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Orientational order at finite temperature on undulated surfaces

We study the effect of thermal fluctuations in the XY-model on a surface with non vanishing mean curvature and zero Gaussian curvature. Unlike Gaussian curvature that typically frustrates orientational order, the extrinsic curvature of the surface can act as a local field that promotes long range order at low temperature. We find numerically that the transition from the high temperature isotropic phase to the true long range ordered phase is characterized by critical exponents consistent with those of the flat space Ising model in two dimensions, up to finite size effects. Our results suggest a versatile strategy to achieve geometric control of liquid crystal order by suitable design of the underlying curvature of a substrate or bounding surface.