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REVIEW 3 major objections 4 minor 96 references

On the Wiedemann-Franz law violation in Graphene and quark-gluon plasma systems

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2501.00490 v3 pith:VTZW4Y6O submitted 2024-12-31 cond-mat.str-el cond-mat.stat-mechnucl-th

Ashutosh Dwibedi , Subhalaxmi Nayak , Sathe Subodh Kiran , Sabyasachi Ghosh , Sesha Vempati This is my paper · ORCID
classification cond-mat.str-elcond-mat.stat-mechnucl-th
keywords Wiedemann-FranzlawLorenzratiographeneDiracfluidquark-gluonplasmaenthalpypercarrierBoltzmanntransportequationrelaxationtimeapproximationfluid-to-nonfluidcrossover
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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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 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.

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 (3)
  1. [Sec. II A, Eqs. (28)–(32)] 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.
  2. [Sec. II B, Eqs. (36)–(38)] 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.
  3. [Abstract and Sec. III] 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.
minor comments (4)
  1. [Sec. III, final paragraph before Sec. IV] 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.
  2. [Sec. III, discussion of Fig. 6(a)] 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.
  3. [Eqs. (25)–(26) and (28)] 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.
  4. [Fig. 6(a)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Lorenz-ratio derivation is algebraically self-contained and the comparison to Crossno et al. is an external benchmark, not a fitted input.

full rationale

The central result, Eq. (32), follows by explicit algebra from the covariant Boltzmann equation in the Anderson-Witting relaxation-time approximation. The distribution-function deviations in Eqs. (23)-(24) are obtained from the BTE plus conservation equations, and the conductivities in Eqs. (30)-(31) share a common prefactor involving the relaxation time, which cancels identically in the ratio L = kappa/(sigma T). No experimental datum enters the derivation: the only inputs are the assumed dispersion, degeneracy, and constant relaxation time, and the final ratio is independent of the relaxation time. The comparison with Crossno et al. in Fig. 6(a) is described by the authors as qualitative ('we highlight the qualitative divergence tendency'), so it is an external test rather than a fitted input. Self-citations to Refs. [36], [80], and [90] are methodological or background citations; the derivation itself is written out in the manuscript, and the cited prior work is not used to define or force the target result. The concern raised by a skeptical reader, that the measured thermal conductivity requires the open-circuit condition J=0 whereas the paper identifies kappa with the diagonal coefficient a22, is a physical-modeling or benchmark-validity issue, not a circularity: it does not make the predicted Lorenz ratio equivalent to the paper's own inputs by construction. Since no load-bearing step reduces to a fitted parameter, a self-citation chain, or a definitional identity, the circularity score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central formula L = (h/(k_BT))^2 (k_B^2/e^2) is derived from relativistic BTE with RTA, requiring the assumptions listed. The only fitted numbers are the hand-drawn domain boundaries; tau_c cancels in L. No new entities are introduced.

free parameters (2)
  • Domain boundaries n_fluid and n_Ohmic = 8x10^9 cm^-2 and 1.5x10^10 cm^-2
    Chosen by eye to separate fluidic, mixed, and Ohmic regions in Fig. 6(a); not derived from the model and no uncertainty is quoted.
  • Relaxation time tau_c
    Assumed equal for electrons and holes (and quarks/antiquarks) and energy-independent; cancels in the Lorenz ratio but sets the scale of sigma and kappa.
assumptions (4)
  • domain assumption Momentum non-conserving scatterings are negligible in the temperature window of interest, so the collision kernel conserves momentum, energy, and charge.
    Invoked in Sec. II A to justify hydrodynamic electron flow; without it the RTA collision term would not preserve the fluid conservation laws.
  • domain assumption Anderson-Witting form of RTA with a single, energy-independent relaxation time tau_c for both electron and hole bands.
    Stated in Sec. II A; the tau independence of L makes the ratio robust, but the same-tau assumption is not derived from microscopic interactions.
  • domain assumption Landau-Lifshitz hydrodynamic frame is used, so the dissipative part of energy flow vanishes.
    Footnote 1 and Sec. II A; this frame choice underlies the expression q = -(E+P)/rho J and the diagonal identification of transport coefficients.
  • domain assumption QGP is modeled as single-flavor up/anti-up quark plasma with only spin degeneracy (no color or flavor counting).
    Sec. II B; affects quantitative magnitudes but not the qualitative L formula.

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Pith. "Pith review of On the Wiedemann-Franz law violation in Graphene and quark-gluon plasma systems." pith.science (2026). https://pith.science/paper/VTZW4Y6O

@misc{pith2026250100490,
  author       = {Pith},
  title        = {Pith review of: On the Wiedemann-Franz law violation in Graphene and quark-gluon plasma systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VTZW4Y6O}},
  note         = {Machine review of arXiv:2501.00490}
}
abstract

A comparative study of the thermodynamic and transport properties of the ultra-relativistic quark-gluon plasma produced in heavy ion collisions with the "quasi-relativistic" massless electron-hole plasma in graphene sample has been performed. We observe that the enthalpy per net charge carriers emerges as a useful physical quantity determining the transport variables in hydrodynamic domain. Lorenz ratio is defined as thermal to electrical conductivity ratio, normalized by temperature and Lorenz number $L_{0}=\frac{\pi^{2}}{3}\left(\frac{k_{B}}{e}\right)^{2}$. The validity of the Wiedemann-Franz law can be checked by evaluating the Lorenz ratio, which is expected to be unity. We investigate the validity of the Wiedemann-Franz law by examining whether the Lorenz ratio equals unity or deviates from it. Our findings indicate that, within the fluid-based framework, the Lorenz ratio consistently leads to a violation of the Wiedemann-Franz law. This is attributed to the proportional relation between Lorenz ratio and enthalpy per net charge carriers in the fluid. Based on the experimental observation, graphene and quark-gluon plasma, both systems at a low net carrier density, violate the Wiedemann-Franz law due to their fluidic nature. However, graphene at a relatively high net carrier density obeys the Wiedemann-Franz law, followed by metals with high Fermi energy or electron density. It indicates a fluid to the non-fluid transition of the graphene system from low to high carrier density. In this regard, the fluid or non-fluid aspect of quark-gluon plasma at high density is yet to be explored by future facilities like Compressed Baryonic Matter and Nuclotron-based Ion Collider fAcility experiments.

Figures

Figures reproduced from arXiv: 2501.00490 by the authors.

Figure 1
Figure 1. FIG. 1: Representation of different physical systems starting from CMP system, graphene, and metals to HEP [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: (a) The number density and (b) energy density and pressure of graphene with respect to [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: (a) The number density and (b) energy density and pressure of QGP with respect to [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: (a) Enthalpy density per particle [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: (a) Enthalpy density per particle [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: (a) The dependence of [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

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