Instability of point defects in a two-dimensional nematic liquid crystal model

TL;DR

This paper investigates symmetric critical points representing topological defects in a 2D nematic liquid crystal model, proving their existence and showing that most are unstable except for specific degrees.

Contribution

It introduces a variational 2D Landau-de Gennes model for nematic liquid crystals and demonstrates the instability of symmetric defect solutions with certain topological degrees.

Findings

Existence of symmetric critical points representing topological defects.

Most symmetric defect solutions are unstable when the degree is not
or .

The stability depends on the topological degree of the defect.

Abstract

We study a class of symmetric critical points in a variational $2D$ Landau - de Gennes model where the state of nematic liquid crystals is described by symmetric traceless $3\times 3$ matrices. These critical points play the role of topological point defects carrying a degree $\frac k 2$ for a nonzero integer $k$. We prove existence and study the qualitative behavior of these symmetric solutions. Our main result is the instability of critical points when $k\neq \pm 1, 0$.