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Precursor behavior, bag formation, and lattice instabilities of confined hopfions in chiral magnetic films

Andrey O. Leonov* and Takayuki Shigenaga

  • Department of Chemistry, Faculty of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan and International Institute for Sustainability with Knotted Chiral Meta Matter (WPI-SKCM2), Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8531, Japan

  • *Contact author: leonov@hiroshima-u.ac.jp
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Phys. Rev. Research 8, 023193 – Published 21 May, 2026

DOI: https://doi.org/10.1103/7nly-96st

Abstract

Topological particlelike excitations such as skyrmions and hopfions offer rich opportunities for spintronic and photonic applications. While skyrmions have been extensively studied, the stabilization mechanisms and phase behavior of three-dimensional hopfions remain largely unexplored. Here, we investigate the formation, stability, and interactions of hopfions in thin chiral magnetic films with surface anchoring, using three-dimensional micromagnetic simulations within a material-independent framework applicable to both magnetic and liquid crystalline systems. We identify four distinct types of isolated hopfions, generated by rotating bimeron and fingerlike solitons around a central axis. The metastability regions of these precursor textures closely follow the boundaries of modulated finger phases, enabling their size to be continuously tuned through anisotropy-driven inflation and collapse. Remarkably, we demonstrate that hopfions near their inflation threshold possess energies comparable with the homogeneous state, allowing them to enclose regions of modulated phases or other solitons, forming higher-order, baglike domains. In contrast, periodic hopfion lattices remain intrinsically unstable under confinement, spontaneously relaxing into finger phases. These findings establish general principles for stabilizing, tuning, and assembling three-dimensional topological solitons in confined chiral systems, suggesting experimentally accessible routes for texture engineering in liquid crystals via electric-field control.

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References (52)

  1. N. Manton and P. Sutcliffe, Topological Solitons (Cambridge University Press, Cambridge, 2004).
  2. Y. M. Shnir, Topological and Non-Topological Solitons in Scalar Field Theories (Cambridge University Press, Cambridge, 2018).
  3. R. Rajaraman, Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory (North-Holland, Amsterdam, 1982).
  4. G. E. Volovik and V. P. Mineev, Particle-like solitons in superfluid He3 phases, Zh. Eksp. Teor. Fiz. 73, 767 (1977) [Sov. Phys. JETP 46, 401 (1977)].
  5. A. O. Leonov and K. Nakamura, Skyrmionium meta-matter: A topologically heterogeneous magnetic crystal with emergent hybrid dynamics, Phys. Rev. Mater. 10, 036001 (2026).
  6. A. Bogdanov and A. Hubert, Thermodynamically stable magnetic vortex states in magnetic crystals, J. Magn. Magn. Mater. 138, 255 (1994).
  7. N. Nagaosa and Y. Tokura, Topological properties and dynamics of magnetic skyrmions, Nat. Nanotechnol. 8, 899 (2013).
  8. See Supplemental Material at http://link.aps.org/supplemental/10.1103/7nly-96st for a glossary of solitonic objects and a ready-to-use mumax3 script for generating the initial hopfion configurations.
  9. L. Faddeev and A. Niemi, Stable knot-like structures in classical field theory, Nature (London) 387, 58 (1997).
  10. R. Bott and L. W. Tu, Differential Forms in Algebraic Topology (Springer, New York, 1995).
  11. A. Fert, N. Reyren, and V. Cros, Magnetic skyrmions: Advances in physics and potential applications, Nat. Rev. Mater. 2, 17031 (2017).
  12. C. Back et al., The 2020 skyrmionics roadmap, J. Phys. D: Appl. Phys. 53, 363001 (2020).
  13. P. Oswald and P. Pieranski, Nematic and Cholesteric Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments (CRC Press, Boca Raton, FL, 2005).
  14. A. Fert, V. Cros, and J. Sampaio, Skyrmions on the track, Nat. Nanotechnol. 8, 152 (2013).
  15. I. I. Smalyukh, Review: Knots and other new topological effects in liquid crystals and colloids, Rep. Prog. Phys. 83, 106601 (2020).
  16. S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. Böni, Skyrmion lattice in a chiral magnet, Science 323, 915 (2009).
  17. X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z. Zhang, S. Ishiwata, Y. Matsui, and Y. Tokura, Near room-temperature formation of a skyrmion crystal in thin-films of the helimagnet FeGe, Nat. Mater. 10, 106 (2011).
  18. A. O. Leonov, A. N. Bogdanov, and K. Inoue, Toggle-switch-like crossover between two types of isolated skyrmions within the conical phase of cubic helimagnets, Phys. Rev. B 98, 060411(R) (2018).
  19. A. Duzgun and J. V. Selinger, Comparing skyrmions and merons in chiral liquid crystals and magnets, Phys. Rev. E 97, 062706 (2018).
  20. A. O. Leonov, Swirling of horizontal skyrmions into hopfions in bulk cubic helimagnets, Magnetism 3, 297 (2023).
  21. K. L. Metlov, Elliptical stability of hopfions in bulk helimagnets, Phys. Rev. B 111, 174450 (2025).
  22. J. H. C. Whitehead, An expression of Hopf’s Invariant as an Integral, Proc. Natl. Acad. Sci. USA 33, 117 (1947).
  23. J. Gladikowski and M. Helmund, Static solitons with nonzero Hopf number, Phys. Rev. D 56, 5194 (1997).
  24. K. Guslienko, Magnetic Hopfions: A review, Magnetism 4, 383 (2024).
  25. P. Sutcliffe, Hopfions in chiral magnets, J. Phys. A: Math. Theor. 51, 375401 (2018).
  26. N. Kent, N. Reynolds, D. Raftrey, I. T. G. Campbell, S. Virasawmy, S. Dhuey, R. V. Chopdekar, A. Hierro-Rodriguez, A. Sorrentino, E. Pereiro, S. Ferrer, F. Hellman, P. Sutcliffe, and P. Fischer, Creation and observation of hopfions in magnetic multilayer systems, Nat. Commun. 12, 1562 (2021).
  27. C. Saji, R. E. Troncoso, V. L. Carvalho-Santos, D. Altbir, and A. S. Núñez, Hopfion-driven magnonic Hall effect and magnonic focusing, Phys. Rev. Lett. 131, 166702 (2023).
  28. B. Göbel, C. A. Akosa, G. Tatara, and I. Mertig, Opological Hall signatures of magnetic hopfions, Phys. Rev. Res. 2, 013315 (2020).
  29. Z. Khodzhaev and E. Turgut, Hopfion dynamics in chiral magnets, J. Phys.: Condens. Matter 34, 225805 (2022).
  30. X. Yu, Y. Liu, K. V. Iakoubovskii, K. Nakajima, N. Kanazawa, N. Nagaosa, and Y. Tokura, Realization and current-driven dynamics of fractional hopfions and their ensembles in a helimagnet FeGe, Adv. Mater. 35, 2210646 (2023).
  31. P. J. Ackerman and I. I. Smalyukh, Static three-dimensional topological solitons in fluid chiral ferromagnets and colloids, Nat. Mater. 16, 426 (2017).
  32. A. O. Leonov, Surface anchoring as a control parameter for shaping skyrmion or toron properties in thin layers of chiral nematic liquid crystals and noncentrosymmetric magnets, Phys. Rev. E 104, 044701 (2021).
  33. T. Yamamoto, T. Nozaki, H. Imamura, S. Tamaru, K. Yakushiji, H. Kubota, A. Fukushima, and S. Yuasa, Voltage-driven magnetization switching controlled by microwave electric field pumping, Nano Lett. 20, 6012 (2020).
  34. M. T. Johnson, P. J. H. Bloemen, F. J. A. den Broeder, and J. J. de Vries, Magnetic anisotropy in metallic multilayers, Rep. Prog. Phys. 59, 1409 (1996).
  35. X. C. Hu, X. S. Wang, C. Wang, and X. R. Wang, Hopfions in chiral magnetic trilayers, Phys. Rev. B 111, 094436 (2025).
  36. C. Donnelly and V. Scagnoli, Imaging three-dimensional magnetic systems with X-rays, J. Phys.: Condens. Matter 32, 213001 (2020).
  37. S. V. Shiyanovskii, D. Voloschenko, T. Ishikawa, and O. D. Lavrentovich, Director structures of cholesteric diffraction gratings, Mol. Cryst. Liq. Cryst. 358, 225 (2001).
  38. S. V. Shiyanovskii, I. I. Smalyukh, and O. D. Lavrentovich, Computer simulations and fluorescence confocal polarizing microscopy of structures in cholesteric liquid crystals, in Defects in Liquid Crystals: Computer Simulations, edited by O. D. Lavrentovich, P. Pasini, C. Zannoni, and S. Zumer (Kluwer Academic Publishers, Dordrecht, 2001), pp. 229–270.
  39. J. V. Selinger, Interpretation of saddle-splay and the Oseen-Frank free energy in liquid crystals, Liq. Cryst. Rev. 6, 129 (2018).
  40. A. Mertelj, D. Lisjak, M. Drofenik, and M. Copic, Ferromagnetism in suspensions of magnetic platelets in liquid crystal, Nature (London) 504, 237 (2013).
  41. Q. Liu, P. J. Ackerman, T. C. Lubensky, and I. I. Smalyukh, Biaxial ferromagnetic liquid crystal colloids, Proc. Natl. Acad. Sci. USA 113, 10479 (2016).
  42. O. D. Lavrentovich, Ferroelectric nematic liquid crystal, a century in waiting, Proc. Natl. Acad. Sci. USA 117, 14629 (2020).
  43. A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyenberge, The design and verification of MuMax3, AIP Adv. 4, 107133 (2014).
  44. P. J. Ackerman and I. I. Smalyukh, Diversity of knot solitons in liquid crystals manifested by linking of preimages in torons and hopfions, Phys. Rev. X 7, 011006 (2017).
  45. I. I. Smalyukh, B. I. Senyuk, P. Palffy-Muhoray, O. D. Lavrentovich, H. Huang, E. C. Gartland, Jr., V. H. Bodnar, T. Kosa, and B. Taheri, Electric-field-induced nematic-cholesteric transition and three-dimensional director structures in homeotropic cells, Phys. Rev. E 72, 061707 (2005).
  46. P. F. Bessarab, V. M. Uzdin, and H. Jónsson, Method for finding mechanism and activation energy of magnetic transitions, applied to skyrmion and antivortex annihilation, Comput. Phys. Commun. 196, 335 (2015).
  47. Y. Togawa, T. Koyama, K. Takayanagi, S. Mori, Y. Kousaka, J. Akimitsu, S. Nishihara, K. Inoue, A. S. Ovchinnikov, and J. Kishine, Chiral magnetic soliton lattice on a chiral helimagnet, Phys. Rev. Lett. 108, 107202 (2012).
  48. A. Kovács, J. Caron, A. S. Savchenko, N. S. Kiselev, K. Shibata, Z.-A. Li, N. Kanazawa, Y. Tokura, S. Blügel, and R. E. Dunin-Borkowski, Mapping the magnetization fine structure of a lattice of Bloch-type skyrmions in an FeGe thin film, Appl. Phys. Lett. 111, 192410 (2017).
  49. D. Foster, C. Kind, P. J. Ackerman, J.-S. B. Tai, M. R. Dennis, I. I. Smalyukh, Two-dimensional skyrmion bags in liquid crystals and ferromagnets, Nat. Phys. 15, 655 (2019).
  50. J. Tang, Y. Wu, W. Wang, L. Kong, B. Lv, W. Wei, J. Zang, M. Tian, and H. Du, Magnetic skyrmion bundles and their current-driven dynamics, Nat. Nanotechnol. 16, 1086 (2021).
  51. J.-S. B. Tai and I. I. Smalyukh, Static Hopf solitons and knotted emergent fields in solid-state noncentrosymmetric magnetic nanostructures, Phys. Rev. Lett. 121, 187201 (2018).
  52. A. O. Leonov, T. L. Monchesky, N. Romming, A. Kubetzka, A. N. Bogdanov, and R. Wiesendanger, The properties of isolated chiral skyrmions in thin magnetic films, New J. Phys. 18, 065003 (2016).

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Phys. Rev. Research 8, 023193– Published 21 May, 2026
  • Received 27 July 2025
  • Accepted 3 May 2026
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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.

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