  In this letter we study the Hall conductance for a non-Hermitian Chern
insulator and quantitatively describe how the Hall conductance deviates from a
quantized value. We show the effects of the non-Hermitian terms on the Hall
conductance are two folds. On one hand, it broadens the density-of-state of
each band, because of which there always exists a non-universal bulk
contribution. On the other hand, it adds decay term to the edge state, because
of which the topological contribution also deviates from the quantized Chern
number. We provides a simple formula for the topological contribution for a
general two-band non-Hermitian Chern insulator, as a non-Hermitian version of
the Thouless-Kohmoto-Nightingale-de Nijs formula. It shows that the derivation
from quantized value increases either when the strength of the non-Hermitian
term increases, or when the momentum dependence of the non-Hermitian term
increases. Our results can be directly verified in synthetic non-Hermitian
topological systems where the strength of the non-Hermitian terms can be
controlled.
