All Topics  
Anyon

 

 
   Email Print
   Bookmark   Link
 

 

Anyon
 
 
  Topics Anyon Topic Home

In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 and physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, an anyon is a type of particle that occurs only in two-dimensional systems. It is a generalization of the fermion
Fermion

In particle physics, fermions are subatomic particle which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In contrast to bosons, which have Bose-Einstein statistics, only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle....
 and boson
Boson

In particle physics, bosons are subatomic particle which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein....
 concept.

mathematical concept becomes useful in the physics of two-dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
al systems such as sheets of graphene
Graphene

Graphene is a one-atom-thick planar sheet of sp2 bond carbon atoms that are densely packed in a honeycomb crystal lattice. It can be viewed as an chicken wire made of carbon atoms and their bonds....
 or the quantum Hall effect
Quantum Hall effect

The quantum Hall effect is a quantum mechanics version of the Hall effect, observed in 2DEG subjected to low temperatures and strong magnetic fields, in which the Hall Electrical conductivity s takes on the quantized values...
.

In space of three or more dimensions, particle
Subatomic particle

A subatomic particle is an elementary particle or composite particle particle smaller than an atom. Particle physics and nuclear physics are concerned with the study of these particles, their interactions, and non-atomic QCD matter....
s are restricted to being fermions
Fermion

In particle physics, fermions are subatomic particle which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In contrast to bosons, which have Bose-Einstein statistics, only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle....
 or bosons
Boson

In particle physics, bosons are subatomic particle which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein....
, according to their statistical behaviour.






Discussion
Ask a question about 'Anyon'
Start a new discussion about 'Anyon'
Answer questions from other users
Full Discussion Forum



Encyclopedia


In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 and physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, an anyon is a type of particle that occurs only in two-dimensional systems. It is a generalization of the fermion
Fermion

In particle physics, fermions are subatomic particle which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In contrast to bosons, which have Bose-Einstein statistics, only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle....
 and boson
Boson

In particle physics, bosons are subatomic particle which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein....
 concept.

In physics

This mathematical concept becomes useful in the physics of two-dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
al systems such as sheets of graphene
Graphene

Graphene is a one-atom-thick planar sheet of sp2 bond carbon atoms that are densely packed in a honeycomb crystal lattice. It can be viewed as an chicken wire made of carbon atoms and their bonds....
 or the quantum Hall effect
Quantum Hall effect

The quantum Hall effect is a quantum mechanics version of the Hall effect, observed in 2DEG subjected to low temperatures and strong magnetic fields, in which the Hall Electrical conductivity s takes on the quantized values...
.

In space of three or more dimensions, particle
Subatomic particle

A subatomic particle is an elementary particle or composite particle particle smaller than an atom. Particle physics and nuclear physics are concerned with the study of these particles, their interactions, and non-atomic QCD matter....
s are restricted to being fermions
Fermion

In particle physics, fermions are subatomic particle which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In contrast to bosons, which have Bose-Einstein statistics, only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle....
 or bosons
Boson

In particle physics, bosons are subatomic particle which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein....
, according to their statistical behaviour. Fermions respect the so-called Fermi-Dirac statistics
Fermi-Dirac statistics

Fermi-Dirac statistics is a part of the science of physics, that applies to a system comprised of many particles that obey the Pauli Exclusion Principle....
 while bosons respect the Bose-Einstein statistics. In the language of quantum physics this is formulated as the behavior of multiparticle states under the exchange of particles. This is in particular for a two-particle state (in Dirac notation):

(where the first entry in is the state of particle 1 and the second entry is the state of particle 2. So for example the left hand side is read as "Particle 1 is in state and particle 2 in state "). Here the "+" corresponds to both particles being bosons and the "-" to both particles being fermions (composite states of fermions and bosons are not possible).

In two-dimensional systems, however, quasiparticles can be observed which obey statistics ranging continuously
Continuous function

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous....
 between Fermi-Dirac and Bose-Einstein statistics, as was first shown by Jon Magne Leinaas and Jan Myrheim of the University of Oslo
University of Oslo

The University of Oslo is the List of oldest universities in continuous operation#Oldest Universities by Region .28post 1500.29, largest and most prestigious university in Norway, situated in the Norwegian capital of Oslo....
 in 1977. In our above example of two particles this looks as follows:

With "i" being the imaginary unit
Imaginary unit

In mathematics, physics, and engineering, the imaginary unit is denoted by  or the Latin   or the Greek iota . It allows the real number system, to be extended to the complex number system,   Its precise definition is dependent upon the particular method of extension....
 from the calculus of complex numbers and a real number. Recall that and as well as . So in the case we recover the Fermi-Dirac statistics (minus sign) and in the case the Bose-Einstein statistics (plus sign). In between we have something different. Frank Wilczek
Frank Wilczek

Frank Anthony Wilczek is an United States theoretical physics and Nobel laureate. He is currently the Herman Feshbach Professor of Physics at the Massachusetts Institute of Technology....
 coined the term "anyon" to describe such particles, since they can have any phase when particles are interchanged.

Topological basis

In more than two dimensions, the spin-statistics connection
Spin-statistics theorem

In quantum mechanics, the spin-statistics theorem relates the spin of a particle to the particle statistics obeyed by it. The spin of a particle is its intrinsic angular momentum ....
 states that any multiparticle state has to obey either Bose-Einstein or Fermi-Dirac statistics. This is related to the first homotopy group of SO(n,1) (and also Poincaré(n,1)
Poincaré group

In physics and mathematics, the Poincar? group, named after Henri Poincar?, is the group of isometry of Minkowski spacetime. It is a 10-dimensional compact space Lie group....
) with , which is (the cyclic group
Cyclic group

In group theory, a cyclic group or monogenous group is a group that can be generating set of a group by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g ....
 consisting of two elements). Therefore 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)
Poincaré group

In physics and mathematics, the Poincar? group, named after Henri Poincar?, is the group of isometry of Minkowski spacetime. It is a 10-dimensional compact space Lie group....
) is Z (infinite cyclic). This means that Spin(2,1) is not the universal cover
Universal covering group

In mathematics, a covering group of a topological group H is a covering space G of H such that G is a topological group and the covering map p : G ? H is a continuous group homomorphism....
: it is not simply connected. In detail, there are projective representation
Projective representation

In the mathematics field of representation theory, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G towhere GL is the automorphism group of invertible linear transformations of V over F and F* here is the normal subgroup consisting of mult...
s of the special orthogonal group
Generalized orthogonal group

In mathematics, the indefinite orthogonal group, O is the Lie group of all linear transformations of a n = p + q dimension of a vector space real vector space which leave invariant a nondegenerate form, symmetric bilinear form of signature of a quadratic form ....
 SO(2,1) which do not arise from linear representations of SO(2,1), or of its double cover, the spin group
Spin group

In mathematics the spin group Spin is the covering space of the special orthogonal group SO, such that there exists a short exact sequence of Lie groups...
 Spin(2,1). These representations are called anyons.

This concept also applies to nonrelativistic systems. The relevant part here is that the spatial rotation group is SO(2), which has an infinite first homotopy group.

This fact is also related to the braid group
Braid group

In mathematics, the braid group on n strands, denoted by B'n, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group S'n....
s well known in knot theory
Knot theory

In mathematics, knot theory is the area of topology that studies knot s. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs drastically in that the ends are joined together to prevent it from becoming undone....
. 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
Symmetric group

In mathematics, the symmetric group on a Set X, denoted by SX, or Sym, is the group whose underlying set is the set of all bijective function s from X to X, in which the group operation is that of Function composition, i.e., two such functions f and g can be composed to yield a new bijective function ,...
  (2-dimensional) but rather the braid group (infinite dimensional).

A very different approach to the stability-decoherence problem in quantum computing
Quantum computer

A quantum computer is a device for computation that makes direct use of quantum mechanical phenomena, such as quantum superposition and quantum entanglement, to perform operations on data....
 is to create a topological quantum computer
Topological quantum computer

A topological quantum computer is a theoretical quantum computer that employs two-dimensional quasiparticles called anyons, whose world lines cross over one another to form braid theory in a three-dimensional spacetime ....
 with anyons, quasi-particles used as threads and relying on braid theory
Braid theory

In topology, braid theory is an abstract geometry theory studying the everyday braid concept, and some generalisations. The idea is that braids can be organised into group s, in which the group operation is 'do the first braid on a set of strings, and then follow it with a second on the twisted strings'....
 to form stable logic gates.

See also

  • plekton
    Plekton

    In physics, a plekton is a theoretical kind of elementary particle, which obeys a different style of statistics with respect to the interchange of identical particles....
  • fractional quantum Hall effect
    Fractional quantum Hall effect

    The fractional quantum Hall effect is a physical phenomenon in which a certain system behaves as if it were composed of particles with charge smaller than the elementary charge....


Further reading

  • , , Steven H. Simon, Ady Stern
    Ady Stern

    Ady Stern is a Israeli physics and an Associate Professor at the Weizmann Institute of Science. He is best known for his research relating to the theoretical aspects of the fractional quantum Hall effect...
    , Michael Freedman, Sankar Das Sarma, 2007


External links