Superconductivity at the onset of spin-density-wave order in a metal

TL;DR

This paper investigates superconductivity near a quantum-critical point in a 2D metal, revealing that superconductivity can occur even at weak coupling due to diverging subleading logarithmic effects, with T_c influenced by both Fermi-liquid and non-Fermi-liquid fermions.

Contribution

It demonstrates that superconductivity at the QCP arises without a threshold, driven by subleading logarithmic divergences, and provides a detailed analysis of the full gap equation including both Fermi-liquid and non-Fermi-liquid effects.

Findings

Superconductivity can emerge at small coupling due to diverging subleading logs.

Both Fermi-liquid and non-Fermi-liquid fermions contribute to pairing.

Computed T_c aligns well with numerical results.

Abstract

We revisit the issue of superconductivity at the quantum-critical point (QCP) between a 2D paramagnet and a spin-density-wave metal with ordering momentum (\pi,\pi). This problem is highly non-trivial because the system at criticality displays a non-Fermi liquid behavior and because the effective coupling constant \lambda for the pairing is generally of order one, even when the actual interaction is smaller than fermionic bandwidth. Previous study [M. A. Metlitski, S. Sachdev, Phys.Rev.B 82, 075128 (2010)] has found that the renormalizations of the pairing vertex are stronger than in BCS theory and hold in powers of \log^2 (1/T), like in color superconductivity. We analyze the full gap equation and argue that, for QCP problem, summing up of the leading logarithms does not lead to a pairing instability. Yet, we show that superconductivity has no threshold and appears even if the coupling is set to be small, because subleading logarithmical renormalizations diverge and give rise to BCS-like \log(1/T_c) \propto 1/\lambda. We argue that the analogy with BCS is not accidental as at small coupling superconductivity at a QCP predominantly comes from fermions which retain Fermi liquid behavior at criticality. We computed T_c for the actual \lambda \sim 1, and found that both Fermi-liquid and non-Fermi liquid fermions contribute to the pairing. The value of T_c agrees well with the numerical results.