Eigenpairs of Toeplitz and disordered Toeplitz matrices with a Fisher-Hartwig symbol

TL;DR

This paper analyzes eigenvalues and eigenvectors of Toeplitz matrices with Fisher-Hartwig symbols, including effects of disorder, classifying non-perturbative behaviors, and quantifying eigenvector localization.

Contribution

It derives eigenpairs for singular Fisher-Hartwig Toeplitz matrices, studies disorder effects, and classifies non-perturbative eigenvalue behaviors with localization analysis.

Findings

Bulk eigenvalue behavior well captured by second order perturbation theory.

Identification of two classes of eigenvalue 'runaways' with distinct localization properties.

Eigenvectors of type II runaways are highly localized, with super-exponential decay.

Abstract

Toeplitz matrices have entries that are constant along diagonals. They model directed transport, are at the heart of correlation function calculations of the two-dimensional Ising model, and have applications in quantum information science. We derive their eigenvalues and eigenvectors when the symbol is singular Fisher-Hartwig. We then add diagonal disorder and study the resulting eigenpairs. We find that there is a "bulk" behavior that is well captured by second order perturbation theory of non-Hermitian matrices. The non-perturbative behavior is classified into two classes: Runaways type I leave the complex-valued spectrum and become completely real because of eigenvalue attraction. Runaways type II leave the bulk and move very rapidly in response to perturbations. These have high condition numbers and can be predicted. Localization of the eigenvectors are then quantified using entropies and inverse participation ratios . Eigenvectors corresponding to Runaways type II are most localized (i.e., super-exponential), whereas Runaways type I are less localized than the unperturbed counterparts and have most of their probability mass in the interior with algebraic decays. The results are corroborated by applying free probability theory and various other supporting numerical studies.