Fisher-Hartwig expansion for Toeplitz determinants and the spectrum of a single-particle reduced density matrix for one-dimensional free fermions

TL;DR

This paper verifies the Fisher-Hartwig expansion for Toeplitz determinants with a sine kernel, related to the spectrum of free fermions in 1D, and derives related eigenvalue and entropy expansions.

Contribution

It provides a detailed order-by-order verification of the Fisher-Hartwig expansion for the spectral determinant of a sine kernel Toeplitz matrix, including coefficient calculations.

Findings

Verified the Fisher-Hartwig expansion up to tenth order.

Derived eigenvalue and entanglement entropy expansions.

Supported analytical results with numerical examples.

Abstract

We study the spectrum of the Toeplitz matrix with a sine kernel, which corresponds to the single-particle reduced density matrix for free fermions on the one-dimensional lattice. For the spectral determinant of this matrix, a Fisher--Hartwig expansion in the inverse matrix size has been recently conjectured. This expansion can be verified order by order, away from the line of accumulation of zeros, using the recurrence relation known from the theory of discrete Painleve equations. We perform such a verification to the tenth order and calculate the corresponding coefficients in the Fisher-Hartwig expansion. Under the assumption of the validity of the Fisher-Hartwig expansion in the whole range of the spectral parameter, we further derive expansions for an equation on the eigenvalues of this matrix and for the von Neumann entanglement entropy in the corresponding fermion problem. These analytical results are supported by a numerical example.