So, thanks to everyone that had a chance to take a look at Mr. Bates. He's sitting in the back. He wanted to listen into the conversation, so I appreciate anybody's input from what they saw and their thoughts on the exam. His initial exam is about a year and a half ago he presented to me with, let's say, acute vision loss, but relatively sub-acute vision loss with a sublux in the bag, single piece acrylic toric intraocular lens. He had some slit-lap finding suggestive of pseudo-exfoliation. His gonoscopy shows just a very mild SEM-PEL-SE line. The exam also demonstrated basement membrane dystrophy in both eyes. He had normal intraocular pressures at that point. His optic nerve exam showed some mild cup-to-discase symmetry, but there wasn't any notching or thinning thought to be present at that point. But obviously the emphasis or focus of the exam was on the assessment of the subluxated lens and surgical preparation and planning for repositioning of that lens. So surgery was performed. It was scleral fixated via a technique that's been popularized by Gary Condon. Those of you who use this technique know that the needle point on that is not necessarily ideal for this technique. Ideally, it would be just sharp at the tip and then smooth along the edges. There is a little bit of a tapered component to the needle that's currently used with this. And a little bit of cheese wiring of the inferior pass of the suture. Not enough to necessarily abort the technique, but it resulted in a modest decentration of the intraocular lens. It's very well fixated. It doesn't move. There's no denesis or other abnormalities with that. But that did result in some mild decentration of the lens. It was very stable. It was uncorrected vision. Even with a toric lens, we were able to get that reasonably well positioned relative to the original targets. And so it's uncorrected vision is 2030 with manifest refraction of spherical equivalent less than a half diopter. And his astigmatism is consistently less than a half diopter on manifest refraction. His IOP continued to be normal through this process. But despite this recentration of the intraocular lens, he still felt like the vision quality in the left eye was never as good as the right eye. And so it was somewhat of a conundrum for me trying to figure out, well, what is the issue here? Is it this modestly-decentred lens? Is this really causing this type of visual complaint? Could it be the basement membrane dystrophy causing some of these issues? So we did a soft contact lens over refraction, which improved his vision to 2020. At this point, we decided, let's do some medical therapy for the ocular surface and see if we improve that ocular surface. Maybe that'll help improve the visual complaints. And then down the road, we could certainly consider secondary surgical intervention for either the basement membrane dystrophy or the de-centered intraocular lens. In the meantime, so we did that medical therapy to this point. It really hasn't made any difference. In the meantime, started to notice an elevation in his intraocular pressure, and that triggered a decision to form a glaucoma workup. In that, we'll go through the testing and why I think the glaucoma may be a significant issue in terms of his visual complaints. But he still feels very strongly that that intraocular lens issue is a significant component of his visual complaints. He felt like the vision was fairly normal or essentially normal before the lens de-centered. And so one of the reasons we're here is to listen into the discussion and get some input from other individuals to allow for that discussion to be helpful in terms of your opinion and assessment of the cause of his visual complaints. So I'll show you just a few things here. So we've got his topography. You can see there's some irregularity of the anterior surface. This is the OPD-3 for those who aren't familiar with it. The axial topography there in the top center. I see just, again, a little bit of inferior steepening and some irregularity related to the ABMD. This is present in both eyes and probably has a similar impact on vision in both eyes based on topographic analysis and his soft-content lens refraction. So this is the original RNFL that was performed when his pressure was spiking up. You can see he's got significant thinning inferiorly in that left eye. The right eye, fortunately, looks pretty good. It's hard to appreciate when you look at the nerves. He's got small nerves with a lot of nerve tilt and inferior peripapillary atrophy, which makes it a little bit more challenging to assess the nerve on exam. This is 24-2, which was performed just a few months ago. You can see the left eye, again, a significant superior field loss. It's sort of a variant arcuate, a dense arcuate change. There are hemifilte change. And this is a 10-2, showing the central impact of that visual field cut that's present in that left eye. We've got him started on Latanopros, which has lowered his pressure from the mid to high 20s down to the mid teens, 16 or 17. The question in the discussion here is, what impact do you think, with regard to his visual complaints, what impact or contribution are we getting from the basement membrane dystrophy from any of the IOL decentration versus that glaucoma field cut? And based on that assessment, would you recommend anything treatment-wise for the basement membrane dystrophy or surgery for the IOL? Or do we believe that glaucoma field cut is the primary cause of the vision complaint and focus specifically on treatment for that and minimizing the risk of progression or further loss of visual function? Based on the testing, my working diagnosis is that glaucoma field cut is the cause for the vision change. But I don't want to necessarily take that without listening into some of your thoughts. So does anybody have any comments for those who were able to see the patient and assess him? Jeff and then Susan. All right, we'll ask Dr. Churna. Well, one thing I did notice is he does have an NAPD on the lip. Does small nerves or you start out with a small nerve. You don't have much tissue there. Any amount of cupping is significant, I think, if you look at both, compare the nerves. I think there is a lot more cupping there on the lip. And then glaucoma is notoriously difficult for patients to describe the symptoms of. And he will tell you that himself. He wishes that we could see what he could see. It's like looking through wax paper, the nation flying kind of a thing. And so all of that together makes me also agree kind of with your assessment that I think the glaucoma probably is a large part of what he's never seen. I appreciate that. That is something that he's certainly, and I've heard that multiple times, when you try and give him to describe it, it's difficult to describe it, which you can look through and to see what it is that I'm seeing. That's a better appreciate. Yeah, I completely agree. And we know that in eyes that don't get good signal from glaucoma that optics, at least optical abnormalities, aberrations, they have a greater effect. I mean, there's a reason why glaucoma is a contraindication for multifocals. People do horribly when they don't get that crisp, clear image in one focal point right on the retina. And so certainly, I think glaucoma is the main issue here, but anything that could be done to correct optics could potentially help. No, I wouldn't mess with the IOL. I mean, that's in my mind, almost a home run as far as getting that electricity, at least in an axis that is helpful. The ABMD is interesting. I mentioned upstairs, perhaps letting him walk around at the contact lens for a couple of weeks to see if that might make a difference for him if he's functionally noticing a change. Then I'd consider some sort of ABMD treatment, but short of an improvement with the contact lens trial, I wouldn't do anything with the optics. Well, I think that this really illustrates the fact that pseudotelation has on the zonus or his dislocation of the IOL within the capsule bag in person. And we've done lots of studies in our laboratory showing how pseudotelation is the most common cause of this spontaneous dislocation. But interestingly, we had a whole bunch of specimens that Lillian and I received from Germany, where they were sent from a very reputable place with the German people who are very German and really do good examinations, and they found exhalation in a third of the patients. And when we actually did the topology of the specimens we found in two-thirds of them had exhalation. So even in the hands of a really good observer, you can still miss exhalation. So whenever you see spontaneous dislocation within the capsule bag, you really gotta take exhalation first of all. But the second thing is in this case with the toroid lens, I'm really impressed that you've got the results that you've got. Because with those toroid lenses, when you're trying to suture them in position, even if they're off 10 degrees, 20 degrees, you can lose a lot of the effect of the touristy plus the tilt of it. And no matter how you suture them, they're still gonna tilt. I wouldn't touch that lens because chances are not only of not matching this result, but of making things worse by trying to reposition the lens that's in them. So I would leave it alone and treat it like all men and attempt to treat the base of memory disease. So I like that contact lens with the 2020 vision. I suppose with that, that is one more thing to do, but get used daily lens and put it in the days you want it. Don't worry about it the other days. That's a simple approach. Yeah. Make something a lot of sense. Thanks, Joe. As you're saying, when you're stressing the, I guess the great air crowd here is that contact lens is great. And it makes a big difference. And the softness of it, and the hardness really, that surface and the hard provides. And yeah, the physics is sharpened right up. They say, I love this. You know, it's a hassle to put a hard contact. They will pay you to do that with that visual quality. So I think that's really easier than the rest of the approaches that you could try to contact Lundra for. Bill, with the ABMD, it seemed like it wasn't really that central. I think it's pretty peripheral changes. Yeah, I agree.  And you don't really know how much sill you're gonna induce and aberrations. I mean, you assume it's gonna be better, but if you could shift a couple diapters in this refraction, then it's uncrected visual drop. So yeah, it can take six to 12 months for it. Yeah, hard to know. Stabilize. Yeah, it's not an ideal option, but certainly not rushed into that. Any other thoughts? I appreciate your input, and Mr. Bates also appreciates your input to be able to hear that conversation. Thank you.


late so I had to make up some ground and I knew the only way to get him I was to dive so I'm happy it worked. What did the guy say to you when you go back to the dugout? That I look like an athlete out there but they don't know that I am an athlete you know under his body there's there's a guy that that's athletic. As far as pitching is concerned did you feel like you had it from from the first pitch today? I thought I did pretty well commanding the zone at times you know there's some flashes where I lost a little bit of the zone obviously I'll walk two guys which is not ideal but I battled back and I was able to give six trynings. Were you in the dugout for the labors game winning it? No I was in here already getting treatment. What's your takeaway what's your shortstop so far this season? He's been great I mean obviously he did great things in Texas and getting him over here you know obviously we we were looking for a shortstop and he kind of fit the deal and he's been great so far hopefully he can continue. Going back to that night thinning what are your emotions while you're in here watching that frame unfold? Yeah I mean I'm watching I'm watching the night thinning here in the training room and obviously we we we get a guy on and Timmy Steele's and Timmy Steele's and you know we just make it work and it's fun to see that. 99% of pitchers are just gonna say I'll get him next time at first base but do you even have time to think about diving to giving yourself up like that? Yeah like I said I felt like the only chance I had was to dive I know he's he's really really fast especially from the left side and I was a little late to come off the mound so I feel like that was that was a play and and I honestly if I don't do that he's he's safe 100% of the time. I'm not really sure after that when I came in I was trying to watch what I did and I thought I tagged the base but it looks like I might have tagged him before because it looked like a bang bang play after I watched the replay.  You know last year when I pinned Trent in Miami I had that opportunity to dive back to second base and I didn't so I said I need a dirty uniform this time.  I mean it's great it's like I said it's fun to compete here I love pitching at home obviously when you're when you're pitching well it's it's a little nicer just it's fun to come out and have that ovation hopefully I can continue to do so. Now that you've done this for half season last year early on this year what do you think your cable blows as a starter? I tell myself if I'm able to stay healthy the whole year I think I can do big things like I said I go one inning at a time try to give as many quality innings as I can it's fun to go six and seven innings into into every game that I pitch but if I can give four or five innings you know with our bullpen and our hating I think I think it always gives us a shot.


We are nearing the final half an hour mark and of course I want to bring in the audience in just a bit not quite yet Not quite yet. I told your horses I will come to you in just a second because I can't imagine there'll be a number But I want to round up this this round before we are opening up and I'm gonna come to the three of you to To wrap it up Let me start with mark as I say you wrote a very comprehensive book on the war on terror and they are US Generals who are saying with the return of the Taliban now? We are seeing a return of al-Qaeda al-Qaeda is going to be is active in Afghanistan They're saying an imminent threat Perhaps you'd an imminent attack on the US could be realistic and feasible within the next five six months Is that your understanding? Is that is that your assessment as well? Joe Biden said that al-Qaeda was gone from Afghanistan, which is not true So we know that al-Qaeda is still there and we know that the ties have been maintained over the past 20 years And actually there are proofs of that for instance There was a 20 19 strike on the Taliban hideout in Musa cala and actually the emir of Al-Qaeda in the Asian subcontinent was killed in this operation So today again the Taliban have to walk the talk on this topic as well We cannot just believe what they say they signed the doha agreement and in the doha agreement They said that they would break ties but we have no control over that So I mean we need some evidence at some point and here obviously I mean al-Qaeda is different from ISIS because the Taliban and ISIS are at war and on the question of ISIS the Our question is not basically if the Taliban are willing to combat to confront this group If they are able to eradicate them in ISIS as approximately between 1,000 and 500 fighters and 2,000 and 200 that's quite a lot and if you if they managed to Control some part of Afghanistan then perhaps thing they can also export their violence to the west or to south a year right Tatiana there's no doubt that the hasty withdrawal on the part of the United States is a damage to the reputation Of if not credibility of the US of the West of NATO. I think there can be no no doubts about that Again, we talked about the aspect of shard and Freud perhaps that is coming from Moscow and Beijing looking at this How how lasting do you think this reputational damage truly is to the United States? Judging from where you are based Perçu par les Russes comme une opportunité géopolitique Je crois que c'est Zaki l'Aidi qui avait employé le premier jour le terme de une crise de l'adrige augmenter les leviers d'influence l'influence sur russe dans la région mais aussi dans les pays qui sont voisinant des régions donc évidemment c'est un grand dommage pour la répétation américaine dont les russe souhaiterait et espèrent bénéficier Avec la Chine évidemment c'est ce qu'on voit en premier le tandem qui pourrait éventuellement surcharger du problème et peut-être dans le cadre de l'organisation de coopération de Shanghai de fait l'Afghanistan qui a le statut d'observateur dans cette organisation à tous les pays les mitrofs sur le sauf le Turkmenistan qui sont membres de cette organisation mais on sait aussi les faiblesses de cette organisation qui est déchirée par les contradictions et les blocages notamment entre l'Inde et le Pakistan mais il y en a aussi d'autres donc la question aujourd'hui je vois dans les milieux d'expertise en matière de relation à Karin Sena la Russie le sujet qui est brassé c'est peut-on avoir une sorte de multilatéralisme à la carte au sein de l'organisation de coopération de Shanghai pour gérer ce problème-là mais le problème qui va se poser c'est le problème entre l'efficacité donc créer un groupe plus petit pour gérer le souci de l'Afghanistan et proposer de solution et la légitimité il faut que ça puisse être accepté par tous les autres mais en tout cas le dernier sommet de l'organisation de coopération de Shanghai au moins de septembre c'est saisis du problème et a formulé son souhait de voir l'Afghanistan notre et pacifique donc on va voir qu'elles sont à même temps ils n'ont pas il n'y a pas de pistes je vois pas de pistes concrètes qui se dégagent il est juste question de retravailler cette idée de création d'un groupe au sein de l'organisation de coopération de Shanghai mais rien de concrètes pour l'instant et puis je voulais juste dire deux mots vous avez dit qui connaît la réalité du terrain très honnêtement je pense que les russes c'était pas mal au courant de ce qui se passait il depuis 2017 il anticipait en fait la chute éventuelle du gouvernement gagne et c'était dans l'espace public en fait ces craintes là sur la fragilité de ce régime et je pense que c'est l'un des facteurs qui a poussé les russes à discuter avec les talibans se pragmatisme cette vision de la réalité du terrain dépourvue au fond de l'idéologie dans la meilleure optique possible encore une fois pour les russes le vrai problème c'est moins les talibans que leur soutien éventuel ou pas à différents gripes gris pescule basés sur leur territoire juste pour appel la dimanche poutine est arrivée au pouvoir en pleine guerre de chichini et les franges les plus radicales de combattants au nord cocase était soutenu depuis le territoire de l'arganistan et l'arganistan sous le premier rame des talibans était le seul pays au monde avoir reconnu l'hitchkeery indépendante l'hitchkeery c'est le terme pour la chichini donc c'est aussi la question quelle attitude des talibans vis-à-vis des mouvements séparatistes dans les pays avoisinant right so russia already had at least in terms of vision and tactics the leg up you're saying they were already arranging themselves for a new reality and China as we know you mentioned of course unlike the u.s. Has a somewhat pragmatic approach when it comes to conducting foreign policy So they will be dealing and with the taliban perhaps in a very different manner Let's let's round it up by bringing in mk. Narayan one more time before I handed it over to to the audience I know you wanted to jump into no I want to be The devil's advocated this fine time. I think this isn't an issue of I've of Whether the European Union or Russia or all the Americans are we have a problem There is a huge Implosion that is taking place in Afghanistan the jihadists across the world have Been electrified as a result. We already see the thing Al-Qaeda and its actualites like the Lashkar-e-Taybah and the Jaish-e-Mohammed in in the in the South Asian Context I have become why we see is a great deal of revival of the is is particularly the Coruscant group of the isis So these are the these are the issues The there is another problem that people are not talking about the opium trade has more or less Doubled or tripled in the course of the last few weeks. There's nobody's talking about it. What are we talking about? We have to stem that we Nation that deal with it we can't wait for Governments to be established before you do we need to deal with that problem then finally that is you have the problem of How do you deal with it now Tatiana was talking of the SEO now you start off by saying the SEO doesn't include the United States So I mean we cannot that's why I refer to a global concert We require the world to look at this problem because what is going to happen in Afghanistan in the next Year or so is going to dictate. What is the course of events across the world in there? I do I do you wanted to deliver the world to to the terrorist there is the Taliban is is what are called Extracent on the on the face of the of the earth in that sense So we need to deal with it and I think we read I think we should have a global concert a group of nations We've had that in the past it should include therefore not Not only the countries of Europe and this and that because that but certainly I mean One of the prime candidates for that would be the UAE because it has been done and it's got a now I'll record of doing many of these things certainly a country like India over there How do you keep Pakistan out of it will have to be a major factor because it is the So please look at this problem from this perspective Let us not allow ourselves to be driven by older kind of thing But the United States in particular needs to show ideological diversity As has been mentioned here or where you can't say we will not deal with this one We will not look at them because of kind of thing So I leave that because I think we've run out of time A lot more to say but that doesn't matter most most definitely and we will have more to say because we're going to the audience now, I think your point that Afghanistan is one of the most prominent geopolitical challenges awaiting us in the 21st century I think there can no be no two opinions about this Which is why the world policy conference has prominently featured this panel here on the subject matter


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.  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.