Combined Topological and Landau Order from Strong Correlations in Chern Bands

We present a class of states with both topological and conventional Landau order that arise out of strongly interacting spinless fermions in fractionally filled and topologically nontrivial bands with Chern number C = +/- 1. These quantum states show the features of fractional Chern insulators, such as fractional Hall conductivity and interchange of ground-state levels upon insertion of a magnetic flux. In addition, they exhibit charge order and a related additional trivial ground-state degeneracy. Band mixing and geometric frustration of the charge pattern place these lattice states markedly beyond a single-band description.