Quantum criticality in Hofstadter-Chern bands of graphene superlattices

In the conventional quantum Hall effect, the magnetic flux per unit cell is orders of magnitude smaller than the magnetic flux quantum h/e, giving rise to degenerate Landau levels characterizing topological states of electrons in the continuum. Recent progress in the fabrication of superlattices with nanometer scale unit cells has led to the experimental realization of Hofstadter-Chern insulators with large magnetic fluxes per unit cell, where the interplay between lattice effects and electron topology extends beyond the Landau level regime. In this setting, we present an analytical framework to classify the hopping-tuned topological critical points in graphene superlattices subject to a background external magnetic field, which are characterized by multi-component Dirac fermions and large changes in Chern number, as opposed to conventional quantum Hall transitions. We then describe an intimate relationship between the energy scale of such quantum phase transitions and van Hove singularities in Chern bands. Our work provides a route to critical phenomena beyond conventional quantum Hall plateau transitions, and uncover integer and fractional states characterized by strong coupling of electron topology with the lattice.