We present a theoretical analysis of magnetic toroidal moments in periodicsystems, in the limit in which the toroidal moments are caused by a time andspace reversal symmetry breaking arrangement of localized magnetic dipolemoments. We summarize the basic definitions for finite systems and address thequestion of how to generalize these definitions to the bulk periodic case. Wedefine the toroidization as the toroidal moment per unit cell volume, and weshow that periodic boundary conditions lead to a multivaluedness of thetoroidization, which suggests that only differences in toroidization aremeaningful observable quantities. Our analysis bears strong analogy to themodern theory of electric polarization in bulk periodic systems, but we alsopoint out some important differences between the two cases. We then discuss theinstructive example of a one-dimensional chain of magnetic moments, and we showhow 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: BaNiF_4, LiCoPO_4, GaFeO_3, and BiFeO_3.
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