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

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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.