Persistent currents, whirlpools, and local Chern markers in twisted TMD Chern insulators

Recent materials advances have made it possible to fabricate twisted transition metal dichalcogenide homobilayers. These systems have been shown to host integer and fractional Chern insulating states. Because of spontaneous time reversal symmetry breaking, their ground state harbors intriguing spin-polarized currents with whirlpools on the moir\'e length scale that can be measured by scanning probe methods. We first provide a quantitative analysis of these persistent currents and then show that the maximum of the amplitude of the current density in the bulk of the sample is an accurate tracker of topological order. We conclude by calculating how the quantization of the Hall conductance is affected by finite-size effects.