Cooperative orbital ordering and Peierls instability in the checkerboard lattice with doubly degenerate orbitals
aa r X i v : . [ c ond - m a t . s t r- e l ] M a y Cooperative orbital ordering and Peierls instability in the checkerboard lattice withdoubly degenerate orbitals
R.T. Clay, H. Li, S. Sarkar, S. Mazumdar, and T. Saha-Dasgupta Department of Physics and Astronomy and HPC Center for Computational Sciences,Mississippi State University, Mississippi State MS 39762 Department of Physics, University of Arizona, Tucson, AZ 85721 S. N. Bose National Centre for Basic Sciences, Kolkata, India (Dated: October 30, 2018)It has been suggested that the metal-insulator transitions in a number of spinel materials withpartially-filled t g d -orbitals can be explained as orbitally-driven Peierls instabilities. Motivated bythese suggestions, we examine theoretically the possibility of formation of such orbitally-driven stateswithin a simplified theoretical model, a two-dimensional checkerboard lattice with two directionalmetal orbitals per atomic site. We include orbital ordering and inter-atom electron-phonon inter-actions self-consistently within a semi-classical approximation, and onsite intra- and inter-orbitalelectron-electron interactions at the Hartree-Fock level. We find a stable, orbitally-induced Peierlsbond-dimerized state for carrier concentration of one electron per atom. The Peierls bond distortionpattern continues to be period 2 bond-dimerization even when the charge density in the orbitalsforming the one-dimensional band is significantly smaller than 1. In contrast, for carrier densityof half an electron per atom the Peierls instability is absent within one-electron theory as well asmean-field theory of electron-electron interactions, even for nearly complete orbital ordering. Wediscuss the implications of our results in relation to complex charge, bond, and orbital-orderingfound in spinels. PACS numbers: 71.30.+h, 71.28.+d, 71.10.Fd
I. INTRODUCTION
Transition-metal spinel compounds AB X , where X isS or O, have been for many years the subject of intenseexperimental and theoretical activity. The B sublattice ofthe spinel structure forms corner-sharing tetrahedra, giv-ing rise to a geometrically frustrated pyrochlore lattice.In general, oxides and chalcogenides of transition met-als exhibit complex charge- and spin-ordering that areoften coupled with orbital-ordering. Coupled orbital-charge-spin orderings have been investigated widely forcompounds of the late transition metal ions with active e g orbitals within cubic geometry, such as the cuprates andmanganites. In contrast, such orderings in spinel com-pounds are only beginning to be studied . Spinel systemsare considerably more complicated than the cuprates ormanganites due to two distinct reasons. First, the B-ionsin the spinels, which are in an octahedral environment ofthe X anions, often possess partially-filled t g d -orbitals.Examples include V ions in the vanadates of Zn ,Mn , and Cd ; V . in AlV O ; Ti in MgTi O ;Ir . in CuIr S ; and Rh . in LiRh O . Thethreefold degeneracy of the t g orbitals, along with theweaker tendency to Jahn-Teller (JT) distortions in t g -based systems lead to more complicated physics com-pared to e g -based systems. Second, the geometric frus-tration in the spinel structure, mentioned above, pre-cludes simple orderings .Very recently, an exotic orbitally-induced Peierls insta-bility has been proposed as the mechanism behind themetal-insulator transitions in the spinels like CuIr S .Charge-segregation of Ir into formally Ir and Ir , accompanied by the formation of an octamer of Ir ions with alternate short and long bonds is observedin CuIr S . This unusual charge-ordering pattern hasbeen explained within the context of an effective one-dimensionalization and Peierls instability driven by or-bital ordering (OO). The spinel structure of CuIr S con-sists of criss-cross chains of Ir-ions with strongest over-laps between orbitals of the same kind (i.e., d xy with d xy , d yz with d yz ). Within the proposed mechanism, OO inIr . ions leads to completely filled degenerate d xz and d yz orbitals and effective one-dimensional (1D) quarter-filled d xy bands. The latter now undergoes a Peierls in-stability that is accompanied by period 4 charge-ordering · · · Ir /Ir /Ir /Ir · · · and long-intermediate-short-intermediate bonds, as seen experimentally . The OOand Peierls instability are thought to be coupled, as op-posed to independent. Closely related phenomenologieshave been proposed to explain the metal-insulators tran-sitions in MgTi O and LiRh O .While the proposed scenario does provide qualita-tive explanations, it raises many interesting questions.For example, the bond or charge periodicities that resultfrom Peierls distortion in real quasi-1D systems dependstrongly on the bandfilling. Naively then, one might beled to believe that unless the OO is complete and leadsto integer occupations of the individual t g orbitals, ex-actly commensurate distortion periodicities, as suppos-edly observed in CuIr S and MgTi O , are not ex-pected. The exactly period 4 distortion in CuIr S andMgTi O is therefore a puzzle. Secondly, the role ofelectron-electron (e-e) interactions, which can have im-portant consequences on these transitions, is not clear.For example, it has been suggested that the bond distor-tion in MgTi O causes nearest-neighbor Ti pairs toform a spin-singlet state, giving a drop in the magneticsusceptibility at low temperature . Furthermore, e-e in-teractions can strongly affect the distortion periodicitiesin 1D chains with fillings different from 1 electron peratom . Addressing these issues requires explicit calcula-tions based on Hamiltonians containing all the necessarycomponents, that to the best of our knowledge do notexist currently.We report here the results of explicit calculations basedon a simplified model system that displays co-operativeOO and Peierls instability. Our model system is atwo-dimensional projection of the pyrochlore lattice, acheckerboard lattice with two degenerate directional or-bitals per atom. As we discuss in the next Section, themodel captures the effective one-dimensionalization thatoccurs in the spinel lattice. There exist also subtle differ-ences between the our model and the theoretical picturethat has been employed for the spinels , that we discussin Section IV. We have considered two different electrondensities, 1 electron per atom and electron per atom.We find that a stable Peierls state with 1D order occursfor an electron density of 1 electron per atom. Impor-tantly, even for significant deviations of orbital occupan-cies from integer values, the Peierls distortion is purelybond dimerization, as would be true in the precisely -filled 1D band. We speculate that this is a consequence ofinteractions between the effective 1D chains of our model,which are not entirely independent. For the case of electron per atom we did not observe an orbitally-drivenPeierls state even for integer orbital occupancies in theabsence of e-e interactions. For the same model in thelimit of infinite e-e interactions, however, a stable bonddimerization occurs. This suggests the second main re-sult of our paper, namely that e-e interactions are im-portant in stabilizing orbitally-induced Peierls states forthe case of -electron per atom within the orbitally de-generate checkerboard lattice. We will discuss possibleimplications of this result for the spinel lattice.The paper is divided in following subsections: in Sec-tion II we introduce our model Hamiltonian; in SectionsIII A and III B we present numerical results for 1 and electron per atom respectively; and in Section IV we con-clude and further discuss the relationship of our theoryto spinels and related materials. II. THEORETICAL MODEL
As discussed above, no quantitative calculations ex-ist of coexisting orbital and Peierls order in spinel lat-tices. This is due to the huge complexity of these systems,which include three dimensionality, triply degenerate t g metal atom orbitals, possible JT distortions and frustra-tion, in addition to e-e interactions. We therefore con-struct a simpler model based on a checkerboard latticethat is easier to handle, but that is expected to bear simi- xxx z y t x+y t y−x x y (a)(b)(c) yz +Q+Q−Q −Q +Q−Q+Q+Q−Q +Q −Q +Q−Q−Q+Q −Q t y t x FIG. 1: (color online) (a) 2D checkerboard lattice withorbital-ordering pattern given by the H OO term in Eq. 3.Filled dots denote metal atom positions, with two orbitalsper atom, xz and yz . Hopping terms are included along x , y , x + y , and y − x directions as defined in Eq. 5–Eq. 8. Hoppingalong the t x and t y directions, indicated by dashed lines, ismodulated by the inter-atomic e-p coupling. (b) The ( ddπ )overlap of xz , shown in green (light grey), and yz , shown inblue (dark grey), along the x- and y-directions. Note thatalong these directions, non-zero hopping matrix elements ex-ist only along x - ( xz - xz ) or y - ( yz - yz ) direction. (c) The( ddπ ) overlap of xz and yz orbitals along the y − x direction.The overlap along x + y is similar apart from a change of signfor the inter-orbital terms. larities with the spinel problem. Note that although thereexists a large body of literature on the consequences offrustration within the checkerboard lattice with a single atomic orbital per site , we are unaware of similar cal-culations on the present model which deals with doublydegenerate orbitals at each site. An important propertyof orbitally-driven Peierls order in these materials is thatthe OO leads to Peierls order in several different crystallattice directions . In order to incorporate orbital de-generacy, frustration, and the possibility of Peierls orderin multiple directions we consider the following Hamil-tonian for a checkerboard lattice with doubly degeneratemetal orbitals at each lattice site. H = H SSH + H OO + H ee (1) H SSH = X i, a ,γ,γ ′ t a γγ ′ (1 + α a ∆ i,i + a )( d † iγσ d i + a γ ′ σ + h.c. ) + 12 X i a K SSH ∆ i,i + a (2) H OO = g X i,γ = γ ′ Q i ( n iγ ′ − n iγ ) + 12 K OO X i Q i (3) H ee = U X i,γ n iγ ↑ n iγ ↓ + U ′ X i,γ = γ ′ n iγ n iγ ′ (4)The Hamiltonian in Eq. 1 consists of, (i) H SSH thatcontains the kinetic energy and the inter-ion electron-phonon (e-p) coupling (Eq. 2), (ii) an OO term H OO (Eq. 3), and (iii) e-e interaction H ee (Eq. 4) that includesshort-range e-e interactions within each site. We describeeach of these terms separately below. H SSH includes electron hopping between same as wellas different orbitals. In Eq. 2, d † iγσ creates an electron ofspin σ in the orbital γ of atom i . γ and γ ′ correspondto d xz and d yz orbitals, which have lobes that are ori-ented perpendicular to each other at each site as shownin Fig. 1. The inter- and intra-orbital hopping matrix el-ements t a iγγ ′ are based on Slater-Koster parametrizationof hopping integrals connecting t g orbitals . a denotesa unit vector along the x , y , x + y , or y − x directions.Each bond indicated in Fig. 1 connects two orbitals ateach metal ion site with two orbitals at another site, andis hence written as a 2 × (cid:18) t xz,xz t xz,yz t yz,xz t yz,yz (cid:19) y − x = (cid:18) − − − − (cid:19) (5) (cid:18) t xz,xz t xz,yz t yz,xz t yz,yz (cid:19) x + y = (cid:18) −
12 1212 − (cid:19) (6) (cid:18) t xz,xz t xz,yz t yz,xz t yz,yz (cid:19) x = (cid:18) − (cid:19) (7) (cid:18) t xz,xz t xz,yz t yz,xz t yz,yz (cid:19) y = (cid:18) − (cid:19) (8)All hopping integrals are in units of the ( ddπ ) matrixelement (set to -1) involving the two d -orbitals of themetal atoms and mediated by the p -orbital of the an-ion (not shown explicitly in the figure) in between. Inthe above, the small ddδ hoppings have been neglected.Note that electron hoppings along the x ( y ) direction in-volve only the d xz ( d yz ) orbitals. Thus the π -bondingamong the orbitals, as opposed to the σ -bonding inspinels, does not preclude the orbitally-induced effectiveone-dimensionalization .The inter-ion e-p coupling in H SSH is written in theusual Su-Schrieffer-Heeger (SSH) form . Here α a is thee-p coupling constant corresponding to the bond betweenatoms at i and i + a , ∆ i,i + a the deviation of this bond yx FIG. 2: (color online) Pattern of the orbital-ordering andbond-order modulation for one electron per atom. Squares(circles) represent atoms with predominant xz (yz) orbitaloccupation. Bonds alternate in strength along diagonal di-rections, with solid (dashed) lines indicating strong (weak)bonds. from its equilibrium length, and K SSH the correspondingspring constant. We include nonzero e-p couplings onlyalong the x and y -directions, in keeping with the Peierlsdistortions in CuIr S and MgTi O involving only or-bitals of the same kind, and assume e-p couplings of equalstrength ( α x = α y = α ). We take all spring constants K SSH = 1.The OO term H OO breaks the orbital degeneracy be-tween the two orbitals at each individual site. n iγσ = d † iγσ d iγσ is the number of electrons of spin σ on orbital γ of site i , and n iγ = n iγ ↑ + n iγ ↓ . The coordinate Q i couples to the charge density difference between the twoorbitals on the site, with a corresponding coupling con-stant g and spring constant K OO . We fix K OO to thevalue 1. As written in Eq. 3, the relative phase of the Q i at each site is unrestricted. Within the model of ref-erence 13 the effective one-dimensionalization is a con-sequence of OO alone and in principle will occur evenin the absence of Peierls distortion. We have used thisto determine the preferred Q i mode for both 1 and electron per site by calculating the orbital occupationsin several large lattices with open boundary condition(OBC). In a large OBC lattice, the charge densities andbond orders far from the lattice edges spontaneously as-sume the pattern that would occur in the infinite latticefor 0 + coupling limit . An alternate approach to deter-mine the dominant OO mode is to calculate in a periodiclattice the energies corresponding to each mode for fixeddistortion amplitude; the dominant mode is simply theone with the lowest total energy. In the present case wehave performed both sets of calculations for both 1 and electron per site and have determined that the preferredOO mode in both cases is the “checkerboard” pattern ofFig. 1(a), which can be parametrized in terms of a singleamplitude | Q | with Q i = ( − i x + i y | Q | , where i x and i y are the x and y coordinates of the i th atom.The third term in the Hamiltonian, H ee (Eq. 4), in-cludes short-ranged Coulomb repulsions. U ( U ′ ) is theon-atom Coulomb repulsion for electrons in same (differ-ent) orbitals. While exact diagonalization has been usedsuccessfully for many 1D, quasi-1D, and two dimensional(2D) lattices involving both e-e and e-p interactions, inthe present model with two orbitals per metal site theHilbert space is too large to treat any meaningful sizecluster within exact diagonalization. We will thereforeconsider first the non-interacting ( U = U ′ = 0) sys-tem, and then consider the effect of U and U ′ within theunrestricted Hartree-Fock (UHF) approximation, withno assumption of the periodicity of the UHF wavefunc-tion. Specifically, we replace the interaction terms in H ee (Eq. 4) by U X i,γ,σ,σ ′ n iγσ h n iγσ ′ i − U X i,γ h n iγ ↓ ih n iγ ↑ i + U ′ X i,γ = γ ′ ,σ,σ ′ ( n iγσ ( h n iγ ′ σ i + h n iγ ′ σ ′ i ) − ( d † iγ ′ σ d iγσ + H.c. ) h d † iγ ′ σ d iγσ + H.c. i− h n iγσ ih n iγ ′ σ i − h n iγσ ih n iγ ′ σ ′ i + h d † iγ ′ σ d iγσ + H.c. i ] h n iγσ i and h d † iγ ′ σ d iγσ + H.c. i are obtained using a com-bination of self-consistency and simulated annealing forfinding their ground state values. The UHF approxima-tion often gives unphysical results for large interactionstrengths and we will primarily focus on small U and U ′ .We treat the OO and e-p interactions using a standardself-consistent approach derived from the equations ∂ h H i ∂Q = 0 ∂ h H i ∂ ∆ i,i + a = 0 . (9)The self-consistency equations derived from Eq. 9 areused iteratively given an initial starting distortion. Inthe infinite system the OO or the bond distortion wouldoccur for infinitesimally small coupling constants g and α . In finite-size clusters, however, due to the finite-sizegaps between successive energy levels, nonzero couplingconstants are required before the symmetry-broken stateappears. In the following we consider g and α close to theminimum values needed for the broken-symmetry stateto occur.We performed calculations for lattices up to 16 × Q , ∆ i,i + a , and UHF average charge densities)to be determined self-consistently, the calculations oftenbecame trapped in local minima before reaching the trueground state. In all cases we have taken care that thetrue ground state was reached. Below we summarize ournumerical results. These are divided into two subsec-tions that discuss average charge densities of 1 electronper atom and electrons per atom, respectively. < n + > α ∆ B
16 periodic lattice. (a)Majority charge density (see text) in the orbitals forming thequasi-1D chains following orbital ordering. Here and in (b)circles, diamonds and squares correspond to g = 0.6, 0.8 and1.0, respectively. (b) ∆ B along the diagonal chain directions(see text and Fig. 2) as a function of α . (c) Bond alternationplotted as a function of charge density. Circles show the effectof increasing g with constant α =1.3. Diamonds show effectof increasing α with constant g =0.8. Lines are guides to theeye. III. RESULTSA. 1 electron per atom
The OO term in our model makes the orbital occu-pancy of the xz and yz orbitals unequal at each site. Tomeasure the degree of OO quantitatively, we calculatethe majority charge density h n + i , defined as the chargedensity in the xz orbitals at + | Q | sites (see Fig. 1(a)).These orbitals form quasi-1D chains in the x direction(sites denoted by squares in Fig. 2). With one electronper site, h n + i ranges from 0.5 to 1, with h n + i = 1 indicat-ing complete OO, and h n + i = 0 . x and y directions are identical–thecharge density in the yz orbitals at −| Q | sites (denotedby circles in Fig. 2) is also h n + i .As an order parameter for the Peierls distortion, wemeasure the modulation of the bond order, B i,i + a ,γ = X σ h d † i + a ,σ,γ d i,σ,γ + H.c. i . (10)The bond order we are interested in (Eq. 10) is the expec-tation value of charge-transfer between orbitals of samesymmetry belonging to neighboring atoms. The charge-transfer is directly coupled to the bond distortion ∆ i,i + a in Eq. 2, and hence for nonzero ∆ i,i + a the charge trans-fer across consecutive bonds shows periodic modulation.The extent of modulation of B i,i + a ,γ is therefore a di-rect measure of the SSH distortion strength. As shownin Fig. 2, we find bond order modulation along x ( y ) di-rection to involve xz ( yz ) orbitals only. The modulationis purely period-2 (dimerization) with alternating strongand weak bonds, and hence we use ∆ B , the difference between the calculated strong and weak bond orders in-volving orbitals of a particular symmetry, as the orderparameter for the SSH distortion. Because of symmetry,the amplitudes of the bond order modulations involvingthe xz orbitals along the x -direction, and the yz orbitalsalong the y -directions are identical.As discussed above, whether or not a co-operativeorbitally-induced Peierls instability occurs for h n + i < U = U ′ = 0). The co-operativenature of the OO and bond dimerization is shown inFig. 3, where we show the results of our self-consistentcalculations for 16 ×
16 lattices. As seen in Figs. 3(a) and(b), the orbitally-induced Peierls state appears only for h n + i ≥ .
8, with the bond distortion a pure bond dimer-ization regardless of h n + i . As expected for a co-operativetransition, h n + i increases with g , as seen in Fig. 3(a). Foreach g there exists an α c beyond which there occur si-multaneous jumps in h n + i and ∆ B (the jump in h n + i becomes progressively smaller as g increases.) The mag-nitude of α c decreases with increasing g (see Fig. 3(b))To further show the cooperative effect, in Fig. 3(c) weshow the effects of (i) increasing α at constant g , and (ii)increasing g at constant α . While it is to be expectedthat the orbital order parameter h n + i increases with g ,or that the bond alternation parameter increases with α ,we find that either of the coupling constants enhancesboth h n + i and ∆ B .Next we consider the correlated case with nonzero U and U ′ . As for U = U ′ = 0, the orbitally-driven Peierlsstate is again bond-dimerized. In Fig. 4 we plot h n + i normalized by its value in the uncorrelated system as afunction of U for several values of U ′ . Within UHF, thecombined effect of U and U ′ can be to either weakenor strengthen the distortion: for fixed U ′ , U tends toweaken the OO and the bond distortion, while for fixed U , U ′ strengthens both order parameters. Within theUHF approximation for one electron per atom, the effectsof U and U ′ cancel exactly when U ′ = U .From the Hamiltonian, the consequence of U ′ is tominimize the intrasite inter-orbital Coulomb repulsion, < n + > / < n + ( U = U ’ = ) > FIG. 4: Majority charge density as a function of U and U ′ for one electron per atom, normalized with respect to thesame quantity for the uncorrelated system. Results shownare for for 16 ×
16 periodic lattices with g = 0 . α = 1 .
2, and K OO = K SSH = 1. Circles are for U ′ = 0 . U , diamonds arefor U ′ = 0 . U , and triangles are for U ′ = 0 . U . Lines areguides to the eye. which is achieved by orbital ordering. It is thus not sur-prising that U ′ has the same effect as g in Fig. 4. Theeffect of U , as seen in Fig. 4, is however an artifact ofthe UHF approximation. In the case of the strictly 1D -filled band chain with one orbital per site, exact diag-onalization and quantum Monte Carlo calculations haveshown that the Peierls bond-alternation is enhanced bythe Hubbard U . In contrast, the UHF approximationpredicts incorrectly that U destroys the bond alternationin the above case . Had we been able to perform ex-act diagonalization in the present case, we would havefound similar enhancement of the bond dimerization by U . This would have had a profound effect on our overallresult, reducing significantly the α c or the threshold h n + i at which the bond dimerization appears.The most important conclusion that follows from theabove is that an orbitally-induced Peierls instability canoccur even for incomplete OO ( h n + i ∼ . h n + i significantly less than 1. In-deed, it is conceivable that the threshold value of h n + i at which the bond dimerization appears can be evensmaller than 0.8 for nonzero e-e interactions. We havefound no other periodicity or evidence for soliton for-mation in our calculations. An interesting aspect of theOO driven bond distortion here is the phase relationshipbetween the bond order wave states involving the d xz and d yz orbitals in Fig. 2. The short bonds along the x and y -directions occupy the same plaquettes, yieldinga structure that is reminiscent of (but different from)the valence bond crystal obtained within the Heisenbergspin-Hamiltonian for the checkerboard lattice . Fur-thermore, the bond dimerizations along any one direc-tion but on different diagonals of the checkerboard lat-tice are strictly “in-phase”. Both of these indicate thatwhile the bond dimerizations are consequences of effec-tive one-dimensionalization, there exist strong 2D in-teractions in between both the criss-cross and parallelchains. We ascribe the persistence of the bond dimeriza-tion for h n + i < < n + > α ∆ B
16 periodic lattice. (a) Majoritycharge density in the orbitals forming the 1D chains. Hereand in (b) circles, diamonds and squares correspond to g =0.6, 0.8 and 1.0, respectively. (b) ∆ B along the diagonal chaindirections (see Fig. 2) as a function of α . (c) Bond alterna-tion versus majority charge density. Circles show the effectof increasing g with constant α =1.3. Diamonds show effectof increasing α with constant g =0.8. Lines are guides to theeye. from the 2D interactions. B. electron per atom In CuIr S and MgTi O , the distortion along thechain directions is not bond dimerization but a period4 distortion . This is as expected for a Peierls transitionin a 1D chain with carrier density 0.5. We have thereforeperformed self-consistent calculations within our modelHamiltonian also for density 0.5.Not surprisingly, we do obtain self-consistently an or-bitally ordered state here for nonzero g with α = 0. Evenwithin an essentially 100% orbitally-ordered state (mi-nority orbital charge density . α , we were unable to obtain a stable Peierls-distorted state. In all cases our self-consistent simula-tions converged to states with disordered bond distor-tions and charge densities, indicating vanishing bond-charge distortion in the thermodynamic limit. We ob- tained similar results after including U and/or U ′ withinthe UHF approximation. We conclude that for electrondensity away from 1 electron per atom, orbitally-drivenPeierls ordering does not occur within our model in thenon-interacting limit or within the mean-field approachto e-e interactions.We ascribe the absence of bond-charge distortion hereto the important role played by the interaction among thecrisscrossing chains within the checkerboard lattice. Wehave already pointed out in Section IIIA that the stabi-lization of the perfect period 2 distortion for the case of 1electron per atom, even in the absence of complete OO, isa signature of such a 2D interaction. Similar 2D interac-tions should be relevant also for carrier density 0.5. Sincewithin mean field theory the Peierls instability in 2D islimited to carrier density of 1, the absence of the bond-charge distortion in the present case is to be anticipated.On the other hand, we have recently shown in a seriesof papers that specifically for this carrier density andone orbital per site, nonzero e-e interactions can stronglystabilize bond-charge ordered states in 2D lattices .In the case of the checkerboard lattice with a single or-bital per site, plaquette spin-singlet formation (thoughwithout charge-ordering) has similarly been found for thesame carrier concentration . It is conceivable that sim-ilar effects of e-e interactions persist in the present casewith two degenerate orbitals per site.Unfortunately, performing a realistic calculation withfinite U that goes beyond the UHF approximation in thepresent case is beyond our computational capability. Wehave therefore investigated our model Hamiltonian Eq. 1in the limit of U → ∞ , where we assume band orbitaloccupancy corresponding to that for spinless fermions.Fig. 5 shows the same order parameters as in Fig. 3 forthe spinless fermion case. While the transition occurshere for a slightly larger value of the coupling constant α , it is otherwise identical to the transition with 1 elec-tron per atom, viz., bond dimerization occurs along thediagonal directions, and OO and bond distortion rein-force each other cooperatively. IV. DISCUSSION
In summary, we have carried out numerical studies onthe 2D checkerboard lattice with two degenerate direc-tional orbitals per site - a model system that like thespinel compounds can in principle exhibit OO-drivenPeierls bond distortions and charge ordering in multi-ple directions. In addition to OO and bond modulationterms, our model Hamiltonian includes both intra- andinter-orbital e-e correlations that were treated within theUHF approximation. Although some of our results havestrong implications for the spinels, it is useful to pre-cisely understand the differences between the spinel andcheckerboard lattices such that the applicability as wellas limitations of our model can both be understood. Onedifference between the two lattices is that the plaquettesin the 2D checkerboard lattice do not correspond to thetetrahedra in the spinel lattice because of the differencebetween horizontal and vertical bonds in Fig. 1(a) on theone hand and the diagonal bonds on the other . Whatis more important in the present context is that the OOin our model is not driven by a band JT transition thatdestroys the degeneracies of the atomic orbitals in themodel of 13. Within our model the two orbitals of dif-ferent symmetries on a given atom are both potentiallyactive orbitals.We believe that our demonstration of the co-operativeinteraction between OO and the Peierls instability in Sec-tion III A, where each broken symmetry enhances theother, is of direct relevance to the t g -based spinel sys-tems, where qualitative discussions have suggested simi-lar results . Similarly, our observation that the period2 bond distortion persists for 1 electron per atomic siteeven for incomplete OO, with majority charge densityas low as 0.8 per orbital, may also be of significance forthe spinels. This should be particularly true for nonzeroonsite Hubbard interaction, which will tend to decreasethe amplitude of the OO. Complete OO in the real sys-tems CuIr S and MgTi O requires that the energy gapdue to the JT distortion is significantly larger than theHubbard interaction. It is at least equally likely that thecommensurate charge and bond distortions found in theexperimental systems are not due to complete OO butare consequences of the complex interactions between thecrisscrossing chains in the spinel lattice, as in the checker-board lattice.The implication of the absence of the Peierls instabilityfor the case of an electron per atom in the checkerboardlattice within one-electron theory is less clear. One possi-ble implication is that our results for the 2D checkerboardlattice are irrelevant for the three-dimensional (3D) spinellattice because of the fundamental difference betweenthem that has already been pointed out in the above. InCuIr S , the only active orbital following the OO is the d xy orbital , which has been excluded within our model.It is thus conceivable that the 1D character of the activeorbitals in CuIr S following OO is much stronger thanin the checkerboard lattice, and this is what drives themetal-insulator transition in the real system. It is, how-ever, equally likely that the 3D interactions between the d xy -based chains are as strong as the 2D interactions inthe checkerboard lattice (recall, for example, that com-mensurate periodicity for independent 1D chains requirescomplete OO, see above). In this case our null result forthe uncorrelated checkerboard lattice would imply non- negligible contribution of e-e interaction to the metal-insulator transitions in CuIr S and LiRh O . Furthertheoretical work based on the 3D pyrochlore lattice aswell as experimental work that determines the extent ofOO in the real systems will both be necessary to com-pletely clarify this issue.Finally, assuming that e-e interactions play a role,which is subject to further investigations as discussedabove, this raises an interesting question, viz., what ul-timately is the driving force behind the metal-insulatortransitions in CuIr S and LiRh O ? Three of us haveargued elsewhere that for carrier concentration precisely0.5, there is a strong tendency to form a paired-electroncrystal (PEC), in which there occur pairs of spin-singletbonded sites separated by pairs of vacancies . This ten-dency to spin pairing is driven by nearest neighbor an-tiferromagnetic (AFM) correlations (as would exist in alarge Hubbard-U system) and is enhanced in the pres-ence of lattice frustration. Although the original calcu-lations are for the anisotropic triangular lattice with asingle orbital per site, the same tendency to spin-singletformation can persist also in the spinel lattice. If theinsulating state in the spinels CuIr S and LiRh O canbe understood as a PEC with spin-singlet pairing drivenby AFM correlations, it may further indicate that e-einteractions play an important role in superconductivityfound in several structurally-related spinels. Whether ornot e-e interactions play a role in the observed supercon-ductivity in the spinels LiTi O , CuRh S , CuRh Se has remained a lingering question . If the insulatingstate in this class of materials is indeed a PEC, the super-conducting spinels should perhaps be included among thesystems in which superconductivity is driven not entirelyby BCS electron-phonon coupling. V. ACKNOWLEDGMENTS
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