$d$-wave altermagnets have magnetic octupoles as their order parameters [S. Bhowal and N. A. Spaldin, Ferroically ordered magnetic octupoles in $d$-wave altermagnets, Phys. Rev. X 14, 011019 (2024)]. We theoretically show that magnetic octupoles injected from outside generate torque on the $d$-wave altermagnets. The injection can be achieved by the magnetic octupole Hall effect in an adjacent layer. We calculate the magnetic octupole Hall conductivity of the heavy metal $\alpha \text{−}\mathrm{W}$ and find a sizable value comparable to its spin Hall conductivity. Our work generalizes the spin Hall phenomenology (generation by heavy metals and detection by torque in ferromagnets) to the magnetic octupole Hall phenomenology (generation by heavy metals and detection by torque in altermagnets), which can be utilized to electrically control magnetic configurations of altermagnets.

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• Altermagnetism
• Hall effect
• Magnetism
• Spintronics
• Linear response theory