Symmetry classes of dissipative topological insulators with edge dark states

Abstract

We classify the dissipative topological insulators (TIs) with edge dark states (EDSs) by using the 38-fold way of non-Hermitian systems in this paper. The dissipative dynamics of these quadratic open fermionic systems is captured by a non-Hermitian single-particle matrix which contains both the internal dynamics and the dissipation, referred to as the damping matrix $X$. The dark states in these systems are the eigenmodes of $X$ in which the eigenvalues' imaginary part vanishes. However, there is a constraint on $X$, namely, that the modes in which the eigenvalues' imaginary parts are positive are forbidden. In other words, the imaginary line-gap of $X$ is ill-defined, so the topological band theory classifying the dark states cannot be applied to $X$. To reveal the topological protection of the EDSs, we propose the double damping matrix $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{X}=\text{diag}(X,{X}^{*})$, where the imaginary line-gap is well-defined. Thus, the 38-fold way can be applied to $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{X}$, and the topological protection of the EDS is uncovered. Different from previous studies of EDSs in purely dissipative dynamics, the EDSs in the dissipative TIs are robust against the inclusion of Hamiltonians. Furthermore, the topological classification of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{X}$ not only reflects the topological protection of EDSs in dissipative TIs but also provides a paradigm to predict the appearance of EDSs in other open free fermionic systems.