A general nonlinear Hall current in time-reversal breaking insulators

It is well known that the quantum anomalous Hall effect (QAHE) can only arise in insulators with nontrivial topology. Here, we show that at nonlinear order, currents emerge in a time reversal breaking insulator even when their Chern number is zero. Three mechanisms can be identified which create such a Hall response: an interband quasi-particle velocity, a wavepacket center shift, and a renormalization of the Berry curvature. We show that in strained twisted bilayer graphene the new nonlinear contribution is comparable with measured deviations of the QAHE from perfect quantization. Our results establish a fundamental difference between anomalous Hall currents in periodic systems as compared to 2DEGs, and might have implications for the use of the QAHE as a resistance standard.