Bound states, Cooper pairing, and Bose condensation in two dimensions

Abstract

For a dilute gas of fermions interacting via an arbitrary pair potential in d=2 dimensions, we show that the many-body ground state is unstable to s-wave pairing at T=0 if and only if a two-body bound state exists. We further obtain, within a variational pairing Ansatz, a smooth crossover from a Cooper-paired state (${\ensuremath{\xi}}_{0}$${\mathrm{k}}_{\mathrm{f}}$\ensuremath{\gg}1) to a Bose condensed state of tightly bound pairs (${\ensuremath{\xi}}_{0}$${\mathrm{k}}_{\mathrm{f}}$\ensuremath{\ll}1). We briefly discuss non--s-wave superconductors. Insofar as this model is appli- cable to the high-${\mathrm{T}}_{\mathrm{c}}$ materials, they are in the interesting regime with the coherence length ${\ensuremath{\xi}}_{0}$ comparable to the interparticle spacing ${\mathrm{K}}_{\mathrm{f}}^{\mathrm{\ensuremath{-}}1}$.