Zero temperature momentum distribution of an impurity in a polaron state of one-dimensional Fermi and Tonks-Girardeau gases

TL;DR

This paper derives an exact zero-temperature momentum distribution for an impurity in a 1D polaron state within Fermi and Tonks-Girardeau gases, revealing decay and peak features and connecting to anyon correlations.

Contribution

It provides a Fredholm determinant representation of the impurity momentum distribution in a Bethe ansatz solvable model, including strong interaction limits and connections to anyon correlations.

Findings

Fourth power decay at large momentum

Weakly divergent peak at finite momentum

Distribution expressed via anyon correlation functions in strong interaction limit

Abstract

We investigate the momentum distribution function of a single distinguishable impurity particle which formed a polaron state in a gas of either free fermions or Tonks-Girardeau bosons in one spatial dimension. We obtain a Fredholm determinant representation of the distribution function for the Bethe ansatz solvable model of an impurity-gas $\delta$-function interaction potential at zero temperature, in both repulsive and attractive regimes. We deduce from this representation the fourth power decay at a large momentum, and a weakly divergent (quasi-condensate) peak at a finite momentum. We also demonstrate that the momentum distribution function in the limiting case of infinitely strong interaction can be expressed through a correlation function of the one-dimensional impenetrable anyons.