Bulk-Boundary Correspondence in a Non-Hermitian System in One Dimension with Chiral-Inversion Symmetry

Asymmetric coupling amplitudes effectively create an imaginary gauge field, which induces a non-Hermitian Aharonov-Bohm (AB) effect. Nonzero imaginary magnetic flux invalidates the bulk-boundary correspondence and leads to a topological phase transition. However, the way of non-Hermiticity appearance may alter the system topology. By alternatively introducing the non-Hermiticity under symmetry to prevent nonzero imaginary magnetic flux, the bulk-boundary correspondence recovers and every bulk state becomes extended; the bulk topology of Bloch Hamiltonian predicts the (non)existence of edge states and topological phase transition. These are elucidated in a non-Hermitian Su-Schrieffer-Heeger model, where chiral-inversion symmetry ensures the vanishing of imaginary magnetic flux. The average value of Pauli matrices under the eigenstate of chiral-inversion symmetric Bloch Hamiltonian defines a vector field, the vorticity of topological defects in the vector field is a topological invariant. Our findings are applicable in other non-Hermitian systems. We first uncover the roles played by non-Hermitian AB effect and chiral-inversion symmetry for the breakdown and recovery of bulk-boundary correspondence, and develop new insights for understanding non-Hermitian topological phases of matter.

Introduction.-Topological theory has been well established in condensed matter physics [1-31] and recent experimental progresses in optics boost the development of topological photonics [32][33][34][35][36][37][38][39]. The existence of gapless edge states of a system under open boundary condition (OBC) is predictable from the change of topological invariants associated with the bulk topology of system under periodical boundary condition (PBC), known as the (conventional) bulk-boundary correspondence, which is ubiquitously applicable in Hermitian systems.
Remarkably, the bulk-boundary correspondence is invalid in certain non-Hermitian topological systems [117,118]: The systems under PBC and OBC have dramatically different energy spectra, and all the eigenstates localize near system boundaries (the non-Hermitian skin effect) [119,120]. These have received great research interests in non-Hermitian systems of asymmetric Su-Schrieffer-Heeger (SSH) model, topological insulators, and nodal-line semimetals [119][120][121][122][123][124][125]. To predict the (non)existence of edge states, the biorthogonal bulk-boundary correspondence is established using a coined biorthogonal polarization [119]; alternatively, a non-Bloch topological invariant defined on the generalized Brillouin zone is suggested as the non-Bloch bulk-boundary correspondence [120]. In contrast, the non-Hermiticity does not inevitably destroy the bulkboundary correspondence [75,76,[91][92][93][94][95], and its validity is verified in a PT -symmetric non-Hermitian extension of SSH model with staggered couplings and losses [83][84][85][86][87][88][89][90]. The existence of topologically protected states at the interface of media with distinct topologies is predicted from the bulk topology, characterized by a geometric phase related winding number [87,89]. However, why bulkboundary correspondence fails in certain non-Hermitian systems but remains valid in some other non-Hermitian systems? What role the non-Hermiticity plays in the breakdown of bulk-boundary correspondence and the appearance of non-Hermitian skin effect? Whether symmetry protects non-Hermitian topological phases and how to characterize the topological properties?
In this Letter, we report that symmetry plays an important role for the validity of bulk-boundary correspondence in non-Hermitian systems and non-Hermiticity may alter the system topology. We elucidate that the breakdown of bulk-boundary correspondence in non-Hermitian systems attributes to the absence of chiralinversion symmetry in the presence of non-Hermiticity, the introducing of which directly induces a topological phase transition and the non-Hermitian skin effect. Fixing the chiral-inversion symmetry through alternatively introducing the non-Hermiticity, the energy spectrum under OBC is not altered; remarkably, the bulk-boundary correspondence recovers. The band touching degeneracy (exceptional) points are topological defects with vortex or antivortex in a vector field associated with the Bloch Hamiltonian, and the vorticity is a topological invariant. The change of vorticity correctly predicts the (non)existence of topological edge states. Non-Hermitian system.-We first consider a non-Hermitian SSH model as depicted in Fig. 1(a) [126]. Under PBC, the topological system is translational symmetric, the Bloch Hamiltonian is where σ x,y are the Pauli matrices. The intercell coupling is t 2 . Set µ = t 1 − γ and ν = t 1 + γ for convenience, the asymmetric intracell coupling strength (µ = ν * ) raises the non-Hermiticity. System a has the chiral symmetry. The eigenvalues are symmetric E a,± = ± t 2 2 + µν + t 2 (µe ik + νe −ik ) [ Fig. 2(a)]. The wave vector is k = πm/n, m ∈ [1, 2n] (m, n are positive integers) for the discrete lattice size N = 4n.
The chiral-inversion symmetry holds when non-Hermiticity is alternatively introduced in system b [ Fig. 1 , ⊗ is the Kronecker product, and P 2n is a 90 degrees rotation of the 2n × 2n identical matrix I 2n . Under symmetry protection, introducing non-Hermiticity does not directly induce a topological phase transition, and two degeneracy points can move in the parameter space without splitting into EP pairs [Figs. 2(b) and 2(d)]. The eigenstates under OBC are symmetric or antisymmetric. Figure 3(b) depicts the averaged IPR for the bulk eigenstates of system b, which is inversely proportional to the system size. This indicates that all the bulk states are extended and the non-Hermitian skin effect vanishes, even in regions that most bulk states are complex.
Systems a and b under OBC possess identical energy spectra as depicted in Fig. 2(c) [144], but with significantly different eigenstates: All the eigenstates of system a localize near system boundary; in contrast, all the bulk states in system b are extended, and only the edge states localize near system boundary (Fig. 3). These reflect the distinct topologies of systems a and b, and manifest that the way of non-Hermiticity appearance affects the system topology. Notably, the non-Hermiticity solely induces nontrivial topology at t 1 = t 2 .
Bulk-boundary correspondence.-The bulk-boundary correspondence recovers in both the regions with entirely real spectra and with complex spectra [Figs. 2(b) and 2(c)]. The recovery of bulk-boundary correspondence enables predicting the (non)existence of topologically protected edge states from the bulk topology of system b, the Bloch Hamiltonian of which has a four-site unit cell The µν , which is for system a with all the asymmetric intracell couplings substituted by the symmetric couplings √ µν and taken two unit cells as a compound one [144]. This equivalent system has a two-site unit cell and the Bloch Hamiltonian is , being identical with the eigenvalues of system b. The bulk topology of h b (k) correctly predicts the (non)existence of edge states in both systems a and b under OBC because they possess identical energy spectra [144]. Notably, h b (k) is identical with that alternatively found in Ref. [120] through solving H a under OBC. For γ = |r|e iθ (−π ≤ θ ≤ π), the band gap closes at and cos 2 (k) = [t 2 2 +t 2 1 −|r| 2 cos (2θ)]/(2t 2 2 ). For real µ and ν at θ = 0, the band touching points are degeneracy (exceptional) points at t 2 1 = + (−) t 2 2 + γ 2 [119,120], being topological defects carrying integer (half-integer) vorticity. The band touching EPs only appear for γ 2 > t 2 2 . Topological invariant.-The topology invariants are recently constructed in the non-Hermitian systems [92, 104-106, 109, 120]. The Chern number defined via Berry curvature [92,105], the vorticity defined via the complex energy [106], and the generalized Berry phase defined via the argument of effective magnetic field [104,105,109] are quantized. The vorticity of the topological defects in a vector field associated with the Bloch Hamiltonian is a bulk topological invariant to characterize the topological properties [145][146][147], which can be generalized to non-Hermitian systems. A two-component vector field F(k) = ( σ x , σ y ) is defined through the average values of the Pauli matrices under the eigenstates of h b (k).
The topological defects in the vector field are associated with vortices (red dots) or antivortices (blue dots). The winding number w = L (2π) −1 (F x ∇F y −F y ∇F x )dk characterizes the vorticity of the topological defects and 2πw accounts the varying direction of F(k) in the closed loop L in the parameter plane k = (k, t 2 ), whereF x(y) = Figs. 4(a) and 4(b); the winding number obtained from the vector field defined in the parameter space is in accord with that defined in the Brillouin zone of a twodimensional (2D) brick wall lattice in the momentum space k = (k x , k y ), along one direction of the 2D lattice is system b [144]. The varying direction of the vector field F(k) accumulated in the loop L is ±2π (±π) in Fig. 4(a) [ Fig. 4(b)] if L encircling a topological defect, the plus (minus) sign corresponds to the vortex (antivortex); otherwise, if L does not encircle a topological defect, the varying direction is zero.
The phase diagram is plotted in Fig. 4(c) for real γ. For µν > 0, the degeneracy points are at t 2 2 − µν = 0. As the non-Hermiticity increases, the band gap inside two high order EPs [46,73] with complex spectrum diminishes and closes at t 1 = 0 when γ 2 = t 2 2 . The high order EPs are at The red (blue) circles indicate the topological defects with vortices (antivortices), which appear at (k, t2) = (0, − t 2 1 − γ 2 ) or (±π, t 2 1 − γ 2 ) in (a) and at (k, t2) = (±π/2, ± γ 2 − t 2 1 ) in (b). (c) Phase diagram for real γ, two topological zero edge states exist in the blue region −t 2 2 < µν < t 2 2 for one intercell coupling t2 missing. (d) Zero edge states for systems a and b under OBC. The system parameters are N = 40, t1 = 1/4, γ = 1/2, and t2 = 1. µν = 0 (t 1 = ±γ), where half of the eigenstates are twostate coalesced at energy t 2 and −t 2 , respectively. For µν < 0, t 2 2 + µν = 0 yields another boundary for the zero edge states determined from the band touching EPs. Two topological zero edge states exist in the regions γ 2 − t 2 2 < t 2 1 < γ 2 + t 2 2 for one intercell coupling t 2 vanishing under OBC [148]. As γ increases, the region with edge states expends when γ 2 t 2 2 , but shrinks when γ 2 > t 2 2 . Topological edge states.-The bulk topology relates to the (dis)appearance of edge states at the interfaces where topological invariant (w) changes. We consider that the unit cells are complete (N = 4n), and one intercell coupling t 2 vanishes (the Supplemental Material provides the cases of defective unit cell at system boundary [144]). In system b, two edge states localize on the left and right boundaries, respectively in all blue regions of Fig. 4(c). In system a, the left and right edge states localize on the left and right boundaries, respectively only in region V; and both two edge states localize on the right (left) boundary in regions I and III (II and IV). For system b, the left edge state localized on the left boundary is ψ 2j = 0 and at large system size limit (N 1). The right edge state is a left-right spatial reflection of the left edge state under symmetry protection. Anomalous edge states localize in single unit cell at system boundary at the high order EPs (t 2 1 = γ 2 ) [104,116,117]. At t 1 = −γ, the left (right) edge state is ψ 1 = 1 (ψ N = 1); at t 1 = γ, the left edge state is ψ 1 = − (+) ψ 3 = 1 and the right edge state is ψ N = − (+) ψ N −2 = 1 when t 1 /t 2 > 0 (t 1 /t 2 < 0).
Discussion.-The non-Hermitian asymmetric coupling can be realized through introducing a synthetic imaginary gauge field [149,150]; in an array of coupled resonators consisting of primary resonators that evanescently coupled with the assistance of auxiliary resonators, the auxiliary resonators have half perimeter gain and half perimeter loss, leading to the amplification and attenuation for the coupling amplitudes in opposite tunneling directions. Alternatively, the asymmetric coupling can be realized via synthetic real gauge field and a pair of gain and loss [151]; the symmetric and anti-symmetric supermodes enable an asymmetric coupling between them. Implementation of asymmetric coupling with ultracold atoms in optical lattice is possible [121].
The finite size effects appear in the discrete systems. The band touching at t 1 = 0 for γ = ±t 2 is subtle, two disconnected regions appear on both sides of k = 0 in the discrete systems [117], the disconnection vanishes as N → ∞ [120]. For system b at complex γ, the momenta k for band touching in the energy spectra are no longer 0, ±π/2, or ±π and may not be seen in the discrete system with small size [144]. The nonvanishing gap in |E| diminishes (vanishes) as system size increasing (N → ∞).
Conclusion.-Non-Hermiticity may alter the system topology by breaking the combined chiral-inversion symmetry of the Hermitian system, induce a topological phase transition, and lead to the breakdown of (conventional) bulk-boundary correspondence. When the chiralinversion symmetry is fixed via alternatively introducing the non-Hermiticity, the bulk-boundary correspondence is valid. The band touching degeneracy (exceptional) points, protected by symmetry, are the topological defects with vortex or antivortex in the vector field that associated with the Bloch Hamiltonian. The vorticity of topological defects is a topological invariant, which characterizes the bulk topology and its change correctly predicts the (non)existence of topological boundary states in the non-Hermitian systems. We first reveal the roles played by symmetry and non-Hermiticity, and settle the fundamental problem of the validity of bulk-boundary correspondence in non-Hermitian systems. Our findings shed light on the non-Hermitian topological phases of matter.
We acknowledge the support of NSFC (Grant Nos. 11605094 and 11874225). * jinliang@nankai.edu.cn and where I N is the N × N identical matrix. The determinant D N (ρ) for system ρ is they satisfy a recursion relationship for integer m from 2 to 2n. Equations

Topological characterization
In system b, the bulk topological properties relate to the (non)existence of edge states. Here we calculate the bulk topological invariant of system b, which is capable of characterizing the topologies of both systems a and b under OBC. The Bloch Hamiltonian of system b is a 4 × 4 matrix; after a similar transformation, the Bloch Hamiltonian can be expressed in the form of B · σ with a two-site unit cell, then we define a vector field F (k) that associated with the Bloch Hamiltonian. The topological defects with vortices or antivortices in the vector field indicate the phase transition points. The vorticity of the topological defects is a topological invariant. We consider that t 1 and γ are real numbers µ = t 1 − γ and ν = t 1 + γ, discussions on other cases are similar following the same procedure below.
For µν > 0, µ, ν, and √ µν are positive real numbers. The Bloch Hamiltonian H b (k) under a similar transformation ν that only consists of diagonal elements, yields which equals to the Bloch Hamiltonian of system a with all the asymmetric couplings µ and ν replaced by the symmetric coupling √ µν and taken two unit cells as a compound unit cell. The Bloch Hamiltonian of equation (12) is rewritten in the form of h b (k) = B · σ, where the effective magnetic field is Notably, in the discrete system with lattice size N = 4n, the wave vector k is k = 2πm/n, m ∈ [1, n] (m, n are positive integers) for the Bloch Hamiltonian with a four-site unit cell, and the wave vector k is k = πm/n, m ∈ [1, 2n] for the Bloch Hamiltonian with a two-site unit cell. We define a vector field F (k) = ( σ x , σ y ) to characterize the topology of h b (k). The eigenstates associated with The average values of the Pauli matrices associated with the two components effective magnetic field σ x,y ± = ψ ± (k)| σ x,y |ψ ± (k) are i.e., ( σ x ± , σ y ± ) = (B x , B y ) /E ± ; thus, ( σ x ± , σ y ± ) reflects the topological properties of the Bloch bands and the system. The vector field F (k) under either eigenstate yields the same winding number w = L (2π) −1 (F x ∇F y − F y ∇F x )dk in the parameter plane k = (k, t 2 ), whereF x(y) = F x(y) / F 2 x + F 2 y and ∇ = ∂/∂k. The phase transition occurs at (k, t 2 ) = 0, − √ µν or ±π, √ µν , which are the band touching degeneracy points. They are topological defects in the vector field possessing integer topological charges (vortices and antivortices) as depicted in Fig. 4(a) in the Letter. The winding number w characterizing the vorticity of the topological defects, is a topological invariant. For µν < 0, −µ, ν, and √ −µν are positive real numbers. The Bloch Hamiltonian H b (k) under a similar transfor- which is the Bloch Hamiltonian of system a with all the asymmetric couplings µ and ν replaced by the symmetric coupling i √ −µν and taken two unit cells as a compound unit cell. The Bloch Hamiltonian can be rewritten as where the effective magnetic field is the eigenvalues are E ± (k) = ± (i √ −µν + t 2 e −ik ) (i √ −µν + t 2 e ik ); correspondingly, the eigenstates are where ∆ = t 2 2 − 2 √ −µνt 2 sin k − µν + t 2 2 + 2 √ −µνt 2 sin k − µν. The average values of σ x,y ± = ψ ± (k)| σ x,y |ψ ± (k) are at large system size limit (N 1). For the anomalous edge states at the high order EPs (t 2 1 = γ 2 ), they are localized at single unit cell at system boundary. For t 1 = −γ, the left (right) edge state is ψ 1 = 1 (ψ N = 1); and for t 1 = γ, the right edge state is ψ N = 1; the left edge state is ψ 1 = − (+) ψ 3 = 1 for t 1 /t 2 > 0 (t 1 /t 2 < 0).
For system b with an odd site number, the energy spectra are depicted in Supplementary Figures 6(a)−6(c), the edge state is depicted in Supplementary Figures 6(d)−6(f). Only one zero edge state exists in this situation. Considering that the unit cell at the right boundary is defective. In the situation that |µν| < t 2 2 , the edge state localizes at the left boundary, the wave function is Eq. (5) in the Letter; in the situation that |µν| > t 2 2 , for the system with site number N = 4n − 1, the edge state localized at the right boundary is ψ 2j = 0 and for the system with site number N = 4n − 3, the edge state localized at the right boundary is ψ 2j = 0 and For the anomalous edge states at the high order EPs (t 2 1 = γ 2 ) and the systems with site numbers N = 4n − 1 and 4n − 3, the left edge state is ψ 1 = − (+) ψ 3 = 1 for t 1 /t 2 > 0 (t 1 /t 2 < 0) at t 1 = γ and ψ 1 = 1 at t 1 = −γ. In contrast, for system a with an odd site number has a defective unit cell at the right boundary, only one zero state exists. The right boundary state is ψ 2j = 0 and ψ N −2j = (−t 2 /ν) ψ N +2−2j when |t 2 | < √ µν. At the high order EPs, the zero state localizes at one site on the left boundary ψ 1 = 1 for t 1 = −γ; and the zero state is extended, being ψ 2j = 0 and ψ 2j−1 = − (+) ψ 2j+1 for t 1 = γ at t 1 /t 2 > 0 (t 1 /t 2 < 0).
Energy spectra for complex asymmetric coupling For system b at γ = 1/2e iπ/4 , the phase transition points are t 1 = ± 4 3/4 ≈ 0.93 and |cos (k)| = (2 + √ 3)/2. The band touching degeneracy points may not be seen in the discrete system with small system size due to the finite number of discrete k. The energy spectra for γ = 1/2e iπ/4 are depicted in Supplementary Figure 7 for N = 40. The nonvanishing gap in |E| shown inside the green circles in Supplementary Figure 7(a) is a finite size effect of the discrete system; as N increases, the gap vanishes and the band touching degeneracy points reveal.