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

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mo>−</a:mo><a:mn>2</a:mn></a:math>, which have the same local structure but very different global structures, underscoring the need for a more nuanced topological understanding of nematic systems.