 And so this is a work that I've been doing in collaboration with the Paris group. And Giuseppe is not anymore in Paris, but he was in Paris for some time. So the group is the group of Laurent Sanchez Palencia. And this work originated as a follow-up of the content of the PhD thesis of Lorenzo Cevolani. And so most of the work indeed has been done by these two young people here under the supervision of Lorenzo. That is now a polytechnic. You can find it in this preprint here. OK, so let me start with an outlook. I will briefly describe. So since there has been a school and we have already heard a lot of talk about auto-equilibrium dynamics, I will not motivate why we are interested in that. But I would rather specify which kind of auto-equilibrium dynamics I will deal with in this talk. So I will talk about relevant Hamiltonians and Quentches. And I will give you the kind of framework in which we can understand these kind of Quentches. There is a framework introduced by Cardin Calabrese around 2006. And I will then, so in this framework, there will be some objects that we call pseudo-particle or quasi-particle, depending on the community. And so I will introduce what they are. And then I will pass to describe the results that we obtained in this picture. One that we obtained already a few years ago with Philippe that is here in the audience about identifying different classes of long-range system. But then we'll see that this was kind of model-specific. And in this new paper, we give a recipe that should tell you what is model-specific and what is universal in this long-range system. And so I think it's a nice kind of wrap around the many results that this paper has kind of triggered. And then once I have introduced this universal picture, we'd apply it both to short-range system, where we rediscover some of the things we already know and we will see some new phenomenology that could be observed in the experiment. And then, indeed, into this long-range system, it will allow us to wrap up all the results that were more or less already known in the literature and systematize them. OK, so let's start from the background. So in this talk, I will focus on Quentum Quentches. So the Quentum Quench is a process on a Quentum system. And so we first introduced the class of Hamiltonian we are interested in. And we are basically interested in Hamiltonian that are translational invariant in such a way that the eigenstate are extended. And there are some of two-body terms, not necessarily nearest neighbors. They could be long-range, but there are two-body terms. So they have kind of a few-body interaction, and then they are translational invariant. So for example, you can think of this dot as a quantum spin chain. And so the position of the spin is denoted by r and r prime in this case. And then you will find a term in the Hamiltonian that gives you the interaction between r and r prime. A typical example are all the prototypical Hamiltonian that we use. For example, in this case, we have the Bose-Habbert Hamiltonian which a and a dagger are creation and annihilation operator of bosons. And so here you have a term that describes the hopping of a boson from one side to another one from r to r prime. And then we have a term that tells you how much energy cost to have two boson or n boson on the same side. OK, so once we have the Hamiltonian, the protocol that we use to put the system out of equilibrium is to quench the Hamiltonian. So we start from an equilibrium state. For example, the ground state of an Hamiltonian described by set of parameters. And now we put the system out of equilibrium by changing the Hamiltonian. For example, changing the parameters in my Hamiltonian. And then following the Schrodinger equation, you can write the evolution of your state in this way. And I have no time to go into that. But in general, these are exponentially hard problems. So we don't know how to do it exactly. And what you observe, so in some specific case, for example, one this system, we have some numerical algorithm that allow us to perform the evolution for some time. But what we observe typically is that if you look at some correlation as a function of time, and here is the position along your, for example, one-dimensional chain, you see that they change with time. So they are out of equilibrium. And so you see these beautiful pictures. So here, if you have a bit of imagination, you can recognize it here. I don't know if you can, but I can. Another question is, how do we explain the things? Since we cannot, we cannot, this was timed, no, Pasquale? No, it was timed, yes. So how do we explain these auto-equilibrium dynamics? And it turns out that the natural setup was introduced in it by Pasquale and John Cardi in this paper. And the idea is that we put the system out of equilibrium. We inject some energy. Since the eigenstates are extended, as I told you before, what happens is that this energy radiates in a form of pseudo-particles. And so how do we see this radiation? So for example, if we look at the system on a circle and we consider a region of size L, the correlation between L and the rest of the system can be described in general through the entanglement entropy. And so we see that as the time evolved, this entanglement entropy increased, meaning that the correlation that were originally localized in the region L. For example, if you were in the ground state, we know that the ground state of local Hamiltonian have a short-range correlation, they start to spread. So that's the idea. And they spread in a very specific way. So they are kind of transported by the pseudo-particle that originally, when they were created close enough, they were originally entangled. And so they transport correlation. And so this picture was introduced here in this paper and then it was later used in several contexts. But let me try to give you an idea on what it's a pseudo-particle. And so in general, you can imagine that you have a state, for example, the ground state, so an equilibrium state. And as we said, the Hamiltonian should have extended eigenstate. And so one possibility is to think of this extended eigenstate as kind of plane waves obtained by acting by some localized operator, more or less localized operator, onto your ground state, and then build up a plane wave out of it. And so if you do that, these things is, depending on the system, can approximate quite well, can be even the exact solution of your eigenstate equation. And the idea is that there will be energy bands. So the energy will be kind of labeled by the momentum of these plane waves. And you can now plot the energy as a function of the momentum. And this ak is called a pseudo-particle state. Because it's kind of the idea that you created locally in the same way you created a particle in second quantization. And so the idea is that now we have these energy as a function of momentum. And what will be important in the following are two concepts. One of them is the idea of the group velocity. So from your quantum mechanics course, you should know that if you create a superposition of several plane waves, the center of mass of the wave packet will move with the velocity that is given by the derivative of the dispersion relation with respect to k at that specific k around which you create this packet. And so this is this green line. And then there is another concept that is the phase velocity that is indeed the slope, so the y divided by x. So the energy at k divided by k. So in general, you don't have to coincide. It's important. These are the two concepts that I will use in the following. So let me stress again that both of these concepts depend on the specific momentum. So they vary depending on the momentum of your pseudoparticle. So now that I introduce all the context, I can tell you about the result. And so the first result is to try to get a quantitative picture in quenches using the pseudoparticle picture. And one of the interesting results from my point of view when was one of the first in this area of lattice system was when we tried to understand what happened if you introduced long range interaction in a spin chain. And so we studied this prototypical spin chain in which you have anizing spins and they interact with the sigma x operator. These are polymatrices. So we have a 1D chain with these polymatrices at each site. And so you see the spins interact with polymatrices, but their interaction is long range. It's not only first neighbor, but it decays as a power law of the interaction. And then we have these transverse fields that is local. And then we have these parameter theta that allows us to decide if the long range part is predominant or if it's not. And then we were doing numerical calculation based on kind of generalization of the MRG. And depending on this value of alpha, we would see these types of plots. And it was very hard to kind of understand when we would pass from something like this to something like this to something like this. They obviously look different quantitatively and qualitatively, but they need to have a real quantitative understanding of what's going on was difficult from simple numerics. And then, Philippe had the idea to, OK, let's check what happened with the spin waves. And spin wave is another way of introducing the pseudoparticle. And so that's an approximate solution to the problem because we approximate the eigenstate assuming that they are a kind of pseudoparticle. And what he was able to do, he was able to go to much larger times, indeed, because it's an approximate technique that allows you, it's basically analytical, and that allows you to go as far as you want in time, differently from this one that is numerical and has some limitation. And so the interesting thing is that, again, you see, once you have a kind of longer time picture, these things start to look very different. Not only, but you can now study the dispersion regulation of this pseudoparticle. And here we have three examples. And you see that as you vary alpha, so for alpha equal to 3, you see that the dispersion relation is continuous, and also its derivative is continuous, and it's bounded. So the derivative, remember, is the group velocity, whether if you go below 2, the dispersion relation is still continuous, but the derivatives start to kind of diverge. And then if you go below 1, you see that you have these peaks. And furthermore, now you can look at the maximal group velocity, so the maximal of these things as a function of your system size. And again, here we have alpha, this power law exponent. And here we have different system size, so lighter means smaller, darker means larger. And now we plot, for example, in this insert here, we plot the maximal velocity as a function of the system size. And you see that for alpha larger than 2, the velocity doesn't depend on the system size. For alpha in between 1 and 2, it starts to depend on the system size, and it keeps going for alphas smaller than 1. But now if you divide by the system size, now you get an idea of the time of an inverse time of arrival of your excitation. And so what you see is that, indeed, as you expect for alpha in between 2 and 3, the larger the system, the smaller the inverse arrival time is, meaning the larger the arrival time is for a perturbation that is created at some place in the chain and has to propagate. But then as we go in between 2 and 1, the same is still true. The time still increases with larger system time, even if less dramatically. But then there is a crossing. And for alpha smaller than 1, the larger is the system size. The shorter is the time that the excitation takes to get to the boundary of your system. So we have a clear picture on where the transition points are. And this was, I think, an interesting result. But it was, indeed, a result on a specific model. So then there have been a huge amount of work, including some experiments in which Philip took part on the theory side and generalization of this idea to x, x, z model, long range fermionic model. There are plenty of references. So if some of you has worked in this and I didn't cite, please accept my apologies. It's not meant to be a representative of things, but rather to cite the stuff that the people in my group have been working on or my collaborators. And now the idea is we started to get contradicting messages because it looked like in certain model, if you wanted to bound on this alpha were different in certain long range model. And so at some point we decided that we needed to go back and check what was going on, why we had this discrepancy, what was different in different models. And so this brings me to this universal picture that we have kind of unveiled recently. And the idea is the following. So this is, again, contained in this paper here. The idea is the following that in the pseudoparticle language, so either in the mean field approximation or in the spin wave approximation or in free fermions or in boson, so you can always write your correlation function in quench in such a way where you have a time independent piece that we are not interested in, but then you have an integral over your brilliance on over some density of pseudoparticle times this factor here. And this factor here has to do with the pseudoparticle energy and the momentum. So these are just plain waves. And so you can immediately recover these expressions here. And so the important thing is that we have these two pieces. So we have this density here, the pseudoparticle density, and we have these kind of oscillating piece in space and time. Now once you can do, so indeed this was derived in many contexts, but you can really find this expression there. So the new things that we have done is just to take seriously this integral and compute approximate solution to it. And so one immediate thing that you can do once you have an integral with an oscillating piece and what everyone would do is to do the stationary phase approximation. So if you do the stationary phase approximation, you get some condition on the phase. So you get that the derivative with respect to the momentum of these objects should be 0. And you immediately get that to remember that the derivative with respect to momentum of the energy is the group velocity. So you immediately get that twice the group velocity should be equal to plus or minus r divided by t. So this is the condition to have a stationary phase solution. Now what does it mean? So let's consider, for example, short range system. So if we have short range system, remember that twice the group velocity should be equal to r divided by t. But if we have short range system, we saw that if the system is regular enough, the group velocity is bounded. So it's bounded, and it means that if we take r divided by t too big, bigger than the maximum group velocity that here I call vg star, then there is no stationary solution. And what does it mean that there is no stationary solution? And you expect there is a kind of Lib Robinson code. So the integral is exponentially suppressed for value of r divided by t larger than the maximum group velocity. So we immediately recover a very simple result. So in this case, the integral is exponentially small. So there are stationary phases only for r divided by t smaller than twice the group velocity. And so now let's have a look on how this stationary phase solution look like. So if there is a solution, so if you are inside the Lib Robinson code, you can just compute the stationary phase value of your integral. Now k is the stationary phase k. So for every k, you get this contribution. And we can analyze these things. So first of all, let's assume for the moment that this factor here is more or less uniform in k. And let's focus on this piece here. And we see that this piece here is oscillating as a cosine. And it has several features, maxima and minima. And this feature now move at a speed that is given by, so you have to take this thing as a constant. And it tells you that these things move at a characteristic speed that is twice the phase velocity. So we have features inside of our light cones that move at a different velocity than the light cone itself. So let's make an example. When is it relevant? So for example, if you go in the superfluid regime of the Haber model, the model that I introduced before, the superfluid regime can be described in some approximation with a Bogolubov theory, also a mean field treatment. And so for example, if you do a quench by u i n equal j to u f n equal 0.5 j, n is the average occupation. So you are deep in the mean field regime. And what you see is that your dispersion relation look like this. So we have a group velocity in green, and we have a phase velocity in blue. And you obviously see that the group velocity is larger than the, I hope that you can see that the group velocity is larger than the phase velocity. So now here we plot space, and here we plot time, and we plot this correlation function. Remember, we take out the piece that is not time dependent because we are not interested in that. So we consider the correlation function of creating an operator at time t in position r and destroying it at time at position 0 at time t. And what you see is that immediately that we do have a Lib Robinson cone that is in green, that indeed propagate at the maximum group velocity. But then we have all these features that are this maxima that start from the Lib Robinson cone because then after that they are exponentially suppressed and go inside. So we have this other feature, this maxima, that propagates much slower than the group velocity. And in order to show you that this is not a feature of this specific quench, here we have plotted the, so we have measured, so this is a numeric simulation. I forgot to tell you, I've done with the kind of generalization of the MRG. So you see the time scale is around 10, similar to the upper panel that I showed you in the previous figure. And so we did several quenches, starting always from this u n equal to j, to different u n divided by j. And you see that there is a systematic, so the blue lines are obtained by fitting the maxima of this figure. And these green lines are obtained by the so-called epsilon method. I can tell you later what it is. But basically it is a method in which you fix an epsilon and you ask yourself, when is it that this function exceed the epsilon? And then this gives you a kind of an arrival time, the kind of liberalism time. And so obviously what you see is that these are numerical, these are the analytical prediction from the mean field approximation. And the dots are the numerical extraction with fits of this velocity. We see that they agree quite well and we see that there is a systematic feature in which the phase velocity that is in blue is always smaller than the group velocity. And so we expect this kind of secondary maxima and minima that's spread from the light cone. A different scenario is obtaining the completely opposite phase where we take the Bose-Habbert model and check what happened if you quench from u infinity in the mod phase to u equal 18j. So we are deep in the mod phase. And what happened that there is a gap, obviously. So you see that your dispersion relation is gapped. But in this case, the phase velocity is smaller than the group velocity. Sorry, is larger than the group velocity because of the presence of the gap. So the group velocity is smaller than the phase velocity. And so what you see in it is in the same kind of plot. Again, you see the scale of time ordered 8j, meaning that we are doing again the MRG. So from this space and this time, you see again this green line where again we fix an epsilon and decide the arrival time. And so we get this green line in agreement with the maximum group velocity. But then we see this fastest inner structure in which maximum propagate faster than the group velocity. Obviously, then they die exponentially out of the group velocity, out of the Libre Robinson cone. But still you have this kind of very strange inner structure. And again, this hasn't been seen in experiment because a typical quench were done very close to the transition from a multi-insulating phase to close to the transition from multi-insulating to superfluid where the two velocity almost coincide because the gap is going to zero and so they tend to become closer and closer. But in principle, it should be observable in experiment and it has not yet been observed. And so now I go to the long range system that it was a bit the class of system that triggered this study despite the fact that we see that we have a new feature in short range system. And the idea is that long range system are a bit more complicated. So for example, I will consider, for example, what is the X and Z model in which the typical things that is called delta we decided to call epsilon. And so you have a long range X and Y interaction and also Z interaction. And now we focus in the regime where on one side the dispersion relation is bounded and on the other side the group velocity diverges. So it was that regime that in the ISM model was between one and two. But we'll see that here it becomes different. So imagine that, so this means that we can write the group velocity as possibly a gap plus okay, so this piece at which the group velocity diverges we fix it to K equals zero. So it's just a gauge transformation. You can do it. There is only one point. So around that point this is kind of the gap if there is a gap and then this is the leading term of your dispersion relation close to K equals zero. So a power law. And then we also assume that the observable so if you remember this was the weight of the pseudoparticle also as a power law behavior. So with these two ingredients, so this is something that depends only on the model because it's a dispersion relation. Obviously there's something that depends on the observable that you're studying which correlation function you're studying. So this I call it universal and this I call it somehow observable dependent even if can be universal. So critical exponents are observable dependent and can be universal. But in this case what I mean is that it depends on the observable that you consider. And so obviously if you assume that the Z is between zero and one so because we want this to be continuous but we want the derivative to be divergent, the group velocity takes this form, okay? And it diverge at K equals zero. So if we repeat again we take the integral and we do the stationary phase approximation since the group velocity now diverges we will always find the solution and the solution, the stationary phase K as this form here and then if you go and kind of substitute your solution into your integral you get this contribution here. And here again we have several features that I want to show you. First of all, you have to see that T is elevated at some power that depends both on U and Z. U remember depends on the observable and Z depends only on the dispersion relation and at the same time R also is kind of, is raised to an exponent and depends on both variable. So the first thing is that now if you ask yourself when this quantity is order one before we found the kind of Lib Robinson cone that was linear with velocity now you see that there is, this quantity is order one if you go along trajectory in which T is proportional to B at some power, no? Sorry to R at some power beta where beta is the ratio between these two exponents and so first of all you will see an algebraic cone rather than a linear cone and it's something that come out from this very simple argument and now we can kind of check also what happens to the inner structure and again the inner structure now depends on several things. First of all if, so remember everything the cone depends on the observable and the inner structure for example depends on the presence or absence of a gap here because if you have a gap you can kind of ignore the things here and if you don't have a gap then you cannot ignore the thing here. So let's assume first of all that we don't have a gap then obviously we go back here we have this feature again and we want to know when they become constant and obviously this is along this line here where T is proportional to R to some beta and beta is exactly the dynamical critical exponent. So if you want the inner structure so the outer structure is this green line here so first of all here you see there is a spin wave approximation because the scale of time is larger it's a log plot. So power law became straight line but this is a power law so it's an algebraic cone. So first of all the green line is an algebraic cone the white line is just a linear thing a guide to the eye to see when the power is one and so the first thing we see that the cone is slower than linear so it means that we have subbalistic propagation so it propagates slower than in the short range system but the inner structure you see depend on Z and obviously this Z is a dynamical critical exponent in the gapless case so for example by looking at the inner structure you can extract this dynamical critical exponent and it is a universal feature it doesn't depend on the observable whether this cone here does depend on the observable but it turns out that if you go to the expression here gamma divided by, chi divided by gamma is always smaller than one because of the expression. Interestingly you can see that this alpha now is 2.3 so now this is the region in which for the ISI model we said that everything was like short range but for this model here since we said that the cone depends on the observable and these observable that we are looking at then we see that even in this so-called short range regime we have a larger bright cone rather than a linear cone but again this depend on the observable now when there is a gap interestingly the outer cone is still subbalistic but the inner cone becomes ballistic yes and I think with this more or less that's it so here again you see scale of time is kind of much longer than what you can reach with the MRG so it's again in a spin wave approximation this is the guide to the I remember this is now this is the transverse ISI model the same model we studied with Philip so you see again that the outer cone is subbalistic so it goes lower than short range but the inner cone is ballistic as we expect in this case for the parameter that we are starting. Okay so hopefully I managed to give you an overview of what you can extract by using taking seriously this pseudoparticle picture and the idea is that you can extract some specific quantitative result and you can extract also a unifying picture in which you understand what do you expect to be universal and what do you expect to be model dependent and in which sense it is model dependent and in our case we have seen that for short range system everything nothing on what I discussed depend on the observable it only depends on the dispersion relation on the ratio between group velocity and phase velocity so depending on this ratio you can see either this minima starting inwards from your Lib Robinson cone or this minima going outwards from the minima and maxima going outwards where for long range system the situation is much more complex the slope of the outer cone is always subbalistic but the inner cone depends for example on the presence or absence of the gap and the exponent of the cone so of the outer cone also depend on the observable and once you have a gapless dispersion relation from this maxima you can extract the dynamical critical exponent and with this I would like to conclude and I want to point your attention that we are hiring so if you know of interested people please tell them to apply, thank you very much.