Myriad phases of the Checkerboard Hubbard Model
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Myriad phases of the Checkerboard Hubbard Model
Hong Yao, Wei-Feng Tsai,
1, 2 and Steven A. Kivelson Department of Physics, Stanford University, Stanford, CA 94305 Department of Physics and Astronomy, University of California, Los Angeles, CA 90095 (Dated: November 11, 2018)The zero-temperature phase diagram of the checkerboard Hubbard model is obtained in the solvable limit in which it consists of weakly coupled square plaquettes. As a function of the on-siteCoulomb repulsion U and the density of holes per site, x , we demonstrate the existence of at least
16 distinct phases. For instance, at zero doping, the ground state is a novel d -wave Mott insulator( d -Mott), which is not adiabatically continuable to a band insulator; by doping the d -Mott statewith holes, depending on the magnitude of U , it gives way to a d -wave superconducting state, atwo-flavor spin-1/2 Fermi liquid (FL), or a spin-3/2 FL. The phase diagram of weakly correlated metals tendsto be relatively simple—superconductivity can occur atlow temperatures induced by weak attractive interac-tions and spin-density waves (SDW) and/or charge den-sity waves (CDW) occur under special circumstanceswhen the Fermi surface is sufficiently well nested. How-ever, with strong interactions, there is no reason not to have multiple ordered phases. Inverting this logic, onemight expect competing phases to be a generic featureof strongly interacting systems. So it becomes increas-ingly important to find a simple and solvable model ofstrong interactions, which could serve as a paradigmaticexample showing these complexities of competing orders.The Hubbard model is the simplest model of a stronglyinteracting electron gas, but alas, no well-controlled so-lution exists in more than one dimension. Here we studythe Hubbard model, Eq. (1), on a checkerboard latticewith hopping matrix element t between nearest-neighborsites on elementary square plaquettes and t ′ between siteson neighboring plaquettes. (See Fig. 1.) For t ′ = t ,this model reduces to the usual (still unsolved) Hubbardmodel on a square lattice. For t ′ ≪ t , where it is a crys-tal of weakly coupled “Hubbard clusters,” we are able toestablish a number of features of the zero temperaturephase diagram, even for strong interactions, U/t >
1, us-ing t ′ /t as a small parameter. (See Fig. 2.) Particularlystriking is the large number of zero temperature phases;we have established the existence of at least 16 distinct t' t r FIG. 1: (Color online) The schematic representation of thecheckerboard Hubbard model. The hopping amplitudes are t = 1 on the solid bonds (blue) and t ′ ≪ r labels sites and R the plaquettes. The lat-tice spacing between nearest neighboring sites is set to 1 forsimplicity. phases and there are undoubtedly more in the portionsof the phase diagram for which we have not yet obtaineda solution. For instance, at x = 0 and U > O ( t ′ ), theground state is a novel d -wave Mott insulator ( d -Mott),which is a true new state of matter. More generally, thedetails of the phase diagram depend sensitively on thechoice of clusters. For example, the phase diagram of thedimerized Hubbard model, largely consists of a singleFermi liquid (FL) phase, plus a band-insulating (BI) andspin-1/2 antiferromagnetic (AF) insulating phase. How-ever, multiple competing phases appear to be a commonfeature of the strong interacting limit.Recently, there have been a number of cluster-dynamical mean-field theory (DMFT) studies of corre- x U
1/ 2
1/ 4 c U t U AF( =1/2)+Orbital Nematic s AF(s=3/2) -BEC d SU(2) -BCS d -Mott d SDW
AF(s=1/2) FL ∞ BI FL-III ( =3/2) s FL-II ( =1/2, two bands) s -CDW d -BEC d PSPS d U PS PS × FL-III ( =3/2) s FL-II ( =1/2, two bands) s × PS PS -BCS d × × FIG. 2: (Color online) Phase diagram of the checkerboardHubbard model for 0 ≤ x ≤ / U >
0. Abbre-viations: “FL”=Fermi liquid; “ s = n/
2” = spin- n/
2; “PS” =phase separation; “AF” = antiferromagnet; “WC” = Wignercrystal; “ d -BCS” = d -wave superconductor; “ d -Mott,” “ d -BEC,” and “ d -CDW” are phases made of d -wave two-particlebound-states (hard-core bosons) which are, respectively, aBose-Mott insulator, a superfluid, and a charge density wave;“SDW”= spin-density wave; and “BI”= band insulator. Thevarious phases are described in the text. lated electrons. The complexity of the phase diagram ofthe present model in the t ′ ≪ t limit, and the strong de-pendence on the precise type of clusters raise questionsconcerning the validity of this approach. Conversely,solvable cluster models in the small t ′ limit can serveas interesting benchmark tests for such approximate ap-proaches, and for future analog simulations with coldfermionic atoms in optical lattices. Model Hamiltonian . The checkerboard Hubbardmodel, originally studied in Ref. 2, has Hamiltonian H = − X h rr ′ i ,σ t rr ′ c † r σ c r ′ σ + U X r [ˆ n r − , (1)where c † r σ creates an electron on site r with spin polar-ization σ = ↑ , ↓ and ˆ n r = P σ c † r σ c r σ . Here t rr ′ is the hop-ping matrix element from site r ′ to r and h rr ′ i denotesnearest-neighbor sites. t rr ′ = t or t rr ′ = t ′ ≪ t when h rr ′ i are a pair of sites connected, respectively, by a solidbond or a dashed bond shown in Fig. 1. We set t = 1as our energy units. Note that the model with uniformon-site repulsion U preserves the point group symmetry, C v , of the square lattice. The density of electrons persite is defined to be n el ≡ − x where x is the density of“doped holes” per site. Since the model is particle-holesymmetric, we restrict our discussion to 0 ≤ x ≤ t ′ as a small parameter permits us to solvethis model using perturbation theory. In the unperturbed t ′ = 0 problem, the 2D lattice consists of decoupled four-site square plaquettes. The Hubbard model of a four-site plaquette is exactly solvable. The eigenstates of thedecoupled 2D system are direct products of the eigen-states on each plaquette. For most densities, x , the un-perturbed ground-state is degenerate, so we use degen-erate (or near-degenerate) perturbation theory to derivean effective Hamiltonian in the low energy state space. Ground states of an isolated plaquette . The eigenstatesof a single plaquette can be specified by the number ofdoped holes, Q h , the total and z component of the spin,and the familiar orbital labels “ s ” (even under C —i.e.,90 ◦ rotation), “ p x ± ip y ” (phase changed by ± π/ C ), and “ d ” (odd under C ). For a single plaquette,the Q h = 0 ground state is unique for any positive U and has d -wave symmetry. In the Q h = 1 sector, theplaquette ground state for U < U t ≈ . p x ± ip y orbital symmetry, i.e., it is four-fold degen-erate corresponding to spin polarizations s = ± / τ = ± /
2. However, for Q h = 1and U > U t , the ground state is spin 3/2 and orbital s wave, so it is still four-fold degenerate. When there aretwo holes ( Q h = 2), the ground state is unique and has s -wave symmetry for all U . The Q h = 3 ground state is(trivially) spin 1/2 and s wave.In adding holes to the system, the issue arises whetherit is energetically cheaper to add two holes to one pla-quette or one hole to each of two plaquettes. Thisis determined by the sign of the pair binding energy ∆ ≡ E (0)+ E (2) − E (1), where E ( Q h ) is the ground state energy of one plaquette with Q h holes. When U < U c ≈ .
6, ∆ is negative which indicates that dopedtwo holes prefer to stay in the same plaquette, effectivelyforming a hard-core boson. When
U > U c , ∆ is pos-itive; i.e., two holes repel each other. Note, in Fig. 2,that the critical values of U at which level crossings oc-cur for the isolated plaquette figure prominently in thephase diagram of the perturbed system, as well. Effective Hamiltonians . Starting from these states, forvarious ranges of x and U , we can derive the effective lowenergy Hamiltonian in powers of t ′ . Although this pro-cedure reduces the number of dynamical degrees of free-dom, it still leaves us with a non-trivial many-body prob-lem, which is only solvable in certain cases; the phasesexhibited in Fig. 2 are those whose existence we haveestablished, but there are compelling reasons to expectadditional phases to exist in the portions of the x - U planethat we have not fully analyzed. We will provide moredetails of the analysis, and a discussion of the regions ofthe phase diagram that have only been partially analyzedin a future publication. For 0 < U ≪ O ( t ′ ), the interactions are weak, so thezeroth order description is in terms of bands (see below),and except at x = 0 [where the Fermi surface (FS) isnested] and x = 1 /
2, where there is a BI, we expect a FLdescription to be valid.For 0 < x < / O ( √ t ′ ) < U < U c − O ( t ′ ), 2 x ofthe plaquettes are occupied by a pair of holes, and (1 − x )have no holes. Identifying hole pairs as hard-core bosons,the effective Hamiltonian is H (1) = − t (1) X h RR ′ i b † R b R ′ + V (1) X h RR ′ i ρ R ρ R ′ , (2)where the bosonic creation operator b † R creates a holepair on plaquette R , ρ R = b † R b R is the number operator,and there is an implicit no-double occupancy (hard-core)constraint. Because the zero-hole state has d -wave sym-metry and the two-hole state has s -wave symmetry, b R is a charge 2 e field which transforms like a d -wave under C . Here t (1) is the effective hopping of bosons and V (1) the repulsion between nearest neighbor bosons, both oforder t ′ . Their explicit dependences on U are somewhatcomplicated, but we have computed them exactly. For 0 < x < / U c − O ( t ′ ) < U < U c + O ( t ′ ),both singly charge and doubly charged plaquettes occur,so the problem maps onto a rather complicated versionof the Boson-Fermion model, as discussed in Ref. 2.For 0 < x ≤ / U c + O ( t ′ ) < U < U t , the low en-ergy states are a mixture of no-hole and one-hole plaque-ttes, where the one-hole states are further distinguishedby two possible total-spin polarizations s = ± / τ = ± / p x ± ip y . Conse-quently, the effective Hamiltonian is a two-flavor versionof the t - J - V model: H (2) = − t (2) X h RR ′ i ,s,τ φ R , R ′ f † R ,s,τ f R ′ ,s, − τ + H (2 , , (3)where f † R ,s,τ creates a fermion with spin polarization s = ± / τ = ± /
2, and there is a no-doubleoccupancy constraint which we have left implicit. Here φ R , R ′ is +1 ( −
1) if the effective bond RR ′ is along theˆ x (ˆ y ) direction. The hopping parameter t (2) is order of t ′ , while H (2 , refers to terms of order t ′ : H (2 , = J (2) X h RR ′ i S R · S R ′ + V (2) X h RR ′ i n R n R ′ (4)+ X h RR ′ i (cid:2) J x τ x R τ x R ′ + J y τ y R τ y R ′ + J z τ z R τ z R ′ (cid:3) , + X h RR ′ i S R · S R ′ (cid:2) J ′ x τ x R τ x R ′ + J ′ y τ y R τ y R ′ + J ′ z τ z R τ z R ′ (cid:3) , where S R , τ R , and n R are spin, pesudo-spin and densityoperators on plaquette R respectively. Strictly speaking,there are additional “pair-hopping” terms, which we havecomputed but do not display; for x = 1 /
4, where H (2 , is the leading term in the effective Hamiltonian, the pair-hopping terms vanish.For U > U t , the one-hole ground state of a singleplaquette has spin-3/2 so the effective Hamiltonian for0 < x ≤ / t - J - V model for spin-3/2 fermions H (3) = − t (3) X h RR ′ i ,s f † R s f R ′ s (5)+ J (3) X h RR ′ i S R · S R ′ + V (3) X h RR ′ i n R n R ′ , where f † R s is the plaquette fermion creation operatoron plaquette R with spin polarization s = ± / , ± / S R and n R are corresponding spin and density opera-tors on plaquette R . In this case, the effective hopping t (3) is order of t ′ while J (3) and V (3) are order of t ′ .Consequently, this model always occurs in what, for thespin-1/2 model, is considered an unphysical limit J (3) , V (3) ≫ t (3) .For U > U c + O ( t ′ ) and 1 / < x < /
2, the effectiveHamiltonian is, again, of the form presented in Eqs. (3)and (4) (for
U < U t ) or Eq. (5) (for U > U t ), but withdifferent values of the couplings. Moreover, whereas for0 ≤ x ≤ / x = 0state with 4 electrons per plaquette, so the mean densityof fermions per plaquette is 4 x ; for 1 / ≤ x ≤ /
2, thevacuum state has 2 electrons per plaquette and the meandensity of fermions is (2 − x ).When x > /
2, the low energy degrees of freedom arealways plaquettes fermions, each of which is just an or-dinary electron. The effective Hamiltonian is the t - J - V model, having the same form as Eq. (5), but for spin-1/2 fermions with different effective parameters t (4) , V (4) ,and J (4) . Here, t (4) is order of t ′ , while J (4) and V (4) areorder of t ′ . So t (4) ≫ V (4) , J (4) . Phase diagram . Much of the structure of the phasediagram is obvious from the effective Hamiltonian. Wenow sketch some less obvious aspects of the analysis.
Zero doping . Because the zero-hole ground state of asingle plaquette is unique and there is a finite gap, at x = 0 the unperturbed ground state in the limit t ′ → t ′ produces only per-turbative corrections which do not change any qualita-tive properties of the ground state. This is an insulat-ing phase with no broken symmetry. However, despitethe fact that there are four electrons per unit cell, thisstate is not adiabatically connected to a BI state, sinceit transforms according to a non-trivial representation ofthe point group: from the d -wave character of the single-plaquette wave function, it follows that the many-bodywave function changes sign under 90 ◦ rotation about aplaquette center. One can think of this as a Mott insulat-ing state with one d -wave boson per plaquette; hence, wecall it a “ d -Mott” state. In terms of macroscopic observ-able properties, this phase has at least two identifyingfeatures: (i) The pair-field pair-field correlation function h c † ↑ c † r ↓ c R ↓ c R + r ′↑ i , although it falls exponentially withdistance | R | , has an asymptotic d -wave symmetry (forlarge | R | ) upon 90 ◦ rotation of r or r ′ separately. (ii)It is an orbital paramagnet. This d -Mott phase is a genuine new state of matter, in contrast with the similarstate in ladder systems where there is no C symmetryto unambiguously distinguish it from a band insulator.The fact that the d -Mott phase is not adiabaticallyrelated to a BI implies that, even for x = 0, there mustbe a phase transition as a function of decreasing U . Forfixed, small t ′ , when U gets small enough, the gap inthe isolated plaquette is no longer large compared to t ′ .Specifically, when U ≪ t ′ the kinetic energy is dominantand the U term can be treated through a weak-couplingHartree-Fock approximation. Since there are four sitesper unit cell, there are for U = 0 four bands as follows: ǫ k = ± p ( t − t ′ ) + 4 tt ′ cos k x ± q ( t − t ′ ) + 4 tt ′ cos k y . The top and bottom bands are well separated from thetwo middle bands ǫ k , ± ≈ ± t ′ (cos k x − cos k y ) by agap of approximately 2 t . Particle-hole symmetry fixesthe Fermi energy at 0 for x = 0, so that the FS coincideswith the lines cos k x = ± cos k y where the two bandstouch. Consequently, the FS is perfectly nested, and anyweak positive U induces an SDW ground state orderingat ( π/ , π/ For x = 1 / U = 0. For x =1, there are no electrons, which is trivially an insulatingstate. The hard-core boson model in Eq. (2) has been stud-ied extensively numerically, and its T = 0 phase dia-gram is known. For most x ∈ (0 , /
2) (i.e., for bosonconcentration between 0 and 1), it has a uniform super-fluid phase, which inherits the d -wave symmetry of thebosons, but has no nodal quasiparticles; this is labeled d -BEC in Fig. 2. At x = 1 /
4, the boson density is 1/2per plaquette; in this case, it is easy to see (by map-ping the problem to an equivalent XXZ model) that, for V (1) /t (1) >
2, the ground state is a √ ×√ d -wave bosons, while for V (1) /t (1) <
2, the ground stateis superfluid. At the critical point separating these twophases, V (1) /t (1) = 2, the effective Hamiltonian has anemergent SU (2) symmetry. In the present case, we findthat at U = U c and U = U s ≈ . V (1) /t (1) = 2, andthat V (1) /t (1) > U s < U < U c and V (1) /t (1) < U < U s . Around the d -CDW line in the phase diagramshown in Fig. 2, there is a small two-phase coexistence orphase separation (PS) region because the transition fromthe d -CDW state to the d -BEC is first-order. The boson-fermion model which applies for U ∼ U c and 0 < x < / x & x . /
2. By increasing U , there exists a crossover(the dotted line in Fig. 2) from the d -BEC phase to aBCS-like superconducting phase ( d -BCS without nodalquasiparticles). For even larger U , a phase transition intoa spin-1/2 FL phase with two flavors (bands) of fermions(FL-II) is expected for the effective model in Eq. (3) atsmall x . The spin-3/2 t - J - V model , Eq. (5), is similar to thespin-1/2 model in the weak hopping limit; this modelwas studied in detail. The ground state depends onthe ratio of J (3) /V (3) . In the present problem, we ob-tain values of V (3) an order of magnitude larger than J (3) . Thus, the ground state is a spin-3/2 FL (FL-III)for x < /
8. For x = 1 / √ ×√ × √ ×√ x = 1 /
4, every plaquette is occupied by a singlefermion, whose spins order to yield a √ ×√ x = 1 /
5, there is a concentration 4(1 / − /
5) = 1 / √ ×√ √ ×√ √ ×√ / < x < / For U c < U < U t and x = 1 /
4, there is one fermionper plaquette, so the only terms in the effective Hamilto-nian that operate are those in H (2 , . This is a complexmodel with a spin and pseudospin on each plaquette.We have solved it by approximating the ground-state asa direct product of spin and pseudospin factors. Sincethe spin interactions are AF and isotropic, they form thewell-understood N´eel ground state of the spin-1/2 AF, inwhich h S R · S R ′ i ≈ − / R and R ′ . Then, the effective psueduo-spin Hamiltonian is H (2 , = X h RR ′ i X α = x,y,z ¯ J α τ α R τ α R ′ , (6)where ¯ J α = J α + J ′ α h S R · S R ′ i . The ordering of thepseudo-spins is determined by the ¯ J α with largest abso-lute value. When U < U n ≈ . − ¯ J x > − ¯ J y > ¯ J z > x direction. This or-bital ordering corresponds to the fact that the electrondensity spontaneously breaks the C rotational symme-try to C with no breaking of translational symmetry,so this is an “electron nematic” or “ orbital nematic”phase. While for U n < U < U t , there is no such ne-matic ordering. For x = 1 /
2, with two electrons per plaquette, theinsulating ground state is adiabatically connected to theBI state at U = 0. Since the plaquette fermion hopping isthe dominant term, for 1 / < x < / / < x < | x − / | < O ( t ′ )] is a spin-1/2 FL, while at x = 3 /
4, theground state is a spin-1/2 AF.
Finite temperature . The finite T phase diagram is alsointeresting and worth future study. For instance, at x = 1 / U c < U < U n , the T = 0 phase is aspin-1/2 AF and orbital nematic. At any finite T , thespin order is lost, leaving a pure (Ising) nematic phaseup to a nonzero critical temperature.We thank D. J. Scalapino for helpful suggestions. Thiswork was supported, in part, by the NSF DMR-0531196(S.A.K.) and the DOE DE-FG02-06ER46287 (H.Y. andW.T.). E. Altman and A. Auerbach, Phys. Rev. B , 104508(2002), have proposed that a local effective Hamiltonianfor the uniform model ( t ′ = 1) can be constructed in anexpanded basis of low energy of states decoupled squares,despite the absence of a small parameter. W. F. Tsai and S. A. Kivelson, Phys. Rev. B , 214510(2006); W. F. Tsai and S. A. Kivelson, Phys. Rev. B ,139902 (2007). S. White, S. Chakravarty, M. P. Gelfand, and S. A. Kivel-son, Phys. Rev. B , 5062 (1992). H. Yao, W. F. Tsai, and S. A. Kivelson, (unpublished). D. J. Scalapino and S. A. Trugman, Philos. Mag. B ,607 (1996). D. J. S. A. Moreo, R. L. Suger, S. R. White, and N. E.Bickers, Phys. Rev. B , 2313 (1990). H. H. Lin, L. Balents, and M. P. A. Fisher, Phys. Rev. B , 1794 (1998). C. Wu, W. Liu, and E. Fradkin, Phys. Rev. B , 115104(2003). F. H´ebert, G. G. Batrouni, R. T. Scalettar, G. Schmid,M. Troyer, and A. Dorneich, Phys. Rev. B , 14513(2001). G. G. Batrouni and R. R. Scalettar, Phys. Rev. Lett. ,1599 (2000). S. A. Kivelson, V. J. Emery, and H. Q. Lin, Phys. Rev. B , 6523 (1990). V. J. Emery, S. A. Kivelson, and H. Q. Lin, Phys. Rev.Lett. , 475 (1990). S. A. Kivelson, E. Fradkin, and V. J. Emery, Nature ,550 (1998). Extensive finite- T analysis of this model at t ′ = 0 (con- sisting of decoupled plaquettes) has been carried out for t ′ = 0 but T >74