Quantum theory of the nonlinear Hall effect

The nonlinear Hall effect (NHE) is characterized by a second-order transverse response, producing a voltage (or current) at zero or double frequency in response to two ac fields, and unlike conventional Hall effects it does not require time-reversal symmetry breaking but does require inversion symmetry breaking. Because the NHE is highly sensitive to discrete and crystal symmetries, it serves as a probe for symmetry-breaking phase transitions (e.g., ferroelectric order or hidden orders). The NHE has a quantum origin via the Berry curvature dipole, and experiments typically operate in the near-dc regime (10–1000 Hz), underscoring the need for a quantum transport treatment. Prior to this work, theoretical descriptions were largely semiclassical Boltzmann approaches under a relaxation-time approximation, with emerging attempts toward quantum formulations and new side-jump terms without semiclassical correspondence. Nonetheless, a systematic quantum theory that incorporates disorder effects explicitly remained lacking. This study addresses that gap by formulating a full quantum, diagrammatic theory for the NHE.

Semiclassical theories relate the NHE to the Berry curvature dipole in time-reversal-invariant, inversion-broken systems (e.g., Sodemann & Fu 2015), and have been applied to materials like WTe2 and MoTe2. Extensions included disorder-induced NHE mechanisms and modified semiclassics via quantum kinetics, identifying side-jump and skew-scattering analogs and even contributions without semiclassical counterparts. Related nonlinear transport phenomena (gyrotropic Hall, Magnus Hall, nonlinear Nernst) and optical second-order responses have been explored, often employing diagrammatic or Kubo-type formalisms in optical contexts. However, a comprehensive quantum transport framework for the low-frequency NHE, systematically including disorder via diagrammatics and ensuring gauge invariance and well-defined dc limits, was missing. The present work builds on many-body and transport diagrammatics (e.g., Kubo-Streda, non-crossing approximation) to fill this gap.

• Formulation of second-order response: The nonlinear current response J_a^(2) to electric fields E_b, E_c at frequencies ω_b, ω_c is expressed via a third-rank conductivity tensor with zero- and double-frequency components. The experimentally accessible double-frequency response χ^e_abc is the focus.
• Gauge-invariant Kubo formalism and effective diagrammatics: To maintain gauge invariance up to quadratic order in fields, the vector potential is expanded in the Peierls-substituted Hamiltonian H(k + eA/ħ) to third order, introducing higher-order velocity vertices v_a, v_ab, v_abc. After careful treatment of all vertices on equal footing, the dc-limit quadratic conductivity splits into three contributions χ^e_I, χ^e_II (Fermi-surface terms corresponding to triangular and two-photon diagrams, respectively), and χ^e_III (Fermi-sea terms). The latter is subleading in the weak-disorder, low-frequency transport regime and is neglected. The final expressions are written in terms of retarded/advanced Green’s functions, their energy derivatives, and velocity vertices [Eqs. (2)–(4) in the paper].
• Diagrammatic construction: In the weak-disorder limit and with time-reversal symmetry, the leading contributions scale with the impurity concentration to order n_i^1. All such diagrams are constructed by adding non-ladder scattering events to the basic triangular and two-photon topologies within the non-crossing approximation. A total of 69 diagrams are identified: intrinsic, side-jump, intrinsic skew-scattering, and extrinsic skew-scattering classes. Of these, 64 involve disorder scattering and vertex/edge corrections.
• Generic two-band model: The electronic structure is modeled by H = h_0 + h_x σ_x + h_y σ_y + h_z σ_z, describing two bands ε = h_0 ± ħω with ħω = (h_x^2 + h_y^2 + h_z^2)^{1/2}. Disorder is modeled by short-range δ-function impurities V_imp(r) = V δ(r−R) with random positions; Gaussian (〈V〉=0, 〈V^2〉=V^2) and non-Gaussian (〈V^3〉=V^3) disorder correlations are included to capture two- and three-event correlations leading to side-jump and skew-scattering processes.
• Mechanism identification and simplifications: Intrinsic terms are tied to Berry curvature; side-jump terms to off-diagonal scattering processes; skew scattering contains intrinsic (Gaussian disorder, asymmetric processes from correlated gradients) and extrinsic (non-Gaussian disorder) parts. A simplified effective diagrammatic representation introduces modified velocities (diagonal, off-diagonal, side-jump, skew-scattering) to reduce the number of explicit diagrams while preserving physics, highlighting qualitative differences from anomalous Hall counterparts and the role of indecomposable self-energies Σ in going beyond non-crossing approximations.
• Application: 2D tilted Dirac model: A minimal model for NHE is implemented by choosing h_0 = t k_x, h_x = v k_x, h_y = v k_y, h_z = m, with parameters t (tilt), v (Dirac velocity), and m (gap). Time-reversal symmetry is restored by including the time-reversed partner (m→−m, t→−t) in the Brillouin zone, which contributes equally by symmetry. Within the non-crossing approximation and at T=0, χ components (e.g., χ_xyz^xx in the figure) are computed numerically. Parameters used in the case study: t = 0.1 eV Å, v = 1 eV Å, m = 0.1 eV, n_V = 10^2 eV^2 Å^2, n_V^3 = 10^6 eV^3 Å^4.
• Symmetry analysis: Based on the tensor structures emerging from the diagrams, the extrinsic χ^ex_abc is symmetric under b↔c, whereas intrinsic χ^in_abc, linked to the Berry curvature dipole, is antisymmetric under exchanges involving the current index with a field index. Using these constraints, allowed tensor elements are classified for all 32 crystal point groups (2D in Table 1; 3D in Supplementary Table 2).

• Quantum diagrammatic theory: A complete quantum framework for the nonlinear Hall effect is developed using Kubo-type diagrammatics, identifying 69 leading-order Feynman diagrams (weak-disorder limit). Of these, 64 are disorder-related, underscoring the decisive role of disorder in low-frequency electronic transport.
• Formal results: Gauge-invariant, dc-limit expressions for χ^e_abc are derived, separating Fermi-surface (triangular and two-photon) and subleading Fermi-sea contributions. Intrinsic terms reduce to Berry-curvature dipole physics in the weak-disorder limit, while disorder yields side-jump and skew-scattering corrections without simple semiclassical counterparts.
• Scaling: Despite qualitative differences from semiclassical theories, the scaling with impurity concentration for each mechanism is preserved; thus previously observed scaling laws remain valid.
• 2D tilted Dirac model case study (T=0): The intrinsic contribution from the quantum theory matches the semiclassical Berry-curvature-dipole result. However, side-jump and skew-scattering quantum contributions have opposite signs compared to semiclassical predictions. Consequently, the total quantum χ shares the sign and similar lineshape of the intrinsic part but with enhanced magnitude. This contrasts with semiclassical totals and helps reconcile experiments where scaling suggests sizable extrinsic contributions yet the overall response follows Berry-curvature-dipole trends (e.g., bilayer WTe2).
• Symmetry predictions: The extrinsic tensor χ^ex has more allowed nonzero elements than the intrinsic χ^in due to different index-symmetry constraints. For 2D systems, pure disorder-induced NHE (χ^in=0 but χ^ex≠0) is allowed in point groups C3, C3h, C3v, D3h, D3. For 3D systems, point groups supporting pure disorder-induced NHE include T, T_h, C3v, D3h (per the detailed symmetry analysis).

The work addresses the lack of a quantum, disorder-explicit theory for the nonlinear Hall effect by constructing a gauge-invariant, diagrammatic formalism valid in the dc and low-frequency limits. The findings demonstrate that while intrinsic (Berry curvature dipole) physics persists, disorder plays a dominant and qualitatively different role than semiclassics suggest: two-photon processes and ω-dependent vertex corrections contribute at leading order, and side-jump and skew-scattering terms differ in sign from semiclassical expectations. In the 2D tilted Dirac model, the total quantum response aligns in sign with the intrinsic component and is enhanced, explaining why experiments that indicate comparable intrinsic and extrinsic magnitudes can still be qualitatively described by Berry-dipole phenomenology. The symmetry analysis clarifies when disorder alone can generate an NHE (certain point groups), guiding material searches and interpretation of measurements in systems where Berry curvature dipoles are symmetry-forbidden. Overall, the theory links microscopic disorder scattering to macroscopic nonlinear transport, advancing beyond semiclassical paradigms.

This study establishes a systematic quantum diagrammatic theory for the nonlinear Hall effect, deriving gauge-invariant dc-limit formulas and classifying all leading disorder-induced mechanisms (intrinsic, side-jump, intrinsic and extrinsic skew-scattering). The framework reveals qualitative deviations from semiclassical predictions while preserving scaling behaviors. In a prototypical 2D tilted Dirac model, quantum extrinsic contributions reverse sign relative to semiclassics, yielding a total response that retains the intrinsic sign with enhanced magnitude. A comprehensive symmetry analysis predicts pure disorder-induced NHE regimes for specific 2D and 3D point groups, providing actionable criteria for experiments and materials design. Future work should: (i) go beyond the non-crossing approximation, including crossed diagrams and indecomposable self-energies; (ii) incorporate full-band electronic structures and realistic disorder beyond δ-correlated models; (iii) explore frequency dependence beyond the dc limit; and (iv) establish general rules governing competition between intrinsic and extrinsic mechanisms across materials classes.

• Weak-disorder and non-crossing approximations are assumed; crossed diagrams and certain self-energy structures (indecomposable Σ) are omitted though they may contribute in some regimes.
• Fermi-sea contributions are neglected as subleading in the weak-disorder, low-frequency limit; this may not hold at higher frequencies or stronger disorder.
• The generic two-band model and δ-function disorder are idealized; real materials have multi-band structures and complex disorder statistics.
• Case-study results are for a specific 2D tilted Dirac model and parameter set at T=0; generality to other systems requires further validation.
• Accurate dc-limit treatment relies on including multi-photon vertices; low-energy linear-k effective models may be insufficient without full-band information.