Quantum kinetic theory of quadratic responses

Recent work has revealed a general procedure for incorporating disorder into the semiclassical model of carrier transport, whereby the predictions of quantum linear response theory can be recovered within a quantum kinetic approach based on a disorder-averaged density-matrix formalism. Here, we present a comprehensive generalization of this framework to the nonlinear response regime. In the presence of an electrostatic potential and random impurities, we solve the quantum Liouville equation to second order in an applied electric field and derive the carrier densities and equations of motion. In addition to the anomalous velocity arising from the Berry curvature and the Levi-Civita connection of the quantum metric tensor, a host of extrinsic velocities emerge in the equations of motion, reflecting the various possibilities for random interband walks of the carriers in this transport regime. Furthermore, several scattering and conduction channels arise, which can be classified in terms of distinct physical processes, revealing numerous mechanisms for generating nonlinear responses in disordered condensed matter systems.