Density correlation functions and the spatial structure of the two-dimensional BEC-BCS crossover

TL;DR

This paper analyzes the spatial structure of a two-dimensional fermionic superfluid across the BEC-BCS crossover using mean-field theory, deriving analytical correlation functions and exploring their large-distance behavior.

Contribution

It provides analytical expressions for density correlation functions and correlation lengths in 2D, revealing detailed spatial structure and differences from 3D cases.

Findings

Correlation functions exhibit exponential decay and oscillations at large distances.

Derived closed-form expressions for correlation lengths and mean pair radius.

Identified differences in scattering length dependence between 2D and 3D cases.

Abstract

The spatial structure of a two-dimensional homogeneous mixture of fermionic atoms in two hyperfine states is analyzed throughout the BEC-BCS crossover. Within the BCS-Leggett mean-field model we consider three functions: the pair wave function and the density-density correlation functions between atoms of the same and of different hyperfine states. For the correlation functions we derive analytical expressions which allow to unveil the rich spatial structure of the superfluid. Mainly, we are able to study the large-distance behavior of the three functions, which exhibits an exponential decay and a well-defined oscillatory behavior. We report closed-form expressions for the correlation lengths and mean pair radius. Differences and similarities emerge when comparing with the three dimensional case. Particularly, we find an expression for the large-distance correlation length, in terms of the chemical potential and the gap, valid in two and three dimensions, but whose dependence on the corresponding scattering lengths differ significantly.