Hopfion

Model of magnetic hopfion in a solid. B_em 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] 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 skyrmions, which exhibit similar topological 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]

$H=(\partial {\bf {n}})^{2}+(\epsilon _{ijk}{\bf {n}}\cdot \partial _{i}{\bf {n}}\times \partial _{j}{\bf {n}})^{2}$

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[]

^ ^a ^b Faddeev, L.; Niemi, Antti J. (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.
^ Manton, Nicholas; Paul Sutcliffe (2004). Topological solitons. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511617034. ISBN 0-511-21141-4. OCLC 144618426.
^ ^a ^b Kent, Noah; Reynolds, Neal; Raftrey, David; Campbell, Ian T. G.; Virasawmy, Selven; Dhuey, Scott; Chopdekar, Rajesh V.; Hierro-Rodriguez, Aurelio; Sorrentino, Andrea; Pereiro, Eva; Ferrer, Salvador; Hellman, Frances; Sutcliffe, Paul; Fischer, Peter (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.
^ Faddeev, Ludvig; Niemi, Antti J. (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.
^ 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. arXiv:cond-mat/0106152. Bibcode:2002PhRvB..65j0512B. doi:10.1103/PhysRevB.65.100512. S2CID 118910995.
^ Rybakov, Filipp N.; Garaud, Julien; Babaev, Egor (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.
^ Sutcliffe, Paul (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.
^ 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].
^ 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. arXiv:2004.10109. Bibcode:2020PhRvL.125e7201V. doi:10.1103/PhysRevLett.125.057201. PMID 32794865. S2CID 216036015.
^ https://newscenter.lbl.gov/2021/04/08/spintronics-tech-a-hopfion-away/ The Spintronics Technology Revolution Could Be Just a Hopfion Away – ALS News
^ Kent, Noah; Reynolds, Neal; Raftrey, David; Campbell, Ian T. G.; Virasawmy, Selven; Dhuey, Scott; Chopdekar, Rajesh V.; Hierro-Rodriguez, Aurelio; Sorrentino, Andrea; Pereiro, Eva; Ferrer, Salvador; Hellman, Frances; Sutcliffe, Paul; Fischer, Peter (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.

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[:0-1] Faddeev, L.; Niemi, Antti J. (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] Manton, Nicholas; Paul Sutcliffe (2004). Topological solitons. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511617034. ISBN 0-511-21141-4. OCLC 144618426.

[kent-3] Kent, Noah; Reynolds, Neal; Raftrey, David; Campbell, Ian T. G.; Virasawmy, Selven; Dhuey, Scott; Chopdekar, Rajesh V.; Hierro-Rodriguez, Aurelio; Sorrentino, Andrea; Pereiro, Eva; Ferrer, Salvador; Hellman, Frances; Sutcliffe, Paul; Fischer, Peter (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.

[4] Faddeev, Ludvig; Niemi, Antti J. (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.

[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. arXiv:cond-mat/0106152. Bibcode:2002PhRvB..65j0512B. doi:10.1103/PhysRevB.65.100512. S2CID 118910995.

[6] Rybakov, Filipp N.; Garaud, Julien; Babaev, Egor (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.

[7] Sutcliffe, Paul (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.

[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. arXiv:2004.10109. Bibcode:2020PhRvL.125e7201V. doi:10.1103/PhysRevLett.125.057201. PMID 32794865. S2CID 216036015.

[10] ttps://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; Sorrentino, Andrea; Pereiro, Eva; Ferrer, Salvador; Hellman, Frances; Sutcliffe, Paul; Fischer, Peter (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.

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Hopfion

Experimental observation[]

See also[]

References[]