Boltzmann scaling of spontaneous Hall current and nonequilibrium spin-polarization

We extend the semiclassical Boltzmann formalism for the anomalous Hall effect (AHE) in nondegenerate multiband electron systems to the spin Hall effect (SHE) and unconventional Edelstein effect (UEE, cannot be accounted for by the conventional Boltzmann equation, unlike the conventional Edelstein effect). This extension is confirmed by extending the Kohn-Luttinger density-matrix transport theory in the weak disorder-potential regime. By performing Kubo linear response calculations in a prototypical multiband model, the Boltzmann scaling for the AHE/SHE and UEE is found to be valid only if the disorder-broadening of bands is quite smaller than the minimal intrinsic energy-scale around the Fermi level. Discussions on this criterion in various multiband systems are also presented. A qualitative phase diagram is proposed to show the influences of changing independently the impurity density and strength of disorder potential on the AHE/SHE and UEE.