# Physics:Hopfion

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

A hopfion is a topological soliton.[1][2][3] It is a stable three-dimensional localised configuration of a three-component field $\displaystyle{ \vec{n}=(n_x,n_y,n_z) }$ with a knotted topological structure. They are the three-dimensional counterparts of skyrmions, which exhibit similar topoligical properties in 2D.

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]

$\displaystyle{ H= (\partial {\bf n})^2 + (\epsilon_{ijk}{\bf n}\cdot\partial_i {\bf n}\times \partial_j{\bf n})^2 }$

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][3]

## Experimental observation

Hopfions have been observed experimentally[10] in Ir/Co/Pt multilayers using X-ray magnetic circular dichroism.[11]

## References

1. Faddeev, L.; Niemi, Antti J. (1997). "Stable knot-like structures in classical field theory". Nature 387 (6628): 58–61. doi:10.1038/387058a0. Bibcode1997Natur.387...58F.
2. Manton, Nicholas; Paul Sutcliffe (2004). Topological solitons. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511617034. ISBN 0-511-21141-4. OCLC 144618426.
3. Kent, Noah; Reynolds, Neal; Raftrey, David; Campbell, Ian T. G.; Virasawmy, Selven; Dhuey, Scott; Chopdekar, Rajesh V.; Hierro-Rodriguez, Aurelio et al. (2021). "Creation and observation of Hopfions in magnetic multilayer systems". Nature Communications 12 (1): 1562. doi:10.1038/s41467-021-21846-5. PMID 33692363. Bibcode2021NatCo..12.1562K.
4. Faddeev, Ludvig; Niemi, Antti J. (1999). "Partially Dual Variables in SU(2) Yang-Mills Theory". Physical Review Letters 82 (8): 1624–1627. doi:10.1103/PhysRevLett.82.1624. Bibcode1999PhRvL..82.1624F.
5. Babaev, Egor; Faddeev, Ludvig D.; Niemi, Antti J. (2002). "Hidden symmetry and knot solitons in a charged two-condensate Bose system". Physical Review B 65 (10): 100512. doi:10.1103/PhysRevB.65.100512. Bibcode2002PhRvB..65j0512B.
6. Rybakov, Filipp N.; Garaud, Julien; Babaev, Egor (2019). "Stable Hopf-Skyrme topological excitations in the superconducting state". Physical Review B 100 (9): 094515. doi:10.1103/PhysRevB.100.094515. Bibcode2019PhRvB.100i4515R.
7. Sutcliffe, Paul (2017). "Skyrmion Knots in Frustrated Magnets". Physical Review Letters 118 (24): 247203. doi:10.1103/PhysRevLett.118.247203. PMID 28665663. Bibcode2017PhRvL.118x7203S.
8. Rybakov, F. N.; Kiselev, N. S.; Borisov, A. B.; Döring, L.; Melcher, C.; Blügel, S. (2019). "Magnetic hopfions in solids". arXiv:1904.00250 [cond-mat.str-el].
9. Voinescu, Robert; Tai, Jung-Shen B.; Smalyukh, Ivan I. (2020). "Hopf Solitons in Helical and Conical Backgrounds of Chiral Magnetic Solids". Physical Review Letters 125 (5): 057201. doi:10.1103/PhysRevLett.125.057201. PMID 32794865. Bibcode2020PhRvL.125e7201V.
10. https://newscenter.lbl.gov/2021/04/08/spintronics-tech-a-hopfion-away/ The Spintronics Technology Revolution Could Be Just a Hopfion Away – ALS News
11. Kent, Noah; Reynolds, Neal; Raftrey, David; Campbell, Ian T. G.; Virasawmy, Selven; Dhuey, Scott; Chopdekar, Rajesh V.; Hierro-Rodriguez, Aurelio et al. (2021). "Creation and observation of Hopfions in magnetic multilayer systems". Nature Communications 12 (1): 1562. doi:10.1038/s41467-021-21846-5. PMID 33692363. Bibcode2021NatCo..12.1562K.