I'm trying to understand the basics of anyons physics. I understand there is neither a Fock space they live in (because Fock is just the space of (anti-)symmetrized tensor product state, see e.g. Wikipedia), nor a (pseudo / fictitious) commutation relation for them, as discussed at this Phys.SE post. But still I've a few questions regarding their statistics:

  • Can we associate a creation / destruction operator to an anyon mode ? Does it make sense to talk about mode of anyons?

  • Is there a general occupation function like Fermi-Dirac or Bose-Einstein for fermions or bosons ? Is it model dependent, i.e. does it depend on the type of anyon ? Does it make sense to discuss number of anyons?

  • What is the ground state of anyons, like a Fermi sea or a Bose-Einstein condensate for fermions or bosons ? Does it make sense to talk about a ground state of a gas of anyons?

I believe this bunch of questions can all be contracted to

  • Does it make sense to talk of the anyons as particles ?

Because in principle a particle exists independently of the Fock space construction, isn't it ? We could still construct a space of the tensor product of non (anti-)symmetrised states.

I realised that a perhaps better approach on the question would be:

  • To which extend is the anyon statistic a (quantum) statistic ?

provided the two other quantum statistics I know (Bose and Fermi) provide a ground state, an occupation number, and some second-quantised version of operators.

Post-scriptum : This Phys.SE question is partially related to mine.

  • $\begingroup$ @Qmechanic Thanks for the corrections, it's really easier to read now :-) $\endgroup$
    – FraSchelle
    May 23, 2013 at 22:11
  • $\begingroup$ Have you found yourself the answer? This question is really old, and I'm curious myself; if you've found the answer already, please write it, because I really want to know more about this topic. $\endgroup$
    – Vendetta
    Oct 30, 2015 at 18:35
  • $\begingroup$ @Vendetta Unfortunately not, otherwise I would have written the answer. Thanks for your interest in this question. I've tried to add an other tag, to see if more people will see it :-) $\endgroup$
    – FraSchelle
    Oct 31, 2015 at 11:35

1 Answer 1


Let me partially address the question as to whether annoys and be considered particles or not.

As you have rightful pointed out the anyonic nature of the braiding relation means that a conventional Fock representation as creation and annihilation operators acting on a vacuum is difficult to use.

One should however consider anyons as proper particles in the sense that they have (usually) a finite energy gap to the ground state out of which they are created and also has a finite lifetime.

An easy example here is the creation anyons in the quantum Hall effect at $\frac13$ filling for instance. Here pairs of anyons can be created out of the background quantum hall fluid. These excitations have fractional electric changes (this is related to them being anyons) but cost a finite energy and have a finite life time.

Anyons can also be created in the Quantum Hall system when the number of flux quantum $N_\phi$ compared to the number of electrons $N_e$ is not exactly $N_e/N_\phi=\frac13$. Essentially for each extra magnetic flux an anyonic quasi-hole is created and these are stable as they make up the ground state at that filling fraction.


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