Monday, 16 April 2012

Majorana Fermions

A Dutch team of researchers are claiming to have seen evidence for the existence of Majorana fermions in superconductors:
Majorana fermions are particles identical to their own antiparticles. They have been theoretically predicted to exist in topological superconductors. We report electrical measurements on InSb nanowires contacted with one normal (Au) and one superconducting electrode (NbTiN). Gate voltages vary electron density and define a tunnel barrier between normal and superconducting contacts. In the presence of magnetic fields of order 100 mT, we observe bound, mid-gap states at zero bias voltage. These bound states remain fixed to zero bias even when magnetic fields and gate voltages are changed over considerable ranges. Our observations support the hypothesis of Majorana fermions in nanowires coupled to superconductors. 

(That's the paper's abstract).  I originally read of this at a reasonable article over at the BBC, but I felt there were a few things I wanted to say.  One thing that could easily mislead the lay person is that Majorana fermions are far from the first particle to be its own antiparticle.  Photons, the particle of light, are a simple example.  Rather, Majorana fermions are there own antiparticle and have not been discovered yet.

 Let's start by discussing what a fermion is.  The name comes from the Italian-American physicist Enrico Fermi, also noted for asking how many piano tuners there are in Chicago and building a nuclear reactor in a squash court.  As such, fermion should really be capitalised, but it usually isn't.  I'm getting a little ahead of myself, however.

Fermions are an intrinsically related to quantum mechanics.  However, we don't need to delve very deep to get sufficient understanding.  In the usual presentation of quantum mechanics, the most fundamental object is something called the wavefunction, commonly represented by the Greek letter psi: $\Psi (x, t)$.  I have written x and t in brackets to denote that $\Psi$ explicitly depends on space and time.  It is one of the axioms of quantum mechanics that the wavefunction contains the maximum information about a quantum system that it is possible to obtain, even in principle.

The other thing we need to know is that all measurements are, roughly speaking, related to the square of $\Psi$.1  This is cleanest with the case of a single particle.  If we measure the position of that particle, the square of the wavefunction at a particular point in space gives the likelihood that we will measure the particle to be there.  (Here we see the intrinsic randomness of quantum mechanics, that we can only speak in terms of probabilities of measurement outcomes.)

Now, imagine that I have two identical particles; two photons or two electrons, say.  If these particles are truly identical, then I can exchange their locations and other characteristics (speeds, directions etc) and you could never tell.  Swappng the two particles does not change the physical state, nor the predictions for any experiment.  But the predictions for experiments are given by the square of the wavefunction.  So this must be unchanged when we interchange the two particles.

Note, however, that this does not mean that the wavefunction itself is unchanged.  That can happen, of course, but alternatives do exist.  If swapping the particles changes the sign of the wavefunction, its square is the same.  Are there any other possibilities?  Well, swapping the particles twice must leave the wavefunction unchanged, because I literally haven't done anything.  The only two numbers that square to one are plus one and minus one, so there aren't.

So, we have these two cases.  What makes all this useful is that for each type of particle, it always either leaves the wavefunction unchanged, or flips its sign, when we exchange two of them.  The proof of this is a bit too tricky for this post, but its not too hard to convince yourself that it's believable.

The particles where the wavefunction is unchanged are called bosons, after the Indian physicist Satyendra Bose.  The consequences of this property were developed with some unknown German, and are interesting but the topic for another time.  Fermions are the other types of particles, where the wavefunction changes sign; Enrico Fermi and  developed many of the implied properties.  (In very simple terms, bosons like to be together and fermions like to be apart.  The latter statement is responsible for the entirety of chemistry.)

Now, many particles we have discovered are fermions; arguably most.  Among them are electrons, protons, quarks and neutrinos.  However, with the possible exception of neutrinos, all of these are Dirac fermions.  (The status of neutrinos is one of the big unanswered questions in particle physics.)  This means that they are different from their antiparticles.  Two electrons can never annihilate one another; we need an electron and an anti-electron or positron for that.

The difference between Dirac and Majorana fermions really just comes down to this antiparticle question.  The names represent the historical development; in particular, Dirac wrote down the Dirac equation that describes Dirac fermions first, trying to find a way to unify quantum mechanics and special relativity without the problems of the Klein-Gordon equation.  He did this by imposing extra constraints and structure on the Klein-Gordon equation; in the same way, Majorana discovered the fermions that bear his name by imposing constraints on Dirac's equation.

A modern understanding realises that quantum field theory is pretty much the only feasible way to unify quantum mechanics and special relativity; and a systematic analysis in quantum field theory shows that both Dirac and Majorana fermions should exist (as well as the massless Weyl fermions, and some other exotics).  But all this is with the benefit of hindsight; it should not be seen as undermining the brilliance of Dirac and Majorana's explorations of the unknown theoretical landscape.

Finally, to return to the original paper briefly, it's worth noting that the Majorana fermions the Dutch researchers observed are not `fundamental'.  Rather, they are quasi-particles, states made up from the collective motion of the entire superconductor.  That doesn't make this work any less valuable, but its another thing the BBC article seemed fuzzy on.  We do think fundamental Majorana fermions might exist; they are nearly essential in supersymmetry, a popular extension of particle physics that the LHC is looking for right now; they would be very different to what this experiment found, however.

1 For those of you who know complex numbers, be aware that this should strictly be the absolute value squared, as $\Psi$ is itself complex.

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