Anyon
[1] In general, the operation of exchanging two identical particles, although it may cause a global phase shift, cannot affect observables.
Abelian anyons, detected by two experiments in 2020,[2] play a major role in the fractional quantum Hall effect.
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.
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.
[5] Microsoft has invested in research concerning anyons as a potential basis for topological quantum computing.
[6] Anyons circling each other ("braiding") would encode information in a more robust way than other known potential quantum computing technologies.
[7] Like so many deep ideas in physics, the topological underpinnings of anyons can be traced back to Dirac.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.
[9] 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.
[10] 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.
[11] 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.
It is important to note that there is a slight abuse of notation in this shorthand expression, as in reality this wave function can be and usually is multi-valued.
Frank Wilczek in 1982 explored the behavior of such quasiparticles and coined the term "anyon" to describe them, because they can have any phase when particles are interchanged.
Mathematical models of one-dimensional anyons provide a base of the commutation relations shown above.
[16]: 22 The fact that the homotopy classes of paths (i.e. notion of equivalence on braids) are relevant hints at a more subtle insight.
The Feynman path integral can be motivated from expanding the propagator using a method called time-slicing,[17] in which time is discretized.
In 2020, two teams of scientists (one in Paris, the other at Purdue) announced new experimental evidence for the existence of anyons.
[2] In April, 2020, researchers from the École normale supérieure (Paris) and the Centre for Nanosciences and Nanotechnologies (C2N) reported results from a tiny "particle collider" for anyons.
"[22][23] As of 2023, this remains an active area of research; using a superconducting processor, Google Quantum AI reported on the first braiding of non-Abelian anyon-like particles in an arXiv article by Andersen et al. in October 2022,[24] later published in Nature.
[25] In an arXiv article released in May 2023, Quantinuum reported on non-abelian braiding using a trapped-ion processor.
[26] In 1988, Jürg Fröhlich showed that it was valid under the spin–statistics theorem for the particle exchange to be monoidal (non-abelian statistics).
When there is no degeneracy, this subspace is one-dimensional and so all such linear transformations commute (because they are just multiplications by a phase factor).
When there is degeneracy and this subspace has higher dimension, then these linear transformations need not commute (just as matrix multiplication does not).
Gregory Moore, Nicholas Read, and Xiao-Gang Wen pointed out that non-Abelian statistics can be realized in the fractional quantum Hall effect (FQHE).
As of 2012, no experiment has conclusively demonstrated the existence of non-abelian anyons although promising hints are emerging in the study of the ν = 5/2 FQHE state.
[needs update][30][31] Experimental evidence of non-abelian anyons, although not yet conclusive and currently contested,[32] was presented in October, 2013.
As a rule, in a system with non-abelian anyons, there is a composite particle whose statistics label is not uniquely determined by the statistics labels of its components, but rather exists as a quantum superposition (this is completely analogous to how two fermions known to have spin 1/2 are together in quantum superposition of total spin 1 and 0).
The essential point is that one braid can wind around the other one, an operation that can be performed infinitely often, and clockwise as well as counterclockwise.
[35][36] Fractionalized excitations as point particles can be bosons, fermions or anyons in 2+1 spacetime dimensions.
