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Fundamental Relations as the Leading Order in Nonlinear Thermoelectric Responses with Time-Reversal Symmetry

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Pith's one-line read For time-reversal-symmetric materials, where linear Hall effects vanish, disorder-induced nonlinear charge and heat currents obey fixed second-order Mott and Wiedemann-Franz relations—with side-jump ratios universal and skew-scattering rati

desk verdict Plausible extension of Mott/WF laws to disorder-induced nonlinear transport, but the central Table II relations rest on a hidden derivation and on omitted terms the authors admit could shift the prefactors — worth a peer review, with a flag. read the letter →

arxiv 2601.19625 v3 pith:JFNFCA6I submitted 2026-01-27 cond-mat.mes-hall

Ying-Fei Zhang , Zhi-Fan Zhang , Hua Jiang , Zhen-Gang Zhu , Gang Su This is my paper · ORCID
classification cond-mat.mes-hall
keywords nonlinearthermoelectrictransportsecond-orderMottrelationWiedemann-Franzlawside-jumpscatteringskewtime-reversalsymmetrytopologicalinsulatorsurfaceCoulombscreening
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper develops a semiclassical theory of disorder-induced second-order nonlinear transport in time-reversal-symmetric systems, where linear Hall responses vanish, and derives universal relations between the second-order electrical, thermoelectric, and thermal coefficients. For side-jump scattering, the theory predicts a material-independent second-order Mott relation and a corresponding Wiedemann-Franz law, with the Lorentz number appearing as a fixed prefactor. For skew scattering, both relations hold with a prefactor that depends only on the dimensionless Coulomb screening parameter. The paper demonstrates these relations in a hexagonally warped topological-insulator surface model with screened Coulomb impurities, where the intrinsic Berry-curvature-dipole contribution vanishes by symmetry and skew scattering dominates. If correct, these are the first leading-order fundamental relations connecting nonlinear charge and heat transport in the presence of disorder, and they provide experimentally testable fingerprints of the dominant scattering mechanism.

What carries the argument

The semiclassical wave-packet transport theory, in which the second-order response coefficients are written as energy integrals of kernel functions X^in, X^sj, and X^sk weighted by Fermi-function derivatives. The side-jump and skew-scattering kernels each factor into a band-structure-dependent integral times a Fermi-function derivative; the Mott and Wiedemann-Franz relations arise because the coefficients are different moments of the same kernel, yielding fixed ratios. The topological-insulator surface Hamiltonian with hexagonal warping (C3v symmetry) and screened Coulomb impurities supplies the concrete model, with q_s = q_TF/k_F the only parameter governing the skew-scattering ratio P.

What would settle it

Compute or measure the full second-order side-jump coefficients without dropping the Hessian term and the interband Berry-connection terms; if the ratio α^sj_xxy/σ^sj_xxy deviates from -L/3 by more than the leading-order correction even in the clean side-jump regime, the claimed universality fails. Experimentally, measure α_xxy and σ_xxy in a time-reversal-symmetric material where side-jump is the dominant scattering mechanism, identified via the ε_F^2 scaling, and test the ratio against -L/3.

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Extended reading notes

Core claim

The central discovery is that for time-reversal-symmetric systems, disorder-induced second-order coefficients obey the second-order Mott relation and Wiedemann-Franz law, with distinct forms for the two extrinsic scattering channels. Side-jump transport satisfies α^sj_xxy = −(1/3)L σ^sj_xxy and ∂κ^sj_xxy/∂ε_F = (L/(3e)) σ^sj_xxy, independent of the impurity potential and model details. Skew scattering satisfies α^sk_xxy = L P σ^sk_xxy and ∂κ^sk_xxy/∂ε_F = −(L/e) P σ^sk_xxy, where P is a dimensionless function only of the Thomas-Fermi screening parameter q_s, approaching ≈0.43 in the q_s→0 limit. These relations emerge because the side-jump and skew-scattering kernels in the semiclassical wav

Load-bearing premise

The derivation keeps only the highest-order relaxation-time terms for side-jump and skew scattering, drops the Hessian term (inverse effective mass) under the large-effective-mass approximation, and omits interband Berry-connection contributions; if any of these neglected terms are comparable to the kept terms, the exact prefactors -L/3 and L/(3e) and the q_s-only dependence of P can change.

Editorial extensions

If this is right

  • In any time-reversal-symmetric material where side-jump dominates the second-order response, the nonlinear Nernst signal is predicted to be exactly -L/3 times the nonlinear Hall conductivity, regardless of impurity details, warping strength, or band structure—a directly measurable fingerprint.
  • The second-order Wiedemann-Franz law means the Fermi-level derivative of the nonlinear thermal conductivity is locked to the nonlinear charge conductivity, allowing cross-checks between independent thermoelectric and transport measurements.
  • For skew scattering, all ratios collapse onto a single function of the dimensionless Coulomb screening parameter q_s; experimentally varying carrier density and thus q_s should trace out the predicted P(q_s) curve.
  • In the q_s→0 limit, the skew-scattering Mott ratio approaches approximately 0.43L, providing a quantitative zero-screening benchmark for experiments.
  • The model predicts distinct power-law scalings—σ^sk ∝ ε_F^4 versus σ^sj ∝ ε_F^2, and κ^sk ∝ ε_F^5 versus κ^sj ∝ ε_F^3—so Fermi-level sweeps alone can separate the two scattering channels before checking the ratio laws.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the side-jump ratio -L/3 is truly universal beyond the leading-order approximation, it could serve as a diagnostic: any measured deviation in a nominally side-jump-dominated sample would signal the neglected interband Berry-connection contributions or electron-electron interaction effects, placing quantitative bounds on those corrections.
  • The q_s-only dependence of P suggests a new way to extract the Thomas-Fermi screening length from macroscopic nonlinear transport—complementing STM and ARPES—by measuring the ratio α_xxy/σ_xxy at several Fermi energies and fitting P(q_s(ε_F)).
  • The framework is expected to extend to other symmetries: under PT symmetry or with magnetic order, the quantum metric dipole contributes, and the kernel structure suggests modified but still constrained relations—perhaps with additional metric-dipole-dependent terms, which future density-matrix or diagrammatic calculations could test.
  • The mixed second-order response to simultaneous electric field and temperature gradient, which the paper explicitly defers, may obey its own generalized Mott-Wiedemann-Franz relations; the kernel formalism here provides the template to derive them.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper develops a semiclassical wave-packet framework for disorder-induced second-order electric, thermoelectric, and thermal conductivities in time-reversal-symmetric (TRS) systems. Applying it to the hexagonal-warping surface states of topological insulators with screened Coulomb impurities, the authors obtain explicit Fermi-level scalings (e.g., σ_sk ∝ ε_F^4, σ_sj ∝ ε_F^2) and propose universal leading-order relations in Table II: for side-jump, α_sj_xxy = −(1/3)L σ_sj_xxy and ∂κ_sj_xxy/∂ε_F = (L/3e) σ_sj_xxy; for skew scattering, α_sk_xxy = L P σ_sk_xxy and ∂κ_sk_xxy/∂ε_F = −(L/e) P σ_sk_xxy, with P claimed to depend only on the Thomas-Fermi screening ratio q_s. The paper also compares the skew-scattering Mott-like relation qualitatively with a recent experiment [48].

Significance. If correct, Table II constitutes novel disorder-induced second-order analogues of the Mott relation and Wiedemann-Franz law, providing testable universal constraints on nonlinear charge and heat currents in TRS materials. The authors are explicit about the model and parameters, and the scaling laws offer concrete predictions. The qualitative agreement with the experimental observation in [48] is a point in favor of the framework, and the paper provides first theoretical predictions for the remaining three relations. However, the central universality claim is currently conditional: the main derivations are deferred to the supplementary material, and the paper itself lists omitted contributions that could change the prefactors.

major comments (4)
  1. [Final paragraph (limitations)] The load-bearing relations in Table II are derived in the supplementary text [61], not in the main text. The main text only gives the kernel forms in Eqs. (4a)–(5c), which are insufficient for the reader to verify the q_s-only dependence of P or the independence of the side-jump prefactors from the scattering potential. In particular, P is introduced in Table II as a dimensionless quantity determined by q_s, but no explicit expression is given. Please include the derivation (or a detailed outline) in the paper itself, at least for P and for the cancellation steps leading to the exact prefactors −1/3 and 1/3.
  2. [Eq. (4b) and large-effective-mass approximation] The paper explicitly states: 'We only take into account the contribution of the highest-order relaxation time for sj and sk scattering. We note that there are other contributions related to the interband Berry connection that has no counterpart in linear response, which may also affect the relations.' This caveat directly undermines the claim of universality. For side-jump, the claimed prefactor −L/3 is independent of q_s, but the omitted interband Berry-connection terms likely carry q_s dependence and appear at the same order in τ (τ^2); if so, the independence from the scattering potential is lost. Similarly, the skew-scattering P(q_s) could absorb such corrections. The authors must either include these terms in the calculation or demonstrate that they vanish or are subdominant in the TRS model with Coulomb impurities.
  3. [Eq. (1)] The Hessian term Γ in Eq. (4b) is neglected by assuming a large effective mass (Γ→0). For the topological-insulator surface Hamiltonian in Eq. (6), the dispersion is gapless and not characterized by a large effective mass near the Dirac point; the Fermi energies considered in Fig. 2 may extend to regions where the effective mass is not large. Please provide a quantitative estimate of the relative size of the Γ term compared to the retained terms in Eqs. (4a) and (4c) for the parameters used, and state the resulting error in the Table II relations.
  4. [None] The response framework in Eq. (1) omits mixed electric-field and temperature-gradient terms (terms proportional to E·∇T). The relations in Table II connect pure E^2 and pure (∇T)^2 coefficients. Since thermoelectric experiments often involve both drives simultaneously, the mixed terms could contribute to the measured second-order responses and could in principle modify the effective relations. The authors mention these mixed terms are deferred to future work, but the scope of the claimed universality should be explicitly restricted to the diagonal second-order coefficients, and the possible effect of mixed terms should be discussed.
minor comments (5)
  1. [Final paragraph] Typo: 'and and the second-order thermal effect' should read 'and the second-order thermal effect'.
  2. [Fig. 2] Typo: 'obervation' should be 'observation'.
  3. [Text near Fig. 2] The caption says 'ratio α... in unit of L', but the text and Table II refer to P as the skew-scattering prefactor. Please label the vertical axis explicitly (e.g., α_sk_xxy/(L σ_sk_xxy)) and include the numerical value P(0)≈0.43 in the figure or caption. Also 'gray dot line' should be 'gray dotted line'.
  4. [Reference [66]] The text says 'an increase in the wrapping term'—should be 'warping term'.
  5. [None] The title of Ref. [66] has a typo: 'Be2Te3' should be 'Bi2Te3'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the second-order Mott/WF relations are computed from stated semiclassical response kernels, not installed as inputs.

full rationale

The paper's load-bearing claim is Table II, which follows from the semiclassical expressions in Table I and the explicit topological-insulator surface-state calculation. The side-jump relations carry universal prefactors (-L/3 and L/(3e)) that are not fit parameters; the skew-scattering factor P is presented as a function of the Coulomb screening parameter q_s with a stated limiting value (~0.43 at q_s to 0), and the relation is therefore a theory-derived proportionality rather than a tautological definition in the main-text derivation. The self-citations (e.g., refs. [14] and [15]) are used as background for intrinsic BCD thermal relations, not as the basis for the new disorder-induced relations, so they are not load-bearing. The final paragraph explicitly lists omitted interband Berry-connection and other contributions ('may also affect the relations'); that is an admitted approximation and a robustness/correctness caveat, not a circular step, because the claimed relations are not redefined to absorb the omitted terms. The paper also checks the skew-scattering Mott-like trend against an external experimental observation (ref. [48]). Without access to the supplementary derivation, the main text does not exhibit a reduction of any predicted relation to its own fitted input. Hence no circularity is established.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted to the target relations; q_s is a physical input and P is the derived ratio. The main burden is model assumptions rather than hidden fitted constants, but the truncation to leading relaxation-time order and neglect of interband Berry terms are real restrictions on the claimed universality.

free parameters (4)
  • q_s = q_TF/k_F (Thomas-Fermi screening ratio)
    Controls P in Table II; treated as a model control parameter, not fitted to the target relations. For Bi2Se3 experiments the paper cites q_s ≲ 0.4.
  • tau (relaxation time) = 0.1 ps
    Chosen for numerical plots in Fig. 2; cancels in the ratio relations of Table II because all compared coefficients within a mechanism share the same tau scaling.
  • lambda (hexagonal warping strength) = 80 and 250 eV Å^3
    Material parameters for Bi2Se3 and Bi2Te3, taken from literature; not fitted to the transport coefficients.
  • v (Dirac velocity) = 3.291 eV Å
    Material input for the TI surface model; not fitted.
assumptions (6)
  • domain assumption Semiclassical Boltzmann transport with independent intrinsic/side-jump/skew-scattering channels fully captures the second-order disorder-induced response.
    Used throughout; no comparison to quantum kinetic or density-matrix methods is made in the main text.
  • ad hoc to paper Only the highest-order relaxation-time contribution for side-jump and skew scattering is retained; interband Berry-connection terms are neglected.
    The final paragraph admits these omitted terms 'may also affect the relations,' so the table of universal relations is conditional on this truncation.
  • domain assumption Screened Coulomb impurities with Thomas-Fermi screening, V(q)=2παv/(q+q_TF), are the relevant disorder; delta-correlated impurities produce no skew scattering.
    Stated in the text after Eq. (6); this choice is what makes the skew-scattering contribution nonzero.
  • ad hoc to paper Large effective mass: the Hessian term Γ in Eq. (4b) is negligible.
    The authors state that under the large-effective-mass approximation the contribution of this term can be neglected.
  • domain assumption C3v symmetry and time-reversal symmetry hold, so the Berry curvature dipole vanishes and only the xxy-type extrinsic components survive.
    Model Hamiltonian Eq. (6) and the symmetry analysis justify focusing on σxxy, αxxy, κxxy.
  • domain assumption The response current formula Eq. (1) excludes mixed E·∇T second-order terms.
    Explicitly noted in the text as a direction for future work, but the central relations are derived without those mixed terms.

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Cite this review

Pith. "Pith review of Fundamental Relations as the Leading Order in Nonlinear Thermoelectric Responses with Time-Reversal Symmetry." pith.science (2026). https://pith.science/paper/JFNFCA6I

@misc{pith2026260119625,
  author       = {Pith},
  title        = {Pith review of: Fundamental Relations as the Leading Order in Nonlinear Thermoelectric Responses with Time-Reversal Symmetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JFNFCA6I}},
  note         = {Machine review of arXiv:2601.19625}
}
abstract

In recent years, nonlinear transport phenomena have garnered significant interest in both theoretical explorations and experiments. In this work, we utilize the semi-classical wave packet theory to calculate disorder-induced second-order transport coefficients: second-order electrical ($\sigma$), thermoelectric ($\alpha$), and thermal ($\kappa$) coefficients, capturing the interplay between side-jump and skew-scattering contributions in systems with time-reversal symmetry. Using a topological insulator model, we quantitatively characterize the Fermi-level dependence of these second-order transport coefficients by explicitly including Coulomb impurity potentials. Furthermore, we elucidate the relationships between these coefficients, establishing the second-order Mott relation and the Wiedemann-Franz law induced by disorder. This study develops a comprehensive theoretical framework elucidating the nonlinear thermoelectric transport mechanisms in quantum material systems.

Figures

Figures reproduced from arXiv: 2601.19625 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Illustration of the intrinsic and two other extrin [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Crystal structure of the surface of the topological [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

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