Why any fermion can be written as a combination of two Majorana fermions? Is there any physical meaning in it? Why Majorana fermion can be used for topological quantum computation?


I put an extra answer, since I believe the first Jeremy's question is still unanswered. The previous answer is clear, pedagogical and correct. The discussion is really interesting, too. Thanks to Nanophys and Heidar for this.

To answer directly Jeremy's question: you can ALWAYS construct a representation of your favorite fermions modes in term of Majorana's modes. I'm using the convention "modes" since I'm a condensed matter physicist. I never work with particles, only with quasi-particles. Perhaps better to talk about mode.

So the unitary transformation from fermion modes created by $c^{\dagger}$ and destroyed by the operator $c$ to Majorana modes is $$ c=\dfrac{\gamma_{1}+\mathbf{i}\gamma_{2}}{\sqrt{2}}\;\text{and}\;c{}^{\dagger}=\dfrac{\gamma_{1}-\mathbf{i}\gamma_{2}}{\sqrt{2}} $$ or equivalently $$ \gamma_{1}=\dfrac{c+c{}^{\dagger}}{\sqrt{2}}\;\text{and}\;\gamma_{2}=\dfrac{c-c{}^{\dagger}}{\mathbf{i}\sqrt{2}} $$ and this transformation is always allowed, being unitary. Having doing this, you just changed the basis of your Hamiltonian. The quasi-particles associated with the $\gamma_{i}$'s modes verify $\gamma{}_{i}^{\dagger}=\gamma_{i}$, a fermionic anticommutation relation $\left\{ \gamma_{i},\gamma_{j}\right\} =\delta_{ij}$, but they are not particle at all. A simple way to see this is to try to construct a number operator with them (if we can not count the particles, are they particles ? I guess no.). We would guess $\gamma{}^{\dagger}\gamma$ is a good one. This is not true, since $\gamma{}^{\dagger}\gamma=\gamma^{2}=1$ is always $1$... The only correct number operator is $c{}^{\dagger}c=\left(1-\mathbf{i}\gamma_{1}\gamma_{2}\right)$. To verify that the Majorana modes are anyons, you should braid them (know their exchange statistic) -- I do not want to say much about that, Heidar made all the interesting remarks about this point. I will come back later to the fact that there are always $2$ Majorana modes associated to $1$ fermionic ($c{}^{\dagger}c$) one. Most has been already said by Nanophys, except an important point I will discuss later, when discussing the delocalization of the Majorana mode. I would like to finnish this paragraph saying that the Majorana construction is no more than the usual construction for boson: $x=\left(a+a{}^{\dagger}\right)/\sqrt{2}$ and $p=\left(a-a{}^{\dagger}\right)/\mathbf{i}\sqrt{2}$: only $x^{2}+p^{2} \propto a^{\dagger} a$ (with proper dimension constants) is an excitation number. Majorana modes share a lot of properties with the $p$ and $x$ representation of quantum mechanics (simplectic structure among other).

The next question is the following: are there some situations when the $\gamma_{1}$ and $\gamma_{2}$ are the natural excitations of the system ? Well, the answer is complicated, both yes and no.

  • Yes, because Majorana operators describe the correct excitations of some topological condensed matter realisation, like the $p$-wave superconductivity (among a lot of others, but let me concentrate on this specific one, that I know better).
  • No, because these modes are not excitation at all ! They are zero energy modes, which is not the definition of an excitation. Indeed, they describe the different possible vacuum realisations of an emergent vacuum (emergent in the sense that superconductivity is not a natural situation, it's a condensate of interacting electrons (say)).

As pointed out in the discussion associated to the previous answer, the normal terminology for these pseudo-excitations are zero-energy-mode. That's what their are: energy mode at zero-energy, in the middle of the (superconducting) gap. Note also that in condensed matter, the gap provides the entire protection of the Majorana-mode, there is no other protection in a sense. Some people believe there is a kind of delocalization of the Majorana, which is true (I will come to that in a moment). But the delocalization comes along with the gap in fact: there is not allowed propagation below the gap energy. So the Majorana mode are necessarilly localized because they lie at zero energy, in the middle of the gap.

More words about the delocalization now -- as I promised. Because one needs two Majorana modes $\gamma_{1}$ and $\gamma_{2}$ to each regular fermionic $c{}^{\dagger}c$ one, any two associated Majorana modes combine to create a regular fermion. So the most important challenge is to find delocalized Majorana modes ! That's the famous Kitaev proposal arXiv:cond-mat/0010440 -- he said unpaired Majorana instead of delocalised, since delocalization comes for free once again. At the end of a topological wire (for me, a $p$-wave superconducting wire) there will be two zero-energy modes, exponentially decaying in space since they lie at the middle of the gap. These zero-energy modes can be written as $\gamma_{1}$ and $\gamma_{2}$ and they verify $\gamma{}_{i}^{\dagger}=\gamma_{i}$ each !

To conclude, an actual vivid question, still open: there are a lot of pseudo-excitations at zero-energy (in the middle of the gap). The only difference between Majorana modes and the other pseudo-excitations is the definition of the Majorana $\gamma^{\dagger}=\gamma$, the other ones are regular fermions. How to detect for sure the Majorana pseudo-excitation (zero-energy mode) in the jungle of the other ones ?

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    $\begingroup$ An extra point: since $c{}^{\dagger}c=\left(1-\mathbf{i}\gamma_{1}\gamma_{2}\right)$ is the only correct number operator, it means that $\mathbf{i}\gamma_{1}\gamma_{2}$ is either $\pm 1$. So the two Majorana modes are either absent or present at the same time. One can not discuss $\gamma_1$ or $\gamma_2$ separately. This remarks may help understanding the delocalisation and its topological property. $\endgroup$ – FraSchelle Mar 23 '13 at 19:07
  • $\begingroup$ How can one see the connection between a state lying at the middle of the gap and its exponential decay in space? $\endgroup$ – Arnab Barman Ray Sep 7 '17 at 20:06
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    $\begingroup$ @ArnabBarmanRay Well, a state lying in a gap must have an exponential decay, from basic dispersion relation property and the definition of the gap. Hence such a state can not be a bulk state, and must lies on the surface of the materials. You can check en.wikipedia.org/wiki/Surface_states for instance. Note that not all surface states have topological origin. $\endgroup$ – FraSchelle Sep 9 '17 at 11:34

Majorana fermions are fermions which are their own antiparticles. As a result, they only have half the degrees of freedom as a regular Dirac electron. One physical interpretation, at least for Majorana fermion quasiparticles in condensed matter systems, is that they can be thought of a superposition of an electron and hole state.

Only Majorana bound states can be used to do topological quantum computation. If you have a system with $2N$ well-separated Majorana fermions then you have a $2^N$-fold degenerate ground state. You can perform quantum computation on this system by performing a sequence of exchanges between these $2N$ Majorana fermions. These exchanges are known as "braiding" operations. Additionally, the order in which you perform the braiding operations matters. Hence the system is said to possess nonabelian statistics.

It is important to understand what it means to do a computation. Since Majoranas are sort of like half-fermions we can't really measure them directly. We can infer their existence. Another way to understand the trouble with measuring a Majorana fermion is an ambiguity in its unique identification! Say we have a system with $2N$ Majorana fermions and consequently $N$ regular fermions. You need to pair up two Majoranas to get a regular fermion (which we can measure). But there are more than one ways of doing this! You can do this in $$\frac{(2N)!}{2!(2N-2)!}$$ number of ways. Say we agree on a convention and decide to pair up or fuse two Majoranas in a certain way. For example, in a 1-D lattice we decide to only pair up nearest-neighbor Majoranas. This, in fact, is the best choice of pairing them. So now you do a bunch of braiding operations and in the end fuse the Majoranas according to the agreed convention and then measured the resulting regular fermion states. That's when we have done the computation. Just exchanging them is not enough. You won't able to tell if they were actually exchanged without fusing them! You can read more on this in section 3 of this excellent review article:


Finally, I'll comment on what this whole "topological" business is. One of the most fascinating and counterintuitive property of a system containing Majoranas is that you can have nonlocal states in your system. As I mentioned above, you can pretty much fuse any two Majoranas to get a regular electron. It doesn't matter if those two Majorana fermions are spatially far apart. The resulting (regular) electronic state by the fusion of these two Majoranas is highly nonlocal. The fact that this electronic state is nonlocal means that any local perturbations cannot destroy this state. Hence such systems are immune to decoherence which is one of the biggest problems faced by other quantum computation schemes. This is one of the biggest appeals of topological quantum computation.

There is, however, one catch to this interesting story. You cannot perform universal quantum computation with Majorana fermions. Two additional processes are necessary for that: the $\pi/8$ phase gate and a way to read the eigenvalue of the product of 4 Majorana operators without measuring the eigenvalues of individual pairs (in that group of 4). Unfortunately current ways implementing such processes do not enjoy topological protection. But still some topological protection is better than none!

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    $\begingroup$ @Heidar what's the most accurate terminology? My knowledge of high energy is limited; my area of study is primarily topological materials. So it's possible I don't know something about Majoranas in full generality. $\endgroup$ – NanoPhys Mar 17 '13 at 17:52
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    $\begingroup$ Well, "Majorana fermion" is the correct terminology in the sense that this is the term most often used in the field. My problem is that this is often very misleading, since these objects are NOT fermions at all but actually non-abelian anyons. You explain this very correctly and clearly above, my problem was just that you say they are fermions in the first sentence (which is not really correct). On the contrary, the high-energy Majorana fermion IS actually a fermions and therefore in a certain sense less exotic than the condensed matter version. (continued) $\endgroup$ – Heidar Mar 17 '13 at 19:57
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    $\begingroup$ It also seems that it's a little misleading to say they are their own anti-particles. This is true for the high-energy Majorana, but in the condensed matter case it just means that $c^{\dagger}=c$ (which I don't think is the same thing). Moreover the identity $c^{\dagger}=c$ makes sure the Majoranas don't satisfy fermi statistics, which is not the case for the particle physics Majorana. (continued) $\endgroup$ – Heidar Mar 17 '13 at 19:57
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    $\begingroup$ The term "Majorana fermion" often lead to other misunderstandings too. I often see popular articles saying the Majorana is a candidate for dark matter and it has been found in condensed matter systems. This is problematic since: 1) two different objects are confused with each other which only share a name and a few mathematical details, but are otherwise physically VERY different. (continued) $\endgroup$ – Heidar Mar 17 '13 at 19:58
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    $\begingroup$ Sorry for this way too long comment. To summarize: I think this is a really nice and accurate answer, my only problem was just a minor detail in the first sentence which is not correct and is caused by the misleading terminology. The fact that they are NOT fermions is why we are interested in them in the first place. $\endgroup$ – Heidar Mar 17 '13 at 20:06

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