Fractional defect charges in $p$-atic liquid crystals on cones

TL;DR

This paper investigates the behavior of $p$-atic liquid crystals on conical surfaces, revealing fractional defect charges at the apex and the influence of geometry on ground states, supported by numerical validation.

Contribution

It introduces a theoretical framework for understanding fractional defect charges in $p$-atic liquid crystals on cones, including predictions for ground states and metastability.

Findings

Ground states characterized by fractional defect charges at the apex

Agreement between theoretical predictions and numerical simulations

Identification of metastable states depending on cone angle and symmetry

Abstract

Conical surfaces, with a delta function of Gaussian curvature at the apex, are perhaps the simplest example of geometric frustration. We study two-dimensional liquid crystals with $p$-fold rotational symmetry ($p$-atics) on the surfaces of cones. For free boundary conditions at the base, we find both the ground state(s) and a discrete ladder of metastable states as a function of both the cone angle and the liquid crystal symmetry $p$. We find that these states are characterized by a set of fractional defect charges at the apex and that the ground states are in general frustrated due to effects of parallel transport along the azimuthal direction of the cone. We check our predictions for the ground state energies numerically for a set of commensurate cone angles (corresponding to a set of commensurate Gaussian curvatures concentrated at the cone apex), whose surfaces can be polygonized as a perfect triangular or square mesh, and find excellent agreement with our theoretical predictions.