pith. machine review for the scientific record.
sign in

arxiv: 2509.20120 · v1 · submitted 2025-09-24 · ❄️ cond-mat.mtrl-sci

Multipole analysis of spin currents in altermagnetic MnTe

Ryosuke Hirakida , Karma Tenzin , Chao Chen Ye , Berkay Kilic , Carmine Autieri , Jagoda S{\l}awi\'nska This is my paper

Pith reviewed 2026-05-18 14:39 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords altermagnetMnTemagnetic spin Hall effectmultipole analysisspin currentsNéel vectorspin-momentum locking
0
0 comments X

The pith

In altermagnetic MnTe the combination of intrinsic spin-orbit coupling and altermagnetic spin splitting produces a magnetic spin Hall angle reaching 16 percent.

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

The paper investigates charge transport and spin currents in the altermagnet α-MnTe by combining symmetry analysis in the multipole framework with fully relativistic first-principles Kubo calculations. It shows that Néel vectors aligned along the y or x directions create distinct magnetic order parameters that produce spin-momentum locking of opposite parities and magnetic spin Hall effects with different directional anisotropies. The calculations find that this combination yields a magnetic spin Hall angle as large as 16 percent, comparable to or larger than the value in platinum. The anisotropy of the effect supplies a practical experimental signature for distinguishing the type of order parameter present. Through the multipole lens the same method is presented as a general route for analyzing transport in the wider family of altermagnetic materials.

Core claim

Using multipole symmetry analysis together with relativistic Kubo transport calculations, the authors establish that in α-MnTe the choice of Néel-vector direction (N̂ ∥ y versus N̂ ∥ x) selects different order parameters; these in turn generate spin-momentum locking of opposite parities and magnetic spin Hall effects whose anisotropies differ, with the largest magnetic spin Hall angle reaching 16 percent when intrinsic spin-orbit coupling acts together with the altermagnetic spin splitting.

What carries the argument

The multipole framework that classifies the distinct order parameters arising from different Néel-vector orientations and links them to the resulting spin currents and magnetic spin Hall conductivities.

If this is right

  • Néel-vector orientation directly controls both the magnitude and the anisotropy of the magnetic spin Hall effect in MnTe.
  • The magnetic spin Hall angle of 16 percent makes MnTe competitive with heavy metals for efficient spin-current generation at zero net magnetization.
  • Measuring the anisotropy of the magnetic spin Hall effect provides an experimental handle to identify which order parameter is realized.
  • The multipole classification supplies a systematic way to predict spin-transport properties across other altermagnets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Device architectures could exploit the predicted anisotropy to electrically detect or switch between different altermagnetic states.
  • Similar large magnetic spin Hall angles may appear in other altermagnets once their multipole order parameters are mapped.
  • The same symmetry-plus-Kubo workflow can be applied to predict spin currents in altermagnetic heterostructures or under strain.

Load-bearing premise

The first-principles Kubo calculations correctly capture the altermagnetic spin splitting and the spin-orbit-driven transport without large errors from the exchange-correlation functional or finite-size approximations.

What would settle it

Experimental measurement of the magnetic spin Hall angle in oriented MnTe films that yields a value well below or far above 16 percent for the predicted Néel-vector orientations would falsify the calculated magnitude.

Figures

Figures reproduced from arXiv: 2509.20120 by Berkay Kilic, Carmine Autieri, Chao Chen Ye, Jagoda S{\l}awi\'nska, Karma Tenzin, Ryosuke Hirakida.

Figure 1
Figure 1. Figure 1: FIG. 1. Crystal and magnetic structure of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Spin-momentum locking of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Scattering rate Γ dependence of the dissipative electric conductivity [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: shows the energy dependence of the magnetic SHC σ s(J),s ij . Several general trends can be observed for both Nˆ ∥ y and Nˆ ∥ x. Although the band gap of α￾MnTe lies between 0.0 and 0.6 eV, SHC vanishes in an even larger region extending till approximately 0.8 eV; the finite magnitudes emerge only beyond this energy value. This behavior is likely due to the very small Fermi velocity and Berry curvature of … view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Magnetic spin Hall angle [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Spin-polarized band structure along the high-symmetry lines with spin polarization [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Dissipative electric conductivity [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Intrinsic SHC [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
read the original abstract

Altermagnets, a class of unconventional antiferromagnets where antiparallel spins are connected by combined rotational and translational symmetries, have recently emerged as promising candidates for spintronic applications, as they can efficiently generate spin currents while maintaining vanishing net magnetization. Here, we investigate charge transport and spin currents in $\alpha$-MnTe, a prototypical altermagnet, using symmetry analysis within the multipole framework and fully relativistic first-principles calculations using the Kubo formalism. Our results show that different magnetic configurations with N\'eel vectors $\hat{N}\parallel y$ and $\hat{N}\parallel x$ in MnTe induce distinct order parameters. This distinction gives rise to spin-momentum locking with different parities and magnetic spin Hall effects (magnetic SHEs) with different anisotropies. Strikingly, our calculations show that the combination of intrinsic spin-orbit coupling and altermagnetic spin splitting yields a large magnetic spin Hall angle of up to 16 \% rivaling or exceeding that of heavy metals such as Pt. On the other hand, the anisotropy of the magnetic SHE provides a practical means to identify the type of order parameter. This establishes, through the powerful framework of multipoles, a general approach for studying transport phenomena that extends to a broader class of altermagnets beyond MnTe.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript applies multipole symmetry analysis and fully relativistic first-principles Kubo-formula calculations to charge and spin transport in the altermagnet α-MnTe. It shows that Néel vectors along x versus y produce distinct magnetic order parameters, different-parity spin-momentum locking, and anisotropic magnetic spin Hall effects, with the combination of intrinsic SOC and altermagnetic splitting yielding a magnetic spin Hall angle reaching 16 %.

Significance. If the numerical results are robust, the work demonstrates that altermagnets can generate large spin currents without net magnetization and with efficiencies comparable to or exceeding those of heavy metals such as Pt. The multipole framework supplies a general, symmetry-based route to classify and predict such transport phenomena across the wider family of altermagnets.

major comments (1)
  1. Computational methodology section: The headline result is a magnetic spin Hall angle of up to 16 % obtained from the fully relativistic Kubo linear-response formula. No convergence data are reported for k-mesh density or the broadening parameter η. Because the altermagnetic spin splittings are only tens of meV, both parameters can shift the conductivity by tens of percent when the mesh fails to resolve Fermi-surface features or when η is comparable to the splitting; this directly affects the quantitative claim that the angle rivals Pt.
minor comments (2)
  1. Abstract: The maximum value of 16 % should be tied explicitly to a particular Néel-vector orientation or current direction so that the anisotropy statement can be checked against the later figures.
  2. Figure captions: Several panels lack explicit labels for the Néel-vector direction or the spin-current component being plotted, making it difficult to connect the plots directly to the multipole order parameters discussed in the text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comment regarding the computational methodology. We address the point below and will revise the manuscript to incorporate additional convergence information.

read point-by-point responses

  1. Referee: Computational methodology section: The headline result is a magnetic spin Hall angle of up to 16 % obtained from the fully relativistic Kubo linear-response formula. No convergence data are reported for k-mesh density or the broadening parameter η. Because the altermagnetic spin splittings are only tens of meV, both parameters can shift the conductivity by tens of percent when the mesh fails to resolve Fermi-surface features or when η is comparable to the splitting; this directly affects the quantitative claim that the angle rivals Pt.

    Authors: We agree that explicit convergence tests are important to substantiate the quantitative value of the magnetic spin Hall angle, especially given the scale of the altermagnetic splittings. In the revised manuscript we will expand the Computational Methodology section with a new paragraph (or subsection) presenting convergence data for both the k-mesh density and the broadening parameter η. These tests confirm that the reported spin Hall angle remains stable to within a few percent once the k-mesh exceeds the density used in the main calculations and for η values well below the altermagnetic splitting. This addition will directly address the referee’s concern and strengthen the reliability of the 16 % figure. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results from independent first-principles Kubo + multipole symmetry analysis

full rationale

The paper computes the magnetic spin Hall angle (up to 16%) via fully relativistic DFT and the Kubo linear-response formula applied to the electronic structure of α-MnTe, combined with symmetry analysis in the multipole framework. These steps rely on external computational codes and standard first-principles methodology rather than fitting parameters to the target observable or reducing via self-citation chains. No equation or claim in the abstract or described methodology reduces the reported value to an input by construction; the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard relativistic density-functional theory and linear-response Kubo transport; no new free parameters, ad-hoc entities, or non-standard axioms are introduced in the abstract.

axioms (2)
  • domain assumption Fully relativistic first-principles calculations within the Kubo formalism accurately describe spin currents in the presence of altermagnetic order.
    Invoked when the abstract states that calculations yield the reported spin Hall angles.
  • domain assumption Multipole symmetry analysis correctly classifies the distinct order parameters for Néel vectors along x versus y.
    Stated as the framework that gives rise to different parities and anisotropies.

pith-pipeline@v0.9.0 · 5791 in / 1497 out tokens · 30491 ms · 2026-05-18T14:39:58.698894+00:00 · methodology

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

67 extracted references · 67 canonical work pages

  1. [1]

    ˇSmejkal, R

    L. ˇSmejkal, R. Gonz´ alez-Hern´ andez, T. Jungwirth, and J. Sinova, Crystal time-reversal symmetry breaking and spontaneous Hall effect in collinear antiferromagnets, Sci. Adv.6, eaaz8809 (2020)

    work page 2020

  2. [2]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova, and T. Jungwirth, Beyond Con- ventional Ferromagnetism and Antiferromagnetism: A Phase with Nonrelativistic Spin and Crystal Rotation Symmetry, Phys. Rev. X12, 031042 (2022)

    work page 2022

  3. [3]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova, and T. Jungwirth, Emerging Re- search Landscape of Altermagnetism, Phys. Rev. X12, 040501 (2022)

    work page 2022

  4. [4]

    Hayami, Y

    S. Hayami, Y. Yanagi, and H. Kusunose, Momentum- dependent spin splitting by collinear antiferromagnetic ordering, J. Phys. Soc. Jpn.88, 123702 (2019)

    work page 2019

  5. [5]

    Hayami, Y

    S. Hayami, Y. Yanagi, and H. Kusunose, Bottom-up de- sign of spin-split and reshaped electronic band structures in antiferromagnets without spin-orbit coupling: Proce- dure on the basis of augmented multipoles, Phys. Rev. B 102, 144441 (2020)

    work page 2020

  6. [6]

    L.-D. Yuan, Z. Wang, J.-W. Luo, and A. Zunger, Predic- tion of low-Z collinear and noncollinear antiferromagnetic compounds having momentum-dependent spin splitting even without spin-orbit coupling, Phys. Rev. Mater.5, 014409 (2021)

    work page 2021

  7. [7]

    K.-H. Ahn, A. Hariki, K.-W. Lee, and J. Kuneˇ s, Antifer- romagnetism in RuO 2 as d-wave Pomeranchuk instabil- ity, Phys. Rev. B99, 184432 (2019)

    work page 2019

  8. [8]

    I. I. Mazin, K. Koepernik, M. D. Johannes, R. Gonz´ alez- Hern´ andez, and L.ˇSmejkal, Prediction of unconventional magnetism in doped FeSb 2, Proc. Natl. Acad. Sci.118, e2108924118 (2021)

    work page 2021

  9. [9]

    ˇSmejkal, A

    L. ˇSmejkal, A. B. Hellenes, R. Gonz´ alez-Hern´ andez, J. Sinova, and T. Jungwirth, Giant and tunneling mag- netoresistance in unconventional collinear antiferromag- nets with nonrelativistic spin-momentum coupling, Phys. Rev. X12, 011028 (2022)

    work page 2022

  10. [10]

    Bhowal and N

    S. Bhowal and N. A. Spaldin, Ferroically ordered mag- netic octupoles in d-wave altermagnets, Phys. Rev. X14, 011019 (2024)

    work page 2024

  11. [11]

    Mazin, Altermagnetism in MnTe: Origin, predicted manifestations, and routes to detwinning, Phys

    I. Mazin, Altermagnetism in MnTe: Origin, predicted manifestations, and routes to detwinning, Phys. Rev. B 107, L100418 (2023)

    work page 2023

  12. [12]

    Fakhredine, R

    A. Fakhredine, R. M. Sattigeri, G. Cuono, and C. Autieri, Interplay between altermagnetism and nonsymmorphic symmetries generating large anomalous Hall conductivity by semi-Dirac points induced anticrossings, Phys. Rev. B 108, 115138 (2023)

    work page 2023

  13. [13]

    Q. Cui, Y. Zhu, X. Yao, P. Cui, and H. Yang, Giant spin- Hall and tunneling magnetoresistance effects based on a two-dimensional nonrelativistic antiferromagnetic metal, Phys. Rev. B108, 024410 (2023)

    work page 2023

  14. [14]

    Q. Cui, B. Zeng, P. Cui, T. Yu, and H. Yang, Efficient spin Seebeck and spin Nernst effects of magnons in alter- magnets, Phys. Rev. B108, L180401 (2023)

    work page 2023

  15. [15]

    R. M. Fernandes, V. S. De Carvalho, T. Birol, and R. G. Pereira, Topological transition from nodal to nodeless Zeeman splitting in altermagnets, Phys. Rev. B109, 024404 (2024)

    work page 2024

  16. [16]

    Hariki, A

    A. Hariki, A. Dal Din, O. Amin, T. Yamaguchi, A. Badura, D. Kriegner, K. Edmonds, R. Campion, P. Wadley, D. Backes,et al., X-ray magnetic circular dichroism in altermagneticα-MnTe, Phys. Rev. Lett. 132, 176701 (2024)

    work page 2024

  17. [17]

    E. W. Hodt and J. Linder, Spin pumping in an altermagnet/normal-metal bilayer, Phys. Rev. B109, 174438 (2024)

    work page 2024

  18. [18]

    M. Naka, S. Hayami, H. Kusunose, Y. Yanagi, Y. Mo- tome, and H. Seo, Anomalous Hall effect inκ-type organic antiferromagnets, Phys. Rev. B102, 075112 (2020)

    work page 2020

  19. [19]

    Gonzalez Betancourt, J

    R. Gonzalez Betancourt, J. Zub´ aˇ c, R. Gonzalez- Hernandez, K. Geishendorf, Z. ˇSob´ aˇ n, G. Springholz, K. Olejn´ ık, L.ˇSmejkal, J. Sinova, T. Jungwirth,et al., Spontaneous Anomalous Hall Effect Arising from an Un- conventional Compensated Magnetic Phase in a Semi- conductor, Phys. Rev. Lett.130, 036702 (2023)

    work page 2023

  20. [20]

    Kluczyk, K

    K. Kluczyk, K. Gas, M. Grzybowski, P. Skupi´ nski, M. Borysiewicz, T. Fas, J. Suffczy´ nski, J. Domagala, 10 K. Grasza, A. Mycielski,et al., Coexistence of anoma- lous Hall effect and weak magnetization in a nomi- nally collinear antiferromagnet MnTe, Phys. Rev. B110, 155201 (2024)

    work page 2024

  21. [21]

    Takagi, R

    R. Takagi, R. Hirakida, Y. Settai, R. Oiwa, H. Tak- agi, A. Kitaori, K. Yamauchi, H. Inoue, J.-i. Yamaura, D. Nishio-Hamane,et al., Spontaneous Hall effect in- duced by collinear antiferromagnetic order at room tem- perature, Nat. Mater.24, 63 (2025)

    work page 2025

  22. [22]

    Aoyama and K

    T. Aoyama and K. Ohgushi, Piezomagnetic properties in altermagnetic MnTe, Phys. Rev. Mater.8, L041402 (2024)

    work page 2024

  23. [23]

    M. Naka, Y. Motome, T. Miyazaki, and H. Seo, Nonrel- ativistic piezomagnetic effect in an organic altermagnet, J. Phys. Soc. Jpn.94, 083702 (2025)

    work page 2025

  24. [24]

    Takahashi, C

    K. Takahashi, C. R. W. Steward, M. Ogata, R. M. Fer- nandes, and J. Schmalian, Elasto-Hall conductivity and the anomalous Hall effect in altermagnets, Phys. Rev. B 111, 184408 (2025)

    work page 2025

  25. [25]

    M. Naka, S. Hayami, H. Kusunose, Y. Yanagi, Y. Mo- tome, and H. Seo, Spin current generation in organic an- tiferromagnets, Nat. Commun.10, 4305 (2019)

    work page 2019

  26. [26]

    Gonz´ alez-Hern´ andez, L.ˇSmejkal, K

    R. Gonz´ alez-Hern´ andez, L.ˇSmejkal, K. V´ yborn´ y, Y. Ya- hagi, J. Sinova, T. Jungwirth, and J. ˇZelezn´ y, Effi- cient Electrical Spin Splitter Based on Nonrelativistic Collinear Antiferromagnetism, Phys. Rev. Lett.126, 127701 (2021)

    work page 2021

  27. [27]

    ˇZelezn´ y, Y

    J. ˇZelezn´ y, Y. Zhang, C. Felser, and B. Yan, Spin- Polarized Current in Noncollinear Antiferromagnets, Phys. Rev. Lett.119, 187204 (2017)

    work page 2017

  28. [28]

    Kimata, H

    M. Kimata, H. Chen, K. Kondou, S. Sugimoto, P. K. Muduli, M. Ikhlas, Y. Omori, T. Tomita, A. H. Mac- Donald, S. Nakatsuji,et al., Magnetic and magnetic in- verse spin Hall effects in a non-collinear antiferromagnet, Nature565, 627 (2019)

    work page 2019

  29. [29]

    C. C. Ye, K. Tenzin, J. S lawi´ nska, and C. Autieri, Domi- nant orbital magnetization in the prototypical altermag- net MnTe (2025), arXiv:2505.08675

    work page arXiv 2025

  30. [30]

    K. D. Belashchenko, Giant strain-induced spin splitting effect in MnTe, ag-wave altermagnetic semiconductor, Phys. Rev. Lett.134, 086701 (2025)

    work page 2025

  31. [31]

    S. Lee, S. Lee, S. Jung, J. Jung, D. Kim, Y. Lee, B. Seok, J. Kim, B. G. Park, L. ˇSmejkal,et al., Broken Kramers Degeneracy in Altermagnetic MnTe, Phys. Rev. Lett. 132, 036702 (2024)

    work page 2024

  32. [32]

    Krempask´ y, L

    J. Krempask´ y, L. ˇSmejkal, S. D’souza, M. Hajlaoui, G. Springholz, K. Uhl´ ıˇ rov´ a, F. Alarab, P. Constantinou, V. Strocov, D. Usanov,et al., Altermagnetic lifting of Kramers spin degeneracy, Nature626, 517 (2024)

    work page 2024

  33. [33]

    Osumi, S

    T. Osumi, S. Souma, T. Aoyama, K. Yamauchi, A. Honma, K. Nakayama, T. Takahashi, K. Ohgushi, and T. Sato, Observation of a giant band splitting in alter- magnetic MnTe, Phys. Rev. B109, 115102 (2024)

    work page 2024

  34. [34]

    Jeong, S.-H

    D. Jeong, S.-H. Kang, and Y.-K. Kwon, Magnetic and Crystal Symmetry Effects on Spin Hall Conductivity in Altermagnets (2025), arXiv:2508.07639

    work page arXiv 2025

  35. [35]

    Momma and F

    K. Momma and F. Izumi, VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data, J. Appl. Crystallogr.44, 1272 (2011)

    work page 2011

  36. [36]

    Kriegner, H

    D. Kriegner, H. Reichlova, J. Grenzer, W. Schmidt, E. Ressouche, J. Godinho, T. Wagner, S. Martin, A. Shick, V. Volobuev,et al., Magnetic anisotropy in antiferromagnetic hexagonal MnTe, Phys. Rev. B96, 214418 (2017)

    work page 2017

  37. [37]

    M. I. Aroyo, J. M. Perez-Mato, C. Capillas, E. Kroumova, S. Ivantchev, G. Madariaga, A. Kirov, and H. Won- dratschek, Bilbao Crystallographic Server: I. Databases and crystallographic computing programs, Z. Kristallogr. Cryst. Mater.221, 15 (2006)

    work page 2006

  38. [38]

    M. I. Aroyo, A. Kirov, C. Capillas, J. Perez-Mato, and H. Wondratschek, Bilbao Crystallographic Server. II. Representations of crystallographic point groups and space groups, Acta Cryst. Sect. A: Found. Adv.62, 115 (2006)

    work page 2006

  39. [39]

    Kresse and J

    G. Kresse and J. Furthm¨ uller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B54, 11169 (1996)

    work page 1996

  40. [40]

    P. E. Bl¨ ochl, Projector augmented-wave method, Phys. Rev. B50, 17953 (1994)

    work page 1994

  41. [41]

    J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996)

    work page 1996

  42. [42]

    V. I. Anisimov, J. Zaanen, and O. K. Andersen, Band theory and Mott insulators: Hubbard U instead of Stoner I, Phys. Rev. B44, 943 (1991)

    work page 1991

  43. [43]

    V. I. Anisimov, F. Aryasetiawan, and A. Lichtenstein, First-principles calculations of the electronic structure and spectra of strongly correlated systems: the LDA+ U method, J. Phys. Condens. Matter.9, 767 (1997)

    work page 1997

  44. [44]

    Buongiorno Nardelli, F

    M. Buongiorno Nardelli, F. T. Cerasoli, M. Costa, S. Cur- tarolo, R. De Gennaro, M. Fornari, L. Liyanage, A. R. Supka, and H. Wang, PAOFLOW: A utility to construct and operate on ab initio Hamiltonians from the projec- tions of electronic wavefunctions on atomic orbital bases, including characterization of topological materials, Com- put. Mater. Sci.143, ...

    work page 2018

  45. [45]

    F. T. Cerasoli, A. R. Supka, A. Jayaraj, M. Costa, I. Siloi, J. S lawi´ nska, S. Curtarolo, M. Fornari, D. Ceresoli, and M. Buongiorno Nardelli, Advanced modeling of materials with PAOFLOW 2.0: New features and software design, Comput. Mater. Sci.200, 110828 (2021)

    work page 2021

  46. [46]

    J. R. Yates, X. Wang, D. Vanderbilt, and I. Souza, Spec- tral and Fermi surface properties from Wannier interpo- lation, Phys. Rev. B75, 195121 (2007)

    work page 2007

  47. [47]

    S. D. Ganichev, E. L. Ivchenko, V. V. Belkov, S. A. Tarasenko, M. Sollinger, D. Weiss, W. Wegscheider, and W. Prettl, Spin-galvanic effect, Nature417, 153 (2002)

    work page 2002

  48. [48]

    Tenzin, A

    K. Tenzin, A. Roy, H. Jafari, B. Banas, F. T. Cera- soli, M. Date, A. Jayaraj, M. Buongiorno Nardelli, and J. S lawi´ nska, Analogs of Rashba-Edelstein effect from density functional theory, Phys. Rev. B107, 165140 (2023)

    work page 2023

  49. [49]

    A. Roy, M. H. D. Guimar˜ aes, and J. S lawi´ nska, Uncon- ventional spin Hall effects in nonmagnetic solids, Phys. Rev. Mater.6, 045004 (2022)

    work page 2022

  50. [50]

    Kubo, Statistical-Mechanical Theory of Irreversible Processes

    R. Kubo, Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems, J. Phys. Soc. Jpn. 12, 570 (1957)

    work page 1957

  51. [51]

    Freimuth, S

    F. Freimuth, S. Bl¨ ugel, and Y. Mokrousov, Spin-orbit torques in Co/Pt(111) and Mn/W(001) magnetic bilayers from first principles, Phys. Rev. B90, 174423 (2014)

    work page 2014

  52. [52]

    H. Li, H. Gao, L. P. Zˆ arbo, K. V´ yborn´ y, X. Wang, I. Garate, F. Doˇ gan, A. ˇCejchan, J. Sinova, T. Jung- wirth,et al., Intraband and interband spin-orbit torques in noncentrosymmetric ferromagnets, Phys. Rev. B91, 134402 (2015). 11

    work page 2015

  53. [53]

    Zhang, Y

    Y. Zhang, Y. Sun, H. Yang, J. ˇZelezn` y, S. P. Parkin, C. Felser, and B. Yan, Strong anisotropic anomalous Hall effect and spin Hall effect in the chiral antiferromagnetic compounds Mn 3X(X= Ge, Sn, Ga, Ir, Rh, and Pt), Phys. Rev. B95, 075128 (2017)

    work page 2017

  54. [54]

    Gonz´ alez-Hern´ andez, P

    R. Gonz´ alez-Hern´ andez, P. Ritzinger, K. V´ yborn´ y, J. ˇZelezn´ y, and A. Manchon, Non-relativistic torque and Edelstein effect in non-collinear magnets, Nat. Commun. 15, 7663 (2024)

    work page 2024

  55. [55]

    Hayami and H

    S. Hayami and H. Kusunose, Microscopic Description of Electric and Magnetic Toroidal Multipoles in Hybrid Or- bitals, J. Phys. Soc. Jpn.87, 033709 (2018)

    work page 2018

  56. [56]

    Hayami and H

    S. Hayami and H. Kusunose, Classification of atomic- scale multipoles under crystallographic point groups and application to linear response tensors, Phys. Rev. B98, 165110 (2018)

    work page 2018

  57. [57]

    Kusunose, R

    H. Kusunose, R. Oiwa, and S. Hayami, Complete Multi- pole Basis Set for Single-Centered Electron Systems, J. Phys. Soc. Jpn.89, 104704 (2020)

    work page 2020

  58. [58]

    Yatsushiro, H

    M. Yatsushiro, H. Kusunose, and S. Hayami, Multipole classification in 122 magnetic point groups for unified un- derstanding of multiferroic responses and transport phe- nomena, Phys. Rev. B104, 054412 (2021)

    work page 2021

  59. [59]

    Hayami and H

    S. Hayami and H. Kusunose, Unified Description of Elec- tronic Orderings and Cross Correlations by Complete Multipole Representation, J. Phys. Soc. Jpn.93, 072001 (2024)

    work page 2024

  60. [60]

    Winkler and U

    R. Winkler and U. Z¨ ulicke, Standard model of electromagnetism and chirality in crystals (2025), arXiv:2405.20940

    work page arXiv 2025

  61. [61]

    L. Liu, T. Moriyama, D. Ralph, and R. Buhrman, Spin- torque ferromagnetic resonance induced by the spin Hall effect, Phys. Rev. Lett.106, 036601 (2011)

    work page 2011

  62. [62]

    C.-F. Pai, L. Liu, Y. Li, H. Tseng, D. Ralph, and R. Buhrman, Spin transfer torque devices utilizing the giant spin Hall effect of tungsten, Appl. Phys. Lett.101, 10.1063/1.4753947 (2012)

    work page doi:10.1063/1.4753947 2012

  63. [63]

    Liu, C.-F

    L. Liu, C.-F. Pai, Y. Li, H. Tseng, D. Ralph, and R. Buhrman, Spin-torque switching with the giant spin Hall effect of tantalum, Science336, 555 (2012)

    work page 2012

  64. [64]

    Laczkowski, J.-C

    P. Laczkowski, J.-C. Rojas-S´ anchez, W. Savero-Torres, H. Jaffr` es, N. Reyren, C. Deranlot, L. Notin, C. Beign´ e, A. Marty, J.-P. Attan´ e,et al., Experimental evidences of a large extrinsic spin Hall effect in AuW alloy, Appl. Phys. Lett.104, 10.1063/1.4870835 (2014)

    work page doi:10.1063/1.4870835 2014

  65. [65]

    P. E. Faria Junior, K. A. de Mare, K. Zollner, K.-h. Ahn, S. I. Erlingsson, M. van Schilfgaarde, and K. V´ yborn´ y, Sensitivity of the MnTe valence band to the orientation of magnetic moments, Phys. Rev. B107, L100417 (2023). Appendix A: Band structures The spin-polarized band structure forα-MnTe is shown in Fig. 6. While the band structure for ˆN∥yhas a...

    work page 2023

  66. [66]

    We note that the dissipative and magnetic REE vanish inα-MnTe, which is a direct consequence of the in- version symmetry of the crystal

    Symmetry-imposed shape of response tensors Table VII summarizes the response tensor forms, for both ˆN∥yand ˆN∥x, as derived from the active mul- tipoles. We note that the dissipative and magnetic REE vanish inα-MnTe, which is a direct consequence of the in- version symmetry of the crystal. The dissipative EC and intrinsic SHE exhibit the same multipole d...

    work page

  67. [67]

    7 and Fig

    Dissipative electric conduction and intrinsic spin Hall effect For the dissipative EC and intrinsic SHE shown in Fig. 7 and Fig. 8, the trends are similar to those of the AHE and magnetic SHE; hole doping produces a much stronger response than electron doping. Note that the intrinsic SHE of MnTe has six independent components of the SHC tensor, in contras...

    work page