Hall viscosity and nonlocal conductivity of gapped graphene

Abstract

We calculate the Hall viscosity and the nonlocal (i.e., dependent on wave vector $\bm q$) Hall conductivity of "gapped graphene" (a non-topological insulator with two valleys) in the presence of a strong perpendicular magnetic field. Using the linear-response theory at zero temperature within the Dirac approximation for the Landau levels, we present analytical expressions for both valley and total Hall viscosity and conductivity up to $\bm q^2$ at all frequencies. Although the final formulas for total Hall viscosity and conductivity are similar to the ones previously obtained for gapless graphene, the derivation reveals a significant difference between the two systems. First of all, both the Hall viscosity and the Hall conductivity vanish when the Fermi level lies in the gap that separates the lowest Landau level in the conduction band from the highest Landau level in the valence band. It is only when the Fermi level {\it is not} in the gap that the familiar formulas of gapless graphene are recovered. Second, in the case of gapped graphene, it is not possible (at least within our present approach) to define a single-valley Hall viscosity: this quantity diverges with a strength proportional to the magnitude of the gap. It is only when both valleys are included that the diverging terms, having opposite signs in the two valleys, cancel out and the familiar result is recovered. In contrast to this, the nonlocal Hall conductivity is finite in each valley. These results indicate that the Hoyos-Son formula connecting the Hall viscosity to the coefficient of $q^2$ in the small-$q$ expansion of the $q$-dependent Hall conductivity cannot be applied to each valley, but only to the system as a whole. The problem of defining a "valley Hall viscosity" remains open.