Fermionic full counting statistics with smooth boundaries: from discrete particles to bosonization

TL;DR

This paper investigates the full counting statistics of free fermions in one dimension, exploring the transition from discrete particle counting to continuous density measurement using analytical and numerical methods.

Contribution

It introduces a combined analytical and numerical approach to study the crossover in full counting statistics with smoothing, connecting discrete fermion counting to bosonization results.

Findings

Fisher--Hartwig expansion describes discrete particle counting

Bosonization results emerge in the continuous limit

The study relates full counting statistics to orthogonality catastrophe and Fermi-edge singularity

Abstract

We revisit the problem of full counting statistics of particles on a segment of a one-dimensional gas of free fermions. Using a combination of analytical and numerical methods, we study the crossover between the counting of discrete particles and of the continuous particle density as a function of smoothing in the counting procedure. In the discrete-particle limit, the result is given by the Fisher--Hartwig expansion for Toeplitz determinants, while in the continuous limit we recover the bosonization results. This example of full counting statistics with smoothing is also related to orthogonality catastrophe, Fermi-edge singularity and non-equilibrium bosonization.