Monte Carlo Studies of the XY Model on Two-Dimensional Curved Surfaces

Abstract

To explore the interaction between topological defects and curvature in materials with orientational order, we perform Monte Carlo studies of the two-dimensional XY model on the surface of curved substrates. Each curved surface is patterned with a random lattice constructed via random sequential absorption, and an XY spin is positioned at each lattice site. Spins lie in the plane locally tangent to the surface and interact with neighbors defined via a distance cutoff. We demonstrate that the relative phase associated with vortices is significant in curved geometries and plays a role in microstructural evolution. We also observe that any nonuniform curvature, e.g., on the surface of a torus, induces spontaneous segregation of positive and negative vortices and promotes the formation of deeply metastable defect microstructures. Though qualitative in nature, these observations provide novel insights into the patterning of topological defects in curved geometries and suggest that the Kosterlitz-Thouless transition may be altered in geometries with nonuniform curvature.