{"id":"3f70e4c8-e1e0-4cc0-8c57-0eae7cc66a18","arxiv_id":"1908.00901","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Six trigonal space groups enforce accordion-like Weyl points and three enforce Weyl nodal lines; tellurium and Te16Si38 are candidate realizations.","lead":"The paper classifies which trigonal crystal symmetries force electron bands to cross, identifying Weyl points and nodal lines that must exist in any material with those space groups. It then finds real materials, most notably tellurium, where these crossings sit near the Fermi energy and could be studied in experiments.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The paper's central claim is a group-theoretic existence statement: the nonsymmorphic symmetries force band crossings regardless of material details. I read the derivation as internally consistent. Equation (2.2) gives three C3 eigenvalue branches whose Kramers pairings at Γ and A differ in a p-dependent way (Table II), forcing a connected group of six bands with at least two crossings on Γ–Δ–A. The additional C2 screws in SGs 151–154 conjugate the C3 eigenvalue in the same way as time reversal, so they do not change the pairings and do not remove the crossings. For the nodal lines, the glide eigenvalues ±i e^{-ikz/2} flip the Kramers-partner assignment between the kz=0 and kz=π TRIMs, which forces crossings on any path connecting blue to red TRIMs. I found no internal inconsistency in these band-connectivity arguments. The reader's concern about DFT-level Fermi energies is real but secondary: the paper itself reports a 1.1 eV mBJ gap versus a 1.78 eV experimental gap for Cu2SrSnS4, and a 2.8 eV gap for Ag2HPO4, so 'near the Fermi energy' in the abstract should be read as a DFT-level statement. That caveat does not weaken the symmetry-enforced existence claim, so the verdict should remain unchanged.","tokens_in":20809,"tokens_out":37847,"duration_ms":364321,"concrete_test":"Use the Bilbao Crystallographic Server to compute the double-valued irreps at Γ, Δ, and A for SGs 144, 145, 151–154 and at Γ, M, D, A, L for SGs 158, 159, 161, then verify that the compatibility relations reproduce the connectivity diagrams in Figs. 1 and 4 and that no C2-screw-induced 4D irrep at A or Z removes the enforced crossings.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection to the central claim. The paper's proof that C3,p screws enforce Weyl crossings on Γ–Δ–A (Eq. 2.2, Table II) and that glide mirrors enforce nodal lines (Eq. 3.1) is internally consistent; the extra C2 screws in SGs 151–154 conjugate C3 eigenvalues in the same way as time reversal, so they do not alter the endpoint pairings. The only genuine caveat is the abstract's 'near the Fermi energy' claim for candidate materials, which rests on PBE/mBJ without quasiparticle corrections; the paper itself reports a 1.1 eV mBJ gap versus 1.78 eV experimental for Cu2SrSnS4 and a 2.8 eV gap for Ag2HPO4. This affects the illustrative material list, not the symmetry-enforced existence or location of the crossings.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper classifies nonsymmorphic band crossings in trigonal space groups with strong spin-orbit coupling. Using the kz-dependence of screw-rotation and glide eigenvalues together with compatibility relations of double-valued irreps, it argues that nonmagnetic materials in SGs 144, 145, 151, 152, 153, and 154 must exhibit Weyl points with accordion-like dispersion along the Γ–Δ–A line, while SGs 158, 159, and 161 must exhibit Weyl nodal lines in glide-mirror-invariant planes. It additionally derives filling constraints and reports DFT band structures for Cu2SrSnS4, Cu2SrGeS4, Ag2HPO4, Te, Se, and Te16Si38 that reproduce the predicted connectivity, including surface-state calculations for Ag2HPO4 and Te.","tokens_in":20913,"tokens_out":16463,"duration_ms":159877,"significance":"If correct, the central theorem is a strong and falsifiable result: the crossings are mandatory for any nonmagnetic material in the listed space groups, and their location in the Brillouin zone is fixed by symmetry independent of chemistry. The derivation is parameter-free and uses standard external irrep tables, and the DFT band structures independently reproduce the predicted connectivity in every candidate material, including surface arc states for Ag2HPO4 and Te. This is a clean extension of the hexagonal classification of Ref. [38] to trigonal systems and provides concrete targets for experimental searches. The material-realization claim that these crossings are 'near the Fermi energy' is, however, overstated for the large-gap compounds and depends on DFT accuracy.","major_comments":[{"comment":"The abstract and conclusions state that the identified materials realize the crossings 'near the Fermi energy', but for the gapped compounds this is not supported by the numbers in the text. In Sec. V A a the mBJ gap of Cu2SrSnS4 is quoted as about 1.1 eV, Appendix A says about 1.5 eV, and the experimental gap is 1.78 eV, so the symmetry-enforced Weyl points along Γ–Δ–A are at least about 1 eV away from E_F. The paper itself notes that probing these crossings is 'very challenging'. Please either qualify the abstract to say that the crossings are symmetry-enforced and material-independent while their proximity to E_F is a per-material quantitative question, or provide a consistent energy scale showing that the crossings lie within a few tens of meV of E_F.","section":"Abstract; Sec. V A (Cu2SrSnS4, Cu2SrGeS4)"},{"comment":"The database search is reported in a single sentence ('We look for suitable compounds in the ICSD database, the AFLOW database, and the Materials Project database, which yields four materials with accordion Weyl points and one material with Weyl nodal lines'), with no search criteria, no list of candidate compounds in the relevant space groups, no count of false positives, and no explanation for why SGs 153, 154, 158, and 159 have no examples. This makes the materials-realization part of the abstract non-reproducible. Please add a table or appendix describing the search workflow, the number of candidates per SG, and the screening conditions used to select the five reported compounds.","section":"Sec. V A, first paragraph"},{"comment":"The claimed minimal Weyl-point multiplicities (four for SGs 144, 145, 151, and 153; twelve for SGs 152 and 154) are supported only by an explicit construction of a symmetry-allowed configuration; no lower-bound proof is given that rules out smaller configurations, for example a four-point configuration in SG 152. Since the abstract and Table I do not state multiplicities, the central existence claim does not depend on this, but the text should either label these values as constructive counts that are consistent with symmetry and the fermion-doubling theorem, or provide a rigorous minimality argument.","section":"Sec. II C (Fig. 2)"}],"minor_comments":[{"comment":"The statement that 'out of the eighteen trigonal space groups with hexagonal lattice system (P-trigonal) there are only six space groups without inversion symmetry' is incorrect, because SGs 149, 150, 156, and 157 are also non-centrosymmetric. The intended statement is that only six of these have a nonsymmorphic threefold screw and lack inversion; please rephrase.","section":"Sec. II A, paragraph 4"},{"comment":"Tellurium is called metallic in Sec. V A c ('Since tellurium is metallic...'), while Appendix B describes surface states that 'lie within the bulk band gap'. Please clarify whether Te is a small-gap semiconductor or a metal in the PBE calculation and reconcile the wording.","section":"Sec. V A c vs Appendix B"},{"comment":"The reported gap of Cu2SrSnS4 is inconsistent: 1.1 eV in Sec. V A a, 1.5 eV in Appendix A, and 1.78 eV as the experimental value. State clearly which functional gives which value and how the experimental comparison is made.","section":"Sec. V A a; Appendix A"},{"comment":"The VASP calculations are not described with sufficient detail (pseudopotentials, energy cutoff, k-point sampling, convergence criteria); providing these parameters, perhaps in an appendix, would improve reproducibility.","section":"Sec. V A (computational methods)"},{"comment":"There are several typos that should be corrected in proof: 'Nonsymmoprhic' in the abstract, 'derivation form' in Sec. II B, 'exibits' in the caption of Fig. 10, and 'crystalizes' in Sec. V A c.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The central symmetry-enforcement proof is sound and the paper is within the journal's scope. I recommend major revision mainly to correct the overstated near-Fermi claim and to make the database search reproducible; both are fixable without changing the main theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read it. The central result is the exhaustive classification: among the 25 trigonal space groups, six enforce accordion Weyl points on Γ–Δ–A and three enforce glide-mirror Weyl nodal lines, for any nonmagnetic, strong-SOC material. That is new and, as far as I can tell, correct. The p=1 versus p=2 Kramers pairing distinction (Table II) is genuinely useful group-theoretic bookkeeping, and the compatibility-relation derivation in Secs. II–III is consistent with the eigenvalue argument. The filling constraints also match known results, which is a good sanity check. I agree with the reader's verdict: ACCEPT, with no critical flaw in the band-connectivity logic.\n\nCredit where due: the paper does not hand-wave the symmetry part. The irreps at Γ, Δ, and A for SG 144, and the glide-mirror cases 158, 159, and 161, are laid out concretely. The DFT band structures reproduce the forced connectivity in all the claimed candidates, which is exactly the right role for computation here (illustration, not proof). The Te surface-state calculation is a nice bonus.\n\nSoft spots, in order of seriousness. First, the material-realization claim in the abstract is overstated. Cu2SrSnS4 has a ~1.1 eV mBJ gap versus 1.78 eV experimental, and the paper itself says the Weyl points are \"very challenging\" to probe; Ag2HPO4 is called \"experimentally impossible.\" So \"realized near the Fermi energy\" should be read as \"occur in the band structure of existing compounds,\" not \"sit at the Fermi level.\" That is a framing fix, not a scientific error, but it should be fixed. Second, the database screening is underreported: one sentence says ICSD, AFLOW, and Materials Project were searched and yielded four materials, with no search criteria, hit lists, or exclusion counts. For a paper whose pitch includes \"database search,\" that is thin. Third, the minimal Weyl-point multiplicity arguments in Sec. II.C are constructive rather than fully proven: they show a configuration that satisfies symmetry and chirality cancellation, but not that no smaller configuration exists. I think the claim is right, but it is asserted more strongly than it is demonstrated. Fourth, Se's spin-orbit splitting is only ~50 meV, so the accordion crossings there are borderline; the paper does acknowledge this.\n\nThe citation pattern is fine: Ref. 38 is a methodological self-citation, and the trigonal results are not assumed there. The literature context, including Refs. 45–47, is properly positioned.\n\nWho is this for? People working on nonsymmorphic semimetals and on cataloguing enforced crossings will want this table. It is a genuinely citable classification. It deserves a serious referee; I would send it to review, and I would expect revision on the reporting of the materials screening and the Fermi-level language rather than on the core math.","headline":"A clean symmetry-forced crossing inventory for trigonal space groups, with DFT backing for candidate materials; the classification is the contribution, not the Fermi-level claims.","tokens_in":21467,"tokens_out":1965,"would_cite":true,"duration_ms":20089,"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 nonmagnetic trigonal crystals with strong spin-orbit coupling, six space groups force accordion Weyl points and three force Weyl nodal lines, independent of chemistry.","keywords":["symmetry-enforced band crossings","nonsymmorphic symmetries","Weyl points","Weyl nodal lines","trigonal space groups","spin-orbit coupling","accordion dispersion","topological semimetals"],"falsifier":"Compute the band structure of any nonmagnetic, strong-spin-orbit material in space group 144 along $\\Gamma$--$\\Delta$--$A$: if the bands do not form connected groups of six with at least two crossings, the enforcement claim fails; for SG 158, a mirror plane with no nodal line separating the two types of time-reversal-invariant momenta would be an equally decisive counterexample.","tokens_in":20647,"feed_emoji":"⚛️","tokens_out":16134,"duration_ms":131854,"temperature":0.7,"pith_summary":"This paper classifies the band crossings that nonsymmorphic crystal symmetries force in nonmagnetic trigonal materials with strong spin-orbit coupling. It shows that six of the 25 trigonal space groups (Nos. 144, 145, 151, 152, 153, 154) necessarily produce Weyl points with an accordion-like dispersion on the threefold-screw-invariant line Γ–Δ–A, and three space groups (Nos. 158, 159, 161) necessarily produce Weyl nodal lines in glide-mirror-invariant planes. Because the crossings are enforced by symmetry rather than by material details, they occur in every band of any material in these space groups; only their energy relative to the Fermi level is material dependent. The paper identifies existing compounds — tellurium, selenium, Cu2SrSnS4, Cu2SrGeS4, and Ag2HPO4 for the Weyl points, and Te16Si38 for the nodal lines — and confirms the predicted connectivity with density-functional band structures and surface-state calculations.","feed_headline":"Six space groups force accordion Weyl points; three force nodal lines","feed_subtitle":"Screw rotations and glide mirrors fix the crossings' location, so every crystal in these space groups hosts them.","key_machinery":"The central machinery is the momentum dependence of nonsymmorphic symmetry eigenvalues. For the screw rotation $C_{3,p}:(x,y,z)\\to(-y,x-y,z+p/3)$, the eigenvalue is $e^{i\\pi(2m+1)/3}e^{-ipk_z/3}$ with $m=0,1,2$; because the Kramers pairing changes between $\\Gamma$ and $A$, the six-band group along $\\Gamma$--$\\Delta$--$A$ cannot avoid crossings. For the glide mirrors $M$ with $M^2=-\\hat{z}$, eigenvalues $\\pm i e^{-ik_z/2}$ give $\\pm1$ at some time-reversal-invariant momenta and $\\pm i$ at others, and the forced switching of Kramers partners generates nodal lines in the mirror plane. Compatibility relations between double-valued irreducible representations (the symmetry labels appropriate for spinful electrons) at high-symmetry points and lines prove the same connectivity and show why the effect disappears when inversion or symmorphic rotations are present, which is why only the six and three space groups in Table I qualify.","core_discovery":"On the paper's own terms, the central discovery is a systematic classification of nonsymmorphic band degeneracies in trigonal space groups under time-reversal symmetry and strong spin-orbit coupling. Along the $\\Gamma$--$\\Delta$--$A$ line, the eigenvalue of the threefold screw rotation $C_{3,p}$ winds with $k_z$; the Kramers pairs at $\\Gamma$ and $A$ pair different eigenvalue labels (for $p=1$: $(0,0)$ with $(1,2)$; for $p=2$: $(0,1)$ with $(2,2)$), so the six bands in the connected group cannot be disentangled and must cross at least twice, producing Weyl points with accordion dispersion. In the glide-mirror planes of SGs 158, 159, and 161, the mirror eigenvalue is $\\pm i e^{-ik_z/2}$; at some time-reversal-invariant momenta the Kramers partners share the same eigenvalue while at others they have opposite eigenvalues, so any path connecting the two types forces a crossing, and the crossings form Weyl nodal lines. The paper further derives filling constraints ($6\\mathbb{N}$ for the screw-rotation groups, $4\\mathbb{N}$ for the glide-mirror groups) and identifies tellurium, selenium, Cu$_2$SrSnS$_4$, Cu$_2$SrGeS$_4$, Ag$_2$HPO$_4$, and Te$_{16}$Si$_{38}$ as materials realizing these crossings, with surface arcs and drumhead states following from Chern numbers and $\\pi$-Berry phases.","pith_inferences":["The classification can be used as a screening rule: any compound found in one of these nine space groups automatically qualifies as a candidate for symmetry-enforced Weyl physics, and the only material-specific question is where the Fermi energy sits.","Because the near-Fermi claim is not protected by symmetry, the wide-gap candidates (Cu$_2$SrSnS$_4$, Cu$_2$SrGeS$_4$, Ag$_2$HPO$_4$) should be re-examined with quasiparticle or hybrid-functional calculations; if those shift band energies by a few hundred meV, these compounds remain topological but are no longer near-Fermi examples.","The same eigenvalue-pairing logic may predict enforced crossings in other crystal families, and the contrast with hexagonal systems suggests a general criterion based on how nonsymmorphic operations act on time-reversal-invariant momenta.","A direct experimental check of the 'in all bands' statement would be angle-resolved photoemission on a cleaved tellurium surface: the predicted arc states from both the H-point Weyl points and the accordion Weyl points should cross the entire surface Brillouin zone between valence and conduction bands."],"forward_implications":["Any nonmagnetic material with strong spin-orbit coupling in space groups 144, 145, 151, 152, 153, or 154 must contain accordion Weyl points along $\\Gamma$--$\\Delta$--$A$; no symmetry-preserving perturbation can remove them.","Any material in space groups 158, 159, or 161 must contain Weyl nodal lines in the glide-mirror planes, separating time-reversal-invariant momenta of opposite glide-mirror eigenvalue type.","The crossings imply observable topological surface states: arc states for the Weyl points and drumhead states for the nodal lines, tied to nonzero Chern numbers and a $\\pi$-Berry phase.","Band insulators in these space groups are allowed only at specific electron fillings — $6\\mathbb{N}$ for the accordion Weyl groups and $4\\mathbb{N}$ for the nodal-line groups — so other fillings force metallic behavior.","Density-functional calculations for tellurium, selenium, Cu$_2$SrSnS$_4$, Cu$_2$SrGeS$_4$, Ag$_2$HPO$_4$, and Te$_{16}$Si$_{38}$ show the predicted crossings in their calculated band structures."],"supporting_citations":[{"why":"Establishes the eigenvalue-evolution and compatibility-relation method for hexagonal analogues that this paper extends to trigonal space groups.","marker":"[38]"},{"why":"Supplies the double-valued irreducible representations and compatibility relations used to prove band connectivity for SGs 144 and 158.","marker":"[50]"},{"why":"Provides the representation-theory framework for forming time-reversal-invariant Kramers pairs from complex irreps.","marker":"[51, 52]"},{"why":"Gives the fermion doubling theorem used to fix the minimal number and chirality balance of Weyl points.","marker":"[53]"},{"why":"Derives filling constraints for nonsymmorphic space groups, with which the paper's $6\\mathbb{N}$ and $4\\mathbb{N}$ constraints agree.","marker":"[44]"},{"why":"Prior first-principles identifications of Weyl points at the H point in tellurium, which the paper extends to the accordion Weyl points on the $\\Gamma$--$\\Delta$--$A$ line.","marker":"[75, 76]"}],"fun_headline_variants":["Trigonal symmetry forces accordion Weyl points and nodal lines","Accordion Weyl points and nodal lines from nonsymmorphic trigonal symmetry","Screw rotations and glides dictate Weyl crossings in trigonal crystals","Accordion Weyl points in tellurium and Cu2SrSnS4, nodal lines in clathrate"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The protection argument assumes the crystal truly has the listed nonsymmorphic space group with time-reversal symmetry and strong spin-orbit coupling; the near-Fermi claim assumes the density-functional band energies are accurate, which the paper does not check with higher-level corrections.","fun_headline_variants_meta":{"raw":{"variants":["Trigonal symmetry forces accordion Weyl points and nodal lines","Accordion Weyl points and nodal lines from nonsymmorphic trigonal symmetry","Screw rotations and glides dictate Weyl crossings in trigonal crystals","Accordion Weyl points in tellurium and Cu2SrSnS4, nodal lines in clathrate"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000183,"raw_usage":{"total_tokens":1397,"prompt_tokens":1108,"completion_tokens":289,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":724,"completion_tokens_details":{"reasoning_tokens":199}},"tokens_in":724,"tokens_out":289,"duration_ms":3119,"temperature":1.0,"reasoning_tokens":199,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:30:01.409008+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the band structure of any nonmagnetic, strong-spin-orbit material in space group 144 along $\\Gamma$--$\\Delta$--$A$: if the bands do not form connected groups of six with at least two crossings, the enforcement claim fails; for SG 158, a mirror plane with no nodal line separating the two types of time-reversal-invariant momenta would be an equally decisive counterexample.","supporting_citations":[{"cited_title":"Zhang, Y.-H","cited_arxiv_id":null,"evidence_quote":"Establishes the eigenvalue-evolution and compatibility-relation method for hexagonal analogues that this paper extends to trigonal space groups."},{"cited_title":"Elcoro, B","cited_arxiv_id":null,"evidence_quote":"Supplies the double-valued irreducible representations and compatibility relations used to prove band connectivity for SGs 144 and 158."},{"cited_title":"Watanabe, H","cited_arxiv_id":null,"evidence_quote":"Derives filling constraints for nonsymmorphic space groups, with which the paper's $6\\mathbb{N}$ and $4\\mathbb{N}$ constraints agree."}],"review_version":1}