• Open Access

Diversity of Knot Solitons in Liquid Crystals Manifested by Linking of Preimages in Torons and Hopfions

Paul J. Ackerman and Ivan I. Smalyukh
Phys. Rev. X 7, 011006 – Published 23 January 2017

Abstract

Topological solitons are knots in continuous physical fields classified by nonzero Hopf index values. Despite arising in theories that span many branches of physics, from elementary particles to condensed matter and cosmology, they remain experimentally elusive and poorly understood. We introduce a method of experimental and numerical analysis of such localized structures in liquid crystals that, similar to the mathematical Hopf maps, relates all points of the medium’s order parameter space to their closed-loop preimages within the three-dimensional solitons. We uncover a surprisingly large diversity of naturally occurring and laser-generated topologically nontrivial solitons with differently knotted nematic fields, which previously have not been realized in theories and experiments alike. We discuss the implications of the liquid crystal’s nonpolar nature on the knot soliton topology and how the medium’s chirality, confinement, and elastic anisotropy help to overcome the constraints of the Hobart-Derrick theorem, yielding static three-dimensional solitons without or with additional defects. Our findings will establish chiral nematics as a model system for experimental exploration of topological solitons and may impinge on understanding of such nonsingular field configurations in other branches of physics, as well as may lead to technological applications.

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  • Received 4 September 2016

DOI:https://doi.org/10.1103/PhysRevX.7.011006

Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Condensed Matter, Materials & Applied PhysicsPolymers & Soft MatterGeneral Physics

Authors & Affiliations

Paul J. Ackerman1,2 and Ivan I. Smalyukh1,2,3,4,*

  • 1University of Colorado, Department of Physics, Boulder, Colorado 80309, USA
  • 2University of Colorado, Department of Electrical, Computer and Energy Engineering, Boulder, Colorado 80309, USA
  • 3University of Colorado, Liquid Crystal Materials Research Center and Materials Science and Engineering Program, Boulder, Colorado 80309, USA
  • 4University of Colorado, Renewable and Sustainable Energy Institute, Boulder, Colorado 80309, USA

  • *ivan.smalyukh@colorado.edu

Popular Summary

Carl Friedrich Gauss postulated that knots in the field lines of magnetic or electric fields could behave like particles, provided that the field lines were not allowed to cross. Later, mathematicians rigorously demonstrated that, indeed, knotted fields could be smoothly embedded in a uniform far-field background, a result that attracted strong theoretical interest in physics and cosmology. However, experimental realizations and demonstrations of topological solitons with knotted fields have been hindered by a lack of three-dimensional spatial imaging of the physical fields. Moreover, according to the Hobart-Derrick theorem, because of energetic reasons, physical systems cannot host the static three-dimensional solitons described within the simplest field theories, and the challenge of realizing topological solitons has accordingly persisted for decades. Here, for the first time, we describe the spontaneous appearance and facile laser generation of a series of topologically nontrivial solitons with differently knotted fields in chiral nematic liquid crystals, similar to the ones used in electronic displays.

We use three-dimensional nonlinear optical imaging and numerical modeling to reveal the structure and topology of knotted field lines. We consider these fields within liquid crystals, which we prepare by adding very low levels of dopants with chiral properties. We theoretically show how the medium’s chirality helps to overcome the constraints of the Hobart-Derrick theorem, yielding static three-dimensional solitons that we manipulate using low-power (2–5 mW) optical tweezers. Our findings establish chiral liquid crystals as a model system for experimental exploration of static three-dimensional solitons, and our work advances the study of other solitons, such as the two-dimensional skyrmions.

We expect that our findings will inform the development of next-generation bistable information displays and other technological breakthroughs.

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Vol. 7, Iss. 1 — January - March 2017

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