Topological Physics – Phenomena in physics controlled by the topology (often: the homotopy theory) of the physical system.
General theory:
In electromagnetism:
quantum algorithms:
In solid state physics, many or all anyon-species of (potential) practical interest (such as for topological quantum computation) are thought to be characterized by affine Lie algebras (at some level ), in that their wavefunctions are, essentially, -conformal blocks and their braiding is described by -Chern-Simons theory at level (possibly fractional, see at logarithmic CFT here).
If here , then one also speaks of “SU(2)-anyons” (with varying conventions on capitalization, etc.). With “Majorana anyonons” () and “Fibonacci anyons” () this class subsumes most or all anyon species which seem to have a realistic chance of existing in nature.
Notably the ‘Majorana anyons (in the guise of “Majorana zero modes”) are (or were until recently, see arXiv:2106.11840v4, p. 3) at the focus of attention of an intense effort to finally provide a practical proof of principle for the old idea of topological quantum computation (see the plan of Das Sarma, Freedman & Nayak 15 and the latest informal announcement Nayak 22, after a setback in 2021 and again in 2022).
On the other hand, Majorana anyons are known not to be universal (not all quantum gates may be approximated with Majorana braiding). The simplest universal -anyon species are the Fibonacci anyons at level (e.g. Simeon 2021).
Early consideration of -anyons is implicit in the context of Laughlin wavefunctions due to
Early discussion of topological quantum computation in -Chern-Simons theory:
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)
More concrete discussion of these phenomena in terms of anyons:
S. Trebst, M. Troyer, Z. Wang and A. W. W. Ludwig, A short introduction to Fibonacci anyon models, Prog. Theor. Phys. Supp. 176 384 (2008) arXiv:0902.3275, doi:10.1143/PTPS.176.384
C. Gils, E. Ardonne, S. Trebst, D. A. Huse, A. W. W. Ludwig, M. Troyer, and Z. Wang, Anyonic quantum spin chains: Spin-1 generalizations and topological stability, Phys. Rev. B 87 (2013), 235120 doi:10.1103/PhysRevB.87.235120, arXiv:1303.4290
Sankar Das Sarma, Michael Freedman, Chetan Nayak, Majorana zero modes and topological quantum computation, npj Quantum Inf 1 15001 (2015) doi:10.1038/npjqi.2015.1
E. G. Johansen, T. Simula, Fibonacci anyons versus Majorana fermions – A Monte Carlo Approach to the Compilation of Braid Circuits in Anyon Models, PRX Quantum 2 010334 (2021) arXiv:2008.10790
Discussion of Fibonacci anyons:
Gerbold C. Ménard, Andrej Mesaros, Christophe Brun, François Debontridder, Dimitri Roditchev, Pascal Simon, Tristan Cren, Isolated pairs of Majorana zero modes in a disordered superconducting lead monolayer, Nat Commun 10 2587 (2019) doi:10.1038/s41467-019-10397-5
Chetan Nayak, Microsoft has demonstrated the underlying physics required to create a new kind of qubit, Microsoft Research Blog (March 2022)
Pasquale Marra: Majorana nanowires for topological quantum computation: A tutorial [arXiv:2206.14828]
Relating anyonic topologically ordered Laughlin wavefunctions to conformal blocks:
Gregory Moore, Nicholas Read, Section 2.2 of: Nonabelions in the fractional quantum hall effect, Nuclear Physics B 360 2–3 (1991) 362-396 doi:10.1016/0550-3213(91)90407-O, pdf
Xiao-Gang Wen, Yong-Shi Wu, Chiral operator product algebra hidden in certain fractional quantum Hall wave functions, Nucl. Phys. B 419 (1994) 455-479 doi:10.1016/0550-3213(94)90340-9
Specifically for logarithmic CFT:
Victor Gurarie, Michael Flohr, Chetan Nayak, The Haldane-Rezayi Quantum Hall State and Conformal Field Theory, Nucl. Phys. B 498 (1997) 513-538 arXiv:cond-mat/9701212, doi:10.1016/S0550-3213%2897%2900351-9
Michael Flohr, §5.4 in: Bits and pieces in logarithmic conformal field theory, International Journal of Modern Physics A, 18 25 (2003) 4497-4591 doi:10.1142/S0217751X03016859, arXiv:hep-th/0111228
Specifically for su(2)-anyons:
Kazusumi Ino, Modular Invariants in the Fractional Quantum Hall Effect, Nucl. Phys. B 532 (1998) 783-806 doi:10.1016/S0550-3213(98)00598-7, arXiv:cond-mat/9804198
Nicholas Read, Edward Rezayi, Beyond paired quantum Hall states: Parafermions and incompressible states in the first excited Landau level, Phys. Rev. B 59 (1999) 8084 doi:10.1103/PhysRevB.59.8084
Last revised on July 11, 2022 at 03:03:19. See the history of this page for a list of all contributions to it.