Interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal. I: The $\gamma$-model and its phase diagram at $T=0$. The case $0<\gamma <1$

Abstract

We analyze a class of quantum-critical models, in which momentum integration and the selection of a particular pairing symmetry can be done explicitly, and the competition between non-Fermi liquid and pairing can be analyzed within an effective model with dynamical electron-electron interaction $V({\Omega }_m)\sim 1/|{\Omega }_m{|}^{\gamma }$ (the $\gamma$-model). In this paper, the first in the series, we consider the case $T=0$ and $0<\gamma <1$. We argue that tendency to pairing is stronger, and the ground state is a superconductor. We argue, however, that superconducting state is highly non-trivial as there exists a discrete set of topologically distinct solutions for the pairing gap ${\Delta }_n({\omega }_m)$ ($n=0,1,\mathrm{2...},\infty$). All solutions have the same spatial pairing symmetry, but differ in the time domain: ${\Delta }_n({\omega }_m)$ changes sign $n$ times as a function of Matsubara frequency ${\omega }_m$. The $n=0$ solution ${\Delta }_0({\omega }_m)$ is sign-preserving and tends to a finite value at ${\omega }_m=0$, like in BCS theory. The $n=\infty$ solution corresponds to an infinitesimally small $\Delta ({\omega }_m)$. As a proof, we obtain the exact solution of the linearized gap equation at $T=0$ on the entire frequency axis for all $0<\gamma <1$, and an approximate solution of the non-linear gap equation.We argue that the presence of an infinite set of solutions opens up a new channel of gap fluctuations. We extend the analysis to the case where the pairing component of the interaction has additional factor $1/N$ and show that there exists a critical $N_{cr}>1$, above which superconductivity disappears, and the ground state becomes a non-Fermi liquid.We show that all solutions develop simultaneously once $N$ gets smaller than $N_{cr}$.

Replaced with the published version

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• Upper bound on $T_c$ in a strongly coupled electron-boson superconductor
• Dynamical vortices in electron-phonon superconductors
• Phonon Induced Energy Relaxation in Quantum Critical Metals