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 $V\left({\mathrm{\Omega }}_m\right)\propto 1/{\left|{\mathrm{\Omega }}_m\right|}^{\gamma }$ (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 exists a discrete, infinite set of topologically distinct solutions for the superconducting gap, all with the same spatial symmetry. The gap function ${\mathrm{\Delta }}_n\left({\omega }_m\right)$ for the $n\mathrm{th}$ 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 exists an infinite set of pairing instability temperatures, $T_{p,n}$, and the eigenfunction ${\mathrm{\Delta }}_n\left({\omega }_m\right)$ changes sign $n$ times as a function of a Matsubara number $m$. We argue that ${\mathrm{\Delta }}_n\left({\omega }_m\right)$ retains its functional form below $T_{p,n}$ and at $T=0$ coincides with the $n\mathrm{th}$ 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 $N_{\text{cr}}=O\left(1\right)$. The gap function vanishes at $T\to 0$ for $N>N_{\text{cr}}$ and remains finite for $N<N_{\text{cr}}$. This is consistent with the $T=0$ analysis.

• Received 4 June 2020
• Revised 9 July 2020
• Accepted 13 July 2020

DOI:https://doi.org/10.1103/PhysRevB.102.024525