nLab
braid group statistics

Idea

In quantum physics one speaks of the “statistics” of a particle species or other object when referring to the irreducible representation of the group which exchanges identical such particles that it belongs to. Typically this is the symmetric group and accordingly one speaks of boson and fermion statistics. But in special cases, namely whenever the codimension in space of the objects in question is 1 (e.g. point particles in 2-dimensional quantum field theories), accordingly in this situation these may have what is then called braid group statistics and one speaks of anyons.

Related concepts

• boson, fermion, anyon

• parastatistics,

• quantum Hall effect,

• topological quantum computation

References

General

The concept of anyons satisfying braid group statistics goes back to:

• J. M. Leinaas, J. Myrheim, On the theory of identical particles, К теории тождествениых частиц, Nuovo Cim B 37, 1–23 (1977) (doi:10.1007/BF02727953)

• Frank Wilczek, Magnetic Flux, Angular Momentum, and Statistics, Phys. Rev. Lett. 48, 1144, 1982 (doi:10.1103/PhysRevLett.48.1144)

Rigorous discussion in terms of superselection sectors in algebraic quantum field theory:

• Klaus Fredenhagen, Karl-Henning Rehren, Bert Schroer, Superselection sectors with braid group statistics and exchange algebras – I: General theory, Comm. Math. Phys. Volume 125, Number 2 (1989), 201-226. (euclid:cmp/1104179464)

• Klaus Fredenhagen, Karl-Henning Rehren, Bert Schroer, Superselection sectors with braid group statistics and exchange algebras – II: Geometric aspects and conformal covariance, Reviews in Mathematical PhysicsVol. 04, No. spec01, pp. 113-157 (1992) (doi:10.1142/S0129055X92000170 pdf)

• Jürg Fröhlich, F. Gabbiani, Braid statistics in local quantum theory, Reviews in Mathematical Physics, Volume 2, Issue 03, pp. 251-353 (1990) (doi:10.1142/S0129055X90000107)_

Anyons in the quantum Hall liquids

References on anyon-excitations (satisfying braid group statistics) in the quantum Hall effect (for more on the application to topological quantum computation see the references there):

The prediction of abelian anyon-excitations in the quantum Hall effect (i.e. satisfying braid group statistics in 1-dimensional linear representations of the braid group):

• B. I. Halperin, Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States, Phys. Rev. Lett. 52, 1583 (1984) (doi:10.1103/PhysRevLett.52.1583)

  Erratum Phys. Rev. Lett. 52, 2390 (1984) (doi:10.1103/PhysRevLett.52.2390.4)

• Daniel Arovas, J. R. Schrieffer, Frank Wilczek, Fractional Statistics and the Quantum Hall Effect, Phys. Rev. Lett. 53, 722 (1984) (doi:10.1103/PhysRevLett.53.722)

The original discussion of non-abelian anyon-excitations in the quantum Hall effect (i.e. satisfying braid group statistics in higher dimensional linear representations of the braid group, related to modular tensor categories):

• Gregory Moore, Nicholas Read, Nonabelions in the fractional quantum Hall effect, Nucl. Phys. 360B(1991)362 (pdf, doi:10.1016/0550-3213(91)90407-O)

Review:

• Ady Stern, Anyons and the quantum Hall effect – A pedagogical review, Annals of Physics Volume 323, Issue 1, January 2008, Pages 204-249 (doi:10.1016/j.aop.2007.10.008, arXiv:0711.4697)

Anyons in topological superconductors

On anyon-excitations in topological superconductors.

via Majorana zero modes:

Original proposal:

• Nicholas Read, Dmitry Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect, Phys. Rev. B61:10267, 2000 (arXiv:cond-mat/9906453)

Review:

• Sankar Das Sarma, Michael Freedman, Chetan Nayak, Majorana Zero Modes and Topological Quantum Computation, npj Quantum Information 1, 15001 (2015) (nature:npjqi20151)

• Nur R. Ayukaryana, Mohammad H. Fauzi, Eddwi H. Hasdeo, The quest and hope of Majorana zero modes in topological superconductor for fault-tolerant quantum computing: an introductory overview (arXiv:2009.07764)

Further development:

• Meng Cheng, Victor Galitski, Sankar Das Sarma, Non-adiabatic Effects in the Braiding of Non-Abelian Anyons in Topological Superconductors, Phys. Rev. B 84, 104529 (2011) (arXiv:1106.2549)

via Majorana zero modes restricted to edges of topological insulators:

• Biao Lian, Xiao-Qi Sun, Abolhassan Vaezi, Xiao-Liang Qi, and Shou-Cheng Zhang, Topological quantum computation based on chiral Majorana fermions, PNAS October 23, 2018 115 (43) 10938-10942; first published October 8, 2018 (doi:10.1073/pnas.1810003115)

Experimental detection of anyons

While the occurrence of anyon-excitations in the quantum Hall effect is a robust theoretical prediction (see the references above), and while the quantum Hall effect itself has long been established in experiment, the actual observation of anyons in these systems is subtle:

An early claim of the observation of non-abelian anyons seems to remain unconfirmed:

• Sanghun An, P. Jiang, H. Choi, W. Kang, S. H. Simon, L. N. Pfeiffer, K. W. West, K. W. Baldwin, Braiding of Abelian and Non-Abelian Anyons in the Fractional Quantum Hall Effect (arXiv:1112.3400)

The claimed observation of abelian anyons is apparently more securely established:

• H. Bartolomei, M. Kumar, R. Bisognin, A. Marguerite, J.-M. Berroir, E. Bocquillon, B. Plaçais, A. Cavanna, Q. Dong, U. Gennser, Y. Jin, G. Fève:

  Fractional statistics in anyon collisions, Science 368, 173-177 (2020) (arXiv:2006.13157)

• James Nakamura, Shuang Liang, Geoffrey C. Gardner, Michael J. Manfra, Direct observation of anyonic braiding statistics, Nat. Phys. 16, 931–936 (2020). (arXiv:2006.14115, doi:10.1038/s41567-020-1019-1)

• Bob Yirka, Best evidence yet for existence of anyons, PhysOrg News July 10, 2020 (phys.org/news/2020-07)

Topological quantum computation with anyons

The idea of topological quantum computation via the Chern-Simons theory of anyons (e.g. in the quantum Hall effect) is due to:

• Alexei Kitaev, Fault-tolerant quantum computation by anyons, Annals Phys. 303 (2003) 2-30 (arXiv:quant-ph/9707021, doi:10.1016/S0003-4916(02)00018-0)

• Michael Freedman, Alexei Kitaev, Michael Larsen, Zhenghan Wang, Topological quantum computation, Bull. Amer. Math. Soc. 40 (2003), 31-38 (arXiv:quant-ph/0101025, doi:10.1090/S0273-0979-02-00964-3, pdf)

• Michael Freedman, Michael Larsen, Zhenghan Wang, A modular functor which is universal for quantum computation, Communications in Mathematical Physics. 2002, Vol 227, Num 3, pp 605-622 (arXiv:quant-ph/0001108)

• D. Melnikov, A. Mironov, S. Mironov, A. Morozov, An. Morozov, Towards topological quantum computer, Nucl. Phys. B926 (2018) 491-508 (arXiv:1703.00431, doi:10.1016/j.nuclphysb.2017.11.016)

Textbook accounts:

• Zhenghan Wang, Topological Quantum Computation, CBMS Regional Conference Series in Mathematics 112, AMS 2010 (ISBN-13: 978-0-8218-4930-9, pdf)

• Tudor D. Stanescu, Part IV of: Introduction to Topological Quantum Matter & Quantum Computation, CRC Press 2020 (ISBN:9780367574116)

Review:

• Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, Sankar Das Sarma, Non-Abelian Anyons and Topological Quantum Computation, Rev. Mod. Phys. 80, 1083 (2008) (arXiv:0707.1888)

• Ady Stern, Netanel H. Lindner, Topological Quantum Computation – From Basic Concepts to First Experiments, Science 08 Mar 2013: Vol. 339, Issue 6124, pp. 1179-1184 (doi:10.1126/science.1231473)

• Ville Lahtinen, Jiannis K. Pachos, A Short Introduction to Topological Quantum Computation, SciPost Phys. 3, 021 (2017) (arXiv:1705.04103)

• Eric Rowell, Zhenghan Wang, Mathematics of Topological Quantum Computing, Bull. Amer. Math. Soc. 55 (2018), 183-238 (arXiv:1705.06206, doi:10.1090/bull/1605)

• Bernard Field, Tapio Simula, Introduction to topological quantum computation with non-Abelian anyons, Quantum Science and Technology 2018 (arXiv:1802.06176, doi:10.1088/2058-9565/aacad2)

Realization in experiment:

• Daniel Nigg, Markus Mueller, Esteban A. Martinez, Philipp Schindler, Markus Hennrich, Thomas Monz, Miguel A. Martin-Delgado, Rainer Blatt,

  Experimental Quantum Computations on a Topologically Encoded Qubit, Science 18 Jul 2014: Vol. 345, Issue 6194, pp. 302-305 (arXiv:1403.5426, doi:10.1126/science.1253742)

  (for quantum error correction)

See also:

• Mohamed Taha Rouabah, Compiling single-qubit braiding gate for Fibonacci anyons topological quantum computation (arXiv:2008.03542)

Braid group representations (as topological quantum gates)

On linear representations of braid groups (see also at braid group statistics and interpretation as quantum gates in topological quantum computation):

Review:

• Camilo Arias Abad, Introduction to representations of braid groups, Rev. colomb. mat. vol.49 no.1 (2015) (arXiv:1404.0724, doi:10.15446/recolma.v49n1.54160)

in relation to modular tensor categories:

• Colleen Delaney, Lecture notes on modular tensor categories and braid group representations, 2019 (pdf, pdf)

See also:

• R. B. Zhang, Braid group representations arising from quantum supergroups with arbitrary $q$ and link polynomials, Journal of Mathematical Physics 33, 3918 (1992) (doi:10.1063/1.529840)

As quantum gates for topological quantum computation with anyons:

• Louis H. Kauffman, Samuel J. Lomonaco, Braiding Operators are Universal Quantum Gates, New Journal of Physics, Volume 6, January 2004 (arXiv:quant-ph/0401090, doi:10.1088/1367-2630/6/1/134)

• Samuel J. Lomonaco, Louis Kauffman, Topological Quantum Computing and the Jones Polynomial, Proc. SPIE 6244, Quantum Information and Computation IV, 62440Z (12 May 2006) (arXiv:quant-ph/0605004)

  (braid group representation serving as a topological quantum gate to compute the Jones polynomial)

• C.-L. Ho, A.I. Solomon, C.-H.Oh, Quantum entanglement, unitary braid representation and Temperley-Lieb algebra, EPL 92 (2010) 30002 (arXiv:1011.6229)

• Louis H. Kauffman, Majorana Fermions and Representations of the Braid Group, International Journal of Modern Physics AVol. 33, No. 23, 1830023 (2018) (arXiv:1710.04650, doi:10.1142/S0217751X18300235)