Theory of Liquid Crystal Ground States on Hyperbolic Cones

Building on work by Vafa et al. [1], we generalize the analytic theory and simulation model for liquid crystal ground states on conventional cones with positive apex curvature, to study liquid crystal ground states on hyperbolic cones which have negative apex curvature in the family of conical geometry. While both the local apex curvature on a conventional cone and a hyperbolic cone appear as a fixed unquantized pseudodefect in the conformal domain, behaving like regular topological defects with opposite charges, the fundamental difference between them is stressed, referred to as a violation of charge parity in a liquid crystal phase. To demonstrate the consequences of the broken charge parity in the pseudodefect, we exploit two simple examples: (a) p-atic liquid crystals on a hyperbolic cone with free boundary conditions at the cone base. (b) p-atic liquid crystals on a hyperbolic cone with tangential boundary conditions at the cone base. In the case of p=1 liquid crystals on a hyperbolic cone with tangential boundary conditions, it turns out that the positive pseudocharge caused by the apex curvature could even be stably bound with a topological charge of the same sign regardless of their repulsive interaction.

[1] F. Vafa, G. H. Zhang, and D. R. Nelson. "Defect ground states for liquid crystals on cones and hyperbolic cones." Journal of Physics A: Mathematical and Theoretical 58, no. 22 (2025): 225003