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Morse Theory and Meron-Mediated Interactions Between Disclination Lines in Nematic Materials

Joseph Pollard1,2,* and Richard G. Morris1,2,3,†

  • 1School of Physics, UNSW, Sydney, NSW 2052, Australia
  • 2EMBL Australia Node in Single Molecule Science, School of Medical Sciences, UNSW, Sydney, NSW 2052, Australia
  • 3ARC Centre of Excellence for the Mathematical Analysis of Cellular Systems, UNSW, Node, Sydney, NSW 2052, Australia

  • *Contact author: joe.pollard@unsw.edu.au
  • †Contact author: r.g.morris@unsw.edu.au
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Phys. Rev. X 15, 021099 – Published 23 June, 2025

DOI: https://doi.org/10.1103/lbbj-txnx

Abstract

The topological understanding of nematic liquid crystals has traditionally centered on defects and their classification via homotopy invariants. Here, we draw upon several aspects of Morse theory to describe a finer approach to topology that characterizes the isotopy invariants within a given homotopy class. This approach unifies various disparate ideas in liquid crystal physics, including the homotopy and Volterra classifications of defects, the winding and self-linking of disclinations, and the Pontryagin-Thom construction. Specifically, we use singularity theory and dividing curves to characterize nematic textures via tomography, as well as to provide a full topological classification of the rewiring and crossing of disclination lines. This process reveals the central role that nonsingular but nontrivial topological solitons—so-called merons—play in mediating interactions between disclination lines. We also clarify the notion of the charge of a disclination line, give a simple method for computing it, and apply this to examples drawn from experiment. In particular, this method shows that restricting a disclination line’s charge modulo 2 is incorrect; we construct examples of disclinations with point defect charges 0 and −2, which have the same local structure but very different global structures, underscoring the need for a more nuanced topological understanding of nematic systems.

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Popular Summary

Nematic liquid crystals—materials made of rodlike molecules that flow like a liquid but retain some directional order—have played important roles in many areas of science, from display technologies to biological systems. In two dimensions, the topology of these materials is well understood through the study of pointlike defects, where the molecular alignment breaks down. But in three dimensions, defects form lines and loops, and existing theories from 2D do not fully explain the complex behaviors observed. In this study, we use ideas from the mathematical field of Morse theory—which relates the topology of a space to the behavior of smooth functions on that space—to better understand the topology of 3D nematics.

The key insight is to move away from a defect-centric approach. We instead highlight the importance of merons—smooth, nonsingular topological features that do not disrupt the material like defects do. We find that merons play a key role in organizing and changing the behavior of defect lines in 3D nematics. Specifically, they govern how defect lines can link, cross, or rearrange, and we provide methods for identifying and classifying these features.

This framework offers a more complete picture of how nematic structures behave in three dimensions. By understanding how merons and defects interact, we can design better materials and devices that make use of the rich topology of liquid crystals.

Article Text

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Phys. Rev. X 15, 021099– Published 23 June, 2025
  • Received 8 August 2024
  • Accepted 30 May 2025
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