Three-dimensional (3D) topological solitons are continuous but topologically nontrivial field configurations localized in 3D space and embedded in a uniform far-field background, that behave like particles and cannot be transformed to a uniform state through smooth deformations. Many topologically nontrivial 3D solitonic fields have been proposed. Yet, according to the Hobart–Derrick theorem, physical systems cannot host them, except for nonlinear theories with higher-order derivatives such as the Skyrme–Faddeev model. Experimental discovery of such solitons is hindered by the need for spatial imaging of the 3D fields, which is difficult in high-energy physics and cosmology. Here we experimentally realize and numerically model stationary topological solitons in a fluid chiral ferromagnet formed by colloidal dispersions of magnetic nanoplates. Such solitons have closed-loop preimages—3D regions with a single orientation of the magnetization field. We discuss localized structures with different linking of preimages quantified by topological Hopf invariants. The chirality is found to help in overcoming the constraints of the Hobart–Derrick theorem, like in two-dimensional ferromagnetic solitons, dubbed ‘baby skyrmions’. Our experimental platform may lead to solitonic condensed matter phases and technological applications.
This is a preview of subscription content, access via your institution
Open Access articles citing this article.
Nature Communications Open Access 29 July 2023
Nature Communications Open Access 04 July 2023
Nature Communications Open Access 17 December 2022
Subscribe to this journal
Receive 12 print issues and online access
$259.00 per year
only $21.58 per issue
Rent or buy this article
Prices vary by article type
Prices may be subject to local taxes which are calculated during checkout
Manton, N. & Sutcliffe, P. Topological Solitons (Cambridge Univ. Press, 2004).
Kauffman, L. H. Knots and Physics (World Scientific Publishing, 2001).
Hopf, H. Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche. Math. Ann. 104, 637–665 (1931).
Heisenberg, W. Einführung in die einheitliche Feldtheorie der Elementarteilchen (Hirzel, 1967).
Derrick, G. H. Comments on nonlinear wave equations as models for elementary particles. J. Math. Phys. 5, 1252–1254 (1964).
Hobart, R. H. On the instability of a class of unitary field models. Proc. Phys. Soc. 82, 201–203 (1963).
Skyrme, T. H. R. A non-linear field theory. Proc. R. Soc. A 260, 127–138 (1961).
Faddeev, L. & Niemi, A. J. Stable knot-like structures in classical field theory. Nature 387, 58–61 (1997).
Battye, R. A. & Sutcliffe, P. M. Knots as stable soliton solutions in a three-dimensional classical field theory. Phys. Rev. Lett. 81, 4798–4801 (1998).
Smalyukh, I. I., Lansac, Y., Clark, N. A. & Trivedi, R. P. Three-dimensional structure and multistable optical switching of triple-twisted particle-like excitations in anisotropic fluids. Nat. Mater. 9, 139–145 (2010).
Chen, B. G., Ackerman, P. J., Alexander, G. P., Kamien, R. D. & Smalyukh, I. I. Generating the Hopf fibration experimentally in nematic liquid crystals. Phys. Rev. Lett. 110, 237801 (2013).
Hall, D. S. et al. Tying quantum knots. Nat. Phys. 12, 478–483 (2016).
Bolognesi, S. & Shifman, M. Hopf Skyrmion in QCD with adjoint quarks. Phys. Rev. D 75, 065020 (2007).
Gorsky, A., Shifman, M. & Yung, A. Revisiting the Faddeev-Skyrme model and Hopf solitons. Phys. Rev. D 88, 045026 (2013).
Acus, A., Norvaišas, E. & Shnir, Y. Hopfions interaction from the viewpoint of the product ansatz. Phys. Lett. B 733, 15–20 (2014).
Thompson, A., Wickes, A., Swearngin, J. & Bouwmeester, D. Classification of electromagnetic and gravitational hopfions by algebraic type. J. Phys. A 48, 205202 (2015).
Kobayashi, M. & Nitta, M. Torus knots as hopfions. Phys. Lett. B 728, 314–318 (2014).
Mertelj, A., Lisjak, D., Drofenik, M. & Čopič, M. Ferromagnetism in suspensions of magnetic platelets in liquid crystal. Nature 504, 237–241 (2013).
Zhang, Q., Ackerman, P. J., Liu, Q. & Smalyukh, I. I. Ferromagnetic switching of knotted vector fields in liquid crystal colloids. Phys. Rev. Lett. 115, 097802 (2015).
Rößler, U. K., Bogdanov, A. N. & Pfleiderer, C. Spontaneous skyrmion ground states in magnetic metals. Nature 442, 797–801 (2006).
Bogdanov, A. N. & Hubert, A. Thermodynamically stable magnetic vortex states in magnetic crystals. J. Magn. Magn. Mater. 138, 255–269 (1994).
Romming, N. et al. Writing and deleting single magnetic skyrmions. Science 341, 636–639 (2013).
Yu, X. Z. et al. Real-space observation of a two-dimensional skyrmion crystal. Nature 465, 901–904 (2010).
Fert, A., Cros, V. & Sampaio, J. Skyrmions on the track. Nat. Nanotech. 8, 152–156 (2013).
Nagaosa, N. & Tokura, Y. Topological properties and dynamics of magnetic skyrmions. Nat. Nanotech. 8, 899–911 (2013).
Dzyaloshinskii, I. E. Theory of helicoidal structures in antiferromagnets. I. Nonmetals. JETP 19, 960 (1964).
Chaikin, P. M. & Lubensky, T. C. Principles of Condensed Matter Physics (Cambridge Univ. Press, 2000).
Ackerman, P. J. & Smalyukh, I. I. Reversal of helicoidal twist handedness near point defects of confined chiral liquid crystals. Phys. Rev. E 93, 052702 (2016).
Liu, Q., Ackerman, P. J., Lubensky, T. C. & Smalyukh, I. I. Biaxial ferromagnetic liquid crystal colloids. Proc. Natl Acad. Sci. USA 113, 10479–10484 (2016).
Evans, J. S., Ackerman, P. J., Broer, D. J., van de Lagemaat, J. & Smalyukh, I. I. Optical generation, templating, and polymerization of three-dimensional arrays of liquid-crystal defects decorated by plasmonic nanoparticles. Phys. Rev. E 87, 032503 (2013).
Mertelj, A., Osterman, N., Lisjak, D. & Čopič, M. Magneto-optic and converse magnetoelectric effects in a ferromagnetic liquid crystal. Soft Matter 10, 9065–9072 (2014).
Hietarinta, J. & Salo, P. Faddeev-Hopf knots: dynamics of linked un-knots. Phys. Lett. B 451, 60–67 (1999).
Ackerman, P. J., van de Lagemaat, J. & Smalyukh, I. I. Self-assembly and electrostriction of arrays and chains of hopfion particles in chiral liquid crystals. Nat. Commun. 6, 6012 (2015).
We thank Q. Liu for the assistance with synthesizing ferromagnetic nanoplates and H. O. Sohn and Q. Zhang for discussions. We acknowledge support of the National Science Foundation Grant DMR-1410735.
The authors declare no competing financial interests.
About this article
Cite this article
Ackerman, P., Smalyukh, I. Static three-dimensional topological solitons in fluid chiral ferromagnets and colloids. Nature Mater 16, 426–432 (2017). https://doi.org/10.1038/nmat4826
This article is cited by
Nature Communications (2023)
Nature Communications (2023)
Nature Communications (2022)
Nature Communications (2022)