  An interesting route to the realization of topological Chern bands in
ultracold atomic gases is through the use of optical flux lattices. These
models differ from the tight-binding real-space lattice models of Chern
insulators that are conventionally studied in solid-state contexts. Instead,
they involve the coherent coupling of internal atomic (spin) states, and can be
viewed as tight-binding models in reciprocal space. By changing the form of the
coupling and the number $N$ of internal spin states, they give rise to Chern
bands with controllable Chern number and with nearly flat energy dispersion. We
investigate in detail how interactions between bosons occupying these bands can
lead to the emergence of fractional quantum Hall states, such as the Laughlin
and Moore-Read states. In order to test the experimental realization of these
phases, we study their stability with respect to band dispersion and band
mixing. We also probe novel topological phases that emerge in these systems
when the Chern number is greater than 1.
