{"id":"dc1c56eb-9ffe-414b-b4d6-f9b376bed753","arxiv_id":"2501.00490","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A covariant Boltzmann equation with one relaxation time yields L/L0 = (3/pi^2)(h/(k_BT))^2 for both graphene and QGP, so the Wiedemann-Franz law fails as the net carrier density approaches zero.","lead":"This paper derives the Lorenz ratio (thermal over electrical conductivity) for graphene and for the quark-gluon plasma from one relativistic fluid equation, and shows it grows as the inverse square of the net carrier density. It connects two very different quantum fluids and proposes enthalpy per net carrier as the single quantity that explains the Wiedemann-Franz law breakdown.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The measured thermal conductivity requires the open-circuit condition J=0, but Eqs. (28)-(29) imply q^mu is strictly proportional to J^mu, so the open-circuit thermal conductivity vanishes; Eq. (32) is a diagonal projection, not the experimental Lorenz ratio.","rationale":"The paper gives a clean, parameter-free derivation of Eq. (32) from the stated covariant BTE in RTA, and the algebra is self-consistent. The load-bearing weakness is the step from the calculated response coefficients to the experimentally measured thermal conductivity. The reader's verdict correctly identifies the off-diagonal/open-circuit issue as the weakest assumption; my check strengthens it: because q^mu is exactly proportional to J^mu, the open-circuit thermal conductivity is zero, not merely a perturbation of a22. This invalidates the quantitative comparison to Crossno et al., although it does not necessarily destroy the more qualitative claim that a fluid-based diagonal ratio violates the Wiedemann-Franz law. The reader already assigned CONDITIONAL, and this concern is consistent with that verdict, so no adjustment is needed. The paper has no machine-checked proof or reproducible code, but that is not the main issue; the main issue is the physical interpretation of a22 as the thermal conductivity. The proposed analytic test directly settles whether the comparison is meaningful.","tokens_in":24808,"tokens_out":15165,"duration_ms":172017,"concrete_test":"Analytic check: write the 2x2 response matrix from Eqs. (28)-(29) as J = C X and q = C (E+P)/(e n) X, with X = -rho/(E+P) E-tilde + T^{-1} grad T. Impose the experimental open-circuit condition J=0 and evaluate q. If q is identically zero for any grad T, then kappa_OC = 0, and the Lorenz ratio plotted in Fig. 6(a) is not the experimental one. Numerically, recompute Fig. 6(a) using kappa_OC instead of a22 and show that the claimed agreement with the Crossno data disappears. A useful diagnostic is the degenerate limit mu >> k_B T: the diagonal formula gives L/L0 ~ (3/pi^2)(mu/k_B T)^2, whereas a Fermi liquid should return L/L0 -> 1; checking whether kappa_OC restores that limit would distinguish a frame artifact from a physical Wiedemann-Franz violation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Sec. II A defines J^mu and q^mu with the same thermodynamic force (Eqs. (28)-(29)). Comparing the two equations gives q^mu = (E+P)/(e n) J^mu, so the response matrix has rank one. The paper then identifies thermal conductivity with a22, obtaining Eq. (32), but the quantity measured in thermal-transport experiments is the heat current under the open-circuit condition J=0. Imposing J=0 in the same matrix gives q^mu=0: the open-circuit thermal conductivity is identically zero in this model, not a22. Equivalently, kappa_OC = a22 - a21 a12/a11 = 0 with the appropriate Onsager sign convention. Thus the off-diagonal thermoelectric terms do not merely add a comparable correction near charge neutrality; they exactly cancel a22. Consequently, comparing Eq. (32) with the Crossno et al. data in Fig. 6(a) compares a diagonal coefficient, not the measured thermal conductivity. The algebraic derivation is internally consistent, but the identification of a22 with the experimental thermal conductivity is unsupported, and the claimed agreement rests on that identification.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript derives thermoelectric transport coefficients for a two-dimensional massless electron-hole plasma (graphene) and a three-dimensional quark-antiquark plasma (QGP) from the relativistic Boltzmann equation in Anderson-Witting relaxation-time approximation. The authors express thermodynamic variables and currents in terms of Fermi integrals, identify electrical conductivity with the diagonal coefficient a11 and thermal conductivity with a22, and obtain the Lorenz ratio L = κ/(σT) = [(E+P)/(n k_B T)]^2 (k_B^2/e^2) for graphene, with an analogous expression involving Q_u for QGP. They compare the graphene result with the experimental data of Crossno et al. in Fig. 6(a), identify fluidic, mixed, and Ohmic density domains, and conclude that the fluid-based framework explains Wiedemann-Franz law violation near the Dirac point and its restoration at high doping, with QGP as the high-energy analogue.","tokens_in":24906,"tokens_out":9441,"duration_ms":101915,"significance":"The algebraic derivation is self-contained and transparent: the Lorenz-ratio expression is independent of the relaxation time, requires no fitted parameters in the ratio itself, and the unified representation (D+1) f_{D+1}(A)/f_D(A) for the enthalpy term is a useful observation. If the coefficient identified as thermal conductivity were the quantity measured in thermal-transport experiments, the result would be a compact, parameter-free demonstration that enthalpy per net carrier controls the Lorenz ratio in both Dirac fluids. However, the central identification is not valid for transport measurements (see major comments), so the paper's significance as an explanation of the Crossno et al. data is not established; the remaining value is a formal two-band RTA calculation of the diagonal response ratio, which is not the experimentally measured Lorenz ratio.","major_comments":[{"comment":"The identification of the thermal conductivity with the diagonal coefficient a22 is the load-bearing step of the paper, and it is unsupported. Equations (28) and (29) show that q^μ is strictly proportional to J^μ, because both are proportional to the same thermoelectric force X^μ with coefficients that differ only by the factor (E+P)/(e n). The 2×2 response matrix therefore has rank one, and a22 is not the thermal conductivity measured under the open-circuit condition J=0. Imposing J^μ=0 in Eqs. (28)–(29) forces X^μ=0 and hence q^μ=0; equivalently, κ_open = a22 − a21 a12/a11 = 0, not a22. The sentence after Eq. (31), \"we will focus only on the diagonal components of the matrix a,\" drops exactly the off-diagonal terms needed to obtain the measured Lorenz ratio. Consequently, Eq. (32) is a ratio of two diagonal response coefficients, and the comparison with the Crossno et al. data in Fig. 6(a) does not establish the claimed agreement.","section":"Sec. II A, Eqs. (28)–(32)"},{"comment":"The QGP analogue inherits the same problem. Equation (38) is obtained from the diagonal coefficients a-tilde_22 and a-tilde_11 of a response matrix that is again rank-one, because the heat flow is again proportional to the charge flow in the same RTA. Thus L-tilde/L0 is not the Lorenz ratio that would be measured under the open-circuit condition J-tilde=0. The statement in Sec. III that Fig. 6(b) supports Wiedemann-Franz law violation due to the fluid aspect of quark matter therefore rests on the same unsupported identification rather than on a transport measurement or a well-defined theoretical open-circuit coefficient.","section":"Sec. II B, Eqs. (36)–(38)"},{"comment":"The claim that experimental observation shows Wiedemann-Franz law violation in both graphene and quark-gluon plasma overstates the QGP evidence. The cited references [45–49] are model calculations and phenomenological estimates of κ/σ for hot QCD or hadronic matter, not direct experimental determinations of the Lorenz ratio in heavy-ion collisions. The comparison in Fig. 6(b) is therefore a comparison of one model class with other model estimates, not with experimental data, and the abstract's wording should be corrected.","section":"Abstract and Sec. III"}],"minor_comments":[{"comment":"The sentence \"n > 1.5 × 10^9 cm^-2\" is inconsistent with the stated Ohmic-domain threshold \"n > 1.5 × 10^10 cm^-2\" given earlier in the same section; please correct the exponent or the domain definition.","section":"Sec. III, final paragraph before Sec. IV"},{"comment":"The text first says the theoretical curve is \"in good agreement with the data from S1\" and then states that the comparison is only qualitative and no quantitative matching is attempted; these two statements should be reconciled so the reader knows what claim is being made.","section":"Sec. III, discussion of Fig. 6(a)"},{"comment":"The symbol E-tilde is used both for the comoving electric field and later for the combination E-tilde^μ + (1/ρ)∇^μ P, both in the same notation; please distinguish these quantities explicitly to avoid confusion.","section":"Eqs. (25)–(26) and (28)"},{"comment":"The experimental data points from Ref. [65] are shown without error bars and without a stated criterion for selecting S1 rather than S2 for the claimed agreement; since sample cleanness is invoked, specifying how the comparison was made would increase the transparency of the qualitative match.","section":"Fig. 6(a)"}],"recommendation":"major_revision","confidential_remarks":"The formal derivation is internally consistent, but the main experimental claim turns on a textbook distinction between the diagonal coefficient a22 and the open-circuit thermal conductivity. If the authors can compute the full response-matrix combination a22 − a21 a12/a11 and show that it reproduces the Crossno et al. data, the paper would be publishable after revision; if not, the experimental comparison should be removed and the paper reframed as a formal diagonal-response calculation. I would also ask the editor to ensure that the overlap with the authors' earlier works (Refs. [36] and [80]) is addressed after the scope is clarified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear X,\n\nQuick take: this is a clean, self-contained derivation of L = (h/kBT)^2 (kB/e)^2 for graphene and the QGP analogue from a covariant BTE in RTA, and a useful side-by-side comparison of the two systems. If you work on Dirac fluids, the paper is a readable account of how enthalpy per net carrier enters transport. But the central quantitative claim—agreement with Crossno et al.—rests on identifying the measured thermal conductivity with the diagonal coefficient a22, and that identification does not survive close reading.\n\nWhat is new is the unified treatment: the same thermodynamic variable h/n appears in both systems, the Lorenz ratio diverges near charge neutrality, and the fluid/mixed/Ohmic classification in the n–L/L0 plane is a helpful organizing picture. The RTA algebra is presented in enough detail to be checked, no parameters are fitted into the ratio itself, and the authors are honest that the agreement with sample S1 is qualitative and that the mixed-domain boundaries are not derived. Credit where due: they cite Refs. 39, 41, and 42 for the core formula, so they do not overclaim novelty, and they explicitly flag the diagonal-projection choice in the text.\n\nThe soft spot is load-bearing. Equations (28)–(29) imply q^mu is proportional to J^mu, so the 2x2 response matrix is rank one. The experimentally relevant thermal conductivity is measured with J = 0; imposing that condition makes the open-circuit heat current vanish in this model. The kappa plotted in Fig. 6(a) is a22, a coefficient defined at zero electric field, not the measured quantity. The off-diagonal thermoelectric terms are not a small correction near charge neutrality—they exactly cancel a22. So the comparison to Crossno et al. does not establish quantitative or even qualitative agreement in the fluidic domain; the divergence in the plotted curve is not the divergence seen in experiment. A proper treatment would need separate relaxation channels or a momentum-conserving collision term. The QGP half is less affected because there is no direct data comparison, but the same open-circuit issue applies to the definition of kappa.\n\nThis paper is worth a referee: the derivation is clean, the comparison is instructive, and the flaw is specific and fixable by revising the identification and tempering the claims about Fig. 6. I would not cite it for the quantitative result, but it could be a good reading-group example of why boundary conditions matter in thermoelectric transport.\n\nMy recommendation: send it to peer review, but with a referee who will push hard on the open-circuit condition and the Fig. 6 comparison.","headline":"A clean, self-contained RTA derivation of a known Lorenz-ratio formula and a useful graphene/QGP comparison, but the comparison to Crossno et al. rests on a diagonal coefficient that is not the measured open-circuit thermal conductivity.","tokens_in":25646,"tokens_out":3518,"would_cite":false,"duration_ms":39838,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives a common fluid formula for the Lorenz ratio in graphene and quark-gluon plasma and shows that the Wiedemann–Franz law fails whenever the enthalpy per net carrier grows without bound.","keywords":["Wiedemann-Franz law","Lorenz ratio","graphene Dirac fluid","quark-gluon plasma","enthalpy per carrier","Boltzmann transport equation","relaxation time approximation","fluid-to-nonfluid crossover"],"falsifier":"Measure thermal conductivity in graphene under a genuine open-circuit condition ($J=0$) while sweeping the gate voltage through the Dirac point, and compare the resulting $\\kappa/(\\sigma T)$ with the diagonal-coefficient formula $(h/k_B T)^2 (k_B^2/e^2)$; if the open-circuit ratio does not diverge at low net density, the claimed violation is an artifact of the diagonal truncation. A less experimental version is to evaluate the full $2\\times2$ thermoelectric matrix from the same $\\delta f_{e,h}$ and impose $J=0$: if $\\kappa/(\\sigma T)$ at $J=0$ differs materially from the diagonal expression, the identity fails.","tokens_in":24492,"feed_emoji":"⚛️","tokens_out":10462,"duration_ms":90948,"temperature":0.7,"pith_summary":"The paper tries to establish that the breakdown of the Wiedemann–Franz law observed in graphene and quark–gluon plasma is a consequence of fluid behavior, not a separate anomaly. Solving the covariant Boltzmann equation in relaxation-time approximation for an electron–hole plasma, the authors derive a Lorenz ratio $L = \\kappa/(\\sigma T) = \\left(\\frac{E+P}{n k_B T}\\right)^2 \\frac{k_B^2}{e^2} = \\left(\\frac{h}{k_B T}\\right)^2 \\frac{k_B^2}{e^2}$, where $h$ is the enthalpy per net charge carrier. Because $h/(k_B T)$ grows without bound as the net carrier density $n$ goes to zero, the ratio $\\kappa/(\\sigma T)$ diverges, violating the Wiedemann–Franz law near the charge-neutrality point of graphene and at low quark chemical potential in QGP. At high carrier density $h \\approx \\mu$ and the standard Lorenz number is recovered, marking a fluid-to-non-fluid crossover. The authors compare their graphene curve with experimental data and argue that the same mechanism operates in both systems.","feed_headline":"One fluid formula explains Wiedemann-Franz breakdown","feed_subtitle":"Graphene and quark-gluon plasma share the same ratio: Lorenz number scales as enthalpy per net carrier squared.","key_machinery":"The load-bearing object is the enthalpy per net charge carrier, $h = (E+P)/n$, which appears squared in the Lorenz ratio. It enters through the heat-current definition and the Gibbs–Duhem relation that eliminate time derivatives from the out-of-equilibrium distribution functions; the ratio $h/(k_B T)$ is what turns the diagonal thermal and electrical conductivities into the compact formula $L = (h/k_B T)^2 (k_B^2/e^2)$. For graphene the phase-space integrals are Fermi integrals $f_j(A)$, and the same derivation with dimension 3 and speed $c$ yields the QGP analogue.","core_discovery":"The central claim is an identity: for a two-dimensional gas of massless Dirac carriers, relativistic kinetic theory in the relaxation-time approximation gives $L = \\left(\\frac{h}{k_B T}\\right)^2 \\frac{k_B^2}{e^2}$ with $h = (E+P)/n$, and the identical structure, with $c$ replacing $v_F$, quark charge $Q_u$ replacing $e$, and three-dimensional phase space, gives $\\tilde L = \\left(\\frac{\\tilde h}{k_B T}\\right)^2 \\frac{k_B^2}{Q_u^2}$ for a quark–antiquark plasma. The derivation builds electron and hole distribution functions from the covariant Boltzmann equation, forms the charge and heat currents, and identifies the diagonal transport coefficients. The authors state that within this fluid-based framework the Lorenz ratio consistently implies a violation of the Wiedemann–Franz law, because $h/(k_B T)$ diverges as the net carrier density approaches zero; the experimental graphene data of Ref. [65] show the same divergent trend in the Dirac-fluid regime.","pith_inferences":["Editorial inference: imposing the open-circuit condition $J=0$ would mix the off-diagonal thermoelectric coefficients into the thermal conductivity; near charge neutrality those terms are comparable to the diagonal one, so the quantitative comparison with the experimental data in Fig. 6(a) likely shifts even if the divergent trend survives.","Editorial inference: the same enthalpy ratio may govern Lorenz ratios in other linearly dispersing two-component plasmas, such as electron–positron plasma or candidate Dirac materials beyond graphene, where the net carrier density can be tuned through zero.","Editorial inference: real samples contain charge puddles that prevent $n$ from reaching zero, so the predicted divergence should be cut off; the experimental data's finite but growing ratio is consistent with this cutoff, and a clean measurement of the saturation value would test the formula quantitatively."],"forward_implications":["Any clean two-dimensional Dirac fluid at low net carrier density should show $\\kappa/(\\sigma T)$ rising as $n^{-2}$ as $n \\to 0$, matching the observed divergent trend in graphene.","At high chemical potential, $h \\simeq \\mu$, so the Lorenz ratio returns to $L_0$; the crossover window defines a density range where fluid and Ohmic behavior mix, which gate-tuned transport measurements can resolve.","The same formula, with $c$ and quark charge in place of $v_F$ and $e$, predicts a Wiedemann–Franz violation in quark–gluon plasma at low net quark density and a return to $L_0$ at high density, where future high-baryon-density heavy-ion experiments could test the fluid-to-non-fluid transition.","The relaxation time $\\tau$ cancels in the ratio $\\kappa/(\\sigma T)$, so the violation is a thermodynamic property of the fluid rather than a detail of scattering."],"supporting_citations":[{"why":"It supplies the two graphene-device datasets whose divergent Lorenz ratio the fluid curve is compared against.","marker":"[65]"},{"why":"It is an earlier hydrodynamic treatment of the same experimental violation that the paper extends.","marker":"[39]"},{"why":"It provides the relaxation-time closure of the Boltzmann equation on which the whole derivation rests.","marker":"[87]"},{"why":"It is a thermodynamic Fermi-integral treatment of the Lorenz ratio for massless Dirac fermions, used as a reference for the divergence.","marker":"[41]"},{"why":"It gives the bipolar-diffusion and band-gap explanation of the violation, the main alternative baseline.","marker":"[42]"},{"why":"It supplies the relativistic dissipative hydrodynamics at finite chemical potential that underlies the QGP transport framework.","marker":"[45]"},{"why":"It gives the procedure for eliminating time derivatives in relaxation-time hydrodynamics, used to obtain the gradient form of the distribution functions.","marker":"[93]"}],"fun_headline_variants":["Graphene and QGP share a fluid law that breaks Wiedemann-Franz","One ratio explains Wiedemann-Franz violation in graphene and QGP","Enthalpy per carrier dictates Wiedemann-Franz breakdown in fluids","Fluid Dirac systems violate Wiedemann-Franz: graphene and QGP","Lorenz ratio scales with enthalpy per carrier, breaking Wiedemann-Franz"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes the experimentally measured thermal conductivity is given by the diagonal coefficient alone, without enforcing the open-circuit condition of zero electric current, even though off-diagonal thermoelectric coefficients also contribute to the heat current at $J=0$ and are comparable near the charge-neutrality point.","fun_headline_variants_meta":{"raw":{"variants":["Graphene and QGP share a fluid law that breaks Wiedemann-Franz","One ratio explains Wiedemann-Franz violation in graphene and QGP","Enthalpy per carrier dictates Wiedemann-Franz breakdown in fluids","Fluid Dirac systems violate Wiedemann-Franz: graphene and QGP","Lorenz ratio scales with enthalpy per carrier, breaking Wiedemann-Franz"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000618,"raw_usage":{"total_tokens":2941,"prompt_tokens":1093,"completion_tokens":1848,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":709,"completion_tokens_details":{"reasoning_tokens":1746}},"tokens_in":709,"tokens_out":1848,"duration_ms":15299,"temperature":1.0,"reasoning_tokens":1746,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:50:08.835857+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure thermal conductivity in graphene under a genuine open-circuit condition ($J=0$) while sweeping the gate voltage through the Dirac point, and compare the resulting $\\kappa/(\\sigma T)$ with the diagonal-coefficient formula $(h/k_B T)^2 (k_B^2/e^2)$; if the open-circuit ratio does not diverge at low net density, the claimed violation is an artifact of the diagonal truncation. A less experimental version is to evaluate the full $2\\times2$ thermoelectric matrix from the same $\\delta f_{e,h}$ and impose $J=0$: if $\\kappa/(\\sigma T)$ at $J=0$ differs materially from the diagonal expression, the identity fails.","supporting_citations":[{"cited_title":"Crossno, J","cited_arxiv_id":null,"evidence_quote":"It supplies the two graphene-device datasets whose divergent Lorenz ratio the fluid curve is compared against."},{"cited_title":"Lucas, J","cited_arxiv_id":null,"evidence_quote":"It is an earlier hydrodynamic treatment of the same experimental violation that the paper extends."},{"cited_title":"Anderson and H","cited_arxiv_id":null,"evidence_quote":"It provides the relaxation-time closure of the Boltzmann equation on which the whole derivation rests."},{"cited_title":"Rycerz, Wiedemann–franz law for massless dirac fermions with implications for graphene, Materials 14, 10.3390/ma14112704 (2021)","cited_arxiv_id":null,"evidence_quote":"It is a thermodynamic Fermi-integral treatment of the Lorenz ratio for massless Dirac fermions, used as a reference for the divergence."},{"cited_title":"Tu and S","cited_arxiv_id":null,"evidence_quote":"It gives the bipolar-diffusion and band-gap explanation of the violation, the main alternative baseline."},{"cited_title":"Jaiswal, B","cited_arxiv_id":null,"evidence_quote":"It supplies the relativistic dissipative hydrodynamics at finite chemical potential that underlies the QGP transport framework."}],"review_version":1}