The Contact in the BCS-BEC crossover for finite range interacting ultracold Fermi gases

Using mean-field theory for the Bardeen-Cooper-Schriefer (BCS) to the Bose-Einstein condensate (BEC) crossover we investigate the ground state thermodynamic properties of an interacting homogeneous Fermi gas. The interatomic interactions modeled through a finite range potential allows us to explore the entire region from weak to strong interacting regimes with no approximations. To exhibit the thermodynamic behavior as a function of the potential parameters in the whole crossover region, we concentrate in studying the contact variable, the thermodynamic conjugate of the inverse of the s-wave scattering length. Our analysis allows us to validate the mean-field approach across the whole crossover. It also leads to predict a quantum transition-like in the case when the potential range becomes large. This finding is a direct consequence of the k-dependent energy gap for finite interaction range potentials.