PHYSICAL REVIEW B 70, (2004) (Received 24 February 2004; revised manuscript received 17 September 2004; published 3 December 2004)

Transcription

1 PHYSICAL EVIEW B 70, (2004) Charge orderng n extended Hubbard models: Varatonal cluster approach M. Achhorn, 1 H. G. Evertz, 1 W. von der Lnden, 1 and M. Potthoff 2 1 Insttut für Theoretsche Physk, Technsche Unverstät Graz, Petersgasse 16, A-8010 Graz, Austra 2 Insttut für Theoretsche Physk und Astrophysk, Unverstät Würzburg, Am Hubland, D Würzburg, Germany (eceved 24 February 2004; revsed manuscrpt receved 17 September 2004; publshed 3 December 2004) We present a generalzaton of the recently proposed varatonal cluster perturbaton theory to extended Hubbard models at half-fllng wth repulsve nearest neghbor nteracton. The method takes nto account short-range correlatons correctly by the exact dagonalzaton of clusters of fnte sze, whereas long-range order beyond the sze of the clusters s treated on a mean-feld level. For one dmenson, we show that quantum Monte Carlo and densty-matrx renormalzaton-group results can be reproduced wth very good accuracy. Moreover we apply the method to the two-dmensonal extended Hubbard model on a square lattce. In contrast to the one-dmensonal case, a frst order phase transton between spn densty wave phase and charge densty wave phase s found as functon of the nearest-neghbor nteracton at onste nteractons U 3t. The snglepartcle spectral functon s calculated for both the one-dmensonal and the two-dmensonal system. DOI: /PhysevB PACS number(s): w, a, h, b I. INTODUCTION In recent years an ncreasng number of theoretcal and expermental studes n condensed matter physcs have focused on the descrpton and understandng of quas-one- and quas-two-dmensonal strongly correlated electronc systems. Several fascnatng propertes of these materals are due to the competton between dfferent phases wth longrange order. Hgh-temperature superconductvty n cuprates s one of the most famous examples whch s not yet understood n a satsfactory way. ealstc models that are used n ths context consst of a knetc part whch accounts for the electron moton and an nteracton part whch s of the same order of magntude. The smplest model that can be constructed under these assumptons s the tght bndng Hubbard model. It conssts of a knetc energy part, where the electrons can only hop between nearest-neghbor stes and the Coulomb nteracton U whch acts only locally on each ste. Although ths model was used wth great success for the descrpton of a wde class of materals, there are nterestng physcal questons whch requre an extenson. The ncluson of the nearest-neghbor Coulomb nteracton, for example, s necessary for the study of nhomogeneous phases, such as the charge-densty wave (CDW). Ths leads to the so-called extended Hubbard model (EHM). But knowng the approprate model for the descrpton of a materal s only the frst step on the way to understandng the physcs. Already for the smple Hubbard model wthout nonlocal Coulomb nteracton, an exact calculaton of statc and dynamc propertes s possble n very specal cases only and one must be content wth approxmate methods n general. For the nterestng case where the Coulomb nteracton U s of the same order of magntude as the bandwdth W, the conventonal perturbatve approach must fal. Ths s expected for weak-couplng perturbaton theory but also for the complementary approach wth exact treatment of the nteracton part and perturbatve treatment of the knetc energy. 1 3 Numercal methods are more promsng, such as quantum Monte Carlo (QMC), 4 exact dagonalzaton (ED), and densty-matrx renormalzaton group (DMG). 5 They are able to gve essentally exact results at least for lmted system szes or (DMG) for the one dmensonal case. Another nonperturbatve approach s the mean-feld method and, n the context of the Hubbard model, the dynamcal mean feld theory (DMFT), 6 n partcular. Whle the DMFT drectly works n the thermodynamc lmt of nfnte system sze, t must be regarded as a strong approxmaton snce spatal correlatons are neglected altogether. Cluster generalzatons of the DMFT nclude at least short-range correlatons va the exact treatment of a small cluster nstead of consderng a sngle mpurty only. Both, a recprocal-space (dynamcal cluster approxmaton, DCA 7 ) and a real-space constructon (cellular dynamcal mean feld theory, C-DMFT 8 10 ) have been suggested. Essentally the same dea s followed wth the cluster perturbaton theory (CPT), whch s a cluster extenson of the strong-couplng expanson for the Hubbard model: The lattce s dvded nto small clusters whch are solved exactly whle the hoppng between adjacent clusters s treated perturbatvely. The lowest order of the strong-couplng expanson n the ntercluster hoppng yelds the CPT. Short range correlatons on the scale of the cluster are taken nto account exactly, for nstance by the Lanczos technque at zero temperature, whle correlatons on a scale larger than the cluster sze are neglected. The CPT s a systematc approach wth respect to the cluster sze,.e., the method becomes exact n the lmt N c, where N c s the number of stes wthn a cluster. It allows for the calculaton of the sngle-electron Green s functon at arbtrary values of the wave vector k. Ths s a consderable mprovement compared to standard Lanczos calculatons for small clusters, where only a few k ponts are avalable. The CPT has been successfully used to descrbe spectral propertes of the hgh-t C materals, and has already been extended to fnte temperatures. 17 ecently a new method has been proposed whch explots a general varatonal prncple for the self-energy of a system of nteractng fermons. Ths self-energy-functonal approach (SFA) 18 approxmates the self-energy of the orgnal system n the thermodynamc lmt by the self-energy of an exactly solvable reference system wth the same nteracton part. The self-energy s vared by varyng the sngle-partcle parameters of the reference system. Choosng the reference system /2004/70(23)/235107(13)/$ The Amercan Physcal Socety

2 AICHHON et al. PHYSICAL EVIEW B 70, (2004) to be a cluster of fnte sze yelds a nonperturbatve and consstent cluster approach. It has been shown 19 that wthn ths framework the CPT as well as the C-DMFT appear as specal approaches dependng on the number of addtonal uncorrelated ( bath ) stes taken nto account: The optmum number of bath stes s actually a free parameter whch can be determned from the general varatonal prncple. It has been ponted out 19 that at least for one-dmensonal models a large cluster wthout bath stes must be preferred. The use of a reference system wthout bath stes represents a generalzed CPT n whch the sngle-partcle parameters of the fnte cluster are optmzed accordng to the varatonal prncple. Ths varatonal CPT (V-CPT) has successfully been used n a recent study for the nvestgaton of the symmetrybroken antferromagnetc phase of the two-dmensonal Hubbard model. 20 So far a consstent formulaton of the (varatonal) clusterperturbaton approach could be acheved for lattce models wth on-ste nteractons only. The reason for ths restrcton s that wthn the SFA the reference system must be chosen wth the same nteracton as the orgnal model. As detaled n ef. 18, ths ensures that functonals gven by the skeleton-dagram expanson are the same for both, the orgnal and the reference model. In case of the EHM the nteracton couples the dfferent stes of the lattce. Thus there s no reference system wth the same nteracton whch conssts of decoupled subsystems of fnte sze. The motvaton of the present paper s therefore to extend the deas of the CPT and V-CPT to the nvestgaton of the EHM ncludng nearestneghbor Coulomb nteracton. It s shown that a mean-feld decouplng of the ntercluster nearest-neghbor nteracton yelds a systematc and relable cluster approach. The paper s organzed as follows: In Sec. II we gve a short descrpton of the V-CPT method, Sec. III shows how to decouple clusters n the case of the EHM. In Sec. IV and V we present results for one two dmensons, respectvely. The conclusons are gven n Sec. VI. II. VAIATIONAL CPT Let us consder a system of nteractng fermons on a lattce wth Hamltonan H, n general consstng of a snglepartcle part H 0 and an nteracton part H 1. The lattce s then dvded nto clusters, where t s of crucal mportance for the dervaton of the method that those clusters are connected by H 0 only. The Hamltonan can then be wrtten as H = H c 0 + H 1 + H 0, where denotes the ndvdual clusters, H c 0 s the part of the sngle-partcle term that acts only nsde a sngle cluster, H 1 s the nteracton part nsde the cluster, and the ntercluster hoppng s gven by H 0 = T a,b c,a c,b, 2 a,b where the hoppng matrx T a,b s nonzero only for hoppng processes across the cluster boundares. The ndces a and b 1 are general quantum numbers wthn a cluster, e.g., poston and spn ndex, and c,a creates an electron wth quantum number a n cluster. The quantty of nterest s the sngle partcle Green s functon G,a,,b = c,a ;c,b. Usng translatonal nvar- ance at the level of the superlattce vector, the Green s functon becomes dagonal wth respect to the wave vector Q from the reduced Brlloun zone correspondng to the superlattce. The resultng Green s functon n recprocal space s a matrx G Q wth elements G Q,a,b and a,b quantum numbers wthn a cluster. Wthn the CPT approxmaton ths Green s functon G Q can be expressed n terms of Green s functons of the decoupled clusters G, agan matrces n the quantum numbers a and b, and the ntercluster hoppng T a,b by the expresson G Q = G 1 T Q 1 wth the Fourer-transformed ntercluster hoppng T Q,a,b = 1 T L a,b e Q. For the detals of the dervaton of the CPT formulas we refer the nterested reader to efs. 12 and 13 and references theren. We want to menton that one can transform Eq. (3) nto a Dyson-type equaton G Q = G Q 0 1 1, where G Q 0 s the free Green s functon of the nfnte lattce, and s the cluster self-energy. In other words CPT conssts of approxmatng the self-energy of the nfnte system by the self-energy of a cluster of fnte sze. Note that CPT s based on the exact evaluaton of small clusters wthout any self-consstency procedure, and thus does not allow for the occurrence of symmetry-broken phases. Ths restrcton s overcome wth the V-CPT method. 19,20 The observaton underlyng V-CPT s that the Hamltonan Eq. (1) s nvarant under the transformaton H 0 c H 0 c + O, H 0 H 0 O, wth an arbtrary ntracluster sngle-partcle operator O = a,b a,b c,a c,b, whch can for nstance be a fcttous symmetry-breakng feld, thus allowng for broken symmetry already on a fnte system nstead of only n the thermodynamc lmt. The queston of what choce for = a,b wll optmze the results can be answered by the SFA Wthn ths approach, the optmal value of s determned from the statonary pont of the functon

3 CHAGE ODEING IN EXTENDED HUBBAD PHYSICAL EVIEW B 70, (2004) 1 = + T tr ln n,q G 0 Q n 1, n LT tr ln G, n, 8 n where s the grand potental of the decoupled cluster, whch serves as reference system. The frequency sum runs over dscrete Matsubara frequences n, L s the number of clusters or Q ponts, respectvely, T gves the temperature, and bold symbols denote matrces n the cluster ndces a and b. Note that the fracton n the frst lne n Eq. (8) s the CPT Green s functon, Eq. (5). The sngle-partcle parameters can nclude all sngle-partcle parameters of the orgnal Hamltonan or only part of t, as well as addtonal terms, e.g., a fcttous staggered feld. The actual choce and number of parameters depends on the problem under consderaton. For more detals of the dervaton of the method see ef. 20. A necessary condton for the applcablty of the method s that the clusters are coupled by sngle-partcle operators only. At ths pont t s easy to see that a straghtforward applcaton of the method to the EHM where the clusters are also coupled by Coulomb nteractons s not possble. However, we wll show n Sec. III how one can decouple the lattce nto clusters approprate for the applcaton of CPT even n the case of the EHM. III. DECOUPLING THE CLUSTES We start from the Hamltonan of the extended Hubbard model H = T,j c j, c j + U n n + V n n j n, where, j ndcate the poston n the lattce, and for convenence we use a constant value V,j V for all nearestneghbor bonds. Accordng to Eq. (1) we decouple the lattce nto clusters yeldng H = H c 0 + H c U + H c V + H 0 + H V, 10 where the frst row ncludes only terms of a sngle cluster and the second row couples dfferent clusters. By comparng the second row wth the correspondng term n Eq. (1) one can see that the term causng problems n the case of the EHM s the nteracton term H V = V n n j, 11 whch s of two-partcle type. The symbol ndcates that the sum runs only over bonds connectng nearest neghbors n dfferent clusters. For nearest-neghbor nteractons ths means that the ndces n must belong to the cluster boundares of two adjacent clusters. For the applcaton of the method derved n Sec. II the couplng term must be of 9 sngle-partcle type, whch can be acheved by a mean-feld decouplng of the nteracton term Eq. (11). Hence we get H V,MF = V n n j + n n j V n n j. 12 Due to the translatonal nvarance wth respect to the superlattce vector, the mean-feld parameters n and n j are ndependent of and and wll be denoted by and j, respectvely. Wth these abbrevatons we get H V,MF = V,j = V = n j + n j j n j + n j j H V,MF. 13 The double sum over and reduces to a sngle sum, because for fxed values of,, and j only one term of the sum over contrbutes due to the fact that two-ste nteractons couple at most two dfferent clusters. One must be careful n order to avod double countng of the bonds. For nstance, for a one-dmensonal cluster of length N, Eq. (13) reduces to V n 1 N + n N 1 1 N, 14 because the only decoupled bond connects stes 1 and N of dfferent clusters. By ths mean-feld decouplng, two parameters are ntroduced for each decoupled bond, e.g., 1 and N n one dmenson, and n general all these parameters are ndependent of each other. But as we wll see below, the number of mean-feld parameters can be strongly reduced n specal cases. The decoupled nteracton Eq. (13) s of sngle-partcle type and can be ncluded n the ntracluster hoppng term H c 0, leadng to a modfed ntracluster sngle-partcle term H 0 c, = H c 0 + H V,MF,, 15 where we explctly denoted the dependence on the parameters. After mean-feld decouplng we fnally get the Hamltonan, H MF = H 0 c, + H c U + H c V + H 0, 16 for whch the method descrbed n Sec. II s applcable. From the decouplng of the clusters we have addtonal parameters whch are external parameters to the Hamltonan Eq. (16) and must be determned n a proper way. For ths purpose we propose two dfferent procedures

4 AICHHON et al. PHYSICAL EVIEW B 70, (2004) () One can get the parameters from a self-consstent calculaton on an solated cluster. That means that one starts wth a certan guess for the, whch are the expectaton values of the electron denstes on stes. Then the groundstate wave functon of an solated cluster s calculated, gvng new values for the. In ths step open boundary condtons (OBC) are used n order to be consstent wth the OBC necessary for the calculaton of the cluster Green s functon n Eqs. (3) and (8). These new values serve as parameters n the Hamltonan for the next determnaton of the ground state, and the whole procedure s terated untl convergence of the s acheved. Ths procedure may work qute well for the EHM n the case of a frst order phase transton between a dsordered and an ordered phase, because (due to an avoded level crossng) the transton pont,.e., the crtcal Coulomb nteracton V c, s almost ndependent of the cluster sze. 21 For second order phase transtons we expect that ths method wll not gve satsfactory results, because here we face a dscrepancy between the parameters calculated on the solated cluster and the parameters that would gve the optmal result n the thermodynamc lmt. () The shortcomng n the case of second order phase transtons can be overcome n the followng way: As we show n the Appendx, the self-consstent calculaton of mean-feld parameters s equvalent to the mnmzaton of the free energy F. Snce the relaton =F N holds at T =0, ths mnmzaton can be done at the same tme as the optmzaton of the sngle-partcle parameters n the SFA formalsm, and we can use Eq. (8) for the determnaton of the parameters, too. Note that all quanttes n Eq. (8) whch depend on the snglepartcle parameters are dependent on the mean-feld parameters as well. To keep the calculatons smple we consder only half-flled systems, where t s suffcent to use only two dfferent values for the, namely A =1 and B =1 on sublattces A and B, respectvely. Under ths assumpton we have only one meanfeld parameter, and the grand potental s =,. The general procedure s now, that for each value of the statonary pont wth respect to must be found as requred by the SFA formalsm, yeldng a functon =. By fndng the mnmum of ths functon one can determne the optmal value for. Conceptually, the latter method () of determnng the mean-feld parameters s superor to the procedure () descrbed frst as t uses nformaton on the Green s functon n the thermodynamc lmt for the calculaton of. However, one must keep n mnd that for each choce of the Green s functon G of the solated cluster must be calculated many tmes to evaluate Eq. (8) whch s much more tme consumng than the self-consstency procedure on the solated cluster. IV. ONE DIMENSION The Hamltonan of the one-dmensonal EHM s gven by H = t c, c +1, + H.c. + U n n, + V n n +1 n. 17 Throughout the paper we set t as the unt of energy. Although FIG. 1. Schematc phase dagram of the one-dmensonal EHM, followng ef. 22. Smlar phase dagrams have been reported n efs. 21, 24 26, 28, and 38, but wth dfferent extensons of the BOW phase n the U-V plane. The thck lne marks the frst order phase transton, and the dashed lne marks U=2V. ths model has been studed ntensvely, the ground-state phase dagram s stll under some dscusson. We use ths model as a testng ground for our method, because many results are avalable for comparson. The chemcal potental s =U/2+2V due to partcle-hole symmetry at half-fllng. In one dmenson at half-fllng, the phase dagram of the EHM ncludes spn densty wave (SDW) and charge densty wave (CDW) phases. By weak-couplng renormalzatongroup (G) technques ( g-ology ) 29,30 the phase boundary between SDW and CDW phase was determned to U=2V, whch actually concdes wth strong-couplng calculatons for large U and V usng second order perturbaton theory. 29,31 For ntermedate couplng the boundary was found to be shfted from the U=2V lne, enhancng the SDW phase, by QMC calculatons 21 23,32 and strong-couplng calculatons up to fourth order. 33 Moreover the nature of the transton s dfferent n the two couplng regons, wth a second order transton at weak couplng and a frst order transton at strong couplng. The multcrtcal pont, where ths change takes place, was nvestgated ntensvely n the past. Cannon and Fradkn 34 obtaned U m 1.5 by feld-theoretcal technques, whereas recent QMC studes 21,22 gave U m = The latter value s n good agreement wth results based on bosonzaton and G. 26,28,35 Other estmates for the multcrtcal pont are U m 3.7 (DMG 27 ) and U m =3.5 5 from fnte-sze extrapolatons of Lanczos results. 36 Only recently Nakamura 24,25 has proposed an addtonal phase between the SDW and CDW phases, the so-called bond order wave phase (BOW). The exstence of ths phase has afterwards been confrmed by several studes. 21,22,26 28,38 A schematc phase dagram ncludng Nakamuras BOW s depcted n Fg. 1. There s good agreement on the exstence of the BOW phase, but ts extenson n the U-V plane has not yet been clarfed n detal. A. Frst order phase transton For a frst test of our method we studed the onedmensonal EHM at U=8, whch s well above the mult

5 CHAGE ODEING IN EXTENDED HUBBAD PHYSICAL EVIEW B 70, (2004) FIG. 2. Grand potental as a functon of the mean-feld parameter at U=8 calculated on a cluster wth N c =8 stes as reference system. Upper panel, V=4.1. Lower panel, V=4.2. Sold lnes, wth optmzaton of a staggered feld. Dashed lnes, wthout optmzaton of a staggered feld. FIG. 3. Ground state energy E 0, knetc energy E kn, and order 2 parameter m CDW of the one-dmensonal EHM at U=8 after fnte sze scalng. Lnes are gudes to the eye only. crtcal pont. The phase transton s then of frst order wthout any BOW phase between SDW and CDW phases. As reference system H accordng to Sec. II, we used decoupled clusters of dfferent lengths consstng of N c =8, 10, and 12 stes, respectvely. For the determnaton of the mean-feld parameter we used the method () descrbed n Sec. III, where s calculated from the mnmum of the energy of the system. For the SFA optmzaton of the sngle-partcle parameters, we were guded by a recent study of the Hubbard model, 19 whch showed that t s not necessary to use the hoppng n the cluster or a staggered magnetc feld as varatonal parameter. Snce here we study charge-orderng effects, we used as varatonal parameter a staggered feld coupled to the charge denstes gven by Eq. (7) wth a,b = a,b e Q a, 18 where Q= s the wave vector of staggered orderng and s the staggered-feld strength. The grand potental obtaned n ths way s shown n Fg. 2 at two values of the nterste Coulomb nteracton. For comparson, calculatons wthout optmzaton of the staggered feld are shown as dashed lnes n Fg. 2. As one can see, the optmzaton gves only mnor changes to. The optmal staggered-feld strengths n these calculatons vared between opt =0.0 at =0.0 and opt 0.05 at =1.0 at both values of V. From the shape of one can drectly nfer the order of the transton. If three mnma occur at =0 and =± CDW, t s of frst order, whereas t s of second order f has only two mnma at =± CDW and a maxmum at =0. As one can easly see n Fg. 2, we have clear evdence for a frst order phase transton at U=8 wth an SDW mnmum at =0.0 and two degenerate CDW mnma at =± CDW.AtV =4.1 the SDW phase s realzed, 0 CDW, whereas at V=4.2 we have 0 CDW and the CDW phase s the stable one. Thus we can state that the crtcal value V c for the phase transton s located between V=4.1 and V=4.2. For a more accurate determnaton of the phase boundary V c, we have calculated the grand potental at several values of V and cluster szes N c =8, 10, and 12. In addton to the grand potental and the ground state energy E 0 = + N e wth N e the number of electrons n the system, we calculated the order parameter m CDW = 1 N c n j n e Q j, 19 j where Q=, N c s the number of cluster stes, and the knetc energy E kn. Both propertes can be extracted from the spectral functon. 12,13,20 Wthn our approach t s necessary to use the Lehmann representaton for the cluster Green s functon wth small but fnte Lorentzan broadenng. Whereas the grand potental Eq. (8) shows only mnor dependence on ths broadenng, the dependence of the order parameter and the knetc energy s consderably larger and one must do an extrapolaton to =0. 13 Although the formalsm apples to the thermodynamc lmt, results show a fnte sze dependence due to the fnte sze of the clusters servng as reference system. We found that the order parameter exhbts the strongest fnte-sze effects, whch were of the order 2 2 m CDW,Nc =10/m CDW,Nc = at all values of V. Lnear fnte-sze scalng to N c = s easly done and the results are shown n Fg. 3. Our results should be compared to Fg. 10 of ef. 21 whch shows excellent quanttatve agreement wth a devaton of less than 2% for the calculated quanttes at all values of V. From our calculatons we get V c = , agan n agreement wth the prevous studes. 21,27 In order to provde a complete pcture of the method we also performed calculatons wth mean-feld parameters obtaned by a self-consstent procedure on an solated cluster, see method () n Sec. III. For nstance for N c =12 and V =4.1 one fnds self-consstent solutons for =0 and for SC =0.832, whch dffers only slghtly from the value extracted from the grand potental, CDW = For ths reason the calculaton of the ground-state energy, knetc energy, and order parameter usng SC nstead of CDW gves practcally the same results as n Fg. 3. In the present case t s therefore suffcent to calculate the mean-feld parameter from an solated cluster whch s much faster than fndng the mnmum of the grand potental

6 AICHHON et al. PHYSICAL EVIEW B 70, (2004) FIG. 4. Densty plot of the spectral functon A k, of the one-dmensonal EHM at U=8, calculated on a cluster of sze N c =12 wth Lorentzan broadenng =0.1. Darker regons represent larger spectral weght. Coulomb nteracton V as ndcated n the plots. Whte lnes are fts to a Hartree-Fock SDW/CDW dsperson (see text). Whereas the propertes we have shown so far are well known for the one-dmensonal EHM, we addtonally calculated for the frst tme the spectral functon for arbtrary wave vector k. In Fg. 4 results are shown at U=8 and selected values of V wth a reference system consstng of N c =12 cluster stes, and the mean-feld parameter calculated selfconsstently by method (), see Sec. III. We want to menton that the strped structure, partcularly vsble n the regons marked by C n Fg. 4, occurs because the decouplng nto clusters breaks the translatonal nvarance of the system. The spectral functon at V=2.0 s very smlar to the spectral functon of the Hubbard model V=0 12,13 wth splttng of the low-energy band nto a spnon and an holon band, whch are marked n Fg. 4 by A and B, respectvely. Ths smlarty could have already been expected based on the full Hartree-Fock soluton decouplng of all nteracton terms n the Hamltonan where one has no dependence on V at all n the SDW phase. But ths smple pcture holds only away from the transton pont V c as can be seen n Fg. 4 n the plot at V=4.0. At ths pont, n the vcnty of the phase transton V c =4.14, the gap s consderably smaller than at V = 2.0, a clear devaton from the Hartree-Fock predcton. Ths ndcates that charge fluctuatons become very mportant n ths regme, whch are completely neglected by the Hartree-Fock approxmaton, but are taken nto account on the length scale of the cluster n our approach. But although we found ths devaton, one can stll see resduals of the splttng of the low-energy band, a sgnature for spn-charge separaton. For ths reason we nfer that spn-charge separaton s present up to the transton pont. The whte lnes n Fg. 4 correspond to fts of the holon branch to a Hartree- Fock dsperson E k =± 2 + k 2. The ftted values for the hoppng matrx element t ft and the gap ft are denoted n Table I where we ncluded the values at V=0 for completeness. One fnds that the gap ft s almost constant from V = 0 to V = 2 and, as mentoned above, consderably decreases near the the phase transton V=4. The hoppng matrx element t ft shows the opposte behavor and ncreases when approachng the transton pont from below. Ths s due to the fact that n the vcnty of V c, doubly occuped and sngly occuped stes becomes close n energy, whch enhances the movement of the electrons. The actual value of the matrx element t ft s very large compared to the orgnal value t =1 n the Hamltonan. A ft to the spnon band would gve a smaller value closer to t=1, but whereas fttng to the holon band s consstent over the whole range of momentum vectors k, the spnon band s only present for k /2 for 0 (and k /2 for 0, respectvely). The spectral functon n the CDW phase shows a qualtatvely dfferent behavor. At V=4.5 we found a gap consderably larger than n the SDW phase, and ths gap ncreases very fast wth ncreasng V, as can be seen n the plot at V = 6. Moreover, no evdence for spn-charge separaton can be seen n the spectral functons. By comparng the ftted value ft wth the Hartree-Fock soluton HF, one can see that the agreement at V=4.5 s better than at V=4, and that t becomes stll better wth ncreasng V. For ths reason we conclude that charge fluctuatons whch are neglected n the Hartree-Fock approxmaton play a mnor role n the CDW phase. The values for t ft and ft gven above are determned by calculatons wth a N c =12 cluster as reference system. An analyss of the fnte-sze dependence of these propertes shows that fnte-sze effects are almost neglgble n the

7 CHAGE ODEING IN EXTENDED HUBBAD PHYSICAL EVIEW B 70, (2004) FIG. 5. Grand potental as functon of the mean-feld parameter at U=3 calculated on a cluster wth N c =8 stes as reference system. Upper panel, V=1.6. Lower panel, V=1.7. Sold lnes, wth optmzaton of a staggered feld. Dashed lnes, wthout optmzaton of a staggered feld. The arrow marks the CDW mnmum at V=1.7. SDW phase well below V c. However, n the vcnty of the transton pont, these effects ncrease consderably, especally for ft. For nstance, at V=4.0 we found t ft =2.49 and ft =1.60 for the N c =8 cluster. Ths means that the values gven n Table I underestmate the hoppng and overestmate the gap n the vcnty of V c. In the CDW phase, the fntesze effects become smaller agan, but are stll larger than n the SDW phase (e.g., t ft =1.77 and ft =7.38 for V=6 and N c =8). B. Second order phase transton So far all calculatons were done at U=8, where the system shows a frst order phase transton. In the followng, we study the EHM at U=3, where the model exhbts a second order transton nto the charge ordered CDW phase. 21,27 In ths paper we do not consder the BOW, snce t has been argued that the SDW-BOW transton s of Kosterltz- Thouless type. 25 For an analyss of ths type of transton the avalable cluster szes are far too small and do not allow a clear dstncton between SDW and BOW phase. We calculate the grand potental n the same way as n Sec. IV A n order to determne. The result of a calculaton on a cluster consstng of N c =8 stes s shown n Fg. 5. One can easly see a strkng dfference between the grand potental at U=8, Fg. 2, and at U=3. In the latter case there s only a sngle mnmum. It s located at =0 for V V c. Wth ncreasng V the curve for becomes flatter n the regon around =0 and fnally two degenerate CDW mnma occur at =± CDW for V V c. Note that here changes contnuously when crossng V c, whereas t shows a dscontnuty n the case of a frst order phase transton. We fnd that now t s ndeed mportant to use a staggered feld, Eq. (18), as a varatonal SFA parameter. In Fg. 5, results are shown wth such an optmzaton (sold lnes) and wthout (dashed lnes). Whereas at V = 1.6 both calculatons show only the SDW mnmum at =0, they dffer at V=1.7 where the system should already be n the charge-ordered phase. 21,24,25,27 Wthout optmzaton of the staggered feld, FIG. 6. Ground state energy E 0, knetc energy E kn, and order 2 parameter m CDW of the one-dmensonal EHM at U=3 for cluster szes N c =8 (dotted), N c =10 (dashed), and N c =12 (sold lne). we would stll fnd the SDW mnmum at =0, but wth optmzaton the mnmum shows up for a fnte value of = ±0.31 characterstc for the CDW phase. For the determnaton of the crtcal value V c, we calculated the ground state energy E 0, knetc energy E kn, and the order parameter m CDW at several values of V, whch are contnuous across the transton, shown n Fg. 6. The cluster szes are too small for a systematc fnte-sze scalng. From the knetc energy and the order parameter calculated on a cluster of sze N c =12, we extract a crtcal value of V c = , whch s n good agreement wth the crtcal value V c 1.65 obtaned by QMC 21 and dagonalzaton methods, 24,25,36 and wth V c = from DMG calculatons. 27 The slght dfference s lkely due to remanng fnte-sze effects. Moreover we made use of a sngle varatonal parameter only, namely the staggered feld Eq. (18), and t can be expected that ncludng more snglepartcle parameters n the SFA optmzaton procedure would gve even more accurate results. We would lke to pont out that n the present case of a second order phase transton, the most accurate way of calculatng the mean-feld parameter s to fnd the mnmum n the grand potental ncludng SFA optmzaton of snglepartcle parameters. Calculatons on a cluster of sze N c =12 showed that wthout optmzaton the crtcal value would be V c = Compared to V c = ths s further away from the values obtaned by other methods as gven above. Calculatons wth obtaned self-consstently on an solated cluster are nsuffcent. In ths case one would get V c = for the N c =12 cluster. Ths means that for a second order phase transton should be determned by mnmzng the grand potental, whereas for frst order transtons the self-consstent determnaton was suffcent. The spectral functon A k, at V=1.0, 2.0, and 3.0, whch has not been calculated prevously, s depcted n Fg. 7. We found that the spectral functon at V=1.0 shows only mnor dfferences to the spectral functon of the Hubbard model V=0. The whte lnes n Fg. 7 are fts to a Hartree- Fock SDW/CDW dsperson. The parameters t ft and ft can be read off from Table. II. In the SDW phase at V=0 and

8 AICHHON et al. PHYSICAL EVIEW B 70, (2004) TABLE I. Ftted values for the hoppng matrx element t ft, gap ft, and gap HF of the full Hartree-Fock approxmaton at U=8. Ftted values from results for the N c =12 cluster. t ft ft HF V= V= V= V= V= systems, especally n the context of hgh-temperature superconductvty. But dfferent from the one-dmensonal case, where many sophstcated methods have been used to nvestgate the extended Hubbard model as descrbed n Sec. IV, only few studes have been done for the two-dmensonal EHM. One reason for ths s that many modern methods such as DMG or fermonc loop-update QMC are dffcult to apply to more than one spatal dmenson. However, wthn our present approach, the extenson to two dmensons s straghtforward. The two-dmensonal EHM s defned by the Hamltonan H = t c,, c j, + H.c. + U n n + V n n j n, 20 FIG. 7. Densty plot of the spectral functon A k, of the onedmensonal EHM at U=3 calculated on a cluster of sze N c =12 wth Lorentzan broadenng =0.1. Darker regons represent larger spectral weght. From top to bottom, V=1.0,2.0,3.0. Whte lnes are fts to a Hartree-Fock SDW/CDW dsperson (see text). V=1.0, the gap ft s constant. Smlar to the case U=8 the agreement between ft and HF s better n the CDW phase than n the SDW phase. The hoppng parameter t ft ncreases when approachng the phase transton from below, smlar to Table I, but the ftted values for t ft are consderably smaller than n the case U=8. For the fnte-sze effects of these propertes the same behavor was found as for U=8. V. TWO DIMENSIONS The two-dmensonal Hubbard model s one of the most ntensely dscussed models for strongly correlated electron where connects nearest neghbors and the chemcal potental s =U/2+4V at half-fllng. Early QMC studes 39 showed that ths model has a SDW-CDW transton smlar to the one-dmensonal case wth transton pont V c U/4. But due to numercal dffcultes t was mpossble to determne the exact poston and the order of the phase transton. For repulsve nteractons, calculatons wthn the Hartree- Fock approxmaton showed two stable phases for the Hamltonan Eq. (20) at half-fllng, the SDW and CDW phase, separated by a phase boundary at V c =U/4. The same value for the crtcal nteracton was obtaned by the fluctuaton-exchange approxmaton (FLEX). 43 For the applcaton of the method presented n Sec. II, the two-dmensonal square lattce must be decoupled nto clusters of fnte sze. Three possble tlngs wth dfferent numbers of cluster stes N c are shown n Fg. 8. Some care must be taken concernng the staggered orderng. Whereas for clusters wth N c =8 and N c =10 shown n Fg. 8, the staggered orderng ndcated by open and full crcles s commensurate over the cluster boundares, a straghtforward decou- TABLE II. Same as Table I, but for U=3. t ft ft HF V= V= V= V=

9 CHAGE ODEING IN EXTENDED HUBBAD PHYSICAL EVIEW B 70, (2004) FIG. 9. Grand potental calculated on a cluster of sze N c =8 at U=8, V=2.1 (upper panel) and U=3, V=0.76 (lower panel). FIG. 8. Possble tlngs of the two-dmensonal square lattce nto clusters that allow for staggered orderng, N c =8 (bottom rght), N c =10 (bottom left), supercluster wth N c =48 (top). plng nto clusters of sze N c =12 s not possble. As one can easly see, a supercluster wth N c =48 consstng of four N c =12 cluster must be constructed n order to take nto account the staggered orderng correctly. The Green s functon of the supercluster can be calculated by swtchng off the hoppng processes that connect the sngle N c =12 clusters, n other words on bonds across the dotted lnes n Fg. 8. Ths gves a block-dagonal Hamltonan whch can be treated by the Lanczos algorthm. The swtched off hoppng processes are then ncorporated agan perturbatvely, that means by ncludng the correspondng hoppng terms n the matrx T a,b n Eq. (4). Note that here the vectors and denote the superclusters and not the sngle N c =12 clusters. Of course there are many other possble tlngs lke the 4 3 cluster used n efs. 12, 13, and 44, but also n that case a supercluster of N c =24 must be used. We start the analyss of the two-dmensonal EHM wth the determnaton of the order of the phase transton. For ths purpose we use the N c =8 cluster shown n Fg. 8 and calculate the grand potental n the vcnty of the transton pont at U=8.0 and U=3.0 as descrbed n method () n Sec. III. Here we dd not use a staggered feld as varatonal parameter, because t does not change the qualtatve shape of (see Fgs. 2 and 5) and s therefore not necessary for the determnaton of the order of the transton. The result of ths calculatons s shown n Fg. 9. At both valus of U we found three mnma, located at =0 and =± CDW. Ths ndcates a frst order phase transton, dfferent from the one-dmensonal EHM, where at U=3.0 the transton s of second order. We checked that ths dfferent behavor s not lkely to be a fnte sze effect due to the small lnear dmenson of the two-dmensonal N c =8 cluster by calculatng for the one-dmensonal model wth N c =4 whch stll shows clear evdence of a second-order phase transton at U=3. The fact that the system shows frst order transtons at both U=8.0 and U=3.0 smplfes the subsequent calculatons. As dscussed n the precedng secton, one gets good results n the case of a frst order transton by usng a meanfeld parameter determned self-consstently on an solated cluster, as descrbed n method () n Sec. III. Ths procedure s much faster than the calculaton of the grand potental for many values of, whch makes t possble to use the N c =48 supercluster shown n Fg. 8. We want to menton at ths pont that the calculaton of the grand potental for the twodmensonal system s much more tme consumng than for one dmenson because of the larger number of Q ponts requred n Eq. (8). For one dmenson L 40 s suffcent for convergence, whereas L 500 s necessary for two dmensons. Nevertheless t s of crucal mportance to use a cluster as large as possble, because the rato of bonds treated exactly to mean-feld decoupled bonds ncreases wth ncreasng cluster sze, especally pronounced for the twodmensonal square lattce. After havng determned the mean-feld parameter for the CDW phase self-consstently, we also performed an SFA optmzaton of a staggered feld, Eq. (18). A few more words must be sad about calculatons n the SDW phase =0. ecent studes 20 of the pure Hubbard model revealed that t s mportant to take nto account the long-range magnetc order for the accurate descrpton of salent features of the system. Ths can be acheved by usng a staggered magnetc feld as varatonal parameter, gven by a,b = h a,b z e Q a, 21 wth z = ±1 for spn projecton =,, and h the strength of the feld. Addtonally t was argued that due to the connecton of the hoppng parameter t and the magnetc exchange constant J, results could be further mproved by lettng the hoppng n the clusters be of strength t and optmzng the staggered magnetc feld and t smultaneously. Therefore we use a,b = h a,b z e Q a ab, 22 where the symbol ab s equal to one for nearest-neghbor bonds nsde the cluster and zero otherwse. The feld

10 AICHHON et al. PHYSICAL EVIEW B 70, (2004) FIG. 10. Ground state energy E 0, knetc energy E kn, and order 2 parameter m CDW of the 2D EHM at U=8.0 (left) and U=3.0 (rght). Calculatons were done on a N c =48 supercluster. strength h and =t t are the varatonal parameters n the optmzaton procedure. To sum up, the followng steps are performed n the analyss usng the N c =48 supercluster: () Frst we determne the mean-feld parameter CDW n the CDW phase selfconsstently on an solated cluster and () use a staggered feld Eq. (18) for an SFA optmzaton procedure. () In the SDW phase =0 the staggered magnetc feld Eq. (21) and the ntracluster hoppng t are optmzed smultaneously. (v) After determnaton of the SFA varatonal parameters we calculate the quanttes we are nterested n. The results for the ground state energy, knetc energy, and order parameter are shown n Fg. 10. At both U=8.0 and U=3.0, the behavor of a frst order transton can be seen, where the change n the slope of E 0 s much stronger at U=8.0 than at U=3.0. Ths change at U=8.0 s even more pronounced than for the one-dmensonal model at U = 8.0. From Fg. 10 we can extract the crtcal value V c of the phase transton by fttng E 0 to a straght lne n the vcnty of the transton pont, and for the N c =48 supercluster we fnd V c = at U=8.0, and V c = at U=3.0. These values of V c are much closer to the Hartree-Fock result V c =U/4 than for one dmenson. Wthn our approach we cannot clarfy whether ths s an ntrnsc feature of the twodmensonal model or t s an artfact of the approxmaton due to the larger number of mean-feld decoupled bonds. The SFA varatonal parameters n the SDW phase near the phase transton pont are found to be almost ndependent of the nteracton V. At U = 8, the optmzaton resulted n t 1.1 for the ntracluster hoppng and h 0.14 for the staggered magnetc feld. The optmzaton of just one sngle parameter leads to t 1.03 h=0 and h 0.12 t =t, and the value of also dffers sgnfcantly from the value obtaned by the smultaneous optmzaton of t and h. Ths means that due to the strong connecton between the magnetc orderng and the hoppng matrx element t s mportant to optmze t and h smultaneously n order to get the best approxmaton for the physcs n the thermodynamc lmt. In the charge ordered phase the dependence of the varatonal parameter, Eq. (18), on the nteracton V s larger wth =0.08 at V =2.01 and =0.22 at V=2.1. A smlar behavor can be found FIG. 11. Densty plot of the spectral functon A k, of the two-dmensonal EHM at U=8 calculated on a N c =48 supercluster wth broadenng =0.1. Darker regons represent larger spectral weght. Top, V=1.0. Bottom, V=3.0. Whte lnes are fts to Hartree- Fock dspersons. For the meanng of the black lnes at V=1.0 see text. at U=3: In the SDW phase the varatonal parameters t 1.61 and h 0.15 are almost ndependent of V. In the CDW phase we get = 0.03 at V=0.76 and = 0.18 at V =0.84. Whereas the applcaton of the magnetc staggered feld exhbts the symmetry h h, ths s not the case for the staggered feld Eq. (18), because the symmetry s already broken by the mean-feld decouplng. We found no statonary pont of for fnte h n the CDW phases. The spectral functon at U=8 n the SDW phase V =1.0 and n the CDW phase V=3.0 s shown n Fg. 11. We found that the spectral functon at V=1.0 s very smlar to the spectral functon of the Hubbard model V=0. 20 One can see that the spectrum manly conssts of four features, two hgh-energy Hubbard bands and two low-energy quaspartcle bands, separated by a gap n the spectrum. The dsperson of these low-energy exctatons n the SDW phase dffers sgnfcantly from the Hartree-Fock shape shown as whte lnes n the upper panel of Fg. 11, whch does not account for the splttng nto coherent low-energy bands and hgh-energy Hubbard bands. The ft parameters were t ft =1.34 and ft =2.51. The wdth of the coherent bands X 1.25 s rather set by the magnetc exchange J, consstent wth QMC calculatons at V=0. 45,46 The black lnes are fts to E k =± +J/2 cos k x +cos k y 2 whch ac

11 CHAGE ODEING IN EXTENDED HUBBAD PHYSICAL EVIEW B 70, (2004) counts better for the dsperson of the low-energy bands than the Hartree-Fock dsperson. 45 The ft parameters were ft =2.69 and J ft = 0.63, whch s n good agreement wth the second-order perturbaton theory result J= 4t 2 / U V = In the CDW phase the whte lnes correspond to Hartree-Fock dspersons wth ft parameters ft =7.69 and t ft =1.16, and dfferent from the SDW phase they agree well wth the exctatons of A k,. Note that here the fnte-sze dependence of the ftted parameters s hard to nvestgate, snce the clusters dffer not only n the number of stes, but also n ther shape, dfferent from one dmenson. Hence we could not extract a conclusve fnte-sze behavor from our results. Fgure 12 dsplays the spectral functon at U=3 at nteractons V = 0.5 and V = 1.0, respectvely. The whte lnes agan correspond to Hartree-Fock dspersons wth ft parameters t ft =1.06, ft =0.64 at V=0.5, and t ft =1.07, ft =1.82 at V=1.0, respectvely. As n the case U=8 the dsperson of the coherent low-energy bands n the SDW phase dffers from the Hartree-Fock predcton, but n ths case the devaton s much smaller. We dd not fnd an accurate functonal form n order to ft the low-energy exctatons, but nevertheless we can extract the value of J from the bandwdth of the coherent bands yeldng J= 1/2 X Ths value s agan n good agreement wth the perturbaton theory result J= 4t 2 / U V = 1.6. We would lke to menton that our results at U=3 n the SDW phase are qualtatvely dfferent from QMC results at U=3, V=0, and nverse temperature =3t, 47 where the spectral functon shows metallc behavor wth no gap around the Ferm energy. Ths dfference may be due to temperature effects or due to poor resoluton of the maxmum-entropy nverson of QMC correlaton functons. At both U=8 and U=3, one can easly see that agreement of the Hartree-Fock dspersons wth the low-energy exctatons of A k, s better n the CDW phase than n the SDW phase. In addton the gap HF calculated wthn the Hartree- Fock approxmaton s much closer to the ftted gap ft n the CDW phase (e.g., HF =7.76, ft =7.69 at U=8, V=3) than n the SDW phase (e.g., HF =3.57, ft =2.51 at U=8, V=0). Therefore we conclude that n the CDW phase charge fluctuatons play only a mnor role compared to the SDW phase, smlar to the one-dmensonal system. FIG. 12. Same as Fg. 11, but at U=3. Top, V=0.5. Bottom, V =1.0. VI. CONCLUSIONS The ncluson of nonlocal Coulomb nteractons n quantum cluster approaches s of general nterest n current condensed matter theory. In ths paper we have presented a generalzaton of the varatonal cluster perturbaton theory to extended Hubbard models at half-fllng. The method s based on the self-energy-functonal approach (SFA) whch uses dynamcal nformaton of an exactly solvable system (reference system H ) n order to approxmate the physcs n the thermodynamc lmt. For the applcaton of ths method, a mean-feld decouplng of the ntercluster part of the nearest-neghbor Coulomb nteracton s performed frst. After ths step, one s left wth a Hamltonan whch couples the dfferent clusters va the hoppng only and whch can be treated by the known (varatonal) CPT procedure. The mean-feld decouplng yelds effectve onste potentals on the cluster boundares as external parameters of the Hamltonan. These parameters are determned ether selfconsstently on an solated cluster (suffcent for the study of frst order phase transtons) or by determnaton of the mnmum of the SFA grand potental. In order to test the accuracy of our approach we appled the method to the extended Hubbard model n one dmenson, because results from other methods lke QMC and DMG are avalable for comparson. At U=8 the results for the crtcal nteracton V c, the ground-state energy, knetc energy, and charge order parameter showed excellent quanttatve agreement wth prevous QMC studes. At U=3 our method predcted a second-order phase transton wth transton pont V c = agan n good agreement wth prevous studes. In addton we calculated the spectral functon for several values of the nteracton V, whch has not been done prevously. At both U=8 and U=3, we found evdence for spncharge separaton n the SDW phase, but not n the CDW phase. By fttng the bands by Hartree-Fock dspersons we found that the hoppng parameter s strongly renormalzed. The agreement between the ftted value of the gap and the value wthn the Hartree-Fock approxmaton was much better n the CDW phase than n the SDW phase gvng rse to the concluson that charge fluctuatons play a mnor role n the CDW phase. Whereas the applcaton of sophstcated methods lke DMG or fermonc loop-update QMC to more than one

12 AICHHON et al. PHYSICAL EVIEW B 70, (2004) dmenson s dffcult, ths extenson s straghtforward wthn the present approach. We were thus able to perform the frst nonperturbatve study of the two-dmensonal extended Hubbard model on a square lattce at half-fllng and zero temperature beyond Hartree-Fock. We found frst order transtons at both U=8 and U=3 wth transton ponts V c = and V c = for an N c =48 supercluster, respectvely. The spectral functon n the SDW phase shows coherent low-energy quaspartcle exctatons wth bandwdth set by the magnetc exchange constant J, and an ncoherent background, consstent wth prevous QMC studes for the Hubbard model at V=0. The Hartree-Fock predcton dffers sgnfcantly from the low-energy feature and does not descrbe the splttng nto coherent quaspartcle bands and ncoherent background. In the CDW phase the Hartree-Fock dspersons account much better for the exctatons, and no addtonal low-energy features caused by a magnetc orgn could be found. Smlar to one dmenson the agreement between the Hartree-Fock approxmaton and the low-energy exctatons obtaned by the present method s much better n the CDW phase, confrmng that charge fluctuatons are less mportant n the charge-ordered phase than n the SDW phase. In ths paper we appled our method to half-flled systems only, but one can study orderng phenomena at other fllngs, too, as long as the possble order patterns are commensurate wth the shape of clusters used as reference system. Wth some effort t s also possble to study phases wth long wavelength charge densty waves by couplng several clusters to a supercluster and applyng approprate contnuty condtons between the ndvdual clusters wthn the supercluster. In addton the applcaton to systems wth lattce geometry dfferent from the two-dmensonal square lattce, e.g., ladder materals, s an nterestng subject for further studes. Work n ths drecton s n progress. ACKNOWLEDGEMENTS Ths work has been supported by the Austran Scence Fund (FWF), projects P15834 and P One of the authors (M.A.) s supported by a doctoral scholarshp program of the Austran Academy of Scences. The authors gratefully acknowledge useful and stmulatng dscussons wth M. Hohenadler, W. Koller, E. Arrgon, and C. Dahnken. The authors thank A. Sandvk for useful comments on ths paper. APPENDIX: MEAN-FIELD SOLUTION AND FEE ENEGY In ths secton we show that a mean-feld soluton obtaned self consstently s drectly connected to a mnmum n the free energy. For smplcty let us assume that we have only two dfferent mean-feld parameters A =1 and B =1+, see also Sec. III. We can wrte the mean-feld decoupled Hamltonan Eq. (16) as H MF = H 0 MF + H 1 MF,, A1 0 where H MF ncludes all terms ndependent of the meanfeld parameters. Accordng to the thrd lne n Eq. (13) H 1 MF, s gven by 1, = V H MF n B + n j A A B = V n 1+ + n j 1 1 2, A2 where we assumed wthout loss of generalty that the bonds connect stes on sublattce A wth stes j on sublattce B. The free energy of the system s gven by F = 1 ln Z = 1 ln tr e H MF = 1 ln tr exp H MF 0 H 1 MF,. Takng the dervatve wth respect to yelds F = V n n j +2. A3 A4 All clusters are equvalent, therefore we suppress the ndex n the followng. Settng ths dervatve to zero we get the self-consstency condton n n j +2 =0. A5 For one dmenson, Eq. (A5) s gven by n N n 1 =2, A6 because n ths case we have only one decoupled bond 1N wth ste 1 N belongng to sublattce A B, respectvely. To conclude, one can state that f self-consstency, Eq. (A5), s fulflled, then the free energy has an extremum wth respect to the mean-feld parameter. By thermodynamc stablty arguments ths extremum must always be a mnmum. 1 W. Metzner, Phys. ev. B 43, 8549 (1991). 2 S. Parault, D. Sénéchal, and A.-M. S. Tremblay, Phys. ev. Lett. 80, 5389 (1998). 3 S. Parault, D. Sénéchal, and A.-M. S. Tremblay, Eur. Phys. J. B 16, 85(2000). 4 E. Dagotto, ev. Mod. Phys. 66, 763 (1994). 5 S.. Whte, Phys. ev. Lett. 69, 2863 (1992); Phys. ev. B 48, (1993). 6 For a revew see A. Georges, G. Kotlar, W. Krauth, and M. J. ozenberg, ev. Mod. Phys. 68, 13(1996)