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(ωm) 1/|ωm|γ (the γ model). In paper I [A. Abanov and A. V. Chubukov, Phys. Rev. B 102, 024524 (2020)2469-995010.1103/PhysRevB.102.024524], two of us analyzed the γ model at T=0 for 0<γ<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 Δn(ωm) for the nth 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, Tp,n, and the eigenfunction Δn(ωm) changes sign n times as a function of a Matsubara number m. We argue that Δn(ωm) retains its functional form below Tp,n and at T=0 coincides with the nth 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 Tp,0 remains finite at large N due to special properties of fermions with Matsubara frequencies ±πT, but all other Tp,n terminate at Ncr=O(1). The gap function vanishes at T→0 for N>Ncr and remains finite for N<Ncr. This is consistent with the T=0 analysis.
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We thank I. Aleiner, B. Altshuler, E. Berg, D. Chowdhury, L. Classen, R. Combescot, K. Efetov, R. Fernandes, A. Finkelstein, E. Fradkin, A. Georges, S. Hartnol, S. Karchu, S. Kivelson, I. Klebanov, A. Klein, R. Laughlin, S.-S. Lee, G. Lonzarich, D. Maslov, F. Marsiglio, M. Metlitski, W. Metzner, A. Millis, D. Mozyrsky, C. Pepin, V. Pokrovsky, N. Prokofiev, S. Raghu, S. Sachdev, T. Senthil, D. Scalapino, Y. Schattner, J. Schmalian, D. Son, G. Tarnopolsky, A.-M. Tremblay, A. Tsvelik, G. Torroba, E. Yuzbashyan, and J. Zaanen for useful discussions. The work by A.V.C. was supported by NSF-DMR Grant No. 1834856.