Finite-temperature statistical mechanics of liquid crystals with p-fold rotational symmetry on cones and hyperbolic cones

Models of liquid crystals with p-fold rotational symmetry (p-atics) have direct implications for problems in both colloidal science and developmental biology. While the defect-free ground state of such systems on flat surfaces with free boundary conditions is well understood, the introduction of substrate curvature and novel boundary conditions can introduce defects even at zero temperature. The simplest surfaces that have both non-zero Gaussian curvature and an edge on which to implement boundary conditions are cones and their close cousins, hyperbolic cones, whose Gaussian curvatures have, respectively, positive and negative delta functions at the apex. We build on previous work on p-atic ground states on cones, and first study ground state defect textures in p-atics on hyperbolic cones. We then numerically and theoretically study these systems at finite-temperature. In particular, we perform Monte Carlo simulations to investigate the finite temperature behaviors of topological defects and the apex charge above and below the defect-unbinding transition expected in flat space. We find that charged defects that appear due to thermal fluctuations can efficiently screen the apex charge only at sufficiently high temperatures. Our findings demonstrate the importance of boundary conditions and geometry on the ground state and phase behavior at finite temperature of p-actics on curved surfaces.