# Topology and the Nobel Prize

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.)

# Justin Trudeau and quantum computing

You’ve probably seen already that clip of Justin Trudeau, the Prime Minister of Canada, explaining to a surprised group of journalists why quantum computing excites him so much. In case you haven’t seen it, here is a link. A number of things strike me about this. Firstly, of course, he’s right: If we can get quantum computing to work then that would be a really, really big deal and it’s worth being excited about! Second, it’s a bit depressing that a politician having a vague idea about something scientific is a surprising exception to the rule. Thirdly, while his point about storage of information is right, there’s a whole lot more that quantum computers can do that he didn’t mention. Of course, that’s fair enough because he wasn’t trying to be comprehensive, but it gives me an opportunity to talk about some of the stuff that he missed out.

Before that, let’s go over exactly what a quantum computer is. As the Prime Minister said, a normal (or “classical”) computer operates using only ones and zeroes which are represented by current flowing or not flowing through a small “wire”. (However, as you might have already read, this might have to change in the future!) A quantum computer is completely different because instead of these binary bits, it has bits which can be in state that is a mixture of zero and one at the same time. This like the electron simultaneously going through both slits in the two slit experiment, or Schrödinger’s famous cat being alive and dead “at the same time”: It’s an example of a quantum mechanical superposition of states. A quantum computer is designed to operate on these quantum states and to take advantage of this indeterminacy, changing them from one superposition to another to do computations. If you can get the quantum bits to become entangled with each other (meaning that the quantum state of one bit will be affected by the quantum state of all the others that it is entangled with) then you can do quantum computing! Exactly how this would work from a technological point of view is a big subject which I’ll probably write about another time, but options that physicists and engineers are working on include using superconducting circuits, very cold gases of atoms, the spins of electrons or atomic nuclei, or special particles called majorana fermions.

A big field of study has been to find algorithms that allow this quantum-ness to be used to do things that classical computers can’t. There are a few examples that would really change everyday life if they could be implemented. The first sounds a bit boring on the face of it, but quantum algorithms allow you to search a list to determine if an item is in a list or not (i.e. to find that item) in a much shorter time than classical algorithms. So, if you want to search the internet for your favourite web site, a quantum google will do this much faster than a classical google. Quantum algorithms can also tell quickly whether all the items in a list are different from each other or not.

Another application is to solve “black box” problems. This has nothing to do with the flight data recorders in aircraft, but is the name given to the following problem. Say you have a set of inputs to a system and their corresponding output, but you don’t know what the system does to turn the input into the output. The system is the black box, and the difficult problem is to determine what operations the system does to the input. This is important because these black box problems occur in many different areas of science including artificial intelligence, studies of the brain, and climate science. For a classical computer to solve this exactly would require an exponential number of “guesses”, but a quantum computer could do this in just one “guess”!

But perhaps the most devastating use of a quantum computer is to break the internet. Let me explain this a bit! There is a mathematical theorem which says that every number can be represented as a list of prime numbers multiplied together, and that for each number there is only one such list. For example, $30=2\times 3\times5$, or $247=13\times19$. This matters because most digital security currently depends on the fact that this is a very difficult thing for classical computers to start with a big number and work out what the prime factors are. The way that most encryption on the internet works is that data is encoded using a big number that is the product of only two prime numbers. In order to decrypt the information again, you need to know what the two prime numbers that give you the big number are. Because it’s hard to work out what the two prime numbers are, it is safe to distribute the big number (your public key) so that anyone can encode information to send to you securely. But only you can decode the information because only you know what the two primes are (this is your private key). But, if it suddenly becomes easy to factorise the big number into the two primes then this whole mode of encryption does not work! Every interaction that you have with your bank, your email provider, social media, and online stores could be broken by someone else. The internet essentially wouldn’t be private! Or at least, it wouldn’t be private until a new method for doing encryption is found. This is the main reason why security agencies are working so hard on quantum computing.

Finally, I want to quickly mention one application is a bit more specialised to physics: Quantum computers will allow us to simulate quantum systems in a much more accurate way. Currently, the equations that determine how groups of quantum mechanical objects behave and interact with each other pretty much can’t be solved exactly, in part because the quantum behaviour is difficult to model accurately using classical computing. If you have a quantum computer, then part of this difficulty goes away because you can build the quantum interactions into the simulation in a much more natural way, using the entanglement of the quantum bits.

So in summary, Prime Minister Trudeau was right: Quantum computers have the potential to be absolutely amazing and to change society and are really exciting (and possibly slightly scary!) But storing information in a more compact manner is really only the tip of the iceberg.