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We propose a modified Boltzmann nonlinear electric-transport framework whichdiffers from the nonlinear generalization of the linear Boltzmann formalism bya contribution that has no counterpart in linear response. This contributionfollows from the interband-coherence effect of dc electric-fields duringscattering and is related to the interband Berry connection. As an application,we demonstrate it in the second-order nonlinear Hall effect of the tiltedmassive Dirac model. The intuitive Boltzmann constructions are confirmed by aquantum kinetic theory, which shows that arbitrary $n$th-order nonlinear dcresponse up to the first three leading contributions in the weak disorderpotential is handled by the same few gauge-invariant semiclassical ingredients.

Theory of nonlinear Hall effects: renewed semiclassics from quantum kinetics