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

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.