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

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.