  Chern insulators are band insulators exhibiting a nonzero Hall conductance
but preserving the lattice translational symmetry. We conclusively show that a
partially filled Chern insulator at 1/3 filling exhibits a fractional quantum
Hall effect and rule out charge-density wave states that have not been ruled
out by previous studies. By diagonalizing the Hubbard interaction in the
flat-band limit of these insulators, we show the following: The system is
incompressible and has a 3-fold degenerate ground state whose momenta can be
computed by postulating an generalized Pauli principle with no more than 1
particle in 3 consecutive orbitals. The ground state density is constant, and
equal to 1/3 in momentum space. Excitations of the system are fractional
statistics particles whose total counting matches that of quasiholes in the
Laughlin state based on the same generalized Pauli principle. The entanglement
spectrum of the state has a clear entanglement gap which seems to remain finite
in the thermodynamic limit. The levels below the gap exhibit counting identical
to that of Laughlin 1/3 quasiholes. Both the 3 ground states and excited states
exhibit spectral flow upon flux insertion. All the properties above disappear
in the trivial state of the insulator - both the many-body energy gap and the
entanglement gap close at the phase transition when the single-particle
Hamiltonian goes from topologically nontrivial to topologically trivial. These
facts clearly show that fractional many-body states are possible in topological
insulators.
