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 Nozi\`eres and Schmitt-Rink, and calculate the phase coherence length ξₚₕₐₛₑ (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-condensed 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 ξₚₕₐₛₑ is compared with the coherence length ξₚₐᵢᵣ for two-electron correlation, which is relevant to distinguish the weak- (kF{{ξ}}ₚₐᵢᵣ1) from the strong- ({k}Fξₚₐᵢᵣ{}1) coupling limits (kF being the Fermi wave vector) as well as to follow the crossover in between. It is shown that ξₚₕₐₛₑ coincides with ξₚₐᵢᵣ down to kF{{ξ}}ₚₐᵢᵣ10,{{ξ}}ₚₐᵢᵣ$ in turn coinciding with the Pippard coherence length. In the strong-coupling limit we find instead that ${{ξ}}ₚₕₐₛₑ{{ξ}}ₚₐᵢᵣ$, with ${{ξ}}ₚₐᵢᵣ$ 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 ${{ξ}}ₚₐᵢᵣ$ with the ``range'' of the residual boson-boson interaction, which is physically the only significant length associated with the dynamics of the bosonic system. {} 1996 The American Physical Society.