Theory of nonlinear Hall effects: Modified semiclassics from quantum kinetics

Abstract

We propose a modified Boltzmann nonlinear electric-transport framework which differs from the nonlinear generalization of the linear Boltzmann formalism by a contribution that has no counterpart in linear response. This contribution follows from the interband-coherence effect of dc electric fields during scattering and is related to the interband Berry connection. As an application, we demonstrate it in the second-order nonlinear Hall effect of the tilted massive Dirac model. The intuitive Boltzmann constructions are confirmed by a quantum kinetic theory, which shows that the arbitrary $n\mathrm{th}$-order nonlinear dc response up to the first three leading contributions in the weak disorder potential is handled by the same few gauge-invariant semiclassical ingredients.