Topological Physics – Phenomena in physics controlled by the topology (often: the homotopy theory) of the physical system.
General theory:
In electromagnetism:
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.
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)_
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):
Review:
On anyon-excitations in topological superconductors.
via Majorana zero modes:
Original proposal:
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:
via Majorana zero modes restricted to edges of topological insulators:
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:
The claimed observation of abelian anyons is apparently more securely established:
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, (phys.org/news/2020-07)
The idea of topological quantum computation via the Chern-Simons theory of anyons 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)
Further discussion:
Samuel J. Lomonaco Jr., 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)
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)
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)
Review:
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)
Last revised on February 17, 2021 at 00:51:37. See the history of this page for a list of all contributions to it.