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Using mean-field theory for the Bardeen-Cooper-Schriefer (BCS) to theBose-Einstein condensate (BEC) crossover we investigate the ground statethermodynamic properties of an interacting homogeneous Fermi gas. Theinteratomic interactions modeled through a finite range potential allows us toexplore the entire region from weak to strong interacting regimes with noapproximations. To exhibit the thermodynamic behavior as a function of thepotential parameters in the whole crossover region, we concentrate in studyingthe contact variable, the thermodynamic conjugate of the inverse of the s-wavescattering length. Our analysis allows us to validate the mean-field approachacross the whole crossover. It also leads to predict a quantum transition-likein the case when the potential range becomes large. This finding is a directconsequence of the k-dependent energy gap for finite interaction rangepotentials.

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