Orientational order on curved surfaces - the high temperature region

TL;DR

This paper investigates how orientational order behaves on curved surfaces at high temperatures, deriving the density of zeros in vector fields and comparing it with disclination charge densities from Coulomb gas models, relevant for 2D materials.

Contribution

It introduces a method to compute the average density of zeros of Gaussian vector fields on curved surfaces and compares it with disclination densities from Coulomb gas models.

Findings

Derived the average density of zeros of Gaussian vector fields on curved surfaces.

Compared zero density with disclination charge density from Coulomb gas models.

Applicable to disordered states of 2D materials with orientational degrees of freedom.

Abstract

We study orientational order, subject to thermal fluctuations, on a fixed curved surface. We derive, in particular, the average density of zeros of Gaussian distributed vector fields on a closed Riemannian manifold. Results are compared with the density of disclination charges obtained from a Coulomb gas model. Our model describes the disordered state of two dimensional objects with orientational degrees of freedom, such as vector ordering in Langmuir monolayers and lipid bilayers above the hexatic to fluid transition.