Topological Orders and Phase Transitions in Hofstadter-Chern Bands

The quantum Hall effect provides a realization of topological phases of matter in the presence of an external magnetic field. In conventional semiconductors, the achieved magnetic flux per unit cell is orders of magnitude smaller than the magnetic flux quantum h/e, giving rise to degenerate Landau levels enabling strong electronic correlations. Motivated by the realization of two-dimensional moiré superlattices with unit cell of few to hundreds of nanometers in linear size, we discuss a lattice composite fermion theory that accounts for topological states in the opposite regime where the magnetic flux per unit cell is of the order of the magnetic flux quantum, which gives rise to Hofstadter-Chern bands that can be partially filled by electrons. Through analytical and numerical methods, we uncover classes of candidate topological states beyond the Landau level regime, which are characterized by strong coupling of electronic states with the lattice. We explore this setting to establish the existence of topological phase transitions mediated by modulations of the lattice potential. Our study, therefore, identifies new topological orders and how to manipulate them in two-dimensional Hofstadter-Chern lattices.