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REVIEW 2 major objections 5 minor 71 references

Phenomenology of altermagnets

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read All altermagnet effects can be described with an ordinary vector antiferromagnetic order parameter, without spin space groups or magnetic multipoles.

desk verdict 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. read the letter →

arxiv 2506.01823 v1 pith:VLVNFUUV submitted 2025-06-02 cond-mat.mtrl-sci

Maxim Mostovoy This is my paper · ORCID
classification cond-mat.mtrl-sci
keywords altermagnetsNéelvectorweakferromagnetismspinsplittingHalleffectspacegroupsLandautheorynon-collinearmagnetism
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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

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.

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 (2)
  1. [Non-collinear altermagnets (Eq. (4) and Table I)] 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.
  2. [Reduction to S1,S2,S3 and inversion assumption] 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).
minor comments (5)
  1. [p. 3] The word 'aproach' should be 'approach'.
  2. [Table I, CrSb/MnTe row] 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.
  3. [Eq. (2) and Table I] 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.
  4. [Eq. (7)] 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.
  5. [p. 6] The phrase 'non-nonlinear SHE effect' appears to be a typo; 'non-linear SHE effect' is presumably intended.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: symmetry invariants are constructed from magnetic space groups, not fitted to or defined by the target effects.

full rationale

The paper is a symmetry phenomenology. Its central items—Eq. (1) for non-relativistic spin splitting, Eq. (3) for the spin current, and the WFM/spin-splitting/SHE rows of Table I—are obtained by invariant-theoretic construction under the Turov sign-factor action of the magnetic space group (e.g., "Under a crystal symmetry operation, R, (l·s) is multiplied by σR. Invariance of δεkm implies g(Rk)=σR g(k)"). No material-specific coefficient (A, B, λ) is fitted; no quantity called a prediction is first used as an input. The non-collinear reduction S1+S2+S3=0, S_nbar=S_n, and the later V1,V2,V3 parametrization are changes of variables, not target results imported as assumptions. The self-citations ([52], [68], [69]) are contextual examples (hexagonal-manganite vortices, Cr2O3 magnetoelectricity, even/odd magnetic-order arguments) and are not load-bearing for the altermagnet classification. The one substantive caveat is completeness, not circularity: for the non-collinear rows of Table I (section "A similar symmetry analysis can be applied to non-collinear AFM conductors...") the paper asserts that the listed invariants exhaust all symmetry-allowed effects but gives no explicit enumeration against spin-space-group operations; a missed invariant would weaken the "all effects" claim. That is an omitted proof/correctness risk, not a reduction of the derivation to its own inputs. Hence no circular step is quotable, and the score is 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted numerical parameters; its constants A, B, lambda are material-specific coupling coefficients whose values are irrelevant to the symmetry classification. The framework relies on standard Landau symmetry analysis and on the domain assumption that spin-orbit coupling can be neglected for non-relativistic effects. For non-collinear orders, the parametrization by V1, V2 is a specific modeling choice.

free parameters (3)
  • A and B (spin-splitting and spin-current amplitudes) = not determined
    Appear in Eqs. (1)-(3), (6)-(8) as unspecified material-dependent coefficients; their values are not needed for the symmetry classification.
  • lambda (weak ferromagnetic coupling) = not determined
    Zeeman-type coupling coefficient for RuO2/MnF2; a known phenomenological constant taken from prior literature, not fitted here.
  • K (magnetic anisotropy) = K > 0
    Used to fix domains in Mn3Ir and alpha-MnTe; taken from prior literature, not derived in this paper.
assumptions (4)
  • standard math Macroscopic responses (gyration vector, Hall vector, spin currents) are fully determined by symmetry invariants built from the magnetic order parameter and applied fields.
    Core assumption of Landau phenomenological symmetry analysis, used throughout the paper.
  • domain assumption In the absence of spin-orbit coupling, spin is conserved along the Néel vector for collinear magnets, so spin splitting has the form (l.s)g(k).
    Used for Eq. (1); breaks down when spin-orbit coupling is significant.
  • ad hoc to paper For non-collinear 120-degree orders, the three spins S1,S2,S3 with sum zero can be represented by two orthogonal vectors V1,V2, and inversion maps sites n to -n with S_bar_n = S_n.
    This parametrization underlies Table I for Mn3Ir/Mn3Sn; it is a modeling choice specific to this paper's reduction of non-collinear to vector order.
  • standard math Time-reversal Onsager relations hold, including g(-k)=g(k) for spin splitting and reciprocal relations for conductivity.
    Standard symmetry principle invoked when constructing spin-splitting and transport invariants.

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Pith. "Pith review of Phenomenology of altermagnets." pith.science (2026). https://pith.science/paper/VLVNFUUV

@misc{pith2026250601823,
  author       = {Pith},
  title        = {Pith review of: Phenomenology of altermagnets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VLVNFUUV}},
  note         = {Machine review of arXiv:2506.01823}
}
read the original abstract

Altermagnets have recently emerged as a new class of magnetic materials sharing properties of both antiferromagnets and ferromagnets. Despite very small net magnetization, they show phenomena usually associated with ferromagnetism, such as the Faraday, Kerr and Anomalous Hall effects, resulting from the relativistic spin-orbit coupling, as well as the spin splitting of electron bands and Spin Hall Effect of non-relativistic origin. Spin space groups and magnetic multipoles are used to explain symmetry properties of altermagnets. Here, I show that the conventional phenomenological description in terms of a vector antiferromagnetic order parameter can be applied to all effects observed in altermagnets with collinear and non-collinear spin orders. I also discuss non-relativistic effects in non-altermagnets.

Figures

Figures reproduced from arXiv: 2506.01823 by the authors.

Figure 1
Figure 1. FIG. 1: (a) Non-collinear 120 [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗

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Works this paper leans on

71 extracted references · 68 canonical work pages

  1. [1]

    Nakatsuji, N

    S. Nakatsuji, N. Kiyohara, and T. Higo, Nature 527, 212 (2015)

    2015

  2. [2]

    A. K. Nayak et al., Sci. Adv. 2, e1501870 (2016)

    work page 2016

  3. [3]

    in Mn3Ir. The term, HxSx 1 + HySy 2 + HzSz 3, describing weak ferromagnetism, is invariant under 3 [111],2[001] and 2[110] rotations, which together with inversion and translations (not broken by the magnetic ordering) form generators of Pm¯3m space group. Similarly, the spin texture in reciprocal space near Γ-point (in non-collinear magnets spin is not c...

    work page

  4. [4]

    Guo and T.-C

    G.-Y . Guo and T.-C. Wang, Phys. Rev. B96, 224415 (2017)

    work page 2017

  5. [5]

    Ikhlas et al., Nat

    M. Ikhlas et al., Nat. Phys. 13, 1085 (2017)

    work page 2017

  6. [6]

    Li et al., Phys

    X. Li et al., Phys. Rev. Lett. 119, 056601 (2017)

    work page 2017

  7. [7]

    Higo et al., Nat

    T. Higo et al., Nat. Photonics 12, 73 (2018)

    work page 2018

  8. [8]

    Wu et al., Appl

    M. Wu et al., Appl. Phys. Lett. 116, 132408 (2020)

    work page 2020

Show all 71 references
  1. [9]

    Hayami, Y

    S. Hayami, Y . Yanagi, and H. Kusunose, J. Phys. Soc. Jpn.88, 123702 (2019)

    2019

  2. [10]

    L.-D. Yuan, Z. Wang, J.-W. Luo, E.I. Rashba, and A. Zunger, Phys. Rev. B 102, 014422 (2020)

    2020

  3. [11]

    Libor ˇSmejkal et al., Sci. Adv. 6, eaaz8809 (2020)

    2020

  4. [12]

    Bai et al., Phys

    H. Bai et al., Phys. Rev. Lett. 128, 197202 (2022)

    2022

  5. [13]

    Ding et al., Phys

    J. Ding et al., Phys. Rev. Lett. 133, 206401 (2024)

    2024

  6. [14]

    Reimers et al., Nat

    S. Reimers et al., Nat. Commun. 15, 2116 (2024)

    2024

  7. [15]

    Lee et al., Phys

    S. Lee et al., Phys. Rev. Lett. 132, 036702 (2024)

    2024

  8. [16]

    Krempask ´y et al., Nature 626, 517–522 (2024)

    J. Krempask ´y et al., Nature 626, 517–522 (2024)

    2024

  9. [17]

    Osumi et al., Phys

    T. Osumi et al., Phys. Rev. B 109, 115102 (2024)

    2024

  10. [18]

    Naka et al., Nat

    M. Naka et al., Nat. Commun. 10, 4305 (2019)

    2019

  11. [19]

    Gonz ´alez-Hern´andez et al., Phys

    R. Gonz ´alez-Hern´andez et al., Phys. Rev. Lett. 126, 127701 (2021)

    2021

  12. [20]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X12, 040501 (2022)

    2022

  13. [21]

    B. B. Krichevtsov, K. M. Mukimov, R. V . Pisarev, M. M. Ruvinshtejn, JETP Lett.34, 379–382 (1981)

    1981

  14. [22]

    A. V . Zenkov et al., Sov. Phys. JETP 69, 792-796 (1989)

    1989

  15. [23]

    K. B. Vlasov, et al., Fizika Tverdogo Tela 22 1656–1659 (1980)

    1980

  16. [24]

    A. V . Kimel, Th. Rasing, and B. A. Ivanov, J. Magn. Magn. Mater.598, 172039 (2024). 10

    2024

  17. [25]

    B. H. Rimmler, B. Pal, and S.S.P. Parkin, Nat. Rev. Mater. 10, 109–127 (2025)

    2025

  18. [26]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova and T. Jungwirth, Phys. Rev. X12, 031042 (2022)

    2022

  19. [27]

    P. Liu, J. Li, J. Han, X. Wan and Q. Liu, Phys. Rev. X 12, 021016 (2022)

    2022

  20. [28]

    Suzuki, T

    M.-T. Suzuki, T. Koretsune, M. Ochi and R. Arita, Phys. Rev. B 95, 094406 (2017)

    2017

  21. [29]

    Hayami, Y

    S. Hayami, Y . Yanagi and H. Kusunose, Phys. Rev. B102, 144441 (2020)

    2020

  22. [30]

    Electrodynamics of continuous media,

    L. D. Landau and E. M. Lifshitz, “Electrodynamics of continuous media,” (Pergamon, New York, 1984)

    1984

  23. [31]

    Symmetry and physical properties of antiferromagnetics,

    E. A. Turov, “Symmetry and physical properties of antiferromagnetics,” (Cambridge International Science Publishing, 2010)

    2010

  24. [32]

    Berlijn et al., Phys

    T. Berlijn et al., Phys. Rev. Lett. 118, 077201 (2017)

    2017

  25. [33]

    Z. H. Zhu et al., Phys. Rev. Lett. 122, 017202 (2019)

    2019

  26. [34]

    Feng et al., Nat

    Z. Feng et al., Nat. Electron. 5 735–743 (2022)

    2022

  27. [35]

    Liao, Y .-C

    C.-T. Liao, Y .-C. Wang, Y .-C. Tien, S.-Y . Huang, and D. Qu, Phys. Rev. Lett.133, 056701 (2024)

    2024

  28. [36]

    Bai et al., Phys

    H. Bai et al., Phys. Rev. Lett. 130, 216701 (2023)

    2023

  29. [37]

    Mukuda et al., Phys

    H. Mukuda et al., Phys. Rev. B 60, 12279 (1999)

    1999

  30. [38]

    Keßler et al., npj Spintronics, 2, 50 (2024)

    P. Keßler et al., npj Spintronics, 2, 50 (2024)

    2024

  31. [39]

    Smolyanyuk, I

    A. Smolyanyuk, I. I. Mazin, L. Garcia-Gassull, and R. Valent ´ı, Phys. Rev. B 109, 134424 (2024)

    2024

  32. [40]

    Liu et al., Phys

    J. Liu et al., Phys. Rev. Lett. 133, 176401 (2024)

    2024

  33. [41]

    Kiefer et al., J

    L. Kiefer et al., J. Phys.: Condens. Matter 37, 135801 (2025)

    2025

  34. [42]

    Plouff et al., npj Spintronics 3, 17 (2025)

    D. Plouff et al., npj Spintronics 3, 17 (2025)

    2025

  35. [43]

    G. P. Felcher and R. Kleb, Europhys. Lett. 36, 455 (1996)

    1996

  36. [44]

    N. F. Kharchenko, O. V . Miloslavskaya, and A. A. Milner, Low Temp. Phys.31, 825–830 (2005)

    2005

  37. [45]

    Higuchi and M

    T. Higuchi and M. Kuwata-Gonokami, Nat. Commun. 7, 10720 (2016)

    2016

  38. [46]

    Baruchel et al., J

    J. Baruchel et al., J. Phys. Colloques 49 C8-1895-C8-1896 (1988)

    1988

  39. [47]

    Takei, D

    W. Takei, D. E. Cox, and G. Shirane, Phys. Rev. 129, 2008 (1963)

    1963

  40. [48]

    Kunitomi, Y

    N. Kunitomi, Y . Hamaguchi, and S. Anzai, J. Phys. France 25, 568-574 (1964)

    1964

  41. [49]

    Kluczyk et al., Phys

    K. Kluczyk et al., Phys. Rev. B 110, 155201 (2024)

    2024

  42. [50]

    R. D. Gonzalez Betancourt et al., Phys. Rev. Lett. 130, 036702 (2023)

    2023

  43. [51]

    O. J. Amin et al., Nature 636, 348 (2024)

    2024

  44. [52]

    T. Choi, Y . Horibe, H. Yi, W. Wu and S. W. Cheong, Nat. Mater.9, 253–258 (2010)

    2010

  45. [53]

    Artyukhin, K

    S. Artyukhin, K. T. Delaney, N. A. Spaldin and M. Mostovoy, Nat. Mater. 13, 42–49 (2014). 11

    2014

  46. [54]

    Fedchenko et al., Sci

    O. Fedchenko et al., Sci. Adv. 10, eadj4883 (2024)

    2024

  47. [55]

    Bose et al., Nat

    A. Bose et al., Nat. Electron. 5, 267–274 (2022)

    2022

  48. [56]

    Karube et al., Phys

    S. Karube et al., Phys. Rev. Lett. 129, 137201 (2022)

    2022

  49. [57]

    ˇZelezn´y, Y

    J. ˇZelezn´y, Y . Zhang, C. Felser, and B. Yan, Phys. Rev. Lett.119, 187204 (2017)

    2017

  50. [58]

    D. F. Shao and E. Y . Tsymbal, npj Spintronics2, 13 (2024)

    2024

  51. [59]

    Szunyogh, B

    L. Szunyogh, B. Lazarovits, L. Udvardi, J. Jackson, and U. Nowak, Phys. Rev. B 79, 020403 (2009)

    2009

  52. [60]

    E. F. Bertaut, D. Fruchart, J. P. Bouchaud and R. Fruchart, Solid State Commun. 6, 251–256 (1968)

    1968

  53. [61]

    Fruchart, E

    D. Fruchart, E. F. Bertaut, R. Madar, G. Lorthioir, and R. Fruchart, Solid State Commun.9, 1793–1797 (1971)

    1971

  54. [62]

    Iwaki et al., Appl

    H. Iwaki et al., Appl. Phys. Lett. 116, 022408 (2020)

    2020

  55. [63]

    B. E. Zuniga-Cespedes et al., New J. Phys. 25, 023029 (2023)

    2023

  56. [64]

    Zhang et al., Adv

    K. Zhang et al., Adv. Funct. Mater. 2424472 (2025)

    2025

  57. [65]

    Boldrin et al., ACS Appl

    D. Boldrin et al., ACS Appl. Mater. Interfaces 10, 18863 (2018)

    2018

  58. [66]

    Tomiyoshi and Y

    S. Tomiyoshi and Y . Yamaguchi, J. Phys. Soc. Jpn.51, 2478–2486 (1982)

    1982

  59. [67]

    Jianpeng and L

    L. Jianpeng and L. Balents, Phys. Rev. Lett. 119, 087202 (2017)

    2017

  60. [68]

    Reichlova et al., Nat

    H. Reichlova et al., Nat. Commun. 10, 5459 (2019)

    2019

  61. [69]

    Mostovoy, A

    M. Mostovoy, A. Scaramucci, N. A. Spaldin, and K. T. Delaney, Phys. Rev. Lett.105, 087202 (2010)

    2010

  62. [70]

    Mostovoy, npj Spintronics 2, 18 (2024)

    M. Mostovoy, npj Spintronics 2, 18 (2024)

    2024

  63. [71]

    Hayami, Y

    S. Hayami, Y . Yanagi, and H. Kusunose, Phys. Rev. B101, 220403 (2020). 12

    2020

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.