  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.
