Towards a microscopic theory of toroidal moments in bulk periodic crystals

We present a theoretical analysis of magnetic toroidal moments in periodic systems, in the limit in which the toroidal moments are caused by a time and space reversal symmetry breaking arrangement of localized magnetic dipole moments. We summarize the basic definitions for finite systems and address the question of how to generalize these definitions to the bulk periodic case. We define the ``toroidization'' as the toroidal moment per unit cell volume, and we show that periodic boundary conditions lead to a multivaluedness of the toroidization, which suggests that only differences in toroidization are meaningful observable quantities. Our analysis bears strong analogy to the ``modern theory of electric polarization'' in bulk periodic systems, but we also point out some important differences between the two cases. We then discuss the instructive example of a one-dimensional chain of magnetic moments, and we show how to properly calculate changes of the toroidization for this system. Finally, we evaluate and discuss the toroidization (in the local dipole limit) of four important example materials: $\mathrm{Ba}\mathrm{Ni}{\mathrm{F}}_{4}$, $\mathrm{Li}\mathrm{Co}\mathrm{P}{\mathrm{O}}_{4}$, $\mathrm{Ga}\mathrm{Fe}{\mathrm{O}}_{3}$, and $\mathrm{Bi}\mathrm{Fe}{\mathrm{O}}_{3}$.