 Thank you very much, Nick, for the kind introduction and it has been a pleasure for me to be here at OIST for the last few months. Thank you all very much for coming today, and I'd like to tell you a little bit about my work on topological phases of matter and beyond. So I'm a theoretical condensed matter physicist and when I tell people that the first reaction is often cool, but really sorry, what does that mean? Well, fundamentally, condensed matter physics studies phases of matter. So you may say that's a very familiar concept. We have solids, liquids, and gasses. So what all is there to study? Well, we know that solids are, for example, a situation where the atoms or molecules that make up a material are arranged in a nice orderly fashion like so. Whereas in liquids, the molecules or atoms are close together, but are able to move around freely, but they tend to interact with each other a lot, bump into each other. Whereas in a gas, the molecules or atoms tend to be free streaming, free flowing, except for occasional collisions with each other. And so you say, okay, well, we understand solids, liquids, and gasses. What is left for us to study? And the key thing is that in condensed matter physics, we often study phases of matter, which are not just phases of atoms or molecules. In particular, we can study phases of other objects. So for example, we can consider the phases of an electron in a material. In a metal, the electrons are free to move and they slosh around and behave much like a liquid. Whereas in an insulator, like diamond, the electrons are frozen in place and don't move very much and behave much more like a solid. So by considering phases of things other than atoms or molecules, we get these new ideas like metals and insulators rather than just the solid, liquid, or gas form of the material that those electrons live in. We can also consider phases of matter of spins. So electrons in a material all have some angular momentum. They all spin. And when the direction of that spinning is aligned between all of the electrons in a material, you get a magnet or a ferromagnet, as we call it. On the other hand, if the electrons tend to spin around randomly and orient themselves in various directions, you get something that is non-magnetic, like a paperclip. So all of these different things, ferromagnet, insulator, metal, paramagnet, all of these things are different phases of matter that we get by investigating the phases of things other than just the atoms and molecules themselves. So in that case, why do we want to study new phases of matter? Fundamentally, new phases of matter mean that we get new physical phenomena to study. So for example, a familiar one, a magnet, has a very interesting physical phenomenon associated with it. If you throw it at your refrigerator, it sticks. But we can also use these physical phenomena for a lot of practical purposes as well. For example, the physics of magnetism plays a critical role in looking at solid state hard drives. So in general, there's a wide range of phases of matter, all of which have their own associated physical phenomena with them. Just talked about magnetism. Superconductors have the property that they can conduct electricity with zero electrical resistance. And this also leads to interesting things like this levitation here. And then there's other phases, which I will, this one I will talk about at length in my talk today, the quantum hull effect. So I'll get into what this plot means. And high temperature superconductivity, which means, which is the same idea, but these conduct at much higher temperatures. Still far below room temperature at this point. So I just want to bring up a historical note that the way this field has tended to progress is that experimentalists have gone out there and seen a physical phenomenon, and then eventually theorists have come around and explained them. So magnetism, this is kind of dramatic. We've known magnets existed since ancient times under the term loadstones. And this was explained in the 1800s, 1900s. Superconductivity was observed by Camerling von Onnes in 1911 and explained by Bardeen Cooper and Triefer in 1957. The integer quantum hull effect was observed in 1980 and explained in 1983, and high temperature superconductors were observed in 1986. And there's not really a set, an agreed upon explanation for them in the field just yet. One of the interesting trends in condensed matter physics, though, that makes it exciting to be a theorist there, is that now theory is driving experiments. So there have been all these different phases of matter, which have been predicted before they have been observed experimentally, which go under various names. And you'll notice that sometimes it takes only a couple of years for these things to be observed. Sometimes it takes 25 years or about 40, 50 years. And in some cases, it's controversial as to whether we've even observed these predictions at all. Along this trend, though, there have been an explosion of new phases, which have been predicted by theorists, which have not yet been realized in any experimental system. And I've written down a bunch of names here. This is essentially jargon for the purposes of this talk. But the thing to notice is that in a lot of these, there's this word topological. And so this is fundamentally that these sorts of phases of matter are what I study and what I'm interested in doing. What I'm interested in describing to you today. So as a condensed matter physicist, what do we actually want to do as a theorist in particular? We want to figure out what phases might exist and develop tools and frameworks to describe these new phases, propose experimental realizations of them, and then in the process hopefully learn things about other areas of physics. So what I'm going to be describing to you today is working towards these goals for topological phases. So what are topological phases? And this is essentially explaining this slide will be what I'm trying to do for the rest of this talk. Topological phases of matter have physical properties that are quantitatively robust throughout the phase. And they're enabled by quantum mechanics and described by the mathematics of topology. So there's a wide range of phases of matter. And I want to show you in this talk how they host a highly unusual new phenomena physical phenomena and rather universal physics that doesn't really depend on the details of the material. And I also want to give you some insight into how beautiful mathematics describes these topological phases. So the rest of the talk is going to go like so. So I'm going to start with a brief introduction to the topology apologies to all the mathematicians in the audience. As I will be taking a very physicist centric view point of this. And then I will go into a fair bit of detail on some example topological phases the integer quantum Hall effect. And I will try to explain the physical phenomena associated with this these phases and how we describe them and where the topology comes into play. And then I'll focus on another set of physical phenomena which are quite interesting they're anions and these are types of emergent particles within a material that do not exist as fundamental particles in nature as far as we as far as any of our current understanding goes and we would not expect them to be able to exist as a fundamental particle but in materials and in these topological phases they can exist. And then at the end I will talk a little bit about some recent and future directions. I should mention that the focus of these of these two bullet points is not going to be anything that's particularly new this was done mostly work done in the late mid to late 80s and 90s. But it kind of is the and nor is it my work. I was not a physicist in those days. And but it's kind of the it gives the qualitative idea of what this field is about. And it underpins essentially all of the work that's ongoing today in top in this field of topological phases. So with that, let me proceed to a physicist's introduction to topology. So let's we'll take a map interlude. There's a at this point fairly tired joke if you are familiar with topology at all and a not very meaningful joke if you are not familiar with topology that a topologist can't tell the difference between a doughnut and a coffee cup. Well that's not a terribly useful statement on its own. So what does that really mean? So the point is that within the framework of topology you can turn a coffee cup into a doughnut. That's an allowed transformation that we want to study. So what do I mean by that? Well, let me show you one way that I could turn a coffee cup into a doughnut. Here's a coffee cup. Here's a doughnut. Look, I turned a coffee cup into a doughnut. Great. Let's declare victory. Obviously, this is not a very interesting process to study. I just replaced one with the other. Instead, what we want to do is consider a process more like this. So you'll notice that we start with the coffee cup and then we start squishing the coffee, stretching the coffee cup and deforming it in different ways. So you know, bring up the ground, the bottom of it into here and then smooth it out a little bit and you know, deform it further and further and further and then we get to a doughnut. So the difference between these two processes where I just instantly replaced one with the other and the one here is that we want to consider smooth modifications with apologies to the mathematicians for abusing the technical term smooth to use it in a colloquial term. I mean continuous, but I'm going to keep saying smooth for the rest of this talk. So in particular, the field of topology characterizes properties of objects that are unchanged or invariant under smooth modifications. There's no tearing of anything allowed. And in particular, you'll notice, for example, that a coffee cup and a doughnut both have one hole. There's a hole here and a hole here. And that and if you want to add or subtract holes from this shape, you're going to need to tear the shape. In particular, the number of holes is what's called a topological invariant. This is also known as the genus G, but it's the number of holes in a two-dimensional surface. So a sphere has no holes. And so its genus is zero. A doughnut has one hole. Its genus is one. And then this shape here, maybe a tug of war toy for a dog, has genus two because it has two holes here. And the key property here is that under smooth, smooth changes, this number of holes cannot possibly change. In particular, if you have two surfaces with different numbers of holes, that means in order to transform one into the other, you must tear the surface. I cannot possibly take a sphere and turn it into a doughnut without ripping the sphere open and reconnecting it in some way. It's also important to notice that this topological invariant is an integer, zero, one, two, three, and so on and so forth. There's not really any meaning to having a hole or sorry, half a hole or a hole and a half or a third of a hole in a system in one of these things. That's really just not even very sensible. So I've shown you a little bit about topological invariance of surfaces. But other objects can have topological invariance as well. So I'm going to show you how a path can have a topological invariant. So here's the typhoon. And what we are going to do is look at the direction that the wind is blowing in this typhoon. So here's some arrows showing the wind direction here. So of course, at this part of the typhoon, the wind is blowing to the west, at this part it's blowing to the east, and so on and so forth. That's all that's shown here. Now the eye of the typhoon is calm, so there's not actually any wind blowing there. And so this arrow that I've drawn doesn't really make sense at the center. Keep that in mind. I'll be coming back to that point in just a moment. So what we're going to do is we're going to take our sailboat and we're going to sail in some path around the typhoon. I don't recommend doing this in real life, but we're going to consider the thought experiment anyways. So let me take away the sailboat just to make the picture a little bit clearer. So what I'd like you to observe is how the wind direction changes along this path. So at the top of the path, at the start, the wind is blowing to the west. Over here, it's blowing to the south, east, north, so and then when we get all the way back to the top, it's blowing to the west again. And in particular, the wind direction is winding all the way around the compass rose as we travel on this path. So it starts on the west, it ends on the west, but we make a full loop all the way around. Now, as long as you take a closed path, this property actually really doesn't depend on very much about the path that you take. Oh, and sorry, the number of times that this winds around this wind direction winds, I'm going to call w and that's one for this path. This is going to be our topological invariant. Let me explain what I mean by topological invariant here. So if you take this path and you start shifting it around, no matter where it is, you'll find that the same thing happens. You wrap all the way around the compass rose exactly once. So here maybe the wind is starting off to the north northwest, but if you track this direction as you go around the circle, it's going to stay point, it's going to wind all the way around. Even if the details of how that does, how it winds changes a little bit in this process. Except there is a way to change this. If we start deforming this path so that it crosses through the eye of the hurricane or the typhoon, something interesting starts to happen, because there there is no wind direction. This this arrow doesn't make any sense, because the wind isn't blowing at the center. It's calm. So once the path passes through the eye of the typhoon, this winding number can change. And since this arrow doesn't make sense there, what we're saying is that tearing the path includes pushing it through this eye of the typhoon, where the arrow doesn't make any sense. And you'll see that once we move this path over here so that it no longer encloses the eye of the typhoon, now the winding number has changed. The wind starts out pointing to the northwest here, and then it's west, and then it's north, and it's west, or a northwest again, and it never points east along this path. So we haven't wound all the way around this the compass rose. We've only kind of hung out on one half of the compass rose on this path. So here the arrows don't wind, and so the winding number is zero. So this winding number is a topological invariant. It only changes when we do something that is essentially ripping the path. Here we didn't actually tear the path itself apart, but we passed through this very singular point where something unusual is happening with the wind direction. So that's a brief introduction to what topological phases are and what topological invariants are. So I'll pause for a moment for questions before I proceed to the question of what does topology have anything to do with phases of matter and physics for that matter. Please don't be shy, but please do use a microphone. Okay, so if there are no questions at this point, let's proceed to physics. So what I'm going to explain to you is that different topological phases have different topological invariants. And these topological invariants describe the physical behavior of the electrons in a material or more generally spins or some other degrees of freedom in a material, but I'm going to focus on electrons. So if we're going to use topology to describe the behavior of electrons, we had better agree on what the ground rules are. So I told you that topology characterizes properties of objects that are unchanged under smooth modifications. There's no tearing allowed. What does smooth mean when we're talking about the behavior of electrons? How do we modify the behavior of electrons smoothly? And for that matter, what does it mean to tear the behavior of electrons? That doesn't seem obviously well defined. So what I mean by these terms is the following. So first of all, we're going to start out with a ground rule. We're going to talk about electrical insulators only. No metals allowed. Sorry to the heavy metal fans. And so then we're going to say to stay in the same topological phase, you're allowed to make changes to the material in the following way. First of all, they have to be smooth. So if you make a big change to the material, you need to be able to build up that change from small changes. So instead of just instantaneously making a big change to it, you need to be able to take small steps from your starting point to reach the end point. And also, there's no tearing. What that means in this context is that the material is going to stay an insulator throughout the process of any changes that you make. In particular, if you make a big change as a sequence of small changes, it needs to stay an insulator through this entire process. So those are the ground rules. And this is what I mean when we're talking about smooth changes to a material and not tearing the behavior of the electrons. And so that brings me to my main example of a set of topological phases, actually, the integer quantum Hall effect. So what we're going to do is consider a two-dimensional material. You can think of this as an abstraction, but there are in fact two-dimensional materials in existence today. Graphene is a single atomic layer of carbon atoms. So it's basically as thin as physically possible in our universe. It's only one atom thick. And so that is for all practical purposes, a two-dimensional material. And there are some other examples as well, but the details of what material we're considering won't be important for me. And we're going to consider electrons moving in this two-dimensional material. We're also going to apply a magnetic field. So now I'm going to describe to you a physical experiment that you can do and people have done in a lab, which is going to end up detecting a topological invariant. The details aren't too important, but the idea is that we're going to take a battery of some voltage v, and we're going to hook it up with wires to two ends of the material. The positive terminal for this end, the negative terminal for this end. And what this is going to do is this is going to cause the electrons to move around in the material. And because of the presence of the magnetic field, what it's actually going to do is cause the electrons to move in this direction. It's not going to cause the electrons to move in this direction. It's going to move in this direction, but you get some electric current I. And so you can measure that current, and then you can compute the ratio of the voltage you applied to the electric current you get out. And this is a number called the Hall resistance. For those familiar with Ohm's law, this would normally be called the resistance, but because of the directions of the electric field here and the current, it's got this H subscript. It's Hall. Again, I'm just describing the experiment, but let's not get into that detail. When you do this measurement, what you get is a plot that looks like this. So as you increase the magnetic field in the sample, the Hall resistance generally increases, and it increases in this pattern. So when the magnetic field is low, the Hall resistance just goes up. But when the magnetic field gets bigger and bigger, some strange features start appearing. And in particular, you start getting these plateaus. And so what this is meaning is that when you change the magnetic field over a pretty big region or regime, the Hall resistance doesn't change very much. And in particular, if you look at this three, this plateau labeled three here, it's actually extremely flat. So what we're going to do is measure the Hall resistance in units of fundamental constants. So H here is Planck's constant and tells us about quantum mechanics. And E is the electron charge. And there's some number N that comes out in front here. And as it turns out, each of these plateaus here correspond to a different topological phase. Each of them has some topological invariant, or there's a topological invariant, which changes value when you go between them. So the way to think about this plot is that you have one topological phase here, and then you transition to a different one. And you stay in the phase here, and you transition to a different one, and so on and so forth. And in particular, what you find experimentally is that this N is quantized. It's equal to an integer, so minus 1, 0, 1, 2, 3, so on and so forth, to an accuracy that is incredibly precise, one part in 10 to the fifth. This is in fact so precise that it's used now in metrology experiments to define the unit of electrical resistance. So this is a really remarkable thing. It's very unusual that you would go around and find some quantity, which as you can see can vary continuously, but the measurement you get is saying it is 3, not 2.99, not 3.01, 3. So how is it then that this N here is so well quantized? Why is that? That's a really remarkable phenomenon. And this is where the topology comes into play. In fact, what I'm going to show you is that N is a topological invariant. So for this, let's take our two-dimensional, remember we were talking about an electron, or electrons, living in a two-dimensional system. So I'm just going to take this two-dimensional system and curl it up on the surface of a cylinder for the moment. And I'm actually going to apply a slightly different magnetic field. I'm going to run that magnetic field down the axis of the cylinder. And I'm going to forget about all the details of the material here. I don't care what the radial magnetic field is. I don't care what this is made of. All I really care about is that this material is going to be an insulator. When it's an insulator, that means the electrons in this material are arranged in a nice regular pattern, and they can't really move about much. So what we're going to do is a thought experiment where we're going to take the magnetic field that goes down this axis here and increase it from 0 to 2 pi and 2 pi inappropriate units of the flux quantum. And we're going to do this smoothly. So what's going to happen? So we're changing the physical environment that the electrons live in. So this process of changing this magnetic field here is a path through different physical environments for the electrons. Now that's kind of a more abstract space to think about moving around in, then say on the surface of the earth around a typhoon or something like that, but it's still a path that you can consider taking. And so the question is, does this path have some sort of meaningful, physically meaningful topological invariant? So let's see what actually, let's unpack what happens a little bit more. So we're going to start with this magnetic field off, and then we're going to slowly increase it. So when we increase it a little bit, the electrons are going to notice something is going to happen. Maybe this electron will wiggle around a little bit over here, and this one will move a little bit here, and this one moves a little bit to the right, and so on and so forth. But in general, this is going to be some complicated process that depends on all the details of the material. So we don't really know what happens in general when we increase the magnetic field a little bit. However, it turns out, this is just a fact that you'll have to accept for me from quantum mechanics, that electrons can't tell the difference between B equals zero and B equals two pi in this context. So what that means is that when we end, we're in basically the same physical situation as the start. And so we expect the electrons to basically sit there and do exactly the same thing they were at the start. However, these folks up here, in particular Loughlin, and this was elaborated on by these folks here later, showed that there's a different possibility. The other possibility is that everywhere away from the edges of the cylinder, the electrons do nothing, but in this process, where we go from zero to two pi, an electron or a few move from the bottom to the top. And you can think of this as, for example, the electron here moving up a little bit, and this electron moving up a little bit, and this one moving up a little bit. So all the individual electrons are making very small movements, but the net result is that you lose one from the bottom edge, and you get an electron at the top edge. And so n electrons move from the bottom to the top. Now, what n is, is an integer because it's some number of electrons, and it depends on the material and the radiomagnetic field and all sorts of other things. But if you start with a material where n is one electron, that's not going to change to a half because you can't rip an electron apart or anything like that. And also, if you make slow or smooth changes to the material, then we shouldn't expect that this n can jump. It doesn't make sense to go or to make a small change to the material and then suddenly go from one electron to two electrons as long as this thing stays insulated. You would, you would have to maybe go through half an electron in between, but you can't do that. You can't rip an electron apart. So a prelude to Nobel Prize worthy fact was, was shown by Laughlin that actually this haul resistance, this physical measurement that I talked about before, I said that there was this n in here. This n and the n I just talked about, the number of electrons that move from bottom to top are the same number. And so that's why the haul conductivity is quant, or the haul resistance is quantized because it's, what it's physically telling you is that you're moving one electron or two electrons or three electrons from the bottom to the top in a particular physical process and you can't move half an electron. A Nobel Prize worthy fact is to actually show that this n is mathematically a topological invariant. In particular, this is something called the first turn number of a principle you want bundle. Whatever that means for, or what, what exactly that means is not important for the purposes of this talk. The point is that you can actually compute something that you can prove mathematically is a topological invariant. And that thing is exactly equal to something that you can go out and measure in the lab. So the conclusion of all of this is that each plateau in the quantum hull effect is its own topological phase. It's characterized by a topological invariant, which is a physical, physically measurable quantity. And topology is why that measurement is so well quantized. Physically, the topology is coming into play because you can't break an electron into pieces. You can't have half of an electron. So I do want to point something important out though. Have I lied to you? So I've been talking about this physical process where we turn on this magnetic field and some electrons started out at the bottom here and they end up at the top. So some electrons were moving. That sounds a little bit sketchy because I told you that moving electron, that metals, aren't allowed in this context if we're staying in the same topological phase. Didn't we ban metals from consideration here? And the answer is yes, we did. But the key point here is that this material and sorry, metals are what generally allow electrons to move. But the key point here is that in this material it is metallic, but only at the edge. And in fact, this is a consequence of topology, not a problem. Let me give you a brief argument as to why that's the case. So here's our quantum hull state. It's an insulator. It's a topological phase. It's interesting. We now are going to put an edge on this system. And so we're going to put it up next to empty space, which is a boring old phase of matter in which nothing very interesting happens. So what we've done is we've taken something which is in a topological phase and put it next to something that is not in a topological phase. And I told you that the only way you can change a topological invariant is to rip the behavior of the electrons in some way. And where is that rip occurring here? It's occurring at the edge of the system because that's where the phase is changing. And so what this means is that the rip, that is the metallic behavior, is stuck on the edge and it has to exist. So the topology is actually forcing the edge to metallic. It's telling us that I have set up a situation where I am changing a topological invariant. So there must be a rip somewhere. And that has to be where the topological invariant changes. That is at the edge. And this is a very general phenomenon in topological phases. Their edges or surfaces in higher dimensions tend to have very interesting physics that is guaranteed to exist by the topology of the system. So this is a beautiful story. We found that because you have an integer number of electrons that move in some physical process, we get these nice plateaus. And then the experimentalists came along and ruined it for us in the best possible way. So there was a measurement which was done later. So I will explain what's going on in this plot. So we have the same measurement that I was talking about earlier. This is the magnetic field and this is the Hall resistance here. And just focus on this line here. So you'll notice that the magnetic field is very large. It goes up to 10, 20, 30 Teslas. That's a huge magnetic field that takes a lot of work to create in the lab. And so as you increase the magnetic field, you start seeing these plateaus that I talked about. So here's a plateau and here's a tiny plateau and here's a plateau. And if you identify the Hall resistance of these plateaus, you find that this one is n equals one and this one is n equals two. And these are the things that we said we expected to find based on the previous experiments. But up here is another very flat plateau and this one corresponds to n equals a third. So this is dramatic. This is spectacular. This is as spectacular as me saying that I have a lab in which I can measure holes very well in surfaces. And I measured a surface really carefully and did a really, really good experiment. And I found that it has exactly a third of a hole in it. And you would say, Danny, what does that mean? How could you possibly have a third of a hole in your surface? I would say, I don't know, but I measured it and there really is something that looks like there's a third of a hole in my surface. That's what's happening here. We're measuring something where a third of an electron is doing something. I don't know what it means to rip an electron into thirds, but it really, really looks like we've ripped our electron into three pieces. So I'm not actually going to go into the details of how you can effectively rip an electron into three pieces. But I'm going to talk a bit about the physical consequences of this measurement and how we understand this phase of matter. And this is the concept of anions. This is a rather dramatic concept of topology. So what I'm going to do is I'm going to describe for you how some topological phases violate a long-standing belief. And this belief is that all quantum particles are either bosons or fermions. And I'll explain exactly what I mean by that in a moment. So what I'm going to do is I'm going to argue why we expect only bosons or fermions to exist. And I'll tell you what exactly I mean by bosons and fermions. And then I'll find the loophole that allows anions to exist. And topology is going to play a key role in this argument. So let me start with an extremely abridged view of many body quantum mechanics. So imagine that we have a quantum system that has two particles in it, which I'm going to call particle one and particle two. So the way that we describe a quantum mechanical system is with a function called a wave function. And it has this name psi of x comma y. And the meaning of this is that this first argument here is particle one's position. And the second one is particle two's position. And so given a position for particle one and a position for particle two, this thing is going to spit out a complex number. And the magnitude squared of that complex number is going to have the physical interpretation of the probability of finding particle one at position x and the particle two at position y. And so as we vary x and y, we're going to find different probabilities for different points in space for each of these particles. So now what happens if particle one and two are identical? So for example, electrons are identical particles. Every electron in the universe, as far as we know, are exactly the same. And it would really destroy pretty much all of our understanding of modern physics if it turned out that this was not true. So now if we have particle one at position x and particle two at position y, that has some wave function amplitude, some probability associated with it. But and just reminder that the left argument is particle one's position, the right argument is particle two's position. But now imagine that we look at switching the particles. We're going to exchange them. So the probability for this to happen is given by the wave function with y for particle one and x for particle two. Now I told you that these two particles are absolutely identical. So we couldn't possibly tell the difference between whether particle one's on the left and particle two's on the right or particle two's on the left and particle one's on the right. They're identical. And so what that tells us is that the probability of these two configurations has to be exactly the same because otherwise we would be able to tell whether particle one was on the left or particle two was on the left despite the fact that these particles are identical. So you can take this equation and what you find is that this tells you that the probability for particle or that the wave function for particle one at y and two at x is the same thing for x and y but with a number out front, which I'm just going to call z. It's just some name for the number. And it turns out this z is some complex number, it's modulus squared is one. So all that's doing is saying that particle one and two are identical. Now I can tell you what bosons and fermions are. z is one for bosons and it's minus one for fermions. And those are two solutions to this equation here. Now all known fundamental particles are either bosons or fermions as it turns out. Okay, so why do we care? As it turns out, this has really profound physical consequences. For example, two identical fermions are not allowed to be in the same quantum state. This is actually a very famous principle called the Pauli exclusion principle. And when applied to electrons, it is the basis for all of chemistry as we understand it, that two electrons can't be in the exact same quantum state. This property is not obeyed by bosons. So if all of our matter was made out of bosons, chemistry would completely break. It would not work anything like it does in our universe. So it's very important to know whether particles are bosons and fermions for exactly this reason. So why is it then that all known particles are fermions or bosons? As it turns out, there's many solutions to this equation. z could be a whole continuum of different numbers. Let me give you the physical explanation for this. So we're going to take particles one and two, and remember we said that z shows up when you exchange one and two. So you swap which particles on the left and which particles on the right. Now I just did that instantaneously before, but this is a talk about topology. Let's do things smoothly. So what we can do is we can take particle two and physically move it to the left side of particle one and then drag the particles back. This is a physical process that implements this exchange that I talked about before. So exchanging is really a physical process of dragging particles around. Now if we exchange the particles twice, so we start with one on the left and two on the right, we put two on the left and then we put two back on the right. Well, we got back to the same thing we started with. I'm going to ignore the sliding for the moment. It doesn't really play a role in what's to come. So we started and ended with exactly the same configuration, particle one on the left and particle two on the right. In this process, we exchange the particles twice, and so that means we're going to get two factors of Z. So one exchange gives us a factor of Z and then exchanging again gives us another factor of Z. But now we really have the same physical configuration. We didn't even swap, have the particles swapped. So in particular, these two need to be exactly equal to each other because this is exactly the same configuration that we started with. And that's the thing that I'm going to show you why those things need to be exactly equal to each other. But the key point here is that physically, if you take particle two and exchange it twice, we are making a loop around particle one. And in particular, this Z squared here that we got shouldn't really depend on how you make the loop. An electron's an electron. It's not going to suddenly turn into a boson because it's moving this way, as opposed to moving this way. So let's see what happens as we change the path that particle two takes. So this Z squared is going to be totally unchanged as we change the path here. So we're going to start with this path, and I've just drawn on some axes and a plane here just to make it a little clearer what I'm about to do. So one thing you can do is change the path just by a little bit. So I've moved this path out of the plane that it started in. And you're still going to get the same Z squared. Going to change it a little more. Still the same Z squared. And a little more. And a little more. And a little more. But now, absolutely nothing is happening. I'm not even moving particle two at all. So what that tells us is that exchanging twice doing this loop is exactly the same thing physically as doing nothing whatsoever. And so that tells us that these two things must be exactly equal to each other. This Z, and which tells us that this Z squared has to be one. And that tells us that Z is plus or minus one. And one corresponds to bosons and minus one corresponds to fermions. Great. So this has shown you that we expect that only bosons or fermions can exist. And this is tied to a beautiful piece of mathematics. This is a topological fact, which goes like this. The fundamental group of three-dimensional space minus a point is trivial. For an equation, it looks something like this. But all this is saying topologically is that this process, if I pick a loop like this, then I can always deform it down to doing absolutely nothing without crashing into this other particle. That's all that topological fact is saying math. That's a mathematical, sorry. This statement is just a mathematical statement of the deformation process that I just talked about. And so we come to our conclusion. All quantum particles are either bosons or fermions. But I just told you that anions should be able to exist. So let's find the loophole in that argument. And the loophole is I was talking about a three-dimensional universe. So imagine a two-dimensional universe. So now the Z direction doesn't even exist. And we have some identical particles moving around here. So you can run exactly the same argument as I just did. You say, okay, well, if I exchange these particles twice, that's the same thing physically as taking particle two and dragging it in a loop around particle one. And so we said that we then argued that this meant that this two exchanges is the same as doing nothing. But how did we argue that? We took this loop and pushed it out of the plane. But now we're in a two-dimensional universe. So this is not a process that is possible. There is no Z direction. How could you possibly have a particle moving in a direction that doesn't exist? And so what that is actually telling us is that our argument is breaking. And in fact, you can prove that in two dimensions, particle two's path cannot be shrunk to nothing without crashing into particle one, which is not allowed according to the rules of this process. And so what this means is that the consequence that we derive from that argument is invalid because the argument for it broke. And in fact, it turns out that anions can exist in two dimensions. When you do a double exchange of these two particles in general in two dimensions, you get some number out here, which turns out to be a topological invariant of the phase. And this has physical consequences. For example, you end up with modified versions of the poly exclusion principle. But there's a lot of other forms of really interesting physical consequences that come out as well, some of which there's a lot of hope and work on turning those consequences into a quantum computer. But that's perhaps for another talk. So anions are possible in two dimensions because of a topological fact, which goes like this. The fundamental group of two-dimensional space minus point is non-trivial. So instead of having zero on the right hand side of this equation, we have something else. But all this equation is saying is that when you take this loop in two dimensions, you cannot shrink it down to nothing without crashing into particle one. So the math rabbit hole here goes very deep. There's a lot of really elaborate mathematics that gets used to describe anions in their full generality along with a variety of other related phases of matter. But real materials really exhibit anions. In fact, it was shown in 2020 by two rather incredible experiments that anions actually exist. This was suspected for probably 20 or 30 years at this point. But this was the first really solid direct experimental evidence that in fact anions do exist in real materials. So that's anions. There are particles which don't exist in our three-dimensional space and therefore don't exist as fundamental particles in our universe, which can emerge from the behavior of electrons inside a material. And these quasi-particles, as we call them, exhibit really unusual phenomena. For example, strange generalizations of the Pauli exclusion principle. So let me just conclude my talk with some recent directions and in particular tease around some of the things that I work on more directly. So anions can get even weirder. In particular, anions can have a fraction of an electron charge, even though they appear in a system that is built out of electrons. Now this is a really amazing and remarkable fact. And the result of this goes under and the set of phases in which these fractional charges show up, it goes under the name Symmetry Enriched Topological Phases or SETs. And one, I have done a lot of work on SETs. It turns out that certain SETs in two dimensions can only live on the surface of special three-dimensional systems. And I've worked to understand this phenomenon quite thoroughly. Now another interesting recent direction is another loophole in all of this stuff that I was talking about. So all quantum particles are either bosons or fermions in a 3D universe. There's another loophole, which is that we should change the statement to all mobile particles. So what do I mean by mobile particles? So imagine running our same argument that we had before and saying and just imagine now that we've got these identical particles, but for some weird reason, we're in three-dimensional space, but particles one and two aren't allowed to move in the z direction. Now you can try to run the same argument. You say okay, two exchanges of these particles means particle two makes a loop around particle one. And we argued that this is the same thing, it means that two exchanges is the same thing as doing nothing. But again, we argued that by taking this loop and deforming it out of the plane. And if this particle two is not allowed to move in the z direction, then this process isn't allowed. So again, we can't shrink the path down. And so the consequence that two exchanges is doing the same as doing nothing again breaks down. And so what this and so having these weird restrictions on how particles can move actually gives you some generalization of anions in three dimensions instead of two dimensions. And these movement constraints actually occur in certain things called fracton phases. And so for example, you have a particle which is allowed to move in the y direction and allowed to move in the x direction, but can't move in the z direction. And by having these constraints, you sidestep the rule, the usual rules of topological phases. And it allows you to have unusual things that are close to anions in three dimensions. And one of my research goals is to understand and characterize these fracton phases. So that brings me to the end of my talk. To summarize, topological phases are a beautiful mix of mathematics and physics. Topological invariance give physical properties that are quantitatively unchanged throughout the phase of matter. And we get by considering these topological invariance and these topological phases, we get a lot of really unusual new physical phenomena including quantized conductivity or resistance, as I was talking about it earlier, and anions and as it turns out, much, much more. And in the future, we're still searching both theoretically and experimentally for new topological phases and their realizations and signatures as well. So thank you very much for your attention and I'm happy to answer more questions. Thanks, Dany, for a really clear and interesting talk. Questions? Now please do use the microphone for the benefit of this online. Thank you for a very nice talk. It's a question about your graph, about the institute of optimal effect. Yes. If we lower the magnetic field, then the amount of electrons sitting at the edge increases. Is there some fundamental limit to how many electrons can actually move through the edge? Maybe set by some temperature scale or something when the magnetic field is not strong enough? So let me see here. So you're saying, for example, in this process, we start shoving electrons from one side to the other. Is there some limit on how much that can be? Yes, because when the magnetic field is becoming smaller, it seems like more electrons are moving to the edge. Good. In principle, the answer is that if you imagine a situation where you have infinitely many electrons in your system because you take a very, very, very large system, then there's really not any fundamental limit. But in any realistic system, then yes, there will be a limit and that will be set probably by the electric field that builds up as you transfer more electrons from one side to the other. So you're going to have a net charge on top and on bottom. So you're going to build up some electric field there. That is actually going to end up balancing the balance of the electric field from the charge distribution and the electric fields that you're applying and magnetic fields that you're applying is going to that balance is what's going to fix the number of electrons that move. So at zero magnetic field, it's actually kind of tricky that all the resistance will be actually zero as well. Correct. Well, I should caveat that that there are some situations in which there is a non-zero, sorry, zero, you can apply zero magnetic, external magnetic field and get a quantum Hall effect still. But yeah, we could talk a little bit about that offline. Further questions? Thank you very much. So if the 2D universe is a sphere or a torus, what kind of a part cruise do you expect? So it's still anions that you get there. And in fact, whether you have anions or not doesn't depend on what two-dimensional surface you take. However, you do point out that there are these topologically distinct two-dimensional universes you can consider. Does that have physical consequences? And it turns out it does. It's not in what types of particles can exist, but it's in how many quantum ground states the system has. As it turns out, when you have anions, the genus of the surface tells you how many, it tells you information about how many ground states the system has. And those ground states, there's a lot of protected interesting properties of those ground states. Are they the experimental tryout to consider non-planar to the universe? Not as far as I'm aware. It seems very difficult to realize in a lab. The most natural place I would imagine looking for it is in cold atom experiments, but that still seems quite challenging to realize a torus or a sphere. I believe that one-dimensional rings have been realized, but not two-dimensional non-trivial surfaces like that. Thank you. Further questions? People online can also ask questions. Okay, let me ask one in the interim. You hinted that quantum computers have something to do with all this. Would you care to say more? Sure. This actually gets close to the point that we were just talking about. As it turns out, there are certain types of anions that if you have many of these anions, the quantum state of the whole system is encoded in the behavior of those anions, but if you mess with just one of the anions, nothing happens to that quantum state. The idea is that if you can control what quantum state you construct with these many anions, then that quantum state will be resistant to noise. Generally speaking, noise happens when you mess with the system at a single point. It's a local process, but the state of the system that you're using is encoded in widely separated objects. The idea then is that by choosing this state, you can encode information. This information will be robust to noise because it's spread out this way. Then once you have information encoded in the state of the system, you can start manipulating that information. As it turns out, these exchange processes that I was talking about are the way that you manipulate that information in these anion-related quantum computing schemes. These anion quantum computing schemes are being worked on by Microsoft, as far as I'm aware, as the only company that's directly working in this direction. It's still a ways off before we realize it, but in practice it could be a very useful platform for building a quantum computer. Thanks. Another question? If not, I suggest we thank Benny again for a great talk.