Boltzmann approach to spin-orbit-induced transport in effective quantum theories

In model studies of the spin/anomalous Hall effect, effective Hamiltonians often serve as the starting point. However, a complete effective quantum theory contains not only the effective Hamiltonian but also the relation linking the physical observables to the canonical ones. In this work we construct the semiclassical Boltzmann transport framework in the weak disorder-potential regime directly in the level of the effective quantum theory, and confirm this construction by formulating a generalized Kohn-Luttinger density matrix transport theory also in this level. The link and difference between the present theory and previous phenomenological Boltzmann, quantum kinetic and usual Kubo-Streda theories are clarified. In the application to a two-dimensional Rashba electron effective model, a nonzero spin Hall effect important in the case of strong Rashba coupling but neglected in previous theories is found.