{"id":"ce7d40db-e905-4b71-8b27-3b82b4f0ff63","arxiv_id":"2608.06964","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Spin-group symmetry predicts the spin-orbit-coupling order at which each Edelstein and spin-orbit-torque component appears in collinear ferromagnets.","lead":"A symmetry framework sorts current-induced spin torques in ferromagnets by how strongly they depend on spin-orbit coupling. The authors derive which torque forms appear at each order and test them against first-principles calculations in several material stacks.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Third-order SOC fits in PtMnSb exceed second-order and C4v angular fits show poor residuals, so the quantitative SOC-order claims are not established by the same-data fitting used.","rationale":"This stress-test pass starts from the paper's goal: to classify Edelstein-effect and SOT tensor components by SOC order using spin-group symmetry, with first-principles validation. The spin-group derivation is a genuine contribution: it derives allowed tensor forms from symmetry constraints rather than assuming the conventional FL/DL forms. The layer-resolved analysis and the PtMnSb symmetry argument are physically interesting and may well be correct. The weak point is not the symmetry formalism but the quantitative validation. The SOC-vector expansion is fitted to first-principles data with no independent convergence test, and the manuscript's own reported numbers undercut the truncation: PtMnSb third-order coefficients are comparable to or larger than second-order ones, and the Ti/Ni angular-dependent torkance has a fit with R²≈0.56 for one component with an admitted need for higher-order terms. Therefore the strongest claim, as phrased quantitatively, is not established. This is not a reason to reject the symmetry classification; the first-order-vanishing prediction in PtMnSb can be tested directly at small SOC, and if it holds the formal SOC-order assignment stands. The conditional verdict is therefore unchanged, but the acceptance conditions should include an out-of-sample or low-ξ test of the polynomial expansion and a report of coefficient stability under higher-order fits.","tokens_in":16557,"tokens_out":7334,"duration_ms":85511,"concrete_test":"Re-fit the PtMnSb m//x Edelstein/torkance data of Fig. 8 with quadratic, cubic, and quartic polynomials in ξ/ξ0 and report whether the quadratic coefficient changes by more than 20% when cubic terms are included; then compute χ_y^y and t_z^y at ξ/ξ0=0.1 and 0.2 on a 200×200×1 k-mesh and test whether log-log slope is consistent with 2 (first order absent) rather than 1. If the quadratic coefficients are unstable under added cubic terms, the second-order truncation is not quantitatively validated; if the low-ξ slope is inconsistent with 2, the symmetry-based vanishing of first-order terms is wrong.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing assumption is that the SOC-vector expansion χ=χ^(0)+αO+βOO+... is a valid, convergent perturbation series whose truncation at second order is quantitatively accurate at physical SOC. This is not established, and the paper's own data supply counterexamples. In PtMnSb (Fig. 8), the fitted third-order coefficient for χ_y^y is +1.932, larger in magnitude than the second-order coefficient -0.308, and for t_z^y the third-order coefficient (-0.113) is comparable to the second-order one (-0.140). Thus at ξ=ξ0 the 'leading second-order SOC' statement is true only in a formal symmetry sense, not as a quantitative description of the response. For C4v, the Ti/Ni angular fit in Fig. 4 has R²≈0.56 for t_x^x, and the text concedes that a higher-order SOC expansion is required for t_x^x and t_x^y. Since the expansion coefficients are obtained by fitting the same data later presented as validation, the quantitative agreement claimed in the abstract is not an independent test. The symmetry statement that first-order terms vanish in PtMnSb may survive, but the stronger quantitative claim of a universal SOC-order classification is not secured.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a spin-group symmetry framework to classify the Edelstein effect and spin-orbit torques (SOTs) in collinear ferromagnets by expanding the response tensors in powers of spin-orbit coupling (SOC). It derives symmetry-allowed forms for C4v and C3v point groups, shows that conventional damping-like and field-like torques arise at first order in SOC with additional second-order terms, and analyzes PtMnSb where first-order SOC contributions vanish by symmetry. The analytic forms are fitted to first-principles KKR calculations for Ti/Ni, Pt/CoFe, Pt/Co(111), and PtMnSb, and field-free switching is illustrated for the 3m torque via LLG simulations.","tokens_in":16886,"tokens_out":2423,"duration_ms":29603,"significance":"If the central classification is correct, the paper offers a useful conceptual advance: it ties the conventional FL/DL torque phenomenology to a definite SOC perturbation order and provides symmetry-allowed functional forms that are independently checkable. The symmetry derivation is the strongest part and appears plausible, and the paper explicitly makes falsifiable predictions (e.g., vanishing first-order Edelstein terms in PtMnSb). However, the quantitative claims of 'excellent agreement' are not established because the expansion coefficients are fitted to the same data used for validation and because the paper's own PtMnSb and C4v results show that third-order SOC terms can be comparable to or larger than second-order terms. The classification is nonetheless worth publishing after the quantitative claims are appropriately reframed or supported by additional evidence.","major_comments":[{"comment":"The expansion χ=χ^(0)+αO+βOO+... is introduced from Ref. [7] without proof that it is a convergent order-by-order perturbation series whose symmetry-allowed terms are complete at each order. The paper never specifies its radius of validity in ξ, and the Pt/CoFe and PtMnSb fits show that higher-order terms are not small at physical SOC. This is load-bearing because the central claim of a universal SOC-order classification depends on this expansion. The authors should state the assumptions under which the expansion is controlled, or explicitly restrict the conclusions to a leading-order symmetry classification rather than a quantitative one.","section":"Section II, SOC expansion of Edelstein effect"}],"minor_comments":[{"comment":"Many equations are typeset with inconsistent subscripts/superscripts (e.g., χ_x^x vs. χ_y^y, t_x^y vs. t_y^x) and the transformation rules for the α and β tensors appear to contain typographical errors in the determinant factors. A careful revision of all displayed equations is needed.","section":"Throughout"},{"comment":"The method section does not state how the SOC strength ξ/ξ0 is varied in the KKR calculations (e.g., rescaling the speed of light and which terms are scaled), nor how the fitting to polynomials in ξ/ξ0 is performed. Reporting the fitting procedure and uncertainties would strengthen the quantitative claims.","section":"Section III, computational methods"},{"comment":"The matrix forms for the PtMnSb Edelstein tensors are garbled in the text and hard to read; the definition of the coordinate frame and the chosen m//x direction should be stated explicitly before presenting the matrices.","section":"PtMnSb subsection"},{"comment":"The LLG simulation results in Fig. 7 lack details such as the damping constant, the current pulse shape, and the anisotropy model. These details are needed for reproducibility.","section":"Section III, LLG simulation"},{"comment":"Reference [7] is used as the basis for the SOC-vector expansion, but the precise relation between the present formalism and Ref. [7]'s derivation should be clarified, especially since the manuscript says the expansion is 'introduced' rather than derived.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The symmetry framework is novel and fits the journal's scope, and the paper ships an extensive set of first-principles calculations. My main concern is that the quantitative validation is circular and the data themselves show convergence problems at third order in PtMnSb and poor fits for some C4v torkances. These issues are fixable by reframing the claims as a symmetry classification rather than a quantitative SOC-order expansion, and by adding independent validation. I recommend major revision rather than rejection, because the central symmetry derivation appears sound and the over-claims can be corrected within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague — this paper's real contribution is a symmetry-based ordering of Edelstein and SOT terms by power of spin-orbit coupling. The spin-group derivation for C4v and C3v is a legitimate, non-circular advance: the functional forms at each order follow from symmetry, not from the numerics, and it gives a clean interpretation of why conventional DL and FL torques appear at first order and the 3m torque at second. The PtMnSb prediction — that zeroth- and first-order contributions vanish by symmetry — is the sharpest new claim, and it is worth testing experimentally.\n\nThe paper does honest work in places. The Ti/Ni and Pt/CoFe SOC-scaling fits show the expected hierarchy for a weak-SOC versus a strong-SOC system. The layer-resolved analysis gives a plausible orbital-to-spin conversion story for Ti/Ni. The field-free switching discussion for C3v is a nice application even if the LLG simulation is only illustrative.\n\nThe soft spots are real and concentrated in the quantitative claims. The \"excellent quantitative agreement\" in the abstract is not an independent test: the expansion coefficients are fit to the same first-principles data that are then presented as validation. The C4v angular fits are visibly imperfect — the t_x^x fit has R² about 0.56, and the text concedes higher-order terms are needed for some components. The PtMnSb case is more damaging: the fitted third-order coefficient for χ_y^y is +1.93 versus -0.31 for second order, and the t_z^y third-order coefficient is comparable to the second-order one. So the statement that the leading SOT emerges at second order is only true in a formal symmetry sense; quantitatively, the second-order truncation is not reliable at physical SOC in this material. The paper itself acknowledges the sizable third-order terms, but the abstract does not.\n\nThere are also presentation weaknesses: the second-order derivations are relegated to Supplementary Note 1, and no code or data are shipped, so the reproducibility of the fits is limited. The SOC-vector expansion is taken from Ref. [7], and the paper does not justify convergence or the truncation order for strong-SOC systems.\n\nWho is this for? People working on SOT symmetry and material search will get real value from the ordering scheme and the PtMnSb idea. It deserves a serious referee — conditional acceptance at best, with the quantitative claims softened and the PtMnSb truncation discussed honestly. I would not cite it for the quantitative scaling, but I would cite the symmetry classification if it survives review.","headline":"A genuinely useful symmetry-based ordering of SOT by SOC order, but the quantitative claims outrun the evidence because the fits and the validation use the same DFT data, and the PtMnSb fit itself shows the second-order truncation failing.","tokens_in":17450,"tokens_out":2103,"would_cite":false,"duration_ms":21506,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Spin-group symmetry assigns field-like and damping-like torques to specific orders of spin-orbit coupling, and predicts a ferromagnet where the leading torque appears only at second order.","keywords":["spin-orbit torque","Edelstein effect","spin-group symmetry","collinear ferromagnets","spin-orbit coupling order","field-like torque","damping-like torque","first-principles calculations"],"falsifier":"Measure or compute the SOC-strength dependence of the torkance in PtMnSb at small SOC scaling: if any torque component grows linearly with SOC strength, the claimed vanishing of first-order SOC is wrong, whereas quadratic leading behavior supports the classification.","tokens_in":1718,"feed_emoji":"🧲","tokens_out":2189,"duration_ms":69572,"temperature":0.7,"pith_summary":"This paper tries to give a symmetry-based account of the Edelstein effect and spin-orbit torque in collinear ferromagnets by expanding the response tensors order by order in spin-orbit coupling strength. It argues that the conventional field-like and damping-like torques are not independent phenomena but carry specific perturbation orders: both appear at first order in spin-orbit coupling, and both receive second-order corrections. First-principles calculations on Ti/Ni and Pt/CoFe bilayers are used to validate the predicted SOC scaling and magnetization-angle dependence. The paper further shows that in the cubic ferromagnet PtMnSb, symmetry forces the zeroth- and first-order SOC contributions to vanish identically, leaving second-order SOC as the leading torque mechanism. If correct, the work gives experimenters a practical rule for when conventional torque forms apply and when they fail.","feed_headline":"Field-like and damping-like torques now have fixed spin-orbit orders","feed_subtitle":"A symmetry expansion in spin-orbit coupling explains torque data in Ti/Ni, Pt/CoFe, and PtMnSb.","key_machinery":"The load-bearing object is the SOC-vector expansion of the Edelstein tensor together with spin-group transformation rules for the expansion coefficients. The SOC vectors $O_a$ encode how spin-orbit coupling breaks spin symmetry in each spin-space and orbital channel; the spin-group operation $\\{U||R\\}$ acts on the tensor coefficients, and enforcing the spin-only group plus the nontrivial spin group of the crystal point group leaves only the symmetry-allowed terms at zeroth, first, and second order in SOC. This machinery converts the microscopic transport problem into a finite list of allowed tensor forms, each labeled by its power of SOC, which the paper then compares with first-principles transport data.","core_discovery":"The central claim is that, in collinear ferromagnets, the Edelstein tensor and torkance can be organized by a spin-group symmetry expansion in SOC vectors, $\\chi = \\chi^{(0)} + \\alpha O + \\beta O O + \\cdots$, where the spin group constrains which tensor components survive at each order. For the $4mm$ ($C_{4v}$) point group, the constraints reduce the first-order time-reversal-odd coefficients to a single independent parameter and the time-reversal-even coefficients to two, so the induced spin density and torque take the conventional field-like and damping-like forms at first order in SOC, with additional torque forms appearing at second order. The paper further shows that in cubic $-43m$ PtMnSb, zeroth- and first-order SOC contributions vanish identically by symmetry, so the leading Edelstein effect and spin-orbit torque emerge at second order in SOC. First-principles KKR calculations on Ti/Ni, Pt/CoFe, Pt/Co(111), and PtMnSb reproduce the predicted SOC-scaling and angular dependence of the Edelstein tensors and torkances.","pith_inferences":["The same symmetry logic likely extends to other spin-charge conversion responses, because the spin Hall conductivity and the Edelstein tensor obey identical spin-group constraints; this could unify spin Hall, orbital Hall, and Rashba-Edelstein descriptions under one SOC-order classification.","The second-order truncation may become insufficient for very strong spin-orbit coupling, since the paper itself finds sizable third-order SOC contributions in PtMnSb; a testable extension is whether Pt/CoFe also develops non-negligible third-order terms at realistic SOC strengths.","The symmetry criterion for field-free switching can be used as a materials-screening rule: point groups with two perpendicular mirror planes or high rotational symmetry suppress the needed second-order torque, while lower-symmetry groups admit it."],"forward_implications":["In $C_{4v}$ and $C_{\\infty v}$ bilayers, the conventional damping-like and field-like torques are first-order SOC effects, so they should scale approximately linearly with SOC strength in weak-SOC systems, with deviations revealing second-order contributions.","In strong-SOC systems such as Pt/CoFe, second-order SOC terms are comparable in magnitude to first-order ones, so quantitative torque modeling must include them; the paper provides fitted first- and second-order coefficients for Ti/Ni and Pt/CoFe.","In $C_{3v}$ systems, a symmetry-allowed second-order torque known as the 3m torque appears, and together with the conventional damping-like torque it enables deterministic field-free switching of perpendicular magnetization, as shown by LLG simulation for Pt/Co(111).","In PtMnSb, conventional field-like and damping-like torque forms are forbidden at zeroth and first order in SOC, so any observed spin-orbit torque must arise from second- and higher-order SOC, giving a concrete system where the standard SOT phenomenology breaks down."],"supporting_citations":[{"why":"Introduces the SOC vectors $O_a$ and the spin-group expansion used to order the Edelstein tensor by powers of SOC.","marker":"[7]"},{"why":"Provides the expansion of Edelstein and torkance tensors in powers of the magnetic order parameter m, whose T-even/T-odd transformation rules the paper adapts.","marker":"[3]"},{"why":"Defines the conventional damping-like and field-like torque forms whose SOC-order assignment is the paper's central result.","marker":"[1]"},{"why":"Establishes the Edelstein effect as the linear-response spin density induced by an electric field.","marker":"[2]"},{"why":"Identifies Ti/Ni as orbital-Hall-dominated and Pt/CoFe as spin-Hall-dominated, the two C4v systems used for first-principles validation.","marker":"[6]"},{"why":"Supplies the KKR Green's function formalism used to compute Edelstein tensors and torkances.","marker":"[8]"},{"why":"Provides the alloy-analogy finite-temperature model used to evaluate torkances at 300 K.","marker":"[12]"}],"fun_headline_variants":["Spin-group symmetry fixes torque orders in ferromagnets","Torque orders set by spin-group symmetry, not just spin Hall","Symmetry forces torque into second order in PtMnSb","Spin-group theory pinpoints SOC order for each torque","Edelstein effect and torques: now pinned to SOC order"],"cache_read_input_tokens":19456,"weakest_assumption_plain":"The paper assumes the SOC expansion of the Edelstein tensor is a valid, order-by-order perturbation series whose symmetry-allowed terms at each order are complete, and that stopping at second order is accurate even when spin-orbit coupling is strong.","fun_headline_variants_meta":{"raw":{"variants":["Spin-group symmetry fixes torque orders in ferromagnets","Torque orders set by spin-group symmetry, not just spin Hall","Symmetry forces torque into second order in PtMnSb","Spin-group theory pinpoints SOC order for each torque","Edelstein effect and torques: now pinned to SOC order"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000789,"raw_usage":{"total_tokens":3553,"prompt_tokens":1094,"completion_tokens":2459,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":710,"completion_tokens_details":{"reasoning_tokens":2375}},"tokens_in":710,"tokens_out":2459,"duration_ms":19156,"temperature":1.0,"reasoning_tokens":2375,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T17:30:03.519050+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure or compute the SOC-strength dependence of the torkance in PtMnSb at small SOC scaling: if any torque component grows linearly with SOC strength, the claimed vanishing of first-order SOC is wrong, whereas quadratic leading behavior supports the classification.","supporting_citations":[{"cited_title":"Calculating linear-response functions for finite temperatures on the basis of the alloy analogy model","cited_arxiv_id":null,"evidence_quote":"Establishes the Edelstein effect as the linear-response spin density induced by an electric field."}],"review_version":1}