{"id":"64da9487-d22d-43ec-bc24-0c60e205d778","arxiv_id":"2607.08648","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":2,"one_line_summary":"Valley-resolved Hall viscosity is finite in gapped Dirac materials, regularizing a previously identified divergence and extending the Hoyos-Son formula to individual valleys.","lead":"The paper computes a valley-resolved Hall viscosity for gapped Dirac materials like graphene on boron nitride, finding it finite where prior work found it divergent. If correct, it extends the Hoyos-Son viscosity-conductivity relation to individual valleys and identifies TMDs as the best platform for measuring valley Hall viscosity.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"Finiteness of valley-resolved Hall viscosity rests on an unproven Hilbert-space locality claim and an externally referenced formula; the equivalence with Kubo formalism makes the disappearance of the divergence unexplained.","rationale":"The reader correctly identified the most load-bearing concern: the regularization mechanism is unexplained and the result is hedged. I agree with the reader's assessment that the CONDITIONAL verdict with MODERATE confidence is appropriate. The concern is real and central — if the Hilbert-space locality claim fails, the paper's main result (Eq. 13) and its downstream consequence (the valley-resolved Hoyos-Son formula, Eq. 17) both collapse. However, I would not escalate beyond CONDITIONAL because: (1) the Wigner-Weyl formalism is a legitimate and well-established framework for computing transport coefficients, (2) the valley-summed result matches prior work (modulo the normalization factor), providing partial validation, and (3) the structural similarity of the valley-resolved Hoyos-Son formula to the Galilean case is a non-trivial consistency check. The paper acknowledges its own limitations honestly (hedging language, acknowledgment that the bilayer calculation is only leading-order, acknowledgment that the Hoyos-Son extension is empirical). The lack of provided code for the symbolic calculations is a reproducibility concern but not by itself a correctness argument. The experimental predictions being 2-4 orders of magnitude below current signals is a practical limitation, not a theoretical flaw. The verdict should remain CONDITIONAL pending either a rigorous proof of the Landau-level truncation property or independent numerical verification of Eq. (13).","tokens_in":14897,"tokens_out":2473,"duration_ms":117158,"concrete_test":"Independently expand the Wigner-Weyl expression for N_η (Eq. 11) in the Landau level eigenbasis of the massive Dirac Hamiltonian, keeping the full sum over all Landau levels. Check whether the sum over intermediate states genuinely truncates to a two-level neighborhood of the chemical potential as claimed. If the sum does not truncate — i.e., if higher Landau levels contribute non-negligibly — the finiteness of the valley-resolved Hall viscosity is an artifact of implicit truncation in the symbolic computation, and the central claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that the valley-resolved Hall viscosity is finite, contradicting Sherafati & Vignale (PRB 2019) who found a divergence using Kubo formalism. The paper describes its Wigner-Weyl approach as 'an equivalent Green function formulation' to Kubo. If the two formalisms are genuinely equivalent, a divergence in one should appear in the other. The paper does not explain this contradiction. Instead, it states that 'only a two Landau level neighborhood of the chemical potential modulo particle-hole partner states is relevant to evaluate them' and calls this 'a new finding,' but provides no derivation of this locality property in the main text. The formula enabling the calculation (Eq. 11 and its eigenfunction representation) is referenced to another preprint by the same author [38/42], not derived here. The abstract itself hedges with 'seems to be regularized.' If the Landau-level sum in the Wigner-Weyl expression does not actually truncate to a finite number of states, the finite result could be an artifact of the symbolic Python computation implicitly truncating the sum. Additionally, the factor-of-four discrepancy with [25] is resolved by asserting their stress tensor normalization is wrong ('which we consider to be twice as large as it should be'), without reference to a standard definition or independent check. If the normalization is wrong in the opposite direction, the valley-resolved Hoyos-Son formula (Eq. 17), which is validated by comparison with [25]'s nonlocal conductivity, would also fail.","agreement_with_reader":"agree"},"referee_report":{"model":"glm-5.2","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.","tokens_in":15670,"tokens_out":1174,"duration_ms":250228,"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":[{"comment":"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","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null}],"minor_comments":[{"comment":"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.","section":null},{"comment":"The notation s(ζ) is defined on page 4 but could be confused with spin; a brief clarifying note would help.","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a genuine gap in the literature and the results, if correct, are interesting. However, the central claim about regularization of the divergence is insufficiently justified — the hedging language in the abstract is unusual for a paper claiming a new result that contradicts prior work. The reliance on symbolic Python for the key calculation, with intermediate steps in an external preprint, makes independent verification difficult. I would recommend the author strengthen the presentation by including the derivation of Eq. (13) and explicitly addressing the equivalence with the Kubo formalism before the paper is accepted."},"author_rebuttal":null,"desk_editor":{"model":"glm-5.2","letter":"The main thing to know: this paper claims that the valley-resolved Hall viscosity for Semenoff-massive Dirac materials in integer quantum Hall phases is finite (Eq. 13), directly contradicting Sherafati and Vignale (PRB 2019), who found it divergent in Kubo formalism. The author also derives a valley-resolved Hoyos-Son formula (Eq. 17) and computes the valley Hall viscosity for biased Bernal bilayer graphene (Eq. 22). All three are genuinely new results not present in the prior literature, and the Wigner-Weyl Green function framework is a legitimate approach. The cross-check against [25]'s independently computed nonlocal Hall conductivity is a real consistency test, not circular reasoning — the structural identity of the valley-resolved Hoyos-Son formula with the Galilean case is a non-trivial observation earned by explicit computation. Credit is due for the algebraic work and for transparency about the bilayer approximation's limitations. The experimental estimates, while sobering (signals 2–4 orders of magnitude below already-measured Hall viscosity), are honest and useful for the community. The identification of group-VI TMDs as the most promising platform is a concrete, falsifiable prediction. Now the soft spots. The central issue is the one the stress-tester flags: the paper calls the Wigner-Weyl approach 'an equivalent Green function formulation' to Kubo, yet one formalism diverges and the other doesn't. The abstract hedges with 'seems to be regularized,' and the main text states that 'only a two Landau level neighborhood of the chemical potential' is relevant, calling this 'a new finding' — but provides no derivation of this locality property. The enabling formula (Eq. 11) is referenced to the author's own preprint [38/42], not derived here. If the Landau-level sum doesn't actually truncate and the symbolic Python computation is implicitly truncating it, the finite result could be an artifact. This is the load-bearing concern. The factor-of-four discrepancy with [24,25] is resolved by asserting their stress tensor is 'twice as large as it should be,' confirmed only by a non-relativistic limit check. That's reasonable but not airtight — if the normalization is wrong in the other direction, the Hoyos-Son comparison with [25] would also break. The bilayer calculation is on weaker footing: the chiral fermion approximation breaks Lorentz invariance, so Eq. 11 captures only the leading piece, which the paper acknowledges. No code is provided for the symbolic calculations, which is a reproducibility gap. These concerns are real but proportionate — the paper is not wrong, it's under-proven. The right move is a serious referee who can either verify the Hilbert-space locality claim or demand the author prove why the Wigner-Weyl sum truncates where the Kubo sum doesn't. The factor-of-four normalization needs an independent check against a standard stress-tensor definition. If those hold up, this is a solid and useful contribution to valleytronics and geometric transport.","headline":"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.","tokens_in":15640,"tokens_out":1978,"would_cite":false,"duration_ms":130946,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["73.43.-f","72.80.Vp","67.10.Bf"],"model":"glm-5.2","headline":"Per-valley Hall viscosity is finite in massive Dirac materials","keywords":[],"falsifier":"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.","tokens_in":15107,"feed_emoji":"🌀","tokens_out":991,"duration_ms":267082,"temperature":0.7,"pith_summary":"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.","feed_headline":"Per-valley Hall viscosity is finite in massive Dirac materials","feed_subtitle":"A Wigner-Weyl reformulation resolves a divergence that blocked valley-resolved viscous transport, enabling a per-valley Hoyos-Son formula.","key_machinery":"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 ","core_discovery":"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.","pith_inferences":[],"forward_implications":["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."],"fun_headline_variants":["Wigner-Weyl calculus resolves valley Hall viscosity divergence","Valley-resolved Hall viscosity is finite in massive Dirac systems","Per-valley Hall viscosity finite via Wigner-Weyl Green functions","Divergence in valley Hall viscosity regularized by eigenfunction method","Valley-resolved Hoyos-Son formula holds for integer quantum Hall phases"],"cache_read_input_tokens":0,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Wigner-Weyl calculus resolves valley Hall viscosity divergence","Valley-resolved Hall viscosity is finite in massive Dirac systems","Per-valley Hall viscosity finite via Wigner-Weyl Green functions","Divergence in valley Hall viscosity regularized by eigenfunction method","Valley-resolved Hoyos-Son formula holds for integer quantum Hall phases","Each valley's Hall viscosity is finite and well-defined","Wigner-Weyl reformulation yields finite per-valley Hall viscosity","Valley Hall viscosity divergence resolved in massive Dirac materials","Per-valley Hoyos-Son relation holds in Dirac quantum Hall phases","Massive Dirac materials have finite valley-resolved Hall viscosity"]},"model":"glm-5.2","effort":"low","cost_usd":0.0,"raw_usage":{"total_tokens":1321,"prompt_tokens":621,"completion_tokens":700,"prompt_tokens_details":null},"tokens_in":621,"tokens_out":700,"duration_ms":41565,"temperature":1.0,"reasoning_tokens":573,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T03:38:17.070374+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"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.","supporting_citations":[],"review_version":1}