The Effect of Repulsion on Superconductivity at Low Density

We examine the effect of repulsion on superconductivity in a three-dimensional system with a Bardeen-Pines-like interaction in the low-density limit, where the chemical potential is much smaller than the phonon frequency. We parameterize the strength of the repulsion by a dimensionless parameter f. We find that the superconducting transition temperature T_c approaches a nonzero value in the limit of zero density as long as f is below a certain threshold f^*. We show that T_c goes to zero as a power of f^*- f, in contrast to the high density limit, where T_c goes to zero exponentially quickly as f approaches f^*. At the same time as T_c goes to zero, the nodes of the gap function along the Matsubara axis approach zero also as a power of f^*- f.