Non-Hermitian second-order skin and topological modes

Yongxu Fu,∗ Jihan Hu, and Shaolong Wan†

Department of Modern Physics, University of Science and Technology of China, Hefei, 230026, China

The skin effect and topological edge states in non-Hermitian system have been well-studied, andthe second-order skin effect and corner modes have also been proposed in non-Hermitian systemrecently. In this paper, we construct the nested tight-binding formalism to research the second-ordercorner modes analytically, which is a direct description of the generic non-Hermitian tight-bindingmodel without other assumptions. Within this formalism, we obtain the exact solutions of second-order topological zero-energy corner modes for the non-Hermitian four-band model. We validate thenested tight-binding formalism in the hybrid skin-topological corner modes for the four-band modeland a non-Hermitian two-dimensional (2D) extrinsic model. In addition, we exactly illustrate thecorner modes induced by second-order skin effect for a simplest 2D non-Hermitian model by thenested tight-binding formalism.

I. INTRODUCTION

Beyond the conventional hotspot for topological insula-tors and superconductors [1–8] and their classification [9–18] in condensed physics past decades, it rapidly ramifiesinto two patulous fields which involve higher-order topo-logical phases [19–38] and non-Hermitian topological sys-tems [39–60] in recent years. An nth-order topological in-sulator, which originates from the topological crystallineinsulators [34], has topologically protected gapless statesat a boundary of the system of co-dimension n [20, 33],but is gapped otherwise. For example, a two-dimensionalsecond-order topological insulator has topological cornerstates but a gapped bulk and no gapless edge states. Thenon-Hermitian Hamiltonians are widely used in describ-ing open systems [61–66] and wave systems with gainand loss [67–78] (e.g., photonic and acoustic), etc. Ofall properties in non-Hermitian systems, the existence ofexceptional points [45, 52, 79] and the skin effect [47–50, 53, 57] are the most intriguing. The exceptionalpoints are the points where complex energy bands co-alesce, while the skin effect describes the localized bulkstates in non-Hermitian systems. We call the localizedbulk states in non-Hermitian systems the skin bulk statesin this paper. Recently, the higher-order states of thenon-Hermitian systems have been studied [80–86] andtwo novel states, the second-order skin (SS) and skin-topological (ST ) states [84], have been proposed.

The abundant localized states in first-order non-Hermitian systems exploit more possible second-order lo-calized states. The contribution from two directions withtopological edge (T ) states or skin bulk (S) states inducesthree possible types of second-order localized cornermodes: second-order topological (TT ), skin-topological(ST ), and second-order skin (SS) modes, which havebeen numerically calculated in Ref. [84]. However, theanalytical forms of these corner modes are still not ob-tained. The meaning and configuration of these corner

∗ yancyfoy@mail.ustc.edu.cn† slwan@ustc.edu.cn

modes are also not clear enough in Ref. [84]. In this pa-per, we investigate the three types of corner modes anddeduce their localization behavior analytically in non-Hermitian systems. Based on the nested tight-bindingformalism constructed in Sec. III A, we exactly researchthe TT , ST , and SS corner modes, and clarify the mean-ing and configuration of these corner modes. By thisformalism, we can analytically study the generic tight-binding model without any other assumptions. We ob-tain the analytical solutions of TT corner modes from theeffective Hamiltonian in the subspace of the edge-states,generated from the generic two-dimensional (2D) tight-binding Hamiltonian. Although not protected by bulk-energy band topology, the nonzero-energy edge statesstill contribute to the second-order corner modes. Ac-tually, the gapped edge-localized states are protectedby Wannier band topology in Hermitian systems withhigher-order topological phases [21], and the gaplessedge-localized states are protected by bulk-energy bandtopology. Hence we do not distinguish the zero- andnonzero-energy edge states when we research the second-order corner modes. In this sense, the definitions of STand TT [84] modes are reasonable. In principle, the pos-sible higher-order cases can be obtained from the first-and second-order cases. Hence we mainly concentrate onthe second-order corner modes in this paper.

This paper is organized as follows. In Sec. II, in-spired by the topological origin of the skin effect [57], westudy two typical one-dimensional (1D) non-Hermitianmodels with first-order skin effect. Then we illustratethe second-order skin effect for the simplest 2D non-Hermitian model [84]. In Sec. III, we construct the nestedtight-binding formalism and investigate the TT and STcorner modes. Utilizing this formalism, we study thefour-band model [84] with TT and ST corner modes andthe 2D model [86] with extrinsic corner modes. Finally,the conclusion and discussion are given in Sec. IV.

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II. WINDING NUMBER AND SECOND-ORDERSKIN EFFECT

The skin effect, which is a remarkable difference be-tween complex energy spectra under periodic boundarycondition (PBC) and those under open boundary con-dition (OBC), is the most charming property in non-Hermitian systems. There are extensive number ofskin bulk modes localized at arbitrary boundaries. InSec. II A, after a brief review of the topological origin offirst-order skin effect [57], we emphasize the differencebetween winding number protecting first-order topologi-cal edge states and that protecting skin effect, and studytwo typical 1D non-Hermitian models with first-orderskin effect. In addition, the second-order skin effect isinvestigated for the simplest 2D non-Hermitian model inSec. II B.

A. Winding number and first-order non-Hermitianskin effect

The first-order skin effect, which originates from intrin-sic point-gap topology of non-Hermitian systems [57], isdetermined by the winding number of the complex energycontour for a 1D Hamiltonian. For simplicity, we refer theskin effect and edge states to the first-order cases andspecify the order for higher-order cases hereafter. Thetopological invariant for point-gap is the winding num-ber of complex spectra under PBC around the referenceskin mode point E

W (E) =1

2πi

∫ 2π

0

dkd

dklog det[H(k)− E]. (1)

We should distinguish the meaning of the winding num-ber protecting first-order topological edge states fromthat protecting skin effect. The conventional windingnumber of a (2n+1)-dimensional Hermitian HamiltonianH(k) with chiral symmetry S, which protects topologicaledge states at the 2n-dimensional surface, comes from ahomotopy map: BZ2n+1 → U(N),

W2n+1 =n!

2(2πi)n+1(2n+ 1)!

∫BZ2n+1

tr(SH−1dH)2n+1.

(2)In addition, the conventional winding number has been

generalized to the winding number of non-Bloch Hamilto-nian H(β) in 1D non-Hermitian systems recently [47, 51],where β is in the generalized Brillouin zone (see Ap-pendix A). However, the winding number protecting theskin bulk part of the spectra under OBC [Eq.(1)] is calcu-lated from the complex energy spectra under PBC for asystem with point-gap. The winding number of the skineffect vanishes for the Hermitian Hamiltonian since theenergy spectra are always real. Note that, the conven-tional winding number, which protects the edge states ofa 1D Hermitian Hamiltonian Hh with chiral symmetry, is

-2 -1 0 1 2

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Re(Ε)

Im(Ε)

(a)

-4 -2 0 2 4 6

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Re(Ε)

Im(Ε)

(b)

-4 -2 0 2 4

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Re(Ε)

Im(Ε)

(c)

FIG. 1. (a) The complex energy spectra for the non-HermitianSSH model with t1 = 1, t2 = 1, γ = 4/3. There are two degen-erate topological edge modes located exactly at zero energy.The complex energy spectra for the two-band model, Eq. (5)with t0 = 1, t− = 2, t+ = 1, w0 = 1, w− = 1, w+ = 3, c = 1are plotted in (b) and those with t0 = 1, t− = 2, t+ = 1, w0 =−1, w− = 1, w+ = 3, c = 1 are plotted in (c). The spectraunder PBC are plotted as orange or cyan loops, while thespectra under OBC are plotted as black parts.

actually the winding number of the chiral non-Hermitianblock Hamiltonian

Wh1 =

1

2πi

∫ 2π

0

dkd

dklog det[h(k)], Hh =

[0 h(k)

h†(k) 0

].

(3)Moreover, the value of winding number W (E) counts thedegenerate skin modes at reference energy E [59]. Whenwe study the skin effect for a generic 1D multiple-bandsystem, we should sum over all the winding numbers foreach band Eµ(k)

W (E) =1

2πi

q∑µ=1

∫ 2π

0

dkd

dklog[Eµ(k)− E]. (4)

3

Firstly, we consider the typical non-Hermitian Su-Schrieffer-Heeger (SSH) model HnSSH(k) = (t1 +t2 cos k)σx + (t2 sin k + iγ/2)σy [47]. The energy spec-tra of this model under PBC form two energy bandsE±(k) = ±

√(t1 + t2 cos k)2 + (t2 sin k + iγ/2)2. Each

band forms a semicircle [cyan and orange semicircles inFig. 1(a)] in the complex plane. The winding number foreach skin mode Es under OBC [point on the black linesin Fig. 1(a)] is

W (Es) = W+(Es) +W−(Es) = 1.

Therefore each point on the black lines in Fig. 1(a) is aneigenenergy of one skin mode localized at one boundaryfor the Hamiltonian under OBC. However, the modes atorigin in Fig. 1(a) are not skin modes, which contain twodegenerate topological edge states.

Secondly, we consider the model with two energybands [87], and the Hamiltonian reads

H2(k) =

[h1(k) cc h2(k)

], (5)

where h1(k) = t0 + t−e−ik + t+e

ik and h2(k) = w0 +w−e

−ik + w+eik. The two energy bands are E±(k) =

h+(k)±√c2 + h2−(k), where h±(k) = (h1(k)±h2(k))/2.

In Figs. 1(b) and 1(c), the complex energy spectra underPBC and OBC are plotted as orange loops and blackparts respectively. The skin modes (black parts) onlyexist in the area with nonvanishing winding number.

B. The second-order skin effect

Consider the simplest 2D non-Hermitian model [84]possessing second-order skin effect. The Hamiltonian inmomentum space is

H2D(~k) = tx+e−ikx + tx−e

ikx + ty+e−iky + ty−e

iky , (6)

where tx,y± = tx,y±γx,y are the real nonreciprocal hoppingterms inducing non-Hermiticity. This Hamiltonian re-

spects time-reversal symmetry TH2D(−~k)T−1 = H2D(~k)and T is the complex conjugation operator. Hence, H2D

belongs to class AI with point gap [9, 14, 60], whichis topologically trivial in 2D resulting in the absence offirst-order edge states. It follows that the pure first- andsecond-order skin effect are not protected by the conven-tional topological invariant but protected by the point-gap topology.

From the simplest 2D model mentioned above, the sin-gle y-layer Hamiltonian [see Sec. III A] Hs, which is theHatano-Nelson model [88], reads

H2Ds =

∑x

[c†x+1,ytx+cx,y + c†x−1,yt

x−cx,y]. (7)

We can obtain βx =

√tx+tx−eik (k ∈ [0, 2π]) forming the cir-

cular generalized Brillouin zone (Appendix A). The en-ergy spectrum under OBC is ε(k) = 2

√tx+t

x− cos k, which

-4 -2 0 2 4

-3

-2

-1

0

1

2

3

Re(Ε)

Im(Ε)

(a) (b)

-2 -1 0 1 2

-3

-2

-1

0

1

Re(Ε)

Im(Ε)

(c) (d)

FIG. 2. Complex energy spectra of the simplest 2D model inEq. (6). The number of unit cells is 30× 30 with parameterstx = ty = 1, γx = γy = 0.8. (a) Spectra under double-PBC (cyan), x OBC/y PBC (orange) and full OBC (black)respectively. (b) Spectra under double-PBC (cyan) and xOBC/y PBC (orange) are plotted in E-ky space. (c) Theloops projected from (b) for a fixed energy band under x PBC(cyan, kx = π/2) and x OBC (orange, k = π/2) respectively.The orange loop surrounds its corresponding spectra underfurther taking y OBC (black points). (d) A typical second-order skin mode with ESS = −2.38769 localized at one corner.

is derived in Ref. [51] by non-Bloch band theory, whilethat under PBC is εP (kx) = tx+e

−ikx + tx−eikx forming

a loop, which is obtained by Fourier transformation ofH2Ds . The former lies in the interior of the latter, indica-

tive of skin effect along x direction. Since the internal de-gree of freedom is 1 in this model, we obtain the effectiveHamiltonian for second-order skin effect [see Sec. III A]

Heff (ky) =∑k

(ty−eiky + ε(k) + ty+e

−iky ). (8)

This effective Hamiltonian, as a function of ky, is ex-pressed in one skin-mode subspace along x direction foreach k value. Consequently, we obtain the second-orderskin modes (SS modes) under further taking y OBC forHeff (k). The meaning and configuration of SS cornermodes is that the corner modes under full OBC are con-tributed from skin modes along x direction (skin-modesubspace) and y direction (further taking y OBC). Foreach fixed k value, the complex energy spectrum forms aloop C(k), for which ε(k) assigns the loop center varyingin[−2√tx+t

x−, 2

√tx+t

x−].

We illustrate the second-order skin effect of this modelin Fig. 2. The full OBC energy spectra (black) lie withinx OBC/y PBC energy spectra (orange), which in turnlie within double-PBC energy spectra (cyan) [Figs. 2(a)and 2(b)]. The loops [Fig. 2(c)] projected from Fig. 2(b)for a fixed x PBC (cyan, kx = π/2) and x OBC (orange,k = π/2) energy band indicate the skin effect along y di-rection and second-order skin effect, respectively. All the

4

cyan and orange loops, with varying kx and k, respec-tively, form the corresponding cyan and orange energyspectra in Fig. 2(a). As the topological origin of the skineffect clarified by Ref. [57], each loop C(k) surrounds itscorresponding spectra under further taking y OBC [blackpoints in Fig. 2(c)], which are the SS modes localizedat one corner under full OBC [Fig.2(d)]. Therefore thesecond-order skin effect indeed originates from the point-gap topology along each of the two directions with first-order skin effect respectively. The conventional windingnumber does not protect the SS modes and there areno edge states of the simplest 2D non-Hermitian model,Eq. (6). The SS modes [black points in Fig. 2(c)] areprotected by the winding number of C(k) [orange loopin Fig. 2(c)] around corresponding SS modes, i.e., thepoint-gap topology of Heff (ky) for fixed k = π/2.

III. NESTED TIGHT-BINDING FORMALISMAND SECOND-ORDER CORNER MODES

A. The nested tight-binding formalism

One of the simplest perspectives to give the second-order corner modes is working out the localized statesin turn along two related directions. It means that weput the localized information of one direction into theother directions, for which we call the nested process. Forthe lattice tight-binding model, our general formalismfor second-order phase is called the nested tight-bindingformalism.

A generic tight-binding 2D Hamiltonian, with Lx, Lylattice sites, Rx, Ry the hopping range along x, y direc-tions, respectively, and q the internal degrees of freedomper unit cell, is

H =

Lx∑x=1

Ly∑y=1

q∑µ,ν=1

[ Rx∑i=−Rx

cµ†x+i,ytxi,µν c

νx,y

+

Ry∑j=−Ry

cµ†x,y+jtyj,µν c

νx,y

]. (9)

We first deal with a fixed single y layer

Hy =

Lx∑x=1

q∑µ,ν=1

Rx∑i=−Rx

cµ†x+i,yTxi,µν c

νx,y, (10)

where T x0,µν = tx0,µν + ty0,µν and T xi,µν = txi,µν (i 6= 0). Wecan formally give qLx right eigenstates with eigenenergiesεµ(βα) for the above Hamiltonian,

|ΦR,µα,y 〉 =

Lx∑x=1

N∑j=1

βxα,j |φRj,µα,y 〉 |x〉 :=

Lx∑x=1

q∑ν=1

φR,µναx |ν〉 |x〉 ,

(11)where α = 1, 2, . . . , Lx and µ = 1, 2, . . . , q. We denotethat φR,µναx contains all the contributions from solutions

βj with its multipliers sj , of which the detail is given inRef. [89]. Focusing on the general forms of the solutions,we do not elaborate βj with its multiplier sj here. Ifwe impose PBC along x direction, we consider the stan-dard Bloch theorem with kx := −i log βα = 2π

Lxα [α =

0, 1, . . . , (Lx−1)], while if imposing OBC we extend thatto the generalized Bloch theorem [89]. In non-Hermitiansystems, |βα| 6= 1 does indicate the skin effect of thecontinuous bulk bands.

Using biorthogonal relation of the eigenstates, we candiagonalize the single-particle Hamiltonian of Hy to diag-onal eigenenergy matrix {εµ(βα)} in the right eigenstatebasis

{|ΦR,µα,y 〉

}(see Appendix B for details)

ε = U†L ·Hy · UR. (12)

The remaining inter-layer hopping terms along the y di-rection of the total Hamiltonian are thus similarly trans-formed by

Tyj = U†LTyj UR, (13)

where (Tyj )αµ,βν =∑Lxi=1

∑qρ,σ=1 φ

L,µρ∗αi (tyj )ρσφ

R,νσβi with

α = 1, . . . , Lx and j = −Ry, . . . , 0, . . . , Ry (see Ap-pendix B),

T yj =

tyj . . . 0...

. . ....

0 . . . tyj

Lx×Lx

, tyj =

tyj,11 . . . tyj,1q...

. . ....

tyj,q1 . . . tyj,qq

q×q

.

The entry below ˆ means excluded. We finally ob-tain a 1D effective Hamiltonian along y direction, in thebiorthogonal basis along x direction, which is given as

Heff =

Ly∑y=1

Ry∑j=−Ry

Lx∑α,β=1

q∑µ,ν=1

ΦR,µ†α,y+j · (Tyj )αµ,βν · ΦL,νβ,y ,

(14)

In this Hamiltonian, ΦR,µ†α,y =∑Lxx=1

∑qν=1 φ

R,µναx cν†x,y and

ΦL,νβ,y is the annihilation operator of the corresponding

biorthogonal left eigenstate [see Appendix B].

Here we give the difference between the constructionof topological phases via coupled layers and our nestedtight-binding formalism. For the former, people cou-ple the fermionic operators (c) between different layers;

for the latter, we couple the operators (ΦR, ΦL) (corre-sponding biorthogonal eigenstates of a single layer underOBC) between different layers. In other words, we firstsolve the eigenstates for a single-layer Hamiltonian (suchas y-layer Hamiltonian Hy(kx)) under OBC (x OBC),and then couple the fermionic operators of these eigen-states between different layers in the perpendicular di-rection (y direction), by which we obtain the 1D effec-

tive Hamiltonian Heff . The internal degrees of freedomof this effective Hamiltonian are exactly the eigenstatesof the single-layer Hamiltonian (Hy(kx)) under OBC (xOBC). After further solving this effective Hamiltonian

5

under OBC (y OBC), we arrive at the results underfull OBC. By our formalism, we can clarify the mean-ing and configuration of TT , ST , and SS corner modes.Actually, our formalism also validates more general 2Dtight-binding Hamiltonian, which contains the hopping

ti,j between c†x+i,y+j and cx,y for any integer values i, j,i.e., the coupling between kx and ky in momentum space.Subsequently, T yj (j = 1, 2, . . .) is no longer block diago-

nal (banded block) and Heff is more intricate. To eluci-date the nested tight-binding formalism, we research themodels with nearest-neighbor hopping. The more generalmodels will be studied in future work.

The nested tight-binding formalism is valid to inves-tigate TT and ST modes when the edge-state subspacepart of Heff is independent from the bulk part, in otherwords, the degrees of freedom of topological edge eigen-states along x direction are not coupled with that ofbulk eigenstates in Heff . In next part, this formalismwill be further confirmed for the four-band model (com-

plete block diagonalization of Heff for typical parameterchoices) to obtain TT and ST corner modes analytically,and the meaning and configuration of these corner modeswill be clarified. Moreover, the block diagonal result alsoapplies to the 2D model with extrinsic second-order cor-ner modes [86]. When the skin bulk block part of Heff

is independent from the edge-state subspace part, theHeff induces the pure second-order skin effect, which isthe combination of skin bulk eigenstates along the x di-rection (internal degrees of freedom of Heff ) and the skin

effect of Heff along the y direction. In other words, skin

bulk block of Heff also has nontrivial point-gap topol-ogy indicating the existence of the skin effect along they direction. The simplest 2D model [Eq. (6)] with pureSS modes has already been given in Sec. II, of whichthe effective Hamiltonian is easily obtained as Eq. (8).Although it is cumbersome to analyze the SS modes fora more complicated model due to the complexity of skinbulk states, the numerical result also can indicate the SSmodes. Hence, we focus on the generally analyzable STand TT modes hereinafter.

B. The four-band model

Consider a 2D non-Hermitian four-band model [82, 84]

H(~k) =

0 0 H1,− −H4,−0 0 H∗3,− H∗2,−

H∗1,+ H3,+ 0 0−H∗4,+ H2,+ 0 0

, (15)

where Hj,± = tx ± δj + λeikx for j = 1, 2 and Hj,± =ty ± δj + λeiky for j = 3, 4, setting tx = ty = t forsimplicity. The Hermitian counterpart of this model(δj = 0, j = 1, 2, 3, 4) has already been investigated inRefs. [21, 29]. Without any other parameter assignments,the Hamiltonian of this model only preserves sublattice

symmetry S−1H(k)S = −H(k) with S = τz. We setδ1 = −δ2 = −δ3 = δ4 = γ for simplicity, from whichwe consider the model investigated in Ref. [82] with netnonreciprocities for both x and y directions, i.e.

H(~k) = (t+ λ cos kx) τx − (λ sin kx + iγ) τyσz

+ (t+ λ cos ky) τyσy + (λ sin ky + iγ) τyσx. (16)

Besides sublattice symmetry, this Hamiltonian also pre-serves mirror-rotation symmetry M−1xy H(kx, ky)Mxy =H(ky, kx) with Mxy = C4My, while its Hermitiancounterpart preserves both mirror symmetries Mx =τxσz,My = τxσx and four-fold rotational symmetry C4 =[(τx − iτy)σ0 − (τx + iτy)(iσy)]/2.

Applying our nested tight-binding formalism, we studya single x-layer Hamiltonian

Hs =∑y

(c†ym0cy + c†yt+y cy+1 + c†y+1t

−y cy), (17)

where

m0 = t(τx + τyσy) + iγ(τyσx − τyσz),

t+y =λ

2(τyσy − iτyσx),

t−y =λ

2(τyσy + iτyσx). (18)

As usual, we assume the eigenstate of the Hamiltonianunder OBC is

|ψ〉 =

Ly∑y=1

βy |y〉 |φ〉 ,

where |φ〉 is a four-component column vector representingthe internal degrees of freedom. From the eigen-equationHs |ψ〉 = ε |ψ〉, the secular equation of the bulk equationreads

det(t−y β−1 +m0 + t+y β − ε) = 0,

which gives

1

β2[λ(t+ γ)β2 + (2t2 − 2γ2 + λ2 − ε2)β + λ(t− γ)]2 = 0.

The four nonzero finite bulk solutions satisfy the relation

βb1βb2 = βb3β

b4 =

t− γt+ γ

.

As derived in Refs. [47, 51], the continuous conditiongives

|βb1| = |βb2| = |βb3| = |βb4| =

√∣∣∣∣ t− γt+ γ

∣∣∣∣,which we call the skin effect indicator (left-localized when|t| > |γ|) along the y direction (same for the x direction).In momentum space, the Hamiltonian of this model is

Hs(ky) = t(τx + τyσy) + iγ(τyσx − τyσz)+λ cos kyτyσy + λ sin kyτyσx. (19)

6

The above Hamiltonian possesses four nonzero-energyedge states under OBC, which contribute to the second-order corner-localized modes. We emphasize that thegapped edge states of Eq. (16) under y OBC/x PBC (orx OBC/y PBC) are not protected by bulk-energy bandtopology due to the vanishing Chern number [82]. Weneed to research the second-order topological modes.

Firstly, we solve the left-localized edge states of theHamiltonian Hs under OBC [40, 89]. The bulk andboundary equations are

(t−y β−1 +m0 + t+y β) |φ〉 = ε |φ〉 , (20)

(m0 + t+y β) |φ〉 = ε |φ〉 . (21)

We can obtain |φ〉 the kernels of t−y , which are

|u1〉 = u1 |σ〉 ,|u2〉 = u2t |σ〉 , (22)

where u1 = (0, 0, 0, 1), u2 = (0, 1, 0, 0). We denote|σ〉 = (|1〉 , |2〉 , |3〉 , |4〉)T as the internal degrees of free-dom. Substituting the linear combination of |u1,2〉 intothe bulk equation, we obtain two solutions as

|φ±L 〉 = |u1〉 ± r |u2〉 := φ±L |σ〉 , (23)

where r =√

t+γt−γ . Accordingly, the two left-localized

solutions with energies ε± = ±√

(t+ γ)(t− γ), are

|ψ±L 〉 =

Ly∑y=1

βy1 |y〉 |φ±L 〉 , (24)

where β1 = − t−γλ . Additionally, the left-localized con-dition |β1| < 1 guarantees the above solutions automat-ically satisfying the right boundary equation for largeenough Ly.

Secondly, the right-localized edge states are given as

|ψ±R〉 =

Ly∑y=1

β−Ly+y2 |y〉 |φ±R〉 , (25)

with respective energies ε±, where β2 = − λt+γ and

|φ±R〉 = |v1〉 ± r−1 |v2〉 := φ±R |σ〉 , (26)

with

|v1〉 = v1 |σ〉 ,|v2〉 = v2 |σ〉 , (27)

v1 = (0, 0, 1, 0), v2 = (1, 0, 0, 0).

We find that the numerical results of U†LTxUR and

U†LT†xUR are both block-diagonal, of which each block

is a 4 × 4 matrix in this model. Therefore we can dealwith the edge-state subspace independently. However,we have to find the corresponding left eigenstates of the

right eigenstates |ψ±L,R〉 due to the biorthogonal relationof the non-Hermitian Hamiltonian. So we solve the edge

states for eigen-equation HTs |ψ

′〉∗

= ε |ψ′〉∗. With the

same procedure solving right eigenstates, we have theleft eigenstates

|ψ′±L 〉∗

=

Ly∑y=1

β−y2 |y〉 |φ′±L 〉 ,

|ψ′±R 〉∗

=

Ly∑y=1

βLy−y1 |y〉 |φ

′±R 〉 , (28)

where

|φ′±L 〉 = |u1〉 ± r−1 |u2〉 := φ

′±L · |σ〉 ,

|φ′±R 〉 = |v1〉 ± r |v2〉 := φ

′±R · |σ〉 . (29)

We construct the biorthogonal diagonalized matrices inthe edge-state subspace as

UedgeR =

((φ+L)T , (φ−L )T , (φ+R)T , (φ−R)T

),

Uedge†L =

((φ′+L )T , (φ

′−L )T , (φ

′+R )T , (φ

′−R )T

)T. (30)

After biorthogonally normalizing of the right and lefteigenstates, we finally arrive at the effective Hamiltonianin edge-state subspace,

Hedgej =

Lx∑x=1

(φj†x ε0φ′jx + φj†x t

+j φ′jx+1 + φj†x+1t

−j φ′jx ), (31)

where j = L,R corresponding to left- or right-localizededge-state subspace and

φj†x = (φj+†x , φj−†x ),

φ′jx = (φ

′j+x , φ

′j−x )T . (32)

The fermionic operators in above equations are

φL±†x =

Ly∑y=1

N yLβ

y1 (c1†x,y, c

2†x,y, c

3†x,y, c

4†x,y) · (φ±L )T ,

φR±†x =

Ly∑y=1

N yRβ

y−Ly2 (c1†x,y, c

2†x,y, c

3†x,y, c

4†x,y) · (φ±R)T ,

φ′L±x =

Ly∑y=1

N yLβ−y2 φ

′±L · (c

1x,y, c

2x,y, c

3x,y, c

4x,y)T ,

φ′R±x =

Ly∑y=1

N yRβ−y+Ly1 φ

′±R · (c

1x,y, c

2x,y, c

3x,y, c

4x,y)T ,(33)

where the biorthogonally normalized coefficientsN yj (j =

7

L,R) are given as

N yL = [2

Ly∑y=1

(β1β−12 )y]−1/2,

N yR = [2

Ly∑y=1

(β−11 β2)−Ly+y]−1/2. (34)

The hopping matrices are given by

ε0 =√

(t+ γ)(t− γ)σz

and

t± =1

2Uedge†L · t±x · U

edgeR =

[t±L 00 t±R

],

where

t±L =λ

2r±[

1 ∓1±1 −1

](35)

and

t±R =λ

2r±[

1 ±1∓1 −1

]. (36)

This effective Hamiltonian Eq. (31) is the main result ofapplying our nested tight-binding formalism to the four-band model. Here, we clarify the meaning and config-uration of TT and ST corner modes. The superscriptedge means this Hamiltonian is in the edge-state sub-space along the y direction, which contributes the topo-logical edge (T ) modes. Therefore, combining with theskin bulk (S) modes and topological edge (T ) modes de-

duced from Hedgej under x OBC, we obtain the ST and

TT corner modes under full OBC respectively.

The edge effective Hamiltonian, Eq. (31), in momen-tum space is

Hedgej (kx) = t−j e

−ikx + ε0 + t+j eikx , (37)

and the energy spectra under PBC read as

ε2j (kx) = t2 − γ2 + λ2 + λ[(t+ γ)eikx + (t− γ)e−ikx ],

where j = L,R. They form two orange loops localized onboth sides of the imaginary axis in the complex energyplane [Fig. 3(b)], which are exactly projected from thekx dependent x PBC/y OBC edge-state subspace spec-tra [isolated orange lines in Fig. 3(a)]. These two loops

depict the skin effect of Hedgej under OBC along the x

direction leading to the ST modes [84] under full OBC,which are plotted as black lines lying within the orange

loops in Fig. 3(b). The skin effect indicator for Hedgej is

also |ρ| =√| t−γt+γ |, which implies the localization of all

bulk states at the left side when |t| > |γ|. Together withthe edge-state subspace along the y direction, the fourzero TT modes are localized at the four corners and the

(a)

-2 -1 0 1 2-0.4

-0.2

0.0

0.2

0.4

Re(Ε)

Im(Ε)

(b)

(c) (d)

FIG. 3. Complex energy spectra of the four band model inEq. (16) with parameters t = 0.6, λ = 1.5, γ = 0.4. Thenumber of unit cells is 20 × 20. (a) Spectra under double-PBC (cyan) and x OBC/y PBC (orange) are plotted in theE-ky space. (b) The two orange loops, which are projectedfrom the isolated edge spectra in (a), deduce the ST modes(black lines) lying within the orange loops and four degener-ate zero-energy TT modes (black point on the origin). Thetypically localized zero-energy TT mode and nonzero-energyST mode with energy EST = −1.06094 are plotted in (c) and(d), respectively.

ST modes are localized at the low-left and up-left cornerswhen |λ| > |t − γ|, |t + γ|. The four zero-energy cornermodes localized at low-left (LL), low-right (LR), up-left(RL) and up-right (RR) can be written as

|ΨLL〉 = N xLN

yL

Lx∑x=1

Ly∑y=1

βx1βy1

[|φ+L〉 − |φ

−L 〉]|x〉 |y〉 ,

|ΨLR〉 = N xLN

yR

Lx∑x=1

Ly∑y=1

βx1βy−Ly2

[|φ+R〉+ |φ−R〉

]|x〉 |y〉 ,

|ΨRL〉 = N xRN

yL

Lx∑x=1

Ly∑y=1

βx−Lx2 βy1[|φ+L〉+ |φ−L 〉

]|x〉 |y〉 ,

|ΨRR〉 = N xRN

yR

Lx∑x=1

Ly∑y=1

βx−Lx2 βy−Ly2

[|φ+R〉 − |φ

−R〉]|x〉 |y〉 ,

(38)

where the normalized coefficients read (δ = x, y)

N δL = [2

Lδ∑δ=1

(β1β−12 )δ]−1/2,

N δR = [2

Lδ∑δ=1

(β−11 β2)−Lδ+δ]−1/2.

Noteworthily, |ΨLL〉 and |ΨRR〉 are invariant undermirror-rotation transformation when Lx = Ly, while|ΨLR〉 and |ΨRL〉 are transformed to each other.

8

However, the TT modes are all numerically localizedat the low-left corner [Fig. 3(c)], while the ST modesat the low-left corner with larger amplitude, low-rightand up-left corners with smaller amplitude [Fig. 3(d)].In addition, the pure SS modes are also all localized atthe low-left corner by numerical result. The analyticaland numerical results are seemingly inconsistent, but wenotice that the linear combinations of energy degeneratestates are also the eigenstates of the Hamiltonian withthe same energy. Based on this consideration, we caneliminate this inconsistence, on which we will elaboratein the following.

Let us focus on the 1D Hamiltonian, Eq. (31), to ex-plore the difference between analytical and numerical re-sults. Following the procedure of Eqs. (20)-(27), we fig-

ure out the topological zero edge modes for Hedgej under

OBC analytically. Writing the two zero modes of HedgeL

as an example,

ψ0,L = N xL

Lx∑x=1

(− t− γλ

)x(1,−1)T ,

ψ0,R = N xR

Lx∑x=1

(− λ

t+ γ)x−Lx(1, 1)T . (39)

Requiring |λ| > |t − γ|, |t + γ|, the two solutions are lo-calized on the left and right sides along the x directionrespectively. However, the two numerical edge states arelocalized only on left side when we set parameters ast = 0.6, γ = 0.4, λ = 1.5. After carefully comparing thesesolutions, we find that the numerical solutions are pre-cisely the linear combination of the two analytical zeromodes

ψ0 = ±αLψ0,L − αRψ0,R,

but the coefficient αR is much smaller than αL, leadingto the two zero modes both localized on the left side. Inaddition, the two combination solutions are not orthogo-nal normalization since they satisfy biorthogonal relationin non-Hermitian system.

Motivated by the 1D case, we obtain the four second-order zero modes localized at the low-left corner by lin-ear combination of the analytical four zero-energy cornermodes

|Ψk〉 =∑

i,j=L,R

αkij |Ψij〉 , (40)

where k = 1, 2, 3, 4 denotes the four zero-energy cor-ner modes. The domination of the coefficient αLL in-duces the final four zero modes all localized at low-leftcorner, which are indeed consistent with the numericalresult [Fig. 3(c)]. Although the difference between an-alytical and numerical results exists, the second-ordertopological invariant, which is constructed in Ref. [82]by utilizing mirror-rotation symmetry Mxy, character-izes the number of TT zero-energy corner modes not thelocalization behavior of those. Additionally, based on

our tight-binding formalism, the point-gap topology of

the edge-state subspace effective Hamiltonian Hedgej (kx)

protects ST modes, i.e., the winding numbers of orangeloops in Fig. 3(b) around corresponding ST modes [blacklines in Fig. 3(b)].

Due to the mirror-rotation symmetry, we can also ob-tain ST modes analytically by first considering a single y-layer tight-binding model along the x direction. Then weobtain low-left and low-right localized ST modes, degen-erate with the preceding low-left and up-left localized STmodes (the result by first considering the single x layerHamiltonian). By properly combining these ST modeswith degenerate energy (i.e. larger coefficient for low-leftlocalized ST modes and smaller coefficients for up-leftand low-right localized ST modes), we can obtain the STmodes consistent with the numerical result [Fig. 3(d)]. Inaddition, the SS modes, all localized at the low-left cor-ner, are obviously induced by the left-localized skin effectalong both directions.

It is analyzable when we take |δ1| = |δ2| and |δ3| = |δ4|.In general, the coupling terms between neighbor latticescan also be different, i.e., λ1, λ2 for x and y directionsrespectively. Following our nested tight-binding formal-ism, we solve the Hamiltonian for a single y-layer with net

nonreciprocity. It is well known that√| tx−δ1tx−δ2 | < (>)1

indicates the skin bulk states localized on left (right) sidealong the x direction. Moreover, the localization be-havior of analytical edge states is determined by β1 =− tx−δ1λ1

and β2 = − λ1

tx−δ2 . As derived in Ref. [47], themerging-into-bulk condition yields the topological phase-transition points

|β1| = |β2| =√| tx − δ1tx − δ2

|, (41)

resulting in (tx−δ1)(tx−δ2) = ±λ21. Noticing the nonre-ciprocity condition δ1 = −δ2 = γ1, we obtain the phase-transition edge for the x direction t2x − γ21 = ±λ21. Fol-lowing the above derivation of phase-transition edge, weobtain the similar result t2y − γ22 = ±λ22 for the effec-tive Hamiltonian [Eq. (31)] in the edge-state subspace.Therefore we recover the phase diagram with boundaryt2 − γ2 = ±λ2 in Ref. [82] taking tx = ty = t andγ1 = γ2 = γ. Moreover, we introduce other parameterchoices for the four-band model in Appendix C.

C. The 2D model with extrinsic ST modes

We further consider a 2D model possessing extrinsicsecond-order corner modes, of which the second-ordertopological invariant has been given in Ref. [86]. How-ever, the ST modes and TT modes have not been dis-tinguished, to which we apply our nested tight-bindingformalism. The simple Hamiltonian [86] of this model

9

-2 -1 0 1 2-2.0-1.5-1.0-0.50.00.51.01.52.0

Re(Ε)

Im(Ε)

(a)

-2 -1 0 1 2

-2

-1

0

1

2

Re(Ε)

Im(Ε)

(b)

FIG. 4. (a) Complex energy spectrum forHeff (kx) in Eq. (45)under PBC (orange) surrounds that under OBC (black). (b)

Complex energy spectra under full OBC for He(~k) in Eq. (42).The number of unit cells is 25×25 and the parameters are thesame as Ref. [86]: tx = 1, gx = 0.9, ty = 0.8, gy = 0.7. Thereare two degenerate TT corner modes located exactly at zeroenergy with odd number of lattice sites.

has two internal degrees of freedom and reads

He(~k) = 2tx cos kxτ0 − 2igx sin kxτz

−2ity cos kyτy − 2igy sin kyτx, (42)

where ty > gy > 0 and tx > gx > 0 without loss ofgenerality.

The complex energy spectrum of single y-layer Hamil-tonian Hx(kx) forms a loop, indicative of skin effect,while those of single x-layer Hamiltonian Hy(ky) formpure imaginary lines, suppressing skin effect. For simplic-ity, we start from Hy(ky) with two localized zero topo-logical states,

Hy(ky) = −2ity cos kyτy − 2igy sin kyτx. (43)

We can easily work out the two localized zero modestaking odd lattice sites (for even sites and details in Ap-pendix D)

|ψL〉 =

(Ly+1)/2∑y=1

β2y−1 |2y − 1〉φL,

|ψR〉 =

(Ly+1)/2∑y=1

βLy−2y |2y − 1〉φR, (44)

where |β| =√

ty−gyty+gy

and φL = (0, 1)T , φR = (1, 0)T .

Hence, we obtain the effective Hamiltonian in edge-statesubspace, which is block independent with the bulk andconsistent with the numerical result. Actually, the effec-tive edge Hamiltonian is exactly the transposition of Hx

under OBC, which in momentum space reads

Heff (kx) = 2tx cos kxτ0 + 2igx sin kxτz. (45)

The complex energy spectrum of Heff (kx) underPBC [orange loop in Fig. 4(a)] surrounds the skin bulkcomplex spectrum under OBC [black part in Fig. 4(a)],which is identical with the second-order corner-localizedmodes under full OBC [central part in Fig. 4(b)]. The

effective Hamiltonian Heff (kx) is actually two decoupledHatano-Nelson models [88] with opposite nonreciprocity.Hence, thanks to the left (right) localization behavior ofthe skin modes for the two Hatano-Nelson models, we canexactly deduce the low-left (up-right) corner modes [86].We emphasize that these corner modes in this modelare categorized into hybrid x-skin and y-topological STmodes [84]. Nevertheless, the TT zero modes localizedat the same corners appear if the lattice site number isodd, since the zero edge state exists in the Hatano-Nelsonmodel with odd number of lattice sites. The two TT zeromodes localized at the low-left and up-right corners readas

|ψLL〉 =

Lx+12∑

x=1

Ly+1

2∑y=1

ρ2x−1β2y−1 |φL〉 |2x− 1〉 |2y − 1〉 ,

|ψRR〉 =

Lx+12∑

x=1

Ly+1

2∑y=1

ρLx−2xβLy−2y |φR〉 |2x− 1〉 |2y − 1〉 ,

(46)

where ρ = β = i√

tx−gxtx+gx

and |φL〉 = |2〉 , |φR〉 = |1〉are the internal degrees of freedom. Similarly, the twonumerical TT corner modes are a linear combination oftwo analytical solutions. In addition, the extended Her-

mitian Hamiltonian of He(~k) is topologically character-ized only by chiral symmetry (for details in Ref. [86]).

This leads to the trivial topology of He(~k) (vanishingwinding number of spectra under full PBC) and non-trivial topology of edge-state subspace effective Hamilto-nian Heff (kx) (non-vanishing winding number of orangeloop in Fig. 4(a)), thus extrinsic feature of ST cornermodes. That is because the edges are topologically non-trivial while the bulk is trivial in the extrinsic higher-order topological phase [86].

IV. CONCLUSION AND DISCUSSION

In this paper, we construct the nested tight-bindingformalism, within which we deduce the second-ordercorner-localized modes in non-Hermitian systems. Uti-lizing this formalism, we have strictly illustrated pureSS modes for the simplest 2D model [Eq. (6)], the STcorner modes for the four-band model [Eq. (16)], and theextrinsic ST corner modes for the 2D model [Eq. (42)].Additionally, we have obtained the analytical solutionsof zero-energy TT corner modes for the four-band model[Eq. (16)] and the 2D model [Eq. (42)]. We also clarifythe meaning and configuration of TT , ST , and SS cornermodes. Not distinguishing the zero- and nonzero-energyedge states, we conclude that the corner modes are classi-fied into three types: (i) The pure second-order skin effect(SS) modes are the result of contribution from two di-rections with first-order skin effect. (ii) The pure second-order topological (TT ) corner modes, inherited from Her-

10

mitian counterpart, are the result of contributions fromtwo directions with topological edge states. Note that weshould distinguish the topology of edge states from thatof skin effect, in which the former is inherited from Her-mitian counterpart and the latter is a pure non-Hermitianconsequence. (iii) The most charming skin-topological(ST ) modes are result of contribution from two direc-tions with topological edge states and skin effect, respec-tively; in other words, the Hermitian ramification andpure non-Hermitian consequence for each of two direc-tions, respectively.

The gapped edge-localized states of the Benalcazar-Bernevig-Hughes (BBH) model (Hermitian counterpartof the non-Hermitian four-band model) are protectedby Wannier band topology instead of bulk-band topol-ogy, which reveals a new symmetry-protected topologi-cal (SPT) phase in higher-order systems [21]. The non-Hermitian extension of this new SPT phase will be givenin other work. Our nested tight-binding formalism fornon-Hermitian higher-order topological insulators natu-rally applies to a Hermitian system, such as the BBHmodel, of which we can easily obtain the four second-order corner modes. In addition, a 2D non-Hermitianmodel given in a recent related work [90], can also beanalytically studied by our nested tight-binding formal-ism. The study of a more general model utilizing ourformalism is left for future work. We believe that thereexists the mixture of bulk and edge in one direction dueto the failure of block-diagonal of effective HamiltonianHeff for a more general 2D model, and it will lead tomore possible corner and edge states under full OBC.

ACKNOWLEDGMENTS

The authors thank Haoshu Li, Shuxuan Wang and Zhi-wei Yin for helpful discussions. This work was supportedby NSFC Grant No.11275180.

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,

H =∑ij,µν

c†iµHij,µν(k2, ...kd, λ′s)cjν , (A1)

where k2, ...kd, λ′s are all parameters. The Hamiltonian

of a 1D tight-binding model, with range of hopping Rand internal degrees of freedom q per unit cell, is

H =

L∑n=1

R∑i=−R

q∑µ,ν=1

cµ†n+iti,µν cνn. (A2)

We assume the solution as

|Φ〉 =

L∑n=1

|φn〉 |n〉 =

L∑n=1

q∑µ=1

βnφµ |µ〉 |n〉 , (A3)

and the Schrodinger equation is H |Φ〉 = E |Φ〉. We ob-tain the bulk equation

q∑ν=1

H(β)µνφν :=

q∑ν=1

R∑i=−R

ti,µνβiφν = Eφµ (A4)

and the secular equation

det(

R∑i=−R

ti,µνβi − E) = 0. (A5)

From the above linear equation set of φ′s, we can linearlyexpress the (q − 1) φ′s by the remaining one

φµ = Jνµ(β)φν , µ = 1, 2, . . . ν, . . . , q; ν = 1, 2, . . . , q,(A6)

The secular equation of the bulk equation can be solved,resulting in 2qR roots of β in general. We briefly ig-nore the multiple roots case (it has been well studied inRef. [89]). Now the full solution is

|Φ〉 =

L∑n=1

q∑µ=1

|φnµ〉 |n〉 =

L∑n=1

q∑µ=1

2qR∑j=1

βnj φjµ |µ〉 |n〉 .

(A7)Imposing the boundary condition both on the left andthe right boundaries,

R∑i=−s

ti |φs+i+1〉 = E |φ1+s〉 ,

s∑i=−R

ti |φL−s+i〉 = E |φL−s〉 , (A8)

where s = 0, 1, . . . , (R− 1). They can reduce to [91]

|φ0〉 = |φ−1〉 = . . . = |φ−R+1〉 = 0,

|φL+1〉 = |φL+2〉 = . . . = |φL+R〉 = 0. (A9)

We obtain

2qR∑j=1

β−sj φjµ = 0; s = 0, 1, . . . , (R− 1);µ = 1, 2, . . . , q,

2qR∑j=1

βL+sj φjµ = 0; s = 1, . . . , R;µ = 1, 2, . . . , q. (A10)

Using Eq. (A6) for any ν, we obtain

2qR∑j=1

fsµ(βj , E)φjν = 0; s = 0, 1, . . . , (R− 1);µ = 1, 2, . . . , q,

2qR∑j=1

gsµ(βj , E)βLj φjν = 0; s = 1, . . . , R;µ = 1, 2, . . . , q,(A11)

11

where

fsµ(βj , E) = Jνµ(βj)β−sj ,

gsµ(βj , E) = Jνµ(βj)βsj . (A12)

We can denote the 2qR functions fsµ and gsµ asfj , gj , j = 1, 2, . . . , qR respectively. The boundary condi-tions require [51]

det

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

f1(β1, E) . . . f1(β2qR, E)...

. . ....

fqR(β1, E) . . . fqR(β2qR, E)g1(β1, E)βL1 . . . g1(β2qR, E)βL2qR

.... . .

...gqR(β1, E)βL1 . . . gqR(β2qR, E)βL2qR

∣∣∣∣∣∣∣∣∣∣∣∣∣∣= 0. (A13)

We number the solutions satisfying |β1| 6 ... 6 |βqR| 6|βqR+1| 6 ... 6 |β2qR| and take limit L → ∞. If |βqR| <|βqR+1|, only one leading term survives in Eq. (A13),

F (βi∈P1, βj∈Q1

, E) := det

∣∣∣∣∣∣∣f1(β1, E) . . . f1(βqR, E)

.... . .

...fqR(β1, E) . . . fqR(βqR, E)

∣∣∣∣∣∣∣×det

∣∣∣∣∣∣∣g1(βqR+1, E) . . . g1(β2qR, E)

.... . .

...gqR(βqR+1, E) . . . gqR(β2qR, E)

∣∣∣∣∣∣∣ = 0, (A14)

where P1 = {β1, . . . , βqR} , Q1 = {βqR+1, . . . , β2qR}. The

above equation gives discrete β′s, deducing the edge

states isolated from the continuous bulk states.

If |βqR| = |βqR+1|, two leading terms survive. Let P0 ={β1, . . . , βqR−1, βqR+1} , Q0 = {βqR, βqR+2 . . . , β2qR},then the continuous β

′s are given [51]

− F (βi∈P1, βj∈Q1

, E)

F (βi∈P0 , βj∈Q0 , E)=

(βqRβqR+1

)L. (A15)

We can obtain the bulk band spectra (or continuous bandspectra) and generalized Brillouin zone (GBZ) [87] as

Ebulk = {E ∈ C : |βqR(E)| = |βqR+1(E)|} ,Cβ = {β ∈ C : ∀E ∈ Ebulk, |βqR(E)| = |βqR+1(E)|} .

(A16)

We emphasize that the GBZs depend on Riemann en-ergy spectra sheets (i.e., complex energy bands) Eµ withµ = 1, 2, . . . , q in general. In other words, there are qGBZs Cµβ one-to-one corresponding to q Riemann energyspectra sheets Eµ. However, the multiple GBZs are de-generate in some simple model, e.g., the non-HermitianSSH model [47]. In this paper, we only consider the de-generate GBZs or the single-band model and leave themultiple GBZs for numerical calculation in other work.

The above process to solve the eigenstates in non-Hermitian system is the non-Bloch band theory withoutany symmetry constraint proposed in Ref. [51], whichhas been extended to symplectic class [91, 92] and Z2

skin effect [57] recently.

Appendix B: Biorthogonal diagonalization of thesingle y-layer Hamiltonian

The qLx eigenvalue solutions can also be written asfermionic creation operators

ΦR,µ†α,y =

Lx∑x=1

q∑ν=1

φR,µναx cν†x,y, (B1)

where φR,µναx is the xν-th column component of αµ-thright eigenstate for generic non-Hermitian system. Wedefine the right eigenstate matrix

UR =

φR,1111 . . . φR,11Lxq...

......

φR,Lxq11 . . . φR,LxqLxq

(B2)

and

cy = (c11,y, . . . , cq1,y, . . . , c

1Lx,y, . . . , c

qLx,y

)T ,

c†y = (c1†1,y, . . . , cq†1,y, . . . , c

1†Lx,y

, . . . , cq†Lx,y),

ΦR†y = (ΦR,1†1,y , . . . , ΦR,q†1,y , . . . , ΦR,1†Lx,y, . . . , ΦR,q†Lx,y

),

ΦLy = (ΦL,11,y , . . . , ΦL,q1,y , . . . , Φ

L,1Lx,y

, . . . , ΦL,qLx,y)T , (B3)

then

ΦR†y = c†yUR. (B4)

From the eigenequation of H†, we can obtain the lefteigenstates with the equations

ΦL†y = c†yUL,

ΦLy = U†Lcy (B5)

and the biorthogonal relation

URU†L = ULU

†R = 1. (B6)

The inverse relation between two fermionic operators is

c†y = ΦR†y U†L,

cy = URΦLy . (B7)

The result transformed to the biorthogonal basis of thesingle y-layer Hamiltonian Hy is

ε = U†LHyUR,

Hy = ΦR†y εΦLy . (B8)

12

where

Hy =

T x0 . . . T xRx . . . 0...

. . ....

. . ....

T x−Rx . . . T x0 . . . T xRx...

. . ....

. . ....

0 . . . T x−Rx . . . T x0

,

T xi =

Txi,11 . . . T xi,1q...

. . ....

T xi,q1 . . . T xi,qq

,

ε =

ε1(β1) . . . 0 . . . 0 . . . 0...

. . ....

. . ....

. . ....

0 . . . εq(β1) . . . 0 . . . 0...

. . ....

. . ....

. . ....

0 . . . 0 . . . ε1(βLx) . . . 0...

. . ....

. . ....

. . ....

0 . . . 0 . . . 0 . . . εq(βLx)

.(B9)

Note that the lowest q eigenvalues are edge states energieswhich deduce the ST and TT corner modes. For Hermi-tian cases, the biorthogonal relation reduces to U†L = U−1Rand the diagonalization process reduces to the standardone in linear algebra, ε = U−1HyU .

Appendix C: Other parameter choices for thefour-band model

Single nonreciprocity case: δ1 = δ2 = −δ3 = δ4 = γ.The net nonreciprocity only exists along the y direction.The Mxy is broken in this case. The corner modes con-tain: ST modes and four TT zero modes, while the SSmodes are absent. The forms of edge states read

|φ±L 〉sn = |u1〉 ± r−1 |u2〉 ,|φ±R〉sn = |v1〉 ± r−1 |v2〉 . (C1)

The edge effective Hamiltonian is then deduced

t±L =λ

2r∓[

1 ∓1±1 −1

], (C2)

t±R =λ

2r±[

1 ±1∓1 −1

]. (C3)

Double reciprocity case: δ1 = δ2 = −δ3 = −δ4 = γ.The Mxy is also broken in this case. The numerical re-sults are provided in Ref. [84] for the TT and ST modes,while the SS corner modes are absent.

Asymmetry case: δ1 = δ2 = 0 or δ3 = δ4 = 0 while theother direction is nonreciprocal. The mirror symmetryMx or My is restored. The TT and ST corner modes arepresent while the SS modes absent.

Hermitian case: δ1 = δ2 = δ3 = δ4 = 0. Both Mx andMy are restored as well as the fourfold rotation symme-try C4; the only existent corner modes are the TT zeromodes.

Non-Hermitian case with on-site gain and loss [82]:δ1 = δ2 = δ3 = δ4 = 0 but with the additional term−iuτz. The C4 is restored and the only existing cornermodes are the in-gap TT modes.

Appendix D: Edge states for 2D extrinsic model

The zero edge modes have very simple forms forHy(ky)in the main text when the number of lattice sites is odd.The bulk equation for the Hamiltonian is

t+y φy+1 + t−y φy−1 = (t+y β + t−y β−1)βyφ = 0, (D1)

where

t+y =

[0 −ty − gy

ty − gy 0

],

t−y =

[0 −ty + gy

ty + gy 0

]. (D2)

With the boundary conditions

t+y φ2 = 0,

t−y φLy−1 = 0, (D3)

the amplitudes for exact zero edge states are destroyed oneven lattice sites, which is consistent with the numericalresults. A similar amplitude destruction is also found inRef. [46]. Utilizing the bulk equation, we obtain β =

i√

ty−gyty+gy

and two edge states given by Eq. (44) in the

main text. In addition, the edge states form [Eq. (44)] isvalid for the Hamiltonian Hx(kx) along the x directionunder OBC for odd sites.

For an even number of lattice sites, the edge solutionsof Hy are the linear combination of two localized edgestates, of which the left- (right-)-localized edge state isdestroyed on even (odd) lattice sites. Consequently, weobtain the effective edge-state subspace Hamiltonian

Heff (kx) = 2tx cos kxτ0 + 2igx sin kxτx. (D4)

From the above effective Hamiltonian, we find that theST modes are present, while the TT zero modes are ab-sent with an even number of lattice sites.

[1] M. Z. Hasan and C. L. Kane, “Colloquium: Topologicalinsulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).

[2] Xiao-Liang Qi and Shou-Cheng Zhang, “Topological in-sulators and superconductors,” Rev. Mod. Phys. 83,

13

1057–1110 (2011).[3] Liang Fu and C. L. Kane, “Topological insulators with

inversion symmetry,” Phys. Rev. B 76, 045302 (2007).[4] Xiao-Liang Qi, Taylor L. Hughes, and Shou-Cheng

Zhang, “Topological field theory of time-reversal invari-ant insulators,” Phys. Rev. B 78, 195424 (2008).

[5] Chao-Xing Liu, Xiao-Liang Qi, HaiJun Zhang, Xi Dai,Zhong Fang, and Shou-Cheng Zhang, “Model hamilto-nian for topological insulators,” Phys. Rev. B 82, 045122(2010).

[6] Jeffrey C. Y. Teo and C. L. Kane, “Topological defectsand gapless modes in insulators and superconductors,”Phys. Rev. B 82, 115120 (2010).

[7] Roger S. K. Mong and Vasudha Shivamoggi, “Edge statesand the bulk-boundary correspondence in dirac hamilto-nians,” Phys. Rev. B 83, 125109 (2011).

[8] Taylor L. Hughes, Emil Prodan, and B. Andrei Bernevig,“Inversion-symmetric topological insulators,” Phys. Rev.B 83, 245132 (2011).

[9] Andreas P. Schnyder, Shinsei Ryu, Akira Furusaki, andAndreas W. W. Ludwig, “Classification of topologicalinsulators and superconductors in three spatial dimen-sions,” Phys. Rev. B 78, 195125 (2008).

[10] Takahiro Morimoto and Akira Furusaki, “Topologicalclassification with additional symmetries from clifford al-gebras,” Phys. Rev. B 88, 125129 (2013).

[11] Ching-Kai Chiu, Hong Yao, and Shinsei Ryu, “Classi-fication of topological insulators and superconductors inthe presence of reflection symmetry,” Phys. Rev. B 88,075142 (2013).

[12] Ken Shiozaki and Masatoshi Sato, “Topology of crys-talline insulators and superconductors,” Phys. Rev. B 90,165114 (2014).

[13] Ching-Kai Chiu and Andreas P. Schnyder, “Classificationof reflection-symmetry-protected topological semimetalsand nodal superconductors,” Phys. Rev. B 90, 205136(2014).

[14] Ching-Kai Chiu, Jeffrey C. Y. Teo, Andreas P. Schnyder,and Shinsei Ryu, “Classification of topological quantummatter with symmetries,” Rev. Mod. Phys. 88, 035005(2016).

[15] Ken Shiozaki, Masatoshi Sato, and Kiyonori Gomi,“Topology of nonsymmorphic crystalline insulators andsuperconductors,” Phys. Rev. B 93, 195413 (2016).

[16] Jorrit Kruthoff, Jan de Boer, Jasper van Wezel,Charles L. Kane, and Robert-Jan Slager, “Topologicalclassification of crystalline insulators through band struc-ture combinatorics,” Phys. Rev. X 7, 041069 (2017).

[17] Ken Shiozaki, Masatoshi Sato, and Kiyonori Gomi,“Topological crystalline materials: General formulation,module structure, and wallpaper groups,” Phys. Rev. B95, 235425 (2017).

[18] Eyal Cornfeld and Adam Chapman, “Classification ofcrystalline topological insulators and superconductorswith point group symmetries,” Phys. Rev. B 99, 075105(2019).

[19] Robert-Jan Slager, Louk Rademaker, Jan Zaanen, andLeon Balents, “Impurity-bound states and green’s func-tion zeros as local signatures of topology,” Phys. Rev. B92, 085126 (2015).

[20] Frank Schindler, Ashley M. Cook, Maia G. Vergniory,Zhijun Wang, Stuart S. P. Parkin, B. Andrei Bernevig,and Titus Neupert, “Higher-order topological insula-tors,” Science Advances 4, eaat0346 (2018).

[21] Wladimir A. Benalcazar, B. Andrei Bernevig, and Tay-lor L. Hughes, “Electric multipole moments, topologicalmultipole moment pumping, and chiral hinge states incrystalline insulators,” Phys. Rev. B 96, 245115 (2017).

[22] Sheng-Jie Huang, Hao Song, Yi-Ping Huang, andMichael Hermele, “Building crystalline topologicalphases from lower-dimensional states,” Phys. Rev. B 96,205106 (2017).

[23] Hassan Shapourian, Yuxuan Wang, and Shinsei Ryu,“Topological crystalline superconductivity and second-order topological superconductivity in nodal-loop mate-rials,” Phys. Rev. B 97, 094508 (2018).

[24] Motohiko Ezawa, “Magnetic second-order topological in-sulators and semimetals,” Phys. Rev. B 97, 155305(2018).

[25] Max Geier, Luka Trifunovic, Max Hoskam, and Piet W.Brouwer, “Second-order topological insulators and su-perconductors with an order-two crystalline symmetry,”Phys. Rev. B 97, 205135 (2018).

[26] Eslam Khalaf, “Higher-order topological insulators andsuperconductors protected by inversion symmetry,”Phys. Rev. B 97, 205136 (2018).

[27] Flore K. Kunst, Guido van Miert, and Emil J. Bergholtz,“Lattice models with exactly solvable topological hingeand corner states,” Phys. Rev. B 97, 241405 (2018).

[28] Akishi Matsugatani and Haruki Watanabe, “Connectinghigher-order topological insulators to lower-dimensionaltopological insulators,” Phys. Rev. B 98, 205129 (2018).

[29] Linhu Li, Muhammad Umer, and Jiangbin Gong, “Di-rect prediction of corner state configurations from edgewinding numbers in two- and three-dimensional chiral-symmetric lattice systems,” Phys. Rev. B 98, 205422(2018).

[30] Mao Lin and Taylor L. Hughes, “Topological quadrupolarsemimetals,” Phys. Rev. B 98, 241103 (2018).

[31] Ryo Okugawa, Shin Hayashi, and Takeshi Nakanishi,“Second-order topological phases protected by chiralsymmetry,” Phys. Rev. B 100, 235302 (2019).

[32] Yutaro Tanaka, Ryo Takahashi, and Shuichi Murakami,“Appearance of hinge states in second-order topologicalinsulators via the cutting procedure,” Phys. Rev. B 101,115120 (2020).

[33] Josias Langbehn, Yang Peng, Luka Trifunovic, Felix vonOppen, and Piet W. Brouwer, “Reflection-symmetricsecond-order topological insulators and superconduc-tors,” Phys. Rev. Lett. 119, 246401 (2017).

[34] Liang Fu, “Topological crystalline insulators,” Phys. Rev.Lett. 106, 106802 (2011).

[35] Zhida Song, Zhong Fang, and Chen Fang, “(d − 2)-dimensional edge states of rotation symmetry protectedtopological states,” Phys. Rev. Lett. 119, 246402 (2017).

[36] Moon Jip Park, Youngkuk Kim, Gil Young Cho, andSungBin Lee, “Higher-order topological insulator intwisted bilayer graphene,” Phys. Rev. Lett. 123, 216803(2019).

[37] Zhongbo Yan, Fei Song, and Zhong Wang, “Majoranacorner modes in a high-temperature platform,” Phys.Rev. Lett. 121, 096803 (2018).

[38] Qiyue Wang, Cheng-Cheng Liu, Yuan-Ming Lu, andFan Zhang, “High-temperature majorana corner states,”Phys. Rev. Lett. 121, 186801 (2018).

[39] Kenta Esaki, Masatoshi Sato, Kazuki Hasebe, andMahito Kohmoto, “Edge states and topological phases innon-hermitian systems,” Phys. Rev. B 84, 205128 (2011).

14

[40] Kohei Kawabata, Ken Shiozaki, and Masahito Ueda,“Anomalous helical edge states in a non-hermitian cherninsulator,” Phys. Rev. B 98, 165148 (2018).

[41] V. M. Martinez Alvarez, J. E. Barrios Vargas, andL. E. F. Foa Torres, “Non-hermitian robust edge states inone dimension: Anomalous localization and eigenspacecondensation at exceptional points,” Phys. Rev. B 97,121401 (2018).

[42] Tony E. Lee, “Anomalous edge state in a non-hermitianlattice,” Phys. Rev. Lett. 116, 133903 (2016).

[43] Ye Xiong, “Why does bulk boundary correspondence failin some non-hermitian topological models,” Journal ofPhysics Communications 2, 035043 (2018).

[44] Daniel Leykam, Konstantin Y. Bliokh, Chunli Huang,Y. D. Chong, and Franco Nori, “Edge modes, degenera-cies, and topological numbers in non-hermitian systems,”Phys. Rev. Lett. 118, 040401 (2017).

[45] Huitao Shen, Bo Zhen, and Liang Fu, “Topological bandtheory for non-hermitian hamiltonians,” Phys. Rev. Lett.120, 146402 (2018).

[46] Flore K. Kunst, Elisabet Edvardsson, Jan Carl Budich,and Emil J. Bergholtz, “Biorthogonal bulk-boundary cor-respondence in non-hermitian systems,” Phys. Rev. Lett.121, 026808 (2018).

[47] Shunyu Yao and Zhong Wang, “Edge states and topo-logical invariants of non-hermitian systems,” Phys. Rev.Lett. 121, 086803 (2018).

[48] Shunyu Yao, Fei Song, and Zhong Wang, “Non-hermitian chern bands,” Phys. Rev. Lett. 121, 136802(2018).

[49] Ching Hua Lee and Ronny Thomale, “Anatomy of skinmodes and topology in non-hermitian systems,” Phys.Rev. B 99, 201103 (2019).

[50] S. Longhi, “Topological phase transition in non-hermitian quasicrystals,” Phys. Rev. Lett. 122, 237601(2019).

[51] Kazuki Yokomizo and Shuichi Murakami, “Non-blochband theory of non-hermitian systems,” Phys. Rev. Lett.123, 066404 (2019).

[52] Kohei Kawabata, Takumi Bessho, and Masatoshi Sato,“Classification of exceptional points and non-hermitiantopological semimetals,” Phys. Rev. Lett. 123, 066405(2019).

[53] Fei Song, Shunyu Yao, and Zhong Wang, “Non-hermitian skin effect and chiral damping in open quan-tum systems,” Phys. Rev. Lett. 123, 170401 (2019).

[54] Ken-Ichiro Imura and Yositake Takane, “Generalizedbulk-edge correspondence for non-hermitian topologicalsystems,” Phys. Rev. B 100, 165430 (2019).

[55] Nobuyuki Okuma and Masatoshi Sato, “Topologicalphase transition driven by infinitesimal instability: Majo-rana fermions in non-hermitian spintronics,” Phys. Rev.Lett. 123, 097701 (2019).

[56] Dan S. Borgnia, Alex Jura Kruchkov, and Robert-JanSlager, “Non-hermitian boundary modes and topology,”Phys. Rev. Lett. 124, 056802 (2020).

[57] Nobuyuki Okuma, Kohei Kawabata, Ken Shiozaki, andMasatoshi Sato, “Topological origin of non-hermitianskin effects,” Phys. Rev. Lett. 124, 086801 (2020).

[58] Haoran Xue, Qiang Wang, Baile Zhang, and Y. D.Chong, “Non-hermitian dirac cones,” Phys. Rev. Lett.124, 236403 (2020).

[59] Zongping Gong, Yuto Ashida, Kohei Kawabata, KazuakiTakasan, Sho Higashikawa, and Masahito Ueda, “Topo-

logical phases of non-hermitian systems,” Phys. Rev. X8, 031079 (2018).

[60] Kohei Kawabata, Ken Shiozaki, Masahito Ueda, andMasatoshi Sato, “Symmetry and topology in non-hermitian physics,” Phys. Rev. X 9, 041015 (2019).

[61] Simon Malzard, Charles Poli, and Henning Schome-rus, “Topologically protected defect states in open pho-tonic systems with non-hermitian charge-conjugation andparity-time symmetry,” Phys. Rev. Lett. 115, 200402(2015).

[62] H. J. Carmichael, “Quantum trajectory theory for cas-caded open systems,” Phys. Rev. Lett. 70, 2273–2276(1993).

[63] Hui Cao and Jan Wiersig, “Dielectric microcavities:Model systems for wave chaos and non-hermitianphysics,” Rev. Mod. Phys. 87, 61–111 (2015).

[64] Tony E. Lee and Ching-Kit Chan, “Heralded magnetismin non-hermitian atomic systems,” Phys. Rev. X 4,041001 (2014).

[65] Youngwoon Choi, Sungsam Kang, Sooin Lim, WookraeKim, Jung-Ryul Kim, Jai-Hyung Lee, and Kyung-won An, “Quasieigenstate coalescence in an atom-cavityquantum composite,” Phys. Rev. Lett. 104, 153601(2010).

[66] Tony E. Lee, Florentin Reiter, and Nimrod Moiseyev,“Entanglement and spin squeezing in non-hermitianphase transitions,” Phys. Rev. Lett. 113, 250401 (2014).

[67] K. G. Makris, R. El-Ganainy, D. N. Christodoulides, andZ. H. Musslimani, “Beam dynamics in PT symmetricoptical lattices,” Phys. Rev. Lett. 100, 103904 (2008).

[68] S. Longhi, “Bloch oscillations in complex crystals withPT symmetry,” Phys. Rev. Lett. 103, 123601 (2009).

[69] Shachar Klaiman, Uwe Gunther, and Nimrod Moiseyev,“Visualization of branch points in PT -symmetric waveg-uides,” Phys. Rev. Lett. 101, 080402 (2008).

[70] S. Bittner, B. Dietz, U. Gunther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schafer, “PT symmetry andspontaneous symmetry breaking in a microwave billiard,”Phys. Rev. Lett. 108, 024101 (2012).

[71] A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti,M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N.Christodoulides, “Observation of PT -symmetry break-ing in complex optical potentials,” Phys. Rev. Lett. 103,093902 (2009).

[72] M. Liertzer, Li Ge, A. Cerjan, A. D. Stone, H. E. Tureci,and S. Rotter, “Pump-induced exceptional points inlasers,” Phys. Rev. Lett. 108, 173901 (2012).

[73] Zin Lin, Hamidreza Ramezani, Toni Eichelkraut,Tsampikos Kottos, Hui Cao, and Demetrios N.Christodoulides, “Unidirectional invisibility induced byPT -symmetric periodic structures,” Phys. Rev. Lett.106, 213901 (2011).

[74] B. Peng, S. K. Ozdemir, S. Rotter, H. Yilmaz,M. Liertzer, F. Monifi, C. M. Bender, F. Nori,and L. Yang, “Loss-induced suppression and re-vival of lasing,” Science 346, 328–332 (2014),https://science.sciencemag.org/content/346/6207/328.full.pdf.

[75] Liang Feng, Zi Jing Wong, Ren-Min Ma, Yuan Wang,and Xiang Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014),https://science.sciencemag.org/content/346/6212/972.full.pdf.

[76] Kohei Kawabata, Yuto Ashida, and Masahito Ueda,“Information retrieval and criticality in parity-time-

15

symmetric systems,” Phys. Rev. Lett. 119, 190401(2017).

[77] Tomoki Ozawa, Hannah M. Price, Alberto Amo, NathanGoldman, Mohammad Hafezi, Ling Lu, Mikael C.Rechtsman, David Schuster, Jonathan Simon, Oded Zil-berberg, and Iacopo Carusotto, “Topological photonics,”Rev. Mod. Phys. 91, 015006 (2019).

[78] Stefano Longhi, “Parity-time symmetry meets photonics:A new twist in non-hermitian optics,” EPL (EurophysicsLetters) 120, 64001 (2017).

[79] W D Heiss, “The physics of exceptional points,” Journalof Physics A: Mathematical and Theoretical 45, 444016(2012).

[80] Elisabet Edvardsson, Flore K. Kunst, and Emil J.Bergholtz, “Non-hermitian extensions of higher-ordertopological phases and their biorthogonal bulk-boundarycorrespondence,” Phys. Rev. B 99, 081302 (2019).

[81] Motohiko Ezawa, “Non-hermitian boundary and inter-face states in nonreciprocal higher-order topological met-als and electrical circuits,” Phys. Rev. B 99, 121411(2019).

[82] Tao Liu, Yu-Ran Zhang, Qing Ai, Zongping Gong,Kohei Kawabata, Masahito Ueda, and Franco Nori,“Second-order topological phases in non-hermitian sys-tems,” Phys. Rev. Lett. 122, 076801 (2019).

[83] Zhiwang Zhang, Marıa Rosendo Lopez, Ying Cheng, Xi-aojun Liu, and Johan Christensen, “Non-hermitian sonicsecond-order topological insulator,” Phys. Rev. Lett.122, 195501 (2019).

[84] Ching Hua Lee, Linhu Li, and Jiangbin Gong, “Hybridhigher-order skin-topological modes in nonreciprocal sys-

tems,” Phys. Rev. Lett. 123, 016805 (2019).[85] M. Michael Denner, Anastasiia Skurativska, Frank

Schindler, Mark H. Fischer, Ronny Thomale, TomasBzdusek, and Titus Neupert, “Exceptional topologi-cal insulators,” (2020), arXiv:2008.01090 [cond-mat.mes-hall].

[86] Ryo Okugawa, Ryo Takahashi, and Kazuki Yokomizo,“Second-order topological non-hermitian skin effects,”Phys. Rev. B 102, 241202 (2020).

[87] Zhesen Yang, Kai Zhang, Chen Fang, and JiangpingHu, “Non-hermitian bulk-boundary correspondence andauxiliary generalized brillouin zone theory,” Phys. Rev.Lett. 125, 226402 (2020).

[88] Naomichi Hatano and David R. Nelson, “Vortex pinningand non-hermitian quantum mechanics,” Phys. Rev. B56, 8651–8673 (1997).

[89] Abhijeet Alase, Emilio Cobanera, Gerardo Ortiz, andLorenza Viola, “Generalization of bloch’s theorem for ar-bitrary boundary conditions: Theory,” Phys. Rev. B 96,195133 (2017).

[90] Kohei Kawabata, Masatoshi Sato, and Ken Shiozaki,“Higher-order non-hermitian skin effect,” Phys. Rev. B102, 205118 (2020).

[91] Kohei Kawabata, Nobuyuki Okuma, and MasatoshiSato, “Non-bloch band theory of non-hermitian hamilto-nians in the symplectic class,” Phys. Rev. B 101, 195147(2020).

[92] Yifei Yi and Zhesen Yang, “Non-hermitian skin modesinduced by on-site dissipations and chiral tunneling ef-fect,” Phys. Rev. Lett. 125, 186802 (2020).