Interplay between superconductivity and non-Fermi liquid behavior at a quantum critical point in a metal. III. The γ model and its phase diagram across γ =1

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 (Ω_m) ∝1 /|Ω^{_m| 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 et al. 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 Δ_n(ω_m) , all with the same spatial symmetry, and (ii) each Δ_n evolves with temperature and terminates at a particular T_{p ,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 ∫d ω_mV (Ω_m) , 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 Δ_n(ω_m) 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 Δ_n(ω_m) changes qualitatively, and the spectrum of condensation energies E_{c ,n} becomes continuous at T =0 . We introduce different extension of the model, which is free from singularities for γ >1 .