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A topological soliton occurs when two adjoining structures or
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
s are in some way "out of phase" with each other in ways that make a seamless transition between them impossible. One of the simplest and most commonplace examples of a topological soliton occurs in old-fashioned coiled telephone handset cords, which are usually coiled clockwise. Years of picking up the handset can end up coiling parts of the cord in the opposite counterclockwise direction, and when this happens there will be a distinctive larger loop that separates the two directions of coiling. This odd looking transition loop, which is neither clockwise nor counterclockwise, is an excellent example of a topological soliton. No matter how complex the context, anything that qualifies as a topological soliton must at some level exhibit this same simple issue of reconciliation seen in the twisted phone cord example. Topological solitons arise with ease when creating the crystalline semiconductors used in modern electronics, and in that context their effects are almost always deleterious. For this reason such crystal transitions are called topological defects. However, this mostly solid-state terminology distracts from the rich and intriguing mathematical properties of such boundary regions. Thus for most non-solid-state contexts the more positive and mathematically rich phrase "topological soliton" is preferable. A more detailed discussion of topological solitons and related topics is provided below. In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, a topological soliton or a topological defect is a solution of a system of
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...
s or of a
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
homotopically distinct from the vacuum solution.


Overview

The existence of a topological defect can be demonstrated whenever the
boundary conditions In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
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 In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
which is preserved in
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s; 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
perturbation Perturbation or perturb may refer to: * Perturbation theory, mathematical methods that give approximate solutions to problems that cannot be solved exactly * Perturbation (geology), changes in the nature of alluvial deposits over time * Perturbat ...
s, 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.


Formal 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 In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
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 In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
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 A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
for ''G''/''H'' then, it can be shown that π''n''(''G''/''H'') = π''n''−1(''H''), where π''i'' denotes the ''i''-th
homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
. 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 In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wo ...
of π1(''R'') 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(''R'').


Examples

Topological defects occur in
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...
s and are believed to drive
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states ...
s in
condensed matter Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the su ...
physics. The authenticity of a topological defect depends on the nature 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 In quantum field theory, a false vacuum is a hypothetical vacuum that is relatively stable, but not in the most stable state possible. This condition is known as metastable. It may last for a very long time in that state, but could eventually ...
and a true vacuum, respectively.


Solitary wave PDEs

Examples include the
soliton In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the me ...
or solitary wave which occurs in
exactly solvable model In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first i ...
s, such as * screw dislocations in crystalline materials, *
skyrmion In particle theory, the skyrmion () is a topologically stable field configuration of a certain class of non-linear sigma models. It was originally proposed as a model of the nucleon by (and named after) Tony Skyrme in 1961. As a topological sol ...
in quantum field theory, and *topological defects of the
Wess–Zumino–Witten model In theoretical physics and mathematics, a Wess–Zumino–Witten (WZW) model, also called a Wess–Zumino–Novikov–Witten model, is a type of two-dimensional conformal field theory named after Julius Wess, Bruno Zumino, Sergei Novikov and Ed ...
.


Lambda transitions

Topological defects in lambda transition universality class systems including: *screw/edge-dislocations in
liquid crystals Liquid crystal (LC) is a state of matter whose properties are between those of conventional liquids and those of solid crystals. For example, a liquid crystal may flow like a liquid, but its molecules may be oriented in a crystal-like way. The ...
, *magnetic flux "tubes" known as
fluxons In physics, a fluxon is a quantum of electromagnetic flux. The term may have any of several related meanings. Superconductivity In the context of superconductivity, in type II superconductors fluxons (also known as Abrikosov vortices) can for ...
in
superconductors Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
, and *vortices in superfluids.


Cosmological defects

Topological defects, of the cosmological type, are extremely high-energy phenomena which are deemed impractical to produce in Earth-bound physics experiments. Topological defects created during the universe's formation could theoretically be observed without significant energy expenditure. In the
Big Bang theory The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...
, the universe cools from an initial hot, dense state triggering a series of
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states ...
s much like what happens in condensed-matter systems such as superconductors. Certain
grand unified theories A Grand Unified Theory (GUT) is a model in particle physics in which, at high energies, the three gauge interactions of the Standard Model comprising the electromagnetic, weak, and strong forces are merged into a single force. Although this ...
predict the formation of stable topological defects in the early
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the univers ...
during these phase transitions.


Symmetry breakdown

Depending on the nature of symmetry breakdown, various solitons are believed to have formed in the early universe according to the Kibble-Zurek mechanism. The well-known topological defects are: * Cosmic strings are one-dimensional lines that form when an axial or cylindrical symmetry is broken. *
Domain wall A domain wall is a type of topological soliton that occurs whenever a discrete symmetry is spontaneously broken. Domain walls are also sometimes called kinks in analogy with closely related kink solution of the sine-Gordon model or models with pol ...
s, 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 Foams are materials formed by trapping pockets of gas in a liquid or solid. A bath sponge and the head on a glass of beer are examples of foams. In most foams, the volume of gas is large, with thin films of liquid or solid separating the ...
, dividing the universe into discrete cells. * 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 monopole In particle physics, a magnetic monopole is a hypothetical elementary particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magneti ...
s"). * Textures form when larger, more complicated symmetry groups are completely broken. They are not as localized as the other defects, and are unstable. *
Skyrmions In particle theory, the skyrmion () is a topologically stable field configuration of a certain class of non-linear sigma models. It was originally proposed as a model of the nucleon by (and named after) Tony Skyrme in 1961. As a topological solit ...
*
Extra dimension In physics, extra dimensions are proposed additional space or time dimensions beyond the (3 + 1) typical of observed spacetime, such as the first attempts based on the Kaluza–Klein theory. Among theories proposing extra dimensions are: ...
s and higher dimensions. Other more complex hybrids of these defect types are also possible. As the universe expanded and cooled, symmetries in the laws of physics began breaking down in regions that spread at the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
; topological defects occur at the boundaries of adjacent regions. The matter composing these boundaries is in an
ordered phase In physics, the terms order and disorder designate the presence or absence of some symmetry or correlation in a many-particle system. In condensed matter physics, systems typically are ordered at low temperatures; upon heating, they undergo on ...
, which persists after the phase transition to the
disordered phase In physics, the terms order and disorder designate the presence or absence of some symmetry or correlation in a many-particle system. In condensed matter physics, systems typically are ordered at low temperatures; upon heating, they undergo on ...
is completed for the surrounding regions.


Observation

Topological defects have not been observed by astronomers; however, 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 deviations from what astronomers can see. Because of these observations, the formation of defects ''within the observable universe'' is highly constrained, requiring special circumstances (see
Inflation (cosmology) In physical cosmology, cosmic inflation, cosmological inflation, or just inflation, is a theory of exponential expansion of space in the early universe. The inflationary epoch lasted from  seconds after the conjectured Big Bang singulari ...
). On the other hand, cosmic strings have been suggested as providing the initial 'seed'-gravity around which the
large-scale structure of the cosmos The observable universe is a ball-shaped region of the universe comprising all matter that can be observed from Earth or its space-based telescopes and exploratory probes at the present time, because the electromagnetic radiation from these obj ...
of matter has condensed. Textures are similarly benign. In late 2007, a cold spot in the
cosmic microwave background In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all spac ...
provided evidence of a possible texture.


Condensed matter

In condensed matter physics, the theory of
homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homot ...
provides a natural setting for description and classification of defects in ordered systems. 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 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 Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas and the first in the noble gas group in the periodic ta ...
-3.


Stable defects

Homotopy theory In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolo ...
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. Ferromagnetic materials have regions of magnetic alignment separated by domain walls.
Nematic Liquid crystal (LC) is a state of matter whose properties are between those of conventional liquids and those of solid crystals. For example, a liquid crystal may flow like a liquid, but its molecules may be oriented in a crystal-like way. Th ...
and bi-axial nematic liquid crystals display a variety of defects including monopoles, strings, textures etc.


Images


See also

* GN Drive *
Condensed matter Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the su ...
*
Differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
*
Dislocation In materials science, a dislocation or Taylor's dislocation is a linear crystallographic defect or irregularity within a crystal structure that contains an abrupt change in the arrangement of atoms. The movement of dislocations allow atoms to s ...
*
Quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
* Quantum topology * Quantum vortex * Topological entropy in physics * Topological excitations *
Topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout math ...
* Topological order *
Topological quantum field theory In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathe ...
* Topological quantum number *
Topological string theory In theoretical physics, topological string theory is a version of string theory. Topological string theory appeared in papers by theoretical physicists, such as Edward Witten and Cumrun Vafa, by analogy with Witten's earlier idea of topological ...
*
Topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
*
Vector soliton In physical optics or wave optics, a vector soliton is a solitary wave with multiple components coupled together that maintains its shape during propagation. Ordinary solitons maintain their shape but have effectively only one (scalar) polarization ...


References


External links


Cosmic Strings & other Topological Defects
* http://demonstrations.wolfram.com/SeparationOfTopologicalSingularities/ {{DEFAULTSORT:Topological Defect Large-scale structure of the cosmos Inflation (cosmology)