Optical conductivity of a Dirac-Fermi liquid

A Dirac-Fermi liquid (DFL)--a doped system with Dirac spectrum--is an important example of a non-Galilean-invariant Fermi liquid (FL). Real-life realizations of a DFL include, e.g., doped graphene, surface states of three-dimensional (3D) topological insulators, and 3D Dirac/Weyl metals. We study the optical conductivity of a DFL arising from intraband electron-electron scattering. It is shown that the effective current relaxation rate behaves as $1/\tau_{J}\propto \left(\omega^2+4\pi^2 T^2\right)\left(3\omega^2+8\pi^2 T^2\right)$ for $\max\{\omega, T\}\ll \mu$, where $\mu$ is the chemical potential, with an additional logarithmic factor in two dimensions. In graphene, the quartic form of $1/\tau_{J}$ competes with a small FL-like term, $\propto\omega^2+4\pi^2 T^2$, due to trigonal warping of the Fermi surface. We also calculated the dynamical charge susceptibility, $\chi_\mathrm{c}({\bf q},\omega)$, outside the particle-hole continua and to one-loop order in the dynamically screened Coulomb interaction. For a 2D DFL, the imaginary part of $\chi_\mathrm{c}({\bf q},\omega)$ scales as $q^2\omega\ln|\omega|$ and $q^4/\omega^3$ for frequencies larger and smaller than the plasmon frequency at given $q$, respectively. The small-$q$ limit of $\mathrm{Im} \chi_\mathrm{c}({\bf q},\omega)$ reproduces our result for the conductivity via the Einstein relation.