Third-Order Nonlinear Anomalous Hall Effect

• Third-Order Nonlinear Anomalous Hall Effect is a high-order quantum transport phenomenon that generates cubic Hall currents through Berry curvature quadrupoles and Berry connection polarizability.
• Experimental detection employs multi-harmonic lock-in techniques to measure the third-harmonic voltage, confirming symmetry-dictated tensor components and cubic current scaling.
• This effect is observed in diverse materials like Dirac semimetals and ferromagnets, paving the way for advanced topological electronics and programmable spintronic devices.

The third-order nonlinear anomalous Hall effect (NLAHE) is a high-order quantum transport phenomenon in which a transverse Hall current, cubic in the applied electric field, arises in materials with specific symmetry and topological properties. Unlike the linear or second-order Hall effects, the NLAHE is governed by higher Berry curvature multipoles—specifically, the Berry curvature quadrupole or Berry connection polarizability—yielding sharp symmetry selection rules and unique dependencies on band structure, disorder, and geometry. The effect has been verified in a diverse array of quantum materials, including time-reversal-symmetric Dirac semimetals, ferromagnets, and multivalent Weyl semimetals, and is accessible by lock-in measurement of the third-harmonic Hall voltage under an ac drive. This article consolidates theoretical foundations, symmetry conditions, microscopic mechanisms, experimental approaches, and current research frontiers in the third-order NLAHE.

1. Theoretical Foundations and Formulation

The core observable of the third-order NLAHE is the appearance of a transverse Hall current density $j_a^{(3\omega)}$ under an oscillating electric field $E_b(\omega)$, such that

$$j_a^{(3\omega)} = \chi^{(3)}_{abcd}\;E_b(\omega)\,E_c(\omega)\,E_d(\omega)$$

where $\chi^{(3)}_{abcd}$ is the rank-4 third-order nonlinear Hall susceptibility tensor. The leading microscopic origin in both time-reversal-broken and time-reversal-symmetric regimes is the Berry curvature quadrupole $Q_{bc}$ or, equivalently, a field-induced Berry connection polarizability $G_{ij}$, with

$$\chi_{yxxx}^{(3)} \propto \int d^d k\ f_0(k)\ \frac{\partial^2}{\partial k_x^2}\Omega_z(\mathbf{k}),$$

for a current applied along $x$ and Hall response along $y$ (Esin et al., 12 Feb 2025, Dai et al., 23 Apr 2026, Zhang et al., 2020).

In centrosymmetric Dirac semimetals (e.g. NiTe$_2$) with preserved inversion ($E_b(\omega)$0) and time-reversal ($E_b(\omega)$1) symmetry, the second-order response is symmetry-forbidden, so the NLAHE provides the leading nonlinear Hall channel (Esin et al., 12 Feb 2025).

Crucially, in systems with vanishing Berry curvature monopole and dipole (e.g., by symmetry), the third-order term set by the quadrupole becomes the dominant intrinsic contribution (Korrapati et al., 2024, Zhang et al., 2020).

2. Symmetry Constraints and Berry Curvature Multipole Hierarchies

The existence and dominance of the third-order NLAHE are tightly constrained by the following symmetry and topological criteria:

• Inversion and Time-Reversal: The second-order NLAHE requires broken inversion. In contrast, the third-order Hall effect is symmetry-allowed in bulk Dirac semimetals with $E_b(\omega)$2 and $E_b(\omega)$3 (Esin et al., 12 Feb 2025, Zhang et al., 2020).
• Magnetic Point Groups: Analysis of all magnetic point groups shows that in 3D, 66 groups allow a nonzero Berry curvature quadrupole, and for 15 of them (including groups such as $E_b(\omega)$4 or $E_b(\omega)$5 in 2D), the quadrupole is the lowest nonvanishing Berry multipole (Zhang et al., 2020).
• Magnetic Order: In time-reversal-broken materials (e.g., ferromagnetic Fe$E_b(\omega)$6GaTe$E_b(\omega)$7 or antiferromagnetic monolayer SrMnBi$E_b(\omega)$8), the quadrupole contribution survives even when the monopole and dipole vanish by symmetry (Dai et al., 23 Apr 2026, Korrapati et al., 2024).
• Crystal Symmetry: The number and relationships among independent components of $E_b(\omega)$9 are further reduced by space group constraints (e.g. $$j_a^{(3\omega)} = \chi^{(3)}_{abcd}\;E_b(\omega)\,E_c(\omega)\,E_d(\omega)$$0 in NiTe$$j_a^{(3\omega)} = \chi^{(3)}_{abcd}\;E_b(\omega)\,E_c(\omega)\,E_d(\omega)$$1), dictating both the angular response and tensor structure in experiments (Esin et al., 12 Feb 2025, Yang et al., 12 Jun 2025).

3. Microscopic Mechanisms: Intrinsic and Extrinsic Contributions

The third-order NLAHE arises from both intrinsic and extrinsic mechanisms, whose relative significance depends on disorder, temperature, and material purity.

• Intrinsic (Berry-Geometry–Driven) Contribution: For pristine, high-mobility samples at high temperature (or weak disorder), the dominant channel is through the Berry connection polarizability, where the field-induced distortion of the Bloch wavefunctions generates a nonequilibrium Berry curvature quadrupole. The resulting transverse current is cubic in the electric field:

$$j_a^{(3\omega)} = \chi^{(3)}_{abcd}\;E_b(\omega)\,E_c(\omega)\,E_d(\omega)$$2

with $$j_a^{(3\omega)} = \chi^{(3)}_{abcd}\;E_b(\omega)\,E_c(\omega)\,E_d(\omega)$$3 and $$j_a^{(3\omega)} = \chi^{(3)}_{abcd}\;E_b(\omega)\,E_c(\omega)\,E_d(\omega)$$4 the relaxation time (Korrapati et al., 2024).

• Extrinsic (Disorder-Mediated) Contributions:
  • Skew scattering: Arises from asymmetric impurity-induced scattering, scaling as $$j_a^{(3\omega)} = \chi^{(3)}_{abcd}\;E_b(\omega)\,E_c(\omega)\,E_d(\omega)$$5 and manifesting strong dependence on carrier mobility and Fermi surface structure. In ultraclean materials, this can overpower the intrinsic part at low temperatures (Barman et al., 2024, Ye et al., 2022).
  • Side-jump: In time-reversal-invariant metals, the leading ($$j_a^{(3\omega)} = \chi^{(3)}_{abcd}\;E_b(\omega)\,E_c(\omega)\,E_d(\omega)$$6) side-jump vanishes for third-order Hall conductivity, with subleading ($$j_a^{(3\omega)} = \chi^{(3)}_{abcd}\;E_b(\omega)\,E_c(\omega)\,E_d(\omega)$$7) corrections parametrically small (Barman et al., 2024).

The crossover between intrinsic and extrinsic regimes can produce sign reversals in the third-harmonic Hall voltage as a function of temperature, conductivity, or gate bias, as observed in TaIrTe$$j_a^{(3\omega)} = \chi^{(3)}_{abcd}\;E_b(\omega)\,E_c(\omega)\,E_d(\omega)$$8 (Yang et al., 12 Jun 2025) and modeled for WTe$$j_a^{(3\omega)} = \chi^{(3)}_{abcd}\;E_b(\omega)\,E_c(\omega)\,E_d(\omega)$$9 (Ye et al., 2022).

4. Experimental Measurement Approaches and Signatures

The defining characteristic of third-order NLAHE in transport is the emergence of a transverse Hall voltage $\chi^{(3)}_{abcd}$0 at triple the driving frequency under an ac current:

• Lock-in Detection: Multi-harmonic lock-in amplifiers are employed to resolve $\chi^{(3)}_{abcd}$1 for $\chi^{(3)}_{abcd}$2, isolating cubic scaling through $\chi^{(3)}_{abcd}$3 (Esin et al., 12 Feb 2025, Dai et al., 23 Apr 2026).
• Angular and Frequency Dependence: Angle-resolved studies exploit the crystal anisotropy to distinguish between tensor components and identify symmetry-allowed terms. Independence from external magnetic field (in, e.g., NiTe$\chi^{(3)}_{abcd}$4) confirms the geometric/topological origin (Esin et al., 12 Feb 2025).
• Scaling and Hysteresis: Scaling analysis, plotting $\chi^{(3)}_{abcd}$5 vs. $\chi^{(3)}_{abcd}$6 or third-harmonic Hall resistance vs. $\chi^{(3)}_{abcd}$7, is critical for disentangling intrinsic/extrinsic channels and establishing the link to Berry curvature quadrupole (Dai et al., 23 Apr 2026).
• Domain and Magnetization Reversal: In magnetic systems, $\chi^{(3)}_{abcd}$8 tracks the coercive field of the linear AHE, and reverses under domain flip by Onsager reciprocity (Dai et al., 23 Apr 2026, Yu et al., 2024).
• Nonlinear Magnetoelectric Probes: Third-harmonic signals are intrinsically sensitive to antiferromagnetic order and its reversal, allowing unbiased probes of Neél order and domain state (Yu et al., 2024).

5. Material Platforms and Quantitative Characteristics

Significant third-order NLAHE has been detected across various material classes:

Material/System	Symmetry/Order	Dominant Mechanism	Notable Features
NiTe$\chi^{(3)}_{abcd}$9	$Q_{bc}$0 Dirac semimetal	Intrinsic (BCP, quadrupole)	Cubic, $Q_{bc}$1-independent, $Q_{bc}$2 A·m/V$Q_{bc}$3 (Esin et al., 12 Feb 2025)
Fe$Q_{bc}$4GaTe$Q_{bc}$5	Ferromagnet, D$Q_{bc}$6	Intrinsic (Berry quadrupole) $Q_{bc}$7 skew	Persists up to $Q_{bc}$8 K, tracks domain (Dai et al., 23 Apr 2026)
TaIrTe$Q_{bc}$9	Weyl semimetal, nonsymmorphic	BCP at high $G_{ij}$0, skew at low $G_{ij}$1	Electric-field tunability, sign reversal at $G_{ij}$2 K (Yang et al., 12 Jun 2025)
WTe$G_{ij}$3	Low-symmetry semimetal	BCP + orbital skew-scattering	Angle-dependent signal, ratio analysis (Ye et al., 2022)
Antiferro. SrMnBi$G_{ij}$4	MPG $G_{ij}$5	Pure quadrupole (no monopole/dipole)	Fourfold angle signatures, $G_{ij}$6-independent (Zhang et al., 2020, Korrapati et al., 2024)
Cr$G_{ij}$7O$G_{ij}$8	Antiferromagnetic, $G_{ij}$9	Nonlinear magnetoelectric (free-energy)	$$\chi_{yxxx}^{(3)} \propto \int d^d k\ f_0(k)\ \frac{\partial^2}{\partial k_x^2}\Omega_z(\mathbf{k}),$$0 scaling, domain reversal of sign (Yu et al., 2024)

Measurements typically yield third-order conductivities $$\chi_{yxxx}^{(3)} \propto \int d^d k\ f_0(k)\ \frac{\partial^2}{\partial k_x^2}\Omega_z(\mathbf{k}),$$1 in the range $$\chi_{yxxx}^{(3)} \propto \int d^d k\ f_0(k)\ \frac{\partial^2}{\partial k_x^2}\Omega_z(\mathbf{k}),$$2–$$\chi_{yxxx}^{(3)} \propto \int d^d k\ f_0(k)\ \frac{\partial^2}{\partial k_x^2}\Omega_z(\mathbf{k}),$$3 A·m/V$$\chi_{yxxx}^{(3)} \propto \int d^d k\ f_0(k)\ \frac{\partial^2}{\partial k_x^2}\Omega_z(\mathbf{k}),$$4, producing microvolt-scale signals under strong ac drive and high mobility.

6. Temperature, Electric Field, and Disorder Dependence

• Temperature: In the intrinsic regime, the third-order response shows minimal $$\chi_{yxxx}^{(3)} \propto \int d^d k\ f_0(k)\ \frac{\partial^2}{\partial k_x^2}\Omega_z(\mathbf{k}),$$5-dependence; skew scattering dominates at low $$\chi_{yxxx}^{(3)} \propto \int d^d k\ f_0(k)\ \frac{\partial^2}{\partial k_x^2}\Omega_z(\mathbf{k}),$$6, leading to sign reversals and strong field-sensitivity (Yang et al., 12 Jun 2025, Ye et al., 2022).
• Electric Field Control: Application of an in-plane dc bias enables real-time tuning of both the magnitude and sign of the NLAHE, shifting crossovers between intrinsic and extrinsic regimes and enabling programmable electronics (Yang et al., 12 Jun 2025).
• Disorder Effects: The intrinsic (BCP) term scales as $$\chi_{yxxx}^{(3)} \propto \int d^d k\ f_0(k)\ \frac{\partial^2}{\partial k_x^2}\Omega_z(\mathbf{k}),$$7 in time-reversal-symmetric metals; the extrinsic skew-scattering scales as $$\chi_{yxxx}^{(3)} \propto \int d^d k\ f_0(k)\ \frac{\partial^2}{\partial k_x^2}\Omega_z(\mathbf{k}),$$8 in leading order, with side-jump negligible (Barman et al., 2024). Anisotropy and Fermi-level tuning further modulate the response.

7. Outlook, Challenges, and Applications

The third-order NLAHE stands as a highly symmetry-selective probe of band geometry, Berry multipoles, and quantum topology. Prospects and current initiatives include:

• Nonlinear electronics and topological logic: Use of nonlinear Hall rectification in THz detection, power conversion, or memory elements (Esin et al., 12 Feb 2025, Dai et al., 23 Apr 2026).
• Programmable spintronics and valleytronics: Tuning via gate voltage, dc bias, or strain provides new avenues for topological device engineering (Yang et al., 12 Jun 2025).
• Domain and transition sensors: Hall signals that correlate directly to magnetic order—sensitive to domain reversals and phase transitions—offer applications in antiferromagnetic spintronics (Yu et al., 2024).
• Open problems: Controlling disorder, optimizing symmetry breaking, and quantifying higher multipole moments remain central to pushing measurement sensitivity and realizing new phenomena such as frequency quadrupling or hexapole Hall effects (Zhang et al., 2020).

The field has achieved rapid experimental progress, but systematic explorations across magnetic, topological, and low-symmetry quantum materials—coupled with precision symmetry engineering—will be required to fully realize and deploy the third-order NLAHE in functional devices and quantum sensors.