 Okay, so you are starting recording, but I think we are in time and probably we can start. So, okay, so first of all, thank you very much, Maxime, to be here and to give this ICTPC webinar, say, due to this situation. As I told you, we will be happy to invite you physically when everything will come back to the normal situation. Probably it's making it really sense to introduce you to everybody because I think most of you guys know Maxime. As far as I remember, you were in the US, now it's a couple of years, you are in Austria in IST, so you are an expert on non-equilibrium thermalization or more effects related to anomalous thermalization or not thermalization, many body quantum systems, many body localization and today I'm very excited about the talk because you will talk about something, so many body quantum scars and so please go ahead. Thank you, it's very nice to be, to visit at least virtually, especially since I would say that my work on quantum scars owes to its existence to ICTP conference, I will mention that in a couple of slides. But today, I will discuss most recent results, again, there is a number of publications and I will focus on the latest couple of works which actually discuss relation between scars, anomalous thermalization and phenomena of mixed space, which is kind of very common in classical systems and I kind of discovered it in a weird way by writing to it through quantum systems. So let me start just to set up a basic question, everybody probably is familiar with it, but still to be on the same page, we are interested in what is the fate of isolated quantum system under the Unitary Evolution and the system as a whole remains in a pure state, but what is happening is that subsystems like I'm showing here may become thermal in the sense that their density metrics may resemble more and more thermal density metrics. And now, how this happens, there is one specific route to such thermalization which goes under the name of Eigenstate thermalization hypothesis, which suggests that individual Eigenstates are thermal. And if your individual Eigenstates are thermal, right, then dynamics, what dynamics does to you, it defaces, it screws up phases of different Eigenstates with respect to each other and you arrive very naturally to a thermal state. It's a very powerful hypothesis, has a lot of consequences, but of course it's natural to ask do all quantum systems follow this hypothesis and use it kind of set and store. And of course, no, right, many people I guess in your community work on integrable systems, there is another example that I'm just going to briefly contrast, right, many body localized systems and expectations from thermalizing systems shown here on the left include ballistic growth of entanglement, right, there is this quasi-particle picture of light cone spreading, which works even in the case when transport is not ballistic, right, even if transport is diffusive, entanglement spreads ballistically. Eigenstates which I showed here, I'm trying to sketch an Eigenstate and these guys need some sort of entanglement structure and you would see that entanglement scales are the volume law, so if I do a cut, right, entanglement of the subsystem is going to increase proportional to its size and again, it's believed that these systems are governed by Eigenstate thermalization hypothesis. Now in presence of strong disorder, totally different regime emerges, entanglement dynamics becomes logarithmically slow, right, Eigenstates become much weaker entangled, it's governed by quasi-local integrals of motion and why, partially why it is interesting is because it's an example of quantum coherent dynamics on a much longer time scale, right, because in this case, this ETH practically in many models it kicks in after very short microscopic time and practically in experiment, you would see entanglement increases, right, and then saturates very quickly, right, as a function of time and after this time scale, right, experiments are going to see just stuff that has relaxed. Now localized systems offer much longer time scales and in that sense they would also, they can be helpful for just been marking your experimental settings. But now what about disorder, right, so disorder is kind of believed to be necessary for MBL, even though again there are also some discussions about MBL without disorder, but the question is, can we go without disorder? And now again, I'm pretending that this is question I asked, right, but in fact, this system basically fall down on my head when I and my collaborators visited 3S in 2017, just when I started to ICTP, so we started at IST, it was very pleasant drives, there was a conference in 3S, and that's where we heard this talk by Misha Lukin, who was showing his latest results from a Rydberg chain. And I'm not going to go into details of experiment, for me, I'm just to put everybody in the same page, just describe kind of description of this Rydberg chain from a theorist point of view, right? So they managed to trap a bunch of Rydberg atoms, which I'm showing here, right? Each atom is an individual laser. And now I would think about atom as a two level system, it has either excited Rydberg state or ground state. And now the physics is following that if atom is in a Rydberg state, if this guy is in a Rydberg state, its neighbors, this guy, right, it will have such a strong energy shift that it's off resonance. So it cannot reach Rydberg state simultaneously with its neighbor. And that's, again, very cartoon picture, right? There is an energy scale involved here, and in fact, Hamiltonian, but for now I'm ignoring it all, right? And I'm operating in condition of Rydberg blockade, right? I'm saying that such configurations aren't principle allowed. But what is happening is that in this system all atoms try to do oscillations between ground state and excited state because of laser shining on them. And now if one atom is in a Rydberg state, another one is so much detuned that it's not resonating. But that's how we arrived to this Rydberg blockade, right? And experimental surprise was that for some states like this one, I would call it Z2 state, there exists coherent long time oscillations. And for some states like this one, which is, looks even more uniform, right? There is no density wave here, just everybody's in the ground state. System would relax much faster. And experiment kind of motivated us to look deeper into this model, and that's where after this conference all the story kind of began for me. And if you look at this model, right, the effective model that I described, I write the Hamiltonian here, I would call it PXP model, right? So it's a paramagnet, it's a collection of free spins. But in presence of projectors, right, these free spins become not trivial. And this projector, right, again, it enforces this constraint that no two Rydberg atoms are excited at the same time if they're nearest neighbors. And now you see that this constraint is actually does already something interesting or even at the level of Hilbert space, right? For example, it destroys a product state, product structure of a Hilbert space. Instead of, normal here was paid for two spins, we have four dimensions, as I'm showing here, right? But this guy is dead because of Rydberg blockade. And now we have a three dimensional Hilbert space and it would be scaling as some combination of the number that depends on the boundary conditions in your problem. So here, I'm assuming so far, periodic boundary conditions. And now this model, right, is again starting with a free paramagnety, striped out some configurations by this projection, right? But and yet the remaining matrix elements are always one in this metric. So I'm not gonna use it extensively throughout the talk, but it's actually interesting to think about this model as a simulating a ginormous graph, right? In a many body Hilbert space, I can label all vertices of this graph by corresponding product state of Rydberg atoms with a constraint, right? And now what my Hamiltonian does, it comes, let's say, in this configuration. And what it tries to do, it tries to flip every possible spin, right? And if it flips this guy, it goes here, right? If it flips this one, it goes there. If it flips this one, it goes here, right? And it cannot flip any white dot here because of Rydberg blockade and so on, right? So this basically, I'm trying to sketch how this graph corresponds exactly one to one to this big matrix, right? The adjacency matrix of this graph is my quantum Hamiltonian. And now it's a very modest graph I draw here. But if you believe in an experiment with 51 spin, right? There would be much, much bigger graph, right? It would have a lot of vertices. And then as I show in this next slide, the puzzle is that I prepare a system on one given vertex, let's say I prepare a system here, right? And if I expect a good density in a quantum dynamics, right? I would expect that the system spreads out throughout the graph and never comes back. Instead, as we are gonna see in this slide, what is happening in the system if you simulated numerically, right? And I'm looking now at the right plot, right here. If I simulate numerical, I can start system as I promised in this Z2 state and track return probability. So I initialize it on one vertex of the graph and try to ask is there any sizable probabilities that it will return there, right? It will be with a high amplitude in the same product state. And surprisingly, you see the same a typicality as they saw in experiment, numerical simulation, right? You see that for Z2 state- Sorry, Maxim. Sorry, is this probability surviving the thermodynamic limit? I mean, it's something that- No, this probability is expected to go Z to zero in the thermodynamic limit, right? But if you wish something that is meaningful, right? It is a fidelity revival per spin, right? So you can map, write f small minus log f divided by a system size. Right. And that thing, you would see that it is pretty much constant. Okay, so the density, okay, yes. And it converges very, for very, very small system sizes, it's already converged, right? So, but of course, yeah, due to orthogonality catastrophe, you don't expect, you know, this fidelity here, right? This would strictly go to zero in the thermodynamic limit, of course, right? But what I'm excited about is this, either this measure fidelity per spin, or, you know, in some, in other way, you can see stronger typicality here, right? Let's say Z2 state does have revivals, Z3 state does have revivals. And Z4 state is kind of barely visible line. You can barely see this magenta line. It's fiddling there at the bottom. So Z4 state doesn't care. It says, oh, no, you know, I don't want to revive, goodbye. Right. And the same you see on the left, on the left is actually ITV dissimulation. So there is no boundary, right? It's an infinite system, but of course limited time evolution. And you see that entanglement for all initial states, for zero states, Z2, Z4, and so on, it grows ballistically, right? You see a linear slope of entanglement growth here. But this linear slope is very, very different, again, as you notice. Entanglement for zero states shoots very quickly, right? We cannot simulate after time 10 because of too much entanglement. But for Z2 state, entanglement still keeps moderately low. And there is this puzzling entanglement oscillations that I'm showing here, right? That's entanglement subtracted from a linear slope. You see it oscillates. You see oscillations in local observables. Again, it's an infinite system. There is no boundary effects. And you see that up to time, you know, 20 and probably up to time hundreds, there are still going to be oscillations which haven't decayed, right? So that's kind of was the puzzle. Why some states revive, some states don't. And where is the specificity comes from in this model that was kind of initial striking thing that looks at you when you try to understand experiments by doing numerical simulations? And, you know, there are now two approaches. One can think about eigenstates of this system. And that's something that I'm mostly not going to speak today about, right? And let me just briefly mention what is happening in eigenstates. And not surprisingly, if you have an oscillations and very strong revivals, you expect that your original product states has a very unusual expansion of an eigenstate. And that's what you indeed confirm numerically. You, I plot here amplitudes of overlaps of all eigenstates with a given product state. And you see that, you know, most of the states are here, right? But still, there is this number of atypical eigenstates, right? This model has a particle whole symmetry, e to minus c is kind of a symmetry. So they have partners on the right. But there are these special eigenstates which have, especially here, very strong overlaps with Neil, right? And moreover, they're approximately equally spaced in energy. And here, my revivals come, right? I expand my Neil state, I expand the two guys, mostly over this few states. And they have approximately periodic energy spacing. And they're going to, you know, they're going to be this many body revival. Now, surprisingly, these very same states, right? You would guess that they must be somewhat anomalous because they don't have typical overlaps with Neil. And if you plot entanglement entropy, you see that these are the same seven states. And that smells like a pretty serious violation of ETH. Of course, it's violation of ETH for very few eigenstates, right? Most of eigenstates, still shown by color, are here. Are at the top of this parabola, as you expect from ETH, right? This is the center of my many body energy band, right? Maximum density of states, maximum entanglement. Yeah, everybody is there. But now these very small, pretty much seven states, they manage to escape, right? ETH parabola. And they hang out down there, right? And, you know, part of the, you know, very quickly, we started to understand and started to explore what's happening and why these special eigenstates exist, how to explain them, right? And we attributed them at some point to hidden embedded as you two represent age. And I will mention this a bit towards the end of my talk. But, you know, there are two kinds of people, right? There are people who like eigenstate and there are people who say, you know, eigenstates aren't physical, very hard to prepare, don't care about eigenstates. And this is kind of my approach today. I also will say, okay, this model has some bunch of special eigenstates, we'll come back there later, but let's think about physics without invoking eigenstates, right? Without thinking in the eigenbasics. And, you know, Kiehlberg space picture is kind of complicated as I was showing, right? There are these special eigenstates, right? And these special eigenstates, they would correspond, they would help system to go from this vertex of a graph to this one and back and forth. But dynamically, if you forget about Hilbert space, right? Your system looks very simple. You oscillate between two Niel states. You oscillate between Niel and Niel shifted by one side, right? Niel and anti-Niel state. And it's a coherent two unit cell ribosilations. And at the top, by the way, I'm showing experimental data, right? You see it's not as perfect, but you still see that this pattern, right, is displaced by one unit cell and then again displaced back and so on, right? This is experimental data from initials looking group and you really see the same Niel collective oscillations. So now what I want to kind of discuss in more details in my talk is actually how to understand this stuff and I will be using variational principle, right? For me variational principle is just some device, right? And you feed in your quantum Hamiltonian to this device, you must specify some variational state and it gives you classical equations of motion. Of course, there are many makes and models. People are working to develop new machines, mean field, truncated beginner, some Gaussian state projection. I will be actually using metrics product states and time dependent variational principle, right? And that allows to put an explicit parallels, right? It, as I will show, right? What I'm gonna be doing, I will be mapping my quantum system, projecting it on some manifold and obtaining classical equations of motion. So now why do I need metrics product states? Actually, it's mostly because of constraint, if you want. Because of course the best thing to do or the easiest things to do would be to do a product state, right? But the product state has a problem. It violates a Rydberg constraint, right? Because if I take a product state with a unit cell size two as I'm trying to sketch, right? That's my unit cell. I would repeat this unit cell again and again, right? There would be a problem if this pin points here and the next pin points there, right? In my spin language, there is a constraint that such configuration is absolutely prohibited in my wave function. And this wave function is not obeying such a constraint, but there is a very easy trick how to make it obey such constraint, right? There is a magic following metrics if you like sigma plus or sigma minus. And this power matrix, which is written here, it squares to zero, right? So basically the trick of MPS here is that MPS is encoding very, pretty much the same physics. It's encoding a product state of spins where odd spins, they point at some angle theta odd, even spins point at some angle theta even. But now MPS implements projection because if your spin is pointing up, you assign an amplitude that is proportional to sigma plus. And by the same, by the matrix multiplication in this matrix product states, if you have a up here and a up here, you would multiply and you would get zero. Your wave function is gonna vanish. And then a down or a ground state basically follows by normalization, right? So a down has to be some other metrics, which, again, there is only one parameter here that's theta, which specifies tilt angle of your spin. And now, again, this is, I hope I persuaded you that this is nothing else but a bit more glamorous version of a mean fuel, right? And that is really necessary because of this constraint. But otherwise it's a very simple answer and what already Pernia group did, they projected quantum dynamics onto the sun's out. And projected this dynamics basically by the means of time dependent variational principle. So they found the best representation, right? The close approximation to the quantum dynamics in this variational manifold. And it gives you equation of motion for these two variables, right? It gives equation of motion for these two parameters, theta A and theta B or theta odd and theta even, where we are assuming that unit cell is size two and everything is periodic because that's what we want for our neon state, right? Our neon state has unit cell size two in the internet system, it's periodic thing. And now surprisingly in this nonlinear system of equation, what when they and others found, they found this periodic trajectory that is shown here by red line. I'm trying to, you know, to circle it. And this trajectory goes exactly between two states, neon and anti-neon. So once you project this quantum dynamics under such a Hamiltonian onto this variational manifold, you recover pretty much same physics now in the classical equations of motion. And that's actually what gave name to SCARS, right? Because now let me kind of briefly recap, if you wish, right? SCARS have been known in the context of single particle billiards and in single particle billiards, right? If I start with a generic initial condition, for example, I prepare a particle here and at some angle, what will happen, this particle after a long time, it will visit every point of my static. But of course, there is a set of measure zero of initial conditions. For example, if I set my particle bouncing vertically, right? My particle will just bounce like this forever, right? And that will be it. It will never escape. Now this all such trajectories are unstable. And here I'm actually showing what happens if you prepare a particle at some tilt angle, which is very small. So the particle will hang for a really long time around the same trajectory and then it will escape. Turns out that once this is a classical system, now you can ask, what about quantum system? And quantum system is really kind of trivial. Everybody in the kindergarten can do this. You're looking at the spectrum of your Schrodinger equation in this manifold. It's a free system. So you're basically looking at a spectrum of Laplacian, right? You're solving number of psi is equal e psi with the proper boundary conditions. Psi at the boundary is zero. And you're asking, let's say how eigenstate number K1512 looks. And the eigenstate with K number 1512 looks like this. It looks like a mass or looks like a collection of plane waves. But now if you keep calculating this eigenstate, at some point you would plot an amplitude of wave function and you find maybe that state K number 400 or 4,000 would look very unusual. That would be a state which doesn't look at all like a mass collection of a plane waves. Moreover, this state looks exactly like a standing wave between these two borders, right? And Eric Heller already in 84 attributed this state, the set of unusual states to unstable periodic orbits that exist in the stadium in a large number actually, right? This is the simplest unstable periodic orbit. There are more complicated ones. There would be the both I one can do such a periodic orbit and so on. So there are many periodic orbits and some of them will have counterparts in a quantum system. Now this is actually, now you can see the analogy, hopefully, right? So here when they found some unstable periodic trajectory and this unstable periodic trajectory seems to be also related to the presence of a set of unusual icons. And again, that's what named scars. And I hope it's kind of clear right here. I can also show a bit more analogy. Analogy is actually not very obvious because we are dealing in one case on here, we are dealing with a single particle system with a very easy way to visualize. In a Rydberg chain, we're dealing with a many body Hilbert space. We are dealing with a discrete degrees of freedom. So analogy is not so straightforward, but again, many things really are similar. For example, here I'm showing participation ratios and they say that while typical eigenstates behave as expected, these scarred eigenstates up to very large system sizes, they do have a normal participation ratio, right? They're more concentrated. And that's again similar to this bow tie, for example, right? I'm drawing here a bow tie trajectory. You can see a bow tie, right? And you can see a quantum wave function which is concentrated around this bow tie trajectory, right? So that's again, that's analogy. But now, of course, we can ask how much further we can bring it, right? So this, again, these scars in single particle systems they have been explored in many different models, many different geometries and so on. Let me now kind of try to a bit explore what's happening in this Rydberg chains. So one thing that we did is actually is that these trajectories they do generalize to arbitrary dimensions. For that you need a bit of a trick. So you need not the matrix product states, you need the 10s or three ansatzas and you need to violate your lattice geometry. So basically you cannot, it's hopeless to try to do something analytical if the lattice has a loops. But what we managed to do, right? We managed to map this square lattice geometry to a three geometry and it goes wrong because it doesn't capture loops. So there is no loop here, right? This loop is kind of cut. I'm trying to show you the parallel, but still it captures local physics. And in this case, you can derive generalized equations of motion. They would still have a periodic trajectory. It would still correspond to a quantum dynamics reasonably well, right? On a small four by four lattice, I'm showing how TDVP, TDVP here is the solid lines, right? TDVP oscillates forever. It's a periodic trajectory classical system. Now quantum system, of course, decays, right? But still does it quite slowly. Oscillation period agrees quite well and so on. So it's kind of generalizes to higher dimensions. Surprisingly, generalizes also to more complicated lattices. That's something that is not possible in one dimension. So here it's most recent preprint. I really like the story of decorating a lattice in a way that it has different connectivities. Now it has two type of sub lattices which are kind of imbalanced. And now what's happening is that your trajectory, if you don't introduce any tuning parameter, your trajectory actually ends up in a singularity. So without tuning parameter, when this omega would be equal to one, your trajectory is not gonna reach the two prime state, it's gonna fall down into this singularity. However, if you introduce this tuning parameter, you can find that there's a special value of this tuning parameter, which I call here omega C. There would be a periodic trajectory which connects the two and the two prime states. Of course, again, this is a classical equation of motion. So you can now go back to quantum system and ask, does quantum system care about it or no? And you see that quantum system, of course, it's not as single, right? You see that here I'm plotting negative logarithm of fidelity per lattice side. So it's a quality of revivals. You see that this quality collapses between two different system sizes. So, you know, we believe that this is kind of means that we don't have strong finite size effects. And now you see that the smaller this guy, the better it is, right? Because it's negative log of fidelity. And we see that the quantum system reaches minimum, not exactly where a classical trajectory avoids singularities, but quite close, right? Again, we don't have any other mean at this point to improve this prediction from a classical thing, right? But you see that it kind of works, right? Again, one would be here, and at one fidelity is quite a lot worse. You know, it looks like some tiny factor, but you multiply it by 40 lattice sites and put it into exponent, and you would get a huge suppression, right? So in principle, this means that quantum dynamics knows about this trajectory, right? Quantum dynamics never gonna be as abrupt and you're never gonna have exactly infinity at this point just because quantum trajectory doesn't exist. You are kind of smooth, right? But still it does care. However, you know, something else that you learn if you do classical dynamical system is a two-dimensional phase space, which I was playing with here, is actually quite non-generic, right? And that's something that promotes to do slightly different ansatz and try to look at different kind of states. So let's say we want to understand revivals and dynamics of initial states, which has periods three, right? So I have a unit cell, which we call it K of size three here. And for that, two-site unit cell ansatz is not enough, right? But there is a very trivial extension instead of two-site unit cell ansatz, you insert now three angles. Now I need three angles, TTA, TTA, TTA, TTA, TTA. And these three angles would be enough to describe, let's say, Z3 state and many other states, right? It's an ansatz, which describes periods three, translational and variant initial conditions. Now you repeat the same machine, right? You insert TXP model, you get very terrible system of dynamic alterations of motion, but you don't care how terrible it is if you know how to analyze such systems using contrast actions, right? And the idea is that if such a system has a periodic trajectory, as is shown here by the black line, I can track when, let's say given parameter TTA B goes through zero, right? So I'm evolving my system for all initial conditions that I'm asking when TTA B is equal to zero. And that corresponds to a plane. Now you can understand that if there is a period trajectory, you're gonna cross this plane only in, say, two points, right? Maybe you're gonna cross this plane here and there. And that would be kind of a fixed point of your point-corner map. If you also fix derivative, right? You would get rid of this guy because you would say that derivative must be positive, so you would see a fixed point of your point-corner map. Now, of course, if you displace a bit away from your periodic trajectory, what KM would say you, that you would expect to be winding on a torus around this periodic trajectory. And then you would expect a circle of dots on this plane. And now on the right, I'm showing here really simulated numerically upon cross-section of that terrible system on the previous line. And what you see here is that, first of all, there is a bunch of periodic trajectories which you can see as the centers of this big tori. And they're all symmetry-related. There is a lot of symmetries in this problem. So I'm just choosing one of them, which I show by star. And now, again, around this trajectory, you see a big stable torus, right? That's what something would be called KIM torus, right? It surrounds periodic trajectory. And it says that if I initialize my system here, I would always be in the vicinity of this very trajectory. Again, it's so far classical, right? I would come back here, here, here, here, and so on. And that would be this circle in my point cross-sections. Now you see also lots of smaller tori. There is a lot of tori in this picture. But also you see something that resembles chaotic scene, right? You see that maybe if I go here, then I would start exploring all around this plane and I would kind of be unstable, right? So now, again, this is actually a much more generic situation because two-dimensional phase spaces, I promised this kind of a bit pathological very often. And now we find this KIM torus and this very stable trajectory. Now we can ask, what about quantum system? Does it care or no? And actually it turns out that it cares quite a lot. Because again, here I'm showing comparison of Poincaré sections and classical stuff. To now, this is a quantum dynamics. This is ITBD. So again, no boundary effects, infinite system. Limited time evolution. And you see that in this case, it is a trajectory which has the slowest growth of entanglement, right? You see that Z3 product state, which would be somewhere where this green dot, which, okay, I just colored. So I'm not gonna color it. So this is Z3 state, which is close to torus, but not exactly at the torus and looks like it's maybe even in a chaotic region. It has faster entanglement growth, right? And now this small torus, right? I'm just choosing some other point. It has even faster entanglement growth, right? That would be the small torus. And the best one seems to be the start trajectory, which lives in the fastest possible torus. Again, all other big fat torus are relatives of the same start trajectory. Just because you have three-side unit cell plus inversion and translations, you can do a lot of rearrangements, right? And that would give rise, you know, basically all these big torus are equivalent to each other, modular shuffling of your initial lattice size, right? So that's, and that's again, that's kind of interesting news, right? That quantum dynamics does care about most stable region and looks like, right? It's actually, at this point, it would be more correct not to call it scars, but to call it so-called regular eigenstates, right? Because in some sense, again, rigorously speaking, scars must come from an unstable trajectories, right? And people also point some name for eigenstates which come from the stable trajectories called regular eigenstates. But of course, again, translating this from a single particle billiards to many body system is absolutely open problem. Now, not surprisingly, the same mixed phase space also appears in two-dimensional system. So that's another example of two-dimensional system I'm using again, the tensor three. And now, before I used just two angles, right? I would use that theta A and theta B. Now I introduce the chemical potential. Chemical potential breaks certain symmetry and says that one angle per spin is not enough anymore. So now I need phi A and phi B. And that's how I get to a three-dimensional phase space because now there is also energy conservation. So H would be constant. And this gives rise, again, I have four variables, right? Four variables minus one is three-dimensional phase space. I do a contrast section here and I again see this big stable region around the periodic trajectory. And this periodic trajectory again works with variational, agrees with your quantum dynamics, right? So again, here, solid lines of periodic trajectories are never decaying, the solid line, it's always perfect oscillations. Quantum system begins to decay, begins to deviate, right? But still it follows this periodic trajectory for fairly long time. Now, probably many of you again have this question that in plots that I'm showing, each plot has probably 50 to 30 different periodic trajectories. If you are carefully zooming in into small tori, right? This would be another periodic trajectory and this one and this one and maybe this one and so on. So it looks like a complete mess. So why should we care or how do we know that this star is most important one and other ones are maybe not so important? So Maxi, I have a question on this. If I look at these bunches of trajectories, do they correspond to states which have the same structure but they are somehow eigenstates of the momentum operator with this different momenta? So far I actually fixed, I fixed my translational symmetry, right? So, but you still can have an untrigger momenta, right? With the translational, the two translational symmetry you can have k equal zero and k equal five, right? They're still kind of there. That's actually something we haven't explored. Thank you for the question. So I don't, to be honest, I don't know. And now that's actually something that you can introduce, right? So in order to understand if quantum trajectory is important or no, you can ask how much weight would I lose if I assume that when I project, right? I'm doing a variational principle. So each time my quantum dynamics brings me out of my variational manifold and I'm projecting something which I can quantify. This is this gamma and gamma quantifies how much state do I throw away at each time step? Of course, you can assume that quantum system never comes back. We can check this assumption later, right? But if you assume that this norm, this leakage is irreversible, you can, turns out that you can bound quantum fidelity revivals. So it's not a rigorous mathematical bound because there is a bit of an issue with thermodynamic limit in this bound, but we still think that maybe it can be made rigorous, right? So I'm trying to bound fidelity revival per spin, right? So that's something that again scales well in the thermodynamic limit. I'm trying to bound it by some quantity which is actually pretty much classical, right? This quantity is calculated on classical equations of motion, right? So this quantity doesn't care too much about quantum Hamiltonian. It knows a bit more about quantum Hamiltonian compared to TGVP because this quantity involves a norm. And if you wish, norm will have Hamiltonian squared. So it knows a bit more about energy fluctuations, but still, right? It's a classical animal, right? Doesn't know about full complexity of quantum dynamics. Turns out that actually in some cases when we check, this bound really works. So here I'm showing this fidelity by blue lines. This is fidelity extracted from numerics. And you see that, again, different system sizes collapse into each other. So this is a well-defined quantity. And now red line is a bound from TGVP. And you see that it's another of magnitude wrong, which actually is a good news. It says that my quantum system, it's not only going away from my manifold, right? It's probably also coming back. If you wish, if this is my variational manifold, then the quantum system probably does something like this, right? You know, it kind of goes away from it. And then at some point, it returns back, right? And then, and so on, right? So it's not, you know, this leakage is not always kind of destructive. And that's why there is such a big distance between these two curves, but still you see that quantum dynamics is bounded by this classical quantity. And namely, the criterion would be that over my period, I don't want to lose too much, right? If I lose too much over a period, then kind of no way function remains. And this criterion, again, qualitatively works. So let's say if I say that this guy is order one, right? That's about where this guy, classical guy becomes order one. And that's where, again, maybe it's incidental. You don't have a proof yet, but that's where we see that quantum stuff also begins to behave in a weird way, right? We start stop seeing collapses in the quantum stuff and so on, right? So it looks like that this leakage is a good criterion, right? That we want to have a low leakage regime. And in a low leakage regime, if gamma T is much less than one, we would expect that trajectory has a lot to do, right? With real quantum dynamics, because basically, you know, you know that your quantum dynamics deviates very little at each time step. So now in the remaining few minutes, let me actually ask, okay, maybe leakage is large. Maybe we go somewhere here. Do we care at all, right? Is it, you know, is everything hopeless or no? And that's, you know, a bit of kind of explorative study. So we did a different model. We did a thermalizing ising model, transferred field ising model, which, you know, if you compare this to a PXP model, right, that's entanglement of eigenstates of PXP. And this is entanglement of ising in our parametric. You see, you know, no outliers, right? All these guys, which are very unusual, unusual and low entangled states, and they stare at you, they're all gone. There is empty, nobody is here, right? Nice, you know, terminal spectrum. We choose non-trivial, one-dimension two MPS. We choose some energy density, which is large and near the center of the spectrum. And we just ask, okay, let us run the same machine on ising. And turns out that you can find still periodic trajectories. And what happens is that, you know, of course leakage is very bad. So what's happening is a TDVP, and you know, your periodic trajectory, system stops following your periodic trajectory after time, or the one. Because again, you know, it's thermalizing ising, entanglement grows like crazy. Still what we seem to see is that close, being close to trajectory has an impact, even at late times, right? Even once we evolve to time equal five, when entanglement again reaches five, you still see that this trajectory cares about, or entanglement cares about trajectory. And we also see that, let's say there is a very strong beyond kind of error bars correlation between leakage rate and entanglement at late times. And that's again, you may say it's surprising, you may say it's not so surprising. I find it quite surprising because again, TDVP with bond dimension two is ridiculous approximation for the sizing model. Yet, TDVP with bond dimension two is able to tell us something about, is able to provide still some information at time equal five, when entanglement, you know, when the bond dimension needed to describe this model would be assault, right? Why and how this happens? It's an open question. So now let me kind of come to a conclusion. And that's actually what I'm showing a slide, which I used to show in my talks about many body localization, right? I would argue that, you know, there are two classes in classical as a robotic and integrable systems and quantum wise NBL is like a counterpart, right? Of an integrable system in a quantum, but actually, you know, in the classical world, right? At least few body systems, they're never completely chaotic, right? Hamiltonian, few body systems typically have some things that have a mixed phase space, right? And that's something that, you know, principle you shouldn't ignore. And now after the study of PXP model looks like this mixed phase space, it has one-to-one counterpart, or it may have quite a strong effect in quantum systems and its effect in a quantum system suggests that it may lead to a slow thermalization in a broad classes of models, right? Even in icing, you are able to, by initializing it in a MPS one-dimension two state with very low entanglement, we're able to slow down entanglement at time equal five. And again, icing is probably as bad as it gets. And this suggests that again, that systems with mixed phase space classically, they do have counterparts, even in many body quantum systems without any obvious classical limit and so on, right? So, and this opens again interesting question, which I'm kind of already mentioned is how we can understand this kind of classical systems that come from a variational principle. And that's of course on going work, many other groups are working, but it would be something very interesting to try and make much more specific connections, right? On the quantum models, we have entanglement growth, you have thermalization rate, you have some spreading velocity, and it looks like this classical manifold that you get from TVPs, they must have some very intricate organization, right? They must have a memory of them coming from a quantum system. And I'm here showing a Russian doll, right? Because in principle, nobody told, nobody can fix a particular one dimension, right? They're all in principle, there should be democracy, right? And you can just, if you think about running one dimension seven, yeah, you can run one dimension seven, right? But now they should know about each other, right? And they should somehow know about entanglement spreading and locality in your system. So that's kind of a bit of a different problem from a classical dynamical systems. Instead of one dynamical system, each of these Russian dolls would give you a new dynamical system, which incorporates all previous ones, plus a lot of additional degrees of freedom. And to me, the interesting question is, how staff at left-hand side, how would it say be Robinson or locality of entanglement spreading? What does it fix for this hierarchical time-dependent variational principle systems? And something else that again, I just promised to briefly mention is the algebraic approach. And what we found some time ago is that you can add an exponentially decaying deformation to the PXP model. And as a result of this exponentially decaying deformation, you see that your eigenstates kind of become much better, right? As I show here by the red line, in presence of deformation, all other guys kind of go down. And these eigenstates become better and better decoupled, right? And we understand now this deformation as a emergence of this hidden embedded SU2 representation. So this bunch of eigenstates, right, would be realising of realization of SU2 representation and would be decoupled from all other states. Now, this is again, algebraic picture and would be very interesting to understand how this is related to the TVP that I was mentioning. Does it mean that in this, the form model trajectory becomes perfect? Can you have a perfect trajectory in principle or no? Or what is it doing, right? So this is kind of so far, I would say that this deformation is the closest thing that has happened to analytical solution of this model. But now, of course, it's still, first of all, only ansatz, but also it's not clear how to connect this two-time dependent variational principle. And surprisingly, you know, just to understand that this deformation doesn't have anything to do this quantum integrability with better ansatz integrabilities, the same deformation exists in two-dimensional systems, which again, to my view, probably rules out relation of this deformation and this model to integrable physics, right? So now I'm here showing again the same PXP model in a square lattice and with a weak deformation, which is also 5%, you see that PXP model on a very big lattice is much more thermal. You see that there is very few low-lying entangled states and so on. And after deformation, you see that these guys become much better separated, right? And you see that there is still some weakly entangled, much weaker entangled states decoupled from the rest, right? So again, deformation is there, how to understand it compared to the trajectory and generally how to understand this hidden SU2 representations which seem to emerge in this model still remains an open question. And with this, let me just try to summarize. So I was discussing a particular many body models that follows variational dynamics, right? This PXP model. And generally I found that projection of this model on variational manifold produces mixed phase space, equation of motion with mixed phase space. And now it's not enough to have periodic trajectories, right? In this mixed phase space, this quantum leakage or this leakage is important parameter and if it's low, we have special more properly, I would call them regular eigenstates. I think scars is also okay, but because it's so much nicer, but again, I think rigorously, people would say that these are regular eigenstates and fidelity revivals, strong leakage, right? Looks like it still has some effect, right? In a form of slower criminalization. And now again, I mentioned some of the open questions. I know people at the CTP-3S also work on it. Relation to supersymmetry is something exciting. I guess there were some recent results from your groups and some of the more open questions, right? If one can try to link this kind of classical TDP to a quantum formulation of KM and if one can engineer trajectories, which would be strongly entangled, that would be of course, most likely that's impossible or that would be something very exciting, right? That system produces entanglement and then kind of disentangles back. And that's already something that is happening in this trajectory, right? If you think about what I was showing for this restate, right? This oscillations of entanglement, you see that entanglement grows a bit, right? And then comes down. Not, it doesn't come down exactly perfectly to its original value, but it entangles and kind of a bit disentangles. But the question is, can you produce much bigger amount of entanglement pretty much in the same way? With this, let me just show collaborators. Alex was, you know, spinning the handle of this heat grinder and doing variational analysis. Chris was doing a lot of exagonalization and stuff on deformations. Dima and Zlatko, main collaborators and this deformation stuff was also was done that we gathered with the Harvard team of Michel Vauquin when Ways from Bono and Thomas Pickler. That's where I'm sitting right now, somewhere around this red arrow is. And with this, let me just stop and ask you four questions. Okay, thank you, Maxim. Yes, thank you for the talk. Actually, I think Antonello ready. I want to ask for a question. Okay, I also see some messages in chat, but probably you can handle that, right? I shouldn't do that. Yeah, yeah, yeah, so Antonello, I think. Okay. Yeah, hi, Maxim. How are you? This is Antonello speaking. Can you hear me? Yes, I can hear you. Okay, good. So I was wondering actually, so there is a classical representation of this Hamiltonian in the sense that you can just write all the projectors and X in terms of spin variables and then consider them as a classical spins, right? So typically this does not happen. For example, if you take the Eisenberg model, you've got it as a classical spin system. It's not integrable, it's chaotic. Yes, as a classical. So, but have you, because the question is that the variables that you are using here, the t-tas, they don't have a classical interpretation. I mean, they are a mean field on the Schrodinger equation, so you cannot think about quantizing them. When people talked about scars in billiards, they were actually looking at classical variables and you know how to quantize them somehow. So have you tried looking if the corresponding classical Hamiltonian has any strange feature that you, I mean, this will be a, you know, the projectors are, the projectors are two spins, two spins objects, right? And then X, so this Hamiltonian will be like, with the five spin terms, essentially. But let me just try to understand what you mean because so you want basically to take a semi-classical limit of a spin size going to infinity. Yeah, exactly. So in principle, I mean, that would be the corresponding classical Hamiltonian. Then you can quantize with the Berry phase if you want to give a given spin representation, right? So in principle, if you really want to parallel with the scars in billiards, you should look at the classical Hamiltonian whether there are periodic orbit in the classical Hamiltonian. Okay, so let me, I would say that again, I post kind of agree and also slightly disagree because I think, yeah, I haven't looked at the classical Hamiltonian, but I know that in the paper of Benway, this paper with trajectories, they do a spin-ass generalization of this model. So there is a spin-ass generalization of this model and I believe that from there, you probably can go all the way to both infinite and classical case. And I would mainly expect that probably trajectory survives, but at the same time, I also expect that it's very probable that leakage may increase and that the phenomenon may disappear. But okay, that's actually a very interesting direction. But now I would like also slightly disagree because I think that in principle, one can try to quantize and that's people, people have began thinking about, right? About formulating path integrals over NPS states, that's something that for example, N-degree pasteurized. So in principle, to me, it looks like that maybe not equally promising approach is try to go back to this variational principle with just two angles and try to induce some fluctuations on them and try to think about going back, right? To kind of re-quantizing it from there. So, and that approach, to be honest, seems to be a bit more natural because some things that I never kind of, I mean, maybe you can explain this to me. There's not so many people and we can have a discussion. That's something that I never understood is that, you know, in turning S to infinity, you are doing a sequence of Hamiltonians, in a sense, right? And that's why to me, it's not obvious, right? That this kind of Russian doll construction, right? For example, this Russian doll construction that I'm drawing here, it allows you to increase number of variables by keeping Hamiltonians the same. And that's what interesting for me, that, you know, you can again, take a Heisenberg model or take an Ising model and you can make spin bigger and bigger and bigger. And then you are going through a sequence of really different Hilbert space in different Hamiltonians. Or, right, I can take the same Ising model and do chi, increase chi. And then I'm keeping the same Ising Hamiltonians the same two-side Hilbert space, but I'm incorporating into my theory variables which know about longer range correlations. And now again, I would be very interested if somebody can comment in any way on the distinction between these two approaches and on any, you know, kind of, to me, it looks like very different limits. Which one has more of these? Of course, they are very different limits, I think, because in one case, you are stuck in the spin one-off representation, for example, you're just increasing the description of your Schrodinger equation. Yes. In the other case, you're really changing your system, the Hamiltonian of your system because you're changing the representation of the algebra. Yes. And, I mean, as I said, in principle, you can quantize any spin Hamiltonian by writing the classical Hamiltonian and then doing an integral with the very face with the appropriate value. But, I mean, that's typically useless and says you cannot do anything because you have these phases that they have to, so it's not like, it's much less useful than a semi-classical representation on a PX Hamiltonian, like a billion, for example. So I was wondering whether in this case, you had thought about that and you didn't proceed because it was too difficult? Well, to be honest, I know that there is, again, there is even a good starting point for that, but I never, you know, I never even started trying. So I think it's, probably it's gonna give something interesting because, again, there is even a sequence of Hamiltonians that is in the state here, by the way. So if you're interested in it, I'm happy to discuss with you more. Yeah, we can discuss with you. Again, but by my answer is that, no, I haven't tried this kind of semi-classical limit here. Okay. Is there any other questions? I have one question in while people take time to present. So Maxime, is it possible to distill a principle out of this semi-classical analysis that tells you, given a Hamiltonian, before computing its properties by just looking at some of these generic features, if this is a candidate system to have the eigenstates which have these properties or not? I, so far, I don't know it, right? So far it's a bit more of a stereo in a sense that I think, you know, if you prohibit me doing diagonalization and just allow me to do to DVP, I can give you an answer, right? Because what I would do, I would find semi-classical equation, to DVP equations of motion, find trajectories, find leakage, and use this criterion, and the small leakage, I think it's guaranteed to give you revivals, right? But this should give you scars. But this you can do as long as the scars are the good representation in terms of the variational wave functions that you use for the DVP. Well, for that, yeah. For that, you have to be lucky, right? You have to be, to have a correct unit cell or you have to know correct degrees of freedom. And yeah, unfortunately, I don't understand, you know, in some sense, it's like you have to try it, right? And then if leakage is small, you would, again, you are promised to get revivals, but I don't, right? Leakage is, you know, it's some calculation which involves manual trivial ingredients. It would be good actually to indeed understand generic cases when leakage is small, right? And obviously some generic cases would be near classical limit, right? Or, you know, if it's pretty clear that if I take, I know some model with a very low entanglement production, right? I think I should stay very close to classical trajectory. But I think it's interesting, right? To try and think about more examples like this XP when you actually create entanglement, right? As again, as I was showing in this trajectory is you do create some entanglement and then you kind of come back, right? And that's somehow is a lot more than trivial. Thank you. So is there some message in the chat or? Oh, yes, Silvia. Hi. So I have a question regarding the mixed space space for the BXP. So you were showing that mixed space space commenting on KAM theory, which arises when you add an integrability break in term from starting from an integrable aminsonium. So I was wondering, do you have ideas of what is this integrable aminsonium and what are the integrals of motion and, okay. Yeah, thank you. I mean, excellent question. You see, right? It's, it would be interesting, right? But this problem is equivalent to ungrinding your meat, right? In some sense. To what? Can you repeat that? In some sense, you are asking for something, right? That would be in an everyday business equivalent to ungrinding your meat, right? You put your meat in a grinder and you get ground beef, right? And in the same way you put your quantum system into this machine, right? You put here your BXP model and you get some out, right? I get this terrible system. So now, even if somebody comes and says that, you know, here, there is this smart function of teta a and teta b that is gonna make my system integrable, right? There is no promise that you can stick it back here and obtain a quantum Hamiltonian, which is local. So, you know, but in principle, again, it's a very good question, but I don't know how to approach it if it is within this framework, right? It would be very interesting, right? To find maybe examples of non-integrable quantum systems that maybe give integrable projections onto certain manifolds. But again, because of this nonlinearity and the whole machine involved hidden in this guy, right? And it's not very complicated, but there is a bunch of things, right? You have to really do the Gram-Schmidt stuff and all these norms and all the metrics of your quantum states. So it knows about quantum geometry. So, and I mean, maybe one can, but again, it's interesting to think about it, but the short answer, right? That as also in many cases when people refer to KAM, they say that integrable system is somewhere there, right? Generically somewhere not too far, but where it is, so far I don't have any constructive answer. I would just like to comment a little bit on the discussion on the dolls inside the doll. So the classical limit. So somehow, I mean, when you do a classical limit and a natural way to have the hierarchy would be to do a classroom in field, right? So I mean, field is strictly speaking as to infinity, whereas you could improve that by clustering and the set of... So you would say that again, I would have, you know, in a classroom in field, I would have a unit cell of size one, that would be K for one unit cell and then I would put S to infinity. So would because a classroom in field, some sort of two-site unit cell. Yeah. Because that then smells somewhat similar to, you know, what is happening. Yeah, yeah, yeah. I guess in that case, the biggest difference is that you allow entanglement only on a... On a shorter range. Say on blocks, yeah, on shorter blocks. Well, then that's probably something that can be also implemented with NPS, because NPS, right, that's what NPS is doing. In some sense, NPS is encoding some... One dimension two allows for someone really on nearest neighbor correlation, right? And be fairly strong with that. But yeah, okay, that's an interesting question. Or an interesting question. Thank you. Thanks to you. Any other questions? Okay, Max, I may ask you one things about the icing model, because probably I was not... Okay, yes, yes, the previous slide. Okay, actually it's about both slides, but I'm curious because you say that this model, okay, so the model you are relating here is an easy model with both transverse and longitudinal fields. And when you say that your entanglement entanglement is growing linearly, what was your initial state? I want to be... For me, it's pretty unclear. So, you know, we're kind of trying to do the same approach as to PXP model, right? We chose this and that, right, which is shown here, which is bond dimension two ansatz, actually not even most generic bond dimension two ansatz, but now we sample states from this ansatz. So at most, these are some weakly entangled states, right? They are not exactly product states. Product states are obviously embedded here, but there are non-trivial variables which would be this xi and chi, and these variables give a bit of local entanglement. And now we project flow under Ising model onto this variational ansatz, so we get the same, you know, dynamical system pi dot is equal to something chi dot and so on, and we find some periodic trajectory. And this is shown by a fat dashed line. This dashed line shows me a periodic trajectory. And now again, if it would be a TDVP, you know, your entanglement on this periodic trajectory would kind of oscillate like this forever, right? You know, that would be TDVP prediction for this trajectory. Yes. Of course, right, you see that quantum model doesn't care about TDVP prediction pretty much because entanglement and, you know, because it's chaotic, but now looks like this, you know, you can ask what is the best product state, right? To initialize my Ising model at a given energy density to have the smallest entanglement growth, right? And you see that this weekly entanglement state outperforms it, right? So you can produce a bit of local correlations by this one dimension two ansatz. And you see entanglement here, it's actually not exactly zero, it's like point one or point one, two. So there is very weak initial entanglement. And as a result of that very weak initial entanglement, somehow by being close to trajectory, you still gain almost one unit of entanglement at late times. And, you know, you can also try to attribute this to some initial T star. And here I'm doing some final size scaling, align integrable system. So I scale entanglement by, this is entanglement of different regions. This is bipartite cut. This is entanglement of a region of size L also in ITVD. And I'm doing scalings of different subregions. And I account for some small constant time shift to bring them kind of to the same point, right? And you see that looks like there are maybe even different velocities, right? You see that this is the game. This is the topmost state, that's the fastest state. And that's the slowest state. And they're all in the same bond dimension two ansatz. So I'm sampling only the states with the bond dimension two originality. So does it- I see, okay. Does it make sense? Yes, yes, okay, I see. Okay, thank you. Is there any other question? I mean, someone, I don't think so. Okay. Well, Maxim, I mean, thank you very much for your talk. So let's thank again, even though I mean, I will thank you. And yes, I mean, it has been a pleasure to have you in this webinar. And let's hope everything will come back to normal and we can have you physically in Trieste. So- Okay, yeah, that would be great. You know, again, I would love to come back. I haven't been there for three years now, so. Thank you. Okay.