Standard model of electromagnetism and chirality in crystals

We present a general, systematic theory of electromagnetism and chirality in crystalline solids. Symmetry is its basic guiding principle, enabling us to consider macroscopic multipole densities without reference to any specific microscopic configurations. We use a formal analogy between space inversion $i$ and time inversion $\theta$ to 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) embody the ways in which electromagnetic multipole order can be realized in solids, thus expanding the familiar notion of electric dipolarization in ferroelectrics and magnetization in ferromagnets to higher-order multipole densities. The five categories of chirality (parachiral, electrochiral, magnetochiral, antimagnetochiral, and multichiral) extend the notion of enantiomorphism--conventionally associated with the lack of spatial mirror symmetries--to include all possibilities for creating non-superposable images by applying the inversions $i$, $\theta$, and $i\theta$. In particular, multichiral systems lack all inversion symmetries and thus have four different enantiomorphs. Each category of chirality arises from particular superpositions of electric and magnetic multipole densities. Jointly, the categories of polarizations and chirality yield a classification of all 122 magnetic point groups into 12 types that exhibit distinct physical properties and are identifiable by characteristic features in the electronic band structure that we elucidate in detail. The classification makes the formal equivalence of $i$, $\theta$, and $i\theta$ explicit and reveals striking correspondences between apparently dissimilar systems and their physical properties.