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

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 $V(Ω_m) \propto 1/|Ω_m|^γ$ (the $γ$ model). In paper I [A. Abanov and A. V. Chubukov, Phys Rev B. 102, 024524 (2020)] two of us analyzed the $γ$ model at $T=0$ for $0<γ<1$ 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 $Δ_n (ω_m)$ 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 $Δ_n (ω_{m})$ changes sign $n$ times as a function of a Matsubara number $m$. We argue that $Δ_n (ω_{m})$ 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 $\pm πT$, 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.