Thermodynamic formulation of the spin magnetic octupole moment in bulk crystals

The discovery of unconventional antiferromagnets, such as altermagnets, has drawn significant attention to higher-rank magnetic multipoles, particularly magnetic octupoles. Despite the advances in research, attempts to understand their microscopic properties remain limited due to the unbounded nature of the position operator in bulk crystals. In this paper, we address this problem by using a well-known thermodynamic approach and derive a formula for the spin magnetic octupole moment (SMOM) that can be used in bulk crystals. The resulting formula is gauge invariant and satisfies St\v{r}eda formulas that relate the SMOM to the spin magnetoelectric dipole-quadrupole susceptibilities. Furthermore, we apply this formula to several models and examine the fundamental properties of the SMOM. For example, in $d$-wave altermagnets, the nonrelativistic component of the SMOM, which is independent of spin-orbit coupling, is larger than the relativistic component, which is induced by spin-orbit coupling. These nonrelativistic SMOMs have the same microscopic origin as the nonrelativistic spin splitting that characterizes $d$-wave altermagnetism. Moreover, they exhibit a N\'{e}el vector dependence consistent with Landau theory for $d$-wave altermagnetism [Phys. Rev. Lett. $\textbf{132}$, 176702 (2024)].