Projective fixed points for non-Fermi liquids: A case study of the Ising-nematic quantum critical metal

In metals, low-energy effective theories are characterized by a set of coupling functions. Among them, the angle-dependent Fermi momentum specifies the size and shape of Fermi surface. Since the size of Fermi momentum grows incessantly relative to an energy scale that is lowered under the renormalization group (RG) flow, a metallic fixed point is defined only modulo a rescaling of Fermi momentum. In this paper, we discuss the physical consequences of this projective nature of fixed points for non-Fermi liquids with hot Fermi surfaces. The first is the absence of a unique dynamical critical exponent that dictates the relative scaling between energy and momentum across all low-energy observables. The second is mismatches between the scaling dimensions of couplings and their relevancy. Nonetheless, each projective fixed point is characterized by a few marginal and relevant coupling functions, and the notion of universality survives. We illustrate our findings by charting the space of projective fixed points and extracting their universal properties for the Ising-nematic quantum critical metal beyond the patch theory. To approach the interacting theory in two space dimensions from a controlled limit, we use the dimensional regularization scheme that tunes the codimension of Fermi surface as a control parameter. Near the upper critical dimension, two exactly marginal coupling functions span the space of stable projective fixed points: one specifies the shape of the Fermi surface and the other sets the angle-dependent Fermi velocity. All other coupling functions, including the Landau functions and the universal pairing interaction, are fixed by those two marginal functions. As the space dimension is lowered, vertex corrections alter the fate of the four-fermion coupling in a channel-dependent way. To the leading order, the forward scattering remains irrelevant while the pairing interaction becomes relevant near two dimensions. In two dimensions, it is expected that the universal superconducting fluctuations lower the symmetry of the non-Fermi liquid realized above the superconducting transition temperatures from the loop U (1) group to a proper subgroup.