The thermodynamic origin of the Contact and its relation to the gap in the BEC-BCS crossover

TL;DR

This paper reveals that the contact parameter in ultracold gases is thermodynamically conjugate to the inverse scattering length, linking it to the gap in the BEC-BCS crossover and deriving key known results.

Contribution

It identifies the contact as conjugate to the inverse scattering length and relates it to the gap in the BEC-BCS crossover using mean-field theory.

Findings

Contact is conjugate to inverse scattering length

Contact proportional to the square of the gap

Derived known results about the contact

Abstract

As can be inferred from present experiments in ultracold gases, the scattering length is a quantity that determines the thermodynamic state of the gas. As such, there exists a conjugate thermodynamic to it. Here, we show that the recently introduced "contact" is the conjugate of the inverse of the scattering length. We find that this identification allows for a derivation of essentially all the known results regarding the contact. Using the mean-field theory for the Bose-Einstein (BEC) to Bardeen-Cooper-Schriefer (BCS) crossover, we also find that the contact is proportional to the square of the gap. We analyze in detail both a homogenous balanced mixture of fermions and its inhomogenous counterpart in a harmonic trap.