Superconductivity versus 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 Tc as a function of N . Our results show an anomalous scaling between the zero-temperature gap and Tc. For N greater than a critical value, we find that Tc 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.

Non-vanishing zero-temperature normal density in holographic superfluids

“…It is worth contrasting our results with those obtained in a qualitatively different large N limit, where the effects of a gapless boson (a proxy for critical order parameter fluctuations) on a Fermi surface are investigated, [65][66][67][68]. There, the fermions transform in the fundamental representation of some internal flavor symmetry group SU(N ) while the bosons transform in the adjoint.…”

“…By contrast, in the scale-covariant geometries studied in [19] and in appendix C, there is a nonzero T c with a finite condensate. This condensate only sources an irrelevant deformation of the IR geometry, which retains the same scaling properties as the underlying normal phase -and hence is not a naked metallic quantum critical point in the sense of [65][66][67][68]. As we have described in some detail, the scaling of a number of thermodynamic or transport observables in the superfluid phase remain controlled by the normal phase.…”

How non-Fermi liquids cure their infrared divergences

“…First, it will be interesting to analyze how the thermal scale ξ T ∼ (g 2 T ) −1/2 modifies thermodynamic and transport properties. Furthermore, while we have argued that superconductivity becomes irrelevant above N c = 8, we plan to present a detailed analysis at finite temperature in future work, finding the solutions to the corrected Eliashberg equations and revisiting the role of the first Matsubara frequencies [15,23,24]. Another direction recently explored in [16,17] is the relevance of thermal effects to numerical quantum Monte Carlo results [28][29][30][31], something that deserves further study both at finite but small temperature and including the effects of pairing interactions.…”

“…1 In contrast, much less is known about infrared problems at finite density. Work in this area, both in the condensed matter and high energy fields, includes [6][7][8][9][10][11][12][13][14][15][16][17].…”

Solvable Strong-Coupling Quantum-Dot Model with a Non-Fermi-Liquid Pairing Transition

“…The pairing problem in various large N models for a Fermi surface (FS) coupled to critical bosons have been intensively studied. Interestingly, in a class of these models [2,15,35], as a function of N , the system at T = 0 can either be in a pairing phase, or remain at the normal state, separated by a quantumcritical point. The latter situation is particularly striking -contrary to BCS theory where even an infinitesimal attractive interaction drives a Fermi liquid to a superconducting state, the incoherence of the nFL state destroys superconductivity, even if the attractive pairing interaction is strong.…”