Topological defect

In mathematics and physics, solitons, topological solitons and topological defects are three closely related ideas, all of which signify structures in a physical system that are stable against perturbations. Solitons do not decay, dissipate, disperse or evaporate in the way that ordinary waves might. The stability arises from an obstruction to the decay, which is explained by having the soliton belong to a different topological homotopy class or cohomology class than the base physical system. More simply: it is not possible to continuously transform the system with a soliton in it, to one without it. The mathematics behind topological stability is both deep and broad, and a vast variety of systems possessing topological stability have been described. This makes categorization somewhat difficult.

Overview

The original soliton was observed in the 19th century, as a solitary water wave in a barge canal. It was eventually explained by noting that the Korteweg-De Vries equation, describing waves in water, has homotopically distinct solutions. The mechanism of Lax pairs provided the needed topological understanding.
The general characteristic needed for a topological soliton to arise is that there should be some partial differential equation having distinct classes of solutions, with each solution class belonging to a distinct homotopy class. In many cases, this arises because the base space, three-dimensional space or four-dimensional space, can be thought of as having the topology of a sphere, obtained by one-point compactification: adding a point at infinity. This is reasonable, as one is generally interested in solutions that vanish at infinity, and so are single-valued at that point. The range of the variables in the differential equation can also be viewed as living in some compact topological space. As a result, the mapping from space to the variables in the PDE is describable as a mapping from a sphere to a sphere; the classes of such mappings are given by the homotopy groups of spheres.
To restate more plainly: solitons are found when one solution of the PDE cannot be continuously transformed into another; to get from one to the other would require "cutting", but "cutting" is not a defined operation for solving PDE's. The cutting analogy arises because some solitons are described as mappings, where is the circle; the mappings arise in the circle bundle. Such maps can be thought of as winding a string around a stick: the string cannot be removed without cutting it. The most common extension of this winding analogy is to maps, where the first three-sphere stands for compactified 3D space, while the second stands for a vector field. Such maps occur in PDE's describing vector fields.
A topological defect is perhaps the simplest way of understanding the general idea: it is a soliton that occurs in a crystalline lattice, typically studied in the context of solid state physics and materials science. The prototypical example is the screw dislocation; it is a dislocation of the lattice that spirals around. It can be moved from one location to another by pushing it around, but it cannot be removed by simple continuous deformations of the lattice. The mathematical stability comes from the non-zero winding number of the map of circles the stability of the dislocation leads to stiffness in the material containing it. One common manifestation is the repeated bending of a metal wire: this introduces more and more screw dislocations, making the bent region increasingly stiff and brittle. Continuing to stress that region will overwhelm it with dislocations, and eventually lead to a fracture and failure of the material. This can be thought of as a phase transition, where the number of defects exceeds a critical density, allowing them to interact with one-another and "connect up", and thus disconnect the whole. The idea that critical densities of solitons can lead to phase transitions is a recurring theme.
Vortices in superfluids and pinned vortex tubes in type-II superconductors provide examples of circle-map type topological solitons in fluids. More abstract examples include cosmic strings; these include both vortex-like solutions to the Einstein field equations, and vortex-like solutions in more complex systems, coupling to matter and wave fields. Tornados and vorticies in air are not examples of solitons: there is no obstruction to their decay; they will dissipate after a time. The mathematical solution describing a tornado can be continuously transformed, by weakening the rotation, until there is no rotation left. The details, however, are context-dependent: the Great Red Spot of Jupiter is a cyclone, for which soliton-type ideas have been offered up to explain its multi-century stability.
Topological defects were studied as early as the 1940's. More abstract examples arose in quantum field theory. The Skyrmion was proposed in the 1960's as a model of the nucleon and owed its stability to the mapping. In the 1980's, the instanton and related solutions of the Wess–Zumino–Witten models, rose to considerable popularity because these offered a non-perturbative take in a field that was otherwise dominated by perturbative calculations done with Feynmann diagrams. It provided the impetus for physicists to study the concepts of homotopy and cohomology, which were previously the exclusive domain of mathematics. Further development identified the pervasiveness of the idea: for example, the Schwarzschild solution and Kerr solution to the Einstein field equations can be recognized as examples of topological gravitational solitons: this is the Belinski–Zakharov transform.
The terminology of a topological defect vs. a topological soliton, or even just a plain "soliton", varies according to the field of academic study. Thus, the hypothesized but unobserved magnetic monopole is a physical example of the abstract mathematical setting of a monopole; much like the Skyrmion, it owes its stability to belonging to a non-trivial homotopy class for maps of 3-spheres. For the monopole, the target is the magnetic field direction, instead of the isotopic spin direction. Monopoles are usually called "solitons" rather than "defects". Solitions are associated with topological invariants; as more than one configuration may be possible, these will be labelled with a topological charge. The word charge is used in the sense of charge in physics.
The mathematical formalism can be quite complicated. General settings for the PDE's include fiber bundles, and the behavior of the objects themselves are often described in terms of the holonomy and the monodromy. In abstract settings such as string theory, solitons are part and parcel of the game: strings can be arranged into knots, as in knot theory, and so are stable against being untied.
In general, a field configuration with a soliton in it will have a higher energy than the ground state or vacuum state, and thus will be called a topological excitation. Although homotopic considerations prevent the classical field from being deformed into the ground state, it is possible for such a transition to occur via quantum tunneling. In this case, higher homotopies will come into play. Thus, for example, the base excitation might be defined by a map into the spin group. If quantum tunneling erases the distinction between this and the ground state, then the next higher group of homotopies is given by the string group. If the process repeats, this results in a walk up the Postnikov tower. These are theoretical hypotheses; demonstrating such concepts in actual lab experiments is a different matter entirely.

Formal treatment

The existence of a topological defect can be demonstrated whenever the boundary conditions entail the existence of homotopically distinct solutions. Typically, this occurs because the boundary on which the conditions are specified has a non-trivial homotopy group which is preserved in differential equations; the solutions to the differential equations are then topologically distinct, and are classified by their homotopy class. Topological defects are not only stable against small perturbations, but cannot decay or be undone or be de-tangled, precisely because there is no continuous transformation that will map them to a uniform or "trivial" solution.
An ordered medium is defined as a region of space described by a function f that assigns to every point in the region an order parameter, and the possible values of the order parameter space constitute an order parameter space. The homotopy theory of defects uses the fundamental group of the order parameter space of a medium to discuss the existence, stability and classifications of topological defects in that medium.
Suppose R is the order parameter space for a medium, and let G be a Lie group of transformations on R. Let H be the symmetry subgroup of G for the medium. Then, the order parameter space can be written as the Lie group quotient R = G/''H.
If G'' is a universal cover for G/''H then, it can be shown that πn'' = π_n−1, where π_i denotes the i-th homotopy group.
Various types of defects in the medium can be characterized by elements of various homotopy groups of the order parameter space. For example,, line defects correspond to elements of π₁, point defects correspond to elements of π₂, textures correspond to elements of π₃. However, defects which belong to the same conjugacy class of π₁ can be deformed continuously to each other, and hence, distinct defects correspond to distinct conjugacy classes.
Poénaru and Toulouse showed that crossing defects get entangled if and only if they are members of separate conjugacy classes of π₁.