Classification of topological phases in one dimensional non-hermitian systems

Topological phases in non-Hermitian systems became a fascnating subject recently. In this paper, we attempt to classify topological phases in 1D non-Hermitian systems. We begin with the non-Hermitian generalization of Su-Schrieffer-Heeger(SSH) model and discuss its many body topological Berry phase, which is well defined for any quasi hermitian systems(non-Hermitian systems that have real energy spectrum).
We then demonstrate that the classifications of topological phases for quasi hermitian systems are exactly the same as their hermitian counterparts. Moreover, we find that unitarity actually can emerge for fixed point partition function describing topological phases in 1D non-Hermitian systems with local interactions. Thus we conjecture that for 1D non-Hermitian system, the classification of topological phases are exactly the same as Hermitian systems.