Defect absorption and emission for $p$-atic liquid crystals on cones

TL;DR

This paper studies how topological defects in $p$-atic liquid crystals behave on conical surfaces, revealing defect absorption/emission at the apex and describing ground state configurations depending on cone angle and boundary conditions.

Contribution

It introduces an analytical framework for defect interactions on curved surfaces, especially cones, and predicts defect absorption and emission phenomena based on geometric parameters.

Findings

Defects behave as if they have an effective topological charge influenced by cone geometry.

Ground state defect arrangements depend on cone angle and boundary conditions.

Numerical simulations confirm analytical predictions with high accuracy.

Abstract

We investigate the ground state configurations of $p$-atic liquid crystals on fixed curved surfaces. We focus on the intrinsic geometry and show that isothermal coordinates are particularly convenient as they explicitly encode a geometric contribution to the elastic potential. In the special case of a cone with half-angle $\beta$, the apex develops an effective topological charge of $-\chi$, where $2\pi\chi = 2\pi(1-\sin\beta)$ is the deficit angle of the cone, and a topological defect of charge $\sigma$ behaves as if it had an effective topological charge $Q_\mathrm{eff} = (\sigma - \sigma^2/2)$ when interacting with the apex. The effective charge of the apex leads to defect absorption and emission at the cone apex as the deficit angle of the cone is varied. For total topological defect charge 1, e.g. imposed by tangential boundary conditions at the edge, we find that for a disk the ground state configuration consists of $p$ defects each of charge $+1/p$ lying equally spaced on a concentric ring of radius $d = (\frac{p-1}{3p-1})^{\frac{1}{2p}} R$, where $R$ is the radius of the disk. In the case of a cone with tangential boundary conditions at the base, we find three types of ground state configurations as a function of cone angle: (1) for sharp cones, all of the $+1/p$ defects are absorbed by the apex; (2) at intermediate cone angles, some of the $+1/p$ defects are absorbed by the apex and the rest lie equally spaced along a concentric ring on the flank; and (3) for nearly flat cones, all of the $+1/p$ defects lie equally spaced along a concentric ring on the flank. Here the defect positions and the absorption transitions depend intricately on $p$ and the deficit angle which we analytically compute. We check these results with numerical simulations for a set of commensurate cone angles and find excellent agreement.