Modified Szego-Widom Asymptotics for Block Toeplitz Matrices with Zero Modes

TL;DR

This paper extends the Szego-Widom theorem to account for zero modes in block Toeplitz matrices, inspired by topological superconductor models, providing a new asymptotic determinant formula.

Contribution

It introduces a modified Szego-Widom asymptotic formula for block Toeplitz matrices with zero eigenvalue modes, bridging mathematical theory and topological physics.

Findings

Derived a new asymptotic expression for determinants with zero modes

Connected zero mode phenomena to topological superconductor models

Enhanced understanding of Toeplitz matrix asymptotics in physical systems

Abstract

The Szego-Widom theorem provides an expression for the determinant of block Toeplitz matrices in the asymptotic limit of large matrix dimension n. We show that the presence of zero modes, i.e, eigenvalues that vanish as \alpha^n, |\alpha|<1, when n \to \infty, require a modification of the Szego-Widom theorem. A new asymptotic expression for the determinant of a certain class of block Toeplitz matrices with one pair of zero modes is derived. The result is inspired by 1-dimensional topological superconductors, and the relation with the latter systems is discussed.