{"id":"7fb18c7a-e53d-417f-a51c-e75735b82e94","arxiv_id":"2411.17156","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"NdAlSi shows a Hall Lorenz number up to about 2 L0, an unusually enhanced violation of the Wiedemann-Franz law, which the authors attribute to energy-dependent elastic scattering off localized 4f moments.","lead":"Researchers measured heat and charge flow in the magnetic semimetal NdAlSi and found the Hall Lorenz number rises to about twice the standard Sommerfeld value, instead of falling as in most metals. The effect appears below about 30 K, persists outside the magnetically ordered state, and is attributed to Kondo-type scattering between localized 4f electrons and itinerant electrons.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The electronic-origin claim hinges on applying a 'universal' phonon/magnon Hall-angle bound to NdAlSi's strongly spin-phonon-coupled CEF-resonant phonon system; if that bound is violated, the excess κ_xy and the enhanced Lxy could be non-electronic.","rationale":"The reader identifies the phonon/magnon Hall exclusion as the weakest assumption; I agree. The absolute 10-15% calibration uncertainty cannot explain a factor-of-two enhancement, so that is not the bottleneck. The Kondo model's unconstrained parameters are a real but secondary interpretive concern: even if the microscopic mechanism is not established, the observation of an enhanced Lxy would remain interesting. However, if the charge-neutral thermal Hall exclusion fails, the central phenomenon would no longer be an electronic Lorenz anomaly at all. The quantitative linchpin is the empirical Hall-angle bound, and the observed angle exceeds it by roughly a factor of 30. The paper does not demonstrate that NdAlSi's phonon system, with strong resonant CEF scattering and magnetoelastic coupling, lies in the bound's validity domain. The controls (LaAlSi, SmAlSi) do not fully control for this because their 4f configurations and spin-phonon couplings differ. Hence the verdict should remain CONDITIONAL, with the proposed check determining whether the phonon interpretation can be definitively closed.","tokens_in":18,"tokens_out":18178,"duration_ms":466878,"concrete_test":"Independently re-derive the phonon-Hall upper bound from the Boltzmann equation for NdAlSi using the CEF-resonant phonon scattering parameters extracted from the modified Callaway fit (Supplementary Note 2) and the skew-scattering amplitude calibrated from the materials in Supplementary Table 1. If the re-derived maximum possible phonon κ_xy at 5.5 T and 10 K remains below ~0.1 W/m/K, then the observed 0.4-0.5 W/m/K excess cannot be phononic and the bound application is safe; if the bound is breached or becomes material-dependent, the phonon interpretation reopens.","verdict_should_be":"UNCHANGED","load_bearing_attack":"To claim an electronic Lorenz-number enhancement, the paper must exclude phonon and magnon thermal Hall. It does so primarily by invoking the empirical upper bound |κ_xy/(μ0Hκ_xx)| < 2×10^-3 T^-1 for phonon Hall and ~10^-3 for magnon Hall (Discussion; Supplementary Note 10, Table 1). The observed ratio at 5.5 T and 10-12 K is |κ_xy/κ_xx| ≈ 35%, i.e., ≈ 6.4×10^-2 T^-1, a factor ~30 above the bound, with excess κ_xy ≈ 0.4-0.5 W/m/K. The bound, however, was established in insulating or weakly spin-orbit-coupled phonon systems; NdAlSi is a metal whose phonons are resonantly scattered by CEF levels (≈4.1 meV) and by magnetic excitations, with κ_xx(H) changing by more than 200% at low T. If this strong spin-phonon scattering changes the phonon skew-scattering ratio, the bound may not apply. The T-dependence argument (excess between 5-25 K with no corresponding feature in κ_xx) is suggestive but not quantitative; the nonmagnetic analog LaAlSi and the SmAlSi controls do not have the same CEF level scheme and phonon coupling as NdAlSi. Therefore the central electronic interpretation is conditional on an unvalidated application of the bound. The Kondo model is a further interpretive step, but the phonon/magnon exclusion is the more load-bearing gate.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15726,"tokens_out":8083,"duration_ms":78139,"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":[{"comment":"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.","section":"Discussion; Supplementary Note 10; Fig. 5a"},{"comment":"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.","section":"Discussion; Supplementary Note 9; Fig. 5b"},{"comment":"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.","section":"Discussion; Supplementary Note 10; Fig. 3"}],"minor_comments":[{"comment":"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.","section":"Abstract; Results; Methods"},{"comment":"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.","section":"Fig. 5b; Supplementary Note 1"},{"comment":"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.","section":"Discussion; Supplementary Note 10"},{"comment":"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.","section":"Methods; Code availability"}],"recommendation":"major_revision","confidential_remarks":"This is a well-executed transport study with a striking empirical result that is likely to be of broad interest. My main concern is that the paper's headline interpretation—an electronic Lorenz-number enhancement caused by Kondo scattering—depends on excluding phonon and magnon thermal Hall through an empirical bound that may not apply in this strongly spin-phonon-coupled material. I would not reject the manuscript on that basis, because the observation itself is valuable and the authors can address the issue with additional analysis or measurements, but I would not accept without that being resolved. The Kondo model also needs to be presented with enough parameter detail to be assessed. The manuscript is otherwise carefully written and the data availability statement is a strength."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a new, carefully measured observation—Lxy up to about 2L0 in NdAlSi, reproduced in four samples and absent in the LaAlSi and SmAlSi controls. The authors deserve credit for the multi-sample reproducibility, the explicit exclusion of trivial backgrounds, and the public data deposition. If the observation holds, it is a rare counterexample to the usual finite-temperature suppression of the Lorenz number.\n\nThe soft spot is the exclusion of phonon and magnon thermal Hall. The paper leans on a published upper bound for the phonon Hall angle, |kappa_xy/(mu0 H kappa_xx)| < 2e-3 per tesla. The measured ratio at 5.5 T is about 6e-2 per tesla, thirty times above that bound. That bound was established in materials without NdAlSi's resonant CEF-phonon scattering and strong field-dependent spin-phonon coupling, so applying it here is an assumption. The T-dependence argument against phonon Hall is suggestive but not quantitative; SmAlSi and LaAlSi are useful controls, but they do not share NdAlSi's CEF scheme, so they do not directly rule out a phonon origin specific to NdAlSi.\n\nThe Goff/Kondo interpretation is sensible and the authors are appropriately cautious, but the red fit in Fig. 5b depends on a tau(epsilon) minimum whose parameters are not given in the main text. It tracks the trend, not the scatter. The 10-15% absolute calibration uncertainty is minor for a factor-of-two effect.\n\nMy take: the central observation is solid enough to take seriously; the electronic interpretation is conditional. I would send this to a serious referee. It deserves careful scrutiny, and revision should push for a more quantitative exclusion of phonon Hall in this material, or a parameter-free prediction from the model.","headline":"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.","tokens_in":16256,"tokens_out":3150,"would_cite":true,"duration_ms":30610,"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":"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.","keywords":["Wiedemann–Franz law","Hall Lorenz number","thermal Hall effect","Weyl semimetal","NdAlSi","Kondo scattering","4f electrons","quasiparticle relaxation time"],"falsifier":"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.","tokens_in":15171,"feed_emoji":"🧲","tokens_out":11831,"duration_ms":92306,"temperature":0.7,"pith_summary":"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.","feed_headline":"Heat flow in NdAlSi doubles the standard metal bound","feed_subtitle":"A rare upward violation of the Wiedemann–Franz law points to Kondo-type scattering off 4f moments.","key_machinery":"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 $L_{xy}$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the method of extracting the Lorenz ratio from the thermal Hall conductivity rather than the longitudinal one.","marker":"[16]"},{"why":"establishes NdAlSi as a Weyl semimetal with RKKY-mediated helical ferrimagnetism, setting the 4f/5d exchange context.","marker":"[18]"},{"why":"provides the magnetic phase diagram and the characteristic fields Hm and Hp that locate the enhanced Lxy regions.","marker":"[20]"},{"why":"reports an unconventional Nernst effect linked to energy-dependent relaxation time, a competing mechanism considered in the discussion.","marker":"[23]"},{"why":"supplies the universal upper bound on the phonon Hall angle used to exclude phonon carriers.","marker":"[32–34]"},{"why":"supplies the magnon Hall angle benchmark used to exclude magnon carriers.","marker":"[35]"},{"why":"provides the model in which an energy-symmetric dip in conductivity raises the Lorenz number.","marker":"[41]"},{"why":"links Kondo scattering of conduction electrons to an increased Lorenz number in (La,Ce)Al2.","marker":"[46]"},{"why":"supplies the phase-diagram argument for coexistence of Kondo and RKKY couplings in NdAlSi.","marker":"[49]"}],"fun_headline_variants":["NdAlSi breaks Wiedemann–Franz law with doubled heat flow","Kondo scattering boosts Hall Lorenz number in NdAlSi","Magnetic Weyl semimetal shows rare Lorenz number enhancement","Hall Lorenz number hits 2× Sommerfeld value in NdAlSi","Upward Wiedemann–Franz violation tied to Kondo effect in NdAlSi"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["NdAlSi breaks Wiedemann–Franz law with doubled heat flow","Kondo scattering boosts Hall Lorenz number in NdAlSi","Magnetic Weyl semimetal shows rare Lorenz number enhancement","Hall Lorenz number hits 2× Sommerfeld value in NdAlSi","Upward Wiedemann–Franz violation tied to Kondo effect in NdAlSi"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000823,"raw_usage":{"total_tokens":3645,"prompt_tokens":1035,"completion_tokens":2610,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":651,"completion_tokens_details":{"reasoning_tokens":2514}},"tokens_in":651,"tokens_out":2610,"duration_ms":18334,"temperature":1.0,"reasoning_tokens":2514,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:26:23.081598+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"P ., Xu, Z","cited_arxiv_id":null,"evidence_quote":"supplies the method of extracting the Lorenz ratio from the thermal Hall conductivity rather than the longitudinal one."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"establishes NdAlSi as a Weyl semimetal with RKKY-mediated helical ferrimagnetism, setting the 4f/5d exchange context."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the magnetic phase diagram and the characteristic fields Hm and Hp that locate the enhanced Lxy regions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"reports an unconventional Nernst effect linked to energy-dependent relaxation time, a competing mechanism considered in the discussion."},{"cited_title":"& Tokura, Y","cited_arxiv_id":null,"evidence_quote":"supplies the magnon Hall angle benchmark used to exclude magnon carriers."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the model in which an energy-symmetric dip in conductivity raises the Lorenz number."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"links Kondo scattering of conduction electrons to an increased Lorenz number in (La,Ce)Al2."},{"cited_title":"The Kondo lattice and weak antiferromagnet ism","cited_arxiv_id":null,"evidence_quote":"supplies the phase-diagram argument for coexistence of Kondo and RKKY couplings in NdAlSi."}],"review_version":1}