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

TL;DR

This paper explores the phase diagram and behavior of a quantum-critical model with a tunable interaction parameter gamma, revealing how the system transitions across gamma=1 and the effects of an additional parameter N on the superconducting gap functions.

Contribution

It extends previous gamma-model analyses by examining the transition at gamma=1 and introduces a regularized model for gamma > 1, highlighting new behaviors for N ≠ 1.

Findings

For N=1, divergencies cancel and gap functions evolve smoothly through gamma=1.

For N≠1, divergencies do not cancel, leading to qualitative changes in gap functions and a continuous spectrum of condensation energies.

The model extension removes singularities for gamma > 1, enabling analysis of the system's behavior in this regime.

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 $V(\Omega_m) \propto 1/|\Omega_m|^\gamma$ (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 $\gamma \to 1$ 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 $N \neq 1$, 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 $\gamma >1$.