Symmetry classes of dissipative topological insulators with edge dark state

TL;DR

This paper classifies dissipative topological insulators with edge dark states using a 38-fold non-Hermitian classification, revealing their topological protection and robustness against Hamiltonian inclusion.

Contribution

It introduces a double damping matrix approach to apply the 38-fold classification to dissipative TIs with edge dark states, uncovering their topological protection.

Findings

Topological protection of edge dark states in dissipative TIs is established.

Edge dark states are robust against Hamiltonian perturbations.

A new classification paradigm for open fermionic systems is proposed.

Abstract

We classify the dissipative topological insulators (TIs) with edge dark states (EDS) 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, refereed to as damping matrix $X$. And the dark states in these systems are the eigenmodes of $X$ 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 can not be applied to $X$. To reveal the topological protection of EDS, we propose the double damping matrix $\tilde{X} = \text{diag}\left( X, X^* \right)$, where the imaginary line-gap is well defined. Thus, the 38-fold way can be applied to $\tilde{X}$, and the topological protection of the EDS is uncovered. Different from previous studies of EDS in purely dissipative dynamics, the EDS in the dissipative TIs are robust against the inclusion of Hamiltonians. Furthermore, the topological classification of $\tilde{X}$ not only reflects the topological protection of EDS in dissipative TIs but also provides a paradigm to predict the appearance of EDS in other open free fermionic systems.