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

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 $V(\Omega_m) \propto 1/|\Omega_m|^\gamma$(the$\gamma$-model). We analyze both the original model and its extension, in which we introduce an extra parameter $N$ to account for non-equal interactions in the particle-hole and particle-particle channel. In two previous papers(arXiv:2004.13220 and arXiv:2006.02968), we considered the case $0<\gamma <1$ and argued that (i) at $T=0$, there exists an infinite discrete set of topologically different gap functions, $\Delta_n (\omega_m)$,allwiththesamespatialsymmetry,and(ii)each$\Delta_n$evolveswithtemperatureandterminatesataparticular$T_{p,n}$. In this paper, we analyze how the system behavior changes between $\gamma <1$ and $\gamma >1$, both at $T=0$ and a finite $T$. The limit $\gamma \to 1$ is singular due to infra-red divergence of $\int d{\omega }_{m}V({\mathrm{\Omega }}_m)$, 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 $\Delta_n (\omega_m)$graduallyevolvethrough$\gamma=1$bothat$T=0$andafinite$T$. For $N\mathrm{\ne }1$, divergent terms do not cancel, and a qualitatively new behavior emerges for $\gamma >1$. Namely, the form of ${\mathrm{\Delta }}_n({\omega }_m)$ 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 $\gamma >1$.