Anyon
Template:Statistics (stat. mech.) In mathematics and physics, an anyon is a type of projective representation of a Lie group.
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.
Topological basis
The topological reason behind the phenomenon is this: 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. On the other hand, for n > 2, for SO(n,1) and Poincaré(n,1), it's only Z2 (cyclic of order 2); meaning that the spin group is simply connected.
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 is no longer the symmetric group (n!-dimensional) but rather the Braid group (infinite dimensional).
In physics
This mathematical concept becomes useful in the physics of two-dimensional systems such as sheets of graphite or the quantum Hall effect. In space of three dimensions (or more), elementary particles have tightly constrained quantum numbers and, in particular, are restricted to being fermions or bosons. In two-dimensional systems, however, quasiparticles could be observed whose quantum states range continuously between fermionic and bosonic, taking on any quantum value in between. Frank Wilczek coined the term "anyons"[1] to describe such particles, since they can have "any" phase when particles are interchanged.
Let's say we have two identical particles on a plane. If we interchange both particles so that each particle travels counterclockwise for half a cycle around the center of both particles, the wave function of the system changes by a factor of where θ is an angle which only depends upon the type of particle in question. If θ is zero, we have a boson and if θ is π we have a fermion, and in 3 dimensions or more, this is the only consistent solution. For any other value, we have an anyon, and in 1977, Jon Magne Leinaas and Jan Myrheim of the University of Oslo showed[2] this to be consistent in 2 dimensions. If we have two particles and , which may or may not be identical, then their mutual statistics is the change in the phase factor, which is picked up after particle is rotated counterclockwise around particle for one full cycle. The mutual statistics may be completely unrelated to the interchange angle between two identical particles.
