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 a 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/τJ ∝ ω 2 + 4π 2 T 2 3ω 2 + 8π 2 T 2 for max{ω, T } µ, where µ is the chemical potential, with an additional logarithmic factor in two dimensions. In graphene, the quartic form of 1/τJ competes with a small FL-like term, ∝ ω 2 +4π 2 T 2 , due to trigonal warping of the Fermi surface. We also calculated the dynamical charge susceptibility, χc(q, ω), outside the particle-hole continua and to one-loop order in the dynamically screened Coulomb interaction. For a 2D DFL, the imaginary part of χc(q, ω) scales as q 2 ω ln |ω| and q 4 /ω 3 for frequencies larger and smaller than the plasmon frequency at given q, respectively. The small-q limit of Imχc(q, ω) reproduces our result for the conductivity via the Einstein relation.