Topological defect

See also topological excitations and the base concepts: topology, differential equations, quantum mechanics and condensed matter physics.

In mathematics and physics, a topological soliton or a topological defect is a solution of a system of partial differential equations or of a quantum field theory homotopically distinct from the vacuum solution; it can be proven to exist because the boundary conditions entail the existence of homotopically distinct solutions. Typically, this occurs because the boundary on which the boundary 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 (homotopically) to a uniform or "trivial" solution.

Examples include the soliton or solitary wave which occurs in many exactly solvable models, the screw dislocations in crystalline materials, the skyrmion and the Wess–Zumino–Witten model in quantum field theory.

Topological defects are believed to drive phase transitions in condensed matter physics. Notable examples of topological defects are observed in lambda transition universality class systems including: screw/edge-dislocations in liquid crystals, magnetic flux tubes in superconductors and vortices in superfluids.

The authenticity of a topological defect depends on the authenticity of the vacuum in which the system will tend towards if infinite time elapses; false and true topological defects can be distinguished if the defect is in a false vacuum and a true vacuum, respectively.

Cosmology

Certain grand unified theories predict topological defects to have formed in the early universe. According to the Big Bang theory, the universe cooled from an initial hot, dense state triggering a series of phase transitions much like what happens in condensed-matter systems.

In physical cosmology, a topological defect is an (often) stable configuration of matter predicted by some theories to form at phase transitions in the very early universe.

Symmetry breakdown

Depending on the nature of symmetry breakdown, various solitons are believed to have formed in the early universe according to the Higgs–Kibble mechanism. The well-known topological defects are magnetic monopoles, cosmic strings, domain walls, skyrmions and textures.

As the universe expanded and cooled, symmetries in the laws of physics began breaking down in regions that spread at the speed of light; topological defects occur where different regions came into contact with each other. The matter in these defects is in the original symmetric phase, which persists after a phase transition to the new asymmetric phase is completed.

Types of topological defects

Various different types of topological defects are possible, with the type of defect formed being determined by the symmetry properties of the matter and the nature of the phase transition. They include:

• Domain walls, two-dimensional membranes that form when a discrete symmetry is broken at a phase transition. These walls resemble the walls of a closed-cell foam, dividing the universe into discrete cells.
• Cosmic strings are one-dimensional lines that form when an axial or cylindrical symmetry is broken.
• Monopoles, cube-like defects that form when a spherical symmetry is broken, are predicted to have magnetic charge, either north or south (and so are commonly called "magnetic monopoles").
• Textures form when larger, more complicated symmetry groupsTemplate:Which are completely broken. They are not as localized as the other defects, and are unstable. Other more complex hybrids of these defect types are also possible.
• Extra dimensions and higher dimensions.

Observation

Topological defects, of the cosmological type, are extremely high-energy phenomena and are likely impossible to produce in artificial Earth-bound physics experiments, but topological defects that formed during the universe's formation could theoretically be observed.

No topological defects of any type have yet been observed by astronomers, however, and certain types are not compatible with current observations; in particular, if domain walls and monopoles were present in the observable universe, they would result in significant deviationsTemplate:Which from what astronomers can see. Because of these observations, the formation of these structures within the observable universe is highly constrained, requiring special circumstances (see: inflation). On the other hand, cosmic strings have been suggested as providing the initial 'seed'-gravity around which the large-scale structure of the cosmos of matter has condensed. Textures are similarly benign. In late 2007, a cold spot in the cosmic microwave background was interpreted as possibly being a sign of a texture lying in that direction.[1]

Condensed matter

Classes of stable defects in biaxial nematics

In condensed matter physics, the theory of homotopy groups provides a natural setting for description and classification of defects in ordered systems.[2] Topological methods have been used in several problems of condensed matter theory. Poénaru and Toulouse used topological methods to obtain a condition for line (string) defects in liquid crystals can cross each other without entanglement. It was a non-trivial application of topology that first led to the discovery of peculiar hydrodynamic behavior in the A-phase of superfluid helium-3.[2]

Classification

An ordered medium is defined as a region of space described by a function f(r) 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.[2]

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[3] R=G/H.

If G is a universal cover for G/H then, it can be shown[3] that πn (G/H)=πn-1 (H), where πi 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, (in three dimensions), line defects correspond to elements of π1 (R), point defects correspond to elements of π2 (R), textures correspond to elements of π3 (R). However, defects which belong to the same conjugacy class of π1 (R) can be deformed continuously to each other,[2] and hence, distinct defects correspond to distinct conjugacy classes.

Poénaru and Toulouse showed that[4] crossing defects get entangled if and only if they are members of separate conjugacy classes of π1 (R).

Stable defects

The homotopy theory is deeply related to the stability of topological defects. In the case of line defect, if the closed path can be continuously deformed into one point, the defect is not stable, and otherwise, it is stable.

Unlike in cosmology and field theory, topological defects in condensed matter can be experimentally observed.[5] Ferromagnetic materials have regions of magnetic alignment separated by domain walls. Nematic and bi-axial nematic liquid crystals display a variety of defects including monopoles, strings, textures etc.[2] Defects can also been found in biochemistry, notably in the process of protein folding.

Images

A soliton and an antisoliton colliding with velocities ±sinh(0.05) and annihilating.