{"id":"a5dbd420-f343-4fbf-84db-171d43c1913c","arxiv_id":"1908.09447","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Phonons in alpha-quartz are predicted to form a triangular Weyl complex, with one double Weyl point and two single Weyl points connected by long surface arcs.","lead":"Using symmetry theory and computer simulations, this paper predicts that quartz hosts a triangular arrangement of special vibrational points, mixing two types of topological charge. If correct, the result offers a simple material where these phonon Weyl features and their long surface signals could be observed and used.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Surface-arc claim rests on an unvalidated phonon Wannier/Green's-function model; no parameters, convergence checks, or comparison to direct DFT phonon bands are provided.","rationale":"The reader's weakest assumption identifies the same load-bearing point: the surface-arc claim is computed from a Wannier-interpolated tight-binding model whose faithfulness to the underlying first-principles phonon spectrum is asserted but not demonstrated. My reading of the bulk argument does not uncover a more serious internal inconsistency. The symmetry analysis around Eqs. (1)–(6), including the role of the screw eigenvatives and the product C₃zT, is internally coherent, and the reported Wilson-loop Chern numbers are the appropriate check for the bulk Weyl charges. The band structure also matches earlier theoretical and experimental phonon data for α-SiO₂, which gives independent support to the bulk dispersion. What is genuinely missing is validation of the surface-state construction: no Wannier parameters, no phase handling for phonon eigenvectors, no Green's-function convergence data, and no comparison with a direct slab or with the original VASP dynamical matrices. Because the abstract highlights the arcs and the absence of trivial surface states as headline results, this is not a cosmetic reproducibility complaint but a load-bearing gap. The proposed slab calculation would directly test whether the surface arcs are intrinsic to the first-principles model or are introduced by the interpolation. My recommendation is therefore to keep the reader's CONDITIONAL verdict unchanged: the paper is plausible and well-founded in its bulk claims, but the surface-state claim should be treated as conditional until that check is supplied.","tokens_in":7950,"tokens_out":27455,"duration_ms":276745,"concrete_test":"Build a slab of at least 20 atomic layers of α-SiO₂ and compute the (001) surface phonon LDOS at ω = 16.22 THz by directly diagonalizing the slab dynamical matrix from the same second-order force constants, without Wannier interpolation. If the resulting iso-frequency map does not show the two arcs connecting the Γ-bar projection to the two K-bar projections exactly as in Fig. 4(c), the Wannier-interpolated surface-state claim is a model artifact. As a companion check, overlay the Wannier-interpolated bulk bands on the direct VASP phonon bands along Γ–K–H–A and at several generic q-points; require agreement within 0.05 THz.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract's two most distinctive claims—that the phonon surface arcs span the entire surface Brillouin zone and that only nontrivial surface states exist at the iso-frequency surface—are not obtained directly from the first-principles phonon spectra in Fig. 2. They come from a 'phonon tight-binding Hamiltonian in the Wannier representation from second-order interatomic force constants' (Figs. 4a–4d). The main text provides no construction details, no Wannier spread or truncation parameters, no interpolation error, and no overlay of the Wannier bands on the VASP phonon bands at generic q. The iterative Green's function method cited (Refs. 48,50) is an electronic-structure technique; adapting it to phonons requires the (ω²−D) Green's function and a consistent phase convention for phonon eigenvectors, and neither is described. In addition, α-SiO₂ is polar, so the nonanalytical term correction mentioned for q→0 cannot be represented by a short-range Wannier model unless the long-range contribution is separately subtracted and readded; this is not discussed. If the Wannier model misses a band, has a phase inconsistency, or truncates force constants too aggressively, the arcs in Fig. 4(c) and 4(d) can be numerical artifacts even though the bulk Weyl points at Γ, A, K, and H are correctly identified. The bulk topology is supported by Wilson-loop charges and by agreement with earlier phonon calculations, so the surface-state part of the central claim is the least secure link.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a symmetry-based classification of a 'triangular Weyl complex' in phonon spectra of non-centrosymmetric trigonal or hexagonal crystals. Using a two-band effective Hamiltonian constrained by screw rotation C3z and time reversal (Eqs. (1)-(6)), it argues that single Weyl phonons with linear dispersion appear at K/H points and double Weyl phonons with quadratic dispersion appear at Γ/A points. First-principles phonon calculations for α-SiO2 (space group P3_221, No. 154) identify crossings of branches 16/17 at K and A and of branches 17/18 at H and Γ; Wilson-loop and Berry-curvature analysis assigns chiral charges C=∓1 and C=±2. The authors then construct a Wannier-interpolated tight-binding Hamiltonian from second-order force constants and use iterative Green's functions to compute surface LDOS, claiming that two surface arcs connect the double-Weyl projection to two single-Weyl projections and that these arcs span the entire surface Brillouin-zone, with no trivial bulk states at the chosen iso-frequency. A hexagonal-lattice candidate, YPt2B, and the list of possible space groups are relegated to the Supplemental Material.","tokens_in":8236,"tokens_out":16097,"duration_ms":174848,"significance":"If the surface-state results are correct, the paper reports a genuinely novel phenomenon: a local Weyl complex in which one double Weyl point and two single Weyl points of opposite chirality coexist, while the global Nielsen-Ninomiya sum rule is still satisfied. The symmetry analysis is clean, parameter-free, and internally consistent, and the bulk α-SiO2 phonon spectra match earlier theoretical and experimental results, which gives confidence in the first-principles part. The paper would be of interest to the topological-phonon and bosonic-topology communities. However, the most distinctive claims in the abstract and conclusion, namely the surface-arc connectivity, the full-BZ span of the arcs, and the absence of trivial bulk states at the iso-frequency, rest on numerical details that are not presented in the main text and therefore cannot currently be verified.","major_comments":[{"comment":"The two most distinctive claims, that the phonon surface arcs connect the double-Weyl projection to two single-Weyl projections and that there are only nontrivial surface states at the iso-frequency, are obtained from a Wannier-interpolated tight-binding Hamiltonian and iterative Green's functions, but the main text gives no construction details: no Wannier spread or truncation radius, no number of Wannier functions, no interpolation error, no comparison of the Wannier-interpolated bulk bands with the VASP/finite-displacement phonon bands along generic paths, no surface Green's-function k-point sampling, and no convergence tests. As presented, one cannot exclude the possibility that the arcs in Fig. 4(c,d) are numerical artifacts of the interpolation or of the Green's-function implementation. Please provide these details and include an overlay of the Wannier-interpolated and first-principles bulk bands over a substantial portion of the Brillouin-zone, including around Γ, A, K, and H.","section":"Fig. 4 and the paragraph beginning 'To illustrate this'"},{"comment":"The paper does not specify how the iterative Green's-function method of Refs. [48,50], which is an electronic-structure technique, is adapted to phonons; in particular, the form of the retarded phonon Green's function (e.g., G(ω)=(ω^2−D+i0^+)^−1), the eigenvector phase convention, and the construction of the semi-infinite surface self-energy are not described. Additionally, α-SiO2 is polar, and the nonanalytical term correction mentioned in the main text is needed for the long-range dipole-dipole interaction and the direction-dependent q→0 limit. A short-ranged Wannier model built from finite-displacement force constants cannot reproduce this long-range contribution unless it is separately subtracted and re-added. The manuscript does not state how this was handled, and the surface arcs near the Γ and A projections could be affected.","section":"Fig. 4 and the computational-methods paragraph"},{"comment":"The abstract and conclusion claim that there are only nontrivial surface states across the iso-frequency surface. This claim cannot be checked from the presented data: the iso-frequency maps in Fig. 4(c,d) are shown at a single frequency and are not overlaid with the projected bulk continuum, and the LDOS panels in Fig. 4(a,b) follow selected high-symmetry lines only. Please provide complete surface-BZ iso-frequency maps with the bulk projection shaded, and state the frequency range over which the 'only surface states' statement holds, since the wording suggests a much broader claim than a single iso-frequency slice.","section":"Iso-frequency discussion after Fig. 4"}],"minor_comments":[{"comment":"The sentence 'There are six K points in the first BZ and each K point is shared by three neighbor Wigner-Seitz cells in momentum space' is ambiguous: while the six corners of the hexagonal BZ are counted, each is shared by three neighboring BZs, so the effective number of K points per primitive cell is two. Please rephrase to make the fractional counting explicit and to clarify that the total chiral charge of the full BZ is zero while the local complex has unequal numbers of opposite-chirality WPs.","section":"Wilson-loop paragraph following Fig. 3"},{"comment":"The LDOS panels in Fig. 4(a,b) lack a color bar and the surface features are not annotated; please add a color bar and mark the surface branches so that the reader can distinguish them from the projected bulk continuum.","section":"Fig. 4"},{"comment":"The Wilson-loop spectra used to assign the Chern numbers are only described verbally; please include the actual Wilson-loop spectra for loops around K and A (and around H and Γ) in the main text or in the Supplemental Material so that the C=∓1 and C=±2 assignments can be verified.","section":"Wilson-loop paragraph following Fig. 3"},{"comment":"The derivation of Eq. (5) using the product operator C3zT should state explicitly the anti-unitary action of time reversal and the complex conjugation of d(q); without this the symmetry constraint is not fully checkable from the main text.","section":"After Eq. (5)"},{"comment":"The statement that 'the longest phonon arcs can provide entire modes of topological phonon surface states in a robust nontrivial one-way phonon propagation channel' is speculative; no group-velocity or transport calculation is presented, so this sentence should be moved to an outlook or supported by additional calculations.","section":"Conclusion"},{"comment":"The reference to the Supplemental Material contains two typos: 'computional' should be 'computational' and 'inlcudes' should be 'includes'.","section":"Ref. [30]"}],"recommendation":"major_revision","confidential_remarks":"I reviewed the manuscript without access to the Supplemental Material. If the SM contains the Wannier construction parameters, interpolation errors, the surface Green's-function details, and the YPt2B results, then several of the major comments reduce to requests for better cross-referencing and a concise summary in the main text. If not, the surface-state claims are not currently supported. The main text should also be made self-contained for the key numerical validation, and the abstract's strong statements about only nontrivial surface states need to be backed by the projected bulk continuum. The bulk symmetry analysis and the first-principles phonon dispersion are the strongest parts of the paper; the revision should focus on making the surface-state calculation reproducible and verifiable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The one thing you should know is that this paper has a real idea: a triangular Weyl complex where one double Weyl point and two single Weyl points of opposite chirality coexist, protected by screw symmetry in non-centrosymmetric trigonal/hexagonal lattices. That combination is not in the older literature, which only pairs opposite-chirality WPs of the same order. The symmetry analysis is the strongest part: it works with symbolic coefficients, no fitting, and shows why the K and H points host linear single WPs while Gamma and A host quadratic double WPs. The Wilson-loop charges and the Berry-curvature distribution in the kx-kz plane give a consistent picture, and the DFT phonon spectra match earlier calculations and experiment for alpha-quartz.\n\nThe soft spot, as you might guess, is the surface-state part. The claims that the arcs span the whole surface BZ and that only nontrivial surface states appear at the iso-frequency surface come from a Wannier-interpolated tight-binding Hamiltonian built from second-order force constants, with the iterative Green's function method. The main text gives no Wannier spread or truncation parameters, no interpolation error, no overlay of Wannier bands on the DFT bands, and no discussion of how the nonanalytical term correction for polar alpha-SiO2 is treated in a short-range Wannier model. The cited Green's function method is electronic; adapting it to phonons needs a consistent phase convention for eigenvectors and the (omega^2 - D) form. These are legitimate concerns, and they directly affect the abstract's most distinctive claims.\n\nThat said, I would not sink the paper on this. The bulk topology does not depend on the Wannier model; the symmetry argument is self-contained and the Wilson-loop charges are direct. If the supplemental material actually provides the missing details and shows the Wannier bands reproducing the first-principles phonons, the surface arcs are very likely correct. The authors do say computational details are in the SM, so the omission in the main text may be just a space choice. The second example, YPt2B, is also confined to the SM, which is a minor weakness for the claim that this applies broadly.\n\nThis is a worthwhile contribution for the topological phonon and bosonic Weyl community. I would bring it to the reading group and would cite it once the surface-state methodology is either verified in the SM or in follow-up work. It deserves a serious referee: the bulk part is solid, and the surface part needs careful checking rather than dismissal.","headline":"A genuinely new triangular Weyl complex in phonons, with a clean bulk symmetry argument for alpha-quartz, but the flashy surface-arc claims rest on a Wannier model that the main text barely documents.","tokens_in":8761,"tokens_out":1879,"would_cite":true,"duration_ms":21234,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"In inversion-breaking trigonal and hexagonal lattices, screw symmetry forces single and double Weyl phonons into a triangular complex whose surface arcs span the whole Brillouin zone.","keywords":["Weyl phonons","double Weyl points","triangular Weyl complex","phonon surface arcs","screw rotational symmetry","chiral charge","alpha-quartz","topological phonons"],"falsifier":"A direct test would be a high-resolution measurement of the 16-19 THz optical phonon branches of $\\alpha$-SiO$_2$ along the $\\Gamma$-$K$ and $\\Gamma$-$A$ lines, for example by inelastic X-ray or neutron scattering. If the crossing at $K$ is not linear, or if the dispersion around $\\Gamma$ is not quadratic in the $k_x$-$k_y$ plane and linear along $k_z$, then the claimed single and double Weyl phonons are not realized. Equivalently, a Wilson-loop calculation on an independently converged phonon model that gives Chern numbers other than $\\pm1$ at $K/H$ and $\\pm2$ at $\\Gamma/A$ would refute the triangular complex.","tokens_in":7760,"feed_emoji":"🔺","tokens_out":13351,"duration_ms":116262,"temperature":0.7,"pith_summary":"Conventional Weyl points are supposed to come in opposite-chirality pairs, but the paper argues that trigonal and hexagonal crystals without inversion symmetry can break that rule. Because a nonsymmorphic screw rotation along the c-axis is combined with time-reversal symmetry, a single Weyl phonon with linear dispersion and Chern number $C = -1$ and a double Weyl phonon with quadratic dispersion and Chern number $C = +2$ are forced to coexist at high-symmetry points, forming a triangular Weyl complex with zero total chiral charge. First-principles phonon calculations on $\\alpha$-SiO$_2$ confirm the pattern at $K$, $H$, $\\Gamma$, and $A$, and surface Green's function calculations show that the surface arcs connect one double Weyl projection to two single Weyl projections and span the entire surface Brillouin zone. The significance is a concrete material where topological phonon surface states cover a full iso-frequency surface with no trivial bulk states in the way.","feed_headline":"Triangular Weyl phonons span the whole surface Brillouin zone","feed_subtitle":"In alpha-SiO2, one double Weyl point links to two single ones and the surface arcs span the full zone.","key_machinery":"The load-bearing object is the nonsymmorphic screw rotation $\\tilde{C}_{3z}=\\{C_{3z}\\,|\\,(0,0,c/3)\\}$ combined with time-reversal symmetry $T$, acting on a two-band effective Hamiltonian $H(\\mathbf{q})=d(\\mathbf{q})\\sigma_+ + d(\\mathbf{q})^*\\sigma_- + f(\\mathbf{q})\\sigma_z$ referenced to the Weyl-point frequency. The symmetry constraint $\\tilde{C}_{3z}H(\\mathbf{q})\\tilde{C}_{3z}^{-1}=H(R_{3z}\\mathbf{q})$ fixes how $d(\\mathbf{q})$ can depend on $q_\\pm=q_x\\pm iq_y$: at $K$ or $H$, the lowest allowed order is linear, $d(\\mathbf{q})=a_+q_+ + a_-q_-$, giving a single Weyl point with Chern number $\\pm1$; at $\\Gamma$ or $A$, the additional constraint from $\\tilde{C}_{3z}T$ forces the quadratic form $d(\\mathbf{q})=b_+q_+^2 + b_-q_-^2$, giving a double Weyl point with Chern number $\\pm2$. The same effective Hamiltonian is then realized in $\\alpha$-SiO2 from a Wannier-interpolated tight-binding phonon Hamiltonian built from second-order interatomic force constants, and surface states are extracted with the iterative Green's function method.","core_discovery":"The central discovery is a symmetry-enforced coexistence of two kinds of Weyl phonons: single Weyl phonons with linear dispersion and chiral charge $C = -1$ at the $K$ and $H$ points, and double Weyl phonons with quadratic in-plane dispersion and chiral charge $C = +2$ at the $\\Gamma$ and $A$ points. In $\\alpha$-SiO$_2$ these four high-symmetry points split the three optical branches 16, 17, and 18 into two triangular complexes, one with a $C=+2$ source at $A$ and $C=-1$ sinks at $K$, the other with opposite charges at $\\Gamma$ and $H$. The paper verifies the chiral charges with Wilson-loop calculations and shows by iterative Green's function calculations that the phonon surface arcs on both (001) and (010) surfaces start at the double Weyl projection and end at the single Weyl projections. Because the Weyl points are pinned to high-symmetry points by the screw symmetry, the arcs cross each half of the surface Brillouin zone and therefore span the entire first surface Brillouin zone.","pith_inferences":["Editorial inference: the symmetry analysis yields a screening recipe: for any noncentrosymmetric trigonal or hexagonal crystal, compute the phonon Chern numbers at Gamma, A, K, and H from first-principles force constants to identify new triangular Weyl materials, no new experiment required.","Editorial inference: because the complex has zero total chiral charge only as a whole, the minimal topologically neutral unit in these space groups is a triangle rather than a pair; other polygons mixing Weyl points of unequal Chern numbers may be constructible under screw rotations of higher order.","Editorial inference: the only-nontrivial-surface-states property suggests a concrete application as a topological phonon waveguide; a finite-element simulation of a phononic crystal engineered with the same screw symmetry would provide a direct test of the surface-arc geometry."],"forward_implications":["Any trigonal or hexagonal noncentrosymmetric material with the same screw-rotation and time-reversal symmetries should host the triangular Weyl complex, so alpha-SiO2 is a first example rather than an isolated case.","Because the surface arcs span the entire first surface Brillouin zone, the topological phonon surface states provide a complete one-way propagation channel across the iso-frequency surface.","The absence of trivial bulk states at the iso-frequency surface means the surface arcs can in principle be detected by surface-sensitive probes without a bulk background.","The same symmetry mechanism should generate triangular Weyl complexes in photonic and phononic crystals, and the reasoning carries over to fermionic systems with identical space-group symmetries."],"supporting_citations":[{"why":"Supplies the no-go theorem that the total chiral charge in a periodic system must vanish, which the triangular complex satisfies by combining a +2 charge with two -1 charges.","marker":"[25]"},{"why":"Extends the no-go theorem and underlies the usual expectation that Weyl points come in opposite-chirality pairs that this paper overcomes.","marker":"[26]"},{"why":"Shows that fourfold or sixfold rotational symmetry can protect quadratic double or cubic triple Weyl points, the higher-order Weyl class used here.","marker":"[27]"},{"why":"Demonstrates that screw rotational symmetry can protect double Weyl points, the key symmetry ingredient for the Gamma and A degeneracies.","marker":"[14]"},{"why":"Provides the Wilson-loop method used to compute the Chern numbers of the single and double Weyl phonons.","marker":"[48]"},{"why":"Supplies the iterative Green's function method used to compute the phonon surface states and surface arcs.","marker":"[50]"},{"why":"Provides the finite-displacement scheme used to obtain the second-order interatomic force constants and phonon dispersions.","marker":"[39]"},{"why":"Supplies the first-principles plane-wave code used for the density-functional calculations of alpha-SiO2.","marker":"[38]"}],"fun_headline_variants":["Triangular Weyl complex yields surface arcs spanning full Brillouin zone","Symmetry-protected triangular Weyl complex with zone-spanning arcs","Single and double Weyl phonons form symmetry-protected triangular complex","Phonon surface arcs from triangular Weyl complex span entire Brillouin zone"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the Wannier-interpolated phonon Hamiltonian built from second-order interatomic force constants faithfully represents the real $\\alpha$-SiO$_2$ phonon spectrum over the whole Brillouin zone, so the predicted surface arcs are real and not artifacts of that interpolation.","fun_headline_variants_meta":{"raw":{"variants":["Triangular Weyl complex yields surface arcs spanning full Brillouin zone","Symmetry-protected triangular Weyl complex with zone-spanning arcs","Single and double Weyl phonons form symmetry-protected triangular complex","Phonon surface arcs from triangular Weyl complex span entire Brillouin zone"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000628,"raw_usage":{"total_tokens":2928,"prompt_tokens":994,"completion_tokens":1934,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":610,"completion_tokens_details":{"reasoning_tokens":1857}},"tokens_in":610,"tokens_out":1934,"duration_ms":14025,"temperature":1.0,"reasoning_tokens":1857,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:10:55.365701+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct test would be a high-resolution measurement of the 16-19 THz optical phonon branches of $\\alpha$-SiO$_2$ along the $\\Gamma$-$K$ and $\\Gamma$-$A$ lines, for example by inelastic X-ray or neutron scattering. If the crossing at $K$ is not linear, or if the dispersion around $\\Gamma$ is not quadratic in the $k_x$-$k_y$ plane and linear along $k_z$, then the claimed single and double Weyl phonons are not realized. Equivalently, a Wilson-loop calculation on an independently converged phonon model that gives Chern numbers other than $\\pm1$ at $K/H$ and $\\pm2$ at $\\Gamma/A$ would refute the triangular complex.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the no-go theorem that the total chiral charge in a periodic system must vanish, which the triangular complex satisfies by combining a +2 charge with two -1 charges."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Extends the no-go theorem and underlies the usual expectation that Weyl points come in opposite-chirality pairs that this paper overcomes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows that fourfold or sixfold rotational symmetry can protect quadratic double or cubic triple Weyl points, the higher-order Weyl class used here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Demonstrates that screw rotational symmetry can protect double Weyl points, the key symmetry ingredient for the Gamma and A degeneracies."},{"cited_title":"Dorner, H","cited_arxiv_id":null,"evidence_quote":"Provides the Wilson-loop method used to compute the Chern numbers of the single and double Weyl phonons."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the first-principles plane-wave code used for the density-functional calculations of alpha-SiO2."}],"review_version":1}