Chern numbers and localization by non-Hermitean operators

TL;DR

This paper extends the concept of Chern numbers from Hermitean quantum systems to non-Hermitean localization problems, revealing how topological invariants manifest in more general, non-Hermitean contexts.

Contribution

It introduces a generalized definition of Chern numbers applicable to non-Hermitean operators, connecting topological invariants with localization phenomena in diverse physical systems.

Findings

Point eigenvalue degeneracies become loops with branch cuts in non-Hermitean systems.

Chern number remains quantized and is associated with Berry flux in these systems.

Non-quantized contributions arise when loops intersect the integration surface.

Abstract

In Hermitean quantum mechanics, extended current-carrying states are distinguished from localized ones by their non-zero Chern number. We generalize the notion of Chern number to non-Hermitean localization problems such as tiltedflux lines in type-II superconductors with line defects and passive scalar transport in random vorticity fields. We find that the usualpoint eigenvalue degeneracies that occur in the Hermitean case are now loops bounding a branch cut sheet with the same quantized total Berry flux, and hence Chern number, as the original point. If a loop cuts the integration surface non-quantized contributions to the total flux result.