Topological defect

Solitons won't decay, dissipate, disperse or evaporate in the way that ordinary waves (or solutions or structures) might.

It was eventually explained by noting that the Korteweg-De Vries (KdV) equation, describing waves in water, has homotopically distinct solutions.

The range (codomain) of the variables in the differential equation can also be viewed as living in some compact topological space.

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.

One common manifestation is the repeated bending of a metal wire: this introduces more and more screw dislocations (as dislocation-anti-dislocation pairs), making the bent region increasingly stiff and brittle.

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 (fracture) the whole.

Vorticies in superfluids and pinned vortex tubes in type-II superconductors provide examples of circle-map type topological solitons in fluids.

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.

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.

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.

[1] 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.

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.

Various types of defects in the medium can be characterized by elements of various homotopy groups of the order parameter space.

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 created during the universe's formation could theoretically be observed without significant energy expenditure.

In the Big Bang theory, the universe cools from an initial hot, dense state triggering a series of phase transitions much like what happens in condensed-matter systems such as superconductors.

grand unified theories predict the formation of stable topological defects in the early universe during these phase transitions.

Depending on the nature of symmetry breaking, various solitons are believed to have formed in cosmological phase transitions in the early universe according to the Kibble-Zurek mechanism.

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 at the boundaries of adjacent regions.[how?]

In particular, if domain walls and monopoles were present in the observable universe, they would result in significant deviations from what astronomers can see.

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.

[clarification needed] In late 2007, a cold spot in the cosmic microwave background provided evidence of a possible texture.

[5] In condensed matter physics, the theory of homotopy groups provides a natural setting for description and classification of defects in ordered systems.

Poénaru and Toulouse used topological methods to obtain a condition for line (string) defects in liquid crystals that 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.

Unlike in cosmology and field theory, topological defects in condensed matter have been experimentally observed.