Pairing instabilities of a Non-Fermi liquid in the presence of nematic and gauge fluctuations

In the absence of Fermi-liquid starting point, instabilities of non-Fermi liquids are theoretically challenging problems. Here we note that a non-Fermi liquid state occurring at $\nu=1/2$ may be a promising concrete case for theoretical investigation of the issue for two reasons. Firstly, exotic ordered states observed in half-filled Landau levels, namely the FQH state at $\nu=5/2$ which is most likely best described as a paired state, and the quantum Hall nematic state at $\nu=9/2$, present a compelling possibility that the non-Fermi liquid state with gauge fluctuations at $\nu=1/2$ is close to instabilities towards these ordered states. Secondly, a recent theoretical progress [Metlitski et al., arXiv:1403.3694] offers a scheme for a controlled renormalization group study of the problem. We will discuss competition between the two fluctuations in promoting or suppressing a superconducting instability, based on the phase diagram we obtain from a renormalization group calculation.