Quantised Bulk Conductivity as a Local Chern Marker

A central property of Chern insulators is the robustness of the topological phase and edge states to impurities in the system. Despite this, Chern number cannot be straightforwardly calculated in the presence of disorder. Recently, work has been done to propose a local analog of the Chern number, called local markers, that can be used to characterise disordered systems. However, it was unclear whether the proposed markers represented a physically-measurable property of the system. Here we propose a local marker starting from a physical argument, as a local cross-conductivity measured in the bulk of the system. We find the explicit form of the marker for a non-interacting system of electrons on the lattice and show that it corresponds to existing expressions for the Chern number. Examples are calculated for a variety of disordered and amorphous systems, showing that it is precisely quantised to the Chern number and robust against disorder.