Valley Hall viscosity in gapped graphene with and without a magnetic field

Hall viscosity is conventionally defined through the stress response to time-dependent strain, a perturbation that is difficult to implement in solid-state experiments. We formulate a related viscoelastic response to static, spatially inhomogeneous electric fields and compare it with the strain-based response. For gapped graphene in a perpendicular magnetic field, the two formulations give the same Landau-level response, whose Hall viscosity is asymmetric between the two valleys. At zero magnetic field, a valley-even quantum-metric coefficient combines with the valley-odd Hall conductivity to produce equal and opposite valley-resolved responses; global time-reversal symmetry therefore forces the net Hall viscosity to vanish. In an insulating state, exact particle--hole symmetry eliminates this zero-field response, whereas particle--hole-symmetry breaking generates a cutoff-dependent geometric contribution from the occupied Fermi sea. These results connect valley-dependent viscoelasticity with electromagnetic response in gapped Dirac materials and clarify the conditions under which a valley Hall viscosity can arise.