  Topological phases in non-Hermitian systems have become fascinating subjects
recently. In this paper, we attempt to classify topological phases in 1D
interacting non-Hermitian systems. We begin with the non-Hermitian
generalization of the Su-Schrieffer-Heeger (SSH) model and discuss its
many-body topological Berry phase, which is well defined for all interacting
quasi-Hermitian systems (non-Hermitian systems that have real energy spectrum).
We then demonstrate that the classification of topological phases for
quasi-Hermitian systems is exactly the same as their Hermitian counterparts.
Finally, we construct the fixed point partition function for generic 1D
interacting non-Hermitian local systems and find that the fixed point partition
function still has a one-to-one correspondence to their Hermitian counterparts.
Thus, we conclude that the classification of topological phases for generic 1D
interacting non-Hermitian systems is still exactly the same as Hermitian
systems.
