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Yi-Ming Wu, Ar. Abanov, Yuxuan Wang, Andrey V. Chubukov
Abstract
In this paper we continue the analysis of the interplay between non-Fermi liquid and superconductivity for quantum-critical systems, the low-energy physics of which is described by an effective model with dynamical electron-electron interaction $V({\mathrm{\ensuremath{\Omega}}}_{m})\ensuremath{\propto}1/{|{\mathrm{\ensuremath{\Omega}}}_{m}|}^{\ensuremath{\gamma}}$ (the $\ensuremath{\gamma}$ model). In paper I [A. Abanov and A. V. Chubukov, Phys. Rev. B 102, 024524 (2020)], two of us analyzed the $\ensuremath{\gamma}$ model at $T=0$ for $0<\ensuremath{\gamma}<1$ and argued that there exists a discrete, infinite set of topologically distinct solutions for the superconducting gap, all with the same spatial symmetry. The gap function ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ for the $n\mathrm{th}$ solution changes sign $n$ times as the function of Matsubara frequency. In this paper we analyze the linearized gap equation at a finite $T$. We show that there exists an infinite set of pairing instability temperatures, ${T}_{p,n}$, and the eigenfunction ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ changes sign $n$ times as a function of a Matsubara number $m$. We argue that ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ retains its functional form below ${T}_{p,n}$ and at $T=0$ coincides with the $n\mathrm{th}$ solution of the nonlinear gap equation. Like in paper I, we extend the model to the case when the interaction in the pairing channel has an additional factor $1/N$ compared to that in the particle-hole channel. We show that ${T}_{p,0}$ remains finite at large $N$ due to special properties of fermions with Matsubara frequencies $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}T$, but all other ${T}_{p,n}$ terminate at ${N}_{\text{cr}}=O(1)$. The gap function vanishes at $T\ensuremath{\rightarrow}0$ for $N>{N}_{\text{cr}}$ and remains finite for $N<{N}_{\text{cr}}$. This is consistent with the $T=0$ analysis.