Winding Numbers, Complex Currents, and Non-Hermitian Localization

TL;DR

This paper investigates how winding numbers and complex currents characterize delocalized states in disordered non-Hermitian models, revealing a topological aspect of localization transitions relevant to physics and biology.

Contribution

It introduces the concept of winding numbers to describe extended states in non-Hermitian disordered systems, providing a novel topological perspective on localization.

Findings

Delocalized states are characterized by integer winding numbers.

Eigenvalue trajectories in the complex plane reveal complex currents.

Winding numbers extend to higher dimensions, broadening the theory.

Abstract

The nature of extended states in disordered tight binding models with a constant imaginary vector potential is explored. Such models, relevant to vortex physics in superconductors and to population biology, exhibit a delocalization transition and a band of extended states even for a one dimensional ring. Using an analysis of eigenvalue trajectories in the complex plane, we demonstrate that each delocalized state is characterized by an (integer) winding number, and evaluate the associated complex current. Winding numbers in higher dimensions are also discussed.