nLab su(2)-anyon



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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 𝔤^ k\widehat{\mathfrak{g}}^k (at some level kk), in that their wavefunctions are, essentially, g^\widehat{g}-conformal blocks and their braiding is described by GG-Chern-Simons theory at level kk (possibly fractional, see at logarithmic CFT here).

If here 𝔤=\mathfrak{g} = 𝔰𝔲 ( 2 ) \mathfrak{su}(2) , then one also speaks of “SU(2)-anyons” (with varying conventions on capitalization, etc.). With “Majorana anyons” (k=2k = 2) and “Fibonacci anyons” (k=3k = 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 𝔰𝔲(2)\mathfrak{su}(2)-anyon species are the Fibonacci anyons at level k=3k = 3 (e.g. Simeon 2021).



Early consideration of 𝔰𝔲(2)\mathfrak{su}(2)-anyons is implicit in the context of Laughlin wavefunctions due to

Early discussion of topological quantum computation in SU(2)SU(2)-Chern-Simons theory:

More concrete discussion of these phenomena in terms of anyons:

Discussion of Fibonacci anyons:

  • Ryan Simeon, Universality of Fibonacci anyons in topological quantum computing (2021) [[pdf]]

Experimental realization

(Non-)Observation of Majorana zero modes

The general strategy of realizing Majorana zero modes in supercondocuting/semiconducting nanowires is due to

reviewed in:

  • Pasquale Marra: Majorana nanowires for topological quantum computation: A tutorial [arXiv:2206.14828]

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:


Proposal to realize Fibonacci anyons on quasicrystal-states:

Laughlin wavefunctions as conformal blocks

Relating anyonic topologically ordered Laughlin wavefunctions to conformal blocks:

Specifically for logarithmic CFT:

Specifically for su(2)-anyons:

Last revised on September 11, 2022 at 12:12:41. See the history of this page for a list of all contributions to it.