Dissipation Spectrum and Angular Dynamics of 2+1D Non-Fermi Liquids

The linear-response coefficients of a non-Fermi liquid are encoded in the Bethe-Salpeter kernel. In this work, we study the eigenvalues of the kernel using the Migdal-Eliashberg formalism or the Yukawa-SYK model in the context of a Fermi surface coupled to the Ising-nematic quantum critical point. We find that the generic eigenvalues scale in the same way as the self-energy. For a circular Fermi surface, there are two soft-modes related to the conserved density and momentum of the fermions. The density mode shows a regular diffusion on the Fermi surface, and the momentum mode shows an anomalous angular diffusion dynamics whose diffusion operator contains sixth order derivative. The implications of our formalism include an incoherent quantum critical conductivity and a non-Fermi liquid analog of the tomographic transport regime.