Semiclassical dynamics up to second order in uniform electromagnetic fields : theory and applications

We derive a complete semiclassical theory up to second order with respect to electromagnetic fields, by establishing the equations of motion for the velocity and the force up to second order and modifying the band energy up to second order. With this theory, we are able to calculate magnetoelectric polarizability, nonlinear anomalous Hall conductivity, and the magnetic susceptibility. Finally, we derive a density quantization scheme which is the generalization of Onsager's rule to any polynomial order, and show its theoretical and experimental implications.