Nonlinear Anomalous Hall Effect (NLAHE)

• Nonlinear anomalous Hall effect is a transverse transport response scaling nonlinearly with the applied field due to inversion symmetry breaking and Berry curvature dipole.
• It reveals key quantum geometrical features and topological transitions in materials like 2D TMDs, organic Dirac systems, and magnetic topological compounds.
• Extrinsic mechanisms such as skew scattering, side-jump, and quantum metric contributions also influence NLAHE, offering diverse experimental probes.

The non-linear anomalous Hall effect (NLAHE) refers to transverse electrical or thermal transport responses that depend nonlinearly on the applied electric field or other probes, manifesting even in systems where the conventional (linear) Hall effect is forbidden by symmetry. Unlike traditional Hall effects, which require time-reversal symmetry (TRS) breaking (via magnetization or applied magnetic field), the NLAHE arises predominantly in systems with broken inversion symmetry and can appear at zero external magnetic field or net magnetization. The leading theoretical mechanism is the Berry curvature dipole (BCD), but higher-order Berry multipoles, quantum metric effects, chiral anomaly physics, magnetoelectric couplings, and even classical mechanisms in multicarrier systems can generate or mimic NLAHE. The effect is under intense investigation as a symmetry-tunable probe of quantum geometry, topological order, and dynamical textures in crystalline and magnetic solids.

1. Theoretical Foundations and Unifying Formalism

The NLAHE originates in the interplay of Berry-phase-induced anomalous velocities and nonlinear driving by external electric fields. For a Bloch band with dispersion $\varepsilon_n(k)$ and Berry curvature $\Omega_n(k)$, the semiclassical electron dynamics under a weak field $\mathbf{E}$ yield an anomalous velocity $$({e}/{\hbar})\,\mathbf{E}\times\mathbf{\Omega}_n(k)$$. When inversion symmetry is broken, an out-of-equilibrium occupation develops a net Berry curvature dipole, producing a rectified Hall response quadratic in the field, even when TRS is unbroken (Du et al., 2021, Kang et al., 2018).

The nonlinear transverse current takes the form

$j_a^{(2)} = \chi_{abc}\,E_b E_c$

where $\chi_{abc}$ is the rank-3 nonlinear Hall conductivity tensor. The leading intrinsic contribution at low frequencies and in the relaxation time ($\tau$) approximation is

$$\chi_{abc} = -\frac{e^3\tau}{2\hbar^2}\epsilon_{adc}\,D_{bd}$$

with $$D_{bd} = \sum_n\int[dk]\,f_n^0\,\partial_{k_d}\Omega_{n, b}(k)$$, the Berry curvature dipole (BCD), which measures the first moment of Berry curvature on the Fermi surface (Du et al., 2021, Zhang et al., 2018).

Time-reversal ($\mathcal{T}$) symmetry implies $\Omega_n(k) = -\Omega_n(-k)$, so the linear Hall response vanishes unless $\mathcal{T}$ is broken. However, the BCD can survive under broken inversion ($\mathcal{P}$) even when $\mathcal{T}$ is preserved, making NLAHE a hallmark of non-centrosymmetric crystals.

2. Symmetry Constraints and Berry Multipole Classification

The existence, form, and selection rules for NLAHE are tightly constrained by crystalline point group symmetries. Table I summarizes leading contributions:

Hall Effect Order	Berry Curvature Multipole	Frequency Response	Key Symmetry Condition
Linear (1st)	Monopole ($\int \Omega$)	$\omega$	$\mathcal{T}$-breaking; $\mathcal{P}$ optional
Quadratic (2nd)	Dipole ($\int \nabla_k\Omega$)	$2\omega$	$\mathcal{P}$-breaking; $\mathcal{T}$ optional
Cubic (3rd)	Quadrupole ($\int \nabla_k^2\Omega$)	$3\omega$	$\mathcal{T}$ and $\mathcal{P}$ assigned by point group (Zhang et al., 2020)

Symmetry can enforce the vanishing of all lower moments, so certain magnetic or crystallographic classes have leading Hall responses only in higher order. For instance, in higher-wave-symmetric unconventional magnets and altermagnets, both the linear and intrinsic second-order AHE vanish by point group selection rules, but a symmetry-breaking electric field can induce a nonzero BCD via quantum metric couplings (Korrapati et al., 23 Oct 2025). Symmetry analysis across 2D and 3D magnetic point groups allows a full classification of expected NLAHE orders (Zhang et al., 2020).

3. Mechanisms Beyond Intrinsic Berry Dipole

While the intrinsic BCD remains the canonical NLAHE mechanism in low-disorder, inversion-broken, TRS solids (as in monolayer WTe$_2$ and MoTe$_2$ (Zhang et al., 2018, Kang et al., 2018)), several additional (and sometimes dominant) mechanisms have been identified:

• Extrinsic Disorder Contributions: Skew scattering and side-jump processes generate NLAHE terms that can overwhelm the intrinsic BCD around the Fermi level peak, especially in disordered, PT-symmetric systems (Atencia et al., 2023). These terms are $\tau^0$ (independent of scattering lifetime), while the intrinsic BCD term scales as $\tau$.
• Quantum Metric Dipole: Field-induced quantum metric dipole ($G^{ab}$) shifts give rise to $\tau^0$ nonlinear Hall currents even when the Berry dipole is symmetry-forbidden, as in even-layer MnBi$_2$Te$_4$ (Kaplan et al., 2022) and symmetry-driven multipolar systems (Korrapati et al., 23 Oct 2025).
• Magnetoelectric Nonlinear Coupling: Nonlinear Hall terms can arise from even (or odd) order magnetoelectric coupling tensors $\chi^{(n)}$, e.g., $$\sigma_{\alpha\beta}(E) = \epsilon_{\alpha\beta\gamma}\chi^{(2)}_{\gamma\delta\epsilon}E_\delta E_\epsilon$$. This enables NLAHE in magnets under multi-axial electric fields, controlled via the magnetic order parameter or Néel vector (Yu et al., 18 Sep 2024).
• Chiral Anomaly and Tilt-Induced CNLAHE: In Weyl semimetals with tilted cones, the synergy between anomalous velocity and chiral anomaly (when $E\cdot B \neq 0$) generates NLAHE scaling as $j\sim E^2 B$. The effect can exhibit strong sign reversals with internode scattering and distinguishes between Weyl (linear-in-B, sign-changing) and spin-orbit metals (quadratic-in-B, no sign reversal) (Ahmad et al., 4 Sep 2024, Li et al., 2020).
• Orbital and Magneto-NLHE: In quantum magnets, a further nonlinear Hall response proportional to $E B$ arises from the anomalous orbital polarizability and magnetic field perturbation of the Berry connection. This "magneto-NLHE" is large near Weyl points and is confirmed experimentally in kagome ferromagnets (e.g., Fe$_3$Sn$_2$) (Wang et al., 7 Mar 2024).

4. Experimental Realizations and Key Results

NLAHE has been validated across a diverse set of materials and symmetry classes:

• 2D TMDs and Broken Inversion: Quadratic Hall voltage scaling ($V_H \propto I^2$) is observed in monolayer and few-layer Td–WTe$_2$ (Kang et al., 2018, Zhang et al., 2018). The signal's angular dependence and switchable sign by gate field or strain agree with BCD theory predictions.
• Organic Dirac Systems: In $\alpha$-(BEDT-TTF)$_2$I$_3$, NLAHE appears in the charge-ordered phase just below the critical pressure, with the Hall voltage vanishing as inversion symmetry is restored at higher pressures (Kiswandhi et al., 2021).
• Magnetic Topological Materials and Altermagnets: In EuCd$_2$As$_2$, a giant intrinsic NLAHE arises from spin-canting-driven Berry curvature hotspots. 97% of the total Hall response in certain regimes is nonlinear, tracking the orientation of the magnetic moments (Cao et al., 2021).
• Antiferromagnets and Néel-vector Sensing: In compensated AFMs such as CuMnSb, the BCD and thus the NLAHE are strong functions of the Néel vector direction – providing an all-electrical probe for AFM order-parameter readout (Shao et al., 2019).
• Twisted Bilayer and Moiré Materials: In TDBG, the Berry curvature dipole flips sign as the system undergoes a topological transition (change in valley Chern number) tuned by a perpendicular electric field, directly reflected in NLAHE measurements (Chakraborty et al., 2022).
• Higher-Order Effects: Third- and fourth-order NLAHEs (proportional to $E^3$, $E^4$) have been predicted and modeled in systems with suitable Berry quadrupole/hexapole moments, including surface states of TIs and antiferromagnets (Zhang et al., 2020).
• Classical Mechanism Crossovers: In multi-carrier semimetals such as ZrTe$_5$, classical Drude-Lorentz effects can produce nonlinear Hall signals ($\rho_H^{NL}\sim B^3$) that mimic quantum NLAHE, diverging near charge neutrality and highlighting the need for careful interpretation (Yamada et al., 19 Jun 2025).

5. Probes of Quantum Geometry and Future Directions

The NLAHE serves as a unique window into the quantum geometry of Bloch bands. Measuring the second- or higher-order Hall response allows direct access to:

• Berry Curvature Dipole: The spatial asymmetry of Berry curvature and its dependence on Fermi level, strain, and gating (Du et al., 2021).
• Quantum Metric: The field-derivative of Berry connection, revealed especially in higher-wave unconventional magnets and even-layer magnetic TIs, where only metric dipole contributions survive (Korrapati et al., 23 Oct 2025, Kaplan et al., 2022).
• Topological Phase Transitions: Sign reversals in NLAHE have been demonstrated to track topological transitions in moiré materials and Dirac semiconductors (Chakraborty et al., 2022).
• Magnetization and Néel Vector: In magnetic systems, the current-induced or field-induced nonlinear Hall signals can diagnose the orientation and magnitude of magnetic order parameters (Shao et al., 2019, Yu et al., 18 Sep 2024).

The rapid emergence of dynamical and ultrafast techniques, angular-resolved and lock-in protocols, and device-level applications (e.g., rectifiers, frequency doublers, terahertz detectors) suggests NLAHE will remain a frontier for quantum sensing, topological characterization, and next-generation magnetoelectric devices (Du et al., 2021, Wang et al., 7 Mar 2024). Ongoing work aims to develop unified quantum kinetic frameworks encompassing all intrinsic and extrinsic contributions, expand to higher-rank Berry multipoles, and realize robust optical, thermal, and spintronic functionalities in both quantum and classical material platforms.

6. Common Misconceptions and Distinguishing Mechanisms

Several key misconceptions have been clarified in recent literature:

• Nonlinear Hall Signals Can Have Classical Origins: In multicarrier systems, the classical Lorentz force can yield strong nonlinear Hall resistivities indistinguishable in magnitude and even sign from those explained by Berry curvature dipole physics. Zero-magnetic-field second-harmonic Hall signals, and symmetry controls, are essential for unambiguous identification of quantum NLAHE (Yamada et al., 19 Jun 2025).
• Disorder-Driven Effects May Dominate: Side-jump, skew scattering, and disorder-renormalized BCD terms can produce NLAHE signals an order of magnitude above intrinsic contributions in disordered or PT-symmetric systems. Frequency, temperature, and impurity control experiments are critical to isolate these channels (Atencia et al., 2023, Du et al., 2021).
• Band Structure Evolution Under External Fields Can Mimic Chiral or Real-Space Effects: Berry curvature hotspots and NLAHE peaks in certain materials (e.g. EuCd$_2$As$_2$) can result from field-induced band inversions, not only from real-space spin chirality or topological Hall physics (Cao et al., 2021).

7. References to Notable Papers

• "Nonlinear Hall Effects" (Du et al., 2021)
• "Observation of the nonlinear anomalous Hall effect in 2D WTe2" (Kang et al., 2018)
• "Giant nonlinear anomalous Hall effect induced by spin-dependent band structure evolution" (Cao et al., 2021)
• "Disorder in the non-linear anomalous Hall effect of $\mathcal{P}\mathcal{T}$-symmetric Dirac fermions" (Atencia et al., 2023)
• "Nonlinear anomalous Hall effects probe topological phase-transitions in twisted double bilayer graphene" (Chakraborty et al., 2022)
• "Electric field induced Berry curvature dipole and non-linear anomalous Hall effect in higher wave symmetric unconventional magnets" (Korrapati et al., 23 Oct 2025)
• "Anomalous Hall effect from nonlinear magnetoelectric coupling" (Yu et al., 18 Sep 2024)
• "Pseudo anomalous Hall effect in semiconductors and semimetals: A classical perspective" (Yamada et al., 19 Jun 2025)

The NLAHE thus provides an intricate, symmetry-dependent landscape linking real and momentum-space geometry, topological invariants, disorder physics, and device engineering.