 Okay, let's start. Good. Excellent. So, good afternoon to everybody. Welcome to my CTP and to our activity. This is a highlight of our CTP's activities in the year. I think today we are celebrating the unity of science, the unity of physics awarding this direct medal to three high-level scientists with a totally different origin backgrounds that are contributing to the field of theoretical physics at the highest possible level. And it's a pleasure to have all of them here. We are professors, from Harvard, the professor Dantan Song from Chicago, and professor Xiao Gang Wen from MIT. So the direct medal in 2018 was awarded to the three of them for their independent contributions to our understanding novel phases in a strongly interacting media body systems, introducing original cross-disciplinary techniques. So in that sense, that's a beautiful combination of three different researchers with different backgrounds and different for culture and so on contributing to physics, but in the sense that we are contributing to the unity of physics. So let me say a few words about the direct medal. It was started in 1985. So it's one of the probably most prestigious awards in theoretical physics. It was started by Abdul Salam and they decided to name it after Paul Dirac, not only because he was a big figure of a big role model for all theoretical physicists, but also because he was very close to Abdul Salam. When Salam was a student in Cambridge, Dirac was a professor and helping him a lot. And also, since 1968, Salam was a frequent visitor to ICTP, so it was very closely related to ICTP and its mission. So there are many stories about Dirac visiting Trieste and it's beautiful to read some of the letters that they exchanged between Salam and Dirac in the time when he was coming here. The standard letter was that he was coming for an activity, asking for a room with a view to the sea. For instance, there's a nice beautiful handwritten letters. And so it's an honor for me to host this activity. This is the tenth time I do it and this is the last time also. As Stefano was reminding me a few minutes ago. So it's a real pleasure to participate on this because this is theoretical physics at its highest level. And also the level of the award is also guaranteed by the level of the people in the committee. And the people in the committee, this year there were Michael Green from Cambridge, David Gross from Santa Barbara, Bert Halperin from Harvard, Giorgio Parisi from Rome, Martin Rees also from Cambridge and Ashok Sen from India from HRI in Alahabad. So they take this duty very seriously and they were all very pleased with the selection and well we will proceed later on to give the award to each of the awardees and they will give a presentation. But before we start with this official award ceremony, my colleague and dear friend David Tong from Cambridge, he had kindly agreed to give an introductory presentation, putting together the research subjects of the three awardees. David, probably you all know, besides being a top scientist, he's also one of the best communicators in the field. Many students, without telling anybody, go and read his lecture from classical mechanics to solitons and so on. So I know by talking to students that he's one of the most popular physicists in the world and not only his lecture notes are very good, but also his way of presenting things is excellent. So I was very pleased when I asked and David agreed to give this presentation. So let's start welcoming David. Thank you, Fernando, for that very warm and kind introduction. It's the microphone working. Can everyone hear okay? I'll take that as a yes. Well it's a real pleasure and honor to be here to introduce three physicists, three physicists who have influenced my work and my thinking very much over the years and three physicists who I know have influenced the work of many in the community and as I hope to make clear as this talk progresses, many people in many different communities and that's really going to be one of the themes that runs through this talk. We're here to recognize their work and the work actually of many people in developments in quantum matter and quantum field theory and I would like in part to try and explain some of the achievements of our medallists but also in part just to try and set the scene to try and tell you the kind of questions that people have been asking and what sort of the broader developments have been over the past 25 to 30 years that Subir, Jalgang and Son have been involved in. Fernando already touched upon I think one of the main themes of this prize which is the unity of physics and I think it's something I'll touch upon over and over again in the lecture but we'll see that there's really a cross-disciplinary aspect to the developments that have happened and so to put this in perspective I'll go through each of them. Jalgang did his PhD with Whitton in string theory and then very quickly changed into condensed matter physics and is now working on topics which are really at the cutting edge of pure mathematics. I know this because last week I went to a conference on the work that Jalgang initiated full of mathematicians I didn't understand a word of what they were saying. Son is trained as a nuclear physicist he then moved into doing pioneering work in string theory and in holography and is currently doing equally pioneering work in condensed matter physics that we'll hear about. Subir started life as a condensed matter physicist he then stayed as a condensed matter physicist but today we're going to hear about how condensed matter physics and new developments in condensed matter physics have really made a major impact into our understanding of black holes so there really is these many many different themes and threads that build into what I'm going to tell you about today. Alright I should grab the clicker. Good so let me tell you the broad perspective the topic that we're celebrating has many names it's sometimes called many body theory it's sometimes called quantum matter it's sometimes called quantum field theory. The upshot is that if you take quantum mechanics and locality then those two things combine give you a framework which is known as quantum field theory and quantum field theory is an extremely powerful tool it can be used in many many different areas from understanding the central the interior of neutron stars to understanding what's happening at the LHC to understanding many different condensed matter experiments so it's a broad framework in which we would like to ask and address broad questions and there are many questions you could ask it's huge swathes of physics are covered within this but it turns out it's fruitful to ask very very simple questions in order to get a broad understanding of what's going on and so the simplest question or perhaps the most useful question I should say that the one can ask about quantum field theory is the following take your favorite quantum field theory and look at low energies where low energies depends on your preference and ask at low energies what's happening and in particular how many degrees of freedom are there at low energies so what's going on so to give you some examples if you're a particle physicist low energy means 10 to the 12 electron volts okay that's the energy that we're probing at the LHC it doesn't sound particularly low it's the highest energy we've ever reached in experiments here on earth but it's low compared to the Planck scale which is it's really the relevant comparison in contrast if you're a condensed matter physicist low energy means well maybe 10 to the minus 3 electron volts or 10 to the minus 4 electron volts if you're interested in say quantum Hall systems so it can be a huge difference in energy scales nonetheless the questions are roughly the same and questions are what kind of things can be happening at those energies and there are three possibilities to this it's a classification that I learned from John McGreevy that I like very much the three possibilities for what can happen in any quantum field theory at low energies is the following the number of degrees of freedom can be none it can be some or it can be lots and so what I'd like to do is just explain what I mean by none some and lots and give you some examples the way this is going to work is as follows I'm firstly going to tell you the classic view on the subject where the classic view is basically much of 20th century physics up to around about 1990 and then I'm going to tell you the developments that have happened in the last 25 or 30 years in large part due to due to our medallists today so the classic view has a name the name is called the Landau Wilson paradigm I'll maybe explain a little bit more about where this name comes from as we go forward but this is the classic view if a system has no degrees of freedom at low energies what does that mean it means that your system has particles all the particles are massive it costs an energy equals MC squared to excite those particles but you don't have that energy available so all you have is the vacuum and life is just tedious and boring there's nothing interesting to say about life when there's nothing going on that that a spoiler by the way that it turns out is a big fat lie in a way that we will see later but this is the view up to about 1990 if there's no degrees of freedom there's nothing interesting you can say all right some degrees of freedom is more interesting so how do you get some degrees of freedom well you you have massless particles in your system or perhaps you have particles that aren't quite massless but a light compared to some particular energy scale and you'd like to know what the properties of these particles are and how these particles behave well it turns out light particles are rather rare and precious in quantum field theory quantum field theory doesn't want the particles to be light somehow it wants the particles to have a rather large mass and whenever you have a particle which is massless or at least lighter than you might have anticipated there's some story to tell and the stories that we tell about this are really the great stories of 20th century physics and each of these has a major breakthrough let me tell you some examples perhaps most familiar particle which is massless is the photon the graviton of which we've obviously only seen the classical version gravitational waves is another very famous example roughly speaking there's a reason these are massless and the reason is gauge symmetry roughly speaking there's lots of caveats to that but that's one way to think about it there are other examples of particles that are massless that appear more commonly in condensed matter systems sound waves and solids or phonons are another good example again there's a reason why these particles are massless the reason goes back to the work of Nambu and Goldstone and it's because the world has translational symmetry and solids do not respect that translational symmetry they they break it by the lattice and Goldstone and I should say as I'm in this place Goldstone's theorem was proven by Goldstone Weinberg and importantly Salam Goldstone's theorem says that whenever you have a broken symmetry you have a massless particle that's associated to it so it explains sound waves rather remarkably just to give you some sense about how rare massless particles or light particles are there's a particle in nature called the pion the pion is not massless in the slightest the pion has an enormous mass of 100 million electron volts so if you're a condensed matter physicist this is an enormous mass nonetheless it's light enough for us to worry that there should be some reason why it's not heavier and indeed there is a fantastic reason it's precisely Goldstone's theorem as Nambu himself realized in in the early 1960s there are various other reasons why particles could be massless chiral fermions is one that I won't go into but one important one that that I will just spend some time explaining is is called a critical point so when we're undergraduate physicists we learn about critical points but not in the context of quantum field theory or quantum mechanics we we usually learn about critical points in the context of statistical mechanics so let me just just give you a reminder about what critical point is take a pot of water and put it on the stove and boil it and when it reaches 100 degrees it goes through this very violent transition where the water changes into steam now do the same experiment but put a lid on the on the pot and if you hold the lid really really tight so that it's above something like 200 times atmospheric pressure you need some muscles to really really push it down you'll find that that violent phase transition doesn't happen you'll find that the water turns just very smoothly into steam now there's a point where the violent stops and the smoothness begins and that point is called the critical point and at the critical point we get what's called a second-order phase transition and what happens at a second-order phase transition is you have fluctuations in water bubbles of steam but bubbles of steam at all different sizes now if you take that same story and you translate it into the quantum world those critical points become massless particles they become massless particles because you have fluctuations of all sizes that corresponds to wavelengths of all sizes and the only particles which have wavelengths with no characteristic size are a massless ones massive ones have a Compton wavelength which which is special so these are roughly the four different ways the last one notice is not generic for the last one someone has to be there dialing the knobs and holding the lid on the steam the others are generic the last one is is rather special finally let me tell you about the third classification of stuff the third classification is when you have lots of degrees of freedom and lots of degrees of freedom means a Fermi surface so what why is it lots it's because you go back to the other one just massless particles as you look at smaller and smaller energies with massless particles you have fewer and fewer degrees of freedom so stated correctly the density of states scales to zero as the energy scales to zero I'll just go back quickly we have this picture of the Dirac cone here in graphene and you can see that the zero states zero energy states are right at the tip there is you go to closer and closer to zero you have fewer and fewer states in contrast for a Fermi surface as you go closer and closer to zero energy which is the edge of the Fermi surface you have a finite number of states which which remain so this is lots of degrees of freedom when we're undergraduates we learn about Fermi surfaces but we learn about Fermi surfaces for free electrons well there's a very deep and beautiful result that goes back to Landau which says that even though the interactions are important among electrons there the free electron picture that you have of the Fermi surface is basically right in in most respects so it's called Landau Fermi liquid theory the more modern perspective is due to Joe Polchinski and Shankar from the 1990s and if there's any students in the audience I whether you're high-energy students or condensed matter students I absolutely recommend that you read Joe's and Shankar's approach to the Fermi surface in part because it's beautiful science but in part because they're two of the most beautifully pedagogical articles in physics that I've ever read so if you want to learn some physics and learn how to explain physics correctly Polchinski and Shankar's approach to the Fermi surface is the right way all right so this was the status of physics up until the 1990s as I said all of these are major accomplishments in 20th century physics I think going back to this list Nobel Prizes were one for every single one of these with the exception of perhaps the most important one gauge symmetry and the reason I think a Nobel Prize wasn't one for this is because probably because of anti-Semitism because because Einstein didn't get it for relativity he got it for for the photoelectric effect even though everyone knew that he really deserved it for relativity so this is life up to 1990s let me tell you some of the things that have happened since then in large part due to due to people here firstly if you have nothing going on if you have no degrees of freedom whatsoever you have an extraordinarily rich and subtle system where there's an awful lot of physics to explore and it was Yao Gang I think who was the really the first person to appreciate what was happening in systems with nothing going on at all it's now one of the most interesting areas in theoretical physics I think Yao Gang will talk about topological order perhaps also about the idea of symmetry protected topological phases which which he initiated I can give you some example of this so I think one of the most interesting developments in the last decade or so is the subject of topological insulators so firstly what's an insulator an insulator is an example of a boring material but firstly there's an energy gap so if you look at suitably low energy is nothing going on and the way insulators manifest themselves as being boring is if you do something to them like you drop a voltage across them they just don't do anything they just sit there because they're dull and they're lifeless and inert all the electrons are locked in place and you know there's just nothing interesting they can do well about 10 years ago a number of physicists realized I think something we all knew deep down that there's lots of different ways you can be boring and there's a classification of boringness and in particular a classification of of these topological insulators and a classification of lots of other boring things that the Yao Gang sure can tell you about and the wonderful thing is that you take one boring object on its own and it's just dull but if you take two of these boring objects and put them together then on the interface between the two something magical happens and the interface becomes alive with massless particles described by the Dirac equation and many other wonderful things that I think we'll be hearing about later so there's a whole enormous story to tell here meanwhile the examples of massless particles well again we have a new one on the list which is the edge of topological phases as Yao Gang first realized for the Quantum Hall effect long ago in in the 1990s but we also have many more interesting examples and Subir has been one of the people really pushing this forward understanding what those critical points like boiling water are in the Quantum language and some novel kinds of critical points in particular something called de-confined criticality and more importantly what you do with these critical points and to appreciate that once you have a critical point even though it's very finely tuned it only exists at one particular point in the phase diagram nonetheless it exists things you care about like transport transport of heat transport of charge in wide regimes of the phase diagram and so these points are very special and really dominate the physics finally experimentally we know that despite the theorems of Landau and Joe Polchinsky and Shankar there are in nature Fermi liquids that are not described accurately by free fermions they're called rather uninspiringly non Fermi liquids we don't really know what they are there's been a lot of theoretical work over the last few decades trying to understand them I think these are the ones that we're furthest still from understanding but it's become clear through the work of many people including Subea that trying to understand these non Fermi liquids this is the phase diagram of high temperature superconductors by the way trying to understand these non Fermi liquids is related to trying to understand these quantum critical points that exist somewhere in the phase diagram okay so this is the big picture we have very very basic questions what can stuff do and we've learned in the last 25 years that there are things that stuff can do that is surprising and subtle and exciting and exciting because of what we're going to do with this so to move forward I would just like to just give you two examples where I'll delve into a little more detail and tell you a little bit more about the kind of novelties that that have arisen so the first oh before I do I should just say that every one of our medalists has worked on every single one of of these so for example some of song's beautiful early work was understanding the Fermi surface in the interior of neutron stars where it's the quarks forming the Fermi surface rather than the electrons and in fact understanding how the Fermi surface turns into a superconductor in in neutron stars and we'll hear later today son talk about another Fermi surface this time in the half field Landau level in condensed matter experiments so 10 orders of magnitude between the energy but exactly the same kind of physics that's that's emerging that should give you some idea of the power behind behind these concepts all right let me give you an example of of nothingness so this is an example of the first kind of system that I mentioned it's one of my favorite things in physics it's the quantum Hall effect it's it's just a gem I mean it's it's beautiful experiments but where all the highlights of mathematical physics from the past 30 years somehow appear in the laboratory by which I mean everything from Witton's fields meddled winning work on not invariance in the Jones polynomial in churn assignments through to the effect of disorder and dirt on the way particles move through systems and it's it's all there and it's all there in in this this picture so that the system is unbelievably simple you take a bunch of electrons you restrict them to move on the plane and you put a magnetic field that way and miraculous things happen so I'll explain some of these miraculous things this picture shows two things it shows the whole resistivity that's this line and it shows the longitudinal resistivity that that's this line so the whole resistivity is you put an electric field that way and you measure the resistance for a current to move in that direction and the longitudinal resistivity is the usual thing where the current goes in the same direction as as the electric field it's a two line classical calculation to convince yourself that as you crank up the magnetic field the whole resistivity should grow linearly so it should grow as a straight line which it sort of does except there are these jumps and starts as you as you go along so there are these various plateaus and we should try to understand these various plateaus so there's a bunch of questions one could ask you could ask why are the plateaus there in the first place if I start from a microscopic understanding of electrons moving in this system when do I expect to see plateaus how big will the plateaus be that turns out to be a difficult question but one which in which we've made some progress another question which turns out to be more fruitful is you just put yourself on one of the plateaus and you ask what's the right description of the physics on the plateau now each of these plateaus is individually boring meaning there are no massless excitations if you look below a particular energy gap nothing can happen at all but it's very clear that they're all boring in different ways in particular the the whole conductivity roughly one over the resistivity is sort of true if you invert a two by two matrix the whole conductivity is a bunch of fundamental constants times a number new and this number new is either an integer one two three four or is very specific fractions like one third and two fifths and three sevenths and what's happening in each of these plateaus is very different depending on the fraction so something particularly miraculous that happens if you sit in the one third plateau the electrons in the sample which I remind you are fundamentally indivisible particles divide and the objects that are roaming around the sample and conducting electricity carry one third of the charge of an electron but they're not quarks by the way if anybody's wondering in other plateaus they carry different fractions different fractional charge of of an electron so one question you could ask is how do you characterize these different plateaus what's the right way to think about these particular boring states of nature which very clearly are far from boring and there's work of Landau from the 1950s that tells you how to classify different states Landau says you should think about the symmetries of the states and you should understand which symmetries are manifested and which are not in a given ground state it turns out that doesn't work at all for these systems each of these plateaus has exactly the same symmetries and Landau's classification just fails to work by the way I maybe I'll make a comment if you read popular science and you read new scientist for example every single week without fail new scientist has a cover which says is quantum mechanics wrong or was Einstein wrong just every week for like 40 years now and you know of course the answer is always no but but if they just went down the pantheon of great physicists just sort of three steps they could ask was Landau wrong or was Wilson wrong and the answer would be yes and somehow that strikes me that it'll be a much more interesting read than the nonsense that they publish constantly so this is an example where Landau was wrong and Landau's classification doesn't work and we need a different classification and Xiaogang understood in the early 90s what this classification should be and it's rather surprising so Xiaogang pointed out that these different phases of matter can be distinguished at very low energies but to do that you need to do something rather strange you need to take the system and put it on a manifold with some strange geometry now this is hard to do experimentally not least because if you want to put a magnetic field through this you have to find a magnetic monopole somewhere in the universe before you do it nonetheless it turns out that these systems with no degrees of freedom react differently to different manifolds and in particular the number of ground states that they have is is different and this is the right way to to classify them by the way another side remark I actually took these from my lecture notes on string theory these are pictures of world sheets of the string but I had dinner with Xiaogang last night he told me that the reason he thought about this was because he did a PhD in string theory and that was what led him to think about what from a condensed matter perspective is completely preposterous idea but it turns out this is the idea that that allowed us to make progress this is the idea that allowed us to realize among many other things why there are fractional charges in these systems and what's happening on the edges of these systems and how the edges of these systems capture what's what's happening in the in the middle in a some baby version of holography and moving forward this is what's going to allow us to understand how to build quantum computers from non-abelian quantum Hall states that that are protected in in various ways so this was really a major leap forward to think about this way of of classifying states all right I want to give you one last example which is a rather strange example of gapless matter but it's one that both Son and and Subir have made major contributions to and it dates back to the 1997 proposal of Juan Maldesina of what's sometimes called the ADSCFT correspondence or gauge gravity duality or more simply just just holography I suspect that many of you have heard an introduction to this many times if you haven't I'm just going to be very very brief and then I'm going to tell you a bit of a story Juan's deep idea was the following that in certain circumstances quantum gravity is entirely equivalent to more down-to-earth systems of quantum field theory and the game is the following you take gravity and you put it in a box the box is known technically as anti-desitter space so inside this box there are gravitational things going on there are there are stars forming and there are black holes forming and black holes hawking evaporating away and there are there are gravitational waves propagating there's all the stuff that you learn about when you do gravitational physics and on the edge of this box on the boundary is a quantum field theory and on the edge are the kind of things you learn about when you do particle physics there's quantum particles scattering and doing quantum e superposition e things and the miracle of Maldesina's conjecture is that what's happening on the edge is exactly the same as what's happening in the middle that the the edge of the box is a holographic projection of what's happening in the middle and there's a one-to-one map between what's happening in the middle and what's happening on the edge so Juan suggested this in 1997 I was I was halfway through my PhD in 1997 and I do remember that the string theories communities surprise and shock and excitement at this idea I mean it's such a an astonishing idea that there could be a holographic image of something happening in a higher dimensional space that somehow gravity is the same as non-gravitational systems and for many years all string theories did they dropped everything they more or less worked only on this and the game was try to find evidence for it which means try to calculate very complicated things in a gravitational theory and very complicated things on the the boundary and then be amazed when when the two agree and they try to find more examples of this and this went on for a few years and then in 2001 I was a postdoc in Columbia University Son was a assistant professor in Columbia University we had lunch together almost every day we barely talked but I remember one lunch time you told me you were interested in holography said do I know anything about it well I didn't know too much Son told me he was trying to learn holography and trying to learn string theory and he was a new cliff physicist so I was a bit surprised I thought you know why would he care about this and he told me he was trying to learn it because he wanted to calculate viscosity and I thought who the hell calculates viscosity that must have been done in 1890 something why why would you possibly care about that you know what the cool kids were doing the cool kids were calculating bps correlators in n equals 4 blah blah blah blah blah but Son wanted to calculate viscosity and of course you know rather like Xiao Gang coming to the quantum hall effect with ideas of string world sheets and it bringing something Son knew something which I didn't which was basic physics Son knew that computing viscosity in very strongly correlated quantum fluids was way beyond anything that we could hope to do using traditional methods and in fact these are exactly the questions that Subir was also asking from a different perspective by thinking about high temperature superconductors and trying to understand what could possibly happen when you have those strongly coupled critical points about which we we don't really understand in fact it didn't take long before Son and Subir joined forces they wrote a paper on computing conductivity of black holes I have to say for me this was the turning point taking black holes and deriving Ohm's law I just thought was really cool so that's really what got got me into it but there is a theme here and the theme is that be a two-trick pony more tricks if you can I guess but if if you have skills in one field don't just focus and try and shop yourself around and look in in neighboring fields and talk to your colleagues and see what what they're doing because you know there's really lots of developments that could be made by just having a slightly broader perspective and I think our three medallists here really personify that idea of of moving across fields so it just remains for me to say many congratulations to Jaugang, to Son and Subir it's a very well-deserved Dirac medal thank you very much David this is a wonderful presentation and motivation to the field and introduction to the three medalists work and I don't think it could have been done better thank you so okay so now let's go to the next part of the of the program and so I will call each of the medallists and I will give the award and then each of them give the talk so it's very simple but of course we have to post there so that the photograph can capture the Dirac medal so let me start something that I didn't say before but I think it's very important is that as David emphasized and so on the the three medallists are top scientists doing very important research in the multidisciplinary but also but they have also in common that they all come from different countries and countries that have been very close to the mission of ICTP from China, Vietnam and India and so for us they they represent role models to many many of our young scientists from different countries and especially for developing countries like our own that you can see that doesn't matter where you come from you have opportunities you can excel in physics and science in general and so we have now in front of us three great role models for all of our young scientists in developing countries so I will start with Subir I cannot say anything better than David so I will just give the award so essentially well something else well Subir has a great career we all know and so I can read but it's in the program you can also read it and since we don't have too much time probably I will just say more personal things well I have seen Subir's name evolving over the years and at some point I remember having a conversation with David a few years ago and we said we're talking about string theory, I said well Subir is one of us essentially, Subir is like a one string theory and I would say it's the other way around the string theories are following Subir so somehow you can see now his movies and papers are with the top string theories and so on so in that sense it's great that this unit of physics has been working but Subir has also been very helpful to ICDP because he has been in the scientific council for us and has been very important not only participating in the scientific council and discussions and so on but also in being in the panels to select our top scientists also so we are always very thankful to Subir so and so after saying that I think I will just ask for the medal and I will give the medal to Subir and allow him to give the stock so it's a pleasure to call Subir Sajdev and to give him the RAC medal 2018. Love your talk, why don't you prepare yourself. So the title of the Superstock is Strange Metal and Black Holes. Okay well first of all thank you very much Fernando it's an incredible honor for me to be here it's been coming to ICDP from the very beginning of my career in late 80s and learned a great deal about not only my field but many other fields and many wonderful interactions here and thank you also to David for that incredible presentation that wasn't really me you were talking about but anyway I'll take it thank you for a couple of it all right so I'm going to talk about a topic that David mentioned which is strange metals which are a phase of matter found in high temperature superconductors and I'll present a simple model of these strange metals which is part of the way towards understanding the experiments but quite unexpectedly it's also had some impact in understanding some subtle problems in the physics of black holes but before I begin I need to acknowledge the many students and postdocs who I've worked with over the years I'm sorry I don't have the time to to go through every one of their names but there's my webpage where you can learn more about them I'm most grateful to my students and various postdocs I've had over the years and much of what I'm going to say is come out of work in collaboration with people too numerous to name so all right so I'm going to follow a theme somewhat similar to David's remarkable presentation so I'll fit in very nicely with the way he talked about things so I want to talk about his the third category of quantum phases where there's lots of low energy excitations which as David told us are well described by Landau Fermi liquid theory in before the 90s but now we have strange metals which are not described by this method and amazingly then there's a connection between the strange metals and black holes so let me just begin by talking about ordinary metals just reminding you of some very basics so an ordinary metal is something like silver or gold or copper and has some very basic properties they're shiny they conduct heat and electricity very well and they're very hard and and all of these properties can be well understood from some very simple assumptions that these are all materials in which some of the electrons are delocalized that is they're free to move through their entire crystal and just the Fermi the fact that they're fermions and a few other factors combined with Fermi liquid theory enables us to understand these properties and actually really compute them very accurately so what is the what is the the basis for the success the basis for this success is something called a quasi particle so these electrons as they're moving through the crystal repel each other and the repulsion is not weak it's quite strong and so how is it that you're able to describe you know trillions of particles with strong repulsion in any consistent way well it turns out that for reasons we we thought we understood but don't really completely understand in many many cases each if you just imagine that each elementary particle really becomes a quasi particle where it has a cloud of other particles around it and then you treat this lump as a new particle which for all practical purposes is the same particle except it has a slightly different mass so that's a certain renormalization and then this quasi particle the bonus is that the quasi particles hardly feel each other they interact rather weakly and so then a free particle description becomes valid just after replacing real by quasi and that's really the success of essentially a lot of kinesmer of physics since the 19 until the 1990s and what are the some basic characteristics of quasi particles in particular what would be a formal definition of a quasi particle so here let's imagine the formal you have some many particle systems say of n sites and the total number of many body states is exponentially large in n and we want to have some description of these exponentially large number of states well it turns out if you have a system with quasi particles these exponentially large number of states can be built up one by one you just take a bunch of quasi particle excitations and you only need about n of them and then you start adding them and you can take two elementary excitations and create more excitations hence the word elementary and in fact the energy of the combined excitation will be just the sum of the energy of the quasi particles e alpha is the energy of each quasi particle there could be some interactions between them that's f alpha beta this is a two-body interaction and so on so the net result is that when you have done this you've taken a system which has many many in fact exponentially many low energy excitations and described it in terms of a polynomial number of parameters the e alpha and f alpha beta and so on so it's quite remarkable you can do this and in fact almost always you can do this and it's that assumption that's behind the success of the quasi particle theory but here of course my attention is going to focus on a many body system but this is not possible okay so let's take the Fermi liquid as a simplest example of a quasi particle system and ask some basic questions that an experimentalist might be interested in so we can ask suppose you have a metal and you and you know zap a laser on it and you ask how long does it take before the system equilibrates say at a slightly higher temperature well it turns out for a quasi particle system the answer to that already was there in the work of Boltzmann for a classical system and Boltzmann showed that for a classical gas that this collaboration time is roughly just the mean collision time between two particles so if you compute that for a quantum system with a Fermi surface you get this answer the collaboration time is the square of the interaction strength and also diverges as temperature goes to zero as one over temperature square so this is time diverges and in fact it's a very long time as temperature goes to zero and it's long in the sense that these quasi particles are essentially independent excitations so they live along move along independently for a while so they have to find each other and it takes them a while to find each other before the collaboration starts but you can ask more precisely long compared to what and it turns out the appropriate time and this came from a lot of examples that that we've studied over the years and others have also studied the time that we have to compare this equilibration time to is now being called the Planckian time and it's a very appropriate name because it's a time that just depends on Planck's constant h bar and the value of the temperature kb is the Boltzmann constant which we wouldn't need if you measure temperature units of energy so the the claim is that this which is of course true that this time for a Fermi liquid is much longer than the Planckian time as temperature goes to zero but the strongest statement is that this is true for any system that has quasi particles there are many other examples other than the ones with Fermi surfaces and in some sense there's a kind of a lower bound beyond which you can't have any faster than which you can't have equilibration and that's just the Planckian time okay so so that's my rough quick introduction to the old physics so let's see how this breaks down in in a system in strange metals so it's turned out remarkably just in the last to it's just the last year there've been many papers discovering something new about the strange metal there's a paper in the cuporates by the typhare group there's a paper on the panic tides another set of high-temperature super conductors by Jean-Pierre Paolione's group there's a paper on twisted magic angle graphine by Pablo Arilo Herrero and there's a paper in ultra-cold atom systems from the Princeton group and they all have seen something quite remarkable in that when you look at the resistance of these materials as a function of temperature it's a linear function of temperature so here's an example so let me and why is that strange and what's remarkable about these observations so in the old traditional quasi-particle theory of resistivity you would write the resistivity by this formula where n is the density of electrons e is the charge of an electron m star is the mass of the quasi-particle I told you about and the tau is the time of collisions for each quasi-particles so this is a formula goes back to the 19th century which is very much based on Boltzmann work and Ruder's work so what the experimentalists did well let's just take this formula and deduce from the temperature dependent on the resistivity the temperature dependent on the time tau now remember I told you for a Fermi liquid that time diverges is one over temperature squared and now the other quantities here are roughly temperature independent so you would conclude that resistivity should vanish as temperature squared in fact it doesn't it vanished linearly in temperature so that's a universal characteristic of all of these strange metals even more remarkably if you take the observed measurements which is what these papers recently did and compute the actual value of tau so you measure the resistivity you put in the numbers of these quantities now this is a bit ambiguous because what is m star since you don't have quasi-particles and in each case people determine m star by some kind of lower temperature limit where you can sensibly define it so the experimentalists you know have their own black magic art on how they compute this but what all of these four different groups have found that when you do this within a factor of two and even more closer this time is exactly the Planckian time and remarkably it's independent of the strength of the interactions whereas in a Fermi liquid it would be proportional to the square of the strength of the interaction and this is really the central mystery trying to understand this that's been the focus of a lot of work on strange metal which has become even sharper in recent work whereas become clear that this constant here is pretty close to one okay so I'm not going to solve that mystery but I'm going to give you some recent proposals in our group but I'll also survey some older work okay so the tool that we've been using is now called the SYK model and let me try to describe it so the purpose of these models is to go beyond the land of Fermi liquid theory and put in interaction between the electrons but more importantly not just the interactions but the ability for the electrons to entangle with each other so an entanglement is a very basic property of quantum mechanics where we've heard about the famous EPR paradox where measurement of one particle can change the state of an entangled particle far away but here we're interested in just understanding something about complex entanglement of really an infinite number of particles what what can such a system do especially in a case where it's a metal so here's a simple model which has helped us a lot in understanding this entanglement so we imagine there are n positions now these are not necessarily spatial positions these could be labels of some orbital spd orbital or something like that and in these n positions some fraction are occupied by electrons I'm going to ignore the spin of the electrons in my discussion and now these electrons want to move around and and the one assumption we made about these electrons moving around was we said let's put a little restriction on them the restriction is that no electron can move on its own so every time an electron moves it has to have a partner and or more than one partner but the simplest case has to have one partner and the two of them have to move together so for example I could pick these two electrons and they both are going to together move to this side and that side of course the electrons are indistinguishable so I could equally well say they move that way but that's a process that can happen quantum mechanically and let's say it happens and once it's happened in some sense these electrons have entangled with each other they've moved together and forever they're going to remember that fact okay so that's one possible process but there's many such processes in fact roughly there's n to the power of four processes because you can pick the initial position n squared ways and the final position in squared ways and every one of those is allowed so now you want to understand the state of matter where every one of these processes happens and interferes with each other so everyone this is not real time the pictures i'm showing you these are quantum mechanical tunneling events that are all happening together and we want to understand the many particle wave function which are the sum of all of these different states where the tunneling matrix elements between these states involve two particle motion so the most important property here really is that we forbid one particle motion that's is a very trivial idea that made all of this progress possible all right so so that let me see okay there's a few more examples we'll quickly move on and it turns out that this particular state turns out to be a toy model not only of a strange metal but of a certain type of black hole this kind of entanglement is characteristics of some very novel states of matter in across very different energy scales okay so if you want a few equations here is the representation of the Hamiltonian this is an amplitude for k and electrons from site k and l to go to site i and j and with a certain amplitude what i forgot to mention which i'm now mentioning that this amplitude you take to be a random number that's really the key step now if i give you some Hamiltonian of this type with some fixed set of u i j k l that's impossibly hard to solve because you have to diagonalize a you know two to the n by two to the n matrix say when n is even a hundred that's impossible in any computer today the remarkable thing is that if you take these numbers n squared numbers into the fourth numbers to be random numbers drawn from some random ensemble in the limit of large n you can solve this exactly so randomness turns out to be a powerful tool in understanding a complex system because you then have the ability to use something analogous to the central limit theorem where properties of each site end up looking exactly the same even though the couplings between them are not the same and that central limit theorem really is behind the solution of very complicated many body interacting systems like this but the Hamiltonian itself is as simple as you can imagine i've also added another second term here which is the onsite energy if you wish which is the same on every site we're going to relax that in a minute all right so let me just tell you something about the properties of this model this was in a different version solved already in 1993 when i didn't know when well the ad safety course wasn't around and and i certainly didn't know anything about black holes but it turns out the solution of these type of Feynman graphs if you care about these things and you can compute things like the fermion like the fermion grains function here i just introduced this to introduce this parameter e so this particular correlator tells you what happens when you add a particle at time zero and remove it from the same site at time tau later whereas here you remove a particle and then add it later so there's a particle hole asymmetry in in this process and and that's characterized by this number e the other thing you find from our solution is that this process has an amplitude that decays with the square root of time and that was really the first hint that something bizarre is going on because this time for a Fermi liquid would be one over time not one over square root of time okay now since then of course we can do many other things and among the things that have been computed is the spectral greens function so this is basically the amplitude for adding or removing a particle at some excitation energy omega so when this number onsite number is zero then this is a perfectly symmetric function but more importantly it's a function of not just frequency but frequency divided by temperature so this is one of the most interesting properties here which you will not find a Fermi liquid the the scale over which the the particle is damp where it just loses its coherence is not given by the interaction strength squared as it was given in a Fermi liquid it's just given by universal constant of nature Planck's constant and the temperature nothing else appears in the scale of which this spectral function is damp it's just a function of omega over t so that is then appearance then at least in the spectral function of this Planckian time and of course the big puzzle is why should the Planckian time appear in the resistivity now you can also do this by changing the value of this onsite energy e if you make e positive then you find that the peak is at finite energy because now you know it's easier to remove a particle at positive energy because all of the particles have energy to begin with and if you make e negative you can get a peak at negative energy so now this is the role of the particle of symmetry and this number capital epsilon is a pure dimensionless number which tells you universally again on the scale omega over t over which the spectral function acquires an asymmetry and you can also look at other values okay i've been speaking for how long Fernando i think i may have to keep going no no i have much more to say i'm i'm afraid all right another remarkable property of this model well we figured out in collaboration with my friends in paris Antoine George and Olivia Parker lay is that this model has really a ridiculous number of low energy states so David mentioned uh you know the the third category was stated many low energy excitation where a Fermi liquid had a Fermi surface worth of low energy excitations this has so many that even violates the third law of thermodynamics uh and in fact when we discovered this we were very embarrassed by it we said something is really bizarre about this model and we put this now fundamental result in the appendix to our paper because it didn't want anybody to look at it but we got our wish because nobody looked at it for the longest time but anyway today we understand that this is actually quite a robust and generic properties of this model and it's really a key indicator of the absence of quasi particles because what this entropy implies is that there are this any the spacing between energy levels near the ground state is exponentially small and that's quite exceptional there aren't other systems which do that in any generic manner this does it robustly and generically and there's so many states at low energy that there's no way you can build them up out of n quasi particle states you know Fermi liquid you can take n quasi particle state and just by adding them and you know as a building block you can build all the other states there is no such decomposition here uh and understanding the structure of these low energy state and their correlations is really at the heart in some ways of understanding the strange metal problem in this context all right so now that's a summary of this syk model by itself is not a model of anything it's just a toy model that we can solve of these electrons that are sitting on this island and just hopping from side to side and one of the key features of the model is that this energy e is the same on every side so now we want to generalize this so one way of imagining generalizing this is adding another label alpha beta gamma delta and now allowing the energies of the electrons to have different energies on side e alpha so you diagonalize the one body hopping into a range of energy e alpha with some other random couplings u and what you find when you do this and there's many people who worked on this is that this omega over t scaling the key property the Planckian scaling of the syk model still holds but now only over a certain intermediate temperature range where w is the width of these energies and it's in this intermediate temperature range that you get this Planckian scaling but using this trick now you can actually define a model in which you can compute the resistivity so you can imagine a model like this where each island here is the syk model of some set of electrons we think of this as some big atom with many orbitals and the all the orbitals have some interaction between them and two electrons are on this island and we also allow the electrons to hop between islands in a translation invariant way so notice that the picture of every one of these circles is identical that's supposed to tell you that these couplings u are the same on every island so every atom is identical there's complete translation invariance so this is some kind of generalized Hubbard model too if you may wish to think of it like this and it turns out here's a remarkable lattice model that you can solve exactly and what you find precisely in this intermediate temperature regime i told you about that not only do you have omega over t scaling of the spectral function but the resistivity is linear in t so that's great that it's linear in t that's really hard to get that in any really any other model furthermore the value of the resistivity itself is quite large the unit of the quantum unit of resistance but in some way this is not a fully satisfactory model of the of the experiments it's got many things right but what it doesn't seem to have right one is that the fact that many experiments you can extend this resistivity behavior down to very low temperatures lower than this lower estimate you would make for any real material and furthermore the coefficient of the resistivity is not universal in the sense it's observed in the experiment it does seem to depend on the parameters of the interaction there's a u sitting here for example all right so that's been an open problem now for a year or two and something we're still thinking about i just want to very quickly mention some ideas with my current graduate student Avesh Karpatel on how to solve this problem so it turns out if you make a further assumption on this generalized syk model where you only allow this interaction to be resonant that is you have a bunch of sites or atoms with energy e alpha alpha is really a momentum space label alphas momentum and you only allow degenerate excitations that is where you can serve the energy of the bare quasi particle if you restrict yourself to this subspace and you can argue why that might be reasonable in some work not yet completed we find that the resistivity properties of this model look exactly like the experiment so that's something we're extremely excited about but hold on and there'll be a paper on that soon all right so then the last part of my talk is what is this got to do with black holes and why is this anything connected to black holes so let me just quickly remind you what is a black hole so in Einstein's theory which is a purely classical theory a black hole is a remarkable object where light itself can't leave and so but from the outside a black hole is very boring it's just a sphere a featureless sphere sitting there forever and it has nothing more to say about black hole other than its mass and the size of its horizon but starting in the mid-seventies through the work of Hawking and Beckenstein we learned that when you consider quantum black holes a black hole is an incredibly rich object with lots and lots of quantum states associated with it in particular Hawking showed that each black hole has an entropy and a temperature called the Hawking temperature and this is also in some sense a consequence of quantum entanglement it's a consequence of entanglement between degrees of freedom just inside and outside the horizon so there's always vacuum fluctuations right near the horizon and sometimes you can get a pair of particles created near the vacuum where one falls in and the other falls out even though they're ripped apart they're still entangled and that entanglement leads to an entropy and it's a large entropy that Hawking computed and he found that it was proportional to the surface area of the black hole not the volume like any other quantum system that we learned about before but from this view of the entropy being an entanglement entropy it's easy well it seems natural that it's proportional to the area if you think of it it's just entanglement across the horizon another properties of black holes is that they relax to equilibrium in a certain time that can be computed from Einstein's equations but when you express the times in terms of the Hawking temperature it's quite a remarkable time from a condensed matter point of view it's this Planckian time that I've been emphasizing so many times the Planckian time that appears also in the SYK model okay so all black holes are very independent of many details of their parameters seem to have this Planckian characteristic time in which they settle down to a spherical state okay so now from these remarkable observations of black holes David mentioned the ADS-CFT correspondence how and in some special cases we can understand these properties and we understand this by saying that the black hole theory of quantum gravity is equivalent to an ordinary quantum theory of stuff without gravity on the on a boundary of a certain space which David mentioned with the anti-decider space and so that allows us to understand many things for example the the entropy being proportional to surface area since the quantum system is sitting on the surface anyway that entropy is proportional to its volume so that's exactly right for that particular for this type of mapping and this second property that the relaxed equilibrium in the Planckian time we can now also understand if the system on the boundary is a system without quasi particles it can't be a Fermi liquid or conform well it can be certain types of strong intranet conformal field theory it has to be something that has the Planckian time as a natural time for relaxation and the SYK model is one okay so because of all these correspondences now you can ask is there a black hole that looks like the SYK model and it turns out there is and let me just quickly describe it and conclude my talk so it turns out in this case you just have to take the theory of our universe you take Maxwell's theory of electromagnetism and Einstein's general relativity you just make one tiny change you change the sign of the cosmological constant and then under these conditions you can find solutions of these equations of charged black holes but there's a net charge on this black hole and now we want to understand something about the quantum theory of this black hole but we don't have any complete understanding of such of such the quantum theory of generic black holes like this but if you look at low energies it turns out there we can understand quite a bit so let's zoom in to very low energy right near the horizon along this radial direction zeta and one of the features of Einstein's solutions of these equation solutions of Einstein's equations is that there's a decoupling between the angular modes labeled by x and the radial modes labeled by zeta and you can basically ignore the angular modes because they're higher energy so at low enough energies the quantum description of this black hole becomes a theory in just one spatial direction which is the with this direction here so you have a theory of gravity at very low energies which you want to quantize which has one space and one time dimension well holography that should be equivalent to a quantum system in zero space and one time dimension and what is that system what we know today that system is in fact one example of such a system is the syk model so here's a more careful statement of it for these people to know a little bit about gravity as if you solve Einstein's equations under the conditions I've just mentioned you get a black hole where the metric of the black hole in the radial direction is this ant ads2 factor you ignore the transverse direction x and there's a gauge field or electric field coming out from the surface of black hole which is related to this parameter e that I introduced for the syk model but now it also characterizes the black hole and now if you try to quantize this theory of quantum fluctuations along the radial direction the low energy theory you obtained for that direction turns out to be exactly the same as you get from the syk model now one of the first evidence that this is the case came from examining the entropy of the two systems we have Hawking's rules on how to compute the entropy of this black hole and then we have the computation of the entropy of the of the syk model that was originally done with Antoine George and Olivia Parker-Lay. If you look at the details of these entropy and how they depend on the electric chart that turns out to be exactly the same so you can derive the same sort of equations in particular this key equation here starting from the quantum mechanics of electrons just hopping by in pairs or starting from Einstein's equation in general activity so that's quite remarkable that you can map these two very different systems to each other amazingly in more recent work it's been learned that this mapping just doesn't just hold for the lowest zero temperature limit of the entropy but even the next leading correction in that case there's a common theory that's been derived called the Schwarzian theory which in the in the syk model you can think of as some effective theory for the low energy for these many low energy states and for the black hole you can think of it as some fluctuation of a boundary between the near horizon region and the far horizon region okay let me just quickly state that result which is due to the work of many people and there's a mixture a remarkable mixture of kinetics matter theorists and string theorists in this list and the main result is that two very different system of the same low energy theory one which i've hopefully described almost completely for you because it's so simple it's just models of fermions with the random fair hopping of strength u at low temperatures and the other is just a black hole solution with gen with non-zero chart in a system of Einstein's equations with a negative cosmological constant then both these systems at low enough energies have a common quantum theory and if you want to see the equations here it is here's the path integral which in fact can be derived in two different ways you can start from the model i described of electrons hopping in pairs or you can start from Einstein's equations and quantize them in a semi-classical manner both lead to essentially this description okay so let me just alright i should mention that this this connection is very similar of course and very much inspired by the ADS 50 correspondence but it's certainly different it doesn't work for chart it only works for charged black holes whereas the usual ADS 50 correspondence is typically for neutral black holes and but an advantage of this description although it's much more limited is that both sides of duality are fully solvable that we can quantum mechanically study two sides completely and this is allowed numerous recent studies by malda sena and others of the subtleties of black holes of quantum information in wormhole geometries all right so then we conclude so what i've really talked about my a topic that i've been thinking about for more than two decades is the system of quantum matters that don't have quasi-particle excitations and their characteristic property is this plankian time as the fundamental time for their dynamics uh and there are certain toy models from this syk model that are getting fairly close we believe to a description of the experimental observations another system astrophysical system which has the same property the same fundamental plankian time are black holes and certain black holes charged black holes and the syk model have the same low energy theory which is this short chain theory which i just flashed before you as a result of many people's recent work and let me finally also thank my family and my wife usher for all their support over the years thank you thank you very much super wonderful talk uh maybe we can take a couple of questions if anybody has a question for subir okay it's okay so we can just have more time to clap for her and more for congratulations for the good part shall i have a question yes sure please please yeah so the in the black hole side yeah so right so the single particle hopping uh i think one way to treat that is to take two syk models and have a particle hopping between them and that's in fact been studied by Mulder sailor and Shaolan chi recently and it ends up being a wormhole between two black holes and it's that wormhole geometry which then describes the tunneling within two syk model in fact it's precisely that's configuration that Mulder sailor and Shaolan chi have been using to address subtle properties of quantum information near a black hole in particular you can throw in a qubit on one side of the black hole and then recover it on the other side to the wormhole without uh faithfully uh and and that's something that you can describe both using s5k models or equations of quantum gravity yeah within the single syk model yeah what is the single particle hopping does that mean so this charged black hole is somehow unstable they will turn to something else i don't think so i don't think so i mean uh yeah uh i the i the single particle once you turn on single particle hopping that ends up being uh a system with very strong quantum gravity fluctuations so in other words how do you describe a Fermi liquid using black holes not very easily you have to have very strong geometry geometric fluctuations very good very good okay so let's thank again super okay so just to continue with the program so it's a great pleasure to to to mention Dan and son from the University of Chicago he is also besides being a wonderful physicist as we have already again he's very much committed to to support science in developing countries and to the mission for ctp in particular um i understand that you and have your own blog in vietnamese to support scientists from vietnam to follow the latest discoveries and learn about the syk models and so on which is very good and uh so but but also uh i have seen him very often to come to ctp and participate in our activities which is good recently he had agreed to be in the scientific council of our new center in in china in so that's uh we are also very thankful and uh so uh so it's again a great pleasure for me and an honor to call down son and then give him the direct medal some will give us the title of his talk is from fractional quantum hall effect to filter ethical dualities okay so it is my great uh honor to be here and receive uh the direct medal together with my colleagues subir and shogang today i'm going to tell you about something that i've been working on for the last few years uh trying to understand the fractional quantum whole effect um phenomenon in condensed matter physics that uh david has been telling us about and how this can be related to dualities in quantum field theory something that uh typically high energy physicists are interested in so i will make some general remarks uh and then talk about fractional quantum whole effect uh the notion of the composite fermion and then how this might be related to dualities in quantum field theory one often encounters statements that condensed matter physics is dirty uh complicated sometimes ugly clearly if you look at the picture of the metals in subir's talk you can feel that just pieces of random materials looks quite complicated to describe that would be true if one looks superficially at the system at the microscopic level when you have to worry about interaction between many many particles and this interaction may be quite complicated there are coulombs interaction there are spin orbit interaction etc but emerging phenomena uh there are phenomena that emerges from this uh this sludge this uh dirty mess of particles that show amazing beauty and the quantum whole effect is one of such phenomena at the microscopic level the quantum whole effect arises when we put a system of many electrons many particles charged particles moving in two dimensions restricted uh to be to move your on a plane in a magnetic field that is perpendicular to that plane all that we have to solve is the following Hamiltonian this h is equal to the sum of the kinetic energies of all the particles where the index a run from one to n where n is the number of electrons uh plus the pairwise interaction between the electrons which in this case choose to write it as the sum of the coulombs terms and in the real system sometimes we need to have impurities to uh to to have the desired physical effect so as a high energy physicist one might think that um it's a boring physics because everything is known the Hamiltonian is completely known we don't have to discover new particles or discover new interactions everything is written already written here we have the theory of everything however as dirac has said in 1929 uh at that time he was the quantum mechanics already uh discovered he wrote in one of his paper that the underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known and the difficulty is only that the exact application of these laws leads to equations much too complicated to be uh soluble the problem here is that it's not easy to solve the say the time dependent Schrodinger equation when the wave function is a function of n or three n's of the coordinates where n is some large number say n's larger than a thousand uh say if n is 10 to the 10th it's hopeless to put that problem on the computer because to start the information about the wave function we need more bit the number the number of particles in the universe luckily in many cases uh many aspects uh a lot of time most interesting aspect of this many body system becomes simple in the limit when that number n is large in the case of the fractional quantum hole effect we have seen that picture from david's talk it's a measurement of uh the longitudinal and the whole resistivity in a quantum hole uh system we see that despite the fact that the system is very dirty it shows remarkable uh um features where for example the whole conductive the whole resistivities all conductivities are quantized in the unit of the combination of fundamental constant of nature let's me explain to you a little bit more about the underlying physics of the quantum hole effect and let me start with the simpler version of the quantum hole effect called the integer quantum hole effect so in the integer quantum hole effect one can start by ignoring the interaction between the electrons which makes the problem much much more simpler because we reduce the n particle problem in the combination of one particle problems for one particle the energy level uh are quantized to landau levels and one when some of the landau levels are fully occupied uh say in this picture these are the energy levels of a particle in a magnetic field if we put a chemical potential between two landau level we would have a situation where an integer number of landau levels are filled and that should now to be the key to the explanation of the integer quantum hole effect returning to that picture before these plateaus with number here two three four correspond to a state in which two three or four landau levels are filled the fractional quantum hole effect is much more difficult to explain that's because the feeling factor that is the number of electrons divided by the number of states on one landau level is not an integer number in particular when that feeling factor is less than one then we have less electrons than the number of states available on the lowest landau level and there is without interaction there we would have a very large ground state degeneracy and we don't know theoretically we don't have any idea a priority what are the properties of this ground state and how it depends on this number new uh we have a lot of way to put this to this distribute this particle around the orbital and they would have the same energy so interactions are essential to determine the property of the ground state and what have been learned from experiments is that for some value of new for example when new is equal one third the state ground state is gapped we have a system that uh david described as boring uh and when new is equal one half for example we have another system where the we we know that the system is ungapped and the degrees of freedom seems to be belong to the same class as that of a Fermi liquid the one third is one very uh and uh very nice plateau in this picture and one half correspond to something happening here near new equal one half although in the previous picture there seems to be nothing happening but a more careful determination showed that there are quasi particles that appears to move on a on a straight line that is if you tune the camp the in experiment if you tune the magnetic field to the values when the feeling factor is one half we see the uh resonances that correspond to a quasi particle moving in a circular orbit where the radius of the circle goes to infinity that is very strange because here the magnetic field is non-zero electrons orbit are all very small but some quasi particle have orbits that is very large and the radius diverges when we tune the new to be one half they move in straight line so there are very strange quasi particles at this value of the magnetic field what is the nature of this quasi particle and it has uh been understood uh that this is a type of quasi particle that is completely different from all the quasi particles that has appeared previously in condensed matter physics now it is called for historical reason it's called composite fermions and let me describe to you a traditional picture how this composite fermion arises the standard picture is that the composite fermion is a combination of an electrons and two flux quanta so in this picture if we have us say some number of electrons in a magnetic field i choose to uh to illus to to draw the magnetic field as a combination of lines these arrows each of the arrows correspond to one magnetic flux quantum for new equal one third there will be three arrows per electrons and one imagine that the electron somehow absorbed two of the flux quanta and turn itself into some objects that we now call the composite fermions in that case there would be only one arrows per composite fermion which correspond to an integer quantum whole effect of the composite fermion in a new reduced magnetic field with the feeling factor equal to one if we start with new equal one half where there are only two arrows per electrons then doing this so-called flux attachment procedure that was first talked about in the context of anion superconductivity but then applied to quantum whole effect by these authors around 1990s we would see that the composite fermion now live in zero magnetic field so that object that lives in zero magnetic field can move in a straight line and that would be the quasi particle that we observe in half feelings so there is no leftover magnetic field so that idea of flux attachment and composite fermion has been further formalized into a quantum field theoretical framework by Halperin Lee and Reed who extracted a lot of phenomenological consequences of this theory and this there is this theory is very successful the composite fermion in this picture would be an electron that is dressed by two flux quanta similar to the picture of a horse in in in in a suber stocks it's a horse the electron dressed by two flux quanta there is a problem with this picture and that is a problem of particle hole symmetry the standard picture does not know about a symmetry called the particle hole symmetry which roughly speaking change a particle into a hole that symmetry acts on the on a single Landau level so if you have a single Landau level one can introduce a basis occupation number basis in which each of the orbitals are either filled or empty and one can define a particle hole conjugation operator that flip the occupation number from one to zero and zero to one so that operation moved new into one minus new particle into holes new to one minus new and one feature of that it makes new equal one half half field and our level to map to itself in other words a half field and our level is also half empty can think about that as a half filled by electrons or half filled by whole state and it has been known that this is a symmetry of the problem when we can concentrate on a single Landau level however this flux attachment procedure breaks particle hole symmetry because we have chosen to attach flux to particles so but particle must be equivalent to hole under particle hole symmetries so it this quasi particle somehow somehow the notion of flux attachment itself breaks particle hole symmetry so the lack the particle hole symmetry or more precisely the lack of particle hole symmetry in the HLR the Halperin-Leon Ritt quantum field theory of the half field and our level was a big puzzle in condensed matter physics and the puzzle is seen as a tension between theory I would like to see that as a tension between theory and experiments experiments showed that the quasi particles what we call composite fermion exist and it would move in straight line a half filling but the but the standard picture somehow where the composite fermion is a type of dress electrons cannot be correct because it cannot be completely correct because it can we can also dress the whole and there is only one type of quasi particle since in experiments how can a dress particle being the same as a dress hole how can a lack of a particle be when you put the two magnetic flux quanta to it being the same as the particle with the two magnetic flux quanta around 2014 I was aware of that problem already mostly mostly by some boring job of sitting in search committees for searching for candidates for faculty jobs so I have to read a lot of paper and some of these paper talk about particle whole symmetry and quantum whole effect and then I made a failed attempt half hearted attempt to construct a holographic or gauge gravity duality model of the fractional quantum whole effect but I expected that integer quantum whole effect would be easier to model through gauge gravity duality but that would be somewhat boring the fractional quantum whole effect is much harder and then see through a chance conversation I with the string theories I realized that the electromagnetic duality in the bulk if such a gauge gravity duality would exist then the dual theory must have a gauge field that has an electromagnetic duality in the bulk which I learned from the importance of which I learned from working with Pavel Kapton, Chris Herzog and Subir and that electromagnetic duality can be used to convert integer quantum whole effect into a fractional quantum whole effect and after that the importance of particle vortex duality becomes rather obvious so I drop my attempt and try to think about particle vortex duality. Particle vortex duality is an example where high energy physics and condensed matter physics merge. It was discovered by Peskin thinking about lattice gauge theories and that's Gupta-Harperin in sometime later thinking about condensed matter problem of insulator to superconductor phase transition and in modern language the two sides of the dualities that we believe to be equivalent to each other is the theory of a complex scalar field in two plus one dimension and a billion six model in two plus one dimension and the duality acts in such a way that it maps theory one to theory two in a very non-civil fashion in which the phase in which theory one has a goldstone ball zone corresponds to a phase in theory two where we have a massless photon the phase when symmetry is unbroken in theory one and we have particle and anti-particle correspond to a superconductor superconducting phase in theory two when you have vortices and anti-vortices the duality maps charge density in one theory to a magnetic field in another theory and vice versa the magnetic field that coupled to that field to a charge density in another theory it flips the notion of particle in the magnetic field so then I made a conjecture and Max Mitliski, fish one art and one in center conjectured that underlying the transition from electrons to the composite fermions in the quantum hole effect is a duality a fermionic version of the particle vortex duality in which one side of the theory one side of the duality is the theory of non-interacting theory of Dirac fermions and the theory on the other side is a fermion coupled to a gauge field so this theory on this side side e with the index e indicates electrons what is magnetic field on this side would be a density on the sec on the other side and the density on of the electron would correspond to a magnetic field on the other side so let me demonstrate that for you in a simple calculations if we look at these two actions that supposed to describe the same physics the physical electromagnetic field is coupled minimally to the electrons but coupled to the side field on the other side through a churned simons coupling to a dynamical gauge field small a and if we compute the electromagnetic chart for example in the two theory we see that in one side it is the psi bar gamma not psi of psi dagger psi but on the other side it will be the magnetic field so if the duality is correct then the density of the electrons much must becomes the magnetic fields on the other side and one of the equation of motion also establish the reverse relationship that is the density of this psi which i would call the composite fermion is equal to the magnetic field acting on the electron if that is correct then we have a very interesting situation emerging when we try to put theory one in a magnetic field theory one is a theory of Dirac fermions in a magnetic fields the spectrum also organized into landau levels and one of the landau levels has zero energy and at zero chemical potential we have a large degeneracy of the ground state as in the fractional quantum hole effect we don't know which state here should be considered as a particle and which here considers whole. Dirac told us that we have to feel the Dirac C which consists of the state with negative energy but the state with zero energy is something that is ambiguous in this case so we have exactly the same situation as the situation of the half field landau level we can consider psi e as an electron if duality is correct then the magnetic field correspond to density so this theory too it will be a theory of some fermion at finite density and here we have zero charge density because the chemical potential is zero so this composite fermion would be in zero magnetic field so here we have a Fermi liquid of the composite fermion the duality would explain to us what is the origin of these particles that move in zero magnetic field with you know in a straight line so thus the quasi particle of the half field landau level is some type of a Dirac fermions rather than the non relativistic fermion as in the old HLR theory. This is not the first time Dirac fermion made appearance in condensed metaphysics the most famous previous example was graphene when the low energy spectrum of electrons in the hexagonal lattice can be shown to have Dirac cones where the Hamiltonian is effectively Dirac Hamiltonians of a two components fermions so the fermionic particle vortex duality provides us with an explanation of for the composite fermion near half filling but with an additional bonus that that theory actually has an explicit particle hole symmetry which is related to discrete symmetry of the Dirac fermions. It reproduces good features of the HLR theory but it differs in detail from the HLR theory and many one can compute various physical quantities and show that the two side of the theories do not agree with each other. In particular it predicts that the composite fermion has a very face of pie around the Fermi surface in fact this is the only signal of the diracness of the composite fermion that we can expect because the as Landau told us the notion of the quasi particles only makes sense near the Fermi surface. We cannot go deep in the side the Fermi surface to see the Dirac cones. We have to infer the diracness of the composite fermion by looking at the Berry phase and that Berry phase has been confirmed by numerical simulations of the half field Landau level and it's also this theory the Dirac composite fermion theory also suggests a new quantum phase called the ph Fabian phase. So to understand that phase let's imagine that we have composite fermions forming a Fermi surface but we know from condensed matter physics that sometimes this Landau quasi particle like to pair up with each other in the superconductor if the interaction between the quasi particle are attractive then they would pair form cooper pairs. In the case when the fermion is the Dirac fermions the simplest channels of pairing would be a channel that roughly has orbital angular momentum zero and it can be zero angular momentum because we can make use of the Dirac indices when we construct the wave function of the BCS pair or in other words the Berry phase of pi make it possible for this composite fermion to form a cooper pair with angular momentum zero and still obey the Fermi Dirac statistics. That would be a gap stage if the fermion pair up and that gap state has been seen at a feeling factor two plus a half where a two Landau levels is filled and the next Landau levels is half filled and that has been seen a long time ago and there have been many candidates proposed for that state. The most famous one is the Fabian state that is proposed by More and Reed in 1991. There is also the anti-Fabian state that is the particle hole conjugate of the Fabian state. These states have very similar properties and it's very difficult to distinguish them from each other but one way one can distinguish them is by looking at the thermal hole coefficient the heat conductance in the direction perpendicular to the gradient of the temperature. Recently there was an experiment that has tried to that has measured the coefficients of the thermal hole coefficients in certain units that number seems to match the value of 2.5 two and a half that the theory tells us that it should be the coefficients of the thermal hole conductivity for the pH Fabian state. So the situation is is still under debate. We don't know exactly whether that experiment actually indicates that the ground state of the half of the new world five half state is a pH Fabian state or as some proposal indicates is some kind of a mixture of Fabian and anti-Fabian state in or arranged in certain random fashion especially but the experiment seems to be extremely interesting. So now that these two dualities that I've mentioned before the bosonic and the fermionic particle vortex duality the first being proposed by Peskin, Harperin and Dasgupta in the around 1980s and the fermionic particle vortex dualities proposed recently to explain the fractional quantum hole effect can all be derived from a certain seed duality in which one postulates that a fermion is equivalent to a boson coupled to a transiment sketch field and it turns out that from if this duality is correct then the whole web of new duality can be derived that encompass not only these two dualities but also many many different other dualities some of which are very interesting for example one can show that the theory of two flavor if you have two flavor of fermions coupled to one you one gauge field that theory is self-dual and that is the work by David and Andreas Kahr as well as cyborg central one in Witten. So let me conclude my talk the fractional quantum hole system provides a very interesting case where we have an experimentally realizable example of something that seems to be a deep duality in quantum field theory. Within condensed matter physics we have seen a mysterious appearance of a new type of quasi particle that doesn't seem to fit the standard picture that a quasi particle is a dress particle it's both a dress particle and a dress hole at the same time and we currently lack the intuitive way to understand the nature of that quasi particle you can say that it's some phenomenon of emergence when a new type of quasi particle with completely different quantum numbers as the original particle arises from the interaction between between the the the the original fermions and I hope to demonstrate to you what has been emphasized in by by Fernando and David and Subir that the interaction between high energy physics and condensed matter physics can be extremely fruitful and one should be one should expect that this interaction would give us many interesting results further in the future. So let me give my thanks to my collaborators and colleagues from whom I learned a lot of things new physics to my professor and teacher who give me sorry background in education to my late father mother and sister who have been always encouraging to me and my wife Ikan and son my son Chokson who has been encouraging and patient with me thank you very much thank you very much son it's a wonderful talk also maybe we want taking a couple of questions in case yeah questions from anybody any question this is that a simple way sorry is there a simple way of distinguishing this ph fafian from other fafians which is not based on the thermal or conductance or this is really the only measurements that would be definitely distinguishing between the different scenarios so one thing that distinguished them if I if I can so let me give you some two two two step answer so if I if I can measure the response of the system to gravitational waves then they would have different transport coefficient called the whole the the the whole viscosity so if I have a like a neutron star source and then shine a gravitational wave to such a system I can distinguish some that one so under certain circumstances I believe then that has not been published so I should probably should not I should be careful but I believe under certain circumstances one can do Raman scattering to to distinguish them and we can talk in private about that so if you go to a higher dimension is it possible to construct this particle vortex line or rings or whatever transformations in higher dimensions higher dimension yes we are lucky in three dimensions because a vortex would be a particle yeah so I don't know if there are like in higher dimension with like t-form field what is there any similar duality I I don't know for the graphene for the quantum how the effect in graphene at the philic fraction one-half so this particle symmetry will be more rigorous in the ordinary quantum state when you have a Landau level mixing since that could break it so the question is that for this pie phase shift around the Fermi surface you know how that will be affected by the Landau level mixing whether there's estimate it as a basically my question is that with Landau level mixing whether there's a particle symmetry how rigorous it is in the you know in a more standard semi-connected quantum how states yeah as far as I know so let me answer this so if we have a small particle whole symmetry breaking by Landau level mixing then one expects that the berry phase would be not exactly pie would be different from pie how different is it from pie and how it depends on the ratio of the Coulomb energy scale and the land the the the Landau level energy scale we actually don't know we even don't know what is the sign what if we have this mixing is the berry phase pi plus epsilon or pi minus epsilon that even we don't know so there are a lot of things that we we need to understand yeah thank you for that point I think well please join me to thank Eson again and congratulations for your matter well so too in the the the program it's a pleasure to to to mention Chargan Nguyen so again he's he's has a wonderful career in physics as we have here also from David the others he has been also very keen to participate in activities of ICTP and supporting science and in developing countries in particular in in China and so on and we were talking before that it's there is a nice relationship with the previous awardees of a supervisor and student so his supervisor was the the first direct medallist waiting but also we realized that the supervisor the supervisor was also direct medallist so she has a great degree also with this and you have a challenge for your students now to see if they can achieve it but so well it's again a pleasure and an honor for me to to to call Jiao Gan Nguyen and give the direct medal presentation you can put it here so Jiao Gan is going to give us the the talk will be titled topological order and known a billion statistics okay yeah it's really a great honor to receive this prize so I like to really thank ICTP and it's also you know it's a in the direct matter I know this is a long time and when I look at it's a list and they are all my heroes so it's really it's great honor for me to join this list and today I'm going to discuss topological order and the now billion statistics so we know that we have a very systematic understanding of all kinds of matter states based on the Langdaw's century breaking theory you know as I mentioned by by David so the quantum hall states is a such a wonderful system and and one of its great contributions is that is a counter example for Langdaw's theory so I should still remember when we when I first learned quantum hall states we try to understand which symmetry was broken because at that time we feel a Langdaw's century breaking theory understand every phase of matter and all the phase transition whenever we see the phase of matter we first ask what is order of parameter what is signature is broken you know and then once we know the order of parameter know the broken symmetry we can write down get some law theory then we kind of understand phase transition and and many other politics but however quantum hall states is an exception because we have many different quantum hall states at the different feeling fraction so the feeling fraction actually is an electron density divided by magnetic field but the magnetic field measured by the density of flux the flux quantum so this is some some simple Russian number so it turns out that those plateaus that happen at the feeling fraction which is the integer of some of the simple Russian numbers okay but but however we examine the quantum hall state the more carefully we see that they all have a same symmetry so there's no distinction in their symmetry and so so so that's really a concrete example that we cannot use the Langdaw's point of view to understand the different quantum hall states so you have to come up with some a new way to look at it and one way to do that is this using this so-called topology order which is defined by this Guanty-Diener say on the different topology of a space you know it turns out that the quantum hall state have this very special property when you put on a sphere you have a single Guant state when you put on a torus you may have a three Guant state when you put on genus two Riemann surface you have a nine Guant state and etc so their Guant state Diener say depend on a topology of a space something quite interesting actually magnetically very interesting but the the problem starts why this a magnetical interesting fact have anything to do with a phase of matter because at that time people think the phase of matter are described by so-called thermodynamic properties such as like order parameter symmetry and etc but the Guant state Diener say is defined only for the finite system it also depends on the boundary condition of the topology of a space so it looks like a finite size effect you know something about a finite size but since had nothing to do with the the order or the the the phase of matter which phase are in so this is a big problem and so so how to convince people that yeah this a seemingly finite size fact actually do characterize a phase of matter I think that's the first major obstacle try to convince people that this Guant state Diener say actually describe a phase of matter so it turns out that this this a Guant state Diener say is a very robust actually it's a you can add any perturbations and the Guant state do not change so it's a robust against any perturbation and this is really weird you know we know a lot of Guant state Diener say in quantum mechanics but usually the example we have are come from symmetry you know when the system have a certain symmetry if a Guant state form a so-called non-trivial representation of the symmetry they'll have a degeneracy so the degeneracy always come from symmetry but here there's nothing to do with symmetry so the so the system is a how degeneracy even without any symmetry so this is a wonderful wonder of quantum mechanics so how this is possible now we have a better understanding that is a we have this a many body system and we'll look at the part of a many body system they all look the same but if you look at the whole they are different so this is a key feature which led to the Guant state Diener say because when you add the local interactions they can only see the part of a system but those degenerate Guant states when you look at them as a part of it they're identical but however when you look at them as a whole wave function they are orthogonal they are different states so such a thing is possible in quantum mechanics as a very special feature for many body entanglement so local entanglement is identical but the global entanglement are different so this is the one one thing we realize that it's possible that's a kind of wonderful many body entanglement so that is encoded by this statement that Guant state Diener say are robust against any perturbation theory so so that's like a universal problem we know that to characterize a phase of matter we need some quantum number which do not depend on small perturbation so that's a whole phase of matter share the same feature so that's topology environments and the Guant state Diener say satisfy this another thing is that the Guant state Diener say can change only if you close energy gap you know close energy gap and we open that the Guant state Diener say can change from 1 to 2 so therefore only via phase transition we can change that so so that's really convince us that this Guant state Diener say actually is a new kind of quantum number some kind of kind of like other parameter which can be which can characterize different phase of matter but what is characterize as I David say is the boring phase of matter they all have a gap there's a nothing at the low energy but so but not quite nothing in a sense on the sphere there's nothing we are Diener say equal to 1 on a torus the Diener say not equal to 1 so you have something a little bit of something which really characterize this a boring phase and so actually this the name of a topology order actually also motivated by this the topology quantum field theory which is the first introduced by by Witten in 1989 so certainly the topology order this name really because Guant Diener depend topology space that's a little bit but however this is also coming from this motivation that's a is topology quantum field theory because for quantum Hall states the low energy effective field theory actually is nothing but topology of quantum field theory so it's a topology field theory actually is was introduced by a study of super stream you know actually I don't know why super stream theory started this boring state but it's really it's a it's a byproduct of the super stream theory and so now it have a very physical realization in quantum Hall state so that's why this quantum Hall state such a wonderful system and there's all different field are really highly connected so so so what kind of a system realizes the topology order here by what I mean by a real material in the lab or some toy model in in theory and certainly for central breaking theory you know a long time ago we know there's a magnet so that will be material which you realize a central breaking order actually there's a many many example of a topological order it's amazingly the first example actually superconductor which is discovered in 1904 the superconductor is a have a so-called a little topology order but the most textbook regard a superconductor as a example standard example of central breaking states so what's uh so so what's wrong because uh usually in the condensed metal textbook when we study superconductor we do not include a dynamical electromagnetic field so the textbook superconductor indeed is an example of a central breaking states but however the real superconductor in the lab have a dynamical electromagnetic field we will include the electro dynamical field then it's a real life superconductor it turns out to be not a textbook superconductor it's actually it's a it's a it's a topology order and amazingly that the Landau's central breaking theory developed based try to explain superconductor which is not which is counter example of central breaking things you know that's how the history you know kind of ironic and certainly this uh this is indeed a quantum half and a flash of quantum half states they are all examples of this topology order and during the 19 late 1980s this is high-tech superconductor was discovered so it's a wonderful time uh in condensed metal physics at least for me you know at that time I switch from super string theory to condensed matter which I don't know too much condensed matter but because of high-tech superconductor so there's a there's a period of time anything goes you know whatever theory you have you can publish it so so it's a really good time for me you know and so so we we we introduced we we kind of started this so-called carol spin states which turned out to be one theory for superconductor high-tech superconductor but papers they were published and it's kind of like it's like a deep light at the superconductor for spin lines okay so actually it's a we are this study of this special of spin liquid a carol spin liquid you know largely led to this notion of topology order really from study of a carol spin liquid but quickly this theory is falsified this is not the theory of anything but then later we realized that this topology order actually applied to quantum half states so it's still a theory of something and so uh so then we have this uh uh another example is that uh we can have this got none a billion quantum half states and then there's also a z2 spin liquid so in addition to carol spin liquid and there's a z2 spin liquid actually severe you know did a very important early work in this and so uh then there's an all kind of uh uh uh you know different theory uh in particular there is a p plus p superconductor uh suggested by read and the green which actually turns out to be kind of like a square root of an integer quantum half states so something uh like if you're stacking two p plus p superconductor the physics of that is stuck the system kind of like a single integer quantum half states and uh and this this turns out to be important because later you know the kith half find a hundred con lattice model they get this another type of carol spin states which turn out to be p p plus p superconductor for spin lines so it's like a square root of the this first carol spin states but it's a stacking two of this together you'll get this this t plus id uh carol spin states so so there's a list as kind of continuous but you'll notice that uh this black one are really for the experimental sample which is realizing top-large order and the blue one are basically a theoretical models so it's like there's a there's a big gap you know so we have a lot of a theoretical model have the theories have a lot of fun but uh for the experimentally the progress is extremely agonizingly slow and uh and later say it's all seven uh you know you and me uh they have uh they have a uh some kind of a lot is a sample called the herbis mucide and the sugera maybe it's a z2 spin liquid you know they're first studied uh by the readcetive and myself and also to get a torcal model uh but however the the sample have about 10% impurities so it's not you know 10 years later we're still not totally settled is that really a spin liquid or not and and uh as uh uh dansun have talked about this uh this uh this uh filling function uh five half there's a non-villian uh version of quantum power states uh we we have some experimental measurement there maybe that and more recently this p plus ip kind of spin states uh uh have this uh this uh this kind of sample can realize that uh so again there's a lot of experiments you know hopefully this uh we we see a more uh more examples especially experimental example uh which realize uh this uh highly entangled quantum system so basically top-large order basically something like a it's a many-body quantum system but have a lot of entanglement so that's what is new about it so today uh i'm going to concentrate on on this non-villian quantum power states and uh this uh maybe mention a little bit of this uh this experimental uh detection and so i'm glad you know david and uh dansun have been talked about this uh this quantum power states uh in their talk so uh so first we have this uh interior quantum power states uh the interior quantum power states are really described by this uh the the wave function like this for example if you feel if you feel the first on the level if you feel first on the level you say what is the many-body wave function well if you write write down it just like that okay and so it's like uh it's a first the quantum theory uh for uh for many-body system usually people don't do that they say when i will do uh many-body physics we should immediately go to second quantum theory like a electron equation annihilation operator these advanced techniques but uh but in this case if you're using these advanced techniques uh you'll miss some discovery because if you really write down this first context version uh then you can you can just add an exponent you know there's a few on the level just an exponent here is one but for the same wave function we add this exponent that's what the lawfully did may not get the wave function for fractional quantum power state and the Nobel Prize and so so so therefore this fractional quantum power state can be viewed as an explanation of the just field along the level so if you feel first along the level you'll get this raise to the nth power you've got the fractional quantum power states and so so that is our earlier theory to explain why we have a fractional quantum power states okay then what is the non-villain uh quantum power states so non-villain quantum power states have a non-villain statistics so basically uh like this uh this uh indeed the quantum power states they have an excitation but their excitation are fermions however when your race is power then the excitation probably change fundamentally their excitation become anion have a fractional statistics but theoretically people realize that the anion is not all the possibility there's even more strange exotic possibility that's a there's a particle which carries so-called non-villain statistics which is more strange so i will explain a little bit later in which part is more strange but but how to realize is a theoretical possibility of non-villain statistics it turns out to be quite simple so here we have a lawfully wave function we just raise the power of a one field along the level but when we say y one we can start with the feeling function two in get to a non-state and raise the power well that's what it did so it's like this chi two is a wave function with two longer level field then you raise to the power it turns out that's it that is a non-villain states oh you can raise you can have a chi two to feel a non-level rate to three power that's another non-villain states with different non-villain statistics so then there's another way to do it that's a that's a farthing wave function we can assume this electron pair so have wave function z one minus z two this electron pair have the electron paired wave function but you say oh this wave function is bad because when z one go to z two the diverge where we can add this this factor this is a chi one factor to to kill the divergence then this became a totally legal wave function it turns out that this pairing is another way to produce non-villain statistics yeah amazingly that's a this wave function there's a chi two square wave function and this pairing function they have a same type of a non-villain statistics but this chi two cube wave function they have a a different non-villain statistic it's called the Fibonacci non-villain statistics and and actually this this non-villain statistic is a it's a more useful for quantum computation because they can they can do this a universal quantum computation okay so that's a that's basically the the results in nineteen nineteen one and here try to explain those results you know so basically i try to say so why this kind of wave function have a non-villain statistics you know we try to explain why why we think they have a non-villain statistics and we use a certain theoretical tricks okay so basically we use two theoretical tricks and the the first trick is so called the trick of the effective theory the first trick is the effective theory so what is the effective theory well we first have they let's consider this using an example of the initial quantum statistics so we have an unfilled lambda level so that's the lambda level we just feel two of them so m into two that's a two field around the level and then this system are described as many body Hamiltonians so those are electron equation operator this electron annihilation operator then we have a many body Hamiltonian have this form and this a is a background of my nephew provide my nephew for those electron lambda levels and we can also add some probe field we can have a non-uniform my nephew or a little bit add some potential as a probe field to probe the system okay and what is the effective theory well the effective theory basically is that we do the passing the row using this Hamiltonian okay and when you do the passing the row we get a partition function which is dependent on this our probe field so we write a partition function this is an exponential form so this function here dependent on the probe field will be this effective Lagrangian but usually doing this passing the row is very difficult and so it's a we don't we really don't know how to do it but however in this case we can we can we can we can play some we do some approximations basically we know that this effective Lagrangian have two terms one term is energy term you know the minus total energy so have these are minus energy term but energy term is can be calculated easily as there's one way to calculate energy term basically it's as a less this chi delta a to be the grand state wave function of this many body Hamiltonian that we can compute their average energy so this average energy will be one term under this effective Lagrangian so that is a kind of a simple way to compute the effective Lagrangian without doing passing the row but however we only have this but the compute the grand state energy as a function of this probe probe field or perturbation that's only a part of the effective Lagrangian so what is the other part and the other part will be this we assume this this probe field dependent this have time dependence then the compute the grand state wave function they're overlap at different time so this is a kind of barrier phase term to compute this time dependent the grand state wave function and the because wave function involves there's a barrier phase contribution it turns out that this barrier phase contribution plus this average energy contribution gave us this effective Lagrangian so this is a very simple trick try to compute effective Lagrangian from the very from the simple by the simple method and this effective Lagrangian have this so-called transformant term which actually describes this quantized hog conductance you know so this coefficient of transformant term is the quantized hog conductance and the m appear here because the m is a homony field lambda level we have a more field lambda level this coefficient just increase linearly because the hog conductance also increase linearly with a feeling for action okay so that's the first trick we use to explain to to explain navvelian statistic that the first trick but this trick is not enough so we need a second trick so the second trick is called a projective construction basically we try to answer the question why this wave function have a navvelian is navvelian states so these are so in these uh tricks what you do is that we cut the electron into many many part we call the power times and then we glue the pattern back into electron so we do something since uh just we didn't do anything but uh but this became very important the trick again it developed in this period of time we saw everything goes anything goes in high-tech connection so this called a slave particle or pattern construction developed to compress spin liquidity and we can use the same uh tricks to construct as we understand navvelian quantum states okay so how we do it well so this is a wave function so we have an m field along the level raised to the nth power okay so that's a wave function we want to want to argue this wave function have a navvelian statistics and then we then we write this wave function as an n different product but the view of those variable as independent you know in this wave function all the variable we just have n variable but when we view this nth power as n piece and each piece have the own independent variable that's why we have patterns because each piece is a wave function for the pattern pattern is a fermion they have this integer quantum how anti-symmetric wave function then we have this projection this projection try to say tie all the patterns together so we have a first pattern a second pattern a third pattern then we tie those patterns together back to electron so that's a projection then after projection this a pattern wave function became our target wave function that's wave function we want so that is a second trick and so to to proceed and we we first write that assume pattern independent so we get all this projection assume pattern independent and then write down the Hamiltonian for independent patterns okay and then we say that then from this independent pattern we can obtain each pattern have this wave function and then we have a n different patterns then the total wave function for the all n different pattern is this one then using projection to construct the electron wave function we can using the electron operator electron operator actually is a product of a pattern operator to annihilate this pattern this group pattern one by one and to get back to the electron wave function so this is a more mathematical description of a projection to bring pattern together glue pattern back into electron so why we do that because we can using this method we can add this a probe field so we can add some kind of a this a this a sketch field but because we have patterns so we have n patterns so this sketch field contains term which makes different patterns so actually it's a n by n matrix so this sketch field is now a really engaged field we can add that as a probe field for pattern Hamiltonian but once we add that we obtain a new new wave function for the patterns because of the the Hamiltonian string the ground state this modified Hamiltonian gave us a new wave function and then but however we do the same projection using this new wave function using new wave function we do the same projection to get a new electron wave function okay so so at the end we get this the new electron wave function which depend on these parameters so what do we get what we get actually is that we get an electron wave function depend on certain parameters like we are doing variational approach we want to have some parameters we want to choose the parameters to minimize this ground state or energy for the electron but here our parameter is a little bit strange our parameter is not just one parameter of two parameters it's actually the field this whole now really engaged field is our variational variational parameters and so however because this because this is a product of the however this we have we can we can do this SUM gate transmission you know we can we can rotate the patterns like this it turns out that this the electron operator have this particular form this electron operator are invariant under SUM transformation it's SUM singlet so as a result that the the ground state energy okay the ground state energy are actually even more that this the the wave function itself this the electron wave function are invariant under SUM gate transformation so we can change this variational parameter by this a gate transformation but then we find the exact same manner about the wave function so this variational parameter and the wave function have many to one correspondence so there's a redundant labeling and so so with this we can using this variational parameter to develop effective theory for this variational parameter so now we view this variational parameter as some a field as a fluctuating field as some degree freedom then we can derive effective theory for this degree freedom using the same trick we can compute their average energy and we can compute their biorefics so now this something we can do okay so the result of this calculation is a is a is a key that's a we got this a a transformer term because this a probe field it's kind of like a this a probe magnet field and this a pattern it's like an integer quantum hall states because they have a hall conductance well that's naturally led to a transformer term but however because this a probe field is not a willing gauge field so actually what you get is the so-called not a willing transformer term so it's like a magic you know by by by break electron apart into the patterns by preferable pattern using not a willing gauge field the oligo then we do the project of back electron then compute the effect theory for this degree this not a willing gauge field degree freedom then in this way we got a not a willing gauge field and not a willing gauge field is known to produce not a willing statistics actually that's a witness of work on Jones polynomial and a knot environment which is basically that's really more mathematical way to say that there's not a willing to be able to really describe the different linking environments and not being statistics okay so this is a this is a way for us to understand why this why this particular wave function have anything to do with a not a willing transformer term and then that's from there we get a not a willing statistics but some of you may not be may not be happy you know what is this you know you get a wave function which I understand you derive something transformer theory which I don't understand and you tell me that's a not a willing statistics you know so so basically the point is that's a what is not a willing statistics and so in terms of physics what kind of physical measurement what kind of physics which give us not a willing statistics and certainly a deriving not a willing transformer theory is one way to derive a not a willing statistics because many magnetic work and the physical work shows you the not a willing transformer theory to not a willing statistics but they are they are indirect so what is a more direct experimental feature for not a willing statistics but using this a using this a wave function using this very concrete wave function we can understand not a willing statistics and some physical problems of not a willing statistics so so we say this is a ground state so this is a chi two is a two field on the level we square it we get a not a willing quantum power states so how to see that's a not a willing quantum power states let's create some excitations so how do you create excitations well instead of using field along the level we can use the longer level with a few holes if a longer level is not totally filled we have a few holes then we then the corresponding wave function will be a wave function of a not ground state but a wave function with excitations and so we can put a let's create a four holes we can put all the four holes here in the in the first factor of chi two but that's not so good because if a chi two have a four hole the first factor of chi two have a four hole the second factor don't have a hole the number of a part are different then we cannot do projection because projection means bind the pattern into electrons so we assume they have a number pattern first type pattern and second pattern should be equal in number so to create a four holes we can put a two holes in the first pattern and a two hole in the second pattern is like this you know we have two hole in the in the in the first pattern and two hole in second pattern and then we construct this wave function with a four holes and this this is a wave function with a four excitations one two three four okay and then something very interesting happens even when you fix the position x one x two x three x four fix their position there are several ways to do this because we can put a x one x two in the first pattern three four in the second pattern or we can put the x one x three in the first pattern x two x four in the second pattern there's several ways to do so and here we just list three ways to do to do this yeah so there's a they're definitely to do so and it and what is most important is that far away from those holes those wave functions are locally the same they they because of the they just just just filled the pattern wave function their product so they locally look the same and also even near the hole they look the same because near Wang hole near Wang hole we have a we have a hole in the in the first pattern or in the second pattern but however if you only look at the neighborhood Wang hole it's just a product wave function whether the wave function appear in the first factor second factor they are the same so this is amazing way to confront something what i just mentioned this is a different way to confront a four hole wave function the wave function locally looks the same but somehow globally is different and you may say well we have a three different three orthogonal states but actually it's a little more tricky it turns out there's a three different way to pull the holes only give us two linearly independent many-body refunction so actually we don't have a three orthogonal state we don't have a two orthogonal states so four pattern four four extension give us two degener two four degeneracy even after we fix the position of holes okay and that is a key factor of a nine billion statistics that is when you have a nine billion quasi particle even when you fix their positions there's a robust grand state degeneracy which cannot be lifted by any perturbation and this is also how people use nine billion statistics in quantum computation because the grand state degeneracy can be used as a qubit and these qubits cannot be lifted by any perturbation means the environment do not affect them so they are immune for the environmental perturbation but the trick is after this we have wave function which it looks it looks the same but globally are distinct and from here we can see nine billion statistics by by saying that if if it exchange two particles and then there's a if it changes this blue and red we get this configuration if we change this blue and red we get this configuration so therefore these two these two four degeneracy rotate each other by exchanging particles so we have this a matrix rather than a pure face we have two by two matrix to explain to describe exchanging two particle and that is a now that's a kind of physical definition for nine billion statistics and what is more amazing is that a nine billion statistics actually have a so-called fractionalized degree freedom we heard about the fractionalized charge you know electron carry charge one but the quasi particle carry charge one third so that's looks very strange but even degree freedom can be fractionalized and that's kind of a I believe because wouldn't buy fractionalized degree freedom okay so so here I just mentioned that when you fix the location of a particle there are degeneracy so we try to see where this data come from because the position of particle already fixed so we say maybe this degeneracy coming from this internal degree freedom of each particle like an electron carry spin one half the fixed position of electron the each electron still have a spin degree freedom so we have a four electrons you have a two to the four which is 16 degree freedom okay but here is that we have a four particle we have a two four degeneracy we have two degree freedom in a sense so let's have a six particle eight particle n particle so we have a many many particles we say what is what is degree freedom we have that degeneracy we have n particle degeneracy degeneracy increase when n increase they increase exponentially okay then there's a there's a asymptotic form you know you know if you electron have a two degree freedom you know every time you add one electron the degeneracy increase by factor two here again every time you increase n the degeneracy increase by some factors so what factor this factor called d so this d is given by this this sorry this is n's root there's one over here it's the n's root of this degree freedom that's a factor we all obtained by every time we add a particle how many how many degree freedom increase for electron this d is equal to two but however for this particular uh sorry for this particular wave function d equal to square root of d so this looks like each particle carry half qubits you know electron is one qubits but this particle carry half qubits two particle carry one degree freedom that's a one qubits but each particle carry half qubits okay and there's another another state which i mentioned that's a this is a chi three raised to the sorry chi two raised to third power that's another state we call fubinachi and in fubinachi this number this d is not square root of two it's a it's another thing it's a golden ratio actually golden ratio appear in the fubinachi number so actually uh that's why we call this a chi two raised to third power as a fubinachi statistics because of this uh there there are quantum dimension or degree freedom this golden ratio contains square root of five okay so this is uh i think this is the most dramatic feature of a non-abelian statistic you know usually when you talk about non-abelian statistics people say will it change to particle you have a matrix rather than a face for the most what is more amazing is that for non-abelian particle each particle carry a non-ethnic dear degree freedom some really interesting degree freedom okay yeah so uh so this is basically uh it's a uh it's a uh it's some very basic explanation of non-abelian statistics and so so because of time so i just want to mention that uh as David mentioned that uh the topological states are basically boring states is that have a gap there's nothing but however at the boundary there's a very something very very interesting they'll have very interesting boundary and actually for non-abelian quantum hall states it's the same story the boundary of non-abelian quantum hall states is very interesting it turns out to be it's a it's something like a conformal field theory but not uh it's a it's a pretty uh interesting one it's called a conformal field theory described by kasabuni algebra so so therefore this again the kasabuni algebra of conformal field theory is studied very extensively in stream theory at least in early days you know and nowadays people probably people study other things but in early days people study conformal field theory a lot and actually uh the the conformal field theory really appear on the boundary of quantum hall states so this knowledge from conformal field theory became very useful and uh and as a as a as a feature of non-abelian statistic is that there's a boundary of non-abelian quantum hall states also have something kind of like a fraction degree freedom have a fraction number for mode you know usually when I have an edge state you'll say how many modes are there it's a one wave is one mode a two wave is two modes but what is the number for mode on the boundary it turns out it's not integer it's you know it's kind of this half integer case you know for this uh non-abelian statistic like like this uh uh chi-two square this non-abelian state it have a half a mode and also like this uh farthing states also it's a three over two and like this uh a phi phi uh uh danson talk about it's a one half you know the the number of mode is a one half but the integer part can be uh can be different you know so this uh it turns out that there's a there's a fractional number of mode on the boundary and the fractional statistic in the bulk are directly related if the bulk we don't have a non-abelian statistic if it's all a billion the number of modes on the boundary are integer you have just come to have this wave description but if a bulk have a non-abelian statistic then the boundary would have a fractional number mode you know this uh but this mode have a more formal name you can more feel through it called a central charge center chart it may not be integer and uh maybe the last I just mentioned that this uh this uh this half alone you then make a very heretic effort to really measure this uh central charge it's really ingenious uh experiments and uh let me let me just describe the how they measure central charge and uh so so they have this uh this this a pattern of device so this uh uh uh as a green line are this a quantum edge mode so the electron mode can propagate along this boundary so what they do is that they send in some voltage so there's a blue line the positive voltage and negative voltage so basically they're sending some kind of electron and a hole into this center of junction and they know how much energy they pump into the center because they know the voltage they know the current they know how much energy they inject into the center of this sample and this energy eventually would uh uh would would become a heat and this heat would propagate away by by some edge mode which is going out by this uh this a full edge mode they radiate away by full edge mode and so so therefore the temperature here will not increase indefinitely so temperature would increase to such a number so that the heat transport will be carry away all the energy you're pumping to the system and then they're using this some electron device to measure the noise of the electron noise electrical noise because this this noise depend on the temperature because the measured temperature in the center part is very difficult we don't have the monitor here to measure temperature but they measure the noise and so they're using the noise to measure temperature and then you'll know that how much temperature is here and how much heat so basically they measure how much heat conductance can carry by each edge these turn out to be the central charge that direct measurement of central charge or number for edge mode and actually the this already uh Dan already talked about this so it turns out to be this uh this near this two and a half so looks like this pf of pf of fuffing uh is most likely to describe this particular uh non-abidding states okay so this is uh basically it's a it's a some kind of story that's a it's all effort it's many you know maybe uh almost 30 years effort try to find uh non-abidding states and maybe this sample is a is our best hope but still uh you know experiments still have some uh trickiness in there so so it's not totally confirmed and we hope there's a more experiments maybe better data uh to uh to to really confirm that so then there will be like a discovery of this non-abidding quantum states or non-abidding statistics and so last you know uh like the previous speaker you know i really like to thank my wife you know you know she really uh uh helped me a lot especially during the difficult time you know without her you know i i cannot do anything yeah thanks thank you very much for this wonderful presentation and very interesting physics and mathematics um any questions yes people are getting tired and willing to go for the celebration yes please so that state that you call chi one chi two square yeah the state can it be understood as the pairing of composite fermion in the usual composite fermion picture with some orbital probably yes and uh um but i haven't really from composite fermion view uh to to to look at this and uh so actually i mentioned that this uh so these two states probably uh have closer tie to the composite fermion so one is a p plus ip another p kind of like p minus ip okay and vertically only differ by integer quantum power states they they can all be understood in some pairing state plus different number for integer quantum power states yeah so so that's a so they are really uh very much related and this kawang chi two square is a wave function which occupies first three langdao levels a device where first three level can degenerate this state will be unnatural as a lawfully state so it's uh it's kind of interesting to any mirror device where for example have another one but each band have a turn number is like a three then you have a short drink interaction maybe have a good chance to produce this kind of states yeah but so we need a langdao level degeneracy it's more natural for for these states and in particular i think most interesting is a chi two cube and that's why i have fubinachi statistics which is the which can do so-called universal quantum computation uh the non-linear statistic for the chi two square is too simple they cannot do universal quantum computation but the chi two cube would be more interesting but that requires first a full langdao level to be degenerate very good so before we finish just just let me tell you that just outside there are the refreshments for everybody so you're welcome to to join us for the refreshments except for the medalists because the medalists have to stay here for a photo session and then we we go for the session so let's uh think and congratulate all the medalists for she's a wonderful actor