Spin Hall effects without spin currents in magnetic insulators

The spin Hall effect (SHE) is normally discussed in terms of a spin current, which is ill-defined in strongly spin-orbit-coupled systems because of spin non-conservation. In this work we propose an alternative view of SHE phenomena by relating them to a spin analog of charge polarization induced by an electric field. The spin density polarization is most conveniently defined in insulators, which can have a SHE if they break time-reversal symmetry, i.e. if they are magnetic. The reciprocal of this SHE is a counterpart of the inverse SHE (ISHE), and is manifested in magnetic insulators as a charge polarization induced by a Zeeman field gradient. We use a modified Kane-Mele model to illustrate the magnetic spin Hall effect, and to discuss its bulk-boundary relationship.