lopscience iopscience.iop.org Home Search Collections Journals About Contact us My lOPscience A renormalised Hamiltonian approach to a resonant valence bond wavefunction This article has been downloaded from lOPscience. Please scroll down to see the full text article. 1988 Supercond. Sci. Technol. 1 36 (http://iopscience.iop.Org/0953-2048/1/1/009) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 147.251.57.168 The article was downloaded on 14/03/2012 at 09:37 Please note that terms and conditions apply. Supercond. Sei. Technol. 1 (1988) 36-46. Printed in the UK A renormalised Hamiltonian approach to a resonant valence bond wavefunction F C Zhang, C Gros, T M Rice and H Shibat Theoretische Physik, ETH-Hönggerberg, CH 8093 Zurich, Switzerland Received 1 March 1988, in final form 22 April 1988 Abstract. The effective Hamiltonian of strongly correlated electrons on a square lattice is replaced by a renormalised Hamiltonian and the factors that renormalise the kinetic energy of holes and the Heisenberg spin-spin coupling are calculated using a Gutzwiller approximation scheme. The accuracy of this renormalisation procedure is tested numerically and found to be qualitatively excellent. Within the scheme a resonant valence bond (rvb) wavefunction is found at half-filling to be lower in energy than the antiferromagnetic state. If the wavefunction is expressed in fermion operators, local SU(2) and U(1) invariance leads to a redundancy in the representation. The introduction of holes removes these local invariances and we find that a d-wave rvb state is lowest in energy. This state has a superconducting order parameter whose amplitude is linear in the density of holes. 1. Introduction Very soon after the discovery of high-7c superconductivity Anderson [1] proposed that it was caused by a cooperative condensation of carriers moving in a resonant valence bond (rvb) state of spins. Since then this proposal has been studied extensively and the best account is in Anderson's recent lecture notes [2]. Many other proposals have been made (for a Review see [3]), and there have been questions raised about the use of the simplified effective Hamiltonian derived from a single-band Hubbard model in the atomic limit that forms the starting point of Anderson's treatment. We shall not go into these questions here but just point out that two of us (Zhang and Rice [4]) recently gave an explicit demonstration that a two-band model describing hybridised copper 3d and oxygen 2p states can also be reduced to the same effective Hamiltonian in an appropriate limit. The effective Hamiltonian contains the strict local constraint which forbids double occupancy of any site. This constraint is very difficult to handle analytically. One of the most physically transparent methods to treat this type of problem analytically has been the Gutzwiller approximation which introduces a renormalisation of the quantum mechanical expectation values by a classical weighting factor [5]. Such renormalisation can then be incorporated into a Hamiltonian which may be treated by conventional methods. This approach was used for the heavy fermion problem by Rice and Ueda [6] and was shown to be equivalent to an optimal slave-boson formulation by Kotliar and Ruckenstein [7]. In this paper we will consider this renormilisation t Permanent Address: Institute for Solid State Physics, University of Tokyo, Roppongi, Tokyo 106, Japan. Hamiltonian method for the effective Hamiltonian. Unlike the renormalised Anderson Hamiltonian studied by Rice and Ueda [6], in the present case the renormalised Hamiltonian cannot be simply diagonalised and we must resort to a further mean-field approximation. Mean-field approaches have been considered by many authors [8-10]. Here we choose to formulate the problem in terms of a variational wavefunction. This has several advantages. First it shows us that a consistent mean-field theory must be formulated in terms of two expectation values i.e. one must include particle-hole amplitudes of the form in addition to particle-particle amplitudes of the form . This point has been recently realised by others as well [11]J. The coupled equations to minimise the energy have a wide class of degenerate solutions at half-filling. Secondly a wavefunction formulation is suited to examining the role of the redundancy in the fermion representation which is not present in the spin representation. At half-filling, this redundancy which has its origin in the reduction from 4 to 2 degrees of freedom per site as one goes from fermion to spin representation, appears as a local particle-hole (SU(2)) and gauge (U(l)) invariance. The large degeneracy of the mean-field description arises from this redundancy and it can be shown that it corresponds to the same state in the spin representation. Further the appearance of coherence in the fermion representation is illusory so that there can be no true phase coherence as stressed by Baskaran and Anderson [12]. Thirdly, this formulation allows a direct comparison with the variational Monte Carlo (vmc) results. This allows us on the one hand to check the + In [8], Baskaran and co-workers considered the term as well. However they set it equal to zero in their calculations. 0953-2046/88/010036 + 11 $02.50 © 1988 IOP Publishing Ltd A resonant valence bond wavefunction validity of the renormalised mean-field theory and on the other hand it gives us more insight into the numerical vmc results. Both qualitatively and even quantitatively good agreement is found; for example both point to a d-wave paired state as the most stable and a true superconductivity order parameter which vanishes at half-filling and grows linearly in the deviation from half-filling. The largest discrepancy occurs for the anti-ferromagnetic state which, within this scheme is higher in energy than the rvb state, contrary to the vmc results. Our treatment is essentially limited to zero temperature and the extension to finite temperature will be non-trivial. Some discussion of the problems of calculating excitation energies is given. In particular there are two energy scales of excitations given by the gauge coherence energy (determined by the kinetic energy) and the magnetic coherence energy respectively. Anderson [2] has emphasised this splitting of the charged excitations (holons) and spin excitations (spinons). 2. The model and the renormalised Hamlltonian We study the Hubbard model on a square lattice. In the limit of large on-site Coulomb repulsion U and at one-half, or, slightly less, filling the Hubbard Hamiltonian can be transformed to the form H = Ht + Hs Hx= -t £ c]g Cja + hc ,« HS = J £ SrSj <«J> (1) with the local constraint the number of electrons on any site < 1. This transformation has a long history, and has been used by [13], amongst others. In (1) Ht and Hs are the kinetic and magnetic energies respectively and represent the nearest-neighbour pairs. St are the spin = \ operators and J = 4t2/U. We neglect terms which are higher order in the small parameters t/U and the hole concentration S (= 1 — n; where n is the electron concentration). Since the high-Tc superconducting materials show strong antiferromagnetic (af) spin correlations [14] it is believed that this model contains the essential physics for the high-Tc superconductivity [2, 4]. To study the ground state and the excited states of (1), we use a projected bcs trial wavefunction as suggested by Anderson [1] for a rvb state: \

= P*\ (2) ko> = n("* + (3) k where the Gutzwiller projection operator Pd = II, (1 — nir nu), and |0> is the vacuum state. uk and vk are the variational parameters satisfying the normalisation condition for | : | uk |2 + | vk |2 = 1. Some special forms of (2) have recently been studied numerically. Using the vmc technique, which treats the projection operator exactly [15-19], the energies of these states have been numerically calculated. It is found that in the square lattice, the projected Fermi liquid state (i.e. the state with uk v% = 0) is unstable against d-wave pairing [16]. At half-filling the energy of the d-wave state is found [17] to be very close to the ground-state energy extrapolated from the exact small system calculations [20]. In contrast to the extrapolated exact small system calculations, the d-wave trial wave-function has no long range antiferromagnetic order [17] and may therefore be viewed as an example of a quantum spin liquid. A vmc study of superconductivity has been made independently by Yokoyama and Shiba [19]. They also concluded a possibility of a d-wave superconductivity. The projected bcs wavefunction is a natural generalisation of the usual bcs state to strongly correlated systems. The projection operator, however, makes difficulties for an analytic approach. In this paper we shall use a renormalised Hamiltonian approach to treat the projection operator, and systematically investigate the state (2), carrying out explicitly the variational procedure. In this approach following Gutzwiller [5] the effect of the projection operator on the doubly occupied sites is taken into account by a classical statistical weighting factor which multiplies the quantum coherent result calculated with | cp0y. A clear description of the method has been given by Vollhardt [21]. The hopping energy and the spin-spin correlation of the nearest-neighbour sites in the state |

are related to those in the state | by <4r<>> =0t<4,c,„>o are the expectation values in the states |

0 and |

respectively. The renormalisation factors gt and gs are determined by the ratios of the probabilities of the corresponding physical processes in the states | . In figure 1 we illustrate the possible hopping processes in these two states. The probability of such a process in the state |

is CM1 - "i)nn(l - rij)] 1/2 (a) Figure 1. The possible hopping processes (a) in the non-projected pairing state (3) and {b) in the projected bcs state (2). The spins with broken arrows are optional in the (a) configurations. 37 F C Zhang et al while that in the state | is 1/2 niff are the average electron occupation numbers, (n, = ZffW,-,,) which are the same in the states |

and \(p0), because of the spin symmetry of the wavefunctions. This leads to the result [21] 0, = 23/(1 + S). (4a) To determine gs, we consider the spin exchange process shown in figure 2. The spin-spin interaction occurs only when both sites are singly occupied. The probability for such a process in the state \q>) is (n,t nn nJl nlT)1/2, while in the state | ip0) it is Kt(! - - "(t)M1 - "jtJM1 - "u)]1/2- The same result is obtained for the z component interaction S'Sj. Thus one findst gs = 4/(1 + S)2. (4b) It is important to realise that the projection operator greatly enhances the spin-spin correlations. To further illustrate this point, we list in table 1 all the possible two-site states together with their weights and the contributions to the spin-spin correlation in the half-filled case. Having determined the renormalisation factors, we can define a renormalised Hamiltonian given by H'= g,Ht + gsHs. (5) The energy of the system in the state |

can be evalu- i lie cncigy ui me sysicni 111 me siaic \ tan uc cv ated as the expectation value of H' in the state | W = (H'\. (6) Equations (4)-(6) form the basis of our renormalised Hamiltonian approach, which is analogous to the approach used by Rice and Ueda [6] for the periodic Anderson model with the difference that here we make a further mean-field approximation. This is because in the context of the periodic Anderson Hamiltonian the most important physical effect is the renormalisation of the f-level to the Fermi surface and not the spin interaction, which would make an exact treatment of the effective Hamiltonian impossible. To justify this approach, we have carried out Monte Carlo calculations. Figures 3-5 show the comparisons between the renormalised mean-field theory and the essentially exact mc results for these wavefunctions. The Figure 2. The spin exchange process in the states (2) and (3). t In a systematic series expansion on S and J/t, the higher-order terms of <5 in (4) should be dropped away to be consistent with the effective Hamiltonian (1), where the higher-order terms are not included. This, however, does not change the qualitative physics discussed in this paper. Table 1. This table illustrates the enhancement of the spin-spin correlation in the projected bsc state q>, equation (2), over that in the bcs state cp0, (3), at half-filling. The weight of the configurations actually contributing to S, • S; increases by a factor of four due to the projection. The configurations at each site are denoted by 0 (empty state), ti (doubly occupied state), and a (singly occupied state with spin a). Two-site configurations j No of configurations Weight Contribution to s;s- or s,-s; ? o a a a u Tl Ti o 0 (7 — a 0 a Tl 16 1. 4 1 5 1 5 1 5 1 ts yes quantitative agreement is within 5-15%, while the qualitative agreement is excellent for the wavefunction (2). After replacing the projection operator, the energy of the system can be evaluated analytically. The variational task is to minimise W in (6). This leads to coupled gap equations, which we will derive and solve in the following sections. -1.5 -2.0 -2.5 - < T) -2.0 -2.5 -3.0 "i-1-n—:—7n~ • : /i J// .•IT-" -»H- + -rir /// s<>;/ 0 0.01 o.io 1.00 10.00 Figure 3. A comparison between the renormalised mean-field theory (rmf) (see (4)-(6)) and the Monte Carlo (mc) result for the kinetic energy per hole in the projected bcs state (2). Both were calculated with a total number of sites Ns = 82 and a number of holes Nh = 8,16. The variational parameter A is related to the parameters of the state (2) by A* '*-Mo + IA2 + (ek-^o)2]1/2 where fi0 is a parameter, and ek is given by (fib). In the d-wave pairing state, Ak = A (cos k, — cos ky) and in the s-wave state, Ak = A. The full circles and squares are the mc results for the s- and d-waves respectively. The dotted and broken curves through the mc results are guides for the eyes. The second pair of dotted and broken curves are the results from rmf for s- and d-waves respectively. 38 A resonant valence bond wavefunction -0.20 -0.25 -0.30 -0.15 - -0.20 5 -0.25 -0.10 -0.15' -0.20 Ws=82 s -»H- =16 — 0 0.01 0.10 1.00 A 10.00 Figure 4. A comparison between the renormalised mean-field theory ((4)-(6)) (rmf) and Monte Carlo (mc) results for the nearest neighbour spin-spin correlation in the projected bcs state (2). Both were calculated with a total number of sites N, = 82 and a number of holes Nh = 0, 8,16. The variational parameter A is related to the parameters at the state (2) by "* fifc-Po + Dtf + fa-Po)2]1'2 ' where n0 is a parameter and e,, is given by (8b). In the d-wave pairing state, A,, = A(cos kx — cos kf) and in the s-wave state Ak = A. The full circles and squares are the mc results for the s- and d-waves respectively. The dotted and broken curves through them are guides for the eyes. The second pair of dotted and broken curves are the results from rmf for the s- and d-waves respectively. 3. Gap equations In this section we derive the gap equations for the projected bcs wavefunction within the renormalised Hamil-tonian scheme described in §2. We consider only the even-parity case, i.e., u_kv*k = ukv$, and |u_t|2 = Evaluating (6), we obtain W = 2gt^ek\vu\2 + N^ k x lVk_k,(\vk\2\vk,\2 + ukvtvk,ut) (7) it. *■ where Ns is the total number of sites, and Vk=-ig.Jyk (8a) ek=-tyk (8b) yk = 2(cos(/cx) + cos(*g) (8c) -0.1 Figure 5. The nearest neighbour spin-spin correlation function as a function of electron filling in the projected Fermi liquid state (uk vk = 0 in (2)). The renormalised mean-field theory ((4)-(6)) (rmf), broken curve, agrees well with the Monte Carlo (mc), full circles, result in the entire filling region. Note that ek and Vk have the same functional form, since Hs is derived by kinetic exchange. The electron number operator N = Zt(7 cka cka has expectation value = 2£fc | |2. Let n be the chemical potential of the system, the quantity we want to minimise is ft= o (12) £, = Io (13) with t = x and y, i + t denotes the nn of i in the x direction. Since we consider the even-parity case, £t is real, but At can be complex. Ak and £,k satisfy the following coupled gap equations: k' Zk = lk + n;lY,yk_k.Zk,l(2Ek,). (14) k The first one is the same as the usual bcs gap equation. The second one originates from the particle-hole correlation. From (12), it is clear that ik is related to the pairing in the unprojected state |. It describes the 'smearing' of the pseudo-Fermi surface. However, 2Lk is not the superconducting order parameter in the projected state |

in our theory. Ek turns out to be the quasi-particle excitation energy (in units of %gsJ) in the pairing state as we will show in §5. The coupled gap equations (14) are the basic equations in our approach. They can also be written in the x and y component form: A^n;1 £ costfgA./Efc k Z*= -JVr'I«»(W£4. (15) k The gap equations must be solved simultaneously with the hole concentration equation, <5 = n~1 £k iJEk. Before we discuss the non-trivial solutions, we note that Ak = 0 is a trivial solution of the gap equations. This corresponds to the projected Fermi-liquid state. In this case, £k changes sign at the Fermi surface. The parameters £, = £,,( = 0 is not included, i.e. ix = of (3), rather than the physical states | j remains unchanged under the transformation as required physically. However a singly occupied electron state transforms under SU(2) as ctJ0>j^ei9'cL|0>j. At half-filling, each site is singly occupied. Therefore any half-filled state |

transforms into itself except for an overall phase factor under the SU(2) operator V: i Although does not commute with the projection operator Fd, we observe for the half-filled state | upd\(p0y = pau\(poy. One way to see this is to notice that there is no empty site state also in Pd | 0>. Thus we can rewrite f\ii = n (^t - nu)2i- i The SU(2) transformations all commute with the operator (niT - nnf. Let \

= Pd|0>, and | = U | > then Pi I = Pa U | o> = U | q>y = eie | J>> = eiePd | cp0>. This proves explicitly that the two states | and | except for a phase factor. Therefore all the states in (17) correspond to the same physical state. The RVB ground state is non-degradable. There are also redundancies in the higher energy states in the fermion representation. For instance, the state of Baskaran, Zou and Anderson [8] at half-filled is identical to the projected Fermi liquid state, because the former transforms to the latter under (19) with a = 1, 0 = 0 in one sublattice, and a = 0, 0 = 1 in the other. This equivalence was also pointed out by Yokoyama and Shiba [18] in a different way. 41 F C Zhang et al We now comment on the local gauge symmetry in the original Hubbard model, which in terms of the original fermion operator is HH = -1 X (dl dJa + hc) + U I 4t 4t 4i dn. This Hamiltonian is not invariant under local gauge transformations with respect to the operators dltr. However, up to any finite order in t/U there exists a canonical transformation, which eliminates the doubly occupied sites [13] //„-Heff = eiStfHe-iS. At half-filling, all the odd order terms in t/U vanish. The Hamiltonian is locally gauge invariant with respect to the electron operators in the new representation, i.e. the cj(T of (1) are Wannier operators of the old representation, dia. Therefore the local gauge symmetry holds to any finite order in perturbation theory in t/U. This is the same as saying that the system has undergone a transition to a Mott insulator [13]. We have so far only examined the projected Bcs-type trial wavefunctions. It is likely that the true ground state of the model Hamiltonian (1) at half-filled is the af state [19]. Recently, Yokoyama and Shiba [18, 19] have studied a projected Hartree-Fock-type af state. Using vmc they found the energy per site at half-filled to be -0.642J, slightly lower than —0.636J, the value found by Gros [17] in the d-wave pairing state by using a similar technique. However we may argue that the holes favour the pairing state away from half-filled because of the gain in kinetic energy. We have also applied the renormalised Hamiltonian approach to the af states. Within this approximation, we find that at the half-filled, the af state has higher energy than the rvb state (2), in contrast with the vmc results. We present the derivation and the results in Appendix 1. 5. Non-half-fllled case 5.1. Ground state Firstly we examine the energy needed to create propagating Bloch states. The simplest states for holes have the form l*to> = ctok> which destroys a real electron at site i. We may also make a propagating Bloch state for the hole of the form I = CpJ

- (20) A rigorous calculation is possible at half-filling. We consider any translationally invariant spin singlet state |

at half-filling. Let us denote by a the nn spin-spin correlation in |

, a = <5, • >£,•>,,. Then the magnetic energy loss of a hole in the state Oj(T is — 4aJ, because the four bonds connecting the site i are mixing. Since the matrix of Hs in (2) between any states |4>jo> and | is diagonal, the moving hole state of (20) has the same magnetic energy as in | j(T>. The kinetic energy of the hole in (20) is given by <#,>P = 2t X + S?Si\ exp(i/» • RJt) + hc <>7> where < denotes the expectation value in the half-filled state. Using the fact that >v = 1, we get p = t(l + 4a)(cos (px) + cos (p,)). (21) Since a —0.33 for the ground state, (21) gives a band width for a Bloch hole of 0.64\t\. The minimum energy to remove an electron and create such a Bloch hole is -0.32t-4aJ. The Bloch states are not the lowest energy states of the holes however. We now apply the gap equations to study a system with a few pair of holes. The energy to create a pair of holes is — 2/i by the definition of the chemical potential. Since the parameter £ = 0 at the half-filled, (11) gives the energy per hole to be iVs_1<3H73^>0, a quantity related to the unprojected state | at half-filling. In the presence of holes, the kinetic part of the Hamiltonian explicitly breaks the SU(2) and U(l) gauge symmetries, while the Heisenberg spin part remains invariant under these symmetries. Using (4) and (5), for the states | described by (17), the magnetic energy per hole is i (!)<«■>- «*' a value equivalent to the loss of four bonds in the spin-spin correlations, and it is the same for all these states as a consequence of the SU(2) gauge invariance of the spin part of the Hamiltonian Hs. The kinetic energy per hole in this case is given by T=wM)o=-4t^+^ £t is the particle-hole correlation in | q>0} as defined in (13). When the holes are introduced, a fraction (#,) of this correlation becomes coherent in the state | cp}. Therefore the larger values of £T correspond to the lower kinetic energy of the holes. But £t are subject to (17). The kinetic energy can be written in the following form by using (17): T = -4t(2C2 -\AX + A/)1'2 sgntf, + £,). Different parameters At and an expectation value in the projected state (2). This quantity describes the Cooper pairing in a real space representation. We shall adopt the Gutzwiller method to calculate this quantity. In analogy to the derivation for the hopping energy in §2 we find that the nearest-neighbour sites i and j = ^tO.VJiX). Therefore for nearest-neighbour sites, the order parameter is related to the variational parameter A in the 0 0.1 0.2 6 Figure 7. Variational parameter A and superconducting order parameter Asc as functions of the hole concentration 5 for a choice of t/J = 5 in the d-wave pairing state. gap equations by Asc = 0,A. (22) The value of A^ as a function of 5 is plotted in figure 7 in comparison with A. Asc vanishes linearly near 6 = 0. Asc found in our theory is in good agreement with the Monte Carlo results [17]. The absence of the superconducting order parameter at half-filled obtained from (22) agrees with the discussion in §4 from the viewpoint of the local gauge symmetry. The kinetic energy of holes in the af state is found to be quite high in our analytic approach (see Appendix 1). However, the Gutzwiller approximation we adopted is too rough to determine whether a rvb or af state has lower energy. Numerical results of vmc [17-19] suggest both the spin-spin correlation energy and the kinetic energy of the holes between the d-wave pairing state and the af states are very close. The question which state is more favourable in energy remains unresolved. 5.2. Excited states and finite temperatures We begin by examining the spin degrees of freedom in half-filled and near half-filled cases. An excited state can be created by applying the spin raising operator to a specific site to obtain |¥,.+> = S/Pd|Vo>-We can commute with Pd to obtain \%,+> = ptst\. p. p' This state is therefore a superposition of two independent quasi-particle states similar to a metal where the low energy excitations are made up of superpositions of electron and hole states. The quasi-particle states (spinons) can be denned by l*PPt> = pdch EI («* + 4Tclkl)10>. (23) The quasi-particle energy Ep is denned to be the difference of the expectation values of K = H — in the state |iAP|> and in the ground state |

. We use the Gutzwiller method to calculate the energy of the state (23). The energy difference between the two states contains two parts. One is due to the changes of the renor-malization factors gt and gs, the other comes from the change of the wavefunction itself. The former just cancels exactly the second term in n in (11). Using (7) to calculate the energy difference due to the wavefunction change, we get + 2utt;kNs 1 £ Vk_pufvk. k Applying the gap equations to simplify the expression, we obtain Ep = igsJE„. (24) 43 F C Zhang et al Note this energy is independent of the local SU(2) gauge and does not depend on the particular fermion representation. At the pseudo-Fermi surface, where by definition £p = 0, we have Ep = |Ap |. Since the state (23) breaks a pair of electrons, Ep describes the binding energy of the pair at the pseudo-Fermi surface. The excitation energy depends on the particular rvb. For the Fermi liquid state (uk vk = 0) then Ep = 0 over the whole pseudo-Fermi surface. However in the ground rvb state it vanishes only at four points, e.g. when n = 1, Ep = Ep = 0, if (px, py) = (±7t/2, ±nj2) and the density of spinon states at low energies is - - 2E N(E) = YJS(E-Ep)-+ as£^0. P n(i9s J) We turn now to a brief discussion of the system at finite temperature. The extension of the mean-field gap equations to finite T is not so straightforward. The existence of a finite Ak is controlled by the energy scale of Ek i.e. by J in (24). On the other hand if we consider the limit 6-4 1 there is a very small energy scale ocdt which controls the definition of a coherent gauge. In other words it is only the kinetic energy which allows us to determine the gauge uniquely and at temperatures J > T > St, the gauge coherence will be lost. Yet in this temperature range the magnetic coherence survives since as we have stressed earlier this is independent of the choice of gauge on each site. The properties of the system in this temperature region are clearly very different from Fermi liquid behaviour as Anderson has stressed and these two energy scales should correspond to his 'holon' and 'spinon' energy scales respectively. The thermopower should obey the Heikes formula [24] and we can expect only a low mobility of the holes. However, a more detailed study of this regime is required. energy of order J to break a pair. These pre-existing electron pairs lead to a non-zero superconductivity amplitude upon doping, and the magnitude of this superconducting amplitude or order parameter is shown to be proportional to the hole concentration 6 when 8 is small. The elementary excitations at half-filling are the projected bcs quasi-particle states or spinons, with four point zeros on the pseudo-Fermi surface. Our analytic approach can also be applied to id and systems with dimensionality d ^ 3. We find that lowest energy state in id is the projected Fermi liquid rvb without electron pairing, as shown in Appendix 2. Our theory predicts no superconductivity in a id rvb. For large d. the energy per bond in the rvb pairing state is proportional to l/d, reduces relative to an af. So the pairing state is particularly favourable in 2D. The precise form of the 2d phase diagrams which depends sensitively on the relative energies of the af and d-wave rvb states as a function of <5 is too subtle a question to be settled by the approximation we use here. There are many questions that require further investigation such as the exact relationship between the discussion here in terms of phase coherence among the paired electrons and Anderson's 'holon' [25] concept or the nature of the high-temperature phase where this phase coherence is lost but strong magnetic correlations remain and presumably do not lead to a Fermi liquid that is the usual description of a normal state. Acknowledgments We would like to thank D Poilblanc, R Joynt, M Roos, T K Lee, G Kotliar for many useful discussions. Financial support from the Swiss Nationalfonds is gratefully acknowledged. 6. Discussion We have used a variational method to study a projected bcs trial wavefunction for the square lattice effective Hamiltonian. Using a Gutzwiller approximation to treat the effect of the projection operator, we obtained a renormalised Hamiltonian in which the projection operator is replaced by renormalisation factors. This approximation is shown to be in good agreement with numerical Monte Carlo calculations for such projected wavefunctions. In this mean-field approximation both particle-particle and particle-hole pairing amplitudes must be included. The fermion representation for the ground state at the half-filled band is highly redundant, due to a local SU(2) invariance at exactly half-filling. This redundancy is reflected in an apparent degeneracy of the bcs trial wavefunction before projection. Doping destroys the local SU(2) invariance and splits these degenerate states, and we find that the stable state upon doping is the d-wave pairing rvb state. In this rvb state, electrons are paired even at half-filling and it costs an Appendix 1: The projected spin-density-wave state In this Appendix, we use the renormalised Hamiltonian approach to study the projected spin-density-wave state for effective Hamiltonian (1). That state was proposed and studied using vmc by Yokoyama and Shiba [18]. The generalisation of the Gutzwiller method to the anti-ferromagnetic states for the hopping process was formulated by Ogawa and co-workers [26]. The projected spin-density-wave state [18, 19] is l*> = J\il*o> (Al.l) I *o> = n(«*ck+ <">*<*+ (A1.2) ko where * runs over the Fermi sea, Q = n/a (1, 1), and «fc = [(1 + cos 6J/2]1'2 i>t = [(1 - cos 0*)/2]1/2 cos 6k = yk/(MF + y2)1'2. AAF is a variational parameter, and yk is given by (8c). 44 A resonant valence bond wavefunction In a study of the expectation value in the state (A 1.1), we use the Gutzwiller approximation to replace the projection operator by renormalisation factors. In analogy to the analysis we discussed in §2, we find that 1 - n 9l 'l-ln^n^n 0, = 0 -2ntwl/n)-2 where nt and nl are the spin-up and spin-down electron occupation number of state | \f/0} in one sublattice respectively. The renormalisation factors reduce to (4) in the case nT = nt, and the form for gt agrees with [26]. Within this scheme, we obtain the energy per site at half-filling where w= -2J(a2 + 6Ž>2)/(1 +a2)2 k The spin-spin correlation oo corresponds to the Neel state. The results agree well for large values of AAF, but there are substantial deviations for small AAF. The kinetic energy per hole in our theory is T= -16*6/(1 +a2). For the optimal value of AAF (~0.9), T = — 2.16t, substantially higher than that in the d-wave pairing state. Note that this value is also higher than that found in 1.3- I I | I Ml |-1-1 I | I I I !|-1-1 I | I I II |-1-1 I | I I I I -4 Figure A1. Spin-spin correlation and staggered magnetisation Ms as functions of AAF in the projected spin-density-wave state. The full curves are the results of the renormalised Hamiltonian approach, and the broken curves are the vmc results (extrapolated to the infinite systems) by Yokoyama and Shiba [18, 19]. the vmc calculation [18], where the optimal AAF is found to be much smaller. Appendix 2. rvb in a id system The renormalised Hamiltonian approach can be straightforwardly applied to the model (1) in id. Using the projected bcs wavefunction (2), and the same technique for 2D, we have found that the rvb ground state at the half-filling is described by an equation between £x and &x (defined in (12M13)): with \K\2 + £ = c\ C1=(2iVs)-1Xlcos(fcx)| = 2/7t2 (A2.1) (A2.1) is parallel to (17) in 2d. Similar to the 2D case, different parameters in (A2.1) are related to each other under the SU(2) gauge transformation, and correspond to the same physical state. This state is described by the projected Fermi liquid state, where Ax = 0,