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

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