The semiclassical Boltzmann equation is widely used to study transport effects. However, being semiclassical and borrowing heavily from classical mechanics, the formalism calls for verification from the perspective of quantum mechanics. Although previous works discussed the relation between the quantum density matrix and the semiclassical formalism, direct comparison, especially of disorder effects, including side jumps and skew scattering in the two approaches, has not been fully conducted. In this work, we systematically and directly compare the semiclassical Boltzmann equation and its counterpart arising from the density matrix. We find that there is an additional correction to the side-jump velocity, the longitudinal velocity, which is longitudinal in the leading order, and its resultant current does not require time-reversal symmetry breaking. Moreover, we find the semiclassical side-jump collision integral is an approximation of the quantum result at moderate temperatures, and it also contains a correction induced by the longitudinal velocity. We also show that the scattering rate obtained from the density matrix agrees with the semiclassical results. Our work illuminates the quantum roots of the semiclassical Boltzmann equation.

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• Electrical conductivity
• Quantum transport
• Boltzmann theory