Orientational order at finite temperature on undulated surfaces

TL;DR

This study investigates how the extrinsic curvature of undulated surfaces influences the orientational order of the XY-model at finite temperatures, revealing a transition akin to the 2D Ising model and proposing geometric control of liquid crystal order.

Contribution

It demonstrates that extrinsic curvature can promote long-range order in the XY-model on curved surfaces, offering a new approach to control liquid crystal alignment through surface design.

Findings

Extrinsic curvature acts as a local field promoting order.

Transition exhibits critical exponents similar to 2D Ising model.

Surface curvature can be used to control liquid crystal order.

Abstract

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.