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We study topological properties of one-dimensional non-Hermitian systemswithout chiral symmetry and give phase diagrams characterized by topologicalinvariants $\nu_E$ and $\nu_{total}$, associated with complex energy vorticityand summation of Berry phases of complex bands, respectively. In the absence ofchiral symmetry, we find that the phase diagram determined by $\nu_E$ isdifferent from $\nu_{tot}$. While the transition between phases with different$\nu_{E}$ is closely related to the band-touching point, the transition betweendifferent $\nu_{tot}$ is irrelevant to the band-touching condition. We give aninterpretation for the discrepancy from the geometrical view by analyzing therelation of topological invariants with the winding numbers associated withexception points of the system. We then generalize the fidelity approach tostudy the phase transition in the non-Hermitian system and find that transitionbetween phases with different $\nu_{tot}$ can be well characterized by anabrupt change of fidelity and fidelity susceptibility around the transitionpoint.

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