Abstract

Cones with order in the local tangent plane provide a soft matter analog of the Aharonov-Bohm effect.   We study two-dimensional liquid crystals with p-fold rotational symmetry (p-atics) on the surfaces of cones, and find both the ground state and a ladder of quantized metastable states as a function of both the cone angle and the liquid crystal symmetry p.    These states are characterized by a fractional defect charge at the apex and the ground states are in general frustrated due to effects of parallel transport along the azimuthal direction on the cone. 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 good agreement.    When tangential boundary conditions are applied at the base, the cone apex absorbs and emits quantized defect charges, as a function of cone angle.   Defect emission and absorption events at the apex change dramatically when we consider active nematics on cones.   Grace Zhang, and D. R. Nelson, Phys. Rev. E 105 , 054703 (2022)                                                                      [2] Farzan Vafa, G. H. Zhang, and D. R. Nelson, Phys. Rev. E 106 , 024704 (2022)                                                  [3] Farzan Vafa, D. R. Nelson, and A. Doostmohammadi, arXiv:2304.06739 (2023)