Gauge-invariant absolute quantification of electric and magnetic multipole densities in crystals

Electric and magnetic multipole densities in crystalline solids, including the familiar electric dipole density in ferroelectrics and the magnetic dipole density in ferromagnets, are of central importance for our understanding of ordered phases in matter. However, determining the magnitude of these quantities has proven to be conceptually and technically difficult. Here we present a universally applicable approach, based on projection operators, that yields gauge-invariant absolute measures for all types of electric and magnetic order in crystals. We demonstrate the utility of the general theory using concrete examples of electric and magnetic multipole order in variants of lonsdaleite and diamond structures. Besides the magnetic dipole density in ferromagnets, we also consider, e.g., the magnetic octupole density in altermagnets. The robust method developed in this work lends itself to be incorporated into the suite of computational materials-science tools. The multipole densities can be used as thermodynamic state variables including Landau order parameters.