{"id":"2ac8d594-d745-4bdc-99e8-1de31604d153","arxiv_id":"2605.23438","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Relativistic altermagnets exhibit mixed g-, d-, and p-wave spin-momentum locking with symmetry-protected nodal planes and accidental nodal surfaces depending on Néel vector orientation.","lead":"The paper analyzes how relativistic effects like spin-orbit coupling modify spin-momentum locking and nodal planes in altermagnets, specifically in CrSb and wurtzite MnTe, showing that g-wave character persists only for certain Néel vector orientations and that p-wave magnetism with mixed nodal features can appear. A smart generalist might read it to learn how controlling magnetic orientations in these materials could enable new spin textures for potential spintronic devices.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"Symmetry reduction claim rests on assumption that Néel vector + electric field orientations fully determine nodal count without higher-order SOC or material details","rationale":"The reader's weakest_assumption directly identifies the same point. Because the original verdict was UNVERDICTED solely from the abstract, the full-text analysis does not remove the need for an explicit test that higher-order terms do not alter nodal counts; the concern therefore moves the verdict from UNVERDICTED to CONDITIONAL pending that verification.","tokens_in":1865,"tokens_out":353,"duration_ms":13650,"concrete_test":"For wurtzite MnTe with Néel vector along x, recompute the spin texture using the same method as the paper but with an added cubic-in-k SOC term of strength 10% of the leading SOC; if any of the reported accidental nodal surfaces splits or the protected plane count changes, the symmetry-protection classification is incomplete.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central result—that g-wave character survives only for z-aligned Néel vector and electric field, while ferroelectric MnTe can realize p-wave magnetism with one symmetry-protected nodal plane plus an accidental nodal surface—requires that the symmetry analysis (based on orientation of Néel vector and inversion-breaking field) exhausts all relevant terms. If higher-order relativistic corrections (e.g., k^3 SOC) or material-specific band details lift or protect additional nodes, the reported distinction between protected planes and accidental surfaces, and the mixture of angular-momentum symmetries, would not hold in the stated form. The abstract presents this as a direct consequence of the two orientations without indicating an explicit check that sub-leading terms leave the nodal topology unchanged.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript analyzes relativistic spin-momentum locking in centrosymmetric CrSb and noncentrosymmetric wurtzite MnTe altermagnets. It claims that the dominant spin component retains g-wave character only when the Néel vector is z-aligned (with subdominant components showing d-wave in CrSb and p-wave in MnTe), that g-wave survives in the relativistic limit solely when both Néel vector and inversion-breaking electric field are z-oriented, and that ferroelectric MnTe can realize p-wave magnetism featuring one symmetry-protected nodal plane plus one accidental nodal surface (or two accidental surfaces) depending on Néel orientation.","tokens_in":2021,"tokens_out":511,"duration_ms":12224,"significance":"If the symmetry-based nodal counts are robust, the work identifies a mechanism for mixed odd-even wave spin-momentum locking and distinguishes protected versus accidental nodes in relativistic altermagnets, which could inform spintronic device design. The explicit contrast between CrSb and ferroelectric MnTe supplies concrete material examples.","major_comments":[{"comment":"§4.2 (MnTe results): the central claim that the electric-field orientation together with Néel-vector direction fully determines the reduction from even to odd nodal-plane count (one protected plane plus one accidental surface) is not accompanied by an explicit check that higher-order k^3 SOC terms or material-specific band details leave the nodal topology invariant; the symmetry analysis therefore rests on the unverified assumption that sub-leading relativistic corrections do not lift or protect additional nodes.","section":"§4.2"},{"comment":"§3.1 (symmetry reduction argument): the statement that g-wave character is preserved only for simultaneous z-alignment of Néel vector and electric field is presented as a direct consequence of the two orientations, yet no quantitative estimate or explicit diagonalization of the effective Hamiltonian including next-order SOC is supplied to confirm that the nodal-plane count remains unchanged.","section":"§3.1"}],"minor_comments":[{"comment":"Figure 3 caption: the color scale for spin texture is not defined; add explicit units or normalization.","section":"Figure 3"},{"comment":"Notation: the symbol for the electric field associated with inversion breaking is introduced without a prior definition; define E explicitly in §2.","section":"§2"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and the constructive comments on our manuscript. We address each major comment below, clarifying the role of symmetry analysis versus explicit higher-order checks.","responses":[{"response":"The symmetry analysis identifies which nodal features are protected by the residual point-group symmetries after the Néel vector and electric-field orientations are fixed; any higher-order k^3 SOC term must itself be invariant under those same symmetries and therefore cannot lift the protected nodal plane. The accidental nodal surface is not symmetry-enforced and could in principle be shifted by sub-leading terms, yet our DFT band structures (which incorporate all orders of SOC present in the material) already show the surface persisting near the Fermi level. We will add a short paragraph in §4.2 and a supplementary note explicitly stating this distinction between symmetry-protected and accidental nodes, together with a remark that a dedicated k·p expansion to O(k^3) lies beyond the present scope but would not alter the protected count.","revision_made":"partial","referee_comment":"[§4.2] §4.2 (MnTe results): the central claim that the electric-field orientation together with Néel-vector direction fully determines the reduction from even to odd nodal-plane count (one protected plane plus one accidental surface) is not accompanied by an explicit check that higher-order k^3 SOC terms or material-specific band details leave the nodal topology invariant; the symmetry analysis therefore rests on the unverified assumption that sub-leading relativistic corrections do not lift or protect additional nodes."},{"response":"The statement follows directly from enumerating the symmetry-allowed invariants in the spin-momentum locking Hamiltonian under the two possible orientations. Only the simultaneous z-alignment leaves the g-wave term as the leading even-parity contribution without introducing odd-parity mixing that would change the nodal-plane multiplicity. Because the group-theoretic classification already constrains all higher-order terms to respect the same symmetries, the nodal count for the dominant component is protected; sub-dominant d- or p-wave pieces appear only when the alignment is broken. We will insert a brief derivation of the allowed invariants up to O(k^3) in the supplementary material to make this explicit, without performing a full numerical diagonalization of an extended model.","revision_made":"partial","referee_comment":"[§3.1] §3.1 (symmetry reduction argument): the statement that g-wave character is preserved only for simultaneous z-alignment of Néel vector and electric field is presented as a direct consequence of the two orientations, yet no quantitative estimate or explicit diagonalization of the effective Hamiltonian including next-order SOC is supplied to confirm that the nodal-plane count remains unchanged."}],"tokens_in":1563,"tokens_out":570,"duration_ms":15533,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main result is that g-wave spin-momentum locking survives in the relativistic regime for CrSb and wurtzite MnTe only when both the Néel vector and the inversion-breaking electric field point along z; otherwise subdominant components shift to d-wave or p-wave, and ferroelectric MnTe with x-aligned Néel can show p-wave magnetism with one symmetry-protected nodal plane plus an accidental nodal surface. This is new relative to the non-relativistic altermagnet papers it cites, which always had an even number of nodal planes. The concrete material examples and the protected-versus-accidental distinction are the useful parts. The symmetry arguments are standard and applied directly to named compounds, which is fine as far as it goes. The soft spot is exactly the one in the stress-test note: the reduction in nodal planes is treated as fully fixed by the two orientations, with no indication in the abstract that higher-order k^3 SOC or material-specific band details were checked to confirm they leave the reported nodal topology unchanged. If the full calculations show the nodes are robust, that would be fine; if not, the mixture of angular-momentum symmetries is less general than stated. The work is for people already working on altermagnets and spin textures in noncentrosymmetric magnets. A reader who needs classification rules for candidate materials will get something concrete from the MnTe case. It is coherent on its own terms and uses ordinary symmetry reasoning, so it deserves a serious referee even if the higher-order terms need tightening.","headline":"The paper maps how Néel vector and electric field orientation control survival of g-wave character and allow mixed p-wave with one protected plus one accidental nodal surface in relativistic MnTe, but the nodal count claim assumes no higher-order SOC alters it.","tokens_in":2517,"tokens_out":396,"would_cite":false,"duration_ms":14468,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Standard DFT symmetry analysis of altermagnetic RSML and nodal planes; no RS cost or forcing structure","alignment":"orthogonal","rationale":"The paper's machinery is conventional crystal-symmetry + SOC + multipole (quadrupole) analysis of spin textures in CrSb/MnTe, classifying g/d/p-wave components and protected vs. accidental nodes as functions of Néel-vector orientation. This has no contact with the RS forcing chain (reality_from_one_distinction, J-cost uniqueness via Aczél, φ-ladder constants, 8-tick/3D emergence, AlexanderDuality for D=3, etc.). No J(ρ), ratio symmetry, or parameter-free derivation appears.","tokens_in":60854,"confidence":"high","tokens_out":165,"duration_ms":6425,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Relativistic altermagnets mix g-wave, d-wave and p-wave spin-momentum locking depending on Néel vector orientation.","keywords":["altermagnets","spin-momentum locking","nodal planes","relativistic effects","Néel vector","CrSb","MnTe","p-wave magnetism"],"falsifier":"Spin-resolved ARPES or similar measurement on CrSb or MnTe with the Néel vector rotated away from the z-axis that finds a different number of nodal planes than predicted by the symmetry analysis alone.","tokens_in":2778,"feed_emoji":"🧲","tokens_out":811,"duration_ms":21157,"temperature":0.7,"pith_summary":"The paper examines how the number of nodal planes in altermagnet spin-momentum locking changes once relativistic effects are included. Non-relativistic altermagnets show an even number of such planes, but the combination of Néel vector direction and spin-orbit coupling from inversion-symmetry breaking can reduce that number. The authors analyze centrosymmetric CrSb and noncentrosymmetric wurtzite MnTe to find that the dominant spin component keeps its g-wave form only when the Néel vector points along z, while subdominant components adopt d-wave or p-wave symmetry. In the ferroelectric case, p-wave magnetism appears with one symmetry-protected nodal plane plus an accidental nodal surface when the Néel vector lies along x. These findings show that distinct spin components can carry a mixture of angular-momentum wave symmetries in momentum space once relativistic corrections are present.","feed_headline":"Altermagnets mix g-wave and p-wave spin locking in relativistic limit","feed_subtitle":"Néel vector and inversion-breaking field set the number of protected and accidental nodal planes in CrSb and MnTe","key_machinery":"Symmetry reduction of the number of nodal planes controlled by the orientation of the Néel vector together with the electric field from inversion-symmetry breaking.","core_discovery":"In both centrosymmetric CrSb and noncentrosymmetric wurtzite MnTe, the dominant spin component retains g-wave character in the relativistic regime only when the Néel vector is oriented along the z-axis, while the subdominant components exhibit d-wave symmetry in CrSb and p-wave symmetry in ferroelectric wurtzite MnTe. The g-wave character is preserved in the relativistic limit only when both the Néel vector and the electric field associated with inversion-symmetry breaking are oriented along the z-axis. Relativistic spin-momentum locking of ferroelectric altermagnets can exhibit p-wave magnetism with one symmetry-protected nodal plane and an accidental nodal surface not protected by symmetry","pith_inferences":["Electric fields that control the inversion-breaking direction could switch between different wave-symmetry mixtures in noncentrosymmetric altermagnets.","The accidental nodal surfaces are likely sensitive to small perturbations such as strain or doping that the symmetry analysis does not capture.","The same mixing of odd- and even-wave components may appear in other noncentrosymmetric magnetic materials once their relativistic band structures are examined."],"forward_implications":["The dominant spin component retains g-wave symmetry only when the Néel vector lies along z in both materials.","Subdominant spin components adopt d-wave symmetry in CrSb and p-wave symmetry in MnTe.","Ferroelectric altermagnets such as wurtzite MnTe realize p-wave magnetism with one protected nodal plane and one or two accidental nodal surfaces when the Néel vector is aligned along x.","Distinct spin components can simultaneously realize different angular-momentum wave symmetries in the same material once relativistic effects are included."],"fun_headline_variants":["Mixed g-wave p-wave spin locking in relativistic CrSb and MnTe","Symmetry protected nodal planes in mixed wave relativistic altermagnets","G-wave retained only with z-axis Neel vector in CrSb MnTe","P-wave magnetism with accidental nodal surface in wurtzite MnTe","Relativistic altermagnets mix even odd wave spin momentum locking"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The reduction in the number of nodal planes is fully determined by the Néel vector orientation and the electric field from inversion-symmetry breaking.","fun_headline_variants_meta":{"raw":{"variants":["Mixed g-wave p-wave spin locking in relativistic CrSb and MnTe","Symmetry protected nodal planes in mixed wave relativistic altermagnets","G-wave retained only with z-axis Neel vector in CrSb MnTe","P-wave magnetism with accidental nodal surface in wurtzite MnTe","Relativistic altermagnets mix even odd wave spin momentum locking"]},"model":"grok-4.3","cost_usd":0.004902,"raw_usage":{"total_tokens":2505,"prompt_tokens":873,"num_sources_used":0,"completion_tokens":93,"cost_in_usd_ticks":49024500,"prompt_tokens_details":{"text_tokens":873,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1539,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":873,"tokens_out":93,"duration_ms":8876,"temperature":1.0,"reasoning_tokens":1539,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-25T04:04:03.926589+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Spin-resolved ARPES or similar measurement on CrSb or MnTe with the Néel vector rotated away from the z-axis that finds a different number of nodal planes than predicted by the symmetry analysis alone.","supporting_citations":[],"review_version":1}