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Introduction to topological defects: from liquid crystals to particle physics

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Abstract

Liquid crystals are assemblies of rod-like molecules which self-organize to form mesophases, in between ordinary liquids and anisotropic crystals. At each point, the molecules collectively orient themselves along a privileged direction, which locally defines an orientational order. Sometimes, this order is broken, and singularities appear in the form of topological defects. This tutorial article is dedicated to the geometry, topology, and physics of these defects. We introduce the main models used to describe the nematic phase and discuss the isotropic–nematic phase transition. Then, we present the different families of defects in nematics and examine some of their physical outcomes. Finally, we show that topological defects are universal patterns of nature, appearing not only in soft matter, but also in biology, cosmology, geology, and even particle physics.

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Data Availability Statement

No data associated in the manuscript.

Notes

  1. Note that the choice of the second Legendre polynomial here implicitly assumes three-component \(\vec n\) vectors. For two-component vectors, a different choice is needed [74].

  2. Other possibilities include lyotropic nematics: the external parameter is the concentration of nematogens in a solvent, and the most adequate description is Onsager’s model.

  3. The sign of m can be distinguished by the rotation of the crossed polarizers. Indeed, polarizing microscopy reveals Schlieren patterns, for which the number of dark brushes is \(4|m|\)0. Dark brushes from a positive (negative) defect rotate in the direction the same as (opposite to) that of the polarizers.

  4. For the sake of simplicity, we omitted the time component, but in relativity, one must bear in mind that time and space are put on an equal footing, and the real interval must involve quadratic terms in cdt, with c the speed of light.

  5. Like curvature, torsion is a property of the connection (actually this is the antisymmetric part of the connection and it vanishes in the case of a Levi–Civita connection, this is why general relativity does not deal with torsion). When torsion vanishes, geodesic (“shortest”) lines are also autoparallel (“straightest,” i.e., lines along which the tangent vector is parallel-transported via the connection).

  6. Vectors at a point x of a manifold live in the tangent space at x. The tangent spaces at neighboring points are different vector spaces, and the vector bundle is understood as the collection of them. The connection is the object which allows us to transport or compare vectors at different points [161].

  7. Finding general regular solutions of these equations is still one of the Millennium open problems listed by the Clay Institute.

  8. There is also another set forming the Ericksen–Leslie equations, which are simpler but limited to uniaxial media and to smooth variations of the nematic ordering.

  9. Still, this difference is also the reason why we have used two different notations for vectors, \(\textbf{r}\) for the ordinary space vectors and \(\vec n\) for the order parameter degrees of freedom.

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Acknowledgements

This review is intended for students and researchers working in theoretical physics. It is written for the jubilee of Malte Henkel, our friend and colleague at Université de Lorraine, who has inspired many students during his career.

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Fumeron, S., Berche, B. Introduction to topological defects: from liquid crystals to particle physics. Eur. Phys. J. Spec. Top. 232, 1813–1833 (2023). https://doi.org/10.1140/epjs/s11734-023-00803-x

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