Thank you all for coming. So. I think the topic is, you all know, since presumably that played some small role in persuading you to, come in. Like all topics in physics, you know, if you start to look, there's a precursor of, you know, precursors of ideas that go back a long time. And so topology already showed up in classical physics. You know, it seemed fluid vortices, the great Poincaré used topological ideas and thinking about chaos.
So everything does have a precursor till you get back to the bacteria. Who knows, maybe when they were using topology in some ways that we don't yet understand, but will soon, once the next model from open AI is out. But in my own field of condensed matter physics and that of my colleagues today, we tend to think of sort of the discovery of the quantum Hall effect, as a point where it's condensed matter physicists, you know, really began to have topology, as a main item in their arsenal.
It existed previously in that so-called topological sort of defects that we'll talk about a little bit. But then it really took off. Then it sort of went dormant for a while. And then there was the discovery of these so-called topological insulators, which essentially extended the Hall effect to a much larger class of potential phenomenon. All of this then goes along with this is sort of single particle physics. And then there's a lot of many body physics. And so on.
So meanwhile, you know, there are parallel developments in, in quantum field theory used by particle physicists. Again, you know, starting in a big way in the, in the 70s, terms like soliton, instantons, a lot of actually beautiful mathematics, done at Oxford, with, Michael, and, people around him in parallel with quantum field theory and of course, Roger Penrose in the context of, other problems.
So that's a you should think of today as sort of a my job is to introduce a few ideas, I teach an eight week course in the Masters in Mathematical physics. So this is the compressed version of it. This. Lesson. Right. And then my colleagues, who are much more active and working on things, at the cutting edge at the moment will tell you about things which are, more substantial in terms of, cutting edge physics. Very good.
So with that, I guess all I need is some sort of a, does this work, like to. All right. Okay. So so we start with the, question of, you know, what is what is topology. And so of course, informally. Right. Topology studies properties of objects, ultimately mathematical objects which remain unchanged under continuous, you know, deformations.
So topology is prior to doing calculus, you know, when you have functions that, you know, things which have to be smooth so you can do things like derivatives, topology doesn't demand that you be able to do that. And so of course, the example that everybody, trots out is that a coffee mug can be continuously deformed into, a donut. And here is, from Wikipedia for that particular maneuver.
So as you can see, the first thing you do is you take the coffee mug and you raise the, the bottom to all the way to the top. It's not very useful as a coffee mug anymore. And then you smooth it out and, and there's your. And there's your donut. Now, as I said, you know, the mathematical topological education of physicists, you know, and I belong like him to a generation that this is not standard. So I can't say that I have a, you know, proper understanding of the mathematics.
I don't physicists are often described as recreational mathematicians. I best, but it's not wrong. Pragmatic mathematicians, maybe the other way, which is right. But in any event, there is a question in mathematics. So when you see pictures like this, then these pictures are in our three dimensional space. So they are sort of embedded. Whereas mathematicians also distinguish between what happens if you talk about objects without thinking of them as being embedded in anything. Right.
So there's a sort of more abstract ways of doing it. So to go one step in that direction is that, you know, topology tries to classify, topological spaces up to homomorphisms. So the usual problem of starting to read mathematics, which is that every definition that under the three things that you realize that you don't know, right. So in this case, what sort of a logical space what's a home to a morphism. All right. So homomorphism is is a continuous bijective map of the continuous inverse.
We won't need it today. But a topological space is this. You know some set of points where there's some notion of closeness. But you don't have the notion of a distance. Right? You can't, you know, you don't need to have a metric which says, I get from here to here. And it's, you know, this distance. But simply the things are close and things are far. And once you do that, you can have limits, continuity, connectedness and so on.
So, okay, now you might say, why would what what would this have to do with physics. So obviously in physics units are important. You well you know, maybe you assume perhaps almost all of you have been physics undergraduates. And one thing that would have been drummed into you is units. How come you wrote six? You know what units, right? I mean, you're a physicist. You have to know what units are. And then maybe on top of it, dimensional analysis, right. And so on and so forth.
And why do we think this something may happen at the Planck scale? Well, you know, there are units. You combine them so and so. It seems a little odd to say, let me start talking about stuff in physics which doesn't have units. Right. So you might say, but surely this is irrelevant. But what? So I have to sort of tell you, I tell you that topology is classified spaces. So you might say what spaces are interesting in physics. Well, so it could be real space.
Now of course the space we live in is what it is. There are cosmologists. I think we do wonder periodically whether the universe itself is a giant donut of three dimensions. But we're not talking about that. As a condensed matter physicist, we actually do sometimes ask the question of what would happen if our condensed matter system was put on spaces of different topology. So in the case of the quantum Hall effect, the fractional quantum Hall effect, you say, does it live on a sphere?
Does it live on a Taurus? Does it live on a donut? Two holes? It's not something that would happen in the laboratory, but it's a way of asking a certain question about the system. And it turns out, for example, that the one third quantum Hall state would have one state on the Taurus, three on the sorry, one on the sphere, three on the Taurus, and nine on the next one. And that's a way of, discovering that that state which otherwise looks like a liquid, has something special going on inside.
Okay, the next one up is momentum space. So, this requires that you have been forced to sit through solid state physics. I hope you were, and so you may remember things called Fermi surfaces and filament zones. So the one, the spherical one is that of sodium. And so that just looks like a sphere. And notice that it sits inside this wire cage, which is the balloon zone. So that's the unit in which things are going to repeat. And momentum space. So you have one sphere in a second and a third.
But they won't touch each other. But the next one for copper has this black things which are actually pipes connecting to the next bell, ones which have been cut off to display it. So if we actually plotted the whole thing, you would have this right thing extending through this momentum space with all these tubes connecting neighboring cells. So you can see that the topology of the two is different. In one case that, you know, you can surround the sphere and disconnected from everybody else.
In the other case, they're just pipes all over the place. Looks like the basement of a particularly, you know, difficult building. Okay. Then you could say, well, what about, you know, spaces of trajectories. So this will come up in, in, Steve's talk that you have, you should think of these as identical particles, quantum mechanically identical particles. So here they are at initially and here they are some later time.
And along the way, if we insist that they don't, they cannot be at the same point at the same time, which were fermions of standard. But let me say there are other particles, and indeed these will be the anyons of this talk for which this is true. Then you can sort of see that this is like taking bits of string and, you know, kind of maybe braiding them around another and reconnecting them at the top.
And, you know, you can't simply there's something stable about that, because once you've scrambled them up a bit, braided them a bit, you cannot unbreak them without really taking them off the hooks at the top. And so the space of trajectories can be another one which has topologies. And that will show up today and then up the food chain, the things called fiber bundles.
And while this picture is useless for the purposes of actually doing anything with fiber bundles really, but it is roughly the idea which is you have some base manifold and then these things which have been drawn as fibers, but the main point is there is a space for physics. You know, it could be, for example, real space. And the fiber is something that lives at that point. So you've all done Maxwell theory electric and magnetic fields.
So think of real space as the base manifold and the electric field at the point, or the magnetic field at a point as living in the fiber. And then that's the that's the space. And it could be space. It could be space time. So so this could be a space whose topology we're interested in. Then you could have extended objects in condensed matter physics that a vertex lines, vertex loops, you could even make them up in gauge theories that are things like a Wilson loop operators, extended operators.
And in fact, again and again, Steve may not bring them up, but some of the things we'll talk about, you calculate them mathematically as Wilson loop operators. And you're interested in questions such as do they link or do they not? And actually do they not. Then you know Ode, which is a beautiful circuit. All right. So that's basically to to explain that there are lots of settings in which you would be interested in the topology of some space that is relevant to physics.
That doesn't mean that the physics of distances has gone away of units that's there. So sometimes there is a specific term that you're trying to calculate which is not distance dependent, but is stuff topology dependent. And occasionally you get lucky. And the whole answer depends upon the topology and nothing else. But that's, you know, more rare and of course, more beautiful when that happens. All right. So as a subject in mathematics, topology has all sorts of, you know, tricks up its sleeve.
So, of course, ideally you could just be completely confident whether there's a morphism or not, but topologies, especially that talk about very, very abstractly defined spaces have come up with ways of thinking about them, which are somewhere partial. You can decide whether two spaces are the same or not, depending on asking some more limited question. So one of the techniques, goes under the topic of homotopy groups.
This is actually not the one that turns out to be mathematically the easiest to do, but it's the one that's easiest to explain. So basically in homotopy, what you do is you have your space sitting somewhere that you're interested in, and on the right hand side you put down spheres of various dimensions. Right. So let's start with the simplest, but it's not the simplest one. But let's start with the second simplest one, which is a circle, then the sphere in our dimensions.
And then of course you go up and one more dimension and one more dimension from having done that, you consider a map from this thing that you put down here to the space of interest. Right. So there's some functions you can write, and you want these functions to be continuous. But then you ask the question, how many different types of functions are and what's the definition of type or equivalence class.
It's that if you can deform one function continuously into the other, you say they're the same, right? And if you can't, now you have a new art. So a this is a, you know, sacred maneuver in mathematics that you look at equivalence classes of objects. And having done that, then you're often interested in how do the equivalence classes do they compose, can you multiply them? You know, and so on and so forth.
And in the case of homotopy, which, it will turn out that these equivalence classes will have a group structure. And if you go from the sphere and n dimensions to, space, those are called the that will be called the Anscombe or Tobiko. All right. So we will do use the first and the second, which are and then and then stop. I know a little bit about PC, but after that I have no clue what's going on. All right.
So if you're mapping from a circle to some space, so let's say the space is, you know, sort of I on a map from the circle is to draw a loop on the space that you are going to. That's, that's essentially what it means, because each point is mapping somewhere because the circle closes on itself. Any loop that I draw in the space has to close on itself. So far, so good. So what we're asking is you have this space, unknown space, you call it X. How many different types of loops can you draw on it?
That is what the first homotopy group is going to tell you. Okay, so if we are looking on the plane right. So if let's say on the blackboard, if I draw something, you can always shrink it to zero, right. It doesn't matter what complicated thing it is. Think. You know, it just sort of continuously keeps shrinking is shrinking shrinking. It will be down to a point. Then you can get rid of it. But something interesting happens if you decide to exclude one point from the plane. So here it is.
The black dot is gone. It's a hole in the plane. So now I could have had a curve here that I could shrink to zero. But the moment I've gone once around it, it cannot be shrunk to zero. If I go twice, I can't shrink it to this, I can't deform it to this. And likewise right. So the number of times you think of carrying a piece of rope and walking around and sort of stretching it out, right, and then tying it at the end, as long as you are, you are living on the plane yourself, right?
You're not allowed to jump over it. You know, you can't unwind it, right once you've tied it. So that tells you that the first homotopy group of the plane with the point removed is the integers. Because. So. And then to compose them you just do one and then you do the second following. And that's called multiplication. And in that it looks like addition here. Sorry I seem to have lost. Oh sorry. The last line it says the following.
I said it for a plane with the point removed, but actually most of the plane was irrelevant. I can just shrink the rest of the plane. Basically just talk about a circle, I can expand the whole and so on. And so if you say you're going to draw loops on a circle, that's also the same classification, right? Go around the circle. You know, you don't go around, go around ones go around twice or go around minus one on the other side minus two the other side. So far so good.
So that's the meaning of saying that maps from the circle to the circle are classified by the integers. So this is a simple example of a topological construct. What can we use this one. So we can ask matter. Physicists are often interested in things that look like magnets, either because they are magnets or because we've learned to map all sorts of things on to things that look like magnets. And so often we talk about models. So for example, there's something called the x, y model.
And what the x y model is is x y because there's a spin. And the spin has to live in a plane, which we call the xy plane, therefore the x y model. And so now a ground state, in a magnet ferromagnetic is one in which all the spins agree on what to do, right? So every spin will point in the same direction. So you, the A picture in just a second. But then if the energy is not minimum, the spins will deviate from this exceptionally happy configuration. Right? So many things can happen.
But one of the things that can happen are what are called topological defects, which I'll just describe in a minute. And the claim is in the XY model, the topological defects are classified by pi one of s one. The first homotopy group of the circle. So how does this happen. So here's the ground state right. So everybody's pointing the same way. Now this is an example of a spontaneously broken symmetry.
Which means that while I've drawn it this way really we could globally rotate every single spin. You can just take the picture, do this, do that, do that. And that's an equally allowed configuration, right? I'm imagining rotating the spins, not the space in which they live. So that axis, the fact that they're pointing up is just a particular choice really. They could point in any direction. But the important thing is that in the ground state, they all agree.
The energetics of a magnet of a ferro magnet says to each spin, try and agree with your neighbors. Okay. And then antiferromagnetic. In the simplest case, the instruction to each spin is try and disagree with your neighbors. Which, looks a bit more like American politics. Which which is my country of citizenship. So I'm allowed to. Okay, good. Very good. Now, you could make deformations of this if I said, you know, raise the energy.
You could say it could take a given spin and you could move it a little bit, and you would say, I should expect to end up at a higher energy anyway. But there are particularly interesting things which you can do which are very hard to get rid of. And that's where the topology comes in. Okay. So here's a vertex right. So of course you know them from fluids. But I want to explain what this has to do with the topology that I was describing.
So the first thing is a, a a defect by its name suggests something doesn't look right somewhere. So in this case something doesn't look right at the middle. So let's ignore that. If we go far from the nominal location of the defect, what we expect to find on energetic grounds is that, at least locally, this configuration should look like it's in a ground state, because that's the state of minimum energy.
And indeed, if you look at this picture, you will notice that if you I only gave you a small snapshot of this, you would say the spins are all pointing up. If I said this, you would say to the left down to the right, right. So locally I have something that looks like a ground state, but there's this aspect that globally, even very far the spins are not all agreeing. Right? So they sort of slowly twist. And in fact, we could do it mathematically.
If we go further and further out, we can write down the configuration in which that, you know, energy can be calculated. So given the assumption which is energetically justified, that far from the location of the defect, the system must locally be in the ground state. We can now ask the following question. We can say go very far away and walk around the basics. What do you see? So observe that you're walking in a circle. That's your S1 from which you are mapping locally.
You see a direction of the spin. The spin itself lives on a circle. That's the S1 to which you are mapping. And what do you see? If you walk clockwise, you find that the spin rotates. Sorry, anti-clockwise. The spin rotates anti-clockwise, of course. Clockwise, clockwise. Right. If you go back to the ground state and we do the same thing at infinity, what do we see? The spin doesn't rotate at all. So the ground state is characterized by the trivial map in which every point.
Maybe I should just draw this here. So this is the S1 from which I'm mapping. This is the S1 which I'm mapping. So the ground state consists of mapping every point on this circle to a single point in that circle. But the vertex consists of mapping every point right. And you'll see the same recur to the identical point. Right. And these two are in different topological classes. Small deformations of these maps cannot take you from one to the other. You have to do something pretty major.
You have to take your string around the whole are you don't right? And you have to cut it in order to do it. That so. So what you can say is quite generally that if you have an XYZ system and you are in two dimensions and you have a defect, if you walk around it at very far away, you should be picking up some element of the homotopy group. We've seen, you know, zero and one. And if you want to know what two looks like, well here it is again. Don't worry about what happens in the middle.
Just focus on what happens at the boundary. It's points to the right, left. See that is returned to itself. Although only gone halfway. And then it will repeat. So it will rotate twice for each circle at infinity. And that will become the element corresponding to the second. You know, the instead of being two. And then I can draw pictures in which, you know, you go clockwise and the spin
rotates anti-clockwise and then that will give you negative numbers. So now the reason that the topological stuff is interesting is that because we know that you can't continuously deform one of these maps into the other, it shows you that if you had such a configuration and you looked at it at infinity, you could be confident that you can't just, you know, by flipping a few spins, you can't get to the other one.
You've got to do something very, you know, large scale now in physics, you know, the energy of if you have a system of a certain size, the energy is something finite. So sometimes I like to say that, you know, topology proposes but energetics disposes to just being physics. Right. And that's true. But nevertheless topology gives you a very good idea that some things may be very, very hard to do.
Okay. So that's now there's a much fancier topological theory of defects, which, you know, liquid crystals, things like that. Obviously the utility comes when you use mathematics to find things that, you know, you couldn't just have said by just looking at pictures, and, and trust me that, it's examples like that. All right. Okay. So I want to use this, homotopy for one more thing. And are you all able to see the board more than. No. Because I think you need eyes.
Okay. And where other the, this is good. All right. So let's let's do that. Okay. So so far this was kind of classical, right? Classical in the sense I was just doing energetics. I now want to show you that the same sort of stuff can enter quantum mechanics. So I'm going to consider a particle on a circle. Is this readable. No not really. Okay, okay. What? Okay, so this is why I mostly opted to do slides. And you don't really need this. It's a particle on a circle.
Let me try and do something bigger. All right. How about that? Any better? Yes. Okay. Good. Excellent. When you all this enthusiastic when you were in classes as young people. All right, so here is the location of the particle. Just to simplify, let's take the circle to have radius one okay. So the circumference has lengths two pi. The particle is here.
And we will say well at time t it's at some location theta of t. And then it's going to go at from theta of zero is going to be some, some initial theta and theta of T is going to be some final theta. So we're going to start somewhere and somewhere right okay. So in classical mechanics we're just going to do a free particle. And since the radius is fixed and one you know distance and angle are the same thing. Although the units have to be different.
So when you write your homework solutions I will penalize you for those. But I myself am going to take ignore units. All right. Good. So that's our free particle Lagrangian half mass times the velocity squared. And as you'll recall what we want to do is that once we write the action and then extrema is it subject to these two conditions. Right. So that's that's classical mechanics. Now we can derive the equations of motion from this.
And they're just going to say that the mass times acceleration has to be zero because it's a free particle. All right. Now one thing about the particle living in the circle that we just discussed is that it can start, you know, so let's say this is theta I and this is theta f. So you can certainly do this. But it could also do that or go around a certain number of times. Right.
So actually this is one of these cases where you know, many trajectories are possible for the, same sort of initial and final conditions. In fact, sometimes it's simple to simply take the final condition to be the initial condition. Just for analysis, which gives there's a solution in which you just sit there. That's the bureaucratic solution.
And then there are solutions of the, I just I get the feeling that, this particular audience is not as hostile to bureaucratic, inertia as, some of us have come to be. All right. I can define something called the winding number, which is I just integrate the time derivative right of theta. Now, you know, you might be tempted to say, well, this is just theta f minus theta I, but because this is a periodic variable, you know, this is not true.
It's actually this plus two pi times L, which is the winding numbers. Right. So that's because it's a multiple multiple valued variable. Good. So now I claim that I can change this action to this plus some number l five times w. And that that's absolutely every bit as good a classical action as the one that I wrote down at the start. Okay, so if you remember dimly, you know, you may remember that total derivatives don't matter for action.
If not, the next thing you can say is, well, if I have something like this, which I'm keeping the end point fixed, something that seems to depend only on the end points, you know, can't matter because I can do my variation without this value changing. But in the worst case, you just pull out your Euler-Lagrange equations and you differentiate with respect to theta, not you get a constant differentiate with respect to time you get zero. So literally you get the same equation.
So classical mechanics doesn't care whether I do this action or I add to it something that looks like that. But quantum mechanics isn't indifferent, because in quantum mechanics, what I'm going to do is I'm going to write a path integral which will take me from theta I to theta F right in summertime t, and then I have to let me put in this number alpha because it will matter.
And the reason is that what you're going to do is, you're going to say, well, this is a sum over all parts of E to the I s plus alpha times w. Now what I'll do is I'll take my parts and I'll break them up into parts of a given homotopy right, a given winding number.
And so I can say, well this thing is sum over all winding numbers e to the I alpha times the winding number, sum over the parts that belong to a winding number of e to the I s. And as you can see, this set of numbers right is going to be the same no matter what alpha is. But when I put an alpha, it obviously makes a different. So what I discovered is that when I quantize this fact that my space was not fully connected, suddenly pops up and tells me, you know what?
Your quantum mechanics is ambiguous. You have choices to make. And those choices are sensitive to this winding number. Now, of course, you know this in some other language. If I put a magnetic flux through the ring, right, it's not in the ring, it's not on the ring, but it's in the ring. And that's the wrong effect, which says that, you know, that flux is something that the that the particle sees. But this is something you can discover from topology.
And then there are fancier field theoretic examples with stuff, illogical terms which do not look like putting a flux through anything. But you simply observe that the space is topology. You can write down such a term, and now you know that quantum mechanics can have such things, right?
The most famous example of that is the theta term in QCD, which is supposed to have all sorts of fascinating possibilities, although the state of the discussion involves axioms and so on, which is beyond my ability to go through. Okay, good. So far, so good. So we've done two uses of pi one I guess one. So I don't know how am I doing in time because I may, have five minutes. I'm nowhere close to what was it? Oh, 20 minutes. Okay. Oh, good. Then I'm very good. But I really panicked.
I okay, I didn't practice, so I have no idea how this how this is going to work, but. All right, so let's go up and use the next homotopy. Right. So the next one is we're going to use maps from a two sphere, the usual sphere that we introduced. And we're going to map it into the space of interest to us okay. So. We will make the other space be a sphere. And you'll see how. So we are interested in the second homotopy group of the sphere. So maps from the sphere to the sphere.
And it turns out that's also the group of integers, which by the way also means that the group is abelian. You know, you can add them in whatever order. This is not always true of homotopy groups. You can certainly come up with ones which are which are non-abelian, where the order doesn't matter. Okay. All right. So the index is going to be the number of times the first sphere wraps around the second sphere. So let's take a look. So here's again the trivial map.
So the trivial map consists of simply mapping all points. What do I do with this. Oh all points on the first sphere which is the test sphere onto exactly the same point, which of course I can pick to be anything. And as you can see, the entire area of the first sphere has been mapped onto zero area. On the second sphere. By the way, you know how I produce this. I'm absolutely hopeless.
I went into ChatGPT and said, write me a piece of Mathematica code to do x spattered Mathematica code, put it into Mathematica, produce this right? I did try earlier to say I'm giving a talk tomorrow. On the following topic. And it said in its usual cheery fashion, sure. You know, what would you like me to do? So I dictated, you know, a couple of paragraphs and it produced slides. In fact, I'm using the PowerPoint slides that it started with. Of course I had to because, you know, it wasn't.
It will be by next January. But this and I'm not even kidding. But it was it produced a basic structure. It understood words this that's on and so forth, you know, and kind of stuff. So we're, as I said, a subject for a different talk. Which I don't know if some of you may have noticed, I, I ran a series on AI these days. The first one was the Dominic Cummings. Josh Simons is coming next. And, so that's the parallel track of trying to keep track of
what's going on in the world. That's it. That's how I produced this. But this is a trivial map. This entire area gets mapped on to zero. The next map. Much as for the circle one, is this one in which every point in the first sphere gets mapped to the corresponding point in the second sphere. And you can think of this I took, you know, and basically I've wrapped the first sphere, you know, just around that. And so the index is, you know, it's Iraq.
Once every little area on the first year went to the corresponding little area on the second one, and so on. How do I get the next one? Well, so here's a picture stolen from Wikipedia. So okay, what's the picture? It's really you should these two halves have to be glued together? They've been cut open to show you what's going on. So you took the first sphere and you started wrapping it around the second sphere. But you go around twice before you, before you're done.
So if you go, it was like with the vertex. You went once around, right? And the spin rotated twice. So in this case you go first around the first sphere and you end up going twice around the second sphere on the equator. And in this case you've chosen a particular axis. But of course, if you took the whole thing and rotated this way, that would be a continuous deformation and that would be a map of the same degree. Right?
So maps of the sphere to the sphere have this character, and that's the second home which pick up speed off the sphere. So so far so good. So we're up to pi two of S2. Now the question is what can we use it for. So we're going to use it for something which is close to what we did with the xy vertex. Except it won't be a defect. It will be a solid term. So what's the difference between these x and solid turns. So with the defect, as the term suggests, there's something wrong with it.
And what's wrong with it is the region in the center where, you know, spins are really moving in a way that really wouldn't like to, because energetically it's very costly, but far from the defect. You know, they're they're behaving in the way that they would close to the ground state. Now, the reason defects are important is, is, is the topology, which is if you have one, it's stuck. You can't easily get it out.
So even with solid pieces of metal, right, which are so on and so forth, you know, there's holes when you bend them and you have trouble, you know, bending a piece of metal back. It's because you've introduced defects which are then hard to get rid of. So, so, so defects are very, very important because, you know, they're there. You can't easily get rid of them. Solid towns are not defects. They have often similar sort of, properties. And we'll use the same topology.
But things are everywhere smooth. So in this case we're going to use pi two of S2 to produce a solid on a topologically solid turn. And they've come to be called squirmy arms and condensed matter physics. They were actually nicknamed babies fermions, because the actual skyrmions, due to, Tony's come. I can't work not very far from here. We're actually meant to be in one more dimension and with, you know, a bigger sort of space of spins because he was thinking of how to get.
Neutrons and protons, nucleons. Starting from a field theory of pions. There's no point in getting into why he was trying to do it. It was a brilliant idea. It was ultimately completed by and with me because the big challenge was, how do you get Fermi statistics from things that you start off with, which are which are Bosnich. But anyway, the, the hence the term scum want so. These fermions had the following property. They. So we need two spheres right to map between.
So the first sphere will be the space on which the soliton lives. And the second is going to be the space of fixed length spins, which unlike Z spins now will be spins in three dimensions. So fixed length vectors in three dimensions live on a sphere, right? So you're going to map from a spatial sphere sphere to a spin sphere okay. So this is going to appear in two dimensional space systems and field theories. But now let me walk you through how that works okay. So first thing.
Let's assume that the space on which the spins live is literally a sphere, right. That's not going to be true in a solid. But let's let's start with that. Yeah. By the way, same deals ChatGPT Mathematica. So here's the ground state spins agree everywhere. Right. Again, I picked an arbitrary direction. It could have been I could rotate all the spins in some direction and that would be just as fine.
So this is I've decorated, as it were, the spatial sphere with this, Okay, then this is this isn't quite right. And so just ignore that the way you're supposed to think about it. Focus on this patch is that at each point in the spatial sphere, the spin points radially outwards. So it's like a hedgehog perfect hedgehog. Right? Okay. So this is a map in which each point on the spatial sphere has been mapped to the corresponding point on the spin sphere.
This is topologically distinct small deformations of it I could rotate, you know, nearby spins a little bit this way. That way won't change the fact that it's very far from the ground state, but everybody with points pointing the same. So far, so good. So we should agree. Now this is not a defect. Everything is smooth as you walk along the sphere. Right. There's there's not a point at which you see that anything particularly violent, this is just slowly rotating. So this is a okay.
Now the next thing I want to show you is that we don't have to have the system live literally on a sphere in order for this to, to work, we can actually descend to the plane, which is where you can find systems. And the way to do that is stereographic projection. Right. So stereographic projection is going to tell us that, you know, corresponding points on the sphere will be mapped to points on the plane, and the north pole will eventually be mapped to infinity. Right.
But the thing we need on the when we do the stereographic projection, whatever the spin was doing at the North Pole, will be what this spins will be doing anywhere at the boundary of an extremely large system. So in all directions you should get the same answer. And if you have such configurations on the plane, you can come back to fly the plane, which is to say reverse started vertically projected to the sphere. And so then you can use the topology which was so manifest on the sphere.
So here's the ground state. Everybody points the same way. Go off to infinity. Everybody points the same way. The next map remember was more or less right from the each point. So the North Pole was going to go to the North Pole. The South Pole was going to go to the South Pole. So here's the fermion drawn on the plane. So the South Pole was the point of the origin. It's not perfect, but this thing is pointing down.
And as you can see, as you go to infinity along any direction, the spin goes from being down to being smoothly up and everywhere else, almost up, everywhere at the boundary. Spins are pretty much recovered to being up. So in a magnet in the lab or some system with this kind of, order parameter, what you should then expect to find are excitations of the ground state, which locally have this structure. They look like particles because after a while, you know, they recover.
And these particles are called skyrmions. So for example, probably which most completely, you know, sort of does what you want to do is actually in the quantum Hall effect, something that was my thesis. And you get these particles, they actually have, charge, which in the case of the one third Hall stage is the fractional charge one third of an electron. They have a funny statistic. So they really I think they would have made Tony Scott very happy. They do everything.
So they're genuinely new kinds of particles. But with this kind of description. Okay. So that came out. Now again I said, you know, topology proposes an energetic disposes. In this case, it turns out that the size of this curve means is itself actually can vary. They're not like this. And the particular energetics of systems is important. And the quantum Hall effect, they end up having a nice healthy size, which actually if you do in parameters you can make as big as you want.
And then they are nice objects and lots of their physics can be understood that way. All right. Good. So, next up and the last, thing I'll do is to use the same topology, Pi two of S2, to tell you something about how one gets fancy band insulators. So this is, so solid state 101.
Just to remind you where bands come from, you start with, let's say, the dispersion of a free particle, which is a parabola, and then you put on a weak periodic potential and that has the annoying habit of causing things to mix across a certain right momentum range, which is the inverse of the distance scale that you've introduced. And that causes gaps to open.
And then you fold them and draw them this way, because once you don't have perfect translation invariance and you want only translation invariant up to a certain distance, legally, there's no reason for you to insist that there's something called a momentum. Momentum as a consequence of translation invariance. And if translation invariance is only do there up to a certain, you know, periodicity, then momentum is only defined up with that periodicity.
And there's no meaning to drawing this separate from that. So you might as well draw it this way. Once you do that, what you have is that you have a set of energy bands. Each one of these and the energy bands are a function of momentum, but they have to be periodic. So and then you go up in higher dimensions, they have to be periodic in every direction of momentum.
So if you take the underlying space here, which is momentum space and you insist the things on it have to be periodic, it says if the momentum lives on a torus like this, so the Brillouin zones of solids are to try in one day is just a circle to the it's a more familiar Taurus. And so okay, good. Now this is one day which is very special and kind of not that interesting. But once you move up from one dimension, interesting things can happen.
So for example, you can have the two of these bands are sort of there, and I can deform the Hamiltonian and do things to it. So that states in here mix among each other. Maybe at some stage the gap closes and it opens again. Well, I don't really affect stuff that happens. You know, further up. So for this reason, it's actually perfectly legitimate to think about what happens when you have a finite number of bands living on atoms. Okay. So this is this is when we get to that mathematics.
And then the okay. So so that's the next step is for instance, we could consider two dimensions and we could say just take two bands. And the thing I'm not telling you is that the way that the Hamiltonian of interest will always was for single this a single particle physics independent particle physics will will factor at each point in momentum space. Right. So so because of that, what this formula is telling you is considered a given point.
Okay. Consider a three dimensional vector of unit length and consider the set of poly matrices sigma x, sigma y and sigma z. And then this dot product gives you the most what is all it's really telling you is the most general Hermitian two by two matrix. You can write can be written like this. That's it. And the Hamiltonian has to be here at each point in the language that. Okay. So what have I done?
I'm telling you that the way to think about a two band problem is to think in terms of these vectors and of k, the vectors live on this sphere, but k lives on the tourist. So at this point I should go through a lot of painful topology, to work that out. But actually take my word for it, I can replace d2 by S2 and get the same answer. It's not obvious, right? But but it's true.
So with that, I'm in business because we've already discussed what happens for maps between S2 and S2, so it's sort of funny. I'm applying topology now to classify these band Hamiltonians. I'm saying if there's a two by two, if I have two bands, any single particle Hamiltonian independent electrons not interacting electrons, I put in it will be classified by some map of the sphere to the sphere. So there must be an integers worth of them.
There is some trivial map where all the ends are the same. Maybe point in the z direction. It looks like the polling matrix everywhere, and otherwise there will be things that were win, and in fact these ends will do pretty much what those fermions were doing that are, you know, had in the in the last picture. Okay. So the subject of topological insulators precedes this way. This is the simplest case. You use somewhat fancier version of this called k theory.
Then you have to worry about disorder and so on and so forth. But basically you start from here and you make your way up the, hierarchy of powerful topological tools. Now you might say, okay, this is true. Why do I care? Does this have anything to do with anything? Right. That's an utterly reasonable question. And the answer is, well, if you have a nontrivial winding of the Hamiltonian, it has two bands.
If you fill one of them with electrons, yeah, you're going to find that the whole conductance. So electric field this way current off of it right is actually going to end up being given by the. So the winding number of times E squared over the longitudinal conductance will be zero. Because there's a you know, there's a gap in the energy spectrum.
And so these things end up exhibiting the quantum Hall effect in a lattice, which is not how it was initially, you know, discovered and they are now called Shannon switches. Right. So the same topology allowed us to discover that these Chern insulators can exist with quantized Hall conductance one to and then plus minus one, you know, plus minus one, minus two, one time, which has to do with you literally get the opposite sign of the Hall effect in a in a given sample. All right. So I think.
Okay. So I'll just mention this for those of you who are a glutton for punishment, which is that I talked about the Hamiltonian, but actually, you know, I only failed one band. So somebody may be saying, well, I don't have to think about the Hamiltonian. Why don't I just think about the actual state, the wave function. You can and you can get the same answer that requires a different fiber bundle. And you know, that would require a little more work. Okay. So I'm done.
So there are many uses of topology. Right. All the spaces we talked about and we explored a few defects, all of forms, topological terms and band theory. And then, you know, there's been this huge try to work. As is often the case, these days.
It's not of theory tends to outrun experiments, which are much harder to do, but nevertheless, especially with regard to, you know, single electron physics that's actually been a remarkable amount of experimental work realizations, materials, clever material physics, most of which I haven't even I tried to describe to you. And with that, I will stop. So for the purposes of recording, I'm going to repeat the question. Which is to. So the question, as I said, it makes a difference.
But I didn't explain what what the difference is when we add the topological term to the quantum mechanics. So this is of course easiest to say in the language of passenger intervals. But I could just as well do it in terms of Hamiltonians. The energy spectrum will be different, right. So you'll get, you know, different eigenvalues.
But if you calculate the potential value and said, you know, okay, so this particular one has a certain circular symmetry, let's say the energy spectrum is different which is you get different energy eigenvalues.
So it's so in Hamiltonian language you will end up doing something like either you will explicitly put a vector potential of the flux in the middle and keep periodic boundary conditions, or you are forced to go to boundary conditions in which the wave function has to twist and come back by a factor E to the I alpha. And that is another way of saying why the eigenfunctions will then come with different eigenvalues.
So in the case of QCD, which we can describe better, there's a topological term which will then break symmetries and you know, CP violation, things like that. And that really sort of a descendent of, ascendant of this, simple, simple concern. Okay. So the question is what did I write here? So the first Lagrangian is just one half the mass times the velocity squared zero square. Yeah. And then the action was just that the and plus.
Well let me answer this way, which is to say there there are field theories that people study mathematically which are topological quantum field theories. You know, Steve Stockwell, I don't know if you talk about them, but they will be background music, or maybe invisible, inaudible background music to what he's saying. But those are field theories in which that imaginary quantum mechanical universes in which, yes, everything is topology. There's literally no notion of distance. Right.
So people study, I think there's some moral gravity problems, ideological with that. Well, it's not the world you live in. It's just it's simply not the world you live in. Right? Because if you live in London versus living in Edinburgh, I mean, it's very clear that it takes longer to get there. So, so that's why which is to say, that so it's, it's it's all I mean, it's instructive to study these simple cases. And so to get the, that's not so much fun.
It's so the topological quantum field theory is it's actually spacetime to topological. So both are have no clear meaning. The only thing you know is did you do this or did you not do this right around so so so that the world around us is knocked off? Illogical in that sense.
It's not invariant to arbitrary, you know, changes of position, but nevertheless, I mean, you know, a lot of very beautiful mathematical physics, which has been very insightful, actually has has gone into exactly the universe you would like to, consider I plead guilty, I see the. Yes. No, no. Absolutely, absolutely. Absolutely. William. Not so much. But yeah, I mean, although even in helium, in cold atomic. Sorry, I'm supposed to repeat the question.
I didn't mention, laboratory examples of topological defects. And you're right. Liquid crystals, you know, they're beautiful imaging in called atomic gases where they have image them, you know, you can you can see them. And you're absolutely right. Topological defects, cosmic strings. I think that's still up for discussion. So, but absolutely, it's it's absolutely correct. So the statement is that a vertex anti vertex pair is not, from a topological point of view, is not a stable configuration.
It could disappear. But if there is an isolated vertex sitting there it would require you to do something on the entire size of the system. So topology that was the statement that topology proposes. But of course the details of physics and energetics still remain. Sometimes, you know, it can happen that you make a vertex anti vertex and they get stuck for some reason so that they're forever right.
And so from a physics viewpoint, they're infinitely long lived, even though topology would say there's no reason for them to be there. So topology is really insightful because when you start to deal with complicated order parameter spaces, so liquid crystals, you know, you can have a sphere with amphipods identified. You know, that's not something we're used to thinking in everyday life.
Then these formal mathematical results really come into their into their own, because it allows you to identify defects that you wouldn't have been able to that easily get if you were just sitting and trying to draw pictures. I mean, eventually you would, but it's just easier to look at the book. And go, right. But that's actually.