Topological defect

In

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" (as with scissors), but "cutting" is not a defined operation for solving PDE's. The cutting analogy arises because some solitons are described as mappings ${\displaystyle U(1)\to U(1)}$, where ${\displaystyle U(1)\simeq S^{1}}$ 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 ${\displaystyle S^{3}\to S^{3}}$, where the first

${\displaystyle S^{3}}$

${\displaystyle O(3)}$; the length fixes a point. This has a

${\displaystyle SU(2)}$

${\displaystyle SU(2)\simeq S^{3}}$

A topological defect is perhaps the simplest way of understanding the general idea: it is a soliton that occurs in a

of the map of circles ${\displaystyle S^{1}\to S^{1};}$ the stability of the dislocation leads to

critical density

, allowing them to interact with one-another and "connect up", and thus disconnect (fracture) the whole. The idea that critical densities of solitons can lead to phase transitions is a recurring theme.

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 (neutron or proton) and owed its stability to the mapping ${\displaystyle S^{3}\to SU(2)}$. In the 1980's, the

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 (quantum) field configuration with a soliton in it will have a higher energy than the

The existence of a topological defect can be demonstrated whenever the

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

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).

Topological defects occur in

The authenticity^[

• Cosmic strings are one-dimensional lines that form when an axial or cylindrical symmetry is broken.
• 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.
• Monopoles, cube-like defects that form when a spherical symmetry is broken, are predicted to have magnetic charge,^[why?] either north or south (and so are commonly called "magnetic monopoles").
• which?] are completely broken. They are not as localized as the other defects, and are unstable.^{[clarification needed}
  ]
• Skyrmions
• dimensions
  .

In condensed matter physics, the theory of

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 have been experimentally observed.^[7] 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] In crystalline solids, the most common topological defects are dislocations, which play an important role in the prediction of the mechanical properties of crystals, especially crystal plasticity.

In magnetic systems, topological defects include 2D defects such as skyrmions (with integer skyrmion charge), or 3D defects such as Hopfions (with integer Hopf index). The definition can be extended to include dislocations of the helimagnetic order, such as edge dislocations ^[8][9] and screw dislocations ^[10] (that have an integer value of the Burgers vector)