{"id":"9b13b5cc-829a-4615-a743-ea98d552a41e","arxiv_id":"2608.06787","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Spin-group symmetry yields four local spin-texture types and three hidden-spin categories for noncollinear magnets, identifying hundreds of candidate materials.","lead":"This paper classifies hidden spin polarization in noncollinear magnets using spin-space group symmetry, defining four local spin-texture types and three global hidden-spin categories. The framework is applied to screening hundreds of candidate materials and to predicting spin Hall and layer Hall responses.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sector-partition independence asserted in Appendix B is load-bearing and unproven in the main text; the SST/HSP labels and MAGNDATA counts would become partition-dependent if two valid compensation operations can yield non-conjugate sector stabilizers.","rationale":"The reader's CONDITIONAL verdict identifies the sector-projection assumption as the weakest link; I agree that this is the most load-bearing concern. My formulation differs slightly in emphasis: rather than stressing numerical projection ambiguity for delocalized Bloch states, I focus on the unproven algebraic claim of partition independence when multiple compensation operations O exist. Both concerns target the same underlying issue: sector-resolved spin polarization is not uniquely defined, and the taxonomy's material labels inherit that ambiguity. I do not see a fatal internal inconsistency in the main text: the SST definitions are coherent, the HSP categories follow from the stated symmetry conditions, the MAGNDATA counts are arithmetically consistent, and the nonrelativistic/SOC-free scope is clearly stated. The problem is a missing proof of a central uniqueness claim. The paper explicitly limits itself to the nonrelativistic limit and provides a symmetry-based framework that may well be correct; the issue is that the material survey's assignability depends on Appendix B's deferred proof. Therefore the appropriate verdict remains CONDITIONAL, pending verification of the sector-partition independence claim and reproduction of the survey assignments. I would not reject the paper on this basis, but the condition should be explicit: the authors must either provide the proof or show that all valid O operations in their 279 candidates give identical classifications.","tokens_in":16212,"tokens_out":12963,"duration_ms":141257,"concrete_test":"Independently re-derive the sector-partition independence claim algebraically: for a fixed magnetic spin space group, enumerate all operations O in G_SS that (i) exchange two sublattices and (ii) reverse every symmetry-allowed spin component at the same k. For each such O, construct the two sectors as the orbits of O, compute the sector stabilizer within G_SS, and determine the resulting SST/HSP category. Check whether all valid O yield the same category; if any two yield different stabilizers up to conjugation, the classification is partition-dependent.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that every noncollinear magnet's hidden spin polarization can be classified by sector stabilizer symmetries and the compensation operation O rests on the assertion in Appendix B that 'the existence and category of HSP are fixed by the full SSG, independent of the sector partition.' This is load-bearing because the SST type and the spin dimensionality d_s are defined through sector-resolved spin polarizations S_alpha(k), and the material survey's 133/7/139 counts inherit that dependence. The main text does not prove this independence; it only states it and defers to the SM. The assertion is nontrivial: a given magnetic SSG can admit more than one operation O that exchanges a pair of sectors and reverses the allowed spin components, and different choices of O can lead to different 'natural magnetic sublattices.' The stabilizer of a sector — which fixes d_s and the momentum-parity structure — depends on which sublattice partition is selected. If two valid O operations yield sector stabilizers that are not conjugate under the SSG, the same material would receive different SST and HSP labels. The paper also ships no code or input files, so the survey assignments cannot be independently reproduced. Thus the taxonomy's material-level conclusions rest on an unverified uniqueness assumption, not merely on numerical projection details.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a spin-space-group classification of nonrelativistic hidden spin polarization (HSP) in noncollinear magnets. It defines four local split spin-texture (SST) types from the actions of g_a- and g_b-type spin symmetries, and three HSP categories from the local spin dimensionality d_s together with the sector-compensating operation O. The framework is illustrated with tight-binding models and five representative materials, and a MAGNDATA survey is reported to yield 133, 7, and 139 candidate materials for HSP-1, HSP-2, and HSP-3, respectively. The paper also derives symmetry-allowed spin Hall and layer Hall responses and presents a first-principles calculation for SrFe2Se2O. The abstract and body counts are internally consistent (133 = 122 + 11; 139 = 80 + 59).","tokens_in":16464,"tokens_out":8870,"duration_ms":96712,"significance":"If the central classification is valid, the paper would provide a genuinely general symmetry framework for hidden spin polarization in noncollinear magnets, extending earlier HSP theory beyond nonmagnetic and collinear systems. The main strengths are that the classification is derived from the external spin-space-group formalism rather than fitted to the target taxonomy, the tight-binding parameters are illustrative rather than adjustment knobs, and the predicted response tensors are concrete and symmetry-enforced. The MAGNDATA survey, if reproducible, would be a useful resource. However, the significance is conditional on two currently under-supported points: the asserted independence of the HSP/SST labels from the choice of sector partition, and the operational definition of sector-resolved spin polarization for delocalized Bloch states. Both points affect the material-level assignments and hence the headline counts.","major_comments":[{"comment":"The assertion that 'the existence and category of HSP are fixed by the full SSG, independent of the sector partition' is load-bearing but is not proved. The SST label and the dimensionality d_s are defined through sector-resolved spin polarizations S_alpha(k), and Table 1 explicitly allows multiple admissible compensation operations O for a given HSP type. If two valid O operations exchange different pairs of 'natural magnetic sublattices' and yield sector stabilizers that are not conjugate under the SSG, the same material could receive different SST and HSP labels. Appendix B only states the independence claim and refers to the supplementary material; it does not supply a theorem, a canonical construction of the sector partition, or a worked example with two candidate O operations. Because the MAGNDATA counts (133/7/139) and the individual material assignments inherit this dependence, this is a central gap rather than a numerical detail. I ask the authors to provide a proof or a precise canonical definition of the natural sector partition, and to demonstrate on at least one concrete noncollinear magnet that all admissible choices of O yield the same SST/HSP classification.","section":"Appendix B / 'Hidden spin polarization in noncollinear magnets'"},{"comment":"The classification into exactly four SST types is the foundation of the paper, but the completeness argument is not present in the main text. The text introduces the three operation classes g_a, g_b, and g_c for a generic k and then states that their interplay gives rise to four generic SST types, with the detailed symmetry conditions deferred to Tables S1-S4 of the supplementary material. I could not verify from the manuscript that these classes are exhaustive for all spin-space-group elements at a generic k, nor that the possible combinations of the g_a-determined spin subspace and the g_b momentum parity can produce only the four listed cases. Since every subsequent HSP label and all survey assignments depend on this taxonomy, the authors should state the classification as a theorem with a proof sketch in the main text, or ensure that the supplementary material is part of the review package and that the tables are self-contained and checkable.","section":"Spin-texture prototypes in noncollinear magnets; Tabs. S1-S4"},{"comment":"The first-principles sector-resolved quantities S_alpha(k) are not operationally defined in the main text. For delocalized Bloch states, the projection onto local sectors can be performed in several inequivalent ways (for example, atom-centered Wannier projection, real-space partitioning, or site-resolved spin-density projection), and these choices can change the sector-resolved spin texture. The material examples, including the SST-4/HSP-3 assignment for SrFe2Se2O, therefore need a specification of the exact projector used and a numerical test showing that the reported SST/HSP labels are stable under reasonable projection choices. This is particularly important because the MAGNDATA survey is carried out by symmetry criteria, while the individual material claims are based on first-principles sector-resolved calculations; the two procedures must refer to the same sector definition.","section":"Realization in realistic materials; Figs. 3 and S6"},{"comment":"The survey paragraph does not state whether incommensurate MAGNDATA entries are included or excluded. The cited MAGNDATA papers cover both commensurate and incommensurate cases, and the spin-space-group criterion as formulated in the main text uses a finite fractional translation tau and an ordinary Brillouin zone, which suggests a commensurate assumption. Since the headline results include the total count of 790 noncollinear candidate materials and the 133/7/139 HSP counts, the authors should explicitly report the inclusion rule for incommensurate magnetic structures and, if incommensurate entries are retained, justify how the SSG classification applies to them.","section":"Realization in realistic materials; MAGNDATA survey paragraph"}],"minor_comments":[{"comment":"The definition eta_g = det|U_s| is ambiguous for antiunitary operations: the determinant of the unitary rotation part is +1, yet time reversal reverses the spin vector and should give eta_g = -1. Please define eta_g directly as the sign acquired by the spin vector under the full spin part of the operation.","section":"Eq. (2)"},{"comment":"Reference [80], described as a short review on metal phosphide based 2D nanomaterials, appears unrelated to the USb/U monopnictide context in which it is cited; please verify and replace it with an appropriate reference.","section":"Reference list"},{"comment":"In the HSP-2 row, the entry 'O = [T||P|tau] ([C_2z||E|tau])' is unclear: it should state explicitly whether both operations are always allowed or whether the parenthesized operation is allowed only under additional symmetry conditions.","section":"Table 1"},{"comment":"The phrase 'the total spin polarization are hidden' should be corrected to 'the total spin polarization is hidden'.","section":"Abstract and Introduction"},{"comment":"The sentence 'after excluding collinear magnets and materials with fractional atomic occupancies' should also specify whether the search was restricted to the commensurate subset of MAGNDATA, and how the spin-space-group symmetry of each entry was determined algorithmically.","section":"MAGNDATA survey paragraph"}],"recommendation":"major_revision","confidential_remarks":"The framework is plausible and the paper makes a useful conceptual step, but the sector-partition independence claim is genuinely load-bearing: the survey counts and material assignments inherit it. I do not think the manuscript should be rejected, because the issue is fixable by adding a proof or a canonical construction and by specifying and numerically testing the sector projection. I also strongly recommend that the supplementary material be made available during review, since the central SST tables and the complete MAGNDATA list are referenced there. The reference problem with [80] should be corrected before resubmission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is a real advance, not a repackaging. The four SST types and three HSP categories are new, clearly defined, and internally consistent. The collinear limit correctly reduces to even-parity SST-2, which contains the altermagnetic textures, and SST-3/SST-4 have no static collinear counterpart. The MAGNDATA survey gives a concrete list of candidates, and the SrFe2Se2O example shows how the symmetry rules play out in a computed response.\n\nThe soft spots are real but not disqualifying. The main one: Appendix B asserts that the existence and category of HSP are fixed by the full SSG and independent of the sector partition. That is load-bearing, because the SST label and d_s are defined through sector stabilizers, and the survey counts depend on it. The stress-test concern is legitimate: nothing in the main text rules out two valid compensation operations giving non-conjugate sector stabilizers, and the proof is deferred to the Supplementary Materials, which neither the reader nor I had. That is a proof gap, not a demonstrated error. A referee should require the proof to be brought into the main text or at least fully verified in the SM. The lack of code and input files for the survey is a smaller issue but worth noting; the assignments are not independently reproducible. The SHC numbers are parameter-dependent, which is fine for illustration but should not be read as quantitative predictions.\n\nCitation pattern looks solid: the spin-group refs are the right ones, and the claim that HSP was previously confined to nonmagnetic and collinear systems is accurate.\n\nWho this is for: anyone working on spin symmetries, altermagnetism, or noncollinear spintronics; the candidate list is the practical takeaway. The paper deserves a serious referee. Conditional accept is the right call if the partition-independence proof holds up in the SM; I would engage with it.","headline":"Useful spin-group taxonomy for HSP in noncollinear magnets, but the unproven sector-partition independence keeps the material-level claims conditional.","tokens_in":16990,"tokens_out":3826,"would_cite":true,"duration_ms":37749,"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":"Spin groups classify hidden spin polarization in noncollinear magnets.","keywords":["hidden spin polarization","noncollinear magnets","spin space group","spin texture classification","nonrelativistic magnetism","spin Hall effect","layer Hall effect","magnetic symmetry"],"falsifier":"Take a predicted HSP material such as SrFe2Se2O, compute or measure the momentum-resolved spin polarization projected onto each of the two magnetic sublattices, and check at every $k$ where the sector-exchanging symmetry is preserved whether $\\mathbf{S}_\\alpha(\\mathbf{k})+\\mathbf{S}_\\beta(\\mathbf{k})=0$ exactly. Finding even one symmetry-preserving $k$ with a nonzero sum, or one predicted SST parity (such as odd in momentum) contradicted by the observed texture, would falsify the compensation or the texture classification.","tokens_in":16024,"feed_emoji":"🧲","tokens_out":9359,"duration_ms":84750,"temperature":0.7,"pith_summary":"Hidden spin polarization (HSP) is the situation in which electrons in separate real-space sectors of a crystal carry opposite spin polarizations at every momentum, so the crystal as a whole shows no net spin even though each sector is locally spin-polarized. The paper establishes that in the nonrelativistic, spin-orbit-free limit the spin symmetries of a noncollinear magnet force the momentum-resolved spin polarization of any local sector into one of four split spin-texture (SST) types, and that an additional compensation operation $O$ determines whether those local textures hide globally or produce an observable spin splitting. Combining the SST form with the dimensionality of the local spin texture yields three HSP categories, HSP-1, HSP-2, and HSP-3, illustrated by tight-binding models and by specific compounds. A survey of a public magnetic-structure database finds 133, 7, and 139 candidate noncollinear magnets in the three categories, and the analysis predicts nonzero spin-related response tensors, including a spin Hall conductivity and a sector-resolved layer Hall effect, in many of these materials.","feed_headline":"Spin groups classify hidden spin polarization in noncollinear magnets","feed_subtitle":"Local spin-texture classes determine which noncollinear magnets hide spin globally while keeping it locally.","key_machinery":"The load-bearing object is the spin space group, written $G_{SS}=G_{SO}\\times G_{NS}$, with $G_{SO}$ the spin-only subgroup and $G_{NS}$ operations acting jointly on spin and space. Its operations are divided into three classes, $g_a$ (fixed $\\mathbf{k}$), $g_b$ (maps $\\mathbf{k}$ to $-\\mathbf{k}$), and $g_c$ (maps $\\mathbf{k}$ to other momenta), and the paper's classification is carried by the interplay of $g_a$ and $g_b$. The second ingredient is the compensation operation $O$, a spin-space-group element that exchanges the two local sectors and enforces $\\mathbf{S}_\\alpha(\\mathbf{k})=-\\mathbf{S}_\\beta(\\mathbf{k})$; the allowed form of $O$ depends on the spin dimensionality of the local texture and defines the HSP-1, HSP-2, and HSP-3 labels.","core_discovery":"The central discovery is that a noncollinear magnet's hidden spin polarization is not an accident of electronic structure but is fixed by its spin space group, the symmetry group of a magnetic crystal in the limit where spin rotations act independently of spatial operations. The paper classifies the symmetry operations that constrain a momentum-space spin texture $\\mathbf{S}(\\mathbf{k})$ into three types: those that keep $\\mathbf{k}$ fixed and fix the allowed spin dimensionality $d_s$; those that map $\\mathbf{k}$ to $-\\mathbf{k}$ and fix whether each spin component is even or odd in momentum; and those that relate distinct momenta. The first two types generate exactly four local split spin-texture classes, SST-1 through SST-4, distinguished by parity and dimensionality. HSP then arises when a spin operation $O$ exchanges the two real-space sectors and requires $\\mathbf{S}_\\alpha(\\mathbf{k}) = -\\mathbf{S}_\\beta(\\mathbf{k})$ at every momentum; depending on $d_s$ and on the allowed form of $O$, this yields HSP-1 (collinear local textures), HSP-2 (coplanar local textures), and HSP-3 (noncoplanar local textures). The framework reproduces the known collinear even-parity altermagnetic textures as a special case and predicts, in the nonrelativistic limit, that hundreds of noncollinear magnets host HSP with sizeable spin-dependent response tensors.","pith_inferences":["An implication the authors leave implicit is that the same symmetry criteria could be used as a pre-screen on any magnetic-structure database, so the candidate counts are likely to grow as more noncollinear structures are added, and HSP-2 stays the rarest class because it requires the restrictive coplanar symmetry $[TC_{2z}\\|P]$.","A testable expectation beyond the paper is that the odd-parity SST-3 and mixed-parity SST-4 textures will be the most strongly reshaped by spin-orbit coupling, since their momentum-parity pattern is not anchored to a global spin axis.","The predicted sector-resolved layer Hall effect suggests an experimental route the paper does not develop: a transport probe that couples to one magnetic sublattice should see a finite anomalous Hall response even though the crystal's total response vanishes."],"forward_implications":["Any noncollinear magnet in the nonrelativistic limit can be assigned a definite local SST class (SST-1 through SST-4) from its spin symmetries alone, without computing the full electronic structure.","If the compensation operation $O$ is present, the total spin polarization vanishes at every momentum while each sector stays spin-polarized; if it is absent, a global nonrelativistic spin splitting is symmetry-allowed instead.","The classification puts the previously separate collinear hidden-spin and even-parity altermagnetic textures into the same framework, with odd-parity SST-3 and mixed-parity SST-4 textures existing only in noncollinear magnets.","Candidate noncollinear magnets are abundant: the database survey yields 133 HSP-1, 7 HSP-2, and 139 HSP-3 materials, and many of them are predicted to show nonzero spin Hall conductivity, magnetoelectric effects, or a layer Hall effect even without spin-orbit coupling."],"supporting_citations":[{"why":"Defines spin space groups for magnetic materials with negligible spin-orbit coupling, giving the $G_{SO}\\times G_{NS}$ structure the paper uses.","marker":"[66]"},{"why":"Provides the full classification of spin space groups that licenses the enumeration of allowed operations.","marker":"[67]"},{"why":"Establishes hidden spin polarization in antiferromagnets, including the sector convention and compensation criterion the paper generalizes to noncollinear order.","marker":"[29]"},{"why":"Supplies the magnetic-structure database entries from which the coplanar and noncoplanar candidate counts are drawn.","marker":"[73]"},{"why":"Supplies additional database entries used in the survey.","marker":"[74]"},{"why":"Defines the collinear even-parity textures whose altermagnetic subclasses the SST classification recovers in the collinear limit.","marker":"[70]"},{"why":"Provides the crystal and magnetic structure of SrFe2Se2O used as the worked material example.","marker":"[75]"},{"why":"Gives the interpolation method used to compute intrinsic spin Hall conductivity.","marker":"[94]"},{"why":"Establishes the spin Hall conductivity calculation for magnetic compounds used in the response calculation.","marker":"[95]"},{"why":"Gives the tensor transformation rule that selects the symmetry-allowed spin Hall conductivity components.","marker":"[112]"}],"fun_headline_variants":["Spin groups dictate hidden spin textures in noncollinear magnets","Spin symmetry rules hidden spin in noncollinear magnets","Classifying hidden spin polarization via spin groups","How spin groups expose hidden spin in noncollinear magnets","Hidden spin polarization decoded by spin group symmetry"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole scheme depends on dividing the crystal into two well-defined local sectors, the natural magnetic sublattices connected by the sector-exchanging symmetry, whose spin polarizations are well defined and exactly opposite at every momentum; if that division is ambiguous for delocalized electrons in a first-principles calculation, the labels assigned to real materials become projection-dependent.","fun_headline_variants_meta":{"raw":{"variants":["Spin groups dictate hidden spin textures in noncollinear magnets","Spin symmetry rules hidden spin in noncollinear magnets","Classifying hidden spin polarization via spin groups","How spin groups expose hidden spin in noncollinear magnets","Hidden spin polarization decoded by spin group symmetry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000253,"raw_usage":{"total_tokens":1654,"prompt_tokens":1126,"completion_tokens":528,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":742,"completion_tokens_details":{"reasoning_tokens":454}},"tokens_in":742,"tokens_out":528,"duration_ms":4906,"temperature":1.0,"reasoning_tokens":454,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:48:22.174347+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a predicted HSP material such as SrFe2Se2O, compute or measure the momentum-resolved spin polarization projected onto each of the two magnetic sublattices, and check at every $k$ where the sector-exchanging symmetry is preserved whether $\\mathbf{S}_\\alpha(\\mathbf{k})+\\mathbf{S}_\\beta(\\mathbf{k})=0$ exactly. Finding even one symmetry-preserving $k$ with a nonzero sum, or one predicted SST parity (such as odd in momentum) contradicted by the observed texture, would falsify the compensation or the texture classification.","supporting_citations":[{"cited_title":"Fukaya, K","cited_arxiv_id":null,"evidence_quote":"Defines spin space groups for magnetic materials with negligible spin-orbit coupling, giving the $G_{SO}\\times G_{NS}$ structure the paper uses."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the full classification of spin space groups that licenses the enumeration of allowed operations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the magnetic-structure database entries from which the coplanar and noncoplanar candidate counts are drawn."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies additional database entries used in the survey."},{"cited_title":"Jiang, Z","cited_arxiv_id":null,"evidence_quote":"Defines the collinear even-parity textures whose altermagnetic subclasses the SST classification recovers in the collinear limit."},{"cited_title":"Guo, M.-T","cited_arxiv_id":null,"evidence_quote":"Provides the crystal and magnetic structure of SrFe2Se2O used as the worked material example."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the interpolation method used to compute intrinsic spin Hall conductivity."},{"cited_title":"Zhang, Y","cited_arxiv_id":null,"evidence_quote":"Establishes the spin Hall conductivity calculation for magnetic compounds used in the response calculation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the tensor transformation rule that selects the symmetry-allowed spin Hall conductivity components."}],"review_version":1}