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

Abstract

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.