A Concrete Example of Fractional Chern Insulator

We present a concrete example of fractional Chern insulator whose fermion Hamiltonian consists of hopping and Coulomb repulsive interaction terms. Both of them are of finite range on the square lattice. In a strong coupling limit for the interaction Hamiltonian, we show that the Hall conductance is fractionally quantized to $1/2$ in the sense of the expectation value with respect to the four-fold degenerate ground state at the filling $3/8$. We also present a slightly different example in which there appears a long-range order of charge density wave with the Chern number $1$.