Symmetry Classification of Multipolar Orders in Crystals: Theory, Property Tensors and Automated Analysis with MagSymMultipoles
Many structural and magnetic phases and properties of crystalline materials can be related to the ordering of electric and magnetic multipoles. Examples include ferroelectricity, linear magnetoelectricity, and altermagnetism. Determining the symmetry-allowed multipoles in a crystal is therefore useful for characterizing its ordered phases and physical properties. Here, we present a unified Cartesian framework for decomposing moment tensors of arbitrary rank into ordinary, toroidal, and poloidal multipoles, and for determining, from crystallographic and magnetic symmetry, their ferroic, antiferroic, and noncollinear arrangements within the crystal. We further establish the direct symmetry relationship between multipolar order and the allowed components of associated physical-property tensors in both relativistic and non-relativistic settings. We implement this methodology in MagSymMultipoles (https://mag-sym-multipoles.com), an interactive web application for calculating and visualizing symmetry-adapted electric and magnetic multipoles. The application uses magnetic symmetries both with and without spin-orbit coupling to derive the allowed multipoles as well as the underlying Cartesian moment tensors, making it easy to explore the connection between a given multipolar order and the corresponding physical response tensors. We demonstrate the approach for ferroelectric BaTiO3, antiferroelectric PbZrO3, magnetoelectric Cr2O3, and the altermagnets MnF2, MnTe, and Mn3IrSi. These examples demonstrate how our method allows us to obtain an intuitive picture linking the multipolar order of a material directly to its physical properties, thereby facilitating the interpretation of theoretical and experimental results.