Quantum melting of charge ice and non-Fermi-liquid behavior: An exact solution for the extended Falicov-Kimball model in the ice-rule limit
aa r X i v : . [ c ond - m a t . s t r- e l ] J un Quantum melting of charge ice and non-Fermi-liquid behavior:An exact solution for the extended Falicov-Kimball model in the ice-rule limit
Masafumi Udagawa, Hiroaki Ishizuka, and Yukitoshi Motome
Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan (Dated: August 21, 2018)An exact solution is obtained for a model of itinerant electrons coupled to ice-rule variables onthe tetrahedron Husimi cactus, an analogue of the Bethe lattice of corner-sharing tetrahedra. Itreveals a quantum critical point with the emergence of non-Fermi-liquid behavior in melting ofthe “charge ice” insulator. The electronic structure is compared with the numerical results for thepyrochlore-lattice model to elucidate the physics of electron systems interacting with the tetrahedronice rule.
PACS numbers: 71.10.Fd, 71.10.Hf, 71.20.-b, 71.23.-k
The ice rule is a local constraint observed in a broadrange of systems in condensed matter physics. It im-poses a configurational constraint on two-state variablesdefined at neighboring four lattice sites so that two out offour are in the opposite state to the other two. The mostwell-known material is water ice, in which the two statescorrespond to the configuration of hydrogens [1, 2]. Ananalogy was drawn by Anderson in the cation orderingof Fe and Fe in magnetite Fe O [3]. More recently,a magnetic analogue was found in several pyrochlore ox-ides, the so-called spin ice, such as Ho Ti O [4] andDy Ti O [5].The ice rule enforces local correlations; however, it isunderconstraint and not enough to make the entire sys-tem ordered. The ground-state manifold retains macro-scopic degeneracy, resulting in residual entropy [2, 5].Nevertheless, the ice-rule configuration is not completelydisordered but cooperative in nature: There is a spa-tial power-law correlation in the two-state variables orig-inating from a hidden gauge structure [6]. Considerableprogress on the understanding of such cooperative as-pects has been made in the last decade through the studyof spin ice [7].In contrast to such “localized spin physics”, much lessis known for the role of the ice rule in itinerant systems. Itis intriguing to elucidate how the cooperative nature fromthe ice-rule constraint affects the electronic and trans-port properties. The issue will also be experimentallyrelevant to a wide range of pyrochlore-based compounds,such as mixed-valence compounds with a charge-orderingtendency [10–12] and itinerant d -electron materials in-cluding Ising-like rare-earth moments [13, 14]. Only afew theoretical studies have been carried out so far [8, 9].In this Letter, we address this issue in one of the sim-plest models which describe fermions interacting with ice-rule variables, an extended Falicov-Kimball model. Weobtain an exact solution to this model on the Husimicactus of tetrahedra, i.e., an analogue of the Bethe lat-tice composed of corner-sharing tetrahedra. We clar-ify the ground-state phase diagram including a “chargeice” insulator in which the fermions are localized in theice-rule configuration. The solution reveals that a non-Fermi-liquid behavior appears at a quantum critical point (QCP) where the charge ice melts as the interaction de-creases. By comparison with the numerical results for thepyrochlore lattice, we show that our Husimi cactus modelprovides a cornerstone of itinerant ice-rule problems.We start with the extended Falicov-Kimball model onthe pyrochlore lattice [Fig. 1(a)], H = − t X h i,j i ( c † i c j + H . c . )+ U X i n ci ( n fi −
12 )+ V X h i,j i n fi n fj , (1)where the first term describes the hopping of spinlessfermions c , the second term represents the onsite repul-sion between spinless fermions and immobile particles f , and the last term is the repulsion between immobileparticles. Here, n ci = c † i c i , n fi = 0 or 1 (c number), (a) (b)(c) (d)
012 3
FIG. 1: (color online). A typical ice-rule configuration for (a)pyrochlore lattice and (b) tetrahedron Husimi cactus (THC).The sites with n fi = 1 (0) are shown by filled (open) circles.An example of loop and one-dimensional chain connectingthe sites with n fi = 1 is shown by bold orange lines. (c) Anapparently different ice-rule configuration obtained from (b)by interchanging the upper right and lower right branches.(d) A branch of THC considered in the calculations of g inEq. (3). and the sums h i, j i are taken over the nearest-neighborsites. Hereafter, we focus on the case in which the im-mobile particles satisfy the ice rule; see Fig. 1(a). This isachieved by setting P i n fi = N/ N is the total numberof sites) and V → ∞ .The partition function of the model is calculated by Z = Tr f Tr c exp( − β H ), where Tr f (Tr c ) is the trace overthe immobile-particle (spinless-fermion) degree of free-dom, and β = 1 /k B T is the inverse temperature with theBoltzmann constant k B . For a given configuration of im-mobile particle { n fi } , the Hamiltonian (1) is reduced toa one-body problem given by H ( { n fi } ) = − t X h i,j i ( c † i c j + H . c . ) + X i U i n ci , (2)where U i is the onsite potential determined by the con-figuration { n fi } as U i = U ( n fi − / f in the partition function is replaced by the sumover { n fi } which satisfies the ice-rule constraint: Z = P { n fi } ∈ ice Tr c exp[ − β H ( { n fi } )] . This is, in principle, fea-sible to calculate because Tr c is performed by a diago-nalization of the one-body N × N Hamiltonian (2), butin practice, it is difficult for large system sizes becausethe sum within the ice manifold increases exponentially ∼ . N/ [2].A dramatic simplification is introduced by consideringa modified structure of the pyrochlore lattice, that is, aHusimi cactus of tetrahedra [15]. It is an analogue ofthe Bethe lattice composed of corner-sharing tetrahedra,as shown in Fig. 1(b), which we call here the tetrahe-dron Husimi cactus (THC). THC shares two importantstructural features with the pyrochlore — the tetrahe-dral units and their corner-sharing network. A differenceis in the global connection of tetrahedra: The pyrochlorelattice has loops running across different tetrahedra [seeFig. 1(a)], but THC does not have such global loops. De-spite of this difference, theTHC model captures severalessential features of the pyrochlore, as we will see later.The simplification by considering THC is twofold.First, while the ice-rule configurations in THC are alsomacroscopically degenerate (= 6 × N/ ), they are topo-logically equivalent because one can relabel the site num-bers by interchanging branches which spread from thesame tetrahedron [see Figs. 1(b) and (c)]. Consequently,all the possible ice-rule configurations give an identicalBoltzmann weight, and therefore, the sum over { n fi } canbe suppressed in the calculations of any observable.On top of that, the second crucial point is that forany ice-rule configuration on THC we can obtain one-body Green’s functions exactly by using recursion equa-tions similar to those often used in the Bethe latticeproblems [16–18]. To see this explicitly, let us considerthe T = 0 local retarded Green’s function at the site i , G i ( ε ) ≡ t h i | [ ε − H ( { n fi } ) + iδ ] − | i i ( t is included to make G dimensionless). Similarly, we define the Green’s func-tion g i ( ε ) for a branch given by terminating a half of thetetrahedral network at the site i , as shown in Fig. 1(d). g is formally written by the expansion in terms of thehopping: g = g (0)0 + g (0)0 ( g + g + g ) g − g (0)0 (cid:2) g ( g + g ) + g ( g + g ) + g ( g + g ) (cid:3) g + · · · , (3)where g (0)0 = t ( ε − U ) − is the atomic Green’s function.Here, the second term corresponds to the processes wherea fermion hops from the site i = 0 to one of the otherthree sites in the same tetrahedron, j = 1, 2, or 3, thenpropagates within the branch belonging to the site j ,and returns to the original site i = 0 [see Fig. 1(d)].The next term describes higher-order contributions, e.g.,a hopping process 0 → → → g i depends not explicitly on i butonly on the value of U i ; i.e., g i = g ± corresponding to U i = ± U/
2. The recursive equations are given by g ± (1 − g ∓ ) + 2 g ∓ (1 − g ± )1 + g ∓ (1 − g ± ) = 1 t (cid:0) ε ∓ U (cid:1) − g ± . (4)The full Green’s functions G ± are similarly obtained as G − ± = 2 g ± − t (cid:0) ε ∓ U (cid:1) . (5)Equations (4) and (5) give the exact solution to the localGreen’s functions of the extended Falicov-Kimball model(1) in the ice-rule limit: P i n fi = N/ V → ∞ .By extending the above calculations, it is also possibleto obtain the nonlocal as well as the finite- T Green’sfunctions. The whole procedure can be straightforwardlyapplied to a broader range of models with general ice-ruletype constraints on general cacti of a complete graph.Such extensions will be discussed elsewhere.Now let us discuss the ground-state properties derivedfrom the exact solution. Figure 2 shows the site-resolveddensity of states (DOS) given by ρ ± = − Im G ± /πt andtheir summation ρ = ρ + + ρ − . In Fig. 2(a), we show DOSat U = 0, which is given by the analytic form of ρ ( ε ) = π Re { [ p t − ( ε + 2 t ) ] / [16 t − ( ε + 2 t ) ] } + δ ( ε − t ) . This is composed of the delta-functional peak (flat bands)at ε = 2 t and the broad spectrum for | ε + 2 t | < √ t . Inthe opposite limit of U ≫ t also, a simple analytic formis available: By approximating g ± ≃ ± ε ≃ ∓ U/ ρ ± ( ε ) = π Re { t − [ ε ∓ ( U/ } − / . This is identical to DOS for one-dimensional (1D) tight-binding chains centered at ε = ± U/
2. The coincidencecomes from the fact that THC under the ice-rule con-straint is broken up into 1D chains of the same potential+ U/ − U/
2, as schematically shown in Fig. 1(b). Atypical result is shown in Fig. 2(e) at U = 100 t .In both the limiting cases of U = 0 and U ≫ t , thesystem is insulating at half filling of the spinless fermion, n c ≡ h n ci i = 1 /
2. However, the origins of the two in-sulating states are quite different. The insulating phaseat U = 0 is a simple band insulator, in which the broadspectrum is fully occupied and the flat band is empty. Onthe other hand, the large- U phase is an incompressiblestate with a large gap ∼ U originating from the repul-sive interaction between fermions and immobile particles.We call this correlation-driven insulating state “chargeice”, in analogy with the spin ice [4, 5], since the mobilefermions are localized in the ice-rule configuration whichis composed of the sites with n fi = 0.The question is how the system changes from the bandinsulator to the “charge ice” as U increases. In Figs. 2(b)-(d), we show the change of DOS obtained in the inter-mediate range of U . When switching on U , while thebroad spectrum is almost unchanged, the flat bands areperturbed to be broadened around ε = 2 t , leading to areduction of the gap [Fig. 2(b)]. The gap decreases as U increases and vanishes at U ≃ t [Fig. 2(c)]. Further in-crease of U opens a gap again [Fig. 2(d)], and increases it FIG. 2: (color online). DOS for (a)-(e) THC and (f)-(j) py-rochlore lattice at
U/t = 0, 1, 2, 3, and 100. The results forthe pyrochlore model are calculated for a 3 superlattice of4 × lattice sites [19]. ρ + ( ε ) [ ρ − ( ε )] is shown by the boldred (dotted blue) curves. In (a)-(d) and (f)-(i), the total DOS ρ ( ε ) is plotted by thin black curves. continuously. Critical behavior around U = 2 t is shownin Fig. 3(a). The plot suggests that the gap closes onlyat U = U c = 2 t .The critical behaviors of DOS can be understood an-alytically from Eqs. (4) and (5). We can prove that thedivergence occurs at ε = ε U = t + ( U c − U ) /
2: At ε = ε U ,we obtain g − = − − ( U − U c ) /t , g − − = 1, G + = − t/U ,and G − = ∞ , resulting in ρ + ( ε U ) = 0 (except for U = 0)while ρ − ( ε U ) → ∞ . With regard to the gap, by con-sidering a small deviation from U = U c and evaluating g ± at ε ≃ ε U , we find that the energy gap opens as∆( U ) ≃ t | U − U c | for both U > U c and U < U c .Therefore the energy gap closes only at U = U c . Thisis identified as QCP between the band insulator and thecharge ice (see the phase diagram in Fig. 4).The critical behavior of the gap ∆( U ) ∝ | U − U c | ispeculiar in contrast to the usual linear behavior ∆( U ) ∝| U − U c | in the Mott transition [20]. Actually, QCP ispeculiar also in the sense that the self-energy exhibitsan anomalous power-law behavior. From Eqs. (4) and(5), we can derive that the self-energy Σ ± , defined byΣ ± = g (0) − ± − G − ± , shows the following critical behavior:ReΣ ± ( ε ) = 2 − C ± | ε − µ c | sgn( ε − µ c ) , (6)ImΣ ± ( ε ) = − C ± √ | ε − µ c | , (7)where C − = (4 /t ) , C + = C − /
2, and µ c = t is the crit-ical chemical potential [Figs. 3(b) and (c)]. The anoma-lous power law ∝ | ε − µ c | indicates that the systemshows a non-Fermi-liquid behavior at QCP. DOS alsoshows a singular energy dependence, ρ ± ( ε ) ∝ | ε − µ c | ± at U = U c , resulting in the anomalous T dependenceof thermodynamic quantities at QCP. For example, thespecific heat is predicted to behave as ∝ T at low T .Collecting the results with varying U , we summarizethe exact ground-state phase diagram in Fig. 4. Thereare two metallic regions for 0 < n c < / / < n c <
1, which are separated by the two insulating regions athalf filling n c = 1 /
2, i.e., the band insulator for
U < U c FIG. 3: (color online). (a) DOS around ε = µ c = t at U/t = 1 . . . . . ± [Eq. (7)]. The green dashed linesshow the asymptotic power law ∝ | ε − µ c | . t / UU / t m / U -3.0-2.0-1.00.01.02.0 m / t ( U c , m c )=(2 t , t ) -1+ 3-1- 3 metal < n c < metal < n c < n c = n c = r + r + r - r - charge iceinsulator n c = band insulator n c = U =0 U = U = U c =2 t FIG. 4: (color online). Ground-state phase diagram of theextended Falicov-Kimball model on THC in the ice-rule limit. and the charge ice for
U > U c . All these four phasesmeet at QCP at ( U c , µ c ) = (2 t, t ).As indicated by the bold curve crossing QCP in Fig. 4,the divergence of ρ − at ε = ε U is transferred from theupper-band bottom to the lower-band top [see Fig. 3(a)].The divergence contains a bunch of extended states onthe sites with n fi = 0. The extended states are fullyoccupied in the charge ice state, while they are emptyin the band insulator. This fact leads us to define an“order parameter” to distinguish two insulating states, O Ψ ≡ h c † Ψ c Ψ i for the corresponding extended-state op-erator c Ψ = P i c i ( − i / √ L, where i represents the se-quential site number on a chain composed of n fi = 0 sites( L is the length). O Ψ changes from 0 for the band insula-tor to 1 for the charge ice. The transfer of the extendedstates bears some analogy to the “levitation scenario”proposed for the quantum Hall systems [21]. This anal-ogy suggests a discrete change of the “transport nature”at QCP. Further analysis will be discussed elsewhere.Finally, we return to consider the original pyrochloremodel. As indicated in Fig. 2, we observe many similar behaviors in DOS between THC and pyrochlore mod-els [19, 22]: (i) DOS consists of the flat bands and dis-persive bands at U = 0, (ii) U drives a quantum phasetransition to the charge ice insulator at half filling, (iii)the divergence of DOS transfers through QCP, and (iv)DOS shows a one-dimensional-like form for U ≫ t . Theseare direct consequences from the key features shared be-tween THC and pyrochlore, i.e., the corner-sharing net-work of tetrahedra and the resulting macroscopic ice-ruledegeneracy. Interestingly enough, (iii) suggests a possi-bility that the transition in the pyrochlore case is alsodescribed by an order parameter analogous to O Ψ , ac-companied by similar anomalous critical behavior. Wenote that, in the weak U region, THC is a band insu-lator, whereas the pyrochlore model is metallic at halffilling [Figs. 2(f)-(h)]; however, this difference is ratherirrelevant since our focus is on the strongly correlatedphysics related to the charge ice insulator. The bene-fit from obtaining the exact solution exceeds the minordissimilarity. Thus, our THC solution captures the com-mon essential physics of the itinerant ice-rule systems.Further comparisons, including the effect of global loopsneglected in THC, will be reported separately [22].In summary, we have exactly solved the extendedFalicov-Kimball model on the tetrahedron Husimi cac-tus in the ice-rule limit. The solution reveals a quantumcritical point and associated non-Fermi-liquid behavior inquantum melting of “charge ice” insulator. Furthermore,the results capture many essential features of more real-istic lattices with corner-sharing tetrahedra, such as thepyrochlore lattice. Our exact solution provides a canon-ical reference to the itinerant ice-rule physics, and willopen new avenues of research with wide applicability tocorrelation-induced phenomena under strong frustration.The authors thank I. Maruyama for fruitful dis-cussions. This work was supported by KAKENHI(Nos. 17071003, 19052008, 21740242, and 21340090), theGlobal COE Program “the Physical Sciences Frontier”,and by the Next Generation Super Computing Project,Nanoscience Program, MEXT, Japan. [1] J. D. Bernal and R. H. Fowler, J. Chem. Phys. , 515(1933).[2] L. Pauling, J. Am. Chem. Soc. , 2680 (1935).[3] P. W. Anderson, Phys. Rev. , 1008 (1956).[4] M. J. Harris et al. , Phys. Rev. Lett. , 2554 (1997).[5] A. P. Ramirez et al. , Nature , 333 (1999).[6] D. A. Huse et al. , Phys. Rev. Lett. , 167004 (2003).[7] For a recent review, S. T. Bramwell, M. J. Gingras, and P.C. W. Holdsworth: Chap. 7 in Frustrated Spin Systems ,ed. H. T. Diep (World Scientific, Singpore, 2005)[8] M. S. Chen et al. , J. Chem. 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