Hierarchy of fractional Chern insulators and competing compressible states

We study the phase diagram of interacting electrons in a dispersionless Chern band as a function of their filling. We find hierarchy multiplets of incompressible states at fillings $\nu$ = 1/3, 2/5, 3/7, 4/9, 5/9, 4/7, 3/5 as well as $\nu$ = 1/5, 2/7. These are accounted for by an analogy to Haldane pseudopotentials extracted from an analysis of the two-particle problem. Important distinctions to standard fractional quantum Hall physics are striking: absent particle-hole symmetry in a single band, an interaction-induced single-hole dispersion appears, which perturbs and eventually destabilizes incompressible states as $\nu$ increases. For this reason the nature of the state at $\nu$ = 2/3 is hard to pin down, while $\nu$ = 5/7, 4/5 do not seem to be incompressible in our system.