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.
Valley Hall Viscosity in Gapped Graphene with and without a Magnetic Field
Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences（中国科学院大学 Kavli理论科学研究所）；Center for Spintronics Research Network, Keio University（庆应义塾大学自旋电子学研究网络中心）；Department of Physics, Nagoya University（名古屋大学物理学系）；Department of Physics, Kyoto University（京都大学物理学系）；Institute of Solid State Physics, University of Tokyo（东京大学固体物理研究所）；CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences（中国科学院大学 中科院拓扑量子计算卓越创新中心）；RIKEN Center for Emergent Matter Science (CEMS)（理化学研究所 新兴物质科学中心）；Advanced Science Research Center, Japan Atomic Energy Agency（日本原子力研究机构 先进科学研究中心）
Gaussian-Bond Hall Viscosity and Nonlocal Hall Response of a Sampled Landau Level
Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences（中国科学院大学理论物理研究所）；Institute of Solid State Physics, University of Tokyo（东京大学固体物理研究所）；Department of Physics, Nagoya University（名古屋大学物理学系）；Center for Spintronics Research Network, Keio University（庆应义塾大学自旋电子学研究网络中心）；CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences（中国科学院大学拓扑量子计算卓越创新中心）；RIKEN Center for Emergent Matter Science (CEMS)（理化学研究所新兴物质科学中心）；Advanced Science Research Center, Japan Atomic Energy Agency（日本原子能研究开发机构先进科学研究中心）
Origin of nonlocal resistance in multiterminal graphene on hexagonal-boron-nitride: Fermi surface edge currents rather than Fermi sea topological valley currents
Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；ICTP South American Institute for Fundamental Research（国际理论物理中心南美基础研究所）
