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REVIEW 2 major objections 4 minor 47 references

Hydrodynamics in generalized electronic two-band systems

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Pith's one-line read Hydrodynamics derived for two-band power-law electron systems is neither Galilean nor Lorentz invariant except for a=2, and yields viscosity, Lorenz and Prandtl numbers, and plasmons for every band exponent a.

desk verdict A genuine new framework for two-band hydrodynamics with a broken-boost term, but the Dirac-limit Euler equation (18a) is algebraically inconsistent and the viscosity in Appendix D conflicts with the main text. read the letter →

arxiv 2505.21176 v1 pith:4KJ3OK74 submitted 2025-05-27 cond-mat.str-el

E. Di Salvo , P. Cosme , L. Fritz This is my paper
classification cond-mat.str-el
keywords electronichydrodynamicstwo-bandsystemspower-lawdispersionhydrodynamicmasselementboostinvarianceshearviscosityWiedemann-Franzlawplasmons
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 derives the Euler and Navier-Stokes equations for a two-band electron fluid whose bands disperse as $\omega_\pm = \pm B k^a$ in arbitrary dimension $d$, starting from the Boltzmann equation and its collision moments. The central result is that the momentum equation contains a term $-\frac{2-a}{d}\,u(\nabla\cdot u)$ with no counterpart in ordinary fluids, and that this term removes boost symmetry: the symmetry analysis of the equations shows that the Galilean boost is a generator only for $a = 2$ (parabolic bands), so for generic power-law bands, including Dirac bands with $a = 1$, the macroscopic theory is neither Galilean nor Lorentz invariant. From the same framework the paper computes the shear viscosity (with exactly vanishing bulk viscosity), the thermoelectric conductivities, the Lorenz and Prandtl numbers, and the long-wavelength plasmon dispersion. Real materials, such as bilayer graphene, rhombohedral trilayer graphene, and Weyl semimetals, realize different values of $a$, so the analysis places them in one hydrodynamic description and identifies which transport anomalies are generic to two-band power-law systems.

What carries the argument

The load-bearing object is the hydrodynamic mass element $m_\star$ (Eq. 12), the proportionality constant between the average momentum density $n\langle p\rangle$ and the particle current $n u$; for power-law bands it scales as $m_\star \propto n^{(2-a)/d}$, so its density dependence generates the $-\frac{2-a}{d}\,u(\nabla\cdot u)$ term in Eq. (22) that breaks boost symmetry. The viscous sector rests on two additional pieces: the relaxation-time collision integral $C_\pm = \mp (f^{(+)}/\tau_+ - f^{(-)}/\tau_-)$ of Eq. (25), used inside a Chapman-Enskog expansion, and the Onsager reciprocity condition $\alpha = \bar\alpha$ of Eq. (37), which ties the electron and hole relaxation times $\tau_+$ and $\tau_-$ to a common scale $\tau_0$. The Lie-algebra computation on the reduced shallow-water-type system in Appendix C is what converts the presence of that extra term into the claim that no boost generator exists for $a \neq 2$.

What would settle it

Measure $\eta/s$ and the Lorenz number in a clean sample with known non-quadratic bands (for example rhombohedral trilayer graphene, $a = 3$, or a Weyl semimetal, $a = 1$) while sweeping temperature and chemical potential across the Dirac point: the paper predicts $\eta/s$ independent of $T$ and $\mu$ near charge neutrality and a diverging $L$ as disorder is removed, so observing $\eta/s$ to grow with $T$, or $L$ to saturate at a finite value with fixed small disorder, would falsify the transport sector. Alternatively, in a current-biased two-band sample, the asymmetry of counter-propagating hydrodynamic plasmon modes predicted by Eq. (44) as $(a-2)/(2d)\,u_0 k$ directly tests the boost-breaking term of Eq. (22).

Watch

Extended reading notes

Core claim

The paper's central claim is that all macroscopic transport in a two-band power-law system is governed by a single density-dependent object, the hydrodynamic mass element $m_\star$ defined by $n\langle p\rangle = m_\star n u$ in Eq. (12). Because $m_\star \propto n^{(2-a)/d}$ in the Fermi liquid limit, the momentum equation (22) acquires the anomalous convective term $-\frac{2-a}{d}\,u(\nabla\cdot u)$. A Lie-algebra symmetry analysis of this equation (Appendix C) shows that for $a \neq 2$ the symmetry algebra contains only translations and dilations, and that the boost generator appears only for $a = 2$; the paper concludes that the coarse-grained hydrodynamics is neither Galilean nor Lorentz invariant, and that even linear (relativistic-looking) dispersion does not produce a Lorentz-invariant fluid description. The theory nonetheless closes at the Dirac point ($\mu/k_B T \to 0$) and in the Fermi liquid limit ($\mu/k_B T \to \infty$), and it yields a shear viscosity $\eta = G_d(a)(\tau_+ P_+ + \tau_- P_-)$ with zero bulk viscosity, a Lorenz number that violates the Wiedemann-Franz law and diverges at charge neutrality in the clean limit, and plasmons that are gapped in three dimensions and scale as $k^{1/2}$ in two dimensions for any exponent $a$.

Load-bearing premise

Every quantitative prediction inherits the relaxation-time collision model $C_\pm = \mp(f^{(+)}/\tau_+ - f^{(-)}/\tau_-)$ of Eq. (25) and the Onsager reciprocity condition $\alpha = \bar\alpha$ of Eq. (37) that merges $\tau_+$ and $\tau_-$ into a single scale $\tau_0$, so if the real electron-electron scattering of a two-band system is not captured by those two assumptions, the computed viscosities, conductivities, Lorenz and Prandtl numbers would all shift even though the ideal Euler structure, including the broken boost symmetry, could still be correct.

Editorial extensions

If this is right

  • For any material whose low-energy bands disperse as $|k|^a$ with $a \neq 2$, including Dirac systems with $a = 1$, the momentum equation contains $-\frac{2-a}{d}\,u(\nabla\cdot u)$, so the fluid is not Galilean even at the Euler level and momentum transport differs from ordinary Navier-Stokes flow.
  • The shear viscosity is fixed by band geometry through $\eta = G_d(a)(\tau_+ P_+ + \tau_- P_-)$ with exactly zero bulk viscosity, and near the Dirac point the ratio $\eta/s$ is independent of temperature and chemical potential, so clean low-temperature samples should behave as near-perfect fluids.
  • Near charge neutrality the Lorenz number $L = \kappa/(\sigma T)$ is not universal and diverges in the clean limit ($\tau_{\rm dis} \to \infty$), a direct breakdown of the Wiedemann-Franz law; in the Fermi liquid limit $L \to L_0$ is restored.
  • The Prandtl number at the Dirac point, $\Pr = (2d/a)(\tau/\tau_{\rm dis})\, k_B P/(S_+ + S_-)$, is much smaller than one when electron-electron scattering dominates and much larger than one when disorder dominates, making the ratio of momentum to heat diffusivity a tunable sample-quality diagnostic.
  • Hydrodynamic plasmons are gapped in 3D and gapless with $\omega \sim k^{1/2}$ in 2D for every $a$, but the Doppler shift carries the symmetry-breaking prefactor $(1 + (a-2)/(2d))\,u_0 k$, so counter-propagating plasmon modes split asymmetrically when $a \neq 2$.

Reading between the lines

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

  • Beyond the paper: if the broken boost symmetry is real, the momentum current should fail a center-of-mass conservation test; measuring the streamline pattern around a constriction in a clean two-band sample with $a \neq 2$ and comparing with the standard Navier-Stokes profile would expose the extra $u(\nabla\cdot u)$ term directly.
  • Beyond the paper: the $a$-dependent Doppler prefactor $(a-2)/(2d)$ turns the plasmon dispersion into a dynamical probe of the band exponent, so an asymmetric splitting of counter-propagating plasmon modes in a biased two-band material would be a quantitative, setup-specific signature of Eq. (44) that the paper itself does not propose.
  • Beyond the paper: the construction assumes isotropic power-law bands; extending the hydrodynamic mass element to tilted, anisotropic, or multi-pocket dispersions should produce a tensorial $m_\star$ and further symmetry-breaking terms, and the same Lie-algebra test could classify which of those bands retain any boost-like symmetry.
  • Beyond the paper: because every quantitative number is proportional to $\tau_0$, a first-principles computation of $\tau_+$ and $\tau_-$ from the Coulomb collision integral for a concrete material (say rhombohedral trilayer graphene) would convert the present framework into falsifiable magnitudes.
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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

2 major / 4 minor

Summary. The paper derives Euler and Navier-Stokes-type hydrodynamic equations for two-band fermionic systems with power-law dispersion ω± = ±B k^a in d dimensions, using moment expansions of the Boltzmann equation and a generalized hydrodynamic mass m⋆. It analyzes boost symmetry, computes shear viscosity, thermoelectric transport coefficients, Lorenz and Prandtl numbers, and long-wavelength plasmons, with special attention to the Dirac (a=1) and parabolic (a=2) limits. The central result is Eq. (22), the Fermi-limit velocity equation containing a non-Galilean u(∇·u) term for a≠2, from which the subsequent transport and collective-mode results are built.

Significance. If the derivation is corrected, the paper provides a useful unified framework: it recovers the known shear-viscosity results for graphene (a=1, d=2) and for the dilute parabolic gas (a=2, d=3), it reproduces the Wiedemann-Franz law in the Fermi-liquid limit, and it gives concrete falsifiable predictions for η/s, Lorenz and Prandtl numbers, and plasmon dispersion as functions of the band exponent a and dimension d. The thermodynamic and transport coefficients are computed in closed form from the stated model without fitted parameters, apart from the explicitly phenomenological relaxation times. The main limitation, acknowledged in the text, is that the quantitative transport predictions inherit the relaxation-time ansatz (25) and the Onsager reciprocity condition (37), as the reader's report also notes.

major comments (2)
  1. [Sec. III A, Eq. (18a)] The stated Dirac-limit Euler equation is not the correct reduction of the moment equations. Starting from Eqs. (14a)-(14c) with n⟨p⟩ = m⋆nu and Π = (a/d)n⟨E⟩I, and using ∂tT + u·∇T = -(a/d)T∇·u (which follows from Eq. (18b) and m⋆ ∝ T^{2/a-1}), the momentum equation reduces to ∂tu + (u·∇)u - (2-a)/d (∇·u)u + a/(d m⋆ n)∇[n⟨E⟩] = 0. Eq. (18a) instead has +(1-a)/d (∇·u)u and (1/m⋆)∇[n⟨E⟩]. The two differ at a=1, where the correct coefficient is -1/d rather than 0, and at a=2, where Eq. (18a) predicts a spurious -1/d term in a Galilean-invariant parabolic system. The pressure term also lacks the 1/n factor needed to be dimensionally consistent at charge neutrality. Because Eq. (18a) is used to discuss closure and boost symmetry in the Dirac limit, it must be corrected; the corrected equation is consistent with the general Fermi-limit equation (22).
  2. [Sec. III A, Eq. (23)] As written, the temperature equation ∂tT + u·∇T + (a/d)∇·u = 0 is dimensionally inconsistent unless T is measured with k_BT = 1 throughout. The last term should be (a/d)T∇·u, i.e. ∂tT + u·∇T + (a/d)T∇·u = 0. This relation is used in deriving the sound speed (24), so the authors should verify which form was used and adjust Eq. (24) if necessary.
minor comments (4)
  1. [Appendix B, Eq. (B5)] Eq. (B5) reads ∂tnc - ∇·(ncu) = 0, which differs by a sign from the main-text continuity equation (14a); the '+' sign should be used.
  2. [Sec. II] The symbol α is used both for the band exponent (for example, 'if, and only if, α = 2' after Eq. (12)) and for the shallow-water coefficient in Appendix C. Please use a consistently for the band exponent to avoid confusion.
  3. [Sec. III B] There are minor typographical issues: 'entalphy' should be 'enthalpy', and in Sec. IV 'well-defined ain the absence of disorder' should read 'well-defined in the absence of disorder'.
  4. [Sec. V, Eq. (44)] The figure captions state τ0 = 0.6T exp(-|µ|/T) and τdis = 1, but because σ, κ, Pr and L are all proportional to these relaxation scales, the plotted curves should be explicitly labeled as illustrative model outputs rather than parameter-free predictions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Euler, viscous, transport, and plasmonic results are re-derived from the stated kinetic model, and the only same-group citations are benchmarks or explicitly stated model inputs.

full rationale

The paper's claimed derivation chain is self-contained within the stated kinetic model. The hydrodynamic mass m* is defined from the computed momentum density (Eqs. 11 and 12), the pressure and heat currents follow from equilibrium integrals (Eqs. 15 and 16), and the central Euler equation Eq. (22) is obtained by inserting the Fermi-limit scalings m* prop. n^{(2-a)/d} and P prop. n^{(d+a)/d} into the moment equations (14) and eliminating time derivatives via the continuity equation. The boost-symmetry conclusion is not imported from prior work: Appendix C solves the Lie determining equations for the reduced 1D system (C1) and exhibits the boost generator only for alpha = 1, which by Eq. (22) corresponds to a = 2. The viscous and transport coefficients are derived from the explicitly stated relaxation-time ansatz (25) and the Chapman-Enskog or linear-response procedures; the relaxation times are model assumptions, not fitted to the output, and the paper states that microscopic details are encoded in tau. The same-group citations to [18] and [31] are used for benchmarking known limits and for the illustrative tau0 parameterization, but the central expressions for viscosity, Lorenz number, and plasmon dispersion are re-derived in this paper and do not reduce to those citations. No fitted parameter is renamed as a prediction, no equation is equivalent to its input by construction, and no load-bearing uniqueness theorem is imported from the authors' own prior work. The suspicious Dirac-limit momentum equation (18a) may be algebraically inconsistent with the moment equations, but an algebraic error is a correctness risk, not circularity, and therefore does not raise the circularity score.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a small number of modeling axioms: the power-law two-band dispersion, the collision invariants, a local-equilibrium ansatz, and a relaxation-time collision model. The only numerical inputs are the illustrative τ0 and τdis used for figures. No new particles, forces, or conserved quantities are postulated; the hydrodynamic mass m⋆ is a derived bookkeeping quantity.

free parameters (2)
  • Characteristic collision time τ0 for plots = 0.6 T exp(-|μ|/T) (chosen, not derived)
    Used in Figs. 2-7 to produce curves for MLG/BLG/TLG/WS/2BEG; the paper states 'we employed a relaxation time as τ0 = 0.6T exp(−|µ|/T), assuming Onsager's relation (37) to hold'. It is an input, not a prediction.
  • Disorder relaxation time τdis = τdis = 1 (in units of T or μ) for plots
    Figures 2-7 set a constant τdis = 1; this sets the disorder scale and affects all transport coefficients and dimensionless ratios, but is not derived from material parameters.
assumptions (6)
  • domain assumption The low-energy electronic structure is captured by two bands ω±=±B k^a with a≥0, B>0, in d>1 dimensions.
    Invoked in Sec. I, Eq. (4); all results apply to these power-law band touchings, not to arbitrary band structures.
  • domain assumption The Coulomb collision integral conserves total charge, momentum, and energy, with ∫ p_j(C+ + C-) dp=0; disorder and phonons are neglected in the Euler/Navier-Stokes derivation.
    Sec. III A assumes these collisional invariants; if momentum is not conserved (e.g., umklapp or interband processes), the hydrodynamic equations change.
  • domain assumption Local equilibrium has the boosted Fermi-Dirac form (7) with a single small drift u << v_F, expanded to first order in u.
    Eqs. (7)-(8); the hydrodynamic mass m⋆ is extracted from this linear-response momentum density.
  • ad hoc to paper The relaxation-time collision operator C±=∓(f+/τ+ - f-/τ-) (Eq. 25), with no momentum dependence and no coupling to disorder for the viscous sector.
    Replaces the actual Coulomb integral; the resulting viscosity and transport coefficients are proportional to these timescales.
  • ad hoc to paper Onsager reciprocity α=ᾱ imposes the relation τ+(T++μE+) = -τ-(T-+μE-) (Eq. 37), and τ± are then expressed via a single τ0 following Ref. [31].
    Sec. IV; this is an imposed symmetry condition, not derived from the microscopic collision integral.
  • domain assumption For generic a≠1,2 the hydrodynamic equations close only in the limits μ/T→0 and μ/T→∞; the paper assumes these limits govern the experimentally relevant transport regimes.
    Sec. III A and Conclusions; exact closure for all μ/T is only possible for a=1 and a=2.

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Pith. "Pith review of Hydrodynamics in generalized electronic two-band systems." pith.science (2026). https://pith.science/paper/4KJ3OK74

@misc{pith2026250521176,
  author       = {Pith},
  title        = {Pith review of: Hydrodynamics in generalized electronic two-band systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4KJ3OK74}},
  note         = {Machine review of arXiv:2505.21176}
}
read the original abstract

In this paper, we derive the Euler and Navier-Stokes equations for electronic two-band systems in arbitrary dimension and with generic power-law dispersion relations. We focus on the hydrodynamic transport regime, where such systems offer a unique tunability between a Fermi-liquid type regime at high doping and the inherent two-band physics of the low-density system close to the Dirac-type band-touching point. For a generic dispersion, the absence of Euclidean or Lorentzian invariance leads to novel types of hydrodynamic equations. We characterize these novel hydrodynamic regimes through dimensionless numbers, such as the Prandtl and Lorenz numbers, or the ratio between shear viscosity and entropy density. In all cases, we provide a derivation of the physics of the long-wavelength plasmonic modes.

Figures

Figures reproduced from arXiv: 2505.21176 by the authors.

Figure 1
Figure 1. Schematic illustration of broken boost invariance. Considering a collection of particles which undergo a uniform boost k0 in momentum space. With a non quadratic band structure, the energies/masses (here represented by the radii of the disks) break the boost invariance of the movement of the centre of mass k ′ cm– dark red vs. light red on the right pannel. As such, it is now evident that even in the case of rela￾ti… view at source ↗
Figure 2
Figure 2. We plotted here the value of the ratio between viscosity and entropy (31) against the chemical potential in units of temperature µ/T (panel a) or temperature in units of chemical potential T /µ (panel b) for monolayer (MLG), bilayer (BLG) and trilayer graphene (TLG) in dashed lines, while in thick ones are represented the behaviours for Weyl semimetals (WS) or two-band electronic gases (2BEG). From both figures we c… view at source ↗
Figure 3
Figure 3. We plotted here the value of electric conductivity σ against the chemical potential in units of temperature µ/T (panel a) or temperature in units of chemical potential T /µ (panel b); we employ the same notation and color code as Fig. (2). We notice that the thick lines grow parabolically against the linear growth of dashed ones in terms of chemical potential µ; a similar conclusion can be obtained by studying the d… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: We plotted here the value of thermal conductivity κ against the chemical potential in units of temperature µ/T (panel a) or temperature in units of chemical potential T /µ (panel b); we employ the same notation and color code as Fig. (2). We notice again that the thick…
Figure 6
Figure 6. Figure 6: We plotted here the value of Seebeck coefficient against the chemical potential in units of temperature µ/T (panel a) or temperature in units of chemical potential T /µ (panel b); we employ the same notation and color code as Fig. (2). The value of the coefficient is e…

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    All diagonal elements are equal. We can specify them for the casesn = 2 and n = 4, which are of interest in our coming derivations. The tensor T ab is then proportional to Kronecker delta as long as all its off-diagonal elements are vanishing and the diagonal elements are all ...

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    Charge equation It this case the conserved charge current, we can derive it by computing the contributions by different species, given by the following integrals j± = ± Z Rd dkd (2π)d ∇ωf (±)(ω) (B2) = − Z Rd dkd (2π)d (u · k)∇ω ∂f (±) eq ∂ω (ω) + O(|u|2) . This integral can e...

  38. [47]

    (B6) Now, in order to get the correct convective term for the final equation, we need to expand the perturbed distri- bution function up to the second order

    Momentum equation Being momentum p a vector, the related conserved current is the tensor Π, which we are going to express in terms of its matrix elements Π ij,± Πij,± = ± Z Rd ddk (2π)d (k)i (∇ω)j f (±)(ω) . (B6) Now, in order to get the correct convective term for the final e...

  39. [48]

    Energy equation At last, we go back to energy conservation and the energy current for the species is just: jε ± = Z Rd ddk (2π)d ω∇ωf (±)(ω) = −aB2 Z Rd ddk (2π)d k2a−2k(u · k) ∂f (±) eq ∂ω (ω) + O(|u|2) . (B12) We can prove the charge current can be written as W u using the u...

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Reviewed August 7, 2026 · model on record in the stance chip above.