In physics, an anyon is a type of particle that occurs only in two-dimensional systems, with properties much less restricted than fermions and bosons; the operation of 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.
In space of three or more dimensions, indistinguishable particles are either fermions or bosons, according to their 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):
(where the first entry in |...⟩ 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 ψ1 and particle 2 in state ψ2"). 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).
In two-dimensional systems, however, quasiparticles 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. In our above example of two particles this looks as follows:
with i the imaginary unit and θ a real number. Now |eiθ| = 1, e2πi = 1, and eπi = −1. So in the case θ = π we recover the Fermi–Dirac statistics (minus sign) and in the case θ = 0 (or θ = 2π) the Bose–Einstein statistics (plus sign). In between we have something different. Frank Wilczek coined the term "anyon" to describe such particles, since they can have any phase when particles are interchanged.
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.
Just as the fermion and boson wavefunctions in a three-dimensional space are just 1-dimensional representations of the permutation group (SN of N indistinguishable particles), the anyonic wavefunctions in a two-dimensional space are just 1-dimensional representations of the braid group (BN 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.
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.
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. However, these experiments remain controversial and are not fully accepted by the community.
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.
|Is topological order stable at non-zero temperature?|
In 1988, Jürg Fröhlich[verification needed] 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.  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.
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 d > 2, the group SO(d,1) (which generalize the Lorentz group), and also Poincaré(d,1), have Z2 as their first homotopy group. The cyclic group consisting of two elements, Z2, therefore 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 cover: it is not simply connected. In detail, there are projective representations of the special orthogonal group SO(2,1) which do not arise from linear representations of SO(2,1), or of its double cover, the spin group Spin(2,1). These representations[clarification needed] are called anyons.
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.
This fact is also related to the braid groups 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 S2 (with two elements) but rather the braid group B2 (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 computing is to create a topological quantum computer with anyons, quasi-particles used as threads and relying on braid theory to form stable logic gates.
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