The physics behind high-temperature superconducting cuprates: the 'plain vanilla' version of
RVB
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
2004 J. Phys.: Condens. Matter 16 R755
(http://iopscience.iop.org/0953-8984/16/24/R02)
Download details:
IP Address: 147.251.57.168
The article was downloaded on 14/03/2012 at 09:46
Please note that terms and conditions apply.
View the table of contents for this issue, or go to the journal homepage for more
Home Search Collections Journals About Contact us My IOPscience
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 16 (2004) R755–R769 PII: S0953-8984(04)80644-1
TOPICAL REVIEW
The physics behind high-temperature superconducting
cuprates: the ‘plain vanilla’ version of RVB
P W Anderson1
, P A Lee2
, M Randeria3
, T M Rice4
, N Trivedi3
and
F C Zhang5,6
1 Department of Physics, Princeton University, Princeton, NJ 08544, USA
2 Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
3 Tata Institute of Fundamental Research, Mumbai 400005, India
4 Theoretische Physics, ETH, H¨onggerberg, CH 8093 Zurich, Switzerland
5 Department of Physics, The University of Hong Kong, Hong Kong
6 Department of Physics, University of Cincinnati, Cincinnati, OH 45221, USA
Received 4 May 2004
Published 4 June 2004
Online at stacks.iop.org/JPhysCM/16/R755
doi:10.1088/0953-8984/16/24/R02
Abstract
One of the first theoretical proposals for understanding high-temperature
superconductivity in the cuprates was Anderson’s RVB theory using a
Gutzwillerprojected BCSwavefunction asanapproximategroundstate. Recent
work by Paramekanti et al has shown that this variational approach gives
a semi-quantitative understanding of the doping dependences of a variety of
experimental observables in the superconducting state of the cuprates. In this
paper we revisit these issues using the ‘renormalized mean field theory’ of
Zhang et al based on the Gutzwiller approximation in which the kinetic and
superexchange energies are renormalized by different doping-dependent factors
gt and gS respectively. We point out a number of consequences of this early
mean field theory for experimental measurements which were not available
when it was first explored, and observe that it is able to explain the existence of
the pseudogap, properties of nodal quasiparticles and approximate spin–charge
separation, the latter leading to large renormalizations of the Drude weight
and superfluid density. We use the Lee–Wen theory of the phase transition as
caused by thermal excitation of nodal quasiparticles, and also obtain a number
of further experimental confirmations. Finally, we remark that superexchange,
and not phonons, is responsible for d-wave superconductivity in the cuprates.
Contents
1. Introduction 756
2. The method 758
3. Mean field theory 760
4. Transition temperature 762
0953-8984/04/240755+15$30.00 © 2004 IOP Publishing Ltd Printed in the UK R755
R756 Topical Review
5. Discussion of results 764
6. Conclusion 765
Acknowledgments 767
References 767
1. Introduction
The resonating valence bond (RVB) liquid was suggested in 1973 by Anderson and
Fazekas (Anderson 1973, Fazekas and Anderson 1974) as a possible quantum state for
antiferromagnetically coupled S = 1/2 spins in low dimensions. Their ideas were based
on numerical estimates of the ground state energy. Instead of orienting the atomic magnets on
separate, oppositely directed sublattices, in the liquid they were supposed to form singlet
‘valence bonds’ in pairs, and regain some of the lost antiferromagnetic exchange energy
by resonating quantum mechanically among many different pairing configurations. Such
states form the basis of Pauling’s early theories of aromatic molecules such as benzene (as
well as of his unsuccessful theories of metals), and are a fair description of Bethe’s (1931)
antiferromagnetic linear chain. The S = 1/2 antiferromagnetic Heisenberg model arises
naturally in Mott insulators. Unlike conventional band insulators, Mott insulators have an odd
number of electrons per unit cell and are insulating by virtue of the strong Coulomb repulsion
between two electrons on the same site. Virtual hopping favours anti-parallel spin alignment,
leading to antiferromagnetic exchange coupling J between the spins (Anderson 1959). In
the RVB picture, S = 1/2 is important because strong quantum fluctuations favour singlet
formation rather than the classically ordered N´eel state.
In 1986 the high Tc cuprates were discovered (Bednorz and M¨uller 1986), and it was soon
realized (Anderson 1987a) that the operative element in their electronic structures was the
square planar CuO2 lattice. In the ‘undoped’ condition, where the Cu is stoichiometrically
Cu2+
, the CuO2 plane is just such an antiferromagnetically coupled Mott insulator. In many
instances these planes are weakly coupled to each other. Anderson (1987a, 1987b), in response
to this discovery, showed that an RVB state could be formally generated as a Gutzwiller
projection of a BCS pair superconductingstate. This is a much more convenientand suggestive
representation than those based on atomic spins, and it immediately makes a connection with
superconductivity.
The method of Gutzwiller (1963) was initially proposed as a theory of magnetic metals,
in conjunction with the Hubbard model. His proposal was to take into account the strong
local Coulomb repulsion of the electrons by taking a simple band Fermi sea state and simply
removing, by projection, all (or, in the early version, a fraction) of the components in it which
have two electrons on the same site. When one projects a half-filled band in this way the result
is to leave only singly occupied sites with spins. The new idea is to project a BCS paired
superconducting state; then the spins are paired up in singlet pairs to make a liquid of pair
‘bonds’; see figure 1.
But of course, with exactly one spin at every site, this state is a Mott insulator, not a metal.
Such an RVB liquid state is of rare occurrence in real Mott insulators, which usually exhibit
either antiferromagnetic long range order as in the cuprates, or possibly have ordered ‘frozen’
arrays of bonds. i.e., valence bond crystals rather than liquids. However, the importance of
the RVB liquid was the suggestion that, as one doped this state with added electrons or holes,
the resulting metal would be a high Tc superconductor, retaining the singlet pairs but allowing
them to carry charge and support supercurrents. The motivation for the pairing would be the
antiferromagnetic superexchange of the original Mott insulator.
Topical Review R757
rr
rr rr
2
Figure 1. Snapshot of a resonating valence bond (RVB) configuration showing singlet pairs of
electrons and, in addition, a fraction x of doped holes. The many-body ground state wavefunction
is a linear superposition of such configurations with the spatial dependence of the singlet pairing
amplitudes determined by the function ϕ(r − r ) defined in equation (4).
For over a decade and a half a number of theorists have been trying to implement this
suggestion along a bewildering variety of routes. One main avenue has resulted from the
proposal by several authors (Kotliar and Liu 1988, Suzumuraet al 1988, Gros 1988, Yokoyama
and Shiba 1988, Affleck et al 1988, Zhang et al 1988) that Anderson’s original s-wave BCS be
replaced by an exotic, d-wave state. The d-wave approach in the early days was quantitatively
carried through by Gros (1989) using variational Monte Carlo methods and by Zhang et al
(1988) on a simplified model, and using very rough approximation methods. Recently the
Gutzwiller-RVB wavefunction approach was revived by Paramekanti et al (2001, 2003) who
used careful numerical methods to calculate many quantities of direct experimental relevance.
Their results turn out to correspond remarkably well to the experimental phenomena observed
in the cuprates across a very broad spectrum of types of datum, a spectrum that was simply
not available in 1987–88 when the original work was done. It may be because of this absence
of data at the time that the original paper was for so long not followed up.
All of this work relies on one basic assumption, an assumption which has gone
unquestioned among a large fraction of those theorists concerned with this problem from
the beginning. This is the assumption that the physics of these materials is dominated by the
strong repulsive interactions of a single non-degenerate band of electrons on the CuO2 planes,
and is specifically not at all similar to that of the conventional BCS superconductors. In the
latter the direct electron interactions are heavily screened, and the lattice vibrations play the
dominant role. We feel that the demonstration of d-wave superconductivityin particular makes
phonons as major players difficult to support, even though there are some notable physicists,
such as Mott et al, who disagree. The phonon mechanisms are local in space, extended
in time, making the dynamic screening mechanism emphasized by Schrieffer and Anderson
relevant and leading to s-wave pairing (Schrieffer 1964). This mechanism works better the
more electrons there are per unit cell, and fails for monovalent metals. d-wave pairing, on the
other hand, is essentially non-local in space and deals with strong repulsions by conventional
R758 Topical Review
space avoidance, as suggested by Anderson and Morel (1961) and by Kohn and Luttinger
(1965). Phonon interactions, especially via optical phonons, are local and cannot easily lead
to higher angular momentum pairing.
It has been argued that certain specific phonons in the presence of strong correlation can
enhance d-wave pairing (Shen et al 2002). However, such couplings are reduced for small
doping by the renormalization factor g2
t as discussed later in the article. Even more cogent is
the fact that, as we shall see, the attractive potential for d-wave pairing is more than adequate
without phonons, and even if they contribute positively to it the effect will be minor. (It has
been argued that in some cases the contribution is negative (Anderson 2002).)
Furthermore,it is now known that the energy gap in high Tc superconductors is much larger
than predicted by BCS theory, and can reach a value of order 50 meV. This is comparable to or
exceeds typical phonon frequencies, making it obvious that a phonon cannot be the key player.
We prefer not to further burden this discussion with the equally strong chemical, angle
resolved photoemission spectra and optical evidence for using only a single band; this subject
is treated in, for instance, the paper by Zhang and Rice (1988), or in Anderson’s book (1997).
These considerations suggested the use of models where the strong repulsive correlations
are emphasized, specifically the Hubbard model, which takes as the only interaction a strong
on-site repulsion. The Hubbard model can be transformed by a perturbative canonical
transformation (Kohn 1964)into ablock-diagonalformin which doubleoccupancyisexcluded,
and replaced by an exchange interaction between neighbouring sites as pointed out early on
by Gros et al (1986). This procedure converges well for sufficiently strong on-site interaction
U, but presumably fails at the critical U for the Mott transition; the singly occupied ‘undoped’
case is unquestionably a Mott insulator in the cuprates and this transformation ipso facto
works. The further simplified t–J model is often used; for refined calculations it has been
argued (Paramekanti et al 2001, 2003) that this simplification may be too great, but for the
semi-quantitative purposes of this article we will at least think in terms of that model.
The Mott-insulator-based theory for the cuprates has been expressed in a variety of
forms other than straightforward Gutzwiller projection and we do not claim any great overall
superiority for our method. Early on, Baskaran et al (1987) (see also Anderson 1987b and Zou
and Anderson 1988) introduced the ideas of spin–charge separation (see also Kivelson et al
1987) and of slave bosons and gauge fields introduced to implement the Gutzwiller constraint,
and quite a number of authors (Ruckenstein et al 1987, Weng et al 1998) have followed
this direction, most notably Baskaran (Anderson et al 1987, Baskaran and Anderson 1988),
Fukuyama (Suzumura et al 1988, Fukuyama 1992) Kotliar (Kotliar and Liu 1988), Ioffe and
Larkin (1989) and a series of publications by Lee and co-workers (Nagaosa and Lee 1990,
Wen and Lee 1996). A related method is the Schwinger boson, slave fermion technique which
has been discussed by a number of authors (Wiegmann 1988, Shraiman and Siggia 1989, Lee
1989). Undoubtedly, for discussions of the precise nature of the phase transition and of the
complicated mix of phenomena such as the pseudogap regime which occur above Tc these
theories will be essential, but we here focus on properties of the ground state and of low lying
excitations, which by good fortune includes the basic physics of Tc. We feel that what we can
calculate indicates the correctness of the fundamental Mott-based picture in such a way as to
support the further effort needed to work out these theories.
2. The method
Starting from the Hubbard Hamiltonian (which may be generalized in various ways without
affecting the following arguments)
H = T + U
i
ni↑ni↓ (1)
Topical Review R759
where T is the kinetic energy. We suppose that there is a canonical transformation eiS
which
eliminatesU from the block which contains no states with ni↑ + ni↓ = 2, and which presumably
contains all the low lying eigenstates and thus the ground state; there are no matrix elements
of the transformed Hamiltonian connecting these to doubly occupied states. Thus
eiS
He−iS
= Ht−J = PT P + J
i j
Si · Sj . (2)
Here P = i (1 − ni↑ni↓) is the Gutzwiller projection operator, which projects out double
occupancy. The kinetic energy T is actually modified to include a three-site hopping term,
which we will neglect here, realizing that our Fermi surface and velocity are heuristically
adjusted in any case. The low lying eigenstates of this Hamiltonian are necessarily of the form
P| , where | is a completely general state of the appropriate number of electrons in the
band. Thus Gutzwiller projection is necessary if one is to use the canonical transformation to
eliminate U.
We make the fundamental assumption that the correct | may be approximated by a
general product wavefunction of Hartree–Fock–BCS type, so that
P| = P
k
uk + vkc†
k↑
c†
−k↓
|0 . (3)
In fact, one can simply rewrite P| for a fixed number of electrons (N) as
P| = P
r,r
ϕ(r − r )c†
r↑c†
r ↓
N/2
|0 , (4)
where ϕ(r − r ) is the Fourier transform of vk/uk. This real space wavefunction may be
visualized in terms of a linear superposition of configurations consisting of singlet pairs and
vacancies with no double occupancy. Each valence bond is the snapshot of a preformed pair
of electrons, while the vacancies correspond to doped holes; see figure 1.
In the conventional theory of metals, the Hartree–Fock–BCS ansatz turns out to be
justifiable as the first step in a perturbation series which preserves many of the properties of
the non-interacting particle model, relying on adiabatic continuation arguments in a qualitative
way. We see no reason why it cannot be equally effective in this case. We emphasize that we
are not approximating the actual wavefunction eiS
P| as a product function, but the function
to be projected, | , and we are searching for an effective mean field Hamiltonian which
determines this function. The projected Hamiltonian is a Hermitian operator which acts on
this function, in complete analogy to an ordinary interacting Hamiltonian, and we may treat
it in mean field theory if we so desire. We accept that the wavefunctions are enormously
underspecified by this Hamiltonian, but in fact that makes it more likely, rather than less, that a
simple product will be a fairly good approximation. Similar arguments to those for Koopman’s
theorem in Hartree–Fock theory tells us that the variational mean field equations will give us
approximate single-particle excitation energies.
The philosophy of this method is analogous to that used by BCS for superconductivity,
and by Laughlin for the fractional quantum Hall effect: simply guess a wavefunction. Is there
any better way to solve a non-perturbative many-body problem?
While the main focus of this paper is on the physical properties of the projected
wavefunction, we briefly mention what is known about its energy as a variational state for
the t–J model (Hsu 1990, Yokoyama and Ogata 1996). At half filling, the projected d-wave
BCS state does remarkably well, with an energy of −0.3199 J per bond compared with the
best estimate of −0.3346 J (Trivedi and Ceperley 1989). Interestingly, projecting the BCS
state does just about as well as projecting a spin density wave state which has long range order
R760 Topical Review
(−0.3206 J). This state also has an ordering moment which is much too large (0.9). The best
trial state is obtained by combining the two, which achieves an energy of −0.3322 J and a
staggered magnetization of 0.75, which is close to the best numerical estimates. Upon doping,
AF co-exists with d-wave superconductivity up to x = 0.11 for J/t = 0.3 (Giamarchi and
Lhuillier 1991, Himeda and Ogata 1999, Ogata and Himeda 2003). This is in disagreement
with experimentswhich show that AF orderis destroyed beyond 3–5% doping. However, more
recent work which combines Gutzwiller projection with a Jastrow factor finds that the energy
of the d-wave superconductoris considerably lowered and Sorella et al (2002a) have presented
numericalevidencethat the groundstate of the 2D t–J model has d-wavesuperconductinglong
range order over a wide doping range; see also the work of Maier et al (2000) on the Hubbard
model. This issue is controversial (Zhang et al 1997, Shih et al 1998, White and Scalapino
1999, Lee et al 2002, Sorella et al 2002b) and not easy to settle because of technical difficulties
with fermion simulations. Nevertheless, the most important point from our perspective is that
the superconducting ground state is energetically highly competitive over a broad range of
doping, and thus the variational state whose properties we are describing in this paper will be
a good approximation to the ground state of a model close to the t–J model.
3. Mean field theory
In evaluating the energy of these wavefunctionsZhang et al (1988)used a rough approximation
first proposed by Gutzwiller (1963) which involves assuming complete statistical independence
of the populations on the sites; see also Vollhardt’s (1984) review for a clear explanation. This
is not too bad, since the one-particle states are defined as momentum eigenstates, but not
perfect, as pointed out by Zhang et al (1988) by comparing with Monte Carlo calculations for
a particular case. But in order to understand the results qualitatively we will follow this simple
procedure here. The evaluations in Paramekanti et al (2001, 2003) are carried out without this
approximation.
In the product wavefunction | with the chemical potential fixed so that there are, on
average, 1 − x electrons per site, with x the fraction of holes, the states with zero, one and two
electrons on a given site have probabilities (1+ x)2
/4, (1− x2
)/2 and (1− x)2
/4, respectively.
The corresponding numbers after projection are x, 1 − x and 0. Thus the relative number
of pairs of sites on which a hole can hop from one to the other may be calculated to be
gt = 2x/(1+x), while the relative numberof pairs of sites which can experiencespin exchange
is gS = 4/(1 + x)2
. These are taken to be the renormalization factors for the kinetic energy
and superexchange terms in the t–J Hamiltonian; that is, the Hubbard Hamiltonian is first
transformed into the t–J Hamiltonian, and then its effect on the actual product wavefunction
is estimated in this way. More accurate estimates could be calculated using Monte Carlo
methods, and the extra correlated hopping terms could be included, but we actually doubt
whether the latter change things much.
Essentially, in this approximation all terms of the nature of spin interactions have a single
renormalization factor, gS, while all terms in the kinetic energy are renormalized by a factor
gt . The ratio of these is quite large, being about a factor of eight even at 20% doping. Thus
this method results in an approximate (or quantitative) spin–charge separation, which is as
effective for experimental purposes (Anderson 2000) as the qualitative one of more radical
theories. In reality, the wavefunction will have some correlations of occupancy, but these are
higher order in x—in the limit of small x the holes move independently. Also, in reality the
dispersion relation may not scale perfectly, but again we do not think this is a very large effect.
Thus the renormalized Hamiltonian simply takes the form of a modified t–J Hamiltonian,
Heff = gt T + gS J Si · Sj . (5)
Topical Review R761
0.3
0.2
0.1
0
0 0.2 0 0.2
60
40
20
0
Doping xDoping x
OP
Tc
Gap
(a) (b)
meV
Figure 2. (a) The amplitude of the (dimensionless) d-wave gap (called ˜ in Zhang et al (1988))
and the superconducting order parameter (OP) as functions of hole doping x in the t–J model
for J/t = 0.2 calculated in the renormalized mean field theory of Zhang et al (1988). (b) The
spectral gap (in meV) for Bi2212 as measured by ARPES (Campuzano et al 1999) and Tc as a
function of doping. The x values for the measured Tc were obtained by using the empirical relation
Tc/Tmax
c = 1 − 82.6(x − 0.16)2 (Presland et al 1991) with Tmax
c = 95 K.
(Again, we ignore the three-site hopping terms.) Zhang et al (1988) showed that if we treat
this within the Hartree–Fock–BCS approximation, we arrive at a modified BCS gap equation.
The kinetic energy is renormalized downwards, and the interaction term Si · Sj , which can
be written in the form of four fermion operators c†
c†
cc alike, can be factorized in two ways.
It can be factorized in such a way that it leads to an anomalous self-energy term of the form
J c†
i↑c†
j↓ cj↓ci↑ + h.c., which will lead to a gap; or it can be factorized in such a way as to
give a Fock exchange self-energy χi j = c†
iσ cjσ , with χk its Fourier transform, which is of
nearly the same form as the kinetic energy, and adds to it. Exhaustive study of this form of
wavefunction has led to the conclusion that the optimum gap equation solution is a d wave
of symmetry dx2−y2 (Kotliar and Liu 1988, Suzumura et al 1988, Gros 1988, Yokoyama and
Shiba 1988, Affleck et al 1988, Zhang et al 1988). The outcome is a pair of coupled equations,
one for the anomalous self-energy and the other for the effective particle kinetic energy:
k =
3
4
gS J
k
γk−k
k
2Ek
(6)
which is an orthodox BCS equation, and
χk = −
3
4
gS J
k
γk−k
ξk
2Ek
. (7)
Here ξk = gt εk − µ − χk, εk is the band energy, µ is an effective chemical potential, γk is the
Fourier transform of the exchange interaction, initially simply the nearest neighbour result
γk = 2 cos kx + cos ky (8)
and µ is set to give the right number of electrons Ne, which commutes with the projection
operator. Ek = ξ2
k
+ 2
k
, which has the same form as in the BCS theory.
Zhang et al (1988) gave the result of solving these gap equations in the oversimplified
case where only nearest neighbour hopping is allowed, and we reproduce their figure here as
figure 2. We see that , the magnitude of the d-wave symmetry gap, falls almost linearly with
R762 Topical Review
x from a number of order J, and vanishes around x = 0.3 for J/t = 0.2. The more realistic
model of Paramekanti et al (2001, 2003) gives a similar result. We presume that this quantity
represents the pseudogap, which is known to vary experimentally in this way. (A calculation
by an entirely different method (Anderson 2001) gave the same result.)
Also plotted on this graph is the physical amplitude of the order parameter (OP)
SC = ci↑cj↓ , which is supposed to renormalize with gt . This is actually true but the
argument is more subtle than that given in Zhang et al (1988). It is necessary to recognize that
the two states connected by this operator contain different numbers of particles. The simpler
argument is to realize, as was remarked by Paramekanti et al (2001, 2003), that the physically
real quantity is the off diagonal long range order eigenvalue of the density matrix, which is the
square root of the product of c†
i↑c†
j↓ci+l↑cj+l↓ for large distance l which is renormalized by a
factor of g2
t . This quantity in this early graph, and in the more accurate work of Paramekanti
et al (2001, 2003), bears a striking resemblance to the variation of Tc with doping, and was by
implication suggested to be a measure of Tc; but it was not until 1997 that the Wen–Lee theory
for the renormalization of Tc (to be discussed below) appeared, and it is not quite true that the
order parameter and Tc are identical.
Beforeturning to Tc, webriefly mention resultson nodalquasiparticles(‘nodons’)obtained
from our approach. These are the important low lying excitations in the superconducting state
and dominate low temperature thermodynamics, transport and response functions (Achkir et al
1993, Krishana et al 1995, Zhang et al 2000, Chiao et al 2000), in addition to controlling Tc
(see below). The Gutzwiller projected d-wave superconducting ground state supports sharp
nodal quasiparticle excitations (Paramekanti et al 2001, 2003) whose coherent spectral weight
Z goes to zero as gt but whose Fermi velocity vF is very weakly doping dependent and remains
non-zero as the hole doping x → 0. These results imply that the real part of the self-energy
(k, ω) for the gapless nodal quasiparticles has singular energy and momentumdependences:
Z ∼ x means that |∂ /∂ω| ∼ 1/x which in turn implies ∂ /∂k ∼ 1/x in order to have a
non-zero nodal vF. These predictions are in very good agreement with recent ARPES data as
shown in figure 3, and in addition also explain the remarkable doping dependence of the ‘high
energy’ dispersion of the nodal quasiparticles, above the so-called kink scale (Lanzara et al
2001), which is found to be dominated by ∂ /∂k (Randeria et al 2004).
4. Transition temperature
It is also a consequence of our theory that the electromagnetic response function ρs (the
phase stiffness, or more conventionally 1/λ2
, with λ the penetration depth) renormalizes
with gt , as does the kinetic energy. Lee and Wen (1997) pointed out that the rate of linear
decrease of ρs with temperature, which was the earliest experimental evidence for d-wave
symmetry (Hardy et al 1993), maintains its magnitude independently of doping. They argue
that the decrease is caused by the thermal excitation of quasiparticles near the nodes and is an
electromagnetic response function of these quasiparticles. In the BCS paper it is pointed out
that the electromagnetic response consists of two parts, the diamagnetic current, which is the
acceleration in the field, and the paramagnetic current, which is a perturbative response of the
excited quasiparticles and exactly cancels the diamagnetic term in the normal state (Schrieffer
1964). The number of these quasiparticles in a d-wave state is only proportionalto T2
, because
the density of states is only linear in energy. But the amount of decrease of ρs per quasiparticle
is inversely proportional to its energy, cancelling one factor of T. The key to their argument
is the assumption that the current carried by each quasiparticle is evF. This is the case in BCS
theory, where the quasiparticle does not carry a definite charge because it is a superposition
Topical Review R763
1
0.75
0.5
0.25
0
0 0.2
1
0.75
0.5
0.25
0
0 0.2
0 0.2 0 0.2
(a) EXPT (b) THEORY
(c) EXPT (d) THEORY
Z
3
2
1
0
3
2
1
0
VF
0
Doping x Doping x
VF(eV–Å)low
Figure 3. (a) Doping dependence of the nodal quasiparticle weight Z in Bi2212 extracted from
ARPES data (Johnson et al 2001) with x calculated from sample Tc using the empirical formula
of Presland et al (1991) with T max
c = 91 K. (b) Z(x) predicted from the variational Monte Carlo
calculation of Paramekanti et al (2001). The dashed line is the Gutzwiller approximation result
Z = 2x/(1 + x). (c) The low energy nodal Fermi velocity vlow
F from ARPES data in Bi2212 (open
squares from Johnson et al (2001)) and LSCO (open triangles from Zhou et al (2003)) is nearly
doping independent. (d) Predicted renormalized vlow
F from Paramekanti et al (2001) as a function
of x; the dashed line is the bare band structure Fermi velocity v0
F. This figure is adapted from
Randeria et al (2004).
of an electron and hole, but each of the partners carries the same current evF. Later it was
pointed out by Millis et al (1998) (see also Paramekanti and Randeria 2002) that there can
be a Fermi liquid renormalization of this current to αevF where α is a Fermi liquid parameter
inherited from the normal state. The slope of ρs versus T is now proportional to α2
and we
assume that α is of order unity and relatively insensitive to doping. Thus ρs at T = 0 decreases
proportionally to doping, yet its rate of decrease with temperature does not vanish with x, but
instead remains relatively constant. The decrease of ρs to zero is considered by these authors
to determine Tc. At Tc the system loses phase coherence, but continues to have an energy gap
over much of the Fermi surface for small x. The insensitivity of the linear T slope in ρs(T )
to doping was experimentally demonstrated by Lemberger and co-workers (Boyce et al 2000,
Stajic et al 2003) and verifies our assumption.
As the quasiparticles reduce ρs, eventually there will develop thermally generated vortices
(in truly two-dimensional systems like LSCO and Bi2212) and the actual phase transition takes
place as a Kosterlitz–Thouless(KT) type of phenomenon(Corsonet al 1999). The notion that a
small ρs would lead to strong phase fluctuations which determine Tc was introduced by Emery
and Kivelson (1995) but we must recognize that the ρs which controls the KT transition is not
ρs(T = 0) but the ρs(T) which is greatly reduced by quasiparticle excitations. By combining
these effects, the decrease of ρs(T ) becomes faster than linear, and eventually infinitely steep.
But this happens only quite near to Tc, because the KT ρs is relatively low; thus the quasiparticle
R764 Topical Review
mechanism gives us a good estimate of Tc, as was pointed out by Lee and Wen, and fits various
empirically proposed relationships (Uemura et al 1989). In materials such as YBCO which
are more three dimensional, the transition will be more conventional but is still mediated by
phase fluctuations near Tc, as of course it is in ordinary superconductors but not over as broad
a critical range.
The Lee–Wen mechanism of Tc described above is relevant for the underdoped side of
the phase diagram where it offers a natural explanation for Tc ∼ ρs(0) and holds all the
way up to optimality. On the overdoped side of the phase diagram, ρs(T ) continues to be
linearly suppressed in temperaturedue to thermally excited quasiparticles, but now the stiffness
corresponding to ρs(0) is much larger than the energy gap. Thus superconductivity is lost by
gap collapse and Tc would be expected to scale like the gap for overdoped systems, as in
conventional BCS theory.
5. Discussion of results
The correspondences between the results of our mean field theory and the very unusual
experimental observations on the high Tc cuprate superconductors are so striking that it is
hard to credit that they have had so little general notice, especially considering the fact that
many of them constituted predictions made in 1988 before the experimental situation became
clear, sometimes many years before. The d-wave nature of the energy gap (Kotliar and Liu
1988, Suzumuraet al 1988, Gros 1988, Yokoyamaand Shiba 1988) confirmed only in 1993–94
(Wollman et al 1993, Tsuei et al 1994), is the most striking. The d-wave pairing symmetry
was also predicted by the ‘spin fluctuation theory’ based on a more orthodox structure (Bickers
et al 1987, Monthouxet al 1991). This follows earlier predictions of d-wave superconductivity
in models with strong repulsion in connection with the heavy fermions (Hirsch 1985, Miyake
et al 1986). We emphasize that our theory, though spin based, is by construction not a spinfluctuation
theory, since the latter is based on Fermi liquid theory. Such a Fermi-liquid-based
approach may be relevant to the overdoped side of the cuprate phase diagram, but is unable to
deal with the unusual properties in the vicinity of the Mott insulator.
A second prediction of the RVB approach is the large energy scale represented by ,which
was first observed as a spin gap by NMR at the end of the 1980s (Alloul et al 1989, Walstedt
and Warren 1990, Takigawa et al 1991). Its significance was only slowly recognized by the
mid-1990s and it has come to be called the pseudogap. It merges with the superconducting
gap below Tc, but is visible in many different kinds of density of state measurement far above
Tc (Ding et al 1996, Loesser et al 1996, Renner et al 1998). For well underdoped samples it
expunges the Fermi surface in the anti-nodal direction (Norman et al 1998). Its value has been
studied in detail by Tallon and Loram (2000), and their numbers are in striking agreement with
the calculations of Zhang et al (1988) or Paramekanti et al (2001, 2003), if we leave aside
their claim that it falls to zero in the midst of the superconducting range. The pseudogap is
often associated roughly with a temperature scale ‘T∗
’ below which its effects are first felt. Of
course, in a rigorous sense our mean field theory is a theory of the superconducting phase at
low temperatures, but the pseudogap appears both in the spectra obtained at low temperature
and in the ‘mysterious’ pseudogap state above Tc.
The effects of the renormalization gt on ρs and on the Drude weight, which was shown
by Sawatzky and coworkers (Eskes et al 1991, Tajima et al 1990) to be renormalized with
precisely the factor 2x, is a natural consequence of the RVB based theories, including the mean
field theory described here.
One important observation also postdated the original paper: that the Green function
of the quasiparticles in the superconducting state contains a sharp ‘coherence peak’ at the
Topical Review R765
quasiparticleenergy on top ofaverybroadincoherentspectrum, andARPESexperiments(Feng
et al 2000, Ding et al 2001) have estimated that the amplitude of that peak is proportionalto 2x.
One result has not been previously mentioned in the literature. The renormalization gt
applies to any term in the Hamiltonian which is a one-electron energy. Therefore matrix
elements for ordinary time-reverse invariant scattering are reduced by a factor of about 2x,
and their squares, which enter into such physical effects as the predicted reduction in Tc, or
into resistivity, are reduced by more than an order of magnitude. At the same time the effects
of magnetic scattering are relatively enhanced. Thus the effects of impurities on high Tc
superconductivity—the notorious contrast of the effects of Zn or Ni substitutions in the plane
relative to non-magnetic doping impurities which lie off the plane, (Fukuzumi et al 1996)—
are explained without having any mysterious spin–charge separation in the formal sense. The
same reduction will, on the whole, apply to the effects of electron–phonon scattering which,
like ordinary impurity scattering, seem to have little influence on the resistivity. The electron–
phonon interaction, which enters ordinary BCS superconductivity, is renormalized relative to
the spin interaction by the factor g2
t /gS ∼ x2
and seems unlikely to play a role.
Finally, a word as to the Nernst effect experiments of Ong and coworkers (Xu et al 2000,
Ong and Wang 2003) which measure the electric field transverse to an applied thermal gradient
in the presence of a perpendicular magnetic field. The Nernst signal is expected to bedominated
by the motion of vortices, and the results on two-dimensionalmaterials are very consistent with
expectations for a generalization of the Kosterlitz–Thouless type of transition. What is seen
is a Nernst signal at and below Tc varying at low magnetic field B as B ln B (Ong and Wang
2003), indicating that the underlying ρs of the effective Ginsburg–Landau free energy does not
vanish at Tc; the ln B variation, giving an infinite slope, follows from thermal proliferation of
large vortices whose energies vary as ρs ln B. As B is increased, however, the signal does not
drop to zero until a very large B is reached, indicating a retention of phase stiffness at short
length scales long after superconducting long range order has disappeared. We believe that
this is a natural and probably calculable effect. But with increasing temperature the Nernst
effect disappears well below T ∗
, at least for low fields. In this region we are well out of the
region of applicability of mean field theory, and expect very large fluctuation effects for which
we have no controlled theory.
An additional experimental phenomenon which, we think, supports the essential validity
of a projected wavefunction is the particle–hole asymmetry of the tunnelling conductance as
a function of voltage. We will discuss singe-particle excited states and tunnelling asymmetry
in a forthcoming paper (Anderson and Ong 2004).
6. Conclusion
In broad outline, our basic assumptions as to the physics of the cuprates, together with a
mean field theory which is little less manageable than BCS theory, seem to give a remarkably
complete picture of the unusual nature of the superconducting state. The RVB state is still a
pairing state between electrons. It has its genesis in the BCS state and is smoothly connected to
it, a fact which is made clear in the recent studies of a partially projected BCS state (Laughlin
2002, Zhang 2003). Furthermore, its low lying excitations are well defined quasiparticles
which dominate the low temperature physics. Thus the RVB state is in some ways rather
conventional. What is unusual is the reduction of the superfluid density and the quasiparticle
spectral weight. With increasing degrees of projection, the state evolves from pairing of
quasiparticles to one which is better understood as a spin singlet formation with coherent hole
motion. This evolution has the following dramatic consequence. The BCS pairing is driven
by a gain in the attractive potential at the expense of kinetic energy, since the energy gap
R766 Topical Review
smears out the Fermi occupation n(k). With projection, n(k) is already strongly smeared in
the non-Fermi liquid normal state, and superconductivity is instead stabilized by a gain in
kinetic energy due to coherent hole motion. This picture has been verified by experiments on
undoped samples which monitor the kinetic energy via the optical sum rule (Molegraaf et al
2002).
Why then is the subject so controversial? Aside from purely socio-political reasons, there
is a real difficulty: the proliferation of nearby alternative states of different symmetry. Here we
mention a numberof possibilities that are actively being considered. One important issue is the
evolution to the antiferromagnet at very low doping. On general principles (Baskaran 2000,
Anderson and Baskaran 2001), mesoscopically inhomogeneous states (‘stripes’) are likely to
be stable at low doping on some scale. They show up in some numerical calculations (White
and Scalapino 1999) and a few of the cuprates show indications of them as static (Tranquada
et al 1995) or dynamical excitations (see Stock et al2004). While static stripes are undoubtedly
detrimental to superconductivity, there have been arguments that dynamical stripes may be the
source of pairing (see Carlson et al 2004). We note that in this scenario, the pairing originates
from the ladder structure of the hole-free part of the stripe which also has its origin in RVB
physics. Given the success of the uniformprojected wavefunction,we find these more complex
scenarios neither necessary nor sufficient for the intermediate doping range.
A second class of competing states has its origin in the SU(2) gauge symmetry first
identified for the projected wavefunction at half-filling. The states of an undoped RVB, or in
factany stateoftheMottinsulator,can berepresented by an enormousnumberofwavefunctions
before projection; in fact, as pointed out by Affleck et al (1988) (see also Anderson 1987b,
Zhang et al 1988), it has an SU(2) gauge symmetry. In the undoped state, with exactly one
electron per site, the presence of an up spin is equivalent to the absence of a down spin and
vice versa, thus permitting independent SU(2) rotations at each site. This degeneracy in
the representation of the wavefunction does not imply any true degeneracy; it is merely the
consequence of our using an underdetermined representation.
When we add holes, this gaugefreedomgradually becomesphysical, which we experience
as the development of a stiffness to phase fluctuations which grows from zero proportionally
to x. The fluctuations can actually take place in a larger space of gauge degrees of freedom
which we can represent in terms of staggered flux phases, etc (Affleck and Marston 1988)
as possible Hartree–Fock states, but we expect that these are of higher energy than the
superconducting state for the interesting values of x. However, the energy difference is
small for small x, and Wen and Lee (1996) and Lee and Wen (2001) have proposed that
in the underdoped region SU(2) rotations which connect fluctuations of staggered flux states
and d-wave superconductivity may play a role in explaining the pseudogap phenomenon.
Remarkably, orbital current correlations which decay rather slowly as a power law have been
seen in projected d-wave wavefunctions (Ivanov et al2000). These fluctuations are very natural
in the SU(2) gauge theory but are otherwise unexpected. In a related development, a static
orbital current state, called a d-density wave, has been proposed to describe the pseudogap on
phenomenological grounds (Chakravarty et al 2001).
In this review we have focused our attention on the ground state and low lying excitations
in the underdoped region. Due to the multitude of competing states mentioned above, much
work remains before a full understanding of the pseudogap is achieved. The situation becomes
even worse for doping to the right of the T∗
crossoverline, commonlycalled the ‘strange metal’
phase. Here one sees highly anomalous transport properties such as the linear resistivity which
played such an important role in early thinking. While the RVB theory leads naturally to a
crossover from pseudogap to strange metal and to Fermi liquid as one increases the doping at
a temperature above the optimal Tc, the ideas presented here are no help in understanding the
Topical Review R767
breakdownof Fermi liquid behaviourin the strange metal. Instead of a smooth crossover,many
workers ascribe the anomalous behaviour to a quantum critical point which lies in the middle
of the superconducting dome (Varma 1997, Tallon and Loram 2000, Varma 2003). We simply
remark that the quantum critical point, if it exists, is different from any previous examples in
that there is no sign of a diverging correlation length scale in any physical observable, and it
is difficult to draw lessons from past experience even phenomenologically.
Finally, what about phonons? Of course there is some coupling to optical phonons, which
will influence both the phonons themselves—an influence which will change sign with the
phonon wavevector Q, because of coherence factors—and the dispersion of quasiparticles.
But as remarked, the net effect of an optical phonon on d-wave superconductivity will tend to
cancel out over the Brillouin zone. It certainly will not play a controlling role in a system so
dominated by Coulomb repulsion. In any case, phonon effects on electron self-energies will
tend to be renormalized downwards by the square of gt , as we pointed out before.
We close by remarking thatgreat strides havebeen made in thediscovery ofunconventional
superconductors since 1986. Today, non-s-wave pairing states are almost commonplace in
heavy fermions, organic superconductors and transition metal oxides. Even time reversal
symmetry is not sacrosanct (see the review on Sr2RuO4 by MacKenzie and Maeno (2003)).
The discovery of high Tc has opened our eyes to the possibility that superconductivity is an
excellent choice as the ground state of a strongly correlated system. This may be the most
important message to be learned from this remarkable discovery.
Acknowledgments
We would like to acknowledge collaborations and interactions with many people over the
years, with special thanks to G Baskaran, C Gros, R Joynt, A Paramekanti and X-G Wen. PAL
would like to acknowledge NSF DMR-0201069. MR would like to thank the Indian DST for
support under the Swarnajayanti scheme, and the Princeton MRSEC NSF DMR-0213706 for
supporting his visits to Princeton University.
References
Affleck I and Marston J B 1988 Phys. Rev. B 37 3774
Affleck I, Zou Z, Hsu T and Anderson P W 1988 Phys. Rev. B 38 745
Alloul H, Ohno T and Mendels P 1989 Phys. Rev. Lett. 63 1700
Anderson P W 1959 Phys. Rev. 115 2
Anderson P W 1973 Mater. Res. Bull. 8 153
Anderson P W 1987a Science 237 1196
Anderson P W 1987b Frontiers and Borderlines in Many Particle Physics: Proc. Enrico Fermi International School
of Physics (Varenna, 1987) (Amsterdam: North-Holland)
Anderson P W 1997 The Theory of High Tc Superconductivity (Princeton, NJ: Princeton University Press) chapter 1
Anderson P W 2000 Science 288 480
Anderson P W 2001 Preprint cond-mat/0108522
Anderson P W 2002 Phys. Scr. T 102 10
Anderson P W and Baskaran G 2001 unpublished
Anderson P W, Baskaran G, Zou Z and Hsu T 1987 Phys. Rev. Lett. 58 2790
Anderson P W and Morel P 1961 Phys. Rev. 123 1911
Anderson P W and Ong N P 2004 Preprint cond-mat/0405518
Anderson P W, Randeria M, Trivedi N and Zhang F C 2004 unpublished
Baskaran G 2000 Mod. Phys. Lett. B 14 377
Baskaran G and Anderson P W 1988 Phys. Rev. B 37 580
Baskaran G, Zou Z and Anderson P W 1987 Solid State Commun. 63 973
Bednorz J G and M¨uller K A 1986 Z. Phys. B 64 189
R768 Topical Review
Bethe H A 1931 Z. Phys. 71 205
Bickers N E, Scalapino D J and Scalettar R T 1987 Int. J. Mod. Phys. B 1 687
Boyce B R, Skinta J A and Lemberger T 2000 Physica C 341–348 561
Campuzano J C et al 1999 Phys. Rev. Lett. 83 3709
Carlson E W, Emery V J, Kivelson S A and Orgad D 2004 The Physics of Conventional and Unconventional
Superconductors ed K Benneman and J B Ketterson (Berlin: Springer) at press
Chakravarty S, Laughlin R B, Morr D and Nayak C 2001 Phys. Rev. B 63 94503
Chiao M, Hill R W, Lupien C, Taillefer L, Lambert P, Gagnon R and Fournier P 2000 Phys. Rev. B 62 3554
Corson J, Mallozzi R, Orenstein J, Eckstein J N and Bozovic I 1999 Nature 398 221
Ding H et al 1996 Nature 382 51
Ding H, Engelbrecht J R, Wang Z, Campuzano J C, Wang S C, Yang H B, Rogan R, Takahashi T, Kadowaki K and
Hinks D G 2001 Phys. Rev. Lett. 87 227001
Emery V J and Kivelson S A 1995 Nature 374 434
Eskes H, Meinders M B J and Sawatzky G 1991 Phys. Rev. Lett. 67 1035
Fazekas P and Anderson P W 1974 Phil. Mag. 30 432
Feng D L et al 2000 Science 289 277
Fukuyama H 1992 Prog. Theor. Phys. 108 287
Fukuzumi Y, Mizukashi K, Takenaka K and Uchida S 1996 Phys. Rev. Lett. 76 684
Giamarchi T and Lhuillier C 1991 Phys. Rev. B 43 12943
Gros C 1988 Phys. Rev. B 38 931
Gros C 1989 Ann. Phys. 189 53
Gros C, Joynt R and Rice T M 1986 Phys. Rev. B 36 381
Gutzwiller M C 1963 Phys. Rev. Lett. 10 159
Himeda A and Ogata M 1999 Phys. Rev. B 60 9935
Hirsch J E 1985 Phys. Rev. Lett. 54 1317
Hsu T C 1990 Phys. Rev. B 41 11379
Ioffe L and Larkin A I 1989 Phys. Rev. B 39 8988
Ivanov D A, Lee P A and Wen X G 2000 Phys. Rev. Lett. 84 3958
Johnson P D et al 2001 Phys. Rev. Lett. 87 177007
Kivelson S A, Rokhsar D S and Sethna J P 1987 Phys. Rev. B 38 8865
Kohn W 1964 Phys. Rev. 133 A171
Kohn W and Luttinger J M 1965 Phys. Rev. Lett. 15 524
Kotliar G and Liu J 1988 Phys. Rev. B 38 5142
Krishana K, Harris J M and Ong N P 1995 Phys. Rev. Lett. 75 3529
Lanzara A et al 2001 Nature 412 510
Laughlin R B 2002 Preprint cond-mat/0209269
Lee P A 1989 Phys. Rev. Lett. 63 680
Lee P A and Wen X G 1997 Phys. Rev. Lett. 78 4111
Lee P A and Wen X G 2001 Phys. Rev. B 63 224517
Lee T K, Shih C T, Chen Y C and Lin H Q 2002 Phys. Rev. Lett. 89 279702
Loesser A G et al 1996 Science 273 325
MacKenzie A P and Maeno Y 2003 Rev. Mod. Phys. 75 657
Maier Th, Jarrell M, Pruschke Th and Keller J 2000 Phys. Rev. Lett. 85 1524
Molegraaf H J A, Presura C, van der Marel D, Kes P H and Li M 2002 Science 295 2239
Monthoux P, Balatsky A and Pines D 1991 Phys. Rev. Lett. 67 3449
Millis A J, Girvin S M, Ioffe L B and Larkin A I 1998 J. Phys. Chem. Solids 59 1742
Miyake K, Schmitt-Rink S and Varma C M 1986 Phys. Rev. B 34 6654
Nagaosa N and Lee P A 1990 Phys. Rev. Lett. 64 2450
Norman M R et al 1998 Nature 392 157
Ogata M and Himeda A 2003 J. Phys. Soc. Japan 72 374 (Preprint cond-mat/0003465)
Ong N P and Wang Y 2003 Preprint cond-mat/0306399
Ong N P and Wang Y 2004 private communication
Paramekanti A and Randeria M 2002 Phys. Rev. B 66 214517
Paramekanti A, Randeria M and Trivedi N 2001 Phys. Rev. Lett. 87 217002
Paramekanti A, Randeria M and Trivedi N 2004 Phys. Rev. B at press
(Paramekanti A, Randeria M and Trivedi N 2003 Preprint cond-mat/0305611)
Presland M R et al 1991 Physica C 176 95
Randeria M, Paramekanti A and Trivedi N 2004 Phys. Rev. B 69 144509
Topical Review R769
Renner Ch et al 1998 Phys. Rev. Lett. 80 149
Ruckenstein A, Hirschfeld P and Appel J 1987 Phys. Rev. B 36 857
Schrieffer J R 1964 Theory of Superconductivity (New York: Benjamin)
Shen Z X, Lanzara A, Ishihara S and Nagaosa N 2002 Phil. Mag. B 82 1349
Shih C T, Chen Y C, Lin H Q and Lee T K 1998 Phys. Rev. Lett. 81 1294
Shraiman B and Siggia E 1989 Phys. Rev. Lett. 62 1564
Sorella S et al 2002a Phys. Rev. Lett. 88 117002
Sorella S et al 2002b Phys. Rev. Lett. 89 279703
Stajic J et al 2003 Phys. Rev. B 68 024520
Stock C et al 2004 Phys. Rev. B 69 014502
Suzumura Y, Hasegawa Y and Fukuyama H 1988 J. Phys. Soc. Japan 57 2768
Tajima S et al 1990 Proc. 1989 Tsukuba Conf. vol 2, ed T Shiguro and K Kajimura (Berlin: Springer)
Tallon J L and Loram J W 2000 Physica C 349 53
Takigawa M, Reyes A P, Hammel P C, Thompson J D, Heffner R H, Fisk Z and Ott K C 1991 Phys. Rev. B 43 247
Tranquada J et al 1995 Nature 375 561
Trivedi N and Ceperley D M 1989 Phys. Rev. B 40 2737
Tsuei C C et al 1994 Phys. Rev. Lett. 73 593
Uemura Y J et al 1989 Phys. Rev. Lett. 62 2317
Varma C M 1997 Phys. Rev. B 55 14554
Varma C M 2003 Preprint cond-mat/0312385
Vollhardt D 1984 Rev. Mod. Phys. 56 99
Walstedt R E and Warren W W 1990 Science 248 82
Wen X G and Lee P A 1996 Phys. Rev. Lett. 76 503
Weng Z Y, Sheng D N and Ting C S 1998 Phys. Rev. Lett. 80 5401
White S R and Scalapino D J 1999 Phys. Rev. B 60 753
Wiegmann P B 1988 Phys. Rev. Lett. 60 821
Wollman D A, vanHarlingen D J, Lee W C, Ginsberg D M and Leggett A J 1993 Phys. Rev. Lett. 71 213
Xu Z A et al 2000 Nature 406 486
Yokoyama H and Shiba H 1988 J. Phys. Soc. Japan 57 2482
Yokoyama H and Ogata M 1996 J. Phys. Soc. Japan 65 3615
Zhang F C 2003 Phys. Rev. Lett. 90 207002
Zhang F C, Gros C, Rice T M and Shiba H 1988 Supercond. Sci. Technol. 1 36
Zhang F C and Rice T M 1988 Phys. Rev. B 37 3759
Zhang S, Carlson J and Gubernatis J 1997 Phys. Rev. Lett. 78 4486
Zhang Y et al 2000 Phys. Rev. Lett. 84 2219
Zhou X J et al 2003 Nature 423 398
Zou Z and Anderson P W 1988 Phys. Rev. B 37 627