Nobel lecture: F. Duncan M. Haldane, Nobel Laureate in Physics 2016

Nobel laureates in physics and chemistry laureates and economic sciences members of the Academy ladies and gentlemen the role of science is the question what is regarded as the truth to uncover hidden phenomena to see the possibilities which are of skill to the ordinary mind the passionate observer equipped with imagination and creativity leads us forward to a world with better conditions we stand on the shoulders of giants as Cyril ups new Giants appear and show us the way forward some of them are with us today's the progress of mankind is very much connected to materials the way natural materials can be exploited and used and her new man-made materials can be developed materials have even given the names to the ages of civilization the Stone Age bronze iron silicon ages man is not able to do anything with art materials it is essential to understand the properties and uncover the hidden secrets of materials and unexpected states of matter and to be able to design entirely new and Italian structures constructed using the smallest possible building blocks today's intelligent man-made materials would have been pure science fiction just fifty years ago it is not possible to plan for the unknown new fundamental discoveries that may lead applications which could not be foreseen and the period between a fundamental discovery and eventual applications is often very long compared to the criteria commonly used by governments and funding agencies but with the words taken from venerable the American engineer and science administrator as long as scientists are free to pursue the truth wherever it may lead there will be a flow of new scientific knowledge to those who can apply it to practical problems do you live during the last century we have witnessed an explosive growth of Science and Technology a development nobody could even dream on at the beginning of the last century the French advocate of Enlightenment jean-antoine the contorsi he argued that the expanding knowledge in the Natural Sciences would lead to progressive development of human capabilities and lead to a world of individual freedom and moral compassion since as he believed social evils are the result of ignorance and error rather than an inevitable consequence of the human nature in this respect he seemed to be too optimistic when it comes to help humans treat each other we still see corruption dishonesty and unethical conduct in otherwise well developed societies hiram new theoretical tools for properly designed and formalized negotiations help us to deal with conflicting interests today science and scientific knowledge are needed more than ever we live in a time with widespread resistance to knowledge for even powerful world leaders deny scientific knowledge a new term post truth has been added to the Oxford Dictionary meaning her emotions affect people more than facts we need to trust in science because only knowledge can make the world better Alfred Nobel he recognized the importance of science and those who uncover what has been hidden who explained on mere existence who improve the quality of our lives who opened doors for future findings this week is the big celebration of science and today you are all welcome here to listen to some of those who have increased our knowledge and made a difference to the way we think I will now ask professor tools house Anson member of the Nobel Committee for physics to introduce the Nobel Prize in Physics [Applause] so yeah good morning everybody so I'm very happy for this opportunity to say a few words about the Nobel Prize in Physics 2016 and introduce the laureates who are going to give 2016 Nobel lectures in physics so here they are I'm sure that many of you already know their names it's David Paulus University of Washington Seattle USA it's Duncan hold a Princeton University also u.s.a and it's Michael Kosta leads Brown University United States so the citation of the Academy that motivates the price is for theoretical discoveries of topological phase transitions and topological phases of matter and I will not make any attempt to explain what that means to you yeah you will hear two lectures later that will try to explain to you what that means so I will instead say just a few words about the low rates so I start with Professor David Paulus he was born in the UK 1934 he got his PhD from Cornell University in the United States and he's now the emeritus professor at the University of Washington Seattle and I should tell you that Professor Paulus is a legendary person among condensed matter physicists so he has made really groundbreaking contributions to theoretical physics I will not take x from the lecture to list all his achievements but I should mention that although he's awarded this year's Nobel Prize for his work on topological states so matter in topological phase transitions many physicists my told the view that his verdict when mesoscopic physics very introduced the very central concept without less energy might be as important as the work that we are celebrating today his achievement has earned him many prestigious awards including the Maxwell medal in price the Wolf Prize for Physics the IOP Dirac medal and a large on saw the price now I already mentioned that they I've already read the citation and as you can see it has two pieces to it topological phases of matter and topological phase transitions these two subjects are intellectually closely connected but they are still distinct and they are going to be presented in two lectures by the two other laureates and the first one will be given by professor halt dein and i will now say a few words about him so he was born in London in 1951 got his PhD from Cambridge University but ended up as the Eugene Higgins professor of physics at Princeton University where he is now presently and still very very active and he has worked in many areas of theoretical physics again I won't give any long list but I just mentioned that in addition to the work that we are honoring today yes introduce the important concept in physics the concept of loading a liquid the concept of fractional exclusion statistics and the concept of entanglement spectrum and he has given seminal contributions to the theory of the quantum Hall effect so now it's my pleasure to say also that has been awarded the maxwell medal and prize oliver Buckley condensed matter prize the our CTP Dirac metal and now I will call on professor Hall Dane to come up and give mm first of the 2016 Nobel lectures in physics please prepare to hold a thank you this is a challenging lecture for me to give because usually I give these talks to specialists but I'm going to make an effort to give you some flavor of three kind of surprising feet surprising discoveries that both David fellows who I'm going to talk for for a bit and myself have made and these are all things which I think were not expected at all at the beginning of whatever calculations were done to reach these things and they will emerge your surprises and I heard from Marcel din ice about this David Sir work on this thing which and it's again a similar thing that they didn't understand realize what good stuff was going to happen until almost the end of the calculations okay so I thought I'd give a little bit of background before I do because I'm not sure who I'm talking to but of course you just recall high school chemistry you've got orbitals that get filled you have a set of energy levels in atoms that quantum mechanics [Music] this makes discrete and you could get to fill them up with electrons and the usual case you can put two electrons in each orbital because they have something called spin which goes up and down and I'm going to make an analogy for this because this is actually what's going to happen in the quantum Hall effect and the simple case of the integer quantum Hall effect we have a two dimensional surface on which electrons can move with a large magnetic field going normal to the surface and in a magnetic field electrons go around the little circles just like in the atom but now it's the magnetic force that drives you in a circle the force at right angles to the motion of the electron and in this case it's as the radius of the orbit gets bigger that corresponds to larger kinetic energy of the electrons so in that case we have again a set of an energy level diagram where we can fill up the orbitals but instead of having a as in chemistry 1s orbital or 2s and 2p and then 2d we actually have a large number of orbitals and in the quantum Hall effect the integer quantum Hall effect the number of orbitals is macroscopic is proportional to the area of the system the area in which the electrons are able to move and there is one orbital one independent orbital for every quantum of magnetic flux that passes through the area the system for that in case you haven't seen any of this stuff before so if I'm a strict density of electrons let's choose because the number of the number of orbitals in the in each of these levels is degenerate levels is proportional to the magnetic flux density the magnetic fields we could choose the magnetic fields just right so we can completely fill up the lowest set of orbitals here the lowest Lander level we call these degenerate set of orbitals lando levels and if I say if I take this system where I just tuned things just right so there's exactly the right number of orbitals to be completely filled by the electrons in the lowest lambda level and the others are all empty we get something that appears initially to describe the integer quantum Hall state that turns out to be the earliest of the topological states of matter that were experimentally found and that was found by class 1 kissing in 1919 in 1981 and very quickly because of implications it led to his Nobel Prize in 85 so so when you have this situation you you've got an energy gap if you're a chemist you call it the homo-lumo gap or something and and so this is a kind of rigid system that the that that you filled it just up to this level it's going to cost you energy to put any electrons and he in a higher level and so if I try to squeeze the system where I'm actually changing the number area in which the electron is allowed to move and therefore changing them of orbitals if I compress this thing down I will I will not be able to compress it any further than having fill this lando level up completely because I'd have to if I pushed it together more I'd have to put things in the next round the level which cost me a lot more energy so at first sight it looks that's a very fine-tuned thing that you'd have to get everything just right and the special feature of the Hall effect that from kletzing found is it's very robust but actually this is a topological system and because something has to happen at the edge you don't actually need to fine-tune anything to get this just right because at the edge of the system if I think about the electrons going around in little whatever yeah if they go around in little circles I've drawn them going around in clockwise circles here but if they actually reach the edge of the system and try to go around in a clockwise circle they go bang and they will they will bounce back off the wall so in the one the sort of the orbits that hit the wall will actually precess around the wall in the anti-clockwise direction and that's one of the hallmarks of topological systems they have strange States at the edge of the system and this is what we call the the edge States there counter-propagating edge states that have a throttle feature that they only allow propagation around the edge in a single direction not turned and of course that's because if they can only go one way it means if I looked at this thing backwards in time they'd be going the other way and this is a system with what we call broken time reversal invariance the magnetic field breaks the symmetry of systems under the change of the direction of time so the orbitals at the edge actually go up a little bit and therefore there's a place as a reservoir of electrons at the edge the Fermi level is the highest occupied state the chemical potential and as I change the magnetic field I will just be I will be changing the they read how much I fill this thing up at the edge but I will still maintain the gap in the bulk of the system so this system does have gapless excitations at the edge there's a no gap which is very small it's proportional to the length of the one over the length of the edge so it's a microscope microscopic gap at the edge of the system but I still have in the interior this large bulk gap which is a topologically protected State well what fun cliff singh found he did these measurements of you measured the whole resistance a whole resistivity but i'm going to talk about the whole conductance which is one over that so what he sees in the picture you you change the magnetic field and the whole conductance sticks at various values which turned out to be quantized and it turns out to be an integer times your fundamental constants a squared over H or a squared over 2 pi H bar and being an integer is what finally got understood to be a topological property topology we heard from Hans Hansen that the when the prices were announced about your counting number of holes in in bagels and things and those can't be other than integers these are topological properties that we'll see and this is actually what a very nice result of Davidson's means okay so the Vonk listing system is much dirtier than the theoretical toy model I presented to you it's got all kinds of details of junk inside it and the work at the early 80s focused on these difficult problems of how you treat this order and and randomness and what was going on and but there were kind of tell you about that of Davis is he had the idea to study not the effect of this kind of random dirty system which is difficult but a different Lister a simple periodic system a kind of addict on a crystalline background of his choosing to the to these Landau levels and Marcel denies who worked them they said at the end of the day that was David's brilliance and the thing that he actually chose a tractable problem as opposed to sticking around that the physically relevant problem was apparently the disordered one and we have this sort of situation the the ideal case where it's clean in the middle is perfectly flat and this is a case where Bob Laughlin who won the prize little--it later for the fractional quantum Hall effect had earlier given an extremely powerful argument that says if I have this flat system in the bulk and the only places where there's energy levels at the Fermi level or at the edges then a few applied this electric field that to make things flow the charge would flow across the system within a way that couldn't be affected by any kind of dirt because he it was too far for any scattering between points at one Fermi energy and the other Fermi energy and so here that there was a very clear argument that related a gap in the bulk region of a sample to this perfect conductor perfect quantization of the hall conductance where it's just given by a whole number times from the mental constants so the actual problem that that people trying to be practical or trying to be realistic would have tried to solve is this one here where there's a lot of junk in the middle and a dirty potential but as I said this is very difficult and what David took some very beautiful results from Hofstadter and solve this problem here which should be a perfectly periodic wave in the in the middle the potentials periodic which means some points the the the lowest state in the system will be down here and the highest state will be at this point so the lambo level which in this case is a perfectly flat infinitesimally thin energy level gets smeared out into a range of energies between the top and bottom of this periodic potential and that's this so-called TK n n paper the Fallas kimoto and nightingale and Denise I guess some people trying to be cute put TK and and squared like that but the other three were three postdoctoral collaborators kimoto was Davis postdoc and nightingale and denies where other people's postdoc I guess at Washington University of Washington Seattle anyway they they they together with David they look to this problem of what the presence of this simple periodic potential added to the Lando levels would do to the integer quantum Hall effect and the Soraa the bottom line is they discovered a remarkable formula that they hadn't been looking for and that according to Marcel denies it jumped out of him only at the the very end of the calculation okay they were too concerned with the details but at the end they suddenly saw something something was important so David had been particularly interested in a very and it's very interesting toy model all the people kind of fascinated with at a time it's called it often goes by the name of the Hofstadter butterfly because you had extremely beautiful pattern come out and and Douglas Hofstadter found this in the 1970s and he loved desire aesthetic appeal I guess it's a model of a crystal in a magnetic field and people have gone to town on this model because it's got all kinds of beautiful features in it but basically it's just the same model as before kinetic energy of particles in a magnetic field and then of just a simplest possible periodic potential a cosine potential in the x-direction and a cosine potential in the y-direction added together and this thing is called the butterfly it's usually shown rotate his sideways and I guess it's got tremendous amount of fine structure and fractal structures floating around in it but that's not what sintering interesting us here today it has these very narrow levels which I guess for the purposes of calculating the whole conductivity this was done by Yosi Avron and his former student they - they donated this picture to me so there's no copyright concerns in my lectures and so the numbers here tell you what this integer the should be here at the bottom this thing is not the lander levels but it's very close to the Landau levels structure is for it's almost identical to what you would see for a simple Landau level structure and you've got a lot of a very narrow levels and you fill the first one you get one filled level a second one - but when you look in this other structure you get all kinds of other weird numbers floating around and they got fascinated the team got a fascinating thing first of all how can you calculate these numbers these integers what are they and and so the basic point that Marcel tells me that David pointed out was that Lofton's argument tells you if you've got a gap in the bulk of the system you've got to get this integer quantization coming out and he that lost in a done it for the simple Lando level case but this thing also has gaps in the bulk and if they're coming chemical potentials in there you should get quantization and they were struggling to find what was the formula that get these integers and so the after they solve lots of equations and found the thing they realized they were found how to calculate these integers they found there was had to be a more general principle than just solving what were called daya fans are equations some mathematical structures and basically if you take you have a periodic potential it's actually not the usual kind of the crystal because there's some the periodic the the the unit cell you have to use is not the simple one of the of the cosine potential it's related to how many how many of the unit cells contain an integer number of magnetic flux coins here so it's a rather strange band structure but basically if you've got something periodic there's a block theorem that says you can write the wave function of a I can state of a system in a periodic potential as two things a periodic function times an exponential e to the ir e to the ikx or KL and the details are what's going on in the bands are all contained in this periodic thing so they took a very standard formula a fundamental formula for the conducting calculating the conductivity of a system called the Kubo formula and plug this stuff in so the Cooper formula is usually used for calculating the dissipative part of the resistance but in the case where you're looking this hall is it hole resistance where you're looking at the response to an electric field which is parallel which is at right angles to the motion of the current so it doesn't dissipate any energy you get different formulas well they put this in and they crank the handle and they eventually got a very nice formula they don't actually comment on it in the paper that how nice this was in some sense but they ended up with a formula that depended on the details of the band structure we sure can't of the wave functions of the band structure which are contained in this function here there's no reference anymore to the energy levels of the problem the energy levels are there in the kuba formula but after integrating stuff out and using formulas for the velocity of the system you find the energy levels in when you're calculating this whole conductance this one which is dissipative if not the usual ohmic law conductance the energy levels all drop out so this is the first form they got of a formula for and after they will I'll show you in a minute how this turns into an integer but they found this this formula which gave you the hall conductance which is these numbers a squared over H which is one click sings number of twenty twenty twenty twenty-five thousand three hundred and ten ohms I guess ohms to the minus one sorry and then we have this knot these numbers here which you can do some simple mathematics with Stokes's theorem on so they found this formula and shortly after they published their paper another another another very influential discovery in in mathematics in physics comb was that Michael burry discovered that the adiabatic approach to quantum mechanics had missed out an important geometrical phase and in fact this is another piece of work that people a lot of people think should get the Nobel Prize this is my hint to the committee but it's geometric so I guess you could say it shouldn't have come this year anyway that self so a very simple was a spin in a spin which was polarized along the along some axis and you gradually rotate this and you find that you pick up this additional phase which is very geometrical it's actually the solid angle the spin times e to the I II can spin times a solid angle enclosed by the path which is ambiguous modulo four PI and therefore the spin has to be an integer or a half integer which we knew for other reasons but these are typical of these topological things that have interesting consequences this tells you that spin has to be quantized for example even if you didn't do it by algebra so anyway the mathematician mathematical physicist Barry Simon in 1983 just after the two papers have been published I guess he he he recounts that someone else tipped him off about this connection and he should look at it or two there had to be a connection and he found that the two things matched together very well that they the thallus the T KNN expression was an integral over a curvature associated with the phase of Barry's Sahana me and mathematically it was an integral over a compact manifold which turned out to be the Brillouin zone which is topologically equal to a torus and in fact he pointed out this connection to what is really a mathematical extension of Gauss's famous theorem I agree diem why should I guess he he discovered while he was a foreign member of the Royal Swedish Academy of Sciences because he was joined in 1821 and this is 28th and he thought I think cows gas seems to have thought this was the most most amazing discovery he ever made in mathematics it so what Miguel's discovered was what we heard from Ventura hands hanson about the bagels and the thing so gas discover something that you of probably you can you can easily get the simplest version of it for your from your high school algebra class right you know that the the surface area of a sphere is 4 PI R squared whereas her radius and the Gaussian curvature that Gauss introduced of course for the sphere is is uniform and it's 1 over R squared so if I integrate the gaussian curvature over the surface area of the sphere i get 4 pi but what gauss discovered as this amazing theorem was that i can change the shape of the sphere to anything else like a football or a banana or whatever and they still got 4 pi and this is called the gas when a cat the theorem I guess because cows never wrote it up properly and when they put all the details in later so if I take this thing we get this service by now famous account of the Swedish in the German pretzels but and we also get the account that the famous relation between coffee cups and bagels you have a mug is the same as a ball what's being called a Loving Cup is the same as a Swedish pretzel and I'm not quite sure what kind of Loving Cup this is because it's got three handles but that's the same as a German press although some kind I thought well I'm not sure if that's something like nationality kind of combination but anyway so this is the list apology that you see every year when people giving give talks about topological insulators they love the coffee cup I'd like to see actually the Swedish pretzel changing into the Loving Cup and backwards rather be a change from the coffee cup in the bagel so okay what they what the the formula that they found that that berry simon pointed out that this formula was really the the churn the first churned class integral well basically can divide into two parts it's two things that see the it's the integral of a curvature which you can write as an anti-symmetric tensor but now in momentum space not in real space or case base block space and then this is the formula the other part of the formula is the formula for the the berry curvature and in fact it's kind of interesting fact that again talking about things which had discovered before their time this this formula is in a very correct paper or something like this formula a 3d version which is non topological is in a 1954 paper by car plus and not injure that was denounced as completely wrong and nonsense for many many years only in 1999 was it recognized that it was actually a case of very curvature and somewhat later they was shown to be essentially correct in a lot of situations it being applied to so this again topology takes a formulas and not obvious until some so I guess Barry Simon did a very big contribution in in pointing out the mathematical framework of what David and the other collaborators had found so just quickly in these formulas if we write them down this people make a big correspondence now to think of things like something like electromagnetic fields in in momentum space because the the curvature is kind of like the magnetic fields and the curvature can be written there's something like the curl of a vector potential domain if so which is called a barre connection and the Barry's phase is the integral of the Barre connection around the closed path and it's very much like the bow Maharana phase when you take a electron around the loop and in the presence of a magnetic field and try to calculate what phase it picks up when it goes around that loop and the barre face because the Barre phase is e to the I something this is a number which is only defined modulo 2 pi from this you quite quickly see the mathematics of Stokes theorem tells you that the integral of the curvature over the over something bounded by the by a curve has to be 2 pi has to be 2 pi times an integer so this was the form that that fellows found the actual form they wrote in their paper looked something like this they'd integrated by parts C the first formula and then they found but this thing has to be is when you do the mathematics is it's obviously an integer times e squared over H and that was the thing so that in fact as I said Marcel the nice told me that this were only emerged at the very end of the calculation of kind of afterthought he has some talk about the they realized there was an elephant in the room after looking at their form that had to be something a lot more in their paper that than just the actual values of these numbers in this nice colorful diagram right the numbers of course were not interesting really I mean the only challenge is that you can compute them but then the big picture rather than working rather than being interested in Hopf status by the fly the big picture is to come out with a very fundamental formula that no one had expected existed before and this is this amazing work that he did okay so this is so out of this my own work which was also nation a bit later comes from seeing that this integer quantum Hall effect can actually occur without Lando levels so maybe you could say this was implicit in the Tico name result but I don't know it wasn't noticed the 1982 paper was all about the periodic potential on nando levels and it hadn't really been appreciated that if it's a topological thing in the details surely can't matter and the pacific realization in terms of Lando levels is very complicated and not very practical for for hopefully finding new materials right and so I was again stumbled over this feature that you could actually write down a very simple model that turned out to be very influential because you can't do very simple calculations with it it turned out to be something like graphene it's a modified form of graphene where you our second neighbor electrons can tunnel between second neighbors and they pick up a little bit of a complex phase which is a time reversal breaking effect as they up around so you end up with an extremely simple kind of so-called tight binding band structure and it turns out to have it turns out to really to be the first of the topological insulators this is not a this is a time reversal broken topological insulator which in my view actually has more potential practic practical potential than the normal kind that people are studying a lot now but we'll see what happens with this so this was actually grat it's kind of graphene at the time I we didn't know that someone would invent graphene by taking a scotch tape and on pencil scraping a pencil over something and doing scotch tape so I put in my paper that this would be a single mono layer of graphite but it would be inconceivable for someone to actually realize that and of course I think the lesson is that material science can take us to play but if it can be done someone will do it right however crazy the as long as it's not real black holes and something like that so whatever the crazy thing model you come up with if it's just something interesting it's most likely someone will be able to make something that does it whether it's cold atoms or they otherwise so one of the features of these is that they have they have this one-way channel around the boundary where things move in one direction in the in the first in the Lando level case that that was really coming from the direction of the magnetic field but in graphene if you look at it graphene ickle an edge of graphene there's various edges and there's a nice one called the zigzag edge mode and you find that graphene has itself for the edge if I take a view along the edge there's actually a bound state which comes between two different Dirac points in graphene and exists in part of the surface Brewin zone and it connects see things once you you can open up a gap in graphene by turning it from a semi metal to a semiconductor in two ways by breaking inversion symmetry or breaking time reversal symmetry when you break inversion symmetry you get something boring the these this this state has to be attached to either the top or the bottom band in the boring cases both ends of the state to attach to either the valence band or the conduction band but in the interesting case one in one point ends up can touch the conduction band the other to the valence band and there's a kind of channel a plumbing connection between the two that you can drive States up and down by putting a magnetic field through this flake of graphene like substance and if I fill the system up to the Fermi level is in the gap I find I have these one-way these things that travel one-way this is actually the hallmark of the of the topological states they typically have something very interesting at the boundary and the start of all about to about ten years ago or twelve years ago Eugene and Malayan and Charlie Kane there's something which I'd actually thought of doing and I never did the calculation I thought it obviously won't work they took two copies of my model white and put them on top of each other one with up spin electrons going one way and the bouncing electrons going around the edge the other way and I'd have thought that this would these things would mix and scatter but there was a new a new topological invariant which was something like minus 1 for the churn number that was hadn't been appreciated before so this this was an extra topological invariant not the one that Barry Simon had pointed out with what there's tkn and formula was and that has led suddenly to generalizations for three dimensions and people discovering lots of very interesting topological materials lying in the Bertie shells okay so there was a this toy model which he says somehow a relation to the TK and n story again has a has a very fruitful history it's also been translated into photonics and there's a feel now of topological photonics where you get right to travel one way around the boundary of things okay so the third thing if I have time I'll talk about how much time do I have okay okay is the the magnetic chains which also is about the same time about probably within a month a few months of the of the of the of the quantum hall measurements by eclipsing well that this time things are coming out and and this was actually been being baptized by Shogun when who made who recently made a very exhaustive classification using very fancy mathematics of Ko board ism or Co homology or something to classify all possible subsets of topological things and this time the Sun sense is the hydrogen atom of topological materials it turns out of course I did not know that at the time when I found it but it was something totally unexpected and of course that's the hallmark of topological material say they they differ fundamentally by some number being one or two as opposed to zero which gives them a property that's not present in normal matter and this stuff behaves very differently from what was expected to happen okay so actually this was a interesting case I actually was unable to publish it at the time because these two journals struck dismiss this out of hand based on referee saying you know it's a manifest contradiction two fundamental principles of physics a bit like what happened to car plus and nothing sure that I mentioned and by the time it was published there was actually a more sophisticated argument that that actually removed what I realized had been the connection to the earlier work that we're going to hear next from coastal it's costless phallus transition because there's a very nice mapping from two-dimensional classical systems at finite temperature where there's a Boltzmann factor describing a probability weight to quantum mechanics in one space and one time dimension where the Boltzmann factor gets replaced by an amplitude and the quantum mechanics is richer than classical mechanics because Boltzmann factors are always positive while in this case there may be some time reversal invariance but the amplitude can either be positive or negative any interference effects between one path having a positive amplitude they all have a negative amplitude and therefore the two things can actually cancel some process completely so the thing was in this paper I see that so this my my lost preprint because I didn't have any records of it it was just cited in people's work at the time but GNO sure you had I found had finally kept a copy of it and in fact it's kind of interesting to read it again and it's actually basically using the Khosla Fairless argument translated the quantum mechanics to say why the conventional wisdom of the time was was incorrect you can find it on the archive if you want now I posted it so in magnetic chains we have ferromagnets where the light they like to line up or antiferromagnets where they opposite the case and this might work dealt with a anti ferromagnetic in one dimension okay and the feature there is what was known at the time is that though antiferromagnet the ferromagnets got a conserved angular momentum conserved its total spin so it's kind of stable ground state but the antiferromagnet the quantum fluctuations in one dimension destroy long-range order so people knew that as a mathematical theorem that they sort of but they had a remarkable thing there was a exact solution of this magnetic problem for spin 1/2 system by Hans bethe who incidentally was David's thesis advisor at Cornell and before he went off to tell us how the Sun produces energy he was interested in quantum chains and he did a remarkable piece of work through significance I'm sure he didn't understand because he lucked onto a solution of this model that seemed to work and took about 50 years to discover why it had worked and bata cleared and understand why because he his paper says my next paper will do it in two dimensions and three dimensions but of course that never happened because this is a very specifically one-dimensional thing but in the this was a very obscure difficult piece of work and no one knew how to calculate anything else except energy levels from it but in the mid-1970s another piece of work a remarkable piece of work going back to the 30s a Jordan ring the transformation which Maps spins which are bosons bosons really two fermions by relating relating up spin up to be filled and spin down to be empty states gave a way to translate these spin chain languages into into fermion language and this actually was a lot easier to do calculations with than the beta and SATs and it was it allowed one to translate things into calculations you could do and I guess progress always happens when you can actually turn something into some calculation can actually do on the on a piece of paper or understand then you can understand all the details and this was done by Luther and Peschel Luther was at nor DJ for a long time after that and they turned this thing into a fermion problem and to cut a long story short they got most of it right but they missed out some detail which I discovered later by applying what this lot and your liquid theory that was mentioned and I found that the loot that they didn't really have an understanding of what happened when you went for a easy access to an easy plain easy plane to easy access antiferromagnet whether the spins when the plane are wanting the point out of the plane and looking at the numerical this this new tool gave us a thing to extract numbers from the beta answers calculation they showed immediately some term was missing and these pictures here actually exactly the quantum translation of the cosmos Palace story that Michael will tell Michael tell you about so it turned out that the spin 1/2 system was was special rather than the spin one that one system they both were costly at Silas transition we're driving you for an easy plane to easy access but there was a special cancellation effect which I guess will vaguely say here between the within the spin half case here that you've two isolated what Mike's going to tell you about from a two dimensional space to a spacetime diagram and here there's a tunneling process that takes you between a configuration whether these are compass needles in the plane they kind of say in the plane and as you go around this periodic boundary conditions you go around in the first case there's no kind of winding number the second case there's a twist on it like a mobius strip or sand your twist went in and you can't get from those two things continuously you have to do some kind of breakdown and there's a telling point which is a vortex and it turns out that for the spin half case you only got a double vortex because if you do it you have to do this process of tunneling on one of the bonds between the spins and if you took it you look at the process on two neighboring bonds they should everything should be identical if the bonds were identical strengths except the two pars one in one passes one of the spins rotated 180 degrees clockwise and the other one it rotated 180 degrees anti-clockwise wishes so the difference of the past is rotating a single spin 360 degrees but if it's a half integer spin you get a minus one factor so the quantum mechanics told you something so I'm looking at this way gave you a completely new way to look at things and it turned out that the conventional wisdom was completely wrong and and finally so we came up with this new story which shocked people at the time because it it conflicted with preconceived notions based on on the beta answers equation seem to seem to confirm incorrect ideas about spin ways and finally it's not go through this too much but at the end of the day this turned out this this new state with this gapped integers integer spin liquid state that I found turned out to be one of the simplest topological states with a very non-trivial entanglement there's spin one system and the interesting thing is a spin one can be viewed as two spin has together and in the usual case they just point in some direction but here they the spin won against the fact rationalizing the two spinner house which one on either side and they grab hand they grab hunt they grab on to the spin one half from the others from their partner and the other so next next went along and get entangled into a singlet state and when you do that you get left the two to the two spins at the end of the chain they let left they get left with one arm up stretched out here and the other arm up stretched out here so they're like two half spins at the end this is two unexpected spin one heart at the end of the thing and this is precisely the analog of had the edge States in the quantum Hall state again this this feature you find in all the topological states that there's something strange happening at the boundary where the edges are where things that go from something to normal so anyway so that brings me to the basically the end of the story I had to I couldn't really give you all the technical details and it's probably quite confusing because it's very difficult to explain this to a general audience even a general audience of scientists if they're not condensed matter physicists and but basically the why this I my opinion why my interpretation of why this prize was awarded now is because so much as suddenly in the last 10 years have come out of all this old stuff this these these various little findings which in each turn came from people finding something more amazing than they were looking for just something noticed something strange and it was calm it gave you a new way of looking at things and what's happened is that we've read over the life Tovar over my lifetime in in in condensed matter physics I've seen that the the kind of things we were taught and learned from textbooks like actually off the mermaid and things back in the 70s a lot of the details that were thought to be really important then and really we put them away as boring boring stuff which doesn't relate not important and all kinds of new new things like entanglement and new ideas which were not present in any of his old stuff have turned have become the central kind of questions that we're struggling with and you know this after they after the topological materials started to be found concretely as real things there was also proposals that they could be used as a so-called platform for topological quantum information processing and that of course has stimulated a large amount of interest and there's been a huge amount of people and cold atoms quantum information condensed matter even the string theorist ed ed Witten is now the voting part of its time I guess string theory is you know there's this some spare time to deal with other things now and getting very interested in in extending all this topology to four dimensions so we're now doing we now have tables of topological matter up to eight dimensions so now you can see how the influence for an outside the field has influenced things but it's a very exciting area because we're finding all kinds of well I consider neat stuff about quantum mechanics that we didn't know before and there seems to be plenty of new insights to gain so so that when you can find something very simple and beautiful we all think that's the way that's the truth and maybe we're going to learn a lot about matter and learn a lot about possibly applications of it thank you [Applause]

Channel: Nobel Prize

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Id: 47Ab1izxH9Y

Length: 51min 30sec (3090 seconds)

Published: Sun Dec 11 2016