Electric, magnetic and toroidal polarizations in crystals

Electric and magnetic multipolar order in bulk crystalline solids is characterized by multipole densities that cannot be cleanly defined using the concepts of classical electromagnetism. We use group theory to overcome this difficulty and present a systematic study of electric, magnetic and toroidal multipolar order in crystalline solids [1]. Based on our symmetry analysis, we identify five categories of polarized matter that provide a complete classification of multipolar order in crystals, including insulators and metals. Each category is characterized by distinct features in the electronic band structure. For example, Rashba spin splitting in electropolar materials like wurtzite represents an electric dipolarization, while Dresselhaus spin splitting in zincblende represents an electric octupolarization. We also develop a general formalism of indicators for individual multipole densities that quantify electric and magnetic 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 multipole-ordered variants of hexagonal lonsdaleite and cubic diamond. Our work provides a general framework for classifying and expanding current understanding of multipolar order in complex materials.