nLab su(2)-anyon

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, also Kolganov, Mironov & Morozon 2023).

Related concepts

• parafermion

• Majorana zero mode

• Majorana dimer

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:

• 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:

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

• Ludmil Hadjiivanov, Lachezar S. Georgiev: Braiding Fibonacci anyons [arXiv:2404.01778]

Discussion of universality at higher level $k$ (and also for $SU(N)$-anyons with $N \gt 2$):

• Nikita Kolganov, Sergey Mironov, Andrey Morozov, Large $k$ topological quantum computer, Nuclear Physics B

  987 (2023) 116072 [arXiv:2105.03980, doi:10.1016/j.nuclphysb.2023.116072]

Experimental realization

(Non-)Observation of Majorana zero modes

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:

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

On the general problem of distinguishing the expected effect from noise:

• Sankar Das Sarma, Haining Pan, Disorder-induced zero-bias peaks in Majorana nanowires Phys. Rev. B 103 195158 (2021) [arXiv:2103.05628, doi:10.1103/PhysRevB.103.195158]

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]

Other

Proposal to realize Fibonacci anyons on quasicrystal-states:

• Marcelo Amaral, David Chester, Fang Fang, Klee Irwin, Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing, Symmetry 14 9 (2022) 1780 [arXiv:2207.08928, doi:10.3390/sym14091780]

Laughlin wavefunctions as conformal blocks

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:

• Daniel C. Cabra, Gerardo L. Rossini, Explicit connection between conformal field theory and 2+1 Chern-Simons theory, Mod. Phys. Lett. A 12 (1997) 1687-1697 [doi:10.1142/S0217732397001722, arXiv:hep-th/9506054]

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]