The Effect of Repulsion on Superconductivity at Low Density

Abstract

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 parameterize 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 ∆(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 non-zero value at µ = 0 when f < f * .