Interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal. VI. The $gamma$ model and its phase diagram at $2 < gamma <3$
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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, $Delta_n (omega_m)$, all with the same spatial symmetry, and (ii) each $Delta_n$ evolves with temperature and terminates at a particular $T_{p,n}$. 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 $gamma o 1$ is singular due to infra-red divergence of $int d omega_m V(Omega_m)$, 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 $Delta_n (omega_m)$ gradually evolve through $gamma=1$ both at $T=0$ and a finite $T$. For $N eq 1$, divergent terms do not cancel, and a qualitatively new behavior emerges for $gamma >1$. Namely, the form of $Delta_n (omega_m)$ 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 $gamma >1$.
