Dirac edges of fractal magnetic minibands in graphene with hexagonal moiré superlattices

We find a systematic reappearance of massive Dirac features at the edges of consecutive minibands formed at magnetic fields B p/q =pq ϕ 0 /S providing rational magnetic flux through a unit cell of the moiré superlattice created by a hexagonal substrate for electrons in graphene. The Dirac-type features in the minibands at B=B p/q determine a hierarchy of gaps in the surrounding fractal spectrum and show that these minibands have topological insulator properties. Using the additional q -fold degeneracy of magnetic minibands at B p/q , we trace the hierarchy of the gaps to their manifestation in the form of incompressible states upon variation of the carrier density and magnetic field.