Nonlinear Hall effect: response order, symmetry and hysteresis

Scope

A nonlinear Hall effect is a transverse response beyond first order in the driving electric field. A curved Hall-resistivity trace versus magnetic field can instead arise within ordinary linear-response transport. An anomalous Hall effect, a second-order Hall response, magnetic hysteresis and a superconducting diode signal therefore require separate measurement definitions. The second-order effect can appear in time-reversal-symmetric conductors; a zero-field linear Hall conductivity requires the relevant time-reversal symmetry to be broken. R1 R3 R4

The familiar empirical separation into ordinary Hall and a term proportional to magnetization is not a universal definition of anomalous Hall physics. Noncollinear Mn3Sn provides a counterexample to requiring a large net ferromagnetic moment. Its magnetic symmetry matters even though the net moment is small. A hysteresis loop alone establishes history dependence, not the microscopic origin of a Hall signal. R4 R8

Mathematical assumptions

In a fixed laboratory frame, write $j_a=\sigma_{ab}E_b+\chi_{abc}E_bE_c+\cdots$, summing repeated indices. The spatially uniform electric-dipole second-order tensor vanishes when inversion symmetry is present. Breaking inversion is necessary for this contribution but does not guarantee an allowed nonzero component. Boundaries, finite-wave-vector effects and other response mechanisms require separate treatment. R1

For the illustrative voltage law $V_y=\alpha I+\beta I^2$ and peak-amplitude drive $I=I_0\cos\omega t$, the quadratic contribution is $\beta I_0^2[1+\cos(2\omega t)]/2$. A controlled quadratic DC measurement can also establish second-order transport; detecting $2\omega$ is not an exclusive definition. Conversely, a thermal voltage proportional to dissipated power has the same harmonic structure, so a second harmonic is not sufficient evidence for a Berry-curvature mechanism. Phase and peak-versus-RMS conventions must be reported. R1 R2

In the two-dimensional semiclassical constant-relaxation-time model, $D_a=\sum_n\int d^2k/(2\pi)^2\,f_{n0}\partial_{k_a}\Omega_{n,z}$ has dimensions of length. Time reversal makes Berry curvature odd in momentum, allowing its occupied-state integral to vanish while its dipole remains finite. The model gives a second-harmonic current proportional to $[\tau/(1+i\omega\tau)](\hat{z}\times\mathcal{E})(\mathbf{D}\cdot\mathcal{E})$. Here $\mathcal{E}$ is a complex field amplitude; omitted charge and Planck-constant factors must be restored consistently before numerical comparison. This is a model contribution, not the complete measured nonlinear conductivity. R1 R3

In this two-dimensional setting, a surviving mirror permits a dipole perpendicular to the mirror line, whereas an in-plane twofold rotation forces it to vanish. Thus fourfold-to-twofold nematic symmetry reduction alone does not establish a finite dipole, even before checking inversion. These two-dimensional vector rules should not be transferred unchanged to the three-dimensional dipole tensor. R1

Experimental criteria

Ma's bilayer WTe2 experiment measures transverse second-harmonic voltages while independently tuning two gates. Figure 2d compares transverse and longitudinal contacts at 50 K and 177.77 Hz; Figure 2e uses 100 K, and the systematic Figure 3 gate maps use 10 K. Those temperatures describe different measurements, not superconducting transitions. The methods test several low frequencies and specify a sine-reference phase convention. The gate-dependent hysteresis associated with ferroelectric switching is not a magnetic-field hysteresis measurement. R2

Kang's few-layer WTe2 work combines Hall bars with circular devices for angular measurements. Its five-layer example is measured at 1.8 K and 137 Hz, while an eight-layer disk maps the angular response. Longitudinal and transverse first-harmonic signals help identify geometric mixing, and the disk analysis antisymmetrizes opposing injection angles. Temperature-dependent scaling suggests both intrinsic and extrinsic contributions; layer number, scattering and device geometry prevent assigning one universal nonlinear coefficient to all WTe2 samples. R3

A credible identification records raw longitudinal and transverse channels, amplitude and frequency dependence, contact permutations, crystal axes, thermal stability and sweep history. Quadratic current scaling and allowed angular symmetry are necessary consistency checks for the proposed second-order contribution, but they do not alone eliminate heating or separate every scattering mechanism. Keep fixed-axis current reversal distinct from rotating the injection and voltage-measurement frame; the latter can change a reported transverse component's sign without changing the quadratic field law. R1 R2 R3

Limits and disputes

The superconducting diode effect concerns unequal positive- and negative-direction critical-current magnitudes. Ando's Nb/V/Ta superlattice demonstrates this through DC resistance–current measurements, rather than a transverse Hall channel. A zero-resistance window in one direction does not make the whole switching cycle lossless. Later conventional-film work identifies asymmetric vortex barriers and screening currents as possible diode mechanisms, so nonreciprocity alone does not establish exotic pairing. R5 R9

Romero's selected longitudinal terahertz/DC measurements do not establish a transverse second-order Hall response. Yang's later author manuscript does report Hall resistance nonlinear in magnetic field after subtraction of a linear background, alongside magnetoresistance hysteresis. These are not measurements of the second-order electric-field response defined here. A scattering rate or re-entrant zero-resistance field must not be relabelled a Hall coefficient or Hall switching field; field curvature and history dependence do not by themselves prove ferromagnetic order. R6 R7

Complete scientific acceptance remains open. Selected Ma final figures and methods, Kang author-manuscript methods and figures, and Sodemann's theory were checked; full artifact controls, final-version comparisons and raw-data reproduction remain incomplete. Kang's current-reversal wording additionally requires complete wiring and reference-frame reconciliation before a sign rule is exported. Ando, Nakatsuji and Hou are used at primary-abstract scope. R1 R2 R3 R5 R8 R9

Discovery implications

Design a discrimination experiment before assigning a mechanism: state which response order is measured, which symmetry allows it, and which contact or thermal control could falsify the interpretation. When magnetic or superconducting states coexist, compare magnetization, longitudinal resistance, transverse harmonics and field history on the same documented device. Preserve two-dimensional versus bulk units and distinguish a calculated Berry-curvature dipole from the total experimental response. R1 R2 R3 R9

Related topics

Berry curvature, Nematic order, and Magnetism and re-entrant superconductivity.

Material cases

Values describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.