Topology and Exceptional Points of Massive Dirac Models with Generic Non-Hermitian Perturbations

Topology and exceptional points of massive Dirac models with generic non-Hermitian perturbations

1 2, 3, 1, W. B. Rui, Y. X. Zhao, ∗ and Andreas P. Schnyder † 1Max-Planck-Institute for Solid State Research, D-70569 Stuttgart, Germany 2National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China 3Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China (Dated: June 24, 2019) Recently, there has been a lot of activity in the research field of topological non-Hermitian physics, partly driven by fundamental interests and partly driven by applications in photonics. However, despite these activities, a general classification and characterization of non-Hermitian Dirac models that describe the experimental systems is missing. Here, we present a systematic investigation of massive Dirac models on periodic lattices, perturbed by general non-Hermitian terms. We find that there are three different types of non-Hermitian terms. For each case we determine the bulk exceptional points, the boundary modes, and the band topology. Our findings serve as guiding principles for the design of applications, for example, in photonic lattices. For instance, periodic Dirac systems with non-Hermitian mass terms can be used as topological lasers. Periodic Dirac systems with non-Hermitian anti-commuting terms, on the other hand, exhibit exceptional points at the surface, whose non-trivial topology could be utilized for optical devices.

The fields of non-Hermitian physics and topological have been made with partial success for certain special materials have recently intertwined to create the new re- cases [12–20, 44, 48]. Since most non-Hermitian experi- search direction of non-Hermitian topological phases. As mental systems can be faithfully described by Dirac mod- a result of the joint efforts from both fields, fascinat- els with small non-Hermitian perturbations [2, 3, 5–8, 49– ing new discoveries have been made, both at the fun- 54], a systematic investigation of general non-Hermitian damental level and with respect to applications [1–24]. perturbations of Dirac Hamiltonians would be particu- For instance, topological exceptional points have been larly valuable. This would be not only of fundamental found both in non-Hermitian periodic lattices [13–20] interest, but could also inform the design of new appli- and in non-Hermitian non-periodic systems [25]. Excep- cations. tional rings and bulk Fermi arcs have been discovered in In this Rapid Communication, we present a systematic non-Hermitian periodic lattices with semi-metallic band investigation of massive Dirac Hamiltonians on periodic structures [21–23, 26–31]. At these exceptional points lattices, perturbed by small non-Hermitian terms. We and rings, two or more eigenstates become identical and show that these non-Hermitian Dirac Hamiltonians can self-orthogonal, leading to a defective Hamiltonian with be either intrinsically or superficially non-Hermitian, de- a nontrivial Jordan normal form [32]. These exceptional pending on whether the non-Hermiticity can be removed points have many interesting applications. For exam- by a similarity transformation. According to the Clifford ple, they can be used as sensors with enhanced sensitiv- algebra, general non-Hermitian terms can be categorized ity [1, 33], as optical omni-polarizers [34], for the creation into three different types: (i) non-Hermitian terms that of chiral laser modes [35], or for unidirectional light trans- anti-commute with the whole Dirac Hamiltonian, (ii) ki- mission [36–38]. Furthermore, non-Hermitian periodic netic non-Hermitian terms, and (iii) non-Hermitian mass lattices with nontrivial topology can be utilized as topo- terms. Remarkably, we find a two-fold duality for the logical lasers [2–4]. Moreover, it has been shown that first two types of non-Hermitian perturbations: Dirac non-Hermitian topological Hamiltonians provide useful models perturbed by type-(i) terms are superficially non- descriptions of strongly correlated materials in the pres- Hermitian with periodic boundary conditions (PBCs), ence of disorder or dissipation [39–47]. This has given but intrinsically non-Hermitian with open boundary con- new insights into the Majorana physics of semiconductor- ditions (OBCs). Vice versa, Dirac models with type-(ii) superconductor nanowires [45] and into the quantum os- terms are intrinsically non-Hermitian with PBCs, but su- cillations of SmB6 [46, 47]. perficially non-Hermitian with OBCs. Interestingly, for

arXiv:1902.06617v2 [cond-mat.mes-hall] 20 Jun 2019 Despite these recent activities, a general framework for type-(i) and type-(ii) terms the non-Hermiticity leads to the study and classification of non-Hermitian Dirac mod- (d 2)-dimensional exceptional spheres in the surface and els that describe the aforementioned experimental sys- bulk− band structures, respectively. Type-(iii) terms, on tems is lacking. In particular, a general classification of the other hand, induce intrinsic non-Hermiticity both for exceptional points and topological surface states in non- OBCs and PBCs, but with a purely real surface-state Hermitian systems is missing, although various attempts spectrum and no exceptional spheres. Intrinsic versus superficial non-Hermiticity.— We be- gin by discussing some general properties of non- ∗ [email protected] Hermitian physics. First, we recall that in Hermitian † [email protected] physics only unitary transformations of the Hamiltonian 2 are considered, because only these preserve the reality 2 2 of the expectation values. In non-Hermitian physics, however, the Hamiltonian can be similarity transformed, 1 H V − HV , by any invertible matrix V , which is not necessarily→ unitary but is required to be local. For this 0 0 reason, a large class of non-Hermitian Hamiltonians H can be converted into Hermitian ones by non-unitary similarity transformations, i.e., 2 2 0 0 0 0 1 V − HV = H0,H0† = H0. (1) 2 2 Using this observation, we call Hamiltonians whose non- Hermiticity can or cannot be removed by the above transformation as superficially or intrinsically non-Hermitian, 0 respectively. 0 For non-interacting local lattice models, which is our main focus here, H is a quadratic form, whose entries 0 0 2 2 are specified as H(rα),(r α ) with r the positions of the 0 0 unit cells and α a label for internal degrees of freedom. Correspondingly, the similarity transformation has matrix elements V(rα),(r0α0). By the locality condition, the FIG. 1. (a),(b) Energy spectra of the two-dimensional to- matrix elements H(rα),(r0α0) and V(rα),(r0α0) are required pological insulator HTI,0 perturbed by the type-(i) non- to tend to zero sufficiently fast as r r0 . If Hermitian term iλΓ4 with periodic and open boundary con- H can be converted into a Hermitian| − Hamiltonian| → ∞ by ditions, respectively. Here, we set λ = 0.3 and M = 1.5. a local transformation V , its eigenvalues are necessarily (c),(d) Energy spectra of HTI,0 perturbed by the type-(ii) real. Conversely, any local lattice Hamiltonian with a non-Hermitian term iλΓ2 with periodic and open boundary real spectrum is either entirely Hermitian or superficially conditions, respectively. Here, we set λ = 0.5 and M = 1.5. non-Hermitian [55]. Solid and dotted lines represent bulk and surface states, respectively. The real and imaginary parts of the eigenvalues A characteristic feature of non-Hermitian lattice mod- are indicated in blue and green. Red points represent excep- els is the existence of exceptional points in parameter tional points. space, where two or multiple eigenvectors become identical, leading to a non-diagonalizable Hamiltonian. How- ever, it is important to note that such exceptional points where Γµ denote the gamma matrices that satisfy are not dense in parameter space. That is, there exist Γµ, Γν = 2δµν and M is the real mass parameter. With arbitrarily small perturbations which remove the excep- OBCs{ in} the jth direction and PBCs in all other direc- tional points, rendering the Hamiltonian diagonalizable tions, the Hamiltonian reads [56]. One such perturbation relevant for lattice models are the boundary conditions [19, 57], which modify the 1 1 H (k˜) = (S S†) Γ (S + S†) Γ (3) hopping amplitudes between opposite boundaries. For a 0 2i − ⊗ j − 2 ⊗ d+1 general classification of non-Hermitian Hamiltonians, it is +INj ( sin kiΓi + (M cos ki)Γd+1), b ⊗ b b b− therefore essential to distinguish between different types i=j i=j of boundary conditions, in particular OBCs and PBCs. X6 X6 With PBCs and assuming translation symmetry, we can where k˜ denotes the vector of all momenta except kj, perform a Fourier transformation of Eq. (1) to obtain S = δ is the right-translational operator, and N 1 ij i,j+1 j H0(k) = V − (k)H(k)V (k). Here, V (k) is assumed to be stands for the number of layers in the jth direction. From local in momentum space. It is worth noting that the theb above two equations it is now clear that, according locality in momentum space is essentially different from to the Clifford algebra, there exist only the three types that in real space. Generically, the Fourier transform of of non-Hermitian perturbations discussed above. We will ik (r r0) V (k), Vr,r0 = k V (k)e · − , is not local in general. now study these individually. Non-Hermitian Dirac Hamiltonians.— We now apply Non-Hermitian anti-commuting terms.— P We start the above concepts to non-Hermitian Dirac models of the with non-Hermitian terms of type (i), i.e., terms that form H = H +λU, where H is a Hermitian Dirac Hamil- 0 0 anti-commute with the Dirac Hamiltonian H0. Such non- tonian with mass M, and U a non-Hermitian perturba- Hermitian terms are possible for all Altland-Zirnbauer tion with λ M. Assuming PBCs in all directions, we classes with chiral symmetry [58, 59], in which case they consider the following Hermitian Dirac Hamiltonian on are given by the chiral operator Γ. With PBCs the the d-dimensional cubic lattice Hamiltonian perturbed by these type-(i) terms is ex- d d pressed as H (k) = sin k Γ + (M cos k )Γ , (2) 0 i i − i d+1 i=1 i=1 H(k) = H0(k) + iλΓ, (4) X X 3

d α β α β where Γ is an additional gamma matrix with i.e., dλ ψλ Xi ψλ = 0 with ψλ and ψλ the left and Γ,H (k) = 0 and λ a real parameter. The spec- right eigenstatesh | | ofi H, respectively.h | | i { 0 } trum of H(k) is given by E(k) = d2(k) λ2 with Let us now illustrate the above general considerations 2 2 ± − d (k)I = H (k), which is completely real for all k, pro- by considering as an example, HTI(k) = sin kxΓ1 + 0 p vided that λ is smaller than the energy gap of H . sin kyΓ2 + (M cos kx cos ky)Γ3 + iλΓ4, with Γj the 0 − − Thus, Hamiltonian| | (4) with λ M is only superficially 4 4 Dirac gamma matrices, which describes a topo- × non-Hermitian and we can remove the non-Hermitian logical superconductor in class DIII or a topological in- term by a similarity transformation. The corresponding sulator in class AII [59]. The energy spectra of HTI transformation matrix V (k) can be derived systemati- with periodic and open boundary conditions are shown in cally by noticing that the flattened Hamiltonian H˜0(k) = Figs. 1(a) and 1(b), respectively [see Supplemental Ma- H0(k)/d(k) and Γ form a Clifford algebra and, thus, terial (SM) [60] for details]. We observe that the bulk spectrum is purely real, while the surface spectrum is i[H˜0(k), Γ]/4 generates rotations of the plane spanned by H˜ (k) and Γ. Hence, the explicit expression of the complex with two exceptional points of second order lo- 0 cated at k = arcsin λ . transformation matrix is V (k) = exp[ i H˜ (k)Γη(k)], x 2 0 Non-Hermitian± kinetic| | terms.— We proceed by con- η(k) − with e = (d(k) + λ)/(d(k) λ). From Eq. (1) it sidering non-Hermitian kinetic terms [i.e., terms of type follows that the transformed Hamiltonian− is H (k) = 0 (ii)] added to H0 with PBCs. The effects of these non- 2 2 p 1 λ /d (k)H0(k), which is manifestly Hermitian for Hermitian terms can be most clearly seen by studying − λ M. the continuous version of Eq. (2), namely p With OBCs, on the other hand, type-(i) perturba- d tions lead to intrinsic non-Hermiticity, provided the Dirac H(k) = H0(k) + iλΓj = kiΓi + mΓd+1 + iλΓj, (5) Hamiltonian H0 is in the topological phase. This is be- i=1 cause the topological boundary modes acquire complex X spectra due to the non-Hermitian term iλΓ, even for in- with 1 j d and λ real. The energy spectrum of H(k), ≤ ≤ finitesimally small λ. To see this, we first observe that 2 2 2 E(k) = i=j ki + (kj + iλ) + m , is complex and for any eigenstate ψ0 of H0 with energy E0,Γψ0 is also ± 6 exhibits exceptional points on the (d 2)-dimensional an eigenstate of H0, but with opposite energy E0. Ap- qP − sphere k2 + m2 = λ2 within the− k = 0 plane. plying chiral perturbation theory to H = H0 + iλΓ, we i=j i j 6 find that since iλΓ scatters ψ into Γψ , eigenstates of Hence, H(k) with PBCs is intrinsically non-Hermitian. 0 0 P H can be expressed as superpositions of ψ0 and Γψ0. To study the case of OBCs we consider H(k) in a slab Explicitly, we find that the eigenstates of H are ψ = geometry with the surface perpendicular to the jth di- ± 2 2 rection. The energy spectrum in this geometry is ob- ψ0 + c Γψ0, with c = iE0/λ 1 E0 /λ and en- ± ± ˜ 2 2 ± − tained from H(k, i∂j), i.e., by replacing kj by i∂j in ergy E = E0 λ . This analysis holds in particular − − ± ± − p Eq. (5). Then, it is obvious that the non-Hermitian term also for the topological boundary modes of H0, which are massless Diracp fermions with linear dispersions. Conse- iλΓj can be removed by the similarity transformation λxj λxj ˜ λxj ˜ quently, even for arbitrarily small λ, there exists a seg- V = e . That is, e− H(k, i∂j)e = H0(k, i∂j), − ˜ − ment in the spectrum of H around E = 0 with purely which is manifestly Hermitian. Accordingly, H(k, i∂j) − imaginary eigenergies. has a real spectrum and its eigenstates are related to those of H (k˜, i∂ ) by ψ(x , k˜) = eλxj ψ (x , k˜). We To make this more explicit, we can derive a low-energy 0 − j j 0 j effective theory for the boundary modes, by projecting conclude that continuous Dirac models perturbed by non- the bulk Hamiltonian onto the boundary space. Gener- Hermitian kinetic terms are superficially non-Hermitian with OBCs, but intrinsically non-Hermitian with PBCs. ically, the boundary theory is of the form Hb(k˜) = i The same holds true for lattice Dirac models. i=j kiγ +iλγ, where the first term describes the bound- 6 To exemplify this, we consider the lattice Dirac model ary massless Dirac fermions of H . The matrices γi and P 0 of Eq. (3) perturbed by the non-Hermitian term iλΓ , γ are the projections of Γ and Γ, respectively, onto the j i ˜ ˜ boundary space, and satisfy γi, γ = 0. With this, we i.e., H(k) = H0(k) + I iλΓj. This Hamiltonian can be { } transformed to (see SM⊗ [60] for details) find that the boundary spectrum is E = k˜ 2 λ2, b ± | | − and that there exists a (d 2)-dimensionalq exceptional ˜ 1 1 − H0(k) = (S S†) Γj (S + S†) Γd+1 sphere of radius k˜ = λ in the boundary Brillouin zone, 2i − ⊗ − 2 ⊗ | | which separates eigenstates with purely real and purely 2 2 +bI (b sin kiΓi +b Mb λ Γd+1), (6) complex energies from each other. ⊗ k − i=j q As an aside, we remark that even arbitrarily large non- X6 Hermitian terms iλΓ cannot remove the topological sur- by the similarity transformation V = Nj 1 face state. The reason for this is that iλΓ is a chiral diag 1, α, , α − [(1 + α)I + i(1 α)ΓjΓd+1] . { ··· } ⊗ − operator, which acts only within a unit cell and does not Here α = (M λ)/(M + λ) and M = k − k k couple different sites. In other words, the expectation M i=j cos ki. Equation (6) is manifestly Her- − 6 p value of the position operator Xi is independent of λ, mitian for λ M. As a concrete example, we set in P 4

nian is obtained by Hb(k˜) = P (H0(k˜) + iλΓd+1)P , with k˜ satisfying β = M cos k˜ + iλ < 1. Since Γ | | | − i i | d+1 anti-commutes with Γj, the non-Hermitian perturbation P iλΓd+1 vanishes under the projection. Thus, the effective ˜ i boundary Hamiltonian becomes Hb(k) = i=j sin kiγ , with γi = P Γ P , whose spectrum is manifestly6 real. We i P note that while the effective boundary Hamiltonian is not altered by the non-Hermitian mass term iλΓd+1, the range of k˜ in which the boundary modes exists is changed ˜ to M i cos ki + iλ < 1. To| illustrate− these general| considerations, we consider as an exampleP the two-dimensional topological insulator with a non-Hermitian mass term, i.e., HTI(k) = sin kxΓ1 + sin kyΓ2 + (M cos kx cos ky)Γ3 + iλΓ3. As shown in Fig. 2, the spectrum− of− this non-Hermitian Hamiltonian is complex both with open and periodic boundary conditions. With OBCs there appear surface states within the region M cos kx + iλ < 1, whose spectrum is purely real. | − | FIG. 2. (a),(c) Real and (b),(d) imaginary parts of the energy Discussion.— In summary, we systematically investi- spectra of the two-dimensional topological insulator H TI,0 gated d-dimensional massive Dirac models on periodic perturbed by the non-Hermitian mass term iλΓ3 with periodic and open boundary conditions, respectively. The parameters lattices perturbed by general non-Hermitian terms. We are chosen as M = 1.5 and λ = 0.3. Solid and dotted lines find that there are three different types of non-Hermitian represent bulk and surface states, respectively. terms: (i) non-Hermitian anti-commuting terms, (ii) non-Hermitian kinetic terms, and (iii) non-Hermitian mass terms. Interestingly, we find a two-fold duality for the first two types of non-Hermitian perturbations: Eq. (6) d = 2 with Γi the Dirac gamma matrices, which describes a two-dimensional topological insulator. The With open boundary conditions non-Hermitian anti- energy spectra for this case with periodic and open commuting terms give rise to intrinsic non-Hermiticity, boundary conditions are shown in Figs. 1(c) and 1(d), while non-Hermitian kinetic terms lead to superficial respectively. non-Hermiticity. Vice versa, with periodic boundary Non-Hermitian mass terms.— Finally, we examine the conditions non-Hermitian anti-commuting terms induce effects of non-Hermitian mass terms, i.e., non-Hermitian superficial non-Hermiticity, while non-Hermitian kinetic terms generate intrinsic non-Hermiticity. Importantly, terms of type (iii). For that purpose, we add iλΓd+1 to Eqs. (2) or (3), which is equivalent to assuming that for the anti-commuting and kinetic terms the intrinsic the mass M is complex. Hence, the energy spectrum non-Hermiticity manifests itself by exceptional points in is always complex independent of the boundary condi- the surface and bulk band structures, respectively. Non- tions (see SM [60]). Thus, massive Dirac models per- Hermitian mass terms, in contrast, render the Hamilto- turbed by non-Hermitian mass terms are intrinsically nian always intrinsically non-Hermitian, independent of non-Hermitian, both for open and periodic boundary the boundary condition, but do not induce exceptional conditions. Furthermore, we find that there are no ex- points in the band structure. ceptional points, not in the bulk and not in the surface Our findings can be used as guiding principles for the band structure. Indeed, remarkably, topological bound- design of applications in, e.g., photonic cavity arrays. ary modes are unaffected by type-(iii) perturbations and For example, our analysis shows that topological Dirac keep their purely real energy spectra. systems perturbed by non-Hermitian mass terms exhibit robust surface states with a purely real spectrum [see To demonstrate this explicitly, we solve for the bound- Fig. 2(c)]. In photonic cavity arrays, these surface states ary modes of Eq. (3) perturbed by iλΓd+1. That is, ˜ provide robust channels for light propagation, which are we solve (H0(k) + INj iλΓd+1) ψ˜ = E ψ˜ with ⊗ | ki | ki protected against perturbations and disorder. Impor- ψk˜ the ansatz for the right eigenvector of the bound- tantly, this property can be exploited for the design of | i Nj i ary mode, ψ˜ = β i ξ˜ , where ξ˜ is a efficient laser systems that are immune to disorder. That | ki i=1 | i ⊗ | ki | ki spinor, i labels the latice sites along the jth direction, is, optically pumping the boundary of the cavity array in- and β is a scalar withP β < 1 (see SM [60] for de- duces single-mode lasing in the surface states, with high | | tails). By solving this Schrödingerequation we find that slope efficiency. This was recently demonstrated, in two- β = M cos k˜ + iλ and that the boundary mode dimensional optical arrays of microresonators [2–4]. It − i i is an eigenstate of iΓd+1Γj with eigenvalue +1. Hence, follows from our analysis that these phenomena occur the projectorP P onto the boundary space is given by in a broader class of photonic band structures, namely P = (1+iΓd+1Γj)/2 and the effective boundary Hamilto- in any topological Dirac system [59] perturbed by non- 5

Hermitian mass terms. models and to periodic gapless Dirac models, i.e., to Another possible application are photonic devices that non-Hermitian periodic lattices with semi-metallic band utilize the exceptional points formed by the surface states structures. Second, it would be interesting to study the of Dirac systems with non-Hermitian anti-commuting role of symmetries, specifically, how the symmetries con- terms [see Fig. 1(b)]. While in non-periodic systems the strain the form of the non-Hermitian terms. Third, it creation of exceptional points requires fine tuning, the would be of great value to derive a general and exhaus- surface exceptional points of the discussed lattice Dirac tive classification of exceptional points based on symme- systems are required to exist by topology. Moreover, try and topology [30]. This is particularly important in these surface exceptional points are protected against view of the numerous applications of exceptional points noise and disorder and cannot be removed by symmetry in photonic devices. preserving perturbations (see SM [60] for details). They Acknowledgments.— This work is supported by the could potentially be used for surface sensing. NSFC (Grant No. 11874201), the Fundamental Re- We close by discussing several interesting directions search Funds for the Central Universities (Grant No. for future studies. First, our approach can be general- 0204/14380119), and the GRF of Hong Kong (Grant No. ized in a straightforward manner to non-periodic Dirac HKU173057/17P).

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2 2 Authors: W. B. Rui, Y. X. Zhao, and A. P. Schnyder 0 0

2 2 In this Supplemental Material, we solve the lattice model of the two-dimensional topological insulator with 0 0 0 non-Hermitian anti-commuting terms considering both 0 periodic and open boundary conditions and discuss the robustness of the exceptional points on the surface 2 against symmetry preserving perturbations (Sec. I), de- 2 rive the similarity transformation for the generic Dirac 0 0 model with non-Hermitian kinetic terms for open boundary conditions (Sec. II), and calculate the boundary 2 2 states for the generic Dirac model with non-Hermitian mass terms (Sec. III). 0 0

I. 2D TOPOLOGICAL INSULATOR WITH FIG. S1. The spectra for non-Hermitian 2D topological in- NON-HERMITIAN ANTI-COMMUTING TERMS sulator with large non-Hermitian anti-commuting potentials. 2 2 2 For min dTI(k) < λ < max dTI(k), the bulk spectrum (a) is complex, and the corresponding open boundary spectrum is We start with the Hermitian lattice model of a 2D 2 2 shown in (b); For max dTI(k) < λ , the bulk spectrum (c) is topological insulator, purely imaginary, and the corresponding open boundary spec- (k) = sin k Γ + sin k Γ + (M cos k cos k )Γ , trum is plotted in (d). The dotted lines represent boundary TI,0 x 1 y 2 x y 3 states. The parameter M is chosen to be 1.0. H − − (S1) where the ﬁve gamma matrices are Γ = σ τ ,Γ = σ i i⊗ 1 4 0⊗ τ3,Γ5 = σ0 τ2 (i = 1, 2, 3), with σi and τi denoting the ⊗ constructed below Eq. (4) in the main text, the Hamil- Pauli matrices. The energy eigenvalues are ETI,0(k) = 2 2 2 tonian in Eq. (S3) can be converted to be Hermitian by dTI(k), with dTI(k) = sin kx + sin ky + (M cos kx ± 2 − − the transformation cos ky) . The eigenstates are found to be

1 T 2 +, = sin kx i sin ky, M(k), 0, dTI , 1 λ | ↑i √2dTI − − (k)− TI(k) (k) = 1 2 TI,0(k), (S4) V H V s − dTI(k)H 1 T +, = M(k), sin kx + i sin ky, dTI, 0 , | ↓i √2dTI where (k) = exp[ i (k)Γ η (k)/d (k)], with 1 T V − 2 HTI,0 4 TI TI , = sin k + i sin k ,M(k), 0, d , ηTI(k) x y TI e = (dTI(k) + λ)/(dTI(k) λ). The matrix form |− ↑i √2dTI − of (k) reads, − 1 T V p , = M(k), (sin kx + i sin ky), dTI, 0 , |− ↓i √2dTI − − (S2) ηTI(k) ηTI(k) (k) = cosh i sinh TI,0(k)Γ4/dTI(k). with M(k) = M cos k cos k and the matrix trans- V 2 − 2 H x y (S5) position T. The− system− is topologically non-trivial for M ( 2, 2). Taking open boundary conditions in the y direction ∈ − Including the non-Hermitian anti-commuting pertur- with Ny layers, the real space Hamiltonian reads bation iλΓ4, the Hamiltonian becomes 1 1 TI(k) = TI,0(k) + iλΓ4. (S3) H (k ) = S S† Γ S + S† Γ H H TI x 2i − ⊗ 2 − 2 ⊗ 3 According to our theory, the non-Hermitian perturbation + INy (sin kxΓ1 + (M cos kx)Γ3) + INy iλΓ4. ⊗ b b − b b ⊗ scatters eigenstates ψ0 to Γ4ψ0. The energy eigenvalues (S6) become E (k) = d2 (k) λ2, and the correspond- TI ± TI − ing eigenstates are modiﬁed as ψ = ψ0 + (iETI,0/λ p ± ± Here S and S† are the forward and backward translation 1 E2 /λ2)Γ ψ , with ψ given in Eq. (S2). − TI,0 4 0 0 operators in the y direction, which let S i = i + 1 2 2 | i | i qFor dTI(k) > λ , the spectrum is purely real in momen- and Sb† i =bi 1 , where i labels the ith site in the y tum space. According to the similarity transformation direction.| i The| − correspondingi matrix representationsb of b 2

S and S† read In the main text, the non-Hermitian potential is relatively small and thus can be treated as perturbation. In 0 1 0 0 0 b b0 0 0 0 0 FIG. S1, we increase the strength of the non-Hermitian ··· ··· 1 0 0 0 0 0 0 1 0 0 potential. By increasing λ, the bulk system becomes gap-  ···  0 0 0 1 ··· 0 0 1 0 0 0 ··· less with complex spectrum (a), and gapless with purely S = ··· , S† =  .  , 0 0 1 0 0 0 0 0 0 .. 0 imaginaly spectrum (c). The corresponding boundary . . . ··· .   ......  . . . . .  states are denoted by dotted lines in (b) and (d), which b ......  b ...... 1     exist even when the bulk becomes gapless. 0 0 0 0 1 0 0 0 0 0 0 0        (S7) with the dimension Ny. A. Robustness of the exceptional points on the Though the translational symmetry is broken in the y surface against perturbations direction, it is still preserved in the x direction. We adopt Ny i the ansatz ψkx = i=1 β i ξkx with β < 1 for In this subsection, we discuss the robustness of the sur- the boundary| states.i Solving| i the ⊗ | Schrödingerequationi | | face exceptional points against symmetry preserving per- ˆ P HTI(kx) ψkx = Ekx ψkx leads to the relation turbations. Before going to specific perturbations, it is | i | i necessary to investigate the projection onto the surface 1 sin k Γ + (β β 1)Γ + (M cos k subspace formed by the states of Ψ = Ψ , Ψ corre- x 1 − 2 x { ↑ ↓} 2i − − sponding to the positive eigenvalue of iΓ3Γ2 in Eq. (S10). 1 1 ˆ It is found that after the projection, Ψ Γ1 Ψ = σ1, (β + β− ))Γ3 + iλΓ4 ξkx = Ekx ξkx , −2 | i | i Ψ Γ Ψ = σ , and Ψ Γ Ψ = σ h.| Thus,| i the ef- (S8) 4 3 5 2 fectiveh | | boundaryi − Hamiltonianh | | ini Eq.− (S14) is equivalent in the bulk and to TI,b(kx) = sin kxσ1 iλσ3 in the subspace spanned 1 1 byH Ψ. − sin k Γ + βΓ + (M cos k β)Γ x 1 2i 2 − x − 2 3 The bulk Hamiltonian in Eq. (S3) is invariant un- ˆ 1 ˆ +iλΓ ξ = Eˆ ξ , der the time-reversal symmetry, i.e., − TI( k) = 4 | kx i kx | kx i T Hˆ − T ˆ (S9) TI(k). Here the time-reversal operator is = σ1 τ2 Hwith ˆ the complex conjugation. It isT evident⊗ thatK at the boundary. Taking the difference between Eqs. (S8) K and (S9), we obtain a simpler constraint iΓ1, iΓ4, and iΓ5 are perturbations that preserve time- reversal symmetry. Here is real and relatively small. In

iΓ3Γ2 ξkx = ξkx , (S10) addition to time-reversal symmetry, the space-inversion | i | i 1 symmetry is present if ˆ− TI( k) ˆ = TI(k) with which means the boundary states correspond to the pos- ˆ = iσ τ . The perturbationsP H − P that areH invariant P 3 ⊗ 0 itive eigenvalue of iΓ3Γ2, from which we can construct under both time-reversal and space-inversion symmetry the projector are iΓ4 and iΓ5. Since iΓ4 can be absorbed into the non-Hermitian potential iλΓ , we focus on the perturba- 1 4 P = (1 + iΓ Γ ). (S11) tion iΓ5. After the inclusion of this perturbation, the 2 3 2 effective boundary Hamiltonian becomes (k )0 = HTI,b x Applying this projector to Eq. (S9), we have sin kxσ1 iλσ3 iσ2 and the surface eigenvalues are − 2 − 2 2 E0 = (sin kx λ ). The existence of the ex- ˆ ± − − (sin kxΓ1 + iλΓ4) ξkx = Ekx ξkx . (S12) ceptional points is not affected, but their locations are | i | i shifted from arcsin λ to arcsin √λ2 + 2 . With the relation of Eq. (S10), Eq. (S9) becomes ± | | ± | | ˆ sin kxΓ1 + (M cos kx β)Γ3 + iλΓ4 ξkx = Ekx ξkx , and its difference− with Eq.− (S12) yields | i | i II. SIMILARITY TRANSFORMATION FOR THE GENERIC DIRAC MODEL WITH β = M cos kx. (S13) NON-HERMITIAN KINETIC TERMS FOR OPEN − BOUNDARY CONDITIONS The effective boundary Hamiltonian can be obtained after the projection, With OBCs in the y direction the lattice Dirac model with a non-Hermitian kinetic perturbation reads (k ) = P (k)P = sin k γ1 + iλγ4, (S14) HTI,b x HTI x ˜ 1 1 1 4 H(k) = (S S†) Γj (S + S†) Γd+1 with γ = P Γ1P and γ = P Γ4P , for kx satisfying β = 2i − ⊗ − 2 ⊗ | | M cos kx < 1. The boundary energy eigenvalues are + INj ( bsinbkiΓi + (M b cosb ki)Γd+1 + iλΓj), | − | 2 2 ⊗ − ETI,b(kx) = sin kx λ . For λ < 1, exceptional i=j i=j X6 X6 points emerge± at k = −arcsin λ . | | (S15) p x ± | | 3 which is superficially non-Hermitian. Note that for the III. BOUNDARY STATES FOR THE DIRAC MODEL WITH NON-HERMITIAN MASS TERMS part (M i=j cos ki)Γd+1 + iλΓj in the above Hamil- − 6 tonian, the non-Hermitian term iλΓj can be regarded as P a non-Hermitian anti-commuting perturbation. Hence, With PBCs the generic lattice Dirac model with non- we can first make this part Hermitian by the following Hermitian mass perturbations is given by similarity transformation, d d (k) = sin kiΓi + (M cos ki)Γd+1 + iλΓd+1. 1 2 2 ρ− [(M cos k )Γ + iλΓ ]ρ = M λ Γ , H − i − i d+1 j i k − d+1 i=1 i=1 i=j X X (S24) X6 q (S16) The energy eigenvalues are E(k) = with Mk = M i=j cos Mk. Here ρi = (1 + α)I + i(1 d 2 d 2 − 6 − i=1 sin ki + (M + iλ i=1 cos ki) , which α)ΓjΓd+1, with α = (Mk λ)/(Mk + λ). Its inverse ± − 1 1 P − areq complex in general. is ρ− = [(1 + α)I i(1 α)ΓjΓd+1]. Next we turn P P i 4α p− − With OBCs in the j direction, the real space Hamilto- to the remaining terms with non-trivial spatial parts in nian reads, Eq. (S15), which read ˜ 1 1 1 1 H(k) = (S S†) Γj (S + S†) Γd+1 (S S†) Γ (S + S†) Γ 2i − ⊗ − 2 ⊗ 2i − ⊗ j − 2 ⊗ d+1 +INj ( sin kiΓi+(M cos ki)Γd+1)+INj iλΓd+1. 1 1 1 1 ⊗ b b − b b ⊗ b b= S ( Γj b Γbd+1) S† ( Γj + Γd+1). i=j i=j ⊗ 2i − 2 − ⊗ 2i 2 X6 X6 (S17) (S25) The operatorb ρ acts on the internalb degrees of freedom i Here N is the number of unit cells in the j direction. of the terms in the above equation as j The spectrum of this Hamiltonian is also complex. For both Hamiltonians in Eqs. (S24) and (S25), which obey 1 1 1 1 1 S ρ− ( Γ Γ )ρ = αSˆ ( Γ Γ ), different boundary conditions, adding the non-Hermitian ⊗ i 2i j − 2 d+1 i ⊗ 2i j − 2 d+1 (S18) mass term is equivalent to making the Hermitian mass b term complex, M M + iλ. 1 1 1 1 1 1 → Sˆ† ρ− ( Γ + Γ )ρ = Sˆ† ( Γ + Γ ). Except for the j direction, the translational symmetry ⊗ i 2i j 2 d+1 i α ⊗ 2i j 2 d+1 (S19) in other directions is preserved. We adopt the ansatz Nj i ψ˜ = β i ξ˜ with β < 1 for the boundary | ki i=1 | i ⊗ | ki | | states. By solving the Schödingerequation H(k˜) ψ˜ = We construct the similarity transformation ρS = P | ki 2 N 1 ˆ diag(1, α, α , , α j ) for the spatial part, which en- E˜ ψ˜ we find it gives two constraints, − k| ki ables the following··· transformation 1 1 sin k Γ + (β β− )Γ + (M cos k 1 ˆ i i 2i − j − i ρS− (αS)ρS = S, (S20) i=j i=j X6 X6 1 1 1 1 ρS− ( S†)ρS = S†. (S21) (β + β− ))Γ + iλΓ ξ˜ = Eˆ˜ ξ˜ , (S26) α b − 2 d+1 d+1 | ki k| ki b b Finally, the full similarity transformation operator is and constructed as 1 1 sin k Γ + βΓ + (M cos k β)Γ i i 2i j − i − 2 d+1 2 Nj 1 i=j i=j V = ρS ρi = diag(1, α, α , , α − ) X6 X6 ⊗ ··· + iλΓd+1 ξ˜ = Eˆ˜ ξ˜ . (S27) [(1 + α)I + i(1 α)ΓjΓd+1], (S22) k k k ⊗ − | i | i The difference between the above two equations yields a which converts the non-Hermitian Hamiltonian of simpler relation, Eq. (S15) into

iΓ Γ ξ˜ = ξ˜ . (S28) d+1 j| ki | ki 1 1 1 V − H(k˜)V = (S S†) Γ (S + S†) Γ 2i − ⊗ j − 2 ⊗ d+1 This means the boundary states correspond to the posi- 2 2 tive eigenvalue of iΓd+1Γj, from which we can construct + INb j b( sin kiΓi +b Mb λ Γd+1), ⊗ k − the projector i=j X6 q (S23) 1 P = (1 + iΓd+1Γj) . (S29) with Mk = M i=j cos Mk. − 6 2 P 4

Notice with the relation of Eq. (S28), Eq. (S27) be- tor P , it will vanish after the projection. Thus, the re- comes, sultant eﬀective boundary Hamiltonian is

sin k Γ + (M cos k + iλ β)Γ ξ˜ i i − i − d+1 | ki ˜ i i=j i=j b(k) = sin kiγ , (S33) 6 6 H X X i=j 6 = Eˆ˜ ξ˜ , (S30) X k| ki and under the projection P , Eq. (S27) also becomes, i ˜ with γ = P ΓiP , for k satisfying β = M i=j cos ki + iλ < 1. The localization region| obtained| | − in this6 way is sin k Γ ξ˜ = Eˆ˜ ξ˜ . | P i i| ki k| ki (S31) identical to that obtained by the method of biorthogo- i=j X6 nal bulk-boundary correspondence [S1]. For the effective The difference between the above two equations gives, boundary Hamiltonian the boundary energy eigenvalues ˜ 2 are purely real, i.e., Eb(k) = i=j sin ki. ± 6 β = M cos ki + iλ. (S32) − qP i=j X6 We now calculate the effective boundary Hamiltonian by projection P . Remarkably, since the non-Hermitian mass [S1] F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. term iλΓd+1 anti-commutes with iΓd+1Γj in the projec- Bergholtz, Phys. Rev. Lett. 121, 026808 (2018).