Anyon

In mathematics and physics, an anyon is a type of projective representation of a Lie group.

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In detail, there are projective representations of SO(2,1) which don't arise from linear representations of SO(2,1), or of its double cover, Spin(2,1). These representations are called anyons.

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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.

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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.

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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 are observed whose quantum states range continuously between fermionic and bosonic, taking on any quantum value in between. Frank Wilczek coined the term "anyons" in 1982 to describe such particles.

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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 e^{i heta} 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. For any other value, we have an anyon. If we have two particles a and b, which may or may not be identical, then their mutual statistics is the change in the phase factor, e^{i heta_{ab}} which is picked up after particle b is rotated counterclockwise around particle a for one full cycle. The mutual statistics may be completely unrelated to the interchange angle between two identical particles.

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Physics: Physics (from the Greek, φυσικός (phusikos), "natural", and φύσις (phusis), "nature") is the science of the natural world in the broadest sense, dealing with matter and energy and the fundamental forces of nature that govern the interaction...

Projective representation: In mathematics, in particular in group theory, if G is a group and ρ is a vector space over a field K, then a projective representation is a homomorphism from G to Aut(ρ)/Kx where Kx here is the normal subgroup of Aut(ρ) consisting of multiplications of vectors in ρ by nonzero elemen...

Lie group: In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. Lie groups are important in mathematical analysis, physics and geometry because they serve to describe the symmetry of analytical st...