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| {{Distinguish2|[[anion]], a negatively [[electric charge|charged]] [[ion]]}}
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| {{Statistical mechanics|cTopic=[[Particle statistics|Particle Statistics]]}}
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| In [[physics]], an '''anyon''' is a type of [[particle]] that occurs only in [[two-dimensional|''two''-dimensional]] [[physical system|system]]s, with properties much less restricted than [[fermion]]s and [[boson]]s; the operation of [[exchange symmetry|exchanging two identical particles]] may cause a global phase shift but cannot affect [[observables]]. Anyons are generally classified as ''abelian'' or ''non-abelian'', as explained below. Abelian anyons have been detected and play a major role in the [[fractional quantum Hall effect]]. Non-abelian anyons have not been definitively detected although this is an active area of research.
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| ==Abelian anyons==
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| In space of [[three-dimensional space|three]] or more dimensions, [[indistinguishable particles]] are either fermions or bosons, according to their [[Quantum statistics|statistical behaviour]]. Fermions obey the so-called [[Fermi–Dirac statistics]] while bosons obey the [[Bose–Einstein statistics]]. In the language of [[quantum mechanics]] this is formulated as the behavior of multiparticle states under the exchange of particles. This is in particular for a two-particle state (in [[Dirac notation]]):
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| :<math>\left|\psi_1\psi_2\right\rangle = \pm\left|\psi_2\psi_1\right\rangle</math>
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| (where the first entry in {{braket|ket|...}} is the state of particle 1 and the second entry is the state of particle 2. So for example the left hand side is read as "Particle 1 is in state ''ψ''<sub>1</sub> and particle 2 in state ''ψ''<sub>2</sub>"). Here the "+" corresponds to both particles being bosons and the "−" to both particles being fermions (composite states of fermions and bosons are irrelevant since that would make them distinguishable).
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| In two-dimensional systems, however, [[quasiparticle]]s can be observed which obey statistics ranging continuously between Fermi–Dirac and Bose–Einstein statistics, as was first shown by Jon Magne Leinaas and Jan Myrheim of the University of Oslo in 1977.<ref>{{cite journal
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| | title = On the theory of identical particles
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| | journal = Il Nuovo Cimento B
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| | volume = 37
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| | issue = 1
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| | pages = 1–23
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| | last = Leinaas
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| | first = J.M.
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| | coauthors = J. Myrheim
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| | date = 11 January 1977
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| | doi = 10.1007/BF02727953
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| |bibcode = 1977NCimB..37....1L }}</ref> In our above example of two particles this looks as follows:
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| :<math>\left|\psi_1\psi_2\right\rangle = e^{i\,\theta}\left|\psi_2\psi_1\right\rangle\,,</math>
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| with ''i'' the [[imaginary unit]] and ''θ'' a [[real number]]. Now |''e<sup>iθ</sup>''| = 1, ''e''<sup>2''πi''</sup> = 1, and ''e''<sup>''πi''</sup> = −1. So in the case ''θ'' = ''π'' we recover the Fermi–Dirac statistics (minus sign) and in the case ''θ'' = 0 (or ''θ'' = 2''π'') the Bose–Einstein statistics (plus sign). In between we have something different. [[Frank Wilczek]] coined the term "anyon"<ref>{{cite journal
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| | title = Quantum Mechanics of Fractional-Spin Particles
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| | journal = Physical Review Letters
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| | volume = 49
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| | issue = 14
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| | pages = 957–959
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| | last = Wilczek
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| | first = Frank
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| | date = 4 October 1982
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| | doi = 10.1103/PhysRevLett.49.957
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| | bibcode=1982PhRvL..49..957W}}</ref> to describe such particles, since they can have any phase when particles are interchanged.
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| We also may use θ = 2''πs'' with particle [[Spin (physics)#Spin and the Pauli exclusion principle|spin]] quantum number ''s'', with ''s'' being [[integer]] for bosons, [[half-integer]] for fermions, so that
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| :<math>e^{i\,\theta} = e^{2\,i \pi s} = (-1)^{2\,s}</math> or <math>\left|\psi_1\psi_2\right\rangle = (-1)^{2\,s}\left|\psi_2\psi_1\right\rangle.</math>
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| At an edge, fractional quantum Hall effect anyons are confined to move in one space dimension. Mathematical models of one dimensional anyons provide a base of the commutation relations shown above.
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| Just as the fermion and boson wavefunctions in a three-dimensional space are just 1-dimensional representations of the [[permutation group]] (''S<sub>N</sub>'' of ''N'' indistinguishable particles), the anyonic wavefunctions in a two-dimensional space are just 1-dimensional [[Braid group#Representations|representations of the braid group]] (''B<sub>N</sub>'' of ''N'' indistinguishable particles). Anyonic statistics must not be confused with [[parastatistics]] which describes statistics of particles whose wavefunctions are higher dimensional representations of the permutation group.<ref>{{cite book |title= Fractional Statistics and Quantum Theory |last= Khare |first=Avinash |authorlink= |year= 2005 |publisher= World Scientific |isbn=978-981-256-160-2 |page= 22 |url= http://books.google.com/?id=rBi7zTpvjaAC&dq=Fractional+Statistics+and+Quantum+Theory |accessdate= May 2011}}</ref>
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| ===Experiment===
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| <!--
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| This section below seems mostly redundant with the inline stuff above.
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| A group of [[Theoretical physics|theoretical physicists]] working at the [[University of Oslo]], led by Jon Leinaas and Jan Myrheim, calculated in 1977 that the traditional division between fermions and bosons would not apply to theoretical particles existing in two [[dimension]]s.<ref name=WilczekJan2006>{{cite journal|url=http://physicsworld.com/cws/article/indepth/23894 |title=From electronics to anyonics|first=Frank|last= Wilczek|journal= Physics World|date=January 2006|year=2006| issn= 0953-8585}}</ref> Such particles would be expected to exhibit a diverse range of previously unexpected properties. They were given the name anyons by Frank Wilczek in 1982.<ref>{{cite web|url=http://www.sciencewatch.com/interviews/frank_wilczek1.htm |title=Frank Wilczek on anyons and their Role in Superconductivity|work=Science Watch}}</ref>
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| -->[[File:Laughlin quasiparticle interferometer sample scanning electron micrograph.svg|400px|thumb|[[Robert B. Laughlin|Laughlin]] quasiparticle interferometer [[scanning electron microscopy|scanning electron micrograph]] of a [[semiconductor device]]. The four light grey regions are [[gold|Au]]/[[Titanium|Ti]] gates of un[[Depletion region|depleted]] [[electron]]s, the blue curves are the edge channels from the [[equipotential]]s of these undepleted electrons. The dark grey curves are etched trenches depleted of electrons, the blue dots are the [[Tunnel junction|tunneling junction]]s, the yellow dots are [[Ohmic contact]]s. The electrons in the device are confined to a 2d plane.<ref>{{cite journal|doi =10.1103/PhysRevB.72.075342|title =Realization of a Laughlin quasiparticle interferometer: Observation of fractional statistics|year =2005|last1 =Camino|first1 =F.|last2 =Zhou|first2 =Wei|last3 =Goldman|first3 =V.|journal =Physical Review B|volume =72|issue =7 |arxiv = cond-mat/0502406 |bibcode = 2005PhRvB..72g5342C }}, see ''fig. 2.B''</ref>]]
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| [[Daniel Tsui]] and [[Horst Störmer]] discovered the fractional quantum Hall effect in 1982. The mathematics developed by Leinaas and Myrheim proved to be useful to [[Bertrand Halperin]] at [[Harvard University]] in explaining aspects of it. Frank Wilczek, Dan Arovas, and [[Robert Schrieffer]] verified this statement in 1985 with an explicit calculation that predicted that particles existing in these systems are in fact anyons.
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| In 2005 a group of physicists at [[Stony Brook University]] constructed a quasiparticle [[interferometer]], detecting the patterns caused by interference of anyons which were interpreted to suggest that anyons are real, rather than just a mathematical construct.<ref>{{cite journal|doi =10.1103/PhysRevB.72.075342|title =Realization of a Laughlin quasiparticle interferometer: Observation of fractional statistics|year =2005|last1 =Camino|first1 =F.|last2 =Zhou|first2 =Wei|last3 =Goldman|first3 =V.|journal =Physical Review B|volume =72|issue =7 |arxiv = cond-mat/0502406 |bibcode = 2005PhRvB..72g5342C }}</ref> However, these experiments remain controversial and are not fully accepted by the community.
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| With developments in [[semiconductor]] technology meaning that the deposition of thin two-dimensional layers is possible – for example in sheets of [[graphene]] – the long term potential to use the properties of anyons in electronics is being explored.
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| ==Non-abelian anyons==
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| {{Unsolved|physics|Is topological order stable at non-zero [[temperature]]?}}
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| In 1988, Jürg Fröhlich{{verify source|date=April 2012}} showed that it was possible in a valid particle theory for the particle exchange operation to be non-commutative (non-Abelian statistics). Gregory Moore, Nicholas Read, and Xiao-Gang Wen pointed out that non-Abelian statistics can be realized in [[fractional quantum Hall effect]].<ref>{{Cite journal|doi=10.1016/0550-3213(91)90407-O|url=http://www.physics.rutgers.edu/~gmoore/MooreReadNonabelions.pdf|title=Nonabelions in the fractional quantum hall effect|year=1991|last1=Moore|first1=Gregory|last2=Read|first2=Nicholas|journal=Nuclear Physics B|volume=360|issue=2–3|pages=362|bibcode = 1991NuPhB.360..362M }}; </ref> <ref> [[Xiao-Gang Wen]] [''Non-Abelian Statistics in the FQH states'' http://dao.mit.edu/~wen/pub/nab.pdf]
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| Phys. Rev. Lett. 66, 802 (1991)</ref> While at first non-abelian anyons were generally considered a mathematical curiosity, physicists began pushing toward their discovery when [[Alexei Kitaev]] showed that non-abelian anyons could be used to construct a [[topological quantum computer]]. As of 2012, no experiment has conclusively demonstrated the existence of non-abelian anyons although promising hints are emerging in the study of the ν = 5/2 FQHE state.<ref>{{Cite journal|doi=10.1038/nature08915|title=Non-Abelian states of matter|year=2010|last1=Stern|first1=Ady|journal=Nature|volume=464|issue=7286|pages=187–93|pmid=20220836|bibcode = 2010Natur.464..187S }}</ref><ref>{{Cite arxiv | eprint=1112.3400 | author1=Sanghun An | author2=Jiang | author3=Choi | author4=Kang | author5=Simon | author6=Pfeiffer | author7=West | author8=Baldwin | title=Braiding of Abelian and Non-Abelian Anyons in the Fractional Quantum Hall Effect | class=cond-mat.mes-hall | year=2011 }}</ref>
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| ==Topological basis==
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| {{multiple image
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| | footer = Exchange of two particles in 2 + 1 spacetime by rotation. The rotations are inequivalent, since one cannot be deformed into the other (without the worldlines leaving the plane, an impossibility in 2d space).
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| | width = 150
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| | image1 = Particle exchange 2d anticlockwise.gif
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| | caption1 = Anticlockwise rotation
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| | image2 = Particle exchange 2d clockwise.gif
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| | caption2 = Clockwise rotation
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| }}
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| In more than two dimensions, the [[spin–statistics theorem]] states that any multiparticle state of [[indistinguishable particles]] has to obey either Bose–Einstein or Fermi–Dirac statistics. For any <var>d</var> > 2, the [[Lie group|group]] [[orthogonal group|SO(<var>d</var>,1)]] (which generalize the [[Lorentz group]]), and also [[Poincaré group|Poincaré(<var>d</var>,1)]], have Z<sub>2</sub> as their [[first homotopy group]]. The [[cyclic group]] consisting of two elements, Z<sub>2</sub>, therefore only two possibilities remain. (The details are more involved than that, but this is the crucial point.)
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| The situation changes in two dimensions. Here the first homotopy group of SO(2,1), and also Poincaré(2,1), is '''Z''' (infinite cyclic). This means that Spin(2,1) is not the [[universal covering group|universal cover]]: it is not [[simply connected]]. In detail, there are [[projective representation]]s of the [[generalized orthogonal group|special orthogonal group]] SO(2,1) which do not arise from [[linear representation]]s of SO(2,1), or of its [[Double covering group|double cover]], the [[spin group]] Spin(2,1). These representations{{Clarify|date=August 2010}}<!-- representations for any θ are called so, or only for θ ≠ kπ ? --> are called anyons.
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| This concept also applies to nonrelativistic systems. The relevant part here is that the spatial rotation group is SO(2) has an infinite first homotopy group.
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| This fact is also related to the [[braid group]]s well known in [[knot theory]]. The relation can be understood when one considers the fact that in two dimensions the group of permutations of two particles is no longer the [[symmetric group]] ''S''<sub>2</sub> (with two elements) but rather the braid group ''B''<sub>2</sub> (with an infinite number of elements). The essential point is that one braid can wind around the other one, an operation that can be performed infinitely often, and clockwise as well as counterclockwise.
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| A very different approach to the stability-decoherence problem in [[Quantum computer|quantum computing]] is to create a [[topological quantum computer]] with anyons, quasi-particles used as threads and relying on [[braid theory]] to form stable [[logic gate]]s.<ref>{{cite journal
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| | title = Topological Quantum Computation
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| | journal = Bulletin of the American Mathematical Society
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| | volume = 40
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| | issue = 1
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| | pages = 31–38
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| | last = Freedman
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| | first = Michael
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| | coauthors = Alexei Kitaev, Michael Larsen, Zhenghan Wang
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| | date = 20 October 2002
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| | doi = 10.1090/S0273-0979-02-00964-3}}</ref><ref>{{cite journal|last=Monroe |first= Don |url=http://www.newscientist.com/channel/fundamentals/mg20026761.700-anyons-the-breakthrough-quantum-computing-needs.html |title=Anyons: The breakthrough quantum computing needs? | work=[[New Scientist]] |issue= 2676|date= 1 October 2008}}</ref>
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| ==See also==
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| * [[Plekton]]
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| * [[Flux tube]]
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| * [[Anyonic Lie algebra]]
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| *{{Cite journal|doi = 10.1103/RevModPhys.80.1083| arxiv=0707.1889v2|title = Non-Abelian anyons and topological quantum computation|year = 2008|last1 = Nayak|first1 = Chetan|last2 = Stern|first2 = Ady|last3 = Freedman|first3 = Michael|last4 = Das Sarma|first4 = Sankar|journal = Reviews of Modern Physics|volume = 80|issue = 3|pages = 1083|bibcode=2008RvMP...80.1083N}}
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| *{{cite journal| doi = 10.1103/PhysRevB.65.165113|url=http://dao.mit.edu/~wen/pub/qosl.pdf| title = Quantum orders ''and symmetric'' spin liquids| year = 2002| last1 = Wen| first1 = Xiao-Gang| journal = Physical Review B| volume = 65| issue = 16|arxiv = cond-mat/0107071 |bibcode = 2002PhRvB..65p5113W }}
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| *{{cite journal | doi = 10.1016/j.aop.2007.10.008|arxiv=0711.4697v1|url=http://pitp.physics.ubc.ca/confs/7pines2009/readings/arovas_Stern_2007.pdf | title = Anyons and the quantum Hall effect—A pedagogical review | year = 2008 | last1 = Stern | first1 = Ady | journal = Annals of Physics | volume = 323 | pages = 204 |bibcode = 2008AnPhy.323..204S }}
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| [[Category:Parastatistics]]
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| [[Category:Representation theory of Lie groups]]
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| {{Use dmy dates|date=June 2013}}
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