  A non-Hermitian Chern insulator with gain/loss-type non-Hermiticity shows a
peculiar gap closing; when a nontrivial Chern insulator phase changes to a
gapless phase, conduction and valence bands are combined into one band owing to
destabilization of chiral edge states. A previous recipe of non-Hermitian
bulk-boundary correspondence is insufficient for this system since such a gap
closing is beyond its scope. Here, we revise the recipe by taking the peculiar
gap closing into account and apply it to the non-Hermitian Chern insulator. We
demonstrate that the bulk-boundary correspondence holds in the system including
a destabilization of chiral edge states. A phase diagram derived from the
bulk-boundary correspondence is shown to be consistent with spectra of the
system.
