Quantum fluctuations of one-dimensional free fermions and Fisher-Hartwig formula for Toeplitz determinants

TL;DR

This paper applies the generalized Fisher-Hartwig conjecture to accurately compute the full counting statistics of one-dimensional free fermions, highlighting the importance of discrete charge effects and improving asymptotic formulas.

Contribution

It demonstrates the use of the Fisher-Hartwig formula for fermionic counting statistics, surpassing bosonization methods and proposing higher-precision asymptotics.

Findings

Fisher-Hartwig formula accurately predicts fermionic counting statistics

Numerical evidence shows higher precision than previous proofs

Conjecture of next-order correction to the Fisher-Hartwig formula

Abstract

We revisit the problem of finding the probability distribution of a fermionic number of one-dimensional spinless free fermions on a segment of a given length. The generating function for this probability distribution can be expressed as a determinant of a Toeplitz matrix. We use the recently proven generalized Fisher--Hartwig conjecture on the asymptotic behavior of such determinants to find the generating function for the full counting statistics of fermions on a line segment. Unlike the method of bosonization, the Fisher--Hartwig formula correctly takes into account the discreteness of charge. Furthermore, we check numerically the precision of the generalized Fisher--Hartwig formula, find that it has a higher precision than rigorously proven so far, and conjecture the form of the next-order correction to the existing formula.