Thursday, February 28, 2013

Signatures of Majorana particles

How and when do Majorana fermions arise in a quantum many-body system?
What are their experimental signatures?

For some reason I find Kitaev's discussion somewhat ad-hoc. I do find the following more helpful and illuminating. Perhaps, it is just because it deals with things I am more familiar with.

Start with the transverse field Ising model in one dimension. It describes interacting localised spin-1/2 particles. J is the nearest-neighbour ferromagnetic interaction. h is the transverse magnetic field. At J=h it undergoes a quantum phase transition from a ferromagnetic phase to a paramagnetic phase.
One performs a Jordan-Wigner transformation which maps the spin-1/2 operators onto spinless fermion operators. This is a non-local transformation. The Hamiltonian then becomes quadratic in the fermion operators and so is analytically soluble via a Bogoliubov transformation. This means the "quasi-particles" are spinless fermions.
[For the details see this article which also includes the inhomogeneous case].

This nicely illustrates the profound fact that in a quantum many-body system the emergent quasi-particles  can have quantum numbers and statistics that are different from the underlying constituent particles (see here for more).

Now, it turns out that for an open chain that the spinless fermions defined at the end of the chain are actually Majorana particles. In a sense the end of the chain "splits the fermions in two". I like this because of some similarities to what happens in a Haldane spin-1 chain. The spins at the end are "split in two" into spin-1/2 excitations, as discussed in this earlier post, Edge states define the bulk.

What might be experimental signatures of these unusual edge states?
Brijesh Kumar and Somenath Jalal recently made an important observation based on calculations published by Pfeuty in 1970.

The long-range spin correlations in the bulk of the ferromagnetic phase it scales with p^2 where p is the order parameter.

In striking contrast, for an open chain the correlations between the end spins scales with p^8, a dramatically different dependence. They suggest this is a signature of the Majorana character of the edge excitations.

I thank Brijesh Kumar for explaining his preprint to me. His paper contains some other interesting results about how to experimentally realise this in a chain of cavity QED systems [based on Cooper pair boxes coupled to microwave cavities]. Hopefully, I will blog about that later.


  1. As far as I know the whole hunting game by the condensed matter physicists is for the 'bound' Majorana fermions and that's how they differ from the nuclear physicists (their hope is just the double beta decay). Majorana particles are not stable if they are not kept far apart like at two ends of a 1D superconducting wire, as proposed by Kitaev. I suspect that Brijesh' Majorana modes may not guarantee that.

    1. In addition of the above question, I also want to know what the scale with order parameter p in a short 1D wire?