Generalized Kubo formula for spin transport: A theory of linear response to non-Abelian fields

The traditional Kubo formula is generalized to describe the linear response with respect to non-Abelian fields. To fulfil the demand for studying spin transport, the SU(2) Kubo formulae are derived by two conventional approaches with different gauge fixings. Those two approaches are shown to be equivalent where the non-conservation of the SU(2) current plays an essential role in guaranteeing the consistency. Some concrete examples relating Spin Hall Effect are considered. The dc spin conductivity vanishes in the system with parabolic unperturbed dispersion relation. By applying a time-dependent Rashba field, the spin conductivity can be measured directly. Our formula is also applied to the high-dimensional representation for the interests of some important models, such as Luttinger model and bilayer spin Hall system.