Anyon
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| Maxwell-Boltzmann statistics |
| Bose-Einstein statistics |
| Fermi-Dirac statistics |
| Parastatistics |
| Anyonic statistics |
| Braid statistics |
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In mathematics and physics, an anyon is a type of particle that only occurs in two-dimensional systems. It is a generalization of the Fermion and Boson concept.
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[edit] In physics
This mathematical concept becomes useful in the physics of two-dimensional systems such as sheets of graphene or the quantum Hall effect.
In space of three or more dimensions, particles are restricted to being fermions or bosons, according to their statistical behaviour. Fermions respect the so-called Fermi-Dirac statistics while Bosons respect the Bose-Einstein statistics. In the language of quantum physics 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 not possible).
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[1]. In our above example of two particles this looks as follows:
With "i" being the imaginary unit from the calculus of complex numbers and θ a real number. Recall that | eiθ | = 1 and e2iπ = 1 as well as eiπ = − 1. So in the case θ = π we recover the Fermi-Dirac statistics (minus sign) and in the case θ = 2π the Bose-Einstein statistics (plus sign). In between we have something different. For these types of particles Frank Wilczek coined the term "anyons"[2] to describe such particles, since they can have "any" phase when particles are interchanged.
[edit] Topological basis
In dimensions greater than two, the spin-statistics connection states that any multiparticle state has to obey are either Bose-Einstein or Fermi-Dirac statistics. This is related to the first homotopy group of SO(n,1) (and also Poincaré(n,1)) with n>2, which is Z2 (the cyclic group consisting of 2 Elements). 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 don't arise from linear representations of SO(2,1), or of its double cover, the spin group Spin(2,1). These representations are called anyons.
Actually, this concept also applies to nonrelativistic systems. The relevant part here is that the spatial rotation group is SO(2), which has an infinite first homotopy group.
This fact is also related to the Braid group well known in Knot theory. The relation can be understood when one considers the fact that in 2 Dimensions the group of permutations of 2 particles is no longer the symmetric group S2 (2-dimensional) but rather the Braid group B2 (infinite dimensional).
[edit] References
- ^ J.M.Leinaas, and J.Myrheim, "On the theory of identical particles", Nuovo Cimento B37, 1-23 (1977).
- ^ F.Wilczek, Phys.Rev.Lett. 49, 957 (1982).
