Superconductivity vs quantum criticality: effects of thermal fluctuations

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.

14 pages,. 5 figures

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• Interplay between superconductivity and non-Fermi liquid at a quantum critical point in a metal. II. The $\gamma$-model at a finite $T$ for $0<\gamma <1$
• How non-Fermi liquids cure their infrared divergences
• The special role of the first Matsubara frequency for superconductivity near a quantum-critical point -- the non-linear gap equation below $T_c$ and spectral properties in real frequencies
• The interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal
• Interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal: V. The $\gamma$ model and its phase diagram. The case $\gamma =2$
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• Superconductivity above a quantum critical point in a metal -- gap closing vs gap filling, Fermi arcs, and pseudogap behavior
• Interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal. III: The $\gamma$ model and its phase diagram across $\gamma =1$
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• Instabilities of quantum critical metals in the limit $N_f\to 0$
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