Spin-current-less spin Hall effect in magnetic insulators

The spin Hall effect (SHE) in a nonmagnetic conductor is conventionally described as a transverse spin current induced by a longitudinal electric field or voltage, which in turn leads to accumulations of opposite spin at opposite transverse boundaries. Despite its conceptual attractiveness, this definition of the SHE suffers from fundamental difficulties related to the lack of spin conservation, especially in the strongly spin-orbit-coupled systems where the SHE becomes more prominent. In this work we propose an alternative view of SHE phenomena by relating them to the polarization of spin density, in analogy to the charge polarization, induced by an electric field. The spin density polarization is well defined in insulating systems, and a counterpart of the conventional SHE can be present if there is a transverse response of it to an electric field. A constraint on this response from time reversal implies that in insulating systems it is non-zero only if the systems are magnetic. We use some toy models to illustrate the physical consequences of this magnetic spin Hall effect.