Weak-coupling Treatment of Electronic (Anti-)Ferroelectricity in the Extended Falicov-Kimball Model
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b EPJ manuscript No. (will be inserted by the editor)
Weak-coupling Treatment of Electronic (Anti-)Ferroelectricity inthe Extended Falicov-Kimball Model
Claudia Schneider, Gerd Czycholl
Institute for Theoretical Physics, University of Bremen, D-28334 Bremen, GermanyOctober 26, 2018
Abstract.
We study the (spinless) Falicov-Kimball model extended by a finite band width (hopping t f )of the localized (f-) electrons in infinite dimensions in the weak-coupling limit of a small local interbandCoulomb correlation U for half filling. In the case of overlapping conduction- and f-bands different kinds ofordered solutions are possible, namely charge-density wave (CDW) order, electronic ferroelectricity (EFE)and electronic antiferroelectricity (EAFE). The order parameters are calculated as a function of the modelparameters and of the temperature. There is a first-order phase transition from the CDW-phase to theEFE- or EAFE-phase. The total energy is calculated to determine the thermodynamically stable solution.The quantum phase diagrams are calculated. PACS.
One of the simplest lattice models for strongly correlatedelectron systems, the Falicov-Kimball model (FKM) [1],consists of two types of spinless electrons, namely delo-calized band ( c -) electrons and localized f -electrons, anda local Coulomb (Hubbard) interaction between c - and f -electron at the same site. The FKM was originally intro-duced as a model for metal-insulator and valence transitions[1].It can also be interpreted as a model for crystallization(identifying the ”heavy” f -particles with the nuclei, the c -particles with the electrons and for U < etal. [4] that a novel ferroelectric state could be present inthe mixed-valence regime of the FKM. Whereas a ferro-electric transition is usually connected with a structuralphase transition[5], a purely electronic mechanism wouldlead to this kind of ferroelectricity suggested to occur forthe FKM; it has, therefore, been termed ”electronic ferro-electricity” (EFE)[4]. The origin of EFE is a non-vanishingexcitonic expectation value P cf = h c † f i . In the case ofa vanishing hybridization between c -and f -electron statesand a vanishing electrical (optical) field driving inter-bandtransitions, the existence of a finite P cf = 0 is a kind ofsymmetry breaking, and if the f - and c -states have differ-ent parity, the P cf causes a finite electrical polarizationwithout a driving electrical field, because of which it is”ferroelectricity”. On the other hand, from exact results available for theFKM[2,6,7] one knows that a charge density wave (CDW)phase (”chess board phase”) exists at least for half fillingand in the symmetric case, and no evidence for EFE in theFKM is obtained by these exact treatments. But the possi-bility of CDW ordering was not considered by Portengen et al. [4]. Later calculations[8,9,10,11] could not confirmthe existence of EFE for the FKM. A divergence in thehybridization susceptibility obtained in Ref.[12] does notnecessarily mean that the ground state has spontaneoushybridization P cf = 0[10].More recently it has been shown by Batista et al. [13,14]that the FKM extended by a direct f − f -hopping t f , infact, has the EFE phases with a spontaneous hybridiza-tion. Depending on the sign of t f a ferro- or an antifer-roelectric phase may exist. A CDW phase is also possibledepending on the relative position of c - and f -band andon the value of the Coulomb (Hubbard) correlation U ,and the quantum phase diagrams were obtained in Refs.[13,14]. In this paper we also study the spinless extendedFalicov Kimball model (EFKM) suggested by Batista[13].Batista el al. studied this model in one and two dimen-sions in the strong[13] and the intermediate coupling[14]limit. Here we study an infinite dimensional system in theweak coupling limit. We also obtain electronic ferroelec-tricity (EFE), electronic antiferroelectricity (EAFE) andCDW ordering. We calculate the dependence of the or-der parameters on temperature and on the model param-eters and calculate the total energy. The resulting quan-tum phase diagram is qualitatively very similar to thatobtained previously in the intermediate and strong cou-pling limit[13,14]. We also point out that this EFKM andthe E(A)FE problem and phase is closely related to the Schneider, Czycholl: (Anti-)Ferroelectricity in the Extended Falicov-Kimball Model excitonic insulator phase discussed already about 40 yearsago[15,16,17].The paper is organized as follows. In Sect. 2 we de-scribe the EFKM and point out its connections with otherstandard models of correlated electron systems and solidstate theory. Section 3 describes our weak-coupling ap-proximation. The results are presented in Section 4; the c -and f -electron spectral functions are calculated for differ-ent model parameters and order types, the (CDW, EFE,EAFE) order parameters and the total energy for thesephases are calculated as a function of the model param-eters and the temperature, and the complete quantumphase diagram is presented, before the paper closes inSect. 5 with a short summary and conclusion. The extended Falicov-Kimball model (EFKM)[13] consistsof two types of spinless electrons, here denoted as c -and f -electrons, and a local Coulomb (Hubbard) interaction U between c - and f -electron at the same site. The EFKMHamiltonian reads: H = X R (cid:16) E c c † R c R + E f f † R f R + U c † R c R f † R f R − X ∆ n.n. h t c c † R + ∆ c R + t f f † R + ∆ f R i! (1)= X k (cid:16) ε c ( k ) c † k c k + ε f ( k ) f † k f k (cid:17) + U X R c † R c R f † R f R (2)Here R denotes the sites of a Bravais lattice, ∆ the nearestneighbor lattice vectors, k the wave vectors from the firstBrillouin zone, E c/f are the on-site one-particle matrix el-ements (and thus the band centers) of the c/f -electrons,and the usual nearest neighbor tight-binding assumption(of only nearest neighbor intersite matrix elements t c/f )has been made. Therefore, the c - and f -electron disper-sions are given by: ε c ( k ) = E c − X ∆ n.n. t c e i k∆ , ε f ( k ) = E f − X ∆ n.n. t f e i k∆ (3)Several standard models of solid state theory can beidentified to be certain limiting cases of this EFKM (1,2).In the case of a vanishing f -electron dispersion, i.e. t f = 0,we recover, of course, the standard spinless FKM[1,3]. Inthe case of equal, degenerate c - and f -bands, i.e. E f = E c and t f = t c , one can identify the c -electrons with thespin-up and the f -electrons with the spin-down electronsand obtains the standard Hubbard model[18]. A particle-hole transformation for one kind of electrons, say the f -electrons, leads to an attractive interaction − U ; then inthe case t f = − t c , E c + U = − E f the c -electron bandand the f -hole band are again degenerate, and identify-ing again the c -electrons with the spin-up electrons andthe f -holes with the spin-down electrons one has spin-degenerate fermions with an attractive (short ranged, i.e. k -independent) s-like interaction, i.e. the BCS-model[19]. Finally, if the f -electron band is interpreted as valenceband and the c -electron band as conduction band (of onespin direction), one recovers the standard two-band modelstudied frequently in semiconductor theory[20], in par-ticular to describe optical excitations (excitons etc.) un-der the influence of the Coulomb interaction, only thathere this Coulomb interaction is local (short ranged). Inthis situation the f -(valence) band is narrower than the c -(conduction) band, i.e. | t f | < | t c | , and the f - and c -states have usually a different parity, which in the sim-plest way can be modelled by a different sign of t c and t f (i.e. t c > , t f < We will use the following model assumptions: we measureenergies relative to the origin of the c -band, i.e. we choose E c = 0. Furthermore, we use a semielliptical model densityof states for the unperturbed c -band, i.e. we assume ρ c ( E ) = 1 N X k δ ( E − ε c ( k )) = 2 π p − E (4)for − < E < c -band width asour energy unit. This implies ρ f ( E ) = 1 | t f | ρ c ( E − E f | t f | ) (5)Then we are left with three parameters: the position E f of the (center of the) f -band relative to the (center of the) c -band, the (dimensionless) f -electron hopping t f (i.e. therelation of the f - to the c -band width), and the Coulomb(Hubbard or Falicov-Kimball) correlation U between one f - and one c -electron at the same lattice site.To use the semielliptical model DOS (4) is a standardmodel assumption introduced already more than 40 yearsago by Hubbard[18]. It becomes exact for a Bethe latticein the limit of infinite coordination number. Comparedto the Gaussian model DOS, which becomes exact fora d -dimensional (hyper)cubic lattice in the limit of infi-nite dimensions ( d → ∞ )[21] or coordination number, thesemielliptical model DOS has the advantage that the bandwidth is finite and true band gaps can develop and thatit has the squareroot band edge van Hove singularitiescharacteristic for three dimensional systems.With these additional model assumptions we now ap-ply the generalized (unrestricted) Hartree-Fock approxi-mation (HFA), which becomes correct in the weak-couplinglimit of small U . Within HFA the many-body (interaction)part of the Hamiltonian (1,2) is decoupled according to c † R c R f † R f R = h c † R c R i f † R f R + h f † R f R i c † R c R (6) − h c † R f R i f † R c R − h f † R c R i c † R f R chneider, Czycholl: (Anti-)Ferroelectricity in the Extended Falicov-Kimball Model 3 It is an unrestricted HFA because we also allow for a de-coupling with respect to off-diagonal (excitonic) expec-tation values h c † R f R i and because we allow for a positon-( R -)dependence of the expectation values h c † R c R i , h f † R f R i and h c † R f R i . Within HFA the full Hamiltonian (1) is re-placed by the effective one-particle Hamiltonian H eff = X R (cid:16) ˜ E c R c † R c R + ˜ E f R f † R f R + ˜ V R (cid:16) c † R f R + c.c. (cid:17)(cid:17) − X R X ∆ n.n. (cid:16) t c c † R + ∆ c R + t f f † R + ∆ f R (cid:17) (7)where the effective one-particle parameters˜ E c R = E c + U h f † R f R i ˜ E f R = E f + U h c † R c R i ˜ V R = − U P R cf = − U h c † R f R i (8)have to be determined selfconsistently together with thechemical potential µ for a given total number of electronsper site n = N P R ( h f † R f R i + h c † R c R i ). In this paper westudy the half filled case, i.e. n = 1 electron per site.˜ V R corresponds to an effective, spontaneous hybridizationbetween f - and c -electron states, which exists only if thereis a nonvanishing spontaneous polarization P R cf = h c † R f R i = h f † R c R i 6 = 0 (9)Without loss of generality we assume here that P R cf and˜ V R can be chosen to be real.Concerning the position dependence of the effectiveone-particle parameters ˜ E c R , ˜ E f R , ˜ V R either a homgeneous,translational invariant solution, i.e. no R -dependence, ispossible or an inhomogeneous solution with a periodicmodulation of the expectation values n f R = h f † R f R i = n f + 12 m f cos( Q · R ) n c R = h c † R c R i = n c + 12 m c cos( Q · R ) P R cf = h c † R f R i = P cf cos( Q · R ) (10)will be assumed. Therefore, Hartree-Fock solutions withan additional ordered structure are possible, and the treat-ment allows for the investigation of effects as phase sepa-ration and resulting (structural) phase transitions withinthe (unrestricted) HFA. This is particularly important forinvestigations of the (E)FKM, as it is well known (fromexact results available for two [2,6] or infinite[7] dimen-sions) that for half filling ( n = 1) the chessboard phaseforms the ground state (i.e. an A- and B-sublattice struc-ture with different f − and c − electron occupations on theA- and B- sublattice). Though it is also known (in partic-ular from numerical results for the original FKM in twodimensions[22,23]) that even more complex and interest-ing ordered phases may exist (e.g. striped phases[23]), wewill restrict our investigation to the mentioned chessboardphase. This means that we assume a bipartite lattice which can be decomposed into an A- and B-sublattice and allowfor different expectation values (occupation numbers) onthe A-and B-sublattice (as in the case of antiferromag-netism). Therefore, we restrict the Q -vectors in Eq. 10 tothe nesting vectors Q = π (1 , , . . . ) or cos( QR ) = ± n c,A/B , n f,A/B , P A/Bcf for the expectation values depending on whether R ∈ A or ∈ B sublattice. Then the m c,f introduced inEq. 10 is already the charge density wave (CDW) orderparameter, and a non-vanishing P cf is the order paramaterdescribing spontaneous polarization, i.e. electronic (anti-)ferroelectricity or an excitonic insulator. The selfconsistent solution of the HFA equations alwaysyields a homogeneous, translational invariant solution with-out a spontaneous polarization. Then the f - and c -bandsare simply shifted by the amount U n c and U n f , respec-tively. When the two bands are sufficiently far apart fromeach other, the lower one will be completely filled (i.e. n c/f = 1) and the other one is empty (i.e. n f/c = 0) as inthe case of a conventional semiconductor two-band model.However, when the bands overlap, also other HFA solu-tions are obtained, namely either solutions with a CDWorder parameter or solutions with a spontaneous polar-ization. The HFA solutions with an additional symme-try breaking and order parameter have the lower energy(compared to the homogeneous solution) and, therefore,describe the better and more reliable approximation tothe true ground state.To demonstrate the different types of HFA solutionsobtained, several results for the ( c - and f -electron) spec-tral functions are shown in Fig. 1 for U = 0 . t f = − .
61 and four different values of E f . In the fully (particle-hole) symmetric case E f = 0, for which a half filled c -bandand a half filled f -band centered around U/ n cA = n cB and n fA = n fB on the different A- and B-sublattices. But in this sym-metric situation one has n cA = n fB and vice versa, and P A/Bcf = 0, i.e. no spontaneous polarization. Because ofthe superstructure a CDW gap opens in the spectral func-tions, and as the Fermi energy E F = U/ f -band is shifted compared to the c -band center, a different type of HF solution is obtained,namely one with a non-vanishing cf -polarization or an ef-fective hybridization, as shown for E f = 0 . b) .As the effective hybridization is site-diagonal (local), a hy-bridization gap is formed and the chemical potential fallsinto this hybridization gap. Therefore, for the total filling n = 1 again an insulating ground state is obtained, thistime one with a hybridization gap. This type of insulatoris also termed ”excitonic insulator”[15,16,17], and becauseof the spontaneous cf -polarization P cf it is identical tothe electronic ferroelectric phase[4,13,14] and is sometimes Schneider, Czycholl: (Anti-)Ferroelectricity in the Extended Falicov-Kimball Model a) −1.5 −1 −0.5 0 0.5 1 1.500.511.52 U=0.4, t f =−0.61, E f = 0Energy S pe c t r a l den s i t y f electron, A f electron, B c electron, A c electron, B b) −1.5 −1 −0.5 0 0.5 1 1.500.511.52 U=0.4, t f =−0.61, E f = 0.3Energy S pe c t r a l den s i t y f electron (A=B) c electron (A=B) c) −1.5 −1 −0.5 0 0.5 1 1.5 2 2.500.20.40.60.811.2 U=0.4, t f =−0.61, E f = 1.1Energy S pe c t r a l den s i t y f electron (A=B) c electron (A=B) d) −1.5 −1 −0.5 0 0.5 1 1.5 2 2.500.20.40.60.811.2 U=0.4, t f =−0.61, E f = 1.4Energy S pe c t r a l den s i t y f electron (A=B) c electron (A=B) Fig. 1. c - and f -electron spectral function of theEFKM obtained within the HFA for U = 0 . , t f = − .
61, a temperature T = 0 .
002 and different E f also called ”excitonic Bose-Einstein condensate” (BEC)because of the non-vanishing excitonic expectation value h c † R f R i . There exists another type of solution as shownin Fig.1 c) , which is here obtained for E f = 1 . c - and f -bands. In thiscase both bands are partially filled and the ground stateis, therefore, metallic. Because of the overlapping bands itis a semi-metal. In fact this homogeneous, semi-metallicphase is obtained for all values of E f < | t f | − U asa possible HFA-solution; for low temperature T it is usu-ally not the energetically most favorable solution, but forsufficiently high temperature T > T c , where the possiblyexisting order parameters vanish, it always becomes thestable phase. Finally, if E f is further shifted upwards, asituation is reached, where the two bands no longer over-lap. Then the lower band is totally filled and the upperband is empty, and c − and f − band are separated by agap, because of which one has a conventional band insu-lator. This situation is depicted in Fig. 1 d) for E f = 1 . E f + U − | t f | > U, t f , E f and as a function of tempera-ture T . In Fig. 2 the CDW order parameter m c is plottedas a function of E f for fixed U = 0 . T = 0 .
002 and three values of t f . Obviously, within theHFA a CDW is not only obtained for the symmetric case E f = 0 but in a rather large interval of E f -values around0. The CDW order parameter m c remains constant up toa critical value of E f depending on t f , where m c abruptlydisappears. This means, as a function of E f a first-order(quantum) phase transition is obtained for the CDW or-der parameter m c . The E f -dependence of the ferroelectric
0 0.2 0.4 0.6 0.8 1 0 0.10.20.30.40.50.60.70.8
T=0.002, U=0.8E f m c ( E f ) t f = −1 t f = −0.6 t f = −0.002 Fig. 2.
CDW order parameter m c = | n cA − n cB | for U = 0 . , T = 0 .
002 as a function of E f fordifferent t f chneider, Czycholl: (Anti-)Ferroelectricity in the Extended Falicov-Kimball Model 5 order parameter P cf is shown in Fig.3 for fixed t f = − . U . Obviously, a HFA solution with a non-vanishing P cf = 0 is obtained for all E f smaller than acritical value E fc ( U ), which depends on the value of U . P cf ( E f ) vanishes continuously when approaching E fc , i.e.one has a quantum phase transition of second order fromthe ferroelectric to the homogeneous phase without sym-metry breaking. Altogether all three types of HFA solu-
0 0.2 0.4 0.6 0.8 1 −0.4−0.3−0.2−0.10 t f =−0.4, T=0.002E f P c f ( E f ) U=1U=0.8U=0.6U=0.4U=0.2
Fig. 3.
Ferroelectric order parameter (sponta-neous polarization) P cf for t f = − . , T = 0 . E f for different values of U tions (CDW, spontaneous polarization and homogeneouswithout symmetry breaking) exist for the same parame-ters, at least for sufficiently small values of | E f | . Then onehas to calculate the total energy to decide which HFA-solution is the most stable one and comes closest to thetrue ground state. In Fig. 4 we show the dependence ofthe total HFA energy on E f for the three different possi-ble solutions for U = 0 . , t f = − .
31. One observes thatfor E f = 0 and a small interval around 0 the CDW so-lution is energetically the most stable one. The energyof the CDW state increases linearly with increasing | E f | ,while the energies of the other HFA states increase slowerwith increasing E f . At some value of E f the energy curvescross, and from that critical value E fc on the EFE solu-tion with a spontaneous polarization is energetically themost favorable one up to a second critical E fc , wherethe FE solution merges into the homogeneous unpolarizedHFA solution. At this value E fc the EFE order parame-ter vanishes (continuously, cf. Fig. 3), i.e. there is a secondorder quantum phase transition from the EFE to the ho-mogeneous, unpolarized solution. A CDW-solution existsalso up to E fc , but from E fc on it is no longer the moststable HFA-soution. Therefore, there is a first order quan-tum phase transition from the CDW solution to the EFEsolution at E fc .So far we have presented and discussed solutions fornegative t f . In the case of positive t f > U=0.8, t f =−0.31E f T o t a l E ne r g y not polarized − homogeneousferroelectric − homogeneousCDW Fig. 4.
Total energy of the different HFA-solutions for U = 0 . , t f = − .
31 as a functionof E f ferent (opposite but equal in magnitude) polarizations onthe A- and B-sublattice, i.e. P Acf = − P Bcf = 0. Whenthe phase with a homogeneous spontaneous polarizationis termed ”electronic ferroelectric (EFE)” phase, the cor-responding phase with non-vanishing but opposite polar-izations on neighboring sites must be called ”electronicanti-ferroelectric (EAFE)” phase. For U = 0 . t f > E f . Obviously, exceptfor the different sign the EAFE order parameter behavescompletely analogous as the EFE order parameter in thecase t f <
0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.4−0.3−0.2−0.10 0.1 0.2 0.3 0.4
T=0.002, U=0.8E f P c f A , B ( E f ) t f =0.002 (A) t f =0.002 (B) t f =0.4 (A) t f =0.4 (B) t f =0.8 (A) t f =0.8 (B) t f =1.0 (A) t f =1.0 (B) Fig. 5.
Spontaneous polarization P A/Bcf on the A-and B-sublattice as a function of E f for U = 0 . t f > The temperature ( T -) dependence of the CDW orderparameter m c is depicted in Fig. 6 for E f = 0 , t f = − . U . As it has to be expected from aHFA treatment, the order parameter behaves mean-fieldlike and vanishes continuously (second order phase tran- Schneider, Czycholl: (Anti-)Ferroelectricity in the Extended Falicov-Kimball Model sition) at a critical temperature T c (with a critical indexof ); obviously T c increases with increasing U . Also the T -dependence of the EFE order parameter P cf , shown inFig.7 for U = 0 . , t f = − . E f is mean-fieldlike, and P cf vanishes at a T c depending on E f .
0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.10.20.30.40.50.60.70.8 T m c ( T ) E f =0, t f =−0.4 U=1 U=0.8 U=0.6 U=0.4
Fig. 6.
Temperature dependence of the CDWorder parameter m c for E f = 0 , t f = − . U . U=0.8, t f = −0.4T P c f ( T ) E f =0 E f =0.2 E f =0.4 E f =0.6 Fig. 7.
Temperature dependence of the EFE or-der parameter P cf for U = 0 . , t f = − . E f . The complete phase diagram obtained within the HFAis shown in Fig. 8 for fixed U = 0 . E f − t f -plane.We see that – besides the trivial phases of a completelyfilled f - or c -band (for n = 1) – the CDW phase and theEFE as well as the EAFE phase exist. The CDW phasebecomes broadest for t f = 0, i.e. for the original FKM[1].Otherwise this line t f = 0 is just the phase boundarybetween EAFE and EFE phase. Therefore, just for theoriginal FKM, i.e. for t f = 0, no spontaneous polarizationand no (anti-)ferroelectricity has to be expected. This isin accordance with the fact that for the original FKMthe f -occupation operator f † R f R at each site is an exactlyconserved quantity (commuting with the Hamiltonian),i.e. one has a conservation law for each lattice site R . But this special case t f = 0 is an unstable fixed point: an ar-bitrarily small finite (positive or negative) t f = 0 leadsto an EAFE or EFE phase with a symmetry breakingand a finite order parameter. Altogether this phase dia-gram is in complete agreement with the one obtained pre-viously by different methods in the case of strong[13] andintermediate[14] coupling for one- and two-dimensionalsystems. The same kind of phase diagram is also obtainedin the weak-coupling limit, in which the HFA applied hereis valid. −1.5 −1 −0.5 0 0.5 1 1.5−1−0.8−0.6−0.4−0.200.2 E f t f antiferroelectric(excitonic insul.) antiferroelectric(excitonic insul.) ferroelectric(excitonic insul.) ferroelectric(excitonic insul.) CDW full f bandband insulator full c bandband insulator
Fig. 8.
EFKM phase diagram in the E f − t f -plane for U = 0 . We have investigated the extended Falicov-Kimball model(EFKM), i.e. a spinless two-band model with a finite bandwidth (hopping) of both, the c - and the f -electrons, anda short-ranged (site diagonal, i.e. Hubbard like) Coulombinteraction U between f - and c -electrons. We have con-sidered the weak-coupling limit of small U , in which casethe (unrestricted) Hartree-Fock approximation (HFA) be-comes applicable. For a total number of n = 1 electron persite different ground states are obtained depending on theparameters, namely either the (trivial) cases of completelyfilled c - or f -band, states with a non-vanishing, sponta-neous c − f -polarization P R cf and a CDW-state with anA-B-sublattice structure and different c - and f -electronfillings on the sites of the A- and B-sublattice as in thecase of antiferromagnetism. In both cases there is a sym-metry breaking, and the spontaneous polarization P cf andthe sublattice ”magnetization” (difference of the A- andB-sublattice occupation numbers) m c,f = | n Ac,f − n Bc,f | arethe order parameters. The state with a spontaneous po-larization (and thus hybridization) corresponds to the ex-citonic insulator phase discussed already about 40 yearsago[15,16,17]. A spontaneous polarization is equivalent toa non-vanishing excitonic expectation value h c † R f R i , and chneider, Czycholl: (Anti-)Ferroelectricity in the Extended Falicov-Kimball Model 7 a ground state with an excitonic expectation value is some-times also interpreted as an excitonic Bose-Einstein condensate.[13,14]It may be connected with a spontaneous electric dipolemoment (per site); therefore, the state with a transla-tional invariant ( R -independent) P cf can be interpretedas ”electronic ferroelectricity” (EFE)[4], which is obtainedfor t f <
0. But for t f > P R cf on the A- and B-sublattice is obtained, whichcan analogously be termed and interpreted as ”electronicantiferroelectricity” (EAFE)[13,14]. We have determinedthe phase diagram in the E f − t f -plane, which agrees withresults obtained previously by different methods[13,14].Our conclusions are the following: – Previous results on the EFKM, in particular the phasediagram, obtained in the strong coupling limit by meansof a mapping on an xxz-spin model[13] or in the caseof intermediate U and in two dimensions by means of aconstrained path Monte Carlo method[14], have beenconfirmed. This phase diagram can also be obtainedin the weak-coupling limit of small U within a simple(unconstrained) Hartree-Fock treatment (independentof the dimension). – The excitonic insulator state exists for this kind of two-band model for a wide range of the parameters; it isequivalent to the EFE-state (for t f <
0) or the EAFE-state for t f >
0. It is also equivalent to the BCS-state(if one performs a particle-hole transformation for onekind of the electrons and thus comes to a model withan attractive interband interaction). – Around the symmetric case, i.e. for (almost) coincidingband centers of the c - and f -band, another state is en-ergetically more stable, namely the CDW-state withdifferent and alternating c - and f -occupation on thesites of an A- and B-sublattice. In the older work onexcitonic insulators[16,15,17] and on EFE[4] it has ob-viously not been checked, if a more stable CDW-phaseexists. – There is a first order (quantum) phase transition fromthe E(A)FE state to the CDW state but a second ordertransition from the E(A)FE phase to the homogeneous(translationally invariant) phase (without a symmetrybreaking). – The original FKM[1] with t f = 0 corresponds to thephase boundary between EFE- and EAFE phase. There-fore, neither the EFE- nor the EAFE phase is stableand only the CDW-phase (and probably more complexordered structures or phase separation away from thesymmetric case) are the ordered phases for the origi-nal FKM, in agreement with previous conclusions[8,9].However an arbitrarily small finite t f = 0 leads eitherto the EFE- (for t f <
0) or to the EAFE-state (for t f > Acknowledgments:
This work has been supported by a grant from the”Bereichsforschungskommission Natur-/Ingenieur-wissenschaften (BFK NaWi)” of the University of Bremen (project 01/102/9). We thank Frithjof Anders and ClaasGrenzebach for stimulating discussions.
Note:
After completion (but before submission) of this work,which is an extract from Claudia Schneider’s thesis[24],we became aware of two very recent and related papers onthe EFKM[25,26]. Ref. [25] applies the slave-boson the-ory to an EFKM extended aditionally by a one-particlehybridization term. In Ref. [26] also the HFA has beenapplied (to the EFKM without hybridization and two-and three-dimensional tight-binding bands), and the re-sults are in complete agreement with our results (obtainedfor the semicircular model density of states, i.e. for a Bethelattice in the limit of infinite coordination number).
References
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