Competing chiral and multipolar electric phases in the extended Falicov-Kimball model
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Competing chiral and multipolar electric phases in the extended Falicov-Kimballmodel
B. Zenker and H. Fehske
Institut f¨ur Physik, Ernst-Moritz-Arndt-Universit¨at Greifswald, D-17489 Greifswald, Germany
C. D. Batista
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Dated: November 27, 2018)We study the effects of interband hybridization within the framework of an extended Falicov-Kimball model with itinerant c and f electrons. An explicit interband hybridization breaks theU(1) symmetry associated with the conservation of the difference between the total number ofparticles in each band. As a result, the degeneracy between multipolar electric and chiral orderingsis lifted. We analyze the weak- and strong-coupling limits of the c - f electron Coulomb interaction atzero temperature, and derive the corresponding mean-field quantum phase diagrams at half-fillingfor a model defined on a square lattice. PACS numbers: 71.10.Fd, 71.10.Hf, 71.28.+d, 71.35.-y
I. INTRODUCTION
The Falicov-Kimball model (FKM) was primarily in-troduced to describe the metal-insulator transition of themixed-valence compound SmB . Later on, the modelbecame widely accepted as a minimal Hamiltonian forstudying several strongly correlated electron systems, in particular, heavy fermion compounds. In its origi-nal form, the FKM contains an itinerant c band of elec-trons that interact via a local Coulomb repulsion withlocalized f electrons. The spin degree of freedom of theelectrons is not included. The local f electron number isstrictly conserved and c - f electron coherence cannot beestablished. An explicit hybridization between f and c orbitals provides an opportunity to overcome this short-coming. More recently, it was shown that a finite f electron bandwidth also induces c - f electron coherence,i.e., it can lead to an excitonic condensate even in absenceof an explicit interband hybridization. These extended versions of the FKM were used to sub-stantiate the exciting idea of electronic ferroelectricity(EFE).
The ferroelectric phase only appears whenthe c and f orbitals have opposite parity under spatialinversion. The concomitant spontaneous breaking of in-version symmetry results from a nonvanishing average of h c † f i . Since this expectation value corresponds to (exci-tonic) pairing of electrons and holes from different bands,the appearance of EFE is directly related with the for-mation of an excitonic insulator (EI). The FKM with two dispersive bands, the so-calledextended Falicov-Kimball model (EFKM), was studiedpreviously for describing different properties of the EIphase.
However, as it was shown for the case ofopposite-parity orbitals, the inclusion of a finite in-terband hybridization can be very relevant because it re-moves the U(1) symmetry associated with the conserva-tion of the difference between the total number of parti-cles in each band: N c − N f . In particular, this hybridiza- tion term is certainly relevant when the ground-state ofthe EFKM corresponds to an excitonic condensate. Forthe EFKM with interband hybridization (HEFKM), theexcitonic condensate is in general replaced by Ising-typephases that only break discrete symmetries of the Hamil-tonian.In this paper we present a mean-field study of the influ-ence of an explicit hybridization on the symmetry-brokenstates that can take place for the HEFKM. To determinethe ground-state quantum phase diagram of the HEFKMin the strong- and weak-coupling limits of c - f electroninteraction, we assume that the interband hybridizationamplitudes are small compared to the intraband hopping(transfer) integrals. Given the nature of the discrete sym-metries of the HEFKM, the natural ground-state candi-dates are chiral phases (CHPs) and states with multipo-lar electric orderings. II. MODEL
By expressing the orbital flavor as a pseudospin vari-able, c † i ≡ c † i ↑ and f † i ≡ c † i ↓ , the Hamiltonian takes theform H = X i,σ ε σ c † iσ c iσ + X h ij i ,σ t σ (cid:16) c † iσ c jσ + H . c . (cid:17) + U X i n i ↑ n i ↓ + v X i (cid:16) c † i ↑ c i ↓ + H . c . (cid:17) + v ↑↓ X h ij i (cid:16) c † i ↑ c j ↓ + H . c . (cid:17) + v ↓↑ X h ij i (cid:16) c † i ↓ c j ↑ + H . c . (cid:17) . (1)Here h ij i indicates that i and j are nearest-neighborsites. The fermionic operators c ( † ) iσ annihilate (create)an electron on the spin σ Wannier orbital of the latticesite R i . The lattice has a total number of N sites, and n jσ = c † jσ c jσ is the particle number operator for site j ( σ = {↑ , ↓} ). ε σ denotes the on-site energy for each or-bital, t σ are the intraband hopping amplitudes, U is thelocal interorbital Coulomb interaction strength, and v γ are the interband hybridization amplitudes, where γ = 0for on-site hybridization and γ = {↑↓ , ↓↑} for intersitehybridization. The EFKM is recovered from Eq. (1) bysetting v γ = 0. In this limit, the model has a continuousU(1) symmetry, which is removed by the inclusion of anexplicit hybridization. The discrete symmetries that re-main for the more general HEFKM are spatial inversionand time-reversal invariance.The pseudospin language of Eq. (1) unveils the sim-ilarity of H with other generic many-body Hamiltoni-ans. The EFKM ( v γ = 0) becomes an asymmetric Hub-bard model, i.e., a single band model for electrons witha spin-dependent dispersion. We will still use the name”Falicov-Kimball model” to indicate that the pseudospindegree of freedom represents a physical orbital degree offreedom. From now on, we will consider that H is de-fined on a square lattice and h n i ↑ + n i ↓ i = 1 (half-filledband case). We will also restrict to zero temperature andmeasure all energies in units of t ↑ = 1. Finally, we willassume that the Wannier functions of c and f orbitals, φ ↑ ( r − R i ) and φ ↓ ( r − R i ), are real. III. ORDER PARAMETERS
In the rest of the paper we will refer to the pseudospinsimply as “spin.” The spin representation used in Eq. (1)unveils the SU(2) structure of this internal degree of free-dom. This degree of freedom is the only one that survivesat low energies in the large U/ | t σ | limit. Consequently,the three different local or real-space order parameterscorrespond to the three components of the local spin vari-able, S j = 12 X σ,σ ′ c † jσ σ σσ ′ c jσ ′ , (2)where σ is the vector of the Pauli matrices. More com-plicated (or higher order) real-space order parameters in-volve products of spin operators in more than one unitcell.A real space modulation of hS zj i leads to orbital order-ing . Here we will only consider the ordering wave vector Q = ( π, π ) that leads to staggered orbital ordering (SOO)because the effective interaction is antiferromagnetic be-tween nearest-neighbors and the lattice under consider-ation is bipartite. The corresponding order parameteris δ SOO = X j e i Q · R j hS zj i . (3)If the two orbitals have opposite parity, a nonzero hS xj i implies the presence of a spontaneous local electric po-larization that turns out to be uniform for the HEFKM. This is the EFE that was found in previous works for par-ticular limits of the HEFKM. The uniform electricpolarization is given by h P i = p X j hS xj i (4)with the interband dipole matrix element p = 2 e Z d r φ ↑ ( r ) r φ ↓ ( r ) , (5)where e is the electron charge. This phase breaks thespatial inversion symmetry of H .If the two orbitals have the same parity (for instance s and d orbitals), a nonzero modulation of hS xj i corre-sponds to an electric quadrupole density wave (EQDW)as long as the tensor q νν ′ σσ ′ = e Z d r φ σ ( r ) r ν r ν ′ φ σ ′ ( r ) (6)is nonzero for σ ′ = ¯ σ ≡ − σ ( ν, ν ′ = { x, y, z } ). In casethe tensor q σ ¯ σ [Eq. (6)] vanishes, one has to look forthe lowest order electric multipole that has a nonzeromatrix element between the orbitals φ ↑ ( r ) and φ ↓ ( r ). Inthe rest of this paper, we will assume that the tensor q σ ¯ σ does not vanish. In second quantization, the localelectric quadrupole tensor on the unit cell j is given bythe operator, Q j = q ↑↑ n j ↑ + q ↓↓ n j ↓ + 2 q ↑↓ S xj . (7)We note that q ↑↓ = q ↓↑ . The corresponding quadrupolarorder parameter in momentum space is given by h Q k i = X j e i k · R j h Q j i . (8)Again, for the Hamiltonian under consideration, the wavevector of the electric quadrupolar ordering is k = Q .Equation (7) implies that a nonzero modulation of the x -spin component, hS xj i , corresponds to an EQDW. Whiletranslational symmetry is broken in this phase, time-reversal and spatial inversion symmetries are conserved.We note that the first two terms of Eq. (7) imply thatorbital ordering will also lead to an EQDW. However,as it is also clear from Eq. (7), the quadrupolar ten-sors associated with the ordering along the z and x axesare different. The quadrupolar electric moment that ismodulated under staggered orbital ordering (staggered z component) corresponds to a linear combination of thetensors q ↑↑ and q ↓↓ . On the other hand, the staggeredordering of the x -spin component involves a modula-tion of a quadrupolar electric tensor proportional to q ↑↓ (hybridization-induced quadrupolar electric moment). Inorder to simplify the notation, we will use “EQDW” todenote the staggered ordering of the x -spin componentand SOO for the the staggered ordering of the z -spincomponent.Finally, a nonzero hS yj i implies the spontaneous emer-gence of a current-density distribution between the twoorbitals of the unit cell j . This can be easily verifiedif we neglect the overlap between orbitals that belongto different unit cells. In that case, the current-densityoperator at a point r near R j is given by j j ( r ) = ~ m e S yj X σ σφ ¯ σ ( r − R j ) ∇ r φ σ ( r − R j ) , (9)where m e is the electron mass and the prefactor σ takesthe value +1 for ↑ and − ↓ . We note that this currentdensity flows between the two orbitals of the same unitcell (these are atomic currents when the two orbitals be-long to the same ion), in contrast to the orbital currentsfound in Ref. 28 that flow between different unit cells.This CHP breaks time-reversal symmetry and the physi-cal order parameter is the lowest order nonzero multipoleof the current-density distribution given by Eq. (9). Thechiral ordering is staggered for orbitals with the same par-ity and uniform for orbitals with opposite parity. For in-stance, if we are considering two p orbitals, the staggeredchiral ordering of the y -spin component corresponds toorbital antiferromagnetism, because the current distri-bution given by Eq. (9) generates a net magnetic dipolemoment. Since we will consider the general case of anyarbitrary pair of orbitals, we will use “CHP” to denotethe uniform ordering of the y -spin component (same par-ity orbitals) and “staggered chiral phase” (SCHP) to de-note the staggered ordering. IV. STRONG-COUPLING REGIME
For the case | t σ | , | v γ | ≪ U , we can perform a large- U expansion, thereby reducing the HEFKM [Eq. (1)] to aneffective strong-coupling Hamiltonian, H sc , that repro-duces the low-energy spectrum of the original model.A large on-site Coulomb interaction splits the spec-trum of the HEFKM Hamiltonian into high- and low-energy parts. For t σ = v γ = 0, the lowest-energy sub-space is generated by the 2 N states that have one elec-tron per site. The high-energy subspaces are separatedby energy gaps equal to U n d , where n d is the number ofdouble occupied sites. For nonzero t σ and v γ , the elec-trons are no longer completely localized at their ions, i.e.,an electron can gain kinetic energy by visiting virtually aneighboring site. Since we are considering the half-filledband case (one particle per site), the low-energy effectivemodel becomes a spin Hamiltonian H sc . The expressionfor H sc up to second order in the kinetic-energy terms is H sc = X h ij i ( J xx S xi S xj + J yy S yi S yj + J zz S zi S zj )+ X h ij i ( J xz S xi S zj + J zx S zi S xj + C )+ X i ( B S zi + 2 v S xi ) , (10) where J xx = 4 U ( t ↑ t ↓ + v ↑↓ v ↓↑ ) , (11) J yy = 4 U ( t ↑ t ↓ − v ↑↓ v ↓↑ ) , (12) J zz = 2 U (cid:0) t ↑ + t ↓ − v ↑↓ − v ↓↑ (cid:1) , (13) J xz = 4 U ( t ↑ v ↓↑ − t ↓ v ↑↓ ) , (14) J zx = 4 U ( t ↑ v ↑↓ − t ↓ v ↓↑ ) , (15) C = ε ↑ + ε ↓ − U (cid:0) t ↑ + t ↓ + v ↑↓ + v ↓↑ (cid:1) , (16) B = ε ↑ − ε ↓ . (17)It is well known that the half-filled isotropic Hubbardmodel can be mapped on an effective Heisenberg modelin the limit of a large Coulomb repulsion. For the moregeneral EFKM, the intraband hopping amplitudes andthe different on-site potentials lead to an effective XXZmodel in a magnetic field B along the z axis. B is simplythe energy difference between the two orbitals. As ex-pected, the interband hybridization of the HEFKM gen-erates anisotropic terms that explicitly break the U(1)invariance under uniform spin rotations about the z axis.While the intersite hybridization leads to anisotropic ex-change terms, the on-site hybridization leads to a Zeemancoupling to a uniform field along the x axis.For low enough values of B and no interband hybridiza-tion, the ground-state of H sc exhibits SOO. The simplereason is that J zz ≥ | J xx,yy | , i.e., the effective XXZ modelis easy-axis. If J zz is significantly larger than | J xx,yy | ,the SOO remains robust when the interband hybridiza-tion is included. Clearly, there exists a critical value of B that leads to a spin-flop transition to an ordered phase inthe XY plane with a uniform component along the z axis(canted XY phase). In absence of interband hybridiza-tion, the U(1) invariance of the EFKM implies that spincomponent perpendicular to the applied field can pointalong any direction of the XY plane. In other words,there is a continuous ground-state degeneracy that in-cludes the EQDW (EFE) and SCHP (CHP) for orbitalswith the same (opposite) parity. In this case, the inclu-sion of interband hybridization is very relevant becauseit lifts the continuous degeneracy and stabilizes only oneof the two possible Ising-type orderings (along the x or y spin direction).For orbitals with opposite parity, we have t ↑ t ↓ < v ↓↑ = − v ↑↓ and v = 0. These relationships are de-rived from simple symmetry considerations. In this case J xx = − U ( | t ↑ t ↓ | + v ↑↓ ), J yy = − U ( | t ↑ t ↓ | − v ↑↓ ), and J zx = − J xz . Since J xx , J yy < | J xx | > | J yy | , theenergy is minimized by a ferromagnetic alignment of thespins along the x direction that corresponds to an EFEphase. Since this was previously shown in Ref. 15, fromnow on we will concentrate on the case of equal parityorbitals. In this case, we have t ↑ t ↓ > v ↓↑ = v ↑↓ , and v can be nonzero if the two orbitals belong to different ions.Then J xx > J yy and J zx = J xz . Since J xx > J yy > v ↑↓ favors staggered Ising-type ordering along the x di-rection, while v favors a uniform polarization along the x direction and, consequently, a staggered Ising-type or-dering along the y direction. In other words, the inter-band hybridization can stabilize an EQDW or a SCHPdepending on the ratio between the on-site and intersitehybridization amplitudes.We introduce now the mean-field variational states andthe corresponding energies for the three order parametersthat we introduced in the previous section.(i) SOO . This phase has a staggered spin componentalong the z -direction, and a uniform component alongthe x direction that can be induced by the on-site hy-bridization term v , h S j i θ = S (cid:0) sin θ , , cos θ e i QR j (cid:1) , (18) E SOO0 = − DN S J zz cos θ + DN S J xx sin θ +2 v SN sin θ + DN C . (19)(ii)
SCHP . This phase has a staggered spin componentalong the y direction and uniform polarizations along the x and z directions, h S j i θ ,θ = S (cid:0) sin θ cos θ , e i QR j sin θ sin θ , cos θ (cid:1) , (20) E SCHP0 = DN S (cid:0) ( J xx + J yy ) cos θ (1 − cos θ ) − J yy + ( J yy + J zz ) cos θ + 2 J xz sin θ cos θ cos θ (cid:1) + DN C + N SB cos θ + 2 N Sv sin θ cos θ . (21)(iii) EQDW . In this case the staggered spin componentis aligned along the x direction and there is a uniformcomponent along the z direction induced by B (the y component vanishes), h S j i θ = S (cid:0) e i QR j sin θ , , cos θ (cid:1) , (22) E EQDW0 = DN C + N BS cos θ − DN J xx S sin θ + DN J zz S cos θ . (23)In all cases we have D = 2, Q = ( π, π ), and S = 1 / θ and θ , we determine the quantum phase diagram as afunction of the band-structure parameters.Figure 1(a) shows the effect of a finite intersite hy-bridization amplitude, v ↑↓ , on the quantum phase dia-gram of the EFKM. The intersite hybridization stabilizesthe EQDW relative to the SCHP. On the other hand, theEQDW is also favored relative to the SOO because v ↑↓ decreases the value of J zz and simultaneously increasesthe value of J xx [see Eqs. (11) and (13)]. Figure 1(b)illustrates the effect of a finite on-site hybridization. Inthis case, the SCHP is favored relative to the EQDW. Incontrast to v ↑↓ , v does not change the transition pointbetween the SCHP and the SOO. The simple reason isthat v does not affect the exchange constants. It justgenerates an effective pseudomagnetic field along the x (a)(b)(c)FIG. 1: (Color online) Ground-state phase diagram of the2D EFKM in the strong-coupling regime. Band-structure pa-rameters are ε ↓ = 0 . t ↑ = 1 .
0, and U = 10. Left-hand sidediagrams [in panels (a) and (b)] give results for the nonhy-bridized EFKM ( v = 0 and v ↑↓ = 0), while right-hand sidediagrams show the dependence on the (a) intersite hybridiza-tion v ↑↓ and (b) on-site hybridization v for ε ↑ = 0 .
15. Panel(c) gives the stability region of the staggered chiral phase andthe electric quadrupole density wave in dependence on v and v ↑↓ for ε ↑ = 0 . t ↓ = 0 . axis that leads to a finite canting angle in both phases.Figure 1(c) shows the phase diagram as a function of v and v ↑↓ for a large enough value of B = 0 . v strengthens the SCHP, while the inter-site hybridization v ↑↓ stabilizes the EQDW. Within oursimple mean-field approximation the boundary betweenthese two phases is a straight line. V. WEAK-COUPLING REGIME
It has been shown in Ref. 18 that the mean-fieldground-state phase diagram of the 2D EFKM agrees al-most perfectly with the one obtained by a constrainedpath Monte Carlo technique, even in the intermediatecoupling regime. This agreement motivates us to performa Hartree-Fock decoupling of the HEFKM to explore thequantum phase diagram for small U/ | t σ | .The weak-coupling analysis requires to express the rel-evant order parameters in momentum space. In partic-ular, the Fourier components of hS xj i and hS yj i can berepresented as a complex number,∆ Q = | ∆ Q | e iϕ = UN X k h c † k + Q ↑ c k ↓ i (24)= UN X j e i QR j (cid:0) hS xj i + i hS yj i (cid:1) (25)= U √ N ( hS x Q i + i hS y Q i ) , (26)where c † k σ = 1 √ N X j e i k · R j c † jσ , (27) S ν k = 1 √ N X j e i k · R j S νj . (28)The ordering wave vector Q determines the modulationof the real-space order parameter. According to ourstrong-coupling analysis, we have Q = ( π, π ) for orbitalswith the same parity and Q = (0 ,
0) for orbitals with op-posite parity. Again, the U(1) invariance of the EFKM( v γ = 0) implies that the energy does not depend on ϕ . Consequently, there is an infinite number of ground-states with ∆ Q = 0 that results from the spontaneousU(1) symmetry breaking of the EFKM (excitonic conden-sate). A finite interband hybridization ( v γ = 0) removesthe continuous U(1) symmetry and lifts the ϕ degeneracyof the HEFKM ground-state.For orbitals with opposite parity, the hybridization inmomentum space takes the form v k = 2 iv ↑↓ (sin k x +sin k y ) and the EFE state has a lower energy than theCHP. This is in agreement with the result for the FKMextended by a (small) intersite hybridization in Ref. 14.Therefore, from now on we will focus only on the equalparity case. The Hartree-Fock decoupling suggested byEq. (24) gives H wc = X k ,σ ¯ ε k σ c † k σ c k σ + X k ,σ v k c † k σ c k − σ − X k ∆ Q c † k ↓ c k + Q ↑ − X k ∆ ∗ Q c † k + Q ↑ c k ↓ (29) with ¯ ε k σ = ε σ + U n − σ + 2 t σ (cos k x + cos k y ) , (30) v k = v + 2 v ↑↓ (cos k x + cos k y ) , (31) n σ = 1 N X k h c † k σ c k σ i , (32)∆ Q = UN X k h c † k + Q ↑ c k ↓ i , (33)∆ ∗ Q = UN X k h c † k ↓ c k + Q ↑ i . (34)The mean-field Hamiltonian (29) can be easily diagonal-ized by the canonical transformation C k ,m = u k ,m c k ↑ + v k ,m c k ↓ + ˜ u k ,m c k + Q ↑ + ˜ v k ,m c k + Q ↓ , (35)where m = 1 , , ,
4. The coefficients are solutions of theassociated Bogoliubov de Gennes equations, H wc k Ψ k ,m = E k ,m Ψ k ,m , with H wc k = ¯ ε k ↑ v k − ∆ ∗ Q v k ¯ ε k ↓ − ∆ Q − ∆ ∗ Q ¯ ε k + Q ↑ v k + Q − ∆ Q v k + Q ¯ ε k + Q ↓ (36)and Ψ k ,m = ( u k ,m , v k ,m , ˜ u k ,m , ˜ v k ,m ) T . The energy persite results as E wc0 N = 1 N X k ,m ′ E k ,m f ( E k ,m ) − U n ↑ n ↓ + 1 U | ∆ Q | , (37)where f ( E k ,m ) is the Fermi function containing the newquasiparticle energies E k ,m and the prime denotes thatthe k summation extends over the magnetic Brillouinzone only. The chemical potential µ is determined bythe condition 1 = 1 N X k ,m ′ f ( E k ,m ) . (38)Next we consider the mean-field decoupling that leadsto SOO. In this case, we introduce the possibility of aperiodic modulation in the electronic density with inde-pendent amplitudes for each spin polarization, h n iσ i = n σ + δ σ cos( QR i ) , (39)with δ σ = 1 N X k h c † k σ c k + Q σ i . (40)The associated Bogoliubov de Gennes equations are H SOO k Ψ k ,m = E SOO k ,m Ψ k ,m , with H SOO k = ¯ ε k ↑ v k U δ ↑ v k ¯ ε k ↓ U δ ↓ U δ ↑ ε k + Q ↑ v k + Q U δ ↓ v k + Q ¯ ε k + Q ↓ . (41)The SOO order parameter becomes δ SOO = δ ↑ − δ ↓ . (42)For asymmetric bands, t ↑ = t ↓ , the presence of a nonzeroSOO leads to a secondary charge-density-wave (CDW)order, whose order parameter is given by δ CDW = δ ↑ + δ ↓ . (43)This secondary CDW provides a simple way of detectingthe SOO in real materials. The mean-field energy persite that results from such a kind of SOO is E SOO0 N = 1 N X k ,m ′ E SOO k ,m f ( E SOO k ,m ) − U n ↑ n ↓ − U δ ↓ δ ↑ . (44)By solving the self-consistency Eqs. (33) and (40), andcomparing the corresponding mean-field energies givenby Eqs. (37) and (44), we compute the ground-statephase diagram.Figure 2 is the weak-coupling counterpart of Figure 1.Like for the strong-coupling regime, Fig. 2(a) shows thatan increasing value of v ↑↓ narrows the SOO phase whilethe region of the EQDW phase is enlarged. On the otherhand, the on-site hybridization v favors the SCHP rel-ative to the EQDW and it does not have a noticeableeffect on the transition line between the SOO and theSCHP [see Fig. 2(b)]. This also coincides with the strong-coupling results. Figure 2(c) shows the stability regionsof the EQDW and SCHP as a function of the hybridiza-tion amplitudes for a large enough | ε ↑ − ε ↓ | = 0 .
5. Quali-tatively, the result of our Hartree-Fock approach is simi-lar to the one obtained from the strong-coupling analysis[see Fig. 1(c)]. However, a more quantitative analysisshows that the area of stability for the SCHP is reducedrelative to the strong-coupling result. A large Coulombrepulsion inhibits hopping processes and consequently re-duces the influence of the intersite hybridization v ↑↓ rel-ative to the effect of the on-site hybridization v . VI. CONCLUSIONS
The Hamiltonian that we considered in this work is avery simple extension of the Falicov-Kimball model. Inspite of its simplicity, we have shown that this modelleads to a very rich quantum phase diagram that con-tains all the possible local order parameters (three dif-ferent components of the local spin S j ) considered inSec. III. The ordering wave vector Q is selected by thenesting property of the noninteracting Fermi surface inthe weak-coupling limit and by the antiferromagnetic na-ture of the exchange interactions on a bipartite lattice inthe strong-coupling limit. Most notably, the stability ofthe different broken symmetry states is very sensitive to (a)(b)(c)FIG. 2: (Color online) Ground-state phase diagram of the2D EFKM in the weak-coupling regime. Band-structure pa-rameters are ε ↓ = 0 . t ↑ = 1 .
0, and U = 2. Left-hand sidediagrams [in panels (a) and (b)] give results for the nonhy-bridized EFKM ( v = 0 and v ↑↓ = 0), while right-hand sidediagrams show the dependence on the (a) intersite hybridiza-tion v ↑↓ and (b) on-site hybridization v for ε ↑ = 0 .
15. Panel(c) gives the stability region of the staggered chiral phase andthe electric quadrupole density wave in dependence on v and v ↑↓ for ε ↑ = 0 . t ↓ = 0 . a few band-structure parameters. According to these re-sults, it is necessary to have very accurate informationabout the band-structure properties near the Fermi en-ergy to predict the correct ordered state. In particular, ifthe two orbitals have different angular momentum (like s and d orbitals), the SCHP may remain hidden to most ofthe experimental probes. The simple reason is that thespontaneous current-density distribution given by Eq. (9)has no net magnetic moment. Consequently, this phasecan only be detected by using an experimental probe thatcouples to the lowest nonzero multipole of the current-density distribution. The SCHP becomes stable abovea critical value of the on-site hybridization as long asthe diagonal energy difference between the two orbitals | ε c − ε f | is also larger than a minimal value.Although all the calculations of this work were donefor D = 2, we do not expect any qualitative change for D >
2. The obtained consistency between the weak-and the strong-coupling approaches suggests that ourresults are robust. In particular, the absence of geo-metric frustration in the strong-coupling regime, whoseweak-coupling counterpart is the nesting property of theFermi surface, facilitates the search for the broken sym-metry state that minimizes the energy for each set ofHamiltonian parameters. For orbitals with opposite par-ity under spatial inversion, we confirmed that the ferro-electric phase has always a lower energy than the chi-ral phase. For orbitals with the same parity, we foundthat the stabilization of the electric quadrupole density wave or the staggered chiral phase depends strongly onthe dominant interband hybridization. The on-site hy-bridization, that is only allowed when the two orbitalsbelong to different ions, favors the staggered chiral phase,while a nearest-neighbor interband hybridization favorsthe electric quadrupole density wave.
VII. ACKNOWLEDGMENTS
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