Optical conductivity of a Dirac-Fermi liquid

Abstract

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, ${\chi }_{\mathrm{c}}(\mathbf{q},\omega )$, outside the particle-hole continua and to one-loop order in the dynamically screened Coulomb interaction. For a 2D DFL, the imaginary part of ${\chi }_{\mathrm{c}}(\mathbf{q},\omega )$ 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 $\mathrm{Im}{\chi }_{\mathrm{c}}(\mathbf{q},\omega )$ reproduces our result for the conductivity via the Einstein relation.

28 pages, 10 figures

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