Spin-group theory on Edelstein effect and spin-orbit torque in Collinear Ferromagnets

Jiahao Shentu; Jia Zhang; Jie Li; Qing Zhang; Yizhuo Song

REVIEW 1 major objections 5 minor 2 references

Spin-group theory on Edelstein effect and spin-orbit torque in Collinear Ferromagnets

T0 review · 1 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2608.06964 v1 pith:3VMCBK3N submitted 2026-08-07 cond-mat.mtrl-sci

Yizhuo Song , Qing Zhang , Jiahao Shentu , Jie Li , Jia Zhang This is my paper · ORCID

classification cond-mat.mtrl-sci

keywords spin-orbittorqueEdelsteineffectspin-groupsymmetrycollinearferromagnetscouplingorderfield-likedamping-likefirst-principlescalculations

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

The 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.

There 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.

Who 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.

Referee Report

1 major / 5 minor

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.

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 (1)

[Section II, SOC expansion of Edelstein effect] 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.

minor comments (5)

[Throughout] 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 III, computational methods] 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.
[PtMnSb subsection] 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 III, LLG simulation] 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.
[References] 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.

Circularity Check

2 steps flagged · score 4.0 of 10

Symmetry-derived tensor forms are independent, but the quantitative 'excellent agreement' and first-vs-second-order dominance claims come from least-squares fits to the same first-principles data they are said to validate.

fitted input called prediction [Section III, subsection 'Edelstein tensor and torkance as a function of SOC strength', Fig. 2 and Table I]
"As depicted in Fig. 2, for both bilayer heterostructures, all Edelstein tensors and torkances are well reproduced by an SOC expansion truncated at second order. The corresponding fitting parameters λ and η are tabulated in Table I. In the orbital-Hall-dominated Ti/Ni bilayer, the first-order Edelstein tensors are substantially larger in magnitude than their second-order coefficients."

The coefficients λ and η in Table I are obtained by fitting the same first-principles ξ/ξ0 data shown in Fig. 2; the dashed curves are the fitted polynomials. The conclusion that first-order SOC 'overwhelmingly' dominates in Ti/Ni, and the abstract's claim of 'excellent quantitative agreement with first-principles calculations,' are therefore statements about the fitted parameterization, not independent predictions. The symmetry-derived expansion structure is independent of the data, but the numerical coefficient values and the resulting dominance hierarchy are fit-determined.
fitted input called prediction [Section III, subsection 'Magnetization direction m dependence of Edelstein effect and SOT', Fig. 4 and Fig. 5 captions]
"To validate the symmetry-derived tensor forms for spin density and torkance, we perform angular-dependent calculations on Edelstein tensors and torkances in Ti/Ni(001) and Pt/CoFe(001). ... Open symbols represent the first-principles results, while dashed lines show the corresponding fitting curves according to analytic expressions."

The angular expressions contain free coefficients (a_s, b_s, c_s, a_t, b_t, etc.) that are extracted by least-squares fitting to the very same first-principles angular data displayed as 'open symbols.' The dashed curves are therefore the fitted model, so their agreement measures fit quality rather than an out-of-sample prediction. The functional shapes (e.g., sin2φ, cos²φ) are genuine symmetry predictions and are not circular; however, the numerical 'excellent quantitative agreement' advertised in the abstract is not an independent test of the theory.

full rationale

The spin-group symmetry derivation itself is self-contained: the allowed Edelstein and torkance tensor forms at each SOC order follow from the {U||R} transformation rules, independent of any numerical data. The central classification of conventional DL/FL torques as first-order SOC effects, the additional second-order terms, and the C3v '3m torque' at second order are genuine symmetry predictions. No load-bearing self-citation chain is present; Ref. [6] is a self-citation but is used only to classify the Ti/Ni and Pt/CoFe systems as orbital-Hall- or spin-Hall-dominated and does not support the symmetry derivation. The circularity is confined to the quantitative validation: SOC-scaling coefficients and angular-dependent coefficients are fits to the same first-principles data that are later described as 'excellent quantitative agreement.' The PtMnSb results further show third-order fitted coefficients larger than second-order ones (e.g., λ_yy^(3)=1.932 vs λ_yy^(2)=-0.308), so the 'leading second-order' claim is a formal statement about the lowest nonvanishing order rather than a quantitatively dominant contribution. Overall, the core symmetry framework is independent and non-circular, but the quantitative emphasis in the abstract is weakened by same-data fitting, giving a partial circularity score of 4.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities. It relies on the prior spin-orbit-vector formalism of Ref. [7] and on a set of computational approximations. All quantitative coefficients are fitted to first-principles data, so the free-parameter count is substantial.

free parameters (4)

First- and second-order SOC expansion coefficients for Ti/Ni and Pt/CoFe (Table I: λ_xx, λ_xy, η_xx, η_xy) = 16 values, e.g., λ_xx^(1)=104.86, λ_xx^(2)=-37.40 (a0^2/e); η_xy^(1)=0.055, η_xy^(2)=0.828 (ea0) for Pt/CoFe
Fitted to SOC-scaling first-principles curves in Fig. 2.
C4v angular-dependence coefficients a_s, b_s, c_s, a_t, b_t, c_t and primed variants = e.g., a_t+a_t'=-0.0141, b_t+b_t'=0.145 for Ti/Ni; a_t+a_t'=-0.885, a_t'=-0.0619, b_t+b_t'=0.569, c_t'=-0.137…
Fitted to angular dependence of Edelstein tensors and torkances in Figs. 4 and 5.
C3v torkance parameters a_t+a_t' and a_3m for Pt/Co(111) = -0.880 and 0.104 ea0
Fitted to angular dependence in Fig. 6.
PtMnSb SOC expansion coefficients λ_yy^(2), λ_yy^(3), λ_zy^(2), λ_zy^(3) and corresponding η coefficients = e.g., λ_yy^(2)=-0.308, λ_yy^(3)=1.932
Fitted to SOC scaling curves in Fig. 8.

assumptions (5)

domain assumption The three spin-orbit vectors O_a (a=1,2,3) from Ref. [7] form a complete basis for expanding SOC-induced responses.
Invoked in Section II to expand Edelstein tensors in powers of SOC; no proof of completeness is given in the paper.
domain assumption The spin group of a collinear ferromagnet decomposes as GS = GNS × GSO, with the nontrivial spin group identical to the nonmagnetic point group.
Section II, used to impose symmetry constraints on the α and β coefficients.
ad hoc to paper The response tensors can be Taylor-expanded in SOC strength ξ and truncated at second or third order without missing leading-order symmetry-allowed terms.
Used throughout; justified only by fitting to first-principles data, not by a convergence proof.
domain assumption KKR-LSDA with atomic sphere approximation and lmax=3 gives quantitatively accurate Edelstein tensors and torkances.
Section III, the first-principles method used for all validation.
domain assumption The alloy analogy model at 300 K captures finite-temperature torkance for the systems studied.
Section III, used for torkance evaluation on a 200x200x1 k-mesh.

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Cite this review

Pith. "Pith review of Spin-group theory on Edelstein effect and spin-orbit torque in Collinear Ferromagnets." pith.science (2026). https://pith.science/paper/3VMCBK3N

@misc{pith2026260806964,
  author       = {Pith},
  title        = {Pith review of: Spin-group theory on Edelstein effect and spin-orbit torque in Collinear Ferromagnets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3VMCBK3N}},
  note         = {Machine review of arXiv:2608.06964}
}

read the original abstract

Current-induced spin-orbit torques (SOTs) are central to the electrical manipulation of magnetic order in spintronic devices. In transition-metal/collinear ferromagnet bilayers, field-like and damping-like torques have been described only phenomenologically via the spin or orbital Hall effect, lacking a rigorous symmetry-based foundation. The precise role of spin-orbit coupling (SOC) in both the Edelstein effect and SOTs has remained unresolved. Here we develop a spin-group symmetry theory for the Edelstein effect and SOTs in collinear ferromagnets, treating SOC as a symmetry-breaking perturbation. For 4mm (C4v) point group symmetry, we derive the full forms of field-like and damping-like torques, which arise predominantly from first- and second-order SOC. We further show that SOTs in both orbital-Hall-dominated Ti/Ni and spin-Hall-dominated Pt/CoFe bilayers originate at first-order SOC. Taking the 3m (C3v) torque as a paradigmatic example, we elucidate the role of second- and higher-order SOC torques in field-free switching of perpendicular magnetic anisotropy. Remarkably, in PtMnSb, we demonstrate that SOTs under certain point group symmetries deviate from the conventional form: zeroth- and first-order SOC contributions vanish identically, with the leading SOT emerging at second order. All symmetry-based predictions from spin-group theory are in excellent quantitative agreement with first-principles calculations. Our work establishes a unified symmetry framework for the microscopic understanding of the Edelstein effect and current-induced spin torques in ferromagnetic systems.

Figures

Figures reproduced from arXiv: 2608.06964 by the authors.

**Figure 4.** (a) The angular dependence of Edelstein tensors ,, xxz  x y x , and (b) torkances ,, xxz x y x ttt with magnetic moment rotating within xy plane in Ti/Ni(001) bilayer. ϕ is defined as the angle between the magnetization and x-axis. Open symbols represent the first-principles results, while dashed lines show the corresponding fitting curves according to analytic expressions. For angular dependence calculations of … view at source ↗

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Works this paper leans on

2 extracted references · 2 canonical work pages

[2]

Calculating linear-response functions for finite temperatures on the basis of the alloy analogy model

113zy =−() . Fig. 8 Edelstein tensors z y , y y and corresponding torkances y yt , z yt as a function of SOC scaling for PtMnSb, with magnetization aligned along the x direction. Open symbols represent the first -principles results, while dashed lines show the corresponding fitting curves. Blue curves: y y and z yt fitted up to third -order SOC; red c...

work page 2019
[5]

3m torque

(a) Angular dependence of the Edelstein tensors ,,xxz x y x , and (b) corresponding torkances ,,xxz x y xttt in Pt/CoFe(001) bilayers, with the magnetization confined to the xz plane. θ is defined as the angle between the magnetization and z-axis. Open symbols represent the first-principles results, while dashed lines show the corresponding fitting cur...

work page 2000

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [2]

Calculating linear-response functions for finite temperatures on the basis of the alloy analogy model

113zy =−() . Fig. 8 Edelstein tensors z y , y y and corresponding torkances y yt , z yt as a function of SOC scaling for PtMnSb, with magnetization aligned along the x direction. Open symbols represent the first -principles results, while dashed lines show the corresponding fitting curves. Blue curves: y y and z yt fitted up to third -order SOC; red c...

work page 2019

[2] [5]

3m torque

(a) Angular dependence of the Edelstein tensors ,,xxz x y x , and (b) corresponding torkances ,,xxz x y xttt in Pt/CoFe(001) bilayers, with the magnetization confined to the xz plane. θ is defined as the angle between the magnetization and z-axis. Open symbols represent the first-principles results, while dashed lines show the corresponding fitting cur...

work page 2000