  The interplay between non-Hermitian effects and topological insulators has
become a frontier of research in non-Hermitian physics. However, the existence
of a non-Hermitian skin effect for topological-protected edge states remains
controversial. In this paper, we discover an alternative form of the
non-Hermitian skin effect called the non-Hermitian chiral skin effect (NHCSE).
NHCSE is a non-Hermitian skin effect under periodic boundary condition rather
than open boundary condition. Specifically, the chiral modes of the NHCSE
localize around \textquotedblleft topological defects\textquotedblright
characterized by global dissipation rather than being confined to the system
boundaries. We show its detailed physical properties by taking the
non-Hermitian Haldane model as an example. As a result, the intrinsic mechanism
of the hybrid skin-topological effect in Chern insulators is fully understood
via NHCSE. Therefore, this progress will be helpful for solving the
controversial topic of hybrid skin-topological effect and thus benefit the
research on both non-Hermitian physics and topological quantum states.
