 COVID. So I will tell you about mostly I will focus on thermalization mechanism to avoid it or at least slow it down. So it's breakdown entanglement. So we will see that somehow entanglement would be a very nice way an informative way of thinking about these problems. And in these three lectures, what I will do, I will first give you a bit of motivation, introduce basic questions, then I'll talk about thermalization and state thermalization processes. So something there I'll go relatively fast because I think Anatoly already told you about it in detail in the first couple of lectures. So and from there I will ask how can we prevent system from thermalizing or what are the mechanism of parametrically slow thermalization and that would bring us to three main sets of phenomena. First, many body localization, pre-thermalization in a particular sense that I'll define and quantum many body scars. And first, you know, first probably lecture in the half. So I'll stay pretty sort of pretty basic and we'll go relatively slowly. So please, you know, please don't hesitate. Well, partially because I think after a few intense lectures you need a bit of a change of pace because I'm sure you're tired, right? But please, you know, please stop me and because the goal is not to cover everything but just to give you a flavor of, you know, questions being asked and why they're being asked. So and big part of it, so maybe let me also ask in terms of writing on the board, can you see the whole thing, all of you or should I stick to the center everywhere? Okay, okay, good. So you guys, okay, perfect. Yeah. So in terms of motivation for this area of research, right, it is largely experimental and related to developments in synthetic quantum systems, systems which were largely developed for the goal of quantum simulation, quantum simulators. And there are, there are by now several different platforms. So you have atoms, cold atoms and optical lattices, right? You have superconducting quantum processors where your sort of elementary degrees of freedom are conducting qubits, quantum processors. Then you have Rydberg atom arrays, trapped ions and so on. But those are, those are the main ones. And so, and these systems, right, the great thing about them is that they're very tunable. So for example, if we think about systems of ultra cold atoms and optical lattices, you can, you can control their dimensionality, right? So you can make a one-dimensional system or two-dimensional systems, lattice. So you can, for example, there have been very nice experiment in a slightly different context, realizing honeycomb lattices and the Hall-Bain model on a honeycomb lattice. So you can tune hopping, interactions. So for most of these lectures, I will think about systems which are isolated from environment, but maybe towards the end I'll also tell you some, some recent work about some recent work on engineer dissipation. So here, right in the system, they also, some of them allow you to control dissipation. Perhaps originally driving force for these developments and the, you know, one reason why people, people are developing the systems is that they're very pure in a way, right? You know how, how atoms interact with each other. So they give you a way to realize simple, conceptually simple models, right? And the idea was, okay, maybe we can learn something about strongly correlated physics from, for example, realizing Hubbard model and studying its low temperature physics. So, but of course, and you know, so that's been fruitful, but of course the systems are fundamentally different from, from correlated materials in particular, in that they're to a good degree isolated from, from a thermal bath. So because you have no phonons, for example, right? Of course, they have other forms of dissipation, but it is quite different from, from, right, from what you encounter in, in a solid at low temperature. And on top of that, these systems are quite dilute, right? So if you compare the distance, for example, between atoms and optical lattice, it is really many more orders of magnitude larger than distance between atoms in a, in a solid. And so, and that leads to time scales being very slow, right? So you have slow time scales, let's say milliseconds versus picoseconds in, in solids. And so, and these two properties also also supplemented by the fact that in the systems you have very nice quantum optics tools to, to image them, quantum optics techniques. So all of them actually make, make the systems really, really a very nice platform to study non-equilibrium dynamics, dynamics of isolated, or to a good degree isolated quantum antibody systems. And so, and something that people have not thought maybe that much about in the context of solids where system, you know, time scales are very short and your system falls quickly to thermal equilibrium. So in addition, right, the type of experiments you can do in the systems are quite different compared to what we are used to in condensed matter, right? So in solids, you study things like, you can, you can measure things like conductivity, specific heat, and so on. So these properties that we are used to are kind of hard to measure in synthetic systems. So you can play tricks to get some insight into transport, for example. But instead, you can do other types of experiment. And one is, you know, that I'm sure you already heard about probably even before the school or during the first lectures of the school is a quantum quench, right? So that's the simplest, but very important setup. So where essentially you start your system out in a, in some initial state. So if I think about an optical lattice, I can think of putting atoms, for example, on every other site, right? We can prepare our system in this state and then let atoms interact and evolve under their Hamiltonian, right? And then again, they would hope around, generate, right, they would become entangled due to, due to interactions. And, right, so then you can characterize resulting evolved state psi of t, right? So you can characterize the state psi of t using, you know, using the slow time scale. You can actually, you can see in real time what your system does. And for example, you can do it with single site resolution in cold atomic systems. And of course, if you're now thinking also about, for example, it's protecting quantum processors, you can do even more. You can measure, right? You can measure your system in different bases and really extract a lot of pretty complex observables. And now, you know, so that's roughly, that's the, that's the motivation. And then we will be discussing some basic, you know, basic questions about starting from this type of setup, right? And just another general remark, right? So this field really sort of has developed quite quickly. And one reason is, is that really, you know, it is going to, in a way to, to some very fundamental questions how, how does thermalization occur, right? What can make your system not, not, not thermalize, but also this, this endeavor is related to the desire to find better means of quantum control. So quantum control and developing strategies for really, really controlling many body systems. And another interesting aspect is that as, as we will see, is that many of the non-equilibrium phenomena, actually many, many of the non-equilibrium problems, they have high computational complexity. And so, and therefore, they're, they're a very natural candidate for looking for useful quantum advantage, right? And using, using quantum processors, quantum, quantum computers. So we will broadly, right? Broadly, we will be thinking about simple models either of fermions, right? So for example, let me start with just writing a very simple model to be concrete. So let me consider spinless fermions, a i, a i plus one with nearest neighbor hopping. I will also add on-site disorder, right? So there is an on-site disorder where these numbers are drawn from some distribution. So let's say a flat distribution with, with w. And then we have the nearest, nearest neighbor interaction, right? So that's first, first example of a Hamiltonian. And another, right? In some cases, very much related or even equivalent type of model we will study would be spin, spin models, right? So I can think about spin chains or, for example, two-dimensional spin models. And here, an i is the density on-site i. And so here I can, for example, consider, well, let me just write the simplest spin model, which is actually, actually equivalent in one dimension to this Fermanic model, spinless fermion model, which is Heisenberg model of spin one-half random field in the z-direction, right? So in this case, this is your disorder. And yes, sure. Yeah, yeah, thanks. And what we're interested in, right? So let's start with this quantum quench experiment. So let's say I have an initial state that is a simple unantungled product state. Let's say, for the spin model, this would be, for example, spins pointing randomly in the plus or minus z-direction, right? And then what I am after is understanding this time-evolved wave function, and in particular, understanding the, you know, whether the system reaches a steady state at long times, right? How does it approach, right, approach to the steady state? And broadly, we would like to understand the different regimes, right, or maybe phases of this, right, of this dynamics, right? So what are the, if you wish, universality classes or regimes of dynamics? Of course, out of equilibrium, we are missing some, you know, simple and powerful organizing principle, like in equilibrium, we have symmetry and symmetry breaking, right? Here we don't have that, and we are somehow, we have to find examples and discover mechanisms. Well, so let me be quite loose here. So just qualitatively different behavior, right, qualitatively different approach to the steady state, right, or nature of the steady state. That's what I, any other questions or comments so far? That's an excellent question. So let's, let's get to it a bit later. Yes, because that, that has a different symmetry. And indeed, for example, when we talk about the physics of disorder, symmetry would be quite important. Okay, so now just to set the stage. So those are the questions that somehow motivate our discussion. But now to, you know, just start with a very simple observation, right, to understand, to get an insight into dynamics, we need to focus on a few things. So let me start with some simple observations, right? So to describe dynamics, I, I can start from eigenstates. So let's say I have my Hamiltonian of this form, and I have a set of eigenstates, alpha. I don't need this actually here. So just alpha, right, where this index runs one to the Hilbert space dimension, where Hilbert space dimension is two to the power. And for example, for our spin one half model, and being the number of sites. And, and then I can say, well, my initial state, I can expand in terms of, in terms of the eigenstates, right, with some coefficients, which, okay, I can qualitatively I'll try to understand. And once I've done that, then the time evolution becomes very simple. So I just need to multiply, multiply eigenstates by time-dependent phases generated by the Hamiltonian evolution. Ah, I see. So camera ends here. And, and then so what, you know, one way to characterize the state is by looking at the, at the observables. So I can choose some physical observable described by, by an operator O. For example, could be spin on some site I, spin the projection of the spin on site I. And then I can express expectation value of this operator in the time evolved state, right, by just plugging, plugging the time evolved state. And what I will get, so let me just try the answer, because I think that must have appeared in lectures by, by Natalie. So I've, first I have the diagonal term, right. And, and O alpha, alpha, that's the matrix element of my operator O between, in this case diagonal between, eigenstate alpha. And then I have the O diagonal term here, alpha not equal to beta, c alpha, c beta star, e to the i e beta minus e alpha t O alpha beta, where I've introduced matrix element O alpha beta, which is beta O alpha. And that's the matrix element. So, and this means that to gain an insight into this question. So we need three ingredients, right. We need to have some idea of the eigenstates. Matrix elements are crucial, right. So we also need to understand these guys and how the, how your, your initial state is expanded in terms of eigenstates. So meaning this, this coefficient c alpha, right. And that's roughly, you know, so what we want to do is find examples where, for example, eigenstates and matrix elements would behave, behave qualitatively differently. And let me now jump to this board and just to orient you as to what we, you know, what we will discuss. So different regimes keeping in mind exactly this type of quantum quench setup. So first regime, of course, is by far the most common one is that your system would thermalize, right. In this case you find that it's, it's efficiently long time psi of t effectively will have reaches thermal observables, right. Let's say greater than some thermalization time. And in this case, so once that happened, right, your, your system has reached thermal, thermal equilibrium state, right. And then you can use statistical mechanics to describe it. So, and this, this is the most common disability. Another interesting aspect here is that whatever state you start from, right, what matters is just the global conservation laws. So your, your energy of the state, variance of the energy and things like that, right. And then in this case system forgets, right, forget initial condition, if you wish. Then on the opposite end of the spectrum here, we have a regime that is found at strong disorder, which is many body localization or NBL. So what is crucial here is having strong disorder. In this case, your, what you find is that psi of t remains non-thermal up to time, which is, can be extremely long, right. Up to time, well, let me call it t star. And up to this time, your system actually retains memory of the initial condition. So in this case, behavior of your system in a quantum quench is qualitatively different compared to thermal case. And that's one, one thing we will study, right. So, and in particular, there is a, there is an interesting set of local integrals of motion, which underlies this behavior. Now here, here I should mention, and we will discuss this more. So here, I am saying that this, you know, system does not thermalize up to some very long time. So we know, right, that, let's say the lower bound in some cases of this time, but really, you know, likely higher dimensions system eventually thermalizes, and we will talk exactly about the physics that underlies it. But for, for practical reasons, you know, on, let's experimentally relevant time scales, right. So the system appears non-thermalizing. Yes. Yeah, that's a very interesting question. So the question is whether, well, if your disorder is time dependent, right, whether you only have eventual ergodicity. Well, I think it's a question of time scales. First of all, right, that you would have, I think, for example, if you were to take a finite size, many body localize system and then make disorder time dependent, really, really depends how exactly this happens. And there are other interesting phenomena, maybe we can come back to that. So here also, I should make a comment, you know, thanks for asking the question about, you know, eventual ergodicity. So what, what we will not touch upon is glassiness. Because glassiness, right, that's something, of course, very rich. And that's a way to break ergodicity in a different sense. But that's largely classical phenomenon, right, that comes from having frustration and from developing rough, right, sort of, sort of very, very uneven energy landscape, such that your system gets stuck in some of this minima. So in a way, you'll see that physics we'll talk about is much simpler and, you know, maybe apart from this question about, for example, estimating thermalization time scale in a two-dimensional system with very strong disorder, right, this physics is much simpler than glasses. Now, in a way, what you would see in a, you know, in this course is that many body localized systems are very non-thermalizing, right. They have very little entanglement. They have lots of, um, integrals of motion. So naturally, there are also things that line between, let's say, quickly thermalizing system on one hand and many body localized system on the other hand. And so in particular, there is a set of phenomena which are called pre-thermal phenomena. So the mechanism here is pre-thermalization. So this is a pretty broad, pretty broad set of phenomena. And I will tell you about one particular set of phenomena where your system thermalizes, but really extremely slowly, parametrically slow. And what we will see is in this case, so I'll focus on the range of phenomena where you can actually even, you can be rigorous, so because they're sufficiently general and simple and you will be able to construct a few in this case, a few approximate conservation laws and prove that they would be conserved for a long time, construct approximate integrals of motion. And yet another possibility, right, is having, is to have what is called sometimes weaker electricity braking, prime example being quantum, many body scars. And we will see that in this case, there is a special set of initial conditions and eigenstates which are non-thermal, but you know, for other initial conditions, your system thermalizes. So this is related to special non-thermal eigenstates, eigenstates and initial conditions. And so that's another scenario when your system does not, let's say, thermalize in a quick simple manner. Yeah, please. So there will be a parameter and we will prove a bound that this approximate conservation law is stable up to the time which is exponential in this parameter. Well, it's not a phase transition because it always, if you have a big enough system, it always decays, right, but this time scale can be extremely long and you can, you know, you can control it by tuning this parameter. And this parameter could be, for example, in the, this type of a model, right, could be cranking up V, for example. And then, you know, you would see that your system develops an additional conservation law, right, which leaves time, you know, exponential in V over T, for example. Yes, yes, yes. It's not, I think, you know, I will focus more on the, on the regimes, let's say, right. I think phases, phase, maybe it's too stringent, right. So let us think about, you know, let's be happy to think about phenomena which are quite very different on some long time scale, right, but maybe not on the infinite time scale. And especially, you know, especially because I think we'll get to that, but for example, for example, really learning something about many body localization in the very long time limit and in the very large system limit, it seems to be in this corner where either you have to prove something rigorously, right, or just, you know, you, you cannot get there with experiments or, or numerics. Although, as I said, there are arguments, right, which suggest, suggest stability in two dimensions eventually. Any other questions? Yeah. Okay. We will, we will get to that. We will get to that. So this is more to just orient you, right, give you sort of a map where we are going. Yes, yes. In this case, you can actually, there is a recipe how to construct them. Yeah. Okay. Yes. Yes. Well, it's a good, I mean, pre-thermal phenomena are, they don't require strong disorder, right. And I think we understand in the absence of disorder much better, right, while, you know, with, for example, this argument for destabilizing many body localization to dimensions. So this will involve rare thermal inclusion, which slowly, slowly grows and slowly thermalizes. But there are all sorts of questions which we don't know answers to what should be the size of this inclusion. And right. So that's difficult to answer. So, but, but indeed, I think you would see people using this terminology saying that, oh, this is a pre-thermal regime of MBL, for example. So, so pre-thermalization is a broader, right, broader, broad concept for having some, you know, for some reason, very long termization time scale. Okay. Any other? Yes. That's right. Well, in this case, so, I think this is more related to this type of result that I will describe. So that's an application of this result. So I will focus, you know, indeed, I think a relation of what we'll discuss to, for example, what Norm, Norm talked about in his lectures is that I will focus on mechanisms, right. And I think he focused more on the applications of these mechanisms. Because time, sort of, time crystals, they rely on having many body localization and periodically driven systems, right. That's an example of a phenomenon enabled by many body localization. But under certain conditions, right, you also actually, you can see parametrically long time crystal type behavior without disorder, right, but protected by pre-thermalization. So that's sort of an extension of this idea. For, so people usually, people, to me, it's a here, right. It's some combination of these two. So to me, I mean, of course, Hilbert space fragmentation, this is something special, right. But then if you think about stability of that to perturbations, right, I think you got pretty quickly to pre-thermalization. And in fact, this connection was, was a little bit made in some, in some papers. Okay. Any other questions? Non-nargodic extended, non-nargodic extended state. So are you, what do you have in mind, which model, multi-fractals in many body systems? You mean, are you mean like, for example, having something like random regular graph or, well, those are, sorry, could you repeat the question? So you're asking, not, not in this sense. So I think, I think there it's a little bit of an open question. But, but you know, I think one difference of what I'll talk about from this, this story is that this, this is usually zero-dimensional, right? So like random regular graph. For me, locality will be important. So actually, pre-thermal idea of pre-thermalization that I'll tell you about, so that relies on locality very strongly. Maybe, maybe let's get to that. I think this is, yeah, it's, you know, I'm very happy you're asking questions, but you know, hold on and then we'll get to that. But maybe let me take another, yes, yes. Yes. So this, this is a full care, full care pre-thermalization is a prime example of what I will talk about. Well, in some loose sense, it does require disorder. If, eventually, you know, if you really think about what happens at extremely long times, it is here, right? But, but, but practically it is important because, you know, this conservation laws can be extremely long lived, right? And they give rise to new phenomena. Okay. So good. So that's, that's our, our plan. And I heard something, but if there is a question, please, please ask. Now, there is also, right in the title we had entanglement and let me just spend five minutes on commenting on, on the link between entanglement and complexity, sort of at a very naive level, right? So you can wonder that, you know, let's say I have a spin system of N and spins, right? It has Hilbert space dimension of two to the power N and you can wonder, okay, it's a large Hilbert space. So how come we can actually understand anything at all about many body physics, right? And the answer is that at least a partial answer, it's not, I don't think it's the, there is more to it, but turns out that some states in this Hilbert space are simple to describe classically, right? And often physical states are actually in this simple category. States can be simple and easy to make sense of classically in various ways, but one is related to how, to how quantum they are, how much entanglement do they have. So the way to, so for example, let me take some state psi in this Hilbert space, right? And I can ask how non-classical is it, right? So how far is it from being a product state? One way to quantify it is by computing entanglement. And so let's say I have my system and what I can do, I can take a subsystem A, so which is a block, right, a block of spins, can be a block of adjacent spins or maybe, you know, can be disconnected block of spins. And so let me call my total system S. And if I have a classical state, right, then state, right, state of the degrees of freedom here does not depend on the state of things outside. But if it is a quantum state, right, then, you know, there are quantum correlations and then depending on the, depending on the state of the complement, right, my subsystem finds itself in a different state. So what I can do, I can now look at the, I can trace, trace out everything except for, for my subsystem A, right? So I have a trace over the complement of A of density matrix of my spins. And, you know, if my system is very entangled, then rho A would look very mixed, right? And I can quantify degree of this being, being mixed by computing the von Neumann entropy. So which in this call, in this case becomes the entanglement definition of entanglement entropy of A is just a trace minus trace rho log rho. And then I can look the way to characterize the state is to look at different subsystems, right, and computing their entanglement in this global pure state of the system. And, you know, so for example, maximum amount of entanglement, if I have L spins, then maximum amount of entanglement would be L times log 2, right? So you have log 2 per spin entanglement. And if you have a non entangled state, entanglement entropy would be zero for any division, right, any choice of the subsystem. And the remarkable fact is that states, right, states with, so that the way your entanglement entropy scales with the subsystem size actually tells you something about simplicity or complexity of your state of the system. Right, so what, you know, an important quantity to look at is the scaling of entanglement, right? And if you have a situation where entanglement grows proportional to the volume of the boundary, right, boundary of your subsystem, so in one dimension, it would mean it stays constant, right, because the boundary is just two points in this case. So in this case, you're close to a product state in some sense. And, you know, for all practical purposes, you can compress such states and approximate them using, classically, using only a polynomial, so N to some power, number of parameters. And so in the technology for this compression, so that's tensor networks, I will not probably go into it here, but so, but probably disappears in some following lectures. And these such states are called, said to have area law entanglement or boundary law entanglement. And the terminology comes from, comes from, from Henry physics, from, from black holes. On the other end of the spectrum you have, and so, and of course, you know, importantly, ground states of most many body systems are in this yellow entanglement category, right, and you could say, well, on some level, that's why we can learn things about ground state physics, right, it's still very rich, but you know, in terms of complexity of the wave functions, they're, they're on the, on the easy side. On the, on the other end, you have states where entanglement scales as volume of the subsystem. So these are the volume law states, but well, so that depends on the state. So what is important is that it is polynomial and not exponential in N. So these are volume law states and this, these are states which are very entangled, right, so essentially you have your subsystem, but every degree of freedom is somehow strongly entangled with the outside world. So these states are very far from polynomial states and they cannot be, right, so we don't have a way of compressing them efficiently. And a priori number of parameters goes exponentially, right, exponentially with, with a number of degrees of freedom. And highly excited states, non-equilibrium states are often in this category, right, and that's, that's the origin of the complexity of the computational complexity of phenomena we'll talk about. And in a way, you know, if you want to, right, if you have a quantum computer, right, you want to get into this regime, right, have states which are not easy to approximate classically. Now, of course, not all highly entangled states are non-trivial, right, if I have a highly entangled state which has vanishing physical observables, okay, it's also, it's not very, right, it's not very easy to probe or distinguish from another such state. Right, so here, you know, it's interesting to find examples which are computationally challenging, but at the same time, still have clear, let's say, clear signals in observables, so that's a bit of a challenge. So, and okay, so with this, after this comment, so let us spend the remaining half an hour, or perhaps we can be even faster, briefly talking about the state thermalization hypothesis. So, and in 2D, sorry, could you repeat the question? Yes, yes, so indeed, there are things that's a good comment, so there are, there are examples where it's an area law, right, area law, but you have logarithmic correction, so let me broadly put them into slightly entangled state, state category. So let us talk now about thermalization, and that's something that Anatoli covered already, ETH, so here my goal would be more to just, you know, give you, give you a refresher, right, and maybe make some comments which we would find useful for the later on for discussing mechanisms of non-thermalization. And here, right, there is this quite a remarkable connection that also, you know, you also learned about in Thomas lectures, right, with the, with the random matrix theory, which is, you know, intuitively quite surprising, so random matrix models are just, you have, right, a bunch of levels which are all strongly interacting with each other, so these are zero-dimensional systems, right, and then of course they allow you to get nice analytical result, you have, right, you, you have, you know, that the eigenstates in this case are random vectors in this Hilbert space of your, of your system, right, so you're dealing with eigenstates that are Gaussian random vectors, and then ETH of course, as you learned, right, it, it tells you that important aspects of this random matrix, random matrix theory also show up in, you know, in physical models of the type with local interactions of the type that I wrote, you know, I wrote half an hour ago on the blackboard, so, and the intuition, right, so the intuition that why, why this happens, perhaps one, one way to understand it is as follows, so we have, so let's say we have a system with local interactions, so let me divide it into the left, right, left part and right part, right, so each one has a lot of levels, let me label them by alpha on the left and beta on the right, and, and then, you know, so these are eigenstates of each sub-system, and then I connect them with a local interaction, right, the left, right, intuitively, right, what should happen in this case, your eigenstates of the combined system, of the system, so let me call them N, they should, so what should happen, I should take some energy shell here and here, right, and mix somehow take a, take a superposition of state, right, state of left and right system with, with random coefficients, and that should give me, you know, an idea of what the eigenstates look like, right, so there is, once I connect the system, this, this state in this energy band hybridize, and what we can expect, right, is that eigenstate looks like that, so it's just sum over alpha tensor product beta with some, you know, restriction on energy, delta energy, right, where these coefficients are now random, in a sense, they, they, this picture tells you that your individual eigenstates of the system probe, right, they probe all, all possible configurations, right, of left and right part, so that's the intuition, and we will post-factum, right, once we look at the ETH, and the informatics elements, exactly you see that this is what, what happens, right, so your states, this type of states are, of course, highly entangled, and this whole picture is self-consistent as you go to larger, larger and larger scales, now let us, unless you guys have any questions up to now, alpha and beta are not entangled, no, they're entangled within a subsystem, right, so I'm imagining that I've constructed eigenstates at a certain scale, and then I'm doubling the scale, and I'm asking how do I glue, right, how do I glue the eigenvectors that I, I got at scale L, together to go to scale twice L, what does it mean? Well, it tells me that my, are you asking what does this mean, this, this condition? Well, you know, if you think about a local model, then the norm of this connection is over the one, right, so you know you cannot violate energy conservation by more than one, so that's roughly, that's roughly the intuition, right, that this, this, this hybridization allows you to mix things which differ a little bit in energy, but not, right, let's say not too much, well, here I'm being very loose, but yes indeed, you would, you know, I think there is, of course, there is a, there is a condition on the structure of these eigenvectors and matrix element which would give this form to, to the eigenvectors of the combined system, so, and you're, you know, you're, you're, it takes exactly, you know, I think you cannot prove it in general, right, but you can convince yourself that it is, it is self-consistent, that, that if it holds approximately at this scale, then you can go to the next scale and then we'll only get better. Eigenstates, where, where am I using it or, well, I think, I think thermalization is stronger than that, right, it also, things that are non-local, you know, they also thermalize, of course, you find some interesting phenomena which are a little bit related to a semi-classical limit if you take very long range interactions also, right, and that's, that, that gives you an extra, sort of extra dimension in this phenomena. Okay, so let us now just recall ETH Anzatz, it's an, I assume, so as I said, I assume that Anatoli already covered it, so I'll go fast, but please stop me if it's, you know, something isn't clear, so ETH is essentially supplementing random matrix theory with, with locality, right, and putting, you know, putting this, putting this intuition that I described into, into an Anzatz for matrix elements, for matrix elements of physical operators. So physical here is important and in a second we will discuss exactly what, you know, what physical operator means here, but so let's say I have an operator O, right, then this Sridnitsky, so this is due to Sridnitsky and Deutsch, OMN, which is N matrix element between eigenstates M and N, that looks like that, there is a function, so this now becomes not an operator, but a function, so I'll, you know, just draw, draw a simple O without a hat with E bar, delta MN, plus entropy at the respective energy, F O per omega, R MN, where there is a couple of properties here, so O of E and E bar is the average of the two energies, E N plus E M mean and omega is their difference, there is a couple of extra properties, so O of E bar, smooth function of E bar and it's very close to the micro canonical value of observable O, so MC is micro canonical and it's smooth function of, and same thing is function F O of E bar omega, which enters the off diagonal matrix element, it's smooth and it's actually related to the response function of your system, right, linear response function, so that's the, right, that's the ansatz and you know, what roughly what it tells you is that, so here this factor, right, that's the entropy at your respective energy, so if you are looking at all the eigenstates or at infinite, infinite temperature, then this just becomes one over square root of the Hilbert space dimension and this is a factor which comes, which is a scalar product of two random vectors in your Hilbert space of dimension D, right, that's the origin of this factor here. Otherwise, you know, it tells you that tells you that each eigenstate, it has observables, diagonal observables, which are essentially very close to the micro canonical values, right, so there, all the eigenstates and thermalizing systems look like Gibbs ensembles, if you wish. A priori, you know, if you think about this picture, right, this makes sense because you're effectively probing, right, you're probing all the eigenstates of your subsystem model of energy conservation, right, and therefore it's quite natural to expect that this would hold and this would get better and better as you increase the system size. Now, yeah, cars would violate VTH, that's exactly what's interesting about them, but they would violate both infinite size system or force solvable models, yeah, both, both, both diagonal and off diagonal, yeah, wave function would not be, well, so what we will see is that quantum cars, quantum anybody's cars, are a set of measures zero, so there are quite, quite a few and then other, you know, other eigenstates actually satisfy, satisfy VTH, so in a way, you know, yes, on the set of measures zero this would not be a smooth function, there was a question over there, why is it so you're asking about the origin of this factor, right, well, it comes from this intuition that eigenvectors are kind of like random vectors in the Hilbert space, so, and you have, so imagine I have N and M, right, they're orthogonal, but then I'm acting on them with some local operator, so they're random and orthogonal, but I've acted on it with an operator, so it did something, right, but it still, it stayed largely random, and then this becomes just a scalar product of two random vectors, right, and typically, typically so you can, right, you can estimate this to be one over square root of the Hilbert space dimension, right, so just choose, choose the basis where this is the first vector, and then this guy, or vice versa, this is the first vector, then the scalar product is just the first component of this guy, but it's random, so, okay, so each component is of order one over square root d, we'll, we'll get there, we'll get there, good question, but we'll get there, yeah, okay, yeah, no, no, because then you have to worry about other conservation laws, yeah, exactly, right, I think it's an interesting question within, you know, within a sort of, if you have more, more conservation laws how to modify it, and in some cases you can do it, right, but yes, yes, by, by, by adding more, more conservation laws, yeah, okay, so small comment on what are the physical operators, so turns out that this is quite, quite genetic, so if you take, for example, any operator with a, with a support that is finite, right, so if you take, you know, sigma z or sigma x operator on one side, or you take a product on two sides, or on a few sides, right, this is all good, you can take some, right, some of local operators on different sides also, so that would also satisfy, satisfy each, and you know, what is, those, those are all operators you can measure, right, reasonably easily in experiment, now things that do not satisfy each are, for example, projector onto an eigenstate, many by eigenstate, right, those are, because exactly, so those are in the operator form, they would look completely crazy, right, they would, they would involve a lot of products of different spin operators, so that's something that you cannot reasonably measure in experiment, unless you have sort of exponential precision and time, right, so this is not, this would violate ETH nature, and you know, going back to concrete models, so for example, one simple model that has been, well, simple looking model, because it's, it's actually quite rich, an example of a thermalizing model, for example, you can take transverse fieldizing model, right, and then break integrability, so this is integrable by mapping onto free fermions, and then you can add integrability breaking Z field, and so this, this model would satisfy each, and right, it is, it is thermalizing, and this, right, so as, you know, already appeared in some lectures, right, so this is a very useful tool, so you can, you can do exact normalization, right, and check, check many of those properties, because thermalization in some of these models happens very, very quickly, right, thermalization in general is a very strong, strong tendency, right, due to the fact that your helper space grows very quickly. Now, just maybe, maybe one more comment on, on the page, so, so I'm right assuming that Anatoliy covered, right, covered this in some detail, so there isn't maybe just a couple of comments, first of all, so we know that, you know, this, this ansatz imply, implies thermalization in a quench experiment, right, and you can relatively easily derive that, if you start from a, from an equilibrium state, then at long time it would fall to a microthonical value of observables and the fluctuations, temporal fluctuations around the state would be exponentially suppressed in the system size, right, so ETH implies thermalization in a quantum quench, then this function fo is related to, well, f squared in this case. So it's related to the spectral function of your system and the independent correlator of t of zero, right? So that gives you access to response functions. And that's something that you, you know, that is not provided by th, right? That's, that varies from system to system and structure. So you can use sometimes physical intuition to understand the structure of this function fo. So for example, right, as a simple example, if I have a system which is diffusive, system and I am, for example, I am choosing as my operator o, I am choosing the projection of the spin. So I assume it's conserved and my system shows diffusive transport. So then I can expect that this correlator, z of t, z of zero would decay diffusively. So in one dimension, one dimension it would go as one over square root of t. And so in that, you know, from this we can, we can convince ourselves that this means that f function f squared goes as one over frequency to, to power one half. And of course in a finite system, right? So and especially, especially doing numerics we are, right? We are dealing with finite systems. So if the system sizes l transport in your system saturates one that, you know, once the particle reaches the, reaches the edges of the system. So, so, so the slowest diffusive mode is determined by the tautest time. So here it reaches tautest, which is essentially the time it takes you to diffuse, right, diffuse across the system. And this means that, you know, after this time there isn't much happening. So means that f over saturate becomes constant approximately at time, right? At frequencies that are below tautest frequency, right? Which is the inverse of the tautest time. Now I can sketch the behavior of this function, right? f of omega, right? So we have one over square root omega behavior while in the range where we have diffusion then we have a saturation at the tautest energy. And there is last piece, last piece, what happens at very large frequencies. So at the frequencies which are greater than your, you know, local interaction scale J. So for example, you know, in my, well here, literally you can take this J, right? That's your largest energy in your dynamics. And so you can prove that here, actually you would encounter exponential decay with omega over J. And that is something exactly that would be, we would see that is directly related to floccaprithermalization. So this is very generic and this holds for any local observables in a system with local interactions. Well, that's a good question. So, well, your system, maybe you would say, well, your system settled after this tautest time. It's greater, it's greater, yes, it's greater than tautest time, yeah. Because you would expect a Gaussian or something. Well, so this is just the time it takes you. This is the slowest diffusive mode in your system, right? And then, okay, so if this is your slowest process, that would set thermalization time. Yes, yes, yeah, so that's an assumption, right? This is not, I'm not saying this, this what happens in every system, but indeed, if your system is diffusive, right, that's the slowest mode. Could you speak up, please? Well, yes, you can be things, I mean, first of all, you can have slower transport, right? Or there are examples where thermalization time is longer than that. Yeah, okay, let me come back to this question, maybe let's discuss this, yeah, sure. You mean, so are you referring to the fact that there is a connection between, in terms of level statistics? Yes, yeah, so that's a very good question. So this is a scale, right? This is a scale where your system starts to look like a random matrix, if you wish. There are, I think there are cases where you can, you know, so let's say in one dimension, in a non-interacting system with disorder, so you can establish a relation, a formal relation between diffusion and level statistics, right, so that's been done in the 80s or 90s, right? I think in many body systems, so you expect them to be related, but yeah, I think there isn't such a clean, let's say clean connection. Okay, so I think maybe let me just make a couple more comments before we conclude for the day. So first of all, well, first simple implication of ETH-ansatz is for entanglement. So if I take an eigenstate N, right, then what ETH tells us is that physical observables are, but that tells me that if I take a sufficiently small subsystem, then every observable is thermal, so and that means that the whole state, right, the whole state of the subsystem is actually thermal, so let me call it reduced density matrix of subsystem A in the eigenstate N, right, and that would be actually to look thermal, because you know, if every observable is the same, then the states are essentially the same. And that implies, so we know that thermonuclein entropy scales proportional to the volume at finite, right, finite energy density. So that tells us that entanglement entropy in this case is actually very close to thermodynamic entropy. And that tells us that this should be a volume law, volume law scaling. Now that's an interesting situation where sure, the states are highly entangled, but from the point of view of observables, they're all kind of the same, right? So there is also, there is another axis of simplification you can have, so you can describe them in terms of hydrodynamics, for example, right? Because what you care about is a local, it's somehow local values of conserved quantities, micro-canonical, yeah. You can recalculate, right? You can take your, you know, you can relate your energy to temperature, right? So this comes, I think this follows like usually in statistical mechanics. And maybe, so let me just conclude with coming back to this qualitative picture, right? And just arguing that ETH makes it self-consistent. So, and that follows roughly from the fact that our eigenstates are very fragile to perturbation. So any perturbation leads to lots of resonances and a lot of hybridization. So let me take some Hamiltonian H, which is satisfies ETH, right? So it has some eigenstates H, M with energy E, M. And let me now perturb it. So perturb with some, you know, for example, changing some local field, right? Some local operator, where V would be physical operator, and then let me ask how does this modify the eigenstates? Well, so what I can say, I can go to the basis of eigenstates and look at the matrix elements now. And ETH exactly tells me that V and M, right? So the matrix element of my perturbation between two eigenstates goes as epsilon over square root of two to the power N while smallest energy difference between the energies, right? So EN and EM. So I have two to the power N eigenstates, which are spread over energy that is only proportional to the volume of the system, right? Only proportional to N. So this goes as something like N over two to the power N. So that's the level spacing, right? And what you see that perturbation theory here breaks down miserably, right? Because perturbation theory for this to be, to have only perturbative effect of eigenstates, I need that matrix element epsilon over square root of two to the power N be much smaller than delta of N. And this breaks down already for perturbation, right? Which is exponentially small. So if my epsilon star is greater than one over square root of two N to the power N, I have resonances. So I'm mixing this perturbation leads to, so I have to do degenerate perturbation theory essentially. And so my eigenstates would be reshuffled, right? I'll get a new superposition. And that's exactly, now you can take this argument and think how it would play out here for this setup. And you will see that exactly this would lead you to hybridizing, right? Hybridizing in this finite band of states and that's what leads to this form of the wave function. Then you can leave this argument and take it to the next scale and see that this is all self-consistent. So let me stop here. I think we sort of set, right? Sort of set our maybe objectives and refreshed eigenstates termization from the angle that we would find useful for what we'll follow. So, and then remaining two lectures we will cover, we will next focus on many body localization and then pre-thermalization and quantum scars. Thanks guys, if you have any questions, let's, ah, sorry, reference. Ah, you mean reviews of modern physics, review. So I think that would introduce in particular many body localization, right? There have been interesting developments since then, right, so which I am happy to give you things which happened after. After it, but indeed, I think that's a good reference for many body localization. For thermalization, there is in particular, you know, more detailed review by Natoli and Luca D'Alessio and collaborators in Advances of Physics. These things, they're not in those reviews, so, but I will give you some references once we get there. Or is it more general? Very good to say it again. When we talk about thermalization or realization for many body systems, do you only talk about real space or is it more general? About thermalization, you mean? Yeah. That's a good question. So in the sense of observables or? That's like, yeah, maybe yes. Indeed, you can measure, for example, occupation numbers in the momentum space, but typically they would be expressed back in terms of local in-space operators, right? So you would come back to it, in a way. So I would say, yeah, it comes back to the, yeah, I mean, you need some notion of locality. So locality in real space, or I can talk about locality in momentum space or some other abstract space? I think so, I think so. It's more general. I mean, the range of observables which thermalize, which do thermalize is quite broad, actually. It is, you need to really take very big operators to, not to see thermalization in this system. Okay. Everybody's exhausted. It's like we are exhausted for the day. Thank you.