Spectral Curves of Non-Hermitean Hamiltonians

TL;DR

This paper analyzes the spectral properties of non-Hermitian Hamiltonians in superconductors, revealing a critical transition influenced by the distribution of random site energies and characterizing eigenvalue densities and localization behavior.

Contribution

It provides a simple proof of the spectral curve structure in the maximally non-Hermitian limit and explores how different probability distributions affect spectral transitions and eigenvalue densities.

Findings

Spectral curves are one-dimensional in the maximally non-Hermitian limit.

A critical transition occurs when the average of the log of site energies squared vanishes.

The localization length on the spectral wings diverges linearly near the transition energy.

Abstract

Recent analytical and numerical work have shown that the spectrum of the random non-hermitean Hamiltonian on a ring which models the physics of vortex line pinning in superconductors is one dimensional. In the maximally non-hermitean limit, we give a simple "one-line" proof of this feature. We then study the spectral curves for various distributions of the random site energies. We find that a critical transition occurs when the average of the logarithm of the random site energy squared vanishes. For a large class of probability distributions of the site energies, we find that as the randomness increases the energy at which the localization-delocalization transition occurs increases, reaches a maximum, and then decreases. The Cauchy distribution studied previously in the literature does not have this generic behavior. We determine the critical value of the randomness at which "wings" first appear in the energy spectrum. For distributions, such as Cauchy, with infinitely long tails, we show that this critical value is infinitesimally above zero. We determine the density of eigenvalues on the wings for any probability distribution. We show that the localization length on the wings diverges linearly as the energy approaches the energy at which the localization-delocalization transition occurs. These results are all obtained in the maximally non-hermitean limit but for a generic class of probability distributions of the random site energies.