Fractional defect charges in p-atic liquid crystals on cones

We study two-dimensional liquid crystals with p-fold rotational symmetry (p-atics) on the surfaces of cones with free boundary conditions on the cone edge, and find both the ground state(s) and a ladder of quantized metastable states as a function of the cone angle and the liquid crystal symmetry p. We find that these states are characterized by a fractional defect charge 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 numerically for a set of commensurate cone angles, whose surfaces can be polygonized as a perfect triangular or square mesh, and find exact agreement.