Interplay between superconductivity and non-Fermi liquid at a quantum critical point in a metal. II. The $\gamma$-model at a finite $T$ for $0<\gamma <1$

Abstract

In this paper we continue the analysis of the interplay between non-Fermi liquid and superconductivity for quantum-critical systems, the low-energy physics of which is described by an effective model with dynamical electron-electron interaction $V({\Omega }_m)\propto 1/|{\Omega }_m{|}^{\gamma }$ (the $\gamma$ model). In paper I [A. Abanov and A. V. Chubukov, Phys Rev B. 102, 024524 (2020)] two of us analyzed the $\gamma$ model at $T=0$ for $0<\gamma <1$ and argued that there exist a discrete, infinite set of topologically distinct solutions for the superconducting gap, all with the same spatial symmetry. The gap function ${\Delta }_n({\omega }_m)$ for the $n$th solution changes sign $n$ times as the function of Matsubara frequency. In this paper we analyze the linearized gap equation at a finite $T$. We show that there exist an infinite set of pairing instability temperatures, $T_{p,n}$, and the eigenfunction ${\Delta }_n({\omega }_m)$ changes sign $n$ times as a function of a Matsubara number $m$. We argue that ${\Delta }_n({\omega }_m)$ retains its functional form below $T_{p,n}$ and at $T=0$ coincides with the $n$th solution of the nonlinear gap equation. Like in paper I, we extend the model to the case when the interaction in the pairing channel has an additional factor $1/N$ compared to that in the particle-hole channel. We show that $T_{p,0}$ remains finite at large $N$ due to special properties of fermions with Matsubara frequencies $\pm \pi T$, but all other $T_{p,n}$ terminate at $N_{cr}=O(1)$. The gap function vanishes at $T\to 0$ for $N>N_{cr}$ and remains finite for $N<N_{cr}$. This is consistent with the $T=0$ analysis.

arXiv comment35 pp, 14 figs. Replaced with the published version. The second paper in the series. The first one is arXiv:2004.13220 (Phys Rev B. 102, 024524 (2020))

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