Molecular-orbital representation of generic flat-band models
eepl draft
Molecular-orbital representation of generic flat-band models
T. Mizoguchi and Y. Hatsugai
Department of Physics, University of Tsukuba - 1-1-1 Tennoudai, Tsukuba, Ibaraki 305-8571, Japan
PACS – Theories and models of many-electron systems
PACS – Electronic structure of disordered solids
Abstract – We develop a framework to describe a wide class of flat-band models, with and withouta translational symmetry, by using “molecular orbitals” introduced in the prior work (HATSUGAIY. and MARUYAMA I.,
EPL , , (2011) 20003). Using the molecular-orbital representation, weshed new light on the band-touching problem between flat and dispersive bands. We show thatthe band touching occurs as a result of collapse, or the linearly dependent nature, of molecularorbitals. Conversely, we can gap out the flat bands by modulating the molecular orbitals so thatthey do not collapse, which provides a simple prescription to construct models having a finiteenergy gap between flat bands and dispersive bands. Introduction.-
Flat-band models are a fertile groundfor exotic phenomena in condensed-matter physics. Forfermionic systems, it is a source of a ferromagnetism inthe presence of the interactions [1–7]. In the magneticsystems, the flat band of the magnetic mode in the low-energy indicates the massive degeneracy, resulting fromthe frustrated nature [8–10]. Recently, the combinationof flat bands and topology has also attracted considerableinterests [7, 11–15].So far, lots of efforts have been devoted to seek thesimple tight-binding Hamiltonians having flat bands, suchas a Lieb lattice and related models, [1, 4, 5, 16–18], linegraphs [2,3,7], and partial line graphs [19]. It was found inthose lattices that the quantum interference arising fromthe geometry of the lattices gives rise to localized eigen-states which correspond to the flat bands. For example,in a Kagome lattice, a simple form of the localized eigen-state is given by assigning the staggered weight on theelementary “loops” [20].The construction of localized eigenstates is elegant, butstrongly relies on the geometry of the lattice. This makesit puzzling to consider the models beyond the conven-tional models with nearest-neighbor (NN) hoppings, e.g.,in the presence of disorders and/or the farther-neighborhoppings. In this letter, we present an opposite view ofthe flat-band models, namely, we discuss how to describe dispersive bands in the flat-band models. Extending thenotion introduced in prior works [21, 22], we show thatthe dispersive bands in a wide class of flat-band modelsare spanned by the non-orthogonal basis functions consist of a small number of sites which are referred to as “molec-ular orbitals (MOs)”. Then, the flat bands can be viewedas a complement of that space of MOs, and they are en-forced to have zero energy if the Hamiltonian is writtenonly by MOs. In other words, the flat-band models, withgeneric hoppings including farther-neighbor hoppings, canbe written by MOs. Even in the presence of disorders,where the band picture is not available, the emergence ofmassively degenerate eigenstates is guaranteed as far asthe Hamiltonian is purely composed of MOs.To demonstrate the usefulness of the MO-representation, we revisit the band-touching problem,or, counting the degeneracy of massively degeneratedstates. In the literature, this problem is discussed interms of “topological” nature of the lattices, namely, thenumber of the degenerate modes is associated with thatof “loops” in the lattice [20]; recently, a novel perspectivewas introduced with respect to the singularity of theBloch wave function in the momentum space [23]. Inthe scheme of the MO-representation, the equivalentconclusion is obtained by counting the number of thelinearly independent MOs; more precisely, the number ofthe linearly independent MOs subtracted from the totalnumber of degrees of freedom gives the minimum of thedegeneracy.
Further, the merit of the MO-representationis that, one can easily find how to lift the degeneracybetween flat bands and dispersive bands. To be more spe-cific, we modulate the MOs so that all the MOs becomelinearly independent. We demonstrate this procedurethrough some concrete examples, after introducing thep-1 a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t . Mizoguchi and Y. Hatsugaigeneric formulation. Formulation.-
The construction of flat-band mod-els on the basis of MOs is conceptually similar to the“cell construction” introduced by Tasaki [4, 5]. MO-representation was applied to several models such as gen-eralized pyrochlore models [21] and the minimal model forsilicene [22]. Here we present a general formulation of theMO-representation applicable to both periodic and disor-dered systems. For simplicity, we consider the case wherethe flat band(s) has zero energy. We emphasize that theMO representation is applicable to the cases with multipleflat bands with different energies, too. In such cases, theHamiltonian subtracted by each flat-band energy can bewritten by MOs; see Ref. [22] as an example.Let us consider a lattice model with N sites for spinlessfermions: H = (cid:80) Mi,j =1 C † i h ij C j , (1)where C i is an annihilation operator of a MO i , ( i =1 , · · · , M ) as C i = ψ † i c , (2)with c = ( c , · · · , c N ) T , and ψ i = ( ψ ,i , · · · , ψ N,i ) T .It should be noted that the MOs are not necessarily or-thogonal to each other, i.e., an anti-commutation relation { C i , C † j } = δ i,j does not necessarily hold.Using MOs, the Hamiltonian of Eq. (1) can be writtenin an original fermion basis as H = c † H c ,H = (cid:80) Mi,j =1 ψ i h ij ψ † j = Ψ h Ψ † , (3)where h is an M × M matrix andΨ = ( ψ , · · · , ψ M ) , (4)is an N × M matrix.Now, using a simple formula for N × M and M × N matrices A NM and B MN (see footenote )det N ( I N + A NM B MN ) = det M ( I M + B MN A NM ) , (5)we obtaindet N ( λI N − H ) = det N ( λI N − Ψ h Ψ † )= λ N det N ( I N − λ − Ψ h Ψ † )= λ N det M ( I M − λ − h Ψ † Ψ)= λ N − M det M ( λI M − h Ψ † Ψ) . (6)Then if N > M , there are N − M ( >
0) zero-energy eigen-states. When applying the above argument to the Hamil-tonian matrix in the momentum space, one obtains flat It follows from det N ( I N + A NN B NN ) = det N ( I N + B NN A NN )for generic matrices A NN , B NN when A NN = ( A NM , O N,N − M )and B NN = (cid:18) B MN O N − M,N (cid:19) . bands. One may also apply it to the disordered systemsas well. Condition for additional zero modes.-
Additionalzero modes, or band touchings for translationally invariantsystems, appear whendet M h = 0 , (7)or Ψ i , ··· ,i M = 0 , ( ∀ i < · · · < i M ) (8)whereΨ i , ··· ,i M = det M Ψ i , · · · Ψ i ,M . . .Ψ i M , · · · Ψ i M ,M (9)is an M -th minor of Ψ. It follows from a relationdet M h Ψ † Ψ = det M h (cid:80) | Ψ i , ··· ,i M | . This is a conditionfor the dimension of the projected linear space spanned by M column vectors of Ψ h is less than M . It implies that theprojected nonzero energy bands are collapsed. Intuitively,the condition of Eq. (8) corresponds to the case where theMOs are not linearly independent of each other, thus theexistence of the additional zero modes can be predictedby counting the number of linearly independent MOs. Itshould be noted, however, that the counting of linearly in-dependent MOs cannot capture the number of additionalzero modes. More precisely, we can predict from the condi-tions of Eqs. (7) and (8) whether the reduction of the rankof h Ψ † Ψ occurs, but can not predict the rank itself; seethe case of α - T model for the non-trivial example, wherethe number of additional zero modes increases due to asymmetry. So, the counting of the linearly independentMOs tells us the minimum of the number of additionalzero modes. In the following, we elucidate, by using thespecific models, how to construct the MOs and provide asimple view of the origin of additional zero modes, as wellas a method of how to erase that touching.
Chiral symmetric case.-
Before looking at specificexamples, we present the generic argument on a typ-ical case, namely, chiral symmetric systems with sub-lattice imbalance. The Hamiltonian takes a form of H = (cid:18) O AA D AB D † AB O BB (cid:19) , where D AB is an N A × N B ma-trix ( N A > N B ). It satisfies { H, Γ } = 0 with Γ =diag (1 A , − B ). This H is actually rewritten by a mul-tiplet Ψ, that is a set of M = 2 N B MOs with its totaldimension N = N A + N B , asΨ = (cid:18) D AB O AB O BB B (cid:19) , h = (cid:18) O BB B B O BB (cid:19) By the general argument, it implies that there exist N − M = N A − N B zero modes. This is well known . One may check it directly asdet (cid:18) λ A − D AB − D † AB λ B (cid:19) = det (cid:18) λ A D AB O BA λ B − λ − D † AB D AB (cid:19) = λ N A − N B det B ( λ B − D † AB D AB ) . p-2olecular-orbital representation of generic flat-band models Fig. 1: (a) A bond-disordered Kagome model. The positions ofthe sublattices and primitive vectors are indicated in the figure.(b) The band structure for the translationally invariant case,with t (cid:52) = − . , t (cid:53) = − .
3. (c) The energy spectrum for thedisordered model. The red shade represents the zero-energystates.
Further, additional zero modes appear when det h Ψ † Ψ =( − ) N B det N B D † AB D AB = 0. Example 1:Kagome-lattice models.-
We first con-sider a tight-binding model on a Kagome lattice with L × L unit cells [Fig. 1(a)]. The Hamiltonian considered here iswritten as H = (cid:88) i,j ∈(cid:52) R t (cid:52) R c † i c j + (cid:88) i,j ∈(cid:53) R t (cid:53) R c † i c j , (10)where i and j denote sites on a Kagome lattice, (cid:52) R and (cid:53) R denote the upward and downward triangles on theunit cell R , respectively, and t (cid:52) R and t (cid:53) R are transfer inte-grals.The translationally-invariant case, i.e., t (cid:52) R = t (cid:52) , t (cid:53) R = t (cid:53) for all R , is known as a breathing Kagome model [24–27]. In this model, the band touching between flat anddispersive bands occurs [20, 21, 27] [Fig. 1(b)]. In the dis-ordered case, i.e. t (cid:52) R and t (cid:53) R are chosen randomly, we cannot adopt the band picture. Nevertheless, the analogousphenomenon of the band touching, namely, the additionaldegeneracy of the zero modes, occurs. It should be notedthat the present choice of the randomness is not likely torealize in solid-state systems, since we need a fine tuningbetween the on-site potential and the NN hopping. Nev-ertheless, the present model is of fundamental importancebecause it will bring a novel perspective of disordered flatbands [28–31]. Indeed, the present model is quite differ-ent from the conventional disordered models with randomon-site potentials, in that the degeneracy of zero modes isexactly retained as we will show below.In what follows, we pursue the origin of an additional Fig. 2: Kagome-lattice models with a gapped flat band. (a)A model in Ref. [28], and (b) its band structure for γ = 2. Theorange and purple sites have an on-site potential with γ + 1and 2, respectively. Blue and red triangles denote ˜ C (cid:52) and˜ C (cid:53) , respectively. (c) A model in Ref. [20], and (d) its bandstructure. A hexagonal plaquette colored in magenta denotes¯ C . zero mode by using the MO-representation. We define theMO on each triangle: C R , (cid:52) = c R , + c R , + c R , , (11) C R , (cid:53) = c R + a , + c R + a − a , + c R , , (12)then, the the Hamiltonian of Eq. (10) is written as H = (cid:80) R t (cid:52) R C † R , (cid:52) C R , (cid:52) + t (cid:53) R C † R , (cid:53) C R , (cid:53) . (13)Since the number of sites is 3 × L × L and the numberof MOs is 2 × L × L , the model has at least L × L zeromodes, from the aforementioned argument. In addition,the model has an additional zero mode. We confirm thisnumerically: In Fig. 1(c), we show the energy spectrumfor L = 10 with t (cid:52) R , t (cid:53) R ∈ [ − / , / ×
10 + 1) modes out of 300have zero-energy. The origin of the additional zero modecan be understood as follows. Since a set of all upwardtriangles covers all the sites on a Kagome lattice, and sodoes that of all downward triangles, we have (cid:88) R C R , (cid:52) = (cid:88) R C R , (cid:53) = (cid:88) R ,a =1 , , c R ,a . (14)This implies that one of the MOs is written by the linearcombination of the others, e.g., C R , (cid:52) = (cid:88) R C R , (cid:53) − (cid:88) R (cid:54) = R C R , (cid:52) . (15)Therefore, the dimension of the space spanned by the MOsis not 2 × L × L but 2 × L × L −
1, which gives rise to oneadditional zero mode.p-3. Mizoguchi and Y. HatsugaiThe above argument indicates that the flat band inthe models written by the MOs of (11) and (12) can notbe gapped out. However, in the previous works, severalmodels on a Kagome lattices in which the flat band doesnot touch the dispersive band are proposed [20, 28]. Thismeans that those models are written by the different MOs,and in the following we show the explicit forms of thoseMOs, one-by-one.First, let us consider the model in Ref. [28], shown inFig. 2(a). The MOs describing this model are obtained bymodifying (11) and (12), slightly,˜ C R , (cid:52) = γc R , + c R , + c R , , ˜ C R , (cid:53) = c R + a , + c R + a − a , + c R + a , , (16)with | γ | (cid:54) = 1. Note that ˜ C R , (cid:52) and ˜ C R , (cid:53) are linearly inde-pendent of each other, due to the following reason. Since˜ C ’s are defined in a translationally-invariant manner, onecan perform the Fourier transformation˜ C k , (cid:52) = ψ † k , (cid:52) , c k , ˜ C k , (cid:53) = ψ † k , (cid:53) c k , (17)with c k = ( c k , , c k , , c k , ) T , ψ † k , (cid:52) = ( γ, , ψ † k , (cid:53) = (cid:0) e i k · a , e i k · ( a − a ) , (cid:1) . Clearly, { ˜ C k , (cid:52) / (cid:53) , ˜ C † k (cid:48) , (cid:52) / (cid:53) } = 0 if k (cid:54) = k (cid:48) . Furthermore, ˜ C k , (cid:52) and ˜ C k , (cid:53) are linearly indepen-dent of each other, or, ψ † k , (cid:52) is not parallel to ψ † k , (cid:53) , because ψ † k , (cid:52) × ψ † k , (cid:53) = (1 − e i k · ( a − a ) , e i k · a − γ, γe i k · ( a − a ) − e i k · a ) (cid:54) = , for all k , if | γ | (cid:54) = 1.Now, consider the Hamiltonian H = (cid:80) R ˜ C † R , (cid:52) ˜ C R , (cid:52) +˜ C † R , (cid:53) ˜ C R , (cid:53) . Since all ˜ C ’s are linearly independent as men-tioned above, the band touching does not occur [Fig. 2(b)],as pointed out in Ref. [28]. The present method to gapout the flat band is simple in a sense that it does not re-quire the search for suitable real-space texture of the hop-pings to realize the localized eigenstate on large loops [28].Also, it is easy to construct a model with disorders, as H = (cid:80) R t (cid:52) R ˜ C † R , (cid:52) ˜ C R , (cid:52) + t (cid:53) R ˜ C † R , (cid:53) ˜ C R , (cid:53) .Next, let us consider the second model, shown inFig. 2(c), which contains not only the on-site and NNterms but also next NN and the third NN terms [20]. TheMOs are completely different from (11) and (12). Namely,MOs are defined on the hexagonal loops, rather than thetriangles, as¯ C R = c R , + c R , + c R + a , + c R + a , + c R + a , + c R + a , , (18)that are linearly independent of each other. This choice ofMOs is inferred from the analogous spin model [32]. Then,consider the Hamiltonian H = (cid:80) R ¯ C † R ¯ C R . This Hamilto-nian has two flat bands and one dispersive band, which isconsistent with the fact that the number of linearly inde-pendent MOs is L × L [Fig. 2(d)].As shown above, there are various choices of the MOseven on the same lattices, thus the presence/absence of Fig. 3: (a) A α - T model. The solid (dashed) bonds have thehopping integral 1 ( α ). The positions of the sublattices andprimitive vectors are indicated in the figure. (b) The bandstructures for α = 0 .
3. (c) A modulate α - T model. The bluebonds have the hopping β , instead of 1. (d) The band structurefor α = 0 . β = 0 . the band touching is crudely dependent not on the latticegeometry but on the choice of the MOs. Example 2: α - T model.- Let us move on to the sec-ond example, namely, α - T model [16, 17, 33] [Fig. 3(b)]: H = (cid:88) R (cid:88) n =1 c † R , c R + ξ n , + αc † R , c R + η n , + (h . c . ) , (19)where ξ = 0, ξ = a , and ξ = − a ; η = 0, η = − a ,and η = − a − a . The model is an example of the chiralsymmetric case with sublattice imbalance, and is knownto have a flat band for arbitrary α . The band structurefor α = 0 . (cid:48) points),the number of zero-energy mode is L × L + 4, if L is amultiple of three .For this model, we define two MOs to rewrite the Hamil-tonian, as d R ,a = ( c R , + c R − a , + c R + a , )+ α ( c R , + c R − a , + c R − a − a , ) , (20)and d R ,b = c R , . (21)Notice that d R ,a and d R (cid:48) ,b are orthogonal to each othersince they have the weight on different sublattices. By This condition is necessary in order that the K and K (cid:48) pointsare included in the set of discretized momenta. p-4olecular-orbital representation of generic flat-band modelsusing these MOs, the Hamiltonian of Eq. (19) can bewritten as H = (cid:80) R d † R ,a d R ,b + d † R ,b d R ,a = (cid:80) k d † k ,a d k ,b + d † k ,b d k ,a , (22)where d k ,a/b = (cid:80) R d R ,a/b e − i k · R .Now, let us examine the relation between the numberof linearly independent MOs and the degeneracy of zeromodes. At K point, where the triple band touching occurs,we have d k K ,a = (cid:88) R e − i k K · R [( c R , + c R − a , + c R + a , )+ α ( c R , + c R − a , + c R − a − a , )]= 0 . (23)Similarly, d k K (cid:48) ,a vanishes, too. Thus, the number of lin-early independent MOs is 2 L × L −
2. However, as men-tioned before, the number of zero modes is L × L +4 ratherthan L × L + 2, meaning that there exist two additionalzero modes. The origin of these is chiral symmetry. To bespecific, if we define an operator Γ such that Γ d a Γ = d a ,Γ d b Γ = − d b , the Hamiltonian satisfies Γ H Γ = −H . Con-sequently, d k K ,b and d k K (cid:48) ,b serve as additional zero-energymodes, although they are linearly independent of the otherMOs.How can we lift the additional degeneracy by tuning themodel? To do this, we modify d R ,a slightly, as we did forthe Kagome model:˜ d R ,a = ( βc R , + c R − a , + c R + a , )+ α ( c R , + c R − a , + c R − a − a , ) , (24)with | β | (cid:54) = 1. Since ˜ d R ,a are linearly independent ofeach other, the Hamiltonian given as H = (cid:80) R ˜ d † R ,a d R ,b + d † R ,b ˜ d R ,a , does not have band-touching points [Fig. 3(d)].The corresponding hoppings in the real space is shown inFig. 3(c). A similar method to lift the degeneracy is usedin Ref. [34] for a Lieb lattice. Summary and discussions.-
We have shown that theMOs can express the Hamiltonian of a wide class of flat-band models, and that the subspace spanned by MOs cor-responds to the eigenspace of dispersive modes. Therefore,by counting the dimension of this subspace, we can countthe number of flat modes as well, since the flat bands cor-respond to the co-space of MOs.To demonstrate its usefulness, we have studied the flatband models with and without translational symmetry,focusing on the band-touching problem. From the view-point of MOs, the additional degeneracy of flat bands anddispersive bands appears when the MOs are not linearlyindependent of each other. This indicates that, to gap outthe flat band, one needs a modulation of MOs such thatall the MOs become linearly independent of each other.This offers a simple way to construct models with gappedflat bands. It is worth noting that, although the mathematical for-mulation of the MO-representation is generic and in prin-ciple applicable to any flat-band models , it is not easyto search the MOs for given flat-band models systemati-cally. Nevertheless, it is instructive to see that the choiceof MOs for well-known flat-band models, namely, the NNhopping models on line graphs (e.g. a Kagome latticeand a pyrochlore lattice) and sublattice-imbalanced lat-tices (e.g. a Lieb lattice and an α - T -3 lattice), is inferredfrom their characteristic lattice structures. In the formercase, the natural choice is to define MOs on their duallattices, where the elemental unit of those lattices, e.g.,triangles for a Kagome lattice, are placed. In the lattercase, a typical structure is that a site of “poorer” sublat-tice is surrounded by sites of “richer” sublattice. Then,the MOs can be chosen as a poorer site itself [Eq. (21)for α - T model], and a linear combination of richer sitessurrounding a poorer site [Eq. (20)]. We emphasize thatthe MO representation provides a unified treatment forline graphs and sublattice-imbalanced lattices.It is also worth noting that the MO representation isuseful for identifying the flat-band models with farther-neighbor hoppings. Namely, once the MOs are obtainedfor the NN hoppings, we can examine whether or not thefarther-neighbor hoppings in the model can be written bythe same MOs. For instance, the NN model on a Kagomelattice is written by the “on-site” term of the MOs asin Eq. (13), thus one can easily implement the farther-neighbor hoppings retaining the flat band by introducing,e.g., “the NN hopping term” of the MOs. Such an imple-mentation of the farther-neighbor hoppings was discussedby one of the authors in the context of tuning of the flat-band energy [27]. We hope that the MO-representationsheds light on the physics of flat-band models in variouscontexts. ∗ ∗ ∗ We are grateful to M. Udagawa for fruitful discussionsand careful reading of the manuscript. This work ispartly supported by Grants-in-Aid for Scientic Research,KAKENHI, JP17H06138 and JP16K13845 (YH), MEXT,Japan.
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