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

TL;DR

This paper develops a gauge-invariant expression for the orbital magnetic octupole in crystalline solids and links it to a novel Hall response induced by nonuniform electric fields, revealing new electromagnetic phenomena in altermagnets.

Contribution

It introduces a gauge-invariant formulation of the orbital magnetic octupole moment and connects it to a higher-rank Hall response in altermagnets, expanding understanding of magnetic multipole effects.

Findings

Orbital magnetic octupole expression formulated for periodic crystals.

Identification of a higher-rank Hall response induced by nonuniform electric fields.

Demonstration of Hall response even when conventional anomalous Hall effect is symmetry-forbidden.

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 St\v{r}eda-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.