Brown R., Walet N.R., Guinea F. and 'We have analyzed the nature of edge states in superlattices of gapped graphene on BN. We have shown that nontrivial patterns of Berry curvature are induced in the superlattice Brillouin zone, giving rise to Chern numbers that are typically nonzero, and change from subband to subband',' topological edge modes are thus generic for a Hamiltonian describing modulated fields in monolayer graphene, as is suitable for graphene on BN. The precise value of these numbers depends on details of the superlattice potential, although they are generally present provided that physically reasonable superlattice parameters are used. The existence of finite Chern numbers in the superlattice bands leads to a valley Hall effect. These results are confirmed by real space calculations for superlattice ribbons. We find dispersive bands and crossings near the corners of the Brillouin zone. Currents along the superlattice edges are degraded by short-range intervalley scattering, whereas in clean graphene samples electronic transport is limited by long range, intravalley scattering. The effect of disorder localized at the edges is suppressed by the long decay length of the states, due to the small size of the gaps. Simple estimates of the mean free path and localization length associated with edge disorder give values in the order of microns. This provides an explanation for the low resistivities found in electronic transport measurements of graphene on BN [35] (see also Ref. [17] ). We have demonstrated that the superlattice is of importance to the transport properties of graphene on a substrate such as BN, or as a means to measure the valley Hall effect. We would like to thank M. Ben Shalom, V. Fal’ko, A. Geim, and J. Walbank for useful conversations. 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