Microscopic theory of spin toroidization in periodic crystals

Using the semiclassical theory of electron dynamics, we derive a gauge-invariant expression for the spin toroidization in a periodical crystal. We show that the spin toroidization is comprised of two contributions: One is due to the configuration of a classical spin array, while the other comes from the coordinate shift of the electron as spin carrier in response to the inhomogeneous magnetic field. We then establish a direct and elegant relation between our spin toroidization and the antisymmetric magnetoelectric polarizability in insulators. Finally, we demonstrate our spin toroidization in a tight-binding model and show that it is a genuine bulk quantity.