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We unveil the geometrical meaning of winding number and utilize it tocharacterize the topological phases in one-dimensional chiral non-Hermitiansystems. While chiral symmetry ensures the winding number of Hermitian systemsbeing integers, it can take half integers for non-Hermitian systems. We give ageometrical interpretation of the half integers by demonstrating that thewinding number $\nu$ of a non-Hermitian system is equal to half of thesummation of two winding numbers $\nu_1$ and $\nu_2$ associated with twoexceptional points respectively. The winding numbers $\nu_1$ and $\nu_2$represent the times of real part of the Hamiltonian in momentum spaceencircling the exceptional points and can only take integers. We further findthat the difference of $\nu_1$ and $\nu_2$ is related to the second windingnumber or energy vorticity. By applying our scheme to a non-HermitianSu-Schrieffer-Heeger model and an extended version of it, we show that thetopologically different phases can be well characterized by winding numbers.Furthermore, we demonstrate that the existence of left and right zero-mode edgestates is closely related to the winding number $\nu_1$ and $\nu_2$.

Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems