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A gentle yet comprehensive introduction to the concept of abelian and non-abelian statistics will be much appreciated.

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Before answering, please see our policy on resource recommendation questions. Please write substantial answers that detail the style, content, and prerequisites of the book, paper or other resource. Explain the nature of the resource so that readers can decide which one is best suited for them rather than relying on the opinions of others. Answers containing only a reference to a book or paper will be removed!

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    $\begingroup$ Maybe not the simplest, but certainly clear and worth your time: from the man himself, Kitaev's "Anyons in an exactly solved model and beyond" (arXiv:cond-mat/0506438). The introduction nicely describes abelian anyons/braiding, chapter 8 (and appendix E) describes the non-abelian case. $\endgroup$ – Ruben Verresen Jun 4 '17 at 15:31
  • $\begingroup$ @RubenVerresen how is that not an answer? $\endgroup$ – Wolpertinger Jun 5 '17 at 8:56
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Abelian

  • According to wikipedia, quasiparticles whose statistics range between Fermi-Dirac and Bose-Einstein statistics are called anyons; these particles may only exist in two dimensions. These particles obey fractional statistics, so called because they have fractional spin; see Frank Wilczek's Quantum Mechanics of Fractional-Spin Particles where he also coins the term anyon.
  • In the same vein, see Jon Magne Leinaas and Jan Myrheim's paper On the Theory of Identical Particles where they showed quasiparticles can indeed be observed in two dimensions.

Non-abelian

Other/general

  • Fractional Statistics and Quantum Theory (book) by Avinash Khare covers, as the title indicates, anyons and their statistics. The copyright is 1997, so it won't talk much about the applications of anyons to quantum computing; however, it covers pretty much every topic related to abelian/non-abelian statistics as far as I can tell, so if you're willing to pay money (the Google Books sample is decent, but doesn't include every page, obviously) this is probably your best bet. If you do decide this is what you want, the hardcover copy on Amazon is $25, so not too bad compared to a lot of technical books. Mathematical Reviews said

    The overall style is clear and pedagogical, with emphasis on symmetry and simplicity.

  • If you wish to read about the construction of quantum computation theory using anyons, the paper Topological quantum computation will be of use.
  • A set of lectures transcribed into a paper on arXiv called Introduction to abelian and non-abelian anyons can be found, and they appear to cover all topics of interest.
  • A powerpoint for a talk on Fractional Quantum Statistics also appears to cover everything of interest, though as a powerpoint it may not describe everything in complete detail.

I'll add more papers/books/resources as I find them. Hope these help!

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I'll give you what resources I have on this topic:

General Reviews

  1. Lecture Notes on Anyonic Defects in Condensed Matter

    Lecture notes from a summer school in Florence. A good reference for Majorana zero modes and anyonic defects in condensed matter systems. Large amount of detail--pretty clear.

  2. Braiding with Majorana Fermions

    A good, simple introduction to braiding with anyons and its connection to Majorana fermions. Applications to quantum computing included.

  3. An Anyon Primer

    Excellent introduction to the field with problems, though it can get a little advanced in some places. Applications to the fractional quantum hall effect included.

  4. Quantiki's Introduction to Anyons

    Brief, but possibly a solid starting point. Good links and references as well.

  5. This StackExchange question on non-Abelian anyons in solid state physics might also be useful. See also this StackExchange question for some nice clarifications and links.

Presentations

  1. Fractional Quantum Statistics

    Great presentation on anyons and fractional statistics. Very simple and easily understood. Draws analogies and examples from simple quantum systems. An undergraduate with a background in Griffiths could understand the majority of this presentation.

  2. Fractional Statistics in Two Dimensions: Anyon There?

    Video introducing the concept of anyons to a general audience, from the Perimeter Institute in Waterloo, Ontario. Connections to superconductors are drawn.

Books

  1. Fractional Statistics and Anyon Superconductivity

    Good introduction to the subject by Frank Wilczek, but a bit dated. Has a nice selection of original papers on the topic.

  2. Anyons: Quantum Mechanics of Particles with Fractional Statistics

    By far the best resource on this list. Alberto Lerda's book is the book on anyonic physics, and the best, clearest resource on the subject. Covers a wide range of subject matter, and is understandable at the advanced undergraduate level.

I hope these help. Sorry if there are some duplicates with heather's reply--I tried to list resources that I've found helpful in my own study of fractional statistics.

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