Theory of the Local Hall Conductance of a Disordered Chern Insulator

Topological phases are usually theoretically studied in the context of non-interacting infinite periodic systems whose eigenstates are Bloch modes. Topological invariants are then usually constructed as integral forms of the phase evolution of the Bloch states across sections of this manifold. In any real system discrete translation symmetry is broken, either by the system's edge or by defects in the crystalline structure of the material. This leaves a subset of analytic and numeric techniques to diagnose the usual topology of the system. However in experiments, many topological phases continue to exhibit properties that are absent in these phases' trivial counterparts. Here we studied these experimental signatures in Chern insulators by developing formalism to calculate the local Hall conductance in the absence of translation symmetry and near different types of disordered crystalline defects. We demonstrate how trivial disorder can enlarge the phase space of topologically non-trivial systems.