The Effect of Repulsion on Superconductivity at Low Density

TL;DR

This paper investigates how repulsive interactions influence superconductivity at low electron densities, revealing that a non-zero transition temperature persists below a critical repulsion strength, with the gap function changing sign along the Matsubara axis.

Contribution

It demonstrates that superconductivity can survive at low densities with repulsion, showing a power-law suppression of $T_c$ near a critical repulsion threshold, extending previous models with full dynamical screening.

Findings

$T_c$ approaches a non-zero value as $ ext{μ} o 0$ for $f < f^*$.

The gap function $ ext{Δ}( ext{ω}_m)$ changes sign along the Matsubara axis.

$T_c$ vanishes as a power of $f^* - f$ near the threshold.

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 $\mu$ 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 $f < f^*$.