Majorana particles statistics

Question

What is the influence of Hermitian condition ($\gamma=\gamma^{\dagger}$) 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?

Answer 1

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.

Answer 2

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.