Impurity Green's function of a one-dimensional Fermi-gas

TL;DR

This paper computes the impurity Green's function for a one-dimensional spin-1/2 Fermi gas with delta-function interactions, expressing it as an integral of Fredholm determinants, advancing understanding of impurity dynamics in integrable systems.

Contribution

It provides an exact integral representation of the impurity Green's function for arbitrary interaction strength in a 1D Fermi gas, a novel analytical result.

Findings

Explicit integral formula for impurity Green's function.

Representation as Fredholm determinants of integrable operators.

Applicable to arbitrary interaction strengths.

Abstract

We consider a one-dimensional gas of spin-1/2 fermions interacting through $\delta$-function repulsive potential of an arbitrary strength. For the case of all fermions but one having spin up, we calculate time-dependent two-point correlation function of the spin-down fermion. This impurity Green's function is represented in the thermodynamic limit as an integral of Fredholm determinants of integrable linear integral operators.