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

Abstract

We consider a fermionic system at zero temperature interacting through an effective nonretarded potential of the type introduced by Nozières 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- (kFξpair≫1) from the strong- (kFξpair≪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 kFξpair≃10, ξpair in turn coinciding with the Pippard coherence length. In the strong-coupling limit we find instead that ξphase≫ξpair, with ξpair 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 ξpair 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.

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