Fluctuation Spectrum of 2+1D Critical Fermi Surface and its Application to Optical Conductivity and Hydrodynamics

We extend the kinetic operator formalism developed in the companion paper [H.Guo,arXiv:2311.03455] to study the general eigenvalues of the fluctuation normal modes. We apply the formalism to calculate the optical conductivity of a critical Fermi surface near the Ising-Nematic quantum critical point. We find that the conductivity is the sum of multiple conduction channels including both the soft and non-soft eigenvectors of the kinetic operator, and therefore it is not appropriate to interpret the optical conductivity using extended Drude formula for momentum conserved systems. We also show that the propagation of the FS soft modes is governed by a Boltzmann equation from which hydrodynamics emerges. We calculate the viscosity and it shows clear signature of the non-Fermi liquid physics.