Nonlinear Anomalous Hall Effect

• Non-linear anomalous Hall effect is a second-order transverse response emerging in inversion-breaking systems, primarily driven by Berry curvature dipoles.
• It serves as a sensitive probe of band topology and symmetry, enabling the detection of phase transitions and revealing the influence of intrinsic and extrinsic scattering mechanisms.
• The effect offers promising applications in spintronics and valleytronics with switchable currents and higher-harmonic responses relevant to advanced electronic device designs.

The non-linear anomalous Hall effect (NLAHE) encompasses a suite of second-order (or higher) transverse transport responses in solids, distinct from the familiar linear Hall effect that is restricted by symmetry requirements such as broken time-reversal invariance or the presence of net magnetization. NLAHE arises in systems where inversion symmetry is broken, and its leading mechanism is typically rooted in Berry curvature dipoles or other higher-order moments of Berry curvature—enabling a transverse (Hall) current quadratic or higher order in the applied electric field, and in general allowing such currents in time-reversal symmetric systems. NLAHE serves as a sensitive probe of band geometry and topology, providing diagnostic access to underlying Berry curvature hot spots, phase transitions, and unconventional magnetic or electronic orders.

1. Underlying Mechanisms and Theoretical Framework

The fundamental distinction of NLAHE versus the conventional (“linear”) Hall effect lies in its origin from the asymmetric distribution of Berry curvature in momentum space, enabled when inversion symmetry is broken. In a system with time-reversal symmetry (TRS), the Berry curvature $\Omega(\mathbf{k})$ is odd under $\mathbf{k} \rightarrow -\mathbf{k}$, so its Brillouin zone average vanishes, nullifying the linear anomalous Hall effect (AHE). However, the first moment of Berry curvature—called the Berry curvature dipole $\mathbf{D}$—remains generically nonzero without inversion symmetry and directly produces a second-order Hall current: $$\mathbf{j}^{(2)} \propto \mathbf{E} \times \mathbf{D}$$ where the dipole is defined as

$$D_a = \int_{\mathbf{k}} f_0(\mathbf{k}) \partial_{k_a} \Omega_z(\mathbf{k})$$

with $f_0$ being the Fermi-Dirac distribution. This mechanism generalizes to higher-order derivatives, yielding Berry curvature multipoles (e.g., quadrupoles, hexapoles) that underpin higher harmonic responses (Zhang et al., 2020).

In the semiclassical transport framework, NLAHE is captured by corrections to the electron's velocity and charge density, involving Berry curvature and orbital magnetic moment terms, as well as scattering mechanisms. Symmetry analysis shows that higher-order tensorial responses emerge only when both TRS and inversion are not simultaneously respected.

Extrinsic mechanisms (skew scattering, side-jump, and disorder effects) can also substantially enhance or even dominate NLAHE in real materials (Atencia et al., 2023).

2. Symmetry, Topology, and Material Realizations

The manifestation and leading order of NLAHE in a given material are dictated primarily by crystalline and magnetic point group symmetries. The Berry curvature (a pseudovector) and its multipoles are classified according to their transformation properties, with lower-order responses (dipole, quadrupole, etc.) vanishing unless allowed by the point group symmetry (Zhang et al., 2020). In time-reversal-invariant, inversion-breaking materials (e.g., Td–WTe$_2$), NLAHE is permitted and can be large due to a sizable Berry curvature dipole (Kang et al., 2018).

Topological phase transitions—such as those in twisted double bilayer graphene (TDBG) tuned by an out-of-plane electric field—qualitatively alter the Berry curvature distribution, causing sign change in the NLAHE signal (Chakraborty et al., 2022). The nonlinear Hall coefficients then act as sensitive indicators of phase boundaries and topological indices (e.g., valley Chern numbers).

Symmetry also governs whether the NLAHE can be switched by external means. In systems with ferroelectric transitions or magnetic domain switching, NLAHE currents can exhibit even–odd oscillations or reversibility under order parameter reversal, as shown in few-layer WTe$_2$ and various antiferromagnets (Wang et al., 2019).

3. Experimental Characterization and Key Observations

NLAHE is most directly observed via the detection of second-harmonic (or higher) transverse voltages in AC-driven transport setups, or as quadratic dependencies of transverse voltage on longitudinal current in DC measurements. Key experimental signatures include:

• Quadratic (or higher) scaling: The Hall voltage $V_\perp$ scales as $I^2$ or with higher powers, in sharp contrast to the linear scaling of the ordinary or conventional anomalous Hall effects.
• Angle dependence: The signal’s amplitude and sign depend on the crystal orientation relative to current direction, peaking at generic angles set by crystal symmetry and vanishing along symmetry axes.
• Switchable behavior: In ferroelectric or chiral antiferromagnets, reversal of the order parameter (e.g., ferroelectric polarization, magnetic chirality) flips the NLAHE signal’s sign (Wang et al., 2019, Nayak et al., 2015).
• Current-induced magnetization: In nonmagnetic, non-centrosymmetric metals (e.g., WTe$_2$, MoTe$_2$), in-plane electric fields induce an effective out-of-plane (orbital) magnetization, leading to a nonlinear Hall response (Tiwari et al., 2021, Kang et al., 2018).

Strong numerical NLAHE signals have been observed in noncentrosymmetric T$_d$-MoTe$_2$ and WTe$_2$ (Hall ratios exceeding 2, anomalous Hall conductivities of $8 \times 10^2$ S/m) (Tiwari et al., 2021), as well as in organic quasi-2D Dirac fermion systems (Kiswandhi et al., 2021). In transition metal antiferromagnets such as Mn$_3$Ge and Mn$_3$Ir, large anomalous Hall conductivities are found even at (almost) vanishing net magnetization (Nayak et al., 2015, Chen et al., 2013).

4. Integration of Intrinsic and Extrinsic Mechanisms

While symmetry and band geometry fundamentally enable the intrinsic NLAHE via Berry curvature dipoles and multipoles, disorder and scattering play a crucial quantitative role (Atencia et al., 2023):

• Skew scattering: Arises from asymmetric impurity scattering; the second-harmonic Hall response scales as the cube of the carrier lifetime ($\tau^3$) at high temperature or low impurity concentration.
• Side jump: Lateral displacement during impurity scattering contributes at zeroth order in $\tau$.
• Disorder-enhanced Berry curvature dipole: In $\mathcal{P}\mathcal{T}$-symmetric Dirac systems, disorder not only introduces extrinsic contributions but also modifies the effective BCD, often overwhelming the intrinsic Berry curvature dipole term.

Extrinsic effects typically become dominant at moderate or high scattering rates; thus, careful sample characterization and analysis are essential in experimental studies.

5. Advanced Phenomena: Higher-Order and Magneto-Nonlinear Hall Effects

When lower-order Berry curvature moments are forbidden by symmetry, higher-order multipoles (quadrupole, hexapole, etc.) generate nonlinear Hall responses at higher harmonics of the drive frequency—such as third- or fourth-harmonic voltages (Zhang et al., 2020). This has been proposed and demonstrated in 2D antiferromagnets, topological insulator surfaces, and twisted bilayer systems near quantum anomalous Hall phase boundaries, where quadrupole or hexapole moments are large.

Recent work extends the NLAHE paradigm to magneto-nonlinear Hall effects (mNLHE), where a transverse current arises linearly in both electric and in-plane magnetic fields—a distinct response governed by the orbital properties and quantum geometry of the band structure (Wang et al., 7 Mar 2024). This is observed in kagome ferromagnets such as Fe$_3$Sn$_2$ and is fundamentally controlled by the antisymmetrized dipole of the anomalous orbital polarizability.

Further, chiral-anomaly-induced NLAHE in Weyl semimetals manifests as a current linear in both $E$ and $B$, reliant on cone tilting and chirality relaxation (Li et al., 2020, Ahmad et al., 4 Sep 2024). In spin–orbit-coupled non-centrosymmetric metals, the orbital magnetic moment alone can yield a large quadratic-in-field NLAHE, highlighting the diversity of mechanisms across chiral fermion platforms.

6. Applications, Implications, and Diagnostics

NLAHE significantly broadens the scope for probing and utilizing quantum geometric properties of materials. Specific implications include:

• Band topology diagnostics: Sharp sign reversals and quantifiable changes in NLAHE coefficients can serve as non-invasive signatures of topological phase transitions (e.g., valley Chern transitions in TDBG (Chakraborty et al., 2022), band inversions in Dirac or Weyl systems).
• Device applications: The strong, switchable, and symmetry-selective nonlinear responses present clear opportunities for low-power all-electrical readout of magnetic and ferroelectric orders, rectification devices, non-volatile memory, and frequency-multiplier components.
• Spintronics and valleytronics: NLAHE is directly sensitive to valley-specific Berry curvature distributions, making it applicable to valleytronic information encoding and manipulation.
• Exotic transport:
  • Nonlinear magnetoelectric coupling enables Hall responses that are controlled via multi-axial electric fields in magnetically ordered insulators, accessible even when linear AHE is symmetry-forbidden (Yu et al., 18 Sep 2024).
  • In magnetic Weyl semimetals, the detection of transient, spatially localized NLAHE can be used as a direct probe of domain wall dynamics and magnetization structure (Heidari et al., 2022).

A crucial caveat is that classical multi-carrier effects, mobility anisotropy, and charge compensation can generate significant nonlinear Hall contributions, termed “pseudo-” or “spurious” NLAHE, especially near charge neutrality in semimetals (Yamada et al., 19 Jun 2025). Consequently, classical mechanisms must be rigorously accounted for before attributing nonlinear Hall signals to quantum geometric origins.

7. Mathematical Formalism and Prototypical Expressions

Key expressions that encapsulate NLAHE include:

• Berry curvature dipole–induced second-order conductivity (in 2D):

$$j_a^{(2)} = \epsilon_{abd} \frac{e^3 \tau}{\hbar^2} D_{db} E_b E_c$$

where $\epsilon_{abd}$ is the Levi-Civita symbol, $\tau$ the scattering time, and $$D_{db} = \int [d\mathbf{k}](\partial_{k_b} f_0) \Omega_d$$.

• Chiral anomaly–driven NLAHE in Weyl semimetals:

$j^{(NH)} = \kappa (E \cdot B)(E \times t)$

with tilt vector $t$ and $\kappa$ parameterizing the node-specific anomaly response (Li et al., 2020, Ahmad et al., 4 Sep 2024).

• Higher-order (third/fourth harmonic) nonlinear Hall response:

$$V_{\mathrm{Hall}}^{(3\omega)} \propto Q_{\alpha\beta\gamma} E_\alpha E_\beta E_\gamma$$

with the Berry curvature quadrupole $Q_{\alpha\beta\gamma}$ (Zhang et al., 2020).

Summary Table: NLAHE Mechanisms and Their Key Material Classes

Mechanism	Key Features	Material Examples
Berry curvature dipole	$j_y \sim E_x^2$; switches with order parameter	WTe$_2$, MoTe$_2$, $\alpha$-(BEDT-TTF)$_2$I$_3$
Chiral anomaly–induced	$j \sim (E \cdot B)(E \times t)$; tilt required	Weyl semimetals, SOC-NCMs
Multicarrier classical	$\rho_H^{NL} \sim 1/|\Delta n|$; mobility anisotropy	ZrTe$_5$, general semimetals
Shift dipole (interband)	Optical/CPGE-driven nonlinear Hall	Few-layer WTe$_2$
Magneto-nonlinear (mNLHE)	$j \sim EH$; orbital quantum geometry controlled	Fe$_3$Sn$_2$ (kagome)
Nonlinear ME coupling	High-order $E$-dependent AHE in antiferromagnets	Cr$_2$O$_3$, CoF$_2$

References to Key Results

• Observation in WTe$_2$ (Kang et al., 2018, Tiwari et al., 2021), organic Dirac conductors (Kiswandhi et al., 2021), kagome magnets (Wang et al., 7 Mar 2024)
• Theory of Berry curvature multipole–induced responses (Zhang et al., 2020), disorder effects (Atencia et al., 2023), chiral anomaly–driven NLAHE (Li et al., 2020, Ahmad et al., 4 Sep 2024)
• Device and phase-transition probes using NLAHE (Chakraborty et al., 2022, Heidari et al., 2022, Wang et al., 2019, Yu et al., 18 Sep 2024)
• Classical contrast and necessity for careful background subtraction (Yamada et al., 19 Jun 2025)

Outlook

NLAHE intricately entwines quantum geometric features—Berry curvature, orbital magnetic moment, and their higher moments—with band topology, symmetry, and disorder in a broad array of quantum and classical materials. Its understanding and measurement enable high-fidelity topological diagnostics, programmable nonlinear transport, and new functionalities in spintronics, valleytronics, and quantum electronic systems. Precise symmetry and experimental control, as well as rigorous distinction between intrinsic and extrinsic mechanisms, are essential for both fundamental advances and technological deployment.