Fractional Topological Phases in Generalized Hofstadter Bands with Arbitrary Chern Numbers

We examine similarities and differences between topological flat bands with Chern numbers $C > 1$ and conventional quantum Hall multi-layers. By constructing generalized Hofstadter models that possess ``color-entangled'' flat bands, we provide an intuitive understanding of certain puzzling properties of $C > 1$ flat bands, which can effectively be mapped either to a single-layer or to a multi-layer model depending on the lattice configuration. We identify interacting systems in which the ground state degeneracy depends on whether the system consists of an even or odd number of unit cells along one particular direction, and discuss the relation between these observations and the previously proposed ``topological nematic states.'' Our study also provides a systematic way of stabilizing various fractional topological states in $C > 1$ flat bands.