{"id":"0530a7a7-3e35-4e10-abe5-d22af162d235","arxiv_id":"2602.18132","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Quasisymmetries in crystals result either from sublattice-localized wavefunctions that acquire an emergent mirror or inversion, or from selection rules inherited from a nearby high-symmetry point, quantified by a metric ε.","lead":"This paper identifies two physical reasons why crystals can behave as if they have extra symmetries: electron wavefunctions may be localized on a sublattice with higher symmetry, or selection rules may be inherited from a nearby high-symmetry point. The authors quantify this 'quasisymmetry' with a subspace-invariance metric and apply it to Sn/SiC, TMD monolayers, wurtzites, and AgLa.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Inheritance picture's k·p expansion parameter unquantified for AgLa: the assertion that only δk terms contribute to T-path SOC gaps is not established without checking |δk·p/Δ|≪1.","rationale":"I agree with the reader's weakest assumption and focus on it because the inheritance picture is one of the paper's two central claims and AgLa is its only concrete example. The first (emergent symmetry) picture is supported by direct ϵ matrix calculations and orbital character, though it too would benefit from null baselines. The inheritance picture, however, rests entirely on a k·p approximation that is asserted but not quantitatively checked. A correct computation could confirm the approximation, or show that the T-path gaps have a different origin; in either case the paper's conclusions would need adjustment. The reader already issued CONDITIONAL, which matches my assessment: the concern is addressable by a concrete calculation and does not by itself disprove the framework, but it must be resolved before the inheritance picture is accepted as established. I therefore recommend no change to the reader's verdict. Other issues (missing error bars, Wannier-basis ambiguity, GaP intensity factor) are secondary and do not alter this judgement.","tokens_in":23281,"tokens_out":11488,"duration_ms":105546,"concrete_test":"Perform a first-principles k·p analysis for AgLa. Using DFT band structures at R, extract momentum matrix elements p_{mn} and energy denominators Δ_{nm} (e.g., with DFT2kp). For the actual crossing k-points along T and W, compute the dimensionless ratio r = max_{m≠n} |δk·p_{mn}/Δ_{nm}| for the crossing states α,β,γ and all remote bands within a relevant energy window. If r ≪ 1 (e.g., <0.1) for both T and W, the inheritance approximation is valid and the paper's explanation holds qualitatively. If r is not small, the neglected wavefunction admixtures can produce additional SOC matrix elements, invalidating the δk-only conclusion. An independent check: compute the overlap |⟨u_{n,k}|u_{n,k0}⟩|^2 at the crossing points; values near 1 support the approximation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing concern is the unverified validity of the k·p expansion underlying the inheritance picture (Sec. IIB) when applied to AgLa (Sec. IIIB). The central claim that T-path SOC gaps (7–20 meV) are small because only the δk term survives, while W-path gaps (80 meV) involve the p-term, depends on Eq. (9): u_{n,k} ≈ u_{n,k0} + Σ_{m≠n} (δk·p_{mn}/Δ_{nm}) u_{m,k0}. For this approximation to justify evaluating matrix elements at k0 (Eq. 11) and then retaining only the δk term, one needs |δk·p_{mn}/Δ_{nm}| ≪ 1 for all relevant remote states m. The paper never reports values of δk, p_{mn}, or Δ_{nm} for the actual crossing points along T and W. Symmetry at R may force the specific p_{αβ} to vanish, but the wavefunction correction involves all m; if any coupling is not small, additional contributions to ⟨m,k|H_SOC|n,k⟩ arise that could change the predicted δk-only form. The paper's own condition (|δk·p/Δ|≪1) is precisely what is asserted, not shown. Additionally, the text near the end of Sec. IIB states the δk-term can be neglected and then equates the matrix element to the k0 value, while Eqs. (22)–(23) attribute the T-path gaps entirely to δk terms; this inconsistency underscores that the quantitative role of the δk term is not settled. Since AgLa is the only demonstration of the inheritance picture, this gap in support weakens the paper's central taxonomy of two distinct mechanisms.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes that quasisymmetry (QS) in crystals arises from two distinct physical mechanisms: (i) an emergent symmetry due to strong wavefunction localization on a sublattice (mirror for Sn/SiC and TMDs; spatial inversion for wurtzites), and (ii) a quasisymmetry inherited from a nearby high-symmetry point through k·p selection rules (AgLa). The authors introduce a metric ε (Eq. 13) that quantifies how unitarily a proposed QS operator acts within a selected subspace, and they use group-theoretic selection rules plus first-principles/Wannier calculations to show that QS suppresses first-order spin-orbit coupling. They also apply the inversion QS in wurtzites to explain weak A-transition oscillator strengths in GaP, and they reinterpret the quasi-cubic approximation as a special case of the emergent symmetry picture. The central taxonomy is clear and the presentation is generally careful, but the inheritance picture—which is demonstrated only for AgLa—rests on an unquantified k·p expansion, and there is an internal inconsistency in the treatment of the δk term in the SOC matrix elements.","tokens_in":23714,"tokens_out":5116,"duration_ms":55604,"significance":"If established, the two-picture taxonomy gives physical substance to the formal QS framework of Refs. [91-93], turning it into a predictive tool for near-degeneracies, SOC splittings, and optical selection rules. The ε metric offers a practical, quantitative diagnostic that could enable high-throughput QS identification. The paper also usefully reconciles conflicting interpretations of Sn/SiC and connects the quasi-cubic approximation to the QS paradigm. These are meaningful contributions. The computational work is based on standard, well-tested tools (QE, VASP, Wannier90, DFT2kp, IrRep, Qsymm), and the group-theoretic arguments are internally consistent as far as they go. However, the quantitative support is incomplete in places: the AgLa inheritance analysis does not verify the smallness of the k·p expansion parameter, the role of the δk term is stated inconsistently, and the first-order SOC extraction relies on unvalidated Wannier basis compatibility. These gaps need to be addressed before the central claim can be considered fully established.","major_comments":[{"comment":"The inheritance picture relies on the k·p expansion u_{n,k} ≈ u_{n,k0} + Σ_{m≠n} (δk·p_{mn}/Δ_{nm}) u_{m,k0}, requiring |δk·p_{mn}/Δ_{nm}| ≪ 1 for all relevant remote states m. For the AgLa T and W crossings, the paper never reports values of δk, p_{mn}, or Δ_{nm}. Without such numbers, the claim that only the δk term contributes along T (Eqs. 22-23) is not quantitatively established: if any remote-state coupling is not small, additional contributions to ⟨m,k|H_SOC|n,k⟩ arise. Please provide an explicit check of the expansion parameter for the actual crossing points, or state the symmetry reasons that make all such couplings negligible.","section":"Sec. II.B / III.B, Eq. (9)"},{"comment":"The treatment of the δk term is internally inconsistent. The text after Eq. (11) states that the δk-term contribution is 'typically small and can be neglected,' reducing the matrix element to ⟨m,k|H′|n,k⟩ = ⟨m,k0|H′|n,k0⟩. But Sec. III.B explicitly attributes the small T-path gaps to δk terms (Eqs. 22-23) and contrasts them with the p-term that dominates along W. If the δk term is neglected, the first-order T-path matrix element would vanish (because the p-term vanishes by symmetry at R), leaving the gap unexplained. The paper needs to settle whether the δk term is the leading contribution (and then keep it in Eq. (11)) or is negligible (and then explain the T gaps via second-order processes). The current text cannot have both.","section":"Sec. II.B after Eq. (11); Sec. III.B Eqs. (22)-(24)"},{"comment":"The numerical first-order SOC contributions, λ_C^(1) = 1.78 meV (MoS2) and 10.6 meV (WSe2), are obtained from H_SOC ≈ H_FR − H_SR ⊗ σ0, which presupposes that the SR and FR Wannier bases are exactly compatible. The authors acknowledge this difficulty and list methods (num_iter=0, symmetry-adapted, selectively localized Wannier functions), but they do not report any quantitative measure of basis compatibility for the specific systems. The central claim that QS suppresses the first-order SOC (λ_C^(1) significantly below the total λ_C) depends on these numbers. Please provide a compatibility check (e.g., Wannier spreads, band-structure interpolation errors, or a comparison between different localization schemes) and show that the first-order values are stable under the basis choice.","section":"Sec. II.D / Eq. (14) and TMD results in Sec. III.A.2"}],"minor_comments":[{"comment":"The caption reads 'Sns+p_z'; this appears to be a typo for 's+p_z'. Please correct.","section":"Fig. 2 caption"},{"comment":"The ε metric is presented without uncertainty estimates or a null baseline. Since ε is proposed as a practical diagnostic, it would be helpful to know how far the quoted values (e.g., 0.968, 0.95, 0.93, 0.76) are from a meaningful reference, e.g., a random subspace of the same dimension.","section":"Sec. II.C, Eq. (13)"},{"comment":"The manuscript states that 'all scripts and input files used here are available upon reasonable request.' For reproducibility, consider depositing them in a public repository (Zenodo, GitHub, etc.) and including the DOI in the manuscript.","section":"General / Appendix A"},{"comment":"The notation for the k·p expansion is a bit confusing: the central point is κ≡k with q=0, but earlier κ=k+q. Please clarify the hierarchy of k, k0, δk, and q in one place to help the reader.","section":"Sec. II.B"},{"comment":"Equations (22)-(24) are central but the notation '⟨α|V_y|β⟩_R' is not defined explicitly as the matrix element of ∂V/∂y at R; please define V_ν and the subscript R to avoid ambiguity.","section":"Sec. III.B, Eq. (22)-(24)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a serious contribution with a plausible two-picture taxonomy. My main concern is that the inheritance picture, which is the paper's most novel claim, currently rests on an unverified k·p expansion and contains an apparent inconsistency about the δk term. These are fixable within the manuscript's scope by adding quantitative checks and clarifying the approximation. I would support publication after such a revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The two things you should know: this paper gives the first concrete physical mechanisms for quasisymmetry (QS) — emergent from sublattice localization, or inherited from a nearby high-symmetry point — and it introduces a simple metric epsilon that quantifies QS. The emergent picture, supported by Sn/SiC, TMDs, and wurtzites, is convincing and a real step forward. The inheritance picture, demonstrated only on AgLa, is the weak link: the k·p expansion parameter that the argument depends on is never evaluated, and the text even goes back and forth on whether the δk term is neglected or retained. The paper deserves refereeing, but the inheritance half needs quantitative backup before the taxonomy is taken as established.\n\nWhat is actually new: the two-picture taxonomy is not in the prior QS literature; the epsilon metric gives a workable diagnostic; the paper resolves the M_y vs SU(2) disagreement over Sn/SiC by showing M_y and its C3 partners form a sufficient subset of SU(2) in the RL subspace; and the wurtzite quasi-cubic approximation is reinterpreted sensibly as a compatibility-relation-based QS rather than a true group extension. The DFT and group-theory work is careful, and the selection-rule argument for first-order SOC suppression is internally consistent. Credit is due for stating outright that the Wannier-basis compatibility for Sn/SiC could not be guaranteed.\n\nSoft spots, in rough order of importance. First, the inheritance picture. The claim that T-path gaps (7–20 meV) arise from δk-only terms rests on the approximation u_{n,k} ≈ u_{n,k0} + (δk·p/Δ) u_{m,k0}, but the paper never reports δk, p, or Δ for the actual crossing points. Symmetry may make some p vanish, but the wavefunction correction runs over all remote states, so the smallness condition really needs checking. Also, Sec. IIB says the δk contribution “is typically small and can be neglected,” yet Eqs. (22)–(23) in the AgLa section attribute the T-path gaps to δk terms. That is not necessarily a contradiction, but it is at least a presentation inconsistency that has to be resolved. Second, the GaP optical suppression statement is arithmetically off: Eq. (21) shows the dipole matrix element scales as sqrt(epsilon(1-epsilon)) ≈ 0.3, but the text calls that a ~70% suppression of intensity. Intensity goes as the square, so the suppression is ~90%. Same conclusion, wrong number. Third, epsilon values have no error bars and no null baseline; reporting 0.95 without a comparison to, say, a random subspace makes the metric hard to calibrate. Fourth, the SOC extraction via H_FR − H_SR rests on Wannier-basis compatibility, which the authors themselves note fails for Sn/SiC; for the TMD numbers, a convergence study would be useful. Fifth, there is no data or code repository — “available upon reasonable request” is not enough for a methods paper.\n\nNone of these flaws sink the central qualitative claim. The emergent picture is solid, and the inheritance picture is plausible but unproven. The paper will be of real value to people working on quasisymmetry, spin splittings, and high-throughput symmetry diagnosis. I would cite it, and I would bring it to a reading group. A serious editor should send this to peer review; the referees should push for the missing k·p parameter check and the arithmetic fix, but the core idea is worth engaging with.","headline":"The paper's two-picture taxonomy for quasisymmetry is genuinely useful, and the emergent-symmetry half is well supported; the inheritance half is quantitatively underbuilt and needs more work before the framework is advertised as predictive.","tokens_in":24203,"tokens_out":3285,"would_cite":true,"duration_ms":33378,"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":"Two physical mechanisms generate quasisymmetry in crystals: emergent symmetry from sublattice localization and symmetry inherited from nearby high-symmetry points, supported by first-principles and k·p analysis.","keywords":["quasisymmetry","emergent symmetry","spin-orbit coupling","k·p theory","first-principles calculation","wurtzite semiconductors","transition metal dichalcogenides","sublattice localization"],"falsifier":"For AgLa, compute the full spin-orbit matrix elements along the T path from a method that includes core-state contributions and compare with the δk-only expression. If the p-term contribution is comparable to or larger than the δk-term, or if the T-path gap does not shrink toward R, the inheritance picture is falsified.","tokens_in":23178,"feed_emoji":"⚛️","tokens_out":6663,"duration_ms":70389,"temperature":0.7,"pith_summary":"This paper argues that the puzzling near-symmetries seen in several crystals—dubbed quasisymmetries—are not accidental, but come from two identifiable physical sources. In one picture, when electronic wavefunctions sit almost entirely on one sublattice, that sublattice's own extra mirror or inversion symmetry acts on the states, enlarging the effective symmetry group and forbidding first-order spin-orbit couplings. In the other picture, states at a generic wavevector inherit selection rules from a nearby high-symmetry point through the k·p expansion, so small spin-orbit gaps along certain paths are remnants of that point's stricter symmetry. The authors quantify how good a quasisymmetry is with a single number, the metric epsilon, which measures how unitarily the proposed symmetry acts inside the relevant subspace. If correct, the work turns a classification scheme into a predictive tool for engineering spin splittings, optical selection rules, and near-degeneracies.","feed_headline":"Two mechanisms generate quasisymmetry in crystals","feed_subtitle":"Sublattice-localized wavefunctions and nearby high-symmetry points create near-symmetries that control spin-orbit gaps.","key_machinery":"The central object is the quasisymmetry operator Q acting within a low-energy subspace A of band states, together with the metric ε = sqrt(Tr[P_A Q† P_A Q]/N_A), which equals unity exactly when A is invariant under Q. In the emergent-symmetry picture, Q is a mirror or inversion that leaves the sublattice invariant, and the effective group becomes G_k ⊗ {E, Q}. In the inheritance picture, the k·p approximation u_{n,k} ≈ u_{n,k0} transfers the symmetry constraints of the high-symmetry point k0 to nearby k, making the δk term in the spin-orbit matrix element the only allowed contribution. Perturbative partitioning of the Hamiltonian is used to show that when Q forbids the first-order term, the","core_discovery":"On the paper's own terms: quasisymmetries are exact symmetries of a selected subspace of electronic states, not of the whole crystal. For Sn/SiC and transition-metal dichalcogenide monolayers the subspace is invariant under an emergent mirror (M_y) that the crystal group lacks; for wurtzites it is invariant under spatial inversion. In each case the measured metric epsilon is close to 1 (0.76–0.99), confirming near-exact invariance, and the augmented quasisymmetry group turns an allowed first-order spin-orbit coupling into a forbidden one, pushing the splitting to second order. For AgLa, no such emergent symmetry exists along the T and W paths; instead the crossing states at generic k-points","pith_inferences":["The two pictures are not mutually exclusive; a sublattice-localized state sitting near a high-symmetry point could exhibit both an enhanced ε and inherited selection rules. The paper does not analyze this combined regime, but the machinery it develops would apply.","One could turn ε into a high-throughput descriptor: compute the Q-matrix for candidate materials and rank them by deviation from unity, much as pseudosymmetry searches rank crystal structures—except the search would target band subspaces rather than lattices.","Editorial note: the paper states that for Sn/SiC the basis-set consistency needed for the first-order SOC calculation could not be guaranteed, so the quantitative suppression claim there rests on the TMD examples and on the wurtzite symmetry arguments.","A direct testable extension: if the inheritance picture holds, the T-path gaps in AgLa should decrease monotonically as the path approaches R (δk→0); this could be checked by tracking gap size along the full T path in a single calculation."],"forward_implications":["Near-degenerate gaps that look accidental become explainable: when a quasisymmetry forbids the first-order spin-orbit term, the gap is set by second-order processes, so smallness is a symmetry consequence, not a fine-tuned accident.","The metric ε gives a single computable number that can be used to screen materials for quasisymmetry from first-principles data alone.","In the inheritance picture, avoided-crossing gaps along generic paths scale with the distance δk to the high-symmetry point, so band-structure data along a path can directly indicate which picture applies.","For wurtzites, inversion quasisymmetry does more than the quasi-cubic approximation sometimes fails to: it suppresses the A-transition in GaP by about 70%, matching experiment.","Because the quasisymmetry group is a direct product extension, the framework applies to any perturbation—strain, electric fields, optical transitions—not only spin-orbit coupling."],"fun_headline_variants":["Quasisymmetry's two faces: emergent or inherited","Two origins for quasisymmetry in crystals","How quasisymmetry arises: emergent mirror vs inherited","Crystal quasisymmetry: sublattice-local or inherited","Two mechanisms behind quasisymmetry in materials"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The central claims rest on approximations the paper does not fully verify quantitatively: for AgLa, that the k·p correction is small enough for the δk term to dominate, and for Sn/SiC, that the scalar- and full-relativistic basis sets are consistent enough to compute first-order SOC; if either fails, the corresponding quasisymmetry explanation loses its quantitative footing.","fun_headline_variants_meta":{"raw":{"variants":["Quasisymmetry's two faces: emergent or inherited","Two origins for quasisymmetry in crystals","How quasisymmetry arises: emergent mirror vs inherited","Crystal quasisymmetry: sublattice-local or inherited","Two mechanisms behind quasisymmetry in materials"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000987,"raw_usage":{"total_tokens":4012,"prompt_tokens":726,"completion_tokens":3286,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":470,"completion_tokens_details":{"reasoning_tokens":3208}},"tokens_in":470,"tokens_out":3286,"duration_ms":22155,"temperature":1.0,"reasoning_tokens":3208,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T05:57:36.044192+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For AgLa, compute the full spin-orbit matrix elements along the T path from a method that includes core-state contributions and compare with the δk-only expression. If the p-term contribution is comparable to or larger than the δk-term, or if the T-path gap does not shrink toward R, the inheritance picture is falsified.","supporting_citations":[],"review_version":1}