Electric Quadrupole Hall Effect

• Electric Quadrupole Hall Effect is a class of Hall-like responses driven by rank-2 quadrupolar interactions, observed in systems such as spin-3/2 heavy holes and altermagnets.
• It involves diverse mechanisms including nonlinear anomalous Hall currents, Berry-curvature quadrupoles, and interlayer current patterns that extend conventional Hall physics.
• These phenomena yield measurable signatures like second-order Hall currents in GaAs wells, third-harmonic voltages, and pure quadrupolar responses in twisted multilayers.

Electric quadrupole Hall effect denotes a family of Hall-like transverse responses in which the central object is quadrupolar rather than merely scalar charge or dipolar polarization. Across current literature, the phrase is used for several distinct but related constructions: a nonlinear anomalous charge Hall current generated by an electric-field-induced quadrupolar coupling in spin-$3/2$ heavy holes; a transverse current of electric quadrupole moments in altermagnets; a third-order anomalous Hall response controlled by a Berry-curvature quadrupole in momentum space; an interlayer Hall-current pattern with pure electric quadrupole character in twisted multilayers; and, in broader analogical usage, quadrupole-controlled Hall-viscous or photonic directional responses (Gholizadeh et al., 2022, Ko et al., 1 Aug 2025, Sorn et al., 2023, Xiao et al., 26 Jun 2026, Haldane, 2023, Vázquez-Lozano et al., 2019).

1. Terminology and conceptual scope

The literature does not assign a single universal definition to the term. Instead, “electric quadrupole Hall effect” identifies Hall phenomena in which a rank-2 quadrupolar structure is the operative degree of freedom, but that structure may live in different spaces: internal spin space, orbital space, momentum space, layer space, or hydrodynamic/geometric response.

Setting	Quadrupolar object	Hall observable
2D heavy holes	$\{J_i,J_j\}$ coupling projected to $H_\lambda$	Nonlinear anomalous charge Hall current $j_y=\chi_{yxx}E_x^2$ (Gholizadeh et al., 2022)
$d$-wave altermagnets	Orbital quadrupole $Q_{xy}\propto \{L_x,L_y\}$	Current of electric quadrupole moments $j_i^{Q_{xy}}$ (Ko et al., 1 Aug 2025)
TR-broken metals	Berry-curvature quadrupole $Q_{abc}$	Third-order anomalous Hall response, $V_H(3\omega)$ (Sorn et al., 2023)
Twisted trilayers	Interlayer current distribution $+1,-2,+1$	Pure interlayer electric quadrupole Hall current (Xiao et al., 26 Jun 2026)
Quantum Hall fluids	Bulk quadrupole density $\{J_i,J_j\}$0	Hall-viscous and bound-charge response (Haldane, 2023)
Near-field optics	Electric quadrupole tensor $\{J_i,J_j\}$1	Lateral, unidirectional guided-mode coupling (Vázquez-Lozano et al., 2019)

A recurring source of confusion is that the word “quadrupole” does not refer to the same entity in each setting. In the heavy-hole problem it is a symmetry-allowed electric-dipole coupling to a spin quadrupole; in altermagnets it is an orbital multipole that is itself transported; in Berry-curvature-quadrupole theories it is a second moment of $\{J_i,J_j\}$2; in twisted multilayers it is the out-of-plane multipole structure of layer-resolved Hall currents. These constructions are conceptually adjacent but not interchangeable.

2. Quadrupole-driven nonlinear anomalous Hall effect in spin-$\{J_i,J_j\}$3 heavy holes

A concrete electronic realization appears in a symmetric GaAs (001) quantum well with two-dimensional heavy holes, where the low-energy sector is the $\{J_i,J_j\}$4 heavy-hole doublet projected to an effective pseudospin-$\{J_i,J_j\}$5 description (Gholizadeh et al., 2022). The Hall signal is second order in the in-plane electric field,

$\{J_i,J_j\}$6

and is studied in the presence of an in-plane magnetic field $\{J_i,J_j\}$7 and an out-of-plane Zeeman field $\{J_i,J_j\}$8.

The essential symmetry input is the non-centrosymmetric tetrahedral $\{J_i,J_j\}$9 structure of zincblende GaAs. In the full $H_\lambda$0 basis, the uniform in-plane electric field couples as

$H_\lambda$1

with $H_\lambda$2 and $H_\lambda$3 rank-2 tensor combinations of spin-$H_\lambda$4 operators. After Schrieffer-Wolff projection to the heavy-hole subspace, this becomes

$H_\lambda$5

and for $H_\lambda$6,

$H_\lambda$7

The effect is therefore an electric-field-induced correction to the in-plane $H_\lambda$8-tensor, linear in $H_\lambda$9, quadratic in momentum, and proportional to the perpendicular Zeeman scale $j_y=\chi_{yxx}E_x^2$0. In the language used there, it is an electric quadrupole correction to the $j_y=\chi_{yxx}E_x^2$1-tensor.

This quadrupolar term is not a mere auxiliary detail: the analytic second-order Hall coefficient for $j_y=\chi_{yxx}E_x^2$2,

$j_y=\chi_{yxx}E_x^2$3

is explicitly linear in $j_y=\chi_{yxx}E_x^2$4. Setting $j_y=\chi_{yxx}E_x^2$5 removes the effect in that model. The mechanism is therefore quadrupole-driven at the Hamiltonian level rather than being organized primarily as a Berry-curvature-multipole response.

The same work gives concrete scales. For a $j_y=\chi_{yxx}E_x^2$6 nm GaAs well, $j_y=\chi_{yxx}E_x^2$7. With $j_y=\chi_{yxx}E_x^2$8 nm, $j_y=\chi_{yxx}E_x^2$9, $d$0 ps, $d$1 kV/m, and $d$2 T, the predicted nonlinear Hall current density is $d$3 for $d$4 meV and $d$5 for $d$6 meV; the corresponding Hall voltages across $d$7 are about $d$8 nV and $d$9, respectively. The proposed detection protocol is a low-frequency AC drive $Q_{xy}\propto \{L_x,L_y\}$0, with the Hall response at $Q_{xy}\propto \{L_x,L_y\}$1 isolated by lock-in detection.

3. Hall transport of electric quadrupole moments in $Q_{xy}\propto \{L_x,L_y\}$2-wave altermagnets

In a different and more literal usage, the electric quadrupole Hall effect is a transverse current of electric quadrupole moments generated by an electric field in $Q_{xy}\propto \{L_x,L_y\}$3-wave altermagnets (Ko et al., 1 Aug 2025). The transported quantity is not charge current but the orbital electric quadrupole itself. The central conductivity component is

$Q_{xy}\propto \{L_x,L_y\}$4

describing a current of $Q_{xy}\propto \{L_x,L_y\}$5 in the $Q_{xy}\propto \{L_x,L_y\}$6-direction induced by $Q_{xy}\propto \{L_x,L_y\}$7.

The quadrupole operators are built in orbital space from the $Q_{xy}\propto \{L_x,L_y\}$8-orbital manifold. Starting from the standard electric multipole operator $Q_{xy}\propto \{L_x,L_y\}$9 with $j_i^{Q_{xy}}$0, the Cartesian quadrupoles are represented through symmetrized products of orbital angular momentum operators,

$j_i^{Q_{xy}}$1

The low-energy tight-binding Hamiltonian acts in spin $j_i^{Q_{xy}}$2 sublattice $j_i^{Q_{xy}}$3 orbital space,

$j_i^{Q_{xy}}$4

with

$j_i^{Q_{xy}}$5

The key quadrupolar splitting is

$j_i^{Q_{xy}}$6

which creates antiferroic electric quadrupole order $j_i^{Q_{xy}}$7 in real space and a quadrupole-split band structure in momentum space.

Transport is formulated by a generalized Kubo response,

$j_i^{Q_{xy}}$8

with the quadrupole-current operator

$j_i^{Q_{xy}}$9

The response relation is correspondingly

$Q_{abc}$0

Symmetry is decisive. For the spin point group $Q_{abc}$1, the allowed quadrupole conductivity tensor for $Q_{abc}$2 has only off-diagonal in-plane entries, so the quadrupole Hall effect is symmetry-allowed even though an ordinary anomalous charge Hall effect remains constrained in a centrosymmetric collinear antiferromagnet. The work further states that the electric quadrupole Hall effect is “sizable” and that it always coexists with the spin-splitter conductivity because spin and electric quadrupole splittings are intrinsically linked through the underlying orbital splitting. The material candidates emphasized are rutile $Q_{abc}$3 and $Q_{abc}$4.

This usage is distinct from nonlinear anomalous charge Hall transport. Here the Hall current carries a rank-2 electric multipole, and the closest comparison drawn in the source is to the orbital torsion Hall effect, but in a time-reversal-broken altermagnetic setting rather than a centrosymmetric nonmagnetic one.

4. Berry-curvature quadrupoles and third-order Hall responses

A third line of development identifies the quadrupole not in an internal operator but in momentum-space geometry: a Berry-curvature quadrupole can be the leading source of anomalous Hall response when Berry-curvature monopole and dipole moments vanish by symmetry (Sorn et al., 2023, Korrapati et al., 2024). The defining tensor is

$Q_{abc}$5

or, after integration by parts,

$Q_{abc}$6

In the cubic model for ferro-octupolar order inspired by $Q_{abc}$7, the order parameter $Q_{abc}$8 gaps symmetry-protected line and point degeneracies and produces Berry-curvature hot spots with zero net Berry curvature but nonzero quadrupolar distribution (Sorn et al., 2023). The magnetic point group is $Q_{abc}$9, which forbids linear anomalous Hall conductivity and the second-order Hall effect but allows a third-order Hall tensor with one independent component controlled by $V_H(3\omega)$0. The Hall-like part of the third-order conductivity at $V_H(3\omega)$1 is

$V_H(3\omega)$2

The measurable consequence is a third-harmonic Hall voltage $V_H(3\omega)$3 with cubic current-voltage scaling, $V_H(3\omega)$4. The same analysis emphasizes anisotropy and dissipationlessness, and estimates Berry-curvature quadrupoles $V_H(3\omega)$5.

The thermal extension keeps the same symmetry logic but replaces the electric field by $V_H(3\omega)$6. In systems where Berry-curvature monopole and dipole vanish, the leading anomalous thermal Hall and Nernst coefficients are third order in the longitudinal temperature gradient and are governed by thermal counterparts of the Berry-curvature quadrupole (Korrapati et al., 2024). The paper derives cubic heat and charge currents of the form

$V_H(3\omega)$7

$V_H(3\omega)$8

with all corresponding coefficients proportional to $V_H(3\omega)$9. Their low-temperature asymptotics are

$+1,-2,+1$0

In two dimensions, the magnetic point groups identified as allowing Berry-curvature quadrupole or its thermal counterparts as the leading Berry-curvature moment are $+1,-2,+1$1, $+1,-2,+1$2, and $+1,-2,+1$3.

This Berry-curvature-quadrupole usage differs from the heavy-hole mechanism. There the crucial multipole object is a quadrupolar term in the Hamiltonian and the $+1,-2,+1$4-tensor; here it is a quadrupolar moment of $+1,-2,+1$5 itself.

5. Interlayer electric quadrupole Hall effect in twisted multilayers

In twisted multilayers, the relevant quadrupole is the out-of-plane multipole structure of layer-resolved Hall currents rather than an internal orbital moment or a momentum-space Berry-curvature moment (Xiao et al., 26 Jun 2026). Because time-reversal symmetry enforces zero net linear charge Hall current, the total Hall conductivity satisfies

$+1,-2,+1$6

but individual layers can still carry nonzero transverse currents. This is the layer Hall effect.

Mirror-symmetric twisted trilayer graphene realizes a pure interlayer electric quadrupole Hall effect. Out-of-plane mirror symmetry $+1,-2,+1$7 imposes

$+1,-2,+1$8

and together with $+1,-2,+1$9 gives

$\{J_i,J_j\}$00

The corresponding current pattern is

$\{J_i,J_j\}$01

so the net Hall current vanishes and the interlayer dipole moment cancels, but a finite quadrupole remains.

This is formalized by introducing interlayer electric dipole and quadrupole operators in layer space,

$\{J_i,J_j\}$02

which define dipole and quadrupole Hall conductivities

$\{J_i,J_j\}$03

In mirror-symmetric trilayers, $\{J_i,J_j\}$04 while $\{J_i,J_j\}$05, so the Hall response is purely quadrupolar.

The continuum moiré model further shows that the mirror-even sector reduces to an effective twisted-bilayer problem with enhanced tunneling, while the mirror-odd sector is a decoupled Dirac cone and does not contribute to the layer Hall response. This makes the quadrupole Hall effect intelligible as the superposition of two interface contributions whose dipole pieces cancel and whose quadrupole pieces add.

Quantitatively, for twist angles in the $\{J_i,J_j\}$06–$\{J_i,J_j\}$07 range, layer Hall conductivities often lie around $\{J_i,J_j\}$08–$\{J_i,J_j\}$09 in energy windows where interlayer coupling is strong. At small twist angles, interlayer translation strongly tunes the response; at large angles and low doping, the middle-layer Hall conductivity satisfies

$\{J_i,J_j\}$10

and follows the scaling

$\{J_i,J_j\}$11

with $\{J_i,J_j\}$12 and $\{J_i,J_j\}$13 meV.

The current distribution also generates in-plane magnetic multipoles. For the quadrupole channel,

$\{J_i,J_j\}$14

Using $\{J_i,J_j\}$15 V/m, $\{J_i,J_j\}$16 Å, and $\{J_i,J_j\}$17, the estimated magnitude is $\{J_i,J_j\}$18. The same work proposes that the large-angle interface-decomposition picture extends to more general twisted multilayers, allowing interlayer electric multipole Hall effects beyond quadrupole order.

6. Broader theoretical frameworks and analogues

Several adjacent frameworks do not always use the exact term, but they supply theoretical structures that directly support quadrupole Hall physics. One is the thermodynamic $\{J_i,J_j\}$19-tensor formalism for polarization response in insulators (Onishi et al., 2021). There the quadrupole tensor is defined by

$\{J_i,J_j\}$20

and satisfies

$\{J_i,J_j\}$21

The associated susceptibilities obey

$\{J_i,J_j\}$22

The paper explicitly frames these relations as analogues of the Středa and Mott relations for Hall and Nernst responses. It does not construct a transverse quadrupole Hall conductivity, but a plausible implication is that cross-components or antisymmetric parts of $\{J_i,J_j\}$23 and its derivatives furnish a natural language for such an extension.

A second framework treats incompressible quantum Hall fluids as electric quadrupole fluids (Haldane, 2023). The primitive quadrupole tensor is

$\{J_i,J_j\}$24

and the bulk quadrupole density $\{J_i,J_j\}$25 generates

$\{J_i,J_j\}$26

The central Hall-geometric consequence is the generic Hall-viscosity formula

$\{J_i,J_j\}$27

with $\{J_i,J_j\}$28. In this usage, the “Hall” aspect is a dissipationless transverse stress response governed by an intrinsic electric quadrupole density of the Hall fluid.

A third analogue arises in near-field optics (Vázquez-Lozano et al., 2019). That work does not use the phrase “electric quadrupole Hall effect,” but it generalizes the spin–momentum-locking picture from circular dipoles to electric quadrupoles and higher multipoles. For a pure electric quadrupole,

$\{J_i,J_j\}$29

so the effective source is explicitly momentum dependent. For the tensor with non-zero elements

$\{J_i,J_j\}$30

the $\{J_i,J_j\}$31-polarized evanescent angular spectrum is strongly asymmetric in $\{J_i,J_j\}$32, giving unidirectional coupling to a guided mode. The lateral, spin-dependent energy flow is therefore a photonic quadrupolar analogue of Hall-like directionality.

Taken together, these frameworks show that the electric quadrupole Hall effect is best understood as a class of transverse response phenomena rather than a single transport coefficient. The unifying element is the operative role of a rank-2 electric quadrupole object—embedded in a Hamiltonian, carried by a current, encoded in Berry-curvature geometry, distributed across layers, or promoted to a bulk hydrodynamic field. The principal distinction across subfields is therefore not whether the effect is “Hall,” but what precisely is meant by “quadrupole.”