Majoranas and the Bulk Boundary Correspondence

Some time ago I started, and more recently continued, the background to write about a paper published back in 2017 (see here for a freely available version on the preprint arxiv). This paper is all about ‘Majorana bound states’ and the ‘bulk-boundary correspondence’. The second of these was the topic of the last two blog posts linked above.

Here then I will turn to Majorana bound states, which are a condensed matter equivalent of the theorised Majorana fermions. What are Majorana fermions, and what exactly is a “condensed matter equivalent”? By “condensed matter equivalent” I am referring to a particle which exists as an emergent property of the behaviour of the ions and electrons in a particular solid material, rather than a fundamental particle like an electron or neutrino, those are the preserve of a different type of physicist. These condensed matter Majoranas are not quite like the fundamental version as we shall see. But they do have some analogous properties, and some novel properties, which make them very interesting in their own right.

To understand what a Majorana fermion is it is useful to take a (very) short tour through the history of quantum mechanics. The first relatively complete descriptions of quantum mechanics are the matrix mechanics of Heisenberg and the wave mechanics of Schrödinger. These are actually equivalent and the formulation of Schrödinger remains widely used (with a few additions). However they do not describe everything, and in particular they are not compatible with Einstein’s special theory of relativity. A version which was, and which can be used to describe electrons, was found by Dirac. However this lead to its own issue. Dirac’s equation predicted infinitely many negative energy states! To avoid the problems this would cause he suggested that these states are all filled up. However if one of these filled states is removed, a hole is left which will look much like an electron but with opposite charge, this is the electron’s anti-particle, called a positron. Such a particle was experimentally confirmed in the 1930s and particle-anti-particle pairs are now a standard feature of the standard model of particle physics. A particle and anti-particle destroy each other when they come into contact, leaving behind only light and energy. It was Ettore Majorana who, in 1937 one year before he disappeared mysteriously, found a solution to Dirac’s equation which was its own antiparticle! It is currently still unknown if these exist in nature. All the particles that make up matter that we know of come in particle and antiparticle pairs.

So, how can one get these strange particles in a solid in a laboratory? First we need an equivalent of particles and anti-particles. In solid materials the particles that we are normally looking at are the free electrons moving around which give rise to many of the conducting and thermal properties. These move in a background made by the atomic lattice. The exact make-up of this lattice, its interactions with the electrons, and how the electrons interact with each other, lead to the wide variety of phenomena which can be seen in solid materials. If we go back to our (very) simple picture of the electrons in ‘bands’, see the figure below, we can imagine we start with the lower band completely full of electrons. (The fact that only one electron can be in each state is the famous Pauli exclusion principle which is a fundamental property of electrons.) Now if we give some energy to one electron to raise it up into the upper band we have one electron which is free to move around the upper band, and one ‘hole’ left in the lower band. Shuffling the electrons around the lower band is equivalent to moving the hole around. So we can pretend that this hole is a particle which will have opposite charge to the electrons. Holes are the anti-particles of condensed matter, and we can write a formal transformation between electrons and holes mulch like the particle-antiparticle transformation. So what we need is a particle which is an equal combination of a hole and an electron, how to find one?

Electrons and holes in a simple band picture.

It so turns out that superconductors are systems which have a natural combination of electrons and holes. Due to the properties of the transformation between electrons and holes (our particles and anti-particles) if we want a particle which is both it has to have an energy of zero. If we have the states at the edges of a wire which is topological, as discussed last time, then in fact this has to have zero energy. So we need a topological superconductor, rather than a topological insulator. How to make such a topological superconductor was proposed in several seminal papers some 10 years ago (see for example here or here). Although no conclusive signature of these particles has been found several very promising experiments have been performed on a variety of possible platforms for these exotic particles, see the references here for a partial list.

These emergent Majoranas are interesting for several reasons. One is that they are not in fact fermions. They have the technical name of ‘non-abelian anyons’, which means that if I take two equivalent Majoranas, and swap their positions, I end up with something different than what I started with. If one tries this with bosonic particles the end result is the same as what I started with. If I try it with fermions (like electrons) the result is almost the same, except the function which is used to describe the fermions mathematically is multiplied by a minus sign. This leads to the Pauli exclusion provincials and also makes some other small changes which it is possible to find evidence of, but mostly everything looks the same. This corresponds to what one may expect, swap two things which start the same and nothing should change. This strange property of the Majoranas is behind one push to make a stable quantum computer out of them, which for example Microsoft is currently putting a large amount of effort and funding into.

The possible energies of electrons in one two dimensional topological superconductor. The shaded blue areas show dense regions where there are many possible electron and hole states. The red and green lines show those that exist on the edge, these are the topologically protected states (red and green are travelling in opposite directions). Where they cross at zero energy Majoranas were expected, however they only exist at the central crossing, a point with higher symmetry than elsewhere.

So, finally, what did we actually do? We looked at the topologically protected states at the edges not of wires, but of two dimensional flat crystals. Along their (one dimensional) edges there can also be these special states. In two dimensions instead of having a single state on an edge we find many states forming a “band” all of which exist on the edge. It is expected that if we look in the middle of this band at zero energy we have a Majorana. However, unexpectedly it turns out that this is not always the case and our paper was concerned with understanding when this does and does not happen, see the figure above. This can help us in the search for these strange particles.

We found that whether or not the Majoranas are there is tied closely to the symmetries of the material, and in particular the symmetry it has at the point in the band structure where the Majorana should exist. This point needs a higher symmetry than what surround spot or no Majorana can be present.

The next step is to look for experimental signatures of these states along the edges. Can we tell the difference between the case when too is there and when it is not?

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