nLab
braid group statistics

Contents

Contents

Idea

In quantum physics one speaks of the “statistics” of a particle species or other object when referring to the irreducible representation of the group which exchanges identical such particles that it belongs to. Typically this is the symmetric group and accordingly one speaks of boson and fermion statistics. But in special cases, namely whenever the codimension in space of the objects in question is 1 (e.g. point particles in 2-dimensional quantum field theories), accordingly in this situation these may have what is then called braid group statistics and one speaks of anyons.

References

General

The concept of anyons satisfying braid group statistics goes back to:

Rigorous discussion in terms of superselection sectors in algebraic quantum field theory:

Anyons in the quantum Hall liquids

References on anyon-excitations (satisfying braid group statistics) in the quantum Hall effect (for more on the application to topological quantum computation see the references there):

The prediction of abelian anyon-excitations in the quantum Hall effect (i.e. satisfying braid group statistics in 1-dimensional linear representations of the braid group):

The original discussion of non-abelian anyon-excitations in the quantum Hall effect (i.e. satisfying braid group statistics in higher dimensional linear representations of the braid group, related to modular tensor categories):

Review:

Anyons in topological superconductors

On anyon-excitations in topological superconductors.

via Majorana zero modes:

Original proposal:

  • Nicholas Read, Dmitry Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect, Phys. Rev. B61:10267, 2000 (arXiv:cond-mat/9906453)

Review:

Further development:

  • Meng Cheng, Victor Galitski, Sankar Das Sarma, Non-adiabatic Effects in the Braiding of Non-Abelian Anyons in Topological Superconductors, Phys. Rev. B 84, 104529 (2011) (arXiv:1106.2549)

via Majorana zero modes restricted to edges of topological insulators:

  • Biao Lian, Xiao-Qi Sun, Abolhassan Vaezi, Xiao-Liang Qi, and Shou-Cheng Zhang, Topological quantum computation based on chiral Majorana fermions, PNAS October 23, 2018 115 (43) 10938-10942; first published October 8, 2018 (doi:10.1073/pnas.1810003115)

Experimental detection of anyons

While the occurrence of anyon-excitations in the quantum Hall effect is a robust theoretical prediction (see the references above), and while the quantum Hall effect itself has long been established in experiment, the actual observation of anyons in these systems is subtle:

An early claim of the observation of non-abelian anyons seems to remain unconfirmed:

  • Sanghun An, P. Jiang, H. Choi, W. Kang, S. H. Simon, L. N. Pfeiffer, K. W. West, K. W. Baldwin, Braiding of Abelian and Non-Abelian Anyons in the Fractional Quantum Hall Effect (arXiv:1112.3400)

The claimed observation of abelian anyons is apparently more securely established:

Topological quantum computation with anyons

The idea of topological quantum computation via the Chern-Simons theory of anyons in the quantum Hall effect is due to:

Further discussion:

Review:

Last revised on February 17, 2021 at 00:51:37. See the history of this page for a list of all contributions to it.