Fractional topological phases in generalized Hofstadter bands with arbitrary Chern numbers

We construct generalized Hofstadter models that possess color-entangled flat bands and study interacting many-body states in such bands. For a system with periodic boundary conditions and appropriate interactions, there exist gapped states at certain filling factors for which the ground-state degeneracy depends on the number of unit cells along one particular direction. This puzzling observation can be understood intuitively by mapping our model to a single-layer or a multilayer system for a given lattice configuration. We discuss the relation between these results and the previously proposed topological nematic states, in which lattice dislocations have non-Abelian braiding statistics. Our study also provides a systematic way of stabilizing various fractional topological states in C > 1 flat bands and provides some hints on how to realize such states in experiments.