# nLab su(2)-anyon

Contents

### Context

#### Topological physics

Topological Physics – Phenomena in physics controlled by the topology (often: the homotopy theory) of the physical system.

General theory:

# Contents

## Idea

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).

## References

### General

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

• 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$]$

Early discussion of topological quantum computation in $SU(2)$-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.

#### Other

Proposal to realize Fibonacci anyons on quasicrystal-states:

### Laughlin wavefunctions as 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:

Specifically for su(2)-anyons:

Last revised on November 26, 2022 at 07:31:41. See the history of this page for a list of all contributions to it.