Third-Order Hall Effect in Condensed Matter

• Third-Order Hall Effect is a cubic nonlinear response defined by a rank-4 conductivity tensor that captures intrinsic Berry connection polarizability and extrinsic scattering contributions.
• It serves as a sensitive probe of higher-order quantum geometry, revealing symmetry breaking, CDW transitions, and field-induced Berry curvature modifications in various quantum materials.
• Experimental signatures include a cubic current-voltage law and distinct angular modulations of the third-harmonic voltage, providing insights for device applications and fundamental studies.

The third-order Hall effect is a cubic transverse transport response in which the current contains a term $j_a^{(3)}=\sigma^{(3)}_{abcd}E_bE_cE_d$, or, under an ac drive, a third-harmonic Hall signal such as $V_{3\omega}^{\perp}\propto I_\omega^3$. In contemporary condensed-matter literature, the term covers several closely related phenomena: zero-field nonlinear Hall responses in time-reversal-symmetric conductors governed by Berry connection polarizability, third-order anomalous Hall responses in time-reversal-broken systems governed by higher-order field-induced Berry curvature, and even optical $\chi^{(3)}$ Hall analogues in synthetic gauge fields (Liu et al., 2021, Xiang et al., 2022, Nandi et al., 2 Oct 2025). Across these realizations, the effect is used as a transport probe of higher-order quantum geometry.

1. Definition and tensorial structure

A standard starting point is the field expansion of the current density,

$$j_a(t)=\sigma^{(1)}_{ab}E_b(t)+\sigma^{(2)}_{abc}E_b(t)E_c(t)+\sigma^{(3)}_{abcd}E_b(t)E_c(t)E_d(t)+\cdots,$$

with the third-order Hall component at frequency $3\omega$ given by

$$j_a^{(3\omega)}=\sigma^{(3)}_{abcd}E_b(\omega)E_c(\omega)E_d(\omega)e^{i3\omega t}.$$

In transport experiments this is commonly accessed by driving an ac current $I_\omega$ and detecting a transverse third-harmonic voltage $V_{3\omega}^{\perp}$, which obeys the cubic law $V_{3\omega}^{\perp}\propto I_\omega^3$ when the response is genuinely third order (Chen et al., 25 Jan 2025).

The same content can be written in a more general conductivity notation,

$J_i^{(3)}=\sigma^{(3)}_{ijkl}E_jE_kE_l,$

and the Hall-type part can be isolated by antisymmetrizing the current index with one field index,

$V_{3\omega}^{\perp}\propto I_\omega^3$0

For ac current excitation in a Hall bar, one may also work directly with a third-harmonic voltage. In an 8-nm RuO$V_{3\omega}^{\perp}\propto I_\omega^3$1 Hall bar, for example, the third-harmonic voltage is written as

$V_{3\omega}^{\perp}\propto I_\omega^3$2

with the corresponding third-order conductivity extracted from geometry and current amplitude (Chen et al., 15 Apr 2026).

This tensorial formulation clarifies two points. First, the effect is not restricted to one microscopic mechanism: the same rank-4 conductivity may receive intrinsic geometric and extrinsic disorder-mediated contributions. Second, the experimentally relevant object is usually not the full tensor but a symmetry-selected subset, such as $V_{3\omega}^{\perp}\propto I_\omega^3$3 for an in-plane drive along $V_{3\omega}^{\perp}\propto I_\omega^3$4 and a transverse response along $V_{3\omega}^{\perp}\propto I_\omega^3$5 (Esin et al., 12 Feb 2025).

2. Quantum-geometric mechanisms

In the time-reversal-symmetric nonlinear Hall literature, the central microscopic object is the Berry connection polarizability (BCP). In one common notation, the applied field induces a first-order correction to the Berry connection,

$V_{3\omega}^{\perp}\propto I_\omega^3$6

with

$V_{3\omega}^{\perp}\propto I_\omega^3$7

This $V_{3\omega}^{\perp}\propto I_\omega^3$8 is gauge invariant and symmetric, and it generates a field-induced Berry-curvature correction through $V_{3\omega}^{\perp}\propto I_\omega^3$9 (Liu et al., 2021). In a Boltzmann treatment the third-order conductivity splits into an $\chi^{(3)}$0 contribution controlled by derivatives of $\chi^{(3)}$1 and an $\chi^{(3)}$2 Drude-like term. The same framework was developed for Rashba systems with hexagonal warping, where the conductivity is written as $\chi^{(3)}$3, with $\chi^{(3)}$4 and $\chi^{(3)}$5 (Saha et al., 2023).

A related notation emphasizes a higher-rank BCP tensor $\chi^{(3)}$6. For 1$\chi^{(3)}$7-VSe$\chi^{(3)}$8, the intrinsic part of the third-order conductivity is written as

$\chi^{(3)}$9

with $$j_a(t)=\sigma^{(1)}_{ab}E_b(t)+\sigma^{(2)}_{abc}E_b(t)E_c(t)+\sigma^{(3)}_{abcd}E_b(t)E_c(t)E_d(t)+\cdots,$$0 defined from the field response of the Berry connection. In that formulation the total third-order Hall response is the sum of this intrinsic geometric term and extrinsic skew-scattering and side-jump terms (Chen et al., 25 Jan 2025).

For Weyl and multi-Weyl semimetals, the same BCP physics acquires a topological scaling structure. In a low-energy model with monopole charge $$j_a(t)=\sigma^{(1)}_{ab}E_b(t)+\sigma^{(2)}_{abc}E_b(t)E_c(t)+\sigma^{(3)}_{abcd}E_b(t)E_c(t)E_d(t)+\cdots,$$1, the BCP tensor $$j_a(t)=\sigma^{(1)}_{ab}E_b(t)+\sigma^{(2)}_{abc}E_b(t)E_c(t)+\sigma^{(3)}_{abcd}E_b(t)E_c(t)E_d(t)+\cdots,$$2 forms characteristic multipolar patterns in momentum space, and the overall magnitude of the third-order Hall conductivity grows strongly with $$j_a(t)=\sigma^{(1)}_{ab}E_b(t)+\sigma^{(2)}_{abc}E_b(t)E_c(t)+\sigma^{(3)}_{abcd}E_b(t)E_c(t)E_d(t)+\cdots,$$3; numerically, the dimensionless third-order Hall conductivity grows by roughly one order of magnitude between $$j_a(t)=\sigma^{(1)}_{ab}E_b(t)+\sigma^{(2)}_{abc}E_b(t)E_c(t)+\sigma^{(3)}_{abcd}E_b(t)E_c(t)E_d(t)+\cdots,$$4 and $$j_a(t)=\sigma^{(1)}_{ab}E_b(t)+\sigma^{(2)}_{abc}E_b(t)E_c(t)+\sigma^{(3)}_{abcd}E_b(t)E_c(t)E_d(t)+\cdots,$$5 for the parameters shown in Fig. 6 of the paper (Roy et al., 2021).

Time-reversal-broken systems admit a distinct but related hierarchy. In the generalized semiclassical theory for the third-order intrinsic anomalous Hall effect, the key quantity is the second-order field-dependent Berry curvature arising from the second-order field-induced positional shift. The intrinsic current is written as

$$j_a(t)=\sigma^{(1)}_{ab}E_b(t)+\sigma^{(2)}_{abc}E_b(t)E_c(t)+\sigma^{(3)}_{abcd}E_b(t)E_c(t)E_d(t)+\cdots,$$6

with

$$j_a(t)=\sigma^{(1)}_{ab}E_b(t)+\sigma^{(2)}_{abc}E_b(t)E_c(t)+\sigma^{(3)}_{abcd}E_b(t)E_c(t)E_d(t)+\cdots,$$7

where $$j_a(t)=\sigma^{(1)}_{ab}E_b(t)+\sigma^{(2)}_{abc}E_b(t)E_c(t)+\sigma^{(3)}_{abcd}E_b(t)E_c(t)E_d(t)+\cdots,$$8 is built from the second-order Berry-connection polarizability tensor $$j_a(t)=\sigma^{(1)}_{ab}E_b(t)+\sigma^{(2)}_{abc}E_b(t)E_c(t)+\sigma^{(3)}_{abcd}E_b(t)E_c(t)E_d(t)+\cdots,$$9 (Xiang et al., 2022). In altermagnets this formulation is further sharpened into a Fermi-surface expression involving the second-order BPT and its decomposition into a Berry curvature quadrupole (BCQ), acceleration quantum metric dipole (AQMD), and three-state quantum metric dipole (TQMD); the resonant third-order intrinsic anomalous Hall effect in the Lieb-lattice altermagnet and V$3\omega$0Se$3\omega$1O is reported to be overwhelmingly dominated by BCQ (Xiang et al., 29 Apr 2026).

A further extension appears in centrosymmetric ferromagnets. In Fe$3\omega$2GaTe$3\omega$3, the reported room-temperature third-order Hall response is attributed to the dipole of the second-order Berry-connection polarizability, whose dominant part is identified as a symplectic connection. The working expression is

$3\omega$4

with the symplectic-connection contribution exceeding $3\omega$5 of the SBCP dipole in the first-principles analysis (Cao et al., 22 Apr 2026).

3. Symmetry constraints and symmetry breaking

Symmetry determines whether a third-order Hall tensor element is allowed, which tensor components survive, and what angular harmonics appear in experiment. These constraints are not uniform across all settings.

For nonmagnetic 2D crystals in the BCP framework, time reversal forbids the linear Hall effect, inversion or an in-plane twofold axis forbids the second-order nonlinear Hall effect, and threefold or sixfold symmetry can further eliminate the third-order Hall effect. In that setting the minimal requirement is that no in-plane inversion or threefold symmetry remains (Liu et al., 2021). The surface states of a warped topological insulator provide an explicit example: hexagonal warping alone preserves $3\omega$6, but a tilt term $3\omega$7 lowers the symmetry and allows the third-order Hall effect to become the leading Hall response (Nag et al., 2022).

The 1$3\omega$8-VSe$3\omega$9 case shows how collective order can unlock the effect. In the high-temperature $$j_a^{(3\omega)}=\sigma^{(3)}_{abcd}E_b(\omega)E_c(\omega)E_d(\omega)e^{i3\omega t}.$$0 phase, $$j_a^{(3\omega)}=\sigma^{(3)}_{abcd}E_b(\omega)E_c(\omega)E_d(\omega)e^{i3\omega t}.$$1 rotation and in-plane mirrors forbid second- and third-order Hall response at zero field. Below the incommensurate CDW transition at approximately $$j_a^{(3\omega)}=\sigma^{(3)}_{abcd}E_b(\omega)E_c(\omega)E_d(\omega)e^{i3\omega t}.$$2K, the point group is lowered to a monoclinic subgroup, $$j_a^{(3\omega)}=\sigma^{(3)}_{abcd}E_b(\omega)E_c(\omega)E_d(\omega)e^{i3\omega t}.$$3 is broken, and the BCP tensor acquires nonzero components transforming as $$j_a^{(3\omega)}=\sigma^{(3)}_{abcd}E_b(\omega)E_c(\omega)E_d(\omega)e^{i3\omega t}.$$4 (Chen et al., 25 Jan 2025).

Interface engineering provides a second route. In 2H-NbS$$j_a^{(3\omega)}=\sigma^{(3)}_{abcd}E_b(\omega)E_c(\omega)E_d(\omega)e^{i3\omega t}.$$5 ($$j_a^{(3\omega)}=\sigma^{(3)}_{abcd}E_b(\omega)E_c(\omega)E_d(\omega)e^{i3\omega t}.$$6) and orthorhombic SnS ($$j_a^{(3\omega)}=\sigma^{(3)}_{abcd}E_b(\omega)E_c(\omega)E_d(\omega)e^{i3\omega t}.$$7), symmetry forbids a net third-order transverse current, but in the misfit layer compound $$j_a^{(3\omega)}=\sigma^{(3)}_{abcd}E_b(\omega)E_c(\omega)E_d(\omega)e^{i3\omega t}.$$8 the alternate stacking lowers the in-plane symmetry to $$j_a^{(3\omega)}=\sigma^{(3)}_{abcd}E_b(\omega)E_c(\omega)E_d(\omega)e^{i3\omega t}.$$9 or $I_\omega$0 and breaks mirror constraints, thereby allowing the combinations of BCP indices that feed into a transverse $I_\omega$1 (Li et al., 2024).

Centrosymmetry does not, by itself, eliminate the third-order Hall effect. In NiTe$I_\omega$2, both inversion and time-reversal symmetry are preserved in the bulk, which forces the Berry-curvature dipole to vanish and suppresses the second-order Hall response, but symmetry still allows a nonzero third-order Hall tensor in the $I_\omega$3 point group, including components such as $I_\omega$4 (Esin et al., 12 Feb 2025).

In magnetic systems the symmetry classification changes again. The generalized semiclassical theory identifies 15 time-reversal-broken 3D magnetic point groups that support third-order intrinsic anomalous Hall effect as the leading contribution (Xiang et al., 2022). With spin-orbit coupling included, the spin-group analysis for altermagnets finds ten spin Laue groups in which the third-order intrinsic anomalous Hall effect is generically allowed (Xiang et al., 29 Apr 2026). In centrosymmetric Fe$I_\omega$5GaTe$I_\omega$6 with point group $I_\omega$7, symmetry permits only $I_\omega$8, which leads to the reported current-direction-independent response (Cao et al., 22 Apr 2026).

4. Experimental signatures and scaling analysis

The primary experimental hallmark is the cubic law. In 1$I_\omega$9-VSe$V_{3\omega}^{\perp}$0, NiTe$V_{3\omega}^{\perp}$1, TaIrTe$V_{3\omega}^{\perp}$2, and other systems, the measured transverse third-harmonic voltage scales as $V_{3\omega}^{\perp}$3; in VSe$V_{3\omega}^{\perp}$4 a linear relation between $V_{3\omega}^{\perp}$5 and $V_{3\omega}^{\perp}$6 is also reported (Chen et al., 25 Jan 2025, Esin et al., 12 Feb 2025, Yang et al., 12 Jun 2025). In optical language, the same cubic structure appears in the third-order polarization $V_{3\omega}^{\perp}$7 and current $V_{3\omega}^{\perp}$8 of a $V_{3\omega}^{\perp}$9 medium (Nandi et al., 2 Oct 2025).

Angular dependence is a second diagnostic. Several model systems yield

$V_{3\omega}^{\perp}\propto I_\omega^3$0

which vanishes for $V_{3\omega}^{\perp}\propto I_\omega^3$1 and peaks at intermediate angles (Saha et al., 2023). In the misfit compound $V_{3\omega}^{\perp}\propto I_\omega^3$2, the normalized response

$V_{3\omega}^{\perp}\propto I_\omega^3$3

shows a pronounced twofold modulation and vanishes when the electric field is aligned with the crystallographic $V_{3\omega}^{\perp}\propto I_\omega^3$4 or $V_{3\omega}^{\perp}\propto I_\omega^3$5 axis (Li et al., 2024). In contrast, the Fe$V_{3\omega}^{\perp}\propto I_\omega^3$6GaTe$V_{3\omega}^{\perp}\propto I_\omega^3$7 response is reported to be isotropic with respect to in-plane current direction, consistent with the symmetry restriction to $V_{3\omega}^{\perp}\propto I_\omega^3$8 (Cao et al., 22 Apr 2026). The VSe$V_{3\omega}^{\perp}\propto I_\omega^3$9 CDW phase displays the distinct form

$J_i^{(3)}=\sigma^{(3)}_{ijkl}E_jE_kE_l,$0

despite the underlying trigonal lattice, which is interpreted as evidence for broken threefold rotational symmetry (Chen et al., 25 Jan 2025).

Temperature dependence often tracks the underlying order parameter or scattering regime. In VSe$J_i^{(3)}=\sigma^{(3)}_{ijkl}E_jE_kE_l,$1, the third-harmonic signal persists up to $J_i^{(3)}=\sigma^{(3)}_{ijkl}E_jE_kE_l,$2K, while the normalized response is strongly enhanced below the CDW transition near $J_i^{(3)}=\sigma^{(3)}_{ijkl}E_jE_kE_l,$3K (Chen et al., 25 Jan 2025). In the misfit compound $J_i^{(3)}=\sigma^{(3)}_{ijkl}E_jE_kE_l,$4, $J_i^{(3)}=\sigma^{(3)}_{ijkl}E_jE_kE_l,$5 decreases roughly exponentially with temperature and vanishes above approximately $J_i^{(3)}=\sigma^{(3)}_{ijkl}E_jE_kE_l,$6K (Li et al., 2024). In Fe$J_i^{(3)}=\sigma^{(3)}_{ijkl}E_jE_kE_l,$7GaTe$J_i^{(3)}=\sigma^{(3)}_{ijkl}E_jE_kE_l,$8, the third-order transverse response vanishes above the Curie temperature and survives up to room temperature (Cao et al., 22 Apr 2026). In TaIrTe$J_i^{(3)}=\sigma^{(3)}_{ijkl}E_jE_kE_l,$9, the sign of the normalized third-order nonlinear Hall response reverses near $V_{3\omega}^{\perp}\propto I_\omega^3$00K (Yang et al., 12 Jun 2025).

Scaling analyses are used to separate intrinsic and extrinsic contributions, but the fitted forms are platform-specific rather than universal. In VSe$V_{3\omega}^{\perp}\propto I_\omega^3$01, plotting $V_{3\omega}^{\perp}\propto I_\omega^3$02 against $V_{3\omega}^{\perp}\propto I_\omega^3$03 yields

$V_{3\omega}^{\perp}\propto I_\omega^3$04

with a two-orders-of-magnitude jump in $V_{3\omega}^{\perp}\propto I_\omega^3$05 below $V_{3\omega}^{\perp}\propto I_\omega^3$06, interpreted as a strong enhancement of the intrinsic BCP contribution (Chen et al., 25 Jan 2025). In WTe$V_{3\omega}^{\perp}\propto I_\omega^3$07, the reported scaling law is

$V_{3\omega}^{\perp}\propto I_\omega^3$08

which is used to separate intrinsic and skew-scattering contributions and is tied to the orbital-polarization picture (Ye et al., 2022). In TaIrTe$V_{3\omega}^{\perp}\propto I_\omega^3$09, the fit

$V_{3\omega}^{\perp}\propto I_\omega^3$10

is used to identify a BCP-dominated regime above $V_{3\omega}^{\perp}\propto I_\omega^3$11K and a Drude-like impurity-scattering regime below $V_{3\omega}^{\perp}\propto I_\omega^3$12K (Yang et al., 12 Jun 2025). In Fe$V_{3\omega}^{\perp}\propto I_\omega^3$13GaTe$V_{3\omega}^{\perp}\propto I_\omega^3$14, the reported quartic law

$V_{3\omega}^{\perp}\propto I_\omega^3$15

assigns the $V_{3\omega}^{\perp}\propto I_\omega^3$16-independent intercept to the intrinsic symplectic-connection contribution and the $V_{3\omega}^{\perp}\propto I_\omega^3$17 term to extrinsic skew-scattering-type processes (Cao et al., 22 Apr 2026).

5. Material realizations and representative platforms

The third-order Hall effect has now been reported or modeled across topological semimetals, transition-metal dichalcogenides, Rashba systems, magnetic van der Waals materials, and altermagnets.

System	Reported hallmark	Interpretation
1$V_{3\omega}^{\perp}\propto I_\omega^3$18-VSe$V_{3\omega}^{\perp}\propto I_\omega^3$19 nanosheets (Chen et al., 25 Jan 2025)	$V_{3\omega}^{\perp}\propto I_\omega^3$20 up to $V_{3\omega}^{\perp}\propto I_\omega^3$21K; enhancement below $V_{3\omega}^{\perp}\propto I_\omega^3$22K; twofold angular pattern	CDW-induced symmetry breaking and enhanced intrinsic BCP
$V_{3\omega}^{\perp}\propto I_\omega^3$23 (Li et al., 2024)	$V_{3\omega}^{\perp}\propto I_\omega^3$24 at $V_{3\omega}^{\perp}\propto I_\omega^3$25K	Misfit superlattice lowers symmetry and sharply enhances BCP
NiTe$V_{3\omega}^{\perp}\propto I_\omega^3$26 (Esin et al., 12 Feb 2025)	Negligible $V_{3\omega}^{\perp}\propto I_\omega^3$27; unsaturated $V_{3\omega}^{\perp}\propto I_\omega^3$28; estimated $V_{3\omega}^{\perp}\propto I_\omega^3$29	Centrosymmetric Dirac semimetal with symmetry-allowed third-order tensor
TaIrTe$V_{3\omega}^{\perp}\propto I_\omega^3$30 (Yang et al., 12 Jun 2025)	Sign reversal near $V_{3\omega}^{\perp}\propto I_\omega^3$31K; modulation by in-plane dc field; $V_{3\omega}^{\perp}\propto I_\omega^3$32 suppression at $V_{3\omega}^{\perp}\propto I_\omega^3$33K and $V_{3\omega}^{\perp}\propto I_\omega^3$34	Competition between BCP-like and Drude-like terms; electric-field control
Fe$V_{3\omega}^{\perp}\propto I_\omega^3$35GaTe$V_{3\omega}^{\perp}\propto I_\omega^3$36 (Cao et al., 22 Apr 2026)	Room-temperature response odd in magnetization; isotropic in current direction; $V_{3\omega}^{\perp}\propto I_\omega^3$37	Symplectic-connection-induced effect in a centrosymmetric ferromagnet
(101)-RuO$V_{3\omega}^{\perp}\propto I_\omega^3$38 thin films (Chen et al., 15 Apr 2026)	Room-temperature $V_{3\omega}^{\perp}\propto I_\omega^3$39; tens of $V_{3\omega}^{\perp}\propto I_\omega^3$40V at $V_{3\omega}^{\perp}\propto I_\omega^3$41; $V_{3\omega}^{\perp}\propto I_\omega^3$42–$V_{3\omega}^{\perp}\propto I_\omega^3$43	T-odd third-order Hall signal correlated with altermagnetic order

Few-layer WTe$V_{3\omega}^{\perp}\propto I_\omega^3$44 adds a distinct orbital-polarization realization. There the third-order anomalous Hall effect is reported to be consistent with electric-field-induced polarization of orbital magnetic moment caused by BCP, and the associated orbital polarization is directly detected by polar reflective magnetic circular dichroism spectroscopy (Ye et al., 2022).

Model studies have broadened the platform landscape. Multi-Weyl semimetals host a sizable third-order Hall conductivity whose amplitude increases strongly with monopole charge, with estimated $V_{3\omega}^{\perp}\propto I_\omega^3$45–$V_{3\omega}^{\perp}\propto I_\omega^3$46 and third-harmonic transverse voltages in the $V_{3\omega}^{\perp}\propto I_\omega^3$47–mV range for representative parameters (Roy et al., 2021). Rashba systems with hexagonal warping display nontrivial BCP multipole patterns and a $V_{3\omega}^{\perp}\propto I_\omega^3$48-periodic angular Hall response; for parameters representative of a strong-Rashba surface alloy, the estimated third-order Hall voltage ranges from order $V_{3\omega}^{\perp}\propto I_\omega^3$49 to $V_{3\omega}^{\perp}\propto I_\omega^3$50 depending on warping and gap tuning (Saha et al., 2023). For the surface states of Bi$V_{3\omega}^{\perp}\propto I_\omega^3$51Te$V_{3\omega}^{\perp}\propto I_\omega^3$52-type topological insulators, increasing tilt and hexagonal warping significantly enhances the third-order Hall response, with estimated Hall voltages from $V_{3\omega}^{\perp}\propto I_\omega^3$53 to $V_{3\omega}^{\perp}\propto I_\omega^3$54 in the parameter sets quoted in the study (Nag et al., 2022). In III-V semiconductor heterojunctions with Rashba-Dresselhaus spin-orbit coupling, an infinitesimal Dresselhaus term added to a dominant Rashba term yields a finite third-order Hall response; the 2DEG and 2DHG cases differ in the exchange properties of Rashba and Dresselhaus parameters and in the relative magnitudes of the BCP-induced and band-velocity-induced contributions (Pal et al., 2023).

The effect also survives beyond the bulk semiclassical regime. In a quantum-coherent four-terminal model, the third-order Hall current can show resonant peaks whose magnitudes are up to three orders larger than the first-order Hall current; the enhancement is attributed to quantum interference, is strongly suppressed by dephasing, and can itself be enhanced by weak disorder (Wei et al., 2022). In an all-optical realization in toluene, the antisymmetric part of the third-order conductivity produces a Hall-like probe-photon deflection; the experiment reports a lobe separation consistent with paraxial theory and opposite geometric phases $V_{3\omega}^{\perp}\propto I_\omega^3$55 on the two lobes (Nandi et al., 2 Oct 2025).

6. Conceptual distinctions, sign structure, and outlook

A recurring source of ambiguity is that “third-order Hall” may refer either to a cubic nonlinear Hall response or to the rank of the classical Hall tensor. In the classical constitutive law

$V_{3\omega}^{\perp}\propto I_\omega^3$56

the Hall tensor $V_{3\omega}^{\perp}\propto I_\omega^3$57 is third order as a tensor and is skew-symmetric in its first two indices. Its isotropic invariant theory leads to a minimal integrity basis of ten invariants through the associated second-order tensor $V_{3\omega}^{\perp}\propto I_\omega^3$58 (Liu et al., 2017). This mathematical usage is distinct from the cubic-in-electric-field transport effect discussed in modern quantum materials.

Another point clarified by recent work is that the sign of the third-order Hall coefficient is not fixed as simply as the sign of the linear anomalous Hall coefficient. In a minimal 4-band Dirac model for topological magnets, the sign of the third-order anomalous Hall effect is controlled by the interplay between time-reversal symmetry breaking, magnetization orientation, spin-orbit coupling, and chemical potential; the paper formulates this as a “sign problem” and proposes rotating-field experiments to map the sign diagram (Zhang et al., 2023).

Disorder remains an active part of the theory. In time-reversal-symmetric Dirac materials, semiclassical Boltzmann analysis finds intrinsic BCP-driven, skew-scattering, and side-jump contributions with distinct anisotropy dependences; the study emphasizes that side-jump and skew-scattering terms can materially reshape the total response (Barman et al., 2024). This is consistent with the diverse scaling laws used experimentally to separate intrinsic and extrinsic pieces in WTe$V_{3\omega}^{\perp}\propto I_\omega^3$59, TaIrTe$V_{3\omega}^{\perp}\propto I_\omega^3$60, VSe$V_{3\omega}^{\perp}\propto I_\omega^3$61, and Fe$V_{3\omega}^{\perp}\propto I_\omega^3$62GaTe$V_{3\omega}^{\perp}\propto I_\omega^3$63 (Ye et al., 2022, Yang et al., 12 Jun 2025, Chen et al., 25 Jan 2025, Cao et al., 22 Apr 2026).

The reported applications and future directions are correspondingly broad. Proposed routes include gating or strain control of CDW amplitude and exploration of other $V_{3\omega}^{\perp}\propto I_\omega^3$64 transition-metal dichalcogenides (Chen et al., 25 Jan 2025), high-frequency rectifiers and ac-field detection in interface-engineered misfit compounds (Li et al., 2024), broadband zero-bias terahertz and infrared detection in Dirac semimetals (Esin et al., 12 Feb 2025), room-temperature nonlinear Hall devices in centrosymmetric ferromagnets (Cao et al., 22 Apr 2026), and on-chip Néel-vector readout in altermagnetic candidates such as RuO$V_{3\omega}^{\perp}\propto I_\omega^3$65 (Chen et al., 15 Apr 2026). A plausible implication is that the third-order Hall effect is evolving from a narrowly defined nonlinear transport anomaly into a general diagnostic of high-rank band geometry, symmetry lowering, and coherent transport across both electronic and optical platforms.