Orbital magnetic octupole in crystalline solids and anomalous Hall response to a nonuniform electric field

Abstract: 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 a minimal model of $d$-wave altermagnets. We 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 Středa-type relation. Finally, we demonstrate that such a Hall response can arise even when symmetry forbids the conventional anomalous Hall effect against uniform electric fields, thereby providing an illustrative response characteristic to altermagnets.