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

TL;DR

This paper derives generalized Boltzmann and Eilenberger equations incorporating non-Abelian gauge fields for disordered Fermi gases, revealing how spin and charge dynamics are influenced by gauge fields and external fields.

Contribution

It provides a unified microscopic derivation of transport equations with non-Abelian gauge fields, extending classical results to include spin degrees of freedom.

Findings

Derivation of diffusion equations for charge and spin densities.

Identification of the spin Hall effect as a consequence of SU(2) gauge fields.

Demonstration of in-plane charge currents generated by magnetic and Rashba interactions.

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 non-relativistic 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 give 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.