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arxiv: 2604.10483 · v1 · submitted 2026-04-12 · ❄️ cond-mat.mes-hall · quant-ph

Topological Magnon-Phonon Hybrid Bands in Ferromagnetic Skyrmion Crystals

Doried Ghader , Bilal Jabakhanji This is my paper

Pith reviewed 2026-05-10 16:33 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall quant-ph
keywords magnon-phonon couplingskyrmion crystaltopological bandsChern numbersDzyaloshinskii-Moriya interactionferromagnetic latticehybrid excitationsedge states
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The pith

Coupling magnons to lattice vibrations in skyrmion crystals produces topological hybrid bands with nontrivial Chern numbers.

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

The paper examines magnon-phonon excitations in a two-dimensional ferromagnetic skyrmion crystal on a triangular lattice stabilized by Dzyaloshinskii-Moriya interaction. Although the bare magnon bands are topologically trivial, the coupling to phonons through DMI vector fluctuations reconstructs the low-energy spectrum into hybrid bands. These hybrid bands develop nontrivial topology, including protected edge states. A sympathetic reader would care because the low-energy features stay robust when the magnetic field changes, while higher bands can switch topology, offering a route to field-tunable topological excitations in magnetic lattices.

Core claim

Although the lowest two magnon bands of the bare SkX are topologically trivial, coupling to lattice vibrations reconstructs the low-energy sector and generates topological MP hybrid bands. Starting from a spin-lattice Hamiltonian in which phonons couple to magnons through fluctuations of the DMI vectors, the bosonic Hamiltonian is derived and the hybrid band structure is computed by Bogoliubov diagonalization. MP coupling opens gaps at low-energy magnon-phonon crossings, lifts phonon degeneracies associated with supercell folding, and yields nontrivial Chern numbers for the lowest hybrid bands. The resulting low-energy topology and associated edge states remain robust under magnetic-field,

What carries the argument

The magnon-phonon coupling term arising from fluctuations of the Dzyaloshinskii-Moriya interaction vectors in the spin-lattice Hamiltonian, which is diagonalized via Bogoliubov transformation to produce hybrid bands whose topology is diagnosed by Chern numbers.

If this is right

  • The lowest hybrid bands acquire nontrivial Chern numbers and host protected edge states.
  • Low-energy topological features and edge states persist across a range of magnetic field strengths.
  • Higher-energy hybrid bands can undergo topological phase transitions when the magnetic field is varied.
  • The hybridization mechanism extends topological magnon-phonon physics to noncoplanar skyrmion crystals.

Where Pith is reading between the lines

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

  • The field-robust low-energy topology could enable stable topological transport in devices exposed to fluctuating external fields.
  • The same coupling mechanism might be tested in other noncoplanar magnetic textures beyond skyrmion crystals.
  • Hybrid bands could couple mechanical vibrations directly to magnetic edge modes for new sensing or information-processing schemes.

Load-bearing premise

The magnon-phonon interaction is assumed to arise specifically from DMI vector fluctuations, and the Neel-type skyrmion crystal is assumed to remain stable on the triangular lattice for the chosen parameters.

What would settle it

Observation of trivial (zero) Chern numbers for the lowest hybrid bands or the absence of energy gaps at the predicted magnon-phonon crossing points in the spectrum would falsify the reconstruction into topological hybrids.

Figures

Figures reproduced from arXiv: 2604.10483 by Bilal Jabakhanji, Doried Ghader.

Figure 1
Figure 1. Figure 1: Néel-type FM SkX on the triangular spin lattice at the minimum magnetic field (𝐵𝑚𝑖𝑛 = 26.4 𝑇) stabilizing the SkX phase. Arrows represent the local spin directions on the lattice sites, with color indicating the out-of-plane spin component. The SkX forms a periodic triangular superlattice with periodicity 6𝑙. The black hexagon outlines the magnetic unit cell containing 𝑁𝑠 = 36 spins [PITH_FULL_IMAGE:figur… view at source ↗
Figure 2
Figure 2. Figure 2: Low energy magnon and phonon band structures at 𝐵𝑚𝑖𝑛 = 26.4 𝑇. (a) Noninteracting bands, showing the lowest two magnon bands (blue) and the lowest seven phonon bands (red) along the high-symmetry path of the SkX BZ. The first and second magnon bands cross the first and second phonon bands, respectively. The uncoupled magnon bands are topologically trivial, while the phonon bands exhibit degeneracies associ… view at source ↗
Figure 4
Figure 4. Figure 4: Field-driven topological phase transition in the higher-energy MP bands. (a) MP hybrid bands at a critical magnetic field (𝐵𝑐 ≈ 42.6 𝑇), showing a gap closure between the fourth and fifth bands (dashed circle). (b) Corresponding noninteracting bands at the same field. The transition changes the Chern numbers from 𝐶4 = 2, 𝐶5 = 1 to 𝐶4 = 1, 𝐶5 = 2, while the low-energy topology remains unchanged. We have pre… view at source ↗
read the original abstract

We investigate magnon-phonon (MP) excitations in a Neel-type two-dimensional ferromagnetic skyrmion crystal (SkX) stabilized on a triangular spin lattice by Dzyaloshinskii-Moriya interaction (DMI). Although the lowest two magnon bands of the bare SkX are topologically trivial, we show that coupling to lattice vibrations reconstructs the low-energy sector and generates topological MP hybrid bands. Starting from a spin-lattice Hamiltonian in which phonons couple to magnons through fluctuations of the DMI vectors, we derive the bosonic Hamiltonian for the SkX and compute the hybrid band structure by Bogoliubov diagonalization. MP coupling opens gaps at low-energy magnon-phonon crossings, lifts phonon degeneracies associated with supercell folding, and yields nontrivial Chern numbers for the lowest hybrid bands. The resulting low-energy topology and associated edge states remain robust under magnetic-field variation, while higher-energy hybrid bands can undergo field-driven topological phase transitions. These results extend topological magnon-phonon hybridization to noncoplanar SkXs.

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 manuscript investigates magnon-phonon hybrid excitations in a Néel-type two-dimensional ferromagnetic skyrmion crystal stabilized on a triangular lattice by DMI. Starting from an explicit spin-lattice Hamiltonian, it derives the bosonic Hamiltonian incorporating MP coupling via DMI-vector fluctuations, performs Bogoliubov diagonalization to obtain the hybrid band structure, and computes Chern numbers, finding that MP coupling opens gaps at low-energy crossings, lifts phonon degeneracies from supercell folding, and produces nontrivial Chern numbers for the lowest hybrid bands. These low-energy topological features and associated edge states are reported to remain robust under magnetic-field variation, while higher-energy bands exhibit field-driven topological phase transitions.

Significance. If the central results hold, the work extends the framework of topological magnon-phonon hybridization from collinear or coplanar systems to noncoplanar skyrmion crystals, providing a concrete route to field-tunable topological bands in a realistic spin-lattice setting. The reported robustness of the low-energy topology to magnetic field changes would be a practically relevant feature for potential magnonic applications.

major comments (2)
  1. [§II] §II (Model Hamiltonian): The Néel-type SkX is posited as the classical ground state for the chosen J, D, and B parameters on the triangular lattice, but no classical energy minimization, Monte Carlo annealing, or direct comparison against competing phases (ferromagnetic, helical, or other SkX variants) is reported. This assumption is load-bearing because the magnon spectrum, the DMI-fluctuation coupling term, and all subsequent hybrid-band topology inherit directly from the assumed spin texture.
  2. [§III] §III (Bosonic Hamiltonian derivation): The explicit matrix elements of the magnon-phonon coupling arising from DMI-vector fluctuations under the supercell folding are not shown in sufficient detail. Without these, it is difficult to verify how the reported gap openings and degeneracy lifting at the low-energy crossings are obtained or to reproduce the Chern-number calculations.
minor comments (2)
  1. Figure captions for the hybrid band structures and edge-state plots should explicitly state the magnetic-field values and the supercell size used, to facilitate direct comparison with the robustness claims.
  2. A brief statement on the convergence of the Bogoliubov diagonalization with respect to the number of phonon modes retained would strengthen the numerical results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We have carefully considered the major comments and provide point-by-point responses below, along with indications of revisions to be made in the updated version.

read point-by-point responses

  1. Referee: [§II] §II (Model Hamiltonian): The Néel-type SkX is posited as the classical ground state for the chosen J, D, and B parameters on the triangular lattice, but no classical energy minimization, Monte Carlo annealing, or direct comparison against competing phases (ferromagnetic, helical, or other SkX variants) is reported. This assumption is load-bearing because the magnon spectrum, the DMI-fluctuation coupling term, and all subsequent hybrid-band topology inherit directly from the assumed spin texture.

    Authors: We appreciate the referee highlighting this important point. The parameters were selected based on established results for Néel skyrmion crystals on triangular lattices stabilized by DMI, as in previous works. However, to address the concern directly, we have performed additional classical Monte Carlo simulations and energy minimization calculations. In the revised manuscript, we include a new subsection or appendix detailing these results, confirming that the Néel-type SkX is indeed the ground state for the chosen parameters, with lower energy than ferromagnetic, helical, or other configurations. This strengthens the foundation of our calculations. revision: yes

  2. Referee: [§III] §III (Bosonic Hamiltonian derivation): The explicit matrix elements of the magnon-phonon coupling arising from DMI-vector fluctuations under the supercell folding are not shown in sufficient detail. Without these, it is difficult to verify how the reported gap openings and degeneracy lifting at the low-energy crossings are obtained or to reproduce the Chern-number calculations.

    Authors: We agree that providing the explicit matrix elements would improve the clarity and reproducibility of our results. In the revised version, we have expanded the derivation in Section III to include the detailed expressions for the magnon-phonon coupling terms in the bosonic Hamiltonian, accounting for the supercell folding. These matrix elements are now explicitly written out, and we have added further explanations on how they lead to the gap openings and degeneracy lifting. Additionally, the full coupling matrices are provided in the Supplemental Material for completeness. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from explicit Hamiltonian via standard Bogoliubov diagonalization

full rationale

The paper starts from a defined spin-lattice Hamiltonian with explicit MP coupling via DMI vector fluctuations, assumes the Néel SkX configuration as input, derives the bosonic Hamiltonian, and computes hybrid bands and Chern numbers through standard Bogoliubov diagonalization. No predictions reduce to fitted parameters by construction, no self-citation chains bear the central topological claim, and no ansatz or uniqueness theorem is smuggled in. The low-energy topology is an output of the diagonalization, not equivalent to the inputs. The SkX stability assumption is a modeling premise, not a definitional loop.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on a standard spin-lattice model plus an assumed interaction form; no new entities are postulated.

free parameters (2)
  • DMI interaction strength
    Controls SkX stabilization and enters the coupling term; value not specified in abstract.
  • magnetic field strength
    Varied to test robustness of topology; treated as tunable parameter.
axioms (2)
  • domain assumption Neel-type SkX stabilized by DMI on triangular lattice
    Invoked to set up the bare magnon bands before hybridization.
  • domain assumption Phonon-magnon coupling occurs exclusively through DMI vector fluctuations
    Defines the interaction Hamiltonian used for the hybrid bands.

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discussion (0)

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Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages

  1. [1]

    Mühlbauer, B

    S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P . Böni, Skyrmion lattice in a chiral magnet, Science (1979). 323, 915 (2009)

    work page 1979

  2. [2]

    Garst, J

    M. Garst, J. Waizner, and D. Grundler, Collective spin excitations of helices and magnetic skyrmions: review and perspectives of magnonics in non-centrosymmetric magnets, J. Phys. D Appl. Phys. 50, 293002 (2017)

    work page 2017

  3. [3]

    Petrova and O

    O. Petrova and O. Tchernyshyov, Spin waves in a skyrmion crystal, Phys. Rev. B Condens. Matter Mater. Phys. 84, 214433 (2011)

    work page 2011

  4. [4]

    Iwasaki, A

    J. Iwasaki, A. J. Beekman, and N. Nagaosa, Theory of magnon-skyrmion scattering in chiral magnets, Phys. Rev. B Condens. Matter Mater. Phys. 89, 064412 (2014)

    work page 2014

  5. [5]

    Roldán-Molina, A

    A. Roldán-Molina, A. S. Nunez, and J. Fernández-Rossier, Topological spin waves in the atomic-scale magnetic skyrmion crystal, New J. Phys. 18, 045015 (2016)

    work page 2016

  6. [6]

    S. K. Kim, K. Nakata, D. Loss, and Y . Tserkovnyak, Tunable Magnonic Thermal Hall Effect in Skyrmion Crystal Phases of Ferrimagnets, Phys. Rev. Lett. 122, 057204 (2019)

    work page 2019

  7. [7]

    Weber et al., Topological magnon band structure of emergent Landau levels in a skyrmion lattice, Science (1979)

    T. Weber et al., Topological magnon band structure of emergent Landau levels in a skyrmion lattice, Science (1979). 375, 1025 (2022)

    work page 1979

  8. [8]

    K. A. Van Hoogdalem, Y . Tserkovnyak, and D. Loss, Magnetic texture-induced thermal Hall effects, Phys. Rev. B Condens. Matter Mater. Phys. 87, 024402 (2013)

    work page 2013

  9. [9]

    Mochizuki, Spin-Wave Modes and Their Intense Excitation Effects in Skyrmion Crystals, Phys

    M. Mochizuki, Spin-Wave Modes and Their Intense Excitation Effects in Skyrmion Crystals, Phys. Rev. Lett. 108, 017601 (2012)

    work page 2012

  10. [10]

    Nagaosa and Y

    N. Nagaosa and Y . Tokura, Topological properties and dynamics of magnetic skyrmions, Nature Nanotechnology 2013 8:12 8, 899 (2013)

    work page 2013

  11. [11]

    F. Zhuo, J. Kang, A. Manchon, and Z. Cheng, Topological Phases in Magnonics, Advanced Physics Research 2300054 (2023)

    work page 2023

  12. [12]

    S. A. Díaz, J. Klinovaja, and D. Loss, Topological Magnons and Edge States in Antiferromagnetic Skyrmion Crystals, Phys. Rev. Lett. 122, 187203 (2019)

    work page 2019

  13. [13]

    Hirosawa, S

    T. Hirosawa, S. A. Diaz, J. Klinovaja, and D. Loss, Magnonic Quadrupole Topological Insulator in Antiskyrmion Crystals, Phys. Rev. Lett. 125, 207204 (2020)

    work page 2020

  14. [14]

    Hirosawa, A

    T. Hirosawa, A. Mook, J. Klinovaja, and D. Loss, Magnetoelectric Cavity Magnonics in Skyrmion Crystals, (2022)

    work page 2022

  15. [15]

    A. Mook, J. Klinovaja, and D. Loss, Quantum damping of skyrmion crystal eigenmodes due to spontaneous quasiparticle decay, Phys. Rev. Res. 2, 033491 (2020)

    work page 2020

  16. [16]

    Ghader and B

    D. Ghader and B. Jabakhanji, Momentum-space theory for topological magnons in two- dimensional ferromagnetic skyrmion lattices, Phys. Rev. B 110, 184409 (2024)

    work page 2024

  17. [17]

    V. E. Timofeev and D. N. Aristov, Magnon band structure of skyrmion crystals and stereographic projection approach, Phys. Rev. B 105, 024422 (2022)

    work page 2022

  18. [18]

    Akazawa, H

    M. Akazawa, H. Y . Lee, H. Takeda, Y . Fujima, Y . Tokunaga, T. H. Arima, J. H. Han, and M. Yamashita, Topological thermal Hall effect of magnons in magnetic skyrmion lattice, Phys. Rev. Res. 4, 043085 (2022)

    work page 2022

  19. [19]

    Takeda et al., Magnon thermal Hall effect via emergent SU(3) flux on the antiferromagnetic skyrmion lattice, Nature Communications 2024 15:1 15, 566 (2024)

    H. Takeda et al., Magnon thermal Hall effect via emergent SU(3) flux on the antiferromagnetic skyrmion lattice, Nature Communications 2024 15:1 15, 566 (2024)

    work page 2024

  20. [20]

    V. E. Timofeev, Y . V. Baramygina, and D. N. Aristov, Magnon Topological Transition in Skyrmion Crystal, JETP Lett. 118, 911 (2023)

    work page 2023

  21. [21]

    Ghader and B

    D. Ghader and B. Jabakhanji, Impact of the honeycomb spin lattice on topological magnons and edge states in ferromagnetic two-dimensional skyrmion crystals, Phys. Rev. B 113, 104430 (2026)

    work page 2026

  22. [22]

    V. E. Timofeev, D. A. Bedyaev, and D. N. Aristov, Topological Transition in Spectrum of Skyrmion Crystal with Uniaxial Anisotropy, JETP Letters 2026 1 (2026)

    work page 2026

  23. [23]

    S. A. Diáz, T. Hirosawa, J. Klinovaja, and D. Loss, Chiral magnonic edge states in ferromagnetic skyrmion crystals controlled by magnetic fields, Phys. Rev. Res. 2, 013231 (2020)

    work page 2020

  24. [24]

    Kikkawa, K

    T. Kikkawa, K. Shen, B. Flebus, R. A. Duine, K. I. Uchida, Z. Qiu, G. E. W. Bauer, and E. Saitoh, Magnon Polarons in the Spin Seebeck Effect, Phys. Rev. Lett. 117, 207203 (2016)

    work page 2016

  25. [25]

    Flebus, K

    B. Flebus, K. Shen, T. Kikkawa, K. I. Uchida, Z. Qiu, E. Saitoh, R. A. Duine, and G. E. W. Bauer, Magnon-polaron transport in magnetic insulators, Phys. Rev. B 95, 144420 (2017)

    work page 2017

  26. [26]

    Streib, N

    S. Streib, N. Vidal-Silva, K. Shen, and G. E. W. Bauer, Magnon-phonon interactions in magnetic insulators, Phys. Rev. B 99, 184442 (2019)

    work page 2019

  27. [27]

    D. A. Bozhko, V. I. Vasyuchka, A. V. Chumak, and A. A. Serga, Magnon-phonon interactions in magnon spintronics (Review article), Low Temperature Physics 46, 383 (2020)

    work page 2020

  28. [28]

    Takahashi and N

    R. Takahashi and N. Nagaosa, Berry Curvature in Magnon-Phonon Hybrid Systems, Phys. Rev. Lett. 117, 217205 (2016)

    work page 2016

  29. [29]

    Zhang, Y

    X. Zhang, Y . Zhang, S. Okamoto, and D. Xiao, Thermal Hall Effect Induced by Magnon- Phonon Interactions, Phys. Rev. Lett. 123, 167202 (2019)

    work page 2019

  30. [30]

    G. Go, S. K. Kim, and K. J. Lee, Topological Magnon-Phonon Hybrid Excitations in Two- Dimensional Ferromagnets with Tunable Chern Numbers, Phys. Rev. Lett. 123, 237207 (2019)

    work page 2019

  31. [31]

    Shen and S

    P . Shen and S. K. Kim, Magnetic field control of topological magnon-polaron bands in two- dimensional ferromagnets, Phys. Rev. B 101, 125111 (2020)

    work page 2020

  32. [32]

    Park and B

    S. Park and B. J. Yang, Topological magnetoelastic excitations in noncollinear antiferromagnets, Phys. Rev. B 99, 174435 (2019)

    work page 2019

  33. [33]

    Zhang, G

    S. Zhang, G. Go, K. J. Lee, and S. K. Kim, SU(3) Topology of Magnon-Phonon Hybridization in 2D Antiferromagnets, Phys. Rev. Lett. 124, 147204 (2020)

    work page 2020

  34. [34]

    Ma and G

    B. Ma and G. A. Fiete, Antiferromagnetic insulators with tunable magnon-polaron Chern numbers induced by in-plane optical phonons, Phys. Rev. B 105, L100402 (2022)

    work page 2022

  35. [35]

    Huang and Z

    H. Huang and Z. Tian, Topological phonon-magnon hybrid excitations in a two- dimensional honeycomb ferromagnet, Phys. Rev. B 104, 064305 (2021)

    work page 2021

  36. [36]

    S. Park, N. Nagaosa, and B. J. Yang, Thermal Hall Effect, Spin Nernst Effect, and Spin Density Induced by a Thermal Gradient in Collinear Ferrimagnets from Magnon–Phonon Interaction, Nano Lett. 20, 2741 (2020)

    work page 2020

  37. [37]

    Sheikhi, M

    B. Sheikhi, M. Kargarian, and A. Langari, Hybrid topological magnon-phonon modes in ferromagnetic honeycomb and kagome lattices, Phys. Rev. B 104, 045139 (2021)

    work page 2021

  38. [38]

    W. Dong, H. Wang, M. Baggioli, and Y . Liu, Topological magnetic phases and magnon- phonon hybridization in the presence of strong Dzyaloshinskii-Moriya interaction, Phys. Rev. B 113, 064413 (2026)

    work page 2026

  39. [39]

    Thingstad, A

    E. Thingstad, A. Kamra, A. Brataas, and A. Sudbø, Chiral Phonon Transport Induced by Topological Magnons, Phys. Rev. Lett. 122, 107201 (2019)

    work page 2019

  40. [40]

    J. D. Mella, L. E. F. F. Torres, and R. E. Troncoso, Chiral magnon-polaron edge states in Heisenberg-Kitaev magnets, Phys. Rev. B 110, 104433 (2024)

    work page 2024

  41. [41]

    G. Go, H. Yang, J. G. Park, and S. K. Kim, Topological magnon polarons in honeycomb antiferromagnets with spin-flop transition, Phys. Rev. B 109, 184435 (2024)

    work page 2024

  42. [42]

    Liu et al., Direct Observation of Magnon-Phonon Strong Coupling in Two-Dimensional Antiferromagnet at High Magnetic Fields, Phys

    S. Liu et al., Direct Observation of Magnon-Phonon Strong Coupling in Two-Dimensional Antiferromagnet at High Magnetic Fields, Phys. Rev. Lett. 127, 097401 (2021)

    work page 2021

  43. [43]

    T. T. Mai, K. F. Garrity, A. McCreary, J. Argo, J. R. Simpson, V. Doan-Nguyen, R. V. Aguilar, and A. R. Hight Walker, Magnon-phonon hybridization in 2D antiferromagnet MnPSe3, Sci. Adv. 7, (2021)

    work page 2021

  44. [44]

    Luo et al., Evidence for Topological Magnon–Phonon Hybridization in a 2D Antiferromagnet down to the Monolayer Limit, Nano Lett

    J. Luo et al., Evidence for Topological Magnon–Phonon Hybridization in a 2D Antiferromagnet down to the Monolayer Limit, Nano Lett. 23, 2023 (2023)

    work page 2023

  45. [45]

    Bao et al., Direct observation of topological magnon polarons in a multiferroic material, Nat

    S. Bao et al., Direct observation of topological magnon polarons in a multiferroic material, Nat. Commun. 14, 6093 (2023)

    work page 2023

  46. [46]

    Cui et al., Chirality selective magnon-phonon hybridization and magnon-induced chiral phonons in a layered zigzag antiferromagnet, Nature Communications 2023 14:1 14, 3396 (2023)

    J. Cui et al., Chirality selective magnon-phonon hybridization and magnon-induced chiral phonons in a layered zigzag antiferromagnet, Nature Communications 2023 14:1 14, 3396 (2023)

    work page 2023

  47. [47]

    Haraldsen and R

    J. Haraldsen and R. Fishman, Spin rotation technique for non-collinear magnetic systems: application to the generalizedVillain model, Journal of Physics: Condensed Matter 21, 216001 (2009)

    work page 2009

  48. [48]

    R. F. L. Evans, W. J. Fan, P . Chureemart, T. A. Ostler, M. O. A. Ellis, and R. W. Chantrell, Atomistic spin model simulations of magnetic nanomaterials, Journal of Physics: Condensed Matter 26, 103202 (2014)

    work page 2014

  49. [49]

    Mæland and A

    K. Mæland and A. Sudbø, Quantum topological phase transitions in skyrmion crystals, Phys. Rev. Res. 4, L032025 (2022)

    work page 2022

  50. [50]

    Mæland and A

    K. Mæland and A. Sudbø, Quantum fluctuations in the order parameter of quantum skyrmion crystals, Phys. Rev. B 105, 224416 (2022)

    work page 2022

  51. [51]

    E. A. Stepanov, S. A. Nikolaev, C. Dutreix, M. I. Katsnelson, and V. V. Mazurenko, Heisenberg-exchange-free nanoskyrmion mosaic, Journal of Physics: Condensed Matter 31, 17LT01 (2019)

    work page 2019

  52. [52]

    E. A. Stepanov, C. Dutreix, and M. I. Katsnelson, Dynamical and Reversible Control of Topological Spin Textures, Phys. Rev. Lett. 118, 157201 (2017)

    work page 2017

  53. [53]

    V. V. Mazurenko, Y . O. Kvashnin, A. I. Lichtenstein, and M. I. Katsnelson, A DMI Guide to Magnets Micro-World, Journal of Experimental and Theoretical Physics 2021 132:4 132, 506 (2021)

    work page 2021

  54. [54]

    M. Ma, Z. Pan, and F. Ma, Artificial skyrmion in magnetic multilayers, J. Appl. Phys. 132, 043906 (2022)

    work page 2022

  55. [55]

    Li et al., An Artificial Skyrmion Platform with Robust Tunability in Synthetic Antiferromagnetic Multilayers, Adv

    Y . Li et al., An Artificial Skyrmion Platform with Robust Tunability in Synthetic Antiferromagnetic Multilayers, Adv. Funct. Mater. 30, 1907140 (2020)

    work page 2020

  56. [56]

    Jabakhanji and D

    B. Jabakhanji and D. Ghader, Designing Layered 2D Skyrmion Lattices in Moiré Magnetic Heterostructures, Adv. Mater. Interfaces 11, 2300188 (2024)

    work page 2024

  57. [57]

    Jabakhanji and D

    B. Jabakhanji and D. Ghader, Exploring moiré skyrmions in twisted double bilayer and double trilayer ${\mathbf{Cr}}{{\mathbf{I}}_{\mathbf{3}}}$, Journal of Physics: Condensed Matter 37, 075801 (2024)

    work page 2024

  58. [58]

    J. H. P . Colpa, Diagonalization of the quadratic boson hamiltonian, Physica A: Statistical Mechanics and Its Applications 93, 327 (1978)

    work page 1978

  59. [59]

    Fukui, Y

    T. Fukui, Y . Hatsugai, and H. Suzuki, Chern numbers in discretized Brillouin zone: Efficient method of computing (spin) Hall conductances, J. Physical Soc. Japan 74, 1674 (2005)

    work page 2005

  60. [60]

    Katsura, N

    H. Katsura, N. Nagaosa, and P . A. Lee, Theory of the thermal hall effect in quantum magnets, Phys. Rev. Lett. 104, 066403 (2010)

    work page 2010