Asymptotics of block Toeplitz determinants with piecewise continuous symbols

TL;DR

This paper derives the asymptotic behavior of block Toeplitz determinants with piecewise continuous symbols, providing explicit formulas and new methods, with applications to quantum spin chains and entanglement entropy.

Contribution

It introduces a new localization theorem and a novel method for computing the Fredholm index for Toeplitz operators with piecewise continuous matrix symbols.

Findings

Asymptotic formula for block Toeplitz determinants with piecewise continuous symbols

Development of a new localization theorem for Toeplitz determinants

Application to entanglement entropy in quantum spin chains

Abstract

We determine the asymptotics of the block Toeplitz determinants $\det T_n(\phi)$ as $n\to\infty$ for $N\times N$ matrix-valued piecewise continuous functions $\phi$ with a finitely many jumps under mild additional conditions. In particular, we prove that $$ \det T_n(\phi) \sim G^n n^\Omega E\quad {\rm as}\ n\to \infty, $$ where $G$, $E$, and $\Omega$ are constants that depend on the matrix symbol $\phi$ and are described in our main results. Our approach is based on a new localization theorem for Toeplitz determinants, a new method of computing the Fredholm index of Toeplitz operators with piecewise continuous matrix-valued symbols, and other operator theoretic methods. As an application of our results, we consider piecewise continuous symbols that arise in the study of entanglement entropy in quantum spin chain models.