Theory of electric, magnetic, and toroidal polarizations in crystalline solids with applications to hexagonal lonsdaleite and cubic diamond

Multipolar order in bulk crystalline solids is characterized by multipole densities -- denoted as polarizations in this work -- that cannot be cleanly defined using the concepts of classical electromagnetism. Here we use group theory to overcome this difficulty and present a systematic study of electric, magnetic and toroidal multipolar order in crystalline solids. Based on our symmetry analysis, we identify five categories of polarized matter, each of which is characterized by distinct features in the electronic band structure. For example, Rashba spin splitting in electropolar bulk materials like wurtzite represents the electric dipolarization in these materials. We also develop a general formalism of indicators for individual multipole densities that provide a physical interpretation and quantification of multipolar order. Our work clarifies the relation between patterns of localized multipoles and macroscopic multipole densities they give rise to. To illustrate the general theory, we discuss its application to polarized variants of hexagonal lonsdaleite and cubic diamond structures. Our work provides a general framework for classifying and expanding current understanding of multipolar order in complex materials.