- Physical Review Journals
All JournalsPhysics Magazine

Physical Review X

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Fate of Topological Dirac Magnons in van der Waals Ferromagnets at Finite Temperature

Rintaro Eto1,2,3,4,*, Ignacio Salgado-Linares1,2, Masahito Mochizuki4, Johannes Knolle1,2,5, and Alexander Mook6,3

  • *Contact author: rintaro.eto@tum.de
PDF

Phys. Rev. X 16, 021053 – Published 10 June, 2026

DOI: https://doi.org/10.1103/tbh2-jq9r

Abstract

Dirac magnons, the bosonic counterparts of Dirac fermions in graphene, provide a versatile platform to explore symmetry-protected band crossings and quantum geometry in magnetic insulators while promising high-velocity, low-dissipation spin transport for next-generation magnonic technologies. However, their stability under realistic, finite-temperature conditions remains an open question. Here, we develop a microscopic theory of thermal magnon-magnon interactions in van der Waals honeycomb ferromagnets, focusing on both gapless and gapped Dirac magnons. Using nonlinear spin-wave theory with magnon self-energy corrections and a T-matrix resummation that captures two-magnon bound states, we quantitatively reproduce temperature- and momentum-dependent energy shifts and linewidths observed experimentally in the gapless Dirac magnon material CrBr3, even near the Curie temperature. Our approach provides a consistent interpretation of theoretical predictions and experiment shedding light on the role of bound states in enhancing magnon damping at low temperatures. For gapped Dirac magnon materials such as CrI3, CrSiTe3, and CrGeTe3, we find a thermally induced reduction of the topological magnon gap, while no indication of thermally driven topological phase transitions is observed within the considered parameter range. Classical atomistic spin-dynamics simulations corroborate the gap’s robustness up to the Curie temperature. Furthermore, we establish a practical criterion for observing topological gaps by determining the minimum ratio of Dzyaloshinskii-Moriya interaction to Heisenberg exchange required to overcome thermal broadening throughout the ordered phase, typically around 5%. Taken together, our results elucidate the interplay between thermal many-body effects and topology in low-dimensional magnetic systems and provide a controlled framework for the interpretation of spectroscopic measurements.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Understanding the behavior of Dirac magnons—topological, wavelike magnetic excitations—near the Curie temperature is a major challenge because thermal fluctuations typically degrade the stability of such delicate quantum states. We addressed this by developing a theoretical framework that accounts for finite-temperature interactions, explicitly including complex two-magnon bound states to move beyond standard approximations. Our model successfully replicates experimental observations of energy shifts and damping in chromium tribromide and predicts that the topological magnon gap in materials like chromium triiodide remains robust even as the temperature approaches the magnetic transition point. We also established criteria for experimentalists to identify these topological features by balancing magnetic interaction strengths against thermal effects. These results provide a foundation for designing energy-efficient spintronic devices that leverage topological magnons. Our work offers a predictive framework for exploring thermal stability in low-dimensional magnets and guides future research into the control of excitations in exotic quantum materials.

Article Text

References (76)

  1. J. Fransson, A. M. Black-Schaffer, and A. V. Balatsky, Magnon Dirac materials, Phys. Rev. B 94, 075401 (2016).
  2. S. S. Pershoguba, S. Banerjee, J. C. Lashley, J. Park, H. Ågren, G. Aeppli, and A. V. Balatsky, Dirac magnons in honeycomb ferromagnets, Phys. Rev. X 8, 011010 (2018).
  3. L. Chen, J.-H. Chung, B. Gao, T. Chen, M. B. Stone, A. I. Kolesnikov, Q. Huang, and P. Dai, Topological spin excitations in honeycomb ferromagnet CrI3, Phys. Rev. X 8, 041028 (2018).
  4. B. Yuan, I. Khait, G.-J. Shu, F. C. Chou, M. B. Stone, J. P. Clancy, A. Paramekanti, and Y.-J. Kim, Dirac magnons in a honeycomb lattice quantum XY magnet CoTiO3, Phys. Rev. X 10, 011062 (2020).
  5. A. Scheie, P. Laurell, P. A. McClarty, G. E. Granroth, M. B. Stone, R. Moessner, and S. E. Nagler, Dirac magnons, nodal lines, and nodal plane in elemental gadolinium, Phys. Rev. Lett. 128, 097201 (2022).
  6. S. E. Nikitin, B. Fåk, K. W. Krämer, T. Fennell, B. Normand, A. M. Läuchli, and C. Rüegg, Thermal evolution of Dirac magnons in the honeycomb ferromagnet CrBr3, Phys. Rev. Lett. 129, 127201 (2022).
  7. S.-H. Do, Joseph A. M. Paddison, G. Sala, T. J. Williams, K. Kaneko, K. Kuwahara, A. F. May, J. Yan, M. A. McGuire, M. B. Stone, M. D. Lumsden, and A. D. Christianson, Gaps in topological magnon spectra: Intrinsic versus extrinsic effects, Phys. Rev. B 106, L060408 (2022).
  8. H. Katsura, N. Nagaosa, and P. A. Lee, Theory of the thermal Hall effect in quantum magnets, Phys. Rev. Lett. 104, 066403 (2010).
  9. L. Zhang, J. Ren, J.-S. Wang, and B. Li, Topological magnon insulator in insulating ferromagnet, Phys. Rev. B 87, 144101 (2013).
  10. A. Mook, J. Henk, and I. Mertig, Edge states in topological magnon insulators, Phys. Rev. B 90, 024412 (2014).
  11. S. A. Owerre, A first theoretical realization of honeycomb topological magnon insulator, J. Phys. Condens. Matter 28, 386001 (2016).
  12. S. K. Kim, H. Ochoa, R. Zarzuela, and Y. Tserkovnyak, Realization of the Haldane-Kane-Mele model in a system of localized spins, Phys. Rev. Lett. 117, 227201 (2016).
  13. M. Malki and G. S. Uhrig, Topological magnetic excitations, Europhys. Lett. 132, 20003 (2020).
  14. R. Shindou, R. Matsumoto, S. Murakami, and J. I. Ohe, Topological chiral magnonic edge mode in a magnonic crystal, Phys. Rev. B 87, 174427 (2013).
  15. P. A. McClarty, Topological magnons: A review, Annu. Rev. Condens. Matter Phys. 13, 171 (2022).
  16. K.-z. Du, X.-z. Wang, Y. Liu, P. Hu, M. I. B. Utama, C. K. Gan, Q. Xiong, and C. Kloc, Weak van der Waals stacking, wide-range band gap, and Raman study on ultrathin layers of metal phosphorus trichalcogenides, ACS Nano 10, 1738 (2015).
  17. T. J. Williams, A. A. Aczel, M. D. Lumsden, S. E. Nagler, M. B. Stone, J.-Q. Yan, and D. Mandrus, Magnetic correlations in the quasi-two-dimensional semiconducting ferromagnet crsite3, Phys. Rev. B 92, 144404 (2015).
  18. C.-T. Kuo, M. Neumann, K. Balamurugan, H. J. Park, S. Kang, H. W. Shiu, J. H. Kang, B. H. Hong, M. Han, T. W. Noh, and J.-G. Park, Exfoliation and Raman spectroscopic fingerprint of few-layer NiPS3 van der Waals crystals, Sci. Rep. 6, 20904 (2016).
  19. M.-W. Lin, H. L. Zhuang, J. Yan, T. Z. Ward, A. A. Puretzky, C. M. Rouleau, Z. Gai, L. Liang, V. Meunier, B. G. Sumpter, P. Ganesh, P. R. C. Kent, D. B. Geohegan, D. G. Mandrus, and K. Xiao, Ultrathin nanosheets of CrSiTe3: A semiconducting two-dimensional ferromagnetic material, J. Mater. Chem. C 4, 315 (2016).
  20. B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. Xu, Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit, Nature (London) 546, 270 (2017).
  21. C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G. Louie, J. Xia, and X. Zhang, Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals, Nature (London) 546, 265 (2017).
  22. Q. H. Wang et al., The magnetic genome of two-dimensional van der Waals materials, ACS Nano 16, 6960 (2022).
  23. D. R. Klein, D. MacNeill, J. L. Lado, D. Soriano, E. Navarro-Moratalla, K. Watanabe, T. Taniguchi, S. Manni, P. Canfield, J. Fernández-Rossier, and P. Jarillo-Herrero, Probing magnetism in 2D van der Waals crystalline insulators via electron tunneling, Science 360, 1218 (2018).
  24. D. Ghazaryan, M. T. Greenaway, Z. Wang, V. H. Guarochico-Moreira, I. J. Vera-Marun, J. Yin, Y. Liao, S. V. Morozov, O. Kristanovski, A. I. Lichtenstein, M. I. Katsnelson, F. Withers, A. Mishchenko, L. Eaves, A. K. Geim, K. S. Novoselov, and A. Misra, Magnon-assisted tunnelling in van der Waals heterostructures based on CrBr3, National electronics review 1, 344 (2018).
  25. H. H. Kim, B. Yang, S. Li, S. Jiang, C. Jin, Z. Tao, G. Nichols, F. Sfigakis, S. Zhong, C. Li, S. Tian, D. G. Cory, G.-X. Miao, J. Shan, K. F. Mak, H. Lei, K. Sun, L. Zhao, and A. W. Tsen, Evolution of interlayer and intralayer magnetism in three atomically thin chromium trihalides, Proc. Natl. Acad. Sci. U.S.A. 116, 11131 (2019).
  26. F. Zhu, L. Zhang, X. Wang, F. J. dos Santos, J. Song, T. Mueller, K. Schmalzl, W. F. Schmidt, A. Ivanov, J. T. Park, J. Xu, J. Ma, S. Lounis, S. Blügel, Y. Mokrousov, Y. Su, and T. Brückel, Topological magnon insulators in two-dimensional van der Waals ferromagnets CrSiTe3 and CrGeTe3: Toward intrinsic gap-tunability, Sci. Adv. 7, eabi7532 (2021).
  27. Z. Cai, S. Bao, Z.-L. Gu, Y.-P. Gao, Z. Ma, Y. Shangguan, W. Si, Z.-Y. Dong, W. Wang, Y. Wu, D. Lin, J. Wang, K. Ran, S. Li, D. Adroja, X. Xi, S.-L. Yu, X. Wu, J.-X. Li, and J. Wen, Topological magnon insulator spin excitations in the two-dimensional ferromagnet crbr3, Phys. Rev. B 104, L020402 (2021).
  28. F. J. Dyson, General theory of spin-wave interactions, Phys. Rev. 102, 1217 (1956).
  29. A. B. Harris, D. Kumar, B. I. Halperin, and P. C. Hohenberg, Dynamics of an antiferromagnet at low temperatures: Spin-wave damping and hydrodynamics, Phys. Rev. B 3, 961 (1971).
  30. M. I. Kaganov and A. V. Chubukov, Interacting magnons, Sov. Phys. Usp. 30, 1015 (1987).
  31. M. E. Zhitomirsky and A. L. Chernyshev, Colloquium: Spontaneous magnon decays, Rev. Mod. Phys. 85, 219 (2013).
  32. A. Mook, K. Plekhanov, J. Klinovaja, and D. Loss, Interaction-stabilized topological magnon insulator in ferromagnets, Phys. Rev. X 11, 021061 (2021).
  33. M. Gohlke, A. Corticelli, R. Moessner, P. A. McClarty, and A. Mook, Spurious symmetry enhancement in linear spin wave theory and interaction-induced topology in magnons, Phys. Rev. Lett. 131, 186702 (2023).
  34. J. Bao, M. Gohlke, J. G. Rau, and N. Shannon, Magnon spectra of cuprates beyond spin wave theory, Phys. Rev. Res. 7, L012053 (2025).
  35. S. P. Bayrakci, T. Keller, K. Habicht, and B. Keimer, Spin-wave lifetimes throughout the Brillouin zone, Science 312, 1926 (2006).
  36. S. P. Bayrakci, D. A. Tennant, P. Leininger, T. Keller, M. C. R. Gibson, S. D. Wilson, R. J. Birgeneau, and B. Keimer, Lifetimes of antiferromagnetic magnons in two and three dimensions: Experiment, theory, and numerics, Phys. Rev. Lett. 111, 017204 (2013).
  37. Y.-S. Lu, J.-L. Li, and C.-T. Wu, Topological phase transitions of Dirac magnons in honeycomb ferromagnets, Phys. Rev. Lett. 127, 217202 (2021).
  38. H. Sun, D. Bhowmick, B. Yang, and P. Sengupta, Interacting topological Dirac magnons, Phys. Rev. B 107, 134426 (2023).
  39. S. Banerjee and S. Humeniuk, Interacting Dirac magnons in the van der Waals ferromagnet CrBr3, Phys. Rev. B 112, 094407 (2025).
  40. C. H. Cobb, V. Jaccarino, J. P. Remeika, R. Silberglitt, and H. Yasuoka, Field dependence of the magnetization and spin-wave correlations in ferromagnetic CrBr3, Phys. Rev. B 3, 1677 (1971).
  41. W. B. Yelon and R. Silberglitt, Renormalization of large-wave-vector magnons in ferromagnetic crbr3 studied by inelastic neutron scattering: Spin-wave correlation effects, Phys. Rev. B 4, 2280 (1971).
  42. Y.-M. Li, B. Wei, and K. Chang, Comment on “topological phase transitions of Dirac magnons in honeycomb ferromagnets,” Phys. Rev. Lett. 132, 219601 (2024).
  43. Y.-S. Lu, J.-L. Li, C.-T. Wu, and Lu et al., reply:, Phys. Rev. Lett. 132, 219602 (2024).
  44. Y.-M. Li, X.-W. Luo, and K. Chang, Temperature-induced magnonic Chern insulator in collinear antiferromagnets, Phys. Rev. B 107, 214417 (2023).
  45. H. Zhu, H. Shi, Z. Tang, and B. Tang, Interacting topological magnons in a checkerboard ferromagnet, Chin. Phys. B 33, 037503 (2024).
  46. H. Shi, H. Zhu, B. Tang, and C. Yang, Interacting Floquet topological magnons in laser-irradiated Heisenberg honeycomb ferromagnets, Phys. Lett. A 528, 130054 (2024).
  47. L. Chen, J.-H. Chung, M. B. Stone, A. I. Kolesnikov, B. Winn, V. O. Garlea, D. L. Abernathy, B. Gao, M. Augustin, Elton J. G. Santos, and P. Dai, Magnetic field effect on topological spin excitations in CrI3, Phys. Rev. X 11, 031047 (2021).
  48. J. Habel, A. Mook, J. Willsher, and J. Knolle, Breakdown of chiral edge modes in topological magnon insulators, Phys. Rev. B 109, 024441 (2024).
  49. R. Silberglitt and A. B. Harris, Effect of bound states on the excitation spectrum of a Heisenberg ferromagnet at low temperature, Phys. Rev. Lett. 19, 30 (1967).
  50. R. Silberglitt and A. B. Harris, Dynamics of the Heisenberg ferromagnet at low temperatures, Phys. Rev. 174, 640 (1968).
  51. T. Nomura and J. I. Igarashi, Analysis of resonant inelastic X-ray scattering in La2CuO4, Phys. Rev. B 71, 035110 (2005).
  52. T. Nagao and J. I. Igarashi, Two-magnon excitations in resonant inelastic X-ray scattering from quantum Heisenberg antiferromagnets, Phys. Rev. B 75, 214414 (2007).
  53. B. Fauseweh and G. S. Uhrig, Low-temperature thermodynamics of multiflavored hardcore bosons by the Brückner approach, Phys. Rev. B 92, 214417 (2015).
  54. L. Chen, J.-H. Chung, B. Gao, T. Chen, M. B. Stone, A. I. Kolesnikov, Q. Huang, and P. Dai, Topological spin excitations in honeycomb ferromagnet CrI3, Phys. Rev. X 8, 041028 (2018).
  55. T. Holstein and H. Primakoff, Field dependence of the intrinsic domain magnetization of a ferromagnet, Phys. Rev. 58, 1098 (1940).
  56. V. A. Alyoshin, V. A. Berezin, and V. A. Tulin, rf susceptibility of single-crystal CrBr3 near the Curie temperature, Phys. Rev. B 56, 719 (1997).
  57. E. Rastelli, Statistical Mechanics of Magnetic Excitations: From Spin Waves to Stripes and Checkerboards (World Scientific, Singapore, 2011).
  58. Y. Liu, L. Zhai, S. Yan, D. Wang, and X. Wan, Magnon-magnon interaction in monolayer MnBi2Te4, Phys. Rev. B 108, 174425 (2023).
  59. A. Mook, R. Hoyer, J. Klinovaja, and D. Loss, Magnons, magnon bound pairs, and their hybrid spin-multipolar topology, Phys. Rev. B 107, 064429 (2023).
  60. R. P. Reklis, Numerical calculation of two-spin-wave bound states in some two-dimensional Heisenberg ferromagnets, Phys. Rev. B 9, 4939 (1974).
  61. L. Kecke, T. Momoi, and A. Furusaki, Multimagnon bound states in the frustrated ferromagnetic one-dimensional chain, Phys. Rev. B 76, 060407(R) (2007).
  62. M. Wortis, Bound states of two spin waves in the Heisenberg ferromagnet, Phys. Rev. 132, 85 (1963).
  63. K. Wada, T. Ishikawa, and T. Oguchi, Two-magnon bound states in the triangular and honeycomb Heisenberg ferromagnets, Prog. Theor. Phys. 54, 1589 (1975).
  64. M. Nayak and F. Mila, Possibility to detect the bound state of the Heisenberg ferromagnetic chain at intermediate temperature, Phys. Rev. B 105, 094407 (2022).
  65. R. Silberglitt and J. B. Torrance, Effect of single-ion anisotropy on two-spin-wave bound state in a Heisenberg ferromagnet, Phys. Rev. B 2, 772 (1970).
  66. T. Tonegawa, Two-magnon bound states in the Heisenberg ferromagnet with anisotropic exchange and uniaxial anisotropy energies, Prog. Theor. Phys. Suppl. 46, 61 (1970).
  67. A. Legros, S.-S. Zhang, X. Bai, H. Zhang, Z. Dun, W. A. Phelan, C. D. Batista, M. Mourigal, and N. P. Armitage, Observation of 4- and 6-magnon bound states in the spin-anisotropic frustrated antiferromagnet fei2, Phys. Rev. Lett. 127, 267201 (2021).
  68. C. Franz, S. Säubert, A. Wendl, F. X. Haslbeck, O. Soltwedel, J. K. Jochum, L. Spitz, J. Kindervater, A. Bauer, P. Böni, and C. Pfleiderer, Mieze neutron spin-echo spectroscopy of strongly correlated electron systems, J. Phys. Soc. Jpn. 88, 081002 (2019).
  69. T. Gilbert, A phenomenological theory of damping in ferromagnetic materials, IEEE Trans. Magn. 40, 3443 (2004).
  70. 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, J. Phys. Condens. Matter 26, 103202 (2014).
  71. O. Franke, D. Călugăru, A. Nunnenkamp, and J. Knolle, Thermal spin dynamics of Kitaev magnets: Scattering continua and magnetic field induced phases within a stochastic semiclassical approach, Phys. Rev. B 106, 174428 (2022).
  72. E. M. Smith et al., Case for a U(1)π quantum spin liquid ground state in the dipole-octupole pyrochlore Ce2Zr2O7, Phys. Rev. X 12, 021015 (2022).
  73. J. Geng, I. N. Chan, H. Ai, K. H. Lo, Y. Kawazoe, K. W. Ng, and H. Pan, Magnetic and electronic properties of 2D TiX3 (X=F, Cl, Br and I), Phys. Chem. Chem. Phys. 22, 17632 (2020).
  74. X. Zou, Y. Bai, Y. Dai, B. Huang, and C. Niu, Experimentally feasible CrBr3 monolayer: Electronic and magnonic topological states correlated with rotation symmetry, Innovation Mater. 3, 100109 (2025).
  75. V. V. Gapontsev, D. D. Gazizova, and S. V. Streltsov, Dimerization in α−TiCl3 and α−TiBr3: The DFT study, J. Phys. Condens. Matter 33, 495803 (2021).
  76. S. Zhang, H. J. Changlani, K. W. Plumb, O. Tchernyshyov, and R. Moessner, Dynamical structure factor of the three-dimensional quantum spin liquid candidate NaCaNi2F7, Phys. Rev. Lett. 122, 167203 (2019).

Outline

Information

Phys. Rev. X 16, 021053– Published 10 June, 2026
  • Received 17 September 2025
  • Revised 11 April 2026
  • Accepted 24 April 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation