Abstract
The authors assess the validity of mean-field theories for lattice fermion systems with attractive interactions by comparing variational upper bounds from two plausible mean-field approximations-a Van der Waals (independent particle) approximation and a pairing approximation of the BCS type-for the Helmholtz free energy of a spinless fermion model with attractive nearest-neighbour interactions. It is found that there is a crossover from the Van der Waals approximation giving the best bound to the pairing approximation giving the best bound as t, the hopping integral passes through a critical value tc. This crossover phenomenon exists in all dimensions as well as for the limiting d to infinity model. The authors conclude that there can be no simple mean-field theory which is valid for fermion systems with attractive interactions.
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