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In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, an anyon is a type of
quasiparticle In physics, quasiparticles and collective excitations are closely related emergent phenomena arising when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum. For exa ...
that occurs only in
two-dimensional In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...
systems, with properties much less restricted than the two kinds of standard
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, an ...
s, fermions and
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
s. 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 The fractional quantum Hall effect (FQHE) is a physical phenomenon in which the Hall conductance of 2-dimensional (2D) electrons shows precisely quantized plateaus at fractional values of e^2/h. It is a property of a collective state in which elec ...
. Non-abelian anyons have not been definitively detected, although this is an active area of research.


Introduction

The
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
of large many-body systems obeys laws described by
Maxwell–Boltzmann statistics In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of Classical physics, classical material particles over various energy states in thermal equilibrium. It is applicable when the temperature is high enough or the ...
. Quantum statistics is more complicated because of the different behaviors of two different kinds of particles called
fermions In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
and
bosons In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer ...
. 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 A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strande ...
." "Braiding" two anyons creates a historical record of the event, as their changed wave functions "count" the number of braids.
Microsoft Microsoft Corporation is an American multinational technology corporation producing computer software, consumer electronics, personal computers, and related services headquartered at the Microsoft Redmond campus located in Redmond, Washing ...
has invested in research concerning anyons as a potential basis for
topological quantum computing A topological quantum computer is a theoretical quantum computer proposed by Russian-American physicist Alexei Kitaev in 1997. It employs quasiparticles in two-dimensional systems, called anyons, whose world lines pass around one another to fo ...
. 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

A group of
theoretical physicists The following is a partial list of notable theoretical physicists. Arranged by century of birth, then century of death, then year of birth, then year of death, then alphabetically by surname. For explanation of symbols, see Notes at end of this ar ...
working at the
University of Oslo The University of Oslo ( no, Universitetet i Oslo; la, Universitas Osloensis) is a public research university located in Oslo, Norway. It is the highest ranked and oldest university in Norway. It is consistently ranked among the top universit ...
, led by Jon Leinaas and
Jan Myrheim Jan Myrheim (born 14 February 1948) is a Norwegian physicist. He was born in Røyrvik. He took the cand.real. at the University of Oslo in 1972 and took the dr.philos. degree at the University of Trondheim in 1994. He was then appointed as a pr ...
, calculated in 1977 that the traditional division between fermions and bosons would not apply to theoretical particles existing in two
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
s. Such particles would be expected to exhibit a diverse range of previously unexpected properties. In 1982,
Frank Wilczek Frank Anthony Wilczek (; born May 15, 1951) is an American theoretical physicist, mathematician and Nobel laureate. He is currently the Herman Feshbach Professor of Physics at the Massachusetts Institute of Technology (MIT), Founding Direc ...
published in two papers, exploring the fractional statistics of quasiparticles in two dimensions, giving them the name "anyons."
Daniel Tsui Daniel Chee Tsui (, born February 28, 1939) is a Chinese-born American physicist, Nobel laureate, and the Arthur Legrand Doty Professor of Electrical Engineering, Emeritus, at Princeton University. Tsui's areas of research include electrical prop ...
and
Horst Störmer Horst may refer to: Science * Horst (geology), a raised fault block bounded by normal faults or graben People * Horst (given name) * Horst (surname) * ter Horst, Dutch surname * van der Horst, Dutch surname Places Settlements Germany * Hor ...
discovered the fractional quantum Hall effect in 1982. The mathematics developed by Wilczek proved to be useful to
Bertrand Halperin Bertrand I. Halperin (born December 6, 1941) is an American physicist, former holder of the Hollis Chair of Mathematicks and Natural Philosophy at the physics department of Harvard University. Biography Halperin was born in Brooklyn, New York, ...
at
Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of higher le ...
in explaining aspects of it. Frank Wilczek, Dan Arovas, and
Robert Schrieffer John Robert Schrieffer (; May 31, 1931 – July 27, 2019) was an American physicist who, with John Bardeen and Leon Cooper, was a recipient of the 1972 Nobel Prize in Physics for developing the BCS theory, the first successful quantum theo ...
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 In quantum mechanics, identical particles (also called indistinguishable or indiscernible particles) are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, ...
have the property that exchanging the states of particle  with particle  (symbolically \psi_i\leftrightarrow\psi_j \text i \ne j) 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 \psi_1 and particle 2 in state \psi_2, has state \left, \psi_1\psi_2\right\rangle in
Dirac notation Distributed Research using Advanced Computing (DiRAC) is an integrated supercomputing facility used for research in particle physics, astronomy and cosmology in the United Kingdom. DiRAC makes use of multi-core processors and provides a variety of ...
. Now suppose we exchange the states of the two particles, then the state of the system would be \left, \psi_2\psi_1\right\rangle. These two states should not have a measurable difference, so they should be the same vector, up to a
phase factor For any complex number written in polar form (such as ), the phase factor is the complex exponential factor (). As such, the term "phase factor" is related to the more general term phasor, which may have any magnitude (i.e. not necessarily on th ...
: :\left, \psi_1\psi_2\right\rangle = e^\left, \psi_2\psi_1\right\rangle. Here, e^ is the phase factor. In space of
three 3 is a number, numeral, and glyph. 3, three, or III may also refer to: * AD 3, the third year of the AD era * 3 BC, the third year before the AD era * March, the third month Books * '' Three of Them'' (Russian: ', literally, "three"), a 1901 ...
or more dimensions, the phase factor is 1 or -1. Thus,
elementary particles In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions (quarks, leptons, anti ...
are either fermions, whose phase factor is -1, or bosons, whose phase factor is 1. These two types have different statistical behaviour. Fermions obey Fermi–Dirac statistics, while bosons obey
Bose–Einstein statistics In quantum statistics, Bose–Einstein statistics (B–E statistics) describes one of two possible ways in which a collection of non-interacting, indistinguishable particles may occupy a set of available discrete energy states at thermodynamic ...
. In particular, the phase factor is why fermions obey the
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulat ...
: If two fermions are in the same state, then we have :\left, \psi\psi\right\rangle = -\left, \psi\psi\right\rangle. The state vector must be zero, which means it is not normalizable, thus it is unphysical. In two-dimensional systems, however,
quasiparticle In physics, quasiparticles and collective excitations are closely related emergent phenomena arising when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum. For exa ...
s can be observed that obey statistics ranging continuously between Fermi–Dirac and Bose–Einstein statistics, as was first shown by
Jon Magne Leinaas Jon Magne Leinaas (born 11 October 1946) is a Norwegian theoretical physicist. He was born in Oslo. He took the cand.real. at the University of Oslo in 1970 and the dr.philos. degree at the same institution in 1980. He was a fellow at Nordita, ...
and
Jan Myrheim Jan Myrheim (born 14 February 1948) is a Norwegian physicist. He was born in Røyrvik. He took the cand.real. at the University of Oslo in 1972 and took the dr.philos. degree at the University of Trondheim in 1994. He was then appointed as a pr ...
of the
University of Oslo The University of Oslo ( no, Universitetet i Oslo; la, Universitas Osloensis) is a public research university located in Oslo, Norway. It is the highest ranked and oldest university in Norway. It is consistently ranked among the top universit ...
in 1977. In the case of two particles this can be expressed as :\left, \psi_1\psi_2\right\rangle = e^\left, \psi_2\psi_1\right\rangle, where e^ can be other values than just -1 or 1. It is important to note that there is a slight
abuse of notation In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors a ...
in this shorthand expression, as in reality this wave function can be and usually is multi-valued. This expression actually means that when particle 1 and particle 2 are interchanged in a process where each of them makes a counterclockwise half-revolution about the other, the two-particle system returns to its original quantum wave function except multiplied by the complex unit-norm phase factor . Conversely, a clockwise half-revolution results in multiplying the wave function by . Such a theory obviously only makes sense in two-dimensions, where clockwise and counterclockwise are clearly defined directions. In the case ''θ'' = ''π'' we recover the Fermi–Dirac statistics () and in the case (or ) the Bose–Einstein statistics (). In between we have something different.
Frank Wilczek Frank Anthony Wilczek (; born May 15, 1951) is an American theoretical physicist, mathematician and Nobel laureate. He is currently the Herman Feshbach Professor of Physics at the Massachusetts Institute of Technology (MIT), Founding Direc ...
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. Unlike bosons and fermions, anyons have the peculiar property that when they are interchanged twice in the same way (e.g. if anyon 1 and anyon 2 were revolved counterclockwise by half revolution about each other to switch places, and then they were revolved counterclockwise by half revolution about each other again to go back to their original places), the wave function is not necessarily the same but rather generally multiplied by some complex phase (by in this example). We may also use with particle spin quantum number ''s'', with ''s'' being
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
for bosons, half-integer for fermions, so that :e^ = e^ = (-1)^,   or   \left, \psi_1\psi_2\right\rangle = (-1)^\left, \psi_2\psi_1\right\rangle. At an edge,
fractional quantum Hall effect The fractional quantum Hall effect (FQHE) is a physical phenomenon in which the Hall conductance of 2-dimensional (2D) electrons shows precisely quantized plateaus at fractional values of e^2/h. It is a property of a collective state in which elec ...
anyons are confined to move in one space dimension. Mathematical models of one-dimensional anyons provide a base of the commutation relations shown above. In a three-dimensional position space, the fermion and boson statistics operators (−1 and +1 respectively) are just 1-dimensional representations of the permutation group (''SN'' of ''N'' indistinguishable particles) acting on the space of wave functions. In the same way, in two-dimensional position space, the abelian anyonic statistics operators () are just 1-dimensional representations of the
braid group A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strande ...
(''BN'' of ''N'' indistinguishable particles) acting on the space of wave functions. Non-abelian anyonic statistics are higher-dimensional representations of the braid group. Anyonic statistics must not be confused with
parastatistics In quantum mechanics and statistical mechanics, parastatistics is one of several alternatives to the better known particle statistics models (Bose–Einstein statistics, Fermi–Dirac statistics and Maxwell–Boltzmann statistics). Other alt ...
, which describes statistics of particles whose wavefunctions are higher-dimensional representations of the permutation group.


Topological equivalence

The fact that the
homotopy class In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
es of paths (i.e. notion of equivalence on
braids A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strande ...
) 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 In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
contribute with an appropriate
phase factor For any complex number written in polar form (such as ), the phase factor is the complex exponential factor (). As such, the term "phase factor" is related to the more general term phasor, which may have any magnitude (i.e. not necessarily on th ...
. 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 In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
equivalence class of paths to have different weighting factors. So it can be seen that the
topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
notion of equivalence comes from a study of the Feynman path integral. For a more transparent way of seeing that the homotopic notion of equivalence is the "right" one to use, see
Aharonov–Bohm effect The Aharonov–Bohm effect, sometimes called the Ehrenberg–Siday–Aharonov–Bohm effect, is a quantum mechanical phenomenon in which an electrically charged particle is affected by an electromagnetic potential (φ, A), despite being confine ...
.


Experiment

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 Magazine ''Discover'' is an American general audience science magazine launched in October 1980 by Time Inc. It has been owned by Kalmbach Publishing since 2010. History Founding ''Discover'' was created primarily through the efforts of ''Time'' m ...
s 2020 annual "state of science" issue. In April, 2020, researchers from the
École normale supérieure (Paris) The ''École normale supérieure - PSL'' (; also known as ''ENS'', ''Normale sup, ''Ulm'' or ''ENS Paris'') is a ''grande école'' university in Paris, France. It is one of the constituent members of Paris Sciences et Lettres University (PSL). ...
and the
Centre for Nanosciences and Nanotechnologies The Centre for Nanosciences and Nanotechnologies (Centre de Nanosciences et de Nanotechnologies de l'université Paris-Saclay) or C2N, is a nanotechnology laboratory created as joint research unit (UMR 9001) between the University of Paris-Sac ...
(C2N) reported results from a tiny "particle collider" for anyons. They detected properties that matched predictions by theory for anyons. 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 aluminum 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."


Non-abelian anyons

In 1988,
Jürg Fröhlich Jürg Martin Fröhlich (born 4 July 1946 in Schaffhausen) is a Swiss mathematician and theoretical physicist. He is best known for introducing rigorous techniques for the analysis of statistical mechanics models, in particular continuous symmetry ...
showed that it was valid under the
spin–statistics theorem In quantum mechanics, the spin–statistics theorem relates the intrinsic spin of a particle (angular momentum not due to the orbital motion) to the particle statistics it obeys. In units of the reduced Planck constant ''ħ'', all particles tha ...
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 Nicholas Read is an American physicist, noted for his work on strongly interacting quantum many-body systems. Biography Read was born in Britain in 1958 and did his undergraduate education at Cambridge University. He completed his PhD at the ...
, and
Xiao-Gang Wen Xiao-Gang Wen (; born November 26, 1961) is a Chinese-American physicist. He is a Cecil and Ida Green Professor of Physics at the Massachusetts Institute of Technology and Distinguished Visiting Research Chair at the Perimeter Institute for Theor ...
pointed out that non-Abelian statistics can be realized in the
fractional quantum Hall effect The fractional quantum Hall effect (FQHE) is a physical phenomenon in which the Hall conductance of 2-dimensional (2D) electrons shows precisely quantized plateaus at fractional values of e^2/h. It is a property of a collective state in which elec ...
(FQHE). While at first non-abelian anyons were generally considered a mathematical curiosity, physicists began pushing toward their discovery when
Alexei Kitaev Alexei Yurievich Kitaev (russian: Алексей Юрьевич Китаев; born August 26, 1963) is a Russian–American professor of physics at the California Institute of Technology and permanent member of the Kavli Institute for Theoretical ...
showed that non-abelian anyons could be used to construct a
topological quantum computer A topological quantum computer is a theoretical quantum computer proposed by Russian-American physicist Alexei Kitaev in 1997. It employs quasiparticles in two-dimensional systems, called anyons, whose world lines pass around one another to for ...
. 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.


Fusion of anyons

In much the same way that two fermions (e.g. both of spin 1/2) can be looked at together as a composite boson (with total spin in a superposition of 0 and 1), two or more anyons together make up a composite anyon (possibly a boson or fermion). The composite anyon is said to be the result of the
fusion Fusion, or synthesis, is the process of combining two or more distinct entities into a new whole. Fusion may also refer to: Science and technology Physics *Nuclear fusion, multiple atomic nuclei combining to form one or more different atomic nucl ...
of its components. If N identical abelian anyons each with individual statistics \alpha (that is, the system picks up a phase e^ when two individual anyons undergo adiabatic counterclockwise exchange) all fuse together, they together have statistics N^2 \alpha . This can be seen by noting that upon counterclockwise rotation of two composite anyons about each other, there are N^2 pairs of individual anyons (one in the first composite anyon, one in the second composite anyon) that each contribute a phase e^ . An analogous analysis applies to the fusion of non-identical abelian anyons. The statistics of the composite anyon is uniquely determined by the statistics of its components. Non-abelian anyons have more complicated fusion relations. 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). If the overall statistics of the fusion of all of several anyons is known, there is still ambiguity in the fusion of some subsets of those anyons, and each possibility is a unique quantum state. These multiple states provide a Hilbert space on which quantum computation can be done.


Topological basis

In more than two dimensions, the
spin–statistics theorem In quantum mechanics, the spin–statistics theorem relates the intrinsic spin of a particle (angular momentum not due to the orbital motion) to the particle statistics it obeys. In units of the reduced Planck constant ''ħ'', all particles tha ...
states that any multiparticle state of
indistinguishable particles In quantum mechanics, identical particles (also called indistinguishable or indiscernible particles) are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, ...
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 In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
) and Poincaré(d,1) have Z2 as their first homotopy group. Because the
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
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 A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
: it is not simply connected. In detail, there are
projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group \mathrm(V) = \mathrm(V) / F^*, where G ...
s of the special orthogonal group SO(2,1) which do not arise from
linear representation Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essenc ...
s of SO(2,1), or of its double cover, the
spin group In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a ...
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 group A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strande ...
s well known in
knot theory In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...
. 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 In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
''S''2 (with two elements) but rather the braid group ''B''2 (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 A topological quantum computer is a theoretical quantum computer proposed by Russian-American physicist Alexei Kitaev in 1997. It employs quasiparticles in two-dimensional systems, called anyons, whose world lines pass around one another to for ...
with anyons, quasi-particles used as threads and relying on
braid theory A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strande ...
to form stable quantum logic gates.


Higher dimensional generalization

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 works show that the loop and string like excitations exist for
topological order In physics, topological order is a kind of order in the zero-temperature phase of matter (also known as quantum matter). Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian ...
s in the 3+1 dimensional spacetime, and their multi-loop/string-braiding statistics are the key signatures for identifying 3+1 dimensional topological orders. The multi-loop/string-braiding statistics of 3+1 dimensional topological orders can be captured by the link invariants of particular
topological quantum field theories In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathem ...
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.


See also

* * * * * * * * * * * * * *


References


Further reading

* * * * {{particles Parastatistics Representation theory of Lie groups