into the interval
between (V/t)c ≈ −0.23 [32] and the RK pointV/t = 1 where
the honeycomb RK model supports a triangular covering by
resonating plaquettes (“plaquette” order) depicted approximately
in Fig. 5(f) which is the true VBS order for our model.
The anticipated “columnar” order of Fig. 5(e) only appears
below the (rather close) critical point (V/t)c of the RK model
where the sufficiently large negative potential energy of the
plaquettes wins.
Finally, we conclude the comparison to the Kitaev-like
1D model by a remark that the spin gap behavior for the
honeycomb lattice strongly resembles that of the 1D chain
case [see Fig. 3(c)], i.e., the spin gap gradually closes as we
approach the QCP from both the gas as well as VBS phases.
B. Full honeycomb model
In this section, we explore the phase diagram and to
a limited extent also the excitations of the full model of
Eq. (10) containing both the Kitaev-like interaction K and
the complementary one ˜K. We do not go up to the dominant
˜K regime characterized by strongly interacting quasione-dimensional
condensates hosted by zigzag chains in the
honeycomb lattice [12]. Instead, similarly to the chain case,
we interpolate between the Kitaev-like limit and the isotropic
limit K = ˜K, being both positive as derived. Phase diagram
for arbitrary ˜K/K ratio and positive ET > 0 sector can be
found in Ref. [17].
1. Phase diagram
Figure 6(a) presents the phase diagram of the ET -K- ˜K
model estimated by the method of Sec. IV B, i.e., by tracking
the cluster-size-dependent correlations characteristic to the
individual phases. Due to a larger local basis of four states,
lack of symmetries, and many points to be analyzed, we had
to resort to a combination of small six-site and 12-site clusters.
This makes the phase diagram rather semi-quantitative as
seen for example by comparing the position of the QCP in
the Kitaev-like limit obtained for much bigger clusters in
Sec. V A. Nevertheless, four phases in an arrangement resembling
that of Fig. 4(b) can be identified. Two of them—gas
and VBS phases—were already encountered in the previous
Sec. V A.
The left part of the phase diagram (ET > 0) is a playground
for a competition between the AF correlations and Kitaev-like
frustration of the interactions. In the isotropic K = ˜K limit
[top line in Fig. 6(a)] where the bond interactions handle all
the triplon colors equally, the situation is analogous to the
square-lattice case discussed in the context of Ca2RuO4
[12,14,15]. Since the honeycomb lattice is not geometrically
frustrated, it can easily host an antiferromagnetic phase associated
with a bipartite condensation of triplons captured by
a wave function similar to that of Eq. (23). This AF phase
with “soft” (i.e., far from saturated) van Vleck moments M1
actually takes the largest portion of the entire phase diagram
in Fig. 6(a). Going down to the region with strong Kitaev-like
0.00
0.05
0.10
0.15
0.20
0.25
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
ϕ/π
ϑ / π
d
e
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.1 0.2 0.3 0.4 0.5
nTpersite
ϑ / π
ϕ = π/4
ϕ = 0
ω
Γ′Γ′′
0
1
2
Y Γ M X K Γ Γ′Γ′′
(a)
)c()b(
)e()d(
0
1
2
Y Γ M X K Γ
Γ′′
Γ′
Γ
Y
M
K
X
FQ
AF condensate
VBSgas
e
FIG. 6. (a) Approximate phase diagram of ET -K- ˜K model up
to ˜K =K estimated as in Fig. 4(b) via size-dependent characteristic
correlations. Hexagonal six-site cluster and rectangular 12-site
cluster were used. The model is parametrized as ET = cos ϑ, K =
sin ϑ cos ϕ, ˜K = sin ϑ sin ϕ. Red lines show contours of the increase
of the van Vleck correlations Sx
qSx
−q at the antiferromagnetic (AF)
momentum q = when going from 6- to 12-site cluster. The correlations
detecting the ferroquadrupolar (FQ) phase (blue contours)
were estimated as described in the text. Black square is the extrapolated
position of the QCP for pure Kitaev-like (K only) model [see
Fig. 5(b)]. The dashed line indicates the transition between the gas
phase and AF condensate obtained using Gutzwiller-type treatment
of the hardcore constraint. (b) Comparison of mean-field (solid line)
and ED results (circles, 12-site cluster) for nT at the top (ϕ = π/4)
and bottom line (ϕ = 0) of the phase diagram. (c) Brillouin zone
of the honeycomb lattice (solid) and completed triangular lattice
(dotted) with indicated special points and the path used in the
next panels. (d),(e) Imaginary part of van Vleck spin susceptibility
χzz(q, ω) calculated in the mean-field approximation for the two
points marked by d and e in panel (a) and artificially broadened. The
dotted lines indicate the dispersions of Eq. (33) with α = z.
anisotropy of the interactions and the resulting frustration, the
AF phase gets largely suppressed. One of the highlights of the
soft-moment magnetism based on spin-orbit triplon condensation
is the presence of both transverse magnon modes as well
as an intense amplitude mode (dubbed “Higgs mode” in this
context), that have been observed experimentally in Ca2RuO4
224413-10
252
HIGHLY FRUSTRATED MAGNETISM IN RELATIVISTIC … PHYSICAL REVIEW B 100, 224413 (2019)
[14,15]. It is an interesting and nontrivial problem for a future
study to analyze the effect of the Kitaev-like frustration on
such magnetic excitation spectra. Later in Sec. V B 2, we only
make an attempt to address the gas phase close to the AF phase
boundary, by inspecting the excitation spectrum close to the
point where the condensate is formed.
Focusing now on the large negative ET limit, we can again
notice similarities to the 1D chain case of Sec. IV B. In this
limit the singlets s can be integrated out leaving us with an
effective spin-1 model where the spin now coincides with the
triplon spin −i(T†
× T ). The nature of this model changes
with the ˜K/K ratio. For the strongly anisotropic K-only case,
one formally obtains a biquadratic Kitaev-like model as a
leading term:
HbqK = −
K2
2|ET | i j
Sα
i Sα
j
2
, (28)
with the active component α ∈ {x, y, z} given by the bond
direction as before. Compared to the usual bilinear spin-one
Kitaev model (see, e.g., Ref. [33] and references therein),
the behavior of the biquadratic model is rather trivial. In the
original language, it simply counts the number of proper-color
T dimers and associates an energy gain K2
/2|ET | with each
of them. This selects a large number of degenerate T -dimer
coverings as the model ground state. At this level of approximation
which misses the O(K6
) interactions, the true VBS
ground state cannot be resolved. Furthermore, the low-lying
excitations with the energies ∝K6
are not captured. Hence, in
the Kitaev-like limit, spin-1 is not a suitable elementary object
and bonding-state dimers should be used instead, leading to
the effective quantum dimer model which we extensively
discussed in Sec. V A.
In contrast to that, the isotropic limit K = ˜K can be expected
to be adequately captured by a simple isotropic spin-1
model. Indeed, the preference of the total-singlet T pairs that
may virtually transform into s pairs and back gives rise to
biquadratic interaction described by the effective Hamiltonian
Hbq = −
K2
2|ET | i j
(SiSj )2
(29)
at the isotropic point K = ˜K. Model of this kind is a special
case of bilinear-biquadratic spin-1 models that were
thoroughly explored for various lattices. On a nonbipartite,
geometrically frustrated triangular lattice [34,35] its ground
state shows a ferroquadrupolar (FQ) order [36–38] which does
not break time-reversal symmetry but introduces a preferential
plane in spin space where the spins can be found with a higher
probability. For a nonfrustrated lattice such as square [37,39]
or honeycomb [40–42], the AF phase is more competitive
and the biquadratic model is just on the verge of the FQ and
AF order. This type of order, labeled as FQ in Fig. 6(a) for
simplicity, therefore replaces the Bethe phase of the 1D chain
[compare Fig. 4(b)]. Note that here we refer to the AF phase
of triplon spins −i(T†
× T ); this should not be confused with
the neighboring AF phase of correlated van Vleck moments
M1 which reside on the J = 0 ↔ 1 transitions. The detection
of the “edge-case” FQ/AF order is somewhat complicated,
also due to the incompatible geometry of the two clusters
(six-site hexagon and 12-site rectangle) that we use to check
the size-scaling of the characteristic correlations. To this
end, as the characteristic quantity we take the contribution
to FQ correlations QqQ−q with q = 0 that is carried by
triplon spins at AF momentum q = . More explicitly, we
decompose the various quadrupole operators Q containing
Sα
Sβ
terms (see, e.g., Ref. [34] for their explicit forms) into
a momentum sum Qq ∼ q Sα
q−q Sβ
q and evaluate the fourspin
correlators of the type Sα
q−q Sβ
q Sγ
−q−q Sδ
q constituting
QqQ−q . The contribution with all the momenta being equal
to the AF one is found to dominate and behave well at the reference
K = ˜K |ET | point described by Hbq of Eq. (29). The
resulting correlations obtained as a difference between 12-site
and six-site cluster are shown in Fig. 6(a). They suggest that
FQ/AF phase extends slightly further from its reference point
than the VBS one, though this result may be potentially biased
as the small clusters can not properly accommodate the VBS
state. As a general remark on biquadratic spin models such as
(29), it is worth noticing that while they are typically very
weak in conventional spin systems, they emerge naturally
in singlet-triplet level systems; see yet another example in
Ref. [43].
2. Dynamical Gutzwiller treatment and excitations
As argued in Sec. III C, the Kitaev-like anisotropy of the
interactions should manifest itself by the localized nature of
the dynamic spin response which translates to the flat dispersions
of the spin excitations. Having now covered the entire
interval from the Kitaev-like limit to the isotropic limit, one
might wonder about the corresponding evolution of the spin
excitations. Due to the limited cluster size, ED calculations
do not provide a sufficient resolution to study such effects.
Here we adopt instead a dynamical Gutzwiller treatment
combined with selfconsistent mean-field approximation formulated
for the triplon gas phase. Besides the excitation
spectrum, this enables us to obtain the gas/AF transition point
as a function of non-Kitaev term ˜K. The derivation starts with
a replacement of s implicitly contained in Eq. (10) by the
operator
√
1 − nT which dynamically accounts for the triplon
hardcore constraint. The resulting Hamiltonian is expanded
and a mean-field decoupling is applied leading to the quadratic
Hamiltonian
HMF =
i
(ET + ) nTi
+
i j α
1 − nT KOα
i j + ˜K O¯α
i j + O
¯¯α
i j , (30)
where all the s, s†
operators in O are left out. At this point,
we have already relaxed the hardcore constraint. Two effects
of the hardcore nature of the triplons got captured at the level
of HMF: (i) effective triplon cost is increased by an energy
= −
1
2 δα
K Oα
i,i+δ + ˜K O¯α
i,i+δ + O
¯¯α
i,i+δ (31)
(δ runs through all nearest neighbors) and (ii) the interactions
K and ˜K are reduced by a factor 1 − nT that embodies the
probability of another triplon blocking the interaction process
on the particular bond. After a conversion to momentum
224413-11
253
JI ˇRÍ CHALOUPKA AND GINIYAT KHALIULLIN PHYSICAL REVIEW B 100, 224413 (2019)
space, we obtain
HMF =
kα
E (α†
1kα1k + α†
2kα2k)
+ [καk(α†
1kα2k − α1,−kα2k) + H.c.], (32)
where the unconstrained triplons are labeled by their color
α = x, y, z and the index 1 or 2 referring to the two sites in the
unit cell of the honeycomb lattice. It is convenient to choose
the unit cell for triplons of color α as a bond of direction
α. In this convention, the on-site triplon energy E = ET +
entering Eq. (32) is complemented by the interaction term
καk = 1 − nT (K + ˜Kηαk) with the form factor ηzk given by
ηzk = 2 cos(
√
3
2
kx ) exp(−i3
2
ky), and ηxk and ηyk being simply
rotated by multiples of 2π/3. Note that καk depends on
momentum k via non-Kitaev ˜K term only.
The 4 × 4 problem contained in Eq. (32) can be diagonalized
explicitly and yields the dispersions of the bosonic
quasiparticles
ωBαk = E(E − 2|καk|),
ωAαk = E(E + 2|καk|). (33)
The averages entering all the equations starting from Eq. (30)
have to be calculated self consistently via
nT =
1
4
kα
E − |καk|
ωBαk
+
E + |καk|
ωAαk
− 2 , (34)
=
1
4
kα
|K + ˜Kηαk|
E
ωBαk
−
E
ωAαk
. (35)
The above approach is applicable through the entire gas
phase where the excitations are found to be gapped (ωBαk
and ωAαk > 0). Once the lower-energy ωBαk touches zero at
some point of the Brillouin zone, the triplon condensation
occurs with the condensate structure being similar to the one
of Eq. (23). The corresponding condition E = 2|καk| is first
met at k = 0 which results in the following equation:
ET + = 2 1 − nT (K + 2 ˜K), (36)
determining the points where AF condensate starts to form.
The gas/AF phase boundary obtained this way is presented as
a dashed line in Fig. 6(a). It shows a good overall agreement
with the estimate by ED, correctly capturing the physical
trend of a delayed condensation when the frustration increases
approaching the Kitaev-like limit. As expected, the best agreement
is obtained near the isotropic limit which is also illustrated
in Fig. 6(b) where the isotropic-limit data (ϕ = π/4) of
the self-consistent nT perfectly match the ED values. In the
Kitaev-like limit (ϕ = 0), the deviation is already significant
but still acceptable for our semiquantitative analysis.
With an adequate description of the excitations in the gas
phase at hand, we are now ready to inspect an analogy to
Figs. 4(c) and 4(d) presenting the dynamical spin susceptibility
for the gas phase of the 1D chain model. To this end
we express the Fourier component of the van Vleck moment
operator ∝ −i(s†
Tα − T †
α s) in terms of the unconstrained
triplons as
Sα
q ∝ −i α1q − α†
1,−q + α2q − α†
2,−q e−iq·δα
, (37)
where δα is a unit vector in the direction of α bonds. Next, we
use the eigenspectrum of HMF to find the dynamic correlation
function χαα(q, ω) = Sα
q Sα
−q ω shown in Figs. 6(d) and 6(e)
for two selected points in the phase diagram.
Similarly to the chain case, the vicinity of the Kitaevlike
limit [Fig. 6(d)] is characterized by flat dispersion of
excitations with the modulation being generated by nonzero ˜K
only as it is evident from Eq. (33) and the form of καk. Flat dispersions
are the fingerprints of underlying frustrations, and resemble
the Kitaev-Heisenberg model magnons characterized
by two different energy scales [44]. The excitations in Eq. (33)
have two branches for each triplon color α that cover the
entire Brillouin zone associated with the completed triangular
lattice [dotted hexagon in Fig. 6(c)] by periodic copies of the
smaller Brillouin zone of the honeycomb lattice [full hexagon
in Fig. 6(c)]. The intensity of these excitations in the dynamic
spin susceptibility varies through the Brillouin zone—while
the upper branch dominates around the q = point, the lower
branch is most intense around the AF wave vector q = .
At the latter point (equivalent to q = = 0 in terms of the
bosonic excitations), the magnetic excitations will eventually
touch zero energy signaling the transition into long-range AF
phase as ˜K increases. This is also observed near the isotropic
limit [see Fig. 6(e)] where the modulation of the originally
flat dispersions by the complementary interaction ˜K leads to a
merging of the two excitation branches and the result starts to
resemble the excitonic magnon dispersion. In contrast to the
Heisenberg model at the same lattice, it is characterized by a
maximum at point, as has been seen experimentally in the
square-lattice case of Ca2RuO4 [14].
The observed features of the presented gas-phase spectra
close to the AF transition are expected to be already
quite indicative for the AF phase. After the condensation, an
additional excitation branch corresponding to the amplitude
(Higgs) mode will develop [a hint of this can be noticed in
the chain case when comparing Figs. 4(d) and 4(g)]. Besides
that, there will be also an ongoing redistribution of the spectral
weight in the magnon branch.
VI. CONCLUSIONS
We have studied singlet-triplet models that describe
magnetism of spin-orbit coupled d4
Mott insulators, such
as ruthenium Ru4+
or iridium Ir5+
compounds. Singlettriplet
models appear in various physical contexts (see, e.g.,
Refs. [22,43,45,46]) and are of general interest because they
host—by very construction—a quantum phase transition from
triplon gas to the ordered state of soft moments, when exchange
interactions overcome a singlet-triplet spin gap. The
models considered here bring a new feature into this physics:
a magnetic frustration that originates from bond-directional
nature of orbital interactions [5]. Similar to the case of
spin-orbit pseudospin J = 1/2 Mott insulators with bonddirectional
Ising couplings [8–11] on a honeycomb lattice,
the orbital frustration has a strong impact on magnetism of
singlet-triplet models [12,17].
The main aim of the present work was to understand how a
triplon gas evolves into a dense system of strongly interacting
particles, in particular when bond-directional anisotropy of
the exchange interactions are taken to the extreme as in Kitaev
224413-12
254
HIGHLY FRUSTRATED MAGNETISM IN RELATIVISTIC … PHYSICAL REVIEW B 100, 224413 (2019)
model. We find that this evolution is continuous and results
in a strongly correlated paramagnet smoothly connected to a
triplon gas. Even though this paramagnet misses the defining
features of genuine spin liquids (many-body entanglement and
emergent quasiparticles) [2], it is far from being trivial. In
contrast to a conventional O(3) singlet-triplet systems, spin
correlations here are highly anisotropic and strictly shortranged
even in the limit of strong exchange interactions
where the spin gap is very small. As in the Kitaev model,
these peculiar features of spin correlations follow from the
symmetry properties of the model—an extensive number of
conserved Z2 quantities that decompose the Hilbert space into
subspaces with fixed bond-parity configurations. We have also
shown that the model can be mapped to a bilayer version of
Kitaev model, but with some additional terms in the interlayer
couplings which act to suppress gas-to-liquid phase transition
in a bilayer Kitaev model [18–20]. Exact diagonalization of
the model in 1D-zigzag chain as well as on honeycomb lattice
show that the lowest energy excitations are in the spin sector
(and always gapped). This is different from the Kitaev model
with Majorana bands within the spin gap.
Going away from Kitaev-like symmetry of the exchange
interactions towards isotropic O(3) limit, we find that triplons
condense into AF ordered phase at finite critical value of
non-Kitaev ˜K term. At ˜K < K regime (and close to phase
transition), spin excitations show two distinct branches of
weakly dispersive modes [Fig. 6(d)]; however, this result
is obtained within a mean-field treatment of the hard-core
constraint neglecting any multi-particle scattering processes.
A quantitative description of the highly frustrated magnetic
condensate and its excitations in the regime of ˜K < K remains
an open theoretical problem.
Considering the model at negative ET values, we observe
the links to some exotic models such as biquadratic spin-1 and
quantum dimer models. At negative ET and small ˜K regime,
we find a quantum phase transition from strongly correlated
paramagnetic phase to a plaquette-VBS state of the triplon
dimers.
Apart from a theoretical interest in frustrated singlettriplet
models, this study was partly motivated by magnetic
properties of honeycomb lattice ruthenium compounds, in
particular Ag3LiRu2O6 [47–49]. This compound is derived
from Li2RuO3 by substituting Ag ions for those Li ions
which reside between the Ru-honeycomb planes. As a result,
a structural transition observed in Li2RuO3 [16] is completely
suppressed, suggesting Ag3LiRu2O6 as a nearly ideal
honeycomb lattice system to study the interplay between
spin-orbit coupling and exchange interactions. Current data
[47–49] show that this compound is insulating and has no
magnetic order, which would be consistent with the (correlated)
triplon-gas phase where interactions are either too weak
to overcome the spin-orbit gap, or they are strongly frustrated
preventing triplon condensation. As 4d-electron wave functions
are rather extended in space, a direct overlap processes
can be sizable in ruthenates [50], thus raising the possibility
of bond-directional triplon dynamics in this material. Future
experiments probing magnetic dynamics are necessary to
identify symmetry of the dominant exchange interactions in
Ag3LiRu2O6. Metallic states induced by electron doping of
this material could bring some surprises as well.
Overall, the orbitally frustrated singlet-triplet models show
a rich physics, interesting theoretically and also relevant to
spin-orbit coupled Jeff = 0 Mott insulators based on, e.g.,
ruthenium Ru4+
and iridium Ir5+
ions.
ACKNOWLEDGMENTS
We would like to thank P. Anisimov, M. Daghofer, and T.
Takayama for useful discussions. J.C. acknowledges support
by Czech Science Foundation (GA ˇCR) under Project No.
GA19-16937S and MŠMT ˇCR under NPU II project CEITEC
2020 (LQ1601). Computational resources were supplied by
the Ministry of Education, Youth and Sports of the Czech Republic
under the Projects CESNET (Project No. LM2015042)
and CERIT-Scientific Cloud (Project No. LM2015085) provided
within the program Projects of Large Research, Development
and Innovations Infrastructures. G.K. acknowledges
support by the European Research Council under Advanced
Grant 669550 (Com4Com).
[1] L. Balents, Nature (London) 464, 199 (2010).
[2] L. Savary and L. Balents, Rep. Prog. Phys. 80, 016502 (2017).
[3] M. Vojta, Rep. Prog. Phys. 81, 064501 (2018).
[4] K. I. Kugel and D. I. Khomskii, Sov. Phys. Usp. 25, 231 (1982).
[5] G. Khaliullin, Prog. Theor. Phys. Suppl. 160, 155 (2005).
[6] Z. Nussinov and J. van den Brink, Rev. Mod. Phys. 87, 1 (2015).
[7] A. Kitaev, Ann. Phys. 321, 2 (2006).
[8] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205
(2009).
[9] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett.
105, 027204 (2010).
[10] H. Liu and G. Khaliullin, Phys. Rev. B 97, 014407 (2018).
[11] R. Sano, Y. Kato, and Y. Motome, Phys. Rev. B 97, 014408
(2018).
[12] G. Khaliullin, Phys. Rev. Lett. 111, 197201 (2013).
[13] O. N. Meetei, W. S. Cole, M. Randeria, and N. Trivedi,
Phys. Rev. B 91, 054412 (2015).
[14] A. Jain, M. Krautloher, J. Porras, G. H. Ryu, D. P. Chen,
D. L. Abernathy, J. T. Park, A. Ivanov, J. Chaloupka, G.
Khaliullin, B. Keimer, and B. J. Kim, Nat. Phys. 13, 633
(2017).
[15] S.-M. Souliou, J. Chaloupka, G. Khaliullin, G. Ryu, A. Jain,
B. J. Kim, M. Le Tacon, and B. Keimer, Phys. Rev. Lett. 119,
067201 (2017).
[16] Y. Miura, Y. Yasui, M. Sato, N. Igawa, and K. Kakurai, J. Phys.
Soc. Jpn. 76, 033705 (2007).
[17] P. S. Anisimov, F. Aust, G. Khaliullin, and M. Daghofer,
Phys. Rev. Lett. 122, 177201 (2019).
[18] H. Tomishige, J. Nasu, and A. Koga, Phys. Rev. B 97, 094403
(2018).
[19] U. F. P. Seifert, J. Gritsch, E. Wagner, D. G. Joshi, W. Brenig,
M. Vojta, and K. P. Schmidt, Phys. Rev. B 98, 155101 (2018).
[20] H. Tomishige, J. Nasu, and A. Koga, Phys. Rev. B 99, 174424
(2019).
224413-13
255
JI ˇRÍ CHALOUPKA AND GINIYAT KHALIULLIN PHYSICAL REVIEW B 100, 224413 (2019)
[21] X.-F. Zhang, Y.-C. He, S. Eggert, R. Moessner, and F. Pollmann,
Phys. Rev. Lett. 120, 115702 (2018).
[22] T. Giamarchi, Ch. Rüegg, and O. Tchernyshyov, Nat. Phys. 4,
198 (2008).
[23] C. E. Agrapidis, J. van den Brink, and S. Nishimoto, Sci. Rep.
8, 1815 (2018).
[24] R. Botet, R. Jullien, and M. Kolb, Phys. Rev. B 28, 3914 (1983).
[25] H. J. Schulz, Phys. Rev. B 34, 6372 (1986).
[26] M. den Nijs and K. Rommelse, Phys. Rev. B 40, 4709 (1989).
[27] W. Chen, K. Hida, and B. C. Sanctuary, Phys. Rev. B 67,
104401 (2003).
[28] A. Langari, F. Pollmann, and M. Siahatgar, J. Phys.: Condens.
Matter 25, 406002 (2013).
[29] D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 2376
(1988).
[30] R. Moessner, S. L. Sondhi, and P. Chandra, Phys. Rev. B 64,
144416 (2001).
[31] T. Schlittler, T. Barthel, G. Misguich, J. Vidal, and R. Mosseri,
Phys. Rev. Lett. 115, 217202 (2015).
[32] T. M. Schlittler, R. Mosseri, and T. Barthel, Phys. Rev. B 96,
195142 (2017).
[33] T. Minakawa, J. Nasu, and A. Koga, arXiv:1909.10170.
[34] A. Läuchli, F. Mila, and K. Penc, Phys. Rev. Lett. 97, 087205
(2006).
[35] H. Tsunetsugu and M. Arikawa, J. Phys. Soc. Jpn. 75, 083701
(2006).
[36] H. H. Chen and P. M. Levy, Phys. Rev. B 7, 4267 (1973).
[37] K. Harada and N. Kawashima, Phys. Rev. B 65, 052403 (2002).
[38] B. A. Ivanov and A. K. Kolezhuk, Phys. Rev. B 68, 052401
(2003).
[39] T. A. Tóth, A. M. Läuchli, F. Mila, and K. Penc, Phys. Rev. B
85, 140403(R) (2012).
[40] Y.-W. Lee and M.-F. Yang, Phys. Rev. B 85, 100402(R) (2012).
[41] H. H. Zhao, C. Xu, Q. N. Chen, Z. C. Wei, M. P. Qin, G. M.
Zhang, and T. Xiang, Phys. Rev. B 85, 134416 (2012).
[42] P. Corboz, M. Lajkó, K. Penc, F. Mila, and A. M. Läuchli,
Phys. Rev. B 87, 195113 (2013).
[43] J. Chaloupka and G. Khaliullin, Phys. Rev. Lett. 110, 207205
(2013).
[44] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett.
110, 097204 (2013).
[45] T. Sommer, M. Vojta, and K. W. Becker, Eur. Phys. J. B 23, 329
(2001).
[46] G. Chen, L. Balents, and A. P. Schnyder, Phys. Rev. Lett. 102,
096406 (2009).
[47] S. A. J. Kimber, C. D. Ling, D. J. P. Morris, A. Chemseddine,
P. F. Henry, and D. N. Argyriou, J. Mater. Chem. 20, 8021
(2010).
[48] R. Kumar, T. Dey, P. M. Ette, K. Ramesha, A. Chakraborty, I.
Dasgupta, J. C. Orain, C. Baines, S. Tóth, A. Shahee, S. Kundu,
M. Prinz-Zwick, A. A. Gippius, N. Büttgen, P. Gegenwart, and
A. V. Mahajan, Phys. Rev. B 99, 054417 (2019).
[49] T. Takayama et al. (unpublished).
[50] S. V. Streltsov and D. I. Khomskii, Phys. Usp. 60, 1121 (2017).
224413-14
256
7 Conclusions and outlook
In this thesis we have illustrated the route to unusual kinds of magnetic phenomena where the
spin-orbit coupling plays the role of a key ingredient. Spin-orbit coupling enters in a twofold way
here. First, it rearranges the ionic energy level structure and forms new local degrees of freedom
that serve as a basis for the low-energy collective behavior. In our examples, these were represented
by pseudospins J = 1/2 or the singlet-triplet J = 0, 1 basis. Second, through the spin-orbit
entangled wavefunctions associated with these new degrees of freedom, spin-orbit coupling unifies
the spin and orbital subspaces and imprints the well-known anisotropy of orbital interactions
into the pseudospin interactions. This may in certain cases give rise to highly anisotropic bondselective
interactions that do not appear in conventional magnetic systems based on pure spins.
The systems discussed here were limited to either rigid pseudospin-1/2 moments being subject
to frustrated superexchange interactions or the quasidegenerate singlet-triplet basis, whose members
got dynamically mixed by superexchange in a non-frustrated setting. There are many ways
to go beyond these two situations. Several promising directions that await detailed exploration,
both experimental and theoretical, include:
(i) Combining quantum critical behavior of soft-spin systems with frustration: In the soft-spin
d4
system studied in Sec. 4, the energy gain due to a dynamic mixing of the singlets and triplets
via superexchange interactions competed with the triplet cost, resulting in a quantum critical
behavior. It would be interesting to transfer such a competition into a frustrated setting. This
is readily available thanks to the spin-orbit entangled nature of the singlet-triplet basis states
that produces various kinds of highly anisotropic bond-selective interactions when considering
90◦
metal-oxygen-metal bonds [98, 100]. The case of geometrically non-frustrated honeycomb
lattice where a bosonic model with Kitaev-type frustrations emerges was already inspected in
Ref. [100]. This work can be continued by, e.g., exploration of the effects of trigonal distortion
or extension to other kinds of lattices with 90◦
bonds, like it happened in the case of Kitaev
pseudospin-1/2 systems.
(ii) Lattice control of interactions: As we have seen in Secs. 3.2.1 and 4.2, the spin-orbit
entangled wavefunctions sensitively react to changes in the crystal field induced for instance by
trigonal or tetragonal deformations. By changing the orbital composition, these crystal fields
have a direct impact on the interactions between pseudospin moments. These effects open a
possibility to tune the interactions by e.g. strain in the material to a larger extent than in the
conventional spin systems. Not only the strength of the interactions would be affected but, more
importantly, the balance between the various anisotropic contributions could be changed. In the
soft-spin systems, such manipulations with the crystal field could be used to tune the quantum
critical point.
(iii) Heterostructuring: The materials discussed here are of a natural quasi-2D character enforced
by their crystal structures consisting of active layers sandwiched between “inert” separating
layers. However, such an arrangement can be produced also artificially by growing a superlattice
of two materials [112]. The resulting reduced dimensionality achieved by heterostructuring, as
well as the associated interface effects may bring new physical phenomena not present in the
bulk of the constituent materials. An example is our early proposal [113] of imitating cuprate
physics by heterostructuring nickelates which triggered rather intense experimental activities.
(iv) Doping: Perhaps the richest area of future research are doped variants of the systems
with large spin-orbit coupling. Both the doped carriers as well as the states forming the background
could be complex spin-orbit entangled objects, as it happens for instance in the case of
electron doping of d4
compounds where the mobile objects carry pseudospin-1/2 while the back-
257
258
ground realizes singlet-triplet physics. Such situations lead to highly non-trivial interactions of
the doped carriers with the background and unusual behavior may occur. One of the attractive
options are various forms of unconventional superconductivity that were predicted both for the
doped Kitaev honeycomb systems [114–116] and the square-lattice singlet-triplet systems [117].
Despite being promising theoretically, the preparation of doped materials for the experimental
studies faces severe challenges. The usual chemical doping by element substitution introduces
disorder that may inhibit some more fragile phenomena, unconventional superconductivity included.
Fortunately, there exist clever alternatives, such as those applied to pseudospin J = 1/2
prototype system Sr2IrO4 [118] that is expected to be akin to high-Tc cuprates – surface doping
by potassium deposition [119] on the surface of parent Sr2IrO4 or ionic liquid gating [120].
Bibliography
[1] M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 70, 1039 (1998).
[2] S. Maekawa, T. Tohyama, S. E. Barnes, S. Ishihara, W. Koshibae, and G. Khaliullin,
Physics of Transition Metal Oxides (Springer, Berlin, 2004).
[3] D. Khomskii, Transition Metal Compounds (Cambridge University Press, Cambridge,
2014).
[4] J.R. Schrieffer (editor), Handbook of High-Temperature Superconductivity (Springer, New
York, 2007).
[5] N. Plakida, High-Temperature Cuprate Superconductors (Springer, Berlin, 2010).
[6] M. B. Salamon and M. Jaime, Rev. Mod. Phys. 73, 583 (2001).
[7] D. Khomskii, Physics 2, 20 (2009).
[8] W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, Annu. Rev. Condens. Matter
Phys. 5, 57 (2014).
[9] L. Savary and L. Balents, Rep. Prog. Phys. 80, 016502 (2017).
[10] H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, Nature Reviews
Physics 1, 264 (2019).
[11] P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford University Press, Oxford,
1958).
[12] H. B. Nielsen and S. Chadha, Nucl. Phys. B 105, 445 (1976).
[13] H. Watanabe and H. Murayama, Phys. Rev. Lett. 108, 251602 (2012).
[14] H. J. Liao, Z. Y. Xie, J. Chen, Z. Y. Liu, H. D. Xie, R. Z. Huang, B. Normand, and
T. Xiang, Phys. Rev. Lett. 118, 137202 (2017).
[15] S. R. White and A. L. Chernyshev, Phys. Rev. Lett. 99, 127004 (2007).
[16] J.-F. Yu and Y.-J. Kao, Phys. Rev. B 85, 094407 (2012).
[17] H.-C. Jiang, H. Yao, and L. Balents, Phys. Rev. B 86, 024424 (2012).
[18] A. Kitaev, Ann. Phys. 321, 2 (2006).
[19] E. Ising, Zeitschrift f¨ur Physik 31, 253 (1925).
[20] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009).
[21] J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics (Cambridge University Press,
Cambridge, 2017).
[22] Y. Tanabe and S. Sugano, J. Phys. Soc. Jpn. 9, 753 (1954), ibid. 9, 766 (1954), ibid. 11,
864 (1956).
[23] T. Inui, Y. Tanabe, Y. Onodera, Group Theory and Its Applications in Physics (Springer,
Berlin, 1990).
259
260 BIBLIOGRAPHY
[24] G. Racah, Phys. Rev. 61, 186 (1942), ibid. 62, 438 (1942), ibid. 63, 367 (1943), ibid. 76,
1352 (1949).
[25] J. S. Griffith, The Theory of Transition-Metal Ions (Cambridge University Press, Cambridge,
1971).
[26] J. Kanamori, Prog. Theor. Phys. 30, 275 (1963).
[27] N. W. Ashcroft, N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, New
York, 1976).
[28] P. G¨utlich and H. A. Goodwin (Eds.), Spin Crossover in Transition Metal Compounds I
(Springer, Berlin, 2004).
[29] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics (Pergamon
Press, Oxford, 1982).
[30] L. D. Landau and E. M. Lifshitz, Quantum Mechanics – Non-relativistic Theory (Pergamon
Press, Oxford, 1977).
[31] A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Ions (Clarendon
Press, Oxford, 1970).
[32] S. Sugano, Y. Tanabe, and H. Kamimura, Multiplets of Transition-Metal Ions in Crystals
(Academic Press, New York, 1970).
[33] J. C. Slater and G. F. Koster, Phys. Rev. 94 1498 (1954).
[34] J. B. Torrance, P. Lacorre, A. I. Nazzal, E. J. Ansaldo, and Ch. Niedermayer, Phys. Rev.
B 45, 8209(R) (1992).
[35] J. Zaanen, G. A. Sawatzky, and J. W. Allen, Phys. Rev. Lett. 55, 418 (1985).
[36] J. Zaanen and G. A. Sawatzky, J. Solid State Chem. 88, 8 (1990).
[37] S. Yamada, T. Imamura, T. Kano, Y. Ohashi, H. Matsumoto, and M. Mahida, Journal of
the Earth Simulator 7, 23 (2007).
[38] M. Machida, S. Yamada, Y. Ohashi, and H. Matsumoto, Phys. Rev. Lett. 93, 200402
(2004).
[39] H. A. Kramers, Physica 1, 182 (1934).
[40] P. W. Anderson, Phys. Rev. 79, 350 (1950).
[41] P. W. Anderson, Phys. Rev. 115, 2 (1959).
[42] H. Feshbach, Ann. Phys. 5, 357 (1958).
[43] H. Feshbach, Ann. Phys. 19, 287 (1962).
[44] P. O. L¨owdin, J. Math. Phys. 3, 969 (1962).
[45] P. O. L¨owdin, Phys. Rev. 139, A357 (1965).
[46] H. Eskes and J. H. Jefferson, Phys. Rev. B 48, 9788 (1993).
BIBLIOGRAPHY 261
[47] B. Lake, D. A. Tennant, C. D. Frost, and S. E. Nagler, Nature Materials 4, 329 (2005).
[48] K. Hirakawa and Y. Kurogi, Prog. Theor. Phys. Suppl. 46, 147 (1970).
[49] Z. Nussinov and J. van den Brink, Rev. Mod. Phys. 87, 1 (2015).
[50] J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Annu. Rev. Condens. Matter Phys. 7, 195 (2016).
[51] R. Schaffer, E. K.-H. Lee, B.-J. Yang, and Y. B. Kim, Rep. Prog. Phys. 79, 094504 (2016).
[52] S. Trebst, arXiv:1701.07056 [cond-mat.str-el].
[53] S. M. Winter, A. A. Tsirlin2, M. Daghofer, J. van den Brink, Y. Singh, P. Gegenwart, and
R. Valent´ı, J. Phys.: Condens. Matter 29, 493002 (2017).
[54] M. Hermanns, I. Kimchi, and J. Knolle, Annu. Rev. Condens. Matter Phys. 9, 17 (2018).
[55] Y. Motome and J. Nasu, J. Phys. Soc. Jpn. 89, 012002 (2020).
[56] T. Takayama, A. Kato, R. Dinnebier, J. Nuss, H. Kono, L. S. I. Veiga, G. Fabbris, D. Haskel,
and H. Takagi, Phys. Rev. Lett. 114, 077202 (2015).
[57] K. A. Modic, T. E. Smidt, I. Kimchi, N. P. Breznay, A. Biffin, S. Choi, R. D. Johnson,
R. Coldea, P. Watkins-Curry, G. T. McCandless, J. Y. Chan, F. Gandara, Z. Islam,
A. Vishwanath, A. Shekhter, R. D. McDonald, and J. G. Analytis, Nat. Comm. 5, 4203
(2014).
[58] I. Rousochatzakis, U. K. R¨ossler, J. van den Brink, and M. Daghofer, Phys. Rev. B 93,
104417 (2016).
[59] K. Morita, M. Kishimoto, and T. Tohyama, Phys. Rev. B 98, 134437 (2018).
[60] B. J. Kim, Hosub Jin, S. J. Moon, J.-Y. Kim, B.-G. Park, C. S. Leem, Jaejun Yu,
T. W. Noh, C. Kim, S.-J. Oh, J.-H. Park, V. Durairaj, G. Cao, and E. Rotenberg, Phys.
Rev. Lett. 101, 076402 (2008).
[61] B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita, H. Takagi, and T. Arima, Science
323, 1329 (2009).
[62] H. Liu and G. Khaliullin, Phys. Rev. B 97, 014407 (2018).
[63] R. Sano, Y. Kato, and Y. Motome, Phys. Rev. B 97, 014408 (2018).
[64] H. Liu, J. Chaloupka, and G. Khaliullin, Phys. Rev. Lett. 125, 047201 (2020).
[65] S. M. Winter, Y. Li, H. O. Jeschke, and R. Valent´ı, Phys. Rev. B 93, 214431 (2016).
[66] H. Gretarsson, J. P. Clancy, X. Liu, J. P. Hill, E. Bozin, Y. Singh, S. Manni, P. Gegenwart,
J. Kim, A. H. Said, D. Casa, T. Gog, M. H. Upton, H.-S. Kim, J. Yu, V. M. Katukuri,
L. Hozoi, J. van den Brink, and Y.-J. Kim, Phys. Rev. Lett. 110, 076402 (2013).
[67] J. Nasu, M. Udagawa, and Y. Motome, Phys. Rev. B 92, 115122 (2015).
[68] J. Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moessner, Phys. Rev. Lett. 112, 207203
(2014).
262 BIBLIOGRAPHY
[69] J. Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moessner, Phys. Rev. B 92, 115127
(2015).
[70] J. Nasu, M. Udagawa, and Y. Motome, Phys. Rev. Lett. 113, 197205 (2014).
[71] X. Liu, T. Berlijn, W.-G. Yin, W. Ku, A. Tsvelik, Y.-J. Kim, H. Gretarsson, Y. Singh,
P. Gegenwart, and J. P. Hill, Phys. Rev. B 83, 220403(R) (2011).
[72] F. Ye, S. Chi, H. Cao, B. C. Chakoumakos, J. A. Fernandez-Baca, R. Custelcean, T. F. Qi,
O. B. Korneta, and G. Cao, Phys. Rev. B 85, 180403(R) (2012).
[73] S. K. Choi, R. Coldea, A. N. Kolmogorov, T. Lancaster, I. I. Mazin, S. J. Blundell,
P. G. Radaelli, Y. Singh, P. Gegenwart, K. R. Choi, S.-W. Cheong, P. J. Baker, C. Stock,
and J. Taylor, Phys. Rev. Lett. 108, 127204 (2012).
[74] R. D. Johnson, S. C. Williams, A. A. Haghighirad, J. Singleton, V. Zapf, P. Manuel,
I. I. Mazin, Y. Li, H. O. Jeschke, R. Valent´ı, and R. Coldea, Phys. Rev. B 92, 235119
(2015).
[75] S. C. Williams, R. D. Johnson, F. Freund, Sungkyun Choi, A. Jesche, I. Kimchi, S. Manni,
A. Bombardi, P. Manuel, P. Gegenwart, and R. Coldea, Phys. Rev. B 93, 195158 (2016).
[76] M. Gohlke, G. Wachtel, Y. Yamaji, F. Pollmann, and Y. B. Kim, Phys. Rev. B 97, 075126
(2018).
[77] J. Wang, B. Normand, and Z.-X. Liu, Phys. Rev. Lett. 123, 197201 (2019).
[78] J. M. Luttinger and L. Tisza, Phys. Rev. 70, 954 (1946).
[79] Z. Friedman and J. Felsteiner, Phil. Mag. 29, 957 (1974).
[80] J. Chaloupka and G. Khaliullin, Phys. Rev. B 92, 024413 (2015).
[81] J. Chaloupka and G. Khaliullin, Phys. Rev. B 94, 064435 (2016).
[82] H.-C. Jiang, Z.-C. Gu, X.-L. Qi, and S. Trebst, Phys. Rev. B 83, 245104 (2011).
[83] J. O. Iregui, P. Corboz, and M. Troyer, Phys. Rev. B 90, 195102 (2014).
[84] J. Lou, L. Liang, Y. Yu, and Y. Chen, ArXiv e-prints (2015), arXiv:1501.06990 [cond-
mat.str-el].
[85] J. Rusnaˇcko, D. Gotfryd, and J. Chaloupka, Phys. Rev. B 99, 064425 (2019).
[86] S. H. Chun, J. W. Kim, J. Kim, H. Zheng, C. C. Stoumpos, C. D. Malliakas, J. F. Mitchell,
K. Mehlawat, Y. Singh, Y. Choi, T. Gog, A. Al-Zein, M. M. Sala, M. Krisch, J. Chaloupka,
G. Jackeli, G. Khaliullin, and B. J. Kim, Nature Phys. 11, 462 (2015).
[87] S. M. Winter, K. Riedl, P. A. Maksimov, A. L. Chernyshev, A. Honecker, and R. Valent´ı,
Nature Comm. 8, 1152 (2017).
[88] S. M. Winter, K. Riedl, D. Kaib, R. Coldea, and R. Valent´ı, Phys. Rev. Lett. 120, 077203
(2018).
[89] J. Jakliˇc and P. Prelovˇsek, Phys. Rev. B 49, 5065 (1994).
BIBLIOGRAPHY 263
[90] J. Jakliˇc and P. Prelovˇsek, Adv. Phys. 49, 1 (2000).
[91] M. Aichhorn, M. Daghofer, H. G. Evertz, W. von der Linden, Phys. Rev. B 67, 161103(R)
(2003).
[92] J. Kim, J. Chaloupka, Y. Singh, J. W. Kim, B. J. Kim, D. Casa, A. Said, X. Huang, and
T. Gog, Phys. Rev. X 10, 021034 (2020).
[93] C. S. Alexander, G. Cao, V. Dobrosavljevic, S. McCall, J. E. Crow, E. Lochner, and
R. P. Guertin, Phys. Rev. B 60, R8422 (1999).
[94] S. Nakatsuji, S. Ikeda, and Y. Maeno, J. Phys. Soc. Jpn. 66, 1868 (1997).
[95] G. Cao, S. McCall, M. Sheppard, J. E. Crow, and R. P. Guertin, Phys. Rev. B 56, R2916
(1997).
[96] M. Braden. G. Andr´e, S. Nakatsuji, and Y. Maeno, Phys. Rev. B 58, 847 (1998).
[97] T. Mizokawa, L. H. Tjeng, G. A. Sawatzky, G. Ghiringhelli, O. Tjernberg, N. B. Brookes,
H. Fukazawa, S. Nakatsuji, and Y. Maeno, Phys. Rev. Lett. 87, 077202 (2001).
[98] G. Khaliullin, Phys. Rev. Lett. 111, 197201 (2013).
[99] P. S. Anisimov, F. Aust, G. Khaliullin, and M. Daghofer, Phys. Rev. Lett. 122, 177201
(2019).
[100] J. Chaloupka and G. Khaliullin, Phys. Rev. B 100, 224413 (2019).
[101] A. Jain, M. Krautloher, J. Porras, G. H. Ryu, D. P. Chen, D. L. Abernathy, J. T. Park,
A. Ivanov, J. Chaloupka, G. Khaliullin, B. Keimer, and B. J. Kim, Nature Phys. 13, 633
(2017).
[102] K. I. Kugel and D. I. Khomskii, Sov. Phys. Usp. 25, 231 (1982).
[103] D. Pekker and C. M. Varma, Annu. Rev. Condens. Matter Phys. 6, 269 (2015).
[104] Ch. R¨uegg, B. Normand, M. Matsumoto, A. Furrer, D. F. McMorrow, K. W. Kr¨amer,
H.-U. G¨udel, S. N. Gvasaliya, H. Mutka, and M. Boehm, Phys. Rev. Lett. 100, 205701
(2008).
[105] U. Bissbort, S. G¨otze, Y. Li, J. Heinze, J. S. Krauser, M. Weinberg, C. Becker, K. Sengstock,
and W. Hofstetter, Phys. Rev. Lett. 106, 205303 (2011).
[106] D. Podolsky, A. Auerbach, and D. P. Arovas, Phys. Rev. B 84, 174522 (2011).
[107] S. Gazit, D. Podolsky, and A. Auerbach, Phys. Rev. Lett. 110, 140401 (2013).
[108] A. Ran¸con and N. Dupuis, Phys. Rev. B 89, 180501(R) (2014).
[109] F. Rose, F. L´eonard, and N. Dupuis, Phys. Rev. B 91, 224501 (2015).
[110] P. A. Fleury and R. Loudon, Phys. Rev. 166, 514 (1968).
[111] S.-M. Souliou, J. Chaloupka, G. Khaliullin, G. Ryu, A. Jain, B. J. Kim, M. Le Tacon, and
B. Keimer, Phys. Rev. Lett. 119, 067201 (2017).
264 BIBLIOGRAPHY
[112] H. Y. Hwang, Y. Iwasa, M. Kawasaki, B. Keimer, N. Nagaosa, and Y. Tokura, Nature
Mater. 11, 103 (2012).
[113] J. Chaloupka and G. Khaliullin, Phys. Rev. Lett. 100, 016404 (2008).
[114] T. Hyart, A. R. Wright, G. Khaliullin, and B. Rosenow, Phys. Rev. B 85, 140510(R) (2012).
[115] Y.-Z. You, I. Kimchi, and A. Vishwanath, Phys. Rev. B 86, 085145 (2012).
[116] S. Okamoto, Phys. Rev. Lett. 110, 066403 (2013).
[117] J. Chaloupka and G. Khaliullin, Phys. Rev. Lett. 116, 017203 (2016).
[118] J. Bertinshaw, Y. K. Kim, G. Khaliullin, and B. J. Kim, Annu. Rev. Condens. Matter
Phys. 10, 315 (2019).
[119] Y. K. Kim, O. Krupin, J. D. Denlinger, A. Bostwick, E. Rotenberg, Q. Zhao, J. F. Mitchell,
J. W. Allen, and B. J. Kim, Science 345, 187 (2014).
[120] C. Lu, S. Dong, A. Quindeau, D. Preziosi, N. Hu, and M. Alexe, Phys. Rev. B 91, 104401
(2015).