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 pseudospin 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 pseudospin 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.

中文翻译：

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开放式量子系统中分数量子霍尔态的命运：完整Liouvillian相关拓扑态的表征

尽管先前已经对Lindblad方程描述的开放量子系统进行了广泛的分析，但尚不清楚即使存在跳跃项，相关的拓扑状态（例如分数量子霍尔状态）也是否得以保持。在本文中，我们介绍了Liouvillian的伪旋转Chern数，该伪旋转Chern数是通过仅对加倍的Hilbert空间的子空间之一的边界条件进行扭曲来计算的。这种拓扑不变量的存在说明，即使存在跳跃项，拓扑性质也保持不变，这并不能弥补有效的非埃尔米特哈密顿量（忽略跳跃项而获得的差距）。换句话说，将拓扑属性编码为有效的非Hermitian哈密顿量，而不是完整的Liouvillian。当跳转项可以写为双倍希尔伯特空间中严格块上（下）的三角矩阵时，这特别有用，在这种情况下，跳转项的存在与否均不会影响Liouvillian谱。利用伪自旋Chern数，我们解决了具有两体损失但没有增益的分数量子霍尔态的表征，阐明了即使在存在跳跃项的情况下，非Hermitian分数量子霍尔态的拓扑也得以保留。该数值结果还支持使用非埃尔米特哈密顿量，这大大降低了数值成本。可以扩展类似的拓扑不变量来处理其他空间尺寸和对称性（例如具有反演对称性的一维开放量子系统）的相关拓扑状态，