Bulk and shear viscosities of the 2D electron liquid in a doped graphene sheet

Abstract

Hydrodynamic flow occurs in an electron liquid when the mean free path for electron-electron collisions is the shortest length scale in the problem. In this regime, transport is described by the Navier-Stokes equation, which contains two fundamental parameters, the bulk and shear viscosities. In this Article we present extensive results for these transport coefficients in the case of the two-dimensional massless Dirac fermion liquid in a doped graphene sheet. Our approach relies on microscopic calculations of the viscosities up to second order in the strength of electron-electron interactions and in the high-frequency limit, where perturbation theory is applicable. We then use simple interpolation formulae that allow to reach the low-frequency hydrodynamic regime where perturbation theory is no longer directly applicable. The key ingredient for the interpolation formulae is the "viscosity transport time" $\ddot{I}{„}_{\mathrm{v}}$, which we calculate in this Article. The transverse nature of the excitations contributing to $\ddot{I}{„}_{\mathrm{v}}$ leads to the suppression of scattering events with small momentum transfer, which are inherently longitudinal. Therefore, contrary to the quasiparticle lifetime, which goes as $-1/[T^2\ln (T/T_{\mathrm{F}})]$, in the low temperature limit we find $\ddot{I}{„}_{\mathrm{v}}\sim 1/T^2$.

27 pages, 6 figures, 5 appendices