You may have seen that the Nobel Prize for Physics was awarded this week. The Prize was given “for theoretical discoveries of topological phase transitions and topological phases of matter”, which is a bit of a mouthful. Since this is an area that I have done a small amount of work in, I thought I would try to explain what it means.

You might have seen a video where a slightly nutty Swede talks about muffins, donuts, and pretzels. (He’s my boss, by the way!) The number of holes in each type of pastry defined a different “topology” of the lunch item. But what does that have to do with electrons? This is the bit that I want to flesh out. Then I’ll give an example of how it might be a useful concept.

### What is topology?

In a previous post, I talked about band structure of crystal materials. This is the starting point of explaining these topological phases, so I recommend you read that post before trying this one. There, I talked about the band structure being a kind of map of the allowed quantum states for electrons in a particular crystal. The coordinates of the map are the momentum of the electron.

Each of those quantum states has a wave function associated with it, which describes among other things, the probability of the electron in that state being at a particular point in space. To make a link with topology, we have to look at how the wave function changes in different parts of the map. To use a real map of a landscape as the analogy, you can associate the height of the ground with each point on the map, then by looking at how the height changes you can redraw the map to show how steep the slope of the ground is at each point.

We can do something like that in the mathematics of the wave functions. For example, in the sketches below, the arrows represent how the slope of the wave function looks for different momenta. You can get vortices (left picture) where the arrows form whirlpools, or you can get a source (right picture) where the arrows form a hedgehog shape. A sink is similar except that the arrows are pointing inwards, not outwards.

Now for the crucial part. There is a theorem in mathematics that says that if you multiply the slope of the wave function with the wave function itself at the same point, and add up all of these for every arrow on the map, then the result has to be a whole number. This isn’t obvious just by looking at the pictures but that’s why mathematics is great!

That whole number (which I’m going to call *n* from now on) is like the number of holes in the cinnamon bun or pretzel: It defines the topology of the electron states in the material. If *n* is zero then we say that the material is “topologically trivial”. If *n* is not zero then the material is “topologically non-trivial”. In many cases, *n* counts difference between the number of sources and the number of sinks of the arrows.

### What topology does

Okay, so that explains how topology enters into the understanding of electron states. But what impact does it have on the properties of a material? There are a number of things, but one of the most cool is about quantum states that can appear on the surface of topologically non-trivial materials. This is because of another theorem from mathematics, called the “bulk-boundary correspondence” which says that when a topologically non-trivial material meets a topologically trivial one, there must be quantum states localized at the interface.

Now, the air outside of a crystal is topologically trivial. (In fact, it has no arrows at all, so that when you take the sum there is no option but to get zero for the result.) So, at the edges of any topologically non-trivial material there must be quantum states at the edges. In some materials, like bismuth selenide for example, these quantum states have weird spin properties that might be used to encode information in the future.

And the best part is that because these quantum states at the edge are there because of the topology of the underlying material, they are really robust against things like impurities or roughness of the edge or other types of disorder which might destroy quantum states that don’t have this “topological protection”.

### An application

Now, finally, I want to give one more example of this type of consideration because it’s something I’ve been working on this year. But let me start at the beginning and explain the practical problem that I’m trying to solve. Let’s say that graphene, the wonder material is finally made into something useful that you can put on a computer chip. Then, you want to find a way to make these useful devices talk to each other by exchanging electric current. To do that, you need a conducting wire that is only a few nanometers thick which allows current to flow along it.

The obvious choice is to use a wire of graphene because then they can be fabricated at the same time as the graphene device itself. But the snag is that to make this work, the edges of that graphene wire have to be absolutely perfect. Essentially, any single atom out of place will make it very hard for the graphene wire to conduct electricity. That’s not good, because it’s very difficult to keep every atom in the right place!

The picture above shows a sketch of a narrow strip of graphene surrounded by boron nitride. Graphene is topologically trivial, but boron nitride is (in a certain sense) non-trivial and can have *n* equal to either plus or minus one, depending on details. So, remembering the bulk-boundary correspondence, the graphene in this construction works like an interface between two different topologically non-trivial regions, and therefore there must be quantum states in the graphene. These states are robust, and protected by the topology. I’ve tried to show these states by the black curved lines which illustrate that the electrons are located in the middle of the graphene strip.

Now, it is possible to use these topologically protected states to conduct current from left to right in the picture (or vice versa) and so this construction will work as a nanometer size wire, which is just what is needed. And the kicker is that because of the topological protection, there is no longer any requirement for the atoms of the graphene to be perfectly arranged: The topology beats the disorder!

Maybe this, and the example of the bismuth selenide I gave before show that the analysis of topology of quantum materials is a really useful way to think about their properties and helps us understand what’s going on at a deeper level.

(If you’re really masochistic and want to see the paper I just wrote on this, you can find it here.)