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

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.

arXiv comment12 + 11 pages, 10 figures

References in corpus (27)

• Novel type of phase transition in a system of self-driven particles
• Flocks, herds, and schools: A quantitative theory of flocking
• Spontaneous motion in hierarchically assembled active matter
• Emergence of macroscopic directed motion in populations of motile colloids
• Meso-scale turbulence in living fluids
• Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles
• Long-lived Giant Number Fluctuations in a Swarming Granular Nematic
• Topology and Dynamics of Active Nematic Vesicles
• Topological defect launches 3D mound in the active nematic sheet of neural progenitors
• Two-Dimensional Matter: Order, Curvature and Defects
• Interacting Topological Defects on Frozen Topographies
• Topological defects promote layer formation in Myxococcus xanthus colonies
• Turbulent dynamics of epithelial cell cultures
• Vortices in Superfluid Films on Curved Surfaces
• Defect-mediated morphologies in growing cell colonies
• Tunable structure and dynamics of active liquid crystals
• Long-range nematic order and anomalous fluctuations in suspensions of swimming filamentous bacteria
• Anomalous coupling between topological defects and curvature
• Curvature-Induced Defect Unbinding in Toroidal Geometries
• Role of cell deformability in the two-dimensional melting of biological tissues
• Topological Defects on Fluctuating Surfaces: General Properties and the Kosterlitz-Thouless Transition
• Active nematic defects and epithelial morphogenesis
• Fractional charge and inter-Landau level states at points of singular curvature
• Absence of a Finite-Temperature Melting Transition in the Classical Two-Dimensional One-Component Plasma
• Defect-Mediated Emulsification in Two Dimensions
• Quantization of Superflow Circulation and Magnetic Flux with a Tunable Offset
• Fractional defect charges in $p$-atic liquid crystals on cones

Cited by in corpus (4)

• Gravitational Response of Topological Quantum States of Matter
• Active topological defect absorption by a curvature singularity
• Applications of the Peach-Koehler Force in Liquid Crystals
• Defect Interactions Through Periodic Boundaries in Two-Dimensional $p$-atics