Hall viscosity and electromagnetic response of electrons in graphene

TL;DR

This paper derives an analytic expression for the geometric Hall viscosity of electrons in graphene under a magnetic field, showing its relation to Hall conductivity and proposing a way to measure Hall viscosity experimentally.

Contribution

It extends the connection between Hall viscosity and Hall conductivity to graphene, despite its lack of Galilean invariance, using an effective mass framework.

Findings

Hall viscosity expression derived for graphene electrons.

The Hall conductivity reduces to that of a Galilean-invariant 2DEG with an effective mass.

The effective mass formula predicts Hall conductivity with better than 1% accuracy even near the zero-th Landau level.

Abstract

We derive an analytic expression for the geometric Hall viscosity of non-interacting electrons in a single graphene layer in the presence of a perpendicular magnetic field. We show that a recently-derived formula in [C. Hoyos and D. T. Son, Phys. Rev. Lett. {\bf 108}, 066805 (2012)], which connects the coefficient of $q^2$ in the wave vector expansion of the Hall conductivity $\sigma_{xy}(q)$ of the two-dimensional electron gas (2DEG) to the Hall viscosity and the orbital diamagnetic susceptibility of that system, continues to hold for graphene -- in spite of the lack of Galilean invariance -- with a suitable definition of the effective mass. We also show that, for a sufficiently large number of occupied Landau levels in the positive energy sector, the Hall conductivity of electrons in graphene reduces to that of a Galilean-invariant 2DEG with an effective mass given by $\hbar k_F/v_F$ (cyclotron mass). Even in the most demanding case, i.e. when the chemical potential falls between the zero-th and the first Landau level, the cyclotron mass formula gives results accurate to better than 1$\%$. The connection between the Hall conductivity and the viscosity provides a possible avenue to measure the Hall viscosity in graphene.