We propose a systematical approach to construct generic fractional quantum anomalous Hall states, which are generalizations of the fractional quantum Hall states to lattice models with zero net magnetic field and full lattice translation symmetry. Local and translationally invariant Hamiltonians can also be constructed, for which the proposed states are unique ground states. Our result demonstrates that generic chiral topologically ordered states can be realized in lattice models, without requiring magnetic translation symmetry and Landau level structure. We further generalize our approach to fractional topological insulators, and provide the first explicit wave-function description of fractional topological insulators in the absence of spin conservation.

• Received 2 June 2011

DOI:https://doi.org/10.1103/PhysRevLett.107.126803