Entanglement and Topological Quantum Matter
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Haldane, F. Duncan M.
While the laws of quantum mechanics have been known for almost ninety years, we are still learning what they allow. One surprise in recent years has been the discovery of “topologically-protected” quantum states of matter, in which “entanglement” plays a key role, and ensures that unexpectedly-robust surface states occur at the boundary between “topological” and “ordinary” regions (the so-called “bulk-boundary correspondence”). Information hidden non-locally in the entanglement may provide a route to “topologically-protected” quantum information processing. A recent surprise has been that some non-quantum aspects of these properties can also survive in classical systems involving waves and oscillations (photonics and mechanical systems).
 Music Basically I'm going to tell you about entanglement and how it turns out to become important in condensed matter and how this has finally led to all kinds of beautiful dreams about a new topologically protected quantum computing. So as we know the laws of modern quantum mechanics really haven't changed since 1925 to 32 period. So what has changed is once the laws are written down it takes time to work out what exactly you can do with them. Same thing happened with Maxwell's equations. And all tests have shown quantum mechanics is correct. But what we now have learned is that it permits very cool things to happen that we wouldn't have initially guessed. So starting around 1980 there were new insights into quantum mechanics started to emerge and new unexpected topological states of matter started to be discovered. And since no one had dreamt of such states before they were quite a shock to us. But this has gone on and being combined with a lot of interesting ideas from quantum information theory that Feynman initiated about in 1959 I guess 60 years ago. And people now as we heard on Saturday on Sunday I guess Saturday we're talking about the second quantum revolution. A whole new set of ideas for looking at and interpreting quantum mechanics and coming up with things that you can do with it. So classical information is stored in a simple bit, a switch which is either on or off. But quantum information can be stored in this new idea of the qubit which is a generalization of the idea of a single two state system or an electron spin. And while a spin can point in two directions relative to some axis that axis can be allowed to move around and point in any direction. So there's a lot more subtle ways to incorporate information into the Bloch sphere by an arrow that points on the Bloch sphere. So the central theme in these new approaches to quantum mechanics is really the property of entanglement. And when I started off we really didn't think much about entanglement. Entanglement was a problem for philosophers of science and discussing whether quantum mechanics was compatible with free will and things like that. But now we really recognize it as perhaps one of the central properties of quantum mechanics. And of course it was initially pointed out as important by Einstein. But now we're thinking of it as the fuel which will drive future quantum information processes. So of course Einstein got his Nobel Prize not for relativity but for the photoelectric effect for his early contributions to quantum mechanics. But by 1935 he was really rejecting the quantum theory by thinking it couldn't possibly be a true fundamental theory. And he identified the key property of entanglement which he thought was so strange that it couldn't possibly be right. So he was hoping to consign quantum mechanics to the dustbin of history by pointing out the crazy things it predicted. And he called it of course, spook after fernwirkung, spooky action of the distance. But Schrodinger came up with a much more neutral term that we use which translates into entanglement. So of course one of the fundamental properties of quantum mechanics is the Pauly exclusion principle that says that two electrons which are fermions can't be in the same state. But in addition to position and momentum, the state of electron also depends on this internal degree of freedom, its spin, which is analogous to the spin of a top. And of course the chemical bond which holds this all together is the way that the Pauly principle is overcome because the two electrons can fit in the space between two atoms to form a chemical bond to hold this all together because it has these two different states of the spin. So the poster child of entanglement is the chemical bond which is the maximally entangled state of two electrons where we have one electron is up and the spin and the other electron is down. But the quantum mechanical state is a superposition of the two possibilities up, down and down, up. And this is the maximally entangled state. But of course what Einstein was worried about was if the chemical bond was stretched to large distances maintaining this curious correlation between the electrons that it was a superposition of up, down and down, up, he felt you know if you do a measurement over here on this electron how could it possibly be compatible with relativity that immediately implied that the sort of change in the other electron. So he felt this quantum state of entanglement at large distances couldn't possibly be a true state and it superficially looks like it might violate the principle of relativity that information can't travel faster than the speed of light. But when you analyze this more closely you cannot send information between the two experimentalists by instantaneously by doing a measurement. So it's not really a violation of relativity. So this this puzzle that Einstein, Podolski and Rosen produced got analyzed in great detail by people like David Bohm and then finally John Bell. But finally when the experiments were done, quantum mechanics passes with flying colors. In fact the only let-out clause are still discussions about the free will of experimentalists in choosing which axis to measure. But so anyway the problem is entanglement over large distances is very fragile compared to the chemical bond. And so electrons also interact strongly with the environment so these experiments to prove that Einstein, Podolski, Rosen effect or entanglement could exist at large separations have been done with light. But it's totally convincing now and indeed we just heard on Saturday from Jan Wei Pan about the work in China where we've created entanglement between between the ground and orbits and satellites in orbit. So around about 1980 a condensed matter physicists were kind of arrogant enough to think they had a basic understanding of electronic and magnetic materials. But around that time there were kind of two examples which initially had no obvious connection whatsoever to them each other. Although it turns out they both involved the Berry phase came up and they were both somewhat very unexpected. So the first of the things was of course the discovery of the quantum Hall effect by Klaus von Klitzinger completely unexpected effect which shocked people at the time. And the other thing was some work I did on low-dimensional magnetic systems. And so these can be viewed as the tip of the iceberg in this iceberg metaphor. But in the succeeding decades many many more new kinds of topological materials have been found and the unity which relates these two effects has been found. And of course the most exciting aspect of all this has been probably that the ideas about topological quantum matter have led to a proposed platform for topologically protected quantum computing which at least theoretically provides a nice way to to have scalability with and avoiding decoherence which other non-protected versions of attempts to do quantum computing don't have. So what's topological matter? Topological matter differs from ordinary matter because it has properties which really are probably entanglement properties that can be described by whole numbers like integers like minus one or two etc. So the basic issue is that states of matter which are characterized by an integer can't smoothly change from one to another. Something always has to happen at the boundary and it's usually something pretty interesting that happens on the boundary between different types of matter. So here in the picture we have boring matter which I've assigned which I've assigned the index zero to and some non-trivial matter. Now of course zero is not exactly a boring number it took a long time to be discovered but it's what we might call topologically trivial. So the boundary has to be there and basically that's because this topological properties cannot change continuously and a key aspect of that is it makes the properties robust to small amounts of disorder or dirt in the material which is a contrast to the usual material properties of which our current silicon technologies for information processing are based on which require ultra clean preparation. A single speck of dust or hair or something on the on the silicon chip will ruin it which is why people have to be dressed up in these spacesuits not to protect them from the nasty chemicals in the foundry but to protect the valuable silicon wafers from the people. So the classic example in mathematics is the classification of closed surfaces by what's called the genus or the number of holes in them and this was a metaphor that was being used by the Nobel committee to try to explain to the public what topological matter was probably with great confusion arising but a shape like a sphere with zero holes you can distort a sphere you can distort a football into a rugby ball with small forces but if you want to change it into a donut an object with a hole you have to do a significant amount of violence to it and so holes come in hole numbers you can't you don't have to say as such a thing as 0.36 of a hole so they're easy to you either have it or you don't basically at least of course when you get down to the atomic scale the things are not smooth anymore so there's actually usually a length scale involved where things become topological on the sufficiently large length scale which turns out to be a crucial point. So topology comes from the the great discovery by Gauss who discovered basically that 4 pi r squared the area of a sphere times 1 over r squared is Gaussian curvature was 4 pi. This is a kind of high school formula but what Gauss discovered was if I changed my sphere to the rugby ball or even to a banana that that number stayed the same and in fact you could do a very complicated integral of the curvature which on a non-uniform surface can be positive in some places and negative in others and get this simple number so a very sadistic high school mathematics teacher could give very complicated problems to the students but he would know if they got the right answer or not quite quickly by just checking what multiple of 4 pi they managed to get if they managed to do it. So in this picture we have this famous picture which usually shows the bagel turning into the coffee cup which every experimentalist working on topological matter cannot prevent themselves from showing at the beginning of their talk. This simple idea has been so exciting to people working in the field they they just love the coffee cup metaphor. Well of course in Sweden they were trying to talk about something which wasn't really a pretzel it was actually a Santaluccia bun with two holes in it which would be topologically equivalent to what's called a loving cup but actually when you bake the Santaluccia bun the holes actually fill in so it's really a fake pretzel and the real pretzel of course is the German pretzel with three holes and which is equivalent to this interesting cup here. I'm not sure what it's called but I did hear it suggested to me this was a California loving cup. Anyway the mathematicians of course love to take a simple formula and make it much more general and much more abstract and indeed the Gauss-Borné formula was generalized mathematically from a physical curvature on a real surface embedded surface like the bagel into a much more abstract mathematical curvature on much more abstract surfaces which could occur in any even number of dimensions and so this actually gave rise to the Gauss-Borné-Chern formula. Now the mathematicians don't have a Nobel Prize say the simplest the nearest thing they have to it is the the Fields Medal and maybe there's some Fields Medalists here I don't recall but the Fields Medal is a very kind of a agist thing it's discriminatory because you can't get it however good your math was you can't get it if you're over 40 so they invented a couple of new medals one the Arbel Medal and the other the Cern Medal for kind of old-timers and back on the Cern Medal of course is the Gauss-Borné Cern formula inscribed. Okay so what is the entanglement in a in a quantum mechanical context basically if you take normal matter a trivial matter you can really you can do a measurement on one side of it without really affecting things that happen on the other side and so you can kind of pull it in half quantum mechanically and get a product state but because the topological matter always has these edge states it resists chopping in half and eventually when you do manage to split it the edge state has spread over the the new surfaces so you can't avoid having these edges and so basically topologically trivial states of matter can be in principle be assembled just by bringing the atoms together and stacking them next to each other well if something non-trivial happens the electrons kind of get unbound for the atoms and rearrange themselves somewhere along this path of constructing the matter and in fact once this principle is seen you can see it in old work in fact this is an example going back to 1939 where people shockly managed to do a calculation of toy model of of a one-dimensional chain of atoms and what you see is you bring these things together the energy bands of the which were initially the atomic levels kind of broaden but they they they come together and they split up again and you get these these new topological edge states at the system you have one at each side of the sample and they are very interesting effects such that if you try to fill this with electrons you're more likely to have the chemical potential below or above these two states so you either have one electron too few or one electron too many compared to the original atomic system and that extra electron is split in half one half of it goes at one side and one half goes at the other side that's a very typical thing associated with the topological edge state so this will actually be what we now call a symmetry protected topological insulator so in heavier atoms electrons in a group of atomic orbitals can be locked together so they're always parallel and that gives rise to magnetism the elementary spin is got angular momentum a half h bar and so the total spin will be a whole number which is either even or odd times a half to make an integer and conventional magnetism you just stack these magnetically ordered systems together but there's no real entanglement in them but if you what I discovered by basically looking at this problem from a different viewpoint coming usually one has some history where one's coming to a problem I was working on some other issues in one-dimensional organic conductors and applied some results from that to the magnetic problem so I was shocked to find that the conventional wisdom about what these magnetic systems should do was like 100% wrong it was actually wrong because people have been confirmed in their incorrect ideas by apparent evidence from an exactly solvable mathematical problem but the problem was they completely misinterpreted what that exact mathematics was telling them and it confirmed them in their prejudices so this is I'm sure there's plenty of other things that we believe in which are wrong and we think there's pretty good arguments for them and you may be the one who finds one of you may be the one who finds out that everyone else was wrong and changes our way of thinking okay so this was a very controversial thing and it shocked loads of people and again there was a lot of resistance to it but finally physics is not a question of opinions and experimentalists especially material scientists who managed to motivated to actually make the systems to fit the theorists ideas rather than the other way around they came up with ways to actually make these magnetic systems and finally the experiment showed that the the new kind of states were correct and it led to the actual controversy led to people coming up with lots of new ideas and new methods for studying problems which have turned out to be very important so the metaphor here which you'll see is that we have a line of if we have a line of people who become entangled by holding hands you apparently get left over with a spare pair of independent hands at either end of the system where if a person hadn't had a bad accident you'd come up with the idea that hands always come in close pairs okay and of course this led to this prize business so soon after this discovery the underlying mathematical principles came out this fundamental idea of the Berry phase in quantum mechanics which is really at the heart of all the kind of modern physics that we've been talking about and initially that came up in in quantum spins but then at the same time we had a class from cliff sing had discovered the the the quantum Hall effect which as we've heard has now become the system at the base of the new system of units for replacing the kilogram and this is electrons in a high magnetic field moving around in little circles and it wasn't initially clear what where topology came into this but as was pointed out by Bert Halperin you have things going around in little circles in the interior but if the if you hit the boundary the circular trajectory which leads to quantization is broken and in fact you get these edge states that move around and now we understand that edge states are very typical of topology David Thales was working on this problem with collaborators and they did the important step which I kind of do too which is to replace difficult problems with toy models that you can actually do a calculation on so they replaced a more complicated problem of disorder in a real system with a simple periodic potential which is the kind of thing we can solve and they came up with a beautiful formula which basically showed that the whole whole conductivity was still given by an integer times e squared over h and the mathematician mathematical physicist Barry Simon heard on the one hand about Barry's calculation Barry's phase on the other hand he heard about this numbers that Fowles has formula and he put two and two together and realized it was all the the gas for nature and I think and that was that so it used to be thought that initially that you needed high magnetic fields to get this quantum Hall effect maybe it should have been realized from Fowles's result that that wasn't necessarily true but somehow as an accidental discovery almost like the accidental discovery of the spin chain business realized that you could actually get rid of the magnetic field completely and just keep Barry curvature in the problem and get with standard crystalline materials and get interesting topology including edge states that go around one way on the boundary and so this is actually again not obvious most most interesting things are not obvious even to the people who discover them right but once but that's the nature of things so yeah when one looks at the details you see lots of interesting things going on that I'll have to skip but this actually took shows how long it takes to find something out it was actually predicted in 1988 and it was a theorist toy model for a long time but finally it led to other things and finally that those led to the experiments which were done only in 2013 to show these effects can happen so again without huge magnetic fields the quantum the so-called quantum anomalous Hall effect gives rise to these edge states which move in one direction and actually that seems to have a lot of very interesting technological possibilities in fact one can actually from this edge state aspect of things you can put aside the quantum mechanics because the well the some aspects of the quantum system of course require Planck's constant the the integers don't and the existence of edge states turns out to be to be there in photonics too so the simple toy model I showed you has led to lots of new things both in in photonics and also mechanical systems so there's actually surprising amount of new work going on you know once you realize the principle is there and is quite general it's very beautiful it was course generalized to spin systems without magnetism by Cain and Malie which initially provoked this large revolution because that led to lots of materials actually being initially discovered they were sitting on people's shells but no one had looked at the edges clearly enough to see there was something different about them okay and this led to you know all kinds of new things which I can't go into so where does the quantum computing come from the integer Hall effect was now followed by a much bigger puzzle the fractional Hall effect and here you see the the sample in which the fractional Hall effect was discovered which kind of by looking at you can see the how how the the robustness of topological physics against against disorder right I guess this wouldn't have got Art Gossard well whoever did this probably wouldn't have got a job in the clean room making chips with their soldering skills but anyways so lots of really interesting things happen in the in the fractional quantum Hall effect non-Abelian statistics fractional charges but the most remarkable aspect is actually what we now call non-Abelian quantum statistics which is what this gives us the ability to hide information in the entanglement structure of systems and that's really what was made people so excited about the possibility of a protected form of quantum computing so the proposal for that goes back to Alexei Kitev who looked at work of read and read more and read who came up with the first of these non-Abelian states and basically you do something called braiding where you you move vortices or little defects in this system around the quantum Hall system is like a quantum fluid with little vortices which are little whirlpools in it and the information I'll show you is actually stored in the history of how you prepared the the vortices so basically these little objects can only be created in pairs so you have to create two of them at one place and then pull them apart so here I've got a system which apparently has the same configuration in real space of the four vortices but hidden in it is the history of how these configurations were produced and these are the entanglement patterns and this would represent one state of the qubit the other one would be another state rotated by 90 degrees on this prankery sphere and in fact when you look very closely at a calculation of these systems you so this is again a calculated toy model system with four four of these vortices arranged on a sphere in fact but you can see one state might be the up state of the qubit and the other might be the down state of the qubit or a linear combination of them would be the sideways state and you can see because they're not infinitely far apart from each other they're little interference patterns that distinguish the states so the information is actually an interference pattern and you in principle it's not completely hidden but the wiggles you see become exponentially weaker and weaker as I move these things apart there's always a length scale involved topology happens when things are infinitely separated on on a on a microscopic length scale and here you can see the information becomes more and more delocalized and more and more hidden as you move things apart so it's the magic of how it is is is really encoding an interference pattern so the the top the quantum physics beautiful dream of topologic protected thing is qubits is just one way to maybe overcome this problem of decoherence of course there's lots of other attempts to do quantum computing and everyone is competing so I personally believe that topological quantum computing is a realistic dream but we'll find out the current scheme for topological computing is actually based on super on superconductivity and and so called nano wires again it's very similar to the what I showed before with the spin train again if I put electrons in contact with a superconductor it somehow allows the electron loses its charge into the superconductor and then can actually split up into two kind of half electrons called myerana modes and by joining these things up together in one way you actually can end up with what you think ought to be indivisible an electron state which should contain be able to contain one electron actually split up into two pieces on either end of the sample and that leaves in a string of entanglement between the things so this metaphor of the chain of people is quite similar so the again there's lots of other attempts to do things in quantum computing lots of people are competing so I don't know what will happen but finally I just come to the motivational message again like Mike Kostalitz is you don't actually have to be a you know a great genius like Einstein to get a Nobel Prize I mean of course the really the really satisfying thing is to find the results rather than get these kind of prizes but basically anyone doing research can potentially end up with a Nobel Prize to be it be on the stage because you do need luck in all these discoveries that I've had the fortune to be involved in at least the two of them I showed you need they were looking at some problem from a different viewpoint and allows you to gain some some some aspect of it that wasn't the general the usual people working on it hadn't noticed so but you do need luck to stumble across something often you'll find something which is interesting and you need to take the time to look at something unexpected and this thing most of the time it'll just turn out to be an amusing curiosity but very occasionally it'll turn out to be something that no one had thought about before and no one was looking for so it was out there there was no competition to find it because no one was looking for it of course you do need the preparation to to recognize what you've seen and and some kind of so the preparation is important and you actually do need commitment because typically if anything is sufficiently interesting and you it gets a lot of resistance initially as we heard from Mike and these are probably the same things you hear about in any businessman giving an inspirational talk about how to succeed in business you need luck commitment and and preparation but anyway I think you all have the chance to discover something interesting and sometimes something incredibly amazing obviously it's a very small chance of finding it but you're all got it thank you
 Music  Basically I'm going to tell you about entanglement and how it turns out to become important in condensed matter  and how this has finally led to all kinds of beautiful dreams about a new topologically protected quantum computing.  So as we know the laws of modern quantum mechanics really haven't changed since 1925 to 32 period.  So what has changed is once the laws are written down it takes time to work out what exactly you can do with them.  Same thing happened with Maxwell's equations.  And all tests have shown quantum mechanics is correct.  But what we now have learned is that it permits very cool things to happen that we wouldn't have initially guessed.  So starting around 1980 there were new insights into quantum mechanics started to emerge  and new unexpected topological states of matter started to be discovered.  And since no one had dreamt of such states before they were quite a shock to us.  But this has gone on and being combined with a lot of interesting ideas from quantum information theory  that Feynman initiated about in 1959 I guess 60 years ago.  And people now as we heard on Saturday on Sunday I guess Saturday we're talking about the second quantum revolution.  A whole new set of ideas for looking at and interpreting quantum mechanics  and coming up with things that you can do with it.  So classical information is stored in a simple bit, a switch which is either on or off.  But quantum information can be stored in this new idea of the qubit  which is a generalization of the idea of a single two state system or an electron spin.  And while a spin can point in two directions relative to some axis  that axis can be allowed to move around and point in any direction.  So there's a lot more subtle ways to incorporate information into the Bloch sphere  by an arrow that points on the Bloch sphere.  So the central theme in these new approaches to quantum mechanics is really the property of entanglement.  And when I started off we really didn't think much about entanglement.  Entanglement was a problem for philosophers of science  and discussing whether quantum mechanics was compatible with free will and things like that.  But now we really recognize it as perhaps one of the central properties of quantum mechanics.  And of course it was initially pointed out as important by Einstein.  But now we're thinking of it as the fuel which will drive future quantum information processes.  So of course Einstein got his Nobel Prize not for relativity but for the photoelectric effect  for his early contributions to quantum mechanics.  But by 1935 he was really rejecting the quantum theory  by thinking it couldn't possibly be a true fundamental theory.  And he identified the key property of entanglement which he thought was so strange  that it couldn't possibly be right.  So he was hoping to consign quantum mechanics to the dustbin of history  by pointing out the crazy things it predicted.  And he called it of course, spook after fernwirkung, spooky action of the distance.  But Schrodinger came up with a much more neutral term that we use  which translates into entanglement.  So of course one of the fundamental properties of quantum mechanics is the Pauly exclusion principle  that says that two electrons which are fermions can't be in the same state.  But in addition to position and momentum, the state of electron also depends on this internal  degree of freedom, its spin, which is analogous to the spin of a top.  And of course the chemical bond which holds this all together is the way that  the Pauly principle is overcome because the two electrons can fit in the space between  two atoms to form a chemical bond to hold this all together  because it has these two different states of the spin.  So the poster child of entanglement is the chemical bond  which is the maximally entangled state of two electrons where we have  one electron is up and the spin and the other electron is down.  But the quantum mechanical state is a superposition of the two possibilities up, down and down, up.  And this is the maximally entangled state.  But of course what Einstein was worried about was if the chemical bond was stretched to large distances  maintaining this curious correlation between the electrons that it was a  superposition of up, down and down, up, he felt you know if you do a measurement over here on  this electron how could it possibly be compatible with relativity that immediately implied that  the sort of change in the other electron. So he felt this quantum state of entanglement at large  distances couldn't possibly be a true state and it superficially looks like it might violate the  principle of relativity that information can't travel faster than the speed of light.  But when you analyze this more closely you cannot send information between the two experimentalists  by instantaneously by doing a measurement. So it's not really a violation of relativity.  So this this puzzle that Einstein, Podolski and Rosen produced got analyzed in great detail by people  like David Bohm and then finally John Bell. But finally when the experiments were done,  quantum mechanics passes with flying colors. In fact the only let-out clause are still discussions  about the free will of experimentalists in choosing which axis to measure. But so anyway the problem  is entanglement over large distances is very fragile compared to the chemical bond. And so  electrons also interact strongly with the environment so these experiments to prove  that Einstein, Podolski, Rosen effect or entanglement could exist at large separations have  been done with light. But it's totally convincing now and indeed we just heard on Saturday from  Jan Wei Pan about the work in China where we've created entanglement between  between the ground and orbits and satellites in orbit. So around about 1980 a condensed matter  physicists were kind of arrogant enough to think they had a basic understanding of  electronic and magnetic materials. But around that time there were kind of two examples which  initially had no obvious connection whatsoever to them each other. Although it turns out they  both involved the Berry phase came up and they were both somewhat very unexpected. So the first  of the things was of course the discovery of the quantum Hall effect by Klaus von Klitzinger completely  unexpected effect which shocked people at the time. And the other thing was some work I did on  low-dimensional magnetic systems. And so these can be viewed as the tip of the iceberg in this  iceberg metaphor. But in the succeeding decades many many more new kinds of topological materials  have been found and the unity which relates these two effects has been found. And of course  the most exciting aspect of all this has been probably that the ideas about topological quantum  matter have led to a proposed platform for topologically protected quantum computing which  at least theoretically provides a nice way to to have scalability with and avoiding  decoherence which other non-protected versions of attempts to do quantum computing don't have.  So what's topological matter? Topological matter differs from ordinary matter because it has  properties which really are probably entanglement properties that can be described by whole numbers  like integers like minus one or two etc. So the basic issue is that states of matter which are  characterized by an integer can't smoothly change from one to another. Something always has to  happen at the boundary and it's usually something pretty interesting that happens on the boundary  between different types of matter. So here in the picture we have boring matter which I've assigned  which I've assigned the index zero to and some non-trivial matter. Now of course zero is not  exactly a boring number it took a long time to be discovered but it's what we might call topologically  trivial. So the boundary has to be there and basically that's because this topological properties  cannot change continuously and a key aspect of that is it makes the properties robust to small  amounts of disorder or dirt in the material which is a contrast to the usual material properties  of which our current silicon technologies for information processing are based on which require  ultra clean preparation. A single speck of dust or hair or something on the on the silicon chip  will ruin it which is why people have to be dressed up in these spacesuits not to protect them from  the nasty chemicals in the foundry but to protect the valuable silicon wafers from the people.  So the classic example in mathematics is the classification of closed surfaces by what's  called the genus or the number of holes in them and this was a metaphor that was being used  by the Nobel committee to try to explain to the public what topological matter was probably with  great confusion arising but a shape like a sphere with zero holes you can distort a sphere you can  distort a football into a rugby ball with small forces but if you want to change it into a donut  an object with a hole you have to do a significant amount of violence to it and so holes come in  hole numbers you can't you don't have to say as such a thing as 0.36 of a hole so they're easy to you  either have it or you don't basically at least of course when you get down to the atomic scale  the things are not smooth anymore so there's actually usually a length scale involved where  things become topological on the sufficiently large length scale which turns out to be a crucial  point. So topology comes from the the great discovery by Gauss who discovered basically that  4 pi r squared the area of a sphere times 1 over r squared is Gaussian curvature was 4 pi.  This is a kind of high school formula but what Gauss discovered was if I changed my sphere to  the rugby ball or even to a banana that that number stayed the same and in fact you could  do a very complicated integral of the curvature which on a non-uniform surface can be positive  in some places and negative in others and get this simple number so a very sadistic high school  mathematics teacher could give very complicated problems to the students but he would know if  they got the right answer or not quite quickly by just checking what multiple of 4 pi they  managed to get if they managed to do it. So in this picture we have this famous picture which  usually shows the bagel turning into the coffee cup which every experimentalist working on topological  matter cannot prevent themselves from showing at the beginning of their talk. This simple idea has  been so exciting to people working in the field they they just love the coffee cup metaphor.  Well of course in Sweden they were trying to talk about something which wasn't really a pretzel  it was actually a Santaluccia bun with two holes in it which would be topologically equivalent to  what's called a loving cup but actually when you bake the Santaluccia bun the holes actually  fill in so it's really a fake pretzel and the real pretzel of course is the German pretzel  with three holes and which is equivalent to this interesting cup here. I'm not sure what it's called  but I did hear it suggested to me this was a California loving cup.  Anyway the mathematicians of course love to take a simple formula and make it much more general  and much more abstract and indeed the Gauss-Borné formula was generalized mathematically from a  physical curvature on a real surface embedded surface like the bagel into a much more abstract  mathematical curvature on much more abstract surfaces which could occur in any even number of  dimensions and so this actually gave rise to the Gauss-Borné-Chern formula. Now the mathematicians  don't have a Nobel Prize say the simplest the nearest thing they have to it is the the Fields  Medal and maybe there's some Fields Medalists here I don't recall but the Fields Medal is a very kind  of a agist thing it's discriminatory because you can't get it however good your math was you can't  get it if you're over 40 so they invented a couple of new medals one the Arbel Medal and the other  the Cern Medal for kind of old-timers and back on the Cern Medal of course is the Gauss-Borné  Cern formula inscribed. Okay so what is the entanglement in a in a quantum mechanical context  basically if you take normal matter a trivial matter you can really you can do a measurement on  one side of it without really affecting things that happen on the other side and so you can kind  of pull it in half quantum mechanically and get a product state but because the topological matter  always has these edge states it resists chopping in half and eventually when you do manage to  split it the edge state has spread over the the new surfaces so you can't avoid having these edges  and so basically topologically trivial states of matter can be in principle be assembled just by  bringing the atoms together and stacking them next to each other well if something non-trivial  happens the electrons kind of get unbound for the atoms and rearrange themselves somewhere along  this path of constructing the matter and in fact once this principle is seen you can see it in old  work in fact this is an example going back to 1939 where people shockly managed to do a calculation  of toy model of of a one-dimensional chain of atoms and what you see is you bring these things  together the energy bands of the which were initially the atomic levels kind of broaden  but they they they come together and they split up again and you get these these new topological  edge states at the system you have one at each side of the sample and they are very interesting  effects such that if you try to fill this with electrons you're more likely to have the chemical  potential below or above these two states so you either have one electron too few or one electron  too many compared to the original atomic system and that extra electron is split in half one half  of it goes at one side and one half goes at the other side that's a very typical thing associated  with the topological edge state so this will actually be what we now call a symmetry protected  topological insulator so in heavier atoms electrons in a group of atomic orbitals can be locked  together so they're always parallel and that gives rise to magnetism the elementary spin is  got angular momentum a half h bar and so the total spin will be a whole number which is either even  or odd times a half to make an integer and conventional magnetism you just stack these  magnetically ordered systems together but there's no real entanglement in them but if you what I  discovered by basically looking at this problem from a different viewpoint coming usually one  has some history where one's coming to a problem I was working on some other issues in one-dimensional  organic conductors and applied some results from that to the magnetic problem so I was shocked to  find that the conventional wisdom about what these magnetic systems should do was like 100%  wrong it was actually wrong because people have been confirmed in their incorrect ideas by apparent  evidence from an exactly solvable mathematical problem but the problem was they completely  misinterpreted what that exact mathematics was telling them and it confirmed them in their  prejudices so this is I'm sure there's plenty of other things that we believe in which are wrong  and we think there's pretty good arguments for them and you may be the one who finds one of you  may be the one who finds out that everyone else was wrong and changes our way of thinking okay so  this was a very controversial thing and it shocked loads of people and again there was a lot of  resistance to it but finally physics is not a question of opinions and experimentalists  especially material scientists who managed to motivated to actually make the systems to fit  the theorists ideas rather than the other way around they came up with ways to actually make  these magnetic systems and finally the experiment showed that the the new kind of states were  correct and it led to the actual controversy led to people coming up with lots of new ideas and new  methods for studying problems which have turned out to be very important so the metaphor here which  you'll see is that we have a line of if we have a line of people who become entangled by holding  hands you apparently get left over with a spare pair of independent hands at either end of the  system where if a person hadn't had a bad accident you'd come up with the idea that hands always come  in close pairs okay and of course this led to this prize business so soon after this discovery  the underlying mathematical principles came out this fundamental idea of the Berry phase in quantum  mechanics which is really at the heart of all the kind of modern physics that we've been talking about  and initially that came up in in quantum spins but then at the same time we had a class from  cliff sing had discovered the the the quantum Hall effect which as we've heard has now become the  system at the base of the new system of units for replacing the kilogram and this is electrons in a  high magnetic field moving around in little circles and it wasn't initially clear what  where topology came into this but as was pointed out by Bert Halperin you have things going around  in little circles in the interior but if the if you hit the boundary the circular trajectory which  leads to quantization is broken and in fact you get these edge states that move around and now we  understand that edge states are very typical of topology David Thales was working on this  problem with collaborators and they did the important step which I kind of do too which is to  replace difficult problems with toy models that you can actually do a calculation on so they  replaced a more complicated problem of disorder in a real system with a simple periodic potential  which is the kind of thing we can solve and they came up with a beautiful formula which  basically showed that the whole whole conductivity was still given by an integer times e squared  over h and the mathematician mathematical physicist Barry Simon heard on the one hand about  Barry's calculation Barry's phase on the other hand he heard about this numbers that Fowles  has formula and he put two and two together and realized it was all the the gas for nature and  I think and that was that so it used to be thought that initially that you needed high magnetic  fields to get this quantum Hall effect maybe it should have been realized from Fowles's result  that that wasn't necessarily true but somehow as an accidental discovery almost like the accidental  discovery of the spin chain business realized that you could actually get rid of the magnetic field  completely and just keep Barry curvature in the problem and get with standard crystalline materials  and get interesting topology including edge states that go around one way on the boundary  and so this is actually again not obvious most most interesting things are not obvious  even to the people who discover them right but once but that's the nature of things so  yeah when one looks at the details you see lots of interesting things going on that I'll have to  skip but this actually took shows how long it takes to find something out it was actually  predicted in 1988 and it was a theorist toy model for a long time but finally it led to other things  and finally that those led to the experiments which were done only in 2013 to show these effects  can happen so again without huge magnetic fields the quantum the so-called quantum anomalous Hall  effect gives rise to these edge states which move in one direction and actually that seems to have a  lot of very interesting technological possibilities in fact one can actually from this edge state  aspect of things you can put aside the quantum mechanics because the well the some aspects of  the quantum system of course require Planck's constant the the integers don't and the existence  of edge states turns out to be to be there in photonics too so the simple toy model I showed  you has led to lots of new things both in in photonics and also mechanical systems so there's  actually surprising amount of new work going on you know once you realize the principle is there  and is quite general it's very beautiful it was course generalized to spin systems without  magnetism by Cain and Malie which initially provoked this large revolution because that led to  lots of materials actually being initially discovered they were sitting on people's shells  but no one had looked at the edges clearly enough to see there was something different about them  okay and this led to you know all kinds of new things which I can't go into so where does the  quantum computing come from the integer Hall effect was now followed by a much bigger puzzle the  fractional Hall effect and here you see the the sample in which the fractional Hall effect was  discovered which kind of by looking at you can see the how how the the robustness of topological  physics against against disorder right I guess this wouldn't have got Art Gossard well whoever did  this probably wouldn't have got a job in the clean room making chips with their soldering skills but  anyways so lots of really interesting things happen in the in the fractional quantum Hall effect  non-Abelian statistics fractional charges but the most remarkable aspect is actually what we  now call non-Abelian quantum statistics which is what this gives us the ability to hide information  in the entanglement structure of systems and that's really what was made people so excited  about the possibility of a protected form of quantum computing so the proposal for that goes back to  Alexei Kitev who looked at work of read and read more and read who came up with the first of these  non-Abelian states and basically you do something called braiding where you you move vortices or  little defects in this system around the quantum Hall system is like a quantum fluid with little  vortices which are little whirlpools in it and the information I'll show you is actually stored  in the history of how you prepared the the vortices so basically these little objects can only be  created in pairs so you have to create two of them at one place and then pull them apart so here I've  got a system which apparently has the same configuration in real space of the four vortices  but hidden in it is the history of how these configurations were produced and these are the  entanglement patterns and this would represent one state of the qubit the other one would be another  state rotated by 90 degrees on this prankery sphere and in fact when you look very closely at a  calculation of these systems you so this is again a calculated toy model system with four  four of these vortices arranged on a sphere in fact but you can see one state might be the  up state of the qubit and the other might be the down state of the qubit or a linear combination  of them would be the sideways state and you can see because they're not infinitely far apart from  each other they're little interference patterns that distinguish the states so the information  is actually an interference pattern and you in principle it's not completely hidden but the  wiggles you see become exponentially weaker and weaker as I move these things apart there's always  a length scale involved topology happens when things are infinitely separated on on a on a  microscopic length scale and here you can see the information becomes more and more delocalized and  more and more hidden as you move things apart so it's the magic of how it is is is really encoding  an interference pattern so the the top the quantum physics beautiful dream of topologic protected  thing is qubits is just one way to maybe overcome this problem of decoherence of course there's  lots of other attempts to do quantum computing and everyone is competing so I personally believe  that topological quantum computing is a realistic dream but we'll find out the current scheme  for topological computing is actually based on super on superconductivity and and so called  nano wires again it's very similar to the what I showed before with the spin train again if I put  electrons in contact with a superconductor it somehow allows the electron loses its charge  into the superconductor and then can actually split up into two kind of half electrons called  myerana modes and by joining these things up together in one way you actually can end up with  what you think ought to be indivisible an electron state which should contain be able to  contain one electron actually split up into two pieces on either end of the sample and that leaves  in a string of entanglement between the things so this metaphor of the chain of people is  quite similar so the again there's lots of other attempts to do things in quantum computing  lots of people are competing so I don't know what will happen but finally I just come to the  motivational message again like Mike Kostalitz is you don't actually have to be a you know a great  genius like Einstein to get a Nobel Prize I mean of course the really the really satisfying thing  is to find the results rather than get these kind of prizes but basically anyone doing research  can potentially end up with a Nobel Prize to be it be on the stage because you do need luck  in all these discoveries that I've had the fortune to be involved in at least the two of  them I showed you need they were looking at some problem from a different viewpoint and allows you  to gain some some some aspect of it that wasn't the general the usual people working on it hadn't  noticed so but you do need luck to stumble across something often you'll find something which is  interesting and you need to take the time to look at something unexpected and this thing most of the  time it'll just turn out to be an amusing curiosity but very occasionally it'll turn out to be  something that no one had thought about before and no one was looking for so it was out there  there was no competition to find it because no one was looking for it of course you do need the  preparation to to recognize what you've seen and and some kind of so the preparation is important  and you actually do need commitment because typically if anything is sufficiently interesting  and you it gets a lot of resistance initially as we heard from Mike and these are probably the same  things you hear about in any businessman giving an inspirational talk about how to succeed in  business you need luck commitment and and preparation but anyway I think you all have the chance  to discover something interesting and sometimes something incredibly amazing  obviously it's a very small chance of finding it but you're all got it thank you
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