First-principles theory of spin magnetic multipole moments in antiferromagnets

Abstract

Antiferromagnets with vanishing net magnetization are naturally expected to host higher-order magnetic multipole moments. Understanding and utilizing the multipole degrees of freedom are imperative for novel conceptual designs and applications unique to antiferromagnets. However, a universal, quantitative definition of magnetic multipole moments of antiferromagnetic materials is currently lacking. In this work we provide a unified description of arbitrary-order spin magnetic multipole moments (SM$^3$) of antiferromagnets by introducing a nonlocal spin density in macroscopic Maxwell equations. The formalism makes it transparent how SM$^3$ calculated for translationally invariant bulk systems corresponds to experimental observables when translation symmetry is broken. Through the nonlocal spin density calculated from first principles, we propose a robust scheme to extract arbitrary-order SM$^3$ through symmetry-constrained fitting at long wavelengths. Using this approach, we have calculated SM$^3$ of a few representative antiferromagnets, including $\alpha$-$\rm Fe_2O_3$, Mn$_3$Sn, and Mn$_3$NiN. Moreover, we clarify the role of spin-orbit coupling (SOC) in SM$^3$, especially in the weak SOC limit where clean predictions can be made based on symmetry principles. Our work paves the way for systematically investigating multipolar order parameters of unconventional magnetic materials.