Anyon

This article deals with the quasiparticles, for the similar term negative ions see anion

This article was registered in the quality assurance of the physics editorial team . If you are familiar with the topic, you are welcome to participate in the review and possible improvement of the article. The exchange of views about this is currently not taking place on the article discussion page , but on the quality assurance side of physics.

Anyons (of English any , some ' ) are exotic quasiparticles that are neither bosons (with integer spin ) or fermions are (with half-integer spin). In theoretical solid-state physics , anyons are intensively researched, particularly in connection with the quantum Hall effect . Lately, experimental physicists and computer scientists have also been dealing with it, specifically in connection with so-called " topological quantum computers ". For mathematical reasons, anyons can only exist in two dimensions . As quasiparticles, they are established in two-dimensional systems (e.g. thin layers).

An anyon should not be confused with the chemical term anion .

Appearance and mathematical basis

The existence of these particles is a consequence of the fact that the type of quantum statistics of massive identical particles depends on the dimension of the space: The Hilbert space bears a unitary representation of the fundamental group of the configuration space . For one dimension this is the trivial group and there is no difference between fermions and bosons. For two dimensions this is the Artinian “ braid group ” and for three dimensions and more the symmetrical group . Since the Zopf group only contains the symmetrical group as a quotient, other types of particles are allowed in two-dimensional systems in addition to bosons and fermions.

Example: Fractional (= fractional) quantum Hall effect

The exchange of two elementary excitations with a non-integer charge leads to an Aharonov-Bohm phase due to the attached magnetic flux quanta when rotating through 360 ° , which is neither ( fermions ) nor 0 or ( bosons ), but is characterized by any value ( ). The spin then has the value , so it does not necessarily have to be a whole or half number. ${\displaystyle \pi }$ ${\displaystyle 2\pi }$ ${\displaystyle \theta }$ ${\displaystyle \langle \psi _{1}\psi _{2}\rangle \equiv \langle \psi _{2}\psi _{1}\rangle \,e^{i\theta }}$ ${\displaystyle s=\theta /2\pi }$

In connection with this effect, in particular the connection with the integer and fractional quantum Hall effect , the term “composite fermions” is also relevant (see below under literature).

Applications

Applications concern both real mathematical-abstract aspects like the already mentioned Artin'sche Zopfgruppe as well as objects to be evaluated as speculative at the moment like a promising “topological” realization of the (not yet existing) quantum computer examined by experimental physicists and computer scientists . The so-called non-Abelian anyons , whose commutation relations cannot be described by a phase alone, are particularly interesting for this. Non-Abelian anyons have internal degrees of freedom, so that a system of anyons (at the locations ) has a -fold degeneracy and interchanges among the particles are accompanied by a unitary transformation on -dimensional degenerate space. If these unitary transformations do not all commute with one another , the anyons are called non-Abelian. (The name comes from the fact that the unitary transformations can be understood as a non-Abelian representation of the braid group on which the swap is based.) Topological quantum computing is then realized within the -dimensional space by swapping anyons. For this purpose, the unitary transformations generated by interchanging must be a universal set of quantum gates . ${\displaystyle n}$ ${\displaystyle r_{1},\dots ,r_{n}}$ ${\displaystyle g>1}$ ${\displaystyle n}$ ${\displaystyle g}$ ${\displaystyle g}$

References and footnotes

1. ↑ ^{a b} The essential aspect is that a “braid” can “wrap around” any other as often as desired, so that in two-dimensional not only the position - and thus the permutation behavior - of the singular points is important.
2. ↑ Due to the flow hose grid involved, the system is quasi-two-dimensional; see also the last web link.
3. ↑ Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman , Sankar Das Sarma : Non-Abelian Anyons and Topological Quantum Computation . In: Rev. Mod. Phys. tape 80 , 2008, p. 1083 , doi : 10.1103 / RevModPhys.80.1083 , arxiv : 0707.1889 (English).
4. ↑ Rainer Scharf: Quarter electrons with non-Abelian particle statistics? In: pro-physik.de. April 17, 2008, accessed February 4, 2020 . 

literature

• Frank Wilczek : Fractional Statistics and Anyon Superconductivity. World Scientific Publishing Company, 1990, ISBN 981-02-0049-8 .
• Frank Wilczek: Anyons. In: Scientific American. May 1991.
• Jainendra K. Jain : Composite Fermions. Cambridge University Press, 2007, ISBN 978-0-521-86232-5 .

Web links

Wiktionary: Anyon  - explanations of meanings, word origins , synonyms, translations

• S. Rao: An Anyon Primer , 1992
• Interview with F. Wilczek from 1991 about Anyonen, archived at archive.is ( Memento from March 2, 2008 in the Internet Archive )
• Anyons in Quantum Informatics ( Memento from September 29, 2007 in the Internet Archive ) - seminar paper (winter 2003/04) at the University of Karlsruhe by Pascal Bihler (PDF; 268 kB)