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

Abstract

We find a systematic reappearance of massive Dirac features at the edges of consecutive minibands formed at magnetic fields ${B}_{p/q}=\frac{p}{q}{\ensuremath{\phi}}_{0}/S$ providing rational magnetic flux through a unit cell of the moir\'e 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.