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

TL;DR

This paper calculates the bulk and shear viscosities of a 2D electron liquid in doped graphene, providing microscopic insights and interpolation formulas to connect high-frequency perturbative results with low-frequency hydrodynamic behavior.

Contribution

It offers the first detailed microscopic calculation of viscosities in a 2D Dirac fermion liquid, including a novel interpolation approach using the viscosity transport time.

Findings

Viscosities are computed up to second order in electron-electron interactions.

The viscosity transport time scales as 1/T^2 at low temperatures.

Interpolation formulas connect high-frequency perturbative results to low-frequency hydrodynamics.

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" $\tau_{\rm v}$, which we calculate in this Article. The transverse nature of the excitations contributing to $\tau_{\rm v}$ leads to the suppression of scattering events with small momentum transfer, which are inherently longitudinal. Therefore, contrary to the quasiparticle lifetime, which goes as $-1/[T^2 \ln(T/T_{\rm F})]$, in the low temperature limit we find $\tau_{\rm v} \sim 1/T^2$.