Are non-Fermi-liquids stable to Cooper pairing?

Abstract

States of matter with a sharp Fermi-surface but no well-defined Landau quasiparticles arise in a number of physical systems. Examples include: $(i)$ quantum critical points associated with the onset of order in metals; $(ii)$ spinon Fermi-surface (U(1) spin-liquid) state of a Mott insulator; $(iii)$ Halperin-Lee-Read composite fermion charge liquid state of a half-filled Landau level. In this work, we use renormalization group techniques to investigate possible instabilities of such non-Fermi-liquids in two spatial dimensions to Cooper pairing. We consider the Ising-nematic quantum critical point as an example of a phase transition in a metal, and demonstrate that the attractive interaction mediated by the order parameter fluctuations always leads to a superconducting instability. Moreover, in the regime where our calculation is controlled, superconductivity preempts the destruction of electronic quasiparticles. On the other hand, the spinon Fermi-surface and the Halperin-Lee-Read states are stable against Cooper pairing for a sufficiently weak attractive short-range interaction; however, once the strength of attraction exceeds a critical value, pairing sets in. We describe the ensuing quantum phase transition between $(i)$ $U(1)$ and $Z_2$ spin-liquid states; $(ii)$ Halperin-Lee-Read and Moore-Read states.

v2. Published version. Added a section on pairing near the Mott transition, expanded the introduction and discussion, updated the bibliography. 20 pages, 7 figures

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