Emergent many-body translational symmetries of Abelian and non-Abelian fractionally filled topological insulators

The energy and entanglement spectrum of fractionally filled interacting topological insulators exhibit a peculiar manifold of low-energy states separated by a gap from a high-energy set of spurious states. In the current paper, we show that in the case of fractionally filled Chern insulators, the topological information of the many-body state developing in the system resides in this low-energy manifold. We identify an emergent many-body translational symmetry which allows us to separate the states in quasidegenerate center-of-mass momentum sectors. Within one center-of-mass sector, the states can be further classified as eigenstates of an emergent (in the thermodynamic limit) set of many-body relative translation operators. We analytically establish a mapping between the two-dimensional Brillouin zone for the fractional quantum Hall effect on the torus and the one for the fractional Chern insulator. We show that the counting of quasidegenerate levels below the gap for the fractional Chern insulator should arise from a folding of the states in the fractional quantum Hall system at an identical filling factor. We show how to count and separate the excitations of the Laughlin, Moore-Read, and Read-Rezayi series in the fractional quantum Hall effect into two-dimensional Brillouin zone momentum sectors and then how to map these into the momentum sectors of the fractional Chern insulator. We numerically check our results by showing the emergent symmetry at work for Laughlin, Moore-Read, and Read-Rezayi states on the checkerboard model of a Chern insulator, thereby also showing, as a proof of principle, that non-Abelian fractional Chern insulators exist.