WEBVTT Maybe you're talking about jobs. Thank you, Marcella. Oh, no. You can do what you want, Troy. His girlfriend there. How about Hong Kong? Mainland. No, no, but I'm from Hungary, but accidentally, this word means something in your postdocree. But sometimes people think that even Greek. I have ancestors, or I actually… I mean, my first name is not Greek, so… You don't buy my current name. Dana Korzynski, people think I'm Polish. No, that's just why ex-husband's last name. Ahmad. Okay. Ahmad? Mm-hmm. No. Middle Eastern, Middle Eastern. Middle Eastern, yep. How are you doing on Zoom? Are people getting excited? Um, as far as people attending? No, I… But I'm sure we have plenty of this. Okay, don't worry about it, you're all set up. Yeah. Anybody can hear us, hopefully? Started up time. 00:00:00.000 --> 00:00:02.000 Do that? Great, thank you, okay. Hello, everyone. Welcome back after your spring break, uh, after several talks to the March meeting, maybe. 00:00:02.000 --> 00:00:04.000 Back to our Connect Spider seminar. 00:00:04.000 --> 00:00:10.000 Uh, it's a real pleasure to have, uh, Marge and Hormosh visiting us from Texas University of Budapest. 00:00:10.000 --> 00:00:16.000 Um, and Martin's gonna tell us about his recent work on quantized expense chain. So Martin got… 00:00:16.000 --> 00:00:25.000 his PhD, also in Budapest, but I'm not gonna be able to pronounce the name properly. Utvas University, thank you. And they did a postdoc at Rice. 00:00:25.000 --> 00:00:28.000 Uh, with adoles and member card, and then went to work with… 00:00:28.000 --> 00:00:31.000 Calabusi Cardi… sorry, Calabrucci. 00:00:31.000 --> 00:00:38.000 Uh, Pascual Calabrisi, and then went to his, uh, faculty job at TU Budapest, and we've been collaborating, actually, since I was a graduate student. 00:00:38.000 --> 00:00:43.000 And it's very nice to be actually ready for papers up, you know, more recently, so… 00:00:43.000 --> 00:00:45.000 Please take your win. Thank you. 00:00:45.000 --> 00:00:48.000 Thank you, Jed. Thank you. It's always, uh… 00:00:48.000 --> 00:00:51.000 A pleasure to… to visit Rodgers. 00:00:51.000 --> 00:00:54.000 So, uh… 00:00:54.000 --> 00:00:57.000 So today, I would like to tell you about, uh, uh… 00:00:57.000 --> 00:00:59.000 Actually, two pieces of work. 00:00:59.000 --> 00:01:05.000 Uh, but they're both about, uh, uh, the quantum easing, uh, 00:01:05.000 --> 00:01:12.000 pinching or fear theory. So the first part is based on this publication with my student, Ishtwo Achepani. 00:01:12.000 --> 00:01:17.000 And the second part is, I mean, it's being submitted now, uh, in collaboration with 00:01:17.000 --> 00:01:24.000 Giuseppe Davecchio del Vecchio, it sounds a typo. That's his real… I always get this question, and that's his real, uh… 00:01:24.000 --> 00:01:30.000 last name, let's say, and Benjamin Goyon, okay? So… 00:01:30.000 --> 00:01:33.000 Huh, maybe I have to click here again? 00:01:33.000 --> 00:01:36.000 Yes. So, um… 00:01:36.000 --> 00:01:42.000 Okay, so my outline is very simple. So, after a brief introduction, 00:01:42.000 --> 00:01:44.000 I… I will… 00:01:44.000 --> 00:02:01.000 I would like to tell you about these results, about dynamic correlations in the field theory, so this will… this is the first half will be more field theory, like the scaling limit of the spin chain, but then, because I'm following the chronological order how we got the result, 00:02:01.000 --> 00:02:07.000 Uh, in the second half, I will be talking about the spin chain. 00:02:07.000 --> 00:02:12.000 So, uh, depending on your taste, you can focus on the first or the second half of the talk. 00:02:12.000 --> 00:02:18.000 Uh, well, yeah, let me give you the main message right away, so… The message is this, that, uh… 00:02:18.000 --> 00:02:26.000 So we are… I'm working at finite temperatures. I'm looking at finite temperature, dynamic correlation functions of the order parameter. 00:02:26.000 --> 00:02:35.000 Uh, and, uh, what we found, it seems, uh, it seems that it hasn't been noticed before. The correlation length has some non-analytic dependence on 00:02:35.000 --> 00:02:37.000 essentially any parameter that you can. 00:02:37.000 --> 00:02:43.000 think of in the model. So that will be the main result. 00:02:43.000 --> 00:02:46.000 So let me run through the introduction, even though 00:02:46.000 --> 00:02:55.000 I'm probably most of you or everybody is very familiar with this. So, so my… the model is given by this Hamiltonian, this is the quantum easing spin chain chain, or… 00:02:55.000 --> 00:02:58.000 transverse field easing model. 00:02:58.000 --> 00:03:01.000 Because a quantum critical point at the value of… 00:03:01.000 --> 00:03:09.000 H equals 1, separating a disordered or paramagnetic phase, where, uh… 00:03:09.000 --> 00:03:11.000 the order parameter, CMAX, 00:03:11.000 --> 00:03:20.000 has some finite expectation value, and then there's the ordered phase… uh, sorry, zero expectation, and the ordered phase has a non-zero expectation value. 00:03:20.000 --> 00:03:25.000 Okay, I mean, I didn't say that, like, spin half, uh, spin chain. 00:03:25.000 --> 00:03:30.000 Uh, okay, and as we all know, we can map this 00:03:30.000 --> 00:03:33.000 system to a system of free fermions. 00:03:33.000 --> 00:03:38.000 using the Jordan being a transformation, which is a non-local transformation, so… 00:03:38.000 --> 00:03:41.000 In particular, the order parameter 00:03:41.000 --> 00:03:43.000 has a non-local 00:03:43.000 --> 00:03:47.000 relation with these fermionic operator C. 00:03:47.000 --> 00:03:56.000 and see Dagger. Uh, and then, even after this, we have to work a bit more, we have to do something called a volume rotation. I will not go into the details. 00:03:56.000 --> 00:04:05.000 Uh, but the option is that we can diagonalize the Hamiltonian, so it has this nice quadratic form, and this version relation is given here. Okay, so this is very… 00:04:05.000 --> 00:04:08.000 standard, I believe. Uh… 00:04:08.000 --> 00:04:19.000 And as I said, in the first half, I'll be talking about the field theory. So the field theory corresponds to the scaling limit when we are near the quantum critical point, so H, uh… 00:04:19.000 --> 00:04:29.000 goes to one… the lattice spacing goes to zero, and the coupling J goes to infinity, and of course, capital N also goes to infinity, such that we keep these quantities fixed. 00:04:29.000 --> 00:04:33.000 Uh, so the length… length of the chain, 00:04:33.000 --> 00:04:40.000 the energy gap, so this is the energy gap on the left-hand side here. I write it in this way, MC squared. 00:04:40.000 --> 00:04:47.000 rather you stick way of writing is M is the mass gap, C is the speed of light, which is given by this 00:04:47.000 --> 00:04:52.000 combination. And if you look at this, uh, scaling limit, 00:04:52.000 --> 00:04:55.000 You get a field theory called an easing field theory, 00:04:55.000 --> 00:05:00.000 dispersionation becomes the relative… has this relativistic form. 00:05:00.000 --> 00:05:07.000 And from now, I will set C to 1. So the usual standard unit. 00:05:07.000 --> 00:05:12.000 Uh, okay? And then, as I said, the goal is to compute, uh, 00:05:12.000 --> 00:05:14.000 in the system, 00:05:14.000 --> 00:05:19.000 dining for a two-point functions in equilibrium, so it's finite temperature equilibrium states. 00:05:19.000 --> 00:05:30.000 So, this is the… this is the object that I'm focusing on. Exactly, or asymptotic? Uh, yeah, a very good question. So, uh, mainly I'm interested in the asymptotic behavior of this, uh, 00:05:30.000 --> 00:05:45.000 Long time, yeah, exactly. So, yeah, thank you. So it's, uh, when the separation, spatial separation, both, both the spatial separation and temporal separation goes to infinity in the same way, so it's like a ballistic… 00:05:45.000 --> 00:05:47.000 limit, okay? 00:05:47.000 --> 00:05:52.000 And even though the system can be mapped to three fermions, it's… 00:05:52.000 --> 00:05:54.000 very non-trivial to calculate. 00:05:54.000 --> 00:06:00.000 quarterly, even equal-time correlation functions, uh, let alone dynamic correlation functions, and 00:06:00.000 --> 00:06:05.000 And, uh, let alone, uh, finite temperature. Because of this non-linear mapping. 00:06:05.000 --> 00:06:07.000 Okay, so… 00:06:07.000 --> 00:06:20.000 I quickly, uh, I'm listing some results. Of course, many people and many groups have looked at this different aspects of this problem, or similar systems. 00:06:20.000 --> 00:06:27.000 But somehow this… I believe this hasn't been, uh, studied, what I'm going to talk about. So let me… 00:06:27.000 --> 00:06:30.000 Um, okay, maybe there's just a couple of words, so of course. 00:06:30.000 --> 00:06:35.000 For example, there is a… you can write undifferential equations for the correlators, and then you can study 00:06:35.000 --> 00:06:47.000 Maybe you can try to extract less employees from that, but it has been done for, like, the auto-correlation function, or at zero temperatures, but not in general, I believe. 00:06:47.000 --> 00:06:53.000 Also, there's a very nice, uh, interesting semi-classical approach. I will mention this later in the talk. 00:06:53.000 --> 00:06:57.000 of supervisor and Young. 00:06:57.000 --> 00:07:08.000 Also, you can… you can try to write down some kind of layman perspective representation called form factor series. Actually, this… this… I'm going to talk about, because this will be the starting point. 00:07:08.000 --> 00:07:12.000 And there are other, uh, advanced methods based on 00:07:12.000 --> 00:07:16.000 8,000 dots and so on. But these works were mostly focused on 00:07:16.000 --> 00:07:19.000 on a related models. 00:07:19.000 --> 00:07:22.000 Okay, I'll just, like, those words? 00:07:22.000 --> 00:07:27.000 T is equal to 0, there's, like, a total determinant or something you could do to compute the… 00:07:27.000 --> 00:07:40.000 Right, so actually, numerically, you can calculate things. I will be using that to check the analytic predictions, because it is puffyon technique, this is what I call it. 00:07:40.000 --> 00:07:43.000 So, analytically, you can't do anything at finding time with… 00:07:43.000 --> 00:07:50.000 Uh, yes, um… 00:07:50.000 --> 00:07:55.000 Yeah, I think it's… yeah, it's… I haven't tried, but… 00:07:55.000 --> 00:08:04.000 So, here we are following a different, slightly different route. So, the first part is about the eating, uh, field theory. So here, 00:08:04.000 --> 00:08:07.000 Just to, uh, 00:08:07.000 --> 00:08:16.000 Pretty cool. So yeah, I'm here, I'm interested in this, uh, uh, two-point function. So stigma is now the scaling limit of the sigma X operator. 00:08:16.000 --> 00:08:23.000 Yes. Could we step back a moment and get a big picture? I mean, ultimately, you've just shown it underneath the hood, it's a free field theory. 00:08:23.000 --> 00:08:29.000 Right, right. So, basically, these are exotic, non-local mappings from a field. 00:08:29.000 --> 00:08:31.000 free field theory. 00:08:31.000 --> 00:08:33.000 So why should we expect 00:08:33.000 --> 00:08:40.000 any non-analyticities, any sudden jumps, any discontinuities from a non-interacting field theory. 00:08:40.000 --> 00:08:44.000 And if we see them, shouldn't that be a sign that we perhaps made a mistake? 00:08:44.000 --> 00:08:47.000 Um, right, um… 00:08:47.000 --> 00:08:56.000 I don't even… just big picture, before you… Right. Yeah. Uh, so, um, I don't have a good answer, uh… 00:08:56.000 --> 00:09:02.000 But I guess the quantity that I'm looking at is the correlation length, so it involves a limit. 00:09:02.000 --> 00:09:10.000 Uh, like, something goes to infinity, and maybe, uh, order of limits things, so this is how these discontinuing, or analogic cities can emerge. 00:09:10.000 --> 00:09:13.000 like, okay, this is a technical answer, but uh… 00:09:13.000 --> 00:09:17.000 Obviously, there's a faith tradition there, we understand that as being… 00:09:17.000 --> 00:09:25.000 a source of knowledge. Right, right, and at the very end of the talk, I would like to speculate a little bit about that aspect. 00:09:25.000 --> 00:09:29.000 And it's true that in equal time, you don't… 00:09:29.000 --> 00:09:32.000 It's like, it's… it's coastal. 00:09:32.000 --> 00:09:39.000 If I was really a thermodynamic quantity, but close. So, equal time relations, there are no non-analytic… 00:09:39.000 --> 00:09:50.000 features, but for the dynamical quantities like this, those for which we find, uh, some strange behavior. 00:09:50.000 --> 00:09:56.000 Okay, so here I'm just starting with, just to warm up with zero temperature, which is much easier. 00:09:56.000 --> 00:10:08.000 Uh, okay, and then I measure everything in terms of this, uh, in the units of the mass gap, so… so I can work with the dimension as quantities, and the standard 00:10:08.000 --> 00:10:16.000 way to deal with this, or one of the ways, is to insert a complete set of states, standard layman or spectral representation, 00:10:16.000 --> 00:10:18.000 And then again, this series looks a bit… 00:10:18.000 --> 00:10:27.000 Uh, maybe, uh, ugly, but what happened here is just, I'm going by the number of particles, because the states are multi-particle states. 00:10:27.000 --> 00:10:31.000 So I'm summing over the number of parties in the states, 00:10:31.000 --> 00:10:37.000 But these states are labeled by the moment or rapidities, because we are in relativistic theory of these particles. 00:10:37.000 --> 00:10:43.000 These are just the standard space-time exponential. Here, this is the momentum, this is energy. 00:10:43.000 --> 00:10:50.000 Uh, and then we have the matrix elements of these operators, and these are known exactly in the field theory. They have this very simple form. 00:10:50.000 --> 00:10:54.000 Okay, um… 00:10:54.000 --> 00:11:01.000 So, Msinh theta is the momentum and M push Theta is the… 00:11:01.000 --> 00:11:03.000 Uh, energy. 00:11:03.000 --> 00:11:17.000 Okay? And then… actually, whether we are in the thermomatic or the paramedic phase only shows up here, so this prime means that we are summing over even integers or ordered integers for the paramounting and paramagnetic phase. 00:11:17.000 --> 00:11:20.000 Okay, and then, uh, but… 00:11:20.000 --> 00:11:27.000 Uh, it turns out that you can look at this as being a flat-form determinant, so if you're 00:11:27.000 --> 00:11:31.000 if you're not familiar with this, it's not a problem at all. 00:11:31.000 --> 00:11:35.000 But for those of you who are, you know, 00:11:35.000 --> 00:11:44.000 with this… so this is the kernel associated with this fertilment terminal, but the point is that you can effectively evaluate numerically this, uh… 00:11:44.000 --> 00:11:57.000 This is a flat one determinant, and one more comment is that at zero temperature, we have full Lorentz invariance, so actually, if we know the equal time or the autocorrelation function, we know everything, because everything can only depend on this 00:11:57.000 --> 00:12:00.000 invariant, Lorentz invariant combination. 00:12:00.000 --> 00:12:03.000 Uh, okay, and then, uh… 00:12:03.000 --> 00:12:23.000 Just to check that this threat from determinant numerics works, we checked, uh, various limits, so I'm running through this quickly, so you don't get too tired or bored, but for very small separations, we are supposed to recover the conformal field theory, which is the easing field… easing 00:12:23.000 --> 00:12:30.000 fixed-point behavior, and this is what we demonstrate here. So this power law, uh, behavior. So the method works. 00:12:30.000 --> 00:12:33.000 Uh, for very short, uh, separations. 00:12:33.000 --> 00:12:46.000 Uh, as well. Uh, for large separations, we can… from here, this form factor sees that the first term, and uh… and actually, that integral can either be evaluated, but this is really a large distance, large separation asymptotics. 00:12:46.000 --> 00:12:53.000 But you see that it, uh, it works quite well in ferromagnetic phase, similarly. 00:12:53.000 --> 00:12:58.000 Yeah, and uh… 00:12:58.000 --> 00:13:07.000 Okay, and even in the timeline regime, it turns out that the first term is enough to describe even these oscillating functions. 00:13:07.000 --> 00:13:12.000 And even it's a rather small separations, it's a bit surprising that it's 00:13:12.000 --> 00:13:24.000 for the first time of the four factors series can describe even relatively small separation behavior. Okay, but this was just to check the method, and then eventually we were interested in the finite temperature behavior. 00:13:24.000 --> 00:13:26.000 So now, uh… Can you say a word about… 00:13:26.000 --> 00:13:30.000 And the analytical continuation from spaceflight to time-like. 00:13:30.000 --> 00:13:35.000 Yeah, so here, what, uh, yeah, uh, what… 00:13:35.000 --> 00:13:40.000 So I think if you can prove quite easily that for space size separations, 00:13:40.000 --> 00:13:45.000 it's really, uh, justified to truncate this form factor series. 00:13:45.000 --> 00:13:55.000 Because you can prove these exponential suppression of the terms. For timeline, it's not so clear, because of these oscillations. Instead of the exponential decay. 00:13:55.000 --> 00:13:58.000 Uh, but just… 00:13:58.000 --> 00:14:02.000 So we found that these formulas, if you just 00:14:02.000 --> 00:14:10.000 apply them, uh, to, uh, time-like separations when these, uh, of course, these, uh, uh, expressions become imaginary. 00:14:10.000 --> 00:14:18.000 It works quite fast. This is the real and imaginary part of the collision function, so… 00:14:18.000 --> 00:14:25.000 And then at finite temperature… okay, so my temperature, inverse temperature is beta, so beta is the dimensionless. 00:14:25.000 --> 00:14:32.000 amateur. So here, the starting point was a similar form factor series to the previous case. 00:14:32.000 --> 00:14:40.000 But, so for… okay, let me walk through this. It looks, again, a bit ugly, but if you look at it, 00:14:40.000 --> 00:14:44.000 It's quite similar to the previous one, but there are some differences. 00:14:44.000 --> 00:14:49.000 So, now, again, I'm summing over 00:14:49.000 --> 00:15:02.000 the number of particles, but particles can either come from the thermal trace, if you imagine that you are doing some thermal trace, and probably inserted complete set. So there are two types of particles labeled by just a sign. These are these epsilons. 00:15:02.000 --> 00:15:04.000 for that. Uh… 00:15:04.000 --> 00:15:12.000 And then some statistical factor, temperature-dependent factor, appears here, which wasn't here at zero temperature. 00:15:12.000 --> 00:15:17.000 Uh, and maybe surprisingly, it looks like, uh, like a Bose-Einstein statistics. 00:15:17.000 --> 00:15:23.000 Whereas, we would expect some Ferrmody rock, right? Free Fermionic system. 00:15:23.000 --> 00:15:25.000 But, uh… 00:15:25.000 --> 00:15:27.000 But this disk form is correct. 00:15:27.000 --> 00:15:32.000 Uh, okay, and then finally, the four factors are very similar as before. 00:15:32.000 --> 00:15:36.000 But notice that, uh… 00:15:36.000 --> 00:15:40.000 Two particles coming from 00:15:40.000 --> 00:15:52.000 having a different opposite limb, or opposite sign, have the same rapidities, then they… then there's a pole here. So there… there… we would have a problem, but there's a prescription how to avoid this pause, and to shift the contours. Okay. 00:15:52.000 --> 00:15:57.000 I think this is, um, maybe it was already too… too much technique. 00:15:57.000 --> 00:16:07.000 technical detail, uh, but what I wanted to say that these are… okay, everything is, everything is normalatic in here, but let me just run through this. But the point is that, uh, 00:16:07.000 --> 00:16:14.000 This also can be driven as a photon determinant. This was noticed earlier in these works. 00:16:14.000 --> 00:16:26.000 And now the kernel of this threat will return is slightly more complicated. Actually, it's a 2x2 matrix, but it turns out that everything, uh, can be done again. So a threat from the turn can be evaluated numerically very efficiently. 00:16:26.000 --> 00:16:31.000 independent of, uh… of separation. 00:16:31.000 --> 00:16:40.000 So there's a standard way to evaluate this rapid determinants. Actually, you discretized… so, at the end of the day, you are tackling determinants of a huge matrix. 00:16:40.000 --> 00:16:43.000 And it doesn't need to be very, very, uh, very… 00:16:43.000 --> 00:16:47.000 find a solution or whatever you need. And then you can get quite, uh… 00:16:47.000 --> 00:16:50.000 But what I forgot to mention… 00:16:50.000 --> 00:16:55.000 is that this series makes sense only for space-like separations. 00:16:55.000 --> 00:17:01.000 You can check that if you have thought of timeline separation, then these integrals will blow up. 00:17:01.000 --> 00:17:06.000 Okay, so the terms, one by one, they don't make sense. 00:17:06.000 --> 00:17:09.000 And then came my student, Vishtivan, who… 00:17:09.000 --> 00:17:15.000 Because his idea, I wish it had been mine. He said that, okay, let's try to get around this. 00:17:15.000 --> 00:17:25.000 He said, let's make this parameter, which is a direction in spacetime, like T over X, some kind of inverse velocity, you can call it, like, where we are looking at in spacetime. 00:17:25.000 --> 00:17:28.000 Let's make this parameter complex. 00:17:28.000 --> 00:17:30.000 And then… 00:17:30.000 --> 00:17:42.000 This makes sense up to one, uh, value one. This is the… so if data, sorry, is that a small, uh, less than 1, because that means that we are in a space, like, uh… 00:17:42.000 --> 00:17:50.000 regime, and you cannot do anything in principle, above 1. But he said that if we make this complex, then maybe we can 00:17:50.000 --> 00:17:54.000 Somehow, by analytically continuing, can go around this. 00:17:54.000 --> 00:17:58.000 But this is the Forbidden region, really, this red region. 00:17:58.000 --> 00:18:03.000 Uh, so the problem is that you see that if you want this to be 00:18:03.000 --> 00:18:11.000 very big, like, autocorrelation function would be infinite, because X is equal to 0. Then we would be somewhere here, then this doesn't work. 00:18:11.000 --> 00:18:14.000 For this, you can… 00:18:14.000 --> 00:18:28.000 You can enter, somehow, this forbidden region to some extent, but not everything is accessible. And you can also improve this even by extrapolation. So, you will see in the next slide that actually this method works very, very well. 00:18:28.000 --> 00:18:37.000 Um, and why that… why are there two curves? Because he realized that for the two types of particles, you can… you should pick different… 00:18:37.000 --> 00:18:40.000 So, to ensure convergence. 00:18:40.000 --> 00:18:48.000 different, uh, continuations. So this, I think, I'm really proud of him, because I think this is a very, very, very nice idea. 00:18:48.000 --> 00:18:58.000 And then, again, we, of course, we had to check everything to be sure that we are not doing something stupid. So again, for very small separations, even though we are at finite temperature, we should recover the… 00:18:58.000 --> 00:19:04.000 that come from a field theory, zero times, you come from a field theory result, and this is what we find… found for 00:19:04.000 --> 00:19:11.000 space-like separationism, and these laws prove that his method worked, because even for time-like separations, we recover the… 00:19:11.000 --> 00:19:18.000 the… so you see, that is not too large, but it's in the timeline regime, T is larger than X. 00:19:18.000 --> 00:19:23.000 So, uh, okay. And you see that the agreement with the conformo 00:19:23.000 --> 00:19:26.000 formula is excellent. 00:19:26.000 --> 00:19:34.000 Okay, um, also, in very high temperature, we should recover the finite temperature conformal field theory result, because then the mass does… mass gap doesn 00:19:34.000 --> 00:19:38.000 matter anymore, right? If your temperature is much larger than 00:19:38.000 --> 00:19:51.000 Then the math gap, and this is what we are, we are looking at here. Again, maybe I'm just emphasizing these, that even in the time-like case, we find agreement with this. 00:19:51.000 --> 00:19:53.000 Uh, we saw it. 00:19:53.000 --> 00:19:59.000 Okay, but now let's go get to the point of the whole, uh, project, which was the asymptotic behavior. 00:19:59.000 --> 00:20:01.000 And, uh… 00:20:01.000 --> 00:20:09.000 So for equal time correlations, uh, correlators, there's a very nice exact result by Subhu Sajav back from, from 1996. 00:20:09.000 --> 00:20:14.000 Uh, and quite recently, I mean, a couple of years ago, uh, 00:20:14.000 --> 00:20:17.000 Granny, if I go to MS lab, 00:20:17.000 --> 00:20:22.000 performed, like, a really brute force form factor calculation. 00:20:22.000 --> 00:20:31.000 on the spinchain, uh, and they got some results for timelapse separations. And we checked that we are, we, we are in agreement with them. 00:20:31.000 --> 00:20:36.000 However, we find something interesting. If we are focusing on space, like, so not equal time, 00:20:36.000 --> 00:20:39.000 But Dynamico space-like separations, 00:20:39.000 --> 00:20:49.000 Uh, in the paramedic phase, it turns out that, again, this logic works, that we can truncate the form factor C, this finite tension form factor C, at the first term. 00:20:49.000 --> 00:20:52.000 Which is this. I wrote it out here. 00:20:52.000 --> 00:20:59.000 So we've had restructured, okay, uh, uh, and then it turns out that you can evaluate this by, by, by a residue theorem. 00:20:59.000 --> 00:21:04.000 Okay, so, uh, but there are infinitely many residues, so there's… this is an infinite sum. 00:21:04.000 --> 00:21:09.000 Okay, but then what is the correlational length? Well, correlation length is the smallest 00:21:09.000 --> 00:21:13.000 became given by the small, uh, slowest decaying term, right? 00:21:13.000 --> 00:21:19.000 So, this is what I wrote here. So you can extract the correlational length, uh… 00:21:19.000 --> 00:21:26.000 But it depends on which integer k you have to plug in here depends on which… 00:21:26.000 --> 00:21:31.000 Which gives the smallest number, or the slowest decay. 00:21:31.000 --> 00:21:36.000 And this is why, uh, if you plot a result, you find that there are these 00:21:36.000 --> 00:21:43.000 I don't know if you can call these cusps, or knees, or whatever, so I just wanted to… where the derivatives. 00:21:43.000 --> 00:21:45.000 the derivative jumps. 00:21:45.000 --> 00:21:52.000 So, as a function of the zeta, zeta, let me remind you, Zeta is just this T over X, like, where we are looking at. 00:21:52.000 --> 00:21:55.000 in which direction in spacetime. 00:21:55.000 --> 00:22:02.000 or a fixed data as a function of the temperature, you find these, uh, strange points. 00:22:02.000 --> 00:22:08.000 Where, uh, where there is a change in behavior. This is where these competing exponentials swap. 00:22:08.000 --> 00:22:13.000 Okay, so this is, like, events and variants vote somewhere to get the X. 00:22:13.000 --> 00:22:20.000 be like, X minus omega t, or… So, so at the finer temperature's something going on. 00:22:20.000 --> 00:22:23.000 That K is this correlation length inverse. 00:22:23.000 --> 00:22:26.000 No, just wondering if… 00:22:26.000 --> 00:22:39.000 What happens to the time in that, you know? Ah, sorry, okay. Yeah, yeah, exactly, sorry, yeah, it's a very good question, and I do apologize, man. You know, I'm not explaining this, but because, uh, accent is scaling the same way. 00:22:39.000 --> 00:22:41.000 So you can define correlation length by… 00:22:41.000 --> 00:22:49.000 Either way, so, like, this is why it makes sense to fix Zeta, and then you have, uh, maybe the coefficient of X. 00:22:49.000 --> 00:22:51.000 Well, I don't know if it makes sense, but I'm… 00:22:51.000 --> 00:22:53.000 Sorry, the velocity comes in. 00:22:53.000 --> 00:22:59.000 Yeah, yeah, so… so, for example, let's look at this example. So, I think here what we did… 00:22:59.000 --> 00:23:02.000 Uh, yeah, so the coefficient of X. 00:23:02.000 --> 00:23:13.000 I mean, if you pull out a factor of X, you will have this, and here you have T over X times Q. And this is why this data appears here. 00:23:13.000 --> 00:23:18.000 So this is why, uh, it's that I speaks, then you can use either XRT. 00:23:18.000 --> 00:23:20.000 So this is what I mean. It's a… 00:23:20.000 --> 00:23:26.000 I should have, uh, explained this better, sorry. So, this is what we mean by correlator length. So, it can be coefficient time, or… 00:23:26.000 --> 00:23:31.000 Spacetime. Space-time, yeah, so the rate of decay in that direction. 00:23:31.000 --> 00:23:37.000 which is space-time dependent quantity in that sense. Yeah, the singularity is there if you set things to zero. 00:23:37.000 --> 00:23:46.000 So if you send zettas to 0, this is the equal time, right? No, no, then we recover this, uh, this, uh, such the result, yeah. 00:23:46.000 --> 00:23:53.000 So, for some time, that has resulted as holes, but then it… and then it's… something happens. 00:23:53.000 --> 00:24:00.000 And even you find some non-monotary temperature dependence with it, which is maybe even more counterintuitive. Like, you don't expect the question length to… 00:24:00.000 --> 00:24:03.000 decrease if you increase the… 00:24:03.000 --> 00:24:05.000 Uh, temperature, right? 00:24:05.000 --> 00:24:10.000 Or the other way around. So, I mean, is there some strange, strange… 00:24:10.000 --> 00:24:24.000 features. Okay, but this, uh, if you… if… maybe I stop you for a second for… for… if you have more questions, but now I will, uh, switch to the second half about the spin chain. 00:24:24.000 --> 00:24:30.000 You go back. Yeah. So… 00:24:30.000 --> 00:24:31.000 I want to think about it, right? 00:24:31.000 --> 00:24:34.000 Figure sets of fixed space-time. 00:24:34.000 --> 00:24:38.000 Yes, which is, uh… Very, uh, temperature. 00:24:38.000 --> 00:24:40.000 Right, right. 00:24:40.000 --> 00:24:45.000 Johnny said that these are, like, some sort of actual… they transition the dynamics, sort of dynamic goal? 00:24:45.000 --> 00:24:52.000 Anything going on, or no? Yeah, so we've been thinking of, like, the interpretation of these things, um… 00:24:52.000 --> 00:24:58.000 So maybe, maybe I respond back to this in the associated dynamics architecture dynamics, because you don't have it. 00:24:58.000 --> 00:25:02.000 Yes, yes, so this is… it's very important that this only happens in the… 00:25:02.000 --> 00:25:05.000 time-dependent quantities. 00:25:05.000 --> 00:25:09.000 So, if you start with bosonic field theory, 00:25:09.000 --> 00:25:13.000 Uh-huh. Simple presum, only correct. 00:25:13.000 --> 00:25:20.000 what you are doing is, from this point of view, calculating a very complicated correlation. 00:25:20.000 --> 00:25:24.000 in a bosonic field theory. 00:25:24.000 --> 00:25:32.000 But how… I don't know how to relate this to the bosonic field theory. Innermionic, fermionic, which I can make bosonic loss. 00:25:32.000 --> 00:25:41.000 Uh… yeah, so this is, I think, what Jed was referring to, looks like some huge talk, because it's a multipoint function in the Fermic language, so it's… 00:25:41.000 --> 00:25:48.000 maybe rewritten in terms of topics, maybe pieces of fafiance, or something like that. So in that sense, yes, that's the thing. 00:25:48.000 --> 00:25:52.000 Correct, but it's not easy to extract. So we are trying to extract somehow the… 00:25:52.000 --> 00:25:56.000 the asymptotics. 00:25:56.000 --> 00:25:58.000 of a complicated function. 00:25:58.000 --> 00:26:00.000 That is… Yeah. 00:26:00.000 --> 00:26:05.000 But maybe the issue is with your assertion that the slowest decaying 00:26:05.000 --> 00:26:07.000 Term gives the chlorination there. 00:26:07.000 --> 00:26:12.000 Maybe. That's my definition, yeah. I know, but maybe it doesn't work close to these regions. 00:26:12.000 --> 00:26:15.000 competing terms that are equally important. 00:26:15.000 --> 00:26:19.000 Uh-huh. That means it's space-time, so it decaying. 00:26:19.000 --> 00:26:23.000 So, for fixed beta, uh, like… 00:26:23.000 --> 00:26:27.000 Like, there will always be… 00:26:27.000 --> 00:26:30.000 a one that tweens, I guess, right? 00:26:30.000 --> 00:26:34.000 Um, does the correlation function show any non-alities? 00:26:34.000 --> 00:26:40.000 Not just the length, but two-point function. Mmm, I… no, I don't think so. 00:26:40.000 --> 00:26:43.000 for any finite, uh… 00:26:43.000 --> 00:26:47.000 XMP and final temperature, I think it's a continuous function. 00:26:47.000 --> 00:26:49.000 So this is what I… why I said at this… 00:26:49.000 --> 00:26:53.000 This is… we see more of this limit of that. You're really extracting the… 00:26:53.000 --> 00:26:57.000 close the king term, which… 00:26:57.000 --> 00:27:07.000 give the possibility of having, you know, an analytic, uh… So do you expect for small beta this… the correlation length should be proportional to better? Is there some intuition that for small, better? 00:27:07.000 --> 00:27:13.000 There's something scale that over 1 over T. 00:27:13.000 --> 00:27:18.000 So for large beta, I think this is just… this is the zero temperature result, probably. 00:27:18.000 --> 00:27:21.000 No, I'm asking for small. 00:27:21.000 --> 00:27:23.000 Smaller… 00:27:23.000 --> 00:27:27.000 Sorry, like, this is high temperature, you mean? 00:27:27.000 --> 00:27:29.000 Uh… 00:27:29.000 --> 00:27:31.000 So, high temperatures… 00:27:31.000 --> 00:27:36.000 The next scale should be small. 00:27:36.000 --> 00:27:40.000 Yes? And so, is it proportional? 00:27:40.000 --> 00:27:46.000 Um… yeah, I think yes, because you recover the CFD, I think. 00:27:46.000 --> 00:27:50.000 So CFD at finite temperature, then you have exponential decay, and then the… 00:27:50.000 --> 00:27:55.000 The only scale is the temperature, or inverse temperature. 00:27:55.000 --> 00:27:57.000 I think. These are the formula I showed. 00:27:57.000 --> 00:28:02.000 a couple of slides earlier. 00:28:02.000 --> 00:28:08.000 Okay, so, okay, so this is what, what, uh, sorry, what happens when you go into Euclidean time, imaginary time? 00:28:08.000 --> 00:28:14.000 You get the same results? Um… 00:28:14.000 --> 00:28:17.000 I don't, uh… 00:28:17.000 --> 00:28:20.000 I have a question. Oh, that's why I'm asking. Yeah, yeah. Right. 00:28:20.000 --> 00:28:26.000 It's a good question, uh… 00:28:26.000 --> 00:28:29.000 So, I guess, um… 00:28:29.000 --> 00:28:32.000 I don't… I don't want to say something stupid. 00:28:32.000 --> 00:28:36.000 But I think it would be equipped. 00:28:36.000 --> 00:28:41.000 Yeah, I think probably this theory is maybe even… 00:28:41.000 --> 00:28:47.000 behaving better, actually, in that sense, and then maybe it's even easier to find, uh, 00:28:47.000 --> 00:28:52.000 Okay, I don't know, I don't know, but it's a good point. I should, uh… 00:28:52.000 --> 00:28:54.000 I should, uh, I should have thought of this. 00:28:54.000 --> 00:28:57.000 Yes. What happens is the master mix? 00:28:57.000 --> 00:28:59.000 So, in the massless limit, we sh… 00:28:59.000 --> 00:29:07.000 should, um, recover the conformal, right? So MZ Bureau is really going to the conform of field theory, it is no singularity. 00:29:07.000 --> 00:29:09.000 you know, uh, no, no, sir. 00:29:09.000 --> 00:29:14.000 Okay, so… and in that, uh… wait. 00:29:14.000 --> 00:29:20.000 So M… right, so M appears everywhere, because this is what… these are units, so, uh… 00:29:20.000 --> 00:29:23.000 But yeah, so we should recover… 00:29:23.000 --> 00:29:28.000 Mmm… should you cover the finite temperature CFT we saw that, physically? 00:29:28.000 --> 00:29:33.000 Right, so… which is still exponential decay, which, as Ananda said, 00:29:33.000 --> 00:29:37.000 It only depends on the temperature. 00:29:37.000 --> 00:29:39.000 Okay, but let me, uh… 00:29:39.000 --> 00:29:43.000 Carry on, and uh… 00:29:43.000 --> 00:29:45.000 Uh, sweetly. 00:29:45.000 --> 00:29:55.000 Can you go to the slide? Sorry, I click the extract. So, from that expression here, right, you extract, basically, the correlation length from 00:29:55.000 --> 00:29:57.000 the slowest. 00:29:57.000 --> 00:30:02.000 decaying exponentially. Yeah, exactly. And there is this factor which is always there. 00:30:02.000 --> 00:30:04.000 Yes, and you take the prefactor of the X. 00:30:04.000 --> 00:30:16.000 Right? Yes. I don't see how that depends on time. Yeah, so this was the Jets question. So P and XK are, uh, scale them in uniform in the same way, and Zeta is T over X. 00:30:16.000 --> 00:30:19.000 So if you extract X from here… 00:30:19.000 --> 00:30:28.000 then Zeta appears here, because D over X will appear here, right? It's a correlation function CETA. 00:30:28.000 --> 00:30:37.000 So you can define correlation function in, uh, or question length in various… in different ways. I mean, I could extract T or could extract X, 00:30:37.000 --> 00:30:41.000 It just depends on my taste. 00:30:41.000 --> 00:30:50.000 Well, not really, you know, because at finite temperature, I think you don't have a Lorentzian variance, that sense. So, it's not true that everything depends on your next 00:30:50.000 --> 00:30:56.000 square minus T squared. Well, then you can't assume that it's a correlation point of T. 00:30:56.000 --> 00:30:59.000 Good. Don't you have to have benefits and variants? Substead? 00:30:59.000 --> 00:31:07.000 Correlation length is X, it's the same correlation length. But somehow I, uh… Yes, but you expect… 00:31:07.000 --> 00:31:12.000 You expect, uh… 00:31:12.000 --> 00:31:16.000 I see, I see your point. I know, because you could define a similar thing in your time. 00:31:16.000 --> 00:31:22.000 Because C sub T would have a difference, a different function than this one. 00:31:22.000 --> 00:31:24.000 Because the pre-factor vaccine T. 00:31:24.000 --> 00:31:31.000 No, no, but if, yeah, but if you extract T, the difference with respect to this would be just a factor of the data, sure. 00:31:31.000 --> 00:31:36.000 That wouldn't change the conclusion. 00:31:36.000 --> 00:31:38.000 And you see that they appear, like, on… 00:31:38.000 --> 00:31:44.000 equal footing, right? So there's no, like, social. 00:31:44.000 --> 00:31:50.000 I guess you don't have to chronologists because it's Ed from Data Shop, like, actually… Yeah, yeah. This is a good way to put it. 00:31:50.000 --> 00:31:52.000 But this is what comes out, really. So this is… 00:31:52.000 --> 00:31:58.000 Actually, this part was actually easy, because it was really just the first term of the series, no fret home, whatever, it's just… 00:31:58.000 --> 00:32:00.000 You need to analyze this, uh… 00:32:00.000 --> 00:32:08.000 Uh, so actually, this calculation is not the… wasn't the hardest part. The hard part comes now for the spin chain. 00:32:08.000 --> 00:32:20.000 Okay, so… okay, so at this point, of course, we said very, very nice, and we have had this sidepost paper, but then we were asking ourselves whether it's a field theory artifact, maybe. 00:32:20.000 --> 00:32:22.000 Does this really… 00:32:22.000 --> 00:32:26.000 on the spin chain, which… 00:32:26.000 --> 00:32:29.000 Um, okay, and uh… 00:32:29.000 --> 00:32:35.000 And then we joined forces with Giuseppe and Benjamin. Why? 00:32:35.000 --> 00:32:41.000 the… here, we don't have this finite temperature form factor series, so we couldn't start from the same… 00:32:41.000 --> 00:32:47.000 same point, starting point. But they had a very nice, uh, work, 00:32:47.000 --> 00:32:53.000 On the x-ax screen chain, similar questions. They were asking similar questions. 00:32:53.000 --> 00:32:58.000 Uh, so we thought that we could apply their method 00:32:58.000 --> 00:33:09.000 on the, uh, to the easing, uh, case. Okay, so what is this method? So, again, this is our problem, just to restate the problem. Now I'm leaving field theory, this is really the spin chain. 00:33:09.000 --> 00:33:21.000 What they were looking at was the XXPIN chain. So this is, again, free fermionic, right? You can map it to free fermions. You don't even need to do this book, in some sense, it's even simpler, right? 00:33:21.000 --> 00:33:23.000 30 things. And… 00:33:23.000 --> 00:33:37.000 Okay, now I'm… I will flash the results. I won't, and maybe can't explain everything, but this is just reviewing their results, okay? So where we are really feeling are their method to apply to our… 00:33:37.000 --> 00:33:39.000 Yeah, so this symbol… 00:33:39.000 --> 00:33:50.000 This symbol means, like, asymptotically equal, meaning, uh, okay, I could define it more, uh, precisely, like, up to logarithmic, like, if you take a log of the two sides, uh, 00:33:50.000 --> 00:33:53.000 Mmm… like, okay. 00:33:53.000 --> 00:33:56.000 So, like, power law, correction, up to power law, uh… 00:33:56.000 --> 00:34:08.000 Okay, uh, so what they found is that, asymptotically, uh, this thing, this correlation of business was their correlation of function, which somehow analogous to our T-point function. 00:34:08.000 --> 00:34:12.000 factorizes into two parts. So what are these two parts? 00:34:12.000 --> 00:34:16.000 The second part is a fermionic. 00:34:16.000 --> 00:34:18.000 dynamic collision function. 00:34:18.000 --> 00:34:26.000 As if there was no string. You remember, if you put two of the sigma plus and sigma minus, or sigma sigma, sigmax, sigmax in my case, 00:34:26.000 --> 00:34:31.000 In principle, there is a string between them, right? And there are different times, it's complete. This is why it's complicated. 00:34:31.000 --> 00:34:38.000 But if there was no string, you would naively write down this two-point function, but there's a catch. 00:34:38.000 --> 00:34:43.000 Because you have to evaluate this, not in the normal thermal state, but thermal state, 00:34:43.000 --> 00:34:47.000 in which there's a chemical potential, I pi, 00:34:47.000 --> 00:34:53.000 It's funny. What do I mean by chemical potential? Well, this model has a U1. 00:34:53.000 --> 00:34:56.000 political number is conserved in this model, unlike lacing. 00:34:56.000 --> 00:35:01.000 to the speech chain. So, it makes sense to talk about chemical potential. 00:35:01.000 --> 00:35:09.000 Okay, so this is the second factor. The first factor is even more interesting, I think. 00:35:09.000 --> 00:35:11.000 What is the capital Q? 00:35:11.000 --> 00:35:14.000 So this Q is the integral of… 00:35:14.000 --> 00:35:20.000 charge density and current density along a path in spacetime, gamma, that connects the two 00:35:20.000 --> 00:35:23.000 operator insertions, so… 00:35:23.000 --> 00:35:30.000 The easiest way to think about this is just a straight line connecting, so this is one of my operators is here in the origin. 00:35:30.000 --> 00:35:33.000 The other one is here, this is time, this is space. 00:35:33.000 --> 00:35:40.000 And then you… you integrate this combination along this line. Actually, it doesn't depend on the curve. 00:35:40.000 --> 00:35:45.000 Because of the continuity equation. Turns out that this is an invariant, uh… 00:35:45.000 --> 00:35:50.000 thing. So you can even go like this first, in time and then in space. 00:35:50.000 --> 00:35:53.000 Uh, okay, so this is this quantity. I don't know if it's… 00:35:53.000 --> 00:35:57.000 Yeah, maybe just focus on this simple, okay? So you integrate current… 00:35:57.000 --> 00:36:04.000 Uh… from 0 to T, and then at time t, you integrate charge density. 00:36:04.000 --> 00:36:06.000 on a finite. 00:36:06.000 --> 00:36:10.000 length. I mean, the spatial separation of my… of the two operators. 00:36:10.000 --> 00:36:13.000 Okay, and this… what is this? 00:36:13.000 --> 00:36:17.000 Uh, it looks like, uh… 00:36:17.000 --> 00:36:21.000 Uh… sorry, here. It looks like the generating function of 00:36:21.000 --> 00:36:34.000 If this was an iPad, but if it was Lambda, or iLambda, this would be just the characteristic function of a probability distribution of Q, of this quantity, right? So this has fluctuations. 00:36:34.000 --> 00:36:43.000 So, for this, uh, Benjamin and collaborators developed this theory of ballistic fluctuation theory. This is, uh… 00:36:43.000 --> 00:36:45.000 what they call it. 00:36:45.000 --> 00:36:50.000 And again, for this reason, they wanted to describe large-scale, uh… 00:36:50.000 --> 00:36:52.000 fluctuations. 00:36:52.000 --> 00:37:00.000 of these quantities, integrated charges. In systems that have ballistic transport, like, 00:37:00.000 --> 00:37:03.000 interval systems, typically, okay? 00:37:03.000 --> 00:37:05.000 And then… 00:37:05.000 --> 00:37:11.000 This is really a generating function, right? If you take derivatives with respect to lambda, you get the movements of the distribution of Q. 00:37:11.000 --> 00:37:16.000 And Q can be this general thing, but if you don't… you can just focus on maybe just a… 00:37:16.000 --> 00:37:19.000 This part, this just means that… 00:37:19.000 --> 00:37:23.000 fluctuations of, uh, of the total charge in a finite but large. 00:37:23.000 --> 00:37:27.000 subsystem, but it's an interesting question. 00:37:27.000 --> 00:37:32.000 In some right. Also, if you keep just the current, you say that, okay, I don't care now. 00:37:32.000 --> 00:37:43.000 it is a problem. You define your queue just to involve J, then this is really the pool counting statistics, right? The integrated current up to some finite time. What is the full statistics of that? 00:37:43.000 --> 00:37:48.000 And this is the generating function for that, and what they found is that it behaves, again, asymptotely, 00:37:48.000 --> 00:37:54.000 It behaves in this way, so it's K as with T or X, because it's ballistic, it doesn't matter. 00:37:54.000 --> 00:38:00.000 And this function is not… you can… they were able to write down a function… an exact… 00:38:00.000 --> 00:38:04.000 Not a closed-form expression, but a method how to get this S. 00:38:04.000 --> 00:38:15.000 In the language of, if you're familiar with Breita and Satz, through what thermodynamic playtown Satz, or generalized hydrodynamics language they could use to get this… Okay, so this is just a… 00:38:15.000 --> 00:38:21.000 I'm not gonna say more about this, but they have this very, very nice theory. 00:38:21.000 --> 00:38:24.000 Okay, I think maybe 2 years ago, I was… I was… 00:38:24.000 --> 00:38:28.000 a bit more about that. Okay. 00:38:28.000 --> 00:38:30.000 Now… 00:38:30.000 --> 00:38:43.000 Okay, that was for XX, but now we want to apply this to spin freezing and, uh, spin chain, and we have a problem that we don't have a conserved U1 charge, and this whole formalism depended on this U1 charge, integrated current, blah blah. 00:38:43.000 --> 00:38:54.000 Uh, so the idea, uh, was to do… do some so-called doubling trick. So, it normal thing that you take two copies of the async model, you can build a DROC Fermion. 00:38:54.000 --> 00:39:17.000 So the… you take two copies, so two… let's, let's assume that we… we… GMI runners, right? Yeah, exactly. So these are the first-chain fermetic operators, second chain fermionic operators, and then you build these Majoranas, as you say exactly, and then combining the my runners in some way, you can build an object which is a Dirac fermion. 00:39:17.000 --> 00:39:24.000 Essentially. And then the time of coon is the sum of the, uh… you say the drop Hamiltonian for this, uh, for this object. 00:39:24.000 --> 00:39:30.000 Uh, and then this period, of course, has a U1 charge, the usual Dirac charge. 00:39:30.000 --> 00:39:36.000 So now, the idea is just to… and one more technical detail, is that if you 00:39:36.000 --> 00:39:48.000 I think the product of two string operators in the Jordan-Wigner mapping, there's this string. If you take the copies of two strings, you exactly get E2I pi 00:39:48.000 --> 00:39:50.000 from an integrated charge density. 00:39:50.000 --> 00:40:01.000 So if you see what I'm… and this is really crucial. And then I'm almost done with the technicalities here, so what I… 00:40:01.000 --> 00:40:06.000 Which is that we are looking at a square correlator. The square of the correlator that we are interested in, 00:40:06.000 --> 00:40:11.000 And we view it as a four-point function in the doubles theory. 00:40:11.000 --> 00:40:14.000 Right? Because these are independent copies, essentially. 00:40:14.000 --> 00:40:20.000 And then for this, now we have all the ingredients, so we can use just the method that, um… 00:40:20.000 --> 00:40:23.000 Uh, these gentlemen used for the access spin chain chain. 00:40:23.000 --> 00:40:31.000 So, again, we assume that there's a factorization property, there is this 00:40:31.000 --> 00:40:36.000 to IPI Q that I have been talking about, this integrated charger, whatever, and there's a… 00:40:36.000 --> 00:40:38.000 thermionic two-point function. 00:40:38.000 --> 00:40:41.000 at the clinical potential iPad. Okay. 00:40:41.000 --> 00:40:53.000 Uh, and actually now I can view these four-point function as a square of the easing two-point function, but you can ask, what is I pi in the easing model? I mean, there's no chemical potential in the easing model, right? 00:40:53.000 --> 00:40:55.000 Because there's no conserved E1 charge. 00:40:55.000 --> 00:41:01.000 But actually, exactly if you put IPI, it makes sense, because this is just a firmic parity, which makes sense in the using… 00:41:01.000 --> 00:41:15.000 speed change. So, interestingly going through, maybe there was a big detour, but going through this, uh, all this, we, we, we end, and, uh, end up with this, uh, formula, which actually makes sense in the using spin chain chain. 00:41:15.000 --> 00:41:20.000 Okay, and so the final, after this, so this was the way that we, uh, this was the method. 00:41:20.000 --> 00:41:26.000 that we use, and find the… let me give you the final result. So, we found that, uh… 00:41:26.000 --> 00:41:30.000 asymptotically, at least, there is this factorization, there are two 00:41:30.000 --> 00:41:33.000 uh, factors. The first factor 00:41:33.000 --> 00:41:39.000 If you look at it, it's very similar to that integral that I was analyzing for the field theory. 00:41:39.000 --> 00:41:44.000 is capital F. This is in the paramedic phase. Again, let me emphasize that this has its bosas. 00:41:44.000 --> 00:41:50.000 bourge science statistics. Now, it comes really from this iPad, so disparity insertion. 00:41:50.000 --> 00:41:54.000 It is as if you inserted a… 00:41:54.000 --> 00:42:02.000 A parity projection operator somehow, and then this changes the sign here from Fermi derog to Bose-Einstein. And this is hard to guess if you don't go through this. 00:42:02.000 --> 00:42:06.000 this calculation. This is what not… what… what… 00:42:06.000 --> 00:42:09.000 the aspect that has not been, uh… 00:42:09.000 --> 00:42:12.000 captured by previous studies, I think. 00:42:12.000 --> 00:42:16.000 Uh, okay, this factor, uh, doesn't give… 00:42:16.000 --> 00:42:22.000 Uh, at least to the coalition Act, does not contribute in the ferromagnetic phase. 00:42:22.000 --> 00:42:36.000 And I haven't told you, uh, what is E2IPIQ gives. With this method of policy fluctuation theory, you can calculate it. This is this omega factor here. Okay, so it's exponential. 00:42:36.000 --> 00:42:41.000 of this expression. And actually, but this expression is not new at all. 00:42:41.000 --> 00:42:47.000 Also, it looks, uh, also for the XX pinch change looks like this, but of course, the dispersion relation depends on the model. 00:42:47.000 --> 00:42:53.000 But it was also written down, essentially, by… already by Saga. 00:42:53.000 --> 00:42:56.000 In his semi-classical method, 00:42:56.000 --> 00:43:06.000 He was assuming… he was looking at the problem as, uh, like this. This is a… just a screenshot, a snapshot from his paper. 00:43:06.000 --> 00:43:16.000 Uh, like, classically propagating particles, uh, finite, uh, at low temperatures, low density, or a low density gas of oxygen. 00:43:16.000 --> 00:43:19.000 quasi-particles, and then… 00:43:19.000 --> 00:43:28.000 At low temperature, large beta, you can… this second log tunch just… is just a boat form factor, essentially. And actually, this is what Harry… 00:43:28.000 --> 00:43:39.000 He didn't have, or they didn't have this factor, but they just put the worst one factor, because then the interpretation was that this is the probability of, uh, lower lines of particles crossing this 00:43:39.000 --> 00:43:53.000 this dashed line connecting the two operators. So this was a very nice, very simple, uh, intuitive method. Of course, not… it's not an exact method, because it was supposed to work at temperatures. 00:43:53.000 --> 00:44:04.000 And also, I mentioned this form for brute force tool deforest form factor expansion from 2020. They also… they managed to derive this by re-summing 00:44:04.000 --> 00:44:11.000 Partially resuming the form factor series. So it was a really… I think it's an amazing feat. 00:44:11.000 --> 00:44:13.000 Okay, uh… 00:44:13.000 --> 00:44:20.000 And just let me emphasize one feature of this expression is, in the space, like, uh, for space-like separation, 00:44:20.000 --> 00:44:25.000 Actually, it doesn't depend on time at all. It's very easy to… for symmetry reasons. 00:44:25.000 --> 00:44:31.000 Okay, so in the space-like regime, it gives really only something exponentially in X pure. 00:44:31.000 --> 00:44:35.000 Okay, and then, uh… 00:44:35.000 --> 00:44:43.000 Uh… okay, we wanted to extract the correlational length, so we still need to analyze these two factors, right, asymptotically. 00:44:43.000 --> 00:44:57.000 And, uh, okay, in the timeline for the paramagnetic phase. In the time-like regime, uh, you can do some stationary phase, uh, uh, analysis, but you find algebraic decay, so the correlational length is not modified. 00:44:57.000 --> 00:45:03.000 by this F factor. I'm focusing on this F-factor, this omega is already an exponential. It gives you some 00:45:03.000 --> 00:45:06.000 some, uh, collision length. 00:45:06.000 --> 00:45:14.000 Nowadays, as intermediate region, where we are already in the timeline, uh… sorry, we're already in a space-like case, but we are between these two numbers. 00:45:14.000 --> 00:45:19.000 Okay, this doesn't exist in the field theory, because in a field theory, H goes to 1, right? So this doesn't… 00:45:19.000 --> 00:45:23.000 Because here, you need some… 00:45:23.000 --> 00:45:29.000 more complicated thing, and again, each time they managed to… my student managed to do this analysis, this saddle point analysis, 00:45:29.000 --> 00:45:31.000 had to… 00:45:31.000 --> 00:45:40.000 study this, uh, double-sheeted Riemann surface and whatnot, and he managed to find the correlation length. So there is a contribution from F. 00:45:40.000 --> 00:45:43.000 to the correlation length. 00:45:43.000 --> 00:45:56.000 And even our oscillatory terms, he could obtain. And here is the… what I promise that I'm comparing the results to the numerics based on this Faffian technique. 00:45:56.000 --> 00:46:03.000 And you see that these are the correlation… this is the correlation function itself. 00:46:03.000 --> 00:46:09.000 normalized, uh, somehow, and so you see that the agreement is very, uh, 00:46:09.000 --> 00:46:11.000 Very, very nice. 00:46:11.000 --> 00:46:14.000 And then let me, uh… 00:46:14.000 --> 00:46:18.000 finish with the most interesting regime, which is this, and Zeta is… 00:46:18.000 --> 00:46:25.000 larger, uh, smaller than 1, so we are in the spatial… spatial separation regime. 00:46:25.000 --> 00:46:27.000 It must be a smaller number over H. 00:46:27.000 --> 00:46:33.000 And then it turns out that this settle point analysis became very complicated. 00:46:33.000 --> 00:46:42.000 Uh, the problem was that there are some poles entered the contour, and you can imagine, like, it's a really, uh… 00:46:42.000 --> 00:46:49.000 tedious analysis done by Ishwan, but he managed to get a result. Again, it's not a closed-form result, again, it's… 00:46:49.000 --> 00:46:51.000 The equation of X is given by the maximum of 00:46:51.000 --> 00:46:58.000 a set of, like, depends on which poll wins. Again, it's a bit similar to these exponentials, like, which exponential wins. 00:46:58.000 --> 00:47:02.000 Here, you only have a finite number from which you can select. 00:47:02.000 --> 00:47:07.000 Uh, but again, because it's a competition between different 00:47:07.000 --> 00:47:18.000 pause or settle points. This is what explains mathematically, at least, this behavior. So again, you find these non-analytic, uh, 00:47:18.000 --> 00:47:21.000 features here. 00:47:21.000 --> 00:47:31.000 Okay, so we find similar… we find non-analytic behavior, both as a function of temperature, or as a function of the direction that T over X. 00:47:31.000 --> 00:47:36.000 So this is, like, too deep, also, for the collision and inverse correlation X. 00:47:36.000 --> 00:47:39.000 You can pair the cassis directly? 00:47:39.000 --> 00:47:51.000 Yes? Ah, so this is, this is, this, you say this is one word, C, uh, like, uh, analytical, and then it's PathPN, so it's generics. 00:47:51.000 --> 00:47:55.000 So what was the other question? Sorry? I hit it at the very beginning of the talk, you had the field theory with some… 00:47:55.000 --> 00:47:59.000 You start truncating the series, you're more into… 00:47:59.000 --> 00:48:03.000 You showed us earlier, um, in the field theory. Yes. 00:48:03.000 --> 00:48:06.000 Yeah, just curious if you compared those casinos. 00:48:06.000 --> 00:48:10.000 Not gonna be the same. The field theory under like this, uh… 00:48:10.000 --> 00:48:16.000 So the scale limit, somehow, it should… I think we understood how in the scaling limit, they, uh… 00:48:16.000 --> 00:48:20.000 to each other. But you see that… 00:48:20.000 --> 00:48:30.000 maybe there were some assumptions here, so this is not as rigorous as the field theory, because we have this factorization assumption and ballistic fluctuation theory, whatnot. 00:48:30.000 --> 00:48:36.000 But, uh, but you see that the agreement with numerics is quite convincing. So you see there's some deviations here, but it's… 00:48:36.000 --> 00:48:40.000 Probably just gonna work, so see, right? 00:48:40.000 --> 00:48:45.000 Uh, and now you may ask, okay, so this is the… after all, this is the easing model, so… 00:48:45.000 --> 00:48:51.000 It has been studied to death, so how come nobody really, uh… 00:48:51.000 --> 00:48:54.000 I looked at this, so I think… 00:48:54.000 --> 00:49:07.000 The point is that this Fermi… you remember that there are these two factors. I told you that the second factor appeared, uh, has appeared in the literature a couple of times, but the first one, which I call fermionic, or fluctuation, 00:49:07.000 --> 00:49:10.000 Um, sorry, propagation part, uh… 00:49:10.000 --> 00:49:16.000 was missed, because it was always replaced by something too simplistic, so… 00:49:16.000 --> 00:49:22.000 they put there the… just the zero temperature free fermion propagator. 00:49:22.000 --> 00:49:30.000 or in the form factor calculation, they also put almost by hand, a plausible-looking low-temperature expansion, 00:49:30.000 --> 00:49:35.000 But it turns out that if you take the low temperature expansion of this integral, 00:49:35.000 --> 00:49:40.000 And then calculate the asymptotics, you get a different result. So these limits don't commute. 00:49:40.000 --> 00:49:44.000 So they couldn't possibly get the right answer, because, uh… 00:49:44.000 --> 00:49:52.000 Because they did the hard part, they thought, this omega factor, if you remember. 00:49:52.000 --> 00:49:54.000 they got it, they got that. 00:49:54.000 --> 00:49:57.000 Where is it? Sorry, let me go back, maybe. 00:49:57.000 --> 00:50:05.000 Sorry. Uh, they got this factor, which was a very complicated calculation, and then they multiplied it with some 00:50:05.000 --> 00:50:07.000 some low temperature, 00:50:07.000 --> 00:50:11.000 expansion. Actually, the low-temperature expansion of this integral. 00:50:11.000 --> 00:50:13.000 But as I… as I said, 00:50:13.000 --> 00:50:20.000 The aphletic is not given correctly, if you expand this first, follow up the air temperature. Okay, so I'm about to… 00:50:20.000 --> 00:50:22.000 Finish. So, uh… 00:50:22.000 --> 00:50:26.000 So this is why I think it was missed, and you see that this… 00:50:26.000 --> 00:50:35.000 with Bose Einstein kind of fermionic part was not easy to guess. I mean, everybody expected some Fermi-Dirac distribution and so on, yes? 00:50:35.000 --> 00:50:39.000 We showed us there is some divergence in the correlation. 00:50:39.000 --> 00:50:42.000 Environment, temperature… 00:50:42.000 --> 00:50:45.000 So, what this really happened… 00:50:45.000 --> 00:50:53.000 Did some dynamic of which mission, or…? Uh… It's a physics of that? I was in the field theory plot when the… 00:50:53.000 --> 00:50:55.000 Right there. I know one, yeah. 00:50:55.000 --> 00:50:59.000 that post-everages here. 00:50:59.000 --> 00:51:01.000 It's not diverging, it's just… On the derivative is… 00:51:01.000 --> 00:51:05.000 Here's a derivative jumps. 00:51:05.000 --> 00:51:07.000 And the derivative derivative, uh… 00:51:07.000 --> 00:51:14.000 Anyway, yeah, so, yeah, that's a good question. So let me, let me, uh, C is the derivative of the log. 00:51:14.000 --> 00:51:18.000 Yeah, there's some sense that there's a… 00:51:18.000 --> 00:51:21.000 Yeah, interesting quote. 00:51:21.000 --> 00:51:26.000 Yeah, exactly, and this is what I want to speculate about. This is really my last… 00:51:26.000 --> 00:51:33.000 So… so now, let me plot this. This is the first time I'm plotting this as a function of the magnetic field, okay? 00:51:33.000 --> 00:51:36.000 And even as a function of age, 00:51:36.000 --> 00:51:38.000 For some fixed, uh… 00:51:38.000 --> 00:51:43.000 beta, and okay, for the various values of theta, theta was T over. 00:51:43.000 --> 00:51:52.000 we find this. Okay. So, if that was 0 equals time, there's nothing, right? This is… everything is smooth and continuous, equal time, such the result. 00:51:52.000 --> 00:51:56.000 actually used Analyticity to derive the result. 00:51:56.000 --> 00:52:03.000 Uh, but as soon as you switch on time, so it becomes dynamic acceleration function, this 00:52:03.000 --> 00:52:06.000 break the knee or cusp, or whatever appears. 00:52:06.000 --> 00:52:10.000 And then you increase the R, meaning you are approaching the time-like regime. 00:52:10.000 --> 00:52:12.000 Okay? You arrive here, 00:52:12.000 --> 00:52:18.000 Interestingly, at Zephy equals 1, you hit H equals 1, which is a creepy quantum critical point. 00:52:18.000 --> 00:52:23.000 And then you increase that further, and you stay at the critical point. 00:52:23.000 --> 00:52:27.000 And then this is now really a cost, so the derivative here… 00:52:27.000 --> 00:52:29.000 diverges algorithmically here, so it's really a CASP. 00:52:29.000 --> 00:52:36.000 We, we, we could, we could extract this result. Actually, it's independent of the temperature, as you see. 00:52:36.000 --> 00:52:50.000 Uh, so it seems that this quantity is sensitive… I mean, at least this suggests that this quantity is sensitive to quantum critical point, which is a bit unexpected, because at finite temperature, we don't really expect 00:52:50.000 --> 00:52:52.000 to be able to… 00:52:52.000 --> 00:52:55.000 find a critical point in this way, right? 00:52:55.000 --> 00:52:57.000 So, if… 00:52:57.000 --> 00:53:08.000 What I would suggest is that this dynamical correlation times, coercion functions, coercion length, can be used to detect the quantum critical point, at least in here, in this example for sure. 00:53:08.000 --> 00:53:13.000 But maybe, and we are… we are trying to argue that this structure 00:53:13.000 --> 00:53:20.000 is a hydrodynamic understanding of the asymptotic behavior of this correlation function is general. 00:53:20.000 --> 00:53:26.000 To some extent. But it's a product of a propagation part. 00:53:26.000 --> 00:53:31.000 responsible for quasi-particle creation and annihilation, and there's a fluctuation part. 00:53:31.000 --> 00:53:34.000 Paja and Jungko, this, uh, 00:53:34.000 --> 00:53:41.000 and this classical, if you, if you read, read his book. 00:53:41.000 --> 00:53:46.000 Uh, so the structure is general, and so maybe what we see here 00:53:46.000 --> 00:53:50.000 maybe more general. So this is what I… 00:53:50.000 --> 00:53:54.000 I wanted to, uh, say. So, just to summarize, uh, 00:53:54.000 --> 00:54:00.000 Uh, this is a thing that I was, uh, wanted that I was talking about down in progression functions, 00:54:00.000 --> 00:54:03.000 of the order parameter, a finite temperature in the easing, uh… 00:54:03.000 --> 00:54:06.000 field theory and spin chain chain. 00:54:06.000 --> 00:54:11.000 And uh… so we obtained numerically exact results, uh… 00:54:11.000 --> 00:54:13.000 or in the field theory, 00:54:13.000 --> 00:54:18.000 Even though the Thailand regime we couldn't access, if you remember, by this continuation, and we checked, 00:54:18.000 --> 00:54:21.000 benchmark the method with… in different limits. 00:54:21.000 --> 00:54:27.000 And we found that the interesting thing is where these non-alignant is in the correlation length. 00:54:27.000 --> 00:54:32.000 And then we, in the second half of the talk, I was, uh, I was telling you about the… 00:54:32.000 --> 00:54:39.000 speed change, where you use some kind of hydrodynamic approach, so if you… this is the take-home message. It was some kind of hydrodynamic approach. 00:54:39.000 --> 00:54:44.000 Again, we found a non-analytic behavior in the coalition, length of correlation time, 00:54:44.000 --> 00:54:46.000 Uh, we checked that. 00:54:46.000 --> 00:54:52.000 with numerics, so these results are valid for sure. 00:54:52.000 --> 00:54:59.000 It seems that the question length is sensitive to the quantum critical point, and I think this is the more speculative part of the… 00:54:59.000 --> 00:55:03.000 of the story, it's a very interesting question how, uh… 00:55:03.000 --> 00:55:14.000 In general, this is. And if, of course, it would be interesting to understand all the… if there's, like, a quasi-particle picture or whatever to understand the physical meaning of these cusps. 00:55:14.000 --> 00:55:19.000 Because mathematically, we understand, but it would be nice to have some physical interpretation. 00:55:19.000 --> 00:55:27.000 So, yeah, with that, uh, like, thank you for your attention. 00:55:27.000 --> 00:55:31.000 Ah, okay. I just stood up because I was… 00:55:31.000 --> 00:55:35.000 Then you go back to the last type of Q's theory part. 00:55:35.000 --> 00:55:39.000 moneticity of the… 00:55:39.000 --> 00:55:43.000 consolidation lengths back to the zip. 00:55:43.000 --> 00:55:49.000 Probably this one, right? Uh, ah, you can see it, sorry. 00:55:49.000 --> 00:55:52.000 I know. 00:55:52.000 --> 00:55:58.000 Uh-huh. 00:55:58.000 --> 00:56:03.000 So here? Is this the one? Yeah, so if you imagine your slowest mode to be a wave packet, right? 00:56:03.000 --> 00:56:06.000 This is, like, uh, studying what happens to… 00:56:06.000 --> 00:56:11.000 you know, different parts of this weight buckets. So, going slow, it's also going fast. 00:56:11.000 --> 00:56:15.000 So it's a good faster going with large zita, so it's like going slower going with slow reset. Mm-hmm. 00:56:15.000 --> 00:56:18.000 And this can interpret this moneticity. 00:56:18.000 --> 00:56:23.000 As my wave buggy consisting of different kind of particles that are 00:56:23.000 --> 00:56:29.000 that I'm moving past, because we have more than 0.5 less than 0.5. 00:56:29.000 --> 00:56:32.000 And I can get here a sense of 00:56:32.000 --> 00:56:36.000 It's a group speed of the way back in the… 00:56:36.000 --> 00:56:39.000 Well, I think the problem… 00:56:39.000 --> 00:56:45.000 I think the problem with this is that we are… we are… so all these non-analytic, uh… 00:56:45.000 --> 00:56:47.000 features, uh, 00:56:47.000 --> 00:56:54.000 exist in the space-like separated regime. So, in principle, there is no velocity that can… 00:56:54.000 --> 00:57:00.000 So maybe through some analytic continuation or swapping space and time, you know, there are these… 00:57:00.000 --> 00:57:04.000 these works. So maybe there's a way… so I… 00:57:04.000 --> 00:57:08.000 Zeta is T over X over X. That is T over X? 00:57:08.000 --> 00:57:12.000 So… so here, that is less than 1, so we're in this… 00:57:12.000 --> 00:57:16.000 SpaceLife. Yeah, space life, yes, thank you. So… 00:57:16.000 --> 00:57:31.000 There have been many results, for example, on, like, entanglement dynamics and negativity, and where people could interpret the final result in this quasi-particle picture, like pairs of quasi-particles… 00:57:31.000 --> 00:57:41.000 are created and then… I don't know if you've seen this, but there's a nice… a little bit similar to this Sage Def Semi Classical, um, 00:57:41.000 --> 00:57:47.000 interpretation. So, like, in terms of trajectories, they can… they could, uh, interpret the results, but… 00:57:47.000 --> 00:57:53.000 Here, yeah, this is what I was referring to, that it would be nice to have such an interpretation, but… 00:57:53.000 --> 00:57:57.000 But, uh, yeah, I haven't managed to… 00:57:57.000 --> 00:58:03.000 to find one, and as I said, this is outside of the light cone, so to speak, right? So it's… 00:58:03.000 --> 00:58:06.000 Maybe, maybe some, in some more… 00:58:06.000 --> 00:58:13.000 It's a more complicated way particles could appear, but this is just speculation, really. 00:58:13.000 --> 00:58:18.000 Um, so this cusp comes from, um, 00:58:18.000 --> 00:58:24.000 one term dominating for the correlation length at the series, uh, below a critical beta, and uh… 00:58:24.000 --> 00:58:27.000 mothers are donating after a critical bid? 00:58:27.000 --> 00:58:35.000 Yes, yes. In this case, yes. In the other case, similar, it was just a combination between poles, or saddle points, or whatever. 00:58:35.000 --> 00:58:41.000 at the critical point, do you just have, uh, the critical data? Do you just have, uh… 00:58:41.000 --> 00:58:46.000 two terms that are, uh… Are you swift, or do you have something more interesting? 00:58:46.000 --> 00:58:49.000 So I think these clients, they… 00:58:49.000 --> 00:58:51.000 the, uh… 00:58:51.000 --> 00:58:58.000 If you plot eat them, the lines would cross, right? And then you just follow the lowest one. 00:58:58.000 --> 00:59:04.000 kind of thing. So, indeed, these points, two terms are equal, and then 00:59:04.000 --> 00:59:07.000 The one that used to be the leading one… 00:59:07.000 --> 00:59:12.000 become the top-leading one, and then this is why you start following a different curve. 00:59:12.000 --> 00:59:18.000 Maybe, maybe actually that would be useful to show this in that way, so the different terms that… 00:59:18.000 --> 00:59:24.000 Somehow, uh, compete and cross… the lines cross, and then, uh, if you just 00:59:24.000 --> 00:59:27.000 For the lowest one, of course, you will find these, uh, these… 00:59:27.000 --> 00:59:30.000 brake points, or knees, or, uh, kinks. 00:59:30.000 --> 00:59:35.000 I don't know if this, uh, was laid off. 00:59:35.000 --> 00:59:37.000 Um, and… 00:59:37.000 --> 00:59:44.000 All the models, uh, or older models where you found this, uh, cost integral? 00:59:44.000 --> 00:59:55.000 Yes, uh… yeah, and I forgot to mention that we went back and we checked again the XX spin channel, and we found a similar thing. 00:59:55.000 --> 00:59:59.000 But these are not only integral, these are free fermionic in some… 00:59:59.000 --> 01:00:03.000 lead by disordering there. So… 01:00:03.000 --> 01:00:11.000 I mean, it's more like a question that, uh… so we have some arguments, but they are very vague. 01:00:11.000 --> 01:00:17.000 The same thing can happen in, like, interacting systems, or in… of course, interacting interval, so to speak. 01:00:17.000 --> 01:00:22.000 I would think of interacting in integral systems, and then, uh… 01:00:22.000 --> 01:00:28.000 Okay, more generally, could we maybe even non-interable, but that's, you know, maybe a bit too far. 01:00:28.000 --> 01:00:33.000 Goodbye. Any questions? 01:00:33.000 --> 01:00:37.000 Is this kind of divergency in any other systems? 01:00:37.000 --> 01:00:45.000 You said XX. Yeah, so evening and XX, yes, but they are both, um… Yeah, XXX? 01:00:45.000 --> 01:00:48.000 God bless. Uh, thanks, Henry. 01:00:48.000 --> 01:00:52.000 What do you do with XX to make it? Um… 01:00:52.000 --> 01:00:54.000 You should have a gap list. 01:00:54.000 --> 01:00:57.000 XX, um… 01:00:57.000 --> 01:01:01.000 Yes, what we are at finite temperature, so there we… 01:01:01.000 --> 01:01:05.000 Anyway, it's gonna be exponential decay, right? 01:01:05.000 --> 01:01:09.000 There's no M. 01:01:09.000 --> 01:01:13.000 Yeah, here we have an M, but… 01:01:13.000 --> 01:01:23.000 Yeah, I don't remember that, yeah. Okay, it's a good question. So we should quickly go in there, try and check that, ah, indeed, he found examples for this. So these are free fermionic systems. 01:01:23.000 --> 01:01:32.000 So I don't know how general this is, but I would like to think that it's more general than free-forming systems, so the next logical step… Well, you're having a quantitative point? 01:01:32.000 --> 01:01:38.000 And then… Yeah, I feel like what you're doing is, like, if I draw the fan and temperature and delta… 01:01:38.000 --> 01:01:48.000 I can replace delta with a length, I can replace temperature at the time, and it's like, you just filling through the quantum-critical fan in different ways. If you go into the quantum-critical fan, you hit a constant. 01:01:48.000 --> 01:01:53.000 Yeah, that… yes. Yeah, I think it's somewhat… 01:01:53.000 --> 01:02:06.000 Yeah, I don't know if I believe… that's what I was wondering if… Yeah, so Hume has fails for the second one, but the second one. Do you believe your setting costs for just the first cuss? Listen, here we have infinity money, if I remember. Oh, really? Oh, really? If you keep going, you have more cusps? 01:02:06.000 --> 01:02:21.000 Yes, but in the… as a function of age, the magnetic field on the spin chain, there's only one, so maybe, maybe in that… for that model… That was your interpretation. I think you could… you see what I'm saying? If you replace delta with… 01:02:21.000 --> 01:02:25.000 Like, with one overdue, right? 01:02:25.000 --> 01:02:30.000 then these lines become an interesting curve. Yeah, yeah. Also, temperatures in imaginary times, so… 01:02:30.000 --> 01:02:36.000 Yeah, I was gonna be able to fast and loose. 01:02:36.000 --> 01:02:45.000 Yeah, that might be related, and that's the stuff getting very interesting. Thank you. Okay, let's, uh, thank Martin again. Thank you very much. 01:02:45.000 --> 01:02:52.000 needs. It's more subtle than that. 01:02:52.000 --> 01:02:58.000 Who's next on schedule to your printer? Uh… 01:02:58.000 --> 01:03:04.000 Yes, I have it. I have it. 01:03:04.000 --> 01:03:14.000 Hey, hey, sorry. Yeah, we are… we are going to be caught here, yeah. Okay, cool, very good. Good to have you, bye. 01:03:14.000 --> 01:03:17.000 movements.