SPEAKER 1: Good afternoon, everyone. It's my honor and great pleasure to introduce the Bethe lecture series of this semester. Today, we have the first event of this prestigious three-part lecture series. First, I would like to tell you or perhaps remind you of what a great man Hans Bethe was and why we have this lecture series in honor of him. Then I will introduce today's speaker.
Hans Bethe was truly a giant in 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. Time will permit me to give a full account of his accomplishment. So I would like to provide you a short summary to give you an idea.
Hans Bethe began his career in the 1920s when he was among the first young physicist to explore the application of quantum field 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 this 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 a Nobel Prize for the latter work in 1976-- I'm sorry, 1967. He trained a score of young scientists who have gone on to make important contributions of their own, one of which is David Ellis, who won the 2016 Nobel Prize for his pioneering work on topological phenomenon in condensed matter physics. This is 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. Even at the age of 80, Hans found a solution to the solar neutrino problem that had puzzled astronomers for nearly 20 years. In fact, Hans continued to do research actively, almost right up to his passing in 2005.
You may wonder, how can one sustain such a level of excellence for so long? I didn't have the privilege to interact with Hans personally, but to quote 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.
Throughout his career, Hans was a beacon of courage and personal integrity. Here are a couple of examples. During the anti-communist hysteria of the McCarthy era, he was an early opponent to the development of the hydrogen bomb. He helped protect Cornell physics colleague Philip 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 the atmosphere, and in space. He was a relentless opponent a proponent of peaceful application of nuclear energy. Hans Bethe was truly a great man, and he gave us so much. We are extremely fortunate that he left us with such a wonderful legacy.
Now it is my great pleasure to introduce our Bethe lecture-- Bethe lecturer today, Professor Charlie Kane. Charlie is the Christopher Browne Distinguished Professor of Physics at the University of Pennsylvania. Charlie got his PhD in physics from MIT in the year of 1989. After a post doc at IBM TJ 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 state of matter, including quantum hall, linear liquids, carbon nanotubes, and topological insulators. In particular his trailblazing works on topological insulators and generalizations have given us a transformative new idea about how new quantum phases of matter can emerge 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, the 2012 Oliver Buckley Prize, the 2012 PAM Dirac medal, the 2015 Benjamin Franklin medal, the Breakthrough Prize of 2018, the BBVA Frontiers of Knowledge Award of 2019, and more recently the Fudan-Zhongzhi Science Award of 2020. All right, without further ado, I would like to turn the stage to Charlie and let him tell us a story about symmetry topology and electronic matter. Let's welcome Charlie.
CHARLES KANE: OK, thank you very much, [INAUDIBLE], for that very nice introduction. And what an honor it is to come here and give a series of lectures in the name of Hans Bethe, who, as [INAUDIBLE] very nicely explained, is one of the heroes of physics of our lifetimes.
So I'd like to tell you a story today about symmetry and topology. And these are two of the big ideas that underlie our understanding of matter, underlie our understanding of physics. And so symmetry is a familiar idea when you first study physics. Of course, what symmetry is about is, what can you do to a system that keeps it the same?
And we teach our freshmen about physics. Sometimes I teach freshmen physics. And I try to bring home the point that whenever you're confronted with a problem, the first thing you think about is symmetry because it can simplify things. And it gives you an organizing principle for how to think about it.
Now, one of the powerful things that symmetry can allow-- and this is a deep insight due to Landau-- is that symmetry can give you a principle for distinguishing the phases of matter. So for example, if you think about crystalline matter, then crystalline matter has structure that is reminiscent of these pictures. You can translate them. You can rotate them. You can reflect them. And they can look the same.
And by understanding those symmetries, that gives you a powerful tool for distinguishing distinct crystalline phases of matter. And so this application of symmetry is one of the fundamental things that we have for understanding the phases of matter. But one of the things that has become more and more clear over the last several decades is that it's not the only thing that distinguishes phases of matter.
And so there's another big idea that has deep applications to physics. And that is topology. And this is not something that we tell our freshmen about usually. But it's equally simple. So what topology is about is, what is it that stays the same when things are smoothly deformed?
And the sort of classic example of topology is to think about the sense in which a donut is the same as a coffee cup because if you imagine that the donut is made out of clay, you can stretch it. You can squeeze it in such a way that not very much is happening at every step.
But at the end of the day, the donut has turned into a coffee cup. And of course, the hole in the donut becomes the handle of the coffee cup. And so what mathematicians like to do-- topology has a deep and beautiful branch of mathematics. What mathematicians like to do is to figure out ways of characterizing the sense in which things are the same or the sense in which things are different.
So for example, the mathematicians would introduce what is called a topological invariant. And this is a number that in this simple case just counts the number of holes. We call this the genus of the surface. And so this number is a simple way of distinguishing whether things are the same or different. Now, topology has many applications in physics. And the applications of topology and physics go back many, many decades.
What's relatively new, what is now a part of what I would call 21st century physics is a new appreciation of the interplay between symmetry and topology and how working together they give us something which is sort of bigger than either of them. And so this is the story I'm going to tell you, and in particular in the context of the study of electronic materials. I want to show you how symmetry and topology have this wonderful interplay.
And this is something which happens in the context of what I'll call the band theory of solids. So I'm going to start off with some introduction, which will try to explain to you how it is that topology shows up in thinking about the electronic structure of materials. And then I'll give you a couple of examples of that. I'll talk about topological insulators. I'll talk about a related thing, which is topological superconductivity.
And then towards the end, I'll try to give you some flavor. So this is physics which has sort of grown up over the last 15 years or so. And I'll try to give you some flavor of where we are and where things are going.
And so that's the plan. So I want to talk about electronic materials. And so I always like to start at the beginning with the simplest possible electronic material, the most basic electronic material, which is still very important, which is the electrical insulator. And this is another thing that I tell my freshmen about when I teach them.
And I tell them, in an insulator, the electrons are stuck. They're sort of snapped into place. And they can't do anything. They're stuck and basically nothing happens.
So there's a sense in which the insulating state is the most boring state because every electron is stuck on its home atom and nothing happens. But there's an interesting fact, which is in order to understand why the electrons are stuck, an essential role is played by quantum mechanics because the reason electrons are stuck is because quantum mechanics tells us that the energy levels of an atom are quantized.
And so if you fill up one set of energy levels, then you can have an energy gap for the electron to do anything. So if you have a simple atomic insulator, just a bunch of independent atoms, then you can understand why the electrons are snapped in place, and they're stuck, and they can't do anything. Now of course, maybe as an electronic material an atomic insulator is kind of not so interesting. Not that much happens. Things get more interesting if you bring the atoms closer together so they start interacting with each other and start forming covalent bonds.
And the way we understand this, the language that has been developed to understand this, is called the band theory of solids. And so what happens is that the energy levels, they broaden out a little bit. But you still have occupied energy levels and empty energy levels, which are separated by a gap. And so there's a sense in which it remains electrically inert. In order for the electrons to do anything, they have to overcome this energy gap.
So this language band theory I think is probably one of the most consequential theories that we have in physics. So it's the theory that underlies our understanding of many kinds of electronic materials, in particular semiconductors, like silicon. And one of the triumphs of the physics of the 20th century was the development of a very detailed understanding of materials like silicon. And this forms the foundation for all the kind of technology that we've sort of grown accustomed to in today's information age.
And there are lots of details. I don't want to dwell on what all these wiggles on this graph are about. But there's lots of specific details that we are in complete mastery of for these materials. And so band theory, we have very detailed understanding of these materials. But there's something simple that I want to say about this, which is that there's a sense in which this more complicated electronic structure that silicon has is kind of the same as this very simple electronic structure that an atomic insulator has in the sense of topology. Because what you can imagine is you can imagine in your mind's eye you can imagine smoothly sort of separating the atoms, making them to get further and further apart.
And the electronic structure is going to evolve. And once they get very far apart, it's going to look like this. And along the way, it stays an insulator the whole time. And so there's a sense in the sense of topology that a covalent insulator can be smoothly deformed into a trivial atomic insulator.
Actually, it's also kind of the same as the vacuum. The vacuum is not as empty as you might first have thought. If you think in terms of relativistic quantum mechanics, Dirac's theory of quantum mechanics, then what we learned from Dirac is that electrons have an antiparticle called the positron.
And according to Dirac, a positron is just like a missing electron in the valence band. It's just like a hole in the valence band. Mathematically it's really the same thing. The only difference is that this energy gap for Dirac's vacuum is a million times bigger than the energy gap in silicon. But they're really the same. And so I would like to elevate this sense in which they're the same into a principle. And I want to call this the principle of adiabatic continuity.
So what I'll say is that insulators are topologically equivalent if they can be continuously deformed into each other while staying an insulator the whole time. And one thing I'll say is that the band theory of solids has been around for 90 years. But it was only relatively recently that people phrased this question in this way, because when you say it like this, then it poses the question, are all the insulators the same? Could it be that there are topological phases that are different from the trivial atomic insulator?
Most insulators, like silicon, are the same. So in pictures, if we think of trivial insulators as being like a sphere or the surface of a bowl, which is topologically the same, has no holes in it, is there some analog of the donut? And the remarkable answer to this question is yes. And these topological electronic phases are fascinating phases of matter that I'd like to introduce you to and get used to.
So in order to introduce this, I always like to come up with the simplest version of a problem. And the simplest version of a topological electronic phase that I can think of is what occurs in a one dimensional system, a one dimensional polymer called polyacetylene. So polyacetylene is a very important material in its own right. It's a conducting polymer. It's actually conducting when you add electrons to it.
And so what it is it's just a chain of carbon atoms. And there are some hydrogen atoms attached. But that's not important. The important thing is that there's basically one electron for every carbon atom that's free to move around. And so if it's a pristine chain where every bond is the same as every other one, then it basically forms a half-filled band so it's an electrical conductor. But the thing that's interesting about this is that this is not what polyacetylene wants to do.
What polyacetylene would rather do, it can lower its energy by dimerizing, by having pairs of atoms, alternate pairs of atoms get closer to each other so they formed stronger bonds because if they do that, then what happens is that an energy gap opens up here. And it turns into an electrical insulator, lowering its energy. So polyacetylene is an insulator, unless you add electrons to it.
But the thing is that it's an interesting insulator because there are two ways you can do it. You can either alternate strong, weak, strong, weak, strong, weak. Or you can alternate weak, strong, weak, strong, weak, strong. And so what I would like to argue to you is that these two configurations are topologically distinct in the sense that I introduced because if I call u my displacement-- so up here I've moved the green atoms to the left so u is less than 0. And down here I've moved the green atoms to the right so u is bigger than 0. Then you can't get from u bigger than 0 to u less than 0 smoothly without going through 0.
And u equals 0 is not an insulator. It's a conductor. The gap has gone to 0. So the A phase and the B phase are topologically distinct. And so what we'd like to do is we'd like to come up with a way of characterizing the way in which they're not distinct. Is there some analog of counting the number of-- the genus-- counting the number of holes? And so in fact, there is.
One way you can think about it is to think about the simplest model of polyacetylene in which there's just one state on every atom. And so you can write down a simple-- and there are two atoms in every sort of periodic unit cell. And so if you solve that problem, which is not a hard problem to solve, then the problem that you have is basically there are two bands. And your Hamiltonian as a function of momentum will be just a two by two matrix. And two by two matrices, that's manageable.
And the thing that about two by two matrices is you can write every two by two matrix as a constant piece, which doesn't really-- that's just the constant in the energy-- plus a vector dotted into the Pauli matrices. So any two by two Hermitian matrix can be written like that. And in fact, in this simplest model of acetylene, the Hamiltonian only involves coupling the green atoms to the blue atoms. And what that means is that the only Pauli matrices you use are the ones which are off diagonal. So it's only sigma x and sigma y.
And so the Hamiltonian I can write as some vector in the xy-plane dotted into the sigmas. And so that's one way of characterizing this. So now you can understand this topological distinction because as a function of k-- and k is the one-dimensional momentum. But it lives on a circle.
The circle is called the Brillouin zone. You know the place you learn about Brillouin zones is from Ashcroft and Mermin, which that's where I learned about it at least. In any case, so one can distinguish two cases where as a function of momentum going around the Brillouin zone, either you go around the origin or not.
Or there's a number, which is the number of times it wraps around. That's a winding number. And that is a topological invariant that distinguishes these two phases. Now, have to be a little bit careful here because you have to define the rules of the game, because I told you that any matrix, any two by two matrix can be written as a combination of the Pauli matrices. But then I said that well, we're not using sigma z.
You see, if we use sigma z-- so if there was a z direction coming out towards you here-- then I could get from this to this by just going over the top. And in fact, in this simplest model of polyacetylene sigma z is not there. So you have to ask the question, what is it about sigma z that makes it so that it's not there? And so this is an interesting and important part of the story because this is where symmetry enters. So this polyacetylene has two different kinds of symmetry, which are both kind of interesting.
And so I want to tell you about those two symmetries. The first one is a little bit of a counterintuitive symmetry. So this simplest model of polyacetylene has asymmetry in its spectrum that if I reflect energy to minus energy, it looks the same. So we call this particle hole symmetry. And so this isn't exact symmetry that this simplest tight binding model of acetylene obeys. And that symmetry guarantees that this z component-- the Pauli matrix would violate-- the z Pauli matrix would violate that symmetry.
Now, one thing I'll say, this model system does have this symmetry. Real polyacetylene does not have this symmetry. So it's sort of a model symmetry. It's not a real symmetry. But I want to argue to you that it's interesting to think about this symmetry nonetheless because it has some very specific consequences. And moreover, later on in this talk, I'm going to be discussing a related but slightly different situation that has a model that is just like this, where this particle hole symmetry is something that is intrinsic and exact. So this is worth thinking about.
But it's not a symmetry of real polyacetylene. But real polyacetylene does have symmetries. In particular, it has rotation symmetries. If I rotate by 180 degrees about the center of a bond, then it looks the same. And that symmetry by itself is enough to protect the topological distinction between the A phase and the B phase.
If I don't have that symmetry, if I don't impose that symmetry, then you can smoothly go from one to the other. So if you don't have any symmetry, all of these one-dimensional insulators are the same. But with symmetry they can be different.
And so this is sort of the entry of symmetry into this discussion of topological electronic phases. You get something new. Yes, question?
AUDIENCE: What breaks the energy, the minus energy, symmetry in the real material?
CHARLES KANE: Second neighbor hopping, for example, would break. In general, that will be non-zero. And the thing is that you know that it can't be-- actually, the question was for the Zoom audience, what is it in real polyacetylene that breaks the particle hole symmetry? And so one other thing I would say is that you know that real polyacetylene can't have a particle hole symmetry because there's no zero in energy. There's no magic zero that one can define. So that's the case.
But I want to argue that it's still nonetheless useful to think about it. And for what I have to say next it's going to figure in prominently. So we have this topological distinctiveness. So you can ask, OK, so what? What's the consequence?
And so I want to tell you the consequence, which is one of the, I think, most beautiful things I can think of in physics, which is that if you have a boundary in space between two topologically distinct phases, then you have topologically protected boundary modes. And let me tell you what I mean by that.
So the simplest thing to think about is-- and so this is the Su-Schrieffer-Heeger model of polyacetylene. And so the simplest thing to think about is the most extreme limit where the strong bonds are really, really strong and the weak bonds are zero. And in that case, then you just have a bunch of dimers, independent dimers.
And each dimer is coupled to each other. You start with two levels. You couple them. They split into a bonding and antibonding state separated by a huge energy gap. So the A phase and the B phase are both insulators with a big energy gap.
But notice something interesting happens on the boundary because there's an extra atom here that isn't coupled to anybody. So it's going to stay at zero. So that's a zero mode. So we have a conduction band here and a valence band here. There's going to be a state stuck at zero.
Now, you can ask the question, this is in this extreme limit where I turned off the weak bonds. So that's not so realistic. What if I turn back on the weak bonds? But this is where the particle hole symmetry gives us a very powerful way of thinking about it because if we have the particle hole symmetry, then you could ask, what could happen to this zero mode when I turn it back on, turn the weak bonds back on?
So if I turn them up, it can't-- so the thing about the particle hole symmetry is that every state has a partner at negative it's energy. Now this zero mode is special because it is its own partner. It doesn't violate-- it still respects the symmetry. But it's its own partner, which means that if you turn the weak bonds up, it can't move away from zero because then its partner would have to appear out of nowhere suddenly. And that can't happen smoothly.
So this zero mode is topologically protected. You can't get rid of it. And you can see it has to stay at zero if you have this particle hole symmetry. So this is a fact which has been discovered and rediscovered many times over the years.
And actually, I think the first discovery of this fact was actually field theorists, Roman Jackiw in particular. So they were interested in studying the sort of simple model field theories, and in particular a one-dimensional field theory where you have a one dimensional Dirac Hamiltonian with a mass term. And the mass term changes sign at a domain wall.
And so what Jackiw and [? Rebbe ?] were able to show is that no matter what the shape of this domain wall is, there is always an exact zero energy eigenstate to this Hamiltonian. And for the students in the audience, this is a very rewarding calculation to do, which is to take this Hamiltonian and to see that no matter what the shape of m of x is, as long as it has a domain wall, that this is an exact zero energy eigenstate. So if you haven't done that before, then I'll assign that as a homework problem.
So in any case, what this means is that this zero mode persists even if I turn up the weak bond to be almost the same as the strong bond. It's topologically protected. Now, there's another point that I want to make about this. And this is a point which is going to come up again and again in this talk, is that there are two levels that you can think about this on.
So you can think about it on the level of single particle quantum mechanics, where you're solving the Schrodinger equation for a single electron. And if you do that, you solve the Schrodinger equation, what you find is you have eigenstates with energy, some in the conduction band, some in the valence band. And then there's this zero energy eigenstate.
So in the single particle spectrum, we have this zero energy state. But you see, if we're talking about polyacetylene, the single particle states are not the quantum states of polyacetylene. What we do when you have electronic energy levels is you have to put electrons in those states. So the many body state is the state that you fill up the negative energy states. And you fill them up to the Fermi energy.
So if you think about the many particle states, then something very unexpected happens because you can ask, if I have this zero mode here, then it can either be occupied or not occupied by an electron. I'm going to fill all the states in the valence band. And all the conduction band states are going to be empty. But there are two states, either occupied and empty. But due to the particle hole symmetry, those two states have to have equal and opposite charge because the particle hole symmetry is like replacing the electrons by holes or interchanging electrons and positrons, matter to anti-matter.
So everything changes sign. So these states have opposite charge. But they differ by e, the charge of one electron. So this tells you that the charge on this domain wall is either going to be plus one half of e or minus one half of e. So this domain wall has effectively split the electron in half.
So one way you can think about it is let me imagine that I start with the A phase. Everything's in the A phase. And then I add a single electron to it. That has a charge e. I'm not actually cutting electrons in half. Everything is made out of electrons. But I add this single electron.
But then if I nucleate a B phase in the middle, then I can split that electron in half so I have e over 2 on the ends of it. And these domain walls, they act like particles. They can move around. And so this is an example of charge flag fractionalization, or as I like to say splitting the indivisible. You can't really cut electrons in half. But you can have half of an electron on the boundary between two topological phases. And so that's the thing which is really new here.
OK, so there are many more examples of this. And so let me step it up one. Is there a question?
CHARLES KANE: Please.
AUDIENCE: So if you have like a general crystal [INAUDIBLE] you'll introduce some energy levels in the band gap. So can those always be on the [INAUDIBLE].
CHARLES KANE: So the question is if you just have a defect in the crystal that can bind bound states, whether all of those bound states are topological bound states? I would say no. So if you just have a defect in the crystal, I think you will have-- let's keep the particle hole symmetry-- then you're guaranteed that there's going to be a stated plus e and minus e. There's not going to be a single zero mode. There's going to be two bound states, if you keep the particle hole symmetry.
So these defect modes are really something that's different from the topological modes. And one way I would say it is that if I have the boundary between A and B, A and B are different a long way away. So I can't do a local change to put in this boundary mode. I have to change something everywhere-- I have to change it everywhere to the right. Now, of course, I can make two. And then I can spatially separate them and have them be far apart from each other. But I can't make one locally.
And so that's an important distinction. Thank you. So what I want to do is I want to step it up a dimension. And so I want to think about a one-dimensional electrical conductor. So I have a one-dimensional wire, if you will. And p squared over 2m energy. And I filled them up to the Fermi energy.
And so this is a simple picture for a one-dimensional conductor. And if I think about the states near the Fermi energy, then there are states which have a positive group velocity. They're sort of propagating from left to right. And then I have states with a negative group velocity, which are propagating from right to left.
We give them a fancy name. We call them right and left moving chiral fermions. Now, the thing about it is that one-dimensional conductors are kind of fragile because if you put scattering in, if you put defects that could scatter the electrons, then electron could be moving around, coming along. It scatters. And it scatters back. And then it scatters again. And it's just sort of scattering back and forth. And it doesn't get anywhere.
So in general, if you have a random potential disorder, then the electronic states will become localized. It will be an insulator, not a conductor. Similarly, a simpler version of that is to put on a periodic potential, which is commensurate. Then if I put on a weak periodic potential, then it'll open up a gap and turn it into an insulator. That's sort of what happened in polyacetylene, for example. So the question is, is there any way we can make this one-dimensional conductor less fragile?
And so let me show you how to do this in pictures. So what I want to do is I want to start off now with a bunch of these one-dimensional wires. And what I'm going to try to show you is that I can split these chiral modes apart from each other in the same way that I split an electron in polyacetylene because what I'm going to do is I'm going to take-- here what's happening is I'm pairing up the right-moving chiral mode on one wire with its partner, the left-moving chiral mode on the same wire. But what if I instead pair up the right-moving one on one wire with the left-moving one on the next wire?
So if I do that, then it's going to be a similar story. It looks just like this in the middle here. It's going to be an electrical insulator inside here. There's going to be a gap that opens up. It's an electron insulator down here. This is the same as what I had before. But what you can see from this is that something magic has happened because on the boundary between these two topological phases there is an unpaired chiral mode.
And so I would like to argue that these different pairings form a topologically distinct phase. And we give this one a name. We call it a Chern insulator. And this is related to the quantized Hall effect. But I hope you can see that thinking about this way, you can sort of understand it on the same level as the simple polyacetylene example that I showed you before. It's really the same kind of phenomenon, where what I'm doing is I'm taking something which is-- actually, one other point I want to make is that it's impossible to have just one of these left-moving chiral modes by itself. You can't do that. It's impossible to have it by itself.
So there always got to be two of them. But what I can do is I can separate them. And that's what a topological phase allows us to do. Now, of course, this one-way chiral mode is a remarkable object because if you're an electron in this one-way chiral mode, your only choice is to go forward.
That's the only thing you can do. And so that means that if you add an electron here, you know with 100% certainty it's going to come out the other end. So the transmission in this state is perfect.
Now, again, I want to emphasize that there are two levels that one can think about this phenomenon on. One can think about it on the single particle quantum mechanics level, where again, we're solving the one particle Schrodinger equation for the eigenstates. And there what we find is that there are eigenstates in the conduction band. There are eigenstates in the valence band. And then there's a band of eigenstates that connects the conduction band and the valence band. And this band of states has a positive slope, which means it has a group velocity that's propagating from left to right.
But of course, in a real system, you populate these states with electrons. So you have a ground state, which is these single particle states filled up to the Fermi energy. And then there are going to be low energy excitations about this, where I could promote electrons from just below to just above. And so the many body spectrum is something else that one can contemplate. And so one can develop a low energy description of this, which in this case, is a particularly simple free fermion conformal field theory for a free chiral fermion.
And this free chiral fermion has remarkable quantized properties, which are related to this perfect transmission of these edge states. One is it has a quantized electrical conductance. So if I increase the voltage on this edge so that I fill up the electrons a little bit higher, then those extra electrons are going to be carrying a little bit more current. And you can calculate exactly how much that will be. And it basically is going to be e squared over h times the voltage. And this is the quantized electrical conductance.
Also, if instead of raising the voltage you raise the temperature of the edge, then there are going to be more particle hole pairs thermally created at the edge. And so there will be more heat flowing along the edge because all of those states are moving to the right. And so that leads to a quantized thermal Hall conductance. And one other word that I just want to get across here is I told you the sense in which having this by itself is impossible.
And so having just the right-moving chiral fermion, it can't exist by itself. And so there's a sense in which that theory is ill-defined. It has what field theorists call an anomaly. It has an anomaly.
And there are actually two kinds of anomalies. There's an anomaly associated with the conservation of charge, which is related to the quantized Hall conductance. And there's an anomaly related to the conservation of energy, which is related to a gravitational anomaly.
And so there's a sort of deep connection on this. Now, this story that I'm telling you, I'm telling you a little bit backwards. This physics was not discovered by sort of people sitting in a room by themselves thinking about it. It was actually discovered in the laboratory.
And so the best way to create this in the real world is to put a two-dimensional electron gas in a strong magnetic field. And this leads to the phenomenon of the quantized Hall effect. And this quantized Hall effect precisely gives you this Chern insulator phase, which on the boundary has these one-way chiral modes.
And so it was really the discovery of this by von Klitzing back in 1980 that sort of set all of these ideas in motion. So this was the inspiration for developing these ideas of topology applied to electronic systems. And this quantization of the Hall conductance, where really what you're doing here is you're measuring a topological invariant, this is something that you can measure to one part in a billion. It's incredibly accurately quantized. In fact, in the last couple of years, this is like the best way that you can measure the quantity e squared over h.
And so now the ruling authorities on standards and units have now declared that it's this experiment that defines the kilogram. So this is a very accurate experiment. Is there a question? Yes?
AUDIENCE: Pardon my stupidity here, but does this mean that the interior is not conducting, but the boundary is conducting?
CHARLES KANE: The interior is non-conducting. And the boundary is conducting. Yes, that is exactly what I'm saying, yes. Yes. Yes. And this is the essence of what these-- so I have an insulator on the inside. It's a conductor. But it's not just any conductor. This is a magical conductor that propagates only in one direction.
AUDIENCE: One more quick not.
CHARLES KANE: Yes.
AUDIENCE: If you change that diagram from a square to a simplex and you put simplexes next to each other, can you make a simplex chain moving the conductor across simplexes?
CHARLES KANE: Yes. I think the answer is yes. Now, do you want it to be closed?
CHARLES KANE: OK, you could do that too. You'd have to have magnetic monopoles in the middle though, but yes.
AUDIENCE: If you flip over one of them, then--
CHARLES KANE: I don't think you can make a Mobius strip.
AUDIENCE: That's what I was curious--
CHARLES KANE: No, no. Because then you'd be-- then the chiral modes instead of coming like this, they'd come like this. And you wouldn't be able to erase that scene. Yeah.
AUDIENCE: So orientability is important.
CHARLES KANE: Yeah, orientability is important, yes. Yes. Good. So in any case, this is the quantized Hall effect. And I hope that you can see that when you have these chiral modes you get something that's really, really special, this perfect transmission along the edge.
Now, the drawback, though, is that you need a big magnet in order to create the quantized Hall effect. And so what I want to do is I want to ask if there's another way to do this? And one of the things that you need is it's essential that the system knows which way the electrons can go.
And so it's essential that you break the symmetry under the reversal of time. And so time reversal symmetry is going to play a key role in what comes next. And so let me just say a couple of words about what time reversal symmetry is.
So what time reversal symmetry is about is it is about what happens if you have a motion picture and you run it backwards? Does the backwards motion picture obey the laws of physics or not? So clearly, a magnetic field switches sign when you run the motion picture backwards because a magnetic field is created by electric currents. And when you run the motion picture, they're going the opposite direction.
Likewise, the right mover would turn into a left mover changing the motion picture. It's interesting to think about what happens to the spin of a particle. So electrons have this magical quantity. They're like a little bar magnet, intrinsic angular momentum.
And so if you run the motion picture of that, it's going to sort of spin in the opposite direction. It flips up, spins it down spin. Also another thing that happens is that there's a star here. So it gives you a complex conjugate in the wave function. And there is also this minus sign.
Actually, this minus sign is my favorite minus sign. It's actually the same minus sign that you get if you spin a half electron and you rotate it by 360 degrees, it picks up a minus sign. It's the same minus sign as that. There's also if you have two electrons and you interchange them, they're fermions. They pick up a minus sign. It's the same minus sign as that too. So this is my favorite minus sign.
So in this context, it gives us something very special because if you do time reversal twice, it squares to minus 1. And this fact is what is at the bottom of what's called Kramer's theorem, which says that every eigenstate if you have time reversal symmetry has to come in come in pairs. Now, if spin and orbital degrees of freedom are separate, this is just saying that any state you can put an up spin or a down spin electron in. So it's kind of trivial if you about it that way.
But you see, if you have an interaction between the spin and orbital degrees of freedom, then this has much less trivial consequences. So the question I want to ask-- so this integer quantized Hall state requires broken time reversal symmetry. Is there anything that you can get that doesn't break time reversal? And this is where the 21st century physics starts to enter. So the interplay between topology and time reversal symmetry is what's at the root of topological insulators.
So what I want to do is I want to play the same game of trying to split the indivisible, but now imposing time reversal symmetry. And so now I'm going to take my one dimensional wire. But now, the electrons have spin. So going to the right, I can either have up spins or down spins.
And going to the left, I can either have up spins or down spins. And so what I want to do now is I want to split that in half in a way that looks the same when I run the motion picture backwards. And you can see you can do that because on the top you can have the up spins going to the right and the down spins going to the left, and then the opposite on the bottom.
So if I run the motion picture backwards, it looks the same. Now, in fact, when you have time reversal symmetry, this defines a new kind of topological state because having this by itself is impossible. But you can have it if there is a two-dimensional quantum spin Hall insulator in the middle.
And so what happens is since I have right movers and left movers, I have the up spins are going to the right and the down spins are going to the left, they cross here. And so you can ask the question-- usually in quantum mechanics when you have states that cross, there has to be a good reason because if you add some slight perturbation to it, then the crossing would turn into an anti-crossing.
But we do have a good reason because these two states that are crossing form a Cramer's pair. So they are protected. So they're crossing at zero momentum. And so those two states are the Cramer's pairs. Now, if I go away from zero momentum, then this guy's pair is over here at minus that momentum.
But the crossing is protected, which means that if you have time reversal symmetry, you can't get rid of it. So time reversal symmetry protects these edge states. And these edge states have an anomaly too. It's sort of a z2 anomaly. So it's either 1 or 0. And there's a topological invariant, which is a z2 topological invariant that distinguishes an insulator that has these edge states from an insulator that doesn't.
And the remarkable thing is that these topological insulators have come to life in the real world. The original idea where we sort of conceived of this idea was actually in graphene, which is a two-dimensional material. But actually, the effect is puny there. There's the little gap that opens up. It's a very, very small gap, which is overwhelmed by much else. But other geniuses in physics, like Shoucheng Zhang and Lawrence [? Wallenkop, ?] who's an experimentalist, came up with this idea of studying materials.
So what you need is you need a very strong spin orbit interaction. And that requires that you have atoms that are very close to the bottom of the periodic table, atoms like mercury. And so particularly, mercury cadmium telluride is a well-known semiconductor material. And if you make a sandwich structure out of it with mercury telluride in the middle, then this forms a two-dimensional insulator which has these edge states. And these edge states were observed experimentally by Lawrence [? Wallenkop. ?]
And so this was the genesis of this new field of topological electronic-- it was the realization that these things could actually happen in the real world. And since then the floodgates have opened. And there are many, many, many, many more. So for example, in three dimensions, one can have a three-dimensional topological insulator. And so let me just, again, say it's sort of like splitting the indivisible. I can imagine I have a two-dimensional electrical conductor. Can I split that in half?
And so what you can do is you can split it in half and have a two-dimensional metal with a single what we call Dirac cone in its electronic surface. And this is something which is impossible by itself with time reversal symmetry. But it can happen on the boundary. And this two-dimensional conductor has very specific special properties. And you can't get rid of it. So the most famous material where this arises-- there are many materials where this happens now. But the first one that was really cleanly discovered is a material called bismuth selenide.
And bismuth selenide is a rock basically. You can hold it in your hand. It's a piece of material. But it's an insulator on the inside. And it's a very special electrical conductor on its surface. And one can tell that by-- so there's a famous experiment where Zahid Hasan from Princeton did angle resolve photoemission spectroscopy on the surface. And that's sort of a fancy way of taking a picture of the electronic structure of the surface. And what he observed was what is basically a textbook perfect Dirac cone on the surface of this material.
So let me see. I should probably be finishing up soon? Five minutes? So what I'm going to do-- I have more than enough. I think I'm going to skip the quantum computer bit. And I'm going to just go to the frontier. I mean, this field has grown and bifurcated in so many directions that it's rather amazing. And again, I want to sort of describe two of those directions which are related to these two points of view that I introduced, which was the single particle quantum mechanics point of view, which is basically studying the wave equation, if you will, and the many body situation.
And so in particular, in the context of band theory, which is the single particle quantum mechanics, there are many, many more examples of topological band structures when you have symmetries. And let me just give you an example of that, which is a famous material, graphene. And so graphene, again, is a two-dimensional material.
And the thing that's interesting about graphene is that in its electronic structure the conduction band and the valence band touch at these so-called Dirac points. And so the first question that you can ask-- and actually, this is the question that I asked that got me started on this whole thing many, many years ago-- is you could ask, why? Why do the conduction band and valence band of graphene touch each other?
Because usually when you have a degeneracy, any little perturbation will split it. And so the answer to that is symmetry. So it turns out graphene has symmetry. It has inversion symmetry. It has time reversal symmetry. And since the spin orbit interaction in graphene is very weak, it also has a rotation. You can independently rotate the spins. And those symmetries are enough to understand why this Dirac point in graphene is protected.
And so you could call graphene, if you want, you could call it a symmetry-protected topological semimetal, something where the gap is guaranteed to be zero. But the thing about graphene is it has four Dirac points. So there's k and k prime. There are two valleys. And there's two spins.
And so you can ask the question, could you have less? Now, I already told you that if you have time reversal symmetry, having a single Dirac point is impossible. You can only do it if you have a topological-- it can happen on the boundary of a topological insulator. But otherwise, it can't happen.
But then you could ask the question, could you have two? And in particular, could you have two? And could you have those two protected by some symmetry?
And it turns out the answer to that is no. It's impossible to have two symmetry-protected Dirac points. But that gives us something new because that means that you should be able to have two symmetry-protected Dirac points on the boundary of some new topological phase. And so that's one of the things that we found.
We call this a topological Dirac insulator. And so it's a symmetry-protected two-dimensional system that has on its boundary a single pair of Dirac points. So this is something which is guaranteed by the combination of time reversal symmetry and kind of interesting spatial symmetries that this pattern of wallpaper would have.
And so there's an interesting interplay between symmetry. And there are many more contexts in which this kind of phenomenon can-- it's not just electronic band structures. One can have photonic band structures. One can have acoustic band structures. One can have mechanical systems with vibrational modes.
And so this topological band theory is being applied to all kinds of systems like this. So this is one. And one more thing that I will say is that there's been a huge interplay. With electronic materials one can do calculation. One can calculate the band structures of real materials.
And so there's been this huge interplay between computational physics, the sort of conceptual-minded theoretical physics, like what I do, and experiments. And there's been this very sort of resonant interplay between them. A lot has been going on.
Now, there's another direction one can go, which in a way is a deeper frontier. But it's one which it's harder to make contact with the real world. And so this is combining topology with strong interactions, which one inevitably can have in real many body systems.
And so let me give you an example of that. I told you about splitting the indivisible. I told you that if you have the right and left moving chiral modes of a one-dimensional wire, they can't occur by themselves. But you can split them. So you can split them. And if you split them, then what you get is a chiral mode which has this chiral central charge, which is related to the thermal conductance equal to 1.
So a question asked, can you split it anymore? And one of the things that I didn't have time to tell you, but is true is that if you combine this one-dimensional chiral mode with superconductivity, if you proximitize it with a superconductor, then what can happen is this one-dimensional chiral mode splits into two pieces. And basically, you can rewrite your Dirac fermion as a real part and an imaginary part, where the real and imaginary parts are called Majorana fermions.
And so this is a way of splitting one into one half plus one half. So now, people are trying hard to do this. And there are people who've said they've done it. But there are people arguing about whether it's true. And so it's a little bit of a controversial situation.
But it's not completely crazy. It's something that should work with reasonable systems that are available for experimental study. But let me pose the question again. So let's suppose we've broken it into one half and one half. Can we split it anymore? So what if we have one of these chiral Majorana fermions with c equals one half? Can we split that?
And this equals [INAUDIBLE] is what happens on the boundary of a two-dimensional topological superconductor it turns out. So can we split that anymore? Now, I don't know how to split a single chiral Majorana fermion.
But if you give me seven of them-- so now this is a little bit crazy. But humor me. Humor me a little bit. If you give me seven of them, then I do know how to split this in a non-trivial way. So it turns out that 7/2 splits into 7/10 and 14/5, where these 7/10 and 14/5 are described by rather interesting conformal field theories.
And so a Majorana fermion along this will split into two Fibonacci anions. And Fibonacci anions are something that we would very much like to have, if we could. Now, I don't know if splitting seven Majorana fermions is the best way to do it or not.
Maybe there are better ways. Or maybe it's really just going to be too hard. I don't know. But it would actually be very interesting if we could make these come to life. There are reasons why having Fibonacci anions would be a very, very powerful tool for quantum information.
But this is a situation where a lot more work is going to have to be done to make these come to life in the real world. Maybe some of the younger people in the audience will be the people who actually can make that happen. All right, so let me just conclude. I'm already out of time. So let me just say that these big ideas, symmetry and topology, they form this framework for discovery of these fascinating electronic phases of matter. I told you about some of them. For me, the real test of whether something is interesting and important is whether it comes to life in the real world.
And there are experimentalists here at Cornell who are making things like that happen, which I really admire. And let's hope that there continues to be this glorious interplay between experiments and theory. And with that, I'll thank you very much.
SPEAKER 1: Thank you, Charlie, for the very engaging talk. So I would like to make a couple of announcements before we open for questions. So first thing is that the post-doc undergraduate student and graduate student will have the opportunity to interact with Professor Charlie at PSB 403 after the session. And then this is just the first lecture of the three part series. So we have one more tomorrow, one on Wednesday. And please find the announcement in the ads for that. And so right now, I think it's open for questions.
AUDIENCE: So clearly, you made the right choice to skip the slides about the connection to quantum computing. It would've been another 15 minutes.
CHARLES KANE: Right.
AUDIENCE: I'm wondering if you were going to say something about the near-term prospects for that?
CHARLES KANE: Near term? I mean, I'm not sure near-term is the right word. well, so look, there are different approaches to trying to harness quantum information. And I was going to tell you about one of them, which may not be the best one, at least initially, certainly not in the near term.
So one possibility is to harness topological electronic materials to try to make what is called a topological quantum computer. And there I don't think we're anywhere near the near term for that. And it's questionable whether it's going to work, whether it'll ever work. It's also questionable whether it's the best way to do it.
People controlling qubits in atoms and defects, there's a lot of progress that's been made on that front as well. The question is, can you scale? Can you scale it? So yeah, I don't have any deep insight as to what the best way forward on that is going to be.
Though I will say that, over the years, I've always been completely amazed at how much pessimism is unfounded in the sense that things that I initially thought could never happen can happen. I mean, it's a question of if one's imagination is big enough to really imagine the way things are going to develop. And that I don't know.
But certainly, my view of this is having the quantum information angle is a great motivator. No. And there's a lot of--
AUDIENCE: [INAUDIBLE] great funding source.
CHARLES KANE: Well, let's not even talk about funding. I'm talking about just for scientific progress. There are great science problems to solve in the service of trying to harness quantum information, science problems which are interesting and important even maybe if they don't end up leading to a quantum computer. I mean, who knows? Maybe we'll be making progress.
And then the really useful thing will be something we haven't thought of. But I feel like there's really good science to be done understanding how to harness quantum mechanics, how quantum mechanics works, and how does matter that obeys quantum mechanics behave? And that I think is a really-- there's a lot of good science to be done in that direction.
AUDIENCE: I have a question about-- so you talked a lot about photon symmetry protected states. But we also have some [INAUDIBLE] charge [INAUDIBLE] symmetries. So is there any work on that area besides just time [INAUDIBLE] actual [INAUDIBLE]?
CHARLES KANE: Yes, absolutely. Yeah, yeah. So there are all kinds of symmetries that one can talk about. So the question was, in addition to time reversal symmetry, there's parity, there's charge conjugation, there's all kinds of symmetries. So there's a long answer to this question.
But I will say that combining time reversal symmetry and charge conjugation symmetry leads to a very beautiful mathematical structure. And so this is something which actually sort of generalizes this notion of topological insulators. Now, including parity as well, in addition to parity, which is spatial inversion, one can also think about rotation symmetries in a crystal.
And so combining crystalline symmetries with time reversal symmetry and all those symmetries, that leads to a whole host of what we call topological crystalline insulators. And so there's a lot of understanding that we've developed over the last several years of applying those kinds of symmetries. So the answer is yes.
AUDIENCE: [INAUDIBLE] mentioned [INAUDIBLE] seven [INAUDIBLE] fermion [INAUDIBLE]. I was thinking theoretically what do you need? Do you need a material with seven Dirac points? Or what exactly--
CHARLES KANE: Well, first, you'd need one. So if you could have a two-dimensional material that was a two-dimensional topological superconductor, then you'd have one. So then maybe the simplest thing to do would be have seven layers of that. That would do it. But then you'd have-- then once you have seven, then you have to make them talk to each other in the right way in order to make them do what I want. I don't want to make that sound like it's easy.
AUDIENCE: I assume there are actually ones with-- I mean, the 4/5 and 7/10 there are recognized as the discrete members of the--
CHARLES KANE: Yes, they're exactly what you know about, yes.
AUDIENCE: So I presume there are other numerological coincidences with more than seven as well.
CHARLES KANE: Yeah, I mean, so basically-- and I think this is probably something that I learned from you. So it's related to the coset construction in conformal field theory. That's how this factorization works. So basically, theories that can be factored have this phenomenon that allow for splitting the indivisible. So yeah.
SPEAKER 1: OK, I think in the interest of time, we will end this session. And let's thank Professor Charlie Kane.
CHARLES KANE: Thank you.
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Abstract: Symmetry and topology are two of the conceptual pillars that underlie our understanding of matter. While both ideas are old, over the past several years a new appreciation of their interplay has led to dramatic progress in our understanding of topological electronic phases. A paradigm that has emerged is that insulating electronic states with an energy gap fall into distinct topological classes. Interfaces between different topological phases exhibit gapless conducting states that are protected and are impossible to get rid of. In this talk we will discuss the application of this idea to the quantum Hall effect, topological insulators, topological semimetals and topological superconductors. The latter case has led to the quest for observing Majorana fermions in condensed matter, which opens the door to proposals for topological quantum computation. We will close by surveying the frontier of topological phases in the presence of strong interactions.