Asymptotic behavior of Toeplitz determinants with a delta function singularity

TL;DR

This paper derives the asymptotic behavior of Toeplitz determinants with symbols combining an analytical function and a delta function, crucial for understanding spin correlations in integrable quantum models.

Contribution

It introduces a Wiener-Hopf based method to analyze Toeplitz determinants with delta function singularities, applicable to quantum spin systems.

Findings

Derived explicit asymptotic formulas for Toeplitz determinants with delta singularities.

Applied formulas to compute spin-correlation functions in the quantum XY chain.

Calculated ground state magnetization using the new asymptotic results.

Abstract

We find the asymptotic behaviors of Toeplitz determinants with symbols which are a sum of two contributions: one analytical and non-zero function in an annulus around the unit circle, and the other proportional to a Dirac delta function. The formulas are found by using the Wiener-Hopf procedure. The determinants of this type are found in computing the spin-correlation functions in low-lying excited states of some integrable models, where the delta function represents a peak at the momentum of the excitation. As a concrete example of applications of our results, using the derived asymptotic formulas we compute the spin-correlation functions in the lowest energy band of the frustrated quantum XY chain in zero field, and the ground state magnetization.