  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.
