 OK. So can everybody hear me? Yes. Good. So thanks for the introduction, and welcome everybody to my talk. Today, I will discuss how is it possible to use quantum crunches to explore the anatomy of the fixed point characterizing critical states in perturbalization of quantum antibody systems. Specifically, as the title suggests, what I will show you is that the real-time evolution of observables displays two different scaling regimes, one mirroring a quantum perturbal fixed point, and the other one mirroring a classical perturbal fixed point. Yes. Does it work? Oh, thanks a lot. So actually, this entire discussion can be easily accommodated in a more general picture that you can classify as universality and criticality in and out of equilibrium. So if you think about a classical phase transition, of course, the first example that comes to your mind is a classicalizing model. There, there is a critical temperature that improper dimension is separating a paramagnetic from a paramagnetic phase. And the technical clue that you can use to study the critical point is, in most simplest cases, fit to the fourth theory, field theory. Then, as you know, there are quantum phase transitions that usually are dual to classical phase transitions. So if you consider a quantum antibody system in its ground state, and you change the internal parameter of the Hamiltonian associated to that, you can, again, separate paramagnetic from paramagnetic transition. For instance, in the case of a quantum, I see model. And when I say that are dual, I mean that, indeed, you can map the universality class of the dimensional system to its classical, higher dimensional counterpart where the increase in dimensionality is given by the dynamical critical exponent, z. It's natural to go to the other part of this table and to think about non-equilibrium criticality. Indeed, for the classical case, you start to have a zoo of different instances. Alessio has illustrated the simplest, yet different, case of non-equilibrium critical behavior that is aging, inizing model suddenly changed in temperature. Of course, many of you can be experts of other examples of non-equilibrium classical criticality that range from KPSET model percolation. You can name them. And as you have seen already in the first talk and in the talk by Alessio, the proper formal is to do that is a Martin-Cigarose formalism. So you are complying the cross-current order parameter with a response field that's also inside noise and thermal fluctuations. If you uncover the last window of this table, indeed, what happens has a number of questions because it is a research ongoing field. And you would like potentially to list the zoo of non-equilibrium quantum university classes or ask, for instance, if this D into D plus Z mapping exists. And in general, what will destroy non-equilibrium quantum criticality as temperature does for an equilibrium quantum phase transition? Usually, when you have an ising model in one dimension of final temperature at large times and space, temperature will give you a cutoff, a debris left that will destroy long-range correlations there. And naturally, there is a number of technical questions you can ask as well, so which should be the proper field theory if you can do an RG in order to understand that. And quite surprisingly, I would say these kind of questions have been asked so far in open systems. So typically in driven dissipative quantum many-body systems in different distances, very recently in Kern we have proposed an analog of quantum criticality in Markovian quantum many-body systems. But then we met Alessio and Andrea that suggested that indeed we can also look at that in isolated systems. And we will build up a collaboration that is the goal of this talk. I summarized the message in this single very qualitative slide, and then I will go more focused on the topic. Essentially what I will tell you is that there are, in the pretermal plateau, of a quantum many-body system to non-trivial fixed point, a quantum and a classical one that are extremely sensitive on a specific parameter. This omega-not is the mass of your field theory before the quench. I'm just thinking now that I prepare a system in a Gaussian state with a given mass, omega-not, and then it suddenly changes the mass and that interactions. And when I say that they are controlled by this omega-not is because it acts like a temperature. So if you perform a shallow quench, a small value of this omega-not, you will have a long amount of scales in which the RG flow will linger close to the quantum point while if you do a deep quench, a big value of omega-not, you will suddenly reach in the RG flow with a classical fixed point. As you have realized, and speaking about RG flow, but here, there is time. Because the goal is that such dimensional crossover, such RG crossover between fixed points, I didn't speak about dimensions up to now, will be reflected also in observables. As a less, you probably anticipated that these two zero and a temperature fixed point indeed can host in isolated system an instance of aging. So the same kind of phenomenology that you presented in a classical system. And when I speak about aging, I mean that here you will encounter an instance of quantum aging because the system is isolated. There is no coupling to buff. The system is its own buff. And then instead of this simple picture here, what you will encounter at the end of the talk is a double scaling regime that maybe here is a bit portrayed, but anyway, we'll have a larger picture. There is a double initial time increase with two critical exponent, quantum and classical, that comes from the quantum and classical protermal fixed points. So what you get in the RG flow, you get it in the many body dynamics of real observables. Good. So this is the main message. Of course, it is qualitative. I will make it quantitative. Let me start with this slide that is quite popular here in CISA. And I'm speaking about quantum crunches. For the sake of clarity, the idea is always the same. You prepare a system in the ground state of a quantum many-body Hamiltonian, and you suddenly change some internal parameter of this Hamiltonian. You do it fast, so the new eigenstates in some sense are not the same of the original model. In particular, this one is not an eigenstate. The ground state of the pre-quench Hamiltonian. And so you will have a non-trivial quantum many-body evolutions. If you want, you can rephrase the problem in another way. The quench has populated all these many-body eigenstates in some way, and you want to understand whether the system will relax towards a thermal state in the long-time limit or whether towards some kind of non-equilibrium steady state, which indeed it is what occurred in the first experiments. There is this pioneering work in the group of David Bice where they consider the sudden quench of a one-dimensional, strongly interacting bosigas, and they didn't see relaxation towards a thermal state. This kind of non-equilibrium steady state occurring in experiments can be seen in a modern language, as per Caruso of thermal states. Jörg Schmidt-Meier has shown, considering the current split of quasi-one-dimensional bosigas, that first you have this relaxation towards a non-equilibrium state, a pro-thermal state, if you want to call it in this way. And later there will be a slow departure and approach towards thermalization. I said that the systems are integrable. This means that they have an extensive number of conserved quantities. And if you want to characterize the steady states, you need more than a usual organic canonical ensemble. You may be aware about this generalized Gibbs ensemble that is summarized in some sense in this more theoretical and pictorial way to represent pro-thermalization. So you first are quenching an integrable system more like the first experiment that I discussed. And you have to account for your asymptotic steady state for all the conserved quantities of your model. But in every realistic experiment, as the second kind I was showing, you have always some kind of many-body non-linearity, some kind of inelastic interaction among particles. And this in general can induce a relaxation towards a thermal state. The physics behind this pro-thermalization is given by two scales. Indeed, first you will have a defacing process. This means that your observable can be the composite as the sum of many oscillating terms. Each one has a slightly different frequency and a deterministic limit to the competition and the interference of these object results in a relaxation process. But later, the inelastic scattering term will induce some kind of exchange of energy among the excitations of your system. And this is the only channel through which you can reach a state in which all your excitations have a common temperature. Pro-thermalization has a long history. It was first announced in the field of particle physics. And later there were the first attempts to extend it to condense a bunch of systems. And this inspired Michael and Caroline in 2010 to give the first portrait of this anatomy about pro-thermalization. And indeed, this motivated color to propose that these GGEs occurring in integrable systems are nothing more than these non-equilibrium steady state occurring in pro-thermalization. You can encounter in a number of other situations, these robots to presence of noise, of two coupling to open systems. You can see it in long-ranging interacting systems. People now are attempting more exotic initial state and to go to higher dimensionality. The point is that pre-thermalization, or at least this first plateau, can be easily captured if you work out upper-turbative Dyson equation. But if you want to go further, you need a self-consistent resumption of diagrams. These are the best of my knowledge, has been done in condensate matter recently with equation of motion methods. However, I will focus my entire discussion on this part that is, if you like, the simplest, but is the truly non-equilibrium one, while still you are in a metastable non-equilibrium state. And indeed, the point is that pre-thermalization can ask instances of classical and quantum non-equilibrium criticality. So the fact that this can happen is not such a new big thing what we provided something else, but let me introduce to the point. So there was a pioneering work in 2010 by Calabres and Gambassi, where in prototypical models like an ON interacting field theory, you can see the distances of the non-equilibrium criticality. So here I just provided you the Caledish action. But if you're not familiar with the method, you have just to recognize here the structure of the ON model. There is a mass, and there are quartic interaction. And here is the number of components. We are thinking about a very composite and general quantum quench. So you are simultaneously changing the speed of propagation of quasi-particles, of course, the mass, and you switch on interactions suddenly. This is the generic quench profitable. However, if you want to capture pre-thermalization, what is enough is to do a perturbative one loop, or if you want a one loop self-consistent treatment, it does matter. The point is that if you want, for instance, to get the dressing to your bare mass, you will have a correction mediated by the interaction. And of course, here there will be memory of the initial non-equilibrium condition that you have given to the system. This dressed mass delta of t will be time dependent. But as I said, after some defacing time, it will relax towards a metastable state. And then, if you fine tune your post-quench mass after a critical value, you can make the whole dressed mass to vanish, opening the door to an instance of criticality, which comes together with the emergence of correlation length and typical time scales. What these guys have realized, both with perturbative RG and numerical methods, is that essentially there is aging indeed in this critical quench. So let's just all do that if you do a classical critical quench, you get aging. And here it happens the same, but again, the system is isolated. It acts as its own buff. Nevertheless, this has been quite established, so you will get for correlation function, response function, magnetization, scaling behavior, akin to the one that Alessio was discussing before. The reason for which I say that there is classical and quantum criticality in this paternal state can be guessed quite easily. If you take your Gaussian green functions, for instance, the correlation green function that is a quite close parent to a distribution function, and you perform a deep quench, this will look like a thermal function. While if you do a shallow quench, a leading order, you will get something that reminds correlation functions at zero temperature. The fact that omega naught induces a behavior that is very similar to what a temperature will do, will stimulate you to search in your RG equations, fixed points that are associated to quantum and canonical scaling, so to zero, and high temperature of canonical power counting. Of course, there will be differences, because we are speaking about a non-equilibrium situation. But our goal here in this talk, and in this work that I'm presenting to you, is to write down a unique set of dimensionless flow equations that captures simultaneously the quantum and the classical paternal fixed point. And we use them to characterize the cost-sovere in the RG flow. You then start imagining that you are close to the quantum fixed point, so you decide to adopt the quantum scaling. This is mirrored by the fact that classical and quantum couplings have the same canonical power counting, but that your temperature, quantity, your frequency, value of the mass, omega naught, is scaling dimensionful, like k to the 1. It is the same that will happen in an equilibrium situation. And now the trick is that you have to reabsorb inside a new classical and quantum coupling, some combination of omega naught. This will result in a set of beta functions, of dimensionless beta functions, that has a structure, but I want to focus on few ingredients. First of all, a bit of nomenclature. This is the post-quench mass. And this is the pre-quench mass. Sometimes I use r, sometimes omega. That is r is equal to omega square. But don't worry, that's the same story. And this guy here has a trivial flow. There are no corrections. The only flow is given by its canonical dimension, 1, as I said. And there is the z naught that Alessio already introduced. It is a boundary renormalization for the fields that stay on the boundary. You need them because you're doing a boundary effergy approach. So you're breaking time translation and invariance and will give you the aging exponent. And what I marked here with a red box is the fact that the canonical dimensions get indeed some corrections, non-trivial, because of this transformation here. And now comes the point. The point is the fact that if you are at a quantum fixed point, so if you are at a fixed point of omega naught equal to 0, these corrections are 0. And you get the quantum canonical scaling and you get the quantum critical exponents. But then, since this guy has an unbounded flow, it will start to run and drive you towards the classical fixed point, high temperature point. This omega naught, again, is like t. And then what you will get here, because of this difference of sign reflected by the fact that you multiply in here, you divide, is that you will get different canonical dimensions. The same canonical dimensions you will get for quantum and classical couplings if you perform abinicio, a canonical power counting. So the canonical dimensions of your couplings are flowing with omega naught during the RG flow, driving you from the quantum and classical paternal fixed point. And now if you wonder whether you can capture the temperature of the equilibrium in the same way, the answer is positive, simply the flow equations are much more cumbersome and complicated and there is not the time here to discuss it. Because, as anticipated, my goal is to show you that this kind of behavior will manifest also in real time dynamics. So before doing that, let me spend some words about critical exponents and scales involved in this physics. So as you can see at the linear order in epsilon expansion, you get a co-ordination length critical exponent. There is the same rule that for the equilibrium fixed points. And it fits very well in this quantum classical correspondence, because epsilon is measuring the distance from the upper critical dimensionality. We couldn't say the same for the aging exponent, the one controlling this new phenomenology that was introduced. But essentially what I want more to stress in this slide is that there are two physical scales. The first one is the Ginsburg scale. It tells you when mean field theory where perturbation theory breaks down and when scaling is controlled not anymore by a Gaussian fixed point, but instead being controlled by an interacting fixed point. You can estimate by the breaking of perturbation theory, if you like. And then there is another scale, which is very important. It's an analog of the Delborgi thermal scale, but now omega naught is represented by temperature. It justifies one more time my analogy between the pre-quench value of the mass omega naught and the temperature, t. So this is the quantity in the distance of non-equilibrium quantum criticality is destroying the fixed point. And this qualitative pattern here can be seen if you numerically integrate the dimensionless better functions I was showing before. So what you have to focus here is that we have a coupling, g-classical, one is good as the other. And different colors are corresponding to different values of omega naught. If omega naught is very big, you directly approach the classical pre-terminal fixed point. So you don't have this fine structure just you approach the classical one. Or if you like, the two scale as what in a way that you cannot observe the quantum one. But decreasing omega naught, smaller and smaller, instead you will linger how much as you like and how much as small omega naught close to the quantum fixed point. And then you will go towards the classical one. I heard that I have two minutes. So this is my second last one slide. As anticipated, there is indeed a dynamical crossover that mirrors the RG crossover. In order to see that we benchmark the RG with some exactly solvable model, the infinite model is exactly solvable indeed. And if, for instance, you do some numerics for the scaling of the response function, you will see that these are just collapsed curves for different instances of quenches. You first will have as a function of time scaling controlled by the aging exponent of the quantum proteal matrix at point. And later you will cross over to a scaling in time controlled by the classical aging exponent. So indeed RG flow and non-equilibrium dynamics in these dynamical phase transitions in these instances of aging in isolated systems are two phases of the same model. I could show you also the magnetization, but it would like very similar. Just keep in mind that this double crossover will appear, so you will not just have a single initial slip, but a double one, like here. And the last slide is to tell you that indeed what we are discussing can be done in laboratories. There has been done a recent experiment in the group of Marcus Oberthaler, where they discuss the critical quench of a two component, both a gas. Their effective filtering specifically for the setup they can do is a nice in model, and in some sense the quench is providing a temperature, as I said. So they just see the transition from a quantum scaling to a gap at physics. But it is to say that we are close to an experiment, and in some sense the richer dynamical crossovers we see could be potentially reproduced in other experimental setups. My last slide doesn't deserve a word. It's just an overview. I try to answer the question I posed at the beginning. But just to word you, they are very specific. For this instance of non-equilibrium quantum criticality, if you change quantum criticality out of equilibrium, you get different answers. But if you like, there's the goal to encompass all these instances in a single framework, and we are working on that. Thanks for your attention.