Superconductivity vs quantum criticality: effects of thermal fluctuations

TL;DR

This paper investigates how thermal fluctuations influence the competition between superconductivity and quantum criticality in a non-Fermi liquid system, revealing a critical N beyond which superconductivity vanishes with BKT scaling.

Contribution

It provides a non-Eliashberg solution for the superconducting transition near a quantum critical point, incorporating thermal infrared singularities and analyzing the large N limit.

Findings

Superconducting temperature $T_c$ depends on N with anomalous scaling.

For N above a critical value, $T_c$ vanishes with BKT scaling.

System remains non-Fermi liquid down to zero temperature for large N.

Abstract

We study the interplay between superconductivity and non-Fermi liquid behavior of a Fermi surface coupled to a massless $SU(N)$ matrix boson near the quantum critical point. The presence of thermal infrared singularities in both the fermionic self-energy and the gap equation invalidates the Eliashberg approximation, and makes the quantum-critical pairing problem qualitatively different from that at zero temperature. Taking the large $N$ limit, we solve the gap equation beyond the Eliashberg approximation, and obtain the superconducting temperature $T_c$ as a function of $N$. Our results show an anomalous scaling between the zero-temperature gap and $T_c$. For $N$ greater than a critical value, we find that $T_c$ vanishes with a Berezinskii-Kosterlitz-Thouless scaling behavior, and the system retains non-Fermi liquid behavior down to zero temperature. This confirms and extends previous renormalization-group analyses done at $T=0$, and provides a controlled example of a naked quantum critical point. We discuss the crucial role of thermal fluctuations in relating our results with earlier work where superconductivity always develops due to the special role of the first Matsubara frequency.