Anyon 367654 221329725 2008-06-24T01:29:52Z Yahya Abdal-Aziz 313039 /* Topological basis - grammar and capitalisation */ {{Statistics (stat. mech.)}} 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. ==In physics== This mathematical concept becomes useful in the physics of two-[[dimension]]al systems such as sheets of [[graphene]] or the [[quantum Hall effect]]. In space of three or more dimensions, [[particle]]s are restricted to being [[fermion|fermions]] or [[boson|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]]): <math>\left|\psi_1\psi_2\right\rangle = \pm\left|\psi_2\psi_1\right\rangle</math> (where the first entry in <math>\left|\dots\right\rangle</math> 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 <math>\psi_1</math> and particle 2 in state <math>\psi_2</math>") 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 [[continuous function|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>J.M.Leinaas, and J.Myrheim, "On the theory of identical particles", Nuovo Cimento '''B37''', 1-23 (1977).</ref>. In our above example of two particles this looks as follows: <math>\left|\psi_1\psi_2\right\rangle = e^{i\,\theta}\left|\psi_2\psi_1\right\rangle</math> With "i" being the [[imaginary unit]] from the calculus of [[complex numbers]] and <math>\theta</math> a [[real numbers|real number]]. Recall that <math>|e^{i\theta}|=1</math> and <math>e^{2i\pi}=1</math> as well as <math>e^{i\pi}=-1</math>. So in the case <math>\theta=\pi</math> we recover the Fermi-Dirac statistics (minus sign) and in the case <math>\theta=2\pi</math> the Bose-Einstein statistics (plus sign). In between we have something different. For these types of particles [[Frank Wilczek]] coined the term '''"anyons"'''<ref>F.Wilczek, Phys.Rev.Lett. '''49''', 957 (1982).</ref> to describe such particles, since they can have "any" phase when particles are interchanged. ==Topological basis== In dimensions greater than two, the [[Spin-statistics theorem|spin-statistics connection]] states that any multiparticle state has to obey either Bose-Einstein or Fermi-Dirac statistics. This is related to the [[first homotopy group]] of SO(n,1) (and also [[Poincaré group|Poincaré(n,1)]]) with n>2, which is <math>\mathrm{Z}_2</math> (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 group|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 don't arise from [[linear representation]]s 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]] <math>S_2</math> (2-dimensional) but rather the Braid group <math>B_2</math> (infinite dimensional). ==References== <references /> ==See also == * [[plekton]] * [[fractional quantum hall effect]] ==External links== * [http://www.sciencewatch.com/interviews/frank_wilczek1.htm Interview with Frank Wilczek on anyons and superconductivity] [[Category:Quantum field theory]] [[Category:Representation theory of Lie groups]] [[de:Anyon]] [[pl:Anyon]]