n-atic Order and Continuous Shape Changes of Deformable Surfaces of Genus Zero

TL;DR

This paper studies how n-atic order on deformable genus-zero surfaces influences shape and curvature, revealing a coupling between orientational order, vortex formation, and surface shape changes.

Contribution

It introduces a phenomenological model linking n-atic order to surface curvature and shape, inspired by superconductor vortex theory, to explain shape transitions in deformable surfaces.

Findings

n-atic order expels Gaussian curvature on genus-zero surfaces

Ordered phase contains 2n vortices of strength 1/n

Surface shape becomes nonspherical due to order-curvature coupling

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 with order parameter $\psi = \langle e^{i n \theta} \rangle$. 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 gauge-like coupling between $\psi$ and curvature, and our analysis follows closely the Abrikosov treatment of a type II superconductor just below $H_{c2}$.