Shape Changes of Deformable Spherical Membranes with $n$-atic Order

We present the existence of the Kosterlitz-Thouless (KT) transition for $n$-atic tangent-plane order on a deformable spherical surface and investigate the development of quasi-long range $n$-atic order and the continuous shape changes below the KT transition in the low temperature limit. The $n$-atic order parameter $\psi= \exp[in\Theta]$ describes, respectively, vector, nematic, and hexatic order for $n=1,2,$ and 6. We derive a Coulomb gas Hamiltonian to describe it. We then convert it into the sine-Gordon Hamiltonian and find a linear coupling between a scalar field and the Gaussian curvature. After integrating over the shape fluctuations, we obtain the massive sine-Gordon Hamiltonian, where the interaction between vortices is screened. We find, for $n^{2}K_{n}/\kappa \ll 1/4$, there is an effective KT transition. In the ordered phase, tangent-plane $n$-atic order expels the 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 gauge-like coupling between $\psi$ and curvature and close to the Abrikosov treatment of a type II superconductor.