  In a closed system, it is well known that the time-reversal symmetry can lead
to Kramers degeneracy and protect nontrivial topological states such as quantum
spin Hall insulator. In this letter we address the issue whether these effects
are stable against coupling to environment, provided that both environment and
the coupling to environment also respect the time-reversal symmetry. By
employing a non-Hermitian Hamiltonian with the Langevin noise term and
ultilizing the non-Hermitian linear response theory, we show that the spectral
functions for Kramers degenerate states can be split by dissipation, and the
backscattering between counter-propagating edge states can be induced by
dissipation. The latter leads to the absence of accurate quantization of
conductance in the case of quantum spin Hall effect. As an example, we
demonstrate this concretely with the Kane-Mele model. Our study could also be
extended to interacting topological phases protected by the time-reversal
symmetry.
