Geometry contribution to sound attenuation in double-Weyl semimetals

Avadh Saxena; Shi-Zeng Lin; Varsha Subramanyan

REVIEW 3 major objections 4 minor 43 references

Geometry contribution to sound attenuation in double-Weyl semimetals

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read In double-Weyl semimetals, sound attenuation comes entirely from the strain-induced reshaping of the Fermi surface; the axial coupling that dominates in simple Weyl semimetals is forbidden by inversion symmetry.

desk verdict A genuinely new symmetry-based prediction for sound attenuation in double-Weyl semimetals, but the central cancellation that makes the geometric term the only one is asserted, not shown. read the letter →

arxiv 2607.17064 v1 pith:ENTT6JO2 submitted 2026-07-19 cond-mat.mes-hall

Varsha Subramanyan , Shi-Zeng Lin , Avadh Saxena This is my paper · ORCID

classification cond-mat.mes-hall MSC 82D2082C70 PACS 71.20.-b72.10.-d62.65.+k

keywords double-WeylsemimetalsoundattenuationquantumgeometrictensorFermisurfacegeometrystraincouplingchiralkinetictheoryBoltzmanntransportWeyl

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 argues that sound waves attenuate in double-Weyl semimetals by a mechanism that does not exist in simple Weyl semimetals: the deformation of the Fermi surface's geometry. In a simple Weyl semimetal, strain acts as an axial (pseudo-gauge) field that shifts the two nodes oppositely, and the odd part of the deformation potential drives the dominant anomalous sound attenuation. In a double-Weyl semimetal, inversion symmetry forbids that odd coupling, so the anomalous channel vanishes. The authors show that the only surviving contribution to attenuation, in the low-frequency regime, comes from the sound wave periodically changing the shape of the Fermi surface from isotropic to nematic, quantified through the quantum geometric tensor. The result is a closed formula, Γ = τ*ω²µ/(8πa³)(v1/v2)γ²/(ρv_s²), with no magnetic-field dependence up to second order in strain.

What carries the argument

The quantum geometric tensor (the Fubini–Study metric of the band wavefunctions) carries the argument: the strain-induced energy shift δE = −(C·Z)/E0 is rewritten as a contraction of the strain tensor with the quantum geometric tensor, showing that sound couples to the shape of the Fermi surface. The strained double-Weyl Hamiltonian, H = [(k_x²−k_y²)−γ(u_xx−u_yy)]σ_x + [2k_xk_y−2γu_xy]σ_y + (k_z∓π/2∓γ(π/2)u_zz)σ_z, has strain entering as a director field C that breaks the C4 symmetry and splits the ±2 nodes. The Boltzmann equation with chiral kinetic theory equations of motion, including the generalized Berry curvature S_ij = ∂_{k_j}A^x_i − ∂_{x_i}A^k_j, is used to show that all non-∂_t E te

What would settle it

Compute the full Boltzmann integral for Q in Eq. (12) retaining the anomalous-velocity, Berry-curvature, and S_ij cross terms — without assuming the asserted cancellation — and check whether any term survives in the ωτ* ≪ 1 limit and whether it depends on magnetic field; alternatively, measure the sound attenuation coefficient Γ with and without a weak magnetic field in a candidate double-Weyl semimetal (e.g., HgCr2Se4): the paper predicts Γ is independent of B up to second order in strain, while a measured field dependence would indicate a non-geometric channel.

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Extended reading notes

Core claim

The central claim is that in double-Weyl semimetals, the geometric deformation of the Fermi surface is the only source of sound attenuation. Because strain couples to the double-Weyl Hamiltonian as a vector C that enters the energy only through momentum-squared terms (k_x²−k_y² and 2k_xk_y), the deformation potential has no odd component under inversion, so the anomalous axial mechanism of simple Weyl semimetals is absent. Evaluating the energy dissipation Q through the Boltzmann equation at order ωτ* ≪ 1, all contributions from the strain-induced anomalous velocity, the Berry-curvature terms, and the cross-Berry curvature S_ij vanish identically upon integration, leaving only the term quadr

Load-bearing premise

The load-bearing premise is that every contribution to the dissipation rate from the strain-induced anomalous velocity and Berry-curvature terms vanishes exactly when integrated — a cancellation the authors assert after 'routine (though involved) algebra' but do not display in full; if any such term survived at order ωτ* ≪ 1, the claim that Fermi-surface geometry is the only source of attenuation would fail.

Editorial extensions

If this is right

Sound attenuation in double-Weyl semimetals (e.g., HgCr2Se4) should be dominated by the geometric channel, with a predicted magnitude of order 1.5 kHz for typical parameters.
The attenuation coefficient is independent of magnetic field at low fields — a sharp distinction from simple Weyl semimetals, where the anomaly-driven channel depends on field.
In multi-Weyl semimetals with even winding number, only the geometric relaxation channel exists; in odd-winding materials, both the axial and geometric channels contribute.
Because the geometric channel is expressed through the quantum geometric tensor, ultrasonic attenuation becomes a measurable quantity that directly reflects momentum-space band geometry.
At high magnetic fields, Landau quantization destroys the Fermi-surface geometry, the geometric contribution vanishes, and the out-of-plane u_zz (axial-like) component takes over with a similar ω²τ* scaling.

Reading between the lines

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

If the asserted cancellation of Berry-curvature terms is verified by an explicit computation, the same geometric mechanism should produce anisotropic acoustic responses: sound propagating along different crystal axes would attenuate differently because the nematic deformation pattern (u_xx−u_yy vs u_xy) couples to different components of the quantum geometric tensor — a testable acoustic-birefring
The B-independence at low field is a clean falsifiable prediction: measuring Γ(B) in a candidate double-Weyl material and finding any slope would indicate a missing contribution beyond the geometric channel.
The geometry-only result suggests the ratio of attenuation for longitudinal versus shear sound should be set entirely by how strongly each strain component deforms the Fermi surface, which polarized-transducer experiments could map directly onto the anisotropy of the quantum geometric tensor.
Since δE is written in terms of g_ij, the same framework implies the geometric sound attenuation could track the static-strain-driven nematic transition, with the attenuation coefficient changing character as the C4 symmetry breaks.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this paper identifies a real and previously missed mechanism—sound attenuation via strain-induced deformation of the Fermi surface geometry in double-Weyl semimetals—and the symmetry argument that kills the axial coupling for planar strain is clean. The main result, Eq. (17), is a closed-form, parameter-dependent prediction (Γ ∝ ω², B-independent at low field) that is falsifiable. I think it deserves a rigorous referee, but the referee should push hard on the one load-bearing step.

What's genuinely new: earlier sound-attenuation calculations for simple Weyl semimetals rely on the axial (odd) deformation potential that shifts the two nodes oppositely. For double Weyl nodes, inversion symmetry forbids that odd term, so the authors argue the only remaining relaxation channel is the change in Fermi-surface shape as the sound wave strains the lattice. That intuition is correct for planar strain, and expressing δE in terms of the quantum geometric tensor is a nice touch—it makes the geometric origin explicit. The lattice derivation in Appendix A shows the strained Hamiltonian is not a fit; the formulas are parameter-free up to the usual transport coefficients.

Soft spots, in order of importance. First, the paper's core technical step—the assertion that all non-Ė terms in the Boltzmann equation vanish identically in the limit ωτ*≪1—is not actually shown. The text says 'after some routine (though involved) algebra' and gives only a parity sketch plus the (q_xq_y−q_xq_y) factorization. That is not enough. Those terms include strain-induced velocities and Berry-curvature pieces that can be of the same order in strain and are multiplied by factors like v_F/v_s, which is not small. If any of them survive, Eq. (17) picks up extra, possibly B-dependent, contributions and the central claim 'geometry is the only contributing factor' falls. I would not want to see this published without the full cancellation algebra, or at minimum a clear statement of which symmetry makes each family of integrals vanish.

Second, the abstract and intro overstate the absence of axial coupling. Eq. (8) contains a node-antisymmetric u_zz term (the ±γπ/2 u_zz in the σz term), which is a genuine axial coupling. The authors later say it matters only in the quantum limit and is 'much smaller' than other mechanisms, citing Ref. [23], but they do not quantify that for double-Weyl. The claim should be qualified to planar strain.

Third, the units and labeling in and around Eq. (17) are messy. The V cancellation is written as if V remains on one side, the intermediate expression has a v1/v2 factor that does not obviously come from the Q calculation, and Γ is a temporal decay rate (1/s) while the text at times speaks of spatial attenuation. This is fixable but needs care.

The impurity discussion in Sec. IV is tangential and less rigorous; it reads as a rough estimate and should be clearly separated from the main result.

Bottom line: the symmetry argument is right, the prediction is interesting and testable, and the paper is worth engaging with seriously. It should go to peer review, with the condition that the authors provide the full cancellation algebra or an independent check. I would bring it to my group's journal club, but I would not cite it as a settled result until that step is resolved.

Referee Report

3 major / 4 minor

Summary. The paper argues that sound attenuation in double-Weyl semimetals is fundamentally different from that in simple Weyl semimetals. In simple WSMs, strain acts as an axial (pseudo-gauge) field and produces an anomalous attenuation contribution ∝ |λ^o|^2. The authors claim that in double WSMs this axial coupling is absent for planar strain; instead, strain deforms the Fermi surface into a nematic shape, and the resulting geometric contribution dominates. The central quantitative result is Eq. (17), Γ = τ*ω²µ/(8πa³)(v1/v2)γ²/(ρv_s²), obtained from a Boltzmann calculation in which only the Ė term in δf is retained. The paper asserts that all other Boltzmann terms vanish identically in the limit ωτ*≪1, leaving a magnetic-field-independent attenuation that is quadratic in frequency and strain.

Significance. If the central cancellation is correct, the paper identifies a genuinely new mechanism—geometric sound attenuation driven by strain-induced Fermi-surface deformation—and it makes sharp, falsifiable predictions: Γ ∝ ω², B-independence at low field, and a parameter-free prefactor in material parameters (τ*, γ, v1/v2, µ). A clear strength is that the strained Hamiltonian is derived from a lattice model in Appendix A rather than fitted, and Eq. (17) is a concrete closed-form expression. The symmetry argument that planar strain cannot produce an odd (node-antisymmetric) deformation potential in double WSMs is clean. However, the paper's main claim rests on an unproven cancellation of all non-Ė Boltzmann terms; until that cancellation is demonstrated, the quantitative result and the 'only contributing factor' statement are not established.

major comments (3)

[Sec. III A, Eqs. (12)-(13); Appendix B] The load-bearing step is the assertion that every term in Eq. (12) other than Ė vanishes identically in the limit ωτ*≪1 after 'routine (though involved) algebra.' This is not shown. The term involving ẋ_i ∂_{x_i}E is formally of order q v_F τ* δE^2 relative to the retained Ė term of order ω τ* δE^2, and v_F/v_s is not a small parameter; the k̇_i ∂_{k_i}E terms contain Berry curvature, B^t, S_ij, and possible ω_c τ* prefactors. δE is even in k, so simple parity is insufficient; e.g., (v×B)·∂_k δE can be even. The factorized (q_x q_y − q_x q_y) argument addresses only a subset of the terms in the equations of motion (B10)-(B11). Since Eq. (17) is exactly the claim that all these contributions integrate to zero, the full cancellation algebra, or an independent computation, is required before the central result can be accepted.
[Eq. (8); Appendix A; Sec. V] The statement that axial coupling is 'entirely absent' in double WSMs is not literally true for general strain. The Hamiltonian in Eq. (8) contains the node-antisymmetric term ±γ(π/2)u_zz, which shifts the two Weyl nodes in opposite directions in k_z and is precisely an axial coupling. The same contradiction appears in Appendix A after Eq. (A4), where the text says strain is 'neither a gauge field nor axial' despite the ±γu_zz term in that equation. The Discussion acknowledges the u_zz term but dismisses it as 'much smaller' without a quantitative estimate for double WSMs. The abstract and Sec. V should be qualified to planar strain, or the u_zz contribution should be estimated in the same framework.
[Sec. IV, Eqs. (18)-(22)] The supplementary estimates for the high-field (Landau-level) regime and for µ→0 with donor impurities are order-of-magnitude sketches. They are not load-bearing for the planar-strain result, but the claim in Sec. V that the geometric contribution 'would dominate sound attenuation even when non-planar sound is applied' requires a quantitative comparison with the u_zz contribution discussed above. As written, that claim is not supported by any calculation in the paper.

minor comments (4)

[Sec. II, Eq. (5)] Please check the typesetting of Eq. (5): 'ρmvs' and '(vs.ˆq)^2/D' are hard to parse; the physical notation should be made explicit (e.g., ρ_m v_s and (v_s·q̂)^2/D).
[Sec. IV] There is a stray 'electron-electron (e-e)' in the sentence beginning 'electron-electron (e-e) We finally address...'.
[Sec. III A, Eq. (13)] The angular/energy integral leading to the coefficient 1/(32π) in Eq. (13) is not shown. Since this coefficient enters the final attenuation formula, include the intermediate step (DOS and angular averages over the double-Weyl Fermi surface).
[Appendix B] The expression for Λ in Eq. (B15) is presented, but the text does not trace how Λ is used in the final expression for f_s and why it does not contribute to Q. A brief explanation would help the reader follow the 'other terms vanish' claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the strained Hamiltonian is rederived from a lattice model, and the attenuation coefficient is a parameter-free analytic result.

full rationale

The paper's central claim is not circular. The strained low-energy Hamiltonian (Eq. 8) is not imported by fiat: Appendix A derives it from a lattice model (Eqs. A1-A4) by deforming hopping parameters, and the resulting delta-E (Eqs. 9-10) is an expansion of the band dispersion. The key result Eq. 13 for Q is obtained by keeping only the Ė term in the Boltzmann correction δf=τ*∂E f0(Ė+ẋ∂xE+k̇∂kE) (Eq. 12) and asserting that the remaining terms vanish; the paper even supplies the factorization (∂xCx∂yCy−∂xCy∂yCx)=CxCy(qxqy−qxqy)=0. Whether that cancellation is fully correct is a calculational/completeness issue, not circularity—the assertion does not use Eq. 17 as an input. Γ in Eq. 17 is then a closed expression in material parameters (τ*, µ, γ, v1/v2, ρ, vs) with no parameter fitted to the target attenuation; the predicted scalings Γ∝ω² and B-independence are not imposed. Self-citation of Ref. [29] supplies context and part of the chiral-kinetic framework, but the lattice derivation in Appendix A and the explicit Boltzmann setup in Appendix B make the argument self-contained; removing the citation would not collapse the derivation. The undisplayed 'routine algebra' and the unquantified neglect of uzz are correctness-risk flags, not circular reductions.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central formula is an analytic expression in material parameters; the only inputs chosen by hand are τ*, γ, µ, and v1/v2 for the numerical estimate. The strained Hamiltonian is rederived from a lattice model in Appendix A, and no new physical entities are introduced. The main structural assumption is the validity of the Boltzmann/chiral-kinetic framework and the unshown cancellation of all non-∂_tE terms.

free parameters (4)

effective relaxation time τ* = 10^-11 s (assumed for numeric estimate)
Phenomenological relaxation rate in the Boltzmann equation; enters Γ linearly. Not derived, but a standard transport input, not fit to the target result.
Grüneisen parameter γ = 19/6 (Lennard-Jones value used in estimate)
Material parameter controlling strain coupling in Eq. (8); enters Γ quadratically. Taken from a typical interatomic potential for the order-of-magnitude estimate.
chemical potential µ = 2 meV (assumed doping)
Fermi surface size; enters Γ linearly. Chosen to place the system in the regime µ ≫ k_BT and µ ≫ ℏω_c.
velocity ratio v1/v2 = ≈1 (assumed isotropic cone)
Anisotropy of the double-Weyl dispersion; enters Γ linearly. The paper states 'it is also reasonable to state that the energy scales are such that v1 ∼ v2'.

assumptions (5)

domain assumption The strained low-energy Hamiltonian in Eq. (8) faithfully describes electron-strain coupling in double-Weyl semimetals.
Derived from a lattice model in Appendix A, following the same authors' Ref. [29]. The form of the coupling (director-field, non-minimal) is the basis for the whole analysis.
domain assumption The Boltzmann equation with a relaxation-time approximation and the chiral kinetic theory of Refs. [29,32,42] apply in the regime ωτ* ≪ 1 and qℓ ≪ 1.
Invoked in Section III A and Section IV; the sound wave is treated as a classical field modifying the band structure, with phonon-quantum effects neglected.
standard math The Fermi-surface average of δE, which is proportional to k_x²−k_y² and 2k_xk_y, vanishes by Brillouin-zone symmetry.
Used in Section III A to isolate the δf contribution; valid under C4/lattice symmetry regardless of the form of the average distribution f̄.
domain assumption The k_z direction can be neglected without loss of qualitative generality for planar strain.
Stated in Section III after Eq. (8): 'Including the k_z direction offers no qualitative changes.' The out-of-plane u_zz term is deferred to Section V with an unquantified claim of smallness.
domain assumption Low-temperature distribution f₀ = θ(µ−E) and weak magnetic field (µ ≫ ℏω_c) with no significant Landau quantization.
Stated in Section IV; needed for the Fermi-surface picture and for the absence of lowest-Landau-level physics.

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Pith. "Pith review of Geometry contribution to sound attenuation in double-Weyl semimetals." pith.science (2026). https://pith.science/paper/ENTT6JO2

@misc{pith2026260717064,
  author       = {Pith},
  title        = {Pith review of: Geometry contribution to sound attenuation in double-Weyl semimetals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ENTT6JO2}},
  note         = {Machine review of arXiv:2607.17064}
}

read the original abstract

The axial coupling of strain to the nodes of the simple Weyl semimetals leads to anomalous contributions to sound attenuation in such materials. However, in double Weyl semimetals, there is no such axial coupling. Strain instead couples as a symmetry-breaking director field that deforms the Fermi surface around each Weyl node. In this work, we show that absence of axial coupling in double Weyl semimetals implies a very different mechanism of relaxation due to sound. The deformed geometry of the Fermi surface is the only source of sound attenuation under these conditions. Thus, we identify a geometric contribution to sound attenuation in double Weyl semimetals that is entirely absent in simple Weyl semimetals.

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Since the original Weyl node has split into two nodes of lower winding, let us label the associated electron distribution in each of the nodes f+ andf −

Outline for solving the Boltzmann equation To solve the Boltzmann equation, we first need to estimate the collision integral. Since the original Weyl node has split into two nodes of lower winding, let us label the associated electron distribution in each of the nodes f+ andf ...

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Reviewed August 1, 2026 · model on record in the stance chip above.

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