Flat-band full localization and symmetry-protected topological phase on bilayer lattice systems
aa r X i v : . [ c ond - m a t . d i s - nn ] M a r Flat-band full localization and symmetry-protected topological phase on bilayerlattice systems
Ikuo Ichinose, Takahiro Orito, and Yoshihito Kuno Department of Applied Physics, Nagoya Institute of Technology, Nagoya, 466-8555, Japan Quantum Matter program, Graduate School of Advanced Science and Engineering,Hiroshima University, Higashi-Hiroshima 739-8530, Japan Department of Physics, Graduate School of Science,Tsukuba University, Tsukuba, Ibaraki 305-8571, Japan (Dated: March 3, 2021)In this work, we present bilayer flat-band Hamiltonians, in which all bulk states are localized andspecified by extensive local integrals of motion (LIOMs). The present systems are bilayer extensionof Creutz ladder, which is studied previously. In order to construct models, we employ buildingblocks, cube operators, which are linear combinations of fermions defined in each cube of the bilayerlattice. There are eight cubic operators, and the Hamiltonians are composed of the number operatorsof them, the LIOMs. A suitable arrangement of locations of the cube operators is needed to haveexact projective Hamiltonians. The projective Hamiltonians belong to a topological classificationclass, BDI class. With the open boundary condition, the constructed Hamiltonians have gaplessedge modes, which commute with each other as well as the Hamiltonian. This result comes froma symmetry analogous to the one-dimensional chiral symmetry of the BDI class. These resultsindicate that the projective Hamiltonians describe a kind of symmetry protected topological phasematter. Careful investigation of topological indexes, such as Berry phase, string operator, is given.We also show that by using the gapless edge modes, a generalized Sachdev-Ye-Kitaev (SYK) modelis constructed.
I. INTRODUCTION
Flat-band systems are one of the most attractive top-ics in condensed matter community. Such systems ex-hibit exotic localization phenomena without disorder,currently called disorder-free localization [1–5]. Espe-cially, the system, which is totally composed of flat-bandsand called complete flat-band system, generally possessesextensive number of local conserved quantities called lo-cal integrals of motion (LIOMs) [6–9]. Without inter-actions, complete flat-band systems are, therefore, inte-grable and their dynamics exhibits non-thermalized be-haviors [10–12]. The idea of LIOMs was firstly intro-duced in the study of many-body localization (MBL)[6–9]. There, emergent LIOMs induce localization, non-thermalized dynamics with slow increase of entanglemententropy [13]. On the other hand in the complete flat-bandsystems, the origin of the LIOMs is due to the presenceof the compact localized states (CLS) [14–17], hence theLIOMs are explicitly given in terms of the number oper-ator of the CLS. The origin of LIOMs in these systems isessentially different from that of the emergent LIOMs inthe MBL systems, but both of them play an importantrole concerning to localization.From another point of view, flat-band systems withnontrivial topological bands have attracted many inter-ests. On flat-bands, the kinetic terms are negligibleand interactions play a dominant role in determining thegroundstate. Then, such a system possibly exhibits ex-otic topological phases. Fractional Chern insulator is ex-pected to be realized in nearly flat-band systems [18, 19],and for complete flat-band systems, fractional topologicalphenomena have been reported [20–22]. Also, flat-band system is closely related to frustrated systems, in whichhuge degeneracies exist in the vicinity of the groundstate,and as a result, some kinds of topological phases possiblyemerge there [23–26].As a typical example of such a flat-band system, Creutzladder [27] with a fine-tuning is an interesting system,where the two complete flat-bands and the two types ofCLS appear. Due to the presence of the extensive num-ber of the CLS, the model exhibits explicitly disorder freelocalization phenomena, called Aharanov-Bohm caging[10–12]. It is known that localization tendency surviveseven in the presence of interactions [28–32]. Furthermore,Creutz ladder shows some topological phases with quan-tized Berry phase and zero energy edge states [27, 33–37],and interestingly fractional topological phenomena [22].However, except for Creutz ladder, complete flat-bandmodels with nontrivial topological properties have notbeen studied in great detail so far. Hence, exploring sucha model that goes beyond the above example remains anopen issue.In the present paper, by extending the character of theCLS in Creutz ladder, we propose novel types of flat-bandsystems on bilayer lattice, which can be set in both oneand two-dimensional (2D) lattice geometries. There, theextensive LIOMs are explicitly obtained and all statesare localized in the periodic boundary condition. As atopological aspect, the system Hamiltonians have sym-metries of the BDI class in ten-fold way [38–41]. WithBDI symmetry, the system Hamiltonian set on quasi-1Dlattice explicitly exhibits symmetry protected topological(SPT) phase [42–44]. Also, we find that, in both 1D and2D systems with open boundary conditions, there emergegapless edge modes as a result of chiral symmetry. Thegapless edge modes can be analytically given due to thepresence of the CLS. Therefore, the presence of the edgemodes implies that the present systems can be regardedas a 2D SPT system whose bulk states are full localized.Such kind of models in 1D is studied, e.g., in Ref. [45].This paper is organized as follows. In Sec. II, we pre-pare building blocks called cube operators that are usedfor the construction of the model Hamiltonians. Thereare eight kinds of cube operators, which are linear com-bination of fermions located on eight sites of a cube. Thecube operators located on the same cube commute witheach other, but certain pairs of them located on next-nearest-neighbor (NNN) cubes do not. The cube opera-tors transform with each other by chiral-symmetry trans-formation, and we require the invariance of the Hamilto-nian under chiral-symmetry transformation. In Sec. III,we construct models and their Hamiltonian by using thecube operators. Certain specific location of eight kinds ofthe cube operators is required to obtain exact projectiveHamiltonian. We present two kinds of such Hamiltonian,one of which is defined on a bilayer lattice and the otheron a square prism lattice. Interactions between fermionscan be introduced, which are expressed by the LIOMsand satisfy chiral symmetry. Section IV is devoted forstudy on the edge modes in the above two models. Asa result of chiral symmetry of the bulk Hamiltonian, theedge modes are invariant under the transformation cor-responding to chiral symmetry. Furthermore, effectiveHamiltonian of the edge modes is derived, which is an ex-tension of the SYK model [46–48]. Section V is devotedfor discussion on topological indexes, which characterizenon-trivial topological properties of the emergent eigen-states. Numerical study of the square prism model isgiven to examine the stability of the topological states.Section VI is devoted for conclusion and discussion. Weexplain that flat-band localization by the CLS plays animportant role for topological properties of the systems.
II. CONSTRUCTION OF BILAYER MODELS
In the previous works [30–32], we studied fermionsystems on Creutz ladder, and obtained interesting re-sults concerning to flat-band localization and topologi-cal phase. In this section, we construct fermion systemson the bilayer lattice that exhibits the full-localizationand topological properties. These systems have projec-tive Hamiltonian with time-reversal ( T ), particle-hole ( C )and chiral symmetries ( S = T C ). As a result, they havegapless edge modes under the open boundary condition(OBC), whereas the bulk states are full localized andhave flat-band dispersion. To construct models, we pre-pare eight building blocks with the cubic shape, each edgeof which corresponds to a linear combination of fermionsat two sites of the edge, which is an extension of theCLS in Creutz ladder [27, 33, 34, 36]. Using these build-ing blocks called the cube operators, we can constructbilayer models with various shapes, including a torus, a
FIG. 1. Schematic picture of fermion operators on bilayerlattice. c ~r ’s reside on upper square lattice, and d ~r ’s on lowerlattice, where ~r = ( x, y ). thin cylinder, and a square prism. In the context of thestudy of MBL, each cube operator is also regarded as a ℓ -bit (equivalent to the CLS) [14], and the target Hamil-tonians are obtained by using LIOMs, which are nothingbut the number operator of the ℓ -bits (CLS). A. Eight cube operators
Let us consider a cube, which is a unit cell of the bi-layer lattice, i.e., each vertex of the cube is located ata lattice site, see Fig. 1. We introduce ( x - y - z ) axes,and fermion creation (annihilation) operators c † ~r ( c ~r ) and d † ~r ( d ~r ), where ~r denotes lattice sites, ~r = ( x, y, z ), and z = 1 (2) for c ~r ( d ~r ). Therefore, the fermion c ~r and d ~r are located in the upper and lower layer, respectively,and then we shall use ~r = ( x, y ) hereafter.The eight cube operators are constructed by c ~r and d ~r .To this end, as elementary building blocks, the followingnotations are useful; ω A~r, ˆ i = c ~r +ˆ i + ic ~r , ω B~r, ˆ i = c ~r +ˆ i − ic ~r , (upper layer) , ˜ ω A~r, ˆ i = d ~r +ˆ i + id ~r , ˜ ω B~r, ˆ i = d ~r +ˆ i − id ~r , (lower layer) , ¯ ω A~r, ˆ z = c ~r + id ~r , ¯ ω B~r, ˆ z = c ~r − id ~r , (inter layer) , (1)where ˆ i = ˆ x, ˆ y . It is easily verified, { ω A † ~r, ˆ i , ω B~r, ˆ i } = 0 , etc. By using the above notation in Eq. (1), the eightcube operators are schematically displayed in Fig 2. Itshould be remarked that it is not obvious if configura-tions of ω A , ˜ ω A , etc. shown in Fig. 2 can be constructedconsistently. We give explicit forms of the cube operators FIG. 2. Schematic picture of eight cube operators on bilayer lattice, Q + ~r ∼ ˜ P − ~r . Eight cube operators are linear combinationsof { ω A~r, ˆ i , ˜ ω A~r, ˆ i , ¯ ω A~r, ˆ i } [blue] and { ω B~r, ˆ i , ˜ ω B~r, ˆ i , ¯ ω B~r, ˆ i } [red]. The numbers near vertices of the cube indicates the value of the coefficientof the cube operators. corresponding to those in Fig. 2; Q + ~r = √ [ − d ~r +ˆ x +ˆ y + id ~r +ˆ x + id ~r +ˆ y − d ~r − ic ~r +ˆ x +ˆ y + c ~r +ˆ x + c ~r +ˆ y − ic ~r ] ,Q − ~r = √ [ − d ~r +ˆ x +ˆ y + id ~r +ˆ x − id ~r +ˆ y + d ~r − ic ~r +ˆ x +ˆ y + c ~r +ˆ x − c ~r +ˆ y + ic ~r ] , ˜ Q + ~r = √ [ d ~r +ˆ x +ˆ y − id ~r +ˆ x − id ~r +ˆ y + d ~r − ic ~r +ˆ x +ˆ y + c ~r +ˆ x + c ~r +ˆ y − ic ~r ] , ˜ Q − ~r = √ [ d ~r +ˆ x +ˆ y − id ~r +ˆ x + id ~r +ˆ y − d ~r (2) − ic ~r +ˆ x +ˆ y + c ~r +ˆ x − c ~r +ˆ y + ic ~r ] ,P + ~r = √ [ − d ~r +ˆ x +ˆ y − id ~r +ˆ x + id ~r +ˆ y + d ~r − ic ~r +ˆ x +ˆ y − c ~r +ˆ x + c ~r +ˆ y + ic ~r ] ,P − ~r = √ [ − d ~r +ˆ x +ˆ y − id ~r +ˆ x − id ~r +ˆ y − d ~r − ic ~r +ˆ x +ˆ y − c ~r +ˆ x − c ~r +ˆ y − ic ~r ] , ˜ P + ~r = √ [ d ~r +ˆ x +ˆ y + id ~r +ˆ x − id ~r +ˆ y − d ~r − ic ~r +ˆ x +ˆ y − c ~r +ˆ x + c ~r +ˆ y + ic ~r ] , ˜ P − ~r = √ [ d ~r +ˆ x +ˆ y + id ~r +ˆ x + id ~r +ˆ y + d ~r − ic ~r +ˆ x +ˆ y − c ~r +ˆ x − c ~r +ˆ y − ic ~r ] . By the straightforward calculation, it is verified that alleight operators in Eq. (2) and their hermitian conjugates located at the same cube anti-commute with each otherexcept for the commutators such as, { Q + † ~r , Q + ~r } = 1, etc.However, some of them located at adjacent cubes do notcommute with each other such as { Q + † ~r , Q − ~r +ˆ x +ˆ y } = 1 / B. Time reversal symmetry of eight cube operators
Before going to the model construction, we introducea time-reversal symmetry ( T ) for the second quantizedoperators [39] as follows, which plays an important rolein later discussions; T i T − = − i, T c ~r T − = d ~r , T d ~r T − = c ~r . (3)From Eq. (3), the transformation of cube operators isinduced. It is easily verified, T Q + ~r T − = − iQ + ~r , etc.LIOMs are given as the number operators of the above ℓ -bits, Q + ~r ∼ ˜ P − ~r , K + ~r = Q + † ~r Q + ~r , K − ~r = Q −† ~r Q + ~r , ˜ K + ~r = ˜ Q + † ~r ˜ Q + ~r , ˜ K − ~r = ˜ Q −† ~r ˜ Q + ~r , (4) M + ~r = P + † ~r P + ~r , M − ~r = P −† ~r P + ~r , ˜ M + ~r = ˜ P + † ~r ˜ P + ~r , ˜ M − ~r = ˜ P −† ~r ˜ P + ~r . (5)All the LIOMs in Eq. (5) are invariant under the time-reversal transformation T in Eq. (3). The Hamiltoni-ans are to be constructed via the above LIOMs. We re-quire the target systems to have topological properties.In other words, on the construction of the Hamiltonian,we require to assign the system to some topological classin ten-fold way [40]. To this end, we impose chiral sym-metry on the Hamiltonians, which is discussed in thefollowing subsection. C. Chiral symmetries of eight cube operators
The next step is to assign locations of these cubes todefine Hamiltonian with non-trivial topology. It is pos-sible to construct various models for it by means of theeight cube operators. In the following, we show some ofthem. We shall impose chiral symmetry S to the targetmodels. With a transformation using a unitary operator U , chiral symmetry requires that the (second quantized)Hamiltonian H transforms as [39],( U K ) H ( U K ) − = U H ∗ U − = H, (6)where K is the complex conjugation, O ∗ denotes the com-plex conjugate of the operator O . The chiral operator isgiven by S = U K , which is anti-unitary operator [39].The operator S plays an important role in the construc-tion of topological models.Here, as a candidate of the unitary operator U in S , wecan introduce two unitary operators S x and S y . Under S x for ~r = ( x = even , y ), c ~r → ic † ~r +ˆ x , c ~r +ˆ x → − ic † ~r +2ˆ x ,c ~r +ˆ y → ic † ~r +ˆ x +ˆ y , c ~r +ˆ x +ˆ y → − ic † ~r +2ˆ x +ˆ y ,d ~r → id † ~r +ˆ x , d ~r +ˆ x → − id † ~r +2ˆ x ,d ~r +ˆ y → id † ~r +ˆ x +ˆ y , d ~r +ˆ x +ˆ y → − id † ~r +2ˆ x +ˆ y . (7)In order to construct symmetric Hamiltonians under S x K , the following properties are useful, S x ( Q + ~r ) ∗ S − x = − i ( Q − ~r +ˆ x ) † , S x ( Q − ~r ) ∗ S − x = − i ( Q + ~r +ˆ x ) † , S x ( ˜ Q + ~r ) ∗ S − x = − i ( ˜ Q − ~r +ˆ x ) † , S x ( ˜ Q − ~r ) ∗ S − x = − i ( ˜ Q + ~r +ˆ x ) † , (8)and similarly for { P ~r } ’s. Therefore, S x is a kind of gen-eralized translation operator in the x -direction.Transformation S y can be defined similarly for ~r =( x, y = even), c ~r → − ic † ~r +ˆ y , c ~r +ˆ x → − ic † ~r +ˆ x +ˆ y ,c ~r +ˆ y → − ic † ~r +2ˆ y , c ~r +ˆ x +ˆ y → − ic † ~r +ˆ x +2ˆ y ,d ~r → id † ~r +ˆ y , d ~r +ˆ x → id † ~r +ˆ x +ˆ y ,d ~r +ˆ y → id † ~r +2ˆ y , d ~r +ˆ x +ˆ y → id † ~r +ˆ x +2ˆ y , (9)and under S y , S y ( Q + ~r ) ∗ S − y = − i ( ˜ Q + ~r +ˆ y ) † , etc.As we show in the following sections, we employ S x asa guiding principle for constructing model Hamiltonians.[Chiral symmetry S y is less effective for the construction.Imposing both of them is incompatible for bilayer models.See later discussion.] FIG. 3. Schematic picture of Hamiltonian of bilayer systemgiven by Eq. (11). Unit cell is 2 ×
2, which contains four cubes.
III. MODELS AND THEIR HAMILTONIANA. Models on two-dimensional bilayer lattice
By using transformation properties of { Q } ′ s and { P } ′ s obtained in the previous section [Eq. (8)], we can con-struct various models. As we are interested in the full-localized system with a topological phase, the symme-try S x or S y can be a guiding principle for constructingHamiltonian, which is composed of the LIOMs, i.e., K ± ~r ,etc in Eqs. (4) and (5). In order to construct models, thefollowing facts have to be taken into account;(A). Under the transformation in Eqs. (7), the LIOMstransform suc as K + ~r → Q − ~r +ˆ x ( Q − ~r +ˆ x ) † = − ( Q − ~r +ˆ x ) † Q − ~r +ˆ x + 1 , = − K − ~r +ˆ x + 1 , etc. (10)(B). Commutativity of the LIOMs, { K ~r } and { P ~r } atthe same location does not guarantee that they allcommute with each other. For example, K + ~r doesnot commute with K − ~r +ˆ x +ˆ y .From (A), signs of the LIOMs in the Hamiltonian haveto be determined suitably to cancel the additional con-stant in Eq. (10). From (B), { K ~r } and { P ~r } should becall quasi-LIOMs , although the Hamiltonian is to be con-structed to commute with all of them. There exits cer-tain “selection rule” such that, ( Q + ~r ) † ( Q −† ~r +ˆ x +ˆ y ) | i cannot be an eigenstate of the Hamiltonian. Therefore, suitableassignment of spatial location of the LIOMs is required.There are still various models that satisfy the above re-quirements coming from (A) and (B). One of the genericones defined on the full bilayer lattice is the followingsystem with Hamiltonian such as, H BL = X m,n X ~r =(2 m, n ) (cid:16) τ K [ K + ~r − K − ~r +ˆ x + ˜ K + ~r +ˆ y − ˜ K − ~r +ˆ x +ˆ y ]+ τ M [ M + ~r − M − ~r +ˆ x + ˜ M + ~r +ˆ y − ˜ M − ~r +ˆ x +ˆ y ] (cid:17) , (11)where τ K and τ M are arbitrary real parameters, and( n, m ) are integers. The spatial structure of H BL isschematically shown in Fig. 3. The parameters τ K and τ M can be site dependent as long as they satisfy the sym-metry S x K , such as ( τ K , τ M ) → ( τ K,n , τ
M,n ), but we con-sider the uniform case in this work. Expression of H BL interms of the original fermions, c ~r and d ~r , is obtained bysubstituting Eqs. (2), (4) and (5) into H BL in Eq. (11).With the periodic boundary condition, the number of theLIOMs in H BL is extensive, and all energy eigenstates arelocalized and given as ( Q +2 m, n ) † | i , etc.Here, let us consider the symmetries of the Hamilto-nian H BL . Since the model construction is carried outwith respect to the chiral symmetry S x K , the secondquantized Hamiltonian H BL is chiral symmetric. Also,we easily notice that the Hamiltonian H BL has the time-reversal symmetry T , defined in the previous section.Furthermore, in the usual sense in the topological classi-fication [39], the chiral transformation S x K is to be givenby the product of the time-reversal transformation T anda particle-hole transformation C . Hence, we also can di-rectly notice the presence of the particle-hole transfor-mation C given as T − S x K . Therefore, the Hamiltonian H BL has T , C and S x K symmetries. This fact impliesthat the Hamiltonian H BL belongs to BDI class in theten-fold way [40]. As far as the topological classification[38], this fact indicates that the model H BL can exhibit atopological phase and some gapless edge states for a cer-tain lattice geometry. In later sections, we shall discusssuch topological aspects, some of which are substantiallyrelated to the full-localization properties.Furthermore for the system of H BL , nontrivial interac-tions can be introduced, which preserve the localizationproperties and chiral symmetry S x K . One of them isgiven by, H BLI = g X m,n X ~r =(2 m, n ) (cid:20) K + ~r K − ~r +ˆ x + ˜ K + ~r +ˆ y ˜ K − ~r +ˆ x +ˆ y − K + ~r ˜ K + ~r +ˆ y − K − ~r +ˆ x ˜ K − ~r +ˆ x +ˆ y + ( K → M ) (cid:21) , (12)where g is the coupling constant. The interactions givenby H BLI mostly describe scattering processes of c ~r and d ~r . Another type of interactions can be introduced bythe terms such as, H BLII = g ′ X m,n X ~r =(2 m, n ) (cid:20) ( K + ~r −
12 )( K − ~r +ˆ x −
12 )+( ˜ K + ~r +ˆ y −
12 )( ˜ K − ~r +ˆ x +ˆ y −
12 ) (13)+( K + ~r −
12 )( ˜ K + ~r +ˆ y −
12 )+( K − ~r +ˆ x −
12 )( ˜ K − ~r +ˆ x +ˆ y −
12 ) + ( K → M ) (cid:21) , which is again invariant under S x K .If one discards chiral symmetries but preserves the in-tegrability of models, the interactions with the followingform are possible, i.e., H III = g ′′ X m,n X ~r =(2 m, n ) K + ~r P + † ~r P − ~r +ˆ x + · · · , (14) FIG. 4. Schematic picture of Hamiltonian of square primegiven by Eq. (15). Each cube contains four cube operators, Q + ~r ∼ ˜ Q − ~r . which are composed of K ’s and P ’s. In this case, K ’s areconserved quantities and can be fixed to certain finitevalues. In the specific sector, the model reduces to a freesystem without genuine interaction terms. B. Models on square prism lattice and topologicalproperties
We have shown that the model of H BL belongs to theBDI class. Hence, if one reduces the model to a certainone-dimensional system without changing the symmetryclass, the resultant (quasi-)one-dimensional model has apossibility to exhibit a topological phase characterized bysome bulk topological invariant, e.g., winding number,Berry or Zak phase [49], etc. From this point of view, weconstruct another interesting and also instructive modeldefined on a square prism lattice, whose Hamiltonian isgiven as follows (see Fig. 4), H SP = X ~r =( n, (cid:20) τ [ K + ~r − K − ~r ] + τ [ ˜ K + ~r − ˜ K − ~r ] (cid:21) , (15)where τ and τ are arbitrary real parameters. The model H SP is invariant under the transformation in Eqs. (7) for { c ~r } ’s with the identification c ~r +2ˆ y = c ~r . The above fourLIOMs per unit cube are extensive, and localized single-particle eigenstates are ( Q + ~r ) † | i , · · · , ( ˜ Q − ~r ) † | i . Nontriv-ial interactions can be introduced as in the previous workfor the Creutz ladder [31].In later discussion, we shall study the model in Eq. (15)by numerical methods. To this end, we express theHamiltonian, H SP , in terms of the original fermions. Af-ter some calculation, we obtain H SP = H CL + H IS1 + H IS2 , (16) H CL = X ~r =( n, h i ¯ τ ( c † ~r +ˆ x +ˆ y c ~r +ˆ y − c † ~r +ˆ x c ~r )+¯ τ ( c † ~r c ~r +ˆ x +ˆ y + c † ~r +ˆ y c ~r +ˆ x ) − i ¯ τ ( d † ~r +ˆ x +ˆ y d ~r +ˆ y − d † ~r +ˆ x d ~r ) (17)+¯ τ ( d † ~r d ~r +ˆ x +ˆ y + d † ~r +ˆ y d ~r +ˆ x ) i + h.c. , − x − E ππππ − E FIG. 5. (a) Energy eigenvalues of H SP for τ = 2 and τ = 1for the periodic boundary condition. There exist four flat-bands. The numbers near each band show γ M , the Berryphase. All flat-bands are topologically nontrivial. (b) Energyeigenvalues under the open boundary condition. Four-foldzero-energy edge states appear. H IS1 = X ~r =( n, ∆( c † ~r d ~r +ˆ x + c † ~r +ˆ x d ~r + c † ~r +ˆ y d ~r +ˆ x +ˆ y + c † ~r +ˆ x +ˆ y d ~r +ˆ y ) + h.c. , (18) H IS2 = X ~r =( n, i ∆( d † ~r +ˆ x c ~r +ˆ y + c † ~r +ˆ x +ˆ y d ~r + d † ~r +ˆ y c ~r +ˆ x + c † ~r d ~r +ˆ x +ˆ y ) + h.c. , (19)where ¯ τ = ( τ + τ ) and ∆ = ( τ − τ ). H CL is nothingbut the Creutz ladder Hamiltonian of c ~r and d ~r , and H IS1 and H IS2 mix them and vanish for τ = τ .Let us investigate the topological properties of the non-interacting system of H SP . It is useful to express theHamiltonian Eq. (16) in terms of operators in the mo-mentum space for the x -direction;Φ( k x ) ≡ (cid:16) ˜ c ( k x , y = 0) , ˜ c ( k x , y = 1) , ˜ d ( k x , y = 0) , ˜ d ( k x , y = 1) (cid:17) , where ˜ c ( k x )’s are Fourier-transformed operators and H SP = R dk x Φ † ( k x ) h SP ( k x )Φ( k x ). Explicitly, h SP ( k x ) isgiven as, h SP ( k x ) = τ s ( k ) 2¯ τ c ( k ) 2∆ c ( k ) 2∆ s ( k )2¯ τ c ( k ) − τ s ( k ) − s ( k ) 2∆ c ( k )2∆ c ( k ) − s ( k ) − τ s ( k ) 2¯ τ c ( k )2∆ s ( k ) 2∆ c ( k ) 2¯ τ c ( k ) 2¯ τ s ( k ) , (20)where s ( k ) ≡ sin( k x ) and c ( k ) ≡ cos( k x ).From Eq. (20), the symmetries of h SP ( k x ) is clear [be-sides S x in Eq. (7)]. First, h SP ( k x ) has the time-reversalsymmetry mentioned in Sec. II.B, T b h SP ( k x ) T − b = h SP ( − k x ) ,T b = K (cid:18) (cid:19) , (21)where K is complex conjugate operator and is 2 × T b is anti-unitary. Second, h SP ( k x ) has a particle-hole symmetry, C b h SP ( k x ) C − b = − h SP ( − k x ) ,C b = (cid:18) − iσ y iσ y (cid:19) , (22)where σ y is the y -component of Pauli matrix. Hence, C b is unitary.Third, h SP ( k x ) has a chiral symmetry given by thechiral operator S b = T b C b , S b h SP ( k x ) S − b = − h SP ( k x ) ,S b = K (cid:18) σ y − σ y (cid:19) , (23)where S b is anti-unitary. From these symmetries, h SP ( k x )belongs to the BDI class in ten-fold way [38, 40].Furthermore, the system H SP also has a spatial reflec-tion (inversion) symmetry, which is given by Ih SP ( k x ) I − = h SP ( − k x ) ,I = (cid:18) σ x σ x (cid:19) , (24)where σ x is the x -component Pauli matrix. This reflec-tion symmetry plays an important role for the quanti-zation of Berry (Zak) phase [50, 51], which acts as atopological index in this system.We numerically demonstrate the topological proper-ties of h SP ( k x ). By diagonalizing h SP ( k x ), we can obtainenergy eigenvalues as shown in Fig. 5 (a). Certainly,there appear four flat-bands Here, we calculate Berryphase [49] given by γ M = i R π − π h u ℓ ( k x ) | ∂ k x | u ℓ ( k x ) i dk x ,where | u ℓ ( k x ) i is ℓ -th eigenstate of h SP ( k x ). For each flat-bands, γ M takes π , that is, each bands are non-trivial.This quantization comes from the inversion symmetry I (crystalline topological insulator [52]). We shall show theanother Berry phase obtained by introducing boundarytwist, in later section.Here, we also show the energy spectrum by diagonal-izing H SP under the OBC . The result indicates the ex-istence of the gapless edge modes, which are discussed inSec. IV.We introduce the following “inter-leg” hopping, whichrespects the symmetries of the Hamiltonian H SP . inEq. (7), H v = v X ~r =( n, ( c † ~r +ˆ y c ~r + d † ~r +ˆ y d ~r + h.c.) , (25)where v is an arbitrary real parameter. It should benoted that Eq. (25) does not break the chiral and reflec-tion symmetries. In the previous works on the Creutzladder, we showed that the hopping H v makes all statesextended even for infinitesimal v . However, needless tosay, the symmetries for topological properties are pre-served, thus bulk-band topology does not change with-out gap closing. Also, we expect that the correspondinggapless edge modes are preserved even for finite v .It is interesting and also important to examine if thereexist interactions that respect the S x -symmetry [Eq. (7)].To search them, Eq. (10) is quite useful. There are severalforms of the interactions, and we display typical one; H SPI = λ X ~r h ( K + ~r −
12 )( K + ~r +ˆ x −
12 )+( K − ~r −
12 )( K − ~r +ˆ x −
12 )+( ˜ K + ~r −
12 )( ˜ K + ~r +ˆ x −
12 )+( ˜ K − ~r −
12 )( ˜ K − ~r +ˆ x −
12 ) i , (26)where λ is coupling constant. Here, it should be notedthat the above interaction H SPI [Eq. (26)] is invariantunder transformations Eqs. (21) ∼ (24) as well as S x K .Therefore, we expect that the interaction H SPI plays animportant role for the emergence of gapless edge modesas verified later on. There, we shall also explain that aphase transition takes place as the parameter λ is varied.Discussion on topological properties of the system will begiven as well.In later numerical study on the Hamiltonian H SP , westudy effects of the following interactions, which are in-variant under the transformations Eqs. (21) ∼ (24), H II = V X ~r (cid:20) n c~r n c~r +ˆ y + n c~r n c~r +ˆ x + n c~r +ˆ y n c~r +ˆ x +ˆ y +( c → d ) (cid:21) , (27)where V is an arbitrary real parameter, and n c~r = c † ~r c ~r , n d~r = d † ~r d ~r . The interactions H II in Eq. (27) seem tobreak the S x -symmetry in Eq. (7). In fact under Eq. (7),there appear terms such as P ~r [ n c~r + n c~r +ˆ x + n c~r +ˆ y + n c~r +ˆ x +ˆ y + ( c → d )], in addition to a constant. How-ever as we always consider the system with fixed particlenumber, these terms are irrelevant. Furthermore, theadditional constant in the Hamiltonian does not changewave functions of energy eigenstates, it is also irrelevant.Therefore, we expect the stability of topological prop-erties of the Hamiltonian H SP in the presence of H II ,as long as it does not change the band structure. Thisexpectation will be verified by the numerical calculationlater on. IV. EDGE MODES
In the previous section, we have introduced models H BL and H SP of full localization in the bulk and be-long to BDI class. Then, it is expected that there appeargapless edge modes in the above models in the OBC.Depending on the geometrical structure of the systems,gapless edge modes emerge in a different way. Then, weshall discuss the models H BL and H SP , separately. Edge mode x FIG. 6. (a) Coarse grained schematic picture of bilayersystem with open boundary condition of thin cylinder (disk)shape. (b) There emerge four edge modes, each of whichis one-dimensional and a linear combination of ( ω A ) † ’s or ( ω B ) † ’s. Let us first consider the model H BL in a thin cylinderlattice whose schematic picture is displayed in Fig. 6. Inthe x -direction, the system is periodic, whereas in the y -direction, the boundaries exist. One may expect thatgapless edge modes appear in the boundary surfaces, butthis is not the case. They exist in the four edges of thecylinder [see Fig. 6 (a)].From Fig. 3, these edges of the cylinder are com-posed of edges of a sequence of { K } ’s such as( · · · K + K − K + K − · · · ) or ( · · · ˜ K + ˜ K − ˜ K + ˜ K − · · · ) [andalso sequences of { M } ’s]. Then from Fig. 2, the terms inthe Hamiltonian corresponding to the boundaries of theupper plane is given by h B = τ K ( M ) ′ X ~r ∈C [( ω A † ~r, ˆ x ω A~r, ˆ x ) − ( ω B † ~r +ˆ x, ˆ x ω B~r +ˆ x, ˆ x )] + · · · , where C represents one of the two closed edges in the up-per plane, and P ′ denotes the restricted summation suchas, ~r = ( x = even , y = location of edge). The bound-ary Hamiltonian h B does not contain ω B~r, ˆ x and ω A~r +ˆ x, ˆ x .Therefore, the zero modes are created by the operatorssatisfying the following equation (see Fig. 6 (b)), B C = ′ X ~r ∈C α ~r ω B † ~r, ˆ x = ′ X ~r ∈C α ~r +ˆ x ω A † ~r +ˆ x, ˆ x , (28)where { α ~r } ’s are suitably chosen phase factors as B C commutes with the boundary Hamiltonian h B . [Thecommutativity between B C and the other parts of theHamiltonian H SP is obvious.] It should be remarkedthat this commutativity is preserved even in the pres-ence of the interactions H BLI and H BLII in Eqs. (12) and(13). The above operator on C , B C , is a linear combi-nation of c † ~r with suitably chosen coefficients. For exam-ple for a square edge with four sites (1 , · · · , B †C = c † + ic † + c † + ic † = ( c † + ic † ) + ( c † + ic † ) = i ( c † − ic † ) + i ( c † − ic † ) [See Eq. (28)]. It is obvious thatin general, B C can be constructed consistently with thecondition Eq. (28) because C is composed of an even num-ber of sites. It is also verified that B C transforms suitablyunder chiral symmetry -transformation ( S x K in Eq. (7)); S x ( B † ) ∗C S − x = S x ( c † − ic † + c † − ic † ) S − x = − ic + c − ic + c = ( c † + ic † + c † + ic † ) † = B C . (29)Origin of this one-dimensional gapless edge modes isclosely related to flat-band localization, and will be dis-cussed in detail in Sec. VI, after examining the model of H SP .A few comments are in order. There are four boundarymodes, B a C ( a = 1 , , , a denotes thefour edges of the cylinder. In the previous works [45], itwas argued that gapless edge modes are stable even atfinite temperature if all the other bulk states are local-ized. The present bilayer model has such properties eventhough the gapless edge modes are linear combination ofthe original particles.As discussed in Ref. [53] for one-dimensional models,SYK-type models can be constructed via B C ’s. There,fourth-order terms of them are leading because of chiralsymmetry given in Eq. (29). In the original work of theSYK model, this symmetry was imposed by hand, but itemerges naturally in the present system. Possible effec-tive Hamiltonian is such that H = P V abcd B a † B b † B c B d ,where the edge index ( a ∼ d ) = (1 ∼
4) and the coeffi-cients V abcd are complex random numbers. The classifi-cation of the above model for ordinary complex particles has been already done in Ref. [53].Next, let us consider the model H SP in Eq. (15)with the OBC such as 0 ≤ x ≤ L − w Ax =0 , ˆ y | i , ˜ w Bx =0 , ˆ y | i , w Bx = L, ˆ y | i and ˜ w Ax = L, ˆ y | i [See Fig. 5(b).]. Even in the presence of the interactions H SPI inEq. (26), the four operators ( w Ax =0 , ˆ y , · · · , ˜ w Ax = L, ˆ y ) com-mute exactly with the Hamiltonian H SP + H SPI . Thisindicates that the gapless edge modes survive in the in-teracting system.We also study effects of the interactions H II [Eq. (27)]by numerical methods [54]. In particular as the aboveedge-mode creation operators do not commute with H II ,we are interested in stability of the edge modes. In Fig. 7,we display the density profiles for various values of V for the half-filled + two particles. The additional twoparticles on top of the half-filled state are expected tocorrespond to the gapless edge modes. Numerical cal-culations obviously show the stability of the edge modeseven for large V . We think that this result comes fromthe fact that H II preserves symmetries, as we discussedin the above, and then it enhances homogeneity of thebulk regime. r N un i t r V = 0V = 1V = 2V = 3V = 4V = 5V = 6
FIG. 7. Density profile of half-filled + two particle for variousinteraction strength V . We plot the total density of the unitcell, N unitr = n cr + y + n cr + n dr + y + n dr . Edge modes are quitestable against to the repulsion. The system size is L = 5 (20sites) with 12 particles. In the following subsection, we shall investigate topo-logical indexes corresponding to the model H SP + H SPI in Eqs. (15) and (26).
V. BULK TOPOLOGICAL INDEXES ANDSTRING OPERATOR
In the previous section, we found that the gapless edgemodes emerge under the OBC in the model H SP . Thisfact implies that the present systems include SPT phases.We further investigate topological indexes characterizingtopological properties of the model H SP in Eq. (15), i.e.,Berry phases obtained from a local twist, string order,etc.In the momentum representation, we mention that theBerry phase in the H SP system, γ M , are quantized be-cause of the reflection symmetry in Eq. (24), and take γ M = 0 , π (mod 2 π ). Obtained results of γ M are shownin Fig. 5. Here, we employ another method for calcu-lating Berry phase, which can be used for interactingsystems.To this end, we introduce local twist with θ ∈ ( S :[0 , π )) for all the hopping terms in H SP residing on cer-tain unit cells. Under this local twist, the Hamiltonian H SP depends on θ , that is, H SP ( θ ). For the Hamiltonian H SP ( θ ), if the groundstate is unique and gapped for all θ , then the Z -Berry phase [55–60] from the local twistis given by γ L = i Z π dθ h g ( θ ) | ∂ θ | g ( θ ) i , (30)where | g ( θ ) i is the gapped unique groundstate for H SP ( θ ). The Berry phase γ L can be analytically treatedsince the exact many-body groundstate of H SP is alreadyknown. The groundstate is also useful for the study onthe interacting case with H SPI and H II in Eqs. (26) and(27).To calculate Berry phases practically, we introduce thefollowing twisted hopping in the cube operators at site ~r = ~ , Q + ~ ( θ ) = √ [ − e iθ d ~ x +ˆ y + ie iθ d ~ x + id ~ y − d ~ − ie iθ c ~ x +ˆ y + e iθ c ~ x + c ~ y − ic ~ ] , (31)and similarly for Q − ~ ( θ ) , ˜ Q + ~ ( θ ) and ˜ Q − ~ ( θ ). In factin the Hamiltonian with the twist H ( θ ), the hop-ping terms in the x -direction are changed to K + ~ ( θ ) ≡ ( Q + ~ ( θ )) † Q + ~ ( θ ) ∝ e − iθ d † ~ x +ˆ y d ~ y + · · · . It is easy toverify that the above twisted operators satisfy the samecommutation relations with the operators for θ = 0, i.e., { ( Q + ~ ( θ )) † , Q + ~ ( θ ) } = 1 , { ( Q + ~ ( θ )) † , Q + ~r } = 0 , for ~r = ~
0, etc. Then, operators K + ~ ( θ ), etc, which arecomposed of Q + ~ ( θ ), etc, are LIOMs, and energy eigen-states are given by Q + † ~ ( θ ) | i , etc.As an example, we first consider the grounstate at 1 / τ > τ , whose wave function is given by, | ψ ( θ ) i = [ Q − ~ ( θ )] † Y ~r = ~ Q −† ~r | i . (32)It is easily verified that the energy gap between | ψ ( θ ) i and the excited states does not close for any θ ∈ [0 , π ].Then, iγ L = Z π dθ h ψ ( θ ) | ∂ θ | ψ ( θ ) i = i Z π dθ h | [ − e iθ d ~ x +ˆ y + ie iθ d ~ x − ie iθ c ~ x +ˆ y + e iθ c ~ x ] † × [ − e iθ d ~ x +ˆ y + ie iθ d ~ x − ie iθ c ~ x +ˆ y + e iθ c ~ x ] | i = iπ. (33)Therefore, γ L = π for the groundstate at 1 / H SP (see Fig. 5 (a)) inSec. III.B. On the other hand for the case τ > τ , thegroundstate is given as | ψ ( θ ) i = [ ˜ Q − ~ ( θ )] † Y ~r = ~ ˜ Q −† ~r | i . (34)Similar calculation to the above shows that Berry phaseof | ψ ( θ ) i is γ L = π . Transition between | ψ ( θ ) i and | ψ ( θ ) i takes place at τ = τ . At this transition point∆ = 0, the system is simply two independent Creutz lad-der fermions, and there appears tremendous degeneracy.As a result, Berry phase cannot be defined properly.Let us turn to the half filling case. The groundstatewave function is given as | ψ ( θ ) i = [ Q − ~ ( θ )] † [ ˜ Q − ~ ( θ )] † Y ~r = ~ Q −† ~r ˜ Q −† ~r | i . (35) Berry phase is calculated as iγ L = Z π dθ h | [ Q − ~ ( θ )][ ˜ Q − ~ ( θ )] ∂ θ h [ Q − ~ ( θ )] † [ ˜ Q − ~ ( θ )] † i | i = 116 Z π dθ h | [ D ( θ ) C ( θ )] ∂ θ h [ C ( θ )] † [ D ( θ )] † i | i = 14 Z π dθ h | h C ( θ ) ∂ θ C † ( θ ) + D ( θ ) ∂ θ D † ( θ ) i | i = i π = 0 , (mod 2 π ) , (36)where C ( θ ) = − ie iθ c ~r +ˆ x +ˆ y + e iθ c ~r +ˆ x − c ~r +ˆ y + ic ~r , and D ( θ ) = e iθ d ~r +ˆ x +ˆ y − ie iθ d ~r +ˆ x + id ~r +ˆ y − d ~r . The abovecalculation shows that Q − and ˜ Q − –sectors (and c and d -particles) contribute to Berry phase additively. Berryphases of other states including Q + † ~r , etc are calculatedsimilarly, and similar results are obtained.Let us consider the effects of the interactions H SPI in Eq. (26), which preserve chiral symmetries and thegapless edge modes. As we mentioned in the above,‘phase transition’ between two states | Ψ i = Q ~r ( Q −† ~r ) | i and | Ψ i = Q ~r ( ˜ Q −† ~r ) | i takes place as varying val-ues of τ and τ . At 1/4-filling for τ > τ > λ ( > | Ψ i , and the Berryphase γ L = π as we calculated. As λ increases, theintra-species NN repulsions getting stronger, and at2 λ = τ − τ , there emerge tremendous degeneracies,i.e., states such as | Ψ i , Q ( · · · , Q −† ~r − ˆ x ˜ Q −† ~r Q −† ~r +ˆ x · · · | i , Q ( · · · , Q −† ~r − ˆ x ˜ Q −† ~r +ˆ x Q −† ~r +ˆ x · · · | i , etc. have all the same en-ergy. Because of this degeneracy, the Berry phase is un-defined. Even in this case, the gapless edge modes existbut they cannot be identified because of the tremendousdegeneracy. As the intra-species NN repulsion is gettingstronger [2 λ > τ − τ ], all cubes in one subsystem, sayeven cubes, are occupied by Q − ~r , whereas odd cubes areempty or occupied by ˜ Q − ~r . The number of the emptycubes is equal to that of the doubly-occupied cubes asthere is no inter-species repulsion between Q − ~r and ˜ Q − ~r .When we further add on-cube inter-species repulsion suchas λ ′ P ( K − ~r − )( ˜ K − ~r − ), the degeneracy is resolved.The groundstate is simply doubly-degenerate and has aN`eel-type order, i.e., ( · · · Q −† ~r − ˆ x ˜ Q −† ~r Q −† ~r +ˆ x ˜ Q −† ~r +2ˆ x · · · ) | i .In the thermodynamic limit, these two states are to-tally disconnected, and Berry phase can be defined foreach state as in the ordinary local order parameter suchas magnetization in the Ising model. Each state hasBerry phase γ L = π .In the following, we shall calculate Berry phases γ L forthe system with the additional interaction, H II [Eq. (27)],by numerical methods. In particular, we are interestedin how the Berry phase changes as V is increased. Inorder to have a well-defined Berry phase, the energy gap∆ E ( θ ) between the groundstate and first excited stateshas to be positive for any θ ∈ [0 , π ]. Then, we define∆ E Min ≡ Min[∆ E ( θ )], and calculate it numerically. Theresults are shown in Fig. 8 (a) and (b) for the 1 / V .000.050.10 (cid:1) E M i n V (cid:0) L (a)(b) V (cid:2) E M i n (c) FIG. 8. (a) Berry phase γ L as a function of V for 1 / E Min for 1 / V ∼
3. (c) The behavior of energy gap ∆ E Min for half-filling.At half-filling, the Berry phase remains zero, γ L = 0. Thesystem size is L = 4 (16 sites) with 4 or 8 particles. half-filling cases. For the 1 / γ L as a function of V . At 1 / γ L = π seems stable for V <
3, i.e.., γ L = π and ∆ E Min >
0. However in the very vicinity of V ∼
3, we find that γ L = 0 and ∆ E Min ∼ V >
3, the newphase seems also topological with γ L = π . On the otherhand for the half-filled case in Fig. 8 (c), ∆ E Min ≃ V > .
5. The flat-band gap is destroyed by the repulsion.In the above, we investigated the local quantity, Berryphase, related to topological properties of the systems.As shown in the previous work on the Creutz ladder [31],there is a nonlocal order parameter of Z -topologicalsymmetry, i.e., the string operator [61–63]. We can definea similar quantity in the present bilayer systems, whichwe call Z -order parameter and string operator. Theseoperators are defined in terms of the LIOMs, and for theHamiltonian H SP , Z = ( − P ~r ( K + ~r + K − ~r + ˜ K + ~r + ˜ K − ~r ) . (37)As K − ~r = Q −† ~r Q − ~r , etc, Z -operator in Eq. (37) is closelyrelated to the Berry phase calculated in the above. Underthe OBC considered in the above, the Z operator tendsto Z → ( − P Ln =0 P ~r =( n, ( c † ~r c ~r + c † ~r +ˆ y c ~r +ˆ y + d † ~r d ~r + d † ~r +ˆ y d ~r +ˆ y ) , where we have added the terms such as [ ω B † (0 , , ˆ y ω B (0 , , ˆ y +˜ ω A † ( L, , ˆ y ˜ ω A ( L, , ˆ y ] to make Z = ±
1. On the other hand forthe string operator, O ( ℓ, m ), we define it as follows; O ( ℓ, m ) = ( − P mn = ℓ P ~r =( n, ( K + ~r + K − ~r + ˜ K + ~r + ˜ K − ~r ) . (38)In the above, we studied the ‘phase transition’ causedby the interactions. For small λ , the state has the Berryphase γ L = π . In this state | Ψ i , h K − ~r i = 1 and theother expectation values are vanishing, and therefore hO ( ℓ, m ) i 6 = 0. On the other hand for 2 λ ≥ τ − τ ,there exist large number of degenerate states, in which h K − ~r i = 0 or 1, randomly. Therefore, the micro-canonicalensemble gives hO ( ℓ, m ) i = 0. By adding the on-cubeinter-species repulsion, the degeneracy is resolved exceptthe macroscopic one, and then hO ( ℓ, m ) i 6 = 0. The aboveconsideration of the various states shows that the Berryphase and the string operator give the consistent resultsas topological order parameters.Finally, we study the 1 / Z PR = h Ψ |R PR | Ψ i , (39)where R PR is the PR operator that reflects the siteswithin a segment of lattice with respect to its centrallink(s). Here, we consider the smallest PR segment, i.e.,a single cube located at site ~r = ~r . Then, R PR operatesas [65] c ~r → ic ~r +ˆ x +ˆ y , c ~r +ˆ x → ic ~r +ˆ y ,c ~r +ˆ x +ˆ y → ic ~r , c ~r +ˆ y → ic ~r +ˆ x . (40)Then, the PR overlap of the 1 / Z (1)PR (1 /
4) = h Ψ |R PR | Ψ i . (41)After some analytical calculation, we obtain Z (1)PR (1 /
4) = i e iπ/ / . (42)Similarly for the state | Ψ i , Z (2)PR (1 /
4) = − i e − iπ/ / . (43)The above results of Z PR (1 /
4) indicates that the 1 / Z -topological phase corresponding tothe phase π/ π ) /
4, and a single complex fermionemerges per each boundary in the OBC by the denom-inator 2 [65]. The complex fermion, say at the rightboundary, is given by ( − ω Bx = L, ˆ y − i ˜ ω Ax = L, ˆ y ) † coming from( Q − ~r =( L, ) † . We think that the emergent Z -topologicalphase (not Z -topological phase dictated by BDI class)comes from the four flat-bands structure of the Hamilto-nian H SP as shown in Fig. 5. This point will be discussedfurther in Sec. VI.1 VI. DISCUSSION AND CONCLUSION
In this paper, by making use of the cube operators,which were heuristically found as an extension of the ℓ -bits (CLS) in Creutz ladder, we constructed bilayer flat-band Hamiltonians of the exact projective form. Themodels have extensive numbers of the LIOMs, thus, fulllocalization occurs in the bulk. Since we constructedthe Hamiltonians by imposing certain symmetries, time-reversal and chiral symmetries, the constructed bilayerflat-band Hamiltonians naturally belong to a symmet-ric topological class in ten-fold way. In this work, weexplicitly showed that the constructed bilayer flat-bandHamiltonians belong to the BDI class. From this clas-sification, the constructed bilayer flat-band Hamiltoni-ans exhibit some topological character, i.e., non-trivialbulk topology and presence of the gapless edge modes,in particular in 1D. The model constructed on a quasi-1D lattice (prism lattice), explicitly exhibits SPT phasecharacterized by topological indexes for periodic system,and also the existence of gapless edge modes for the openboundary. This is just bulk-edge correspondence in thecomplete flat-band system.Here, we would like to give a brief discussion on topo-logical properties of strongly-localized states . For the Hamiltonian H SP , the PR overlap shows the existenceof the Z -topology. It is well known that Z -topology ofthe BDI class by the topological classification actuallyreduces to Z [66, 67]. For H SP , the Z -topology seemsquite plausible as the system has four flat-band structure.Not only in prism lattice but also in thin cylinder lattice,gapless edge modes are discovered relying on the form ofthe cube operators. By the topological classification, 2Dsystems in the BDI class have no bulk topological proper-ties. 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