Introduction

In these decades, a variety of novel phenomena have been discovered which arise from topological properties in the bulk1,2,3,4,5,6,7,8,9,10,11,12,13. In particular, the topological ordered phases14 exhibit striking topological phenomena because of correlation effects and the topological properties. One of the representative examples of topological ordered phases is a fractional quantum Hall (FQH) phase15,16,17 or a fractional Chern insulator18,19,20,21,22,23 which hosts anyons obeying fractional statistics due to the topological degeneracy of the ground states. The platforms of topological ordered phases extend to bosonic or spin systems. The toric codes for two-24,25 and three-dimensional systems26 exemplify the emergence of topological ordered phase in spin systems whose relevance of correlated compounds has been discussed recently27,28. These topological ordered phases also attract much attention in terms of application to the quantum computations.

Along with the above progress, recent development of technology has pioneered a new type of topological systems, non-Hermitian topological systems. Extensive studies in these years have discovered various intriguing phenomena described by topological properties of quadratic non-Hermitian matrices. For instance, it has been elucidated that non-Hermiticity may induce a topological phase which does not have its Hermitian counterpart29,30,31,32. Furthermore, non-Hermiticity induces novel gapless excitations in the bulk (e.g., exceptional points33,34,35,36,37,38,39, symmetry-protected exceptional rings40,41,42,43,44,45,46 etc.) which arise from the defectiveness of the Hamiltonian. In addition, non-Hermiticity may induce a unique bulk-boundary correspondence47,48,49,50,51,52 due to the non-Hermitian skin effect.

The above two progresses pose the following crucial question: what are impacts of non-Hermiticity on topologically ordered phases? In particular, it is considered to be significant to elucidate the fate of the topological degeneracy which is source of anyons for Hermitian cases. In spite of such significant open questions, there are few works addressing non-Hermitian topological ordered phases.

In this paper, we address the above issue, providing a new direction in the study of non-Hermitian topological phases. Specifically, we demonstrate the emergence of non-Hermitian FQH states in a two-dimensional system with non-Hermitian interactions where spinless fermions are coupled to Abelian gauge fields. This system is considered to be relevant to cold atoms with two-body loss. We conclude the emergence of non-Hermitian FQH states by combining the following two results: direct computation of the Chern number Ctot indicates that the ground state multiplet is characterized with Ctot = 1; the topological degeneracy is robust against non-Hermitian interactions, which arises from many-body translational symmetry. Furthermore, we discover a novel phenomenon for which non-Hermiticity is essential; the FQH state emerges without the repulsive interactions. We find that this intriguing phenomenon arises from interplay between the kinetic term and the dissipative interactions which is reminiscent of the continuous quantum Zeno effect. This unconventional mechanism of the bulk gap may provide a new way to access exotic topological ordered phases.

Setup and Method

The non-Hermitian Hamiltonian analyzed in this paper is shown in Eq. (3) which is considered to be relevant to cold atoms; the inelastic scattering results in the two-body interactions with the prefactor \(V\in {\mathbb{C}}\) [see Eq. (3c)]. In order to see the details, we start with spinless fermions in Abelian gauge potentials which are decoupled with the environment. After that we take into account the coupling with the environment, yielding a non-Hermitian Hamiltonian.

Firstly, we note that the following square lattice system with Abelian gauge fields may be realized for cold atoms

$${H}_{0}=-\sum _{\langle i,j\rangle }\,{t}_{0}{e}^{i{\varphi }_{ij}}{c}_{i}^{\dagger }{c}_{j}+{V}_{R}\sum _{\langle i,j\rangle }\,{n}_{i}{n}_{j}\mathrm{.}$$
(1)

Here, \({c}_{i}^{\dagger }\) creates a spinless fermion at site i = (ix,iy) of the two-dimensional system. If necessary, we rewrite \({c}_{i}^{\dagger }\) as \({c}_{{i}_{x}{i}_{y}}^{\dagger }\). t0 denotes the hopping between sites. The phase factor ϕij describes the flux penetrating the plaquet. In this paper, we employ the string gauge53 (see Fig. 1). We define the flux density as ϕ \(:\,=\) Nϕ/NxNy where Nϕ denotes the number of flux quanta penetrating the Nx × Ny-square lattice. The filling factor is defined as ν \(:\,=\) Nf/Nϕ where Nf denotes the number of fermions. For fabrication of the above system with cold atoms, the following two ingredients are essential: nontrivial hopping inducing Landau bands and the repulsive interactions. The former ones are introduced by rotating the system54,55,56,57,58 or by optically synthesized gauge fields59. The repulsive interaction (VR > 0) may be fabricated by a Feshbach resonance60,61.

Figure 1
figure 1

Sketch of the model and the string gauge for Nx = Ny = 4. We impose the periodic boundary condition for x- and y-direction. The green arrows in panel (a) represent strings specifying the Peierls phase ϕij = 2πϕnij. Here, nij denotes the number of string penetrating the bond connecting sites i and j, and ϕ denotes the flux density ϕ = Nϕ/NxNy. Panel (b) illustrates the corresponding Peierls phase; when the fermion hops along the blue arrow, it acquires the phase factor ϕij. When ϕ is multiple of Nx−1, the string gauge is reduced to the Landau gauge.

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Now, let us take into account the coupling with the environment induced by inelastic scattering62,63,64,65,66,67,68. The time-evolution of such an open quantum system is governed by the Lindblad equation:

$${\partial }_{t}\rho (t)=-\,i[{H}_{0},\rho (t)]-\frac{1}{2}\gamma \sum _{k}\,({L}_{k}^{\dagger }{L}_{k}\rho (t)+\rho (t){L}_{k}^{\dagger }{L}_{k}-2{L}_{k}\rho {L}_{k}^{\dagger }),$$
(2)

where ρ(t) denotes the density matrix of the system. Lk’s are Lindblad operators describing the loss with the rate γ > 0. Specifically, we set Lk → \({c}_{i}{c}_{i+{{\boldsymbol{e}}}_{x}}\), \({c}_{i}{c}_{i+{{\boldsymbol{e}}}_{y}}\) because the Feshbach resonance induces the two-particle loss62,63,64,65,66,67,68. Here ex(y) denotes the unit vector for each direction, and the lattice constant is set to unity. When we focus on the short-time evolution, the last term describing the quantum-jump is negligible66,67,68,69. In this case, we can see that the time-evolution is described by

$${\partial }_{t}\rho (t)=-\,i({H}_{{\rm{eff}}}\rho (t)-\rho (t){H}_{{\rm{eff}}}^{\dagger }),$$
(3a)
$${H}_{{\rm{eff}}}={H}_{{\rm{kin}}}+{H}_{{\rm{int}}},$$
(3b)

with

$${H}_{{\rm{kin}}}=-\,\sum _{\langle i,j\rangle }\,{t}_{0}{e}^{i{\varphi }_{ij}}{c}_{i}^{\dagger }{c}_{j},\,{H}_{{\rm{int}}}=V\,\sum _{\langle i,j\rangle }\,{n}_{i}{n}_{j}\mathrm{.}$$
(3c)

We note that the interaction strength takes a complex value; \(V={V}_{R}-i\frac{\gamma }{2}\) with VR ≥ 0, which makes the Hamiltonian non-Hermitian \({H}_{{\rm{eff}}}\ne {H}_{{\rm{eff}}}^{\dagger }\). The short-time evolution can be elucidated by diagonalizing the Heff defined in Eq. (3b).

Because treating the large size system is numerically difficult, we simplify the Hamiltonian (3b) with calculating the pseudo-potential70. With this approximation, the Hamiltonian is simplified as

$${H^{\prime} }_{{\rm{eff}}}=-\,\sum _{\langle i,j\rangle }\,{t}_{0}{e}^{i{\varphi }_{ij}}{\tilde{c}}_{i}^{\dagger }{\tilde{c}}_{j}+V\,\sum _{\langle i,j\rangle }\,{\tilde{c}}_{i}^{\dagger }{\tilde{c}}_{j}^{\dagger }{\tilde{c}}_{j}{\tilde{c}}_{i},$$
(4a)

with

$${H}_{{\rm{kin}}}|{\varphi }_{\alpha }\rangle =|{\varphi }_{\alpha }\rangle {\varepsilon }_{\alpha },\,{\tilde{c}}_{i}^{\dagger }=\sum _{\alpha }\,^{\prime} {\varphi }_{i\alpha }^{\ast }{d}_{\alpha }^{\dagger }\mathrm{.}$$
(4b)

Here, |ϕα〉 denotes the eigenstate of Hkin (\(|{\varphi }_{\alpha }\rangle :\,={\sum }_{j}\,{\varphi }_{j\alpha }{c}_{j}^{\dagger }\mathrm{|0}\rangle \)). We label the eigenvalues \({\varepsilon }_{\alpha }\) so that the relation \({\varepsilon }_{1}\)  ≤  \({\varepsilon }_{2}\)  ≤ \(\cdots \) is satisfied. \({d}_{\alpha }^{\dagger }\) creates the fermion of the eigenstate α. ∑′α denotes the summation over states satisfying \({\varepsilon }_{\alpha }\) ≤ \(\varepsilon \)Nkeep; e.g., for Nkeep = Nϕ [Nkeep = 2Nϕ], the summation is taken over the lowest Landau levels (LLs) [the lowest and the second lowest LLs], respectively. For Nkeep = 2Nϕ, we can take into account effects of Landau band mixing which induces an intriguing phenomena as we see below.

For the numerical computation, we set parameters as |V| = t0 = 1, ν = 1/3, and Nx = Ny = N.

Results

Hermitian case

Here we briefly review the results of the Hermitian system (ImV = 0) for ν = 1/3 where FQH states have been observed16,18,19,20,21,22,23,71,72,73,74. In this case, three-fold degeneracy is observed for the ground state multiplet which is separated by the bulk gap. Computing the Chern number Ctot for the ground state multiplet yields Ctot = 1, which characterizes the topology of the FQH state with the Hall conductance σxy = 1/3. For more details, see Sec. I of Supplemental Material.

Non-Hermitian case

Now we introduce the imaginary-part ImV < 0, which makes the system non-Hermitian. Let us start with the definitions of the ground states and the energy gap because the energy spectrum of the non-Hermitian Hamiltonian becomes complex. We define the ground states with the minimum value of the real-part66. For our system, these states also have the longest lifetime τ (~−1/ImE with ImE < 0). Correspondingly, the energy gap is defined as Δ = ReEe − ReEg which is natural extension of the Hermitian case. Here, Eg (Ee) denotes the energy eigenvalue of the ground state (the first excited state), respectively. In the following, we numerically show that the FQH state survives even under non-Hermiticity by setting V = exp(−inθπ/10) with nθ = 0, …, 5.

As a first step, we focus on the case for nθ = 2. In Fig. 2(b), we plot the energy spectrum En where n labels the states such that ReE1 ≤ ReE2 ≤ \(\cdots \) holds. Figure 2(a) indicates that the three-fold degeneracy can be observed even in the presence of non-Hermitian term. The robustness is attributed to many-body translational symmetry, which we discuss below. Besides that, in this figure, we can confirm that the lifetime of the ground states is longer than that of excited states. We note that the energy gap observed in this figure remains finite in the thermodynamic limit, which can be seen in Fig. 2(b).

Figure 2
figure 2

Numerical results for nθ = 2 with V = exp(−inθπ/10). (a) The real- and the imaginary-part of the energy eigenvalues. The data are obtained for nθ = 2, Nϕ = N = 9, and Nkeep = 2Nϕ. (b) The bulk gap as a function of Nf. In this plot, the size of the system is chosen so that the flux density ϕ = Nϕ/N2 satisfies 1/45 ≤ ϕ ≤ 1/40. The data of panel (b) are obtained for Nkeep = Nϕ. However, the difference from the gap for Nkeep = 2Nϕ is less than \(\delta \Delta \lesssim {10}^{-5}{t}_{0}\). (c) The imaginary-part of the Berry curvature trF as a function of θx and θy for Nkeep = Nϕ. For the computation, we divide the two-dimensional space of θ’s into Nθ × Nθ-mesh with Nθ = 14.

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From the above numerical results of the bulk gap and the topological degeneracy, one can expect that the FQH phase survives even in the presence of the non-Hermitian term. To confirm this, we address the characterization of the topology of the ground states by computing the many-body Chern number for the non-Hermitian case which is defined as follows:

$${C}_{{\rm{tot}}}=\int \frac{d{\theta }_{x}d{\theta }_{y}}{2\pi i}{\rm{tr}}F({\theta }_{x},{\theta }_{y}),$$
(5a)
$${F}_{nm}({\theta }_{x},{\theta }_{y})={\epsilon }_{\mu \nu L}{\langle {\partial }_{\mu }{\varPsi }_{n}|{\partial }_{\nu }{\Psi }_{m}\rangle }_{R}\mathrm{.}$$
(5b)

Here, we have imposed the twisted boundary condition: \({c}_{{N}_{x}+\mathrm{1,}{i}_{y}}^{\dagger }={e}^{i{\theta }_{x}}{c}_{\mathrm{1,}{i}_{y}}^{\dagger }\) and \({c}_{{i}_{x},{N}_{y}+1}^{\dagger }={e}^{i{\theta }_{y}}{c}_{{i}_{x}\mathrm{,1}}^{\dagger }\). The integral is taken over 0 ≤ θx(y) < 2π, respectively. F(θx,θy) denotes the Berry curvature defined by twisting the boundary conditions. ∂μ \(:\,=\) ∂/∂θμ. \({\varepsilon }_{\mu \nu }\) (μ, ν = x, y) is an anti-symmetric matrix with \({\varepsilon }_{xy}\) = 1. |Ψn〉R and L〈Ψn| denote ground states with n = 1, 2, 3. The former (latter) ones are right (left) eigenvectors. The summation is taken over repeated indices. We note that the Chern number defined above takes integer (for the proof, see Sec. II of Supplemental Material). This fact indicates that the only imaginary-part of the Berry curvature contributes to the Chern number Ctot; taking the integral, the contribution from the real-part of the Berry curvature generically vanishes. Employing the method introduced in refs 75 and76, we obtain Im[trF]. In Fig. 2(c), we can see that the integrand Im[trF]/2π becomes almost constant. Evaluating the integration, we obtain Ctot = 1.

The above data of the bulk gap, the ground state degeneracy, and the Chern number suggest that the ground state is topologically identical to the FQH state with σxy = 1/3 for the Hermitian case.

In a similar way, we can analyze the system for the other cases of interaction strength. The results are summarized in Fig. 3. This figure indicates that the FQH state observed for nθ = 2 is adiabatically connected to the one for the Hermitian case; Fig. 3(a,b) show that the bulk gap remains finite with decreasing nθ; Fig. 3(c) indicates that the topological properties do not change.

Figure 3
figure 3

(a) Energy spectrum for several values of nθ defining interaction with V = exp(−inθπ/10). Panel (b) shows the magnified data. (c) Chern number as a function of nθ. The data are obtained for N = 9 and Nkeep = 18 where both of the lowest and the second lowest LLs are taken into account. We note that for nθ = 5 the interaction strength V becomes pure imaginary V = −i.

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Intriguingly, Fig. 3(b) indicates that the bulk gap opens even for ReV = 0, implying the potential presence of the FQH state without the repulsive interaction. The details of this issue are addressed below.

Robustness of the ground state degeneracy against non-Hermitian interactions

So far, we have numerically observed the three-fold degeneracy of the ground states even in the presence of the non-Hermitian term [see Figs 2(a) and 3(a)]. This three-fold degeneracy is due to many-body translational symmetry. Namely, the degeneracy multiple of ν−1 = 2m + 1 (\(m\in {\bf{Z}}\)) is observed for arbitrary many-body interaction preserving the translational symmetry. In the following, we discuss the details.

To see this we focus on the case where Nx = Ny and ϕ = nx/Nx (\({n}_{x}\in {\bf{Z}}\)) holds. Due to the latter condition, the string gauge is reduced to the Landau gauge. In this case, the kinetic term Hkin preserves the translation symmetry along the y-axis. Namely, the following condition holds; TyHkinTy−1 = Hkin where Ty is the translation operator satisfying \({T}_{y}{c}_{{i}_{x}{i}_{y}}^{\dagger }{T}_{y}^{-1}={c}_{{i}_{x}{i}_{y}+1}^{\dagger }\).

Because of the translation symmetry of Hkin, one may take the simultaneous eigenstates of Hkin and Ty;

$${H}_{{\rm{kin}}}|{\phi }_{\alpha }({k}_{y})\rangle ={\varepsilon }_{\alpha }|{\phi }_{\alpha }({k}_{y})\rangle ,$$
(6a)
$${T}_{y}|{\phi }_{\alpha }({k}_{y})\rangle ={e}^{-i{k}_{y}}|{\phi }_{\alpha }({k}_{y})\rangle ,$$
(6b)

where ky denotes the momentum along the y-axis (0 ≤ ky < 2π). Here, let us consider the following gauge transformation: \({U}_{G}{c}_{{j}_{x}{j}_{y}}^{\dagger }{U}_{G}^{\dagger }={e}^{-i2\pi \varphi {j}_{y}}{c}_{{j}_{x}{j}_{y}}^{\dagger }\), where UG is an unitary operator. Applying the gauge transformation to the eigenstate |φα(ky)〉, we obtain

$${T}_{y}{U}_{G}|{\phi }_{\alpha }({k}_{y})\rangle ={e}^{-i({k}_{y}-2\pi \varphi )}{U}_{G}|{\phi }_{\alpha }({k}_{y})\rangle ,$$
(7)

which means that applying UG shifts the momentum by Δky := −2πϕ. For the derivation of Eq. (7), see Sec. III of Supplemental Material.

Equation (7) elucidates that the many-body translational symmetry results in the degeneracy multiple of ν−1. This can be seen by noticing the following relation

$$\begin{array}{c}\langle {\phi }_{{\alpha }_{1}}({k}_{y1}),\ldots ,{\phi }_{{\alpha }_{{N}_{f}}}({k}_{y{N}_{f}})|{H}_{{\rm{int}}}|{\phi }_{{\beta }_{1}}({k^{\prime} }_{y1}),\ldots ,{\phi }_{{\beta }_{{N}_{f}}}({k^{\prime} }_{y{N}_{f}})\rangle \\ \,=\,\langle {\phi }_{{\alpha }_{1}}({k}_{y1}),\ldots ,{\phi }_{{\alpha }_{{N}_{f}}}({k}_{y{N}_{f}})|{U}_{G}^{\dagger }{H}_{{\rm{int}}}{U}_{G}|{\phi }_{{\beta }_{1}}({k^{\prime} }_{y1}),\ldots ,{\phi }_{{\beta }_{{N}_{f}}}({k^{\prime} }_{y{N}_{f}})\rangle \\ \,=\,\langle {\phi }_{{\alpha }_{1}}({k}_{y1}+\Delta {k}_{y}),\ldots ,{\phi }_{{\alpha }_{{N}_{f}}}({k}_{y{N}_{f}}+\Delta {k}_{y})|{H}_{{\rm{int}}}|{\phi }_{{\beta }_{1}}({k^{\prime} }_{y1}+\Delta {k}_{y}),\ldots ,{\phi }_{{\beta }_{{N}_{f}}}({k^{\prime} }_{y{N}_{f}}+\Delta {k}_{y})\rangle ,\end{array}$$
(8)

which indicates that the matrix element for the subspace labeled by the total momentum \(K={\sum }_{l}\,{k}_{yl}\) equals to the one for the subspace labeled by K′ = K + ΔkyNf. Because the shift of the momentum is rewritten as ΔkyNf = −2πϕNf = −2πν, we can see that the degeneracy of each eigenvalue is multiple of ν−1.

In the above we have seen the relation between the topological degeneracy and the many-body translational symmetry. In order to support this numerically, we demonstrate that breaking the translational symmetry splits the degeneracy. Specifically, we compute the energy spectrum in the presence of the following disorder

$${H}_{{\rm{dis}}}=\sum _{i}\,{w}_{i}{\tilde{c}}_{i}^{\dagger }{\tilde{c}}_{i},$$
(9)

where wi takes a random value satisfying −w0/2 ≤ wi ≤ w0/2 at each site. In Fig. 4(a) [(b)], the real- [imaginary-] part of the energy eigenvalues are plotted against disorder strength w0, respectively. These figures indicate that breaking the translational symmetry lifts the three-fold degeneracy of the ground states.

Figure 4
figure 4

(a,b) The real- [imaginary-] part of the energy eigenvalues as functions of disorder strength. Turning on disorder w0 splits the three-fold degeneracy observed for w0. The data are obtained for Nϕ = N = 9 and nθ = 2 with V = exp(−inθπ/10). (c) The bulk gap as a function of Nf which is obtained for Hptb [see Eq. (10)]. Energy difference of the ground state multiplet is of the order of 10−13 t0 which is much smaller than the energy gap.

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The above results indicate that the many-body translational symmetry results in the robustness of topological degeneracy against non-Hermiticity. Our numerical data elucidate that the topological degeneracy can be observed for 1/45 ≤ ϕ < 1/40 where the string gauge cannot be reduced to the Landau gauge.

FQH state without the repulsive interaction

Figure 3(b,c) imply that the FQH state emerge without the repulsive interaction. In the following, we elucidate the origin of the FQH state for ReV = 0.

Firstly, we point out that the origin of the gap is the interplay between the kinetic term Hkin and the non-Hermitian interaction Hint (i.e., the mixing between Landau bands). Applying the perturbation theory, we obtain the following Hamiltonian acting on the space spanned by the states in the lowest LLs,

$${H}_{{\rm{ptb}}}={P}_{0}{H}_{{\rm{int}}}{P}_{0}+{P}_{0}{H}_{{\rm{int}}}{P}_{1}\frac{1}{{E}_{g}^{0}-{H}_{{\rm{kin}}}}{P}_{1}{H}_{{\rm{int}}}{P}_{0}\mathrm{.}$$
(10)

Here, Pn denotes the projection operator to the subspace where n-fermions are excited to the second lowest LLs. Eg0 is the ground state energy for V = 0. We have omitted the constant term arising from P0HkinP0. Noticing that the prefactor of the last term is (ImV)2/ħω0 > 0, we can see that the last term serves as the repulsive interaction for ReV = 0. Here ħω0 denotes the energy gap between the lowest LLs and the second lowest LLs for V = 0.

Diagonalizing the effective Hamiltonian (10), we plot the energy gap Δ as a function of Nf in Fig. 4(c). This figure indicates that the energy gap remains finite in the thermodynamic limit. We also note that the three-fold degeneracy of the ground states is also observed. Thus, one may consider that the gapped state is the FQH state, which is confirmed by the numerical computation yielding Ctot = 1 [see Fig. 3(c)].

Therefore, we conclude that the FQH state emerges without the repulsive interaction (ReV = 0) which is adiabatically connected to the FQH state with σxy = 1/3 for the Hermitian case.

We stress that the bulk gap opens due to two-body loss inducing the effective repulsive interaction, which is reminiscent of the continuous quantum Zeno effect66,67,68,77,78,79,80,81. We note that in contrast to the case of ReV = 0, the second term may suppress the bulk gap for ReV > 0. Indeed, for V = VR > 0 the prefactor of the second term of Eq. (10) has the opposite sign [−(ReV)2/ħω0]. Correspondingly, the second order term suppresses the bulk gap induced by the first term of Eq. (10), which can be numerically observed.

Summary and Outlook

In this paper, by focusing on the FQH system at ν = 1/3, we have analyzed impacts of non-Hermiticity on topological ordered phases. We have elucidated the robustness of topological degeneracy against non-Hermitian interactions which arises from many-body translational symmetry. Combining the numerical results of the Chern number Ctot = 1 and the topological degeneracy leads us the conclusion that non-Hermitian Hamiltonian (1b) shows the FQH state. Furthermore, we have discovered that the FQH state emerges without repulsive interactions (ReV = 0). This intriguing behavior arises from the effective repulsion induced by the two-body loss, which is reminiscent of the continuous quantum Zeno effect.

We finish this article with comments on future directions. One of the significant issues is experimental realization of non-Hermitian fractional quantum Hall states. Although more detailed quantitative analysis is desired, we can make a rough estimation. Supposing that hopping is approximately \({t}_{0}/h \sim 1\,{\rm{kHz}}\)82, the bulk gap (Δ) and the lifetime (τ) can be estimated as \(\Delta /h \sim 10\,{\rm{Hz}}\) and \(\tau \sim 0.6\,{\rm{s}}\) for strong gauge field [see Fig. 2(a)]. Concerning the bulk gap, the energy scale is comparable with the temperature range, where the experiments are carried out83. However, the gap is enhanced by the strong repulsive interactions which can be tuned by the Feshbach resonance. Thus, we expect that experimental realization is accomplished for strong repulsive interactions. Furthermore, in Fig. 2(c), we have numerically observed that the Berry curvature is almost independent of θ’s, which implies that the computation of the Chern number may be simplified by defining the non-Hermitian counterpart of the one-plaquet Chern number for the Hermitian case84,85,86,87. We leave the extension of one-plaquet Chern number to non-Hermitian systems as a future work. In addition, we have observed that the interplay between the dissipative two-body interaction and the kinetic term yields four-body interactions which open the bulk gap and yield the FQH state. This unconventional mechanism of gap opening may provide new direction to access exotic topological ordered states induced by many-body interactions higher than two-body (e.g., the Moore-Read state88,89). Hunting such exotic topological ordered states is also left as a significant issue to be addressed.