Hopfion

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Model of magnetic hopfion in a solid. Bem is emergent magnetic field (orange arrows); in a hopfion, it does not align to the external magnetic field (black arrow).

A hopfion is a topological soliton.[1] [2] [3] [4] It is a stable three-dimensional localised configuration of a three-component field with a knotted topological structure. They are the three-dimensional counterparts of 2D skyrmions, which exhibit similar topological properties in 2D. Hopfions are widely studied in many physical systems over the last half century, as summarized here http://hopfion.com

The soliton is mobile and stable: i.e. it is protected from a decay by an energy barrier. It can be deformed but always conserves an integer Hopf topological invariant. It is named after the German mathematician, Heinz Hopf.

A model that supports hopfions was proposed as follows[1]

The terms of higher-order derivatives are required to stabilize the hopfions.

Stable hopfions were predicted within various physical platforms, including Yang-Mills theory,[5] superconductivity[6][7] and magnetism.[8][9][10][4]

Experimental observation[edit]

Hopfions have been observed experimentally in chiral colloidal magnetic materials,[2] in chiral liquid crystals [11], [12] in Ir/Co/Pt multilayers using X-ray magnetic circular dichroism[13] and in the polarization of free-space monochromatic light.[14][15]

In chiral magnets, the hopfion has been theoretically predicted to occur within the spiral magnetic phase, where it was called a "heliknoton".[16] In recent years, the concept of a "fractional hopfion" has also emerged where not all preimages of magnetisation have a nonzero linking.[17][18]

See also[edit]

References[edit]

  1. ^ a b Faddeev L, Niemi AJ (1997). "Stable knot-like structures in classical field theory". Nature. 387 (6628): 58–61. arXiv:hep-th/9610193. Bibcode:1997Natur.387...58F. doi:10.1038/387058a0. S2CID 4256682.
  2. ^ a b Ackerman PJ, Smalyukh II (2017). "Static three-dimensional topological solitons in fluid chiral ferromagnets and colloids". Nature Materials. 16: 426–432.
  3. ^ Manton N, Sutcliffe P (2004). Topological solitons. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511617034. ISBN 0-511-21141-4. OCLC 144618426.
  4. ^ a b Kent N, Reynolds N, Raftrey D, Campbell IT, Virasawmy S, Dhuey S, et al. (March 2021). "Creation and observation of Hopfions in magnetic multilayer systems". Nature Communications. 12 (1): 1562. arXiv:2010.08674. Bibcode:2021NatCo..12.1562K. doi:10.1038/s41467-021-21846-5. PMC 7946913. PMID 33692363.
  5. ^ Faddeev L, Niemi AJ (1999). "Partially Dual Variables in SU(2) Yang-Mills Theory". Physical Review Letters. 82 (8): 1624–1627. arXiv:hep-th/9807069. Bibcode:1999PhRvL..82.1624F. doi:10.1103/PhysRevLett.82.1624. S2CID 8281134.
  6. ^ - Babaev E, Faddeev LD, Niemi AJ (2002). "Hidden symmetry and knot solitons in a charged two-condensate Bose system". Physical Review B. 65 (10): 100512. arXiv:cond-mat/0106152. Bibcode:2002PhRvB..65j0512B. doi:10.1103/PhysRevB.65.100512. S2CID 118910995.
  7. ^ Rybakov FN, Garaud J, Babaev E (2019). "Stable Hopf-Skyrme topological excitations in the superconducting state". Physical Review B. 100 (9): 094515. arXiv:1807.02509. Bibcode:2019PhRvB.100i4515R. doi:10.1103/PhysRevB.100.094515. S2CID 118991170.
  8. ^ Sutcliffe P (June 2017). "Skyrmion Knots in Frustrated Magnets". Physical Review Letters. 118 (24): 247203. arXiv:1705.10966. Bibcode:2017PhRvL.118x7203S. doi:10.1103/PhysRevLett.118.247203. PMID 28665663. S2CID 29890978.
  9. ^ Rybakov FN, Kiselev NS, Borisov AB, Döring L, Melcher C, Blügel S (2019). "Magnetic hopfions in solids". arXiv:1904.00250 [cond-mat.str-el].
  10. ^ Voinescu R, Tai JB, Smalyukh II (July 2020). "Hopf Solitons in Helical and Conical Backgrounds of Chiral Magnetic Solids". Physical Review Letters. 125 (5): 057201. arXiv:2004.10109. Bibcode:2020PhRvL.125e7201V. doi:10.1103/PhysRevLett.125.057201. PMID 32794865. S2CID 216036015.
  11. ^ Ackerman PJ, Smalyukh II (2017). "Diversity of knot solitons in liquid crystals manifested by linking of preimages in torons and hopfions". PHYSICAL REVIEW X. 7: 011006.
  12. ^ https://newscenter.lbl.gov/2021/04/08/spintronics-tech-a-hopfion-away/ The Spintronics Technology Revolution Could Be Just a Hopfion Away – ALS News
  13. ^ Kent N, Reynolds N, Raftrey D, Campbell IT, Virasawmy S, Dhuey S, et al. (March 2021). "Creation and observation of Hopfions in magnetic multilayer systems". Nature Communications. 12 (1): 1562. arXiv:2010.08674. Bibcode:2021NatCo..12.1562K. doi:10.1038/s41467-021-21846-5. PMC 7946913. PMID 33692363.
  14. ^ Sugic D, Droop R, Otte E, Ehrmanntraut D, Nori F, Ruostekoski J, et al. (November 2021). "Particle-like topologies in light". Nature Communications. 12 (1): 6785. doi:10.1038/s41467-021-26171-5. PMC 8608860. PMID 34811373.
  15. ^ Ehrmanntraut, Daniel; Droop, Ramon; Sugic, Danica; Otte, Eileen; Dennis, Mark; Denz, Cornelia (June 2023). "Optical second-order skyrmionic hopfion". Optica. 10 (6): 725–731 – via Optica publishing group.
  16. ^ Voinescu, Robert; Tai, Jung-Shen B.; Smalyukh, Ivan I. (27 July 2020). "Hopf Solitons in Helical and Conical Backgrounds of Chiral Magnetic Solids". Physical Review Letters. 125 (5): 057201. arXiv:2004.10109. doi:10.1103/PhysRevLett.125.057201.
  17. ^ Yu, Xiuzhen; Liu, Yizhou; Iakoubovskii, Konstantin V.; Nakajima, Kiyomi; Kanazawa, Naoya; Nagaosa, Naoto; Tokura, Yoshinori (May 2023). "Realization and Current‐Driven Dynamics of Fractional Hopfions and Their Ensembles in a Helimagnet FeGe". Advanced Materials. 35 (20). doi:10.1002/adma.202210646. ISSN 0935-9648.
  18. ^ Azhar, Maria; Kravchuk, Volodymyr P.; Garst, Markus (12 April 2022). "Screw Dislocations in Chiral Magnets". Physical Review Letters. 128 (15): 157204. arXiv:2109.04338. doi:10.1103/PhysRevLett.128.157204.

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