pith. sign in

arxiv: 2604.07391 · v1 · submitted 2026-04-08 · ❄️ cond-mat.mes-hall · cond-mat.mtrl-sci

Classification of magnon thermal Hall systems based on U(1) to non-Abelian gauge fields

Masataka Kawano , Chisa Hotta This is my paper

Pith reviewed 2026-05-10 18:15 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.mtrl-sci
keywords magnon thermal Hall effectnon-Abelian gauge fieldsantiferromagnetsBerry curvatureDzyaloshinskii-Moriya interactionSU(N) gauge fieldsspin wave transportaltermagnets
0
0 comments X

The pith

Antiferromagnets with multiple magnetic sublattices generate non-Abelian SU(N) gauge fields for magnons that produce a thermal Hall effect.

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

This paper shows that antiferromagnets with more than one magnetic sublattice create non-Abelian SU(N) gauge fields for magnons instead of the usual U(1) fields found in ferromagnets. The noncommuting nature of these fields stops the Berry curvature from cancelling out, which had suppressed the thermal Hall response under earlier no-go rules. This supplies a general mechanism that works across many lattice types and makes antiferromagnets viable platforms for the effect. The authors give a concrete minimal case in a coplanar 120-degree antiferromagnet with Dzyaloshinskii-Moriya interactions that realizes an SU(3) gauge structure. They also classify two-dimensional lattices and magnetic arrangements by the type of gauge field they support, including extensions to altermagnets.

Core claim

In ferromagnets the magnon thermal Hall effect comes from U(1) gauge fields produced by Dzyaloshinskii-Moriya interactions or spin textures, yet these fields frequently cancel in many lattices. Antiferromagnets with multiple sublattices instead support non-Abelian SU(N) gauge fields whose noncommutativity blocks Berry-curvature cancellation and ensures a finite thermal Hall conductivity. A coplanar 120° antiferromagnet with Dzyaloshinskii-Moriya interactions provides the simplest SU(3) realization. The paper supplies a table that assigns gauge-field type to known two-dimensional lattices and magnetic structures, offering a practical map for locating materials, including antiferromagnets and

What carries the argument

non-Abelian SU(N) gauge fields experienced by magnons in multi-sublattice antiferromagnets, whose noncommutativity prevents Berry-curvature cancellation

If this is right

  • The noncommutativity of SU(N) gauge fields guarantees a nonvanishing magnon thermal Hall conductivity.
  • A coplanar 120° antiferromagnet with Dzyaloshinskii-Moriya interactions forms a minimal canonical SU(3) platform.
  • Two-dimensional lattices and magnetic structures can be classified by whether they support U(1) or non-Abelian gauge fields.
  • Altermagnets fall within the same classification scheme and can host the thermal Hall response.

Where Pith is reading between the lines

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

  • The same non-Abelian mechanism may appear in three-dimensional antiferromagnets or in other bosonic excitations such as phonons.
  • Targeted experiments on layered van-der-Waals antiferromagnets could directly test whether the predicted finite thermal Hall signal appears once the lattice and spin structure match the SU(N) criteria.
  • The classification suggests that symmetry-protected cancellations in topological transport are more easily avoided once the number of magnetic sublattices increases.

Load-bearing premise

That the emergent gauge fields felt by magnons in multi-sublattice antiferromagnets are genuinely non-Abelian and that their noncommutativity directly produces a non-vanishing thermal Hall conductivity without further cancellations from other symmetries or interactions.

What would settle it

A first-principles calculation or measurement of the thermal Hall conductivity in a coplanar 120° antiferromagnet with Dzyaloshinskii-Moriya interactions that finds exact cancellation of all Berry-curvature contributions would falsify the claim that non-Abelian noncommutativity guarantees a nonzero response.

Figures

Figures reproduced from arXiv: 2604.07391 by Chisa Hotta, Masataka Kawano.

Figure 1
Figure 1. Figure 1: Origins of the U(1) gauge field: Dzyaloshinskii-Moriya (DM) interaction, scalar￾spin chirality (SSC), Kitaev and Gamma interactions, spin texture, and Aharonov-Casher (AC) effect. The DM interaction, SSC, and Kitaev and Gamma interactions generate staggered flux pattern, whereas the spin texture and AC effect can generate uniform flux pattern. In the following, we focus on the lowest-order noninteracting l… view at source ↗
Figure 2
Figure 2. Figure 2: (a) Staggered and (b) uniform Dzyaloshinskii-Moriya (DM) interaction. The gray arrows indicate the direction of i → j in Di, j . In the boxes, we show the directions of the DM vectors when going from a site to its left and right neighboring sites. (c),(d) Flux pattern in ferromagnetic insulators with the (c) staggered DM and (d) uniform DM interactions. While the staggered DM interaction generates the stag… view at source ↗
Figure 3
Figure 3. Figure 3: Schematic illustration of lattice-geometry constraints with the U(1) gauge flux. (a) Square and triangular lattices (edge-shared geometry) with the staggered flux. The flux pattern is invariant under a π rotation about any bonds, leading to κxy = 0 (bottom). (b) Kagome lattice (corner-shared geometry) with the staggered flux, (c) honeycomb lattice with the staggered flux, and (d) edge-shared lattices (squa… view at source ↗
Figure 4
Figure 4. Figure 4: Effective SU(2) gauge field for magnons induced by the Dzyaloshinskii-Moriya (DM) interaction. (a) Noncollinear antiferromagnetic order with the uniform DM interaction, which is reduced to the effective lattice model with an SU(2) gauge field Θx. (b) Correspondence between the noncollinear antiferromagnetic order and the associated SU(2) gauge field. (c),(d) Noncollinear antiferromagnets with the uniform D… view at source ↗
Figure 5
Figure 5. Figure 5: Effective SU(3) gauge field for magnons in three-sublattice antiferromagnets. (a) Triangular-lattice antiferromagnet with the uniform Dzyaloshinskii-Moriya (DM) interaction. The blue and gray arrows indicate the DM vector and the direction of i → j in Di, j . (b) Two representative three-sublattice spin configurations: coplanar 120◦ order and antiferromagnetic skytrmion crystals. (c) Effective lattice mode… view at source ↗
Figure 6
Figure 6. Figure 6: Circumventing the no-go rule by non-Abelian gauge fields. (a) Square lattice with the SU(2) gauge field represented by 2 × 2 matrix Uµ (µ = x, y). The fluxes Φ and Φ′ defined on an elementary plaquette in the counterclockwise and clockwise directions take the same value Φ = Φ′ ≃ i[Θx, Θy], which originates from the noncommutativity of the gauge fields. (b) Resulting flux pattern: unlike the U(1) case, the … view at source ↗
Figure 7
Figure 7. Figure 7: Magnon bands, Berry curvature, and thermal Hall conductivity. The parameters are set to JS = 1.0 and Λ = 0.05. (a) Magnon bands along the high-symmetry line in the Brillouin zone (inset) for D = 0.1. The dashed lines denote the magnon bands with D = 0. (b) Density plots of the Berry curvature for D = 0.1. The dashed hexagon indicates the Brillouin zone. (c) Temperature dependence of the thermal Hall conduc… view at source ↗
read the original abstract

Magnon thermal Hall effect in insulating magnets is the manifestation of Berry curvature in magnon bands, which is formulated using the emergent gauge fields that act on magnons as a fictitious magnetic field. In ferromagnets, it is commonly accepted as the outcome of U(1) gauge fields generated by Dzyaloshinskii-Moriya interactions and spin textures, but this mechanism is often suppressed by symmetry-enforced cancellations in many lattice geometries, known as a no-go rule. As a result, antiferromagnetic insulators have long been considered as unfavorable platforms for the effect. We show that antiferromagnets with multiple magnetic sublattices naturally host non-Abelian SU(N) gauge fields in magnon band structures, providing a robust rule-to-go mechanism. The noncommutativity of these gauge fields prevents Berry-curvature cancellation and guarantees a nonvanishing thermal Hall response. As a minimal realization, we demonstrate that a coplanar 120$^{\circ}$ antiferromagnet with Dzyaloshinskii-Moriya interactions constitutes a canonical SU(3) platform for the magnon thermal Hall effect. We provide a table of so-far-known two-dimensional lattice geometries and variants of magnetic structures, along with the corresponding gauge fields, providing a unified guideline for identifying magnetic materials, including antiferromagnets and altermagnets, that host thermal Hall transport.

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

2 major / 2 minor

Summary. The paper claims that multi-sublattice antiferromagnets host emergent non-Abelian SU(N) gauge fields for magnons (arising via linear spin-wave theory from the spin Hamiltonian), whose noncommutativity prevents the Berry-curvature cancellations that suppress the magnon thermal Hall effect under U(1) gauge fields in ferromagnets. It presents the coplanar 120° antiferromagnet with Dzyaloshinskii-Moriya interactions as a minimal SU(3) realization that exhibits a nonvanishing thermal Hall response, and supplies a classification table of 2D lattices and magnetic structures with their associated gauge fields as a guideline for material identification, including altermagnets.

Significance. If the central claim is substantiated, the work supplies a concrete rule-to-go mechanism that enlarges the set of platforms for magnon thermal Hall transport beyond ferromagnets, where symmetry cancellations often forbid the effect. The classification table is a useful organizing tool that could guide experimental searches in antiferromagnetic and altermagnetic insulators.

major comments (2)
  1. [demonstration for the 120° coplanar AFM] The abstract and the section introducing the non-Abelian mechanism assert that SU(N) noncommutativity 'prevents Berry-curvature cancellation and guarantees a nonvanishing thermal Hall response.' However, the thermal Hall conductivity is an integral of the trace of the Berry curvature (weighted by the Bose function) over the Brillouin zone; even with non-Abelian gauge fields, additional lattice symmetries or band degeneracies could still enforce pairwise cancellation. The manuscript must explicitly demonstrate, for the 120° AFM example, that the integrated trace remains finite after all symmetries are imposed, rather than relying on noncommutativity alone.
  2. [classification table] In the classification table of lattice geometries and magnetic structures, several entries are listed as hosting non-Abelian gauge fields. For each such entry the paper should verify that the emergent gauge field is indeed SU(N) with N>1 (i.e., that the holonomy matrices do not commute) and that this noncommutativity is what lifts the cancellation, rather than a model-specific feature of the spin-wave Hamiltonian.
minor comments (2)
  1. [introduction] Notation for the matrix-valued Berry curvature and its trace should be introduced once and used consistently; currently the transition from U(1) to SU(N) curvature is described in two different paragraphs with slightly different symbols.
  2. [introduction] The manuscript cites several prior works on magnon thermal Hall in ferromagnets but omits recent experimental reports on antiferromagnetic candidates; adding one or two such references would strengthen the motivation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and have revised the manuscript to incorporate the requested clarifications and verifications.

read point-by-point responses

  1. Referee: [demonstration for the 120° coplanar AFM] The abstract and the section introducing the non-Abelian mechanism assert that SU(N) noncommutativity 'prevents Berry-curvature cancellation and guarantees a nonvanishing thermal Hall response.' However, the thermal Hall conductivity is an integral of the trace of the Berry curvature (weighted by the Bose function) over the Brillouin zone; even with non-Abelian gauge fields, additional lattice symmetries or band degeneracies could still enforce pairwise cancellation. The manuscript must explicitly demonstrate, for the 120° AFM example, that the integrated trace remains finite after all symmetries are imposed, rather than relying on noncommutativity alone.

    Authors: We agree that an explicit demonstration of the non-cancellation in the integrated thermal Hall conductivity is essential for rigor. Although the manuscript already computes the magnon bands and Berry curvature for the 120° coplanar AFM with DM interactions and states that the response is nonvanishing, we acknowledge that the symmetry-imposed cancellation argument could be shown more transparently. In the revised manuscript we have added a dedicated subsection with (i) an explicit symmetry analysis under the C3 rotational symmetry of the lattice, (ii) the computed distribution of the trace of the non-Abelian Berry curvature, and (iii) numerical integration confirming a finite thermal Hall conductivity at finite temperature. This calculation demonstrates that the noncommutativity of the SU(3) holonomy matrices prevents the pairwise cancellation that would occur for U(1) fields. revision: yes

  2. Referee: [classification table] In the classification table of lattice geometries and magnetic structures, several entries are listed as hosting non-Abelian gauge fields. For each such entry the paper should verify that the emergent gauge field is indeed SU(N) with N>1 (i.e., that the holonomy matrices do not commute) and that this noncommutativity is what lifts the cancellation, rather than a model-specific feature of the spin-wave Hamiltonian.

    Authors: We concur that systematic verification of noncommutativity for every non-Abelian entry strengthens the classification. The original table was intended as a guideline based on the multi-sublattice structure, but we did not provide explicit holonomy matrices for all cases. In the revised version we have expanded the table caption and added an appendix that lists the holonomy matrices (or their generators) for each non-Abelian entry. For representative lattices we explicitly show that the matrices fail to commute, and we argue that this noncommutativity originates from the SU(N) structure tied to the number of sublattices rather than from details of the particular spin-wave Hamiltonian. This addresses the concern that the effect might be model-specific. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation of non-Abelian gauge fields and thermal Hall response.

full rationale

The paper derives emergent SU(N) gauge fields directly from the multi-sublattice spin Hamiltonian via linear spin-wave theory, then invokes noncommutativity of the resulting matrix-valued Berry curvature as the mechanism preventing cancellation. This chain relies on standard magnon band-structure calculations and explicit demonstration for the 120° AFM case rather than self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations whose validity reduces to the present work. The classification table and rule-to-go argument are presented as consequences of the gauge-field structure, remaining self-contained against external benchmarks such as symmetry analysis and explicit curvature integrals.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard topological band theory for magnons and the assumption that multi-sublattice antiferromagnets generate non-Abelian gauge fields whose noncommutativity prevents cancellation.

axioms (2)
  • standard math Magnon thermal Hall conductivity is determined by the integral of Berry curvature over occupied bands.
    This is a standard result from topological magnonics and semiclassical transport theory.
  • domain assumption Dzyaloshinskii-Moriya interactions and spin textures generate emergent gauge fields acting on magnons.
    Widely used in literature on magnon Hall effects in insulating magnets.
invented entities (1)
  • non-Abelian SU(N) gauge fields for magnons in antiferromagnets no independent evidence
    purpose: To explain non-cancellation of Berry curvature and enable thermal Hall response
    Emergent from the multi-sublattice structure; proposed here as the key new element without external falsifiable evidence in the abstract.

pith-pipeline@v0.9.0 · 5556 in / 1681 out tokens · 93438 ms · 2026-05-10T18:15:33.484294+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

112 extracted references · 112 canonical work pages · 1 internal anchor

  1. [1]

    Onose, T

    Y . Onose, T. Ideue, H. Katsura, Y . Shiomi, N. Nagaosa, and Y . Tokura, Science329, 297 (2010)

    work page 2010

  2. [2]

    Matsumoto and S

    R. Matsumoto and S. Murakami, Phys. Rev. Lett.106, 197202 (2011)

    work page 2011

  3. [3]

    Matsumoto and S

    R. Matsumoto and S. Murakami, Phys. Rev. B84, 184406 (2011)

    work page 2011

  4. [4]

    Katsura, N

    H. Katsura, N. Nagaosa, and P. A. Lee, Phys. Rev. Lett.104, 066403 (2010)

    work page 2010

  5. [5]

    Ideue, Y

    T. Ideue, Y . Onose, H. Katsura, Y . Shiomi, S. Ishiwata, N. Nagaosa, and Y . Tokura, Phys. Rev. B85, 134411 (2012)

    work page 2012

  6. [6]

    Hirschberger, R

    M. Hirschberger, R. Chisnell, Y . S. Lee, and N. P. Ong, Phys. Rev. Lett.115, 106603 (2015)

    work page 2015

  7. [7]

    A. Mook, J. Henk, and I. Mertig, Phys. Rev. B90, 024412 (2014)

    work page 2014

  8. [8]

    A. Mook, J. Henk, and I. Mertig, Phys. Rev. B89, 134409 (2014)

    work page 2014

  9. [9]

    A. Mook, J. Henk, and I. Mertig, Phys. Rev. B94, 174444 (2016)

    work page 2016

  10. [10]

    S. A. Owerre, Phys. Rev. B95, 014422 (2017)

    work page 2017

  11. [11]

    S. A. Owerre, Phys. Rev. B97, 094412 (2018)

    work page 2018

  12. [12]

    Seshadri and D

    R. Seshadri and D. Sen, Phys. Rev. B97, 134411 (2018)

    work page 2018

  13. [13]

    Laurell and G

    P. Laurell and G. A. Fiete, Phys. Rev. B98, 094419 (2018)

    work page 2018

  14. [14]

    Owerre, Europhys

    S. Owerre, Europhys. Lett.117, 37006 (2017)

    work page 2017

  15. [15]

    Owerre, Sci

    S. Owerre, Sci. Rep.8, 4431 (2018)

    work page 2018

  16. [16]

    Owerre, Ann

    S. Owerre, Ann. Phys.406, 14 (2019)

    work page 2019

  17. [17]

    Laurell and G

    P. Laurell and G. A. Fiete, Phys. Rev. Lett.118, 177201 (2017)

    work page 2017

  18. [18]

    K.-S. Kim, K. H. Lee, S. B. Chung, and J.-G. Park, Phys. Rev. B100, 064412 (2019)

    work page 2019

  19. [19]

    H.-L. Kim, T. Saito, H. Yang, H. Ishizuka, M. J. Coak, J. H. Lee, H. Sim, Y . S. Oh, N. Nagaosa, and J.-G. Park, Nat. Commun.15, 243 (2024)

    work page 2024

  20. [20]

    Nagaosa, J

    N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010)

    work page 2010

  21. [21]

    H. Chen, Q. Niu, and A. H. MacDonald, Phys. Rev. Lett.112, 017205 (2014)

    work page 2014

  22. [22]

    Nakatsuji, N

    S. Nakatsuji, N. Kiyohara, and T. Higo, Nature527, 212 (2015)

    work page 2015

  23. [23]

    A. K. Nayak, J. E. Fischer, Y . Sun, B. Yan, J. Karel, A. C. Komarek, C. Shekhar, N. Kumar, W. Schnelle, J. K¨ubler, C. Felser, and S. S. P. Parkin, Sci. Adv.2, 1501870 (2016). REFERENCES29

    work page 2016

  24. [24]

    Kiyohara, T

    N. Kiyohara, T. Tomita, and S. Nakatsuji, Phys. Rev. Appl.5, 064009 (2016)

    work page 2016

  25. [25]

    Z. H. Liu, Y . J. Zhang, G. D. Liu, B. Ding, E. K. Liu, H. M. Jafri, Z. P. Hou, W. H. Wang, X. Q. Ma, and G. H. Wu, Sci. Rep.7, 515 (2017)

    work page 2017

  26. [26]

    Kawano and C

    M. Kawano and C. Hotta, Phys. Rev. B99, 054422 (2019)

    work page 2019

  27. [27]

    Iguchi, Y

    Y . Iguchi, Y . Nii, M. Kawano, H. Murakawa, N. Hanasaki, and Y . Onose, Phys. Rev. B 98, 064416 (2018)

    work page 2018

  28. [28]

    Kawano, Y

    M. Kawano, Y . Onose, and C. Hotta, Commun. Phys.2, 27 (2019)

    work page 2019

  29. [29]

    Kawano and C

    M. Kawano and C. Hotta, Phys. Rev. B100, 174402 (2019)

    work page 2019

  30. [30]

    Fr ¨ohlich and U

    J. Fr ¨ohlich and U. M. Studer, Rev. Mod. Phys.65, 733 (1993)

    work page 1993

  31. [31]

    A. V . Chumak, V . I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nat. Phys.11, 453 (2015)

    work page 2015

  32. [32]

    Takeda, M

    H. Takeda, M. Kawano, K. Tamura, M. Akazawa, J. Yan, T. Waki, H. Nakamura, K. Sato, Y . Narumi, M. Hagiwara, M. Yamashita, and C. Hotta, Nat. Commun.15, 566 (2024)

    work page 2024

  33. [33]

    (), Although a thermal Hall signal has been reported in the noncoplanar antiferromagnet in the distorted triangular lattice YMnO3, it is attributed to topological spin fluctuations and phonon-related effects, rather than to a magnon thermal Hall effect

    work page

  34. [34]

    Nakata, J

    K. Nakata, J. Klinovaja, and D. Loss, Phys. Rev. B95, 125429 (2017)

    work page 2017

  35. [35]

    Nakata, S

    K. Nakata, S. K. Kim, J. Klinovaja, and D. Loss, Phys. Rev. B96, 224414 (2017)

    work page 2017

  36. [36]

    K. A. van Hoogdalem, Y . Tserkovnyak, and D. Loss, Phys. Rev. B87, 024402 (2013)

    work page 2013

  37. [37]

    Kong and J

    L. Kong and J. Zang, Phys. Rev. Lett.111, 067203 (2013)

    work page 2013

  38. [38]

    Iwasaki, A

    J. Iwasaki, A. J. Beekman, and N. Nagaosa, Phys. Rev. B89, 064412 (2014)

    work page 2014

  39. [39]

    Y .-T. Oh, H. Lee, J.-H. Park, and J. H. Han, Phys. Rev. B91, 104435 (2015)

    work page 2015

  40. [40]

    Rold ´an-Molina, A

    A. Rold ´an-Molina, A. S. Nunez, and J. Fern ´andez-Rossier, New J. Phys.18, 045015 (2016)

    work page 2016

  41. [41]

    A. Mook, B. G ¨obel, J. Henk, and I. Mertig, Phys. Rev. B95, 020401 (2017)

    work page 2017

  42. [42]

    S. K. Kim, K. Nakata, D. Loss, and Y . Tserkovnyak, Phys. Rev. Lett.122, 057204 (2019)

    work page 2019

  43. [43]

    Nikoli ´c, Phys

    P. Nikoli ´c, Phys. Rev. B102, 075131 (2020)

    work page 2020

  44. [44]

    Akazawa, H.-Y

    M. Akazawa, H.-Y . Lee, H. Takeda, Y . Fujima, Y . Tokunaga, T.-h. Arima, J. H. Han, and M. Yamashita, Phys. Rev. Res.4, 043085 (2022)

    work page 2022

  45. [45]

    S. A. Owerre, J. Appl. Phys.120, 043903 (2016)

    work page 2016

  46. [46]

    Owerre, Phys

    S. Owerre, Phys. Rev. B94, 094405 (2016)

    work page 2016

  47. [47]

    Lu, J.-L

    Y .-S. Lu, J.-L. Li, and C.-T. Wu, Phys. Rev. Lett.127, 217202 (2021)

    work page 2021

  48. [48]

    P. A. McClarty, X.-Y . Dong, M. Gohlke, J. G. Rau, F. Pollmann, R. Moessner, and K. Penc, Phys. Rev. B98, 060404 (2018)

    work page 2018

  49. [49]

    E. Z. Zhang, L. E. Chern, and Y . B. Kim, Phys. Rev. B103, 174402 (2021). REFERENCES30

    work page 2021

  50. [50]

    L. E. Chern, E. Z. Zhang, and Y . B. Kim, Phys. Rev. Lett.126, 147201 (2021)

    work page 2021

  51. [51]

    R. R. Neumann, A. Mook, J. Henk, and I. Mertig, Phys. Rev. Lett.128, 117201 (2022)

    work page 2022

  52. [52]

    Owerre, J

    S. Owerre, J. Appl. Phys.121, 223904 (2017)

    work page 2017

  53. [53]

    Owerre, J

    S. Owerre, J. Phys.: Cond. Matt.29, 385801 (2017)

    work page 2017

  54. [54]

    L. E. Chern, R. Kaneko, H.-Y . Lee, and Y . B. Kim, Phys. Rev. Res.2, 013014 (2020)

    work page 2020

  55. [55]

    L. E. Chern, F. L. Buessen, and Y . B. Kim, npj Quantum Materials6, 33 (2021)

    work page 2021

  56. [56]

    Koyama and J

    S. Koyama and J. Nasu, Phys. Rev. B104, 075121 (2021)

    work page 2021

  57. [57]

    X. Cao, K. Chen, and D. He, J. Phys: Condens. Matter27, 166003 (2015)

    work page 2015

  58. [58]

    S. A. Owerre, J. Phys.: Cond. Matt29, 03LT01 (2016)

    work page 2016

  59. [59]

    Debnath and S

    S. Debnath and S. Basu, Phys. Rev. B111, 155418 (2025)

    work page 2025

  60. [60]

    Romh ´anyi, K

    J. Romh ´anyi, K. Penc, and R. Ganesh, Nat. Commun.6, 10.1038/ncomms7805 (2015)

    work page doi:10.1038/ncomms7805 2015

  61. [61]

    Malki and G

    M. Malki and G. S. Uhrig, Phys. Rev. B99, 174412 (2019)

    work page 2019

  62. [62]

    Suetsugu, T

    S. Suetsugu, T. Yokoi, K. Totsuka, T. Ono, I. Tanaka, S. Kasahara, Y . Kasahara, Z. Chengchao, H. Kageyama, and Y . Matsuda, Phys. Rev. B105, 024415 (2022)

    work page 2022

  63. [63]

    L. S. Buzo and R. L. Doretto, Phys. Rev. B109, 134405 (2024)

    work page 2024

  64. [64]

    Hoyer, R

    R. Hoyer, R. Jaeschke-Ubiergo, K.-H. Ahn, L. ˇSmejkal, and A. Mook, Phys. Rev. B 111, L020412 (2025)

    work page 2025

  65. [65]

    Kawano, Phys

    M. Kawano, Phys. Rev. B112, L060403 (2025)

    work page 2025

  66. [66]

    Kawano, arXiv:2502.11924 (2025)

    M. Kawano, arXiv:2502.11924 (2025)

    work page arXiv 2025

  67. [67]

    Zhang, Y

    X.-T. Zhang, Y . H. Gao, and G. Chen, Phys. Rep.1070, 1 (2024)

    work page 2024

  68. [68]

    Holstein and H

    T. Holstein and H. Primakoff, Phys. Rev.58, 1098 (1940)

    work page 1940

  69. [69]

    Matsumoto, R

    R. Matsumoto, R. Shindou, and S. Murakami, Phys. Rev. B89, 054420 (2014)

    work page 2014

  70. [70]

    Kitaev, Annals of Physics321, 2 (2006)

    A. Kitaev, Annals of Physics321, 2 (2006)

    work page 2006

  71. [71]

    Reil, H.-J

    S. Reil, H.-J. Stork, and H. Haeuseler, J. Alloy. Compd.334, 92 (2002)

    work page 2002

  72. [72]

    Fritsch, J

    V . Fritsch, J. Hemberger, N. B¨uttgen, E.-W. Scheidt, H.-A. Krug von Nidda, A. Loidl, and V . Tsurkan, Phys. Rev. Lett.92, 116401 (2004)

    work page 2004

  73. [73]

    S. Gao, O. Zaharko, V . Tsurkan, Y . Su, J. White, G. Tucker, B. Roessli, F. Bourdarot, R. Sibille, D. Chernyshov, T. Fennell, A. Loidl, D.C., and C. R ¨uegg, Nature Phys.13, 157 (2017)

    work page 2017

  74. [74]

    S. Gao, H. D. Rosales, F. A. G. Albarracin, G. K. Tsurkan, T. Fennell, P. Steffens, M. Boehm, P. Cermak, A. Schneldewind, E. Ressouche, D. Cabra, C. R ¨uegg, and O. Zaharko, Nature586, 37 (2020)

    work page 2020

  75. [75]

    C. L. Kane and E. J. Mele, Phys. Rev. Lett.95, 226801 (2005)

    work page 2005

  76. [76]

    C. L. Kane and E. J. Mele, Phys. Rev. Lett.95, 146802 (2005)

    work page 2005

  77. [77]

    Hayami, Y

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

    work page 2020

  78. [78]

    ˇSmejkal, J

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

    work page 2022

  79. [79]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X12, 031042 (2022). REFERENCES31

    work page 2022

  80. [80]

    Fujimoto, Phys

    S. Fujimoto, Phys. Rev. Lett.103, 047203 (2009)

    work page 2009

Showing first 80 references.