Abstract
We consider in mean-field theory the continuous development below a second-order phase transition of n-atic tangent-plane order on a deformable surface of genus zero. The n-atic order parameter ψ = ⟨exp [inθ]⟩ describes, respectively, vector, nematic, and hexatic order for n = 1, 2, and 6. Tangent-plane order expels Gaussian curvature. In addition, the total vorticity of orientational order on a surface of genus zero is two. Thus, the ordered phase of an n-atic on such a surface will have 2n vortices of strength 1/n, 2n zeros in its order parameter, and a nonspherical equilibrium shape. Our calculations are based on a phenomenological model with a gaugelike coupling between ψ and curvature, and our analysis follows closely the Abrikosov treatment of a type-II superconductor just below Hc2.
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