We present a general, systematic theory of electromagnetism and chirality in crystalline solids [1]. Symmetry is its fundamental guiding principle. We identify two complementary, comprehensive classifications of crystals, based on five categories of electric and magnetic multipole order—called polarizations—and five categories of chirality. The five categories of polarizations (parapolar, electropolar, magnetopolar, antimagnetopolar, and multipolar) expand the familiar notion of electric dipolarization in ferroelectrics and magnetization in ferromagnets to higher-order multipole densities. Each of these categories is characterized by distinct features in the electronic band structure that are directly relevant for many applications.
One speaks of chirality when a system exists in two versions (enantiomorphs) that cannot be superposed upon each other by proper (pure) rotations C. Ordinarily, this is tantamount to the case that the enantiomorphs are mirror images of each other, i.e., the enantiomorphs are mapped onto each other by improper rotations iC, where i denotes space inversion. By treating space inversion i, time inversion θ, and their combination iθ on the same footing, we greatly expand the notion of chirality. Besides the usual kind, which we call electrochirality because it arises from the interplay of electric multipole densities, we identify magnetochirality and antimagnetochirality as new categories of chirality characterized by the existence of two distinct enantiomorphs. In multichiral systems, all inversion symmetries (i, θ, and iθ) are absent so that these systems have four distinct enantiomorphs. Each category of chirality arises from distinct superpositions of electric and magnetic multipole densities. We provide a complete theory of minimal effective models characterizing the different categories of chirality in different systems. Jointly these two schemes yield a classification of all 122 magnetic crystallographic point groups into 15 types that have distinct physical properties and characteristic features in the electronic band structure. At the same time, the formal similarities between the inversion symmetries i, θ, and iθ imply striking correspondences between apparently dissimilar systems and their physical properties.
[1] Arxiv:2405.20940.