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?
$\begingroup$
$\endgroup$
1
-
1$\begingroup$ 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. $\endgroup$– DaniMar 22, 2012 at 9:27
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
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 is really confusing.
answered May 27, 2012 at 1:27
-
1$\begingroup$ 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. $\endgroup$– HeidarJun 8, 2012 at 11:25
$\begingroup$
$\endgroup$
1
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:
-
4$\begingroup$ 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 $\endgroup$– Qmechanic ♦Mar 22, 2012 at 18:10