Pith. sign in

REVIEW 2 major objections 4 minor 75 references

Odd-parity p-wave magnets decouple Ising exchange from magnetic beating and produce an intermediate 1/R Dzyaloshinskii-Moriya decay along nodal lines.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-14 09:29 UTC pith:YMUF23KJ

load-bearing objection Solid analytic RKKY paper that cleanly shows odd-parity p-wave order reverses the Ising/Heisenberg beating roles and produces a nonrelativistic DM_z plus a real intermediate 1/R nodal window; soft spots are standard continuum/stationary-phase limits, not load-bearing flaws. the 2 major comments →

arxiv 2607.10757 v1 pith:YMUF23KJ submitted 2026-07-12 cond-mat.mes-hall

Parity-driven RKKY decoupling and anomalous 1/R Dzyaloshinskii-Moriya interaction in p-wave magnets

Morteza Salehi , Tohid Farajollahpour This is my paper
classification cond-mat.mes-hall
keywords p-wave magnetsRKKY interactionDzyaloshinskii-Moriya interactionRashba spin-orbit couplingparity-driven decouplingnodal 1/R decaynon-collinear spintronics
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper shows that the odd-parity spin splitting of a two-dimensional p-wave magnet fundamentally rearranges the indirect RKKY coupling between magnetic impurities once Rashba spin-orbit coupling is present. Because the exchange field is odd under momentum reversal, the out-of-plane Ising channel is insulated from the macroscopic magnetic modulation and oscillates only at the ordinary Fermi wavevector, while the in-plane Heisenberg channels inherit a strong directional beating envelope. The same hybridized bands also generate a three-component Dzyaloshinskii-Moriya interaction: its out-of-plane piece is nonrelativistic and set by the p-wave momentum shift, whereas the in-plane pieces are purely Rashba-driven. Along the nodal directions the competition between vanishing p-wave shift and the residual Rashba gap produces an anomalous intermediate-distance window in which those in-plane chiral components decay as 1/R before recovering the conventional 2D 1/R^{2} tail. The result supplies a concrete microscopic route to directionally tunable non-collinear spin textures that cannot appear in ordinary ferromagnets or even-parity altermagnets.

Core claim

In a 2D p-wave magnet with Rashba spin-orbit coupling the odd parity of the exchange field reverses the usual assignment of spatial beating: the out-of-plane Ising interaction remains free of the macroscopic magnetic wavevector while the in-plane Heisenberg terms carry a strong cos(2k_M R) envelope; simultaneously the hybridized bands produce a three-component DM interaction whose out-of-plane piece is nonrelativistic and whose in-plane pieces exhibit a dimension-reducing 1/R decay along the nodal lines over an extended intermediate window before recovering 1/R^{2}.

What carries the argument

Analytical long-distance real-space Green’s functions obtained by a spin-dependent stationary-phase expansion of the hybridized continuum Hamiltonian; the odd real-space parity of G_z forces a sign reversal in the RKKY trace that routes the magnetic beating exclusively into the in-plane channels and regularizes the nodal DM response with the residual Rashba gap.

Load-bearing premise

The long-distance stationary-phase evaluation of the continuum Green’s functions, together with neglect of sub-leading Rashba phase shifts, remains accurate throughout the intermediate nodal window where the claimed 1/R crossover is supposed to live.

What would settle it

Measure the spatial decay of the in-plane DM interaction between two magnetic impurities placed exactly along a p-wave nodal line: if the envelope fails to follow 1/R over the window 1/Q ≪ R ≲ 1/(2k_R) and then reverts to 1/R^{2}, the central nodal claim is false.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The manuscript presents an analytical theory of the RKKY interaction between magnetic impurities in a two-dimensional p-wave magnet with Rashba spin-orbit coupling. Using a continuum two-band Hamiltonian and a real-space Green’s-function approach based on stationary-phase and residue calculus, the authors derive closed-form asymptotic expressions for the full exchange tensor. They show that the odd parity of the p-wave exchange field produces a structural decoupling: the out-of-plane Ising component J_zz is insulated from macroscopic p-wave beating and oscillates at the shifted Fermi wavevector, while the in-plane Heisenberg components exhibit directionally tunable beating. Hybridization further generates a three-component Dzyaloshinskii–Moriya interaction (with a nonrelativistic out-of-plane piece) together with symmetric off-diagonal anisotropies; along nodal lines the in-plane DM components display an intermediate-distance 1/R decay before recovering the conventional 2D 1/R^{2} asymptote.

Significance. If the analytic results hold, the work supplies a clear, parity-based distinction between odd-parity p-wave magnets and both conventional ferromagnets and even-parity d-wave altermagnets in the RKKY channel. The closed-form envelopes (Eqs. 24–26, 29–34), the explicit nonrelativistic origin of J_DM,z, and the regularized nodal 1/R window constitute concrete, falsifiable spatial predictions that can guide impurity-based probes and the design of anisotropic non-collinear textures. The thorough appendices (A–C) that retain the hybridized poles and supply the unapproximated O(k_R/k_F) corrections are a methodological strength and make the leading-order claims transparent and reproducible.

major comments (2)
  1. [III.D, Eqs. (35)–(37), Fig. 6] Sec. III.D, Eqs. (35)–(37) and Fig. 6: The intermediate 1/R window is a central claim. The linearization sin(2k_R R)≈2k_R R is formally correct for R≪1/k_R, yet the paper should quantify the relative error of this approximation (and of the leading Hankel asymptotics) across the stated window 1/Q≪R≲1/(2k_R). A short comparison of the leading envelope against the full Appendix-C expressions along a nodal cut would confirm that sub-leading phase and amplitude corrections do not erase the dimension-reducing crossover.
  2. [Appendix A, Sec. II.C] Appendix A and Sec. II.C: The stationary-phase evaluation retains only the forward/backward points and the standard 2D prefactor. Near the nodal lines the RSOC-hybridized contours change curvature; a brief remark on whether this modifies the stationary-phase amplitude (or the range of validity of k_F R≫1) inside the intermediate window would strengthen the quantitative reliability of the 1/R claim.
minor comments (4)
  1. [Abstract, Sec. III.C] The adjective “massive” used for the nonrelativistic p-wave shift (abstract, Sec. III.C) is potentially confusing; “macroscopic” or “O(1)” would be clearer.
  2. [Fig. 3] Fig. 3 caption and main text: the geometric quenching of J_xy, J_xz and J_DM,x at φ=0 is stated clearly, but a one-sentence reminder that this is a coordinate choice (not a physical vanishing of all chiral response) would help non-specialist readers.
  3. [Sec. III.A] In the comparison with d-wave altermagnets (Sec. III.A), a more precise citation of the corresponding envelopes from Refs. 70 and 71 would make the claimed “role reversal” sharper.
  4. [Title page] Typographical consistency: the arXiv identifier appears as 2607.10757 while the date line reads July 14, 2026; ensure journal submission metadata match.

Circularity Check

0 steps flagged

No significant circularity: closed-form RKKY tensor follows by direct stationary-phase evaluation of the model Green functions; no fitted parameters or load-bearing self-citations force the parity decoupling or 1/R window.

full rationale

The derivation begins from the continuum Hamiltonian (Eq. 1) with odd-parity Mp_k, constructs the exact 2x2 retarded Green function (Eqs. 4-7), Fourier-transforms under the stationary-phase approximation valid for kF R o o 1 (Appendix A, Eqs. 9-12), and obtains the RKKY tensor by the standard zero-temperature second-order trace (Eq. 13). The parity-driven sign flip Gz(R)Gz(-R) = -Gz^{2} is an immediate algebraic consequence of the odd real-space parity of Gz and is not assumed; it directly produces the role reversal between Jzz (Eq. 26) and the in-plane Heisenberg components (Eqs. 24-25). The three DM components (Eqs. 32-34) and the intermediate-distance nodal 1/R crossover (Sec. III.D, Eq. 36) likewise follow by substituting the hybridized poles κ = kR into the same asymptotic envelopes and linearizing sin(2kR R) for 1/kF o R o 1/(2kR). All numerical envelopes use fixed illustrative ratios (Q/kF, kR/kF) that are never adjusted to external data. Self-citations (e.g., to related altermagnet or Friedel work) supply only background context and do not enter the load-bearing algebra. The calculation is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The central claims rest on a standard continuum two-band model for a 2D p-wave magnet plus Rashba SOC, the usual second-order RKKY formula, and the large-distance stationary-phase evaluation of the Fourier transforms. No new particles or forces are postulated. Free parameters appear only as illustrative dimensionless ratios chosen for the figures; they do not enter the analytic expressions that constitute the claims.

free parameters (3)
  • Q/k_F (effective p-wave exchange strength) = 0.05–0.20 (illustrative)
    Dimensionless ratio used in all numerical plots (e.g., 0.20 or 0.05); chosen by hand to illustrate beating, not fitted to data.
  • k_R/k_F (effective Rashba strength) = 0.01–0.05 (illustrative)
    Dimensionless ratio used in plots (e.g., 0.05 or 0.01); chosen to realize a clear intermediate-distance window, not fitted.
  • impurity angle ϕ and p-wave angle β
    Geometric angles fixed for each figure panel to display directional dependence; free choices of orientation.
axioms (4)
  • domain assumption Effective continuum Hamiltonian Ĥ(k)=α_k σ_0 - M_k^p σ_z + λ(k_y σ_x - k_x σ_y) with odd-parity M_k^p = (J/k_F)(k_x cos β + k_y sin β)
    Taken as the microscopic starting point (Eq. 1); justified by prior literature on p-wave magnets but not re-derived here.
  • standard math Stationary-phase approximation for the 2D Fourier transform at large k_F R ≫ 1, reducing angular integrals to Hankel functions evaluated at the hybridized poles k_± = k_0 ± κ
    Standard asymptotic technique (Appendix A); controls all real-space Green’s functions and therefore every RKKY component.
  • domain assumption Zero-temperature second-order perturbation theory for the RKKY Hamiltonian expressed as the energy integral of the trace over Pauli matrices of G(R) and G(-R)
    Classic RKKY formula (Eq. 13); assumes weak impurity-host exchange J_imp and neglects higher-order Kondo or multi-impurity effects.
  • ad hoc to paper Neglect of sub-leading O(k_R/k_F) amplitude and phase corrections when quoting the leading envelopes in the main text
    Explicitly stated in Sec. III and Appendix C; necessary to isolate the macroscopic beating but limits quantitative accuracy when k_R is not small.

pith-pipeline@v1.1.0-grok45 · 28563 in / 3417 out tokens · 40904 ms · 2026-07-14T09:29:04.187959+00:00 · methodology

0 comments
Cite this review

Pith. "Pith review of Parity-driven RKKY decoupling and anomalous $1/R$ Dzyaloshinskii-Moriya interaction in $p$-wave magnets." pith.science (2026). https://pith.science/paper/YMUF23KJ

@misc{pith2026260710757,
  author       = {Pith},
  title        = {Pith review of: Parity-driven RKKY decoupling and anomalous $1/R$ Dzyaloshinskii-Moriya interaction in $p$-wave magnets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YMUF23KJ}},
  note         = {Machine review of arXiv:2607.10757}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

Unconventional $p$-wave magnets, characterized by an odd-parity momentum-dependent spin splitting, offer a fundamentally distinct paradigm for non-collinear spintronics. Here, we theoretically investigate the Ruderman-Kittel-Kasuya-Yosida indirect exchange in a two-dimensional $p$-wave magnet subjected to Rashba spin-orbit coupling. Using an analytical real-space Green's function formalism, we uncover a parity-driven spatial decoupling in the magnetic response. Because of the odd-parity exchange field, the out-of-plane Ising interaction is structurally insulated from the macroscopic $p$-wave modulation, oscillating isotropically at the shifted Fermi wavevector. Conversely, the in-plane Heisenberg components exhibit pronounced, directionally tunable spatial beating. Beyond collinear exchange, the hybridized bands generate a highly tunable, three-component Dzyaloshinskii-Moriya interaction alongside symmetric off-diagonal anisotropies. We reveal that the out-of-plane chiral twisting is driven by the massive, nonrelativistic $p$-wave momentum shift, while the in-plane chiral components are strictly relativistic. Furthermore, the competition between the $p$-wave nodal geometry and the Rashba gap drives an anomalous, dimension-reducing crossover, in which the in-plane chiral components follow a 1D-like $1/R$ spatial decay along the nodal lines over an extended intermediate-distance window before ultimately recovering the conventional 2D $1/R^2$ asymptote. These findings establish $p$-wave magnets as promising platforms for engineering robust, directionally tunable non-collinear spin textures.

Figures

Figures reproduced from arXiv: 2607.10757 by Morteza Salehi, Tohid Farajollahpour.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Schematic illustration of 2D [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Spatial dependence of the normalized diagonal RKKY interaction components in a two-dimensional [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Evolution of the normalized RKKY interaction ten [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Spatial dependence of the normalized symmetric off-diagonal RKKY interaction components in a two-dimensional [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Spatial dependence of the normalized DM interaction components in a two-dimensional [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Spatial decay of the absolute in-plane DM interac [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

75 extracted references · 16 canonical work pages

  1. [1]

    Marder, M. P. , title =. 2010 , type =

    2010

  2. [2]

    Tsymbal, E.Y. and. Spintronics Handbook, Second Edition: Spin Transport and Magnetism: Volume Two: Semiconductor Spintronics , publisher =. 2019 , type =

    2019

  3. [3]

    , title =

    Blundell, S. , title =. 2001 , type =

    2001

  4. [4]

    L. Phys. Rev. X , volume =. 2022 , doi =

    2022

  5. [5]

    arXiv:2409.10034 , year =

    Jungwirth, Tomas and Fernandes, Rafael M and Sinova, Jairo and Smejkal, Libor , title =. arXiv:2409.10034 , year =. doi:10.48550/arXiv.2409.10034 , type =

    Pith/arXiv arXiv doi:10.48550/arxiv.2409.10034

  6. [6]

    Bender, C. M. and Orszag, S. A. , title =. doi:10.1007/978-1-4757-3069-2 , year =

    doi:10.1007/978-1-4757-3069-2

  7. [7]

    I. I. Mazin , title =. Phys. Rev. X , volume =. 2022 , doi =

    2022

  8. [8]

    Mazin, I. I. , title =. Phys. Rev. B , volume =. 2023 , type =. doi:10.1103/PhysRevB.107.L100418 , url =

    doi:10.1103/physrevb.107.l100418 2023

  9. [9]

    Broken Kramers Degeneracy in Altermagnetic MnTe , journal =

    Lee, Suyoung and Lee, Sangjae and Jung, Saegyeol and Jung, Jiwon and Kim, Donghan and Lee, Yeonjae and Seok, Byeongjun and Kim, Jaeyoung and Park, Byeong Gyu and. Broken Kramers Degeneracy in Altermagnetic MnTe , journal =. 2024 , type =. doi:10.1103/PhysRevLett.132.036702 , url =

    doi:10.1103/physrevlett.132.036702 2024

  10. [10]

    and Souma, S

    Osumi, T. and Souma, S. and Aoyama, T. and Yamauchi, K. and Honma, A. and Nakayama, K. and Takahashi, T. and Ohgushi, K. and Sato, T. , title =. Phys. Rev. B , volume =. 2024 , type =. doi:10.1103/PhysRevB.109.115102 , url =

    doi:10.1103/physrevb.109.115102 2024

  11. [11]

    Orlova, N. N. and Avakyants, A. A. and Timonina, A. V. and Kolesnikov, N. N. and Deviatov, E. V. , title =. JETP Letters , volume =. 2024 , type =. doi:10.1134/S0021364024602926 , url =

    doi:10.1134/s0021364024602926 2024

  12. [12]

    Herasymchuk, Arsen and Hallberg, Karl Bergson and Hodt, Erik Wegner and Linder, Jacob and Gorbar, E. V. and Sukhachov, Pavlo , journal =. 2025 , month =. doi:10.1103/3sw6-y8vf , title =

    doi:10.1103/3sw6-y8vf 2025

  13. [13]

    Das, Sachchidanand and Suri, Dhavala and Soori, Abhiram , title =. J. Phys. Condens. Matter , volume =. 2023 , type =. doi:10.1088/1361-648X/acea12 , url =

    doi:10.1088/1361-648x/acea12 2023

  14. [14]

    2024 , type =

    Theory of Tunneling Spectroscopy in Unconventional p-Wave Magnet-Superconductor Hybrid Structures , journal =. 2024 , type =. doi:10.7566/JPSJ.93.114703 , author =

    work page doi:10.7566/jpsj.93.114703 2024

  15. [15]

    2025 , type =

    Altermagnetism: An unconventional spin-ordered phase of matter , journal =. 2025 , type =. doi:10.1016/j.newton.2025.100162 , author =

    doi:10.1016/j.newton.2025.100162 2025

  16. [16]

    2021 , type =

    Prediction of unconventional magnetism in doped FeSb2 , journal =. 2021 , type =. doi:10.1073/pnas.2108924118 , author =

    doi:10.1073/pnas.2108924118 2021

  17. [17]

    Ding, Jianyang and Jiang, Zhicheng and Chen, Xiuhua and Tao, Zicheng and Liu, Zhengtai and Li, Tongrui and Liu, Jishan and Sun, Jianping and Cheng, Jinguang and Liu, Jiayu and Yang, Yichen and Zhang, Runfeng and Deng, Liwei and Jing, Wenchuan and Huang, Yu and Shi, Yuming and Ye, Mao and Qiao, Shan and Wang, Yilin and Guo, Yanfeng and Feng, Donglai and Sh...

    doi:10.1103/physrevlett.133.206401 2024

  18. [18]

    Anatomy of Dzyaloshinskii-Moriya Interaction at

    Yang, Hongxin and Thiaville, Andr. Anatomy of Dzyaloshinskii-Moriya Interaction at. Phys. Rev. Lett. , volume =. 2015 , type =. doi:10.1103/PhysRevLett.115.267210 , url =

    doi:10.1103/physrevlett.115.267210 2015

  19. [19]

    and Flensberg, K

    Bruus, H. and Flensberg, K. , title =

  20. [20]

    Observation of a spontaneous anomalous Hall response in the Mn5Si3 d-wave altermagnet candidate , journal =

    Reichlova, Helena and Lopes Seeger, Rafael and Gonz. Observation of a spontaneous anomalous Hall response in the Mn5Si3 d-wave altermagnet candidate , journal =. 2024 , type =. doi:10.1038/s41467-024-48493-w , url =

    doi:10.1038/s41467-024-48493-w 2024

  21. [21]

    and Han, L

    Bai, H. and Han, L. and Feng, X. Y and Zhou, Y. J and Su, R. X and Wang, Q. and Liao, L. Y and Zhu, W. X and Chen, X. Z and Pan, F. and Fan, X. L and Song, C. , title =. Phys. Rev. Lett. , volume =. 2022 , type =. doi:10.1103/PhysRevLett.128.197202 , url =

    doi:10.1103/physrevlett.128.197202 2022

  22. [22]

    doi:10.48550/arXiv.2309.01607 , author =

    arXiv:2309.01607 , year =. doi:10.48550/arXiv.2309.01607 , author =. 2309.01607 , archivePrefix =

    Pith/arXiv arXiv doi:10.48550/arxiv.2309.01607

  23. [23]

    Brekke and P

    B. Brekke and P. Sukhachov and H. G. Giil and A. Brataas and J. Linder , title =. Phys. Rev. Lett. , volume =. 2024 , doi =

    2024

  24. [24]

    Nat. Commun. , volume =. 2025 , eprint =. doi:10.1038/s41467-025-62516-0 , author =

    doi:10.1038/s41467-025-62516-0 2025

  25. [25]

    A. A. Hedayati and M. Salehi , title =. Phys. Rev. B , volume =. 2025 , doi =

    2025

  26. [26]

    2026 , type =

    Time reversal preserved spin valve and spin transistor based on unconventional p -wave magnets , journal =. 2026 , type =. doi:10.15302/frontphys.2026.085207 , author =

    work page doi:10.15302/frontphys.2026.085207 2026

  27. [27]

    Purely electrical detection of the spin-splitting vector in p -wave magnets based on linear and nonlinear conductivities , author =. Phys. Rev. B , volume =. 2025 , month =. doi:10.1103/jf1w-wdg8 , url =

    work page doi:10.1103/jf1w-wdg8 2025

  28. [28]

    Coexistence of p -wave magnetism and superconductivity , author =. Phys. Rev. B , volume =. 2025 , month =. doi:10.1103/PhysRevB.111.L220403 , url =

    work page doi:10.1103/physrevb.111.l220403 2025

  29. [29]

    M. A. Ruderman and C. Kittel , title =. Phys. Rev. , volume =. 1954 , doi =

    1954

  30. [30]

    Kasuya , title =

    T. Kasuya , title =. Prog. Theor. Phys. , volume =. 1956 , doi =

    1956

  31. [31]

    Yosida , title =

    K. Yosida , title =. Phys. Rev. , volume =. 1957 , doi =

    1957

  32. [32]

    , title =

    Yafet, Y. , title =. Phys. Rev. B , volume =. 1987 , type =. doi:10.1103/PhysRevB.36.3948 , url =

    work page doi:10.1103/physrevb.36.3948 1987

  33. [33]

    2014 , type =

    RKKY interaction on surfaces of topological insulators with superconducting proximity effect , journal =. 2014 , type =. doi:10.1103/PhysRevB.90.125443 , author =

    work page doi:10.1103/physrevb.90.125443 2014

  34. [34]

    2016 , type =

    Emergent phenomena induced by spin-orbit coupling at surfaces and interfaces , journal =. 2016 , type =. doi:10.1038/nature19820 , author =

    work page doi:10.1038/nature19820 2016

  35. [35]

    2012 , type =

    Dynamics of Dzyaloshinskii domain walls in ultrathin magnetic films , journal =. 2012 , type =. doi:10.1209/0295-5075/100/57002 , author =

    doi:10.1209/0295-5075/100/57002 2012

  36. [36]

    Chesi, Stefano and Loss, Daniel , title =. Phys. Rev. B , volume =. 2010 , type =. doi:10.1103/PhysRevB.82.165303 , url =

    work page doi:10.1103/physrevb.82.165303 2010

  37. [37]

    2007 , type =

    Gate-tunable Ruderman-Kittel-Kasuya-Yosida interaction mediated by low-dimensional electrons with Rashba spin-orbit coupling , journal =. 2007 , type =. doi:10.1063/1.2817405 , author =

    work page doi:10.1063/1.2817405 2007

  38. [38]

    , title =

    Kernreiter, T. , title =. Phys. Rev. B , volume =. 2013 , type =. doi:10.1103/PhysRevB.88.085417 , url =

    work page doi:10.1103/physrevb.88.085417 2013

  39. [39]

    Wang, Shi-Xiong and Chang, Hao-Ran and Zhou, Jianhui , title =. Phys. Rev. B , volume =. 2017 , type =. doi:10.1103/PhysRevB.96.115204 , url =

    work page doi:10.1103/physrevb.96.115204 2017

  40. [40]

    , title =

    Kogan, E. , title =. Phys. Rev. B , volume =. 2011 , type =. doi:10.1103/PhysRevB.84.115119 , url =

    work page doi:10.1103/physrevb.84.115119 2011

  41. [41]

    Shiranzaei, Mahroo and Cheraghchi, Hosein and Parhizgar, Fariborz , title =. Phys. Rev. B , volume =. 2017 , type =. doi:10.1103/PhysRevB.96.024413 , url =

    work page doi:10.1103/physrevb.96.024413 2017

  42. [42]

    Fukaya, Yuri and Lu, Bo and Yada, Keiji and Tanaka, Yukio and Cayao, Jorge , title =. J. Phys. Condens. Matter , volume =. 2025 , type =. doi:10.1088/1361-648X/adf1cf , url =

    work page doi:10.1088/1361-648x/adf1cf 2025

  43. [43]

    JETP Lett

    Properties of a 2D electron gas with lifted spectral degeneracy , author =. JETP Lett. , volume =. 1984 , url =

    1984

  44. [44]

    Dzyaloshinsky , title =

    I. Dzyaloshinsky , title =. J. Phys. Chem. Solids , volume =. 1958 , doi =

    1958

  45. [45]

    Moriya , title =

    T. Moriya , title =. Phys. Rev. , volume =. 1960 , doi =

    1960

  46. [46]

    Fert and P

    A. Fert and P. M. Levy , title =. Phys. Rev. Lett. , volume =. 1980 , doi =

    1980

  47. [47]

    Zhu and D.-X

    J.-J. Zhu and D.-X. Yao and S.-C. Zhang and K. Chang , title =. Phys. Rev. Lett. , volume =. 2011 , doi =

    2011

  48. [48]

    M. T. B. Phys. Rev. B , volume =. 1987 , doi =

    1987

  49. [49]

    Chang, Hao-Ran and Zhou, Jianhui and Wang, Shi-Xiong and Shan, Wen-Yu and Xiao, Di , title =. Phys. Rev. B , volume =. 2015 , type =. doi:10.1103/PhysRevB.92.241103 , url =

    work page doi:10.1103/physrevb.92.241103 2015

  50. [50]

    Observation of time-reversal symmetry breaking in the band structure of altermagnetic

    Fedchenko, Olena and Min. Observation of time-reversal symmetry breaking in the band structure of altermagnetic. Sci. Adv. , volume =. 2024 , type =. doi:10.1126/sciadv.adj4883 , url =

    doi:10.1126/sciadv.adj4883 2024

  51. [51]

    Amundsen and A

    M. Amundsen and A. Brataas and J. Linder , title =. Phys. Rev. B , volume =. 2024 , doi =

    2024

  52. [52]

    Yarmohammadi and U

    M. Yarmohammadi and U. Z. Phys. Rev. B , volume =

  53. [53]

    Sukhachov and J

    P. Sukhachov and J. Linder , title =. Phys. Rev. B , volume =. 2024 , doi =

    2024

  54. [54]

    Bode and M

    M. Bode and M. Heide and K. von Bergmann and P. Ferriani and S. Heinze and G. Bihlmayer and A. Kubetzka and O. Pietzsch and S. Bl. Nature (London) , volume =. 2007 , doi =

    2007

  55. [55]

    and Rader, O

    Bihlmayer, G. and Rader, O. and Winkler, R. , title =. New J. Phys. , volume =. 2015 , type =. doi:10.1088/1367-2630/17/5/050202 , url =

    work page doi:10.1088/1367-2630/17/5/050202 2015

  56. [56]

    S. M. Science , volume =. 2009 , doi =

    2009

  57. [57]

    Nagaosa and Y

    N. Nagaosa and Y. Tokura , title =. Nat. Nanotechnol. , volume =. 2013 , doi =

    2013

  58. [58]

    Fert and V

    A. Fert and V. Cros and J. Sampaio , title =. Nat. Nanotechnol. , volume =. 2013 , doi =

    2013

  59. [59]

    Schmitt and P

    M. Schmitt and P. Moras and G. Bihlmayer and R. Cotsakis and M. Vogt and J. Kemmer and A. Belabbes and P. M. Sheverdyaeva and A. K. Kundu and C. Carbone and S. Bl. Nat. Commun. , volume =. 2019 , doi =

    2019

  60. [60]

    Imamura and P

    H. Imamura and P. Bruno and S. Maekawa , title =. Phys. Rev. B , volume =. 2004 , doi =

    2004

  61. [61]

    Saremi , title =

    S. Saremi , title =. Phys. Rev. B , volume =. 2007 , doi =

    2007

  62. [62]

    A. M. Black-Schaffer , title =. Phys. Rev. B , volume =. 2010 , doi =

    2010

  63. [63]

    Sherafati and S

    M. Sherafati and S. Satpathy , title =. Phys. Rev. B , volume =. 2011 , doi =

    2011

  64. [64]

    Liu and C.-X

    Q. Liu and C.-X. Liu and C. Xu and X.-L. Qi and S.-C. Zhang , title =. Phys. Rev. Lett. , volume =. 2009 , doi =

    2009

  65. [65]

    Manchon and H

    A. Manchon and H. C. Koo and J. Nitta and S. M. Frolov and R. A. Duine , title =. Nat. Mater. , volume =. 2015 , doi =

    2015

  66. [66]

    Heinze and K

    S. Heinze and K. von Bergmann and M. Menzel and J. Brede and A. Kubetzka and R. Wiesendanger and G. Bihlmayer and S. Bl. Nat. Phys. , volume =. 2011 , doi =

    2011

  67. [67]

    S. S. P. Parkin and M. Hayashi and L. Thomas , title =. Science , volume =. 2008 , doi =

    2008

  68. [68]

    L. Sci. Adv. , volume =. 2020 , doi =

    2020

  69. [69]

    R. Gonz. Phys. Rev. Lett. , volume =. 2021 , doi =

    2021

  70. [70]

    2024 , type =

    Altermagnetic lifting of Kramers spin degeneracy , journal =. 2024 , type =. doi:10.1038/s41586-023-06907-7 , author =

    doi:10.1038/s41586-023-06907-7 2024

  71. [71]

    S. S. P. Parkin and D. Mauri , title =. Phys. Rev. B , volume =. 1991 , doi =

    1991

  72. [72]

    Zhou and J

    L. Zhou and J. Wiebe and S. Lounis and E. Vedmedenko and F. Meier and S. Bl. Nat. Phys. , volume =. 2010 , doi =

    2010

  73. [73]

    A. Cr. J. Magn. Magn. Mater. , volume =. 1998 , doi =

    1998

  74. [74]

    Fert and N

    A. Fert and N. Reyren and V. Cros , title =. Nat. Rev. Mater. , volume =. 2017 , doi =

    2017

  75. [75]

    Romming and C

    N. Romming and C. Hanneken and M. Menzel and J. E. Bickel and B. Wolter and K. von Bergmann and A. Kubetzka and R. Wiesendanger , title =. Science , volume =. 2013 , doi =

    2013