Inside the Light Boojums: A Journey to the Land of Boundary Defects

Abstract

We consider minimizers of the energy

in a two-dimensional domain $\Omega$, with weak anchoring potential

This functional was previously derived as a thin-film limit of the Landau-de Gennes energy, assuming weak anchoring on the boundary favoring a nematic director lying along a cone of fixed aperture, centered at the normal vector to the boundary.

In the regime where $s [\alpha^2+(\pi-\alpha)^2]<\pi^2/2$, any limiting map $u_\ast:\Omega\to{\mathbb S}^1$ has only boundary vortices, where its phase jumps by either $2\alpha$ (light boojums) or $2(\pi-\alpha)$ (heavy boojums). Our main result is the fine-scale description of the light boojums.