Superconductivity near a nematic quantum critical point -- the interplay between hot and lukewarm regions

TL;DR

This paper develops a strong coupling dynamical theory for superconductivity near a nematic quantum critical point, analyzing how fluctuations influence the transition temperature and the structure of the order parameter across hot and lukewarm regions.

Contribution

It introduces a fermion-boson model that accounts for both strong and weak fluctuations, providing analytical and numerical solutions for the superconducting transition and order parameter structure near a nematic QCP.

Findings

Superconducting transition temperature T_c is of order λ^2 E_F.

The order parameter varies significantly between hot and lukewarm regions.

s- and d-wave states have nearly the same T_c, with a small difference proportional to λ^2.

Abstract

We present a strong coupling dynamical theory of the superconducting transition in a metal near a QCP towards $Q = 0$ nematic order. We use a fermion-boson model, in which we treat the ratio of effective boson-fermion coupling and the Fermi energy as a small parameter $\lambda$. We solve, both analytically and numerically, the linearized Eliashberg equation. Our solution takes into account both strong fluctuations at small momentum transfers $\sim \lambda k_F$ and weaker fluctuations at large momentum transfers. The strong fluctuations determine $T_c$, which is of order $\lambda^2 E_F$ for both s- and d- wave pairing. The weaker fluctuations determine the angular structure of the superconducting order parameter $F(\theta_k)$ along the Fermi surface, separating between hot and lukewarm regions. In the hot regions $F(\theta_k)$ is largest and approximately constant. Beyond the hot region, whose width is $\theta_h\sim\lambda^{1/3}$, $F(\theta_k)$ drops by a factor $\lambda^{4/3}$. The s- and d- wave states are not degenerate but the relative difference $(T_c^s-T_c^d)/T_c^s\sim\lambda^2$ is small.