to the shifted Fermi surface sums in Eq. (\[eqn:spinandcurrent\]), there is a contribution from a $-k_x$ hole. Now, a hole has opposite spin density to an electron and the spin current density Eq. (\[eqn:scscd\]) contains an additional factor of $k_x$. Therefore, the two spin density vectors in Fig. \[fig:curvature\] [*subtract*]{} to give a net spin density along $\hat{\bf y}$ while the corresponding two spin current density vectors [*add*]{} to give a net spin current density along $\hat{\bf
z}'$. This occurs for all $k_x$ in the sums so ${\bf Q}(x)$ aligns exactly with the local exchange field and thus with the local magnetization.

The opposite situation occurs for fully occupied states below the Fermi energy. The spins of the forward and backward moving electrons combine to produce a net moment aligned with the exchange field, as necessary for self-consistency. Further, the spin currents, with the additional factor of $k_x$ add to give a net spin current along $\hat{\bf y}$, so that its gradient gives the correct form of the phenomenological exchange torque density.

To summarize: an electric current that passes through a spin spiral generates a spin accumulation with a component transverse to the magnetization. The spin current density possesses [*no*]{} 