Mobile impurity in a one-dimensional gas at finite temperatures

TL;DR

This paper analyzes the behavior of a mobile impurity in a one-dimensional fermionic gas at finite temperatures, deriving key properties like the one-body function, momentum distribution, and Tan's contact using Fredholm determinants and effective form factors.

Contribution

It introduces a novel analytical approach combining Fredholm determinants and effective form factors to study impurity properties at finite temperatures in 1D gases.

Findings

Derived the large-distance exponential decay of the one-body function.

Analyzed the impurity's momentum distribution at small momenta.

Computed finite temperature Tan's contact.

Abstract

We consider the McGuire model of a one-dimensional gas of free fermions interacting with a single impurity. We compute the static one-body function and momentum distribution of the impurity at finite temperatures. The results involve averages over Fredholm determinants that we further analyse using the effective form factors approach. With this approach, we derive the large-distance behaviour of the one-body function, which takes the form of an averaged exponential decay. This method allows us to study an experimentally important regime of small momenta of the impurity's momentum distribution. We also consider the one-body function at short distances and compute finite temperature Tan's contact.