A common way to show that anyons exhibit fractional statistics in 2D is by arguing that the paths of two anyons winding round each other cannot be continuously deformed to zero. This seems to assume that the particles cannot pass through each other. Why is this assumption valid?
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$\begingroup$ The configurations in which the particles overlap is singular (e.g. because the interaction energies etc. blow up) so a priori, one isn't allowed to assume that things are smooth around this point. Indeed, for the existence of nontrivial anyons, the point is singular. The burden of proof is on the opposite side than you suggest. If you wanted to prove that there are no anyons because the paths may be deformed - through the coincident positions - you would have to show that it's OK to get through this singular point. In trying to prove so, you would fail because it's not legitimate. $\endgroup$– Luboš MotlCommented Apr 28, 2012 at 9:22
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$\begingroup$ So, does that mean that particles with no mutual interaction cannot exhibit fractional statistics? $\endgroup$– leongzCommented Apr 28, 2012 at 10:04
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1$\begingroup$ Statistics has nothing to do with interactions, just with dimensions. What Lubos is saying [please correct me if I am mistaken] is that if you try to make a path contractible through the point where the particle at time $t$ is, you have to show that contracting through that singular point is a an allowed operation, but it is not. $\endgroup$– DaniCommented Apr 28, 2012 at 10:36
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$\begingroup$ Why is contracting through that singular point not allowed, other than the fact that the interaction energy blows up? $\endgroup$– leongzCommented Apr 29, 2012 at 7:39
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$\begingroup$ It is not just that the interaction energy blows up. The point where the particle is encodes a [phase] singularity and the configuration space is no longer smooth. $\endgroup$– DaniCommented Apr 29, 2012 at 8:34
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1 Answer
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We do not need to make the assumption that "the paths of two anyons winding round each other cannot be continuously deformed to zero".
To define fractional statistics, we only require that the phase of exchanging two particles do not depend on the smooth deformation of the exchange path, as long as two particles are always well separated during the exchange.
Two anyons can coincide with a finite energy cost.
answered May 29, 2012 at 15:03
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$\begingroup$ Would you please elaborate a bit more about your last sentence "Two anyons can coincide with a finite energy cost." If I understand correctly, it would cost infinite energy to put two fermions to coincide (a poor-man Pauli principle, say), zero energy for bosons, and some energy for anyons. Is it dependent on the model, or only on the anionic statistics for instance ? Thanks in advance. $\endgroup$ Commented May 23, 2013 at 21:29