# Winding in Non-Hermitian Systems

## TL;DR

This paper explores the topological winding number properties of eigenfunctions in non-Hermitian systems, extending known interlacing properties from Hermitian systems and examining how these windings change near exceptional points.

## Contribution

It introduces a topological perspective on eigenfunction properties in non-Hermitian systems, linking winding numbers to spectral characteristics and exceptional points.

## Key findings

- Winding numbers of eigenfunctions are well-ordered in Hermitian and PT-symmetric systems.
- Varying system parameters across exceptional points alters eigenfunction windings predictably.
- Nonlinear and higher-dimensional non-Hermitian systems also show these winding properties.

## Abstract

This paper extends the property of interlacing of the zeros of eigenfunctions in Hermitian systems to the topological property of winding number in non-Hermitian systems. Just as the number of nodes of each eigenfunction in a self-adjoint Sturm-Liouville problem are well-ordered, so too are the winding numbers of each eigenfunction of Hermitian and of unbroken PT-symmetric potentials. Varying a system back and forth past an exceptional point changes the windings of its eigenfunctions in a specific manner. Nonlinear, higher-dimensional, and general non-Hermitian systems also exhibit manifestations of these characteristics.