Topological Physics – Phenomena in physics controlled by the topology (often: the homotopy theory) of the physical system.
General theory:
In metamaterials:
topological phononics (sound waves?)
For quantum computation:
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 $\widehat{\mathfrak{g}}^k$ (at some level $k$), in that their wavefunctions are, essentially, $\widehat{g}$-conformal blocks and their braiding is described by $G$-Chern-Simons theory at level $k$ (possibly fractional, see at logarithmic CFT here).
If here $\mathfrak{g} =$ $\mathfrak{su}(2)$, then one also speaks of “SU(2)-anyons” (with varying conventions on capitalization, etc.). With “Majorana anyons” ($k = 2$) and “Fibonacci anyons” ($k = 3$) this class subsumes most or all anyon species which seem to have a realistic chance of existing in nature.
Notably Majorana anyons (in the guise of “Majorana zero modes” in super/semi-conducting nanowires) 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 (following the plan laid out in Das Sarma, Freedman & Nayak 15). After initial claims had to be retracted in 2021 [doi:10.1038/s41586-021-03373-x, doi:10.5281/zenodo.4587841, doi:10.5281/zenodo.4545812, TU Delft press release] and again in 2022 [doi:10.1038/s41586-022-04704-2] (further claims are under criticism, see e.g. doi:10.5281/zenodo.6344447, doi:10.5281/zenodo.6325378 and the list here) there is a new claim of detection by Nayak 22 & MicrosoftQuantum 22, but see Frolov & Mourik 22a, 22b and Frolov 22.
In any case, Majorana anyons are known not to be universal (not all quantum gates may be approximated with Majorana braiding). The simplest universal $\mathfrak{su}(2)$-anyon species are the Fibonacci anyons at level $k = 3$ (e.g. Simeon 2021, also Kolganov, Mironov & Morozon 2023).
Early consideration of $\mathfrak{su}(2)$-anyons is implicit in the context of Laughlin wavefunctions due to
Early discussion of topological quantum computation in $SU(2)$-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:
Simon Trebst, Matthias Troyer, Zhenghan Wang, 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, Simon Trebst, D. A. Huse, A. W. W. Ludwig, Matthias Troyer, Zhenghan 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]
Emil Génetay-Johansen, Tapio Simula, Fibonacci anyons versus Majorana fermions – A Monte Carlo Approach to the Compilation of Braid Circuits in $SU(2)_k$ Anyon Models, PRX Quantum 2 010334 (2021) [arXiv:2008.10790, doi:10.1103/PRXQuantum.2.010334]
Discussion of Fibonacci anyons:
Discussion of universality at higher level $k$ (and also for $SU(N)$-anyons with $N \gt 2$):
987 (2023) 116072 [arXiv:2105.03980, doi:10.1016/j.nuclphysb.2023.116072]
The general strategy of realizing Majorana zero modes in supercondocuting/semiconducting nanowires is due to
Alexei Kitaev, Unpaired Majorana fermions in quantum wires, Physics-Uspekhi, 44 10S (2001) 131-136 [doi:10.1070/1063-7869/44/10S/S29]
Roman M. Lutchyn, Jay D. Sau, Sankar Das Sarma, Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures, Phys. Rev. Lett. 105 (2010) 077001 [doi:10.1103/PhysRevLett.105.077001]
Yuval Oreg, Gil Refael, Felix von Oppen, Helical Liquids and Majorana Bound States in Quantum Wires, Phys. Rev. Lett. 105 (2010) 177002 [doi:10.1103/PhysRevLett.105.177002, arXiv:1003.1145]
reviewed in:
On the general problem of distinguishing the expected effect from noise:
we believe that similar confirmation bias applies to many other topological discovery claims in the literature during 2000–2020 where a precise knowledge of what one is looking for has been the key factor in the discovery claim, with the experimental quantization results themselves not being sufficiently compelling. […] Our results certainly apply to most of the Majorana experiments during 2012–2021 in the literature.
Non-retracted claims of experimental realization of something in the direction of Majorana zero modes:
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)
Microsoft Quantum, InAs-Al Hybrid Devices Passing the Topological Gap Protocol [arXiv:2207.02472]
but see commentary in:
Sergey M. Frolov, Vincent Mourik, We cannot believe we overlooked these Majorana discoveries [arXiv:2203.17060, doi:10.5281/zenodo.6364928, conclusion on: p. 7]
Sergey M. Frolov, Vincent Mourik: Majorana Fireside Podcast, Episode 1: The Microsoft TGP paper live review [video, conclusion at: 1:01:31]
1:01:52 The signal is fully consistent, from what we see, with not having discovered any Majorana or topological superconductivity here. At the same time, the amount of data is extremely narrow.
Sergey M. Frolov, Superconductors and semiconductors, nanowires and majorana, research and integrity [video, general caution: 55:34, concrete criticism: 1:01:41]
1:01:50: The claims of the discovery of Majorana have been overblown and are false. Majorana has not been discovered in nanowires. I don’t believe in any other system it has been discovered either.
On how this could happen:
Elizabeth Gibney, Inside Microsoft’s quest for a topological quantum computer (Interview with Alex Bocharov), Nature (2016) [doi:10.1038/nature.2016.20774]
[Bocharov:] We’re people-centric, rather than problem-centric.
See also:
Sergey M. Frolov, So, You Think You Discovered a New State of Matter?, Physics 14 68 (2021) [physics.aps:v14/68]
Sergey M. Frolov, Quantum computing’s reproducibility crisis: Majorana fermions, Nature 592 (2021) 350-352 [doi:10.1038/d41586-021-00954-8]
Proposal to realize Fibonacci anyons on quasicrystal-states:
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, Non-Abelian statistics in the fractional quantum Hall states, Phys. Rev. Lett. 66 (1991) 802 [doi:10.1103/PhysRevLett.66.802, pdf]
B. Blok, Xiao-Gang Wen, Many-body systems with non-abelian statistics, Nuclear Physics B 374 3 (1992) 615-646 [doi:10.1016/0550-3213(92)90402-W]
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]
Review in the broader context of the CS-WZW correspondence:
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 [doi:10.1016/S0550-3213(97)00351-9, arXiv:cond-mat/9701212]
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]
Eddy Ardonne, Kareljan Schoutens: Wavefunctions for topological quantum registers, Annals Phys. 322 (2007) 201-235 [doi:10.1016/j.aop.2006.07.015, arXiv:cond-mat/0606217]
Ludmil Hadjiivanov, Lachezar S. Georgiev, Braiding Fibonacci anyons [arxiv:2404.01778]
Last revised on May 31, 2024 at 10:11:05. See the history of this page for a list of all contributions to it.