Highly polarized Fermi gases: One-dimensional case

TL;DR

This paper investigates a one-dimensional Fermi gas with a single impurity, calculating its binding energy and effective mass, and validates an approximate method by comparing with exact solutions.

Contribution

It applies and validates an approximate method for impurity problems in one-dimensional Fermi gases, showing strong agreement with exact results across different mass regimes.

Findings

Excellent agreement with McGuire's exact results for equal mass case.

Exact solutions for infinite mass impurity match approximate results.

Validation of the approximate method for one-dimensional systems.

Abstract

We consider the problem of a single particle interacting with $N$ identical fermions, at zero temperature and in one dimension. We calculate the binding energy as well as the effective mass of the single particle. We use an approximate method developed in the three-dimensional case, where the Hilbert space for the excited states of the $N$ fermions is restricted to have at most two particle-hole pairs. When the mass of the single particle is equal to the fermion mass, we find an excellent agreement with the exact results of McGuire. When the mass of the single particle is infinite, we solve exactly the problem and find again excellent agreement between approximate results and exact ones. This overall agreement in one dimension gives a strong validation for the approximate method applied in three dimensions. Moreover it shows that our approximate treatment is excellent for the one dimensional problem in the general case with respect to the mass of the single particle.