| Abstract: | Topological materials may exhibit Hall-like currents flowing transversely to the applied electric field even in the absence of a magnetic field. In graphene superlattices, which have broken inversion symmetry, topological currents originating from graphene's two valleys are predicted to flow in opposite directions and combine to produce long-range charge neutral flow. This effect is observed as a nonlocal voltage at zero magnetic field in a narrow energy range near Dirac points at distances as large as several micrometers away from the nominal current path. Locally, topological currents are comparable in strength with the applied current, indicating large valley-Hall angles. We will discuss topological current mechanisms, and argue that such currents are not hindered by the absence of topologically protected edge modes. In gapped graphene at charge neutrality, the edge states are fragile since they are not enforced by topology or symmetry, and are not protected against backscattering due to roughness on the atomic scale. Naively, this would lead one to conclude that topological currents cease to exist. We will argue that the opposite is true: the absence of conducting edge modes does not present an obstacle since valley currents can be transmitted by the bulk states in the filled Fermi sea beneath the gap. This leads to an interesting behavior: rather than being vanishingly small, the valley currents reach maximum value in the gapped state. We will conclude with discussing requirements for dissipationless valley transport and argue that they can be met under realistic conditions. |