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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 spinnon-conservation. In this work we propose an alternative view of SHE phenomenaby relating them to a spin analog of charge polarization induced by an electricfield. The spin density polarization is most conveniently defined ininsulators, which can have a SHE if they break time-reversal symmetry, i.e. ifthey are magnetic. The reciprocal of this SHE is a counterpart of the inverseSHE (ISHE), and is manifested in magnetic insulators as a charge polarizationinduced by a Zeeman field gradient. We use a modified Kane-Mele model toillustrate the magnetic spin Hall effect, and to discuss its bulk-boundaryrelationship.

Spin Hall effects without spin currents in magnetic insulators