Topological Valley Currents in Gapped Dirac Materials
  Gapped 2D Dirac materials, in which inversion symmetry is broken by a gap-opening perturbation, feature a unique valley transport regime. The system ground state hosts dissipationless persistent valley currents existing even when topologically protected edge modes are absent or when they are localized due to edge roughness. Topological valley currents in such materials are dominated by bulk currents produced by electronic states just beneath the gap rather than by edge modes. Dissipationless currents induced by an external bias are characterized by a quantized half-integer valley Hall conductivity. The under-gap currents dominate magnetization and the charge Hall effect in a light-induced valley-polarized state. 
Yuri D. Lensky, Justin C. W. Song, Polnop Samutpraphoot, Leonid S.   Levitov
cond-mat.mes-hall