{"id":"1d24f3ff-936a-48c2-be9d-cdf3452dffc6","arxiv_id":"2411.16675","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A nonperturbative Boltzmann calculation shows the spontaneous Hall current in inversion-broken, time-reversal-symmetric semimetals crosses over to a quasi-linear response at strong fields and acquires a nonzero 1ω component from field-induced relaxation-time asymmetry.","lead":"The paper predicts that in certain crystals with mirror symmetry but no inversion center, a strong electric field drives a new kind of sideways current that grows nearly linearly with the field and behaves like a time-reversal-broken system, even though the crystal preserves time reversal. It offers a possible explanation for puzzling Hall measurements in Weyl-Kondo semimetals and identifies a concrete strong-field regime for experiments.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 1ω Hall mechanism rests on an assumed sign(Ê·k) relaxation-time asymmetry that is only derived for a 1D toy model with ansatz forms; the E^5 prediction and the odd-harmonic claim are not robust for the 2D model unless the self-consistent τ has exactly the chosen form.","rationale":"The reader's CONDITIONAL verdict correctly identifies the weak link: the odd-harmonic response depends on an assumed field-induced asymmetry of the relaxation time that is not derived for the actual 2D models. The DC and even-AC claims are internally consistent: with τ=τ0, Eq. (4) is solved exactly and the two-band calculations show the breakdown of the Berry-curvature-dipole scaling and a crossover to quasi-linear behavior. Those results do not rely on Eq. (7). The 1ω mechanism, by contrast, is structurally fragile: Appendix A establishes only that a 1D model with chosen ansatz forms produces γ(k)≠γ(−k), and the specific α(E)=α0(E cosωt)^2 used in the main text is one possible fit, not a controlled consequence of the 2D collision integral. The numerical fit in Eq. (A23) even contains a linear term b1E, so dropping it is not justified by the appendix. A direct self-consistent 2D calculation would settle whether the sign(k_x) E^2 form actually emerges; if it does not, the 1ω result and its E^5 scaling are artifacts of the ansatz, while the central FNE Hall crossover would still stand. The manuscript's own stated limitations about multiband and Kondo physics in Ce3Bi4Pd3 further support a conditional rather than unconditional acceptance, but they do not change the assessment of the core model calculation. The proposed numerical test is concrete, feasible, and directly targets the only place where the argument could fail without also undermining the DC result.","tokens_in":65,"tokens_out":31240,"duration_ms":437370,"concrete_test":"Solve Eqs. (A7) and (A9) fully self-consistently on the 2D momentum grid for the Hamiltonian in Eq. (D3) at T>0, without imposing ansatz (A15) or (A25), to obtain the exact γ(k,E). Extract the odd-in-k part γ_odd(k,E) and its leading Taylor coefficients in E; then feed the full γ(k,E) into the harmonic equations (C6) and compute ¯J(1)y. If γ_odd contains a linear-in-E term or has a momentum dependence different from sign(k_x), the E^5 law and the odd-harmonic mechanism are not supported; if γ_odd ≈ α0 E^2 sign(k_x) in the regime of Fig. 3(b), the claim survives.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing input for the odd-harmonic claim is Eq. (7): the paper posits 1/τ(k)=1/τ0+α(E) sign(Ê·k) and then chooses α(E)=α0(E cosωt)^2. Appendix A motivates this only in a 1D parabolic band, using two ansatz forms (A15) and (A25). The numerical fits in Eq. (A23) include a linear-in-E term b1E in the sign-asymmetric part γ1, yet the main text sets b1=0 without deriving that the actual 2D k.p model in Eq. (8)/(D3) produces the same sign(k_x) structure with leading E^2 weight. Since the 1ω current is generated by odd-in-k parts of ¯g0 and ¯g2 (Appendix C), any change in the momentum dependence of γ—a smooth function instead of sign(Ê·k), or a nonzero b1—will alter the odd-harmonic amplitudes and the E^5 scaling shown in Fig. 3(b). The authors themselves state that the precise weak-field scaling of ¯J(1)y is determined by the form of the non-uniformity, which is an admission that Eq. (7) is an assumption rather than a parameter-free derivation. The DC quasi-linear Hall response, computed with τ=τ0, is not affected by this concern; what is at stake is the additional and experimentally motivated claim that a 1ω Hall response appears without breaking microscopic TRS.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops a Boltzmann-equation approach to the spontaneous Hall effect in time-reversal-symmetric, inversion-broken semimetals driven by a strong electric field. The authors solve the DC Boltzmann equation non-perturbatively (with τ = τ0) and find that the Hall current crosses over from the weak-field Berry-curvature-dipole scaling to a quasi-linear E scaling when the field-induced momentum scale kE exceeds the scale kBC set by the Berry curvature distribution. They extend the solution to AC fields and obtain even-harmonic responses; they then introduce a field-induced momentum-space asymmetry in the relaxation time, Eq. (7), which produces a nonzero 1ω Hall response despite microscopic TRS. The results are illustrated with a two-band k.p model and a C4T-symmetric model, and are motivated by experiments on Weyl-Kondo semimetals.","tokens_in":17712,"tokens_out":19538,"duration_ms":167445,"significance":"If correct, the DC and even-harmonic AC results constitute an important advance: they go beyond the weak-field nonlinear Hall paradigm of Sodemann and Fu, provide a concrete breakdown criterion kE ~ kBC, and are supported by an exact solution of the Boltzmann equation (Eq. B3) and a perturbative breakdown analysis (Appendix B). The odd-harmonic mechanism is potentially significant for understanding the 1ω response in Ce3Bi4Pd3, but it is not established at the same level because it relies on an assumed form of the relaxation-time asymmetry; the paper is transparent about the dependence of the E^5 scaling on this choice. The manuscript's strengths include closed-form zero-temperature distribution functions, explicit numerical demonstrations in two models, and a clear separation of the weak-field and FNE regimes.","major_comments":[{"comment":"The 1ω Hall response is a headline result (Abstract, Introduction, Conclusion), yet it rests entirely on the assumed relaxation-time asymmetry 1/τ(k) = 1/τ0 + α(E) sign(Ê·k) with α(E) = α0(E cosωt)^2, Eq. (7). The derivation in Appendix A is performed only for a 1D parabolic band, and the self-consistent fits in Eq. (A23) yield γ1(E) = Σ_{j=1}^4 b_j E^j, which includes a linear-in-E term b1E; the main text sets b1 = 0 and all a_{j≥1} = 0 without showing that the 2D k.p model of Eq. (8)/(D3) produces the same sign(k_x) structure with leading E^2 weight. As shown in Appendix C, the 1ω current comes from the odd-in-k parts of ar{g}_0 and ar{g}_2, so any different momentum dependence of γ—e.g., a smooth function instead of sign(Ê·k), or a nonzero b1—changes both the amplitude and the weak-field scaling of ar{J}_y^{(1)}. The E^5 scaling displayed in Fig. 3(b) is therefore a consequence of the chosen ansatz rather than a derived prediction, a point the authors concede in the sentence 'the precise weak-field scaling of ar{J}_y^{(1)} with E is determined by the form of non-uniformity present in the relaxation time.' To support the abstract's claim that the relaxation time 'generates an 1ω response, even in the presence of microscopic TRS,' the manuscript should either compute τ(k,E) self-consistently for the actual 2D models (numerically, along the lines of Appendix A) or explicitly reframe the odd-ω result as a scenario contingent on a microscopic input that remains to be supplied.","section":"Eq. (7) and Appendix C"},{"comment":"Equation (6) solves the Boltzmann equation as a differential equation in f(k0,t) with a relaxation time written as τ(t), but the relaxation time in Eq. (7) depends on k through sign(Ê·k). As printed, Eq. (6) and the definition A_s(t) = exp{s ∫ dt'/τ(t')} treat τ as independent of k0, in which case the k-asymmetry that generates the odd harmonics would not enter the solution at all. The authors should state explicitly that τ(t) in Eq. (6) denotes τ(k0,t) and specify whether sign(Ê·k) is evaluated at the initial momentum k0 or at the time-dependent momentum k(t) = k0 − x̂(kE/τ0ω) sinωt. This is necessary to reproduce the numerical results in Fig. 3(b).","section":"Eq. (6) and Sec. 'Odd-ω vs. even-ω response'"}],"minor_comments":[{"comment":"The collision integral in Eq. (3b) appears to have the factor [f(k,t) − f0(k)] in the numerator, which would make 1/τ vanish at equilibrium. The expression used in the self-consistent derivation, Eq. (A8), has γ(k,E) g(k,E) = (2π/ħ)∫dk′ δ(...)|U|^2 [g(k)−g(k′)]. Please check the typesetting of Eq. (3b) and ensure it matches Eq. (A8).","section":"Eq. (3b)"},{"comment":"There are several typographical errors: 'spontaneoous' in the abstract and 'dimensioless' near Fig. 1 should be 'spontaneous' and 'dimensionless', respectively; in the Introduction the phrase 'not odd ink' should be 'not odd in k'.","section":"Abstract and text"},{"comment":"The caption states '¯Jy/E ∼ E^4 (dashed curve), which depends on our choice of α(E) = α0E^2 [cf. Eq. (7)].' This is helpful, but the main text should also make clear that this scaling is not a robust prediction of the formalism but a property of the chosen ansatz, as is done in the sentence preceding Fig. 3(b).","section":"Fig. 3(b) caption"},{"comment":"The caption notes that γ± should be expressed as γ±(kx, E, T, µ), but the ansatz in Eq. (A15) depends on kx only through sign(kx); the notation could be simplified or clarified to avoid implying a continuous kx dependence.","section":"Appendix A, Fig. S1 caption"}],"recommendation":"major_revision","confidential_remarks":"The DC and even-AC content is strong and likely publishable; the main reason for major revision is the odd-harmonic claim, which is advertised in the abstract and conclusion but rests on an underevidenced ansatz. If the authors can supply a self-consistent calculation of τ(k,E) for the 2D model or clearly reframe the odd-ω result as a scenario rather than a definitive prediction, the paper could be acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: the DC and even-AC part is the genuine result. Solving the Boltzmann equation without assuming small g, they show the Berry-curvature-dipole expansion breaks down beyond k_BC and the Hall current crosses over to quasi-linear E in two different models (dipole and quadrupole). That is new relative to Sodemann-Fu and Zhang et al. and internally consistent. The 1ω Hall response without broken TRS is also new, but it is not on the same footing. It rests on the field-induced relaxation-time asymmetry 1/τ(k)=1/τ0+α(E) sign(Ê·k), with α(E)=α0(E cosωt)^2 chosen in the main text. Appendix A derives the existence of an asymmetry only for a 1D parabolic band using two ansatz forms, and the numerical fits actually produce both even and odd E terms in γ1; the main text drops the linear-in-E piece. The authors concede the point when they say the precise weak-field scaling of J_y^(1) is set by the form of the non-uniformity. So the E^5 prediction is conditional, not parameter-free. If the real τ asymmetry is weaker or momentum-smooth rather than sign-like, the 1ω amplitude and scaling change. This does not touch the DC/even-AC core, which uses τ=τ0.\n\nMinor issues: Eq. (3b) as printed cannot be dimensionally right, and the \"universality\" language outruns two 2D models, but these are cosmetic.\n\nWho gets value: condensed-matter theorists working on nonlinear Hall, Berry curvature transport, and Weyl-Kondo semimetals. I would cite the DC crossover. The odd-harmonic mechanism should be labeled a proposal, and a referee should demand a self-consistent γ(k,E) calculation for the actual 2D model before publication.\n\nVerdict: send to review. The central claim is solid enough to deserve referee time; the 1ω part needs added derivation.","headline":"The DC and even-harmonic non-perturbative Hall result is solid and worth engaging; the 1ω claim is built on an assumed relaxation-time asymmetry and should be read as a proposal, not a derivation.","tokens_in":18311,"tokens_out":3978,"would_cite":true,"duration_ms":34520,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"For inversion-broken semimetals with microscopic time-reversal symmetry, the spontaneous Hall current crosses over from quadratic to quasi-linear scaling in a fully nonequilibrium regime, and a once-per-cycle response appears without…","keywords":["nonlinear Hall effect","Berry curvature dipole","fully nonequilibrium transport","spontaneous Hall effect","time-reversal symmetry","Berry curvature multipoles","relaxation time asymmetry","Weyl-Kondo semimetal"],"falsifier":"Solve the coupled Boltzmann and collision-integral equations numerically for the 2D model of Eq. (8) with a finite-range impurity potential and check whether $\\gamma(k,E)=1/\\tau(k,E)$ satisfies $\\gamma(k,E)\\neq\\gamma(-k,E)$. In parallel, measure the DC and $1\\omega$ spontaneous Hall current in a candidate TRS-preserving inversion-broken semimetal: the central claim predicts $J_y/E$ flattening for $k_E\\gtrsim k_{BC}$ and a nonzero $1\\omega$ channel growing as $E^5$ at weak fields for $\\alpha\\propto E^2$. If no odd harmonic appears, or if the DC response keeps its weak-field scaling beyond $k_E\\sim k_{BC}$, the fully nonequilibrium mechanism is ruled out.","tokens_in":17069,"feed_emoji":"⚡","tokens_out":9078,"duration_ms":77503,"temperature":0.7,"pith_summary":"This paper tries to establish that in semimetals which preserve time-reversal symmetry but break inversion symmetry, a strong applied electric field produces a spontaneous Hall current that cannot be described by the perturbative Berry-curvature-dipole picture. In the fully nonequilibrium regime, defined by the field-induced momentum scale $k_E=eE\\tau_0/\\hbar$ exceeding the scale $k_{BC}$ at which the Berry curvature peaks sit near the Fermi surface, the DC Hall current crosses over from a weak-field $E^2$ scaling to a quasi-linear dependence on $E$. The same crossover governs the even-harmonic AC response, and the field-driven spatial asymmetry of the relaxation time generates a finite $1\\omega$ Hall response even though microscopic time reversal is preserved. If correct, this gives a mechanism for the large spontaneous Hall effect and its $1\\omega$ component observed in Weyl-Kondo semimetals.","feed_headline":"Hall current in time-symmetric semimetals goes quasi-linear at strong fields","feed_subtitle":"A fully nonequilibrium Berry-curvature theory predicts odd harmonics without breaking time reversal.","key_machinery":"The central object is the fully nonequilibrium (FNE) regime signalled by the ratio $\\epsilon = k_E/k_{BC}$, where $k_E=eE\\tau_0/\\hbar$ is the momentum shift an electric field imparts over one relaxation time and $k_{BC}$ is the shortest distance, along the field, between the extrema of the Berry curvature density and the Fermi surface. The argument is carried by the Boltzmann equation in the relaxation-time approximation, solved directly in $E$ rather than expanded perturbatively, together with a self-consistent collision integral that makes the relaxation time depend on momentum and field. The odd-harmonic response is generated by the minimal asymmetric form $1/\\tau(k)=1/\\tau_0+\\alpha(E)\\,\\mathrm{sign}(\\hat{E}\\cdot k)$, with $\\alpha(E)=\\alpha_0(E\\cos\\omega t)^2$ used for the explicit calculations; this form is motivated in the paper by self-consistent solutions of two 1D ansatz relaxation-time models.","core_discovery":"The central claim is that the spontaneous Hall response in time-reversal-symmetric, inversion-broken semimetals has a fully nonequilibrium regime in which the reference electronic state is itself out of equilibrium and the Hall current is no longer controlled by the Berry curvature dipole. In this regime the occupation of states carrying large Berry curvature density is substantially reorganized by the field, all moments of the Berry curvature contribute comparably, and the DC Hall current scales quasi-linearly with $E$ over an extended range of moderately strong fields. For AC driving, the even harmonics ($0\\omega$ and $2\\omega$) show the same crossover, while the field-induced momentum-space asymmetry of the relaxation time makes odd harmonics, in particular $1\\omega$, nonzero even with microscopic TRS; for the minimal form $\\alpha(E)=\\alpha_0(E\\cos\\omega t)^2$, the weak-field $1\\omega$ current scales as $E^5$. The authors argue that this universality extends to systems where higher Berry multipoles dominate the weak-field response, such as a $C_4T$-symmetric metal whose Berry quadrupole gives $E^3$ at weak fields, and that the mechanism is realized in three-dimensional non-centrosymmetric Weyl semimetals viewed as stacks of 2D planes.","pith_inferences":["Inference: the prediction that the odd-harmonic Hall response is proportional to a positive power of $|\\alpha(E)|$ suggests a new diagnostic, namely that measuring the field and frequency dependence of the $1\\omega$ channel can map the microscopic field-induced anisotropy of impurity scattering.","Inference: because the crossover scale $k_{BC}$ is set by how close the Berry curvature extrema sit to the Fermi surface, chemical-potential tuning should move the crossover field $E^*$; comparing the measured $E^*$ with the band-structure-derived $k_{BC}$ would test the mechanism.","Inference: in moir\\'e materials, whose small Brillouin zones lower the momentum scale $\\Lambda$, modest electric fields may reach $k_E\\sim k_{BC}$, making them candidate platforms for observing quasi-linear nonlinear Hall and odd harmonics at accessible voltages."],"forward_implications":["If the central claim is right, the spontaneous DC Hall conductivity $J_y/E$ of a TRS-preserving, inversion-broken semimetal will be proportional to $E$ only at weak fields; for $k_E\\gtrsim k_{BC}$ it becomes nearly $E$-independent, mimicking an anomalous Hall effect without magnetic order.","The $0\\omega$ and $2\\omega$ AC Hall channels inherit the same crossover, so multi-harmonic transport measurements can locate the FNE regime by the field strength at which the quadratic scaling saturates.","A nonzero $1\\omega$ spontaneous Hall response is predicted to appear without broken TRS once the field-driven relaxation-time asymmetry is present, and its weak-field exponent carries information about how $\\alpha(E)$ depends on $E$.","The same breakdown of the multipole expansion applies regardless of which Berry-multipole moment dominates at weak fields, so the quasi-linear strong-field response is a generic signature of non-centrosymmetric semimetals.","The results connect directly to the observed large spontaneous Hall and $1\\omega$ signals in Weyl-Kondo semimetals and provide a target for strongly correlated topological semimetals."],"supporting_citations":[{"why":"Provides the motivating experimental observation of a large spontaneous Hall effect and an odd-frequency component in a time-reversal-symmetric Weyl-Kondo semimetal.","marker":"[6]"},{"why":"Supplies the perturbative Berry-curvature-dipole theory whose weak-field scaling the paper extends and goes beyond.","marker":"[16]"},{"why":"Establishes the higher-multipole correspondence between Berry-curvature moments and field-scaling that the paper shows breaks down at strong fields.","marker":"[18]"},{"why":"The paper's supplemental material contains the self-consistent derivation of the non-uniform relaxation time, the model Hamiltonians, and the numerical DC and AC response calculations.","marker":"[32]"},{"why":"Provides the Weyl-Kondo semimetal model with small Fermi surfaces and near-Fermi-energy Weyl nodes that motivates the non-perturbative field regime.","marker":"[7]"}],"fun_headline_variants":["Hall current turns quasi-linear at strong fields via Berry curvature","Berry curvature Hall effect works without breaking time reversal at strong fields","Nonequilibrium Hall response skips Berry dipole in strong-field semimetals","Odd harmonics in Hall current from Berry curvature without TRS breaking","Fully nonequilibrium Hall effect from Berry curvature in semimetals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument stands on the assumption that a strong electric field makes the impurity-scattering relaxation time momentum-asymmetric in the two-sided form $1/\\tau(k)=1/\\tau_0+\\alpha(E)\\,\\mathrm{sign}(\\hat{E}\\cdot k)$; this asymmetry is illustrated with 1D toy models and ansatz forms but is not derived from the microscopic collision integral in the actual 2D models, so if the true asymmetry is weaker or has a different momentum dependence, the odd-harmonic prediction changes or disappears.","fun_headline_variants_meta":{"raw":{"variants":["Hall current turns quasi-linear at strong fields via Berry curvature","Berry curvature Hall effect works without breaking time reversal at strong fields","Nonequilibrium Hall response skips Berry dipole in strong-field semimetals","Odd harmonics in Hall current from Berry curvature without TRS breaking","Fully nonequilibrium Hall effect from Berry curvature in semimetals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000244,"raw_usage":{"total_tokens":1548,"prompt_tokens":978,"completion_tokens":570,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":594,"completion_tokens_details":{"reasoning_tokens":480}},"tokens_in":594,"tokens_out":570,"duration_ms":5621,"temperature":1.0,"reasoning_tokens":480,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:50:50.037691+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the coupled Boltzmann and collision-integral equations numerically for the 2D model of Eq. (8) with a finite-range impurity potential and check whether $\\gamma(k,E)=1/\\tau(k,E)$ satisfies $\\gamma(k,E)\\neq\\gamma(-k,E)$. In parallel, measure the DC and $1\\omega$ spontaneous Hall current in a candidate TRS-preserving inversion-broken semimetal: the central claim predicts $J_y/E$ flattening for $k_E\\gtrsim k_{BC}$ and a nonzero $1\\omega$ channel growing as $E^5$ at weak fields for $\\alpha\\propto E^2$. If no odd harmonic appears, or if the DC response keeps its weak-field scaling beyond $k_E\\sim k_{BC}$, the fully nonequilibrium mechanism is ruled out.","supporting_citations":[{"cited_title":"Zhang, X.-J","cited_arxiv_id":null,"evidence_quote":"Establishes the higher-multipole correspondence between Berry-curvature moments and field-scaling that the paper shows breaks down at strong fields."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The paper's supplemental material contains the self-consistent derivation of the non-uniform relaxation time, the model Hamiltonians, and the numerical DC and AC response calculations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Weyl-Kondo semimetal model with small Fermi surfaces and near-Fermi-energy Weyl nodes that motivates the non-perturbative field regime."}],"review_version":1}