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Abstract Magnetic multipoles have been recognized as order parameters characterizing magnetic structure in solids. Recently, magnetic octupoles have been proposed as the order parameters of time-reversal-symmetry-breaking centrosymmetric antiferromagnets exhibiting nonrelativistic spin splitting, which is referred to as “altermagnet”. However, a gauge-invariant formulation of magnetic octupoles in crystalline solids remains elusive. Here, we present a gauge-invariant expression of spin magnetic octupoles in periodic crystals based on quantum mechanics and thermodynamics, which can be used to quantitatively characterize time-reversal-symmetry-breaking antiferromagnets, including d -wave altermagnets. The allowed physical response tensors are classified beyond symmetry considerations, and direct relationships are established for some of them in insulators at zero temperature. Furthermore, our expression reveals a contribution from an anisotropic magnetic dipole, which has the same symmetry as conventional spin and orbital magnetic dipoles but carries no net magnetization. We discuss the relation between the anisotropic magnetic dipole and the anomalous Hall effect.
Sato et al. (2026) studied this question.
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