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Physical Review Letters

Quantum Mechanics of Fractional-Spin Particles

Frank A. Wilczek

Published Oct 4, 1982• 1,693 citations
Publisher

Abstract

Composites formed from charged particles and vortices in (2+1)-dimensional models, or flux tubes in three-dimensional models, can have any (fractional) angular momentum. The statistics of these objects, like their spin, interpolates continuously between the usual boson and fermion cases. How this works for two-particle quantum mechanics is discussed here.

DOI
10.1103/physrevlett.49.957

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Composites formed from charged particles and vortices in (2+1)-dimensional models, or flux tubes in three-dimensional models, can have any (fractional) angular momentum. The statistics of these objects, like their spin, interpolates continuously between the usual boson and fermion cases. How this works for two-particle quantum mechanics is discussed here.
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 θθθ. θ=0,πθ= 0, πθ=0,π correspond to bosons and fermions respectively; at intermediate values we say that we have fractional statistics. In two dimensions, θθθ 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 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. 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 θθθ 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=1/3ν= 1/3ν=1/3 fractional quantum Hall state has been observed both in scattering and in interferometric experiments. Excitations within designed systems, notably including superconducting circuits, can exhibit anyon behavior. Such systems are being developed for possible use in quantum information processing.
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. 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 realize this behavior. More complex charge-flux constructions can involve nonabelian and product groups acting on a spectrum of allowed charges and fluxes, giving rise to nonabelian and mutual statistics. Interchanges of nonabelian anyons implement unitary transformations of the wave function within an emergent space of internal states. 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 unidirectional, 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 in both scattering and interferometric experiments. Excitations within designed systems, notably including superconducting circuits, can exhibit anyon behavior. Such systems are being developed for possible use in quantum information processing.
The fractional angular momentum of a two-dimensional charge-flux composite is a well-established phenomenon usually derived from a semiclassical Hamiltonian. However, when the composite is treated as an isolated system in free two-dimensional space, the fundamental rotational and reflection symmetries of the O(2)O(2)O(2) group demand that its total angular momentum is strictly quantized. We address this conceptual discrepancy by applying a full quantum electrodynamic (QED) approach combined with Noether's theorem. We demonstrate that the interaction between the charge and flux, mediated by the vacuum electromagnetic field, generates an intrinsic interaction angular momentum composed of both field momentum and hidden relativistic momentum. This gauge-invariant interaction angular momentum exactly compensates for the fractional part of the kinetic angular momentum. Consequently, the net angular momentum of the composite strictly follows the integer or half-integer quantization rule. Our formalism clarifies that the conventional fractional spin corresponds to the expectation value of the kinetic angular momentum within the perturbed QED ground state. It also elucidates why the standard classical field angular momentum definition fails to capture this in two dimensions, due to non-vanishing boundary terms.
The quantum statistics of bosons or fermions are manifest through the even or odd relative angular momentum of a pair. We show theoretically that, under certain conditions, a pair of certain test particles immersed in a fractional quantum Hall state possesses, effectively, a fractional relative angular momentum, which can be interpreted in terms of fractional braid statistics. We propose that the fractionalization of the angular momentum can be detected directly through the measurement of the pair correlation function in rotating ultracold atomic systems in the fractional quantum Hall regime. Such a measurement will also provide direct evidence for the effective magnetic field resulting from Berry phases arising from attached vortices, and of excitations with a fractional particle number, analogous to the fractional charge of the electron fractional quantum Hall effect.