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Valley Hall viscosity in the integer quantum Hall phases of (2+1)D Dirac materials

T0 review · 3 major / 5 minor · reviewed 2026-07-10 · glm-5.2

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Pith's one-line read Per-valley Hall viscosity is finite in massive Dirac materials

desk verdict New finite valley-resolved Hall viscosity formula and valley-resolved Hoyos-Son relation; the regularization of the prior divergence is asserted by calculation but not explained mechanistically. read the letter →

arxiv 2607.08648 v1 pith:7DKCTPUH submitted 2026-07-09 cond-mat.mes-hall

M. Selch This is my paper · ORCID
classification cond-mat.mes-hall PACS 73.43.-f72.80.Vp67.10.Bf
keywords hallviscosityvalleyintegerphasesquantumbilayerdirac
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

Prior work using the Kubo formalism found that the Hall viscosity for individual valleys in Semenoff-massive Dirac materials (gapped graphene-like systems) diverges, leaving only the valley-summed quantity well-defined. This paper re-examines the problem using an equivalent Green-function formulation within Wigner-Weyl calculus and finds that, when expressed through energy eigenfunctions and eigenvalues, the per-valley Hall viscosity takes a finite value. The key formula (Eq. 13) gives a closed-form expression for the valley-resolved Hall viscosity coefficient in terms of the Landau level index, the dimensionless mass parameter gamma, and a valley sign. With this result in hand, the author extends the empirical relativistic Hoyos-Son relation, which links the nonlocal correction to Hall conductivity to the Hall viscosity, to individual valleys, showing it is structurally identical to the Galilean-invariant version. The paper also computes the valley-difference Hall viscosity for biased Bernal bilayer graphene and estimates experimental signal strengths for nonlocal transport in graphene-hBN, biased bilayer graphene, and group-VI TMDs, identifying TMDs as the most promising platform for detection.

What carries the argument

The calculation uses Wigner-Weyl calculus, where quantum operators are mapped to phase-space functions (Weyl symbols) with a noncommutative Moyal star product encoding operator ordering. The Hall viscosity coefficient N_eta is written as a topological invariant in differential form notation involving traces of products of the Weyl-symbol Hamiltonian Q_W, its inverse (the propagator G_W), and covariant derivatives. The finite per-valley result emerges because only a two-Landau-level neighborhood of the chemical potential contributes to the valley-resolved viscosity, making the quantity local in Hilbert space. The valley-resolved Hoyos-Son formula is then constructed by combining the computed

What would settle it

An independent Kubo-formalism calculation that reproduces the same finite per-valley result, or a proof that the Wigner-Weyl and Kubo approaches are equivalent for this quantity, would settle the concern. Conversely, if the divergence in the Kubo approach is physical rather than a representation artifact, the finite Wigner-Weyl result would be spurious.

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Extended reading notes

Core claim

The central result is that the per-valley Hall viscosity in integer quantum Hall phases of massive Dirac materials is finite when computed via Wigner-Weyl Green-function methods, with the explicit expression N^zeta_eta = (1/8)[p^2 + (p-1)^2 - s(zeta)((p-1)gamma/sqrt(p+gamma^2) + p*gamma/sqrt(p+1+gamma^2))] per spin degree of freedom. This finite quantity enables a valley-resolved Hoyos-Son formula (Eq. 17) connecting the first nonlocal correction to the Hall conductivity with the Hall viscosity at each valley individually, structurally identical to the known Galilean-invariant relation.

Load-bearing premise

The regularization of the previously identified divergence relies on the Wigner-Weyl Green-function formulation faithfully representing the valley-resolved Hall viscosity. The paper states the divergence 'seems to be regularized' and establishes the finite result by explicit symbolic computation rather than by a general argument proving that the Kubo and Wigner-Weyl approaches must agree or explaining rigorously why they differ.

Editorial extensions

If this is right

  • A valley-resolved Hoyos-Son formula for integer quantum Hall phases in Dirac materials, linking per-valley Hall viscosity to per-valley nonlocal Hall conductivity, is now available for experimental verification.
  • Group-VI TMDs are identified as the most promising platform for detecting valley Hall viscosity via nonlocal transport, with an expected signal roughly two orders of magnitude smaller than the valley-summed Hall viscosity signal.
  • The structural identity between the Galilean and relativistic Hoyos-Son formulas for integer quantum Hall phases suggests a deeper universality of the viscosity-conductivity connection across different symmetry classes.
  • The valley Hall viscosity coefficient for biased Bernal bilayer graphene is given in closed form (Eq. 22), providing a testable prediction for bilayer devices in the hydrodynamic regime.
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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 / 5 minor

Summary. This manuscript computes the valley-resolved Hall viscosity for integer quantum Hall phases of Semenoff-massive Dirac materials (monolayer graphene and group-VI TMDs) using a Wigner-Weyl Green-function formalism. The central result is that the single-valley Hall viscosity, previously found to diverge in the Kubo-formalism treatment of Sherafati and Vignale (PRB 100, 115421, 2019), is rendered finite when expressed in an eigenfunction/eigenvalue representation. The author combines this result with the nonlocal Hall conductivity from [25] to propose a valley-resolved relativistic Hoyos-Son formula. The valley-difference Hall viscosity for biased Bernal bilayer graphene in the chiral fermion approximation is also computed, and experimental prospects in TMD-based devices are discussed.

Significance. The question of whether valley-resolved Hall viscosity is well-defined in massive Dirac systems is timely and relevant to the valleytronics and hydrodynamic-electron communities. The explicit closed-form expressions for valley-resolved viscosity coefficients (Eqs. 13-14, 22) and the proposed valley-resolved Hoyos-Son formula (Eq. 17) are concrete, falsifiable predictions. The experimental signal-strength estimates (Eqs. 26-27) and the identification of group-VI TMDs as the most promising platform are useful for guiding future experiments. The work provides an independent cross-check of the valley-summed results in [24, 25] and extends the framework to bilayer graphene.

major comments (3)
  1. The central claim — that the Wigner-Weyl approach regularizes the divergence found in the Kubo formalism of [25] — is not rigorously justified. The abstract hedges with 'seems to be regularized,' and the main text (page 4) states that 'only a two Landau level neighborhood of the chemical potential modulo particle-hole partner states is relevant' but provides no derivation or proof of this Hilbert-space locality property. Since the Wigner-Weyl formulation is described as 'an equivalent Green function formulation' to Kubo, the disappearance of the divergence requires explanation: either the two formalisms are not fully equivalent in this context, or the divergence in [25] was a representation artifact. The author should clarify which case holds and provide at least a sketch of why the Landau-level sum truncates. Without this, the reader cannot rule out that the finite result is an artifact
  2. The factor-of-four discrepancy with [24, 25] is resolved by asserting that their stress-tensor normalization is 'twice as large as it should be' (page 4), confirmed by a non-relativistic limit. However, no standard reference for the stress-tensor definition is cited, and the non-relativistic check is not shown in the main text. Since the valley-resolved Hoyos-Son formula (Eq. 17) is validated by comparison with [25], a normalization error in either direction would undermine the structural-identity claim. The author should provide the explicit non-relativistic calculation or at minimum cite the specific equation in the supplementary material where this check is performed.
  3. The load-bearing formula for the Hall viscosity coefficient N_η (Eq. 11) and its eigenfunction representation are referenced to the author's own preprint [38/42] rather than derived in the manuscript. While supplementary material is provided, the key steps showing how Eq. (11) reduces to the finite sum yielding Eq. (13) are not present in either the main text or the supplementary appendices included here. The author should either include the derivation of Eq. (13) from Eq. (11) or clearly indicate where in [42] this calculation can be found, so that the result is independently verifiable.
minor comments (5)
  1. The abstract's use of 'seems to be regularized' should be replaced with a more definitive statement or, if the mechanism is not fully understood, the nature of the uncertainty should be specified.
  2. The notation s(ζ) is defined on page 4 but could be confused with spin; a brief clarifying note would help.
  3. In Eq. (12), the relation ν_D = g_sv(p - 1/2) is stated without defining ν_D explicitly as the Dirac filling factor; a brief definition would help.
  4. For bilayer graphene, the validity of Eq. (11) is acknowledged to break down (Appendix D, Eqs. S41-S42), with only the leading piece in filling factor retained. The main text should state more clearly that Eq. (22) is exact only in the large-p limit.
  5. Reference [38] is cited as arXiv:2602.12915 and [42] as arXiv:2606.03932; both appear to be by the same author. The relationship between these preprints and the current manuscript should be clarified to avoid redundancy concerns.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor self-citation of framework formulas; central calculation is independently performed and cross-checked against external results.

  1. self citation load bearing [Eq. (11) and surrounding text; reference [38] (arXiv:2602.12915) and [42] (arXiv:2606.03932)]
    "The coefficient N_eta represents the Hall viscosity topological invariant faithfully for Lorentz (and Galilean) invariant systems and has been introduced in [38] for the first time. ... We refer here to calculations of our previous work [42] relevant for the main text ... Details on Wigner-Weyl calculus, its application to transport and linear response theory may be found in appendices F, G and H, respectively. Finally appendix K exhibits the explicit calculations of N_sigma and N_eta."

    The Hall viscosity coefficient formula (Eq. 11) and the Wigner-Weyl framework details are cited from the same author's prior work [38, 42] rather than derived in this paper. However, this is a framework citation, not a circular derivation: the actual computation of valley-resolved viscosities (Eq. 13-14) is performed here independently via symbolic Python on the Dirac Hamiltonian eigenproblem, and results are cross-checked against the external Hall conductivity results of Sherafati & Vignale [25]. The self-citation provides the computational scaffold but does not assume the target result.

full rationale

The paper's central claim—the finiteness and explicit form of the valley-resolved Hall viscosity (Eq. 13)—is derived by an independent calculation using the Dirac eigenproblem in a magnetic field, not by assuming the result from a prior work. The Wigner-Weyl framework (Eq. 11) is cited from the author's own prior preprints [38, 42], which is a minor self-citation providing the computational scaffold. However, this citation is not load-bearing for the novelty: the valley-resolved computation itself, the regularization of the divergence, and the valley-difference formula (Eq. 14) are new to this paper. The Hoyos-Son formula (Eq. 17) is validated by combining independently computed viscosity (this work) with independently computed nonlocal Hall conductivity from [25] (different authors), making the structural identity a non-trivial cross-check rather than a tautology. The bilayer graphene result (Eq. 22) is also independently computed. No prediction reduces to a fitted input by construction. The self-citation raises a minor concern about verifiability of the framework but does not constitute circularity of the central result.

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

No new physical entities are postulated. The free parameters (γ, γ̃) are dimensionless combinations of known physical quantities, not fitted constants. The axioms are standard domain assumptions from relativistic field theory and Wigner-Weyl calculus, except for the stress-tensor normalization choice which is ad hoc to this paper's disagreement with prior work.

free parameters (2)
  • γ = mv_F² / √(2eB)
    Dimensionless ratio of Semenoff mass to magnetic field energy scale. Not fitted; derived from material parameters and external field.
  • γ̃ = mv_F²γ₁ / (2eB)
    Dimensionless parameter for bilayer graphene combining bias gap, interlayer coupling γ₁≈0.38 eV, and magnetic field. Not fitted.
assumptions (4)
  • domain assumption Wigner-Weyl Green-function formulation is equivalent to Kubo formalism for computing Hall viscosity
    Stated in the abstract and Introduction: 'an equivalent Green function formulation within Wigner-Weyl calculus.' The equivalence is assumed; the difference in divergence behavior between the two approaches is not rigorously explained.
  • domain assumption Lorentz boost symmetry holds for the Dirac fermion theory, enabling the topological representation of N_η in Eq. (11)
    Invoked in the 'Topological robustness' section: 'together with Lorentz boost symmetry, leads to the representation of N_η as given in Eq. (11).' This is standard for the relativistic Dirac theory but explicitly noted to fail for the bilayer chiral fermion approximation.
  • domain assumption Only a two-Landau-level neighborhood of the chemical potential is relevant for valley-resolved Hall viscosity
    Stated in the main text: 'only a two Landau level neighborhood of the chemical potential modulo particle-hole partner states is relevant to evaluate them.' This locality in Hilbert space is what makes the valley-resolved quantity finite, but it is asserted from computation rather than proven generally.
  • ad hoc to paper Stress tensor normalization used in this paper (factor of 2 smaller than [24, 25])
    The paper states: 'we find a Hall viscosity smaller by a factor of four... This follows directly from their definition of the stress tensor which we consider to be twice as large as it should be.' The correctness of this normalization choice is asserted but not independently verified beyond a non-relativistic limit check.

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Pith. "Pith review of Valley Hall viscosity in the integer quantum Hall phases of (2+1)D Dirac materials." pith.science (2026). https://pith.science/paper/7DKCTPUH

@misc{pith2026260708648,
  author       = {Pith},
  title        = {Pith review of: Valley Hall viscosity in the integer quantum Hall phases of (2+1)D Dirac materials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7DKCTPUH}},
  note         = {Machine review of arXiv:2607.08648}
}
read the original abstract

We calculate the valley-resolved Hall viscosity for Lorentz-invariant integer quantum Hall phases in Semenoff-semiconducting graphene-like systems at zero temperature. The Kubo formalism based discussion reported in Phys. Rev. B 100, 115421 (2019) revealed the divergence of single valley viscous Hall contributions for this case with only a valley-summed Hall viscosity being finite and therefore well-defined. Our approach to the Hall viscosity calculation is based on an equivalent Green function formulation within Wigner-Weyl calculus. We find that the previously identified divergence seems to be regularized to a finite value in a proper representation of the valley-resolved Hall viscosity in terms of energy eigenfunctions and eigenvalues. Together with the local Hall conductivity and its first nonlocal correction, reported as well in Phys. Rev. B 100, 115421 (2019), we extend the empirical relativistic Hoyos-Son formula to individual valleys. Both the original Hoyos-Son formula for Galilean invariant fluids and its relativistic extension to Dirac materials are found to be structurally identical for integer quantum Hall phases and expressible in terms of local electric and viscous Hall responses. In addition we evaluate the valley(-difference) Hall viscosity for biased Bernal bilayer graphene in the chiral fermion low energy approximation. Prospects of measuring valley Hall viscosity in nonlocal transport for mono- and bilayer graphene- and group-VI TMD-based devices are discussed.

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