Magneto-Nonlinear Hall Effects

• Magneto-nonlinear Hall effects are transverse transport responses in which the Hall signal becomes nonlinear due to magnetic symmetry breaking and Berry curvature mechanisms.
• They encompass multiple structures including second-order electric responses, bilinear Hall responses, and magnetic field–dependent conductivity, analyzed via semiclassical dynamics and symmetry tensors.
• Applications span from magnetic proximity effects in 2D materials to chiral texture-induced emergent electrodynamics, providing rich platforms to probe quantum geometry in magnetic systems.

Magneto-nonlinear Hall effects are transverse transport responses in which the Hall signal is nonlinear in the applied fields and is enabled, modified, or selected by magnetism, magnetic field, magnetic textures, or magnetic symmetry breaking. In the current literature, this category includes at least three recurring structures: second-order electric responses (j_a=\chi_{abc}E_bE_c), bilinear Hall responses (j_a^{\text{IAHE}}=\chi_{abc}\mu_0 E_b H_c), and Hall conductivities whose nonlinear dependence is written directly as a function of magnetic field, such as (\sigma_H=\mathrm{sgn}(H)\sigma_0+\sigma_1H+\mathrm{sgn}(H)\sigma_2H^2) [2104.06690; 2403.04192; 2502.04920]. The nonmagnetic nonlinear Hall effect in time-reversal-symmetric inversion-broken crystals, governed by the Berry-curvature dipole, provides the formal baseline against which specifically magnetic mechanisms are usually distinguished [2104.06690].

1. Scope and classification

The modern subject is not a single mechanism but a family of Hall responses that share two features: the response is transverse, and linear Hall intuition based only on time-reversal breaking is no longer sufficient. In the baseline nonmagnetic case, a Hall-like current quadratic in electric field can appear in a time-reversal-symmetric metal when inversion symmetry is broken and the Berry-curvature dipole is finite. In semiclassical form,
[
j_a=\chi_{abc}E_bE_c,\qquad
\chi_{\alpha\beta\gamma}=\frac{e^3\tau}{\hbar^2}\epsilon_{\alpha\beta\delta}D_{\gamma\delta},
]
with (D_{\gamma\delta}) the Berry-curvature dipole tensor [2104.06690]. This response is a Fermi-surface quantity, not a quantized filled-band invariant [2104.06690].

Magneto-nonlinear Hall effects extend that framework in several distinct directions. One direction keeps the second-order electric response (j\sim E^2) but makes magnetism the symmetry-breaking actuator, as in layer-selective magnetic proximity in metallic (1H)-Nb(X_2), in ferromagnetic Rashba systems, or in Kane-Mele-type topological insulators with Rashba and Zeeman couplings [2603.24019; 2106.08309; 2108.07860]. A second direction yields responses bilinear in electric and magnetic fields, (j\sim EH), through magnetic-field-induced corrections to Berry connection or Berry curvature, as in the orbital magneto-nonlinear anomalous Hall effect in Fe(_3)Sn(_2) [2403.04192]. A third direction makes the Hall conductivity itself nonlinear in magnetic field, most explicitly in altermagnetic Mn(_5)Si(_3), where the defining contribution is (\mathrm{sgn}(H)\sigma_2H^2) [2502.04920].

A further extension replaces momentum-space inversion breaking by real-space magnetic textures. In that setting, electrons moving through smooth noncoplanar textures acquire emergent electrodynamics, and the leading nonlinear Hall conductivity becomes proportional to an emergent toroidal moment (\mathcal T_a^e) [2409.04638]. Another extension introduces external magnetic field directly into the semiclassical dynamics through the Lorentz force, generating field-induced nonlinear Hall effects that are linear in (B) or controlled by the dipole moment of (\Omega^2) rather than the ordinary Berry-curvature dipole [2201.02505; 2409.03144].

Taken together, these works suggest that “magneto-nonlinear Hall effect” is best understood as an umbrella term for nonlinear Hall transport in which magnetic degrees of freedom are essential either to the symmetry analysis, to the microscopic driving term, or to the experimentally extracted nonlinear coefficient.

2. Symmetry structure and response tensors

Symmetry analysis is decisive because many candidate responses are forbidden in pristine crystals and appear only after magnetic perturbations lower the symmetry. In the time-reversal-symmetric baseline theory, the Berry curvature obeys (\mathbf\Omega(-\mathbf k)=-\mathbf\Omega(\mathbf k)), so the linear anomalous Hall effect vanishes, but the second-order tensor (\chi_{\alpha\beta\gamma}) can survive through the Berry-curvature dipole in noncentrosymmetric metals [2104.06690]. In two dimensions, the nonlinear Hall part transforms as a pseudovector; a single mirror line does not forbid it, but two or more mirror lines force it to vanish. Evenfold rotations (C_2), (C_4), and (C_6) with time reversal force the Berry curvature to vanish pointwise, so the Berry-curvature dipole vanishes as well [2104.06690].

Magnetic symmetry breaking changes those selection rules in controlled ways. In pristine monolayer (1H)-Nb(X_2), time reversal kills the intrinsic anomalous Hall conductivity, while exact (C_3) symmetry forces the in-plane Berry-curvature dipole vector to vanish. Layer-selective magnetic proximity breaks these symmetries independently: one-sided out-of-plane exchange can generate anomalous Hall conductivity while keeping the Berry-curvature dipole exactly zero because (C_3) remains intact, whereas an in-plane exchange component breaks (C_3) and produces a finite (D_y) that is odd and approximately linear in the in-plane exchange scale [2603.24019]. This separation of symmetry requirements is central: linear anomalous Hall conductivity and nonlinear Hall conductivity need not turn on together.

A different symmetry logic appears in the Lorentz-force-induced “type-III” nonlinear Hall effect. There the conductivity takes the form
[
j^{\text{NLH}}\alpha=\sigma^{\text{NLH}}{\alpha\beta\gamma\lambda}E_\beta E_\gamma B_\lambda,
]
and the relevant geometric object is the dipole moment of (\Omega_z^2), not of (\Omega_z) [2409.03144]. Because the integrand scales as (v\,\Omega^2), the effect requires simultaneous breaking of inversion and time-reversal symmetry. The symmetry classification is correspondingly broader: type-III nonlinear Hall effect is allowed in 69 of 122 magnetic point groups, whereas Berry-curvature-dipole and intrinsic type-I nonlinear Hall effects are allowed in 53 of 122 [2409.03144].

In magnetic crystals with nonlinear magnetoelectric coupling, the Hall response can be recast in terms of a Hall vector (\boldsymbol{\mathcal M}\equiv (\sigma_{zy},\sigma_{xz},\sigma_{yx})), which transforms like magnetization. Expanding the effective field conjugate to magnetization in powers of electric field,
[
B^\alpha_{\mathrm{eff}}=\lambda_\alpha+\lambda_{\alpha\beta}E_\beta+\lambda_{\alpha\beta\gamma}E_\beta E_\gamma+\cdots,
]
one obtains electric-field-induced anomalous Hall conductivity components whose lowest allowed order can be linear, quadratic, cubic, uni-axial, bi-axial, or tri-axial depending on magnetic point group [2409.11662]. Type-II magnetic point groups cannot host this magnetoelectric anomalous Hall effect because time reversal remains an independent symmetry operation; type-I and type-III magnetic point groups can [2409.11662].

3. Microscopic mechanisms

The Berry-curvature-dipole mechanism remains the formal starting point. In semiclassical dynamics,
[
\dot{\mathbf r}=\frac{1}{\hbar}\nabla_{\mathbf k}\epsilon(\mathbf k)+\frac{e}{\hbar}\mathbf E\times\mathbf\Omega_{\mathbf k},
]
and the nonlinear Hall current arises because the electric field shifts the nonequilibrium distribution so that opposite regions of Berry curvature are weighted unequally [2104.06690]. The same literature also emphasizes that the dc response scales as (\chi\propto\tau), while high-frequency intraband response becomes effectively (\tau)-independent for (\omega\tau\gg 1) below interband absorption [2104.06690].

An orbital-magnetization formulation rewrites this intrinsic second-order response as a two-step process. First the electric field induces nonequilibrium orbital magnetization,
[
M_i=\chi_{ij}E_j,
]
and then the magnetization current relation
[
J_{H,i}=\epsilon_{ijk}\frac{\partial M_k}{\partial r_j}
]
produces a Hall current (J_{H,i}=\sigma^H_{ijk}E_jE_k). The resulting conductivity formula explicitly contains both the Berry-curvature-dipole term and the quantum-metric-dipole term, thereby unifying the standard intrinsic nonlinear Hall effect with an orbital-magnetization picture [2503.07960]. This suggests a useful bridge between nonmagnetic and magnetic nonlinear Hall phenomena, although the paper explicitly limits that framework to intrinsic band-property contributions in the low-frequency intraband regime [2503.07960].

Magnetically enabled second-order Hall transport appears in several minimal models. In the ferromagnetic Rashba model,
[
H_{\mathbf k}=\frac{\hbar^2k^2}{2m^*}+\alpha^{\rm R}(\mathbf k\times \hat{\mathbf e}_z)\cdot\boldsymbol{\sigma}+\frac{\Delta V}{2}\boldsymbol{\sigma}\cdot\hat{\mathbf M},
]
the nonlinear Hall effect and unidirectional magnetoresistance are different components of the same second-order dc conductivity tensor [2106.08309]. Standard Keldysh and Moyal-Keldysh formalisms give identical numerical results, the nonlinear Hall and UMR conductivities are comparable in magnitude, and the Boltzmann result corresponds not to the intraband term alone but to the sum of the intraband term and an interband term that can be brought into the form of an effective intraband term through the (f)-sum rule [2106.08309].

In a simpler Rashba conductor with an in-plane magnetic field, magnetochiral anisotropy itself generates a nonlinear Hall effect. For (\mathbf E\parallel \mathbf B),
[
j_y^{(2)}\propto E_x^2 B_x,
]
and the predicted Hall response is odd in (B), quadratic in (E), independent of hexagonal warping and cubic Dresselhaus terms, and exactly one third of the longitudinal nonlinear rectification current in the maximized (\mathbf E\perp \mathbf B) geometry [2306.09993]. In Kane-Mele-type two-dimensional topological insulators, the corresponding magneto-nonlinear Hall mechanism requires Rashba coupling and an in-plane Zeeman field simultaneously; the combined perturbation generates a term proportional to (\lambda_R\Delta_B k_y), distorts the Berry-curvature distribution, and produces a finite Berry-curvature dipole with pronounced signatures near topological phase transitions [2108.07860].

Magnetic field can also enter more directly through Berry-curvature-modified Lorentz dynamics. In a time-reversal-symmetric, inversion-broken Weyl semimetal, the interplay of Berry curvature and the magnetic part of the Lorentz force generates a nonlinear Hall conductivity linear in (B), even though the zero-field Berry-curvature-dipole contribution vanishes in the untilted model after summing over nodes [2201.02505]. A related but distinct “type-III” nonlinear Hall effect in magnetic crystals and MnBi(2)Te(_4) thin films scales as (j^{\mathrm H}\propto \tau E^2B) and is governed by the dipole moment of the square of Berry curvature,
[
\mathcal D\alpha^{(2)}=\int \frac{d\mathbf k}{(2\pi)^2}(\partial_{k_\alpha}\Omega_z^2)f_0,
]
rather than the ordinary Berry-curvature dipole [2409.03144].

Real-space magnetic textures produce a different class of nonlinear Hall transport. For conduction electrons coupled to a smooth noncoplanar texture (\mathbf n(\mathbf r)), the emergent magnetic field
[
b_a=\epsilon_{abc}\,\mathbf n\cdot(\partial_b\mathbf n\times\partial_c\mathbf n)
]
defines an emergent toroidal moment
[
\mathcal T_a^e=\frac12\int d^3r\,(\mathbf r\times\mathbf b)a,
]
and the leading nonlinear Hall conductivity obeys (\chi{abb}\propto \mathcal T_a^e) [2409.04638]. This is explicitly identified as the real-space counterpart of the momentum-space Berry-curvature-dipole nonlinear Hall effect [2409.04638].

4. Material platforms and experimental realizations

The experimental landscape now spans magnetic order, magnetic proximity, chiral topology, and altermagnetism. A concise comparison is useful.

Platform	Hall structure	Salient result
Monolayer (1H)-Nb(X_2)	Proximity-enabled AHE and NLHE	(\sigma^{\mathrm{sheet}}_{xy}\sim 10^{-2}(e^2/h)); (
Fe(_3)Sn(_2)	Bilinear in-plane anomalous Hall current	(j_a^{\text{IAHE}}=\chi_{abc}\mu_0E_bH_c); (\chi_{yxy}) about (-4.1) and (-1.8)(\,\Omega^{-1}\text{cm}^{-1}\text{T}^{-1}) at 20 and 50 K
Mn(_5)Si(_3) thin film	Magnetic nonlinear Hall effect in (H)	(\sigma_H=\mathrm{sgn}(H)\sigma_0+\sigma_1H+\mathrm{sgn}(H)\sigma_2H^2), large and unsaturated to 60 T
CoSi	Magnus-type nonlinear Hall effect	(\chi_{\mathrm{exp}}\sim 5\times10^{-3}\,\mathrm{A\,V^{-2}}); odd, approximately linear-in-(B) modulation
WTe(_2) and related nonmagnetic platforms	Baseline BCD NLHE	Flagship nonmagnetic benchmark for comparison with magnetic mechanisms

In metallic monolayer (1H)-Nb(X_2), magnetic proximity is used as a symmetry-design tool rather than merely as a perturbation. Fully relativistic DFT, spinor maximally localized Wannier functions, and Wannier interpolation show that out-of-plane one-sided exchange produces a sizable sheet anomalous Hall conductivity while keeping the Berry-curvature dipole exactly zero; adding an in-plane exchange component or using an orthogonal two-sided texture produces a strongly tunable (D_y), with the strongest response in NbTe(_2). The paper further proposes a dual-interface micron-scale Hall bar in which the signs of the first- and second-harmonic Hall voltages encode two bits of magnetic information using the same contacts [2603.24019].

In kagome ferromagnet Fe(_3)Sn(_2), systematic angular and temperature-dependent transport measurements identify an in-plane anomalous Hall current linear in both applied in-plane electric and magnetic fields. The measured bilinear response is interpreted as an intrinsic orbital magneto-nonlinear anomalous Hall effect dominated by anomalous orbital polarizability rather than by spin contributions. First-principles calculations find the orbital contribution to be 2 to 3 orders of magnitude larger than the spin contribution and at least 1 order of magnitude larger than other terms below 80 K, with Weyl points within 10 meV of the Fermi level acting as hot spots [2403.04192].

In epitaxially strained Mn(_5)Si(_3) thin films, the defining transport law is
[
\sigma_H=\mathrm{sgn}(H)\sigma_0+\sigma_1H+\mathrm{sgn}(H)\sigma_2H^2.
]
The weak-field offset (\sigma_0) is associated with 180° switching of the magnetic structure, (\sigma_1H) is the ordinary Hall contribution, and (\mathrm{sgn}(H)\sigma_2H^2) is identified as the magnetic nonlinear Hall effect. The effect appears in the altermagnetic phase, is absent in the paramagnetic, ferromagnetic, and noncollinear antiferromagnetic control phases, is non-analytic under field reversal, and remains large and unsaturated up to 60 T [2502.04920]. The proposed microscopic origin is a chiral next-nearest-neighbor hopping process that acquires both magnetic-exchange-driven Zeeman energies and Haldane-like chiral flux phases, producing an effective (m^2\tau_z) term in the low-energy Hamiltonian [2502.04920].

The chiral Weyl semimetal CoSi provides a different route. There the conventional Berry-curvature-dipole mechanism is symmetry-forbidden, yet focused-ion-beam-fabricated crossbar devices on the ([111]) surface show a robust second-harmonic Hall voltage at zero magnetic field. The extracted nonlinear Hall conductivity is (\chi_{\mathrm{exp}}\sim 5\times10^{-3}\,\mathrm{A\,V^{-2}}), the zero-field signal changes sign near (T\approx 140) K, and the field-induced correction (\Delta\chi(B)) is odd in magnetic field and approximately linear in (B) up to (\pm 6) T [2604.28091]. The mechanism is attributed to Magnus-type skew scattering of self-rotating electron wave packets, with chirality set by Weyl-node topology rather than by a Berry-curvature dipole [2604.28091].

These magnetic platforms are typically interpreted relative to the nonmagnetic benchmark established by bilayer and few-layer WTe(_2), strained bilayer graphene, and related inversion-broken systems in which the nonlinear Hall effect is already known to arise at (B=0) from a Berry-curvature dipole [2104.06690]. That comparison is methodologically important because it separates genuinely magnetic nonlinear Hall mechanisms from purely structural quantum-geometric ones.

5. Adjacent Hall phenomena and recurrent ambiguities

A central ambiguity in the literature is terminological: not every Hall response that is unusual, zero-field, or nonlinear should be classified identically. The nonmagnetic Berry-curvature-dipole effect is second order in electric field and can exist with preserved time-reversal symmetry, so a nonlinear Hall signal alone is not evidence of magnetic order [2104.06690]. Conversely, several magnetic or field-enabled effects are Hall-like without fitting the standard (E^2) Berry-curvature-dipole template.

The Magnus Hall effect illustrates that distinction. In its original form it is a linear-response Hall effect in a time-reversal-invariant inversion-broken device with a built-in electric field, not a second-order conductivity in the applied bias. Its central conductance,
[
G_H=\frac{e^2}{h}\frac{\Delta U}{2\pi}\int_{v_x(\mathbf k)>0} d^2k\,\Omega(\mathbf k)\,\delta(\epsilon_{\mathbf k}-\mu),
]
measures a directionally restricted Fermi-surface integral of Berry curvature in the ballistic regime [1904.00013]. In three-dimensional topological semimetals the same mechanism can survive even without time-reversal symmetry, remains controlled by built-in field and Berry curvature, and becomes tensorial and strongly anisotropic [2111.05322]. In CoSi, the observed nonlinear Hall effect is described as a Magnus-type skew-scattering response rather than a Berry-curvature-dipole response, which further broadens the usage of “Magnus” within nonlinear Hall transport [2604.28091].

Real-space magnetic inhomogeneity defines another distinct class. In a two-dimensional electron gas subject to an odd magnetic profile (B(-x)=-B(x)), quantum kinetic theory shows that the linear Hall response can vanish by parity while a purely nonlinear Hall response survives. In a periodic real-space magnetic dipole, boundary or snake states near sign changes of (B(x)) dominate the response, and a parametric resonance controlled by the cyclotron ratio and the magnetic-texture width can greatly enhance rectification and second-harmonic generation [2106.02001]. This mechanism does not rely on a Berry-curvature dipole and can occur irrespective of whether the underlying crystal breaks inversion symmetry [2106.02001].

A further source of ambiguity is that some uses of “nonlinear Hall effect” are classical or magnetothermal rather than band-geometric. In a stationary conducting cylinder with radial temperature gradient and axial magnetic field, the Hall current generates an additional axial magnetic field that feeds back into the Hall current itself. The resulting self-consistent suppression of the total field is described as a nonlinear Hall effect in that literature, but it is a magneto-thermal Hall feedback problem rather than a quantum-geometric nonlinear Hall effect of Bloch electrons [2304.13630].

Optical strong-field Hall physics forms yet another neighboring domain. For a topological-insulator surface with broken time-reversal symmetry, intense near-threshold optical driving modifies the dynamical Hall response through coherent interband dressing and the optical Stark effect; the nonlinear Hall signatures are encoded in Faraday and Kerr rotations rather than in dc transverse voltages [1605.02331]. This is a magneto-optical nonlinear Hall effect, not a dc Berry-curvature-dipole response.

6. Methods, limitations, and open directions

The methodological landscape is correspondingly diverse. Semiclassical Boltzmann transport underlies the Berry-curvature-dipole theory, magnetic-field-induced Weyl-semimetal nonlinear Hall transport, and type-III Lorentz-force-induced nonlinear Hall physics [2104.06690; 2201.02505; 2409.03144]. Nonequilibrium Green-function methods, including standard Keldysh and Moyal-Keldysh formalisms, provide fully quantum dc second-order conductivities in ferromagnetic Rashba systems [2106.08309]. Diagrammatic Feynman techniques capture texture-induced nonlinear Hall effects in three-dimensional magnetic metals and reveal their dependence on emergent toroidal moments [2409.04638]. Fully relativistic DFT combined with Wannier interpolation is now standard for material-specific predictions in two-dimensional metallic TMDs and orbital magneto-nonlinear anomalous Hall systems [2603.24019; 2403.04192].

Several limitations recur across the field. The canonical Berry-curvature-dipole theory is a weak-field expansion, is a Fermi-surface theory rather than a quantized filled-band response, and in dc scales linearly with relaxation time even when its tensor structure is geometric [2104.06690]. The ferromagnetic Rashba analysis shows that intraband and interband terms cannot always be cleanly separated by simple Boltzmann intuition [2106.08309]. The texture-induced theory is controlled only in the smooth-texture, weak-disorder, long-wavelength, local-effective-field regime, with (E_F\tau\gg1), (q\ell<1), and ((q\ell)^2<M\tau) [2409.04638]. The layer-selective proximity study in (1H)-Nb(X_2) introduces exchange phenomenologically at the Wannier level rather than self-consistently in DFT, although it does so in a layer- and orbital-selective manner [2603.24019].

A second recurring issue is intrinsic-versus-extrinsic disentanglement. The time-reversal-symmetric nonlinear Hall review explicitly notes side-jump and skew-scattering contributions beyond the constant-(\tau) intrinsic Berry-curvature-dipole term [2104.06690]. CoSi makes this problem more pointed by showing that a sizeable nonlinear Hall effect can be dominated by a skew-scattering mechanism tied to Weyl-node chirality even when the Berry-curvature dipole is symmetry-forbidden [2604.28091]. This implies that scaling analyses, harmonic decomposition, field-angle studies, and symmetry-based device geometries remain essential for interpretation.

The current trajectory of the subject suggests three broad directions. First, magnetic symmetry engineering—through proximity, strain, gating, and altermagnetic order—has become a systematic route to activate nonlinear Hall channels that are forbidden in pristine crystals [2603.24019; 2502.04920]. Second, real-space topology and emergent electrodynamics introduce genuinely three-dimensional magnetic-texture responses that are not reducible to momentum-space Berry-curvature dipoles [2409.04638]. Third, magnetic-field-enabled nonlinear Hall effects increasingly probe geometric objects beyond the ordinary dipole of Berry curvature, including anomalous orbital polarizability, (\Omega^2) dipoles, and effective magnetoelectric invariants [2403.04192; 2409.03144; 2409.11662]. A plausible implication is that magneto-nonlinear Hall transport is evolving from a narrow offshoot of anomalous Hall physics into a broader spectroscopy of magnetic quantum geometry.