Introduction to Defects

The xy model provides a simple description of two-dimensional (2D) polar order. Configurations of the xy model can be classified using the winding number, which is an integer that counts rotations of the orientational order. Configurations may have singularities, such that the winding number is nonzero for any measuring loop that goes around a singularity. In that case, the singularity is a topological defect, and the winding number is a topological charge. The defect is a point in a 2D system, or a line in a 3D system.