N-atic order and continuous shape changes of deformable surfaces of genus zero

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 = (exp [in&]) 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 w-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 φ and curvature, and our analysis follows closely the Abrikosov treatment of a type-II superconductor just below Hc2.