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 $\ensuremath{\mu}$ is much smaller than the phonon frequency ${\ensuremath{\omega}}_{L}$. We parametrize the strength of the repulsion by a dimensionless parameter $f$, and find that the superconducting transition temperature ${T}_{c}$ approaches a nonzero value in the $\ensuremath{\mu}=0$ limit as long as $f$ is below a certain threshold ${f}^{*}$. In this limit, we find that ${T}_{c}$ goes to zero as a power of ${f}^{*}\ensuremath{-}f$, in contrast to the high density limit, where ${T}_{c}$ goes to zero exponentially quickly as $f$ approaches ${f}^{*}$. For all nonzero $f$, the gap function $\mathrm{\ensuremath{\Delta}}({\ensuremath{\omega}}_{m})$ 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 ${T}_{c}$ still saturates to a nonzero value at $\ensuremath{\mu}=0$ when $f<{f}^{*}$.