View source for Anyon

{{Short description|Type of two-dimensional quasiparticle}}
{{Other uses}}
{{distinguish|text=[[anion]], a negatively charged ion}}
{{Statistical mechanics|cTopic=[[Particle statistics|Particle Statistics]]}}
{{Use dmy dates|date=October 2020}}
In [[physics]], an '''anyon''' is a type of [[quasiparticle]] so far observed only in [[two-dimensional]] [[physical system|systems]]. In [[three-dimensional]] systems, only two kinds of [[elementary particle]]s are seen: [[fermion]]s and [[boson]]s. Anyons have statistical properties intermediate between fermions and bosons.<ref name="Bartolomei2020"/> In general, the operation of [[exchange symmetry|exchanging two identical particles]], although it may cause a global phase shift, cannot affect [[observables]]. Anyons are generally classified as ''abelian'' or ''non-abelian''. Abelian anyons, detected by two experiments in 2020,<ref name="DISC-20201212">
{{cite news
 |last=Ornes |first=Stephen
 |title=Physicists Prove Anyons Exist, a Third Type of Particle in the Universe – Physicists give us an early view of a third kingdom of quasiparticles that only arise in two dimensions.
 |url=https://www.discovermagazine.com/the-sciences/physicists-prove-anyons-exist-a-third-type-of-particle-in-the-universe
 |date=12 December 2020
 |work=[[Discover (magazine)|Discover]]
 |quote=This year brought two solid confirmations of the quasiparticles. The first arrived in April, in a paper featured on the cover of ''Science'', from a group of researchers at the École Normale Supérieure in Paris&nbsp;... The second confirmation came in July, when a group at Purdue University in Indiana used an experimental setup on an etched chip that screened out interactions that might obscure the anyon behavior.
 |access-date=2020-12-12
}}</ref> play a major role in the [[fractional quantum Hall effect]].

== Introduction ==
The [[statistical mechanics]] of large many-body systems obeys laws described by [[Maxwell–Boltzmann statistics]]. [[Quantum statistics]] is more complicated because of the different behaviors of two different kinds of particles called [[fermions]] and [[bosons]]. In two-dimensional systems, however, there is a third type of particle, called an anyon.

{{blockquote|In the three-dimensional world we live in, there are only two types of particles: "fermions", which repel each other, and "bosons", which like to stick together. A commonly known fermion is the electron, which transports electricity; and a commonly known boson is the photon, which carries light. In the two-dimensional world, however, there is another type of particle, the anyon, which doesn't behave like either a fermion or a boson. |title="Finally, anyons reveal their exotic quantum properties" |source=Aalto University press release, April 2020<ref name="ExoticQP">
{{cite web
 |url=https://sciencex.com/wire-news/347971706/finally-anyons-reveal-their-exotic-quantum-properties.html
 |title=Finally, anyons reveal their exotic quantum properties
 |publisher=Aalto University
 |quote=They were first proposed in the late 1970s, but direct experimental evidence of their quantum statistics hasn't been conclusively shown until now.
 |date=7 December 2018
 |access-date=2020-09-24
}}</ref>}}

In a two-dimensional world, two identical anyons change their wavefunction when they swap places in ways that cannot happen in three-dimensional physics:

{{blockquote|...in two dimensions, exchanging identical particles twice is not equivalent to leaving them alone. The particles' wavefunction after swapping places twice may differ from the original one; particles with such unusual exchange statistics are known as anyons. By contrast, in three dimensions, exchanging particles twice cannot change their wavefunction, leaving us with only two possibilities: bosons, whose wavefunction remains the same even after a single exchange, and fermions, whose exchange only changes the sign of their wavefunction. |author=Kirill Shtengel |title="A home for anyon?" |source=Nature Physics<ref name="NatPhys2007">
{{cite journal
 |last= Shtengel |first=Kirilli
 |date=2007
 |title=A home for anyon?
 |url= https://www.nature.com/articles/nphys767
 |journal=Nature Physics
 |volume= 3|issue=11|page=763
 |doi=10.1038/nphys767
 |quote=From a physicist's point of view, having two spatial dimensions is special: a pair of particles trading places behave very differently in two dimensions than they do in three. In three dimensions, any two sets of paths taken by two identical particles in the process of exchanging their positions can be continuously morphed into one another. But in two dimensions, particles can wind around each other in two distinct ways, clockwise or anticlockwise. A profound consequence of this observation for quantum mechanics is that in two dimensions, exchanging identical particles twice is not equivalent to leaving them alone.
 |access-date=2020-11-30
 |doi-access=free
}}</ref>}}

This process of exchanging identical particles, or of circling one particle around another, is referred to as "[[Braid group#Formal treatment|braiding]]". Braiding two anyons creates a historical record of the event, as their changed wave functions record the number of braids.<ref name="Yirka2020">
{{cite web
 |url=https://phys.org/news/2020-07-evidence-anyons.html
 |title=Best evidence yet for existence of anyons
 |last=Yirka |first=Bob
 |publisher=Phys.org News
 |quote=If a fermion or a boson were dragged around another of its kind, theory suggests, the action would not produce a record of what had occurred. But because anyons alter wave functions, they would create such a record.
 |date=10 July 2020
 |access-date=2020-11-30
}}</ref>

[[Microsoft]] has invested in research concerning anyons as a potential basis for [[topological quantum computing]].<ref name=":0">{{Cite book |last=Wilczek |first=Frank |title=Fundamentals : Ten Keys to Reality |date=2021 |publisher=Penguin Press |isbn=9780735223790 |location=New York, New York |pages=89–90 |language=en-us |lccn=2020020086}}</ref> They may be useful in quantum computing as a form of memory.<ref name=":0" /> Anyons circling each other ("braiding") would encode information in a more robust way than other known potential [[quantum computing]] technologies.<ref name="Welcome2020">
{{cite journal |last=Castelvecchi |first=Davide |date=3 July 2020 |title=Welcome anyons! Physicists find best evidence yet for long-sought 2D structures |journal=Nature |volume=583 |issue=7815 |pages=176–177 |bibcode=2020Natur.583..176C |doi=10.1038/d41586-020-01988-0 |pmid=32620884 |s2cid=220336025 |quote=Simon and others have developed elaborate theories that use anyons as the platform for quantum computers. Pairs of the quasiparticle could encode information in their memory of how they have circled around one another. And because the fractional statistics is 'topological' – it depends on the number of times one anyon went around another, and not on slight changes to its path – it is unaffected by tiny perturbations. This robustness could make topological quantum computers easier to scale up than are current quantum-computing technologies, which are error-prone. |doi-access=free}}</ref> Most investment in quantum computing, however, is based on methods that do not use anyons.<ref name="Welcome2020"/>

== History ==
{{blockquote |text=Like so many deep ideas in physics, the topological underpinnings of anyons can be traced back to [[Paul Dirac|Dirac]]. |source = The Ancestry of the 'Anyon'<ref>
{{cite journal
 |author=Biedenharn, L. |author-link=Lawrence Biedenharn
 |author2=Lieb, E. |author-link2=Elliott H. Lieb
 |author3=Simon, B. |author-link3=Barry Simon
 |author4=Wilczek, F.
 |date=August 1990
 |title=The Ancestry of the Anyon
 |journal=Physics Today
 |volume=43 |issue=8 |pages=90–91
 |doi=10.1063/1.2810672|bibcode=1990PhT....43h..90B
}}</ref>|author= Biedenharn et al.}}

In 1977, two [[Theoretical physics|theoretical physicists]] working at the [[University of Oslo]], [[Jon Magne Leinaas]] and [[Jan Myrheim]], showed that the traditional classification of particles as either fermions or bosons would not apply if they were restricted to move in only two [[dimension]]s.<ref name=WilczekJan2006>
{{cite journal
 |url=https://physicsworld.com/a/from-electronics-to-anyonics/
 |title=From electronics to anyonics
 |first=Frank |last= Wilczek
 |quote=In the early 1980s I named the hypothetical new particles 'anyons', the idea being that anything goes – but I did not lose much sleep anticipating their discovery. Very soon afterwards, however, Bert Halperin at Harvard University found the concept of anyons useful in understanding certain aspects of the fractional quantum Hall effect, which describes the modifications that take place in electronics at low temperatures in strong magnetic fields.
 |journal= Physics World|date=January 2006|volume=19|pages=22–23|doi=10.1088/2058-7058/19/1/31| issn= 0953-8585
}}</ref> Hypothetical particles, being neither bosons nor fermions, would be expected to exhibit a diverse range of previously unexpected properties. In 1982, [[Frank Wilczek]] published two papers exploring the fractional statistics of quasiparticles in two dimensions, giving them the name "anyons" to indicate that the phase shift upon permutation can take any value.<ref name="SymMag2011">
{{cite web
 |url=https://www.symmetrymagazine.org/breaking/2011/08/31/anyons-anyone
 |title=Anyons, anyone?
 |work=Symmetry Magazine |quote=In 1982 physicist Frank Wilczek gave these interstitial particles the name anyon&nbsp;... 'Any anyon can be anything between a boson or a fermion', Keilmann says. 'Wilczek is a funny guy.'
 |date=31 August 2011
 |access-date=2020-09-24
}}</ref>

[[Daniel Tsui]] and [[Horst Störmer]] discovered the fractional quantum Hall effect in 1982. The mathematics developed by Wilczek proved to be useful to [[Bertrand Halperin]] at [[Harvard University]] in explaining aspects of it.<ref name="Halp1984">{{cite journal
 |title=Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States
 |last=Halperin |first=B. I.
 |journal=Phys. Rev. Lett.
 |volume=52 |issue=18 |pages=1583–1586
 |quote=The appearance of fractional statistics in the present context is strongly reminiscent of the fractional statistics introduced by Wilczek to describe charged particles tied to "magnetic flux tubes" in two dimensions. 
 |date=1984
 |publisher=American Physical Society
 |doi=10.1103/PhysRevLett.52.1583
 |bibcode=1984PhRvL..52.1583H
 |url=https://link.aps.org/doi/10.1103/PhysRevLett.52.1583
|url-access=subscription
 }}</ref> 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.<ref name="1989Khurana">
{{cite journal
 |last=Khurana |first=Anil
 |title=Bosons Condense and Fermions 'Exclude', But Anyons&nbsp;...?
 |url=https://doi.org/10.1063/1.2811205
 |journal=Physics Today
 |quote=In 1984, two years after Wilczek discussed this seemingly arcane possibility, Bertrand Halperin (Harvard University) suggested that the excitations in the theory of fractional quantum Hall effect discussed by Robert Laughlin (Stanford University) behave like anyons. Later Wilczek, Daniel Arovas (University of California, San Diego) and Robert Schrieffer (University of California, Santa Barbara) confirmed the idea.
 |date=7 December 2018
 |doi=10.1063/1.2811205
 |access-date=2020-11-26
|url-access=subscription
 }}</ref><ref>
{{citation |author1=Arovas |first=D. |title=Fractional Statistics and the Quantum Hall Effect |date=1984 |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.53.722 |journal=Physical Review Letters |volume=53 |page=722 |author2=Schrieffer |first2=J. R. |author3=Wilczek |first3=Frank}}</ref>

== Abelian anyons ==
In quantum mechanics, and some classical stochastic systems, [[indistinguishable particles]] have the property that exchanging the states of particle&nbsp;{{math|''i''}} with particle&nbsp;{{math|''j''}} (symbolically {{tmath|\psi_i\leftrightarrow\psi_j \text{  for } i \ne j}}) does not lead to a measurably different many-body state.

In a quantum mechanical system, for example, a system with two indistinguishable particles, with particle&nbsp;1 in state {{tmath|\psi_1}} and particle&nbsp;2 in state {{tmath|\psi_2}}, has state {{tmath|\left{{!}} \psi_1\psi_2 \right\rangle}} in [[Dirac notation]]. Now suppose we exchange the states of the two particles, then the state of the system would be {{tmath|\left{{!}} \psi_2\psi_1 \right\rangle}}. These two states should not have a measurable difference, so they should be the same vector, up to a [[phase factor]]:
: <math>\left|\psi_1\psi_2\right\rangle = e^{i\theta}\left|\psi_2\psi_1\right\rangle.</math>

Here, {{tmath|e^{i\theta} }} is the phase factor.
In space of [[three-dimensional space|three]] or more dimensions, the phase factor is {{tmath|1}} or {{tmath|-1}}. Thus, [[elementary particles]] are either fermions, whose phase factor is {{tmath|-1}}, or bosons, whose phase factor is {{tmath|1}}. These two types have different [[Quantum statistics|statistical behaviour]]. Fermions obey [[Fermi–Dirac statistics]], while bosons obey [[Bose–Einstein statistics]]. In particular, the phase factor is why fermions obey the [[Pauli exclusion principle]]: If two fermions are in the same state, then we have
: <math>\left|\psi\psi\right\rangle = -\left|\psi\psi\right\rangle.</math>

The state vector must be zero, which means it is not normalizable, thus it is unphysical.

In two-dimensional systems, however, [[quasiparticle]]s can be observed that 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
 |last1 = Leinaas |first  = Jon Magne |author-link1 = Jon Magne Leinaas
 |last2 = Myrheim |first2 = Jan       |author-link2 = Jan Myrheim
 |date = 11 January 1977
 |title = On the theory of identical particles
 |journal = Il Nuovo Cimento B
 |volume = 37 |issue = 1 |pages = 1–23
 |url = http://www.ifi.unicamp.br/~cabrera/teaching/referencia.pdf
 |doi = 10.1007/BF02727953
 |bibcode = 1977NCimB..37....1L |s2cid = 117277704
}}</ref> In the case of two particles this can be expressed as
: <math>\left|\psi_1\psi_2\right\rangle = e^{i\theta}\left|\psi_2\psi_1\right\rangle,</math>
where {{tmath|e^{i\theta} }} can be other values than just {{tmath|-1}} or {{tmath|1}}. It is important to note that there is a slight [[abuse of notation]] in this shorthand expression, as in reality this wave function can be and usually is multi-valued. This expression actually means that when particle 1 and particle 2 are interchanged in a process where each of them makes a counterclockwise half-revolution about the other, the two-particle system returns to its original quantum wave function except multiplied by the complex unit-norm phase factor {{nowrap|1=''e''{{sup|''iθ''}}}}. Conversely, a clockwise half-revolution results in multiplying the wave function by {{nowrap|1=''e''{{sup|−''iθ''}}}}. Such a theory obviously only makes sense in two-dimensions, where clockwise and counterclockwise are clearly defined directions.

In the case {{nowrap|1=''θ'' = ''π''}} we recover the Fermi–Dirac statistics ({{nowrap|1=''e''{{sup|''iπ''}} = −1}}) and in the case {{nowrap|1=''θ'' = 0}} (or {{nowrap|1=''θ'' = 2''π''}}) the Bose–Einstein statistics ({{nowrap|1=''e''{{sup|2''πi''}} = 1}}). In between we have something different. [[Frank Wilczek]] in 1982 explored the behavior of such quasiparticles and coined the term "anyon" to describe them, because they can have any phase when particles are interchanged.<ref>
{{cite journal
 |last = Wilczek |first = Frank |author-link = Frank Wilczek
 |date = 4 October 1982
 |title = Quantum Mechanics of Fractional-Spin Particles
 |journal = Physical Review Letters
 |volume = 49
 |issue = 14
 |pages = 957–959
 |quote = If there is a generalized spin-statistics connection, we must expect that the flux-tube-particle composites have unusual statistics, interpolating between bosons and fermions. Since interchange of two of these particles can give ''any'' phase, I will call them generically anyons.
 |url = http://www.ifi.unicamp.br/~mtamash/f689_mecquant_i/prl49_957.pdf
 |doi = 10.1103/PhysRevLett.49.957
 |bibcode=1982PhRvL..49..957W
}}</ref> Unlike bosons and fermions, anyons have the peculiar property that when they are interchanged twice in the same way (e.g. if anyon 1 and anyon 2 were revolved counterclockwise by half revolution about each other to switch places, and then they were revolved counterclockwise by half revolution about each other again to go back to their original places), the wave function is not necessarily the same but rather generally multiplied by some complex phase (by {{nowrap|1=''e''{{sup|2''iθ''}}}} in this example).

We may also use {{nowrap|1=''θ'' = 2''πs''}} with particle [[Spin (physics)#Pauli exclusion principle|spin]] quantum number ''s'', with ''s'' being [[integer]] for bosons, [[half-integer]] for fermions, so that
: <math>e^{i\theta} = e^{2i \pi s} = (-1)^{2s},</math> 
or
: <math>|\psi_1 \psi_2\rangle = (-1)^{2s} |\psi_2 \psi_1\rangle.</math>

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.

In a three-dimensional position space, the fermion and boson statistics operators (−1 and +1 respectively) are just 1-dimensional representations of the [[permutation group]] (S<sub>''N''</sub> of ''N'' indistinguishable particles) acting on the space of wave functions. In the same way, in two-dimensional position space, the abelian anyonic statistics operators ({{nowrap|1=''e''{{sup|''iθ''}}}}) are just 1-dimensional representations of the [[Braid group#Representations|braid group]] (''B<sub>N</sub>'' of ''N'' indistinguishable particles) acting on the space of wave functions. Non-abelian anyonic statistics are higher-dimensional representations of the braid group. Anyonic statistics must not be confused with [[parastatistics]], which describes statistics of particles whose wavefunctions are higher-dimensional representations of the permutation group.<ref name=Khare2005>{{cite book |title=Fractional Statistics and Quantum Theory |last=Khare |first=Avinash |year=2005 |publisher=World Scientific |isbn=978-981-256-160-2 |url=https://books.google.com/books?id=rBi7zTpvjaAC&q=Fractional+Statistics+and+Quantum+Theory}}</ref>{{rp|22}}

=== Topological equivalence ===
The fact that the [[homotopy class]]es of paths (i.e. notion of [[equivalence principle|equivalence]] on [[braids]]) are relevant hints at a more subtle insight. It arises from the [[Feynman path integral]], in which all paths from an initial to final point in [[spacetime]] contribute with an appropriate [[phase factor]]. The [[Feynman path integral]] can be motivated from expanding the propagator using a method called time-slicing,<ref>{{cite book |last1=Lancaster |first1=Tom |last2=Blundell |first2=Stephen J. |title=Quantum Field Theory for the Gifted Amateur |publisher=Oxford University Press |isbn=978-0-19-969932-2 |date=17 June 2014}}</ref> in which time is discretized.

In non-homotopic paths, one cannot get from any point at one time slice to any other point at the next time slice. This means that we can consider [[homotopic]] equivalence class of paths to have different weighting factors.<ref>{{cite book |last1=Schulman |first1=L. S. |title=Techniques and Applications of Path Integration |publisher=Dover Publications |isbn=0-471-76450-7 |date=February 1981}}</ref>

So it can be seen that the [[topological]] notion of equivalence comes from a study of the [[Feynman path integral]].<ref name=Khare2005 />{{Rp|28}}

For a more transparent way of seeing that the homotopic notion of equivalence is the "right" one to use, see [[Aharonov–Bohm effect]].

=== Experiment ===
[[File:Laughlin quasiparticle interferometer sample scanning electron micrograph.svg|upright=1.6|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 junctions]], the yellow dots are [[Ohmic contact]]s. The electrons in the device are confined to a 2d plane.<ref name=FractStat>
{{cite journal
 |doi=10.1103/PhysRevB.72.075342
 |title=Realization of a Laughlin quasiparticle interferometer: Observation of fractional statistics
 |date=17 August 2005
 |last1=Camino |first1=Fernando E.
 |last2=Zhou |first2=Wei
 |last3=Goldman |first3=Vladimir J.
 |journal=Physical Review B
 |volume=72 |issue=7 |page=075342
 |arxiv=cond-mat/0502406
 |bibcode=2005PhRvB..72g5342C
 |s2cid=52245802
 |url=http://quantum.physics.sunysb.edu/MyPapers/PDF/prbFracStat.pdf
 |url-status=dead
 |archive-url=https://web.archive.org/web/20150619152736/http://quantum.physics.sunysb.edu/MyPapers/PDF/prbFracStat.pdf
 |archive-date=2015-06-19
}}, see ''fig. 2.B''.</ref>]]

In 2020, two teams of scientists (one in Paris, the other at Purdue) announced new experimental evidence for the existence of anyons. Both experiments were featured in ''[[Discover Magazine]]''{{'}}s 2020 annual "state of science" issue.<ref name="DISC-20201212"/>

In April, 2020, researchers from the [[École normale supérieure (Paris)]] and the [[Centre for Nanosciences and Nanotechnologies]] (C2N) reported results from a tiny "particle collider" for anyons. They detected properties that matched predictions by theory for anyons.<ref name="Bartolomei2020">
{{citation
 |author1=Bartolomei, H.
 |author2=Kumar, M.
 |author3=Bisognin, R.
 |display-authors=etal
 |url=https://doi.org/10.1126/science.aaz5601
 |title=Fractional statistics in anyon collisions
 |quote=Elementary particles in three dimensions are either bosons or fermions, depending on their spin. In two dimensions, it is in principle possible to have particles that lie somewhere in between, but detecting the statistics of these so-called anyons directly is tricky.
 |journal=[[Science (journal)|Science]]
 |date=10 April 2020
 |volume=368 |issue=6487 |pages=173–177
}}</ref><ref name="Paris">
{{cite web
 |last=Yirka |first=Bob
 |date=10 April 2020
 |url=https://phys.org/news/2020-04-anyon-evidence-tiny-collider.html
 |title=Anyon evidence observed using tiny anyon collider
 |access-date=2020-12-12
 |quote=The work involved creating a very tiny 2-D anyon collider—so small they had to use an electron microscope to observe the action inside of it. The collider consisted of a 2-D plane set between another layered material. More specifically, the collider held a quantum Hall liquid that was kept inside of a strong magnetic field.
 |publisher=Phys.org
}}</ref><ref name="QMag2020">
{{cite web
 |last=Najjar |first=Dana
 |date=12 May 2020
 |url=https://www.quantamagazine.org/milestone-evidence-for-anyons-a-third-kingdom-of-particles-20200512/
 |title='Milestone' Evidence for Anyons, a Third Kingdom of Particles
 |work=Quanta Magazine
 |quote=In 2016, three physicists described an experimental setup that resembles a tiny particle collider in two dimensions. Fève and his colleagues built something similar and used it to smash anyons together. By measuring the fluctuations of the currents in the collider, they were able to show that the behavior of the anyons corresponds exactly with theoretical predictions.
 |access-date=2020-12-12
}}</ref>

In July, 2020, scientists at Purdue University detected anyons using a different setup. The team's interferometer routes the electrons through a specific maze-like etched nanostructure made of [[gallium arsenide]] and [[aluminium gallium arsenide]]. "In the case of our anyons the phase generated by braiding was 2π/3", he said. "That's different than what's been seen in nature before."<ref>{{cite news |last1=Tally |first1=Steve |title=New evidence that the quantum world is even stranger than we thought |quote=One characteristic difference between fermions and bosons is how the particles act when they are looped, or braided, around each other. Fermions respond in one straightforward way, and bosons in another expected and straightforward way. Anyons respond as if they have a fractional charge, and even more interestingly, create a nontrivial phase change as they braid around one another. This can give the anyons a type of "memory" of their interaction. |url=https://phys.org/news/2020-09-evidence-quantum-world-stranger-thought.html |publisher=Phys.org |date=4 September 2020}}</ref><ref name="Nakamura2020">{{cite journal |last1=Nakamura |first1=J. |last2=Liang |first2=S. |last3=Gardner |first3=G. C. |last4=Manfra |first4=M. J. |title=Direct observation of anyonic braiding statistics |journal=Nature Physics |date=September 2020 |volume=16 |issue=9 |pages=931–936 |doi=10.1038/s41567-020-1019-1 |bibcode=2020NatPh..16..931N |url=https://www.nature.com/articles/s41567-020-1019-1 |language=en |issn=1745-2481 |arxiv=2006.14115 |s2cid=220055512 |quote=Anyons are quasiparticles that, unlike fermions and bosons, show fractional statistics when two of them are exchanged.}}</ref>

As of 2023, this remains an active area of research; using a superconducting processor, Google Quantum AI reported on the first braiding of non-Abelian anyon-like particles in an arXiv article by Andersen ''et al.'' in October 2022,<ref>
{{cite arXiv |eprint=2210.10255 |class=quant-ph
 |last1=Andersen |first1=Trond I.
 |last2=Lensky |first2=Yuri D.
 |last3=Kechedzhi |first3=Kostyantyn
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 |last7=Morvan |first7=Alexis
 |last8=Mi |first8=Xiao
 |last9=Opremcak |first9=Alex
 |last10=Acharya |first10=Rajeev
 |last11=Allen |first11=Richard
 |last12=Ansmann |first12=Markus
 |last13=Arute |first13=Frank
 |last14=Arya |first14=Kunal
 |last15=Asfaw |first15=Abraham
 |last16=Atalaya |first16=Juan
 |last17=Babbush |first17=Ryan
 |last18=Bacon |first18=Dave
 |last19=Bardin |first19=Joseph C.
 |last20=Bortoli |first20=Gina
 |last21=Bourassa |first21=Alexandre
 |last22=Bovaird |first22=Jenna
 |last23=Brill |first23=Leon
 |last24=Broughton |first24=Michael
 |last25=Buckley |first25=Bob B.
 |last26=Buell |first26=David A.
 |last27=Burger |first27=Tim
 |last28=Burkett |first28=Brian
 |last29=Bushnell |first29=Nicholas
 |last30=Chen |first30=Zijun
 |display-authors=1
 |date=19 Oct 2022
 |title=Observation of non-Abelian exchange statistics on a superconducting processor
}}</ref> later published in Nature.<ref name="non-abelian2">
{{cite web
 |date=11 May 2023
 |title=Non-Abelian braiding of graph vertices in a superconducting processor
 |url=https://www.nature.com/articles/s41586-023-05954-4
 |access-date=2023-05-17
 |publisher=Nature
}}</ref> In an arXiv article released in May 2023, Quantinuum reported on non-abelian braiding using a trapped-ion processor.<ref name="non-abelian">
{{cite arXiv |eprint=2305.03766 |class=quant-ph
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 |last17=Vishwanath |first17=Ashvin
 |last18=Dreyer |first18=Henrik
 |title=Creation of Non-Abelian Topological Order and Anyons on a Trapped-Ion Processors
 |date=7 May 2023
 |quote=Non-Abelian topological order (TO) is a coveted state of matter with remarkable properties, including quasiparticles that can remember the sequence in which they are exchanged. These anyonic excitations are promising building blocks of fault-tolerant quantum computers. However, despite extensive efforts, non-Abelian TO and its excitations have remained elusive, unlike the simpler quasiparticles or defects in Abelian TO. In this work, we present the first unambiguous realization of non-Abelian TO and demonstrate control of its anyons.
}}</ref>

In May, 2025, researchers at the [[University of Innsbruck]] have observed anyons in a one-dimensional quantum system.<ref>{{Cite journal |last=Dhar |first=Sudipta |last2=Wang |first2=Botao |last3=Horvath |first3=Milena |last4=Vashisht |first4=Amit |last5=Zeng |first5=Yi |last6=Zvonarev |first6=Mikhail B. |last7=Goldman |first7=Nathan |last8=Guo |first8=Yanliang |last9=Landini |first9=Manuele |last10=Nägerl |first10=Hanns-Christoph |date=2025-05-28 |title=Observing anyonization of bosons in a quantum gas |url=https://www.nature.com/articles/s41586-025-09016-9 |journal=Nature |language=en |pages=1–5 |doi=10.1038/s41586-025-09016-9 |issn=1476-4687|doi-access=free |pmc=12137132 }}</ref>

== Non-abelian anyons ==
{{Unsolved|physics|Is [[topological order]] stable at non-zero [[temperature]]?}}
In 1988, [[Jürg Fröhlich]] showed that it was valid under the [[spin–statistics theorem]] for the particle exchange to be monoidal (non-abelian statistics).<ref>
{{cite book
 |last=Fröhlich |first=Jürg
 |chapter=Statistics of Fields, the Yang–Baxter Equation, and the Theory of Knots and Links
 |title=Nonperturbative Quantum Field Theory
 |series=NATO ASI Series
 |location=New York |publisher=Springer
 |year=1988
 |volume=185 |pages=71–100
 |isbn=1-4612-8053-2 |doi=10.1007/978-1-4613-0729-7_4
}}</ref> In particular, this can be achieved when the system exhibits some degeneracy, so that multiple distinct states of the system have the same configuration of particles. Then an exchange of particles can not only contribute a phase change, but also can send the system into a different state with the same particle configuration. Particle exchange then corresponds to a linear transformation on this subspace of degenerate states. When there is no degeneracy, this subspace is one-dimensional and so all such linear transformations commute (because they are just multiplications by a phase factor). When there is degeneracy and this subspace has higher dimension, then these linear transformations need not commute (just as matrix multiplication does not).

[[Greg Moore (physicist)|Gregory Moore]], [[Nicholas Read]], and [[Xiao-Gang Wen]] pointed out that non-Abelian statistics can be realized in the [[fractional quantum Hall effect]] (FQHE).<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
 |date=19 August 1991
 |last1=Moore |first1=Gregory |author-link1=Greg Moore (physicist)
 |last2=Read |first2=Nicholas |author-link2=Nicholas Read
 |journal=Nuclear Physics B
 |volume=360 |issue=2–3 |pages=362–396
 |bibcode=1991NuPhB.360..362M
}}</ref><ref>
{{cite journal |last=Wen |first=Xiao-Gang |author-link=Xiao-Gang Wen |date=11 February 1991 |title=Non-Abelian statistics in the fractional quantum Hall states |url=http://dao.mit.edu/~wen/pub/nab.pdf |url-status=dead |journal=Physical Review Letters |volume=66 |issue=6 |pages=802–805 |bibcode=1991PhRvL..66..802W |doi=10.1103/PhysRevLett.66.802 |pmid=10043904 |archive-url=https://web.archive.org/web/20150326181929/http://dao.mit.edu/~wen/pub/nab.pdf |archive-date=2015-03-26}}</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&nbsp;=&nbsp;5/2 FQHE state.{{Update inline|date=May 2023|reason=See new info in page introduction re: Google Quantum AI and Quantinuum 2022-2023 research.}}<ref>
{{cite journal |last1=Stern |first1=Ady |year=2010 |title=Non-Abelian states of matter |journal=Nature |volume=464 |issue=7286 |pages=187–193 |bibcode=2010Natur.464..187S |doi=10.1038/nature08915 |pmid=20220836 |s2cid=4362827}}</ref><ref>
{{cite arXiv |eprint=1112.3400
 |first1=Sanghun |last1=An
 |first2=P. |last2=Jiang
 |first3=H. |last3=Choi
 |first4=W. |last4=Kang
 |first5=S. H. |last5=Simon
 |first6=L. N. |last6=Pfeiffer
 |first7=K. W. |last7=West
 |first8=K. W. |last8=Baldwin
 |title=Braiding of Abelian and Non-Abelian Anyons in the Fractional Quantum Hall Effect
 |class=cond-mat.mes-hall
 |date=15 December 2011
}}</ref> Experimental evidence of non-abelian anyons, although not yet conclusive and currently contested,<ref>
{{cite journal
 |title=Enhanced Bulk-Edge Coulomb Coupling in Fractional Fabry-Perot Interferometers
 |first1=Curt |last1=von Keyserling
 |first2=S. H. |last2=Simon
 |first3=Rosenow |last3=Bernd
 |volume=115
 |journal=Physical Review Letters
 |arxiv = 1411.4654
 |year=2015 |issue=12 |page=126807
 |doi=10.1103/PhysRevLett.115.126807 |bibcode=2015PhRvL.115l6807V |pmid=26431008 |s2cid=20103218
}}</ref> was presented in October, 2013.{{Update inline|date=May 2023}}<ref>
{{cite journal |author1=Willett |first=R. L. |author2=Nayak |first2=C. |author3=Pfeiffer |first3=L. N. |author4=West |first4=K. W. |date=12 January 2013 |title=Magnetic field-tuned Aharonov–Bohm oscillations and evidence for non-Abelian anyons at ν = 5/2 |journal=Physical Review Letters |volume=111 |issue=18 |page=186401 |arxiv=1301.2639 |bibcode=2013PhRvL.111r6401W |doi=10.1103/PhysRevLett.111.186401 |pmid=24237543 |s2cid=22780228}}</ref> Recent works claim the creation of non-abelian topological order and anyons on a trapped-ion processor <ref name="non-abelian" /> and demonstration of non-abelian braiding of graph vertices in a superconducting processor.<ref name="non-abelian2" />

== Fusion of anyons ==
In much the same way that two fermions (e.g. both of spin 1/2) can be looked at together as a composite boson (with total spin in a [[Clebsch–Gordan coefficients|superposition]] of 0 and 1), two or more anyons together make up a composite anyon (possibly a boson or fermion). The composite anyon is said to be the result of the [[fusion of anyons|fusion]] of its components.

If {{tmath|N}} identical abelian anyons each with individual statistics {{tmath|\alpha}} (that is, the system picks up a phase {{tmath|e^{i \alpha} }} when two individual anyons undergo adiabatic counterclockwise exchange) all fuse together, they together have statistics {{tmath|N^2 \alpha}}. This can be seen by noting that upon counterclockwise rotation of two composite anyons about each other, there are {{tmath|N^2}} pairs of individual anyons (one in the first composite anyon, one in the second composite anyon) that each contribute a phase {{tmath|e^{i \alpha} }}. An analogous analysis applies to the fusion of non-identical abelian anyons. The statistics of the composite anyon is uniquely determined by the statistics of its components.

Non-abelian anyons have more complicated fusion relations. As a rule, in a system with non-abelian anyons, there is a composite particle whose statistics label is not uniquely determined by the statistics labels of its components, but rather exists as a quantum superposition (this is completely analogous to how two fermions known to have spin 1/2 are together in quantum superposition of total spin 1 and 0). If the overall statistics of the fusion of all of several anyons is known, there is still ambiguity in the fusion of some subsets of those anyons, and each possibility is a unique quantum state. These multiple states provide a [[Hilbert space]] on which quantum computation can be done.<ref>
{{cite journal |author1=Nayak |first=C. |author2=Simon |first2=S. H. |author3=Stern |first3=A. |author4=Freedman |first4=M. |author5=Das Sarma |first5=S. |date=28 March 2008 |title=Non-Abelian Anyons and Topological Quantum Computation |journal=Reviews of Modern Physics |volume=80 |issue=3 |pages=1083–1159 |arxiv=0707.1889 |bibcode=2008RvMP...80.1083N |doi=10.1103/RevModPhys.80.1083 |s2cid=119628297}}</ref>

== Topological basis ==

{{multiple image
| 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).
| width             = 150
| image1            = Particle exchange 2d anticlockwise.gif
| caption1          = Anticlockwise rotation
| image2            = Particle exchange 2d clockwise.gif
| caption2          = Clockwise rotation
}}

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>&nbsp;>&nbsp;2, the [[Lie group]]s [[orthogonal group|SO(<var>d</var>,1)]] (which generalizes the [[Lorentz group]]) and [[Poincaré group|Poincaré(<var>d</var>,1)]] have '''Z'''<sub>2</sub> as their [[first homotopy group]]. Because the [[cyclic group]] '''Z'''<sub>2</sub> is composed of two elements, only two possibilities remain. (The details are more involved than that, but this is the crucial point.)

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). Anyons are evenly complementary representations of spin polarization by a charged particle.

This concept also applies to nonrelativistic systems. The relevant part here is that the spatial rotation group SO(2) has an infinite first homotopy group.

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.

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 [[quantum logic gate]]s.<ref>
{{cite journal |last=Freedman |first=Michael |author2=Kitaev |first2=Alexei |author3=Larsen |first3=Michael |author4=Wang |first4=Zhenghan |date=20 October 2002 |title=Topological Quantum Computation |journal=Bulletin of the American Mathematical Society |volume=40 |issue=1 |pages=31–38 |arxiv=quant-ph/0101025 |doi=10.1090/S0273-0979-02-00964-3}}</ref><ref>
{{cite journal
 |last=Monroe |first= Don
 |url=https://www.newscientist.com/channel/fundamentals/mg20026761.700-anyons-the-breakthrough-quantum-computing-needs.html
 |title=Anyons: The breakthrough quantum computing needs?
 |journal=[[New Scientist]]
 |issue=2676
 |date=1 October 2008
}}</ref>

== Generalization to higher dimensions ==
Fractionalized excitations as point particles can be bosons, fermions or anyons in 2+1 spacetime dimensions. It is known that point particles can be only either bosons or fermions in 3+1 and higher spacetime dimensions. However, the loop- (or string-) or membrane-like excitations are extended objects that can have fractionalized statistics.

Current research shows that the loop- and string-like excitations exist for [[topological order]]s in the 3+1 dimensional spacetime, and their multi-loop/string-braiding statistics are the key signatures for identifying 3+1{{nbh}}dimensional topological orders.<ref name="1403.7437">
{{cite journal
 |last1=Wang | first1=Chenjie
 |last2=Levin | first2=Michael
 |date=22 August 2014
 |title=Braiding statistics of loop excitations in three dimensions
 |journal=Physical Review Letters
 |volume=113 | issue=8
 |page= 080403
 |issn= 1079-7114 |doi=10.1103/PhysRevLett.113.080403
 |arxiv=1403.7437 |pmid=25192079 |bibcode=2014PhRvL.113h0403W |s2cid=23104804
}}</ref><ref name="1404.7854">
{{cite journal
 |last1=Wang | first1=Juven
 |last2=Wen | first2=Xiao-Gang
 |title=Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory
 |journal=Physical Review B
 |volume=91 |issue=3
 |page=035134
 |date=15 January 2015
 |issn= 2469-9969 |doi=10.1103/PhysRevB.91.035134
 |arxiv=1404.7854 |s2cid=13893760
}}</ref><ref name="1612.09298">
{{cite journal
 |arxiv=1612.09298
 |last1=Putrov | first1=Pavel
 |last2=Wang | first2=Juven
 |last3=Yau | first3=Shing-Tung
 |title=Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions
 |doi=10.1016/j.aop.2017.06.019
 |volume=384C
 |journal=Annals of Physics
 |pages=254–287
 |bibcode=2017AnPhy.384..254P
 |date=September 2017
 |s2cid=119578849
}}</ref> The multi-loop/string-braiding statistics of 3+1{{nbh}}dimensional topological orders can be captured by the link invariants of particular [[topological quantum field theory|topological quantum field theories]] in 4 spacetime dimensions.<ref name="1612.09298" /> Explained in a colloquial manner, the extended objects (loop, string, or membrane, etc.) can be potentially anyonic in 3+1 and higher spacetime dimensions in the long-range [[Quantum entanglement|entangled systems]].

== See also ==
{{wiktionary}}
* {{annotated link|Anyonic Lie algebra}}
* {{annotated link|Flux tube}}
* {{annotated link|Ginzburg–Landau theory}}
* {{annotated link|Husimi Q representation}}
* {{annotated link|Josephson effect}}
* {{annotated link|Macroscopic quantum phenomena}}
* {{annotated link|Magnetic domain}}
* {{annotated link|Magnetic flux quantum}}
* {{annotated link|Meissner effect}}
* {{annotated link|Plekton}}
* {{annotated link|Quantum vortex}}
* {{annotated link|Random matrix}}
* {{annotated link|Topological defect}}
* {{annotated link|Topological quantum computer|Topological quantum computing}}

== References ==
{{reflist|25em}}

== Further reading ==
{{refbegin}}
* {{cite journal |doi=10.1103/RevModPhys.80.1083 |arxiv=0707.1889 |title=Non-Abelian anyons and topological quantum computation |year=2008 |last1=Nayak |first1=Chetan |last2=Simon |first2=Steven H. |last3=Stern |first3=Ady |last4=Freedman |first4=Michael |last5=Das Sarma |first5=Sankar |journal=Reviews of Modern Physics |volume=80 |issue=3 |pages=1083 |bibcode=2008RvMP...80.1083N|s2cid=119628297 }}
* {{cite journal |doi=10.1103/PhysRevB.65.165113 |url=http://dao.mit.edu/~wen/pub/qosl.pdf |title=Quantum orders and symmetric spin liquids |date=15 April 2002 |last1=Wen |first1=Xiao-Gang |journal=Physical Review B |volume=65 |issue=16 |pages=165113 |arxiv=cond-mat/0107071 |bibcode=2002PhRvB..65p5113W|s2cid=119061254 |archive-url=https://web.archive.org/web/20110609204800/http://dao.mit.edu/~wen/pub/qosl.pdf |archive-date=9 June 2011 |url-status=dead }}
* {{cite journal |doi=10.1016/j.aop.2007.10.008|arxiv=0711.4697 |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 |issue=1 |pages=204–249 |bibcode=2008AnPhy.323..204S |s2cid=15582782 }}
* {{cite journal |url=https://www.quantamagazine.org/milestone-evidence-for-anyons-a-third-kingdom-of-particles-20200512/ |title='Milestone' Evidence for Anyons, a Third Kingdom of Particles |year=2020 |last1=Najjar |first1=Dana |journal=Quanta Magazine }}
{{refend}}

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[[Category:Parastatistics]]
[[Category:Representation theory of Lie groups]]