Orbital magnetic octupole in crystalline solids and characterization of orbital altermagnetism

Magnetic multipole moments beyond dipoles have emerged as key descriptors of unconventional electromagnetic responses in crystalline solids. However, a gauge-invariant bulk expression for orbital magnetic multipole moments has remained elusive, hindering a unified understanding of their physical consequences. Here we formulate a gauge-invariant expression for the orbital magnetic octupole moment in periodic crystals and investigate its behavior in two models with distinct origins of magnetism: a spinless two-orbital model with orbital magnetic order arising from complex hopping and a two-sublattice $d$-wave altermagnetic model based on antiferromagnetic spin order. We also show that the orbital magnetic octupole is naturally linked to a higher-rank Hall response induced by spatially nonuniform electric fields, leading to a generalized St\v{r}eda-type relation. Our results further demonstrate that the orbital magnetic octupole provides a quantitative characterization of \textsl{orbital altermagnetism}.