Екіөлшемді квазибөлшектер: Аниондардың ерекше қасиеттері

Type of two dimensional quasiparticle

In physics, an anyon is a type of quasiparticle so far observed only in two dimensional systems. In three dimensional systems, only two kinds of elementary particles are seen: fermions and bosons. Anyons have statistical properties intermediate between fermions and bosons. play a major role in the fractional quantum Hall effect.

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. In two dimensional systems, however, there is a third type of particle, called an anyon. 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. |title="Finally, anyons reveal their exotic quantum properties" |source=Aalto University press release, April 2020

In a two dimensional world, two identical anyons change their wavefunction when they swap places in ways that cannot 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. |author=Kirill Shtengel |title="A home for anyon?" |source=Nature Physics

This process of exchanging identical particles, or of circling one particle around another, is referred to as "braiding". Braiding two anyons creates a historical record of the event, as their changed wave functions record the number of braids. Microsoft has invested in research concerning anyons as a potential basis for topological quantum computing. They may be useful in quantum computing as a form of memory. Most investment in quantum computing, however, is based on methods that do not use anyons.|author= Biedenharn et al. 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.

The fact that the homotopy classes of paths (i. e. notion of equivalence on braids) are relevant hints at a more subtle insight. It arises from the Feynman path integral, in which all paths from an initial to final point in spacetime contribute with an appropriate phase factor. The Feynman path integral can be motivated from expanding the propagator using a method called time slicing, in which time is discretized. In non homotopic paths, one cannot get from any point at one time slice to any other point at the next time slice. This means that we can consider homotopic equivalence class of paths to have different weighting factors. So it can be seen that the topological notion of equivalence comes from a study of the Feynman path integral.]] In 2020, two teams of scientists (one in Paris, the other at Purdue) announced new experimental evidence for the existence of anyons. Both experiments were featured in Discover Magazines 2020 annual "state of science" issue. In July, 2020, scientists at Purdue University detected anyons using a different setup. The team's interferometer routes the electrons through a specific maze like etched nanostructure made of gallium arsenide and aluminium gallium arsenide. "In the case of our anyons the phase generated by braiding was 2π/3", he said. "That's different than what's been seen in nature before." 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, later published in Nature. In an arXiv article released in May 2023, Quantinuum reported on non abelian braiding using a trapped ion processor.

In 1988, Jürg Fröhlich showed that it was valid under the spin–statistics theorem for the particle exchange to be monoidal (non abelian statistics). In particular, this can be achieved when the system exhibits some degeneracy, so that multiple distinct states of the system have the same configuration of particles. Then an exchange of particles can contribute not just a phase change, but can send the system into a different state with the same particle configuration. Particle exchange then corresponds to a linear transformation on this subspace of degenerate states. 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). While at first non abelian anyons were generally considered a mathematical curiosity, physicists began pushing toward their discovery when Alexei Kitaev showed that non abelian anyons could be used to construct a topological quantum computer. 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. Experimental evidence of non abelian anyons, although not yet conclusive and currently contested, was presented in October, 2013. Recent works claim the creation of non abelian topological order and anyons on a trapped ion processor

In more than two dimensions, the spin–statistics theorem states that any multiparticle state of indistinguishable particles has to obey either Bose–Einstein or Fermi–Dirac statistics. For any d > 2, the Lie groups SO(d,1) (which generalizes the Lorentz group) and Poincaré(d,1) have Z2 as their first homotopy group. Because the cyclic group Z2 is composed of two elements, only two possibilities remain. (The details are more involved than that, but this is the crucial point.) The situation changes in two dimensions. Here the first homotopy group of SO(2,1), and also Poincaré(2,1), is Z (infinite cyclic). This means that Spin(2,1) is not the universal cover: it is not simply connected. In detail, there are projective representations of the special orthogonal group SO(2,1) which do not arise from linear representations of SO(2,1), or of its double cover, the spin group Spin(2,1). Anyons are evenly complementary representations of spin polarization by a charged particle. This concept also applies to nonrelativistic systems. The relevant part here is that the spatial rotation group SO(2) has an infinite first homotopy group. This fact is also related to the braid groups well known in knot theory. The relation can be understood when one considers the fact that in two dimensions the group of permutations of two particles is no longer the symmetric group S2 (with two elements) but rather the braid group B2 (with an infinite number of elements). 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. A very different approach to the stability decoherence problem in quantum computing is to create a topological quantum computer with anyons, quasi particles used as threads and relying on braid theory to form stable quantum logic gates.

Fractionalized excitations as point particles can be bosons, fermions or anyons in 2+1 spacetime dimensions. It is known that point particles can be only either bosons or fermions in 3+1 and higher spacetime dimensions. However, the loop (or string ) or membrane like excitations are extended objects that can have fractionalized statistics. Current research shows that the loop and string like excitations exist for topological orders in the 3+1 dimensional spacetime, and their multi loop/string braiding statistics are the key signatures for identifying 3+1dimensional topological orders. The multi loop/string braiding statistics of 3+1dimensional topological orders can be captured by the link invariants of particular topological quantum field theories in 4 spacetime dimensions. Explained in a colloquial manner, the extended objects (loop, string, or membrane, etc.) can be potentially anyonic in 3+1 and higher spacetime dimensions in the long range entangled systems.