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 $\left[U\left(1\right)\right]$ and spin $\left[SU\left(2\right)\right]$ degrees of freedom is achieved. Within this framework, just as the $U\left(1\right)$ Lorentz force generates the Hall effect, so does its $SU\left(2\right)$ 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.

Figure 1

(Color online) The Rashba spin-orbit coupling constant $\alpha$ is made time dependent by applying a time-dependent gate potential $V_{in}\left(t\right)$. The light (blue) area represents a two-dimensional electron gas inside a heterostructure. When an in-plane magnetic field along $x$, $b^x$, is also switched on, a charge current $j_y^0$ flowing along $y$ and proportional to $\dot{\alpha }$ is generated, its actual sign depending on the sign of $\dot{\alpha }$. The induced voltage drop in the transverse direction $V_{out}\left(t\right)$ can then be used to measure the strength of the Rashba interaction.