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 ${\xi }_{\mathrm{phase}}$ (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-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 ${\xi }_{\mathrm{phase}}$ is compared with the coherence length ${\xi }_{\mathrm{pair}}$ for two-electron correlation, which is relevant to distinguish the weak- (${\mathit{k}}_{\mathit{F}}$${\xi }_{\mathrm{pair}}$≫1) from the strong- (${\mathit{k}}_{\mathit{F}}$${\xi }_{\mathrm{pair}}$≪1) coupling limits (${\mathit{k}}_{\mathit{F}}$ being the Fermi wave vector) as well as to follow the crossover in between. It is shown that ${\xi }_{\mathrm{phase}}$ coincides with ${\xi }_{\mathrm{pair}}$ down to ${\mathit{k}}_{\mathit{F}}$${\xi }_{\mathrm{pair}}$≃10, ${\xi }_{\mathrm{pair}}$ in turn coinciding with the Pippard coherence length. In the strong-coupling limit we find instead that ${\xi }_{\mathrm{phase}}$≫${\xi }_{\mathrm{pair}}$, with ${\xi }_{\mathrm{pair}}$ 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 ${\xi }_{\mathrm{pair}}$ 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.

• Received 27 November 1995

DOI:https://doi.org/10.1103/PhysRevB.53.15168