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). Very often the problems involving defects are too complex for analytical treatment within the framework of an elastic theory. The difficulties arise either from the complexity of the free energy functional (biaxial nematic, smectic C, anisotropic phases of superfluid 3He, etc.) or from the complexity of the defect configuration (e.g., crossing of disclina-tions). Even when the solutions are possible, they rely on certain assumptions and, thus, might be strongly model dependent. An adequate description of defects in ordered condensed media requires introducing a new mathematical apparatus, viz. the theory of homotopy, which is part of algebraic topology. It is precisely in the language of topology that it is possible to associate the character of ordering of a medium and the types of defects arising in it, to find the laws of decay, merger and crossing of defects, to trace out their behavior during phase transitions, and so on. The key point is occupied by the concept of topological invariant, often also called a topological charge, which is inherent in every defect. The stability of the defect is guaranteed by the conservation of its topological invariant. The following simple example of twisted ribbon strips gives a flavor of the concept of topological invariant. 434