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

TL;DR

This work derives a gauge-invariant bulk expression for the orbital magnetic octupole moment $M_{ijk}$ in crystalline solids via a thermodynamic free-energy approach, enabling a bulk description of higher-rank magnetic multipoles. It establishes direct links between $M_{ijk}$ and linear response functions to nonuniform fields, notably the quadrupolar magnetoelectric (QME) effect and the octupolar anomalous Hall (OAH) effect, through a generalized Středa-type relation. Using a minimal two-sublattice $d$-wave altermagnet model with spin-orbit coupling, the authors show finite MO components $M_{zxy}$ and $M_{xyz}$ that grow with SOC, and demonstrate that OAH can be finite even when the conventional anomalous Hall effect is symmetry-forbidden. The framework provides a versatile tool to explore higher-rank multipole physics in altermagnets and related crystalline systems, with potential implications for identifying and characterizing nonuniform-field transport phenomena.

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.

Figures (2)

• Figure 1: SOC dependence of the orbital MO in the minimal AM model. (a) $M_{zxy}$ and (b) $M_{xyz}=M_{yzx}$ as functions of the SOC strength $\lambda$ for metallic ($\mu=-0.4$) and insulating ($\mu=-0.01$) regimes. The parameters are set to $J=0.5$ and $T=0.01$.
• Figure 2: Chemical potential dependence of the orbital MO and its relation to response tensors. (a) Electronic band structure of the minimal model of AM. The red solid (blue dashed) lines represent the bands with up(down)-spin polarization. (b) $\mu$ dependence of $M_{zxy}$ and $M_{xyz}=M_{yzx}$. In (a) and (b), the parameters are set to $\lambda=0.2$, $J=0.5$, and $T=0.01$. The shaded areas correspond to the energy gap.