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[AUDIO LOGO] CHAO MING JIN: Good evening, everybody. My name is Chao Ming Jin. I'm privileged to introduce the beta lecture tonight. First, I would like to begin by telling you a little bit about Hans Bethe, how the wonderful person he is, and what a wonderful legacy he left us. And then, I will introduce our speaker.
To those of you who have attended the colloquium on Monday, apologies. You will hear the same story again. So Hans Bethe was a hero of modern physics. He had a long-lasting career of almost 80 years as an active scientist.
His work helped shape modern physics as we know of today. And with the limited time I have, I won't be able to give a full account of all of his accomplishments, but I would still like to give you a short summary to give you an idea.
Hans began his career in the 1920s when he was among the first young physicists to explore the application of quantum theory. Upon leaving Germany to escape persecution for his Jewish heritage, Hans moved first to Great Britain, where he began his lifelong work in nuclear physics. Almost immediately, he became the world expert in the field.
Hans moved to Cornell in 1935. His research extended across fields as diverse as the quantum theory of solids and the nuclear processes that power the sun, receiving the Nobel Prize for the latter work in 1967.
He trained scores of young scientists, who have gone on to make important contributions of their own, one of which is David Thouless, who won the 2016 Nobel Prize for his pioneering work on topological phenomenon in quantum condensed matter physics, a topic that you will hear about today.
Hans helped build the physics department to the world class status it continues to enjoy today. He fostered an informal collegial environment that remains one of the defining characters of Cornell Physics. Hans retired from Cornell in the mid 1970s after 40 years on the faculty.
The retirement did not put a stop on his career as a scientist. In fact, even at the age of 80, Hans found a solution to the solar neutrino problem that had puzzled the astronomer for nearly 20 years. In fact, Hans continued to do research actively almost right up to his passing in 2005.
Solving mystery in physics was definitely Hans' passion in his entire life. To my colleague, Professor Ira Wasserman, he was brilliant, of course, but physics was fun for him. For his entire life, he was as enthusiastic and forward looking as a new graduate student. So that's always very inspiring to hear and read.
Hans was admired, not just because he was a great physicist, but also because of his courage and personal integrity. Here are a couple of examples. During the anti-communist hysteria of the McCarthy era, he was an early opponent of the development of the hydrogen bomb.
He helped protect Cornell Physics colleague, Phillip Morrison, from being dismissed as a result of his vocal opposition to the Korean War and his purported communist sympathies.
And he defended J Robert Oppenheimer, former head of Los Alamos, in his notorious security clearance hearing. he was a forceful and effective advocate for the Limited Test Ban Treaty, which forbade testing nuclear weapons under water, in atmosphere, and in space.
He was a relentless proponent of peaceful application of nuclear energy. Hans was a great physicist, a great mentor, and a great man. We're extremely fortunate that he left us with such a wonderful legacy.
Now, it's my pleasure to introduce our beta lecturer, Professor Charles Kane. Charlie is the Christopher H Brown Distinguished Professor of Physics at the University of Pennsylvania. Charlie got his PhD in physics from MIT in the year of 1989. And after a post-doc at MIT JT Watson Research Center, he joined the faculty of the University of Pennsylvania in 1991.
He is well known for his work on the characterization of electronic states of matter, including quantum Hall states, Luttinger liquids, carbon nanotubes, and topological insulator. In particular, his trailblazing work on topological insulators and generalizations have given us transformative new ideas about how new quantum phases of matter emerges from the interplay of symmetry and topology.
Charlie is a fellow of the American Physical Society and a member of the National Academy of Sciences. He has won numerous prestigious awards that includes the 2010 Euro Physics Prize, 2012 Oliver Buckley Prize, the Paul Dirac Medal of 2012, Benjamin Franklin Medal in the year of 2015, the Breakthrough Prize of 2018, the BBVA Frontiers of Knowledge Award of 2019, and the Fudan-Zhongzhi Science Award in the year of 2020.
Without further ado, I would like to turn the stage to Charlie and let him tell us the story about emergence of topological quantum matter. Let's welcome Charlie.
[APPLAUSE]
CHARLES KANE: Thank you very much, Chao Ming. It's really a true honor to be able to come to Cornell and to give a lecture in the name of Hans Bethe, a remarkable physicist and a remarkable man. And so thank you so much for the invitation.
And so what I want to do during this hour is to tell you a story about some of the amazing things that matter can do. So matter is the stuff everything is made out of, can do the most ingenious things. And so in this talk, I'm going to be focusing mainly on electronic matter, electronic materials.
And in order to sort of get our bearings and to get a start on thinking about this, I want to go all the way back to the very beginning of the study of electronic materials, all the way back to Benjamin Franklin.
And now, I have to confess, I come from the University of Pennsylvania. Benjamin Franklin was the founder of the University of Pennsylvania. So we talk about Benjamin Franklin a lot. So he's a big deal at Penn but for a good reason.
So of course, Benjamin Franklin was one of the founding fathers of the United States of America. He was a statesman, but in addition to that, he was quite a Renaissance man. So in addition to being a statesman, he was also a scientist. And as a scientist, what Benjamin Franklin is known for is for doing the first observations and experiments on the phenomenon of electricity.
And so he's known for doing these famous experiments where he went outside in a lightning storm and flying a kite. So that's something that I would not recommend that you do at home. That is, it's rather remarkable that he was able to survive that because you see, of course, electricity is the stuff that lightning bolts are made out of. It's also the stuff that makes your socks stick together when you take them out of the dryer.
So it's a rather ubiquitous phenomenon. And Franklin was really the first to think seriously about it and make systematic observations. And so one of the conclusions that he drew from his observations is that matter has a fundamental attribute, which he called electric charge.
And electric charge, he deduced, could either be positive or negative. And charges with the same sign push each other apart, and charges with opposite signs are attracted to each other. And so this is a fundamental aspect of the physical world that we live in.
So he realized another thing too, which maybe is a little bit more practical. He realized that materials can basically fall into two different classes of materials. There are materials that can conduct electricity, in which electric charge can flow in much the same way that water can flow through a pipe.
So these are called electrical conductors. They can transmit electric charge, whereas there are also materials that cannot transmit electric charge. And those are called electrical insulators. And so this is a fundamental dichotomy between different kinds of materials. And so I hope you can see this is something very fundamental that Franklin discovered here.
But in addition to being fundamental, Benjamin Franklin was a very practical man. And so he realized that this is incredibly useful. And so by making this discovery, Benjamin Franklin was able to invent one of the most important technologies of his day, the lightning rod. And of course, the lightning rod is the way you keep your house from burning down in a lightning storm, by having a conductor go around the side so that the electricity from a lightning bolt is shorted around into the ground.
And so, to me, this epitomizes what is best about science. So on the one hand, it's fundamental. It teaches you something fundamental about the physical world. It's beautiful. It's very interesting to think about and understand. But it is also useful.
And it's this interplay between fundamental, beautiful, and useful that has been at the heart of the amazing developments in our understanding of what matter is, how it fits together, and what it can do over the past 250 years. And think about what a ride we've been on over the past 250 years.
So fast forward to the present day, and electronic matter can do the most amazing things. It's like, really, it's like magic. So we all have powerful computers, where we can pick it up and push a couple of buttons and make any telephone on the entire world ring. That's really quite amazing.
And so how did that come about? It came about because of the developments in our understanding of what matter is, how it fits together, and how matter behaves. And so this understanding of electronic matter has enabled all of this technology that we've grown accustomed to in this information age.
And so what I'd like to describe to you in this lecture is I'd like to tell you that actually our understanding of matter, of how it fits together and what it can do, is continuing to evolve. And there are new ideas, new principles, new phases of matter that are being understood and that may well enable new technologies for the current century. And so these are some of the things that I would like to introduce you to in this talk.
And so in order to get started like this, I'm going to be talking about electronic phases of matter. And so let me start off by talking about some simpler phases, some simpler, more familiar phases of matter. And the most familiar phases of matter are the liquid and the solid phases of matter, of water.
And so water can either be a liquid that can flow or it can freeze and be a solid that is rigid. And there's something incredibly amazing about this because the building blocks of water, what water is ultimately made out of, are water molecules. And all water molecules are basically kind of the same, relatively simple building blocks. And yet, those water molecules when you put many of them together, they can exhibit completely different kinds of behaviors.
And so there's much more to the story than just knowing what the fundamental building blocks are. There's an interesting part of the story, which is once you have the fundamental building blocks, how can they behave? And so the liquid and solid are phases of matter.
Another example of phases of matter, for electronic matter, are the conducting and insulating phases of matter. So for instance, if you have a copper wire, that's an electrical conductor. And so electricity can flow through this wire sort of like water through a pipe.
But the rubber that surrounds it, the insulator that surrounds when you have an electrical wire, that is an electrical insulator. And the electricity can't go through the insulator. And so these are also examples of distinct electronic phases of matter.
And so I want to describe to you some new interesting electronic phases of matter. And in order to describe it, I want to start, though, by thinking about what the fundamental building blocks of matter are. And this is the physics that came into focus at the beginning of the last century. The early 20th century, it became understood that matter is composed of fundamental particles. And the most important fundamental particles for our purposes here are the electron, the proton, and the neutron.
And the important thing here is that the electron and the proton have electric charge. So the electron has a charge minus e, and the proton has a charge plus e. And a very important fundamental fact is that every electron has exactly the same charge as every other electron. And it has an exactly equal and opposite charge to the charge of every proton. So the way we say that is we say that electric charge is quantized. It comes in discrete units of e.
So actually, there's a little bit of a funny story about this. So I told you, Benjamin Franklin, he discovered charge. So he was the one who got to decide which is the positive and which is the negative. So he made his choice, and he made the most unfortunate choice that he could have made.
[LAUGHTER]
So the choice that he made, his choice was that the electron is negative. And this is a source of endless confusion for beginning physics students, when they realize that an electric current is flowing this way, but the electrons are moving this way. But we're stuck with Benjamin Franklin's minus sign. So we're just going to have to make do with it.
So matter is made out of electrons, protons, and neutrons. And of course, those fundamental particles arrange themselves into atoms. And an atom is sort of like a solar system, where the electrons are like the planets orbiting around the nucleus, which has the protons and neutrons, which are like the sun.
And so that's actually a pretty good picture of what an atom is, though it is incomplete for another reason, which this is another revolution in physics that happened at the beginning of the last century. And that is quantum mechanics. And so quantum mechanics drastically changed the way we think about the behavior of electrons in atoms.
And so what quantum mechanics tells us is that in an atom, the electrons just occupy discrete quantized energy levels. And so let me tell you what that means. What that means is that an atom, the electrons, it's like they're snapped into place. Once they're snapped in, there's no wiggling around. They're just sort of stuck they're, snapped into place.
It's sort of like LEGOs. You ever play with LEGOs? You snap the LEGOs together, and then there's no play. Everything's rigid. And that's sort of the way electrons are when they're in an atom. And that means that when the electron is snapped in, that in order to unsnap it, you have to give it a finite kick, a finite amount of energy.
And what we say is that an atom has what we call an energy gap. And that energy gap is the amount of energy you have to give it in order to unstick it. So this is a fundamental conclusion that quantum mechanics gives us. But once we understand this, that the electrons are snapped into place and atoms, then this allows us to understand the most basic electronic phase of matter.
And so the most basic electronic phase of matter is the insulating phase of matter. And so the way you can think about it is an insulator is just a bunch of atoms. And here are the atoms. They have the nucleus and the electron going around, and the electrons are snapped into place. And they're stuck there. And they can't go anywhere unless you really give them a big kick.
So in an insulator, basically nothing happens. The electrons are just sitting-- they're just snapped in, and they can't go anywhere. And so nothing can happen. And so there's a sense in which the insulating state is the most boring state.
But actually, I want to convince you, it's actually more interesting than you might first think because actually, one thing you could do, you can imagine you have all these electrons that are snapped in. One thing you could do to an insulator is you could add another electron. So let me add another electron. So here's another electron.
So that's an extra electron. And that extra electron has-- actually, let me say one thing is that atoms basically are electrically neutral. There are just as many electrons as there are protons. So the charge is completely compensated.
But I can add an extra electron to an insulator. And then, I know that I'm going to have an exact charge of minus e, minus because of Benjamin Franklin. Now, once I've added this electron, though, then this is actually kind of interesting because this electron has already paid this energy cost. And so it can move around. So it can hop from one atom to the next.
So when you add electrons to an insulator, then those electrons that you add, they can flow. They can conduct electricity. Now, there's something else you can do that's also interesting because in addition to adding an electron to the insulator, you can take one out. So if you take one out, then there will be a missing electron. We could call it a hole.
Now, this missing electron, you subtracted the charge of the electron. So it's minus, minus e. So the hole has a charge of plus e. But the thing is, is that this hole is like a particle too. It can move around. So watch. So it can move because the electron next to it can hop into its place. And then the hole is moved. And it can move again and again.
So we've discovered something really kind of interesting. So this hole in an insulator is kind of just as fundamental a particle as the electron that we add to an insulator is. But think about it. The hole was not in the original fundamental building blocks that we started with. The hole only can exist inside the insulator. So it's a fundamental particle, but it's a fundamental emergent particle. It's something that is a fundamental excitation of the insulating state.
And so the fundamental excitations of an atomic insulator are charge minus the electrons and charge plus e hole. Now, don't confuse the charge plus e holes with the charge plus e protons, which are sort of there. They're sort of nailed in place at the nuclei of the atoms. They're not moving around. So this is pretty neat.
Now, you might think that if you could add electrons or holes to an insulator so that they can conduct electricity, you might think that might be kind of useful. So in fact, an insulator that it is easy to add electrons and holes to, it's called a semiconductor. And semiconductors are pretty useful.
Like, silicon is a semiconductor. And silicon is the material that is inside all of the computer chips inside your cell phone So that's useful. So silicon is an insulator. And it's more complicated than this cartoon picture.
This cartoon picture, maybe it seems like it's too simple to be true. But it was one of the greatest triumphs of the quantum theory of matter, throughout the 20th century, was the development of our understanding, the detailed understanding of materials like silicon.
So silicon is a crystal. It's a piece of stuff. It's a bunch of-- it's a silicon crystal. It has a specific arrangement of atoms, a specific crystal structure. It has a specific electronic structure. And don't worry about what all these wiggles on this curve is. The point that I want you to get from this is that it's kind of complicated. There are lots of details.
And one of the triumphs of physics in the 20th century was we've mastered those details. And we understand them very, very well. And because we understand them very, very well, that's what has enabled us to take advantage of those properties to do all the amazing things that we can do with them. And that's that.
But there's a thing I want to make there. So it's complicated. But there's a point that I want to make, which we also have this simple cartoon picture that I showed you before. And I would like to argue to you that there is a sense in which that cartoon picture really is right because one thing that I can say is that even in silicon, the complicated silicon, it is still true that the fundamental carriers of electricity, the fundamental charge carriers, are charge minus e electrons and charge plus e holes. That's still true. So somehow, this simple-minded cartoon picture that I showed you got it right.
And so this leads me to what I think is a very deep and beautiful idea in physics. And the way that I like to say it is, I like to say that theoretical physics is like an impressionist painting. So look at this painting. It's a nice painting. But the thing about it is, is that if you go up too close and you look at it really close, it's all wrong. The details are all wrong. The colors are wrong. It's little blobs of paint. It's not a faithful, exact representation of reality.
But if you step back from it and look at it, there is a sense in which it captures the essence of the moment. And there's a sense in which it is right. And I'd like to argue to you that theoretical physics is kind of the same way.
So we write down these simple models, like that simple cartoon picture that I described to you of the insulator. And if you look at it too close in detail, in detail it's not exactly right. But it gets to the essence of what's going on in an insular. And so there's a sense in which it does get it right.
And so what I'd like to try to argue to you is that appreciating the sense in which it gets it right is a very deeply aesthetic experience, which I would like to try to convey to you and get you to appreciate.
And so in order to explain the sense in which it gets it right, I want to introduce another concept, which is the concept of topology. Now, topology is a deep and beautiful branch of mathematics. So it's concerned with the study of objects that can be continuously deformed.
So what I want you to think in your mind about is a piece of clay. And a piece of clay, you can squeeze it, you can stretch it. And you can continuously change its shape. And so what topologists are concerned with is they're concerned with what kinds of shapes can be continuously deformed into one another.
And the classic example for that is to think about the sense in which a coffee cup is the same as a donut. So if you imagine they're made out of clay, you can imagine you start with the donut, and you stretch it a little bit, and you stretch a little bit more and squeeze.
And at the end of the day, you've turned it into a donut, smoothly along the whole way. Of course, the hole in the donut turned into the handle of the coffee cup. So there's a sense, in the sense of topology, that the donut and the coffee cup are the same.
So topologists are interested in understanding the sense in which things are the same. But you see, that doesn't mean that everything is the same. So if you instead think about the surface of a sphere, like the surface of this orange here, so you could continuously deform that. You could step on it and squeeze it down and turn it into a pancake.
But you can't turn it into a donut without poking a hole in it. And you can't poke a hole in it continuously. You have to sort of abruptly go from not having the hole to having the hole. And that's a discontinuous change.
And so the sphere and the donut are different. And so topologists, they like to come up with some simple way of characterizing the sense in which they're different. And it's simple in this case. What you can introduce is what's called a topological invariant, which is basically the number of holes. And we call that the genus of the surface. And so that's a way of distinguishing things.
So you can ask the question, how does this have to do with physics? But what I would like to argue to you is that this cartoon picture of the insulator and the complicated electronic structure of a real material like silicon, they are the same in the sense of topology.
Because what you can imagine doing is you can imagine taking your atoms in real silicon, and you can imagine in your mind's eye, you can imagine pulling them apart very gradually. And once you've pulled them all apart so the atoms are far apart from each other, then it's going to be more like this situation, where you have independent atoms that make up an atomic insulator.
So there's a sense in which they're the same. Now, of course, this picture is much simpler than this picture. So if I think about surfaces, the sphere is the simplest surface that doesn't have any holes in it. It's very simple and beautiful and nice to think about. But that doesn't mean that every surface has to be simple. Surfaces can be very complicated.
So for example, this is Dave the octopus. And Dave the octopus has a more complicated shape. And I think at least if he keeps his mouth shut, then he doesn't have any holes in him either. So his genus is also 0.
So the octopus and the sphere are topologically equivalent. And so maybe there are some things that are simpler to think about, to think about the sphere. But there are properties that are shared by both. And so by thinking about the sphere, you actually learn something about the more complicated shapes. So that's the sense of topology.
Now, one thing I'll say is that most insulators are like the trivial atomic insulator. But when you say it like this, then it invites the question, could there be insulators that are different? So in pictures, if the trivial atomic insulator or silicon is the sphere, then is there something like the donut?
And the remarkable answer to this question is, yes. Now, one thing is the quantum theory of matter, this was developed in the early part of the 20th century. It's been around for many, many, many decades. But it's only been relatively recently that people have thought to ask the question in this way. And once you ask the question in this way, then you are led down this path of discovery, where you find amazing new things. And these are the things I want to tell you about.
So I want to tell you about topological electronic phases. And so in order to do that, in order to motivate, I always like to start with the simplest version of a problem that I can think of. And the simplest version of a problem that I can think of is, what happens in a one-dimensional polymer?
So a polymer is a one-dimensional chain of atoms. And poly acetylene is a particularly simple one-dimensional polymer. It's just a chain of carbon atoms. And then, there's some hydrogen atoms attached there. That's not so important.
The thing that's important about poly acetylene is that it is an insulator. But it's an interesting kind of insulator because what poly acetylene wants to do is it wants the atoms to sort of pair up so that there are strong bonds alternating with weak bonds. And in order to do that, it's able to sort of find its happiest configuration.
But you see, the thing about this is that's interesting is that there are two ways you can do it. You can either alternate strong bond, weak bond, strong bond, weak bond, or you can alternate weak bond, strong bond, weak bond, strong bond.
And I would like to argue to you that these two configurations are topologically different in the sense of topology because if I want to try to continuously go from this to this, I need to go through an intermediate state where both the bonds are the same. And that's not an insulator. And so these two are different.
So now, the impressionist picture that I want you to have in your head for what's going on with it, so I told you in atoms, the electrons are snapped into place. So what's going on here in poly acetylene is that you have these wells that the atoms can sit in, but there are twice as many wells as there are electrons. So the electrons are sort of snapped in every other space. And in the A phase, they're snapped into the blue spaces, and in the B phase, they're snapped into the red phases.
Now, one important thing I want to say about this is that both configurations are electrically neutral, in the sense that the charge of the electrons is exactly compensated by the charge of the nuclei of the atoms. So this is what we have.
So now, you can have-- OK, so these two are different. What sort of consequence might this have? So in order to think about the consequences, I want to show you a magic trick. So here's my magic trick. I'm going to start with-- so I'm in the A phase. So I have all my electrons. They're snapped in every other space. This whole configuration is electrically neutral. There's no electric charge because everything is compensated.
And I told you that insulators are interesting because if you add electrons to an insulator, you can turn it into a conductor. And that's what semiconductors do. So let's add an electron to the poly acetylene. So here we go. There's my electron.
So now I have a situation where I have exactly a charge e, now minus e again. Benjamin Franklin. So now, I've added all the charge that I'm going to add. Now I'm just going to rearrange things a little bit. So watch this. This is my magic trick.
So I'm just going to move things. So what I'm going to do is I'm going to move the electrons over. First, I'm going to move these over. So move them like that. And then, I'm going to do the same thing again. I'm going to take these ones and move them over like that. And I could keep doing that over and over again, sort of get the picture of what's going on.
Now, I'm going to ask you the following question-- what happened to the original charge that I added to the poly acetylene? Now, I know there's a wise guy out there who's saying it's right there.
[LAUGHTER]
Yeah. You're right. But think about it. See in the middle here, in the middle, I'm occupying every other well. So this is electrically neutral in exactly the same reason the original starting point was electrically neutral. There's no charge in the middle there. The only place there's any charge is sort of here and here where the electrons are kind of bunched together. That's where the charge is.
And this place looks exactly like this place. And so what's happened is the added electron has split in half. And it's not, you see, we're not actually cutting an electron in half. You can't do that. That's against the rules.
But you see, this half a charge, it's like a particle. Right? It can move around. I showed you it moving away. I can keep moving it over, and it can move around just the same way that a particle can move around. And it has a charge exactly e over 2. So it's an emergent quasiparticle that's just as good a particle as the hole was.
So I don't know. I think this is truly amazing. And I'm not the only one. So John Preskill. John Preskill is a very distinguished theoretical physicist, somebody who I deeply admire. And so he's a great theoretical physicist. But he moonlights as a poet. And so when he sees something that he thinks is inspiring, he often writes down an interesting poem.
So he learned about this phenomenon, or actually something related to it, from another distinguished theoretical physicist named Leon Balents. And Leon is a good friend. And so he says it like this-- he says, you have to admit it's legit, electrons can split.
So how could this be? How could it be? So electrons are fundamentally quantized in charge of e, and yet, we're able to have half an electron here. Exactly charge e over 2. So what's going on is that you have to-- if you look, the clue to this is if you look in the middle, notice that the electrons are sitting in the red wells, whereas on the outside they're sitting in the blue wells.
So what we have is the boundary between my two different topological phases. And this is the essence of what topological quantum matter is about. The essence is that the impossible, meaning the impossible thing which is cutting an electron in half, that occurs at the boundary between different topological phases. And so if you get this picture, then you understand what topological insulators are all about.
But the point I want to make-- so this is a simple example. There are many, many more examples of this. And these many more examples have come to life in the real world. And the impossible things that they allow are truly amazing. And so I want to show you some of those things.
So the first example that I want to describe to you is it's motivated by a problem. And the problem is, one of the things that you want to do when you make electronics smaller and smaller, there's a problem, which is the motion of electrons in electrical conductors is rather disorganized.
It's sort of like, if you ever go to New York City and go to a Broadway show, you came out after the show, and there's this huge crowd of the people. You're trying to get to the subway, and you're bumping into people, and it's hard to get where you're going. That's sort of how electrons feel when they're trying to make their way through a crowded electrical conductor.
So you can ask the question, is there any way to make that better? And so you might take a clue from how we go about organizing traffic. And so if we want to organize traffic, then a divided highway makes the flow much more organized. And so what I'd like to argue to you is that a topological phase of matter accomplishes precisely this.
And this topological phase of matter, this is the mother of all topological phases. This is the topological phase that we understood first. It's the topological phase that we understand the best. And it has a name. It's called the quantum Hall state.
And what it is, is I would call it a two-dimensional topological instance. What do I mean by two-dimensional? So a piece of paper is two dimensional. It's very thin. And so if you can confine electrons to a two-dimensional plane, which is what you can do if they're sitting on the surface of some material or on an interface between two materials, then you can have a two-dimensional material. And in this two-dimensional topological insulator, in the interior, in the interior, the electrons are snapped into place. It's an insulator. So it's just boring. Every electron is snapped in. It can't get anywhere.
But the thing is, is that on the boundary of this topological insulator, you are guaranteed to have-- and one thing I will say is that this electrical insulator is characterized by a topological invariant, sort of like the genus that I told you about. It can be 0, 1, 2, 3. And that number N guarantees that on the edge, there are N lanes of one-way conductors.
And these one-way conductors are truly remarkable because if you're an electron in this one-way conductor, you have no choice to go forward. It's just like when you're driving down the interstate, no cars are turning around on the interstate. There's no choice but to keep going straight.
Now, I'll say another thing. Let's suppose you could have-- let's say, suppose you could have just this one-way conductor that's propagating from left to right. Let's suppose that's all you had. Let's think about what you could do with that. You know what you could do that with that is you could put the entrance to the one-way conductor inside your refrigerator.
And you could put the exit of the one-way conductor outside your refrigerator. And then, you'd have a way of running your refrigerator without having to plug it in, right? Because this one-way conductor would carry heat from the inside of your refrigerator to the outside. And that's exactly what a refrigerator is supposed to do. It's supposed to pull heat out and to keep it cold inside.
So if you could do that, you'd be rich. You would solve all of the world's energy problems. It's just there's a problem. The problem is, is that in addition to violating the no free lunch principle, having this one-way conductor by itself violates deep principles of physics. It's impossible, just like cutting an electron in half is impossible. But it is possible on the boundary of a topological phase.
Now, of course, if you have a boundary of your topological phase, then there also has to be another boundary over here. So you can't just stick this one in the refrigerator. This one has to go in too. You can't put one in without the other one. So that's how you get around that. That's the reason you're not going to get rich from this.
So now, I'm telling this story a little bit backwards. So this topological phase was not discovered by people sitting in a room by themselves thinking about donuts and coffee cups and things like that. That's not how it happened. It was actually discovered in the laboratory.
And so what was happening, so this was back in the 1980s. And so what was going on at that time, Klaus von Klitzing was an experimental physicist who was studying doing fundamental characterization of the semiconductor materials that now form the components of your cell phone. So he was studying the fundamental properties of that.
And so one of the things that you do when you might want to study the fundamental properties of some material is you study the electrical conduction as a function. You put it in a magnet and study it as a function of the magnetic field.
And what von Klitzing observed was that the resistance is a function of the magnetic field had these steps. And these steps had a remarkably-- they're remarkably flat. And the value of the resistance on these steps has a remarkable quantization in units of these fundamental constant.
So this is Planck's constant, and this is the charge of the electric charge. But the thing that's remarkable about this is that this quantization, if you know these fundamental constants, then this integer N is something that you can measure to one part in a billion. It's like 1.000000000, nine 0s. Experimentally you can measure that.
Another way of saying it is that if you believe the integer is an integer, then you can measure this combination of fundamental constants to one part in a billion. So in fact, this is the best way to measure that combination of fundamental constants. And actually, just in the last couple of years, the ruling authorities that decide what the units are, how big a meter is and how big a kilogram is, the ruling authorities decided that the kilogram should no longer be determined by the mass of some rock that was stored in a vault in France. That's the way it was before, just a couple of years ago. There literally was a rock that was stored in France that was the kilogram.
But the problem is, is that you can only measure the mass of that rock so well. And you can do this experiment better. So the ruling authorities changed it so that this experiment defines the kilogram. It's that good.
So at the time, this was an amazing puzzle. How could it be that-- look at the sample. This is a sample that you do the experiments on. It's not the sample that von Klitzing used. This was from a little bit later. But look at it. It had all these globs of solder on it. And it's dirty, and it's not a perfect sample. It's not a perfect experiment. But somehow, the measurement that you make is virtually perfect. So how could that be?
And so this is what inspired geniuses in theoretical physics, geniuses like Bob Laughlin and David Thouless, who Chao Ming told you was Hans Bethe's student. It inspired these geniuses in theoretical physics to think very deeply about what's going on. And what emerged from that was this idea of topology, that the integer that you're measuring is a topological invariant and that you can't mess it up.
So this messy sample, maybe the messy sample is more like Dave the octopus. But his genus is still precisely equal to 1. And so this is the original topological phase.
Now, you might think this would be kind of useful if you could have this one-way conductor to sort of organize the flow of electricity. You might think that would be a kind of useful thing to be able to do. The only a problem is, in order to get this to go, it requires that you put your sample into a big magnet.
So if you look at this, the magnetic field is like Tesla. And a Tesla, a magnetic field, that's a big, strong magnetic field. You need a big, strong electromagnet in order to get a Tesla of magnetic field. And you don't want to have to carry that around in your pocket.
So you can ask the question, is there any way to get this to go without the big, strong electromagnet? And so this is a hard problem. It's a problem that people are working very hard to try to solve. And one possible route to that direction is to think about another topological phase, which is called a topological insulator. And so I want to introduce you to that.
And so the way I like to say what a topological insulator is, is it's sort of like a chocolate treat that's wrapped in tin foil, in the sense that the inside, the chocolate, is an electrical insulator. It's an electrical insulator where the electrons are snapped in place. They can't do anything. But the surface is an electrical conductor. It's a very special electrical conductor.
Now, this analogy isn't completely right because actually, if a topological insulator was a chocolate treat, it would be a very frustrating chocolate treat because you can't unwrap it. If you tried to cut it open and unwrap it, you would find it's actually still wrapped.
And you see, the thing is, this conducting surface is another example of something that's impossible. And so you can't peel it off because if you peeled it off, then you'd have it by itself, and that's impossible. And so that means that if you have it, then it's impossible to get rid of it. It's protected.
So this conductor is very special. And it has a very specific special property. And I don't want to get into the technical details of this. What we say is that it has a specific signature, which is called a single Dirac cone in its surface electronic structure. And don't worry about what that means. But there's something very specific. It has a very specific property.
Now, the thing about topological insulators, so the remarkable thing is the topological insulators have come to life in the real world. But the story of topological insulators is a little bit different than the story of the quantized Hall effect because I told you that in the quantized Hall effect, it was discovered in the lab, and then that inspired physicists to think deeply about what was going on. And then some understanding emerged from that.
So the topological insulators were conceived of in pure thought. And I played somewhat of a role in this. So what came first was the realization that such a thing could happen. And once that was realized, then we were able to, by pure thought, come up with specific materials where this should happen.
So the first start of this was understanding the electronic structure of specific materials and understanding that they would be topological insulators. Now, if this is all that we've done, if this is all that had happened, then it would be kind of interesting, but it wouldn't be that interesting.
What made it really interesting is that it came to life experimentally in the real world. And what was required for that is, first of all, you had to be able to make the materials. So it required some high-quality growth of crystals. The best crystal that does this is a material called bismuth selenide. So it's a combination. It's a specific crystal arrangement of bismuth and selenium atoms. And it has a specific electronic structure that happens to be a topological insulator.
So the prediction is that this crystal of bismuth selenide has this impossible conductor on its surface. And so the next thing that's required is you have to be able to confirm that. And so what people were able to do is to basically take a fancy picture of the surface. And so the technique they used is called angle resolved photoemission spectroscopy. But basically what that is, is it's taking a fancy picture of the surface that tells you what the electronic structure is.
And I'll just tell you that this picture here shows a textbook-perfect single Dirac cone. So it's exactly what you want to see in order for this to be the case. So these discoveries, they sort of opened the floodgates for all different kinds of materials that you can have. And there's a huge amount of activity doing this.
Before I finish, though, what I want to do is I want to give you one more example of a topological phase. And this final example is motivated by another problem. And the problem that motivates it is the problem of making a quantum computer.
So this is one of the grand challenges of the 21st century. And I don't know whether in the 21st century, we're going to have a quantum computer. But it's a wonderful motivator for doing some great science. And so in order to motivate this, let me remind you what a regular classical computer does.
So a regular classical computer manipulates classical information. And classical information are basically numbers. And numbers can be represented in terms of binary digits, 0s and 1s. So for instance, the number of 5, if you write it in binary, it looks like 101.
And so what a regular computer is, is it's basically a device that takes some bits of the input, does some operation on it, it spits out an answer. Here I added 2 to it and so I got 111, which is the same as 7. That's sort of what a simple-minded classical computer does.
Now, what a quantum computer can do is it takes-- it has bits too, 1s and 0s. But the thing that quantum mechanics allows is it allows a bit to be 1 or 0 or both of them at the same time. Now, don't expect to understand that because I don't understand it.
That doesn't mean-- I know quantum mechanics. I know how to do quantum mechanics. I know how to calculate. I do know that. But that doesn't explain what this means. But it is a fact that a quantum bit can be both at the same time.
And so let's think about what this could allow because if one bit can be 0 and 1 at the same time, then if I have a string of bits, then it could be-- if I have a string of three bits, it could be all of the three-digit binary numbers at the same time. All 2 to the-- if I have N of them, I get 2 to the N numbers at the same time.
So what a quantum computer is, if you could make one, is it's a device that takes quantum bits, does some operation and spits out the answer. And so let's suppose that we send all of the numbers at once in, do one operation, then we get all of the answers at once.
Now, if you're going to do this on a classical computer, you'd have to do it 2 the N times. Here, we just do it once. Now, I actually first heard this in a beautiful lecture that was given by David Mermin. He told the story of how you could do this. And he said, is this too good to be true? Yes. It is too good to be true. It's not this good. But it is still good.
So it is a fact that there are classes of problems that a quantum computer can solve much faster than an ordinary computer. And that fact is a tremendous motivator to make for us to try to make a quantum computer. The problem is, it's hard. And the reason it's hard is because there's a fundamental problem with quantum mechanics, which is if you have this 1 and 0 at the same time, if you look at it, you don't see 1 and 0 at the same time.
What you see is either 1 or it's 0, and which one you get is sort of like flipping a coin. That's what quantum mechanics tells you. Moreover, after you look at it, it's definitely what you saw. So the combination is gone. So if you measure a qubit, you kill it. So that's the fundamental problem.
Now, the practical problem for making a quantum computer is how do you keep the quantum computer from accidentally measuring itself? It doesn't have to be a person that measures it. And so this is the decoherence problem. Quantum information is very fragile. The superposition is very fragile.
And so there are various ideas for how you can get around this problem. So one idea is you want to have your qubits just be very, very well-isolated from everything else so nothing else is interacting with it or looking at it to measure it. So maybe that might be the best way to do it. I don't know.
What I want to tell you about, though, is I want to tell you about another approach of protecting the quantum information, which takes advantage of the topological phase. And so the idea, so a qubit is like-- you can sort of think of it as it's some object that has two states that it can be in. It's some localized object, like an electron, which has a spin that can be up or down. There are various kinds of ways you can imagine having a qubit. But the important thing is that it's sort of localized in one place. It's indivisible. You can't cut a qubit. You can't have half a qubit.
But there exists a topological phase that allows you to have half a qubit on its boundary. Of course, the other half is going to be on the other boundary. And this topological phase is called a topological superconductor.
Now, since I'm probably already too late-- I don't have time to explain to you what superconductor is, let alone what a topological superconductor is, but let me just say that the way we think about it is very similar to the way we think about the other topological phases I did tell you about-- the topological insulators the quantum Hall state, and especially this one-dimensional poly acetylene. It's very close to this. So they're very similar ideas we use to understand topological superconductors.
So the beauty of this, though, is that this qubit, which is sort of split in half, it exists in two places, if you just are able to look locally in one place, then it's impossible to measure it. In order to measure it, you need to look at both places at the same time.
So if you can only do local measurements, which is a reasonable assumption for the kind of accidental measurements that a quantum computer might make on itself, then the quantum information stored in this qubit is topologically protected because it's sort of stored nonlocally in two very distant places.
That's the idea. Split the qubit, and it's exactly the same kind of idea as splitting the electron into e over 2 or splitting an electrical conductor into one-way electrical conductors. It's the same sort of idea.
And so this is something which maybe can happen. But even this is hard. But there has been tantalizing progress. So there are various ideas for what kinds of specific materials this can be. And so one idea is to use a one-dimensional wire made out of this material that's called indium antimonide.
And you put it next to an ordinary superconductor, and so when they do that, they do an experiment where they try to add electrons to the end. And they see a peak here. And this peak is exactly what you would want to see if there was this half of a qubit at the end. And so you might say, ah, happy we're done. Of course, you're not done.
The challenge, and this is something people are still-- it's still controversial. It's still something people are arguing about. The challenge is to convince yourself that this peak, the origin of this peak, is the interesting origin that you hope it is and not some less interesting origin that you need to understand if you're going to understand this experiment.
And so that's a challenge. But nonetheless, this experiment and others like it have spawned a huge amount of effort. And there's lots of really good science that is being done to try to make this vision come to life.
So now, I think I'm out of time. So I want to just leave you with one final thought, which is to emphasize that there's one sort of direction you can go in physics, which is to say that I want to understand the most fundamental, basic building blocks and rules that govern matter. I want to understand what's the most fundamental level on the most deep fundamental level.
So that's one thing, but there's another equally fundamental question that you can ask and this is, once you have the rules, once you know, then how does matter that obeys those rules behave, and what kind of phenomena can emerge from that?
And so let me give you an example, and the example I want to give you is classical mechanics. So Newton. Newton discovered the laws of classical mechanics back in the 1600s. He wrote the Principia. And in the Principia, he has his three laws. You know, F equals ma.
This is what we teach our freshman physics students. And that basically, he nailed it. That is, there's a certain level that's the fundamental rules of classical mechanics. And everything else follows from that. So that's the most fundamental thing.
But I want to say that there's an equally fundamental question, which is, how does matter that obeys Newton's laws behave? And that's a much bigger question. And in order to be able to think about that question, you need organizing principles for how to think about matter. And so as an organizing principle, consider-- so Newton did the Principia in the 1600s.
It wasn't until hundreds of years later that the concept of energy emerged. And so energy is an organizing principle. So energy, of course, energy is contained in Newton's laws. But Newton didn't know about it. His laws did, but he didn't. But energy is an organizing principle, is an organizing principle, not just for physics, but all branches of science and humanity.
So what I'd like to say is that quantum mechanics is in sort of a similar situation. So the fundamental rules of quantum mechanics were laid out in the early part of the 20th century. And they're still right, just in the same sense that Newton is still right.
But today, organizing principles for how matter that obeys quantum mechanics behaves are still being uncovered. Principles like topology. And so this makes it a very exciting time to be a physicist because there are many new things that are developing. And there are many new materials that are being discovered. And again, it's a very exciting time because matter can arrange itself in the most ingenious ways. So thank you very much.
[APPLAUSE]
CHAO MING JIN: So before we open for questions, a quick announcement. There will be a reception afterwards. So for those of you who RSVPed earlier, the reception will be in [INAUDIBLE]. So questions.
AUDIENCE: It was a wonderful talk. Thank you. I'm curious about [INAUDIBLE] bismuth selenide. You said, imagine the topological [INAUDIBLE]
CHARLES KANE: So there's a much longer history of bismuth selenide, which is that it's actually a material that has been known to physicists for a very long time. And so bismuth selenide is actually a very important material because it is a very good thermoelectric material.
And so there's lots of technological applications of bismuth selenide. So it's a material that-- we didn't discover bismuth selenide. And people had computed the electronic structure of bismuth selenide, but they just never asked the right question about it.
And so that's what happened when-- and so it was remarkably satisfying to realize that this material, which is actually already known, actually has this property that nobody had ever thought about. But yeah, it's not an original material. It's something that people have studied for a long time. But they had never studied that particular aspect of it.
AUDIENCE: Yeah. Question. Could you go back to the slide where you originally introduced the [INAUDIBLE]
CHARLES KANE: You mean the magic trick?
AUDIENCE: Yeah. Just to go back a slide or two.
CHARLES KANE: So here we go.
AUDIENCE: Back one more.
CHARLES KANE: Back more. OK.
AUDIENCE: [INAUDIBLE] So just looking at the pictures of A phase and B phase, if I just take the A phase out of the page and rotate it 20 degrees, put it back down.
CHARLES KANE: Yeah?
AUDIENCE: [INAUDIBLE] B phase.
CHARLES KANE: Yeah. That's true.
AUDIENCE: [INAUDIBLE]
CHARLES KANE: That's true. So yeah, or you could just slide it over by one and maybe flip it. It would look the same too. So you're absolutely right. But the rules of the game here are I want to get from this to this without moving anything.
AUDIENCE: Oh.
CHARLES KANE: That's the rules. Because the thing is, I could if I wanted to. So let's imagine what would happen if I tried to get-- I could turn this bond down and turn this bond up. And if I did that, I could get to a picture that looks like this. I could do that. But along the way, I go through a point where both of these two bonds were the same.
And the way I say that is that when both of them are the same, it's no longer an insulator. What has happened is you're going-- at that point, there is a topological quantum phase transition where it ceases to be an insulator. So now it's not an insulator anymore. But then, when you keep going, then you've gone through the phase transition into this distinct topological phase.
AUDIENCE: I just wanted to comment that, if I understood it correctly, your description of the refrigerator was not quite complete because you had a big enough sample or a small enough refrigerator, you could put just half of it in it.
CHARLES KANE: No.
AUDIENCE: And the second one is saved by the circulation.
CHARLES KANE: No. But you can't put half of it in. You see, the point is that these one-way modes, they can't end. They have to keep going. And so that means that if it goes in to the refrigerator, it has to come back out or if it came out of the refrigerator, it had to also go in. And so things can go in both directions. And so that's the paradox. So you would be rich if you could make one of these chiral modes end. But that's the thing you can't do.
AUDIENCE: What happens on the other boundaries?
CHARLES KANE: What happens on what?
AUDIENCE: The other boundaries. You showed us the two boundaries, right? The two sides.
CHARLES KANE: So are we talking about this picture or are we talking about the--
AUDIENCE: No, your 2D.
CHARLES KANE: The 2D? OK. Let's keep going here.
AUDIENCE: Great. You have the N names on the side that each allowed one-directional transition.
CHARLES KANE: Yeah. So what I would say-- so there's one going from left to right on the bottom. There has to be another one going from right to left on the top. And in fact, I drew it like it ends, but what has to happen is this one has to turn around and go around the entire perimeter so they're all connected.
AUDIENCE: That was the circulation plan.
CHARLES KANE: OK. OK. Yeah. Yeah. So I guess I should have drawn it going all the way around, yeah. Good point.
CHAO MING JIN: I think it's a good time to end today's session. Let's thank Professor Charles Kane.
CHARLES KANE: All right. Thank you very much.
[APPLAUSE]
Bethe Cornell Community Lecture by Charles Kane
Host: Chao-Ming Jian
Abstract: Matter can arrange itself in the most ingenious ways. In addition to the solid, liquid and gas phases that are familiar in classical physics, electronic phases of matter with both useful and exotic properties are made possible by quantum mechanics. In the last century, the thorough understanding of the simplest quantum electronic phase - the electrical insulator - enabled the development of the semiconductor technology that is ubiquitous in today's information age. In the present century, new "topological" electronic phases are being discovered that allow the seemingly impossible to occur: indivisible objects, like an electron or a quantum bit of information, can be split into two, allowing mysterious features of quantum mechanics to be harnessed for future technologies. Our understanding of topological phases builds on deep ideas in mathematics. We will try to convey that they are as beautiful as they are fundamental.
