Liquid Crystal Ground States on Cones with Anti-Twist Boundary Conditions

Geometry and topology play a fundamental role in determining pattern formation on 2D surfaces in condensed matter physics. For example, local positive Gaussian curvature of a 2D surface attracts positive topological defects in a liquid crystal phase confined to the curved surface while repelling negative topological defects. Although the cone geometry is flat on the flanks, the concentrated Gaussian curvature at the cone apex geometrically frustrates liquid crystal orientational fields arbitrarily far away. The apex acts as an unquantized pseudo-defect interacting with the topological defects on the flank. By exploiting the conformal mapping methods of F. Vafa et al., we explore a simple theoretical framework to understand the ground states of liquid crystals with $p$-fold rotational symmetry on cones, and uncover important finite size effects for the ground states with boundary conditions that confine both plus and minus defects to the cone flanks. By combining the theory and simulations, we present new results for liquid crystal ground states on cones with anti-twist boundary conditions at the cone base, which enforce a total topological charge of $-1$. We find that additional quantized negative defects are created on the flank as the cone apex becomes sharper via a defect unbinding process, such that an equivalent number of quantized positive defects become trapped at the apex, thus partially screening the apex charge, whose magnitude is a continuous function of cone angle.