The Bulk-Boundary Correspondence

One of my original plans for this blog was to try and explain my own papers to a general audience. There are already three new papers out there for which I have not begun to do this, the first of which concerns ‘Majorana bound states’ and the ‘bulk-boundary correspondence’. (The other two are here and here.) For certain kinds of system these two things are expected to be intimately connected, and the paper looks into some of the details of exactly when we can expect this. It turns out that this really depends on the specifics of what one means by Majoranas. Here, in the first of several blogs, I want to start to explain some of these concepts a little.

The bulk-boundary correspondence is a relation between the bulk properties of a material, by which I mean how a large chunk of it behaves overall, and the possible existence of special particles that on the surface. In fact it says under certain conditions they must exist on the surface, no matter what you do to the surface. To understand how this might work let us consider a simpler system than the one we will turn to shortly. The system we will consider is known to exhibit what is called the quantum Hall effect. Let us imagine we have a two dimensional material in which some of the electrons are free to move around. If we apply a magnetic field perpendicular to the material, then the electrons want to form ‘cyclotron orbits’ around the direction of the field. This is just part of the strange way that charged particles and magnetic fields interact, see figure 1. The size of the radius of the orbit depends on the strength of the magnetic field, and in a material it will also depend on the material properties. Going back to our two dimensional material we have not one, but many electrons moving around in the magnetic field. They will try and set up motion that we can think of as being like in figure 2, with all of them circling around in little orbits. The actual size of the orbits will depend on the strength of the magnetic field, the material parameters, and so on.

Figure 1: An electron in a magnetic field. The field should be understood as uniform in space and everywhere pointing up.

The Hall effect, named after Edwin Hall who discovered it in 1879, is the phenomenon where a applying a voltage drop across such a sample causes a current in a perpendicular direction, see figure 2. As the electrons are now moving preferentially along the direction fo the dashed blue arrow, as opposed to the opposite direction, they tend to be turned more to their left, than to their right. This causes us a current across the sample. The quantum Hall effect, is the quantum analogue of this and in typical quantum style the current becomes quantised into discrete plateaus. In fact there are many other Hall effects too, but we shall leave this here for now, awaiting another time.

Figure 2: Many electrons in a two dimensional material in a magnetic field. The field should be understood as uniform in space and everywhere pointing, in this case, down. The blue arrows show the direction in which electrons will travel when a voltage drop is introduced along the direction of the dashed blue arrow.

Now we can ask what happens at the edges of such a set/up, see figure 3. In this case as the electrons cannot complete the circular orbits they are forced into moving along the edge as shown by the green patch in figure 3. This is what is known as “the bulk boundary correspondence”, i.e. the properties of the bulk of the material tell us something very specific about what will happen at the edge. Not only that but it doesn’t depend at all on the details of the edge! The edge can be as disordered and as messy as possible and the path of the electron will still follow it. This is not confined to the type of system we show here, but happens in several wide classes of materials. The two we are interested in are called topological insulators and topological superconductors. Let’s leave that for next time though…

Figure 3: Looking down on the sample of a two dimensional material in a magnetic field, which is shown in figure 2. The Blue region is the edge of the material. And the green shows the path an electron must take along the edge where it cannot complete the circular orbits.

First there are several simplifications in this picture which I think it is worth mentioning. Note that in a real system the electrons are not localised at particular points and going round these orbits exactly as pictured. Rather they are expended throughout the material, but can be thought of as something like a combination of many of the little orbits we show here. This is related to the fact that for an electron in a crystal the electron will not be confined to a particular place somewhere, but will rather be spread out throughout the entire crystal. It will however have a specific momentum, this is of course related to the fact that no particle can simultaneously have a well defined position and momentum in quantum mechanics. One of the consequences of which is the famous Heisenberg uncertainty principle. This means that looking at pictures in of how things stand in real physical space is not always the best way, and in fact in this field one uses what is called reciprocal space. Instead of asking where everything is, we ask instead what its momentum is. This can take some getting used to, but once one has got used to it then it becomes a very powerful aid in tackling many problems. The reason why it is so useful is also fascinating, but alas this will also have to wait! In most topological insulators and superconductors it is not really a magnetic field which is causing such orbits either, rather they arise out of more complicated physics, but this picture serves as a very useful introduction to these ideas.

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