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Abnormally enhanced Hall Lorenz number in the magnetic Weyl semimetal NdAlSi

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Pith's one-line read In the magnetic Weyl semimetal NdAlSi, heat transport in a magnetic field exceeds the Wiedemann–Franz bound by about a factor of two, and the paper attributes this to Kondo-type scattering of electrons by localized 4f moments.

desk verdict A likely real and well-measured case of enhanced Hall Lorenz number in a 3D metal, but the electronic-origin claim rests on a phonon Hall bound that may not transfer to this strongly spin-phonon-coupled system. read the letter →

arxiv 2411.17156 v1 pith:NXLNZUPP submitted 2024-11-26 cond-mat.str-el

Nan Zhang , Daifeng Tu , Ding Li , Kaixin Tang , Linpeng Nie , Houpu Li , Hongyu Li , Tao Qi
show 4 more authors
Tao Wu Jianhui Zhou Ziji Xiang Xianhui Chen
This is my paper · ORCID
classification cond-mat.str-el
keywords Wiedemann–FranzlawHallLorenznumberthermaleffectWeylsemimetalNdAlSiKondoscattering4felectronsquasiparticlerelaxationtime
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 reports that the Hall Lorenz number $L_{xy}=\kappa_{xy}/(\sigma_{xy}T)$ in the magnetic Weyl semimetal NdAlSi exceeds the Sommerfeld value $L_0$ over a wide range of temperature and field, reaching about $2L_0$ near 8 K and 5.5 T. The deviation starts near 30 K, well above the magnetic ordering temperature $T_m=7.3$ K, and persists until the 4$f$ moments are fully polarized. The paper argues that the excess thermal Hall conductivity is carried by electrons, not phonons or magnons, and attributes it to Kondo-type elastic scattering of itinerant electrons by localized 4$f$ moments, which creates an energy-symmetric minimum in the quasiparticle relaxation time. If correct, this is a rare three-dimensional example of $L>L_0$ and a demonstration that transverse transport measurements can expose the energy dependence of scattering near the Fermi level.

What carries the argument

The central object is the Hall Lorenz number $L_{xy}\equiv\kappa_{xy}/(\sigma_{xy}T)$, compared with the Wiedemann–Franz baseline $\kappa_{xy}^{\mathrm{WF}}=L_0\sigma_{xy}T$. The mechanism is a model, originally applied to chromium, in which a dip in the energy-dependent conductivity $\sigma(\varepsilon)\propto N(\varepsilon)v^2(\varepsilon)\tau(\varepsilon)$ centered at the chemical potential makes the thermal Hall integrand $Q$ draw on more states than the electrical Hall integrand $P$, so heat flow is suppressed less than charge flow and $L_{xy}$ rises. In NdAlSi the dip is produced by Kondo-type elastic scattering of 5$d$ itinerant electrons off localized 4$f$ moments, giving an energy-symmetric minimum in $\tau(\varepsilon)$ that reproduces the measured temperature dependence of $L_{xy}$.

What would settle it

Dilute the Nd 4$f$ moments by substituting nonmagnetic La and remeasure $\kappa_{xy}(H,T)$: if Kondo-type scattering is responsible, the excess heat Hall signal over $L_0\sigma_{xy}T$ should shrink with the Nd concentration and vanish in pure LaAlSi. Alternatively, probe the phonon Hall-angle bound directly in NdAlSi by suppressing the electronic channel, and check whether $|\kappa_{xy}/(\mu_0H\kappa_{xx})|$ exceeds the assumed $2\times10^{-3}$ per tesla.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is an upward violation of the Wiedemann–Franz law in a three-dimensional metal. The measured ratio $L_{xy}=\kappa_{xy}/(\sigma_{xy}T)$ is larger than $L_0$ up to about 30 K, with a maximum near $2L_0$ at $T=8$ K and $\mu_0H=5.5$ T, and the enhancement reproduces across several crystals. The paper rules out charge-neutral carriers (phonons and magnons) using bounds on the phonon and magnon Hall angles, rules out Berry-curvature and phonon-drag mechanisms, and connects the effect to a model in which an energy-symmetric dip in the quasiparticle relaxation time $\tau(\varepsilon)$ suppresses the electrical Hall integrand more strongly than the thermal Hall integrand. It identifies the dip as the fingerprint of Kondo-type scattering between itinerant Nd 5$d$ electrons and localized Nd 4$f$ moments, consistent with the absence of the effect in nonmagnetic LaAlSi, its weakness in SmAlSi, and its suppression once the moments are field-polarized.

Load-bearing premise

The conclusion that the excess heat flow in a magnetic field is carried by electrons rests on the assumption that NdAlSi obeys the published universal upper bound on the phonon Hall angle; if strong spin-phonon coupling violates that bound, part of the measured excess could come from phonons rather than electrons.

Editorial extensions

If this is right

  • If the interpretation is right, the Wiedemann–Franz law can be violated upward in a three-dimensional metal, not only downward as in most finite-temperature cases.
  • The transverse heat-to-charge ratio becomes a quantitative probe of the energy dependence of the quasiparticle relaxation time within a few meV of the Fermi level.
  • Kondo-type coupling between Weyl fermions and local moments should be considered alongside RKKY coupling in magnetic topological semimetals, since the two coexist on the Kondo-lattice phase diagram.
  • The suppression of $L_{xy}$ above the spin-polarization crossover field $H_p$ ties the enhancement to spin-flip scattering of the 4$f$ moments, matching the proposed mechanism.

Reading between the lines

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

  • A direct testable extension: chemical substitution of nonmagnetic La for Nd should suppress the excess $L_{xy}$ in proportion to the 4$f$ moment concentration, and the peak should shift with the Kondo coupling strength, not with $T_m$.
  • The model implies $L_{xy}-L_0$ should exhibit a characteristic $H/T$ scaling in the paramagnetic regime if the scattering is governed by the Zeeman-split Kondo resonance; searching for such scaling would separate Kondo scattering from magnetic critical fluctuations.
  • If the technique works, the same thermal-Hall-plus-Wiedemann–Franz analysis could map $\tau(\varepsilon)$ in other correlated semimetals, though absolute calibration of $\kappa$ and $\sigma$ will limit how small a deviation can be resolved.
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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

3 major / 4 minor

Summary. The paper reports measurements of longitudinal thermal conductivity, thermal Hall conductivity, and electrical Hall conductivity in single crystals of the magnetic Weyl semimetal NdAlSi. The authors find that the Hall Lorenz number Lxy = κxy/(σxyT) exceeds the Sommerfeld value L0 below T ≈ 30 K and over a broad field range, reaching about 2L0 near 8 K and 5.5 T. The enhancement is reproduced in four NdAlSi samples and is absent in SmAlSi and LaAlSi control compounds. The authors argue that the excess κxy is electronic, excluding phonon and magnon thermal Hall contributions mainly on the basis of published bounds on phonon and magnon Hall angles, and they attribute the enhanced Lxy to Kondo-type elastic scattering off Nd 4f moments that produces an energy-symmetric minimum in the quasiparticle relaxation time near the chemical potential. A model calculation based on this scattering mechanism is reported to reproduce the temperature dependence of Lxy.

Significance. If the electronic interpretation is correct, this is a rare demonstration of L > L0 in a three-dimensional metal and provides a concrete example of a scattering mechanism that violates the Wiedemann–Franz law in the opposite direction from the usual inelastic-scattering case. The empirical dataset is substantial: four NdAlSi samples, two control compounds, explicit calibration checks, and supporting specific-heat, NMR, and magnetothermal-conductivity measurements. Data are deposited and fitting codes are stated to be available. The main weakness is that exclusion of phonon and magnon thermal Hall rests on an empirical bound whose applicability to this strongly spin-phonon-coupled material is not established, and the Kondo model is presented with insufficient parameter detail in the main text. These issues do not undermine the reported observation itself, but they do make the central microscopic interpretation conditional.

major comments (3)
  1. [Discussion; Supplementary Note 10; Fig. 5a] The exclusion of phonon and magnon thermal Hall hinges on the empirical bound |κxy/(μ0Hκxx)| < 2×10−3 T−1 for the phonon Hall angle and ~10−3 for the magnon Hall angle. The observed ratio |κxy/κxx| ≈ 35% at 5.5 T and 10–12 K corresponds to about 6×10−2 T−1, roughly 30 times above the bound. However, that bound was compiled largely from systems without the strong resonant CEF scattering and spin-phonon coupling documented here; NdAlSi shows a field-induced change of κxx exceeding 200% at low temperature, and its phonons are resonantly scattered by 4f CEF levels. The comparison to the zero-field κxx(T) in Fig. 1d is also not quantitative, because phonon Hall signals arise from skew scattering and need not produce a corresponding feature in the longitudinal conductivity, and because the excess κxy is reported at 5.5 T rather than zero field. Since the claimed electronic origin of the enhanced Lxy is the central conclusion, the authors need to validate the phonon/magnon bound for this specific material—for example, by a microscopic estimate of phonon skew scattering in the CEF-phonon system, by a measurement that separates electronic and phononic contributions, or by an explicit discussion of why the strongly modified phonon system cannot exceed the bound. The SmAlSi and LaAlSi controls are useful but do not have the same CEF level scheme and spin-phonon coupling, so they do not by themselves resolve this concern.
  2. [Discussion; Supplementary Note 9; Fig. 5b] The red Lxy(T) curve in Fig. 5b is stated to come from a numerical calculation using the integrand functions Q and P described in Supplementary Note 9, with a τ(ε) that has an energy-symmetric minimum at ε = 0. The main text does not give the model parameters, the microscopic form of the Kondo coupling, or any sensitivity analysis, so the reader cannot judge whether the agreement with experiment is a robust prediction or largely reflects adjustable choices. This matters because the title and abstract present Kondo-type scattering as the mechanism behind the enhanced Lxy. Please state the parameters, show how the Weyl-Kondo model produces the assumed τ(ε), and discuss how the computed Lxy(T) depends on those parameters.
  3. [Discussion; Supplementary Note 10; Fig. 3] The argument that the large amplitude of the excess κxy 'invalidates' the phonon and magnon Hall interpretations is only valid if the empirical bound applies to NdAlSi. The paper itself demonstrates that phonon transport in NdAlSi is profoundly affected by CEF resonances and magnetic excitations, with κxx(H) changing by more than a factor of two at low temperature. Because the phonon Hall angle is not a universal constant but depends on the skew-scattering processes available, it is precisely in such a strongly perturbed phonon system that the bound could be violated. The stress-test concern raised about this point therefore lands: without either a direct measurement constraining the phonon contribution in NdAlSi or a calculation showing the bound remains valid under resonant CEF scattering, the electronic-origin claim is not fully established.
minor comments (4)
  1. [Abstract; Results; Methods] There are several typographical errors and OCR artifacts, such as 'electro n heat' in the abstract, 'CFE levels' in the Results section, and 'de vices' in the affiliation line; these should be corrected.
  2. [Fig. 5b; Supplementary Note 1] The error bars in Fig. 5b are described as reflecting only the noise-level uncertainties in ΔTx and ΔTy; the 10–15% absolute calibration uncertainty in κxy and σxy is discussed separately but is not shown on the figure. The caption or the main text should state this explicitly, since the absolute calibration affects the comparison of Lxy values between samples.
  3. [Discussion; Supplementary Note 10] The phrase 'a universal upper limit' for the phonon Hall angle is stronger than what the cited empirical compilation supports; the bound has been observed in a finite set of materials and should be described as a reported upper limit whose range of validity is itself a topic of current research.
  4. [Methods; Code availability] The statement that fitting codes are available from the corresponding author upon request is less reproducible than depositing them in a permanent repository alongside the data; the authors should consider uploading the codes to the figshare deposition as well.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the enhanced Lxy is a directly measured ratio of independent Hall coefficients, and the Kondo-model curve is a post hoc interpretive comparison rather than a load-bearing derivation.

full rationale

The central empirical claim—that Lxy = κ_xy/(σ_xy T) exceeds L0, reaching ≈2L0 near 8 K and 5.5 T—is a direct ratio of separately measured transverse thermal and electrical Hall conductivities. Neither L0 nor the comparison curve κ_WF_xy = L0σ_xyT is fitted to the Hall data, so the enhancement is not an artifact of a fitted parameter renamed as a prediction. The exclusion of phonon and magnon thermal Hall contributions is based on external published bounds on the phonon Hall angle |κ_xy/(μ0Hκ_xx)| < 2×10^-3 T^-1 and on control measurements in LaAlSi and SmAlSi; even if one questions whether that empirical bound transfers to NdAlSi, that is a correctness/risk concern, not circularity. The Kondo-type scattering model enters only after the empirical excess is established: the paper states that a numerical calculation based on the model integrands Q and P 'yields a T-dependent Lxy that satisfyingly tracks the trend' of the data, i.e., an interpretive consistency check, not a derivation of the measured effect from a premise that already contains it. The main-text figure does not quote model parameters, so one cannot exhibit a fitted-parameter-renamed-as-prediction reduction from the text alone. The only self-citation, ref. [20], supplies crystal-growth details and phase-boundary fields, neither of which carries the Wiedemann–Franz violation claim. No load-bearing step is equivalent to its inputs by construction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central observation is direct experimental data, but the proposed Kondo mechanism relies on an assumed form of tau(eps), literature values for phonon and magnon Hall bounds, and standard Boltzmann transport assumptions. No new particles or forces are introduced.

free parameters (1)
  • Kondo model parameters for tau(eps) = not stated in main text (see Supplementary Note 9)
    The red Lxy(T) curve in Fig. 5b is generated from an energy-dependent relaxation time with a minimum at eps=0. Without stated parameter values or constraints, this is an adjustable fit rather than a parameter-free prediction.
assumptions (5)
  • domain assumption Boltzmann transport theory with relaxation time approximation describes transverse electronic transport in NdAlSi.
    Used in Supplementary Note 8 to define kappa_xy and sigma_xy as energy integrals over tau(eps); not independently verified for this material.
  • domain assumption The published universal phonon Hall angle bound applies to NdAlSi.
    Invoked in the Discussion and Supplementary Note 10 to exclude phonon thermal Hall; if strong spin-phonon coupling breaks this bound, the exclusion fails.
  • ad hoc to paper Kondo-type scattering produces an energy-symmetric minimum in tau(eps) at the chemical potential.
    This is the core of the proposed mechanism (Fig. 5c, Supplementary Note 9); it is motivated by Kondo literature but is not independently measured or derived in the main text.
  • domain assumption Anomalous Hall effect and Berry-curvature contributions are negligible.
    Based on the authors' own measurements (Supplementary Fig. 10d) showing no noticeable anomalous Hall effect.
  • domain assumption NdAlSi has Weyl fermions and 4f local moments as described in prior DFT and ARPES work.
    Borrowed from refs 18-21; not rederived in this paper.

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Cite this review

Pith. "Pith review of Abnormally enhanced Hall Lorenz number in the magnetic Weyl semimetal NdAlSi." pith.science (2026). https://pith.science/paper/NXLNZUPP

@misc{pith2026241117156,
  author       = {Pith},
  title        = {Pith review of: Abnormally enhanced Hall Lorenz number in the magnetic Weyl semimetal NdAlSi},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NXLNZUPP}},
  note         = {Machine review of arXiv:2411.17156}
}
abstract

In Landau's celebrated Fermi liquid theory, electrons in a metal obey the Wiedemann--Franz law at the lowest temperatures. This law states that electron heat and charge transport are linked by a constant $L_0$, i.e., the Sommerfeld value of the Lorenz number ($L$). Such relation can be violated at elevated temperatures where the abundant inelastic scattering leads to a reduction of the Lorenz number ($L < L_0$). Here, we report a rare case of remarkably enhanced Lorenz number ($L > L_0$) discovered in the magnetic topological semimetal NdAlSi. Measurements of the transverse electrical and thermal transport coefficients reveal that the Hall Lorenz number $L_{xy}$ in NdAlSi starts to deviate from the canonical value far above its magnetic ordering temperature. Moreover, $L_{xy}$ displays strong nonmonotonic temperature and field dependence, reaching its maximum value close to 2$L_0$ in an intermediate parameter range. Further analysis excludes charge-neutral excitations as the origin of enhanced $L_{xy}$. Alternatively, we attribute it to the Kondo-type elastic scattering off localized 4$f$ electrons, which creates a peculiar energy distribution of the quasiparticle relaxation time. Our results provide insights into the perplexing transport phenomena caused by the interplay between charge and spin degrees of freedom.

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Reviewed August 12, 2026 · model on record in the stance chip above.