Non-Hermitean Localization and De-Localization

TL;DR

This paper investigates localization phenomena in non-Hermitian Hamiltonians inspired by vortex pinning in superconductors, providing analytic descriptions of the density of states and localization length divergence at phase transitions.

Contribution

It introduces simplified models for non-Hermitian localization, revealing non-perturbative structures and mapping the problem to a 2D random walk, advancing understanding of localization in such systems.

Findings

Analytic description of density of states with 'wings' structure

Divergence of localization length at transition points

Mapping of models to 2D random walk problems

Abstract

We study localization and delocalization in a class of non-hermitean Hamiltonians inspired by the problem of vortex pinning in superconductors. In various simplified models we are able to obtain analytic descriptions, in particular of the non-perturbative emergence of a forked structure (the appearance of "wings") in the density of states. We calculate how the localization length diverges at the localization-delocalization transition. We map some versions of this problem onto a random walker problem in two dimensions. For a certain model, we find an intricate structure in its density of states.