Fractional Statistics

Abstract

The quantum-mechanical description of assemblies of particles whose motion is confined to two (or one) spatial dimensions offers many possibilities that are distinct from bosons and fermions. We call such particles anyons. The simplest anyons are parameterized by an angular phase parameter θ (i.e., a complex number e iθ of unit magnitude). θ = 0, π correspond to bosons and fermions respectively; at intermediate values we say that we have fractional statistics. In two dimensions, θ describes the phase acquired by the wave function as two anyons wind around one another counterclockwise. It generates a shift in the allowed values for the relative angular momentum. Composites of localized electric charge and magnetic flux associated with an abelian U(1) gauge group (or a discrete subgroup thereof) realize this behavior. More complex charge-flux constructions can involve non-abelian and product groups acting on a spectrum of allowed charges and fluxes, giving rise to nonabelian and mutual statistics. Interchanges of non-abelian anyons implement unitary transformations of the wave function within an emergent space of internal states. Mutual statistics associates non-trivial transformations with winding of distinguishable particles. Anyons of all kinds are described by quantum field theories that include Chern-Simons terms. The crossings of one-dimensional anyons on a ring are uni-directional, such that a fractional phase θ acquired upon interchange gives rise to fractional shifts in the relative momenta between the anyons. The quasiparticle excitations of fractional quantum Hall states have long been predicted to include anyons. Recently the anyon behavior predicted for quasiparticles in the ν = 1/3 fractional quantum Hall state has been observed both in scattering and in interferometric experiments. The latter demonstrate the defining feature of anyons, i.e. phase accumulation by braiding, quite directly. Anyons are also predicted to occur in spin liquids and other hypothetical states of matter. Excitations within designed systems, notably including superconducting circuits, can exhibit anyon behavior. Such systems are being developed for possible use in quantum information processing.