Magnetic Octupole Hall Effect

• Magnetic Octupole Hall Effect is a class of phenomena where magnetic octupoles act as bulk thermodynamic variables or transported multipoles driven by nonuniform fields.
• It unifies diverse methodologies—from gauge-invariant Bloch-band formulations to composite spin-orbital definitions in altermagnets—with predictions supported by first-principles calculations.
• The effect underpins novel transport signatures in both condensed-matter systems and neutron-star crusts, offering new pathways for torque generation and magnetic field evolution.

“Magnetic octupole Hall effect” denotes a family of Hall-related phenomena in which magnetic octupoles enter either as bulk thermodynamic variables, transported multipolar quantities, order parameters of time-reversal-symmetry-breaking states, or symmetry-resolved components of anomalous Hall response. In periodic crystals the most literal condensed-matter usage is a higher-rank Hall current governed by an orbital magnetic octupole and driven by spatially nonuniform electric fields rather than by $E_k$ itself; in altermagnets and heavy metals it can also mean a transverse current of magnetic octupole moment; in multipolar antiferromagnets it appears through anomalous Hall, anomalous Nernst, Kerr, circular-dichroic, or nonlinear Hall responses controlled by octupolar order; and in neutron-star physics the same phrase refers to Hall-driven magnetic-field evolution toward a dipole-plus-octupole crustal attractor rather than to electronic transport (Sato et al., 30 Dec 2025, Ko et al., 1 Aug 2025, Gourgouliatos et al., 2013).

1. Terminological scope and basic constructions

In current literature the term is not attached to a single universal object. One line of work formulates a magnetic octupole in periodic crystals as a bulk thermodynamic quantity conjugate to the second spatial derivative of magnetic field. In that formulation, the naive Bloch-band expression involving $\hat s_i\hat r_j\hat r_k$ is rejected because it is not gauge invariant, and the spin magnetic octupole is instead defined through the free-energy differential

$$dF=-SdT-Nd\mu-M_i\,dH_i-M_{ij}\,d[\partial_{r_j}H_i]-M_{ijk}\,d[\partial_{r_j}\partial_{r_k}H_i],$$

with

$$M_{ijk}\coloneqq-\left(\frac{\partial F}{\partial[\partial_{r_j}\partial_{r_k}H_i]}\right)_{T,\mu,\mathbf H,\partial_{\mathbf r}\mathbf H}.$$

The same work decomposes the rank-3 axial tensor into irreducible pieces and identifies the anisotropic magnetic dipole

$M'_i\coloneqq \frac{3M_{jji}-M_{ijj}}{\sqrt{10}}$

as a dipolar component with the same symmetry as an ordinary magnetic dipole but with no net magnetization; this quantity is proposed as the order parameter for antiferromagnets that exhibit anomalous Hall effect (Sato et al., 30 Apr 2025).

A second construction, developed for nonrelativistic $d$-wave altermagnets, defines the magnetic octupole as a composite spin-orbital object,

$M_{ijk}=s_iQ_{jk},$

so that the Hall effect is literally a transverse flow of the composite quantity $s_iQ_{jk}$ carried by Bloch quasiparticles (Ko et al., 1 Aug 2025). A third construction shifts the octupole from order-parameter space to response space. In cubic ferromagnets the anomalous Hall conductivity vector is expanded over the sphere of magnetization directions as

$$\sigma^i_{\rm AHE}=\alpha \hat M_i+\beta \hat M_i^3,$$

and the cubic term is termed an octupole in magnetization space (Peng et al., 2024). In multilayers with $C_{4v}$ symmetry, the Hall current is decomposed into odd angular harmonics, where the $\hat s_i\hat r_j\hat r_k$0 harmonic is identified as purely octupolar (Niu et al., 26 Aug 2025).

These uses are related by symmetry rather than by a single microscopic definition. The common element is that a rank-3 magnetic structure, or a response tensor with equivalent transformation properties, becomes the natural descriptor once dipolar language is insufficient.

2. Bulk orbital magnetic octupole and higher-rank Hall response

The most explicit condensed-matter realization of a magnetic-octupole Hall effect is the orbital magnetic octupole theory in periodic crystals. There the bulk free energy is expanded as

$\hat s_i\hat r_j\hat r_k$1

and the orbital magnetic octupole $\hat s_i\hat r_j\hat r_k$2 is defined as the quantity conjugate to magnetic-field curvature. The resulting gauge-invariant bulk expression is built from Bloch-band geometric objects and band-structure derivatives, including the orbital magnetic moment $\hat s_i\hat r_j\hat r_k$3, Berry curvature $\hat s_i\hat r_j\hat r_k$4, and quantum metric $\hat s_i\hat r_j\hat r_k$5, all written with covariant derivatives to preserve $\hat s_i\hat r_j\hat r_k$6 gauge invariance (Sato et al., 30 Dec 2025).

The corresponding Hall response is not the conventional anomalous Hall effect to a uniform electric field. It is an octupolar anomalous Hall effect, or higher-rank Hall response, in which the magnetization generated by a nonuniform electric field satisfies

$\hat s_i\hat r_j\hat r_k$7

and the Hall-like current follows as

$\hat s_i\hat r_j\hat r_k$8

The linear conductivity is

$\hat s_i\hat r_j\hat r_k$9

so the driving field is $$dF=-SdT-Nd\mu-M_i\,dH_i-M_{ij}\,d[\partial_{r_j}H_i]-M_{ijk}\,d[\partial_{r_j}\partial_{r_k}H_i],$$0, not $$dF=-SdT-Nd\mu-M_i\,dH_i-M_{ij}\,d[\partial_{r_j}H_i]-M_{ijk}\,d[\partial_{r_j}\partial_{r_k}H_i],$$1 itself. In the insulating, $$dF=-SdT-Nd\mu-M_i\,dH_i-M_{ij}\,d[\partial_{r_j}H_i]-M_{ijk}\,d[\partial_{r_j}\partial_{r_k}H_i],$$2 limit this obeys the generalized Středa relation

$$dF=-SdT-Nd\mu-M_i\,dH_i-M_{ij}\,d[\partial_{r_j}H_i]-M_{ijk}\,d[\partial_{r_j}\partial_{r_k}H_i],$$3

which directly links Hall transport, the chemical-potential derivative of the orbital magnetic octupole, and the density response to magnetic-field curvature (Sato et al., 30 Dec 2025).

This formalism is designed to distinguish octupolar Hall response from ordinary AHE. The orbital magnetic octupole is a time-reversal-odd rank-3 axial tensor, so it may remain symmetry-allowed in magnetic crystals where the usual rank-2 Hall conductivity vanishes. In the minimal two-sublattice $$dF=-SdT-Nd\mu-M_i\,dH_i-M_{ij}\,d[\partial_{r_j}H_i]-M_{ijk}\,d[\partial_{r_j}\partial_{r_k}H_i],$$4-wave altermagnet model discussed in that work, mirror symmetries forbid the conventional intrinsic AHE to uniform field, yet allow the octupolar response. The symmetry constraints leave only

$$dF=-SdT-Nd\mu-M_i\,dH_i-M_{ij}\,d[\partial_{r_j}H_i]-M_{ijk}\,d[\partial_{r_j}\partial_{r_k}H_i],$$5

nonzero. In the minimal single-orbital model both components vanish at $$dF=-SdT-Nd\mu-M_i\,dH_i-M_{ij}\,d[\partial_{r_j}H_i]-M_{ijk}\,d[\partial_{r_j}\partial_{r_k}H_i],$$6 and increase monotonically with spin-orbit coupling $$dF=-SdT-Nd\mu-M_i\,dH_i-M_{ij}\,d[\partial_{r_j}H_i]-M_{ijk}\,d[\partial_{r_j}\partial_{r_k}H_i],$$7, and inside the insulating gap the orbital magnetic octupole is linear in $$dF=-SdT-Nd\mu-M_i\,dH_i-M_{ij}\,d[\partial_{r_j}H_i]-M_{ijk}\,d[\partial_{r_j}\partial_{r_k}H_i],$$8, so its slope directly supplies the thermodynamic indicator of finite octupolar anomalous Hall conductivity (Sato et al., 30 Dec 2025).

The bulk spin-octupole theory provides a parallel foundation rather than the same transport law. Its main contribution is to show that Hall-active centrosymmetric antiferromagnets can be characterized not by net magnetization but by the anisotropic magnetic dipole embedded בתוך the magnetic-octupole tensor. In that sense the later orbital theory supplies an explicit Hall constitutive relation, while the spin theory clarifies which irreducible octupolar component is symmetry-equivalent to anomalous Hall activity in compensated antiferromagnets (Sato et al., 30 Apr 2025).

3. Multipole-current Hall transport in altermagnets and heavy metals

A distinct usage treats the magnetic octupole Hall effect as a true multipole-current phenomenon. In $$dF=-SdT-Nd\mu-M_i\,dH_i-M_{ij}\,d[\partial_{r_j}H_i]-M_{ijk}\,d[\partial_{r_j}\partial_{r_k}H_i],$$9-wave altermagnets the transported quantity is not charge, pure spin, or orbital angular momentum, but the composite magnetic octupole $$M_{ijk}\coloneqq-\left(\frac{\partial F}{\partial[\partial_{r_j}\partial_{r_k}H_i]}\right)_{T,\mu,\mathbf H,\partial_{\mathbf r}\mathbf H}.$$0. The linear-response framework writes

$$M_{ijk}\coloneqq-\left(\frac{\partial F}{\partial[\partial_{r_j}\partial_{r_k}H_i]}\right)_{T,\mu,\mathbf H,\partial_{\mathbf r}\mathbf H}.$$1

and for $$M_{ijk}\coloneqq-\left(\frac{\partial F}{\partial[\partial_{r_j}\partial_{r_k}H_i]}\right)_{T,\mu,\mathbf H,\partial_{\mathbf r}\mathbf H}.$$2 the current operator is the symmetrized octupole-current operator. Within the spin point group $$M_{ijk}\coloneqq-\left(\frac{\partial F}{\partial[\partial_{r_j}\partial_{r_k}H_i]}\right)_{T,\mu,\mathbf H,\partial_{\mathbf r}\mathbf H}.$$3, symmetry allows several magnetic-octupole conductivity components. The paper distinguishes a first type, which accompanies the spin-splitter effect, from a second type,

$$M_{ijk}\coloneqq-\left(\frac{\partial F}{\partial[\partial_{r_j}\partial_{r_k}H_i]}\right)_{T,\mu,\mathbf H,\partial_{\mathbf r}\mathbf H}.$$4

which remains symmetry-allowed even when the spin-splitter effect is forbidden. This difference is the central claim: magnetic-octupole Hall transport can survive in geometries where pure spin Hall-like transport is symmetry-forbidden, making it a more robust signature of $$M_{ijk}\coloneqq-\left(\frac{\partial F}{\partial[\partial_{r_j}\partial_{r_k}H_i]}\right)_{T,\mu,\mathbf H,\partial_{\mathbf r}\mathbf H}.$$5-wave altermagnetism (Ko et al., 1 Aug 2025).

The heavy-metal literature generalizes this to nonmagnetic sources of magnetic-octupole current. There the atomic magnetic-octupole operator is written as

$$M_{ijk}\coloneqq-\left(\frac{\partial F}{\partial[\partial_{r_j}\partial_{r_k}H_i]}\right)_{T,\mu,\mathbf H,\partial_{\mathbf r}\mathbf H}.$$6

the current as

$$M_{ijk}\coloneqq-\left(\frac{\partial F}{\partial[\partial_{r_j}\partial_{r_k}H_i]}\right)_{T,\mu,\mathbf H,\partial_{\mathbf r}\mathbf H}.$$7

and the constitutive law as

$$M_{ijk}\coloneqq-\left(\frac{\partial F}{\partial[\partial_{r_j}\partial_{r_k}H_i]}\right)_{T,\mu,\mathbf H,\partial_{\mathbf r}\mathbf H}.$$8

This is the octupolar analog of the spin Hall effect. First-principles calculations for Pt give

$$M_{ijk}\coloneqq-\left(\frac{\partial F}{\partial[\partial_{r_j}\partial_{r_k}H_i]}\right)_{T,\mu,\mathbf H,\partial_{\mathbf r}\mathbf H}.$$9

to be compared with

$M'_i\coloneqq \frac{3M_{jji}-M_{ijj}}{\sqrt{10}}$0

so the magnetic octupole Hall conductivity is of the same order as Pt’s spin Hall conductivity (Han et al., 2024).

A wider first-principles survey of $M'_i\coloneqq \frac{3M_{jji}-M_{ijj}}{\sqrt{10}}$1 and $M'_i\coloneqq \frac{3M_{jji}-M_{ijj}}{\sqrt{10}}$2 transition metals finds that the effect is not specific to Pt and traces its microscopic origin to the combined effect of orbital texture and spin-orbit coupling. The largest reported values are

$M'_i\coloneqq \frac{3M_{jji}-M_{ijj}}{\sqrt{10}}$3

in units of $M'_i\coloneqq \frac{3M_{jji}-M_{ijj}}{\sqrt{10}}$4. The same study identifies hcp Zr and hcp Hf as particularly favorable when one wants small spin Hall conductivity but sizable magnetic-octupole Hall conductivity, and fcc Pt, fcc Rh, fcc Pd, and bcc W as platforms where both spin and octupole Hall responses are large (Baek et al., 3 Jul 2025).

The practical motivation is torque generation in altermagnets. In heavy-metal/altermagnet bilayers, injected magnetic-octupole current is predicted to generate a magnetic-octupole torque with symmetry

$M'_i\coloneqq \frac{3M_{jji}-M_{ijj}}{\sqrt{10}}$5

distinct from conventional spin-orbit torque. In the tight-binding bilayer model used to verify this mechanism, the fitted coefficients are

$M'_i\coloneqq \frac{3M_{jji}-M_{ijj}}{\sqrt{10}}$6

in units of $M'_i\coloneqq \frac{3M_{jji}-M_{ijj}}{\sqrt{10}}$7, and the torque persists in configurations where the ordinary SHE-driven nonstaggered torque vanishes (Han et al., 2024).

4. Octupolar order as a source of Hall, Nernst, optical, and nonlinear responses

In several materials the octupole is not itself the transported current but the order parameter controlling Hall-active responses. Eu$M'_i\coloneqq \frac{3M_{jji}-M_{ijj}}{\sqrt{10}}$8Ir$M'_i\coloneqq \frac{3M_{jji}-M_{ijj}}{\sqrt{10}}$9O$d$0 (111) thin films provide a prominent example. The all-in–all-out and all-out–all-in states are described as ferroic magnetic octupole domains with vanishing net magnetic dipole moment, yet they support large zero-field optical Hall responses: magnetic circular dichroism of order $d$1 and Kerr rotation of order $d$2 rad. The paper relates Kerr rotation and circular dichroism to the optical Hall conductivity $d$3, and the domain-by-domain reversal of the CD sign is interpreted as reversal of the sign of $d$4 under AIAO $d$5 AOAI switching (Han et al., 6 May 2025).

A different route appears in hidden-order systems. For a cubic model of ferro-octupolar order motivated by Pr(Ti,V)$d$6Al$d$7, ordinary linear anomalous Hall effect vanishes by symmetry and second-order Hall effect is forbidden by inversion, so the leading Hall signature is third order. The relevant transport quantity is the Berry-curvature quadrupole

$d$8

and the experimental proposal is a third-harmonic Hall voltage

$d$9

in an AC Hall experiment. This response is odd in the ferro-octupolar order parameter $M_{ijk}=s_iQ_{jk},$0, strongly anisotropic, and dissipationless in the transverse sense (Sorn et al., 2023).

When ordinary AHE is symmetry-forbidden, the nonlinear magnetoelectric effect provides a closely related rank-matched probe. The induced magnetization

$M_{ijk}=s_iQ_{jk},$1

is a rank-3, axial, $M_{ijk}=s_iQ_{jk},$2-odd tensor with the same symmetry as magnetic octupoles. It is finite in both a $M_{ijk}=s_iQ_{jk},$3-wave altermagnet and a pyrochlore lattice with all-in/all-out order, and in the pyrochlore magnetic Weyl semimetal phase the intrinsic contribution is strongly enhanced because the response tensor involves the quantum metric, which is enhanced near Weyl points. This work argues that conventional low-rank probes such as AHE are generally ineffective for lowest-rank magnetic octupole order without further symmetry reduction, whereas NMEE and the third-order nonlinear Hall effect are symmetry-natural octupolar responses (Ōiké et al., 2024).

Noncollinear antiperovskites supply direct transport and thermoelectric examples. In Mn$M_{ijk}=s_iQ_{jk},$4Ni$M_{ijk}=s_iQ_{jk},$5Cu$M_{ijk}=s_iQ_{jk},$6N, Hall measurements in a coplanar magnetic-field geometry suppress the conventional dipole contribution and reveal a high-field in-plane anomalous Hall response with $M_{ijk}=s_iQ_{jk},$7 symmetry attributed to the octupole moment, together with a low-field topological-Hall-like feature associated with scalar spin chirality (Rajan et al., 2023). In Mn$M_{ijk}=s_iQ_{jk},$8NiN, scanning anomalous Nernst effect microscopy shows that the sign of the in-plane ANE components is determined by the local octupole macrodomain state, and the combined experimental and micromagnetic analysis suggests an average macrodomain of the order of $M_{ijk}=s_iQ_{jk},$9 (Johnson et al., 2022).

5. Octupoles of response space: ferromagnets, multilayers, and in-plane AHE

Another modern usage does not invoke octupolar magnetic order at all. Instead, the octupole resides in the angular structure of the Hall response. In epitaxial Fe and Ni thin films, the anomalous Hall conductivity vector is expanded in magnetization space as

$s_iQ_{jk}$0

Because $s_iQ_{jk}$1 is generally not parallel to $s_iQ_{jk}$2, the cubic term can generate an anomalous Hall conductivity component perpendicular to the magnetization and thereby produce an in-plane anomalous Hall effect in common cubic ferromagnets. For Fe(103) the predicted angular dependence is

$s_iQ_{jk}$3

for Ni(111) it is

$s_iQ_{jk}$4

and for Fe(001) it vanishes. In a 100-nm Fe(103) film the extracted anomalous Hall conductivities are $s_iQ_{jk}$5 for the conventional linear AHE and $s_iQ_{jk}$6 for the in-plane AHE (Peng et al., 2024).

In magnetic multilayers this response-space viewpoint is reformulated as a harmonic expansion. For $s_iQ_{jk}$7 multilayers with in-plane magnetization $s_iQ_{jk}$8, the Hall current is expanded in odd harmonics, with

$s_iQ_{jk}$9

Here $$\sigma^i_{\rm AHE}=\alpha \hat M_i+\beta \hat M_i^3,$$0 contains both dipole and octupole contributions, whereas $$\sigma^i_{\rm AHE}=\alpha \hat M_i+\beta \hat M_i^3,$$1 is purely octupolar. In $$\sigma^i_{\rm AHE}=\alpha \hat M_i+\beta \hat M_i^3,$$2 multilayers the $$\sigma^i_{\rm AHE}=\alpha \hat M_i+\beta \hat M_i^3,$$3 term accounts for nearly the entire anomalous Hall current and can be up to six times larger than the conventional coefficient $$\sigma^i_{\rm AHE}=\alpha \hat M_i+\beta \hat M_i^3,$$4. The effect is strongly thickness-dependent: for $$\sigma^i_{\rm AHE}=\alpha \hat M_i+\beta \hat M_i^3,$$5 the octupole contribution is more than twice $$\sigma^i_{\rm AHE}=\alpha \hat M_i+\beta \hat M_i^3,$$6, for $$\sigma^i_{\rm AHE}=\alpha \hat M_i+\beta \hat M_i^3,$$7 it is strongest, and for $$\sigma^i_{\rm AHE}=\alpha \hat M_i+\beta \hat M_i^3,$$8 it only slightly exceeds $$\sigma^i_{\rm AHE}=\alpha \hat M_i+\beta \hat M_i^3,$$9. A single Fe$C_{4v}$0Ag interface shows a much weaker octupole contribution, with maximum $C_{4v}$1 about $C_{4v}$2 of the conventional term (Niu et al., 26 Aug 2025).

Trigonal EuCd$C_{4v}$3Sb$C_{4v}$4 thin films provide a field-space analog. On the $C_{4v}$5 principal plane, the in-plane anomalous Hall response near zero field is described by an off-diagonal octupolar tensor component,

$C_{4v}$6

whereas in the forced ferromagnetic phase the dominant term becomes linear,

$C_{4v}$7

The low-field magneto-cubic dependence is observed not only in the paramagnetic phase but also in the antiferromagnetic phase. The corresponding coefficient shows unconventional decay above the magnetic ordering temperature,

$C_{4v}$8

roughly as the inverse temperature to the third power, while the transition to the forced ferromagnetic phase occurs above the in-plane saturation field

$C_{4v}$9

and the linear response persists up to $\hat s_i\hat r_j\hat r_k$00 (Nakamura et al., 29 Jul 2025). Here again the “octupole” is a tensor component of the Hall response rather than a real-space octupole order parameter.

6. Separate astrophysical usage: Hall-driven octupole generation in neutron-star crusts

In neutron-star physics the same phrase has a distinct meaning. The Hall effect there refers to electron-fluid Hall drift in the solid crust, governed by the induction equation with Hall and Ohmic terms, not to charge transport in a crystal. Under axial symmetry the magnetic field is written as

$\hat s_i\hat r_j\hat r_k$01

with $\hat s_i\hat r_j\hat r_k$02 the poloidal flux function and $\hat s_i\hat r_j\hat r_k$03 the toroidal current function. The long-time state is a Hall attractor characterized by electron isorotation,

$\hat s_i\hat r_j\hat r_k$04

analogous to Ferraro’s law (Gourgouliatos et al., 2013).

In that setting, “magnetic octupole Hall effect” does not denote a new Hall conductivity. It means that Hall evolution in the crust robustly generates and sustains an octupolar component of the magnetic field. For an initially dipole-dominated field, the attractor consists mainly of a poloidal dipole $\hat s_i\hat r_j\hat r_k$05, a poloidal octupole $\hat s_i\hat r_j\hat r_k$06, and a weak toroidal quadrupole $\hat s_i\hat r_j\hat r_k$07. The octupole is counter-aligned with the dipole, and for middle-aged neutron stars the predicted octupole-to-dipole ratio is about $\hat s_i\hat r_j\hat r_k$08. For realistic crust parameters, including thickness $\hat s_i\hat r_j\hat r_k$09 km, $\hat s_i\hat r_j\hat r_k$10, $\hat s_i\hat r_j\hat r_k$11, and initial surface field $\hat s_i\hat r_j\hat r_k$12 G, the evolution toward the attractor takes roughly $\hat s_i\hat r_j\hat r_k$13 Myr and can persist for a few Myr while $\hat s_i\hat r_j\hat r_k$14 (Gourgouliatos et al., 2013).

This astrophysical usage is therefore terminologically adjacent but conceptually separate from condensed-matter magnetic octupole Hall physics. It preserves the common idea that Hall dynamics naturally produce octupolar structure, but the octupole is a multipolar component of a macroscopic crustal magnetic field rather than a Bloch-band transport variable.

Across these literatures, the phrase designates not one effect but a cluster of Hall-related phenomena linked by the role of magnetic octupoles as bulk conjugate variables, transported multipoles, symmetry order parameters, or response harmonics. Direct comparison therefore depends on specifying whether the octupole resides in real-space magnetic order, in a Bloch-space current operator, in magnetization or field space, or in an astrophysical multipolar field decomposition.