Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Static three-dimensional topological solitons in fluid chiral ferromagnets and colloids

Abstract

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

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

Prices may be subject to local taxes which are calculated during checkout

Figure 1: 2D and 3D topological solitons.
Figure 2: 3D imaging of topological solitons using 3PEF-PM.
Figure 3: Topological solitons with Q = 1.
Figure 4: Comparison of Q = −1 and Q = 0 solitons.
Figure 5: Twist handedness, free energy density self-assembly and facile switching of hexagonal arrays for Q = 1 topological solitons.

Similar content being viewed by others

References

  1. Manton, N. & Sutcliffe, P. Topological Solitons (Cambridge Univ. Press, 2004).

    Book  Google Scholar 

  2. Kauffman, L. H. Knots and Physics (World Scientific Publishing, 2001).

    Book  Google Scholar 

  3. Hopf, H. Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche. Math. Ann. 104, 637–665 (1931).

    Article  Google Scholar 

  4. Heisenberg, W. Einführung in die einheitliche Feldtheorie der Elementarteilchen (Hirzel, 1967).

  5. Derrick, G. H. Comments on nonlinear wave equations as models for elementary particles. J. Math. Phys. 5, 1252–1254 (1964).

    Article  CAS  Google Scholar 

  6. Hobart, R. H. On the instability of a class of unitary field models. Proc. Phys. Soc. 82, 201–203 (1963).

    Article  Google Scholar 

  7. Skyrme, T. H. R. A non-linear field theory. Proc. R. Soc. A 260, 127–138 (1961).

    Article  CAS  Google Scholar 

  8. Faddeev, L. & Niemi, A. J. Stable knot-like structures in classical field theory. Nature 387, 58–61 (1997).

    Article  CAS  Google Scholar 

  9. 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).

    Article  CAS  Google Scholar 

  10. 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).

    Article  CAS  Google Scholar 

  11. 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).

    Article  Google Scholar 

  12. Hall, D. S. et al. Tying quantum knots. Nat. Phys. 12, 478–483 (2016).

    Article  CAS  Google Scholar 

  13. Bolognesi, S. & Shifman, M. Hopf Skyrmion in QCD with adjoint quarks. Phys. Rev. D 75, 065020 (2007).

    Article  Google Scholar 

  14. Gorsky, A., Shifman, M. & Yung, A. Revisiting the Faddeev-Skyrme model and Hopf solitons. Phys. Rev. D 88, 045026 (2013).

    Article  Google Scholar 

  15. Acus, A., Norvaišas, E. & Shnir, Y. Hopfions interaction from the viewpoint of the product ansatz. Phys. Lett. B 733, 15–20 (2014).

    Article  CAS  Google Scholar 

  16. Thompson, A., Wickes, A., Swearngin, J. & Bouwmeester, D. Classification of electromagnetic and gravitational hopfions by algebraic type. J. Phys. A 48, 205202 (2015).

    Article  Google Scholar 

  17. Kobayashi, M. & Nitta, M. Torus knots as hopfions. Phys. Lett. B 728, 314–318 (2014).

    Article  CAS  Google Scholar 

  18. Mertelj, A., Lisjak, D., Drofenik, M. & Čopič, M. Ferromagnetism in suspensions of magnetic platelets in liquid crystal. Nature 504, 237–241 (2013).

    Article  CAS  Google Scholar 

  19. 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).

    Article  Google Scholar 

  20. Rößler, U. K., Bogdanov, A. N. & Pfleiderer, C. Spontaneous skyrmion ground states in magnetic metals. Nature 442, 797–801 (2006).

    Article  Google Scholar 

  21. Bogdanov, A. N. & Hubert, A. Thermodynamically stable magnetic vortex states in magnetic crystals. J. Magn. Magn. Mater. 138, 255–269 (1994).

    Article  CAS  Google Scholar 

  22. Romming, N. et al. Writing and deleting single magnetic skyrmions. Science 341, 636–639 (2013).

    Article  CAS  Google Scholar 

  23. Yu, X. Z. et al. Real-space observation of a two-dimensional skyrmion crystal. Nature 465, 901–904 (2010).

    Article  CAS  Google Scholar 

  24. Fert, A., Cros, V. & Sampaio, J. Skyrmions on the track. Nat. Nanotech. 8, 152–156 (2013).

    Article  CAS  Google Scholar 

  25. Nagaosa, N. & Tokura, Y. Topological properties and dynamics of magnetic skyrmions. Nat. Nanotech. 8, 899–911 (2013).

    Article  CAS  Google Scholar 

  26. Dzyaloshinskii, I. E. Theory of helicoidal structures in antiferromagnets. I. Nonmetals. JETP 19, 960 (1964).

    Google Scholar 

  27. Chaikin, P. M. & Lubensky, T. C. Principles of Condensed Matter Physics (Cambridge Univ. Press, 2000).

    Google Scholar 

  28. 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).

    Article  Google Scholar 

  29. 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).

    Article  CAS  Google Scholar 

  30. 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).

    Article  Google Scholar 

  31. 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).

    Article  CAS  Google Scholar 

  32. Hietarinta, J. & Salo, P. Faddeev-Hopf knots: dynamics of linked un-knots. Phys. Lett. B 451, 60–67 (1999).

    Article  CAS  Google Scholar 

  33. 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).

    Article  CAS  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

Authors

Contributions

P.J.A. and I.I.S. contributed to all aspects of this work and wrote the manuscript. I.I.S. conceived and designed the project.

Corresponding author

Correspondence to Ivan I. Smalyukh.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Information (PDF 5728 kb)

Supplementary Information

Supplementary movie 1 (AVI 10343 kb)

Supplementary Information

Supplementary movie 2 (AVI 9950 kb)

Supplementary Information

Supplementary movie 3 (AVI 10048 kb)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nmat4826

This article is cited by

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing