Defects in Nematic Shells: a $\Gamma$-convergence discrete-to-continuum approach

TL;DR

This paper rigorously analyzes how topological defects form in nematic shells with complex topology by employing a discrete-to-continuum approach using $ ext{Gamma}$-convergence, revealing regimes with finite defects and defect interactions.

Contribution

It introduces a $ ext{Gamma}$-convergence framework to study defect emergence in nematic shells with non-trivial topology, connecting discrete models to continuum limits.

Findings

Finite number of defects consistent with topology

Identification of a Renormalized Energy governing defect interactions

Two asymptotic regimes for defect formation and behavior

Abstract

In this paper we rigorously investigate the emergence of defects on Nematic Shells with genus different from one. This phenomenon is related to a non trivial interplay between the topology of the shell and the alignment of the director field. To this end, we consider a discrete $XY$ system on the shell $M$, described by a tangent vector field with unit norm sitting at the vertices of a triangulation of the shell. Defects emerge when we let the mesh size of the triangulation go to zero, namely in the discrete-to-continuum limit. In this paper we investigate the discrete-to-continuum limit in terms of $\Gamma$-convergence in two different asymptotic regimes. The first scaling promotes the appearance of a finite number of defects whose charges are in accordance with the topology of shell $M$, via the Poincar\'e-Hopf Theorem. The second scaling produces the so called Renormalized Energy that governs the equilibrium of the configurations with defects.