  Two-band model works well for Hall effect in topological insulators. It turns
out to be non-Hermitian when the system is subjected to environments, and its
topology characterized by Chern numbers has received extensive studies in the
past decades. However, how a non-Hermitian system responses to an electric
field and what is the connection of the response to the Chern number defined
via the non-Hermitian Hamiltonian remain barely explored. In this paper,
focusing on a k-dependent decay rate, we address this issue by studying the
response of such a non-Hermitian Chern insulator to an external electric field.
To this aim, we first derive an effective non-Hermitian Hamiltonian to describe
the system and give a specific form of k-dependent decay rate. Then we
calculate the response of the non-Hermitian system to a constant electric
field. We observe that the environment leads the Hall conductance to be a
weighted integration of curvature of the ground band and hence the conductance
is no longer quantized in general. And the environment induces a delay in the
response of the system to the electric field. A discussion on the validity of
the non-Hermitian model compared with the master equation description is also
presented.
