So far non-abelian quasiparticles have yet to be observed in quantum hall states. However, as far as theory is concerned, in a recent paper a rigorous proof is provided that such quasiparticles *do* exist in quantum hall states. There were many candidate states for this object, starting with the Moore-Read state, however, theoretically there did not yet exist a complete proof of their existence.

This is a really big achievement. Frank Wilzcek describes it as "a landmark proof".

There have been various incompletely verified "sightings" of the $5/2$ state which is the non-abelian state. For results upto 2008 at least, see the RMP paper on topological quantum computation. At the end of this paper various references are given which claim to have observed this state. I don't know what the latest is, but I would have heard of an definitive experimental observation, or so I like to think :)

This post imported from StackExchange Physics at 2014-04-01 17:33 (UCT), posted by SE-user user346