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 μ is much smaller than the phonon frequency ωL. We parametrize the strength of the repulsion by a dimensionless parameter f, and find that the superconducting transition temperature Tc approaches a nonzero value in the μ=0 limit as long as f is below a certain threshold f∗. In this limit, we find that Tc goes to zero as a power of f∗-f, in contrast to the high density limit, where Tc goes to zero exponentially quickly as f approaches f∗. For all nonzero f, the gap function Δ(ωm) changes sign along the Matsubara axis, which allows the system to partially overcome the repulsion at high frequencies. We trace the position of the gap node with f and show that it approaches zero frequency as f approaches f∗. To investigate the robustness of our conclusions, we then go beyond the Bardeen-Pines model and include full dynamical screening of the interaction, finding that Tc still saturates to a nonzero value at μ=0 when f<f∗.