Defect ground states for liquid crystals on cones and hyperbolic cones

Abstract Cones with orientational order in the local tangent plane provide a soft matter analog of the Aharonov–Bohm effect. In this article, we first review recent work on two-dimensional liquid crystals with p -fold rotational symmetry ( p -atics) on cones. By exploiting an analogy with electrostatics, we determine the ground state as a function of both the cone deficit angle and the liquid crystal symmetry p for both free boundary conditions and tangential boundary conditions applied at the cone base. There is an effective topological charge <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>χ</mml:mi> </mml:mrow> </mml:math> at the apex, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> <mml:mi>χ</mml:mi> </mml:mrow> </mml:math> is the deficit angle. The ground states are in general frustrated due to parallel transport along the azimuthal direction on the cone. In the case of tangential boundary conditions, the ground state changes as a function of χ , where the cone apex absorbs and emits quantized defect charges, with intricate dependence on both the deficit angle and p . We check our predictions numerically for a set of commensurate cone angles, whose surfaces can be polygonized as a perfect triangular or square mesh, and find excellent agreement. Cones with both free and tangential boundary conditions can also exhibit metastable states distinguished by quantized screening of the apex charge. We also present preliminary work on hyperbolic cones, where the Gaussian curvature singularity is negative at the apex. When free boundary conditions are applied at the base, the ground states are characterized by an effective topological charge at the apex, similarly to the case of conventional cones. However, when tangential boundary conditions are applied, decreasing the deficit angle (which is now negative) induces neutral defect pair nucleation at the apex followed by emission of a positive defect.