Spin dynamics with non-Abelian Berry gauge fields as a semiclassical constrained Hamiltonian system

The dynamics of observables which are matrices depending on ℏ and taking values in classical phase-space is defined by retaining the terms up to the first order in ℏ of the Moyal bracket. Within this semiclassical approach a first-order Lagrangian involving gauge fields is studied as a constrained Hamiltonian system. This provides a systematic study of spin dynamics in the presence of non-Abelian Berry gauge fields. We applied the method to various types of dynamical spin systems and clarified some persisting discussions. In particular employing the Berry gauge field which generates the Thomas precession, we calculated the force exerted on an electron in the external electric and magnetic fields. Moreover, a simple semiclassical formulation of the spin Hall effect is accomplished.