Classification of non-Fermi liquids and universal superconducting fluctuations

In quantum critical metals, a plethora of different non-Fermi liquids arises depending on the nature of critical fluctuations coupled to Fermi surfaces. In this paper, we classify non-Fermi liquids that arise from q=0 critical fluctuations and characterize their universal superconducting fluctuations. The essential tool is the projective fixed points, which generalizes the notion of fixed points to fixed trajectories that take into account the incessant running of the Fermi momentum under the renormalization group flow. Based on the topology of bundles of projective fixed points, non-Fermi liquids are first grouped into seven superuniversality classes. Each superuniversality class includes multiple universality classes, which are further classified by the universal pairing interactions and emergent symmetries. Despite the pairing interaction generated by critical fluctuations, some non-Fermi liquids remain stable down to zero temperature due to the incoherence of excitations and the lack of scale invariance caused by Fermi momentum. Depending on the strength and span of the universal pairing interaction in momentum space, the emergent symmetry of non-Fermi liquids may or may not be lower than that of Fermi liquids. In non-Fermi liquids that become superconductors at low temperatures, the universal data of the parent metal determine the lower bound for the superconducting transition temperature and the associated pairing symmetry. In superuniversality classes that contain non-Fermi liquids prone to non-s-wave superconducting instabilities, the critical angular momentum above which pairing instability becomes inevitable is sensitive to the Fermi momentum, and the associated superconducting transition temperature oscillates as a function of the density. We use physical examples, as well as a toy model, to elucidate the universal low-energy physics of all superuniversality classes.