Superconductivity with hard-core repulsion: BCS-Bose crossover and $s-/d$-wave competition

We consider fermions on a two-dimensional lattice interacting repulsively on the same site and attractively on the nearest-neighbor sites. The model is relevant, for instance, to study the competition between antiferromagnetism and superconductivity in a Kondo lattice. We first solve the two-body problem to show that in the dilute and strong-coupling limit the s-wave Bose condensed state is always the ground state. We then consider the many-body problem and treat it at mean-field level by solving exactly the usual gap equation. This guarantees that the superconducting wave function correctly vanishes when the two fermions (with antiparallel spin) sit on the same site. This fact has important consequences for the superconducting state that are somewhat unusual. In particular this implies a radial node line for the gap function. When a next-neighbor hopping $t^{\prime }$ is present we find that the s-wave state may develop nodes on the Fermi surface.