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In two and one spatial dimensions, we have the possibility of anyon statistics, that is, quantum statistics of identical particles which interpolates between the canonical choices of fermions and bosons. The mathematical reason is that in a path integral formulation, the group describing interchanges is not the permutation, but the braid group, which, in two dimensions, has a continuum of one-dimensional representation labeled by a U(1) phase parameter θ.
Physically, this statistical parameter describes the phase acquired by the wave function as we interchange two anyons by winding them counterclockwise around each other. This phase leads to a fractional shift in the allowed values for the relative angular momenta.
Anyons can be realized by composites consisting of electric charge and infinitesimally thin magnetic flux tubes. Restrictions for fractional statistics on closed surfaces can be traced back to
Dirac’s monopole condition. In field theory, statistical transmutations, and in particular anyon statistics, can be implemented very naturally by a Chern-Simons term in a fictitious gauge field, which attaches both charge and flux tubes to the particles.
The quasi particle excitations of fractionally quantized Hall states obey anyon statistics. In one dimension, the crossings of anyons on a ring are always uni-directional, such that a fractional phase θ acquired upon interchange gives rise to fractional shifts in the relative momenta between the anyons. In non-Abelian generalizations, anyons span an internal space of (degenerate) states and transform under higher dimensional representations of the braid group as we wind them around each other. Since the internal state vector is insensitive to local perturbations, non-Abelian anyons may prove instrumental in the construction of fault-tolerant quantum computers.
