A gentle yet comprehensive introduction to the concept of abelian and non-abelian statistics will be much appreciated.
- 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.
- According to wikipedia, in 1988, Jurg Frohlich showed that non-abelian statistics existed and were valid, in his paper Statistics of Fields, the Yang-Baxter Equation, and the Theory of Knots and Links (note there's a paywall there).
- Again according to wikipedia, Gregory Moore, Nicholas Read, and Xiao-Geng Wen showed non-abelian statistics can be realized in the fractional quantum Hall effect, and wrote two papers, Non-Abelian Statistics in the Fractional Quantum Hall States (Wen) and Nonabelions In the Fractional Quantum Hall Effect (Read and Moore; see section 2.1 in particular).
- 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!
I'll give you what resources I have on this topic:
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.
A good, simple introduction to braiding with anyons and its connection to Majorana fermions. Applications to quantum computing included.
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.
Brief, but possibly a solid starting point. Good links and references as well.
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.
Video introducing the concept of anyons to a general audience, from the Perimeter Institute in Waterloo, Ontario. Connections to superconductors are drawn.
Good introduction to the subject by Frank Wilczek, but a bit dated. Has a nice selection of original papers on the topic.
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.