Topological Theory of Defects

Abstract

In Chapter 3, we have considered the notion of order parameter, its amplitude and phase. The order parameter is a continuous field (scalar, vector, tensor, etc.) describing the state of the system at each point. Generally, it is a function of coordinates, ψ(r). Distortions of ψ(r) can be of two types: those containing singularities and those without singularities. At singularities, ψ is not defined. For a 3D medium, the singular regions might be either zero-dimensional (points), one-dimensional (lines), or two-dimensional (walls). These are the defects. Whenever a nonhomogeneous state cannot be eliminated by continuous variations of the order parameter (i.e., one cannot arrive at the homogeneous state), it is called topologically stable, or simply, a topological defect. If the inhomogeneous state does not contain singularities, but nevertheless is not deformable continuously into a homogeneous state, one says that the system contains a topological configuration (or soliton).