 It's a pleasure for me to welcome all of you to the joint ICTP C-Cyclochium. It's a nice thing to see real people in real space after such a long break and I think we're beginning to slowly come back to normalcy. I'm very happy that today's speaker is Professor Subir Sachdev, a very distinguished condensed matter physicist. He is currently a Herschel Smith professor at Harvard University. He also has a very long connection with ICTP. He received the Dirac Medal for his important contributions to condensed matter physics broadly but also finding very interesting links with interdisciplinary links with other areas of physics. He was a salam distinguished, he gave a series of salam salam distinguished series of lectures at ICTP and he was also the member of the ICTP scientific council. He has, apart from the Dirac Medal, he has received many other important honors, including the Onsager Prize. And so I'm very happy that Subir is here to tell us about quantum entanglement at all distances. I would like to hand over to Professor Rosario Fazio, who is the section head of the condensed matter physics to be the moderator and to give a more scientific introduction. But before that, I would request Professor Andrea Romagnino, the CSI director, to say a few words. First of all, I'm really happy to be here. First of all, because I'm looking forward to the colloquium, because it's the first time I have the privilege to introduce one of the joint ICTP's colloquia that were initiated during the Director of the Volcker and my predecessor Professor Ruffo. And because I'm really glad that we have this opportunity, at least those of us who are here to attend this colloquium live in person at last despite we will lose the opportunity to switch off the camera and microphone that we want, but I'm sure we won't need it today. So let me thank ICTP for promoting and organizing the colloquium, Professor Sachdev for being here, all of you, and I leave the floor to Professor Fazio for the introduction. And of course, I join in saying that it's a great pleasure to have Professor Sachdev, the contribution that he gave to condensed matter physics are numerous fundamentals. And I should say the title partially characterize a distinct feature of such this contribution to science that is the meeting of several different area of science which sounded the distance going from quantum information to statistical mechanics to gravity strings, but somehow matched together. So it is very, so it's essentially it's possible to list all these contributions I should mention is work on quantum speed liquids. And with phase quantum phase of matter strange metals that brought physics of this order that system with physics for black holes. And to explore the properties of materials without quasi particles which was a paradigm of solid state physics since time of Landau. And what I should mention is contribution to quantum criticality and again here is extension of going beyond the Landau Gisburg approach to quantum critical properties and without further delay I would like to ask to be joined. Thank you. Okay, so my wife, you know, thank you. Thank you very much. And I guess, sorry, for your very kind introductions. It's really wonderful to be back in Trieste as I mentioned I've been coming here for 30 plus years and it's always a very pleasant experience and meet many old friends and learn a lot of wonderful new things. So, I will, this is my, you know, broad title to talk about the role of quantum entanglement in physics, all the way from the microscopic in a little crystal in the laboratory to black holes which could in principle be even of astrophysical size, but a more specific title would be, I'm going to just review some, some unifying features of the quantum mechanics of black holes and strange metals. I should emphasize, I'm aware that many of the audience here are not condensed metaphysicists so I'll try to keep at least the early part of the discussion. Just explain what I'm talking about hopefully and and if you really find that your total loss don't hesitate to raise your hand and ask me to pause. All right, so let me begin by just recalling a foundational contribution by Boltzmann, which will be a central role in everything we're going to talk about. And really the most important work by Boltzmann is emphasized on his gravestone, the famous formula s equals K log omega at K log W. Boltzmann actually died in Trieste over 100 years ago. But this was in 1870, where he proposed this formula linking together statistical mechanics with thermodynamics. So this is really the foundation of something we take for granted today. So on the right hand side is W which is a probability for any system, he was thinking about a gas of molecules to have a certain configuration. On the left hand side is the entropy, which had entered thermodynamics by study of heat engines, and the possibility of making heat engines even more and more efficient people found that there was a quantity quantity called entropy that have to always increase in thermodynamic processes. And Boltzmann linked together probabilistic reasoning and mechanics to do entropy. So the other connection survived the introduction of quantum mechanics. So, in quantum mechanics, you would say that the density of states these are some quantum eigenstates over many body Hamiltonian is the exponential function, we just defined to be the entropy. And then this entropy plays the same role in thermodynamics. So, you know, people in thermodynamics have proposed. Okay, so that's the fundamental equation that I'm going to use throughout my talk. The other fundamental property I'm going to talk about is quantum entanglement. Well, it was probably defined most precisely already in 1935 in a famous paper by Einstein put all skin rosin, who pointed out that the existing principles of quantum mechanics seem to leave to rather contradictory predictions, although in fact there is no contradiction, as they recognize at that point but they still didn't like the conclusion. So, so the central principle of quantum mechanics is the one of superposition that two distinct states of an electron can combine to give a new state. So, a PR proposed that you could do the same thing with more than one electron. So, for example, if you have two electrons in a hydrogen molecule, one on the left and one on the right. They are entangled in this EPR state, where in one possibility the one on the left is spinning up and the one on the right is spinning down, or vice versa. So this is what the quantum mechanics predict roughly in a hydrogen molecule, but they propose a thought experiment where you separated the two electrons without disturbing the spin, something that's in principle possible. And no matter how far apart they are, they are still entangled. So when this is up the other is down or vice versa. No quantum mechanics also says that when you measure something that measured system, then collapses into the, the eigenstate of the observable that you measured. So once you look at this, this left spin, and at that an observable to be down, then since the state is shared by the both electrons, the other electron would at that instant collapse to be up, but not before you looked at it before you looked at it that was still in a superposition and after you looked at it. There seemed to be this instantaneous connection between the states of two electrons no matter how far away. And often this is called spooky action at a distance. And I always thought that was well, you know, where did that phrase come from. It's some very clever way of characterizing what's going on. Surely Einstein never said it. In fact, that's wrong. The phrase spooky action and distance comes from Einstein. It comes from a letter by Albert Einstein to Max Vaughan in 1947. This, I still discovered by armadol Mary and the library the Institute for advanced study in Princeton. So this is the original letter in in German. And this is spooky action and distance in German I won't try to pronounce it. And as you say in the bottom, I cannot seriously believe it, meaning the quantum theory, because the theory cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky actions of a distance. Well, despite Einstein's misgivings there's no question that spooky actions as a distance do occur. It's not really as if a signal is propagating from one one one side of, you know, one electron to the other. It's simply that the electron shared the same quantum state, and that state is neither here nor there. No information has really gone from from one to the other. It's only when you send a classical signal that you can check that the observation that consistent with the entangled. That's the modern understanding of it. And this kind of spooky action and distance of the foundation of many things in physics. It's not really everywhere. Including the next matter physics physics of black holes, quantum computing quantum information is all about controlling and understanding this spooky action at a distance. So let me just now go ahead and jump ahead to black holes, but don't worry I'm not going to assume you know anything about black holes. I certainly never did before some of my my work started, eventually having some connection to black holes. The black holes were discovered as solution the Weinstein's equations already in the 1920s by short child. And he showed that there were some such solutions are in just an empty space, where you assume there's some mass M that's concentrated right to the center of the black hole. So every mass M defines a radius horizon radius. And what happens is that inside the horizon light can never escape light bends back, and it gets attracted to the mass which is presumably right at the center. So this was certainly an ideal theoretical speculation in 1920s. Today we know that black holes are everywhere. And there's one at the center of every galaxy and including our own is a supermassive black hole. And just to get an appreciation of how dense the matter in the center is for the earth's mass you would have to squeeze the earth to the size of a pea before the earth becomes a black hole so that's how dense the matter in a black hole is but there's many of them out there All right, so that's a prediction just a purely classical mechanics as described by Einstein's generalization of Newton's laws, but it doesn't yet involve thanks constant or the uncertainty principle of quantum mechanics or entanglement. That came about due to the work of Beckenstein and Hawking in the 1970s. So again we consider this EPR pair, and imagine that again a thought experiment that you separate the EPR pair, so that one is inside of black hole and the other is outside of black hole. Then quantum mechanics says that they're still entangled. So if you're an observer outside the black hole, and you have this electron in your hand. And you knew ahead of time that they were entangled then you have some information for what's inside the black hole when you look at this. So there's something leaking out from the black hole. It's not completely transparent. So what Hawking argued from by a different sort of arguments, but this is the modern way of saying it or one simple way of saying it. So if you're if you're in position of the possession of this qubit of this electron. Since you have no information about inside the black hole. There's absolutely no way someone inside the black hole and send you a beam of light or some or call you up and tell you what they have. It's lost forever so it just passed the horizon and doesn't exist as far as your concern. So for you this, this electron is just a random state. It's become totally random because you can there's no way you can access its entangled partner. And therefore, if you have something random, well Boltzmann taught us that means it has an entropy randomness is entropy is the fundamental equation of Boltzmann. And through that kind of reasoning and more sophisticated calculations, which I won't go through here Hawking actually computed some numbers. He said that every black hole has a temperature. And this is the Hawking temperature, all you need to know is the mass of the black hole, the velocity of light C and Newton's constant G. And of course Boltzmann constant that's just a way to define the unit of temperature. So this is an entropy, which is proportional to the area of the black hole horizon, and the same fundamental concepts. So these results of, and furthermore, this black hole is slowly evaporating because it's emitting radiation like any other body at this temperature. And so slowly losing energy and becoming lighter and lighter. These are remarkable predictions of Hawking, which, you know, caused a great deal of consternation in physics, a lot of speculation, and a lot of skepticism, but I think today there's no doubt they're all correct. And many of the, the paradoxes that seem to appear from Hawking results are also starting to be understood. But before I go ahead and tell you about a way of understanding these results. There's one more fact about black holes, I want to mention, which is that black holes another interesting property that in some sense they they're very good at coming to thermal equilibrium. So if you were to throw another star into this black hole. It will forever be lost to you, and the mass of the black hole would increase. And as the star is falling in the black hole will distort and so on. But eventually the combined object will become a perfect sphere and have a new temperature, which is given by this, this equation here because the mass has increased. And how long does that take once you throw a black star in how long does it take. Well it turns out that in some fundamental sense, it's as fast as possible as any thermodynamic system. And, in fact, you don't have to take my word for it this has been measured. So this is just they did these these people recently did an analysis of data from LIGO and Virgo of black hole mergers, not a star calling the black hole but two black hole merging into a single one. And then looked at the time it takes for them to to ring down to a perfect sphere. And this is predicted by which feature earlier to be around this value. But if you know I express this purely classical process in terms of the Hawking temperature you get this very simple result it's blanks constant divided by the temperature, measured in units of energy. So that's a very striking result. And, at least when I first learned of it that immediately rang a lot of alarm bells, because that is exactly the property of the type of systems I was studying in connects metaphysics as I'll show you in a few minutes. And from that kind of study and much work since then, we know that this is really like a speed limit no other system can locally reach some of the equilibrium faster than in this time. There are champions in relaxing to thermal equilibrium. All right. So, so all these results of Hawking are very good, but they were obtained in a rather ad hoc manner they're obtained in what's called a semi classical theory, where you do some path integral and imaginary time and then you analytically continue to play a lot of tricks. You don't have any handle on what you normally do when you're doing quantum physics. You know what you're taught in quantum physics is you, there has to be a Hamiltonian you find its eigenstates, it has some discrete energy spectrum. And then once you know the all of those eigenstates you can predict the future. There's no such thing there's not known what the Hamiltonian was. So another way to say it is, can we form actually have a real system with a real Hamiltonian. In other words, a quantum computer, and can we build a quantum computer to simulate the physics of a black hole. So obey the same equation the motion in a different sort of variables and reproduce the approach of the properties of a black hole. Okay, so just from Hawking results and the things I've told you, you come up with a certain set of constraints. First of all, we know from Hawking that the entropy of a black hole is proportion to surface area. And this is a very unusual result, because every other object, we've met in thermodynamics that Boltzmann ever studied the entropy of proportions of volume, not the surface area. But, so it seems like at least in this dependence on the size of the object black holes are very special. The amount of quantum information they have is not, you know, it's not as if there's one qubit everywhere inside the black hole. The number of qubits is roughly the order of the surface area of the black it's only on the surface. Okay, so we'll build a quantum computer. It's only going to have this many qubits which can fit on the surface of a black hole. Of course the size of each qubit is very tiny so that's still a lot of qubits. If there are any qubits involved, they should obey this property, which is that they reach thermal equilibrium in the very fastest possible rate so their couplings must be such that they equilibrate in the fastest possible way. Oops, okay. So how do you do that. Well, as I'll show you what this requires is very strong quantum entanglement. We have built models, which have a very strong quantum entangle between the qubits to allow them to reach thermal equilibrium very fast. And this was the key to connections to other fields of physics that I'm going to now talk about in a few minutes. Okay, and this will lead to, you know, spooky action spooky behavior of these all these qubits entangled with each other over the entire surface. So that's the question we want to ask, is there such a quantum innovation, in particular as density of states should match the Beckenstein-Orkinian. Okay, so I think it's fair to say that we don't have a complete answer this question even today. But for a very special type of black hole with a certain charge, there's been a lot of progress. So if you have a black hole with a net electrical charge. If you have certain special properties are briefly mentioned towards the end of my talk. The answer is yes, that we can write down quantum simulations of a black hole. The first type of this came out from string theory. So string theory gave, you know, quantum system of a charge black hole, and gave you a set of energy levels which in principle, you could compute a lot of things about. What we found in the string theory result is that there was a huge degeneracy of state there were lots and lots of states, all that essentially at zero energy so let's call the ground state of this Hamiltonian zero. And so it became a counting problem for counting all these states here. And so what we did first by Strom and Genwafa, they indeed found a number of states was the exponential of the Hawking entropy. So that was a remarkable result. But it came from counting these highly degenerate states. Today we also know that in fact there's an energy gap, and above the energy gap there's some density of states which is a smooth function of energy. Okay, so so that's the behavior that comes out of string theory, and these black holes have a special feature that they also have supersymmetry at very low energies. So and it's because of the supersymmetry that you're able to count how many states you have. So today we know the answer, I believe, for a generic black hole without low energy supersymmetry. So here's the answer. There's still that exponential factor, but there is no delta function there's no degeneracy. And there's a cinch of square root of energy function here that comes from in fact, work that builds on a model that I'm going to tell you much more about which is the SYK model, which was initially invented to study something in this matter of physics, but it had a lot of features that allowed us to say something about even black hole entropy, which I will tell you a little bit more about and then in the remaining part of my talk. Okay, so that's my introduction to black holes, the summary of what Hawking achieved, and just a statement without any explanation of how we can go beyond Hawking's result, and also understand not just the entropy, but also the full density of states at low energy. Again, everything only depends on the area of the horizon. And these are, you know, in some sense really remarkable formulae, because of the rare physical formula that have both Planck's constant h bar and Newton's constant G in the same thing at the same time. Okay. All right. And so this kind of understanding from the SYK model has also and this I will not be able to talk about has also led to a better understanding of what happens in a black hole evaporates. So in fact Hawking pointed out that there seemed to be a contradiction in the sense that if the radiation coming out of a black hole is purely thermal. And all the information you put in when you when the black hole originally formed. And so it seemed like just falling into a black hole was the way you're violating the unitarity as it's called of quantum mechanics, where all the information is preserved. And I understand how the information is preserved as the black hole evaporates, and all has to do with the fact that the evaporation shouldn't be thought of as just single particles coming out at a time, but really have to follow the many body entanglement of the evaporating state. And that's been possible to do because of this very simple model of black hole dynamics that I'm not going to tell you about. Okay. All right, so before I get to certainly more about black holes. I'll go back to my home base and talk about some materials and quantum materials, and how electrons entangled in quantum materials. And how is it that we first started understanding that there was some highly non trivial entanglement of electrons in, you know, in materials that you can find in any laboratory in physics laboratory in the world. So to do that I have to mention one more key contribution by by Boltzmann I already mentioned his introduction of the entropy, the relationship in entropy and probability in 1870. There was also his work in 1872, where he even told us how a thermodynamic system evolved in time so the entropy, you know this contribution told us how to think about a thermodynamic system like the gas in this room in equilibrium. And this tells us how would this gas evolve. If you, you know, perturbed it if you send in a pressure wave how would it move through the room at what velocity and what temperature and so on. We understand that all in great detail, largely due to the solution of the Boltzmann equation for the dynamics of a gas. So this is a simple way of writing down Boltzmann equation. So it assumes you can describe everything about the gas. In terms of the probability distribution of the particle so Fp is roughly the number of particles the density of particles with momentum P. And so for this density of particle momentum P. The left hand side is just a restatement of Newton's equation, which is DPDT equals F. There's a single particle moving under some force that might be applied by you in the lab and to these molecules in the air. Right, the new term introduced by Boltzmann was the right hand term. It's called the collision term. So here I'm picturing a collision between two particles momentum P and P one scattering interstates P two and P three. So this happens at a certain rate. So to calculate this, the rate at which these collisions happen. I need to know the joint probability distribution their incoming particles momentum P and you are suppose one made the key assumption of what we call sometimes molecular chaos, that we can just write the joint probability of these two particles as a product of the individual probabilities F of P times F of P one. So this basically states that successive collisions are independent of each other. So we have a collision taking place between say oxygen and nitrogen molecules in the air. If two molecules collide, we don't need to know whether they had collided with other molecules before or not, they just to random all the all equivalent. And we just forget about previous collisions to talk about when we talk about subsequent collision. So they happen independently statistically, and, and, and then everything is encoded in the distribution function of these products. All right, I certainly would be mentioning this. If this doesn't work it works extremely well for a classical gas, and really the foundation of, you know, all of condensed matter physics, at least for a dilute classical gas. Okay, but the remarkable thing is that Boltzmann equation also works for a quantum gas, and even for a dense quantum gas so where do we get a dense quantum gas with any metal in any metal like copper or gold or silver. You have electrons moving around, and they're very dense, they overlap with each other a lot. But nevertheless, amazingly, you can just treat the quantum problem. It's exactly the same equation that Boltzmann use for a dilute classical gas. Now the reason is a little more subtle. So the main change you have to make is you have to change the collision term on the right hand side. So classically we have the FP FP one. And now you just have to replace that by this more complicated factor. And what is this telling us this is telling us. We have a quantum state with the particle P here and particle P one here, and they collide into particles P two and P three. But these particles, like electrons away the exclusion principle. So I can't put two particles in state P two at the same time. So this state and this state have to be empty. So that's the one minus F just make sure that P two is empty. And this makes sure that P three is empty. And this is the time reverse process. So you make that one little change, and you come up with a theory of metals. So this, this, this particular theory, this equation is the foundation of really everything in one way or the other that you will find in the kind of physics book, you know, up to the 1980s, everything was very successfully obtained, explained by combining a few more details about the, the equation of motion of electrons in a crystal with this kind of collision process. So in particular, the kind of things you can explain for usual metals by this quantum theory of electrons is their resistivity. So for example in copper, you have a lot of electrons moving freely. You work out by Boltzmann equation, how often did they collide with each other. And if you solve Boltzmann equation you find that this scattering the typical time between collisions is one over temperature squared. So to zero this becomes very long. And the resistivity consequently grows as temperature squared and and this is really observant, essentially any metal at low enough temperatures. Now, the one thing you should now notice is that this collision time is also the order of some kind of equilibration time because without collisions there's no equilibration. And this is much longer than the time I already mentioned for black holes for black holes it was just one over temperature. Here's one over temperature squared so at low enough temperatures. The, you know, the black hole wins it equilibrates very very quickly, whereas electrons in a wire would take a very long time to equilibrate. And this is actually key important for the self consistency of the theory. The collisions have to be rather rare for this assumption of molecular chaos here that Boltzmann made and to be valid so in here, in addition to molecular chaos we're also neglecting quantum interference between successive collisions. And, and that's behind this collision term here so we're just neglecting the fact that the interference can be correlated with each other in other words we're neglecting the entanglement of the electrons. So we're just totally neglecting that the fact that the electrons might entangle, we just say well they collide and then forget about each other. And then we forget any possible entanglement that might grow. And that's required for us to describe the system using Boltzmann equation. All right. So, so that's all fine and this is observed and so nothing more to worry about. So we've changed the discovery of the high temperature superconductors in 1986, 87. So here's a phase diagram of this material. This happens to be yttrium barium copper oxide as a function of some doping density which you change by say changing the density of oxygen and temperature. The most exciting thing about these materials was that they went superconducting at a rather high temperature 100 Kelvin or even above at ambient pressure. And so everyone was extremely excited to do a theory of the superconductivity. I will not do it today. I'm just going to focus on the region above the superconducting temperature. So, I mean and the boy. Well in some ways this is really a prerequisite to have a theory of superconductivity. I'm going to figure out why this is a superconductor. I better know what it was before I called it what's happening up here so that I can figure out why it likes to be a superconductor at low enough temperature. So we want a theory of this metal. And that's now universally called a strange metal, precisely because Boltzmann ideas do not apply. And, and the net conclusion is there must be entangled it. So, for example, if you look at the resistivity of these materials, it's linear in temperature down to very low temperatures, perfectly linear rather than the T square review expected. This kind of behavior is actually much more universal found in many other materials like twisted bilayer graphene. You can also measure the collision time between the particles I won't go into the details of how you measure it. This is the recent paper and in nature. And I just without going to detail you focus on this red line which is a measure of the collision rate. You measure it, and you find, aha, it's just like a black hole, the collision rate or the collaboration time of the system is just h bar divided by Planck's constant with a constant of order of 1.2 out front. So, so somehow, I'm of course telling you things in some a historical order. And it really hints that this was going on in these materials and, and, and this word was originally led me to start thinking about complex entanglement of electrons. You know, of course, it was my lucky day when that that also had some connection to black holes. And the reason that's not, you know, completely ridiculous that they're already you'll see quantitatively is, you know, a striking feature. The interesting this experiment is published in 2021, showing h bar over T is equal to kbt. And in the same year by people who were didn't know in fact I put the two authors in contact with each other. There was this paper on black holes. I'm showing this h bar over kt behavior of the thermalization time. And, and of course the point of my colloquium here this is not an accident, there is a connection between these two systems. Okay, so, there was a. All right, so I've shown you this data. So just this data opens many questions just like Hawking's calculation open many questions and theory of quantum black holes. And most of an equation so well what do you do. We don't know. So you need a theory for current flow in a strange metal. Clearly, the reason we can use most of the equation is because of the entanglement for the electrons. And once you have a theory can you compute the sensitivity. And finally, can you figure out when superconductivity appears. We don't really have completely settled answers with these questions but there's been a lot of exciting progress, and I'll tell you a little bit about that, hopefully by the end of my talk. Okay. All right, so the key to this is something called the SPI came out of these two, the ideas that we've been working on and some of the results and black holes that I've already mentioned. What is this model. Well, before I describe the model. So it's something I'd introduce in a different form in 92, where I was, you know, I was aware of the data on the cuprates. And I said, Well, this is just too complicated. I just want some model with a Hamiltonian for which I know that the Boltzmann picture would break down, meaning that I cannot describe its dynamics in terms of some distribution function of particles. So it's got to be something else. And how would I describe the dynamics of the system. And I said I don't care about modeling this crystal or that crystal. Let's give me any more, give me any more that I can solve and fully understand where I can account for quantum interference between successive collisions, and therefore go beyond the Boltzmann answer, in some reliable way. Before I describe the model, let me go back in time to August calculate, who really in some way, even before the introduction of quantum mechanics introduced us to quantum entanglement. So calculate proposed in 1865 theory of the benzene molecule, where you have six electrons, which can, you know, form a valence bond either this way, or the other way. And so calculate said well maybe they do both. So today we would say he proposed the entanglement of six electrons as a, as a way function for for benzene. And then he said he discovered the ring shape of the benzene molecule after having a daydream of a snake seizing its own tail. So I'll make an analogy with that. And the ancient Indian game of snakes and ladders, where you have snakes going from anywhere to anywhere. And let's say that's driving entanglement of some electrons. So that in a nutshell is what the S like a model is. So you have a bunch of positions. Random issue which, and you put in a whole bunch of electrons on them. And now you allow these electrons to entangle with each other. So for example there's some quantum mechanical process where electrons 11 and 12 go to positions five and 14. And this process is associated with it as a complex number so this is some number which I call you 11 12 514. And for any given Hamiltonian this number is given for you. Okay, so for every such process I have a complex number. So I can have that process and I have a number for that. That process and I have a number for that and so on. So this is the list of these numbers. And that specifies my Hamiltonian and I tell you, okay, tell me what happens how does the system evolve in time. And you will come back and say impossible I can't solve this problem beyond about 20 electrons. There's no way to solve it. It's just too hard, even the fastest computer in the world today couldn't do it. What I noticed was that you could solve it with one additional assumptions. You just have to make the assumption that these numbers are statistically independent of each other. And, and have some root mean square value, then with probability one, in fact, very close to one you can solve the whole thing. And that would limit like phenomenon. You know, where everything is a Gaussian, no matter what the, if you have enough distributions here. There's so many states in the Hilbert space of these and electron this two to the end states that any reasonable property or measure is in fact independent of the precise values of these numbers has a universal structure. I understand today very well. Thanks to the many different works of many people. The original version in 93 was slightly different and you couldn't check it on the computer. The new version you can also solve on the computer proposed by Kateryna and fortunately everything we worked out in 93 and subsequently turned out to be okay. So here's if for those of you know a little bit of many body theory here's the Hamiltonian. There's a bunch of numbers you all for beta gamma delta. And all you need is for these systems to be statistically independent. And then the prop the system is self averaging has essentially the same properties for any choice of these you all beta gamma delta. Okay. So you can, well, let me skip this. So what you can now solve these, this problem in this large and limit, and you can compute a few things. So one of the things you find, which, which is great, you find exactly this blank in time dynamics, and you don't have any quasi particles so in a regime where Boltzmann equation breaks down and you find this h bar over kt number appearing in a very fundamental evolution of this system. You can also compute the entropy. And you find this strange feature which in fact shares with charge black holes that the zero temperature limit of the entropy is non zero. But that doesn't mean there's an exponential degeneracy as I'll show you in a minute. So here you can also so here's your this is so this is the density of states so what you do you took this case about 16 electrons. So there's two to the 16 states and this is a plot of the energy of all of them. In some bins you make a discrete set of bins and this is the density of states. So now, what is what is this function look like well at very low energies. So Boltzmann told us the density of states is exponential of the entropy. So we can compute the entropy I won't show you how you can compute the entropy, and it has a zero temperature value s not that was first computed by George and Barclay and myself, and then a linear in temperature growth of the entropy. So you convert this canonical entropy to a micro canonical entropy and you get this result. And that fits this data quite well. Then when you zoom into the bottom of the band so here was zooming into the very bottom. You see here the actual energy levels. And what you find there's no degeneracy. But there's a lot of levels that are exponentially close to each other exponentially close and the energy level spacing is e to the minus and. And so it's the spacing of these levels which scales is e to the NS not. And, and these are essentially universal results. Now of course, the value of every one of those energy levels depends upon every u alpha beta gamma delta, different set of u alpha beta gamma delta. These energy levels will be all different. They'll be different in a very chaotic way, and any kind of averaging if you look you know e to the minus and is a ridiculously small spacing. If you did some bidding, even a very small energy, meaning of spacing one over and or any power of and you would get the same answer wouldn't depend on the u alpha beta gamma delta. And in fact we know quantitatively what that answer would be. So one of my main messages, in fact, charge black holes have exactly the same density of states at low energy. And we kind of understand that very well by now. I'll try to give you some understanding of where the connection comes from. Okay. Right. So where does the connection come from so the, this density of states is determined by solving this quantum mechanics of this problem. And what you find is that when you look at the path integral of the low energy theory of this is like a model that is a theory of these states down here. You did want to do some statistical theory of these energy levels. What you find that is described by a Feynman path integral, which has a certain symmetry, in particular it's a symmetry, which is reminiscent of the kind of symmetry that Einstein first talked about. Einstein talked about general coordinate invariance yet space time with some coordinate system X and T, and then he said whatever equations you write down for general relativity shouldn't depend on the coordinate system the many different coordinate systems, and, and there's a metric associated with the coordinate system, and you should really think shouldn't depend on the specific metric. They should only depend on things that are curvature invariance, like things like invariance like curvature that don't depend on the coordinate system or the specific metric for that coordinate system. For the quantum mechanics, you would then say, what you have to do as for as Feynman taught us is you have to integrate over all possible metrics all possible reparametization that time and space. And so what you find here is something very reminiscent of that you find the quantum theory of the system has a time reparametization similar not space time, but just time. And you get some action for this time reparametization, which I won't write down but what I just call S yk of F forget about five that has to do with charge fluctuations which I'm just ignore for now. All right and from that you get the theory of, you know these results here, I showed you which it can be checked numerically. So this is you know no fancy stuff this is a thing you can put in the computer. You can check all of these results that come from the Feynman path integral. It gives you this density of states very nicely. But, you know, you'd be really need a computer to get every energy, but if you're satisfied with this little bit of course training, and don't care about all those chaotic energy levels in detail. Then you'll get that from this theory of time reparametization fluctuations. So now let's go to black holes. So what does this got to do with black holes. Amazingly you can make a rather precise connection. I have about 10 minutes left is that correct. All right, so I think that should be enough time. So, this is where the fact I'm talking about charge black holes becomes important. So you take a black hole, which has a net charge. You don't solve just Einstein's equation. There will be electric and the magnetic fields that will be there vacuum. And so you have to sort of like only electric field. And so you have to solve the Einstein Maxwell theory. So the electric field kids some stress energy tensor which then distort the curvature of space time. And then you find out what happens to the short field solution of a black hole. So this was done by Ryzen and not from and you get some solution. If you look very carefully at that solution you notice something quite remarkable. Very far away from the black hole horizon, space is three dimensional, there's three space in one time dimension. But as you come closer and closer to the horizon. So Zeta is this coordinate here where you're following into the black hole. Then you find sort of a factorization. So if you're concerned as you're following into this black hole, all the important things happen only as a function of Zeta, and you can forget about the dependence on Omega, the angular coordinate. So this. So to you, as you're falling into this black hole. You will, it will seem to you as if you're living in a one dimensional world, just falling in one direction provided you ignore the very high energy excitations. You're very fortunate that you're living in a one dimensional world, because quantum gravity in one space in one time dimension is surely simpler than quantum gravity in three space in one time dimensions. But just never that I was going to die in the center of the black hole but anyway you can compute what will happen as you fall there. So you only have to consider quantum gravity in one space and one time dimension. So I take Hawking's result what did Hawking tell us Hawking told us that the entropy is proportional to the surface area in quantum gravity in the black hole. And this means that the, the quantum computer that I have to build lives on the surface. So what's the surface of a one dimensional gravity well the surface of a one dimensional system is a point. In fact, my quantum computer only has to sit on a one point to describe one plus one dimensional gravity. So is there a mapping to some quantum system, which looks like quantum gravity in particular an essential requirement that it had this plank in time dynamics is there a system in zero plus one dimension, which has plank in time Yes, there is that is the chart that is the SIP model. So at least it satisfies certain sanity checks that they should agree. It turns out they exactly agree you can, you can start from so this is kind of a remarkable thing. You can start from Einstein's and Maxwell's theory and write down the low energy theory of this observer falling into the black hole. You can get some results or you can start from this Hamiltonian of the S. Y. K. model, which gave us these energy levels, and write down the low energy theory of this, these levels, and you get the same theory. The various references here I'm not mentioning name this is, you know, I first proposed there was a possible connection but many other people responsible for showing that it holds in great detail. So to summarize then the near the near horizon theory of one of a charge black hole can be assumed to be in one space in one time dimension, sometimes called JT gravity or some variations there of project even title bomb will consider this kind of gravity theory a long time ago. Now, or you can take the S. Y. K. model starting from this Hamiltonian with the u alpha beta gamma data couplings and do a final path integral and the look at the low energy theory. You get essentially the same theater this is equal to this. And there's a very precise connection between these two theories by mapping the boundary value of these observables on the surface to to these time regrammation and the phase fluctuations here. Okay, so so that explains then my earlier claim that I can just borrow the results from the S. Y. K. model and apply them charge black holes. So for example for charge black holes, Hawking taught us that the entropy is proportion of the area of the red and zero temperature, and there's a linear and temperature correction. So you get a correction from just solving the Einstein maximum equation. But amazingly turns out you get the same dependence on temperature from the S. Y. K. model from solving the S. Y. K. model. So now you can go beyond, and you can even look at corrections to Hawking result. And so you get a correction which goes as minus three half of log of temperature in both cases from just the identity of these pattern rules. Entropy you can compute then the density of states as both when defined it, and you get this result I quoted earlier for. So there's these levels which were computed on the computer for the S. Y. K. model at a course grain level also apply to a charge black hole in in our space time. And this is the result for such a black hole that I mentioned earlier this is the exponential factor that comes from the Beckenstein Hawking answer. And this is the correction, which has only been obtained in the last few years, and can be expressed in terms of the area of the horizon. Anyway, so this is not to say that there's actually an S. Y. K. model out there in an actual black hole that high energies you probably there is string theory or some other very complicated theory of elementary particle physics. But if you're interested in the low energy spectrum of a black hole, there's a certain universality of it we now understand. So universality can be modeled by much, much simpler systems like this system of randomly interacting electrons, which you can solve. Okay. All right, so I think I'm going to skip this large part. So the last part was all about closing the loop. If you look at the strange metals that were discovered. Then that motivated me look at what now called the S. Y. K. model, which accidentally happened to be very useful to black hole physicist. But can I now understand strange metals from the S. Y. K. model. This is not as far advanced as I would say today. This is the black hole things have been a lot of rapid progress in that direction, but it's getting there. In particular, the main thing we realize in recent years you have to do is to not just consider models of electrons with random interactions or fermions of random interactions, you have to consider mixtures of fermions and bosons with random interactions. So if you're dealing with models like this, you get what seem to be rather accurate models of strange metals and observations and experiments and this is something really work in progress. And so just leave it at that. So, let me end by just summarizing then I won't tell you more about S. Y. K. models and strange metals. I've talked about various things I've talked about S. Y. K. is it's just, you know, you may not like it or not, but it has some unique features. And many other models like it that have discovered since then. It's a solvable model without particle like excitations which means, you know, at the first step you cannot use the Boltzmann Landau description in terms of some particles or quasi particles. And once you build such a model that it has some other sets of properties which we are now understanding better. One is that it thermalizes very quickly in this blank in time, or another way to say it becomes it's champion and becoming totally chaotic. And this happens in the smallest possible time. Which is, you know, this is a, nothing can thermalize faster than this in the, in certain definitions of thermalizations. It doesn't depend on whether your microscopic coupling constants are electron volts or Giga electron volts. This only depends on the temperature. Okay, so the low energy theory of these clouds of models is a theory of time reprimarizations. And amazingly, the theory of time reprimarization is also that what applies to the boundary of this one plus one dimensional space that's found for charge black holes. And so from this, you can compute new things about black holes for described by Einstein gravity Einstein Maxwell theory, including the low energy spectrum for charge black holes. And this whole class of ideas has led to many other works, mostly not almost all not by me, which have led to some understanding of what's called the page curve of the evolution of the entropy of evaporating black hole with this lower amount of gravity description. And what I'm involved in right now are really applying these ideas to to my favorite materials like the cup rates. And this is working being done with the my former student of Ishkar Patel and current student, how you go. And current postdoc. So thank you. Thank you very much. So, questions. Can you explain a little bit how the picture breaks down if you remove charge. So why is charge so sort of necessary in the picture. Yes, I tried. Since I'm not a gravitational business, but so I go through this picture here. I tried that you start with a three plus one dimensional space time. And then you ended up near the horizon in a regime where you could just focus on the radial direction and forget about the angular direction. It turns out if you don't if you have a neutral black hole, which is almost certainly all the black holes in our universe. And the factorization of the space time into into radial and angular coordinates doesn't happen. And you can't, you have to really consider a full four dimensional theory, even at very low energies and quantum gravity in four dimensions or even conformal field in four dimensional far more complicated than such theories in one plus one dimension that's really I think the main thing. You made in the SPI came on absolutely are this this the model I wrote down has a global you one symmetry is conserved the number of electrons and that's what leads to the phase mode, which is the dual of the electromagnetic field here that comes from the chart. Yeah, so they're connected. I mean, I mean, there's certainly, you know, for the ADC if the correspondence, there's an understanding of at least a super symmetric black holes, you can compute the entropy, but that's done entirely on the gravity side it's not done independently on the field theory side here you can compute it in both sides, at least the energy dependence of the density of states. And what you don't know there also even today is what happens to the energy density down to very low energies so here you know this formula I'm giving you here. Yeah, this gives you the evolution of the density of states down to very low energies. When the cinch becomes you know that becomes a water one. It can become even even about a e to the mind, so that this factor can so the whole d of e can become order one. What's happening at very low energies is not understood in four dimensional shorter black holes. Maybe the strength if I made some wrong statements the string theorists should correct me but I mean, in the S O I K model, there is this average over realization over. Yeah, so these pictures have no ever. This is just one model, but go on. But I wonder, these, the fact that you have to average over and do you think it's something fundamental in the sense that it has something to do also with the gravity side or it turns out to be the easiest way to to get universal behavior. It's the second push I'm almost certainly that's correct. Yeah, so. In fact, that was very much on my mind when I started looking at random models I mean there's a long history in physics, at least for single particle quantum mechanics of using randomness to understand quantum mechanics. I mean, going back to the, you know, the observation by boygas at all that are completely regular symmetric quantum mechanical system, like a particle in the billiards. If you look at the statistics of energy levels that be given by the random matrix theory. And if you want to understand that analytically why that happens the best way to do it is to take a particle moving the random potential and average over the potential and you can reproduce random matrix statistics I guess, or is actually a very important role in making that connection. A long time ago in the 80s. So, that was very much my motivation that you're talking about a system which is extremely chaotic because there are no particle like objects to thermalizes very very fast. So maybe randomness will help you a little bit. Now, if a black holes, at least higher dimensional black holes there aren't enough models that make sense as theories of gravity for us to average over. You know, once you get to three plus one dimensions. You know, this probably just spring theory or some reductions of string theory. But however they're also very chaotic. And I think the current understanding by people much smarter than me is that, even if you take these highly chaotic, completely regular highly super symmetric systems with and do a little bit of averaging say over energy levels or fact are from end to end plus one. You will get theories that are more universal and are described by low energy gravity. So, I think what's certainly the case that if you just take low energy gravity on its own. It describes an ensemble. But that doesn't mean that there actually is an ensemble. It's an ensemble that can be realized by a single system on its own, or another way to say it you know if I want. If I start with a theory of gravity, even in higher dimensions. I think everyone would agree that if I want to have that theory of gravity like Einstein gravity or something very simple will not give me every energy levels. But it can give me something can give me on its own for example the density of space in this case. It's a remarkable thing that's led to the understanding of some understanding of the page curve, it can also give you the entanglement entropy. So the entanglement entropy somehow only depends on the low energy theory of gravity, even though that theory cannot tell me every energy level. And how that, you know distinction happen is something we're learning from the SIP model. Thanks for a very nice talk. Just a first a comment on this last discussion, it's quite, it's quite amusing in some sense because you know this connection between the SIP model and gt gravity, and all the realizations that came from it kind of led to this semi existential crisis. In the last two years right to the point where in strings 2021, there was a poll that was taken do you believe that grouties and ensemble or is it an actual one to one map like we believe in the biography. So, I mean, yeah, I was bemused by it because I didn't see any contradiction but right both sides of the answer but yes. But in some ways you caused it. Thank you. My question actually has to do with something slightly different. You know from from, I guess many of us recollection, going back to the 90s that this idea that black holes are the fastest scramblers comes from this drum that Lenny saskins been beating for many, many years about black holes and being the fastest scramblers. And if you've paid any attention to Lenny's papers recently is he's got a different tune. And also that comes from from the syk model, namely that the double scaling limit of the syk model seems to thermalize, not like a black hole, but like to sit a space like something that's inflating. Yeah, I, what do you think about that. I've seen those papers I haven't, I can make head or kill of them. I'd love to talk to you more. I'm, I, I, yeah, I don't know. I don't understand this is a space it seems like a highly non equilibrium system with the time evolving universe. So my gut feeling would be if you want to understand the city space you got to do some time actual time dependence of x5k models. And my, my gut feeling me you're not going to learn anything about it from equilibrium property that's why I came out. But for my I mean Lenny has been right so many times so let's see I'm willing to learn. A couple of questions. So, for the super symmetric black holes actually the you have exact degeneracy. Yes. Yeah, you want to show that picture. Yeah, because here you're saying that it is not exactly and I see what it's, there is no degeneracy at all. There's no degeneracy also. Okay, it's possible that super symmetric black holes are too special and then that for super symmetric records we do know that there is exactly. Yeah, I think that is what I'm saying here. I recognize all the beautiful work, people have done on super symmetric black hole including you, where you've exactly counted the number of states, you know, in this zero energy limit. And it's quite amazing that that number of states in this limit agrees with the Hawking formula. No question. But the point is that, you know, that agreement independently of whether string theory is the right to the universe of high energies. And it relies on the fact that these black holes of supersymmetry also at low energies, I think that's well understood today, especially this nice papers by was Hawking, Turiachi and Matthew Haderman and others who have looked at the super symmetric case and that's this this result is from their papers. That there's a gap, and then there's a continuum. So in the SYK model, rather than getting this curve. There's a whole continuum. And now the point is that how you know how do you have a large entropy, or do you have an exponentially large entropy about the e to the end or e to the how do you have a large entropy should be here. Yeah. There's a different one. Yeah, there we go. Right, the point is that. Yeah, the, the spacing between these levels is e to the minus and whereas the spacing in that supersymmetric. In fact, it's all a one over and you have e to the end degeneracy. A gap of order one over. So here the gap is e to the minus and there's a world of difference between one over and e to the minus and that's what it's all about. Yeah, one quick question is that I mean you said related to this question that. So, if you can define some notion of a charge. Yeah, because we know that the leading entropy is not should behave like I think you square. So can you. So you know there's some function of q I haven't written it down. Can you check that if it is q square for a charge black hole it should be q square. I think it depends on the dimensionality of space. In four dimensions, it should be q square. I think so. The expression is in my paper or maybe maybe maybe you can use it. No, thanks for the very nice talk but I just wanted to make one minor comment which is, I mean, you know it. Let me just answer artist which is. I don't know the top of my head but that's just from the rise and awesome solution you can look it up. Yeah, yeah. I don't know the lights are not sorry go ahead you. Okay, sorry, sorry. Okay, just, is there any understanding of this nice results via the entanglement entropy. Okay, well I mean so, at least in the s yk model since every site is connected to every site there isn't any sense of space. For example, they have any n over two sites and see what the entanglement entropy compared to the others. And it just goes with the volume, you know it is proportional to n over two, like the entropy also goes as the volume. Let's use my life. This thing is going. Yeah, so, so this entropy is proportional to volume this s not and so is the entanglement entropy. In the details because I won't consent with where the charge of black hole enter for the discussion that we had also yesterday. So what exactly how did I get the answer so well first of all, so this and for the case of an s yk model, where you have a microscopic Hamiltonian, then the entropy is just a pure number. Okay, and it's an exact universal number that we first got in 2001 for a charge black hole that I'm talking about. I don't know what the microscopic theory I only have an effective theory. So how do I get the result well I look at this there's this gamma here the linear and temperature correction. It depends on this gamma, and this gamma is in fact a coupling constant in this common theory so the gamma is a coupling constant this theory and this is here. So I know gamma, which is the linear in temperature term here. So this linear and temperature term is actually already in the back and stand in the Gibbons Hawking paper, if you work it out. So Gibbons and Hawking would have this provision here. So once I know this gamma, I know the coupling constant this theory, and then from that I can then get this result. And actually, I figured out but it's never been written down in this form and until very recently, but the results been implicit in many people's paper including, I think, going back to Maldesina and Stanford and yet, but the fact that it's this number with the factor score to pie. I'm very proud to say I figured out. Very interesting informative talk. Thank you very much for that. I have a few just educational issues. I didn't understand the entanglement in your consideration. It was an assumption, or you found the mechanism generation of entangled states. Well, what what we did. I mean, I didn't talk about entanglement to be people have computed but I was saying the following. I, we solve this problem. So here's a Hamiltonian with this coupling constants. We solve this problem. And then we asked the question, what happens if you perturb the system, how long does it take to collaborate. I had a land off of me liquid picture, and you could have such a picture for this model, which you, you would find that would take a time of order one over temperature squared. And, and so that would tell you that they're quasi particles and once there's quasi particles but my loose definition there is very little entanglement in our normal life is rather seldom rare event, rather than the normal state right. I disagree. Every quantum every measurement is entanglement it's observer entanglement. People spend many times to degenerate the entangled particle. It's very hard to generate the entanglement you want, but there's always entanglement. This is why I said the general assumption. Okay, but my next question. The last question. Yes, of course. There's only one. You mentioned one of your slides you have shown that you're here I didn't catch I'm sorry, or either your theoretical result or experimental results have proven that spooky action at distance doesn't exist. It was written on your slide can you explain once again. Well, there's many, many experiments that have specifically meant done the EPR experiment with light not with electrons in your consideration and asking your considerations. No, it was written. Well, I would say just the fact that you have a many body system without quasi particles that cannot happen without spooky action at a distance. It requires really long range entanglement. More questions. So, then I, question more comment really first of all thanks for the very nice talk, but in relation to some previous questions I just like to state that the black hole needs to be charged only in the lower dimensional theory and the requirement of charge, the the requirement of electric charge can be uplifted to higher dimensions in many different ways. For example, in higher dimensions, the black hole might not have any charge, for example, the black hole can just be rotating without any charge whatsoever. But in the low energy effective description, it looks like the black hole as a charge and it's well described by what's described by the spy camera. This is just a minor comment which is obviously Thank you for the comment. I agree, of course. Thanks. For questions. I can take the chance to have a question. So, any electron problem actually entanglement can manifest in many. Yeah. I, you know, I don't know it but I think there have been a lot of studies of such things. I think Brian swing will particularly have been studying. I haven't fully understood those papers but certainly if you want to make some analytical theories of the black hole entanglement in a regime where there is no quasi particle description. There's probably nothing simpler than the S. Y. K. model. And people have been doing it but I, I mean, I should say in sorrow that these computations of black hole evaporation and how the entanglement entropy of the radiation changes as a black hole evaporates. So people have done those computations so that's a highly non equilibrium situation and how the entropy grows, people have done those calculations. Usually that's why I came out and you know even the simpler gravity models and so. So thank you very much for your talk. I have one question because there is one thing that is not clear to me. Maybe it has been said but I missed it. It's not clear where does the area in the, the horizon, the horizon area comes from in the final formula, because they don't see it from in the S. Y. K. side. Yeah, that's a very good observation. I didn't have to go, you know, I removed some of the analytical calculations. So what do you get from the S. Y. First of all, the Hawking's computation gives you that area term, and the Hawking's computation is just done by a semi classical path integral. You value the saddle point on a cigar geometry and then you'll get that answer. So, so what is the connection. So the connection is the following. So, yeah, I missed it here. So what happens is when you do the S. Y. K. model, you get a pre factor. That's the leading term of the entropy. There's just an overall factor out front, and then you get this theory of time reprimandizations. If you start from Hawking's calculation, you get a pre factor which I should have written out here, which is out here exponential of the area. And then you get a theory of semi classical fluctuations at low energy. So this theory has a coupling constant. This theory also has a coupling constant. What we know is that that coupling constant determines the linear and term linear and temperature dependence of the entropy. Okay, the leading entropy term. So, so basically I have to go back to Hawking's calculation and get this coefficient and that gives me the coupling constant here. Or I can go to the S. Y. K. model and go to my computer and get the coupling constant here. So, so that's sort of the coupling constant depend on the precise microscopic of the system. And there, those are obviously different. But then I get the same theory with a coupling constant as a function of the coupling part and then I can use that to map one from the other I just have to translate this coupling to that couple. Okay, that's that's what I use to to obtain this formula. And then you can check the same square of E dependence by just looking at the numerics here. Okay, so let's thank Professor such that once again. Thank you so good for a beautiful colloquium I think it was illuminating also for experts, but also pitched at a level that all our students can follow. So, as you know, we have a tradition at ICTP that our postgraduate student diploma students stick around without any faculty present to ask questions to colloquium speaker, while everybody goes outside and enjoys refreshment. But then something will be kept also for the students it's not. Thank you.