Magnetic multipoles in altermagnets

Altermagnetism has recently been proposed as a distinct magnetic phase, separate from both ferromagnetism and antiferromagnetism. In this talk, we explore how atomic multipoles can serve as effective tools for identifying altermagnetic phases and their corresponding order parameters. Using symmetry-based arguments and Density Functional Theory we demonstrate the emergence of magnetic multipoles up to the fifth order (l=5) in systems exhibiting compensated collinear magnetic orders, in both the collinear and the non-collinear components of the spin density. We calculate the magnitude of the multipoles around the magnetic ions, and investigate their role in rutile insulator MnF₂, where d-wave altermagnetism arises, and in the hexagonal metal CrSb, which displays g-wave altermagnetism, as a function of spin-orbit coupling. While the spin density along the collinear axis is non-zero at the atomic site, yielding a magnetic dipole, we find that spin-orbit coupling turns on the non-collinear components of the spin density, which have nodes at the atomic sites, but are otherwise non-zero and behave like higher oder multipoles. Additionally, we show that in the perovskite KMnF₃, the onset of g-wave altermagnetism is driven by the rotation of the fluorine octahedra. Our analysis reveals that d-wave altermagnetism is accompanied by a net magnetic octupole in the unit cell, while g-wave altermagnetism corresponds to a higher-order magnetic dotriacontapole (32-pole). These findings illustrate the diverse and complex nature of altermagnetic order and its dependence on symmetry and spin-orbit interactions in crystalline materials.