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It is usually said that anyon statistics are not well-defined if the anyons are massless. If I understand it correctly, the intuition is that any braiding has a finite duration, which means that any such process can always excite arbitarily low $\omega$-modes (if these exist, which is the case for gapless systems), ruining the necessary assumption of adiabaticity.

But what does that for example imply for a system of fermions with a Fermi surface? Are we not allowed to call the excitations 'fermions'?... [Or perhaps we can think even more fundamentally about the massless fermions in our universe.] Surely the fermionic nature must have particular effects, even if 'finite-time braiding' is not terribly well-defined. And if one agrees with that, why are we much more hesitant when it comes to more exotic anyons?

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  • $\begingroup$ This question is misleading: why should fermions be understood as anyons ? Massless fermions are perfectly understood objects, of recent great interest in condensed matter problems. They have even been observed, like e.g. in graphene or in topological insulator. Would you please make your question more clear ? Do you want to know whether massless fermions exist, or only massless anyons ? If I remember correctly, the mathematical structure behind anyons (topological quantum field theory) necessarily generates both a non-trivial statistics and a gap at the same time. $\endgroup$ – FraSchelle Feb 13 '17 at 15:34
  • $\begingroup$ It just means that you can not know the statistics (whether the particle is a boson, a fermion, or anything else) if the mass/gap vanishes, since in that case the theory drops its non-trivial topological character which it needs to describes anyons. Usually, the word anyon is kept for particle which are neither a fermion or a boson, though these two last can be seen as particular cases of anyons. Would you please correct your question in order to make all this clear ? Thank you in advance. $\endgroup$ – FraSchelle Feb 13 '17 at 15:38

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