In physics, an anyon is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than the two kinds of standard elementary particles, fermions and bosons. In general, the operation of exchanging two identical particles, although it may cause a global phase shift, cannot affect observables. Anyons are generally classified as abelian or non-abelian. Abelian anyons (detected by two experiments in 2020) play a major role in the fractional quantum Hall effect.
Introduction
The statistical mechanics of large many-body systems obeys laws described by Maxwell–Boltzmann statistics. Quantum statistics is more complicated because of the different behaviors of two different kinds of particles called fermions and bosons. Quoting a recent, simple description:In the three-dimensional world we live in, there are only two types of particles: "fermions," which repel each other, and "bosons," which like to stick together. A commonly known fermion is the electron, which transports electricity; and a commonly known boson is the photon, which carries light. In the two-dimensional world, however, there is another type of particle, the anyon, which doesn't behave like either a fermion or a boson.
In a two-dimensional world, two identical anyons change their wavefunction when they swap places in ways that can't happen in three-dimensional physics:
...in two dimensions, exchanging identical particles twice is not equivalent to leaving them alone. The particles' wavefunction after swapping places twice may differ from the original one; particles with such unusual exchange statistics are known as anyons. By contrast, in three dimensions, exchanging particles twice cannot change their wavefunction, leaving us with only two possibilities: bosons, whose wavefunction remains the same even after a single exchange, and fermions, whose exchange only changes the sign of their wavefunction.
This process of exchanging identical particles, or of circling one particle around another, is referred to by its mathematical name as "braiding." "Braiding" two anyons creates a historical record of the event, as their changed wave functions "count" the number of braids.
Microsoft has invested in research concerning anyons as a potential basis for topological quantum computing. Anyons circling each other ("braiding") would encode information in a more robust way than other potential quantum computing technologies. Most investment in quantum computing, however, is based on methods that do not use anyons.
History
In 1977, two theoretical physicists working at the University of Oslo, Jon Magne Leinaas and Jan Myrheim, showed that the traditional classification of particles as either fermions or bosons would not apply if they were restricted to move in only two dimensions. Hypothetical particles, being neither bosons nor fermions, would be expected to exhibit a diverse range of previously unexpected properties. In 1982, Frank Wilczek published two papers exploring the fractional statistics of quasiparticles in two dimensions, giving them the name "anyons" to indicate that the phase shift upon permutation can take any value.
Daniel Tsui and Horst Störmer discovered the fractional quantum Hall effect in 1982. The mathematics developed by Wilczek proved to be useful to Bertrand Halperin at Harvard University in explaining aspects of it. Frank Wilczek, Dan Arovas, and Robert Schrieffer verified this statement in 1985 with an explicit calculation that predicted that particles existing in these systems are in fact anyons.
Abelian anyons
In quantum mechanics, and some classical stochastic systems, indistinguishable particles have the property that exchanging the states of particle with particle (symbolically ) does not lead to a measurably different many-body state.
In a quantum mechanical system, for example, a system with two indistinguishable particles, with particle 1 in state and particle 2 in state , has state in Dirac notation. Now suppose we exchange the states of the two particles, then the state of the system would be . These two states should not have a measurable difference, so they should be the same vector, up to a phase factor:
Here, is the phase factor. In space of three or more dimensions, the phase factor is or . Thus, elementary particles are either fermions, whose phase factor is , or bosons, whose phase factor is . These two types have different statistical behaviour. Fermions obey Fermi–Dirac statistics, while bosons obey Bose–Einstein statistics. In particular, the phase factor is why fermions obey the Pauli exclusion principle: If two fermions are in the same state, then we have