Multipole Hall Effect Overview

• Multipole Hall effect is defined as Hall transport phenomena controlled by higher-order Berry curvature moments and magnetic multipoles beyond simple dipolar contributions.
• It employs nonlinear response theory where symmetry determines whether dipolar, quadrupolar, or higher multipoles drive the Hall response in complex materials.
• Experimental studies in materials like Fe3GaTe2 and Mn(Bi,Sb)2Te4 reveal scaling laws and harmonic behaviors consistent with multipolar contributions to electronic transport.

Multipole Hall effect denotes a family of Hall-type transport phenomena in which the Hall-active quantity is controlled by multipolar structure rather than only by the Brillouin-zone integral of Berry curvature or an ordinary magnetic dipole. In current usage, this includes nonlinear anomalous Hall responses governed by Berry-curvature dipoles, quadrupoles, hexapoles, and higher moments; anomalous and spin Hall effects generated by cluster or intersite magnetic multipoles; transverse transport of electric quadrupoles or magnetic octupoles; and layer-resolved Hall counterflows in twisted multilayers that form interlayer electric dipole or quadrupole patterns. A standard transport expansion is

$$j_a = \sigma_{ab} E_b + \chi_{abc} E_b E_c + \xi_{abcd} E_b E_c E_d + \cdots,$$

so that ordinary anomalous Hall transport is linear in $E$, whereas multipolar nonlinear Hall responses appear at higher order in the drive (Zhang et al., 2020, Dai et al., 23 Apr 2026).

1. Conceptual scope and definitions

The term is not used in a single uniform sense across the literature. One major usage concerns momentum-space multipoles of Berry curvature, where the nonlinear Hall hierarchy is organized by successive moments of $\Omega(\mathbf{k})$. A second usage concerns multipolar order parameters in magnetic crystals, especially cluster magnetic multipoles or toroidal multipoles that play the same symmetry role as ferromagnetic dipoles and thereby allow anomalous Hall conductivity even when the net magnetization is zero or nearly zero. A third usage concerns transport of multipole observables themselves, such as electric quadrupole or magnetic octupole current. A fourth usage concerns hidden Hall current structure in layer space, where the net charge Hall current vanishes but the layer-resolved Hall currents form an interlayer electric dipole or quadrupole pattern (Suzuki et al., 2016, Ko et al., 1 Aug 2025, Xiao et al., 26 Jun 2026).

In the Berry-phase formulation of metallic Hall transport, the intrinsic anomalous Hall effect is governed by the Brillouin-zone integral of Berry curvature. In the nonlinear generalization, the relevant geometric object is not only the integral of $\Omega(\mathbf{k})$, but also its spatial distribution in momentum space. This is the sense in which a Berry-curvature dipole can drive a second-order nonlinear Hall effect, a Berry-curvature quadrupole can drive a third-order effect, and higher moments can generate still higher harmonics (Zhang et al., 2020).

In the symmetry-based magnetic literature, the decisive point is that Hall conductivity transforms as a time-odd axial quantity. Cluster multipole theory therefore asks which macroscopic magnetic multipoles belong to the same irreducible representation as the corresponding dipole component. In this language, the anomalous Hall effect in non-collinear antiferromagnets such as Mn$_3$Ir and Mn$_3$Sn/Ge is characterized not by ordinary magnetization but by cluster octupole order (Suzuki et al., 2016).

2. Berry-curvature multipoles and nonlinear Hall hierarchy

A general intrinsic hierarchy was formulated for higher-order nonlinear anomalous Hall effects induced by Berry-curvature multipoles. In this framework, the $(n-1)$-th Berry-curvature moment controls the $n$-th order Hall response. The basic multipoles are

$M_\beta=\int_{\mathbf k} f_0\,\Omega_\beta,$

$$D_{\alpha\beta}=\int_{\mathbf k} f_0\,\partial_\alpha \Omega_\beta,$$

$E$0

$E$1

These correspond, respectively, to the Berry-curvature monopole, dipole, quadrupole, and hexapole (Zhang et al., 2020).

The resulting Hall conductivity at the $E$2-th harmonic was written as

$E$3

with $E$4 the corresponding $E$5-th Berry-curvature moment. This gives the compact rule

$E$6

The familiar special cases are the intrinsic anomalous Hall effect from the monopole, the second-order nonlinear Hall effect from the dipole, the third-order nonlinear anomalous Hall effect from the quadrupole, and the fourth-order effect from the hexapole (Zhang et al., 2020).

Model studies refined this hierarchy in concrete band structures. In Rashba systems with hexagonal warping, the second-order response is governed by a Berry-curvature dipole,

$E$7

while the third-order response is governed by the Berry connection polarizability tensor

$E$8

together with the induced Berry-curvature polarizability

$E$9

The same progression from dipolar to higher geometric response also appears in multi-Weyl semimetals, where the second-order Hall effect is controlled by the Berry-curvature dipole and the third-order effect by the Berry connection polarizability tensor (Saha et al., 2023, Roy et al., 2021).

Symmetry determines which moment is the leading Hall-active one. The magnetic-point-group analysis in the general theory showed that lower moments can be symmetry-forbidden, so that a quadrupole or hexapole becomes the first nonzero Hall-active quantity. This is why third-order or fourth-order Hall harmonics can be the leading Hall response in suitable antiferromagnets, topological-insulator surface states, Weyl systems, and twisted moiré platforms (Zhang et al., 2020).

3. Experimental nonlinear multipole Hall responses

A direct experimental realization of the quadrupolar nonlinear Hall scenario was reported in ferromagnetic Fe$\Omega(\mathbf{k})$0GaTe$\Omega(\mathbf{k})$1. In that system, the third-order nonlinear anomalous Hall effect appears in the third harmonic, the second-harmonic longitudinal and Hall resistances are nearly zero, and the zero-field third-harmonic Hall voltage obeys

$\Omega(\mathbf{k})$2

The third-harmonic Hall signal is hysteretic in field, switches near the same coercive field as the ordinary anomalous Hall effect, remains observable up to the Curie temperature $\Omega(\mathbf{k})$3 K, and shows a scaling behavior consistent with an intrinsic Berry-curvature quadrupole. The scaling analysis uses

$\Omega(\mathbf{k})$4

with $\Omega(\mathbf{k})$5 assigned to extrinsic scattering and $\Omega(\mathbf{k})$6 to the intrinsic Berry-curvature quadrupole contribution. At room temperature the extracted intrinsic term exceeds the scattering term, and the effect persists above room temperature in this ferromagnetic metal (Dai et al., 23 Apr 2026).

An extended odd-order hierarchy was subsequently observed in Mn(Bi$\Omega(\mathbf{k})$7Sb$\Omega(\mathbf{k})$8)$\Omega(\mathbf{k})$9Te$\Omega(\mathbf{k})$0 thin flakes. In that system, third-, fifth-, and seventh-order nonlinear Hall voltages were measured, with

$\Omega(\mathbf{k})$1

The higher odd-order nonlinear Hall voltage exhibits a twofold angular dependence, exists only below the Néel temperature, reaches its maximum near the charge neutral point, decays exponentially as the nonlinear order increases, and is observed in both odd- and even-layer samples with comparable magnitudes. The interpretation given is that the third-, fifth-, and seventh-order responses may arise from the Berry-curvature quadrupole, octapole, and dodecapole, respectively (Li et al., 23 Apr 2026).

At the dipolar end of the nonlinear hierarchy, a robust second-order nonlinear Hall effect has been observed in polycrystalline magnetic multilayers of composition

$\Omega(\mathbf{k})$2

from $\Omega(\mathbf{k})$3 K to $\Omega(\mathbf{k})$4 K. There the measured second-harmonic Hall voltage satisfies

$\Omega(\mathbf{k})$5

is frequency independent from $\Omega(\mathbf{k})$6 to $\Omega(\mathbf{k})$7 Hz, is unchanged for magnetic fields from $\Omega(\mathbf{k})$8 T to $\Omega(\mathbf{k})$9 T, and is accompanied by vanishingly small $_3$0. The scaling relation

$_3$1

yielded a dominant conductivity-independent term $_3$2, which was interpreted as the intrinsic Berry-curvature dipole contribution. Although this work does not explicitly use the term “multipole Hall effect,” it is a clear experimental example of the dipolar sector of the broader program (Kamal et al., 6 Jul 2026).

4. Magnetic multipolar order and Hall transport

Cluster multipole theory supplied an order-parameter language for anomalous Hall conductivity in magnets with little or no net magnetization. The formal cluster multipole moment on cluster $_3$3 is

$_3$4

and the macroscopic multipole is obtained by summing over symmetry-related clusters in the magnetic unit cell. In this framework, the anomalous Hall effect in Mn$_3$5Ir and Mn$_3$6Sn/Ge is characterized by cluster octupole magnetization in the same sense that the anomalous Hall effect in bcc Fe is characterized by dipole magnetization (Suzuki et al., 2016).

The same cluster-multipole logic was extended from $_3$7 orders to modulated magnetic structures with general propagation vector $_3$8. The symmetry-adapted magnetic bases are constructed by mapping cluster multipoles on a virtual cluster to the periodic crystal with the phase factor $_3$9. This makes it possible to identify Hall-active multi-$_3$0 antiferromagnetic structures. In $_3$1-Mn, one of the $_3$2 magnetic structures on the $_3$3 site is a noncollinear antiferromagnet without net magnetization but with the same symmetry as ferromagnetism, so it can support $_3$4. In Co$_3$5S$_3$6 ($_3$7Nb, Ta), single-$_3$8 antiferromagnetic states are Hall-forbidden because time reversal combined with a primitive translation survives, whereas a triple-$_3$9 state generated from $(n-1)$0 bases can induce AHE through an emergent $(n-1)$1, $(n-1)$2 multipole, equivalently described as $(n-1)$3 or $(n-1)$4 in the chiral crystal symmetry (Yanagi et al., 2022).

A related but distinct theoretical route is the spin-cluster Hall mechanism on the pyrochlore lattice. In the spin-ice double-exchange model, a spin-cluster phase made of noncoplanar four-spin tetrahedral molecules spontaneously breaks spatial inversion symmetry while remaining magnetically disordered in the conventional sense. That state was described as an intersite multipole order, and a nonzero spin Hall conductivity was obtained without explicit spin-orbit interaction. The central mechanism is

$(n-1)$5

(Ishizuka et al., 2013).

Altermagnets provided another explicit formulation of transport of multipole observables. In $(n-1)$6-wave altermagnets, a magnetic octupole moment was defined as

$(n-1)$7

that is, a spin-weighted electric quadrupole. Linear-response theory then yields Hall conductivity tensors not only for spin, but also for electric quadrupole and magnetic octupole currents. In this setting, a magnetic octupole Hall effect is a transverse flow of magnetic octupole moment induced by an electric field, and the paper further identified a sizable electric quadrupole Hall effect originating from quadrupole splittings in the band structure. A key result is that one class of magnetic octupole Hall responses remains symmetry-allowed even where the spin-splitter effect is forbidden, making octupole transport a distinct response channel rather than a trivial decoration of spin transport (Ko et al., 1 Aug 2025).

5. Layer-resolved, real-space, and unconventional multipole Hall variants

Twisted multilayers realize a layer-space version of the multipole Hall effect. In that setting the total charge Hall conductivity vanishes by time-reversal symmetry,

$(n-1)$8

yet individual layers carry nonzero Hall currents

$(n-1)$9

For mirror-symmetric twisted trilayer graphene, the relevant interlayer multipole operators are

$n$0

The symmetry-enforced current pattern

$n$1

is precisely a pure interlayer electric quadrupole Hall effect, with no dipole component. The associated in-plane magnetic multipoles follow from the layer-separated Hall currents (Xiao et al., 26 Jun 2026).

A real-space version was proposed for a two-dimensional electron gas under an odd magnetic field profile,

$n$2

Because the Hamiltonian is even under $n$3, $n$4 is odd, and $n$5 is even, the linear Hall response vanishes identically, whereas second-order Hall response survives. The resulting system exhibits a purely nonlinear Hall effect dominated by boundary or snake states near the sign changes of $n$6, rather than by a bulk Berry-curvature dipole. In the periodic structure studied there, the effect is described as a real-space magnetic dipole Hall mechanism and can be resonantly enhanced by matching the characteristic magnetic width $n$7 to the cyclotron scale $n$8 (Huang et al., 2021).

Another field-induced geometric variant is the anomalous planar Hall effect in two-dimensional trigonal crystals. There the linear anomalous planar Hall response is driven by Zeeman-induced Berry curvature, while the nonlinear anomalous planar Hall effect is governed by the Berry-curvature dipole

$n$9

with nonlinear susceptibility

$M_\beta=\int_{\mathbf k} f_0\,\Omega_\beta,$0

This places the nonlinear anomalous planar Hall effect in the dipolar sector of multipole Hall physics, although the underlying control parameter is an in-plane Zeeman field rather than spontaneous crystal multipolar order (Battilomo et al., 2020).

6. Boundaries of the concept, interpretation, and open issues

The literature also marks clear conceptual boundaries. High-order multipole radiation from quantum Hall states in Dirac materials is an optical consequence of Hall-state topology, not a Hall effect in the transport sense. In that work, the observables are OAM-resolved spontaneous-emission rates $M_\beta=\int_{\mathbf k} f_0\,\Omega_\beta,$1, branching ratios, and far-field multipole channels with orders exceeding $M_\beta=\int_{\mathbf k} f_0\,\Omega_\beta,$2; the authors explicitly distinguish this from Hall voltages or multipolar Hall conductivity tensors. It is therefore related to multipolar manifestations of Hall physics, but not itself a multipole Hall effect in the transport sense (Gullans et al., 2017).

A second boundary concerns terminology. Several experimental and theoretical papers analyze dipolar or higher-order nonlinear Hall transport without explicitly using the phrase “multipole Hall effect.” The polycrystalline magnetic multilayer study is a clear example: it is described as Berry-curvature-dipole-driven nonlinear Hall transport rather than with explicit multipole Hall nomenclature, even though the underlying object is the first moment of Berry curvature (Kamal et al., 6 Jul 2026). This suggests that the phrase functions more as a unifying classification than as a universally adopted label.

A third issue is microscopic attribution. In Fe$M_\beta=\int_{\mathbf k} f_0\,\Omega_\beta,$3GaTe$M_\beta=\int_{\mathbf k} f_0\,\Omega_\beta,$4, the Berry-curvature quadrupole assignment is experimentally inferred from cubic current scaling, third-harmonic detection, hysteresis, and conductivity scaling, but the paper does not present an explicit microscopic Brillouin-zone calculation of the quadrupole tensor (Dai et al., 23 Apr 2026). In Mn(Bi$M_\beta=\int_{\mathbf k} f_0\,\Omega_\beta,$5Sb$M_\beta=\int_{\mathbf k} f_0\,\Omega_\beta,$6)$M_\beta=\int_{\mathbf k} f_0\,\Omega_\beta,$7Te$M_\beta=\int_{\mathbf k} f_0\,\Omega_\beta,$8, the higher odd-order responses are likewise interpreted as arising from Berry-curvature multipoles, but the paper phrases that interpretation cautiously as a likely origin rather than a complete microscopic proof (Li et al., 23 Apr 2026). These cases show that multipole Hall assignments often rest on a combination of symmetry analysis, harmonic order, and scaling behavior rather than direct tensor reconstruction.

Taken together, the literature suggests a broad but coherent picture. The multipole Hall effect is not a single mechanism but a symmetry-organized class of Hall responses in which the Hall-active quantity is a multipole: a moment of Berry curvature, a cluster magnetic multipole, a transported octupole or quadrupole, an interlayer electric multipole, or a real-space magnetic multipole texture. The common principle is that transverse transport is controlled by internal structure beyond the lowest dipolar descriptor, and that symmetry can elevate higher multipoles from small corrections to the leading Hall-active objects.