Polaron, molecule and pairing in one-dimensional spin-1/2 Fermi gas with an attractive Delta-function interaction

TL;DR

This paper investigates the behavior of a spin-down impurity in a 1D spin-up Fermi sea with attractive interactions, revealing a crossover from polaron to molecule states using Bethe ansatz solutions.

Contribution

It provides a detailed analysis of the polaron-molecule crossover in 1D fermionic systems with attractive interactions, using exact Bethe ansatz solutions.

Findings

Impurity behaves as a polaron at weak attraction.

Impurity forms a bound molecule at strong attraction.

The crossover from polaron to molecule is universal in 1D systems.

Abstract

Using solutions of the discrete Bethe ansatz equations, we study in detail the quantum impurity problem of a spin-down fermion immersed into a fully ploarized spin-up Fermi sea with weak attraction. We prove that this impurity fermion in the one-dimensional (1D) fermionic medium behaves like a polaron for weak attraction. However, as the attraction grows, the spin-down fermion binds with one spin-up fermion from the fully-polarized medium to form a tightly bond molecule. Thus it is seen that the system undergos a cross-over from a mean field polaron-like nature into a mixture of excess fermions and a bosonic molecule as the attraction changes from weak attraction into strong attraction. This polaron-molecule cross-over is universal in 1D many-body systems of interacting fermions. In thermodynamic limit, we further study the relationship between the Fredholm equations for the 1D spin-1/2 Fermi gas with weakly repulsive and attractive delta-function interactions.