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

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 (${\xi }_0$ ${\mathrm{k}}_{\mathrm{f}}$≫1) to a Bose condensed state of tightly bound pairs (${\xi }_0$ ${\mathrm{k}}_{\mathrm{f}}$≪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 ${\xi }_0$ comparable to the interparticle spacing ${\mathrm{K}}_{\mathrm{f}}^{\mathrm{-}1}$.