Evolution from BCS superconductivity to Bose condensation: Calculation of the zero-temperature phase coherence length

We consider a fermionic system at zero temperature interacting through an effective nonretarded potential of the type introduced by Nozieres and Schmitt-Rink, and calculate the phase coherence length xi(phase) (associated with the spatial fluctuations of the superconducting order parameter) by exploiting a functional-integral formulation for the correlation functions and the associated loop expansion. This formulation is especially suited to follow the evolution of the fermionic system from a BCS-type superconductor for weak coupling to a Bose-condense system for strong coupling, since in the latter limit a direct mapping of the original fermionic system onto an effective system of bosons with a residual boson-boson interaction can be established. Explicit calculations are performed at the one-loop order. The phase coherence length xi(phase) is compared with the coherence length xi(pair) for two-electron correlation, which is relevant to distinguish the weak-(k(F) xi(pair)much greater than 1) from the strong-(k(F) xi(pair)much less than 1) coupling limits (k(F) being the Fermi wave vector) as well as to follow the crossover in between. It is shown that xi(phase) coincides with xi(pair) down to k(F) xi(pair) similar or equal to 10, xi(pair) in turn coinciding with the Pippard coherence length. In the strong-coupling limit we find instead that xi(phase) much greater than xi(pair), with xi(pair) coinciding with the radius of the bound-electron pair. From the mapping onto an effective system of bosons in the strong-coupling limit we further relate xi(pair) with the ''range'' of the residual boson-boson interaction, which is physically the only significant length associated with the dynamics of the bosonic system.