Majorana fermions are described by their topological charge. My question is whether we can see the topological charge of Majorana fermions by braiding a boson or a fermion around it ? Is the only possible way to check the topological charge of Majorana fermion is to braid it around another Majorana fermion which is equivalent to two exchanges with another Majorana fermion ?
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1$\begingroup$ Are you talking about true Majorana fermions (that is, fermions that are their own antiparticles), or about the Majorana states in a condensed matter system, which are, strictly speaking, not fermions, but indeed topological excitations obeying fractional anyon statistics? Also, what precisely do you have in mind by "braiding" another particle around it? $\endgroup$– ACuriousMind ♦Commented Feb 3, 2015 at 12:18
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$\begingroup$ Hi, I am talking about the 'Majorana fermions' which are emergent anyons in condensed mater systems. I want to check what different braidings carry information about topological charge of the Majorana fermion. $\endgroup$– user56199Commented Feb 3, 2015 at 14:01
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1$\begingroup$ "Majorana fermions which are emergent anyons in condensed mater systems". Majorana fermions are fermions with Fermi statistics. They are not anyons. So it is not clear what are you asking? Majorana fermions always have trivial braiding statistics with other fermions and bosons (by definition). $\endgroup$– Xiao-Gang WenCommented Feb 10, 2015 at 21:16
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$\begingroup$ @Xiao-GangWen thanks for the comment. I mistakenly mentioned them as anyons in an attempt to imply that they are different from usual fermions and are found as emergent quasi-particle excitations or bound states. I think I wanted to know if there is any non-trivial signature of a Majorana through its braiding with the bosons and fermions. From your comment, I think there isn't. Thanks again. $\endgroup$– user56199Commented Feb 12, 2015 at 10:48
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1$\begingroup$ "if there is any non-trivial signature of a Majorana". So one needs carefully define what is "Majorana". "Majorana fermions" as defined by wiki always have trivial braiding statistics with other fermions and bosons $\endgroup$– Xiao-Gang WenCommented Feb 12, 2015 at 14:13
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1 Answer
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In the Ising anyon model, there are three topological charges, $1,\sigma,\psi$. $\sigma$ can be thought as carrying a Majorana zero mode, and $\psi$ is an ordinary fermionic excitation. This can also be understood in the context of $p_x+ip_y$ superconductor, where $\sigma$ is the non-Abelian vortex and $\psi$ is the Bogoliubov quasiparticle. Braiding $\psi$ around $\sigma$ results in a $-1$ phase, which can detect the $\sigma$ charge. Of course, braiding $\sigma$ around $\sigma$ also has an interesting consequence: because the S matrix element $S_{\sigma\sigma}=0$, this braiding results in a state that is orthogonal to the initial state. This is useful in designing interferometry experiment to detect non-Abelian anyons.
answered Feb 3, 2015 at 16:41
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$\begingroup$ By braiding $\psi$ around $\sigma$ and getting the -1 phase, are you sure that you have a Majorana fermion ? Can you be sure of a Majorana w/o braiding $\sigma$ around $\sigma$ ? $\endgroup$– user56199Commented Feb 4, 2015 at 15:06
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1$\begingroup$ Once you know that the topological order is described by an Ising anyon model (or if you known this is a p+ip superconductor), the answer is yes, the only anyon charge that gives -1 upon braiding with $\psi$ is $\sigma$. $\endgroup$ Commented Feb 5, 2015 at 4:18
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$\begingroup$ thanks for the comment. No, I do not have an Ising model. I am trying to look whether my anyon has a non-trivial topological charge. Besides this anyon, I have a fermion-like and boson-like charges in my system and am wondering whether I can use them to braid and know the information about my anyon having a non-trivial topological charge or not. Also, can you please clarify t me what is the definition of topological charge of an anyon ? $\endgroup$– user56199Commented Feb 5, 2015 at 4:30
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1$\begingroup$ well, any systems with Majorana zero modes in some way resemble an Ising anyon model, although there are many variants. But regarding your question, generally speaking if your system is fermionic (I mean the system is made out of fermions), then the only excitations that has -1 braiding phase with fermions are the $\pi$ fluxes. This is true regardless of whether the $\pi$ fluxes carry Majorana zero modes or not. So if there is no additional knowledge about the anyon model is available, just the $-1$ braiding phase of fermions do not imply existence of Majorana zero modes. $\endgroup$ Commented Feb 5, 2015 at 5:08
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1$\begingroup$ Also, if you are considering some more complicated topological order (i.e. Moore-Read state or Bonderson-Slingerland states) which contain Ising anyons, things are more involved and maybe one needs to proceed case by case. In my previous comment, "fermions" really mean the fundamental electrons. There could be other fermionic excitations (like the neutral fermions in Moore-Read states). Topological charges is roughly the equivalence classes of localized excitations in a gapped system, where the equivalence is defined by local operators. See arxiv.org/abs/cond-mat/0506438. $\endgroup$ Commented Feb 5, 2015 at 5:14