 Tako, pa tukaj za mladu. Zato se ga se vzelo izgovorila s tem, in tako vzelo izgovorila na konalizaciju koreljače z zelo kvantunem, vsak in mladu, in nekaj je se datu, kvantunem sistemem, z pravimi vse in našelimi interakcijas. Prvom, odlič. O ker, kaj smo vzeli, tudi to je vsezam v sej, odlič kreženom fizika. In te dve zelo v zelo vse. The first one is, so if we drive a system out of equilibrium, how observables time evolve during this, out of observable time evolve during and right after the out of equilibrium protocol, and the second question is how the system reach equilibrium and how can we describe this equilibrium state which is reached a long time after the quench. So there are many ways to drive the system out of equilibrium and the most simple one and the most fluid one is the so-called quantum quench. So a quantum quench is a protocol that consists in preparing a many body quantum system in the ground state of an initial emiltonian and then at t equals zero we sudden change one of the parameters of this emiltonian and we get a new emiltonian HF and there is a known trivial time evolution. So this can be, this seems quite artificial but as we had a lot of tools, really interesting tools on how the system can be realized in co-atomic gases and why co-atomic gases because clearly we can drive the system out of equilibrium in a really controlled way. We can tune different interaction, change the couplings between the, of the interaction between the particle and then we can measure local and global observables during this time evolution with an extremely high accuracy. So we will focus on the time evolution of local observables in particular of correlation. So clearly this is an extremely complicated topic, in particular we need to solve the dynamics and this for many body quantum system in particular in presence of interaction is almost impossible. So there is some universal theorem that arrived to predict some results without solving the microscopic dynamic of the system. And we had a beautiful talk about Michael yesterday which was talking about this and to just keep these results. So it's, if we have a lattice model with short range interactions in generically dimension, if we have a local observable A located at X and a local observable B located on the subset Y, then the time evolution of the commutator can be bounded by this expression here. So even if it seems quite strange, this has really important consequences on the time evolution of the system. In fact, when T is small, clearly since they are local, this is zero or almost zero, so we have clearly that locality imposes the commutation between the two, but if we let T goes up, we let the time evolution, so we see that at the beginning there is an exponential suppression in the distance between the two sets and then just when we arrived at a certain time, which is called activation time T star, we have that correlation are not bounded. So there is a linear increasing region where correlation are exponentially suppressed and a linear and then the other region is where correlation are unbounded. And the problem that the physical meaning of this velocity, which is extremely important, since it is called the light cone velocity in analogy with special relativity, is not determined by the theorem. So it has to be extracted from the microscopic theory. So this is for, even if it is strange, it has been observed in experimental experiments like this one with this snapshot of a one-dimensional system and we see that red atoms that are correlated, they represent a region which is growing linearly in time, and here we have the density-density correlation function following a quantum quench in a bosubard emiltonian, which is a local emiltonian. So the information of the theorem is universal because we don't need to solve the system and every system with short-range interaction respect these bounds, but it's partial in the sense that we just know that here correlation are extremely small. We don't know what happens inside this light cone. Maybe there is an inner structure and we don't know. If we try to extend this result to long-range interaction, we get that it is not possible. There are many propositions. The first one chronologically is the one by ask things that proposed a bound for sufficiently fast decaying interaction, which respect this reproducibility condition, and in this case we can see that the same quantity can be bound this way. So we can notice in this expression a difference. Here we have the expert form of the long-range potential. So we are losing universality because before for the short-range case the result was completely independent from the specific form of the short-range potential. So in this case we have to specify a potential which choose one over r power alpha. In this case we can bound it just if alpha is larger than d and we get a logarithmic light cone. So we see that in this region we have an algebraically suppression of the commutator while here is unbounded. Again this is an upper bound, which means that we don't know exactly what happens here. We just know that here nothing happens. Then there is by Michael, they are ameliorating the bound and in fact for alpha larger than d we can prove that the bound has to be of this type with beta smaller than one. So you can see here we are ameliorating in this sense here we have the logarithmic bound and here we have the algebraic bound for alpha larger than 2d. So the main questions are, are these bound able to describe what happens in the real time evolution of long range interacting many body quantum systems or are just simply bounding their time evolution. What happens from the microscopic point of view and what happens for alpha smaller than d where no bound is present. In order to study this problem we study two different long range interacting systems. The first one is the long range bosabard and miltonium which is a standard bosabard and miltonium with a long range interaction between particles. In this case we study the density-density correlation at equal time and there are several propositions to realize this system in cold atomic gases. Then we study the long range transverse easing chain, which is a spin chain with long range spin exchange term. So the mathematics of the long range part is the same because it's one over r to power alpha but the physics is completely different because here we have more a synetic term while in the other case is exactly an interaction term. So it's a different physics but with the same mathematical bound. So let's start with the easing chain. In this case we had already some results in the literature for the one-dimensional quencher for a local quencher which consists in flipping one spin at the center of the chain and looking at the time evolution of correlation and three regimes have been found. No local regime where correlations are activated at extremely long times for alpha at a large distance in extremely short times for alpha small than one a quasi local regime where correlations spread algebraically for alpha in between one and two and a ballistic regime for alpha larger than one. So the question are is it possible to recover this result from a microscopic point of view? Can we recover this result with a different quencher in particular a global quencher and a different numerical model because here they used the dmrg we want to use a different model and is it possible to recover this result in higher dimensions. So let's start with the numerical results so we used the time dependent variation of Monte Carlo which is where we set an answer for the wave function of the gestro type where all the time dependence is located here and then we can choose different scattering operator here in order to take into account different physics. So the time evolution involves the computation of correlation that are computed using Monte Carlo and then for the light cone we saw that it is possible to describe everything really well in order to take into account two body scatterings. So this method since we can rely on the solution of the differential equation for the alpha parameters and it's a first order differential equation it is extremely stable in time we can access extremely long times we can take into account different scatterings changing the operator OK and it's fine in dimension larger than one. So the result for the transverse switching chain in a global quench so we set the system in its ground state and then we quench one of these parameters are here. So we found the same behaviors as in the local quench regime and we found a non-local regime for alpha smaller than one a quasi-local regime for alpha between one and two and for alpha larger than two we find a local regime. As you can see, our numerical method allows to access extremely long times and long distances which are crucial to extract the power law of the increasing light cone and the velocity in this regime. So it is clear that the microscopic bounds that predicts just are bounded and unbounded region are not able to reproduce these three regions. So we need a microscopic approach to understand what happened. So we did this microscopic approach using the spring wave picture in spring wave analysis and we can do it for d-dimensional systems and in particular we can write the sigma-z, sigma-z correlation in this way here where g-infinite is the termalization value and g is the time-dependent part which is written here. So the physical meaning of this term here is in agreement with the cardical-abreze argument which means that we have two counter-propagating beams of the particle spreading correlation around the system and the initial state is entered just in f. So here is just the final Hamiltonian that enters in the computation and just the final dispersion relations determines the propagation of correlation. So in particular the cardical-abreze argument says that if I want to create correlation on a distance r and since we are like this globe, these local operators that spread with quasi-particles I have to wait a time to create correlation on a distance r which is the distance divided by twice the maximum group velocity. So if the maximum group velocity of this excitation goes to zero I can have an extremely fast propagation because I can create correlation at arbitrary long distance in arbitrarily short times. So it is important to look at the dispersion relation of the excitation that is written here and p of k is the full transform of the long range potential in the dimension. Clearly when k is extremely small this sum here can be divergent if alpha is too small. So this divergence in the energy we create also a divergence in the velocity that creates different propagation. So in particular for alpha, small health and d we have the energy that is divergent algebraically and also the velocity is divergent algebraically. Then for alpha in between d and d plus 1 we have that the energy is a casp so it's finite but with an infinite velocity which again creates propagation which is not ballistic and then for alpha, larger than d plus 1 we have that both the energy and the velocity are bounded. So you can see here they are both bounded for alpha, larger than 3 dimensional system. Here we have a casp and they are both bounded. So this 3 different types of dispersion relations creates 3 different propagation regimes. So for alpha, smaller than d in equal 1, 2 and 3 we find that the propagation is instantaneous. For alpha in between d and d plus 1 we find that there is a ballistic increasing light cone so we can use then our expression to find more analytical results in particular for alpha, smaller than d we have a divergent dispersion relation that creates as you can see here from an exact computation we can find that there is an extremely small time scales. So this extremely small time scales coincide with the first maximum of the correlation function meaning that we have a first signal at extremely small times. So we can extract the scaling of this process and then the scaling of the correlation function for our analytics here and we can compare with the one extracted from the numerics finding that they are in perfect agreement these lines are the prediction and the points are the extracted, the one extracted from the numerics. So for alpha in between d and d plus 1 we can find that there is this casp in the dispersion relation that creates an algebraical increasing light cone and we can extract the power low beta for different value of alpha finding that beta is always larger than 1 while the prediction for the boundary beta is smaller than 1 so we are not evaluating the boundary but we are consistently inside the region predicted where the correlation are not bounded. Then for alpha larger than d plus 1 there is a maximum group velocity and we can compare it with the light cone velocity finding that the cardical approach still holds in a really good approximation. So finally for this model we find three regimes an instantaneous regime fast and ballistic regime and a local regime where they are all just they can be understood just looking at the dispersion relation so the dispersion relation dominates so propagation. So what we can do now is looking at another model the bozabar model and look if this behavior is reproduced again in this system if there is something universal so in this case we look at the density density correlation function at equal time and again we use the time dependent variational Monte Carlo for the one dimensional version of this model and we find that for every value of alpha extremely small like 0.1 we find that the correlation is ballistic in every time. So we can take a look to the dispersion relation because again we found one regime instead of two by the bound so we need again the microscopic theory to understand so we studied the time evolution in the super fluid regime where we have the bogodjub of theory working and we have again the dispersion relation of excitation since we can write the correlation function in the same way as before and here again in V we have the Fourier transform of the long range potential that can introduce some divergence. So if we look at the divergence now we find that for alpha larger than 1 our finite maximum group velocity is present in the spectrum which is consistent with a ballistic propagation and we check that the maximum group velocity is exactly divided by 2 is exactly like on velocity. For alpha smaller than 1 on the other case around k equal to 0 we have extremely fast modes and we have a ballistic spreading of correlation in the numerical data which is a apparent violation of the card and Calabrese argument. So we try to understand why these modes are not contributing and so we take the correlation function and we take the limit and we study it in the stationary phase approximation taking the limit r and t going to infinity together with r divided by t equal to V and this can be written in this way which means that this expression here gives us the the contribution to this expression given by the modes with a fixed velocity V. So instead of summing over the modes this gives just the contribution given by all the modes with a fixed velocity V. The summation is over k alpha which is the solution of the stationary phase approximation and w is given by the prefactor of the stationary phase approximation where now it has been to be noticed that appears f. So f enters in the calculus and f depends on the initial state and in particular on the observables. So different observables give different value of f. So if we look carefully so at the value of w we have a summation some value of w can be extremely small and disappear in this summation. So if we plot in log log scale the dispersion the velocity, the group velocity we found that we can have three solutions of the stationary phase equation like in this case here. The blue one contains the long range part which is the closest to k equal to zero and the other two are determined by the short range physics mostly. If we plot now the function w as function of the velocity in this case we find that the blue line representing the long range part is extremely small compared to the other two meaning that it does not contribute to the correlation function. We can then erase completely this part since its contribution can be neglected and we can predict that velocity of the correlation function will be simply this point here and we can check if this velocity corresponds to the velocity of the Monte Carlo. And in fact we can see for alpha equal to 1.0.5 that the velocity started by the Monte Carlo is extremely close to the velocity found by the quasi-particle approach and this the different is the same order of magnitude as in the other cases where cardin calabres approach work exactly. So here are the conditions the conclusions and we found three genes in the transverse sezing model and everything can be understood simply by the dispersion relation and this computation and agreement both with the time-dependent variation of Monte Carlo and the energy. For the Bose abardementonia we find that there is a ballistic propagation even if we have infinitely faster modes and everything can be understood by dispersion relation but also from other quantity which is observable dependent. All the results does not violate the theorems, the bounds over these quantities but these bounds are not able to reproduce exactly the time evolution. So the microscopic analysis is still fundamental to predict the current behavior. So thank you very much I'm a bit a bit late, sorry.