{"id":"a520e7ad-2dce-4c47-b22e-c1e133406a87","arxiv_id":"2506.01823","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"Altermagnet effects, including spin splitting, anomalous Hall, and spin Hall currents, follow from a conventional vector Néel order parameter plus crystal symmetries.","lead":"This paper shows that altermagnets, a newly recognized class of magnetic materials, can be understood with the same vector order parameters used for ordinary antiferromagnets. It compiles symmetry rules for several compounds and argues that no new theoretical machinery is needed to describe their spin splitting, Hall, and magneto-optical effects.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Completeness of the non-collinear rows of Table I is asserted, not proven: the reduction to S1,S2,S3 assumes no spin-space-group invariant is missed; a symbolic invariant enumeration would settle the central claim.","rationale":"The reader's ACCEPT is well supported: the paper gives a compact symmetry classification, reproduces the observed g-wave spin splitting in CrSb and MnTe, explains domain selection, and grounds the relativistic effects in the standard weak-ferromagnetism invariants. That is real and valuable. However, the abstract's 'all effects' statement is stronger than what is demonstrated for non-collinear orders. The non-collinear rows of Table I are derived by assuming the three site vectors exhaust the order parameter and that the diagonal action S_n -> R S_{R^{-1}n} captures the relevant symmetry. It is plausible, and probably true for the specific 120-degree structures, but it is not proven, and the paper does not show equivalence with the spin-space-group analysis it aims to replace. A missed invariant would not invalidate the individual results, but it would falsify the central completeness claim. The proposed enumeration is a standard, inexpensive symbolic check that settles the issue. I therefore recommend CONDITIONAL acceptance: accept after the completeness check or after the claim is explicitly scoped to 'all effects considered in this paper'. This is not an objection to the overall framework, only to the unproven exhaustiveness that the abstract asserts.","tokens_in":9899,"tokens_out":34484,"duration_ms":410032,"concrete_test":"Run a complete invariant enumeration for the cubic and hexagonal non-collinear altermagnets under the full spin space group, i.e. the pair group (U|R) with U in SO(3)_spin and R in the crystallographic point group (Pm-3m or P6_3/mmc), rather than only the diagonal magnetic-group action. Concretely, use a computer-algebra invariant solver (e.g., GAP with a Reynolds operator) to generate all independent invariants linear in the S_n and linear in E for the T-odd spin current, and linear in the S_n and quadratic in k for the spin texture, while imposing S1+S2+S3=0 and, for the hexagonal cases, verifying S_bar_n=S_n against neutron-diffraction refinements of Mn3Sn/Mn3Ge. If the enumeration reproduces Table I exactly and the inversion check passes, the central claim stands; if extra or missing invariants appear, the classification is incomplete.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is a completeness claim, not just a list of examples: every effect observed in altermagnets with non-collinear spin order is claimed to follow from a conventional vector order parameter. The load-bearing step is the reduction of the 120-degree order to the three site vectors S1, S2, S3 with S1+S2+S3=0 (Eq. 4) and S_bar_n=S_n, and the simultaneous use of the diagonal transformation S_n^a -> R_ab S^b_{R^{-1}n} to enumerate the non-collinear rows of Table I. This transformation is the magnetic-space-group action; it is never shown to coincide with the full spin-space-group action, in which a symmetry can pair a spin rotation U with a lattice rotation R (with U not equal to R). If the spin-space-group of Mn3Sn, Mn3Ge, Mn3Ir, or Mn3GaN contains such operations, the set of invariants built from the diagonal action can be either incomplete (a missing term that would make Table I wrong) or over-restrictive (a listed term actually forbidden). For example, the T-odd spin-current terms linear in the order parameter and linear in E are assumed to have the form j_i^a = A sum_n S_n^a (f_n)_i (E dot f_n); no invariant-theory proof is given that this list is exhaustive, or that no term involving V3 (the chirality vector) appears at this order. A missed term would directly contradict the abstract's 'all effects' statement. The same uncertainty applies to the spin-texture expressions near the Gamma point. The hexagonal entries additionally depend on the assumption S_bar_n=S_n, so a magnetic structure with inversion-odd layers would invalidate that row. This is the weakest load-bearing assumption; the SOC caveat is less serious because the paper explicitly separates relativistic and non-relativistic regimes.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper argues that the phenomenology of altermagnets—including relativistic weak ferromagnetism and magneto-optical/transport effects and non-relativistic spin splitting and spin Hall currents—can be described by a conventional vector antiferromagnetic order parameter: the Néel vector l for collinear systems, and three site spin vectors S1,S2,S3 with S1+S2+S3=0 for 120° non-collinear orders. Using Turov's sign-factor method for collinear magnets and a direct generalization to the Kagome non-collinear magnets, the author derives Table I: symmetry-allowed weak ferromagnetism, band spin splitting near Γ, and T-odd spin Hall currents for RuO2, CrSb/MnTe, Mn3Ir/Mn3GaN, and Mn3Sn/Mn3Ge. The paper also discusses domain selection, nonlinear SHE in non-altermagnets, and experiments that remain controversial (e.g., RuO2). The central claim is the abstract's statement that no spin space groups or magnetic multipoles are needed for these phenomena.","tokens_in":10247,"tokens_out":9739,"duration_ms":109705,"significance":"If the completeness claim is accepted, the paper is valuable: it reconnects altermagnetism with the classical Turov/Landau symmetry framework, gives a single table of falsifiable predictions, and offers practical domain-selection recipes. The collinear analysis (RuO2, CrSb/MnTe) is standard and consistent with published ARPES and transport results. The paper is also commendably candid about experimental controversies, and it produces concrete predictions such as the non-linear SHE in CrSb/MnTe and the T-even chirality-driven SHE in Mn3Sn/Mn3Ge. The main value would be the unification claim, but that claim hinges on the non-collinear classification being exhaustive, which the manuscript does not yet demonstrate.","major_comments":[{"comment":"The completeness of the non-collinear rows is the load-bearing part of the abstract's 'all effects' claim, but it is asserted rather than proved. The transformation S_n^a → R_ab S^b_{R^{-1}n} is a diagonal action of the magnetic space group; in the absence of spin-orbit coupling the full symmetry group of a non-collinear spin arrangement is a spin space group, which can contain operations (U,R) with U ≠ R. The paper never shows that the diagonal action yields the same invariant ring. Concretely, the T-odd spin-current entry j_i^a = A Σ_n S_n^a (f_n)_i (E·f_n) for Mn3Ir/Mn3GaN and Mn3Sn/Mn3Ge is written down without an invariant-theoretic enumeration, and a term involving the chirality vector V3 at the same order is not ruled out; the paper's own Eq. (7) shows that V3-dependent spin currents can be symmetry allowed. A finite symbolic invariant enumeration for these magnetic point groups would settle whether Table I is exhaustive or over-inclusive.","section":"Non-collinear altermagnets (Eq. (4) and Table I)"},{"comment":"The reduction of the hexagonal six-sublattice order to three spins via S_bar n = S_n and the description of the 120° state by two orthogonal vectors V1,V2 plus V3 assumes that no other independent order-parameter component exists. For a general non-collinear magnetic structure the order-parameter space is not automatically spanned by these vectors; the paper should state the conditions under which this reduction is complete or cite the magnetic structure determinations for Mn3Sn, Mn3Ge, Mn3Ir, and Mn3GaN that justify it. Without this, the classification in Table I may miss effects that couple to additional degrees of freedom (e.g., a second chirality or a staggered quadrupole).","section":"Reduction to S1,S2,S3 and inversion assumption"}],"minor_comments":[{"comment":"The word 'aproach' should be 'approach'.","section":"p. 3"},{"comment":"The text and table differ in the weak-ferromagnetism term: the text writes H_x^2 l_x l_y + H_y(l_x^2-l_y^2) l_z while Table I has l_z multiplying the first term as well; these should be harmonized.","section":"Table I, CrSb/MnTe row"},{"comment":"The symbol A is used for the spin-splitting amplitude in Eq. (2) and again for the spin-current coefficients in Table I; distinct symbols would avoid confusion.","section":"Eq. (2) and Table I"},{"comment":"The chirality vector V3 is defined for a triangle in Eq. (5), but for the hexagonal case its orientation should be stated explicitly, since the sign of the T-even SHE depends on it.","section":"Eq. (7)"},{"comment":"The phrase 'non-nonlinear SHE effect' appears to be a typo; 'non-linear SHE effect' is presumably intended.","section":"p. 6"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a well-written single-author phenomenology paper that fits the journal's scope. The main risk is overclaiming completeness for non-collinear altermagnets; I would not object to publication after the completeness issue is addressed by an explicit invariant enumeration or by a softened claim. There is no concern about novelty disclosure."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThe paper is worth your time. Mostovoy does something useful: he shows that the standard Turov/Landau vector order parameter description covers altermagnet effects, so spin space groups and magnetic multipoles are not strictly necessary for the phenomena considered. The main deliverable is Table I, a compact list of symmetry-allowed weak ferromagnetism, spin splitting, and T-odd spin current terms for the canonical altermagnets (RuO2, CrSb/MnTe, Mn3Ir/Mn3GaN, Mn3Sn/Mn3Ge). That table is a practical reference, and it checks out against ARPES and transport data.\n\nThe paper does several things well. It clearly separates relativistic and non-relativistic effects, which resolves some of the confusion in the literature. The use of Turov's sign-factor method is clean and produces invariants without fitting parameters. The explicit discussion of domain selection for Mn3Sn/Mn3Ge and the chirality dependence of the weak ferromagnetism is helpful. I also appreciate that the author flags controversial experimental cases (RuO2 paramagnetism, thin-film spin-splitter current) rather than selling them as settled.\n\nThe soft spot is exactly where the stress test lands: the completeness of the non-collinear rows of Table I. The paper asserts that the vector order parameters S1,S2,S3 (or V1,V2,V3) exhaust the relevant degrees of freedom, and it uses the diagonal action S_n^a -> R_ab S^b_{R^{-1}n} to enumerate invariants. It is never proven that this coincides with the full spin-space-group action, where a spin rotation U could be paired with a lattice rotation R with U≠R. If such operations exist for Mn3Sn/Mn3Ge/Mn3Ir/Mn3GaN, a term could be missing from the table, or a listed term could be forbidden. The same applies to the assumption S_bar_n = S_n for the hexagonal compounds. This is a genuine gap, but it is a gap in proof, not evidence of an error. The listed terms match the known phenomenology, so my expectation is that the table is right at the orders considered.\n\nThe neglect of spin-orbit coupling for the non-relativistic effects is less concerning; the paper explicitly separates the two regimes and that is a legitimate choice.\n\nBottom line: this paper deserves a serious referee. I would send it out. My main request would be to either add a formal invariant-theory check for the non-collinear cases or soften the 'all effects' phrasing to 'the leading-order effects considered here.' That would close the completeness loophole without changing the paper's value. I would probably cite the table in my own work on altermagnets.\n\nBest,\n[You]","headline":"A genuinely useful symmetry table that makes a strong case for vector-order-parameter phenomenology of altermagnets, with the non-collinear completeness claim being the one soft spot.","tokens_in":10796,"tokens_out":3289,"would_cite":true,"duration_ms":35285,"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":"All altermagnet effects can be described with an ordinary vector antiferromagnetic order parameter, without spin space groups or magnetic multipoles.","keywords":["altermagnets","Néel vector","weak ferromagnetism","spin splitting","spin Hall effect","spin space groups","Landau theory","non-collinear magnetism"],"falsifier":"Measure the spin-split band structure of a single-domain crystal of CrSb or MnTe and check whether the splitting tracks $(\\mathbf{l}\\cdot\\mathbf{s})k_z(3k_x^2k_y-k_y^3)$ with $g(-\\mathbf{k})=g(\\mathbf{k})$; a discovered term requiring a different or higher-rank order parameter would disprove the claim that the vector description is complete.","tokens_in":9638,"feed_emoji":"🧲","tokens_out":8098,"duration_ms":80738,"temperature":0.7,"pith_summary":"The paper argues that everything currently attributed to altermagnets — weak ferromagnetism, band spin splitting, and the time-reversal-odd spin Hall effect — follows from the ordinary antiferromagnetic order parameter once it is handled with the right sign conventions, and that spin space groups and magnetic multipoles are not needed for these effects. For collinear two-sublattice orders, a single Néel vector $\\mathbf{l}$ transforms under crystal rotations with a sign factor that records whether the rotation swaps sublattices. For non-collinear 120-degree orders, two orthogonal vectors $\\mathbf{V}_1$ and $\\mathbf{V}_2$ plus their vector chirality $\\mathbf{V}_3=\\mathbf{V}_1\\times\\mathbf{V}_2$ play the same role. The result is a unified table of symmetry-allowed effects for RuO$_2$, CrSb/MnTe, Mn$_3$Ir/Mn$_3$GaN, and Mn$_3$Sn/Mn$_3$Ge, with the non-relativistic spin splitting written as $\\delta\\varepsilon_{\\mathbf{k}m}=(\\mathbf{l}\\cdot\\mathbf{s})g(\\mathbf{k})$ with $g(-\\mathbf{k})=g(\\mathbf{k})$. A sympathetic reader would care because this puts altermagnets inside ordinary Landau theory, making domain selection, strain, and field responses tractable.","feed_headline":"One vector order parameter explains altermagnet effects","feed_subtitle":"Spin space groups and magnetic multipoles are not needed, the paper argues.","key_machinery":"The central object is the Néel vector order parameter $\\mathbf{l}=(\\mathbf{m}_1-\\mathbf{m}_2)/2$ for collinear antiferromagnets, used together with a sign factor $\\sigma_R$: any crystal rotation $R$ either preserves or swaps the two magnetic sublattices, and $\\mathbf{l}$ transforms under $R$ as an axial vector multiplied by $\\sigma_R$. For non-collinear Kagome orders, the same job is done by two orthogonal vectors $\\mathbf{V}_1$ and $\\mathbf{V}_2$ and the vector chirality $\\mathbf{V}_3=\\mathbf{V}_1\\times\\mathbf{V}_2$. The load-bearing identity is the non-relativistic spin-splitting form $\\delta\\varepsilon_{\\mathbf{k}m}=(\\mathbf{l}\\cdot\\mathbf{s})g(\\mathbf{k})$ with $g(-\\mathbf{k})=g(\\mathbf{k})$, which encodes the requirement that the electron spin projection along the Néel vector is conserved without spin-orbit coupling. Invariance under rotations with sign factors then fixes the harmonic forms of $g(\\mathbf{k})$ — d-wave in RuO$_2$, g-wave in CrSb/MnTe, and $k^2$-weighted combinations in the non-collinear magnets — and those forms in turn generate the spin-splitter current.","core_discovery":"The paper shows that the conventional phenomenological description of antiferromagnets in terms of a vector order parameter already contains all effects observed in altermagnets, both collinear and non-collinear. The key move is to track, for each crystal rotation, a sign factor $\\sigma_R$ equal to $-1$ when the rotation interchanges magnetic sublattices and $+1$ otherwise; the Néel vector transforms as a $T$-odd axial vector multiplied by this sign factor. In the absence of spin-orbit coupling, the spin splitting of electron bands takes the form $\\delta\\varepsilon_{\\mathbf{k}m}=(\\mathbf{l}\\cdot\\mathbf{s})g(\\mathbf{k})$ with $g(-\\mathbf{k})=g(\\mathbf{k})$, and the allowed forms of $g(\\mathbf{k})$ for the four representative materials produce the d-wave and g-wave splittings reported experimentally. The T-odd spin Hall (spin-splitter) current is then a direct transport consequence of these $g(\\mathbf{k})$ forms. The same sign-factor analysis, applied to two orthogonal vectors $\\mathbf{V}_1$ and $\\mathbf{V}_2$ and chirality $\\mathbf{V}_3=\\mathbf{V}_1\\times\\mathbf{V}_2$, reproduces the weak ferromagnetism, reciprocal-space spin texture, and spin-splitter currents of cubic and hexagonal 120-degree magnets. The paper also applies the method to non-altermagnet orders, where inversion-odd or translation-breaking orders yield nonlinear spin Hall currents and current-induced magnetization.","pith_inferences":["The paper leaves implicit that the same symmetry argument predicts a measurable nonlinear spin-splitter signal in single-domain CrSb or MnTe samples, of the form $\\mathbf{j}_z\\propto l(3E_x^2E_y-E_y^3)$, which has not yet been reported.","If any future experiment finds an altermagnet effect that requires a spin-order degree of freedom beyond $\\mathbf{l}$ (or $\\mathbf{V}_1$, $\\mathbf{V}_2$, $\\mathbf{V}_3$), the unification would reduce to a coincidence valid only for the listed materials.","The formalism implies that the controversy over whether RuO$_2$ is actually antiferromagnetic is separate from the classification: if RuO$_2$ is paramagnetic, the table entries for it are vacuous but the identical analysis applies to isostructural antiferromagnets with the same magnetic space group.","Because the description is purely symmetry-based, it could be extended to dynamical phenomena, such as finite-frequency Faraday or Kerr spectra and AC spin-current responses, by promoting the scalar coefficients to frequency-dependent functions; the paper does not explicitly do this."],"forward_implications":["Spin space groups and magnetic multipoles are not needed to classify weak ferromagnetism, spin splitting, or the T-odd spin Hall effect in altermagnets; the vector order parameter plus crystal rotations suffices.","Landau theory built on $\\mathbf{l}$ (or $\\mathbf{V}_1$, $\\mathbf{V}_2$, $\\mathbf{V}_3$) can describe how altermagnetic order responds to applied magnetic fields, strains, and stresses, including domain selection by weak ferromagnetism.","The spin-splitter current is a non-equilibrium occupation effect: for d-wave splitting the current is linear in the electric field, while for g-wave splitting a nonlinear T-odd spin Hall effect is symmetry-allowed.","For the 120-degree magnets, the sign of the vector chirality determines whether weak ferromagnetism appears: Mn$_3$Sn and Mn$_3$Ge with negative chirality show it, while positive-chirality Mn$_3$GaN does not.","In non-altermagnets, magnetic orders that break inversion can still produce non-relativistic transport phenomena such as nonlinear spin Hall currents and current-induced magnetization, for example in Cr$_2$O$_3$-like symmetries."],"supporting_citations":[{"why":"Supplies the sign-factor method for expressing weak ferromagnetism through the Néel vector, the foundation of the present classification.","marker":"[30]"},{"why":"Fixes the transformation of gyration and Hall vectors as axial vectors under rotations and inversion, used throughout the symmetry analysis.","marker":"[29]"},{"why":"Predicted the spin-splitter current from spin splitting, the non-relativistic transport effect the paper systematizes.","marker":"[18]"},{"why":"Derived the d-wave spin-splitting expression $g(\\mathbf{k})\\propto k_xk_y$ in RuO$_2$, used as the prototype collinear altermagnet.","marker":"[9]"},{"why":"Reports ARPES observation of spin-split bands in CrSb, used to validate the g-wave form of the splitting.","marker":"[12]"},{"why":"Reports ARPES observation of spin-split bands in MnTe, used to validate the g-wave form of the splitting.","marker":"[14]"},{"why":"Reports the anomalous Hall effect in Mn$_3$Sn, the experimental phenomenon the conventional description must reproduce.","marker":"[1]"},{"why":"Introduced spin space groups as an alternative symmetry framework that the paper argues is not required for these effects.","marker":"[25]"},{"why":"Proposed another spin-space-group formalism for altermagnets, included as a contrast to the vector-order-parameter approach.","marker":"[26]"},{"why":"Identifies the non-relativistic magnetoelectric mechanism in Cr$_2$O$_3$, used as the example of non-altermagnet effects.","marker":"[68]"}],"fun_headline_variants":["One vector order parameter explains all altermagnet effects","Vector Néel order reproduces collinear and non-collinear altermagnets","Altermagnet effects from a single antiferromagnetic vector","Unified vector description for altermagnets, no multipoles needed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The classification assumes that each magnetic order has no symmetry-inequivalent degrees of freedom beyond the Néel vector (or the two orthogonal vectors plus chirality for non-collinear cases), and that spin-orbit coupling is negligible for the non-relativistic effects.","fun_headline_variants_meta":{"raw":{"variants":["One vector order parameter explains all altermagnet effects","Vector Néel order reproduces collinear and non-collinear altermagnets","Altermagnet effects from a single antiferromagnetic vector","Unified vector description for altermagnets, no multipoles needed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000196,"raw_usage":{"total_tokens":1373,"prompt_tokens":969,"completion_tokens":404,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":585,"completion_tokens_details":{"reasoning_tokens":330}},"tokens_in":585,"tokens_out":404,"duration_ms":4348,"temperature":1.0,"reasoning_tokens":330,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:32:18.069115+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the spin-split band structure of a single-domain crystal of CrSb or MnTe and check whether the splitting tracks $(\\mathbf{l}\\cdot\\mathbf{s})k_z(3k_x^2k_y-k_y^3)$ with $g(-\\mathbf{k})=g(\\mathbf{k})$; a discovered term requiring a different or higher-rank order parameter would disprove the claim that the vector description is complete.","supporting_citations":[{"cited_title":"Electrodynamics of continuous media,","cited_arxiv_id":null,"evidence_quote":"Supplies the sign-factor method for expressing weak ferromagnetism through the Néel vector, the foundation of the present classification."},{"cited_title":"Hayami, Y","cited_arxiv_id":null,"evidence_quote":"Fixes the transformation of gyration and Hall vectors as axial vectors under rotations and inversion, used throughout the symmetry analysis."},{"cited_title":"Naka et al., Nat","cited_arxiv_id":null,"evidence_quote":"Predicted the spin-splitter current from spin splitting, the non-relativistic transport effect the paper systematizes."},{"cited_title":"Hayami, Y","cited_arxiv_id":null,"evidence_quote":"Derived the d-wave spin-splitting expression $g(\\mathbf{k})\\propto k_xk_y$ in RuO$_2$, used as the prototype collinear altermagnet."},{"cited_title":"Bai et al., Phys","cited_arxiv_id":null,"evidence_quote":"Reports ARPES observation of spin-split bands in CrSb, used to validate the g-wave form of the splitting."},{"cited_title":"Reimers et al., Nat","cited_arxiv_id":null,"evidence_quote":"Reports ARPES observation of spin-split bands in MnTe, used to validate the g-wave form of the splitting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced spin space groups as an alternative symmetry framework that the paper argues is not required for these effects."},{"cited_title":"ˇSmejkal, J","cited_arxiv_id":null,"evidence_quote":"Proposed another spin-space-group formalism for altermagnets, included as a contrast to the vector-order-parameter approach."},{"cited_title":"Reichlova et al., Nat","cited_arxiv_id":null,"evidence_quote":"Identifies the non-relativistic magnetoelectric mechanism in Cr$_2$O$_3$, used as the example of non-altermagnet effects."}],"review_version":1}