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

TL;DR

This paper explores the geometrical interpretation of winding numbers in one-dimensional chiral non-Hermitian systems, revealing how they characterize topological phases and relate to edge states, including cases with half-integer values.

Contribution

It introduces a geometrical framework for understanding winding numbers in non-Hermitian systems, linking them to exceptional points and topological phase characterization.

Findings

Winding number can be half-integer in non-Hermitian systems.

Winding numbers relate to encircling exceptional points in momentum space.

Topological phases are characterized by these winding numbers and associated edge states.

Abstract

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 $\nu_2$.