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

Abstract

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 $V(Ω_m) \propto 1/|Ω_m|^γ$ (the $γ$-model). We analyze both the original model and its extension, in which we introduce an extra parameter $N$ to account for 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 $0 < γ<1$ 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 $γ\to 1$ is singular due to infra-red divergence of $\int d ω_m V(Ω_m)$, 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 $N \neq 1$, 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 $γ>1$.