What are spin currents in Heisenberg magnets?

Abstract.

We discuss the proper definition of the spin current operator in Heisenberg magnets subject to inhomogeneous magnetic fields. We argue that only the component of the naive “current operator” $J_{ij}{\vec S}_i\times{\vec S}_j$ in the plane spanned by the local order parameters $\langle{\vec S}_i\rangle$ and $\langle{\vec S}_j\rangle$ is related to real transport of magnetization. Within a mean field approximation or in the classical ground state the spin current therefore vanishes. Thus, finite spin currents are a direct manifestation of quantum correlations in the system.