Non-Abelian gauge fields in the gradient expansion: Generalized Boltzmann and Eilenberger equations

We present a microscopic derivation of the generalized Boltzmann and Eilenberger equations in the presence of non-Abelian gauges for the case of a nonrelativistic disordered Fermi gas. A unified and symmetric treatment of the charge [U(1)] and spin [SU(2)] degrees of freedom is achieved. Within this framework, just as the U(1) Lorentz force generates the Hall effect, so does its SU(2) counterpart gives rise to the spin Hall effect. Considering elastic and spin-independent disorder we obtain diffusion equations for charge and spin densities and show how the interplay between an in-plane magnetic field and a time-dependent Rashba term generates in-plane charge currents.