Topological invariants and phase diagrams for one-dimensional two-band non-Hermitian systems without chiral symmetry

We study topological properties of one-dimensional non-Hermitian systems without chiral symmetry and give phase diagrams characterized by topological invariants $\nu_E$ and $\nu_{total}$, associated with complex energy vorticity and summation of Berry phases of complex bands, respectively. In the absence of chiral symmetry, we find that the phase diagram determined by $\nu_E$ is different from $\nu_{tot}$. While the transition between phases with different $\nu_{E}$ is closely related to the band-touching point, the transition between different $\nu_{tot}$ is irrelevant to the band-touching condition. We give an interpretation for the discrepancy from the geometrical view by analyzing the relation of topological invariants with the winding numbers associated with exception points of the system. We then generalize the fidelity approach to study the phase transition in the non-Hermitian system and find that transition between phases with different $\nu_{tot}$ can be well characterized by an abrupt change of fidelity and fidelity susceptibility around the transition point.