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 anyonons” ($k = 2$) 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 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 $\mathfrak{su}(2)$-anyon species are the Fibonacci anyons at level $k = 3$ (e.g. Simeon 2021).

Related concepts

• parafermion

• 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:

• 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 $SU(2)_k$ Anyon Models, PRX Quantum 2 010334 (2021) $[$arXiv:2008.10790$]$

Discussion of Fibonacci anyons:

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

Experimental realization

• 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]

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, 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$]$