Second-Order Topological Phases in Non-Hermitian Systems

Tao Liu,1, ∗ Yu-Ran Zhang,1, 2 Qing Ai,1, 3 Zongping Gong,4, †

Kohei Kawabata,4, ‡ Masahito Ueda,4, 5, § and Franco Nori1, 6, ¶

1Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan2Beijing Computational Science Research Center, Beijing 100193, China3Department of Physics, Applied Optics Beijing Area Major Laboratory,

Beijing Normal University, Beijing 100875, China4Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

5RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan6Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA

A d-dimensional second-order topological insulator (SOTI) can host topologically protected (d−2)-dimensional gapless boundary modes. Here we show that a 2D non-Hermitian SOTI can hostzero-energy modes at its corners. In contrast to the Hermitian case, these zero-energy modes canbe localized only at one corner. A 3D non-Hermitian SOTI is shown to support second-orderboundary modes, which are localized not along hinges but anomalously at a corner. The usual bulk-corner (hinge) correspondence in the second-order 2D (3D) non-Hermitian system breaks down.The winding number (Chern number) based on complex wavevectors is used to characterize thesecond-order topological phases in 2D (3D). A possible experimental situation with ultracold atomsis also discussed. Our work lays the cornerstone for exploring higher-order topological phenomenain non-Hermitian systems.

Introduction.—Recent years have witnessed a surgeof theoretical and experimental interest in studyingtopological phases [1–3] in insulators [4–9], supercon-ductors [10–12], ultracold atoms [13–18] and classicalwaves [19–22]. These topologically nontrivial phasesare characterized by the topological index of gappedbulk energy bands and exhibit gapless states on theirboundaries. Such gapless boundary states cannot begapped out by local perturbations that preserve bothbulk gap and symmetry.

Topological phases have widely been studied inclosed systems, which are described by HermitianHamiltonians featuring real eigenenergies and orthogonaleigenstates. Recently, there has been a great dealof effort in exploring topological invariants of opensystems governed by non-Hermitian operators [23, 24].Non-Hermitian Hamiltonians can find applications in awide range of systems including optical and mechanicalstructures subjected to gain and loss [25–40], and solid-state systems with finite quasiparticle lifetimes [41–45]. In particular, topological phases of non-HermitianHamiltonians have recently been investigated in thesesystems [43–70]. The most prominent feature of non-Hermitian Hamiltonians is the existence of exceptionalpoints (EPs), where more than one eigenstate coalesces[24, 71, 72]. This coalescence of eigenstates at EPs makesthe corresponding eigenspace no longer complete, and thenon-Hermitian Hamiltonian becomes non-diagonalizable.These unique features of EPs can lead to rich topologicalfeatures in non-Hermitian topological systems with nocounterpart in Hermitian cases such as Weyl exceptionalrings [51], bulk Fermi arcs, and half-integer topologicalcharges [57]. Furthermore, the interplay between non-Hermiticity and topology can lead to the breakdown

of the usual bulk-boundary correspondence [50, 52, 58,63, 65–67] due to the non-Bloch-wave behavior of open-boundary eigenstates, where the conventional Blochwavefunctions do not precisely describe topological phasetransitions under the open boundary conditions. Thenon-Bloch winding (Chern) number defined via complexwavevectors in 1D (2D) has recently been introduced tofill this gap [65, 66].

More recently, the concept of topological insulators(TIs) has been generalized to second-order [73–91] andthird-order [74, 92, 93] TIs in Hermitian systems. Incontrast to conventional first-order TIs, a d-dimensionalsecond-order topological insulator (SOTI) only hoststopologically protected (d − 2)-dimensional gaplessboundary states. For example, a 2D SOTI has zero-energy states localized at its corners, and a 3D SOTIhosts 1D gapless modes along its hinges. Therefore,the conventional bulk-boundary correspondence is nolonger applicable to SOTIs. Up to now, studies ofthe second-order and third-order topological phases havebeen restricted to Hermitian systems. We now ask: is itpossible for a non-Hermitian system to exhibit second-order topological phases? If yes, how can we define atopological invariant to characterize them?

In this Letter, we investigate 2D and 3D SOTIsdescribed by non-Hermitian Hamiltonians. Even thoughthe bulk bands are first-order topologically trivialinsulators, there are degenerate second-order boundstates. In contrast to the Hermitian case, these zero-energy states in 2D are localized only at one cornerprotected by mirror-rotation symmetry and sublatticesymmetry. Moreover, the second-order boundary modesin 3D are localized not along the hinges but anomalouslyat a corner. The winding number (Chern number)

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characterizes its second-order topological phase in 2D(3D), where the non-Bloch-wave behavior of open-boundary eigenstates is included due to the breakdownof the usual bulk-corner (hinge) correspondence insecond-order non-Hermitian systems. The proposednon-Hermitian model can experimentally be realized inultracold atoms.

2D SOTI.—We consider a 2D non-Hermitian Hamil-tonian H2D that respects both two-fold mirror-rotationsymmetry Mxy and sublattice symmetry S

MxyH2D (kx, ky)M−1xy = H2D (ky, kx) , (1)

SH (kx, ky)S−1 = −H (kx, ky) , (2)

and [S, Mxy] = 0. Note that the Hermitian counterpartwith the same symmetries was investigated in Ref. [81].Due to the mirror-rotation symmetry in Eq. (1), we canexpress the Hamiltonian H2D on the high-symmetry linekx = ky as

U−1H2D (k, k)U =

(H+ (k) 0

0 H− (k)

), (3)

where U is a unitary operator, and H±(k) acts onthe mirror-rotation subspace. Since H± (k) respectssublattice symmetry S ′ defined in each mirror-rotationsubspace [note that S in Eq. (2) is defined in the entirelattice space], we can define the winding number asfollows:

w± :=

∮BZ

dk

4πiTr

[S ′H−1

± (k)dH± (k)

dk

]. (4)

The topological index that characterizes the second-ordertopological phases in 2D is given by

w := w+ − w−. (5)

We investigate a concrete model of a 2D SOTI ona square lattice, where each unit cell contains fourorbitals and asymmetric particle hopping within eachunit cell is introduced, as shown in Fig. 1(a). The BlochHamiltonian is written as

H2D = [t+ λ cos(kx)] τx − [λ sin(kx) + iγ] τyσz

+ [t+ λ cos(ky)] τyσy + [λ sin(ky) + iγ] τyσx,(6)

where we have set the lattice constant a0 = 1, λ is areal-valued inter-cell hopping amplitude, t ± γ denotereal-valued asymmetric intra-cell hopping amplitudes,and σi and τi (i = x, y, z) are Pauli matrices forthe degrees of freedom within a unit cell. TheHamiltonian H2D can be implemented experimentallyusing ultracold atoms in optical lattices with engineereddissipation [see Fig. 1(b) and Sec. VIII in the

t +γ

λ

λ

3 1

42

t -γ

t - γ

t+γ

(a) (b)

FIG. 1. Non-Hermitian SOTI in 2D. (a) Tight-bindingrepresentation of the model [Eq. (6)] on a square lattice.Each unit cell contains four orbitals (blue solid circles). Theorange lines denote inter-cell coupling, and the red and blacklines with arrows represent asymmetric intra-cell hopping.The dashed lines indicate hopping terms with a negativesign, accounting for a flux of π piercing each plaquette. (b)Schematic illustration of a proposed experimental setup usingultracold atoms [94]. The primary lattice together with apair of Raman lasers gives rise to a Hermitian SOTI, wherethe Raman lasers are used for inducing effective particlehopping. The asymmetric hopping amplitudes are introducedvia coherent coupling to a dissipative auxiliary lattice.

-3

0

3R

e(E)

-0.3

0

0.3

Im(E

)

(a) (b)

0 ky-

(c) (d)

-3

0

3

Re(

E)

-0.3

0

0.3

Im(E

)

0 ky- 0 ky- 0 ky-

FIG. 2. Complex energy spectra of the non-Hermitian SOTIdescribed by Eq. (6) with open boundaries along the xdirection and periodic boundaries along the y direction. Theedge states (red curves) are gapped for (a,b) t = 0.6. No edgestates exist for (c,d) t = 2.0. An EP exists for t = λ+γ = 1.9,where a phase transition occurs. The number of unit cellsalong the x direction is N = 20 with λ = 1.5 and γ = 0.4.

supplemental material [94] for details]. The Hermitianpart of H2D(k) preserves mirror and four-fold rotationalsymmetries with Mx = τxσz, My = τxσx, andC4 = [(τx − iτy)σ0 − (τx + iτy)(iσy)] /2. While theyare broken by asymmetric hopping, H2D stays invariantunder sublattice symmetry S = τz and mirror-rotationsymmetry Mxy = C4My, and [S, Mxy] = 0.Bulk and edge states.—The upper and lower bands

E±(k) of H(k) are two-fold degenerate [94], and thesebands coalesce at EPs with E±(kEP) = 0 for t = ±λ ± γ

or ±√γ2 − λ2. Figure 2 shows the complex energy

spectra with open and periodic boundaries along the xand y directions, respectively. The non-Hermitian systemsupports gapped complex edge states for |t| < |γ| + |λ|,as shown in the red curves in Figs. 2(a) and (b). Onthe other hand, for |t| > |γ| + |λ|, there are no edgestates [see Figs. 2(c) and (d)]. In spite of their existence,edge states can continuously be absorbed into bulk bandsand therefore are not topologically protected. In fact,the bulk bands are topologically trivial, characterized by

3

10

y

1

20(a) (c) -9(b)

-1.3

0

1.3

Re(

E)

-5

0

5

Im(E

)

10

1 15 30n

0 0.5 1

1 15 30n10 20x1 1 10 20x

10

y

1

20(d)

0 0.15 0.3

FIG. 3. Corner states in the non-Hermitian SOTI describedby the Hamiltonian (6). (a) Probability density distributions∑4

i=1 |φR,i,n|2 (n is the index of an eigenstate and R specifiesa unit cell) of four zero-energy states under the open boundarycondition along the x and y directions. The zero-energymodes are localized only at the lower-left corner. (b,c) Realand imaginary parts of complex eigenenergies around zeroenergy. The red dots represent eigenenergies of the cornermodes. The imaginary parts of the bulk eigenenergies of afinite-size sample vanish over a wide range of parameters. (d)Probability density distribution of a typical bulk state underthe open boundary condition along the x and y directions.The bulk state is exponentially localized at the lower-leftcorner. The number of unit cells is 20 × 20 with t = 0.6,λ = 1.5 and γ = 0.4.

zero Chern number (see Sec. I in Ref. [94]) over the entirerange of parameters.

Corner states.—While the bulk bands of H(k) aretopologically trivial, the system with open-boundaryconditions in the x and y directions hosts four zero-energy modes at its corners, as shown in Figs. 3(a-c).Moreover, these zero-energy states are localized only atthe lower-left corner [see Fig. 3(a)]. Note that the mid-gap modes can be localized at the upper-right corner ifthe sign of hopping amplitude t is reversed (see Fig. S1in Ref. [94]). This mid-gap-state localization at onecorner results from the interplay between the symmetryMxy and non-Hermiticity, where each corner mode is asimultaneously topological state of two intersecting non-trivial edges (see Sec. III in Ref. [94]). Furthermore, thesecorner modes are topologically protected against disorderpreservingMxy symmetry and sublattice symmetry (seeSec. IV in Ref. [94]). Note that when the mirror-rotationsymmetry is broken, the mid-gap modes can be localizedat more than one corner, and the sites at which modelocalization occurs can be diagnosed by considering thetype of asymmetric hopping and non-Hermiticity in non-Hermitian SOTIs (see Sec. V in Ref. [94]).

Moreover, the bulk bands of the open-boundary systemare considerably different from those of the periodicsystem. As shown in Figs. 3(b) and (c), the bulkeigenenergies in the case of open boundaries are entirelyreal over a wide range of system parameters as aconsequence of pseudo-Hermiticity of the open-boundarysystem [94], while they are complex in the case of theperiodic boundaries. Furthermore, we find that, incontrast to the Hermitian SOTI, the bulk modes areexponentially localized at the lower-left corner due to thenon-Hermiticity caused by the asymmetric hopping (see

Sec. VI and VII in Ref. [94]), as shown in Fig. 3(d).Topological index.—The topology of the non-Hermitian

Hamiltonian H2D is characterized by the winding numberw [see Eqs.(1-5)]. One of the boundaries of thetopological-phase transition calculated by this index ist = λ+γ = 1.9 (i.e., one of the EPs) using the parametersin Fig. 2. However, numerical calculations for the open-boundary system show that corner states exist only fort <

√λ2 + γ2 ' 1.55. Therefore, this topological index

cannot correctly determine the phase boundary betweentopologically trivial and nontrivial regimes, indicatingthe breakdown of the usual bulk-corner correspondencein non-Hermitian systems. This breakdown resultsfrom the non-Bloch-wave behavior of open-boundaryeigenstates of a non-Hermitian Hamiltonian, as studiedin first-order topological insulators in Refs. [65, 66].To figure out this unexpected non-Bloch-wave behavior,complex wavevectors, instead of real ones, are suggestedfor defining the topological index of non-Hermitiansystems [65, 66]. Here we generalize this idea to the non-Hermitian SOTI (see Sec. VII in Ref. [94] for details).After replacing real wavevectors k with complex ones

k = (kx, ky)→ k = (kx − iln(β0), ky − iln(β0)), (7)

with β0 =√|(t− γ)/(t+ γ)|, the Hamiltonian H± for

H2D in Eq. (3) has the following forms

H±√2

=(t− γ + λβ0eik

)σ∓ +

(t+ γ +

λ

β0e−ik

)σ±,

(8)

where σ± = (σx ± iσy)/2. Note that the location ofthe mid-gap corner modes depends on β0: they arelocalized at the lower-left corners for β0 < 1, and atthe upper-right corners for β0 > 1. Figure 4(a) showsthe topological phase diagram. The number of zero-energy corner modes is counted as 2|w|. Furthermore,the phase boundaries are determined by t2 = λ2 + γ2

and t2 = γ2 − λ2, and the phase diagram contains thetrivial phase (w = 0) and the second-order topologicalphase (w = −2).3D SOTI.—We now consider a 3D non-Hermitian

Hamiltonian H3D that respects two-fold mirror-rotationsymmetry

MxyH3D (kx, ky, kz)M−1xy = H3D (ky, kx, kz) . (9)

Note that the Hermitian counterpart was investigated inRef. [82]. As in the 2D case, due to the mirror-rotationsymmetry in Eq. (9), we can express the HamiltonianH3D along the high-symmetry line kx = ky as

U−1H3D (k, k, kz)U =

(H+ (k, kz) 0

0 H− (k, kz)

), (10)

where H±(k, kz) acts on the corresponding mirror-rotation subspace. We can define the Chern number

C± :=1

2π

∫BZ

Tr [dA± + iA± ∧A±] , (11)

4

-1.5 0 1.5

2

0

-2t

l

= w 0

= w -2

FIG. 4. Topological phase diagram in the 2D non-HermitianSOTI for γ = 0.4. The gray regions represent thetopologically trivial phase with w = 0, while the cyanregions represent the second-order topological phase withw = −2 that hosts corner states. The phase boundaries aredetermined by t2 = λ2 + γ2 and t2 = γ2 − λ2.

where Aαβ± = i⟨χα±(k, kz)

∣∣ ∣∣∣dφβ±(k, kz)⟩

with α and β

taken over the filled bands, and∣∣φα±⟩ (

∣∣χα±⟩) is a right(left) eigenstate of H±(k, kz). This formula is a naturalgeneralization of the single-band non-Hermitian Chernnumber discussed in Ref. [53] to multiple bands. Thenthe topological index that characterizes the second-ordertopological phases in 3D is

C := C+ − C−. (12)

We investigate a concrete model of a 3D non-HermitianSOTI on a cubic lattice described by

H3D =

(m+ t

∑i

cos ki

)τz +

∑i

(∆1 sin ki + iγi)σiτx

+ ∆2 (cos kx − cos ky) τy, (13)

where i runs over x, y and z, and γx = γy = γ0.This Hamiltonian H3D only preserves mirror-rotationsymmetry Mxy (see Sec. IX in Ref. [94]).

When the bulk bands of H3D are gapped and first-order-topologically trivial, it does not support gaplesssurface states, as shown by energy spectra with openboundaries along the y direction in Figs. 5(a) and (b).However, the system with open boundaries in both xand y directions hosts four-fold degenerate second-orderboundary modes, as shown in Figs. 5(c) and (d). Incontrast to the Hermitian case [82], these second-orderboundary modes under the open boundary conditionalong all the directions are localized not along the hingebut anomalously localized at one corner [see Fig. 5(e)].This indicates that the usual bulk-hinge correspondenceis broken for the 3D non-Hermitian SOTI. Moreover,these second-order boundary modes are only localized atthe corners on the x = y plane due to the mirror-rotationsymmetry Mxy (see Fig. S10 in Ref. [94]). In addition,the second-order boundary modes can be localized atmore than one corner when the mirror-rotation symmetryis broken or there exists the balanced gain and loss (seeSec. IX in Ref. [94]).

0 kz--5

0

5

Re(

E)

-0.7

0

0.7

Im(E

)

0 kz-kzkx

Re(

E)Im

(E)

kzkx(b)

(a) (e)

0.4

0.2

0t

γ

0.1 1.1 2

(f)(d)

(c)

0

1

= C 0 = C -2

0

0.05

0.10

0.15

z

30

15

1

z

x y1020

2010

1x y

FIG. 5. Three-dimensional non-Hermitian SOTI describedby Eq. (13). (a,b) Complex energy spectrum under theopen boundary condition along the y direction. (c,d)Complex energy spectrum under the open boundary conditionalong the x and y directions. Red curves denote four-folddegenerate second-order boundary modes. (e) Probabilitydensity distribution |Φn,R|2 (n is the index of an eigenstateand R specifies a lattice site) of mid-gap modes with openboundaries along the x, y and z directions. The mid-gapstates (with eigenenergies of 0.035) are localized only at onecorner. The number of unit cells is 20 × 20 × 30 with t = 1,γ0 = 0.7, γz = −0.2, m = −2, ∆1 = 1.2, and ∆2 = 1.2. (f)Second-order topological phase diagram characterized by thenonzero Chern number.

Due to mirror-rotation symmetry, the second-ordertopological phase in 3D can be characterized by theChern number C [see Eqs. (9-12)]. To generalize thebulk-boundary correspondence in 3D non-Hermitian SO-TIs, we take into account the exponential-decay behaviorof non-Hermitian eigenstates with open boundaries alongall the directions. After considering a low-energycontinuum model of the Hamiltonian H3D to capturethe essential physics of the 3D non-Hermitian SOTI withanalytical results, and replacing real wavevectors k withcomplex ones (see Sec. IX in Ref. [94] for details), theHamiltonian H± for H3D in Eq. (10) can be expressed as

H±(k, kz) =−[m+ 3t− t(k − iα0)2 − t

2(kz − iαz)2

]σz

±√

2 [∆1(k − iα0) + iγ0]σy

−∆1 (kz − iαz)σx, (14)

where

α0 =γ0

∆1, and αz =

γz∆1

. (15)

Figure 5(f) shows the topological phase diagram, wherethe second-order topological phases are characterized bythe nonzero Chern number (C = −2). The number ofhinge states is counted as 2|C|.Conclusions.—In this Letter, we have analyzed 2D and

3D SOTIs in the presence of non-Hermiticity. In spite oftheir first-order topologically trivial bulk bands, second-order boundary modes exist in both 2D and 3D SOTIs.In contrast to the Hermitian cases, the mid-gap states in

5

2D are localized only at one corner protected by mirror-rotation symmetry and sublattice symmetry, and thesecond-order boundary modes are anomalously localizedat a corner in 3D. The winding number (Chern number)defined by complex wavevectors is used to determinetheir second-order topological phases in 2D (3D). Anexperimental realization with ultracold atoms is alsodiscussed. Our study provides a framework to explorericher non-Hermitian physics in higher-order topologicalphases.

T.L. thanks James Jun He for discussions, andYi Peng for technical assistance. T.L. acknowledgessupport from a JSPS Postdoctoral Fellowship (P18023).Y.R.Z. was partially supported by China PostdoctoralScience Foundation (grant No. 2018M640055). Z.G.was supported by MEXT. K.K. acknowledges supportfrom the JSPS through the Program for LeadingGraduate Schools (ALPS). M.U. acknowledges supportby KAKENHI Grant No. JP18H01145 and a Grant-in-Aid for Scientific Research on Innovative Areas“Topological Materials Science (KAKENHI Grant No.JP15H05855) from the JSPS. F.N. is supported in partby the: MURI Center for Dynamic Magneto-Optics viathe Air Force Office of Scientific Research (AFOSR)(Grant No. FA9550-14-1-0040), Army Research Office(ARO) (Grant No. W911NF-18-1-0358), Asian Office ofAerospace Research and Development (AOARD) (GrantNo. FA2386-18-1-4045), Japan Science and TechnologyAgency (JST) (Q-LEAP program, ImPACT program,and CREST Grant No. JPMJCR1676), JSPS (JSPS-RFBR Grant No. 17-52-50023, and JSPS-FWO GrantNo. VS.059.18N), RIKEN-AIST Challenge ResearchFund, and the John Templeton Foundation.

Note added.—After this work was submitted, a relatedpreprint [95] appeared, which focuses on the interplaybetween topological modes and skin boundary modesinduced by non-reciprocity in non-Hermitian higher-order topological phases.

∗ tao.liu@riken.jp† gong@cat.phys.s.u-tokyo.ac.jp‡ kawabata@cat.phys.s.u-tokyo.ac.jp§ ueda@phys.s.u-tokyo.ac.jp¶ fnori@riken.jp

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[94] See Supplemental Material for (I) tight-binding Hamil-tonian, (II) pseudo-Hermiticity, (III) edge theory, (IV)robustness against disorder, (V) effect of a differenttype of asymmetric hopping and non-Hermiticity on thelocalization of corner modes, (VI) degenerate perturba-tion theory, (VII) bulk-state localization and windingnumber, (VIII) possible experimental realization, and(IX) non-Hermitian second-order topological phases in3D.

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1

SUPPLEMENTAL MATERIAL FOR “SECOND-ORDER TOPOLOGICAL PHASES INNON-HERMITIAN SYSTEMS”

I. Tight-binding Hamiltonian

As shown in Fig. 1 of the main text, we consider a minimal model of a second-order topological insulator on a squarelattice, where each unit cell contains four sublattice degrees of freedom and asymmetric particle hoppings within eachunit cell are considered. The tight-binding Hamiltonian in real-space representation is written as

Htb =∑R

[(t− γ)(c†R,1cR,3 + c†R,4cR,2) + (t+ γ)(c†R,3cR,1 + c†R,2cR,4)

+ (t− γ)(c†R,3cR,2 − c†R,1cR,4) + (t+ γ)(c†R,2cR,3 − c

†R,4cR,1)

+λ(c†R,1cR+x,3 + c†R,4cR+x,2 + H.c.) + λ(c†R,3cR+y,2 − c†R,1cR+y,4 + H.c.)], (S1)

where c†R,i (i = 1, 2, 3, 4) is the creation operator of a fermion at sublattice i of unit-cell site R, x and y denote the unitvectors along the x and y directions, λ is the inter-cell hopping amplitude, and t ± γ denote the asymmetric intra-cellhopping amplitudes. In the momentum-space representation, the Hamiltonian Htb is written as H =

∑k ψ†kH2D(k)ψk

with ψk = (ck,1, ck,2, ck,3, ck,4)T . Then, we have

H2D(k) = [t+ λ cos(kx)] τx − [λ sin(kx) + iγ] τyσz + [t+ λ cos(ky)] τyσy + [λ sin(ky) + iγ] τyσx, (S2)

where we have set the lattice constant a0 = 1, σi (i = x, y, z) is a Pauli matrix acting on the sublattice index ofparticles 1 and 2 as well as particles 3 and 4 within a unit cell [see Fig. 1(a) in main text], and τi is a Pauli matrixacting on the space of these two pairs.

The eigenenergies of H2D(k) are

E±(k) = ±[2t2 − 2γ2 + 2λ2 + 2λt cos(kx) + 2λt cos(ky) + 2iλγ sin(kx) + 2iλγ sin(ky)

] 12 , (S3)

where each of the upper and lower energy bands is two-fold degenerate. The upper and lower bands coalesce at EPswith E±(kEP) = 0 for kx = ky = 0 (π) or kx = −ky.

The bulk bands of H2D(k) are first-order topologically trivial in the entire range of parameters, and characterizedby zero Chern number [S1] defined by

N =1

2π

∫BZ

Tr[Fxy(k)] d2k, (S4)

where the trace is taken over the occupied bands, Fxy(k) is the non-Abelian Berry curvature

Fαβxy (k) = ∂xAαβy (k)− ∂yAαβx (k) + i[Ax, Ay]αβ . (S5)

Here Aµ is the Berry connection

Aαβµ (k) = i 〈χαn(k)|∣∣∂µφβn(k)

⟩, (S6)

where |φαn(k)〉 and |χαn(k)〉 are the right and left eigenstates:

H2D(k) |φαn(k)〉 = En |φαn(k)〉 , (S7)

H†2D(k) |χαn(k)〉 = E∗n |χαn(k)〉 , (S8)

and α denotes the band degeneracy. The right and left eigenstates satisfy the following biorthogonal normalizationcondition

〈χαn(k)|∣∣φβm(k)

⟩= δnmδαβ . (S9)

Numerical calculations show that N = 0, indicating that the bulk bands are topologically trivial.

2

II. Pseudo-Hermiticity

In this section, we argue that the real spectrum of our non-Hermitian system, with open boundaries along both xand y directions, results from pseudo-Hermiticity of the real-space Hamiltonian [S2–S5].

We rewrite the real-space Hamiltonian [see Eq. (S1)] as Htb = H1tb +H2

tb +H3tb, where

H1tb =

∑nx,ny

Φ†nx,ny[t(τx + τyσy)− iγ(τyσz − τyσx)] Φnx,ny

, (S10)

H2tb =

∑nx,ny

Φ†nx,ny

[λ

2(τyσy − iτyσx)

]Φnx,ny+1 + H.c., (S11)

H3tb =

∑nx,ny

Φ†nx,ny

[λ

2(τx + iτyσz)

]Φnx+1,ny

+ H.c.. (S12)

Here nx (nx = 1, 2, ..., L) and ny (ny = 1, 2, ..., L) are integer-valued coordinates of unit cells in the x and ydirections, respectively, σi and τi (i = x, y, z) are Pauli matrices for the degrees of freedom within a unit cell, andΦnx,ny

= (cnx,ny,A, cnx,ny,B , cnx,ny,C , cnx,ny,D)T is the column vector of annihilation operators with A,B,C, and Dcorresponding to indexes 1, 2, 3, and 4 in Fig. 1 in the main text and denoting four orbitals within a unit cell. In thebasis Φ = (Φ1,1, Φ1,2, ..., ΦL,L−1, ΦL,L), the Hamiltonian Htb is expressed as

Htb = Φ†H0Φ, (S13)

where H0 is the matrix form of the Hamiltonian Htb.The Hamiltonian Htb is pseudo-Hermitian, which satisfies

ηH†0η−1 = H0, (S14)

where η is a 4L× 4L square matrix, and only contains elements σy at its anti-diagonal sites:

η =

0 0 . . . 0 σy0 0 . . . σy 0...

......

......

0 σy . . . 0 0σy 0 . . . 0 0

. (S15)

While positivity is usually required for the definition of pseudo-Hermiticity [S2–S4], we do not assume the positivityhere. From Eq. (S14), for any eigenenergy En with the eigenequation H0 |φn〉 = En |φn〉, we have

En 〈φn| η−1 |φn〉 = E∗n 〈φn| η−1 |φn〉 . (S16)

Therefore, for 〈φn| η−1 |φn〉 6= 0, we have real eigenenergy En for the open-boundary systems, which holds for a widerange of parameters (see Fig. S1 and Fig. 3 in the main text). Note that the bulk eigenenergies can be complex in acertain range of parameters, as shown in Fig. S2.

III. Edge theory

As we argue in the main text, the mid-gap-state localization at one corner results from the interplay betweensymmetry Mxy and non-Hermiticity, where each corner mode is a mutual topological state of two intersectingnontrivial edges. In this section, we develop an edge theory to explain this result. We label the four edges of a squaresample as I, II, III and IV (see Fig. S3). For the sake of simplicity, we consider the case of minλ, −t, λ+ t, γ > 0,λ maxγ, λ + t, and γ > λ + t. In this case, the low-energy edge bands of the Hermitian part of H2D(k) liearound the Γ point of the Brillouin zone. Therefore, we consider the continuum model of the lattice Hamiltonian [seeEq. (S2)] by expanding its wavevector k to first-order around the Γ = (0, 0) point of the Brillouin zone, obtaining

Hcm = (t+ λ) τx − (λkx + iγ) τyσz + (t+ λ) τyσy + (λky + iγ) τyσx. (S17)

3

10

y

1

20

10 20x

10

y

20

11

10 20x1

(a) (c)

(b) (d)

(e)

(f)-1.5

0

1.5

Re(

E)

-1

0

1

Im(E

)

1 15 30n

0 0.4 0.8

0 0.4 0.8

0 0.4 0.8

0 0.4 0.8

-710

FIG. S1. (a-d) Probability density distributions∑4

i |φR,i,n|2 (n is the index of an eigenstate and R specifies a unit cell) of fourzero-energy states under the open boundary condition along both x and y directions for the 20 × 20 unit cells with t = −0.6,λ = 1.5, and γ = 0.4. All the zero-energy states are localized at the upper-right corner. (e, f) Real and imaginary parts ofcomplex eigenenergies close to zero energy. The red dots represent the eigenenergies of the corner modes. The bulk eigenenergiesfor a finite-size sample are real over a wide range of parameters.

10

y

1

20

10 20x

10

y

20

11

10 20x1

(a) (c)

(b) (d)

(e)

(f)-1.6

0

1.6

Re(

E)

-0.1

0

0.1Im

(E)

1 15 30

0 0.5 1 0 0.5 1

0 0.5 1 0 0.5 1

n

FIG. S2. (a-d) Probability density distributions of four zero-energy states under the open boundary condition along both xand y directions for the 20 × 20 unit cells with t = 0.3, λ = 1.5, and γ = 0.4. All the zero-energy states are localized at thelower-left corner. (e, f) Real and imaginary parts of complex eigenenergies close to zero energy. The red dots represent theeigenenergies of the corner modes. The bulk eigenenergies for a finite-size sample can be complex for the parameters consideredhere.

Ⅰ

Ⅱ

Ⅲ

Ⅳ

FIG. S3. Schematic illustration showing a 2D non-Hermitian SOTI in a square sample. All four zero-energy modes are localizedat the upper-right corner for t < 0, and γ > 0. I, II, III and IV label the four edges of the lattice.

We first investigate the edge I of the four edges. Substituting ky by −i∂y, and treating terms including t + λ andγ as perturbations (which are valid if they are relatively small), we can rewrite the Hamiltonian Hcm into the sum ofthe following two terms:

Hcm,1 = (t+ λ) τyσy − iλ∂

∂yτyσx, (S18)

Hcm,2 = (t+ λ) τx − (λkx + iγ) τyσz + iγτyσx, (S19)

4

where Hcm,1 is Hermitian, and Hcm,2 is treated as the perturbation for λ γ, t+λ. To solve the eigenvalue equation

Hcm,1φIcm(y) = Ecmφ

Icm(y) with Ecm = 0 under the boundary condition φI

cm(+∞) = 0, we can write the solution inthe following form

φIcm(y) = Ny exp(−αIy) exp(ikxx)χI, Re(αI) > 0, (S20)

where Ny is a normalization constant. The eigenvector χI satisfies σzχI = −χI with∣∣χ1I

⟩= |τz = 1〉 ⊗ |σz = −1〉 , (S21)∣∣χ2

I

⟩= |τz = −1〉 ⊗ |σz = −1〉 . (S22)

Then, the effective Hamiltonians for the edge I can be obtained in this basis as

HIedge =

∫ +∞

0

(φI

cm

)∗(y) Hcm,2 φ

Icm(y) dy. (S23)

Therefore, we have

HIedge = (t+ λ) %x + (λkx + iγ) %y, (S24)

where %i (i = x, y, z) are Pauli matrices.The effective Hamiltonian for the edges II, III and IV can be obtained through similar procedures:

HIIedge = − (t+ λ) %x + (λky + iγ) %y, (S25)

HIIIedge = (t+ λ) %x − (λkx + iγ) %y, (S26)

HIVedge = (t+ λ) %x + (λky + iγ) %y. (S27)

It is straightforward to verify that the two zero-energy bound states for edges I and III are localized at their rightends for γ > t + λ (note that each edge exhibits a zero-energy bound state for small γ), while the two zero-energybound states for edges II and IV are localized at their upper ends. Therefore, the zero-energy states are localized atthe upper-right corner for t < 0 and γ > 0 (see Fig. S1).

IV. Robustness against disorder

We now show that the zero-energy corner states are robust against disorder that preserves Mxy symmetry andsublattice symmetry. We consider the following real-space disordered Hamiltonian:

H1tb =

∑nx,ny

Φ†nx,ny

[(t+ d1ξnx,ny

)(τx + τyσy)− i

(γ + d2ζnx,ny

)(τyσz − τyσx)

]Φnx,ny

, (S28)

H2tb =

∑nx,ny

Φ†nx,ny

[1

2

(λ+ d3µnx,ny

)(τyσy − iτyσx)

]Φnx,ny+1 + H.c., (S29)

H3tb =

∑nx,ny

Φ†nx,ny

[1

2

(λ+ d3µnx,ny

)(τx + iτyσz)

]Φnx+1,ny + H.c., (S30)

where ξnx,ny, ζnx,ny

, and µnx,nyare uniform random variables distributed over [−1, 1], while d1, d2, and d3 are the

corresponding disorder strength. As shown in Fig. S4 and Fig. S5 for the different values of disorder strength, thecorner modes are topologically protected against disorder with the Mxy symmetry and sublattice symmetry, unlessthe band gaps close. Moreover, the corner modes are well localized at one corner of a square sample [see Fig. S4(c, f,i)].

V. Effect of a different type of asymmetric hopping and non-Hermiticity on the localization of corner modes

The interplay between the mirror-rotation symmetry and non-Hermiticity leads to the localization of the mid-gapstates only at one corner, as explained in the main text. However, the mid-gap modes can be localized at more than

5

5

0

-5

d1

d2

4

0

-4

0 1 2

0 1 2

5

0

-5

d1

0 1 2

3

0

-3

d2

0 1 2

3

0

-3

d3

0 1 2d3

0 1 2

3

0

-3

Re(

E)R

e(E)

Im(E

)

Re(

E)

Im(E

)

(a) (b)

(d) (e)

(g) (h)

(c)

(f)

(i)

20

10

1

y

x1 10 20

x1 10 20

x1 10 20

0

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

20

10

1

y

20

10

1

y

Im(E

)

FIG. S4. Energy spectra and probability density distributions under the open boundary condition in both x and y directions(a-c) as a function of the disorder strength d1 with d2 = 0 and d3 = 0, (d-f) as a function of the disorder strength d2 withd1 = 0 and d3 = 0, and (g-i) as a function of the disorder strength d3 with d1 = 0 and d2 = 0. The other parameters are chosento be t = 0.6, λ = 1.5, and γ = 0.4. (a, d, g) Real and (b, e, h) imaginary parts of the spectra. Red dots denote zero-energymodes. (c, f, i) Averaged probability density distributions of the four zero-energy states with d1 = 0.8, d2 = 0.8, and d3 = 0.8,respectively.

one corner when the mirror-rotation symmetry is broken. In this section, we consider the case where the mirror-rotation symmetry is broken, and study the effect of a different type of asymmetric hopping and non-Hermiticity(i.e., balanced gain and loss) on the mode localization of the mid-gap states in non-Hermitian SOTIs. Note that theeffect of asymmetric hopping on the higher-order boundary modes in non-reciprocal systems has been investigated inRef. [S6].

We investigate the same lattice mode of the 2D SOTI as the one in the main text. But here we consider differentamplitudes of asymmetric hopping along the x and y directions within each unit cell, i.e., γx 6= γy, as shown inFig. S6(a). The Bloch Hamiltonian is written as

H2D(k) = [tx + λ cos(kx)] τx − [λ sin(kx) + iγx] τyσz + [ty + λ cos(ky)] τyσy + [λ sin(ky) + iγy] τyσx, (S31)

where H2D(k) only respects sublattice symmetry with τzH2D(k)τ−1z = −H2D(k) for γx 6= γy for γx 6= γy. Note that

2D Hermitian SOTIs can exist even in the absence of crystalline symmetries [S7].Figure S6(b) shows the probability density distribution of mid-gap states for the larger hopping amplitude with

(ty +γy) along the y direction and the symmetric hopping along the x direction [see Fig. S6(a)]. In this case, the mid-gap states are localized at both the lower-left and lower-right corners. In contrast, for the larger hopping amplitudewith (tx+γx) along the x direction and the symmetric hopping along the y direction, the mid-gap states are localizedat both the lower-left and upper-left corners, as shown in Fig. S6(c). Moreover, the mid-gap modes are localized onlyat the upper-left corner for the larger asymmetric hopping amplitudes with (tx + γx) and (ty − γy) along the x andy directions, respectively, as shown in Fig. S6(d). Therefore, the localization of corner states relies on the type ofasymmetric hopping for 2D non-reciprocal lattice models.

In addition to the different type of asymmetric hopping, the second-order boundary modes in 2D systems can belocalized at more than one corner in the presence of a different type of non-Hermiticity i.e., balanced gain and loss.

6

5

0

-5

d1 d2

4

0

-40 1 2 0 1 2

5

0

-5

d1

0 1 2

3

0

-3

d2

0 1 2

3

0

-3

d3

0 1 2

d3

0 1 2

3

0

-3

Re(

E)Im

(E)

Re(

E)Im

(E)

Re(

E)Im

(E)

(a)

(b)

(c)

(d)

(e)

(f)

FIG. S5. Energy spectra similar to those of Fig. S4 but with the different values of disorder strength. (a, b) d2 = 0.4 andd3 = 0.4, (c, d) d1 = 0.4 and d3 = 0.4, and (e, f) d1 = 0.4 and d2 = 0.4. The corner modes are topologically protected againstdisorder that preserves Mxy symmetry and sublattice symmetry.

t+γ

λ

λ

3 1

42

t -γ

t - γ

t +γ

(a) (b)

x x

x x

yyy

y

1 10 20x

20

10

1

y

0 0.13 0.26

(c)

1 10 20x

0 0.1 0.2

1 15 30x

30

15

1y

(d)

0 0.3 0.6

FIG. S6. (a) Real-space representation of the 2D model in Eq. (S31) on a square lattice. In contrast to the model shownin Fig. 1(a) in the main text, here we consider different amplitudes of asymmetric hopping along the x and y directions i.e.,γx 6= γy. Probability density distributions of mid-gap states under the open boundary condition along the x and y directions:(b) for γx = 0 and γy = 0.3, (c) for γx = 0.3 and γy = 0, and (d) for γx = 0.3 and γy = −0.3. The number of unit cells is 20 ×20 with tx = ty = 1.0 and λ = 1.5.

t

λ

λ

3 1

42t

(a) (b)

-iu

iu

1 10 20x

20

10

1

y 0 0.04 0.08

x

y

FIG. S7. (a) Real-space representation of the 2D model in Eq. (S32) in the presence of alternating on-site gain and loss (withthe imaginary staggered potentials iu and −iu) within each unit cell. The dashed lines indicate hopping terms with a negativesign. Probability density distributions of mid-gap states under the open boundary condition along the x and y directions. Thenumber of unit cells is 20 × 20 with u = 0.4, tx = ty = 1.0 and λ = 1.5.

As shown in Fig. S7(a), we consider the alternating on-site gain and loss with symmetric particle hopping, where the

7

imaginary staggered potentials are indicated by iu and −iu. The Bloch Hamiltonian is written as

H2D(k) = [tx + λ cos(kx)] τx − λ sin(kx)τyσz + [ty + λ cos(ky)] τyσy + λ sin(ky)τyσx − iuτz. (S32)

Whereas H2D (k) does not respect sublattice symmetry, it still respects pseudo-anti-Hermiticity with τzH†2D(k)τ−1

z =

−H2D(k) [S5]. As a result, the second-order topological phase survives even in the presence of the balanced gainand loss. Figure S7(b) shows the probability density distribution. In the presence of the balanced gain and loss, themid-gap states are localized at four corners in 2D systems. In contrast to non-Hermitian SOTIs with asymmetrichopping, the eigenenergies of mid-gap states have nonzero imaginary parts as in the case for the non-HermitianSu-Schrieffer-Heeger model with the balanced gain and loss [S5].

VI. Degenerate perturbation theory

The bulk-state localization results from the non-Hermiticity due to asymmetric hopping, which can intuitively beexplained using degenerate perturbation theory when γ is small. In this section, we consider a continuum model ofthe lattice Hamiltonian [see Eq. (S2)] by expanding its wavevector k up to second order around the Γ = (0, 0) pointof the Brillouin zone, obtaining Hcm = H1

cm +H2cm, where

H1cm =

[t+ λ− λ

2k2x

]τx − λkxτyσz +

[t+ λ− λ

2k2y

]τyσy + λkyτyσx, (S33)

H2cm =− iγτyσz + iγτyσx. (S34)

Note that H2cm can be obtained from H1

cm as H2cm = H2

cm + ¯H2cm:

H2cm =

iγ

λ

(∂H1

cm

∂kx+∂H1

cm

∂ky

)=γ

λ

([x,H1

cm] + [y,H1cm]), (S35)

¯H2cm = iγ (kxτx + kyτyσy) . (S36)

The degenerate bulk states for the Hermitian part H1cm of the Hamiltonian are denoted by

∣∣φ0i

⟩(i = 1, 2, 3, 4), and

the non-Hermitian part H2cm is considered as a perturbation for λ γ, and k is around the Γ point. By applying

degenerate perturbation theory [? ? ], the first-order correction to the wavefunctions∣∣φ0i

⟩for small k is

∣∣φ1i

⟩=∑

h/∈φ0i

〈h|H2cm

∣∣φ0i

⟩E0d − E0

h

|h〉+∑j 6=i

⟨φ0j

∣∣H2cm |h〉

ui − uj∣∣φ0j

⟩ 〈h|H2cm

∣∣φ0i

⟩E0d − E0

h

'∑

h/∈φ0i

〈h| H2cm

∣∣φ0i

⟩E0d − E0

h

|h〉+∑j 6=i

⟨φ0j

∣∣ H2cm |h〉

ui − uj∣∣φ0j

⟩ 〈h| H2cm

∣∣φ0i

⟩E0d − E0

h

=∑

h/∈φ0i

γλ|h〉 〈h|x

∣∣φ0i

⟩+∑j 6=i

γ2(E0d − E0

h)∣∣φ0j

⟩〈h|x

∣∣φ0i

⟩ ⟨φ0j

∣∣x |h〉λ2(ui − uj)

+

γ

λ|h〉 〈h| y

∣∣φ0i

⟩+∑j 6=i

γ2(E0d − E0

h)∣∣φ0j

⟩〈h| y

∣∣φ0i

⟩ ⟨φ0j

∣∣ y |h〉λ2(ui − uj)

=γ

λ(x− xi)

∣∣φ0i

⟩+∑j 6=i

γ2(E0d − E0

h)[⟨φ0j

∣∣x2∣∣φ0i

⟩− x∗j xi

] ∣∣φ0j

⟩λ2(ui − uj)

+

γ

λ(y − yi)

∣∣φ0i

⟩+∑j 6=i

γ2(E0d − E0

h)[⟨φ0j

∣∣ y2∣∣φ0i

⟩− y∗j yi

] ∣∣φ0j

⟩λ2(ui − uj)

'

γλ

(x− xi) +∑j 6=i

γ2(E0d − E0

h)[⟨φ0i

∣∣x2∣∣φ0j

⟩− x∗i xj

]λ2(uj − ui)

∣∣φ0i

⟩+

γλ

(y − yi) +∑j 6=i

γ2(E0d − E0

h)[⟨φ0i

∣∣ y2∣∣φ0j

⟩− y∗i yj

]λ2(uj − ui)

∣∣φ0i

⟩, (S37)

8

where E0d , E0

h, ui, xi and yi satisfy

H1cm

∣∣φ0i

⟩= E0

d

∣∣φ0i

⟩, (S38)

H1cm |h〉 = E0

h |h〉 , h /∈ φ0i , (S39)

ui =⟨φ0i

∣∣H2cm

∣∣φ0i

⟩=⟨φ0i

∣∣ ¯H2cm

∣∣φ0i

⟩, (S40)

xi =∑m

⟨φ0m

∣∣x ∣∣φ0i

⟩, (S41)

yi =∑m

⟨φ0m

∣∣ y ∣∣φ0i

⟩. (S42)

Then the modified eigenstate is

|φi〉 =∣∣φ0i

⟩+∣∣φ1i

⟩' exp

[γλ

(x+ y − ¯xi − ¯yi)] ∣∣φ0

i

⟩, (S43)

where ¯xi and ¯yi are given by

¯xi = xi −∑j 6=i

γ(E0d − E0

h)[⟨φ0i

∣∣x2∣∣φ0j

⟩− x∗i xj

]λ(uj − ui)

, (S44)

¯yi = yi −∑j 6=i

γ(E0d − E0

h)[⟨φ0i

∣∣ y2∣∣φ0j

⟩− y∗i yj

]λ(uj − ui)

. (S45)

Equation (S43) shows that the bulk states can exponentially be localized in the non-Hermitian case, while they cannotif the perturbation Hamiltonian H2

cm is Hermitian (i.e., if γ is pure imaginary).

VII. Bulk-state localization and winding number

As explained in the main text, the winding number defined by the non-Hermitian Bloch Hamiltonian [see Eqs. (1)-(5)in the main text] cannot correctly describe the bulk-corner correspondence in the second-order topological insulator.This deviation results from the non-Bloch-wave behavior of open-boundary eigenstates of a non-Hermitian Hamiltonian[S8, S9], which leads to the bulk-state localization (see Fig. 3 in the main text, and a quantitative analysis in theprevious section). In order to figure out this unexpected non-Bloch-wave behavior, and precisely characterize thetopological invariants for non-Hermitian systems, modified complex wavevectors, rather than real ones, are proposedfor calculating the winding number [S8, S9]. In this section, we discuss how to modify the topological index based oncomplex wavevectors.

According to Eqs. (S10) – (S13), in the basis Φ = (Φ1,1, Φ1,2, ..., ΦL,L−1, ΦL,L), we solve the real-spaceeigenequation:

H0φ = Eφ, (S46)

where the wavefunction φ is φ = (ϕ1,1, ϕ1,2, ..., ϕL,L−1, ϕL,L)T with ϕnx,ny=

(φnx,ny,A, φnx,ny,B , φnx,ny,C , φnx,ny,D)T . Then, according to Eq. (S46), we have

T †ϕnx−1,ny+1 +M†ϕnx,ny +Rϕnx,ny+1 +Mϕnx,ny+2 + Tϕnx+1,ny+1 = Eϕnx,ny+1, (S47)

where R, M and T are given by

R = t(τx + τyσy)− iγ(τyσz + τyσx), (S48)

M =λ

2(τyσy − iτyσx), (S49)

T =λ

2(τx + iτyσz). (S50)

To derive the eigenequation [see Eq. (S47)], we consider the trial solution

ϕnx,ny = exp(α1nx + α2ny)φ0, (S51)

9

where φ0 = (φA, φB , φC , φD). In order to preserve the symmetry Mxy, we set α1 = α2 = α. According toEqs. (S47)–(S51), we have

(t− γ + βλ)φC − (t− γ + βλ)φD = EφA, (S52)

[λ+ β(t+ γ)]φC + [λ+ β(t+ γ)]φD = βEφB , (S53)

[λ+ β(t+ γ)]φA + β(t− γ + βλ)φB = βEφC , (S54)

− [λ+ β(t+ γ)]φA + β(t− γ + βλ)φB = βEφD, (S55)

where β = exp(α). Therefore, we have

λγβ2 −(γ2 − λ2

)β + t2β − γλ+

(1 + β2

)λt =

β

2E2. (S56)

Equation (S56) has two solutions β1 and β2:

βi =2γ2 − 2λ2 − 2t2 + E2 ±

√(2λ2 − 2γ2 + 2t2 − E2)

2 − 8λ2(t− γ)(γ + t)

4λ(γ + t). (S57)

Moreover, according to Eq. (S56), for E → 0, we have

β1 = − λ

t+ γ, β2 =

γ − tλ

, t ∈ [−√γ2 + λ2,

√γ2 + λ2], (S58)

β1 =γ − tλ

, β2 = − λ

t+ γ, t ∈ [−∞, −

√γ2 + λ2] ∪ [

√γ2 + λ2, +∞]. (S59)

Then, the state vector in Eq. (S51) at each site can be written as

ϕnx,ny = βnx+ny

1 φ10 + β

nx+ny

2 φ20. (S60)

By considering the following boundary conditions

R(β21φ

10 + β2

2φ20) +M(β3

1φ10 + β3

2φ20) + T (β3

1φ10 + β3

2φ20) = E(β2

1φ10 + β2

2φ20), (S61)

T †(β2L−11 φ1

0 + β2L−12 φ2

0) +M†(β2L−11 φ1

0 + β2L−12 φ2

0) +R(β2L1 φ1

0 + β2L2 φ2

0) = E(β2L1 φ1

0 + β2L2 φ2

0), (S62)

and relations

φ(i)A =

βiE(φ(i)C − φ

(i)D )

2(λ+ tβ + γβ), (S63)

φ(i)B =

βiE(φ(i)C + φ

(i)D )

2(t− γ + λβ), (S64)

we have

β2L−12

[2t2 − 2γ2 + 2β1λ(γ + t)− E2

]= β2L−1

1

[2t2 − 2γ2 + 2β2λ(γ + t)− E2

]. (S65)

According to Eq. (S65), we require that β1 and β2 satisfy

|β1| = |β2| (S66)

for a continuum spectrum, where the number of energy eigenstates is proportional to the lattice size L. Otherwise,β1(E) = 0 or 2t2−2γ2 +2β2λ(γ+ t)−E2 = 0 (which is independent of the lattice size L) if |β1| > |β2|, and β2(E) = 0or 2t2 − 2γ2 + 2β1λ(γ + t)− E2 (which is independent of the lattice size L) if |β1| < |β2|.

By combining Eqs. (S57) and (S66), for the bulk states, we have

β0 = |βi| =

√∣∣∣∣ t− γt+ γ

∣∣∣∣. (S67)

10

Then, to account for the non-Bloch-wave behavior [S8, S9], we replace the real vavevector k with the complex one

k = (kx, ky) → k = k + ik′ = (kx + ik′x, ky + ik′y), (S68)

where

k′x = k′y = −ln(β0) = −α0. (S69)

Then the momentum-space Hamiltonian [see Eq. (S2)] can be expressed as

H2D(k) → H(k) = H(k + ik′). (S70)

As shown in the main text, because the Hamiltonian H(k) preserves the mirror-rotation symmetry Mxy , we can

write the Hamiltonian H(k) in a block-diagonal form with kx = ky = k as

U−1H(k, k)U =

[H+(k) 0

0 H−(k)

], (S71)

where H+(k) acts on the +1 mirror-rotation subspace, and H−(k) acts on the −1 mirror-rotation subspace. The theunitary transformation U is

U =

0 0 1 01 0 0 00 1√

20 1√

2

0 1√2

0 − 1√2

, (S72)

and H+(k) and H−(k) are

H+(k) =√

2 [t+ λ cos(k − iα0)]σx +√

2 [λ sin(k − iα0) + iγ]σy, (S73)

H−(k) =√

2 [t+ λ cos(k − iα0)]σx −√

2 [λ sin(k − iα0) + iγ]σy. (S74)

We rewrite Eqs. (S73)–(S74) as

H+(k) =√

2(t+ γ + λβ−1

0 e−ik)σ+ +

√2(t− γ + λβ0eik

)σ−, (S75)

H−(k) =√

2(t− γ + λβ0eik

)σ+ +

√2(t+ γ + λβ−1

0 e−ik)σ−, (S76)

where σ± = (σx ± iσy)/2. The winding numbers for the Hamiltonians H+(k) and H−(k) are expressed as

w+ =i

2π

∫ 4π

0

〈χ+| ∂k |φ+〉〈χ+| |φ+〉

dk, (S77)

w− =i

2π

∫ 4π

0

〈χ−| ∂k |φ−〉〈χ−| |φ−〉

dk, (S78)

where |φ±〉 and |χ±〉 are the right and left eigenstates of H±(k), respectively. The integration over 4π in Eqs. (S77)–(S78) is attributed to the 4π-periodicity of the eigenenergies and eigenstates. Then, the total winding number is

w = w+ − w−. (S79)

VIII. Possible experimental realization

The second-order topological insulator studied here can be experimentally realized in ultracold atoms and photonicsystems. In this section, we propose a possible scheme to realize a non-Hermitian second-order topological insulatorin ultracold atoms in optical lattices. The idea is to combine two state-of-the-art experimental techniques: artificialgauge field and dissipation engineering. The former is used to create a π flux in each unit cell and the latter is usedto make the hopping amplitudes asymmetric.

11

A. General idea

We note that the anti-Hermitian part of the Hamiltonian can be separated into individual four-site blocks [seeEq. (S1)]. By individual we mean that the corresponding local Hamiltonians have no overlap with each other. Tobe specific, we consider the following four-site Hamiltonian (here we omit the notation R in operators for the sake ofsimplicity):

Hsb = (t+ γ)[c†2(c3 + c4) + (c†3 − c†4)c1] + (t− γ)[(c†3 + c†4)c2 + c†1(c3 − c4)], (S80)

whose Hermitian part reads

H0 =1

2(Hsb +H†sb) = t[c†2(c3 + c4) + (c†3 − c

†4)c1 + (c†3 + c†4)c2 + c†1(c3 − c4)]. (S81)

On the other hand, the anti-Hermitian part of Eq. (S80) is given by

1

2(Hsb −H†sb) = γ(c†3c1 + c†1c4 + c†2c3 + c†2c4 −H.c.). (S82)

To engineer this anti-Hermitian part, we follow the method developed in Ref. [S10] to use a combination of thefollowing jump operators that describe the collective loss of two nearest-neighbor sites:

L1 =√

2γ(c3 + ic1), L2 =√

2γ(c1 + ic4), L3 =√

2γ(c2 + ic3), L4 =√

2γ(c2 + ic4). (S83)

At the single-particle or mean-field level, the open-system dynamics of a single block is determined by the non-Hermitian effective Hamiltonian

Heff = H0 −1

2

4∑j=1

L†jLj = Hsb − 2γ

4∑j=1

c†jcj , (S84)

which differs from Eq. (S80) only by a background loss term proportional to 2γ. In the following, we will discussin detail a possible implementation of the above idea with state-of-the-art experimental techniques developed indissipation engineering [S11] and artificial gauge fields [S12].

B. Explicit implementation

We first note that, in the Hermitian limit (γ = 0), there is already an ultracold-atom-based proposal in Ref. [S13].As explained therein, the Hermitian Hamiltonian can be simulated by embedding a superlattice structure into atwo-dimensional π-flux lattice, which can be realized using a setup described in Refs. [S14, S15]. Moreover, we wouldlike to mention that a sharp (box) boundary is also available within current experimental techniques [S16]. This isnecessary for observing the topologically protected corner states.

Now let us move onto the asymmetric hopping amplitudes. Following the theoretical consideration sketched outabove, it suffices to focus on the realization of the jump operators [Eq. (S83)]. We again follow in Ref. [S10] toeffectively engineer a collective loss from a combination of on-site loss of auxiliary states and their coherent couplingto the primary degrees of freedom. Without loss of generality, we now focus on the case of a single block and resonantcouplings. As shown in Fig. S8, the full open-system dynamics in the rotating frame of reference can be written as

ρt = −i[H0 +Ω

2(a†2(c2 + ic4) + a†1(c3 + ic1) + a†3(c2 + ic3) + a†4(c1 + ic4) + H.c.), ρt] + κ

4∑j=1

D[aj ]ρt, (S85)

where aj ’s denote the annihilation operators of the particles in the auxiliary sublattice j and D[L]ρ ≡ LρL†− 12L

†L, ρis the Lindblad superoperator. In the regime κ Ω, we can adiabatically eliminate the fast decay modes in theauxiliary lattice [S17] to obtain the following effective dynamics of the primary lattice degrees of freedom alone:

ρt = −i[H0, ρt] + 2γ

4∑j=1

D[Lj ]ρt, (S86)

12

Auxiliary lattice

Primarylattice

Horizontalcoupling Vertical

coupling

κx

y

x x

y y

κ κ

κ

(a) (b)

(d)

(c)

(e)

FIG. S8. (a) Schematic illustration of a proposed experimental setup. The primary lattice (green) together with a pair ofRaman lasers (not shown) gives rise to a Hermitian second-order topological insulator, where the Raman lasers are used forinducing effective particle hopping. The asymmetry in hopping amplitudes is then introduced via a coherent coupling to adissipative auxiliary lattice (yellow). In particular, the running wave (red arrow) generates the horizontal couplings shown in(b) and (c), which are the cross-sections along the x direction, containing sublattices 2, 4 and 3, 1, respectively. The otherthree standing waves (blue arrows) generate the vertical couplings shown in (d) and (e), which are the cross-sections along they direction containing sublattices 2, 3 and 4, 1, respectively.

where γ = Ω2/(2κ). At the single-particle or the mean-field level, the dynamics reduces to the nonunitary evolutiongoverned by the non-Hermitian Hamiltonian given in Eq. (S84).

However, unlike the simple case discussed in Ref. [S10], which describes a single band in one dimension, here wehave two difficulties to realize the effective dynamics in Eq. (S85): (i) To ensure that an auxiliary site is only coupledto two nearest-neighbor sites, the auxiliary lattice should form a square-octagon pattern in two dimensions; (ii) Wehave to fine-tune the phases of couplings in the presence of a nonzero flux. Let us discuss below possible solutions to(i) and (ii).

To overcome the difficulty (i), we first note that an ideal trap with square-octagon geometry is given by

Vidsqoc(r) ∝∑

(m,n)∈Z2,s=±,ς=0,1

δ(x−

[sς4

+m]a)δ

(y −

[1− ς

4s+ n

]a

), (S87)

which can be expanded into the Fourier series

Vidsqoc(r) ∝∑

(m,n)∈Z2

[im + (−i)m + in + (−i)n] exp

(i2π

a[mx+ ny]

). (S88)

Here a is the lattice constant, which equals λl, the wavelength of the laser that generates the primary lattice [S13].Keeping the terms with |m|+ |n| = 3, 4 in Eq. (S88) followed by dropping the constants and adjusting the coefficients,we can construct a square-octagon-lattice potential as

Vsqoc ∝[cos2 π(2x+ y)

a+ cos2 π(2x− y)

a+ cos2 π(x+ 2y)

a+ cos2 π(x− 2y)

a+ cos2 2π(x− y)

a+ cos2 2π(x+ y)

a

],

(S89)whose profile in a single unit cell is plotted in Fig. S9(a). In practice, this potential can be generated by six standing-wave lasers with amplitude profiles given by

cos2π[cos θ(2x± y)/

√3 + sin θz]

λal, cos

2π[cos θ(x± 2y)/√

3 + sin θz]

λal, and cos

2π[cos θ′(x± y)/√

2 + sin θ′z]

λal, (S90)

where θ = arccos(√

3λal/2a)

and θ′ = arccos(√

2λal/a)

are the tilt angles from the x-y plane. Therefore, we canrather freely choose λal such that the auxiliary lattice only selectively traps a certain metastable state of atoms, suchas the 3P0 state of alkaline-earth atoms. The on-site loss rate κ can be controlled by the strength of an additionallaser that couples the metastable state to a certain unstable state, such as the 1P1 state of alkaline-earth atoms.

13

(a) (b) (c)

FIG. S9. (a) Square-octagon pattern of the auxiliary lattice potential Vsqoc(r) [see also Fig. S8(a)]. (b) Magnitude and (c)phase patterns of the vertical Rabi coupling Ωv(r) given in Eq. (S92). The units in (a) and (b) are set to be the largest Vsqoc(r)and |Ωv(r)|.

To overcome the difficulty (ii), we first note that the effective Rabi coupling Ωtb within the tight-bondingapproximation can be related to the spatial distribution of the Rabi coupling Ω(r) via

Ωtb =

∫d2r Ω(r) w∗a(r − ra) wl(r − rl), (S91)

where ωl and ωa are the Wannier functions of the primary and auxiliary lattices, respectively. Therefore, the horizontalcouplings a†2(c2 + ic4) and a†1(c3 + ic1) can easily be achieved by a running wave Ωh(r) ∝ exp

(i 2πa x)

along the xdirection, with the phase difference between c2 (c3) and c4 (c1) imprinted by the spatial phase variation of Ωh(r). On

the other hand, for the vertical couplings a†3(c2 + ic3) and a†4(c1 + ic4), we cannot simply apply a running wave along

the y direction; otherwise we will obtain a†4(c4 + ic1) instead of a†4(c1 + ic4). To imprint the desired phase information,we can engineer the Rabi-frequency pattern to be

Ωv(r) ∝ cos2πy

a− i sin

2πx

asin

2πy

a, (S92)

which can be created by three standing wave lasers with amplitude profiles

cos2π(cosαy + sinαz)

λcand cos

2π[cosα′(x± y)/√

2 + sinα′z]

λc, (S93)

where α = arccos(λc/a) and α′ = arccos(√

2λc/a)

are the tilt angles from the x-y plane. We plot the spatial patternof the magnitude and phase parts of Ωv [Eq. (S92)] in Figs. S9(b) and (c), where we can clearly find some vorticesindicated by the zeros of |Ωv(r)| accompanied by phase windings. This observation is consistent with the existenceof a π flux in the primary lattice.

IX. Non-Hermitian second-order topological phases in 3D

A. Model

We consider a minimal model of a 3D non-Hermitian SOTI on a cubic lattice:

H3D(k) = [m+ t (cos kx + cos ky + cos kz)] τz + [(∆1 sin kx + iγ0)σx + (∆1 sin ky + iγ0)σy + (∆1 sin kz + iγz)σz] τx

+ ∆2 (cos kx − cos ky) τy, (S94)

where we have set the lattice constant a0 = 1, σi and τi for i = x, y, z are Pauli matrices acting on spin andorbital/sublattice degrees of freedom, respectively, and m, t,∆1,∆2, γ0 and γz are real parameters. Note that theHermitian part of H3D(k) supports chiral hinge modes propagating along the z direction [S18].

14

(b)(a)

x y

z

30

15

1

1020

2010

1x y

z

1

x y

z

30

15

1

1020

2010

1x y

z

1

0

0.05

0.10

0.15

FIG. S10. Probability density distributions |Φn,R|2 (n is the index of an eigenstate and R specifies a lattice site) of mid-gapmodes for a 3D non-Hermitian SOTI with open boundaries along all the directions (a) for m = −2 and γz = −0.2, and (b) form = 2 and γz = 0.2. The mid-gap states (with eigenenergy of 0.035) are only localized at one corner on the x = y plane. Thenumber of unit cells is 20× 20× 30 with t = 1, γ0 = 0.7, ∆1 = 1.2, and ∆2 = 1.2.

For ∆2 = 0, the Hermitian part of H3D(k) is invariant under time reversal T = σyK, where K being the complexconjugation operator, Mx = iσxτz is the x mirror reflection , My = iσyτz is the y mirror reflection, and C4 =exp(−iπσz/4) is the π/2 rotation about the z axis. These symmetries are broken by both non-Hermitian termsincluding γ0 and terms including ∆2. However, H3D(k) is invariant under the mirror-rotation symmetry operationMxy = C4My with

MxyH3D(kx, ky, kz)M−1xy = H3D(ky, kx, kz). (S95)

The bulk energy bands of the Hamiltonian H3D(k) are obtained as

E±(k) =±[−2γ2

0 − γ2z + (t cos kx + t cos ky + t cos kz +m) 2 + 2i∆1 (γ0 sin kx + γ0 sin ky + γz sin kz)

+ ∆22 (cos kx − cos ky) 2 −∆2

1 (cos 2kx + cos 2ky + cos 2kz − 3) /2] 1

2 . (S96)

where the upper and lower branches are two-fold degenerate, respectively. The Hamiltonian H3D(k) is defective atthe exceptional points (EPs) with

E±(kEP) = 0, (S97)

if one of the following conditions is satisfied:(1) EPs appear when kx = 0 and ky = 0, and then Eq. (S97) reduces to

(t cos kz +m+ 2t)2 − (γz − i∆1 sin kz)2

= 2γ20 . (S98)

(2) EPs appear when kx = π and ky = π, and then Eq. (S97) reduces to

(t cos kz +m− 2t)2 − (γz − i∆1 sin kz)2

= 2γ20 . (S99)

(3) EPs appear when kx = 0 (π) and ky = π (0), and then Eq. (S97) reduces to

4∆22 + (t cos kz +m)2 − (γz − i∆1 sin kz)

2= 2γ2

0 . (S100)

(4) EPs appear when kx = −ky with ky 6= 0 and ky 6= π, and then Eq. (S97) reduces to

(2t cos ky + t cos kz +m)2 −∆21(cos 2ky + cos 2kz/2− 3/2)− γ2

z + 2iγz∆1 sin kz = 2γ20 . (S101)

B. B. Localization of second-order boundary states at one corner

As shown in the main text, the mid-gap states in the 3D non-Hermitian SOTI are localized at the right cornerof the x = y plane [see Fig. 5(e) in the main text and Fig. S10(a)]. We note that the mid-gap states can also belocalized at the left corner on the x = y plane when an opposite sign of the parameter m is considered, as shownin Fig. S10(b). The localization of the mid-gap states at one corner on the x = y plane results from the interplaybetween mirror-rotation symmetry Mxy and non-Hermiticity.

15

31

42

(a) (c)(b)3 1

42

-i(m

+γ

) z

i(m-γ

) z

m

-m

(d)

0

0.1

0.2

0.32020 20

1010 10

1010 101010 10

zz z

20

20

20

20 20

20

1

1

1

1

1

1xx xyy y

11 1

FIG. S11. (a) Real-space representation of the 3D model in Eq. (S102), which shows the particle hopping along the z directionbetween 2D layers. This 3D lattice is constructed by stacking the 2D lattice layer shown in Fig. S6(a). Along the z direction,the nearest and next-to-nearest neighbor hoppings are considered. The dashed lines indicate hopping terms with a negativesign. Probability density distributions |Φn,R|2 (n is the index of an eigenstate and R specifies a lattice site) of mid-gap modeswith open boundaries along all the directions: (b) for γx = 0 and γy = 0.5, (c) for γx = 0.5 and γy = 0, and (d) for γx = 0.5and γy = −0.5. The mid-gap states (with eigenenergy of 0.04) are localized at more than one corner for γx = 0 or γy = 0. Thenumber of unit cells is 20 × 20 × 20 with γz = −0.2, m = −1, tx = ty = 1.0 and λ = 1.5.

t

λ

λ

3 1

42t

(a) (c)

31

42

3 1

42

-im

imm

-m

-iu

iu(b) (d)

0

0.01

0.02

0.03

0.04

0.05z

20

20

20

10

10 10

11

1xy

20

20

20

10

10 10

11

1

z

yx

FIG. S12. (a,b) Real-space representation of the 3D model in Eq. (S103). The particle hopping in the (x, y)-plane, with thealternating on-site gain and loss (with the imaginary staggered potentials iu and −iu) within each unit cell, is shown in (a).The particle hopping along the z direction between 2D layers is indicated in (b). This 3D lattice is constructed by stacking the2D lattice layer shown in (a). The dashed lines indicate hopping terms with a negative sign. Probability density distributions|Φn,R|2 (n is the index of an eigenstate and R specifies a lattice site) of mid-gap modes with open boundaries along all thedirections: (c) for u = −0.2, and (d) for u = 0.2. The mid-gap states (with eigenenergy of 0.04) are localized at more than onecorner. The number of unit cells is 20 × 20 × 20 with m = −1, t = 1.0 and λ = 1.5.

C. Effect of a different type of asymmetric hopping and non-Hermiticity on the localization of boundarystates

In the 2D model, we have considered the effect of a different type of asymmetric hopping and non-Hermiticity (i.e.,balanced gain and loss) on the localization of corner states in Sec. V. In this section, we study effect of a different typeof asymmetric hopping and non-Hermiticity on the localization of second-order boundary modes in 3D non-HermitianSOTIs. The non-Hermitian terms of the 3D model studied in our manuscript take the form of an imaginary Zeemanfield [S8, S19]. Therefore, to investigate the effect of a different type of asymmetric hopping on the localizationof second-order boundary modes in 3D SOTIs, we consider a 3D non-reciprocal lattice model by stacking the 2Dlattice layers as shown in Fig. S6(a). The particle hoppings along the z direction between 2D layers are indicated inFig. S11(a). In momentum space, the Hamiltonian has the form

H3D(k) = [tx + λ cos(kx)] τx − [λ sin(kx) + iγx] τyσz + [ty + λ cos(ky)] τyσy + [λ sin(ky) + iγy] τyσx

+m cos(kz)(τx + τyσy) + [m sin(kz)− iγz cos(kz)] τz, (S102)

where the Hermitian part of the Hamiltonian H3D(k) supports four-degenerate chiral hinge modes propagating alongthe z direction. Note that 3D Hermitian SOTIs can be present even in the absence of crystalline symmetries, as inthe 2D case [S7].

16

When the larger hopping strength with amplitude −i(m + γz) (i.e., m and γz have the same signs) along the zdirection is considered [see Fig. S11(a)], the second-order boundary modes are localized at corners of the bottom side(i.e., z = 1 plane), as shown in Figs. S11(b-d). In this case, for the larger hopping amplitude with (ty + γy) along they direction and the symmetric hopping along the x direction, the second-order boundary states are localized at boththe lower-left and lower-right corners of z = 1 plane [see Fig. S11(b)]. In contrast, these second-order boundary modesare localized at both the lower-left and upper-left corners for the larger hopping amplitude with (tx + γx) along thex direction and the symmetric hopping along the y direction [see Fig. S11(c)]. Moreover, the asymmetric hoppingsalong both the x and y directions make the second-order boundary modes localized at one corner [see Fig. S11(d)].Therefore, the localization of the second-order boundary modes in the 3D non-reciprocal lattice model also dependson the type of asymmetric hopping.

In addition to the different type of asymmetric hopping, the second-order boundary modes in 3D systems can belocalized at more than one corner in the presence of a different type of non-Hermiticity i.e., balanced gain and loss.As shown in Figs. S12(a) and (b), we consider the alternating on-site gain and loss with symmetric particle hopping,where the imaginary staggered potentials are indicated by iu and −iu. The Bloch Hamiltonian is written as

H3D(k) = [t+ λ cos(kx)] τx − λ sin(kx)τyσz + [t+ λ cos(ky)] τyσy + λ sin(ky)τyσx

+m cos(kz)(τx + τyσy) +m sin(kz)τz − iuτz. (S103)

Figures S12(c) and (d) show the probability density distributions for u = −0.2 and u = 0.2, respectively. In thepresence of the balanced gain and loss, the second-order boundary modes are localized at more than one corner ofeither the top or the bottom side in 3D systems.

D. Low-energy effective Hamiltonians

The localization of the mid-gap states at one corner on the x = y plane results from the symmetry Mxy and non-Hermiticity, and each mid-gap mode is a mutual topological state of two intersecting surfaces parallel to the z-axis. Inthis section, we present low-energy effective Hamiltonians to explain this effect. We label the four surfaces of a cubicsample as I, II, III, and IV (see Fig. S13). For the sake of simplicity, we consider the case of min−m, t,∆1,∆2 > 0.In this case, the low-energy bands of the Hermitian part of H3D(k) lie around the Γ point of the Brillouin zone.Therefore, we consider a continuum model of the lattice Hamiltonian [see Eq. (S94)] by expanding its wavevector kto second order around the Γ = (0, 0, 0) point of the Brillouin zone, obtaining

Hcm(k) =

[m+ 3t− t

2

(k2x + k2

y + k2z

)]τz + [(∆1kx + iγ0)σx + (∆1ky + iγ0)σy + (∆1kz + iγz)σz] τx

+∆2

2

(k2y − k2

x

)τy. (S104)

We first investigate the surface I. By expressing ky as −i∂y, we can rewrite the Hamiltonian Hcm(k) as

Hcm(kx,−i∂y, kz) = Hcm,1(kx,−i∂y, kz) + Hcm,2(kx,−i∂y, kz), where

Hcm,1(kx,−i∂y, kz) =

(m+ 3t+

t

2∂2y

)τz − i∆1∂yσyτx (S105)

is Hermitian, and

Hcm,2(kx,−i∂y, kz) = ∆1 (kxσx + kzσz) τx + iγ0 (σx + σy) τx + iγzσzτx −∆2

2τy∂

2y (S106)

is treated as a perturbation for max∆2, |γ0|, |γz| max|m+ 3t|, t/2,∆1. Note that we have neglected insignificantterms k2

x and k2z for k around the Γ point.

To obtain the eigenvalue equation Hcm,1φI(y) = EφI(y), with E = 0 subject to the boundary condition φI(0) =φI(+∞) = 0, we write the solution in the following form:

φI(y) = Ny sin(αy)e−βyei(kxx+kzz)χI, Re(β) > 0, (S107)

where the normalization constant is given by Ny = 2√β(α2 + β2)/α2, and the eigenvector χI satisfies σyτyχI = χI.

Then the effective Hamiltonian for the surface I can be obtained in this basis as

HIsurf =

∫ +∞

0

φ∗I (y)Hcm,2φI(y) dy. (S108)

17

Therefore, we have

HIsurf = ∆2

(3 +

m

t

)%z − (∆1kx + iγ0) %x − (∆1kz + iγz) %y. (S109)

The low-energy effective Hamiltonians for the surfaces II, III and IV can be obtained by the same procedures

HIIsurf = ∆2

(3 +

m

t

)%z + (∆1ky + iγ0) %x + (∆1kz + iγz) %y, (S110)

HIIIsurf = ∆2

(3 +

m

t

)%z + (∆1kx + iγ0) %x − (∆1kz + iγz) %y, (S111)

HIVsurf = ∆2

(3 +

m

t

)%z − (∆1ky + iγ0) %x + (∆1kz + iγz) %y, (S112)

for 3 +m/t > 0.For the Hermitian case (i.e., γ0 = 0), the last two kinetic terms of the effective Hamiltonian in Eqs. (S109)-(S112)

describe the gapless surface states, which are gapped out by the first terms with the Dirac mass. To derive the second-order boundary modes, we introduce a new coordinate p along anticlockwise direction within the planes parallel tothe xy plane, and we rewrite Eqs. (S109)-(S112) as

Hsurf = ∆2

(3 +

m

t

)%z − [−i∆(p)∂p+ iγ(p)] %x − (∆1kz + iγz) %y, (S113)

where ∆(p) = ∆1,−∆1,∆1,−∆1, and γ(p) = γ0,−γ0,−γ0, γ0 along the anticlockwise direction of the four surfaces.According to Eq. (S113), it is easy to verify that the Hermitian part (i.e., γ0 = 0) of H3D(k) supports four-folddegenerate gapless hinge modes for kz = 0 (analogous to the Jackiw-Rebbi model [S20]).

In the presence of non-Hermiticity, we first solve the boundary mode along the hinge intersected by the surfaces Iand II. We write this boundary mode in the following form:

ΨIh = eikzz+κI,1pχh,1, Re(κI,1) > 0, p = p− L < 0, (S114)

ΨIIh = eikzz+κII,1pχh,1, Re(κII,1) < 0, p = p− L > 0, (S115)

where L is the width of cubic sample, and χh,1 is the eigenvector. To find the gapless boundary mode along the hingeintersected by the surfaces I and II, the following equations are satisfied:

κI,1 =γ0 ±

√δ2 + (∆1kz + iγz)

2

∆1, κII,1 =

γ0 ±√δ2 + (∆1kz + iγz)

2

∆1, Re(κI,1) > 0, Re(κII,1) < 0, (S116)

where δ = ∆2 (3 +m/t).By applying the same procedures, to have the zero gapless boundary modes along hinges intersected by surfaces II

and III, surfaces III and IV and surfaces IV and I, we have the following equalities:

κII,2 =γ0 ±

√δ2 + (∆1kz + iγz)

2

∆1, κIII,2 =

−γ0 ±√δ2 + (∆1kz + iγz)

2

∆1, Re(κII,2) > 0, Re(κIII,2) < 0,

(S117)

κIII,3 =−γ0 ±

√δ2 + (∆1kz + iγz)

2

∆1, κIV,3 =

−γ0 ±√δ2 + (∆1kz + iγz)

2

∆1, Re(κIII,3) > 0, Re(κIV,3) < 0,

(S118)

κIV,4 =−γ0 ±

√δ2 + (∆1kz + iγz)

2

∆1, κI,4 =

γ0 ±√δ2 + (∆1kz + iγz)

2

∆1, Re(κIV,4) > 0, Re(κI,4) < 0.

(S119)

According to Eqs. (S116)-(S119), it is straightforward to verify that the gapless boundary modes, under the periodicboundary condition along the z direction, appear only along the hinge intersected by the surfaces II and III if|γ0| >

√δ2 − γ2

z with |δ| > |γz| or |γz| > |δ| . In summary, the simple low-energy effective Hamiltonian presentedhere can explain the existence and the localization at one hinge of second-order gapless boundary modes due to theinterplay between mirror-ration symmetry and non-Hermiticity.

18

ⅠⅡ

Ⅲ

Ⅳ

x

y

z

FIG. S13. Schematic illustration of a 3D non-Hermitian SOTI in a cubic sample. I, II, III and IV label the four surfaces of thelattice.

E. Topological index

As shown in the main text, due to mirror-rotation symmetry Mxy, we can write the Hamiltonian H3D(k) into theblock-diagonal form with kx = ky = k as

U−1H3D(k)U =

[H+(k, kz) 0

0 H−(k, kz)

], (S120)

where H+(k, kz) acts on the +i mirror-rotation subspace, and H−(k, kz) acts on the −i mirror-rotation subspace.The unitary transformation U is

U =

0 − 1+i

2 0 1+i2

1+i2 0 − 1+i

2 00 1√

20 1√

21√2

0 1√2

0

, (S121)

and H+(k, kz) and H−(k, kz) are

H+(k, kz) = − [m+ 2t cos(k) + t cos(kz)]σz +√

2 [∆1 sin(k) + iγ0]σy − [∆1 sin(kz) + iγz]σx, (S122)

H−(k, kz) = − [m+ 2t cos(k) + t cos(kz)]σz −√

2 [∆1 sin(k) + iγ0]σy − [∆1 sin(kz) + iγz]σx. (S123)

To account for the non-Bloch-wave behavior of the non-Hermitian systems and to capture the essential physics withanalytical results, we consider a low-energy continuum mode of H3D(k) and H±(k, kz). Then the non-Bloch-wavebehavior can be taken into account by replacing the real vavevector k in Eqs. (S120)-(S123) with the complex oneafter applying the degenerate perturbation theory to Hamiltonian H3D with open boundaries along the x, y and zdirections for min|m+ 3t|, |t|, |∆1| max|γ0|, |γz| [S8]:

k → k + ik′, kz → kz + ik′z, (S124)

with

k′ = − γ0

∆1, k′z = − γz

∆1. (S125)

Then, we rewrite Eq. (S122) and (S123) as

H+(k, kz) = −

[m+ 3t− t

(k − i γ0

∆1

)2

− t

2k2z

]σz +

√2

[∆1

(k − i γ0

∆1

)+ iγ0

]σy −

[∆1

(kz − i

γz∆1

)+ iγz

]σx,

(S126)

19

H−(k, kz) = −

[m+ 3t− t

(k − i γ0

∆1

)2

− t

2k2z

]σz −

√2

[∆1

(k − i γ0

∆1

)+ iγ0

]σy −

[∆1

(kz − i

γz∆1

)+ iγz

]σx.

(S127)

The Chern numbers C+ and C−, corresponding to the H+(k, kz) and H−(k, kz), are defined as

C± =1

2π

∫BZ

F±(k, kz) dkdkz, (S128)

where F±(k, kz) is the Berry curvature

F±(k, kz) = ∂kAkz± (k, kz)− ∂kzAk±(k, kz), (S129)

and Aµ± is the Berry connection

Aµ±(k, kz) = i 〈χ±(k, kz)| |∂µφ±(k, kz)〉 , (S130)

where∣∣φα±(k)

⟩and

∣∣χα±(k)⟩

are the right and left eigenstates of H±(k, kz). Then, the total Chern number is given by

C = C+ − C−. (S131)

∗ tao.liu@riken.jp† gong@cat.phys.s.u-tokyo.ac.jp‡ kawabata@cat.phys.s.u-tokyo.ac.jp§ ueda@phys.s.u-tokyo.ac.jp¶ fnori@riken.jp

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