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What is the influence of Hermitian condition ($\psi=\psi^{\dagger}$) of Majorana fermions operators in their statistical behavior?

A Majorana fermion gas must obey the Fermi-Dirac statistics, or their stastistical behavior may be anyonic?

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What is this $\gamma$? Usually it refers to the $\gamma$ matrices but in the Majorana representation they are purely imaginary, $\gamma^\dagger=-\gamma$ so that Majorana equation can be satisfied by a real field, and not hermitian. –  DaniH Mar 22 '12 at 9:27
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2 Answers 2

Majorana fermions as defined in http://en.wikipedia.org/wiki/Majorana_fermion are really fermions, as its name indicates. So Majorana fermion really have Fermi statistics. It is not proper to say Majorana fermions obey non-abelian statistics, since fermion always obey Fermi statistics by definition.

The thing that people said to have non-Abelian statistics are defects (such as vortices's) that carry a zero-energy mode. Such a thing is not Majorana fermions. Calling a zero mode as a Majorana fermion are really confusing.

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Sadly it is not uncommon to call these vortices or domain walls (with zero modes) for majorana fermions. Some people refer to them as majorana bound states, which might be a better name. –  Heidar Jun 8 '12 at 11:25
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Majorana fermions obey non-abelian statistics and it will be anyonic if your Majorana mode is confined to two dimensions. In $3\text{D}$, you still have the possibility of non-abelian statistics but it is no longer anyonic as the braid group is trivial.

Here are some useful references:

  • Majorana Fermions and Non-Abelian Statistics in Three Dimensions, J. C. Y. Teo and C. L. Kane, Phys. Rev. Lett. 104, 046401 (2010). [1]
  • Majorana Returns, F. Wilczek, Nature Physics 5, 614 (2009) [2] , and references there in. This article can be downloaded also from Wilczek's webpage.
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The first reference is behind a pay wall. In the future, please include link to arXiv abstract page when possible, e.g., arxiv.org/abs/0909.4741 –  Qmechanic Mar 22 '12 at 18:10
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