Definition of the spin current: The angular spin current and its physical consequences

TLDR: In this article, it was shown that the angular spin current (or spin torque) can also induce an electric field, similar to the linear spin current, and the formula for the induced electric field was derived, playing the role of ''Biot-Savart law'' or ''Ampere law''.

Abstract: We find that in order to completely describe the spin transport, apart from spin current (or linear spin current), one has to introduce the angular spin current. The two spin currents, respectively, describe the translational and rotational motion (precession) of a spin. The definitions of these spin current densities are given and their physical properties are discussed. Both spin current densities appear naturally in the spin continuity equation. In particular, we predict that the angular spin current (or the spin torque as called in previous works), similar to the linear spin current, can also induce an electric field $\stackrel{P\vec}{E}$. The formula for the induced electric field $\stackrel{P\vec}{E}$ by the angular spin current element is derived, playing the role of ``Biot-Savart law'' or ``Ampere law.'' When at large distance $r$, this induced electric field $\stackrel{P\vec}{E}$ scales as $1∕{r}^{2}$, whereas the $\stackrel{P\vec}{E}$ field generated from the linear spin current goes as $1∕{r}^{3}$.