Hopfion Explained

A hopfion is a topological soliton.^{[1] [2] [3]} It is a stable three-dimensional localised configuration of a three-component field

\vec{n} = (n _{x,n} _{y,n} _{z)} 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

H = (\partial {\bf n}) ^{2} + (\epsilon _{ijk} {\bf n} ⋅ \partial _{i} {\bf n} x \partial _{j{\bf} n}) ^{2}

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

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

Experimental observation

Hopfions have been observed experimentally in chiral colloidal magnetic materials,^[10] 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, a helical-background variant of 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

• Skyrmion
• Hopf fibration

External links

• Web site: Hopfions in modern physics. hopfion.com.

Notes and References

1. Faddeev L, Niemi AJ . Stable knot-like structures in classical field theory . Nature . 1997 . 387 . 6628 . 58–61 . 10.1038/387058a0. hep-th/9610193 . 1997Natur.387...58F . 4256682 .
2. Book: Manton N, Sutcliffe P . Topological solitons . 2004 . Cambridge University Press . 0-511-21141-4 . Cambridge . 144618426 . 10.1017/CBO9780511617034.
3. Kent N, Reynolds N, Raftrey D, Campbell IT, Virasawmy S, Dhuey S, Chopdekar RV, Hierro-Rodriguez A, Sorrentino A, Pereiro E, Ferrer S, Hellman F, Sutcliffe P, Fischer P . 6 . Creation and observation of Hopfions in magnetic multilayer systems . Nature Communications . 12 . 1 . 1562 . March 2021 . 33692363 . 7946913 . 10.1038/s41467-021-21846-5 . 2010.08674 . free . 2021NatCo..12.1562K .
4. Faddeev L, Niemi AJ . 1999. Partially Dual Variables in SU(2) Yang-Mills Theory. Physical Review Letters. 82. 8. 1624–1627. 10.1103/PhysRevLett.82.1624. hep-th/9807069. 1999PhRvL..82.1624F. 8281134.
5. - 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. 10.1103/PhysRevB.65.100512. cond-mat/0106152. 2002PhRvB..65j0512B. 118910995.
6. Rybakov FN, Garaud J, Babaev E . 2019. Stable Hopf-Skyrme topological excitations in the superconducting state. Physical Review B. 100. 9. 094515. 10.1103/PhysRevB.100.094515. 1807.02509. 2019PhRvB.100i4515R. 118991170.
7. Sutcliffe P . Skyrmion Knots in Frustrated Magnets . Physical Review Letters . 118 . 24 . 247203 . June 2017 . 28665663 . 10.1103/PhysRevLett.118.247203 . 1705.10966 . 29890978 . 2017PhRvL.118x7203S .
8. Rybakov FN, Kiselev NS, Borisov AB, Döring L, Melcher C, Blügel S . 2022. Magnetic hopfions in solids. APL Materials . 10 . 11 . 10.1063/5.0099942 . 1904.00250. 2022APLM...10k1113R .
9. Voinescu R, Tai JB, Smalyukh II . Hopf Solitons in Helical and Conical Backgrounds of Chiral Magnetic Solids . Physical Review Letters . 125 . 5 . 057201 . July 2020 . 32794865 . 10.1103/PhysRevLett.125.057201 . 2004.10109 . 216036015 . 2020PhRvL.125e7201V .
10. Ackerman PJ, Smalyukh II . Static three-dimensional topological solitons in fluid chiral ferromagnets and colloids. Nature Materials. 2017 . 16 . 4 . 426–432 . 10.1038/nmat4826 . 27992419 . 2017NatMa..16..426A .
11. Ackerman PJ, Smalyukh II . Diversity of knot solitons in liquid crystals manifested by linking of preimages in torons and hopfions. Physical Review X. 2017 . 7 . 1 . 011006 . 10.1103/PhysRevX.7.011006 . 1704.08196 . 2017PhRvX...7a1006A .
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, Chopdekar RV, Hierro-Rodriguez A, Sorrentino A, Pereiro E, Ferrer S, Hellman F, Sutcliffe P, Fischer P . 6 . Creation and observation of Hopfions in magnetic multilayer systems . Nature Communications . 12 . 1 . 1562 . March 2021 . 33692363 . 7946913 . 10.1038/s41467-021-21846-5 . 2010.08674 . 2021NatCo..12.1562K . free .
14. Sugic D, Droop R, Otte E, Ehrmanntraut D, Nori F, Ruostekoski J, Denz C, Dennis MR . 6 . Particle-like topologies in light . Nature Communications . 12 . 1 . 6785 . November 2021 . 34811373 . 8608860 . 10.1038/s41467-021-26171-5 . 2107.10810 . 2021NatCo..12.6785S .
15. Ehrmanntraut . Daniel . Droop . Ramon . Sugic . Danica . Otte . Eileen . Dennis . Mark . Denz . Cornelia . Cornelia Denz . June 2023 . Optical second-order skyrmionic hopfion . Optica . 10 . 6 . 725–731 . 10.1364/OPTICA.487989 . 2023Optic..10..725E . Optica publishing group. free .
16. Voinescu . Robert . Tai . Jung-Shen B. . Smalyukh . Ivan I. . Hopf Solitons in Helical and Conical Backgrounds of Chiral Magnetic Solids . Physical Review Letters . 27 July 2020 . 125 . 5 . 057201 . 10.1103/PhysRevLett.125.057201 . 32794865 . 2004.10109 . 2020PhRvL.125e7201V .
17. Yu . Xiuzhen . Liu . Yizhou . Iakoubovskii . Konstantin V. . Nakajima . Kiyomi . Kanazawa . Naoya . Nagaosa . Naoto . Tokura . Yoshinori . Realization and Current-Driven Dynamics of Fractional Hopfions and Their Ensembles in a Helimagnet FeGe . Advanced Materials . May 2023 . 35 . 20 . 10.1002/adma.202210646 . en . 0935-9648. free . 2023AdM....3510646Y .
18. Azhar . Maria . Kravchuk . Volodymyr P. . Garst . Markus . Screw Dislocations in Chiral Magnets . Physical Review Letters . 12 April 2022 . 128 . 15 . 157204 . 10.1103/PhysRevLett.128.157204 . 35499887 . 2109.04338 . 2022PhRvL.128o7204A .