Crystalline Order On Riemannian Manifolds With Variable Gaussian Curvature And Boundary

TL;DR

This paper studies the arrangement of crystalline monolayers on curved surfaces with boundaries, revealing complex defect patterns influenced by variable Gaussian curvature, through analytical and numerical methods.

Contribution

It provides an analytical solution for crystalline structures on paraboloids and explores the effects of curvature and boundary constraints on defect configurations.

Findings

Variable Gaussian curvature induces diverse defect structures.

Boundary constraints significantly influence crystalline arrangements.

Analytical and numerical results agree on defect phenomena.

Abstract

We investigate the zero temperature structure of a crystalline monolayer constrained to lie on a two-dimensional Riemannian manifold with variable Gaussian curvature and boundary. A full analytical treatment is presented for the case of a paraboloid of revolution. Using the geometrical theory of topological defects in a continuum elastic background we find that the presence of a variable Gaussian curvature, combined with the additional constraint of a boundary, gives rise to a rich variety of phenomena beyond that known for spherical crystals. We also provide a numerical analysis of a system of classical particles interacting via a Coulomb potential on the surface of a paraboloid.