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| : ''Also see [[topological excitations]] and the base concepts: [[topology]], [[differential equation]]s, [[quantum mechanics]]'' & ''[[condensed matter]] physics''.
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| In [[mathematics]] and [[physics]], a '''topological soliton''' or a '''topological defect''' is a solution of a system of [[partial differential equation]]s 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 [[homotopy|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 equation]]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 [[wiktionary:perturbation|perturbation]]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.
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| Examples include the [[soliton]] or [[soliton|solitary wave]] which occurs in many [[exactly solvable model]]s, the [[Dislocation#Screw_dislocations|screw dislocation]]s in crystalline materials, the [[skyrmion]] and the [[Wess–Zumino–Witten model]] in quantum field theory.
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| Topological defects are believed to drive [[phase transition]]s 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 tube]]s in [[superconductors]], vortices in [[superfluids]].
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| The authenticity of a topological defect depends on the authenticity of the vacuum in which the system will tend towards if infinite time elapses; we will distinguish false and true topological defects if the defect is in a [[false vacuum]] and a [[true vacuum]], respectively. | |
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| ==Cosmology==
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| 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.
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| In [[physical cosmology]], a topological defect is an (often) stable configuration of matter predicted by some theories to form at [[phase transition]]s in the very early universe.
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| ===Symmetry breakdown===
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| Depending on the nature of [[symmetry breakdown]], various [[soliton]]s are believed to have formed in the early universe according to the [[Higgs–Kibble mechanism]]. The well-known topological defects are [[magnetic monopole]]s, [[cosmic string]]s, [[Domain wall (string theory)|domain wall]]s, [[Skyrmion]]s and [[Texture (cosmology)|textures]].
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| 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.
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| ===Types of topological defects===
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| 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:
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| *[[Domain wall (disambiguation)|Domain wall]]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]], dividing the universe into discrete cells.
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| *[[Cosmic string]]s are one dimensional lines that form when an axial or cylindrical symmetry is broken.
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| *[[magnetic monopole|Monopole]]s, 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]]s").
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| *[[Texture (cosmology)|Texture]]s form when larger, more complicated symmetry groups{{which|date=January 2014}} 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.
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| *[[Extra-dimension]]s and higher [[dimensions]].
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| ===Observation===
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| 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.
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| 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 deviations{{which|date=January 2014}} 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 (cosmology)|inflation]]''). On the other hand, [[cosmic string]]s 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 [[WMAP cold spot|cold spot]] in the [[cosmic microwave background]] was interpreted as possibly being a sign of a [[Texture (cosmology)|texture]] lying in that direction.<ref name="cam">{{cite journal| journal=Science| doi=10.1126/science.1148694| title=A Cosmic Microwave Background Feature Consistent with a Cosmic Texture| first=M.| last= Cruz| coauthors= N. Turok, P. Vielva, E. Martínez-González, M. Hobson| url= http://www.sciencemag.org/cgi/content/abstract/1148694| accessdate=2007-10-25| volume=318| pages=1612–4| year=2007| pmid=17962521| issue=5856|bibcode = 2007Sci...318.1612C |arxiv = 0710.5737 }}</ref>
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| ==Condensed matter==
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| [[Image:Biaxial.png|thumb|180px|Classes of stable defects in [[Biaxial nematic]]s]]
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| In condensed matter physics, the theory of [[homotopy groups]] provides a natural setting for description and classification of defects in ordered systems.<ref name="mermin">{{cite journal |last1= Mermin|first1=N. D.|year= 1979|title=The topological theory of defects in ordered media|journal=Reviews of Modern Physics |volume=51 |issue=3 | doi = 10.1103/RevModPhys.51.591 | bibcode=1979RvMP...51..591M |pages= 591 }}</ref> 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.<ref name="mermin"/>
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| ===Classification===
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| 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.<ref name="mermin"/>
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| 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<ref name="nak">{{cite book |title=Geometry, Topology and Physics |last= Nakahara|first=Mikio|year=2003 |publisher=Taylor & Francis|isbn=0-7503-0606-8|accessdate=2009-12-03}}</ref> ''R=G/H''.
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| If ''G'' is a [[universal cover]] for ''G/H'' then, it can be shown<ref name="nak"/> that ''π<sub>n </sub>(G/H)=π<sub>n-1 </sub>(H)'', where ''π<sub>i </sub>'' denotes the ''i''-th [[homotopy group]].
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| 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 ''π<sub>1 </sub>(R)'', point defects correspond to elements of ''π<sub>2 </sub>(R)'', textures correspond to elements of ''π<sub>3 </sub>(R)''. However, defects which belong to the same [[conjugacy class]] of ''π<sub>1 </sub>(R)'' can be deformed continuously to each other,<ref name="mermin"/> and hence, distinct defects correspond to distinct conjugacy classes.
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| Poénaru and Toulouse showed that<ref>{{cite journal |last1=Poénaru |first1=V.|last2=Toulouse|first2=G.|year=1977 |title=The crossing of defects in ordered media and the topology of 3-manifolds|journal=Le Journal de Physique|volume=38|issue=8}}</ref> crossing defects get entangled if and only if they are members of separate conjugacy classes of ''π<sub>1 </sub>(R)''.
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| ===Stable defects===
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| 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.
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| Unlike in cosmology and field theory, topological defects in condensed matter can be experimentally observed.<ref>{{cite web|url=http://www.damtp.cam.ac.uk/research/gr/public/cs_top.html|title=Topological defects|publisher=Cambridge cosmology}}</ref> Ferromagnetic materials have regions of magnetic alignment separated by domain walls. [[Liquid crystal#Nematic phase|Nematic]] and bi-axial nematic liquid crystals display a variety of defects including monopoles, strings, textures etc.<ref name="mermin"/> Defects can also been found in biochemistry, notably in the process of protein folding.
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| ==Images==
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| [[Image:DoubleWellSoliton.png|frame|none|A static solution to <math>\mathcal{L} = \partial_\mu\phi\partial^\mu\phi - (\phi^2 - 1)^2</math> in 1+1 dimensional spacetime.]]
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| [[Image:DoubleWellSolitonAntisoliton.gif|frame|none|A soliton and an antisoliton colliding with velocities ±sinh(0.05) and annihilating.]]
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| ==See also==
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| * [[Quantum vortex]]
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| * [[Dislocation]]
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| * [[Vector soliton]]
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| *[[Quantum topology]]
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| *[[Topological entropy in physics]]
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| *[[Topological order]]
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| *[[Topological quantum field theory]]
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| *[[Topological quantum number]]
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| *[[Topological string theory]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| *[http://www.damtp.cam.ac.uk/user/gr/public/cs_top.html Cosmic Strings & other Topological Defects]
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| * http://demonstrations.wolfram.com/SeparationOfTopologicalSingularities/
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| {{DEFAULTSORT:Topological Defect}}
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| [[Category:Solitons| ]]
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| [[Category:Large-scale structure of the cosmos]]
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| [[Category:Cosmic inflation]]
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The boy of a , knows determination and determination are important elements in relation to a successful profession- . His to start with album, Continue to be Me, generated the very best strikes “All My Pals “Country and Say” Guy,” while his hard work, Doin’ who has luke bryan toured with (www.hotelsedinburgh.org) Point, luke bryan concert - lukebryantickets.neodga.com - identified the vocalist-about three directly No. 5 men and women: More Calling Is usually a Good Issue.”
In the slip of 2003, Tour: Bryan And that had an outstanding last minute tickets listing of , which includes City. “It’s much like you are receiving a authorization to look to a higher level, affirms these artists that have been a part of the Concert touraround right into a bigger measure of musicians.” It packaged as among the best excursions in its ten-calendar year historical past.
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