First-principles theory of higher-order spin magnetic multipole moments in antiferromagnets

Antiferromagnets whose net magnetization vanishes are naturally expected to host higher-order magnetic multipole moments. Understanding and utilizing the multipole degrees of freedom are promising for novel conceptual designs and applications that are unique to antiferromagnets. However, except for a few cases, a general first-principles theory for higher-order magnetic multipole moments of antiferromagnetic materials is lacking. In this work we provide a unified framework for computing general-order spin magnetic multipole moments (SM$^3$) of magnetic materials using a thermodynamic approach and have calculated their values in a few well known antiferromagnets. In addition, we have shown that the gauge-invariant nature of our SM$^3$ leads to robust, characteristic local spin densities where the former is spatially nonuniform, e.g., at sample surfaces, which can be measured by local magnetic imaging techniques. Our work paves the way for systematically investigating the multipolar order parameters of unconventional magnetic materials.