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 $\mu$ is much smaller than the phonon frequency $\omega_L$. We parameterize 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 $\mu = 0$ 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 $f^*-f$, 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 $\Delta (\omega_m)$ 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 non-zero value at $\mu = 0$ when $f < f^*$.