Renormalization-group approach to the Kohn-Luttinger superconductivity: Amplification of the pairing gap from $\ell^4$ to $\ell$

TL;DR

This paper improves the understanding of the Kohn-Luttinger mechanism for superconductivity by incorporating Kohn anomaly effects into the RG analysis, revealing a much larger pairing gap exponent than previously estimated.

Contribution

It introduces a renormalization group approach that includes Kohn anomaly contributions, significantly enhancing the predicted pairing gap in the Kohn-Luttinger mechanism.

Findings

The gap exponent scales as -ℓ, indicating a larger pairing gap.

RG analysis shows a significant enhancement over previous perturbative estimates.

Comparison with Bethe-Salpeter results validates the RG approach.

Abstract

We revisit the renormalization group (RG) analysis of the Kohn-Luttinger (KL) mechanism for superconductivity. The KL mechanism leads to superconductivity in a system with a repulsive bare interaction. The key ingredient is the screening effect that renders the induced interaction attractive in channels with nonzero angular momentum $\ell \neq 0$, thereby triggering the Bardeen-Cooper-Schrieffer (BCS) instability. According to the original argument, the resulting gap is exponentially small, with its exponent scaling as $-\ell^4$. However, the KL mechanism was originally formulated within perturbation theory, where the series is known to converge poorly in certain cases -- most notably, for the p-wave paring gap induced by a repulsive s-wave contact interaction. This poor convergence may be attributed to a divergent integrand in a specific class of diagrams containing both the BCS logarithm and the Kohn anomaly, suggesting that one must resum the Kohn anomaly contributions separately from the BCS logarithm. In this work, we incorporate the Kohn anomaly contribution into the beta function of the RG equation governing the BCS instability near the Fermi surface. Our solution shows that the KL gap exponent is then proportional to $-\ell$, indicating a significant enhancement of the KL mechanism beyond the previously known result. To illustrate this, we study the spin-triplet p-wave pairing gap arising from a repulsive s-wave contact interaction and compare our RG-based results with those obtained from the Bethe-Salpeter equation in perturbation theory.