Time and temperature-dependent correlation function of an impurity in a one-dimensional Fermi gas as a Fredholm determinant

TL;DR

This paper derives a Fredholm determinant formula for the time-dependent correlation function of an impurity in a one-dimensional Fermi gas, valid for any temperature and interaction strength, unifying previous special cases.

Contribution

It provides a general Fredholm determinant representation for impurity correlation functions at arbitrary temperature and interaction, extending prior specific results.

Findings

Unified representation for zero and finite temperature cases.

Applicable to both repulsive and attractive interactions.

Includes previous special case results as particular instances.

Abstract

We investigate a free one-dimensional spinless Fermi gas, and the Tonks-Girardeau gas interacting with a single impurity particle of equal mass. We obtain a Fredholm determinant representation for the time-dependent correlation function of the impurity particle. This representation is valid for an arbitrary temperature and an arbitrary repulsive or attractive impurity-gas $\delta$-function interaction potential. It includes, as particular cases, the representations obtained for zero temperature and arbitrary repulsion in [Nucl. Phys. B 892, 83 (2015)], and for arbitrary temperature and infinite repulsion in [Nucl. Phys. B 520, 594 (1998)].