Intrinsic spin accumulation in the magnetic spin Hall effect

The magnetic spin Hall effect is a time-reversal-odd phenomenon in which spin current is induced by the charge current. In the presence of a spin-orbit coupling and/or noncolinear magnetism, however, spin current is not uniquely defined. Instead, we study an intrinsic response of spin to an electric field gradient that describes the spin accumulation at the boundaries of magnetic systems. We derive a generic formula expressed by Bloch wave functions and apply it to a minimal model for representative altermagnets, RuO${}_2$ and MnF${}_2$. Our results show that the intrinsic spin accumulation can be nonzero in magnetic insulators in sharp contrast to the magnetic spin Hall conductivity.