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