Series of Abelian and non-Abelian states in C > 1 fractional Chern insulators

We report the observation of a series of Abelian and non-Abelian topological states in fractional Chern insulators (FCIs). The states appear at bosonic filling v = k/(C + 1) (k, C integers) in several lattice models, in fractionally filled bands of Chern numbers C >= 1 subject to on-site Hubbard interactions. We show strong evidence that the k = 1 series is Abelian while the k > 1 series is non-Abelian. The energy spectrum at both ground-state filling and upon the addition of quasiholes shows a low-lying manifold of states whose total degeneracy and counting matches, at the appropriate size, that of the fractional quantum Hall (FQH) SU(C) (color) singlet k-clustered states (including Halperin, non-Abelian spin singlet states and their generalizations). The ground-state momenta are correctly predicted by the FQH to FCI lattice folding. However, the counting of FCI states also matches that of a spinless FQH series, preventing a clear identification just from the energy spectrum. The entanglement spectrum lends support to the identification of our states as SU(C) color singlets, but offers anomalies in the counting for C > 1, possibly related to dislocations that call for the development of alternative counting rules of these topological states.