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 $k$.

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 currently 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 dramatic setback in 2021 and again in 2022).

References

• 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$]$

• Chetan Nayak, Microsoft has demonstrated the underlying physics required to create a new kind of qubit, Microsoft Research Blog (March 2022)