In this paper we continue our analysis of the interplay between the pairing and the non-Fermi liquid behavior in a metal for a set of quantum-critical models with an effective dynamical electron-electron interaction V(ωm) 1/|ωm|γ (the γ model). We analyze both the original model and its extension, in which we introduce an extra parameter N to account for nonequal interactions in the particle-hole and particle-particle channel. In two previous papers [A. Abanov and A. V. Chubukov, Phys. Rev. B 102, 024524 (2020)10.1103/PhysRevB.102.024524 and Y. Wu Phys. Rev. B 102, 024525 (2020)10.1103/PhysRevB.102.024525] we considered the case 0<γ<1 and argued that (i) at T=0, there exists an infinite discrete set of topologically different gap functions Δn(ωm), all with the same spatial symmetry, and (ii) each Δn evolves with temperature and terminates at a particular Tp,n. In this paper we analyze how the system behavior changes between γ<1 and γ>1, both at T=0 and a finite T. The limit γ→1 is singular due to infrared divergence of ∫dωmV(ωm), and the system behavior is highly sensitive to how this limit is taken. We show that for N=1, the divergencies in the gap equation cancel out, and Δn(ωm) gradually evolve through γ=1 both at T=0 and a finite T. For N≠1, divergent terms do not cancel, and a qualitatively new behavior emerges for γ>1. Namely, the form of Δn(ωm) changes qualitatively, and the spectrum of condensation energies Ec,n becomes continuous at T=0. We introduce different extension of the model, which is free from singularities for γ>1.
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We thank I. Aleiner, B. Altshuler, E. Berg, R. Combescot, D. Chowdhury, L. Classen, 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, J. Zaanen, and particularly Y. Wang, for useful discussions. The work by AVC and YW was supported by the NSF DMR-1834856.