Real-space formulation of the Chern invariant and topological phases in a disordered Chern insulator

In this paper, we formulate the real-space Chern number in a supercell framework. In this framework, the overlap matrix between two corners of the Brillouin zone (BZ) is derived from diagonalizing the real-space Hamiltonian with periodic boundary conditions. The path-ordered product of overlap matrices around the BZ boundary forms a Wilson loop, and defines the Chern number in real space. It is analytically shown that the real-space Chern number is quantized at integers for large enough systems and coincides with the Bott index used in the previous studies. The formulation is greatly simplified for the former so that it makes numerical computations more efficient. The real-space formula is used to numerically elucidate topological phases in a disordered Chern insulator. The Chern insulator is modeled by dimensional extension of the Rice-Mele model consisting of two sublattices, and is disordered by including a random onsite potential. As disorder strength increases, the nontrivial-to-trivial phase transition takes place for normal disorder with no sublattice polarization. By contrast, the phase diagram is almost unaffected by polarized disorder, indicating that nontrivial topology persists against disorder. These observations are supported by the linear conductance and the density of bulk states.