Magnetic octupole (MO) currents have recently attracted significant attention as a driving force for the Néel vector dynamics in $d$-wave altermagnets, a unique class of antiferromagnets that exhibit nonrelativistic spin-split band structures. From a symmetry perspective, the MO includes an axial-dipole component analogous to that of the spin, making it essential to clarify how MO currents differ from spin currents. We here investigate the correspondence between MO conductivities and electronic multipoles, which provide a unified and powerful framework for symmetry analysis. We derive the multipole representation of the rank-5 MO conductivity tensor and classify its symmetry-allowed components for all crystallographic point groups, in direct comparison with spin conductivity. We show that time-reversal-even electric-type multipoles give rise to the dissipationless MO current, whereas time-reversal-odd magnetic-type multipoles generate dissipative MO current under an applied electric field. Complementing this macroscopic analysis, the linear-response calculations for a microscopic tight-binding model demonstrate how MO conductivities are activated by symmetry lowering, exemplified by the symmetry reduction from $m\overline{3}m$ to $m\overline{3}$. Our results elucidate the symmetry distinctions between MO currents and spin currents and provide insights into their experimental identification.

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• Altermagnetism
• Spintronics
• Altermagnets
• Group theory
• Linear response theory
• Tight-binding model