Disclination Asymmetry in Deformable Hexatic Membranes and the Kosterlitz-Thouless Transitions

TL;DR

This paper investigates how disclinations in deformable hexatic membranes cause buckling transitions and analyzes the energy differences between positive and negative disclinations, providing estimates for the Kosterlitz-Thouless transition temperature.

Contribution

It calculates the buckling transition and disclination energies, revealing asymmetry between positive and negative disclinations in deformable membranes.

Findings

Negative disclinations have higher energy and lower buckling threshold.

Buckling transition depends on the ratio of bending to hexatic rigidity.

Estimated Kosterlitz-Thouless transition temperature is higher than previous renormalization results.

Abstract

A disclination in a hexatic membrane favors the development of Gaussian curvature localized near its core. The resulting global structure of the membrane has mean curvature, which is disfavored by curvature energy. Thus a membrane with an isolated disclination undergoes a buckling transition from a flat to a buckled state as the ratio $\kappa/K_{A}$ of the bending rigidity $\kappa$ to the hexatic rigidity $K_{A}$ is decreased. In this paper we calculate the buckling transition and the energy of both a positive and a negative disclination. A negative disclination has a larger energy and a smaller critical value of $\kappa/K_{A}$ at buckling than does a positive disclination. We use our results to obtain a crude estimate of the Kosterlitz-Thouless transition temperature in a membrane. This estimate is higher than the transition temperature recently obtained by the authors in a renormalization calculation.