Topological Defects on Fluctuating Surfaces: General Properties and the Kosterlitz-Thouless Transition

TL;DR

This paper explores the Kosterlitz-Thouless transition in hexatic order on fluctuating membranes, deriving Hamiltonians and renormalization relations to understand how bending rigidity influences the transition.

Contribution

It introduces a Coulomb gas and sine-Gordon Hamiltonian framework for the transition, incorporating Gaussian curvature and disclination effects on fluctuating surfaces.

Findings

Transition occurs with decreasing bending rigidity 

Derived gauge-invariant correlation functions for hexatic order

Predicted critical behavior via renormalization group analysis

Abstract

We investigate the Kosterlitz-Thouless transition for hexatic order on a free fluctuating membrane and derive both a Coulomb gas and a sine-Gordon Hamiltonian to describe it. The Coulomb-gas Hamiltonian includes charge densities arising from disclinations and from Gaussian curvature. There is an interaction coupling the difference between these two densities, whose strength is determined by the hexatic rigidity, and an interaction coupling Gaussian curvature densities arising from the Liouville Hamiltonian resulting from the imposition of a covariant cutoff. In the sine-Gordon Hamiltonian, there is a linear coupling between a scalar field and the Gaussian curvature. We discuss gauge-invariant correlation function for hexatic order and the dielectric constant of the Coulomb gas. We also derive renormalization group recursion relations that predict a transition with decreasing bending rigidity $\kappa$.