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 $\nu =k/(C+1)$ ($k,C$ integers) in several lattice models, in fractionally filled bands of Chern numbers $C\ge 1$ 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$\left(C\right)$ (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$\left(C\right)$ 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.