Shape of domains in two-dimensional systems: virtual singularities and a generalized Wulff construction

TL;DR

This paper extends the Wulff construction to two-dimensional domains with position-dependent boundary energy, revealing that large domains develop cusps, aligning with experimental findings.

Contribution

It introduces a generalized Wulff construction for domains with position-dependent boundary energy and solves the texture minimization problem in specific cases.

Findings

Large domains develop cusps under certain boundary energy conditions

The generalized construction applies to XY-like order parameter domains

Exact textures can be described using virtual singularities

Abstract

This is a report on the derivation and application of a generalized version of the Wulff construction in two dimensions. The construction is used to find the shape of a domain containing an XY-like order parameter. In such a domain the energy per unit length of a segment of the bounding curve depends not only on the orientation of the segment, but also on the segment's position, and this is why the Wulff construction must be generalized. When the domain is circular in shape we find that the problem of finding the texture that minimizes the bulk and boundary energies is exactly solvable in certain important cases. Using the generalized Wulff construction and the exact textures, describable in terms of virtual singularities, we find that, given reasonable assumptions concerning the from of the boundary energy, the domain will necessarily develop a cusp if it is sufficiently large. This result is in qualitative agreement with recent experimental observations.