 Thank you for the introduction and thank you very much for the invitation. I find this conference a very good initiative, I have to say. Let me share the screen. Yes. And I think that it's a really good opportunity to meet and start collaborating and find out what other people are doing in the neighborhood of our institute. So, as it was mentioned, I'm currently, I mean, I'm still in Ljubljana, but where I was for the last three years, but I have recently moved to Berlin. I guess you see the full screen, right? You don't see any part of it coming. So I'm referring here to both my present position at FUB, but also the grants and the University of Ljubljana where I did most of this work and the grants that founded my research in the last years. So I'm going to talk about Hamiltonian authentication methods, which is an old method, but we recently revisited it with collaborators and applied it to the dynamics of continuous quantum field theory models. And I think that it's experiencing, revisited the renew with interest because there are many applications of it recently. People from the high energy community are starting to look at it as an alternative to lattice QCD. And from our point of view, it's one of the few methods that is available for the study of cold atom experiments that simulate quantum field theories. So let me pass to the outline of my talk. I will first introduce the motivation that we have in this study. Why do we study the dynamics of quantum continuous quantum field theories? Since most of my talk will be, or all of my talk will be, I will refer to the case of one-plus one-dimensional models. I'm going to explain why we are particularly interested in this. And then in the main part of my talk, in the first part I will talk about some experiments that have recently managed to simulate quantum field theory. So they can play the role of analog quantum field simulators and can help us explore the interesting properties of quantum field theory dynamics and the exotic, I would say, behavior that they have in some cases. And I'm going to explain how do we simulate by means of classical algorithms, these type of experiments. So essentially what we are doing is a classical simulation of a quantum simulator. And I'm going to present the challenges and the method that we are using, which is based on RG theory in a numerical implementation. And in the second part I'm going to present some of the most exciting applications of this method that we have made so far. The first of which is the study of correlations in and out of equilibrium in the sine golden model, which is one of the models, the main model that can be simulated in this type of experiments. I will talk about the study of quantum equilibration and recurrences in the experimental system presenting also the theory behind it and some surprises we found in understanding experiments and matching with theory. And in the last part I'm going to talk about a recent work where we study quantum chaos signatures in the double sine golden model, which is a non-integral model. OK, so first of all, the motivation. Why do we study quantum fields? Well, we know that quantum field theory describes the discouraged nature from the short scale to the large scale. So it has applications to many areas of physics from quantum cosmology to elementary particles and high energy and also to black hole physics. But it also has applications, as we know, to contest matter as an effective theory that describes the macroscopic behavior of lattice systems and statistical systems. So from the theoretical point of view, the motivation of studying quantum fields and quantum statistical mechanics out of equilibrium is that it's related to the problem of quantum equilibration, which is a fundamental and longstanding question in statistical mechanics. Given the microscopic laws of nature, which are described by quantum mechanics, how do we end up in the case of many body systems with the emergence of statistical ensembles, thermal ensembles, et cetera? At the same time, there are some more practical motivation. We want to understand what are the ultimate limits of our classical thermodynamic expectations and whether we can go beyond that and apply this to quantum devices, quantum thermal engines that have been proposed in the last 30 years or something. And of course, we aim to see novel quantum effects at the macroscopic level if possible. Now, one may ask, of course, apart from this motivation from the point of view of quantum technologies and applications, why are we interested now? Why is this problem more feasible to study now than in the past, better than more than in the past? And the answer to that is that on the one hand, there are experimental techniques that allow the study of quantum antibody dynamics in ultra-hot atoms. I'm going to talk about that in a while. And on the other hand, there are very efficient numerical tools like TDMRG, MPS, tensor network-based methods that allow us to simulate quantum dynamics, quantum antibody dynamics and get a glimpse of what is happening in a controllable situation. Now, why do we focus in particular in one spatial dimension? Well, on the other hand, we have exact analytical tools like integrability and exact dualities that allow us to study in principle exactly the dynamics of quantum models in one dimension. And on the other hand, the numerical tools that I mentioned before can be applied to one-plus-one dimensional systems due to the scaling of entanglement and so on. OK, so let me now talk about a specific example of cold-atom experiments that, as I said, can help us simulate continuous quantum field theory models. First of all, what do we mean when we talk about quantum simulation? We know that quantum systems are hard to solve because of the exponential scaling of the Hilbert space with the number of particles. So there is a challenge in solving these models. And the idea of finding one was to use a quantum computer in order to simulate such systems. So by quantum simulation, we mean that we use a quantum system whose Hamiltonian, whose description can be correspond to a model that we are interested in. And we let the system, we make experiments on this system and we see, we get as output the answer to the question that we are interested in for the theoretical model. So this is what has been achieved in these experiments of the Atomship Group in Vienna which are conducted by Professor Spindmeyer. And let me give you some brief explanation of what they are doing. They are using one-dimensional quasi condensate of bosonic particles of atoms which they can be controlled by the magnetic trap produced by an atom tip. So it can be precisely controlled by an electronic device. And this gives them high tuneability in the parameters of the system and allows them to do precise protocols out of equilibrium, dynamics, et cetera, et cetera. In particular, the application that I'm mostly interested in here is the case where they have two parallel quasi one-dimensional condensates which are close to each other and so that there is hoping between the two subsystems. In this case, the hoping plays the role of a solution junction which gives rise in an effective description of the phase and density field of the condensate to the so-called sine Gordon model. So this is what I'm going to talk about in most of the following of the talk. The sine Gordon model is described by this Hamiltonian where phi is the relative phase between the two condensates. And as you can see, it has three parts, this quadratic part. Unfortunately, I can't see my mouth, and it has an interacting part that corresponds, that has a form of this cosine potential. And now in the experiments, they consume the parameters of this model and then they can let the gas expand and through these they get interference patterns. From which they can extract information about the phase profile as a function of the parameter X of the coordinate. And then by repeating the experiment many, many times, hundreds or even thousands of times, they can extract the full distribution of the phase profiles and by doing dynamical protocols like Quentes that I'm going to talk about later, they can see the development of these probability distributions over time. So the dynamics of these fields can be controlled to be either given by the sine Gordon model or if we tune the parameters so that there is no cosine interaction, they are given by simply the quadratic part which in the context of 1D physics it's called Lattenger liquid, and in the context of high energy it's just the simplest, one of the simplest conformal field theories, not of a free Gaussian, a free boson field. Okay, so a few more words about the quantum sine Gordon model. Note that since phi is a phase field, it has a topology of a circle. So it's allowed to take values between 0 and 2 pi, and or in this case, it's actually 2 pi over beta if we express it in this particular form, which means that the ground state of this model is, there is an infinite number of ground states which correspond to the minimum of the potential and therefore there are also solitonic configurations that correspond to field configurations where the field changes from 0 to 2 pi as we go from minus infinity, minus spatial infinity to plus infinity. Now solitons, as I said, can be represented in this form. In the classical case, so the profile that describes a soliton interpolates between the values 0 and 2 pi, and we can have moving solitons, we can have static solitons, and in the classical case, we can see that there are also, we can study the scattering between two solitons and we can see that it's elastic, which means that the full dynamics of a set of solitons can be described by a means of using the tools of integrability and the better answers, because there is never production of more particles than we have initially. So this is a very important advantage because it allows us to solve exactly these models, at least in principle, in the sense that we can construct, I mean, we can construct in the quantum case the folk space describing all the excitations of the system. I should say that the solitons of the classical model persist to the quantum level, so they are stable excitations and they are also elastically scattering in the quantum case, which is the advantage that we have in this model. Now, solitons can also form bound states, bound states, which are called breathers. This is a video of a breather in the classical case. In the quantum case, these pictures, well, should be understood as, rather as a asymptotic solution, so when we consider a breather in the quantum case, it gives a, it corresponds to an excitation, an excited state of the system, where the two solitons are delocalizing all space. But in the classical case, we can imagine like two solitons that are bound to each other through the interaction. Actually, yeah, sorry, the previous animation was about the scattering of two solitons. This is a breather. You can see that the two solitons remain close to each other and don't move away. Okay, so, so now what do we do in, in the theoretical description of this model? How do we see these solitons in the experiment? The, the output of the experiment is an ensemble of phase profiles, like those shown here. And in, in the case where the potential is a parabolic potential, or any potential that has a single minimum, the distribution would look like this in the Gaussian case of a parabolic potential, it would look like it would have a Gaussian distribution when we, after, when we take the distribution of all phases at all points. So we have a field that is simply, that has simply harmonic fluctuations. But in the case that solitons are present, the field can jump from one of the two minima to another one, and possibly also to two times two pi. So integer multiples of two pi. And therefore, the distribution of the phases exhibits a markedly different distribution. It has a bell shaped in the bell shaped distribution in the middle, but it has also these satellite peaks and integer multiples of two pi. And indeed in the experiment through atom interferometry and analysis of these phase profiles, it was possible to show that under, but in certain cases in this fast cooling protocol, the distribution of the phases exhibits these satellite peaks and plus minus two pi. Of course, the higher peaks and larger multiples of two pi are suppressed because it's energetically favorable to excite many, many more solitons. I have one doubt if I can stop you just to understand what this fast and slow cooling mean. I am not familiar with this. So in the experiment, they can do either fast, they can cool the gas either quickly or slowly. In the case that they cool it slowly, the procedure is almost adiabatic, and the system is always at thermal equilibrium. And at thermal equilibrium, the excitations that correspond to solitons are strongly suppressed. So that's why we don't see many of these satellite peaks or strong weights on these peaks. But in the case that they do fast cooling, some of these soliton configurations freeze in the initial state because they start from a configuration where the field has larger fluctuations. So they go quickly to the sign Gordon Hamiltonian with where they are pinned at the minimum of the potential and they get frozen. It's a sort of cubic Zurich mechanism. And for that reason, we have a high probability to have this type of jumps by plus minus two pi. And the bottom part of this plot, you can see one instance of such a soliton in one of the interference patterns that shows the phase profile. So you can see that there are these sort of jumps in the phase. If one tries to draw a line to fit a line at the maximum of this phase coherence factor, which is called ends exactly two pi in this case. Now, having computed the phase correlation, the phase profiles, one can also calculate the correlations between phase at different points, how strongly dependent are is the value of the phase at one point with respect to another point. And in this case, the experiment is can solve in a sense, the unknown properties of the sign Gordon model at equilibrium by measuring the full correlation function, the full set of correlation functions of the phase of any order. So two point three, four point etc. Here, for example, you can see the deviations from Gaussianity when we are in the limit of either a very strong value of the mass parameter in the sign Gordon, the potential becomes very steep. And it can be approximated by a parabolic potential. So fluctuations are really small close to the minimum of the potential. And in this case, we can approximate the theory by a client Gordon theory, which is quadratic. And therefore, weeks theorem applies, which means that if I calculate the four point correlation function, it will be simply factorized into as a sum of products of two point functions. So the experimentalist could compute the connected part of the correlation functions and so that it's, it's small in this case. And so small in the case where there is no interaction at all when the cosine potential vanishes when they tune the parameters to make it zero. While for intermediate values of the interaction, far from the strongly, so far from the strong parameter regime and the weak parameter regime, we get all values or a huge change in the non Gaussianity of the system. And other things that they can do in the experiment is, as I mentioned earlier, they can do, they can study dynamics. And in fact, this was the first experiment or the only experiment that has demonstrated the emergence of a non thermal ensemble of a generalized Gibson sample, when one changes the parameter from the strongly interacting regime to the weakly interacting regime. In this case, the potential changes from a steep parabolic potential to the case where there is no potential at all, the Latin gel liquid model. And by probing the, by following the dynamics of correlations, they were able to show that the correlations cannot be described by a thermal ensemble. One needs more than one effective temperature to describe it, which is precisely the definition of a generalized Gibson sample. Well, or not so precisely, one should check how many exactly are the conserved quantities that are involved, etc. But so what I want to stress is that there is, there are many possibilities in this type of experiments and the theory is still, I mean, it's, it's, we are missing some theoretical tools to, to describe it in full. For that reason, we started thinking about a new medical approach to. Sorry, Spiros, may I ask you a question? Yes, sir. I was wondering whether there is the possibility of describing this amount of this experiment on the dynamics of correlation function with the use of generalized hydro. More specifically, the quantum version to get the behavior of I think that there is definitely interest in this direction and also the group has developed efficient numerical codes for the simulation by means of generalized hydro dynamics. The problem as you can guess is that in the description that we are using here, which is the, the effective description of the phase and density field. The, the, the effective model is the sign Gordon model. And for such models, there is no. Yeah, we should go to the Lattinger limit. Yes. Okay. But of course, of course, one could do something else. The, the actual quantum gas is described by, by the Lieblinger model. So one can use the generalized hydrodynamic theory for the Lattinger, sorry for the Lieblinger model. So to, to, to explain myself, the sign Gordon model is integrable. So generalized hydrodynamic supplies, but we don't know what, how exactly to construct the charges of this model for a general value for general values of the parameter. Why in the original description in terms of bosons describing the cold atom, the system is a Lieblinger gas. And having two such systems parallel to each other corresponds to, well, two Lieblinger gases that are in contact with each other. So of course, this is non integrable if we have the two system cases, because the coupling between them would, would break into the ability, but generalized hydrodynamics can be used or extended in a, in a model. So I think there is some interest in this direction. And some of the experimentalists are working on this. So. Okay. Thank you. Okay. So let me now talk about the methods that we used. And let, let me explain a bit better why we look at, at this. Why we started to think about this method. So as I said, the sign-gold model is integrable due to the fact that the solitons scatter elastically, which, which is a great advantage because combined with relativistic invariance, it gives rise to this, to the application of this S matrix bootstrap, which means that we can calculate properties like the mass spectrum and the S matrix. And with this, with these tools, we can calculate the so-called form factors of local operators, which are essentially matrix elements of local fields in the basis of the solitonic excitation. So we can construct the Fox space as in the free theory by, by acting with creation and annihilation operators of the solitons. And, and this allows us to, to, to practically or not practically, but in principle to solve the, the model. But the progress in this direction is really slow. So it started in 1975 or 79 when the, it was shown that the, it was considered it's exactly solvable in the quantum case as well. And since then, with very slow steps, people were able to calculate the ground state expectation values of local fields or two point correlations in the ground state. Then eventually, thermal correlations is an ongoing project that is, that is obstructed by fundamental conceptual problems of the theory. So there are recent proposals, but the field is still open. And it's not clear how to calculate correlation functions of this model, neither at thermal equilibrium nor in the, in the case of, of quenches out of equilibrium. So if the goal of solving a quantum field theory is to calculate the values of observables and correlation functions, we are still far from, from this by means of analytical tools. So what we thought was to, to cheat a little bit and go faster to the, to the top of the mountain by using a numerical technique that doesn't require facing all of these conceptual problems of the analytical theory. The method that we use is called truncated conformal space approach and some numerical methods for the study of continuous one plus one, mainly dimensional models, irrespective of whether they are integrable or non-integrable. It's based on renormalization group theory and conformal field theory. And in contrast to tension network methods, it's, it's, it's, it's, it's, it's applies to continuous models where the dimensionals of the Hilbert space is from the beginning infinite. Now the positive thing is that it can capture rather efficiently non-perturbative effects. We have seen that in the calculation of the ground state and low energy properties. But the negative aspect is that it doesn't really solve the curse of dimensionality problem. The fact that the Hilbert space of a quantum system grows fast exponentially fast with the number of degrees of freedom. So we are still limited to essentially matrices, to two dimensions that correspond to a matrix problem of dimensions of 10,000 or at least 100,000, which are the limitations of exact diagonalization. Let me explain a little bit more how exactly these methods work. So the problem is to find the spectrum of a continuous quantum field theory, which we, we put in a finite volume so that the spectrum is discrete. Now, one of the simplest ideas one would have would be to express the Hamiltonian of this model as a perturbation, if you want, of, of another Hamiltonian that we, we know its spectrum exactly. It's an exactly solvable Hamiltonian. And then the perturbation, we also need to know how it is represented at the basis of H naught Hamiltonian. So we have known spectrum and eigenvalues and eigenstates for H naught and known matrix elements of delta H, the perturbation, in the basis of H naught. And then if we put the system in finite volume so that the spectrum is discrete and we apply some high energy cutoff to, to make the, to restrict the infinite Hilbert space into a finite truncated Hilbert space. Then we have essentially a matrix problem. We have a matrix that describes the Hamiltonian and we can diagonalize it by numerical means. Of course, in the Sine-Gordon case, the, the, the H naught part, the exactly solvable part is chosen to be this, the free parts of the Sine-Gordon model. So the conformal, the free boson conformal field theory. And the interaction is the cos beta phi. Indeed, it can be expanded easily in the conformal basis using the algebraic tools of conformal theory. Now, if we do this procedure, we'll find out that the Hamiltonian matrix has, in fact, it has a self-similar form. And if we keep increasing the cutoff, the energy cutoff that we use and keep expanding the truncated Hilbert space, we would hope to, to reach convergence. So of course, the, the exact problem is to solve, to find the spectrum in infinite Hilbert space, but this is impossible numerically. So the only thing that we can hope for is to increase the cutoff and see convergence of the numerically obtained spectrum to some spectrum that we would expect to be the exact spectrum. In the cases for, of the numerical implementations that we've done, the trunc, when we change the truncation cutoff, that is the energy of the conformal field theory used for truncation, we see that the number of states increases almost exponentially, a bit slower than exponentially. And this is the maximum truncation cutoff that we have reached. Now, an important warning is that if we apply this procedure, in general, we wouldn't, nothing guarantees that by increasing the cutoff, we would reach convergence. So if I did that for a general model, I could choose the free, the free parts of the Hamiltonian in different ways. And nothing would tell me that by splitting, doing different splitings, I would get convergence of the numerical spectrum. But there is one intuition, very crucial intuition that comes from RG theory, which is that if the perturbation delta 8 is a relevant operator of the conformal field theory, describing the, the, that corresponds to H naught, then we know that the low energy part of this Hamiltonian 8 will be considerably different from that of the conformal field theory of the free part. But at higher energies, it will come closer and closer to that of the free part. So asymptotically at high energies, the system, the model is still free. It's still described by the spectrum of H naught. And it is only a flow of energies that we have strong deviations from the spectrum of H naught. And due to this reason, or this expectation from RG theory, if we keep increasing the cutoff, we will see smaller and smaller corrections coming from the high energy part of the, of the perturbation delta 8. And for that reason, we, we can hope that we will reach convergence. If we chose delta H to be any relevant operator instead, then it would be the high energy part that gets modified while the low energy part doesn't. In that case, the method would just never convert. It would keep changing, the spectrum would keep changing with the cutoff. Okay, so if you have any questions about the methods, please let me know now. Otherwise, I will go to the next part, which is the presentation of some of the main results that we have. So just maybe a precision concerning your previous slide. Yeah, what? No, the one with the matrix. Yeah. Yes. So each, let's say, square, increasingly bigger square is when you, you increase the, the truncation, right? Yes, exactly. But so here it feels, I mean, each square is by increasing one to the, to the, to the, to the amount of truncation that you do, or is it by changing five as was written on the table? This is each one corresponds to increasing by one. So I say that the conformal field theory spectrum is somewhat exceptional, or very exceptional in that it has excessive degeneracies. The conformal field theories correspond to, well, it's probably not the right moment to explain all that, but it's when we, we can wait, if it put you in a, in a embarrassing order for your presentation. We'll just say that when we increase the truncation cutoff by one, there are exponentially many, there is an increasing number of degenerate states that are included in the energy cell. And that's why the size of these boxes increases with, with the cutoff. And what's, that's why exactly the Hamiltonian has this block diagonal, well, not block diagonal form, but it has this pattern with the block structure. The, the, the matrix elements of the sign goal of Hamiltonian in the CFT basis have strong dependence on the, on the cell, the energy cells to which they correspond and, and some internal structure into these blocks. Okay. Okay. Thank you. Okay. So now let me present three or four if there is time. Unfortunately, I can't see my, I can't see the time. I don't know how much time I have. So I would say it's already to 11. If you wind up in 10 minutes, we'll have some time for questions. Ah, great. Okay. I was a bit slow. Okay. So we have explored some effects of the presence of topological excitations, solitons, both in and out of equilibrium. And I will start with the effects on, on ground state correlations and thermal correlations. So here, by using the method that I described before, we construct a spectrum of the sign Gordon model in a finance box. And then we compute correlations of the phase field at different points. And in the massless case, we get a plot like the one on the left side. In the massive case, we have exponential decay of the correlations away from the diagonal, while in the same Gordon case, we have this perhaps surprisingly longer range correlations. And this is due to the fact that the phase field is actually not strictly speaking a local field because, you know, phases in quantum physics are not cannot be measured. But the, but only difference between phase can be measured. So practically what is measured here is the difference of the phase from the edge of the system where it is fixed to zero. And for that reason, it's like a string like object that is the integral of the derivative of the phase from one point to the other. So we have this, this long range looking correlations. We also computed the correlations in thermal stage, which is more interesting for the experiment, because the states of the experimental system are essentially thermal. And we were also able to compute the non-Gosianity of these correlations, so measuring four point functions and measuring the difference of the four point correlation from the weak, weak theorem prediction. We are able to measure a version of the kurtosis and multi point version of the kurtosis of the system. And we were able to explain the experimentally observed behavior that at low temperatures, the system behaves like free because we see essentially the bottom part of the interaction of the potential, which is parabolic at high temperature, the non-Gosianity drops again because we are the energy of the thermal state is far above the potential, the maximum of the potential. So we are in the asymptotically free regime, but at intermediate values, we have a strong deviation from Gaussianity because we see precisely the non-trivial part of the potential. And this matches with, as I said, with the prediction with the measurements of the experimental system. Now the next thing that we started medically was quench dynamics, so we could produce nice or well cool plots of the correlation functions as functions of time and see observe the difference from the free case and also explore the spectral analysis of this. We could also make some nice videos, which is just for visualization purpose of course, but from the theoretical point of view, what we found most exciting is what I'm going to present next, which is what we call the violation of the horizon effect after a quantum quench. So when we do a quantum quench in a relativistic quantum field theory, we usually see that the correlation spread in a light cone form, but because in the initial state, when in the initial state the correlations decay exponentially, distant points are uncorrelated practically and one would have to wait for a sufficient time until the signal of excitations from the middle of the distance between the two points reaches the two observers. And it's only then that we see a change in the correlations between the two points. This was pointed out by Calabres and Cardi when they started talking about quenches and a combined effect of the relativistic invariance and the fact that the initial correlations satisfy clustering. Okay, and since then it has been observed in lattice systems, because also those satisfy the Libre Robinson bound, which can be seen under the analog relativistic invariance in lattice models. And it has also been observed experimentally in various different systems. Now I'm not going to talk about those examples more. I want to go directly to what happens in the case of sine-golden dynamics. So in our simulation, we started with an initial state that is the ground state of the Klein-Golden model that is given by this Hamiltonian, which is quadratic and we know exactly that the correlation length in this ground state is of the order of the inverse of the initial mass of the mass M0. So it has precisely exponential clustering, exponential decay of correlations. And then we switch from this Hamiltonian to the sine-Gordon Hamiltonian and we do the dynamics with the sine-Gordon Hamiltonian. And we observe, we measure the correlations between different points at the same time t. And what we observed was the following that if we do the Klein-Gordon dynamics for these type of points, we see a perfectly nice horizon up to some numerical artifacts that increase when the parameters go far from the CFT point. But when we do the same with the sine-Gordon dynamics and measure the phi-phi correlations, the phase correlations, we don't see this light-con pattern. And on the other hand, for the phi-phi correlations, one may expect that because, as I said, phase is a field that no matter how you calculate it, it's practically a difference between phases at different points because we need some reference point to start measuring the phase. So it's inevitably a non-local field in this sense that we integrate from another point, the actual local field, which is the d-phi. The d-phi corresponds to the density of sultans in the system. And this is the one that can be considered as a local field. Now, when we computed the d-phi-d-phi correlations because, as I said, this is the local field, we still found that there was a strong deviation from what we expected to see. There is some light-con bump, if you want. But we see that correlations spread also outside of that bump and they have a characteristic oscillatory behavior, which increases and the frequency of these oscillations increases when we increase the interaction beta of the sine-Gordon model. So now that was quite peculiar for us, because... Sorry, Spiro. Just one question on the slide before. Does it mean, if I understood well, that if you were to be able to diagonalize fully the Miltonian, which is difficult because the system is not integrable, but assuming you do, and then you look at the excitation in this eigen basis, then you would observe a nice light cone, or is it really impossible for this Hamiltonian? The Hamiltonian is a relativistic invariant, but when we truncate it, as you say, as you probably guessed, we break the Lorentz invariant. So it's only approximately that this holds. And so at first we thought that we were almost sure that this is a numerical effect. But then we increased the truncation cutoff and we found out that the effect doesn't go, it doesn't diminish, and it actually converges. The oscillations converge when we increase the cutoff. So we know that the method is still very strongly limited by the small Hilbert space dimension that we can simulate. And we took very seriously the possibility that there are numerical artifacts. But by increasing the cutoff, we saw convergence of this out of light cone pattern. And we could only think that this is something real. There's one more question. I can't hear. Arithra, if you're talking, we cannot hear you because you're muted. Can you hear me? Yes. So this is at which temperature? So this is an increasing temperature? No, these plots from left to right correspond to increasing the beta parameter, which is the interaction here. Yeah, I should go back. And so what is the Hamiltonian? Well, here, for example, sorry. So the beta parameter controls the interaction, the distance between the minima of the potential. So for small beta, we get to the limit where we can approximate the potential by a parabolic trap because it becomes very steep. But increasing beta, we go away from that approximation. Yeah, sorry. Yeah. And is that the reason? For the first two curves, I don't see much the light cone effect. The periodic pattern is not outside the light cone effect. And then when I increase this delta or beta, the pattern goes out of the effect. Is that the reason because of the trapping? Yes, exactly. So for small beta, we can approximate the Hamiltonian with which we do the dynamics with the Klein-Gordon Hamiltonian with a very large mass. And therefore, in that limit, or even smaller values of delta, we get what we get from the Klein-Gordon simulation. But as we go far from that point, we see the strongly interacting dynamics of the Klein-Gordon model. Okay, thank you. Okay, so yeah, I won't have too much time to go into the details. But in trying to explain this numerical observation, we realize that, as I said before, the horizon effect when it's present, it's a combination of not only the relativistic invariance of the dynamics, but also a property of the initial state, which is that it has exponentially, exponential clustering of correlations. This is a typical property that we always expect from an equilibrium state of a local Hamiltonian. But now the tricky thing is that when we do the dynamics under the Klein-Gordon model, the excitations which describe these dynamics are soliton fields. And as we know from the time when Kodman and Mandelstam looked at the duality between the Klein-Gordon and the massive steering model, these solitons are non-local fields. In the sense that they are given by this expression, if I define the soliton field psi, creating a solvent on position X, it involves a strongly non-local operator, the integral of this pi field from one point up to the point X. So they are string-like operators if you want. So if we require that these non-local fields satisfy clustering, it's a property that we can't expect it to be true for any state like the ground state of the Klein-Gordon that we have here. For the Klein-Gordon ground state, we know that it satisfies clustering with respect to the local field or the Klein-Gordon theory, that these simple linear combinations of the bosonic field. But we have no idea whether this property should be satisfied for this strongly non-local field psi. In fact, we can expect that it doesn't, or we may. So to test this, this is the explanation. If we have a violation of clustering for the soliton fields in the initial state, we exploited the duality of the Klein-Gordon model with the massive steering model which allows us to solve exactly the dynamics of Klein-Gordon quints at the free fermion point. Of course, this from the point of view of the dynamics, this is the most boring point. But on the other hand, it has all the topological ingredients that are necessary to see what we wanted. So using bosonization, and I can give more details since there is interest later, we were able to do an analytical calculation of the two-point correlation function at the free fermion point. And we indeed saw that there is a violation of the horizon. We verified that this has nothing to do with the dynamics and it has everything to do with a violation of clustering in the initial state. So the fermionic operators which describe the solitons of the free fermion point do not satisfy clustering in the initial state which is the Klein-Gordon ground state. Later or very recently actually? Just to remind you, you have just four minutes. So like if you can wind up in five minutes, we can have question answers for two to three minutes again. Yes, great. Okay, so as one paper with further results, very recently, Ivan, one of my collaborators in the previous work, started the Swinger model which is essentially the QED model in one plus one dimension. It describes fermions interacting with the electromagnetic field which by means of bosonization can be mapped to a bosonic theory. And he used the same similar tools like the numerical simulation that we used in the other paper. And he found that the same effect is present also in one plus one dimensional QED that we can discuss later. Now another application that we did recently was to study the dynamics of the experimental system when we start with a strongly interacting initial state, but we do the dynamics with the latent liquid, the conformal field theory corresponding to zero potential, zero interaction. And there we had some results based on, this protocol is one of the problems that can be studied analytically and there were some expectations of what the steady state should be. I won't go into the details since there is not much time, but I can only say that there was a strong surprise in that we were expecting a different type of steady state and the experiments saw that the steady state is different from the theory at first. And then we had to explore what are the special properties of the initial state in this specific experimental protocol and we realized that there is a new mechanism that can give equilibration to a Gaussian state. This is something that we didn't expect to see here. So the experiment really could drive us to the direction to find a new theoretical mechanism. I can talk about that later if there is interest in probably private discussions. And the last point that I want to talk about is recent work that we did with Procen and one student in Ljubljana, Micha Cedriciek, which is about signatures of quantum chaos in non-integrable quantum field theories. So as you may know, when we go from an integrable model to a non-integrable model, there is a very striking change in the spectra of these models in that it starts with level crossings. There are level crossings in the integrable case, but energy level is repulse when we go to the non-integrable case. And this has been checked many times under very different settings in lattice models, but it hasn't been possible so far, apart from one work, I think, very pioneering work by Mussardo and collaborators some years ago. So it has been studied very sparsely in the context of continuous models. And this is what we were trying to do here. So exactly because of this change from level crossing to level repulsion, in the integrable case, the statistics of level spacing is described by a Poisson distribution, which is markedly different than the random matrix theory predictions that hold in the case of non-integrable models. So this is what exactly we did in, we studied in the double sign Gordon model. As I said, the sign Gordon model is integrable, but the double sign Gordon model is non-integrable. And can I just say something? It's already 231. So I mean, it would be nice that we just try to conclude now in next maximum two minutes. So what we observed was that the level spacing statistic is indeed described by the expected Poisson and GOE statistics when we go from the sign Gordon to the double sign Gordon model, and we were able to get the phase diagram corresponding to different parameter values. So the lines, sorry, the straight lines here correspond to exactly the sign Gordon points. But then when we computed the eigenvector component statistics, which is another test of quantum chaos in integrable models, we found that these are markedly different from what we expected to see. We were expecting to see Gaussian distribution, but instead they had the long tails that looked more like power low, so more like power low behavior. Okay. So with that, I would like to conclude the general message is that I think there is much to explore and many surprises in continuous quantum field theories. And I think that this method that I present here has the potential to give us a glimpse of this exotic phenomenon. And at the same time is useful for applications to ground breaking experiments in the field of cold atom, cold atoms in continuous models. So with that, I conclude and I would like to thank you for your attention.