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.
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The datasets generated and/or analyzed during the current study are not publicly available due to the absence of deposition in a public repository, but are available from the corresponding author on reasonable request.
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The numerical calculation codes used in this study are available from the corresponding author upon reasonable request.
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Acknowledgements
The authors gratefully acknowledge J. Oike, K. Shinada, and R. Peters for useful discussions and sharing a related unpublished manuscript prior to its submission. S.H. was supported by JSPS KAKENHI Grants Numbers JP21H01037, JP22H00101, JP22H01183, JP23H04869, JP23K03288, JP23K20827, and by JST CREST (JPMJCR23O4) and JST FOREST (JPMJFR2366).
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T.S. and S.H. conceived the project. T.S. performed the analytical and numerical calculations. Both authors contributed to writing the paper.
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Sato, T., Hayami, S. Quantum theory of magnetic octupole in periodic crystals and application to d-wave altermagnets. npj Quantum Mater. (2026). https://doi.org/10.1038/s41535-026-00865-9
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DOI: https://doi.org/10.1038/s41535-026-00865-9
