Non-Hermitian second-order skin and topological modes
NNon-Hermitian Second-Order Skin and Topological Modes
Yongxu Fu and Shaolong Wan Department of Modern Physics,University of Science and Technology of China, Hefei, 230026, China (Dated: August 25, 2020)The skin effect and topological edge states in non-Hermitian system have already been well studiedin much previous work, while the second-order non-Hermitian edge states and skin effect havealso been proposed recently. We deduce the hybrid skin-topological modes as well as second-ordertopological edge states in a rigorous manner, for which we construct a nested tight-binding formalismin this paper. We also illustrate that the second-order skin effect originates from the existence ofboth two direction first-order skin effect which originates from loop topology of the complex energyspectrum under periodic boundary condition. We conclude that the hybrid skin-topological modeis generated by skin effect and localized edge states for each of two directions respectively, while thesecond-order topological edge states are induced by localized edge states along both two directions.
I. INTRODUCTION
Beyond the conventional hotspot on topological insu-lators and superconductors [1–8] and their classification[9–18] in condensed physics last decades, it rapidly ram-ifies into two patulous fields which involves higher-ordertopological phases [19–38] and non-Hermitian topologicalsystems [39–59] in recent years. An n -th order topologicalinsulator which originates from the topological crystallineinsulators has topologically protected gapless states atthe n -codimension surfaces [20, 33], but is gapped other-wise. For example, a second-order topological insulatorin two dimensions has zero energy states at corners but agapped bulk. Meanwhile the gapless edge states, the sig-nificant symbol of the first-order topological phase, areabsent. Non-Hermitian Hamiltonian describes the wideapplications of open system [60–65] and realizable systemof gain and loss [66–77] such as photonic and phononssystems etc. Of all properties in non-Hermitian systems,the exceptional points [44, 51, 78] at which many com-plex bands coalesce and the skin effect [46, 47, 52, 56]with localized bulk modes are the most intriguing fo-cus. In addition, the combination of higher-order andnon-Hermitian has also been studied [79–83] and two ex-tremely novel states has been proposed that is the second-order skin( SS ) and skin-topological( ST ) state [82].The abundant localized behavior in first-order non-Hermitian system exploits more possible second-order lo-calized states. The interplay between two direction withtopological edge states and skin bulk naturally inducesthree types second-order corner localized behavior cornerstates: topological-topological( T T ), topological-skin( ST or T S ) and skin-skin( SS ), which has been numericallyproposed in Ref [82] and extended to higher-order be-havior. We declare here that the nonzero edge statesis not topological protected in 1D system but they stillcontributes to the second-order corner states, hence weidentify the protected zero edge states and nonzero edgestates isolated from continuous bulk when we search thesecond-order corner states. In this sense, the definednoun for ST and T T [82] is suitable. In principle, afterunderstanding clearly the first and second-order topolog- ical insulator behavior, the higher-order case can be ob-tained by induction, merely more and more complicatedto be strictly unanalysable but can be left for numericalcalculation. Hence we only concentrate on the second-order corner localized behavior in this paper.In this paper, we rigorously depict the second-ordertopological(
T T ) and hybrid skin-topological( ST ) cornermodes. We illustrate this based on the nested tight-binding formalism which is a direct dialog to the generictight-biding model on hand without any additional an-nexing agent. The paper is organized as follows. InSec. II, we revive the topological origin of the first skineffect in previous work, and then elicit the second-orderskin( SS ) effect for a simplest 2D model. In Sec. III, Weconstruct the rigorous general formalism of nested tight-binding formalism. Using this method, we analyse thefour-band model proposing in Ref. [82] and a novel two-band model [83] to investigate the T T and ST cornermodes. Finally, we conclude this paper in Sec. IV. II. WINDING NUMBER AND SECOND-ORDERSKIN EFFECT
The n -th order topological insulator in d -dimensionalsystem is featured by the topologically protected gap-less states at the n -codimension surfaces when we take n -directions open boundary condition(OBC) and theremaining ( d − n )-directions periodic boundary condi-tion(PBC). It indicates that the n -th order topologicalinsulators with arbitrary dimension are all ascribed to n -dimensional Hamiltonian with full-OBC. The remaining( d − n ) parallel momentums k (cid:48)(cid:107) s in d -dimensional systemare just viewed as the parameters generating the hingeor higher dimensional direction for n -th order topolog-ical edge states. Given this, we merely need considera 2D Hamiltonian for second-order skin and topologicalphases, for which we propose the nested tight-bindingformalism to depict this universally in Sec. III. a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug A. Winding number and first-order non-Hermitianskin effect
The first-order skin effect originated from intrinsic non-Hermitian point gap topology [56] is determined by wind-ing number of the complex energy contour for a 1DHamiltonian. For simplicity, we refer the skin effect andedge states to the first-order status and indicate the or-der for higher-order status hereinafter. We emphasizethat the winding number for skin effect is different fromthat for edge states, which characterizing the topologi-cal protected edge states at 2 n -dimensional surface for a(2 n + 1)-dimensional Hamiltonian comes from a homo-topy map: BZ n +1 → U ( N ) W n +1 = n !(2 πi ) n +1 (2 n + 1)! (cid:90) BZ n +1 tr ( H − dH ) n +1 . (1)However, the skin effect winding number is always W for a 1D Hamiltonian, which only characters the skineffect for fixed ( d − d -dimensional system. In addition, the skin effectwinding number is vanishing for Hermitian Hamiltoniansince the energy spectrum is always real in complex plane.Notice that, the topological winding number characteriz-ing the edge states for a 1D chiral symmetric HermitianHamiltonian H h is actually the winding of the chiral non-Hermitian block Hamiltonian W h = 12 πi (cid:90) π dk ddk log det[ h ( k )] , H h = (cid:20) h ( k ) h † ( k ) 0 (cid:21) . (2)In conclusion, the non-Hermitian skin effect of a 1DHamiltonian originates from the point gap topology [56],for which the characteristic topological invariant is thewinding number of the PBC complex spectrum aroundthe reference skin mode point E , while the topology ofedge states inhering from Hermitian counterpart is dif-ferent from that for skin effect W ( E ) = 12 πi (cid:90) π dk ddk log det[ H ( k ) − E ] . (3)It reveals that the nontrivial topology is due to thePBC not OBC spectrum for the point gap, since theOBC spectrum is arcs in complex plane inducing van-ishing winding number. In addition, the value of wind-ing number W ( E ) counts the skin modes degeneracy atreference energy E [58]. In general, we should calcu-late winding number by adding all the winding numberof multiple bands(Riemann energy spectrum sheet) withmultiple Brillouin zones W ( E ) = 12 πi q (cid:88) µ =1 (cid:90) π dk ddk log[ E µ ( k ) − E ] . (4)The Brillouin zones are degenerated at | k | = 1 for PBC,while the generalized Brillouin zones(GBZs) are not de-generated in general cases [84] [see Appendix. A]. - - - - - - ( Ε ) I m ( Ε ) (a) - - - - ( Ε ) I m ( Ε ) (b) - - - - - ( Ε ) I m ( Ε ) (c) FIG. 1. (a)The complex energy spectra for non-HermitianSSH model with t = 0 . , t = 1 , γ = 4 /
3, in which the cyanand orange loop are energy spectra under PBC while blackline and point OBC. The complex energy spectra for two-band model Eq. (5) with t = 1 , t − = 2 , t + = 1 , w = 1 , w − =1 , w + = 3 , c = 1 for (b) and t = 1 , t − = 2 , t + = 1 , w = − , w − = 1 , w + = 3 , c = 1 for (c), in which the orange loopsare energy spectra under PBC for E ± ( k ) while black line andpoint OBC. As a typical model, the energy spectra of non-Hermitian SSH model H nSSH ( k ) = ( t + t cos k ) σ x +( t sin k + iγ/ σ y [46] under PBC sketches two Riemannsheet with ± square of Hamiltonian E ± ( k ) and each sheetencircles half loop(cyan and orange loop in Fig. 1(a)) ofthe energy spectra deducing the winding number for eachskin mode energy E s (points on the black line in Fig. 1(a)) W ( E s ) = W + ( E s ) + W − ( E s ) = 1 . Therefore each point on the black line in Fig. 1(a) whichis divided into two Riemann sheets located at the bothside of imaginary axis is the eigenenergy of one skin moderespect to the Hamiltonian under OBC except the originpoint which contains two degenerated edge modes.Another example has two completely separated bands[84] whose Hamiltonian is H ( k ) = (cid:20) t + t − e − ik + t + e ik cc w + w − e − ik + w + e ik (cid:21) . (5)The two energy bands(Riemann sheets) are E ± ( k ) = h + ( k ) ± (cid:113) c + h − ( k ) with h ± ( k ) = ( h ( k ) ± h ( k )) / h ( k ) = t + t − e − ik + t + e ik , h ( k ) = w + w − e − ik + w + e ik .In Fig. 1(b)(c), the complex energy spectra under PBCare plotted as orange loops for E ± ( k ) while OBC black.The skin modes(black lines) only exists in the area withnon-vanishing winding number. B. The second-order skin effect
Consider the simplest 2D non-Hermitian model [82]possessing second-order skin effect, whose Hamiltonianin momentum space is H D ( (cid:126)k ) = t x + e − ik x + t x − e ik x + t y + e − ik y + t y − e ik y , (6)where t x,y ± = t x,y ± γ x,y are the real nonreciprocal hop-ping inducing non-Hermicity. This Hamiltonian respectstime-reversal symmetry(TRS) T H D ( − (cid:126)k ) T − = H D ( (cid:126)k )with T = K the complex conjugation operator. Thepoint gap reality with T = 1 imposes H D belonging to AI class [9, 14], which is topological trivial for d = 2resulting in the absence of first-order edge states. Itfollows that the pure first and second-order skin effectare not protected by the conventional topological invari-ant consisting with the edge states but protected by thetopology of point gap itself. We emphasize here that theskin modes are continuous bulk part of energy spectrumwhile edge states are isolated from the bulk.They aretopologically irrelevant in this sense but the topologicalinvariant protecting edge states is calculated from theskin bulk bands deducing the modified bulk-boundarycorrespondence [53] in non-Hermitian system. We illus-trate the second-order skin effect of this model in Fig. 2:the full-OBC energy spectrum(blue) lies inside x -OBC/ y -PBC energy spectrum(orange) lying inside double-PBCenergy spectrum(cyan) for varying k y with 3D and 2Dplot in (a) and (b) respectively. The loops [Fig.2(c)]projected from varying k y complex energy for a fixed x -PBC(black) and x -OBC(orange) energy indicate the skineffect along y -direction and second-order skin effect re-spectively. Therefore the second-order skin effect indeedoriginates from two point gap topology along two direc-tion with first-order skin effect.As the simplest 2D model mentioned above , we caneasily read single y -layer Hamiltonian(see Sec. III A) H s from H D in Eq. (6) which is a Hatano-Nelson model [85]ˆ H Ds = (cid:88) x [ˆ c † x +1 ,y t x + ˆ c x,y + ˆ c † x − ,y t x − ˆ c x,y ] . We can solve β x = (cid:114) t x + t x − e ik forming a circular gen-eralized Brillouin zone and the OBC energy spectrum - - - - - Re ( Ε ) I m ( Ε ) (a) (b) - - - - - Re ( Ε ) I m ( Ε ) (c) (d) FIG. 2. Complex energy spectrum illustrations of simplest 2Dmodel in Eq. (6). The full-OBC energy spectrum(Blue) liesinside x -OBC/ y -PBC energy spectrumo(Orange) lying insidedouble-PBC energy spectrum(cyan) for varying k y with 2Dand 3D plotted in (a) and (b) respectively. The loop projectedfrom varying k y complex energy for a fixed band under x -PBC(black) and x -OBC energy(orange) are plotted in (c). Asecond-order skin mode locating at one corner is show in(d).The parameters: t x = t y = 1 , γ x = γ y = 0 . (cid:15) ( k ) = 2 (cid:112) t x + t x − cos k which lies in the PBC energy spec-trum loop (cid:15) P ( k x ) = t x + e − ik x + t x − e ik x indicating the skineffect along x -direction. Since the internal freedom is 1in this model, we can directly obtain the effective Hamil-tonian for second-order skin effect(see Sec. III A) H eff ( k y ) = (cid:88) k ( t y − e − ik y + (cid:15) ( k ) + t y + e ik y ) . (7)For each fixed k value, the complex energy spectrumsketches a loop C ( k ) for which (cid:15) ( k ) assigns the loop cen-ter varying in (cid:8) − (cid:112) t x + t x − , (cid:112) t x + t x − (cid:9) . According to thetopological origin of the first-order skin effect [56], eachloop C ( k ) surrounds the corresponding second-order skinmodes localized on one corner [Fig.2(d)] under both x -OBC/ y -OBC. III. NESTED TIGHT-BINDING FORMALISMFOR SECOND-ORDER PHASEA. The nested tight-binding formalism
A simplest perspective to give the second-order cor-ner states is working out the localized states one-by-onealong two related direction. It means that we put thelocalized information of one direction into the other, forwhich we call the nested process. With the lattice tight-binding model nature, our general formalism for second-order behavior is called nested tight-binding formalism.A generic tight-binding 2D Hamiltonian with L x , L y lattice sites and R x , R y hopping range along x, y directionrespectively and q internal freedom on each site isˆ H = L x (cid:88) x =1 L y (cid:88) y =1 q (cid:88) µν =1 (cid:20) R x (cid:88) i = − R x ˆ c µ † x + i,y t xi,µν ˆ c νx,y + R y (cid:88) j = − R y ˆ c µ † x,y + j t yj,µν ˆ c νx,y (cid:21) . (8)We first deal with a fixed single y -layerˆ H y = L x (cid:88) x =1 q (cid:88) µν =1 R x (cid:88) i = − R x ˆ c µ † x + i,y T xi,µν ˆ c νx,y , (9)where T xi,µν = t xi,µν , i (cid:54) = 0 and T x ,µν = t x ,µν + t y ,µν . Wecan work out qL x right eigenvalue solutions with energies (cid:15) µ ( β α ) for the above Hamiltonian. | Φ R,µα,y (cid:105) = L x (cid:88) x =1 N (cid:88) j =1 β xα,j | φ Rj,µα,y (cid:105) | x (cid:105) := L x (cid:88) x =1 q (cid:88) ν =1 ˜ φ R,µναx | ν (cid:105) | x (cid:105) , (10)where α = 1 , , . . . , L x , µ = 1 , , . . . , q represent the bandindex and we denote that ˜ φ R,µναx contains all the contribu-tions from solutions β (cid:48) j s with its multiplier s j for whichthe detail is in the Ref [86] and we not elaborate heresince we just focus on the general form of the solution. Ifwe impose PBC along x -direction, we reminisce the stan-dard Bloch theorem with k x := − i log β α = πL x α, α =0 , , . . . , ( L x − | β α | (cid:54) = 1 for the continuous bulk bands does in-dicate the skin effect for bulk bands.Using biorthogonal relation of the eigenstates, we candiagnose the single-particle Hamiltonian H y of single y -layer system in Eq. (8) to digonal eigenenergy matrix { (cid:15) µ ( β α ) } in the right eigenstate basis (cid:8) Φ R,µα,y (cid:9) [see Ap-pendix. B for details] (cid:15) = U † L · H y · U R . (11)The remain inter layer hopping terms along y -directionof total Hamiltonian are also similarly transformed by T yj = U † L · T yj · U R , (12)where ( T yj ) αµ,βν = (cid:80) L x i =1 (cid:80) qρ,σ =1 ˜ φ L,µρ ∗ αi ( t yj ) ρσ ˜ φ R,νσβi with α = 1 , . . . , L x and j = − R y , . . . , ˆ0 , . . . , R y [see Ap-pendix. B] T yj = t yj . . . . . . t yj L x × L x , t yj = t yj, . . . t yj, q ... . . . ... t yj,q . . . t yj,qq q × q . We finally arrive at a 1D effective Hamiltonian along y -direction under the biorthogonal basis along x -directionˆ H eff = L y (cid:88) y =1 R y (cid:88) j = − R y L x (cid:88) αβ =1 q (cid:88) µν =1 ˆΦ R,µ † α,y + j · ( T yj ) αµ,βν · ˆΦ L,νβ,y , (13) where ˆΦ R,µ † α,y = (cid:80) L x x =1 (cid:80) qν =1 ˜ φ R,µναx ˆ c ν † x,y and ˆΦ L,νβ,y is the an-nihilated operator of the corresponding biorthogonal lefteigenstate [see Appendix. B].Our nested tight-binding formalism is analyticallyvalid to investigate
T T and hybrid ST modes when theedge-state-subspace block of T yj is independent from thebulk block, in other word, the topological edge statesare not coupling with skin bulk states along x -direction.Fortunately, our nested tight-binding formalism is validfor the typical four-band model(complete block diago-nal for typical parameter choice) to analytically obtain T T and ST corner modes(Sec. III B) which can natu-rally reduce to Hermitian case for second-order topolog-ical corner modes. Moreover, the block diagonal resultalso appears in the 2D model with extrinsic second-order ST states [83]. For the skin bulk block part, the H eff in-duces the pure second-order skin effect which is the resultof combining with skin effect along another y -direction,namely bulk block of H eff also has nontrivial windingnumber topology indicating the existence of the skin ef-fect. The simplest 2D model [Eq. (6)] with pure SS modes has already been given in Sec. II, whose effec-tive Hamiltonian is easily obtained as Eq. (7). Althoughit’s hard to analysis the SS modes for more complicatedmodel due to the complexity of bulk skin states, the nu-merical result also can indicate the SS modes, such as inFig.2. In addition, a deeper sight for SS modes has beenjust proposed in related work [87]. Hence, we focus onthe widely analysable ST and T T modes hereinafter.
B. The four-band model
Consider a 2D non-Hermitian four-band model [80, 82] H ( (cid:126)k ) = H , − − H , − H ∗ , − H ∗ , − H ∗ , + H , + − H ∗ , + H , + , (14)where H j, ± = t x ± δ j + λe ik x for j = 1 , H j, ± = t y ± δ j + λe ik y for j = 3 , t x = t y = t for simplicity. The Hermitian counterpart of this model( δ j = 0 , j = 1 , , ,
4) has already been investigatedin Ref [21, 29]. Without any other parameters assign-ment, this Hamiltonian only preserves sublattice sym-metry with S − H ( k ) S = − H ( k ) , S = τ z . we first set δ = − δ = − δ = δ = γ , from which we reminisce themodel investigated in Ref [80] with net nonreciprocitiesfor both x and y -direction, i.e. H ( (cid:126)k ) = ( t + λ cos k x ) τ x − ( λ sin k x + iγ ) τ y σ z + ( t + λ cos k y ) τ y σ y + ( λ sin k y + iγ ) τ y σ x . (15)Besides sublattice symmetry, this Hamiltonian also pre-serves mirror-rotation symmetry M − xy H ( k x , k y ) M xy = H ( k y , k x ) with M xy = C M y , while the Hermitian coun-terpart of this model preserves both mirror symmetries M x = τ x σ z , M y = τ x σ x and four-fold rotational symme-try C = [( τ x − iτ y ) σ − ( τ x + iτ y )( iσ y )].Using our nested tight-binding formalism, we firststudy a single x -layer Hamiltonianˆ H s = (cid:88) y (ˆ c † y m ˆ c y + ˆ c † y t + y ˆ c y +1 + ˆ c † y +1 t − y ˆ c y ) , (16)where m = t ( τ x + τ y σ y ) + iγ ( τ y σ x − τ y σ z ) ,t + y = λ τ y σ y − iτ y σ x ) ,t − y = λ τ y σ y + iτ y σ x ) . (17)As usual process, we assume the eigenstate of the OBCHamiltonian is | ψ (cid:105) = L y (cid:88) y =1 β y | y (cid:105) | φ (cid:105) , where | φ (cid:105) is a 4-component column vector representingthe internal freedom. From the eigenequation ˆ H s | ψ (cid:105) = (cid:15) | ψ (cid:105) , we can read the characteristic equation of the bulkequation det( t − y β − + m + t + y β − (cid:15) ) = 0 , which gives1 β [ λ ( t + γ ) β + (2 t − γ + λ − (cid:15) ) β + λ ( t − γ )] = 0The four nonzero finite solutions have the relation β β = β β = t − γt + γ . Combining with the continuous condition [46, 50], weobtain | β | = | β | = | β | = | β | = (cid:114) | t − γt + γ | , which indicates the left-localized bulk skin effect along y -direction(the same for x -direction). In momentum space,the Hamiltonian of this model is H s ( k y ) = t ( τ x + τ y σ y ) + iγ ( τ y σ x − τ y σ z )+ λ cos k y τ y σ y + λ sin k y τ y σ x . (18)Although above Hamiltonian possesses four edge statesunder OBC, it’s topological trivial since the edge statescan be continuously absorbed into bulk due to thenonzero energy values but it contributes to the second-order corner-localized behavior. We emphasis that the1D Hamiltonian in Eq. (15) with k x being parameteris also topological trivial, therefore we must search thesecond-order topological behavior for further step. We now find the edge solutions of Hamiltonian H s under OBC and consider the left semi-infinite localizedstates first [40, 86]. The bulk and boundary equationsare ( t − y β − + m + t + y β ) | φ (cid:105) = (cid:15) | φ (cid:105) , (19)( m + t + y β ) | φ (cid:105) = (cid:15) | φ (cid:105) , (20)We can easily obtain | φ (cid:105) the kernels of t − y which are | u (cid:105) = u · | σ (cid:105) , | u (cid:105) = u · | σ (cid:105) , (21)where u = (0 , , , , u = (0 , , ,
0) and we denote | σ (cid:105) = ( | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) ) T as the internal freedom par-ticles. Put again the linear combination of | u , (cid:105) intothe bulk equation, we can obtain β = − t − γλ and (cid:15) ± = ± (cid:112) ( t + γ )( t − γ ) which are corresponding to the solu-tions | φ ± L (cid:105) = | u (cid:105) ± r | u (cid:105) := φ ± L · | σ (cid:105) , (22)where r = (cid:113) t + γt − γ . Therefore we obtain two left-localizedsolution with energies (cid:15) ± respectively, | ψ ± L (cid:105) = L y (cid:88) y =1 β y | y (cid:105) | φ ± L (cid:105) . (23)In addition, the left-localized condition | β | < L y . In the same way,we can solve the right-localized solutions with energies (cid:15) ± respectively, | ψ ± R (cid:105) = L y (cid:88) y =1 β − L y + y | y (cid:105) | φ ± R (cid:105) , (24)where β = − λt + γ and | φ ± R (cid:105) = | v (cid:105) ± r − | v (cid:105) := φ ± R · | σ (cid:105) , (25)with | v (cid:105) = v · | σ (cid:105) , | v (cid:105) = v · | σ (cid:105) , (26)where v = (0 , , , , v = (1 , , , U † L · T x · U R and U † L · T † x · U R are both block-diagonal with each block isa 4 × | ψ ± L,R (cid:105) due tothe biorthogonal normalization of non-Hermitian Hamil-tonian. So we solve the edge states for eigenequationˆ H Ts | ψ (cid:48) (cid:105) ∗ = (cid:15) | ψ (cid:48) (cid:105) ∗ . In the same way with the solving forright eigenstates, we can finally obtain the left eigenstateswith energies (cid:15) ± and left or right-localization | ψ (cid:48) ± L (cid:105) ∗ = L y (cid:88) y =1 β − y | y (cid:105) | φ (cid:48) ± L (cid:105) , | ψ (cid:48) ± R (cid:105) ∗ = L y (cid:88) y =1 β L y − y | y (cid:105) | φ (cid:48) ± R (cid:105) , (27)where | φ (cid:48) ± L (cid:105) = | u (cid:105) ± r − | u (cid:105) := φ (cid:48) ± L · | σ (cid:105) , | φ (cid:48) ± R (cid:105) = | v (cid:105) ± r | v (cid:105) := φ (cid:48) ± R · | σ (cid:105) . (28)Construct the biorthogonal diagonalized matrices of theedge-subspace U edgeR = (cid:18) ( φ + L ) T , ( φ − L ) T , ( φ + R ) T , ( φ − R ) T (cid:19) ,U edge † L = (cid:18) ( φ (cid:48) + L ) T , ( φ (cid:48) − L ) T , ( φ (cid:48) + R ) T , ( φ (cid:48) − R ) T (cid:19) T . (29)After biorthogonal normalizing for right and left eigen-states, we can acquire the effective edge-state-subspaceHamiltonian H edgej = L x (cid:88) x =1 ( ˆ φ j † x (cid:15) ˆ φ j (cid:48) x + ˆ φ j † x t + j ˆ φ j (cid:48) x +1 + ˆ φ j † x +1 t − j ˆ φ j (cid:48) x ) , (30)where j = L, R corresponding to left or right-localizededge-sub-subspace andˆ φ j † x = ( ˆ φ j + † x , ˆ φ j −† x ) , ˆ φ j (cid:48) x = ( ˆ φ j + (cid:48) x , ˆ φ j − (cid:48) x ) T , (31)withˆ φ j ±† x = L y (cid:88) y =1 N j β y − δL y j (ˆ c † x,y , ˆ c † x,y , ˆ c † x,y , ˆ c † x,y ) · ( φ ± j ) T , ˆ φ j ± (cid:48) x = L y (cid:88) y =1 N j β − y + δL y j +1 φ (cid:48) ± j · (ˆ c x,y , ˆ c x,y , ˆ c x,y , ˆ c x,y ) T , (32)and j = 1 , β in the first equation( j = 2 , δ = 0 , j = L, R , N j are the normalized coefficients N L = [2 L y (cid:88) y =1 ( β β − ) y ] − , N R = [2 L y (cid:88) y =1 ( β − β ) − L y + y ] − . (33)The hopping matrices are given by (cid:15) = (cid:112) ( t + γ )( t − γ ) σ z (a) - - - - ( Ε ) I m ( Ε ) (b)(c) (d) FIG. 3. Complex energy spectrum illustrations of four bandmodel in Eq. (15) with parameters t = 0 . , λ = 1 . , γ = 0 . k x dependence x -PBC/ y -PBC(cyan) complex energyspectrum, in which x -PBC/ y -OBC(orange) bulk spectrumlies. The isolated edge spectrum can be projected into twoloops in 2D plane which is plotted in (b), deducing the ST modes(black line) lying in the orange loop and zero T T modes(black point on origin). The localized behavior for typ-ical zero
T T corner mode and ST mode are plotted in (c) and(d) respectively. and t ± = 12 U edge † L · t ± x · U Redge = (cid:20) t ± L t ± R (cid:21) , where we can directly calculate to obtain t ± L = λ r ± (cid:20) ∓ ± − (cid:21) (34)and t ± R = λ r ± (cid:20) ± ∓ − (cid:21) . (35)Transforming the edge effective Hamiltonian Eq. (30)into momentum space H edgej ( k x ) = t − j e − ik x + (cid:15) + t + j e ik x , (36)the energy spectrum under PBC reads as (cid:15) j ( k x ) = t − γ + λ + λ [( t + γ ) e ± ik x + ( t − γ ) e ∓ ik x ] , where j = L, R . They sketch two orange loops inthe complex energy plane locating at both side imag-inary axis [Fig.3(b)], which is exactly projected fromthe k x dependence x -PBC/ y -OBC edge-subspace spec-trum(isolated orange line in Fig.3(a)). It indicates thebulk skin effect for H edgej under OBC along x -directiondeducing the ST mode [82] under full-OBC, which is plot-ted as black line lying in the orange loop in Fig.3(b).The skin effect indicator for H edgej is also | ρ | = (cid:113) | t − γt + γ | which manifests all the bulk states locating on the leftside. Together with the edge-subspace along y -direction,we deduce that the four zero energy T T modes are lo-cated on the four corners and the ST modes are locatedon the low-left and up-left corners when λ > t − γ, t + γ .The four zero corner modes localized at low-left(LL), low-right(LR), up-left(RL) and up-right(RR) can be write as | Ψ ij (cid:105) = N j L x (cid:88) x =1 ρ xi (cid:2) | ψ + j (cid:105) − ( − i + j | ψ − j (cid:105) (cid:3) | x (cid:105) = N j L x (cid:88) x =1 L y (cid:88) y =1 ρ xi β yj (cid:2) | φ + j (cid:105) − ( − i + j | φ − j (cid:105) (cid:3) | x (cid:105) | y (cid:105) , (37)where i, j = L, R (0 ,
1) represents the localized behavioralong x and y -direction respectively and ρ = β , ρ = β .However the T T modes are all numerically localizedon the low-left corner [Fig.3(c)] while the ST modes onlow-left corner with larger amplitude and low-right andup-left corners with smaller amplitude [Fig.3(d)]. In ad-dition, the pure SS modes are also all localized on thelow-left corner by numerical result. The analytical andnumerical results are seemly not consistence but we no-tice that the linear combinations of energy degeneratedstates are also the eigenstates of the Hamiltonian withthe same energy, for which we just perform a basis trans-formation. Based on this consideration, we can makeour analytical and numerical results consistent and weillustrate this manifestation below.Let us focus on the 1D Hamiltonian Eq. (30) to ex-plore the tiny different between analytical and numericalresult. Following the process of Eqs. (19)-(23), we cananalytically figure out the topological zero edge modesfor H edgej under OBC and write the two zero modes of H edgeL as an example ψ ,L = L y (cid:88) y =1 ( − t − γλ ) y (1 , − T ,ψ ,R = L y (cid:88) y =1 ( − λt + γ ) y (1 , T . (38)The two solutions are localized on left and right sidealong x -direction for λ > t − γ, t + γ . However the twonumerical edge modes are dramatically both localized onleft side when we set parameters as t = 0 . , γ = 0 . , λ =1 .
5. After comparing carefully these solutions, it’s foundthat the numerical solutions are indeed the precisely lin-ear combination of the analytical two ψ = ± α L ψ ,L − α R ψ ,R , but the coefficient α R is extremely small comparing with α L so that the two zero modes are both localized on leftside. In addition, the two combination solutions are not necessary orthogonal normalization since they are bio-thogonal in non-Hermitian system. Although the differ-ent between analytical and numerical result, the topolog-ical invariant winding number just characters the numberof zero modes not the localized behavior which dependson the choice of linear combination.Motivated by the 1D case, the four low-left corner lo-calized second-order zero modes of our model can be ob-tained by linear combination of the analytical four cornerlocalized zero modes | Ψ k (cid:105) = (cid:88) i,j = L,R α kij | Ψ ij (cid:105) , (39)where k = 1 , , , α LL induces the fi-nal four zero modes all localized at low-left corner, whichare consistent with the numerical result. For the ST modes, due to the mirror rotation symmetry M xy , we alsocan analytically obtain ST modes by considering single y -layer x -direction tight-binding model first, which ar-rive at low-left and low-right localized ST modes withdegenerated energies for above ST modes. By properlycombining these ST modes with degenerated energy, wecan interpret the localized behavior of the numerical ST modes.In general, it’s analysable when we take | δ | = | δ | and | δ | = | δ | and at least one direction net nonreciprocical.The coupling constant between neighbor lattice can alsobe different in general, i.e. λ , λ for x and y -directionrespectively. Following our nested tight binding formal-ism, we first solve the direction with net nonreciprocityfor a single layer such as x -direction. It’s well known that (cid:113) | t x − δ t x − δ | < ( > )1 indicates the bulk skin modes locatedon left (right) side along x -direction which indeed sug-gests the net nonreciprocity δ (cid:54) = δ relevant to the skineffect. Moreover, the localized behavior of analytical edgestates is determined by β = − t x − δ λ and β = − λ t x − δ under the nonreciprocity condition. As derived in Ref[46], the merging-into-bulk condition is the topologicalphase transition point | β | = | β | = (cid:114) | t x − δ t x − δ | , (40)this gives ( t x − δ )( t x − δ ) = ± λ . Noticing the nonre-ciprocity condition δ = − δ = γ , the topological phaseedge for x -direction is t x − γ = ± λ .Fortunately, the edge-state-subspace block effectiveHamiltonian is independent from bulk subspace as longas one direction is net nonreciprocity for our four-bandmodel. Following the above derivation for phase edge,we can obtain the similar result for edge-state-subspaceblock effective Hamiltonian t y − γ = ± λ . Thereforewe recover the phase diagram t − γ = ± λ in Ref [80]with t x = t y = t and γ = γ = γ . Moreover, we intro-duce another parameters choice for the four-band modelin Appendix. C. - - - - - ( Ε ) I m ( Ε ) (a) - - - - ( Ε ) I m ( Ε ) (b) FIG. 4. (a)Complex energy spectrum for H eff ( k x ) un-der PBC(orange) surrounding that under OBC(black).(b)Complex energy spectrum under full-OBC for H e ( k ). Theparameters are the same as Ref [83]: t x = 1 , g x = 0 . , t y =0 . , g y = 0 . C. The 2D model with extrinsic ST modes We further consider a 2D model possessing extrinsicsecond-order phase whose second-order topological in-variant has been given in Ref [83]. However the ST modes and T T modes has not been distinguished, whichwe still use our nested tight binding formalism to dealwith. The Hamiltonian [83] of this model has simply twointernal freedom H e ( (cid:126)k ) = 2 t x cos k x τ − ig x sin k x τ z − it y cos k y τ y − ig y sin k y τ x . (41)The complex energy spectrum for single y -layer Hamil-tonian H x ( k x ) forms a loop which indicates skin effect,while H y ( k y ) forms pure imaginary lines suppressing skineffect. For simplicity, we start from H y ( k y ) with justtwo localzied zero topological states due to the sublat-tice symmetry and line gap [58, 59] H y ( k y ) = − it y cos k y τ y − ig y sin k y τ x . (42)We can easily work out the two localized zero modeswith odd lattice sites [see Appendix. D] for even sitesand details) | ψ L (cid:105) = L y / (cid:88) y =1 β y − | y − (cid:105) φ L , | ψ R (cid:105) = L y / (cid:88) y =1 β − L y +2 y | y − (cid:105) φ R , (43)where | β | = (cid:113) t y − g y t y + g y with t y > g y and φ L = (0 , T , φ R =(1 , T . Hence, it’s easily obtain the edge-state-subspaceeffective Hamiltonian, which is blocked independent withthe bulk and supported by the numerical result. Actually,the effective edge Hamiltonian is exactly the transposi-tion of H x under OBC after similarity transformationby the biorthogonal edge-state-matrix [see Sec. III B andAppendix. B],therefore H eff ( k x ) = 2 t x cos k x τ + 2 ig x sin k x τ z . (44) The complex energy spectrum of H eff ( k x ) underPBC [orange loop in Fig.4(a)] surrounds skin bulk com-plex spectrum under OBC(black part in Fig.4(a)) whichis also the second-order corner localized modes under full-OBC [center part in Fig.4(b)]. The different between an-alytical and numerical spectrum is due to the finite lat-tice site which is change with different sites number andmust be consistent in the thermodynamic limit. Accord-ing to the left(right) localized behavior of the skin modesof H eff ( k x ) under OBC, we can exactly deduce the low-left(up-right) corner modes [83]. We emphasize that thecorner modes in this model is categorized into hybrid x -skin and y -topological ST modes [82].Nevertheless,the T T modes localized at the same corners appear if thelattice site number is larger enough.In addition, the ex-trinsic feature is due to the extended Hermitian Hamil-tonian [83] preserving only chiral symmetry without anycrystal symmetry leading the termination dependence forsecond-order corner modes.
IV. CONCLUSION AND DISCUSSION
In this paper, we construct the nested tight-bindingformalism to exactly deduce the second-order corner-localized-behavior modes. In the sense of identifying theprotected zero edge states and nonzero edge states iso-lated from continuous bulk, it has been discovered thatthe corner modes are classified to three types [82]: (i)Pure second-order skin effect( SS ) modes which is the re-sult of first skin effect both along two directions. (ii) Thepure second-order topological( T T ) corner modes whichinherits from Hermitian counterpart are interplay be-tween two localized behavior both along two directions.Notice that we should distinguish the topology for edgestates from that for skin effect, in which the former in-herits from Hermitian counterpart and the latter is purenon-Hermitian product. (iii) The most charming hybridskin-topological( ST ) [82] which are the interplay betweenedge states and skin effect induced by nonzero windingnumber, in other word, the Hermitian ramification andpure non-Hermitian product.Utilizing the nested tight-binding formalism,we have strictly illustrated the sim-plest 2D model[Eq. (6)] with pure SS modes,the ST cor-ner modes for four-band model[Eq. (15)] and the extrinsic ST corner modes for a 2D model given in Ref [83].Moreprecisely,we have obtained the complete zero T T cornersolutions for four-band model[Eq. (15)].The typical zero corner modes are numerically unbro-ken for the relevant crystal symmetry M xy for four-bandmodel, while the zero corner modes are unbroken at onlyone symmetry in Hermitian case since M x and M y are an-ticommucation. The numerical unbroken modes are thelinear combination of our analytical zero corner modeslocalized on each corner. The underlying physics for nu-merically unbroken corner states remains to be explored.More precisely, the nonzero edge states for one directionis not topological which can be absorbed into the bulk bycontinuous transformation [80], leading the unstable T T zero corner modes. This instability perhaps is the originof extrinsic second-order corner modes [83] which is leftfor the future work. Moreover,the mechanism of bulk-edge separation after biorthogonal transformation in thenested tight-binding formalism is perhaps related to thecrystal symmetry mathematically, which also remains forfuture work.
ACKNOWLEDGMENT
The authors thank Jihan Hu, Haoshu Li, ShuxuanWang and Zhiwei Yin for helpful discussions.
Note added. —
After completion of this work, we be-came aware of a recent related work [87] which proposesa deeper sight for SS modes. Appendix A: The exact eigenstates of 1Dtight-binding model
Without loss of generality, any first-order tight bindingmodel can be ascribed to a 1D tight-binding model withthe edge parallel momentums k (cid:107) regard as the parame-ters. ˆ H = (cid:88) ij,µν ˆ c † iµ H ij,µν ( k , ...k d , λ ‘ s )ˆ c jν , (A1)where we choose the first axis with OBC, k , ...k d , λ ‘ s are all parameters and omit the internal freedom simi-larly. The Hamiltonian of a 1D tight-binding model withhopping range − R → R and internal freedom q on eachlattice site is ˆ H = L (cid:88) n =1 R (cid:88) i = − R q (cid:88) µ,ν =1 ˆ c µ † n + i t i,µν ˆ c νn . (A2)Assuming the solution is | Φ (cid:105) = L (cid:88) n =1 | φ n (cid:105) | n (cid:105) = L (cid:88) n =1 q (cid:88) µ =1 β n φ µ | µ (cid:105) | n (cid:105) , (A3)with the eigenvalue equation ˆ H | Φ (cid:105) = E | Φ (cid:105) , we obtainthe bulk equation q (cid:88) ν =1 H ( β ) µν φ ν := q (cid:88) ν =1 R (cid:88) i = − R t i,µν β i φ ν = Eφ µ (A4)and the characteristic equationdet( R (cid:88) i = − R t i,µν β i − E ) = 0 . (A5) From the above linear equation set of φ (cid:48) s , we can linearlyexpress the ( q − φ (cid:48) s by the remain one φ µ = J νµ ( β ) φ ν µ = 1 , , . . . ˆ ν, . . . , q ν = 1 , , . . . , q (A6)and J νν = 1 naturally. The characteristic equation ofbulk equation can be solved resulting 2 qR roots of β in general in which we briefly ignore the multiple rootscase(it has be well studies in Ref [86]). Now the full so-lution is | Φ (cid:105) = L (cid:88) n =1 q (cid:88) µ =1 | φ nµ (cid:105) | n (cid:105) = L (cid:88) n =1 q (cid:88) µ =1 2 qR (cid:88) j =1 β nj φ jµ | µ (cid:105) | n (cid:105) . (A7)Imposing the boundary condition both on left and rightboundaries R (cid:88) i = − s t i | φ s + i +1 (cid:105) = E | φ s (cid:105) , s (cid:88) i = − R t i | φ L − s + i (cid:105) = E | φ L − s (cid:105) , (A8)where s = 0 , , . . . , ( R − | φ (cid:105) = | φ − (cid:105) = . . . = | φ − R +1 (cid:105) = 0 | φ L +1 (cid:105) = | φ L +2 (cid:105) = . . . = | φ L + R (cid:105) = 0 (A9)and then we obtain qR (cid:88) j =1 β − sj φ jµ = 0; s = 0 , , . . . , ( R − µ = 1 , , . . . , q, qR (cid:88) j =1 β L + sj φ jµ = 0; s = 1 , . . . , R ; µ = 1 , , . . . , q, (A10)Using Eq. (A6) and fixing a ν , we obtain qR (cid:88) j =1 f sµ ( β j , E ) φ jν = 0; s = 0 , , . . . , ( R − µ = 1 , , . . . , q, qR (cid:88) j =1 g sµ ( β j , E ) β Lj φ jν = 0; s = 1 , . . . , R ; µ = 1 , , . . . , q, (A11)where f sµ ( β j , E ) = J νµ ( β j ) β − sj ,g sµ ( β j , E ) = J νµ ( β j ) β sj . (A12)We can denote the 2 qR functions f sµ and g sµ as f j , g j , j = 1 , , . . . , qR respectively and then the bound-ary requires [50]det (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( β , E ) . . . f ( β qR , E )... . . . ... f qR ( β , E ) . . . f qR ( β qR , E ) g ( β , E ) β L . . . g ( β qR , E ) β L qR ... . . . ... g qR ( β , E ) β L . . . g qR ( β qR , E ) β L qR (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . (A13)0When we sort | β | (cid:54) ... (cid:54) | β qR | (cid:54) | β qR +1 | (cid:54) ... (cid:54) | β qR | and take limitation L → ∞ , the boundary conditionrestricts two type β solutions: discrete and continuoustypes corresponding to edge if exists and bulk states re-spectively. If | β qR | < | β qR +1 | , only one leading orderterm can survives when take L → ∞ in Eq. (A13) F ( β i ∈ P , β j ∈ Q , E ) := det (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( β , E ) . . . f ( β qR , E )... . . . ... f qR ( β , E ) . . . f qR ( β qR , E ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × det (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g ( β qR +1 , E ) . . . g ( β qR , E )... . . . ... g qR ( β qR +1 , E ) . . . g qR ( β qR , E ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , (A14)where P = { β , . . . , β qR } , Q = { β qR +1 , . . . , β qR } . Theabove equation gives discrete β (cid:48) s deducing the edgestates isolated from the continuous bulk states if it exists.If | β qR | = | β qR +1 | , there be two leading order termssurviving. Let P = { β , . . . , β qR − , β qR +1 } , Q = { β qR , β qR +2 . . . , β qR } , then the continuous β (cid:48) s are given[50] − F ( β i ∈ P , β j ∈ Q , E ) F ( β i ∈ P , β j ∈ Q , E ) = (cid:18) β qR β qR +1 (cid:19) L . (A15)Following the above logic, we can obtain the bulk bandspectra(or continuous band spectra) and generalized Bril-louin zone(GBZ) [84] as E bulk = { E ∈ C : | β qR ( E ) | = | β qR +1 ( E ) |} , C β = { β ∈ C : ∀ E ∈ E bulk , | β qR ( E ) | = | β qR +1 ( E ) |} . (A16)We emphasize that the GBZs depends on Riemann en-ergy spectrum sheet µ = 1 , , . . . , q in general. In otherword, there are q GBZs C µβ one-to-one correspondence to q Riemann energy spectrum sheet(i.e. complex energybands) E µ deduced from the q internal freedom. How-ever the multiply GBZs are degenerated in some simplemodel, such as non-Hermitian SSH model [46]. In thispaper, we only consider the degenerated GBZs or sin-gle band model leaving the multiply GBZs for numericalcalculation in future work.The above process to solve the eigenstates in non-hermitian system is the non-Bloch band theory withoutany symmetry constraint proposed in Ref. [50],which hasbeen extended to symplectic class [88, 89] and Z skin ef-fect [54] recently. Appendix B: Biorthogonal Diagonalization of thesingle y -layer Hamiltonian The qL x eigenvalue solutions in main text can also bewrite as creation fermion operatorsˆΦ R,µ † α,y = L x (cid:88) x =1 q (cid:88) ν =1 ˜ φ R,µναx ˆ c ν † x,y , (B1) where ˜ φ R,µναx is the xν -th row component of αµ -th righteigenstate for general non-Hermitian system. We definethe right eigenstate matrix U R = ˜ φ R, . . . ˜ φ R, L x q ... ... ...˜ φ R,L x q . . . ˜ φ R,L x qL x q (B2)and ˆ c y = (ˆ c ,y , . . . , ˆ c q ,y , . . . , ˆ c L x ,y , . . . , ˆ c qL x ,y ) T , ˆ c † y = (ˆ c † ,y , . . . , ˆ c q † ,y , . . . , ˆ c † L x ,y , . . . , ˆ c q † L x ,y ) , ˆΦ R † y = ( ˆΦ R, † ,y , . . . , ˆΦ R,q † ,y , . . . , ˆΦ R, † L x ,y , . . . , ˆΦ R,q † L x ,y ) , ˆΦ Ly = ( ˆΦ L, ,y , . . . , ˆΦ L,q ,y , . . . , ˆΦ L, L x ,y , . . . , ˆΦ L,qL x ,y ) T , (B3)then ˆΦ R † y = ˆ c † y · U R . (B4)Solving ˆ H † with similar manner for ˆ H , we can obtain theleft eigenstates with the equationsˆΦ L † y = ˆ c † y · U L , ˆΦ Ly = U † L · ˆ c (B5)and the biorthogonal relation U R · U † L = U L · U † R = ˆ1 . (B6)The inverse relation between two fermion operators isthen ˆ c † y = ˆΦ L † y · U † L , ˆ c y = U R · ˆΦ Ry . (B7)The result transformed to the biorthogonal basis for thesingle y -layer single-particle Hamiltonian ˆ H y is (cid:15) = U † L · H y · U R , ˆ H y = ˆΦ R † y · (cid:15) · ˆΦ Ly . (B8)where H y = T x . . . T xR x . . . T x − R x . . . T x . . . T xR x ... . . . ... . . . ...0 . . . T x − R x . . . T x ,T xi = T xi, . . . T xi, q ... . . . ... T xi,q . . . T xi,qq ,(cid:15) = (cid:15) ( β ) . . . . . . . . . . . . (cid:15) q ( β ) . . . . . . . . . . . . (cid:15) ( β L x ) . . . . . . . . . . . . (cid:15) q ( β L x ) . (B9)1Notice that the last q eigenvalues are edge states energyfor nontrivial phase which deducing the ST modes and T T corner modes. For Hermitian case, the biorthogonalrelation reduces to U † L = U − R , inducing that the diagonalprocess become standard in linear algebra (cid:15) = U − · H y · U . Appendix C: Other parameters choice for four-bandmodel
If we set δ = δ = − δ = δ = γ , the net nonreciproc-ity only exists along y -direction. The M xy is broken inthis case. The corner modes in this case contain: foursecond-order topological( T T ) zero modes and TS modeswhile the SS modes are absent. This case is almost thesame as the double nonreciprocity case in the edge sub-space, in which the tiny different is the form of edge states | φ ± L (cid:105) sn = | u (cid:105) ± r | u (cid:105) , | φ ± R (cid:105) sn = | v (cid:105) ± r | v (cid:105) . (C1)The edge effective Hamiltonian different from the doublenonreciprocity one is then deduced t ± L = λ r ∓ (cid:20) ∓ ± − (cid:21) , (C2) t ± R = λ r ± (cid:20) ± ∓ − (cid:21) . (C3)The double reciprocity case: δ = δ = − δ = − δ = γ . The M xy is also broken in this case. Unfortunately,the effective Hamiltonian is not block diagonal in numer-ical result. Therefore we cannot give the T T and TSmodes analytically at present, but the numerical plots isobvious in Ref [82]. Meanwhile the SS corner modes arealso absent.Asymmetry case: δ = δ = 0 or δ = δ = 0 while theother direction is nonreciprocity. The mirror symmetry M x or M y is restored. The T T and ST corner modes arepresent while SS modes absent. Hermitian case: δ = δ = δ = δ = 0. Both M x and M y are restored as well as the fourfold rotation symmetry C , the only existence corner modes are T T zero modes.Non-hermitian onsite gain and loss case: δ = δ = δ = δ = 0 with adding term − iuτ z , the C is restoredand the only existence corner modes are T T zero modes.
Appendix D: Edge states for 2D extrinsic model
The zero edge modes has very simply form for H y ( k y )in main text when the lattice site is odd number. Thebulk equation for the Hamiltonian is det ( t + y β + t − y β − ) = 0 , where t + y = (cid:20) − t y − g y t y − g y (cid:21) , (D1)and t − y = (cid:20) − t y + g y t y + g y (cid:21) . (D2)Due to the boundary condition t + y φ = 0 ,t − y φ L y − = 0 , (D3)the amplitude for exact zero edge states are destroyed oneven lattice site which is consistence with the numericalresult. The similar amplitude destruction is also found inRef [45]. Utilizing bulk equation to obtain β = (cid:113) t y − g y t y + g y ,the two edge states are give by Eq. (43) in main text.For even lattice site number, the exact edge solutionshas same form with odd site case when the site number islarge enough. However, the numerical results are tendedto lineally combining the two localized edge states, whichfinal numerically results in the diagonal corners localizedcorner modes. 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