Fate of fractional quantum Hall states in open quantum systems: characterization of correlated topological states for the full Liouvillian

Despite previous extensive analysis of open quantum systems described by the Lindblad equation, it is unclear whether correlated topological states, such as fractional quantum Hall states, are maintained even in the presence of the jump term. In this paper, we introduce the pseudo-spin Chern number of the Liouvillian which is computed by twisting the boundary conditions only for one of the subspaces of the doubled Hilbert space. The existence of such a topological invariant elucidates that the topological properties remain unchanged even in the presence of the jump term which does not close the gap of the effective non-Hermitian Hamiltonian (obtained by neglecting the jump term). In other words, the topological properties are encoded into an effective non-Hermitian Hamiltonian rather than the full Liouvillian. This is particularly useful when the jump term can be written as a strictly block-upper (-lower) triangular matrix in the doubled Hilbert space, in which case the presence or absence of the jump term does not affect the spectrum of the Liouvillian. With the pseudo-spin Chern number, we address the characterization of fractional quantum Hall states with two-body loss but without gain, elucidating that the topology of the non-Hermitian fractional quantum Hall states is preserved even in the presence of the jump term. This numerical result also supports the use of the non-Hermitian Hamiltonian which significantly reduces the numerical cost. Similar topological invariants can be extended to treat correlated topological states for other spatial dimensions and symmetry (e.g., one-dimensional open quantum systems with inversion symmetry), indicating the high versatility of our approach.


INTRODUCTION
Recent extensive studies of non-Hermitian systems have discovered a variety of novel topological phenomena for non-interacting cases 1-4 .For instance, non-Hermiticity enriches topological properties 5 ; it increases the number of symmetry classes and results in two types of the gap, the point-gap 6 and the line-gap 7 .Furthermore, non-Hermiticity may break down diagonalizability of the Hamiltonian which results in non-Hermitian band touching, such as exceptional points 7,8 , symmetryprotected exceptional rings [9][10][11][12][13] etc.In addition, non-Hermitian systems can also show the intriguing bulkboundary correspondence [14][15][16][17][18][19][20][21][22] ; certain topological properties result in the non-Hermitian skin effect which results in extreme sensitivity to the boundary conditions [23][24][25][26] .So far, the above non-Hermitian phenomena for the non-interacting case have been reported in various platforms 8, .
Among them, open quantum systems [49][50][51][52][53][54] also provide a unique platform of the following intriguing issue: the interplay between correlations and non-Hermitian topology [55][56][57][58][59][60][61] .Such systems interact with the environment and may lose energy or particles.Correspondingly, the time-evolution of the density matrix is governed by the Lindblad equation where the coupling between the system and the environment is described by the Lindblad operators L α (α = 1, 2, • • • ).In the previous works 55- 61 , by focusing on the special time-evolution, the correlated topological states have been analyzed for the effective non-Hermitian Hamiltonian H eff := H 0 − i 2 α L † α L α , where H 0 is the Hermitian Hamiltonian of the system; for the short-time dynamics before the occurrence of a jump of the states by Lindblad operators, one can see that the dynamics of the density matrix is described by the effective non-Hermitian Hamiltonian H eff .Recently, it has been pointed out that for non-interacting fermions, the topological properties can survive even beyond the above special dynamics 62 .This is because the gap of the Liouvillian is maintained even when the quantum jump is taken into account.
In spite of the above significant progress in topological perspective on open quantum systems, it is still unclear whether the topological properties for correlated states survive even in the presence of quantum jumps.In order to clarify the stability of correlated topological phases described by H eff against the jump term, topological invariants having the following properties should be introduced: (i) they are quantized as long as the gap of the Liouvillian opens; (ii) in the absence of the jump term, they are reduced to the invariants characterizing the topology of the effective non-Hermitian Hamiltonian H eff .
In this paper, to characterize the correlated states, we introduce a topological invariant having the above two properties by doubling the Hilbert space.Specifically, we define the pseudo-spin Chern number characterizing the correlated topological states for two-dimensional systems without symmetry 63 .This topological invariant can be computed by twisting the boundary conditions for one of the subspaces of the doubled Hilbert space, which is reminiscent of the spin Chern number [64][65][66] .By computing the pseudo-spin Chern number, we demonstrate that even in the presence of the jump term, topological properties of non-Hermitian fractional quantum Hall (FQH) states survive for an open quantum system with twobody loss but without gain.Our results justify the use of the effective non-Hermitian Hamiltonian to topologically characterize the full Liouvillian whose gap does not close even in the presence of the jump term.This is particularly useful for systems where the jump term can be written as a block-upper-triangular matrix in the doubled Hilbert space; in such cases, both the spectral and topological properties are encoded into the effective non-Hermitian Hamiltonian which significantly reduces the numerical cost.We also note that our approach can be extended to characterize correlated topological states for other cases of spatial dimensions and symmetry, indicating the high versatility of our approach.
The rest of this paper is organized as follows.In Sec.II, we briefly review how the effective non-Hermitian Hamiltonian H eff is obtained and provide a detailed description of topological properties which we will discuss in this paper.In Sec.III, we introduce the pseudo-spin Chern number of the Liouvillian.As an application, we demonstrate that for the system with two-body loss but without gain, the topological properties of non-Hermitian FQH states are not affected by the jump term in Sec.IV which is followed by a short summary.The appendices are devoted to the topological characterization of onedimensional open quantum systems with inversion symmetry, topological degeneracy for open quantum systems conserving the number of particles, and technical details.

II. EFFECTIVE NON-HERMITIAN HAMILTONIAN FOR OPEN QUANTUM SYSTEMS
In this section, we briefly review the time-evolution of open quantum systems and concretely explain topological properties on which we will focus in this paper.
Firstly, we note that for open quantum systems, the dynamics is governed by the Lindblad equation, where Here, the Lindblad operators are denoted by a set of L α (α = 1, 2, • • • ) which describes the dissipation arising from coupling to the environment.The density matrix of the system is denoted by ρ(t).The superoperator ) is referred to as the Liouvillian (the jump term).For the details of superoperators, see Appendix A. The operator H 0 denotes the Hamiltonian for the system (H 0 = H † 0 ).For arbitrary operators A and B, the commutation (anti-commutation) relation is written as [A, B] = 0 ({A, B} = 0).
In some previous works 6, [54][55][56][57][58][59][60] on open quantum systems, topological phenomena have been studied for the effective non-Hermitian Hamiltonian, by focusing on the dynamics before occurrence of a jump of the state by L J , which is described by For instance, the Chern number C H eff is computed with the right and left eigenvectors of the non-Hermitian Hamiltonian H eff for a two-dimensional system without symmetry 55 .
Here, in order to elucidate effects of the jump term, let us consider the operator L (λ) interpolating between L 0 and L 0 + L J ; L (λ) := L 0 + λL J (0 ≤ λ ≤ 1).With a slight abuse of terminology, we also call L (λ) "Liouvillian".When the gap-closing of the "Liouvillian" L (λ) does not occur for an arbitrary value of λ, the topological properties are expected to be maintained.[The gap is defined in Eq. ( 5)].However, it remains unclear whether there exists a topological invariant that characterizes the topological properties even in the presence of the jump term.
Previous works [49][50][51][52][53] have addressed how the presence of the jump term affects the topological characterization of open quantum systems in non-interacting cases.We note, however, that topological invariants introduced in these previous works can change without the gap-closing in the spectrum of the Liouvillian L = L 0 + L J .For instance, the topological characterizations proposed in Refs.49, 50, and 52 require the gap in the spectrum of the density matrix, which is not necessary in our framework.

III. PSEUDO-SPIN CHERN NUMBER FOR THE LIOUVILLIAN
In order to clarify whether the topological properties for H eff are maintained even in the presence of the jump term, we introduce the pseudo-spin Chern number for two-dimensional systems without symmetry.
We note that our approach can be extended to characterize correlated topological states for other spatial dimensions and symmetry [e.g., one-dimensional systems with inversion symmetry, (see Appendix B)], although we limit our discussion to the Chern number for the sake of concreteness.

A. Definition
We show that the topological properties of twodimensional open quantum systems can be characterized by the pseudo-spin Chern number C ps := (C KK −C BB )/2 [see Eq. ( 6)] for the doubled Hilbert space.
Firstly, we define "eigenvalues" and "eigenvectors" of the Liouvillian L which can be thought of as a non-Hermitian matrix in a doubled Hilbert space, Ket ⊗ Bra.With the following isomorphism, the density matrix is mapped to a vector in the doubled Hilbert space [67][68][69][70][71][72][73][74][75][76][77][78][79] , where |φ 's are states in the original Hilbert space (or Ket space) generated by acting on the vacuum with creation operators in the real space.The coefficient ρ ij is a complex number.Here, in order to distinguish elements of the doubled Hilbert space from those of the original Hilbert space, we denote a vector in the subspace Ket (Bra) as |φ i K(B) .Correspondingly, L α ρL † α is represented as L α ⊗ L * α |ρ .Therefore, the Liouvillian L can be represented as a non-Hermitian matrix L whose left and right eigenvectors L ρ n | and |ρ n R are defined as with the eigenvalues Λ n , n = 1, 2, • • • , (for more details, see Appendix A).Now, consider the "Liouvillian" L(λ) := L 0 + λL J which interpolates between the two cases, L(0 For λ = 0, the topological invariant can be computed from the eigenvectors of H eff .When the system does not show the gap-closing for 0 ≤ λ ≤ 1, the topological properties are considered to be maintained.Here, we have defined the gap between eigenstates |ρ n R and |ρ n R as 80 The above topological properties can be characterized by the pseudo-spin Chern number C ps = (C KK −C BB )/2 where C σσ (σ = K, B) is defined as Here, the summation n is taken over degenerate states.
The summation is taken for repeated indices µ and ν [µ(ν) = x, y].We have supposed that the right and left eigenvectors satisfy the biorthogonal normalization condition; |ρ n R and L ρ n |, satisfy L ρ n |ρ n R = δ n n for arbitrary integers, n and n .In addition, we have imposed the twisted boundary conditions with (θ x , θ y ) only for the space specified by σ 65,81,82 .The periodic boundary conditions are imposed on the other space.The operator ∂ σ µ denotes the corresponding differential operator acting only on the space specified by σ.For instance, the action of The inner product of vectorized matrices, called the Hilbert-Schmidt inner product, is defined as We have supposed that the eigenvectors of the "Liouvillian" L(λ) shows N 2 d -fold degeneracy for arbitrary λ.For λ = 0, this assumption means that the eigenstates of H eff show the N d -fold degeneracy.
As proven in Sec.III B, the pseudo-spin Chern number C ps elucidates that as long as the gap of the "Liouvillian" L(λ) opens, the topological properties of H eff are maintained even in the presence of the jump term.We note that when the pseudo-spin Chern number changes, the gap-closing should occur in the parameter space of (θ x , θ y ).
The effective non-Hermitian Hamiltonian H eff is particularly useful when L J and L 0 can be written in block-upper-triangular and block-diagonal forms, respectively.This is because in such cases, the effective non-Hermitian Hamiltonian governs not only topological properties but also spectrum of the full Liouvillian 83,84 (see Appendix C), which significantly reduces the numerical cost.

B.
Properties of the pseudo-spin Chern number The pseudo-spin Chern number elucidates that even in the presence of the jump term, topological properties of H eff remain unchanged as long as the gap of the "Liouvillian" L(λ) opens.In order to see this, we note the following three facts.
(i) The pseudo-spin Chern number is quantized even in the presence of the jump term, provided that the gapclosing of L does not occur in the space of (θ x , θ y ) (see Appendix D).
(ii) In the absence of the jump term, C KK is rewritten as with Equation ( 8) is proven in Sec.III B 1. We note that C H eff defined in Eq. ( 9) is nothing but the Chern number of H eff 55 .
(iii) In the absence of the jump term, the Chern number obtained by twisting the boundary conditions only for the subspace Bra (C BB ) satisfies, which is proven in Sec.III B 2. This relation also indicates that for λ = 0, the total Chern number computed by twisting the boundary conditions both for the subspaces Bra and Ket vanishes even when the eigenstates of H eff show topologically non-trivial properties.Based on the fact (i), we can see that the pseudo-spin Chern number is quantized as long as the gap opens.In addition, (ii) and (iii) indicate that the pseudo-spin Chern number C ps = (C KK − C BB )/2 characterizes the topological properties described by the Hamiltonian H eff for λ = 0. Therefore, C ps elucidates that as long as the gap opens, the topology of H eff is maintained even in the presence of the jump term.The effective non-Hermitian Hamiltonian is particularly useful for systems with loss but without gain or vice versa because both the spectral and topological properties are encoded into the effective Hamiltonian H eff which significantly reduces the numerical cost.
In the rest of this section, we prove Eqs. ( 8) and (10).
1. Proof of Eq. ( 8) First, we make the identification 85 where |ρ n R and L ρ n | are right and left eigenvectors of respectively.Vectors |Φ n1 R and L Φ n2 | denote the right and left eigenstates of H eff which satisfy L Φ n2 |Φ n1 R = δ n2n1 .The subscript n denotes the set of integers, n 1 and n 2 , labeling the eigenstates, |Φ n1 R and L Φ n2 |.
We recall that for the computation of the Chern number C KK , the twisted boundary conditions are imposed only on the subspace Ket.In this case, the derivative ∂ K µ acts only on the states in the subspace Ket.Keeping this fact in mind, we obtain the Berry connection A Kµ and the Berry curvature F KK as and Thus, we end up with Eq. ( 8).
2. Proof of Eq. ( 10) For the computation of the Chern number C BB , we impose the twisted boundary conditions only on the subspace Bra, meaning that the derivative ∂ B µ acts only on the states in the subspace Bra.Keeping this in mind, we can see that the Berry connection A Bµ is equal to A * Kµ , which yields , we obtain Eq. (10).
Equation ( 15) also indicates that the total Chern number computed by twisting the boundary conditions both for the subspaces Bra and Ket vanishes; the Berry connection A µ obtained by twisting the boundary conditions both for the subspace satisfies ImA µ = 0, meaning that the relation of ImF := µν ∂ µ ImA ν vanishes.

IV. APPLICATION TO THE FQH STATES FOR AN OPEN QUANTUM SYSTEM WITH TWO-BODY LOSS
By numerically computing the pseudo-spin Chern number, we elucidate that even in the presence of the jump term, the topology of FQH states survives for the following open quantum system with two-body loss.
Let us consider an open quantum system of spinless fermions on a square lattice.We denote by c † i and c i the creation and the annihilation operators of a spinless fermion at site i, respectively.The number operator at i is defined as n i := c † i c i .The system is described by the following Hamiltonian and the Lindblad operators where e µ denotes the unit vector in the µ-direction (µ = x, y).The Lindblad operators L's describe twobody loss (γ > 0).The strength of the nearest neighbor interaction V R is a real number.The summation ij is taken over pairs of neighboring sites i and j.The matrix element h ij = t 0 e i2πφij with real numbers φ ij and t 0 describes hopping between neighboring sites i and j under the gauge field.For the definition of the phase factor φ ij , see Fig. 1 where the string gauge is taken 86 .The number of the flux quanta penetrating the entire system is written as N φ := φL x L y , where L x and L y denote the number of sites along the x-and the y-direction, respectively.This model is considered to be relevant to cold atoms.The Abelian gauge field can be introduced by rotating the system [87][88][89][90][91] or by optically synthesized gauge fields [92][93][94][95][96][97][98][99][100][101][102][103] .The Feshbach resonance 104,105 induces inelastic scattering of two-body loss [106][107][108][109][110] .We address the characterization of non-Hermitian FQH states by the following steps.Firstly, we rewrite the fermionic open quantum system as a closed fermionic system by identifying the Liouvillian as a non-Hermitian Hamiltonian via the isomorphism [see Eq. ( 3)].Secondly, by numerically diagonalizing the mapped fermionic model (18), we elucidate that the topological properties are maintained; the topological degeneracy and the pseudo-spin Chern number are independent of the jump term.

A.
Mapping the fermionic open quantum system to a closed bilayer system Firstly, based on the isomorphism [see Eq. ( 3)], we show that the systems of spinless fermions with two-body loss can be written as a closed bilayer fermionic system with inter-layer couplings.
With the isomorphism, an annihilation operator c i is mapped to a creation operator c † i for the subspace Bra; ρc i ↔ c † i |ρ with {c i , c † j } = δ ij for an arbitrary ρ.Here, a subtlety arises; commutation relations We note, however, that the above commutation relations can be rewritten as the anti-commutation relations by introducing the following operators 112,113 where In terms of the operators d † iσ , the Lindblad equation, which is defined with the Hamiltonian H 0 (16a) and the Lindblad operators (16b), is rewritten as with The number operator is defined as with sgn(σ) taking 1 (−1) for σ = a (σ = b).
The above equation indicates that an open quantum system of spinless fermions can be mapped to a closed bilayer system whose Hamiltonian corresponds to L defined in Eq. (18).Here, we have regarded d † iσ (σ = a, b) as an operator creating a spinless fermion at site i of layer σ.

Overview
We analyze the above bilayer system (18) by introducing a parameter λ (0 ≤ λ ≤ 1), L(λ) := L 0 + λL J .Employing the pseudo-potential approach (see Sec. IV B 2 and Appendix E), we obtain the spectrum and the pseudo-spin Chern number which are shown in Figs. 2 and  Because the open quantum system loses but does not gain particles, the non-equilibrium steady state, the state The black dots are obtained for the subspace labeled by Na − N b , (−1) Na = (0, 1).The Laughlin states with the filling factor ν = 1/3 is denoted by the dots marked with the arrow in panel (b).We note that the Laughlin states denoted with the arrow has a finite lifetime while the vacuum is a non-equilibrium steady state (i.e, its lifetime is infinite).
with the longest lifetime, is the vacuum (|ρ = |0 a ⊗|0 b with |0 σ being the state annihilated by all d iσ ), which is consistent with Fig. 2. Namely, the Laughlin states, which are indicated by dots marked with the arrow, are no longer the states with the longest lifetime.We note, however, that the topology of the Laughlin states is maintained even in the presence of the jump term.Such topological states are considered to be experimentally accessible by observing the transient dynamics of cold atoms.The realization of Laughlin states in cold atoms has been theoretically proposed 89,92,93 .Following these proposals, one can prepare the Laughlin state as the initial state for a sufficiently deep trap potential.Suddenly making the trap potential shallower results in two-body loss.Furthermore, the non-Hermitian FQH states become the first decay modes by tuning the gauge field so that N φ = φL x L y = 6 is satisfied.
As we see below, our numerical results demonstrate that both the spectral and the topological properties are encoded into the effective non-Hermitian Hamiltonian if L J and L 0 are written in block-upper-triangular and block-diagonal forms, respectively.The analysis of H eff is numerically less demanding than that of the full Liouvillian L = L 0 + L J .

Results in the absence of the jump term
Firstly, we discuss the case of L(0) = L 0 which can be understood from the previous work 55 for the effective non-Hermitian Hamiltonian H eff . Let Because the state with the minimum real-part of the energy ReE n1 also shows the longest lifetime, 1/ImE n1 , the pseudo-potential approach is employed where the creation operator c † i is replaced to f † i = n1 ϕ * in1 a † n1 (for more details, see Appendix E).Here, ϕ in1 denotes a state in the lowest Landau level; j h ij ϕ jn1 = ϕ in1 n1 with the energy n1 ∈ R. The operator a † n1 creates a fermion with a state in the lowest Landau level.The summation n1 is taken over states in the lowest Landau level.Diagonalizing H eff for the filling factor ν = 1/3 for the lowest Landau level, we can observe the three-fold degeneracy for the states with the longest lifetime 55,115 , which is the topological degeneracy of the Laughlin states for ν = 1/3.We note that the number of fermions is conserved in the absence of the jump term.For these three-fold degenerate states, the Chern number defined in Eq. ( 9) takes one (C H eff = 1).
With the above facts, we can understand the results of L 0 which can be block-diagonalized into each subsector labeled by (N a , N b ) with N σ denoting the total number of fermions in layer σ = a, b.In Fig. 2, the colored dots represent the spectrum of L 0 which is given by Λ denoting the eigenvalues of H eff .The states indicated by dots marked with the arrow correspond to the Laughlin states at the filling factor ν = 1/3.Here, we note that these states show 9fold degeneracy (N 2 d = 9) because there is topologically protected three-fold degeneracy (N d = 3) for each of the two layers.We also note that the data for N a = 4 is similar to those of N a = 2, which is attributed to the pseudo-potential approach projecting creation operators onto the states in the lowest Landau level 116 .Figure 3 shows that the pseudo-spin Chern number for these 9-fold degenerate states takes three at λ = 0, which is consistent with C H eff = 1.Namely, C ps = N d C H eff = 3 holds with N d = 3 [see Eq. ( 8)].

Results in the presence of the jump term
Let us now analyze the case for a finite value of λ (0 < λ ≤ 1).We show that: (i) topological degeneracy is maintained; (ii) the pseudo-spin Chern number remains one for the non-Hermitian FQH states.
The topological degeneracy (9-fold degeneracy) survives even in the presence of the jump term.This is because the spectrum is not affected by the jump term L J when L J and L 0 can be written in block-uppertriangular and block-diagonal forms, respectively 83,84 (see Appendix C); for the open quantum system with two-body loss but without gain, the jump term L J maps states in the subspace labeled by (N a + 2, N b + 2) to those in subspaces labeled by (N a , N b ), while L 0 is blockdiagonalized for subspaces labeled by (N a , N b ).The numerical data for two-body loss also support the above independence of the spectrum.In Fig. 2, we can see that the eigenvalues of L 0 (colored dots) and those of L = L 0 + L J (black dots) are exactly on top of each other.We note that the spectrum of L is obtained for the subsector labeled by N a − N b and (−1) Na where the "Liouvillian" L(λ) is block-diagonalized.The above numerical data show that the topological degeneracy survives even in the presence of the jump term, which is expected on general grounds.
The pseudo-spin Chern number should not be affected by the jump term, as the gap-closing does not occur.Indeed, Fig. 3 indicates that the pseudo-spin Chern number takes three for an arbitrary value of λ (0 ≤ λ ≤ 1).Noting the relation C ps = 3C H eff [see Eq. ( 8)], we conclude that topological properties of H eff remian unchanged even in the presence of the jump terms.Figure 3 is obtained by employing the method proposed in Ref. 114.In the above, we have confirmed that the topological properties of the Laughlin state are maintained even in the presence of the jump term.Furthermore, the above results elucidate that both the spectral and the topological properties are encoded into the effective non-Hermitian Hamiltonian if L J and L 0 are written in blockupper-triangular and block-diagonal forms, respectively.
We close this section with a remark on the topological degeneracy; for another type of Lindblad operators preserving the charge U(1) symmetry, e.g., the Lindblad operators describing dephasing noise 73,74,76,117,118 , three-fold topological degeneracy can be observed (for more details, see Appendix F).

V. SUMMARY
Despite the previous extensive analysis of open quantum systems, it is unclear whether correlated topological states, such as FQH states, are maintained even in the presence of the jump term.
In this paper, we have introduced the pseudo-spin Chern number computed from the vectorized density matrices in the doubled Hilbert space Ket ⊗ Bra which is akin to the spin-Chern number.The presence of such a topological invariant elucidates that as long as the gap of "Liouvillian" L(λ) = L 0 + λL J opens for 0 ≤ λ ≤ 1, the topology of the full Liouvillian L(1) is encoded into H eff .The effective Hamiltonian is particularly useful for systems where L J and L 0 can be written in block-uppertriangular and block-diagonal forms, respectively.This is because in such systems, both the spectral and topological properties are encoded into the effective Hamiltonian.
As an application, we have addressed the topological characterization of the non-Hermitian FQH states in open quantum systems with two-body loss but without gain.Our numerical results have elucidated that even in the presence of the jump term, topological properties (i.e., the pseudo-spin Chern number and 9-fold topological degeneracy) of the non-Hermitian FQH states are not affected by the jump term.This fact also reduces the numerical cost because the analysis of H eff is numerically less demanding than that of the full Liouvillian L = L 0 + L J .
We note that similar topological invariants can be introduced to characterize correlated topological states for other spatial dimensions and symmetry [e.g., a onedimensional open quantum systems with inversion symmetry (see Appendix B)], indicating the high versatility of our approach.can define the following Berry phase Here ∂ K θ denotes the derivative with respect to θ which acts only on the subspace Ket; for instance, the action of Properties of the Berry phase χKn The Berry phase χ Kn elucidates that as long as the gap of the "Liouvillian" L(λ) opens for 0 ≤ λ ≤ 1, the topological properties of H eff are maintained even in the presence of the jump term, which follows from the following two facts.
(i) The Berry phase is quantized, where the right eigenvector |ρ n (θ 0 ) R is also a right eigenvector of I with an eigenvalue ±1 for θ 0 = 0 or π.Equation (B2) is proven in Appendix B 1 c.(ii) In the absence of the jump term, χ Kn is written as where |Φ n1 R and L Φ n1 | (n 1 = 1, 2, • • • ) are the right and left eigenvectors of H eff (θ), with the eigenvalue E n1 (θ) ∈ C. Equation (B4) is proven in Appendix B 1 c.Equation (B3) indicates that χ Kn is reduced to the Berry phase for H eff in the absence of the jump term.In addition, Eq. (B2) indicates that as long as the gap opens, χ Kn does not change its value even when the jump term is introduced.Therefore, the Berry phase χ Kn elucidates that as long as the gap of the "Liouvillian" L(λ) opens for 0 ≤ λ ≤ 1, the topological properties of H eff are maintained even in the presence of the jump term.
In particular, this fact indicates that the topology of the full Liouvillian is encoded into H eff when L J and L 0 can be written in block-upper-triangular and blockdiagonal forms, respectively.An example of such systems is an open quantum system with loss but without gain, as we have seen in Sec.IV where the two-dimensional system is analyzed.
We note that Berry phases for non-Hermitian systems are defined in several contexts 119,120 .However, it remained unsolved whether there exists a topological invariant that characterizes the topological properties even in the presence of the jump term.
In the rest of this section, we prove Eqs. (B2) and (B3).Proof of Eq. (B2)-.For the inversion symmetric system satisfying IL(θ)I −1 = L(−θ), the following relation holds: with a continuous function c n (θ) taking a complex value c n (θ) = 0. We recall the assumption that the right and left eigenvectors are non-degenerate.By using the above relation, we can obtain This relation simplifies the integral in Eq. (B1a), Namely, c n (0) and c n (π) take 1 or −1.Therefore, combining this fact and Eq.(B7), we obtain Eq. (B2) which indicates the quantization of the Berry phase χ Kn .Proof of Eq. (B3)-.In the absence of the jump term, we can see the following correspondence Here, we recall the assumption that the states are nondegenerate.By using the above correspondence, χ Kn is written as which is the desired Eq. (B3).

SSH model with dephasing noise
In the above, we have introduced the Berry phase for the doubled Hilbert space [see Eq. (B1)].In particular, the Berry phase elucidates that both the spectral and topological properties of the Liouvillian are encoded into the effective non-Hermitian Hamiltonian H eff for open quantum systems whose jump term can be written in a block-upper-triangular form.This is because such a jump term does not affect the spectrum.
In this section, instead of the detailed analysis of such an open quantum system, we address topological characterization of a one-dimensional system with dephasing noise 73,74,76,117,118 , demonstrating that our topological invariant works even when the jump term affects the spectrum of the Liouvillian.Specifically, we analyze the SSH model with dephasing noise whose topological properties have not been analyzed so far.Our analysis elucidates that a non-equilibrium steady state is characterized by the Berry phase taking π in the presence of the jump term although the gap is closed in the absence of the jump term.

a.
Mapping the open quantum system to a closed system Consider the SSH model with dephasing noise described by the Lindblad equation (1a) with Here, c † jα (c jα ) creates (annihilates) a spinless fermion at sublattice α = A, B of site j.Hopping integrals t and t take real values, and γ is a positive number.The number of unit cells is L. We have imposed the periodic boundary condition c † Lα = c † 0α .The above open quantum system is mapped to the closed system which has been discussed for the specific choice of t (t = t) 74,78 .The Liouvillian reads Here, we have used σ =↑ (σ =↓) to specify the subspace Ket (Bra).We denote by d † jασ the creation operator of a fermion with spin σ =↑, ↓ at sublattice α = A, B of site j.The number operator is defined as Now, we derive Eq. (B11).With the isomorphism [see Eq. ( 3)], the following relations hold for an arbitrary density matrix ρ where c iα (c iα ) acts on the vectors in the subspace Ket (Bra).Thus, introducing the following operators, the Liouvillian can be written as with sgn(σ) taking 1 (−1) for σ =↑ (↓).Further applying the particle-hole transformation only for down-spin states, we end up with Eq. (B11).
Here, we define the Liouvillian L(θ) for the SSH model which is necessary to compute the Berry phase.Twisting the hopping between sites j = 0 and j = 1 only for the subspace specified with σ =↑, the Liouvillian L(θ) is written as with By analyzing a simple case for t = 0, we show that in the bulk, the non-equilibrium steady state (i.e., the states with an infinite lifetime) is characterized by the Berry phase π.Correspondingly, for the open boundary condition, edge states result in the charge polarization only at edges.We note that the gap is closed in the absence of the jump term.
(i) Bulk properties-.Let us consider the "Liouvillian" L(λ) = L 0 + λL J under the periodic boundary condition.Here, L 0 and L J are defined in Eq. (B11).This model preserves the total number of particles for each spin.
For t = 0, the problem is reduced to a two-site Hubbard model with the pure-imaginary interaction, Here, let us focus on the half-filled case where the dynamics can be understood by diagonalizing L 2site (λ) for the subsector labeled by (N ↑ , N ↓ ) = (1, 1) with Firstly, we define the basis spanning the subspace labeled by (N ↑ , N ↓ ) = (1, 1).
In this basis, L 2site (λ) is represented as Diagonalizing the matrix L 2site (λ), we can see that the eigenvalues are written as In Fig. 4, the spectrum of "Liouvillian" L 2site (λ) is plotted for 0 ≤ λ ≤ 1.For 4t ≤ γ, an exceptional point appears with increasing λ.However, regardless of the value of γ, the eigenstate with eigenvalue Λ −b is the longest lifetime.In particular, for λ = 1, it is a non-equilibrium steady state, i.e., the lifetime become infinite.From Eq. (B20), we can see that the corresponding left and right eigenstates are L ρ 2site,g | = −2| and |ρ 2site,g R = | − 2 .We note that at λ = 1, the exceptional point can be observed with increasing γ.
For the state |ρ 2site,g R , the Berry phase takes π.To see this, firstly, we note that twisting the hopping t only for the subsector with σ =↑ [see Eq. (B16)] can be accomplished by applying the operator e iθn 1A↑ 121 ; L 2site (θ, λ) = e iθn 1A↑ L 2site (λ)e −iθn 1A↑ . (B22) with −π ≤ θ < π.Here, we note that Eq. (B22) holds only for t = 0. Equation (B22) indicates that the eigenstates of L 2site (θ, λ) can be obtained from those of L 2site (λ); for instance, the eigenstate with the longest lifetime for L 2site (θ, λ) is given by Therefore, computing the eigenvalue of I, yields the Berry phase χ ↑g = π.Here, we have used Eq.(B2).We note that the same result can be obtained by direct evaluation of the integral in Eq. (B1) 122 .
Corresponding to the Berry phase taking π, one may expect the emergence of edge states 123,124 which is discussed at the end of this section.Here, for comparison, we discuss expectation values under the periodic boundary condition.Firstly, we note that the state is written as which we see below.Here, we have normalized the density matrix so that trρ 2site,g = 1 holds.Thus, we obtain Equation (B25) can be seen by a straightforward calculation.As we have applied the particle-hole transformation [see Eq. (B15)], |ρ 2site,g = | − 2 is mapped as which can be rewritten in terms of c iα and ciα as follows: where By normalizing the density matrix so that tr(ρ 2site,g ) = 1 holds, we obtain Eq. (B25).In the above, we have seen that Eq. (B26) holds for the periodic boundary condition.
(ii) Edge properties-.Now, let us analyze the system with edges.We impose the open boundary condition; sites i = 0 and i = L − 1 are decoupled.We again restrict ourselves to the half-filled case.For t = 0, each boundary site is isolated from the bulk.The "Liouvillian" at the edge j = 0 is written as The right eigenvectors and corresponding eigenvalues are easily obtained and written as Here, we note that the states with the longest lifetime are doubly degenerate.Taking into account two edges, we obtain the edge state with an infinite lifetime, with real numbers a and b satisfying a 2 +b 2 = 1.We note that d † 0A↑ d † L−1B↑ |0 is also an eigenstate with the zero eigenvalue.However, we discard this states because we restrict ourselves to the half-filled case, (N ↑ , N ↓ ) = (1, 1) with N σ = n 0Aσ + n L−1Bσ .
As shown below, |ρ edge,g can be rewritten as with a and b are real numbers satisfying a − b = 1.Here, we have renormalized the states so that tr (ρ edge,g ) = 1 holds.Therefore, we obtain This result means that the polarization is observed only at each edge.Namely, we have tr which can be rewritten in terms of c iα and ciα as follows: where . By normalizing the density matrix so that tr(ρ edge,g ) = 1 holds, we obtain Eq. (B31).
In the above, for t = 0, the Berry phase χ ↑g of the non-equilibrium steady states takes π.Correspondingly, while the charge distribution of the bulk is uniform, each edge shows the charge polarization.
We recall that the topological properties remain unchanged as long as the gap does not close.This fact means that for small but finite t , the Berry phase should take π inducing the edge polarization.
In order to see this, let us consider the following square matrix of a block-upper-triangular form, where L (0,0) , L (2,2) , and L (4,4) are non-Hermitian square matrices.Matrices L J(0,2) and L J(2,4) are non-Hermitian and not necessarily square matrices.The spectrum of L(λ) is independent of λ, which can be seen as follows.
The above argument can be straightforwardly extended to a generic case.Thus, we can conclude that the spectrum of the "Liouvillian" L(λ) = L 0 + λL J is independent of λ when L J (L 0 ) is a block-upper-triangular (block-diagonal) matrix.
Secondly, we note that with the pseudo-potential approximation, the operators can be written as follows: With the isomorphism [see Eq. ( 3)], these terms can be identified as follows: Here, we have assumed i = j.By taking into account the above relations, we get Eq.(E1a).
Appendix F: Topological degeneracy for another type of dissipation By a topological argument, we show that the system with the filling factor ν (ν −1 = 1, 3, 5, • • • ) shows at least ν −1 -fold topological degeneracy in the spectrum of the Liouvillian when the Lindblad operators preserve charge U(1) symmetry.We consider fermions in the square lattice (see Fig. 1) with L x = L y = L and φ = 1/L.In this case, the number of states in the lowest Landau level is N φ = L = φ −1 (i.e., the filling factor is ν := N a /N φ = φN a ).
Firstly, let us consider the eigenvectors of the kinetic terms under the Landau gauge: with n 1 = 1, • • • , dim h.Here, we note that the Hamiltonian h ij is invariant under the translation along the ydirection, meaning that the Landau state ϕ jn1 can also be labeled by momentum along the y-direction k y : Here, σ = a (σ = b) specifies the subspace Ket (Bra).The operator d † jxjyσ is the creation operator defined in Eq. (17) where the set of the subscripts j x and j y is denoted by j.Here, T yσ and U σ are defined as With the above relation, we can see that the system shows robust topological degeneracy when the following conditions are satisfied: Therefore, we can conclude that regardless of details of the dissipation, the open quantum system shows at least ν −1 -fold degeneracy as long as both U(1) symmetry [Eq.(F8)] and translational symmetry [Eq.(F9)] are preserved.Namely, in the absence of accidental degeneracy, we have ν −1 -fold degeneracy which is topologically protected.
FIG. 1. (Color Online).Sketch of the model under the periodic boundary conditions.Gray and black circles denote the sites; each site illustrated with a gray circle is identified with the corresponding site illustrated with a black circle on the opposite side.To describe the Abelian gauge field, we have taken the string gauge86 .Green arrows illustrate the phase φij.For hopping parallel to an arrow, φij takes the value shown in the figure.When the fermion hops in the opposite direction, φij takes the values so that φij = −φji is satisfied.The number of the flux quanta penetrating the entire system is written as N φ = φLxLy where Lx and Ly denote the number of sites along the x-and the y-direction, respectively.When φ is multiple of 1/Lx, the string gauge is reduced to the Landau gauge.

3 .
As discussed in Sec.IV B 3, these figures indicate that the topological properties of the non-Hermitian FQH states remain unchanged even in the presence of the jump term; the topological degeneracy and the pseudo-spin Chern number are not affected by the jump term.

FIG. 3 .
FIG. 3. (Color Online).The pseudo-spin Chern number as a function of λ for the Laughlin states indicated by dots marked with the blue arrow in Fig. 2(b).The parameters are set to the same values as those of Fig. 2. For the computation of the Chern number, we have employed the method proposed in Ref. 114.
We have imposed the biorthogonal normalization condition on the right and left eigenvectors of L(θ); |ρ n (θ) R and L ρ n (θ)| satisfy L ρ n (θ)|ρ n (θ) R = δ n n for arbitrary integers, n and n .b.