Microscopic Theory of Spin Toroidization in Periodic Crystals*

Spin toroidization has been proposed to be a primary order parameter in systems that break time reversal and inversion symmetry but preserve the combined symmetries. It has been related to various magnetoelectric phenomenon. However, previous theories of spin toroidization have been based on the multipole expansion in the electromagnetic theory, which is a classical theory by its nature. Here we provide a microscopic quantum-mechanical theory of spin toroidization in periodic crystals using Bloch functions. Our result is gauge invariant, and more importantly, it directly connects to the antisymmetric part of the spin magnetoelectric polarizability. To illustrate our result, we calculate the spin toroidization in a toy model and demonstrate that it is a genuine bulk property.