Conservation of spin currents in spin-orbit-coupled systems

Applying the Gordon-decomposition-like technique, the convective spin current is extracted from the total angular-momentum current. The convective spin current describes the transport properties of the electron spin and is conserved in the relativistic quantum mechanics approach where the spin-orbit coupling has been intrinsically taken into account. Arrestingly, in the presence of external electromagnetic field, the component of the convective spin along the field remains still conserved. This conserved convective spin current is also derived for the first time in the nonrelativistic limit using the Foldy-Wouthuysen transformation.