 So we can get started. It's a pleasure to have Takato Yoshimura, which will talk about collision rate ansatz for quantum integrable systems. And if Takato, can you share yours? Yes, I can share. Sure. Let me. Yes. So, okay. Do you see it? Yeah, please. Go. Okay. So, right. Let me first thank for the organizers for kindly inviting me to give a talk in this workshop. I think this is really a great way to keep the community united. So, yeah, it's really a great pleasure. So what I'm going to talk about in this talk is that my recent work with Haberspon about the collision rate ansatz for quantum integrable systems. So here the collision rate ansatz refers to a particular ansatz for one of the most important quantities in the business of generalized hydrodynamics, which is a hydrodynamic theory for integrable systems. Okay. So let's get started. The plan is as follows. So I think as a first speaker, I'm entitled to have a pleasure of reminding you of some basic tasks, but elastic and diffusive hydrodynamics. So I'm going to talk about it. And then later on, I will move on to some generalized hydrodynamics and a proof of collision rate ansatz in integrable systems. Okay. So hydrodynamics. So this talk will be all about hydrodynamics. It's a simple idea. It's basically an effective theory for describing the long wavelength dynamics of some interacting many body systems. It doesn't have to be quantum or cross-cut. I mean, it either works. And of course, it's just simply so it's considered some one-dimensional interacting many body systems with this number of consolidation laws. And simply stated, the idea of hydrodynamics is local equilibrium and it's stable propagation. But it's simple to say, but there are a lot in this statement. And in practice, this idea is implemented through the continuity equations. So basically you can write down the continuity equations for the average of the charge density and current like this, where this average is taken with respect to some initial ensemble. And this is always correct. And it holds that you want to have a situation or whatever time. And now this idea of hydrodynamics, which is local equilibrium and stable propagation is achieved. Basically, this hydrodynamics allows you to replace this current average with this derivative expansion. So the first guy is describing the ballistic transport. So it's the simplest. But the second one is with wind tricate, which is characterizing diffusive broadening of the plastic transport. And this is a kind of subleading term in the sense of david expansion. And because of this spatial derivative, this can be capable of describing the finer-scaled dynamics than this plastic transport. And you might wonder if you persist this procedure, namely inclusion of higher and higher derivatives, this description, hydrodynamics gets more refined and refined. But that is actually, as far as I know, that's not true. I mean, it results in either the result that is not really physically sensible, or just gives rise to nothing additional. So in general, you basically a truncate after this diffusive correction. I mean, the second term in them, what you get is nothing but the famous nervous talks equation. All right. So I haven't defined this matrix A and D. So this A matrix is simply defined by this differential ratio involving J and Q. And this is just covering the plastic dynamics. Well, this D matrix, I mean, this is really a diffusive matrix, is controlling the diffusive broadening of the plastic trajectories. Another quantity that is equivalent to, actually, this A matrix is the eigenvalues, which we will denote effective velocity VF, is also going to play an important role. And actually, what I'm going to do really is to determine the form of this effective velocity for integral systems. Right. But just these are some equations and definitions, you might not really have good intuition as to what these quantities describe. So to have some intuition, it might be instructive to consider some weak perturbation to the initial background ensemble. So you initially, say, have some homogeneous background ensemble, and you locally perturb that ensemble. And what happens then is that some ripples on top of the homogeneous background ensemble that is propagating with some velocity. And because I'm considering weak perturbation, this velocity is basically constant. And this is nothing but effective velocity. So I'm here considering normal mode. It's kind of a linear transformation of the original physical fluid variables. It's just in transformation. And the good thing about it is that it has a unique velocity. So this normal mode profile is spreading like this, sorry, propagating with effective velocity. And what then this diffusion constant or matrix does is to allow this profile to broaden in the course of time evolution. And this is controlled by this t to the one half basically, it's the broadening. But there are actually, in particular one dimension, the cases that where this diffusion constant for some component is diverging, which results in this broadening, we controlled by a different exponent, not one half, but some other components, exponents like one third or some other fractions. And this is called the most diffusion in general. And I guess this is the topic here and the couple will be talking about I think today's session after our session, I guess. But I'm not going to talk about it. Today I will be really focusing on the body transport and in particular how to determine this form of effective velocity for interval systems. Okay, so far it's all applicable to any interacting many body systems, also in higher dimensions, but from now on I will shift to the interval systems. Okay, so in interval systems, you have a very nice machinery called thermodynamic bed and that's that allows you to do statistical mechanics in a very systematic fashion. And also it allows you to write down these two important ingredients in hydrodynamics, which are charge dense average and current average in terms of causal particle basis. So I guess a lot of you saw this expression already when in the last let me briefly describe. So this role of theta in this expression is called root density, it's characterizing some density of distribution particles, like causal particles, each of theta on the other hand is really simply one particle eigenvalue of the charge qj and by just merely looking at them, you notice that they are almost the same except this the appearance of effective velocity in the current. And its form is actually known from this or initial works and also the works by Bruno and others. And it's really a functional of raw theta. So, but this form is was originally conjecture, I mean we did provide some sketchy proofs, but it was conjectural. But now we have basically two different proofs that establish this the form of effective velocity. So one of them is making use of form factor expansion basically, and it's resumption. Another one is more recent and it was done by the blush by making use of some sort of long range deformation to the interact in integrable spin chains. So we have two different kind of proofs, but either way it gives rise to the same results. So there's really no doubt about this, the form of effective velocity. So today what I'm going to do is to provide you a new proof that relies neither on the form factor expansion or the formations. And I think I okay so I also wrote down this hydro equation. So by just plug in this q and j and their average into this continue equation and then what you see immediately is that this hydro equation at the level of the causal particle must hold. So this is what we normally call generalized hard dynamic equation, and this is what the Bayer equation that describes ballistic transport in integrable systems. Of course, if you want to include diffusive correction, then it's something additional appears in the right hand side, but that is not what I'm going to talk about. Okay, so I think I still have time. Right, so this effective velocity is has a nice allows us to have some nice understanding intuitive understanding from the scattering picture. So basically in a fluid of causal particles, I mean, causal particles is basically the electromagnetic excitations in integrable systems. Effective velocity is the mean velocity of a tracer particle, basically a particle that you are you focus on with the incoming velocity v of theta when traveling over a large distance that acts for large time t dot t. So there is a relation between these three quantities. And here, what we what is intriguing is that you can basically decompose this large distance, travel distance by this tracer particle into three components. So the first term is really simple. It's just describing spatial displacement that is generated by freely propagating tracer particle. But of course, that that is not the end of the story. You would have, I mean, you're considering a fluid of causal particles. So in the course of time evolution, your tracer particle will undergo a lot of collisions with other causal particles. And that effect is encoded through the second third term. And you notice that they're the only difference between these second third terms are the sign in front of the summation. And this is really stemming from the fact that depending on whether your tracer particle hits other causal particles from left or right, you would get different sign. So that is the reason why you have different signs in front of the summation. And then what is summing over, what is being summed over is this jumped instance, which is what I'm noting by phi of theta. This is related to the two particle scatter metrics. Okay, so you can basically consider it as this is a travel distance as an accumulation of the jump that is happening at every collision with other causal particles, plus this freely propagating, I mean, this spatial displacement that is generated by this freely propagating propagation. And by, you can just look at this, for example, cartoon, I drew, for example, right hand side, this is really happening, what is happening when this tracer particle is propagating in the spacetime. So upon the collision, you, so in here, I'm depicting the case where I interpret this collision as a kind of sticking process. I mean, what upon the collision to cause a particle stick each other, and then after some time delay, they start propagating with the same velocity as before. But this resulting in this spatial displacement. I mean, this is microscopically different from the, you know, what I was talking about, I mean, jump this accumulation or jump distance, but microscopically this interpreting as the accumulation for jump distance or accumulation of this time delay are basically equivalent. So I'm depicting this time delay case. Okay, what is important here really is that whatever interpretation you have, what is accumulating is this phase shift, phase shift five. And by carefully calculating these contribution to this distance, what you end up with is this very nice formula for the effective velocity, which is really an integral equation. So you can basically do a standard iterative procedure and then you can solve this interrogation. All right, so this is a kind of picture you can have in your mind when talking about this effective velocity in interval systems. Okay, I can't go to the next slide for some, okay. All right, so the proof. So this proof is going to be really simple. Only two inputs are needed. The first one is the symmetry of B matrix. So I guess you are familiar with charge-charge susceptibility matrix, which is a standard susceptibility matrix involving just two charge densities. But you can also define charge current susceptibility matrices, which are denoted by B like this. It's just differentiating current with respect to some kind of potentials. And I'm of course considering some stationary GGE here. Okay, so obviously this matrix C is symmetric. But what is less obvious is that the B matrix is actually also symmetric. You can kind of show that B matrix is also actually symmetric by invoking some stationarity of the state or clustering for the correlation functions. In general, this symmetry B matrix is satisfied. So that's what we can expect in general. The second one is about the existence of a self-conserved current. So this is also quite often the case. For example, if you consider some Galilean systems that conserved the number of particles, then your particle current is basically conserved. And it equals the momentum. Here, mass is set to 1. On the other hand, for example, in the xxz-spin half-chain, the energy current is always conserved and it coincides with the higher charge q2. So in many cases, you have a self-conserved current. I mean, basically a unique self-conserved current. But there are some known cases where such a situation is not happening. The one of the prominent examples is Fermi Haber model, in which model the energy current is conserved. So it is good to bear in mind that there are exceptions. But my proof is going to work for as long as these two inputs are always satisfied. Okay, so the starting point of the proof is to write this current average in this fashion. So this is always possible just by merely invoking how this charge density average looks like, the first slide, which is nothing but the integration of some functions. Oh, the concept of momentum. And it's also linear in the hj of theta. I mean, the one part going again by the qj. So by just from this fact and also continue equation, it's always guaranteed that you can write this current average in this way. What is just not known is that how, I mean, the part except this hj of theta, this bit is still not fixed by this consideration. So what I'm going to do is to write this integrand in this way. And I'm going to determine, I'm going to claim that V bar has to be indeed be given by no form of effective velocity. That's what I'm going to do. Okay, so let's suppose that we have a pair of indices a and b that gives this bridging pair. So j a is conserved and it coincides with q b. From this, you'd have immediately this identity between c and b matrices. And by using the symmetry of c and b matrices I was talking about, this also implies this second identity, c jb equals b j a. And this is precisely the identity that I'm going to rely on. So the symmetry of the matrix and the existence of a bridging pair admits this identity. I mean, this second identity can be also written in this way. All right, so this extremely simple relation, I mean, it's as if it says almost nothing, but this is actually an retrieve identity. It's going to be a kind of core identity proof. Okay, so the varchy of this identity is that this derivative with respect to chemical potentials directly go through these integrants. I mean, it applies to integrand and also doesn't affect h j of theta because it's not a function chemical potential. So what happens is that this identity can be actually recast into the identity at the level of a particle like this. This is really, I mean, the same argument that we do when deriving the generalized hydrodynamic equations. Okay, so you have this identity. So from this, you can do a little bit of manipulations to the right hand side, and then you can basically show that this right hand side of the identity equals something like this, the derivative of rho times vf with respect to mu a. So this can be shown very easily. And this means that, and also this, you can show explicitly for systems that are known to possess a self-consumption current. And this identity basically means that then this rho times v bar minus vf is constant in mu a. So this is what we can say up to now. Now, depending on the model, the argument might be a little different, but fundamentally they are the same. For instance, in the leaf linear model, which is the gallium variant integrable field theory, you can easily show that rho of theta goes to zero whenever you're taking this limit. I mean, mu a goes to infinity. So this free constant must be zero. And this then claims that v bar has to be the same as the effectiveness, I mean, no form of the effectiveness. So this is it, basically. This is the end of the proof. And it's very simple, like just some few line calculations actually show that. All right. A similar reasoning is also actually possible in the case of xxt spin half chain, and more broadly, in the xyz spin half chain, because they have also a booster operator. Okay. So I think I'm kind of running on time. So the symmetry of BMETR is, like I said, always basically guaranteed. I mean, not maybe not always, but in most cases, as long as your system satisfies some very basic properties. Yeah. But then the question now is, do we always have a self-concept current in integrable systems? And the answer is, like I was mentioning, quite often, yes, but not always. And one exception is frame how about a model? One simple kind of circumstantial evidence that tells you that frame how about a model cannot have a self-concept current is that. So what lies behind this thing, the existence of the self-concept current is the existence of a booster operator that can be written in as a form of the first moment of some consulate trash. And indeed, frame how about a model does have a booster operator, but it cannot be written in that way. So that would be probably the reason why it doesn't have self-concept current. Okay. So I think I should probably conclude. So the main message is that effectability is basically covering the plastic transport in any systems, including integrable systems. But as for the integrable systems, basically by invoking the symmetry of P-metrics and the existence of self-concept current, you can determine the function of form of the effect of velocity. Yes. And also, I guess you noticed that there's no, there's your argument in it. It's really, as the rigor says, time-adventure standards could be. So I think it's a good thing. Also, like I said, frame how about models is an exception that escapes our argument. But I guess there is a way to kind of generalize our approach to establish collision-rate standards for effective velocity in a frame how about a model. Right. And also, I was just talking exclusively about effective velocity or in general, the current generated by Hamiltonian flow. But you might as well also consider the flow generated by other concept charges. And accordingly, we can also define sort of generalized current, so to say, to that flows. And our approach can actually also work for that to establish the form of the generalized current. So we didn't write that in the paper, but it's actually possible. Okay. The final remark is that the boost operator is up, like I was briefly mentioning, it's important in this proof. And actually, also, boost operators appear in the, there is it work by blush, about long range of formation, including interval spin chains. And in doing so, he established also the effect of velocity. So boost operator seems to appear here and there. And I think it's very natural to wonder if there's any universal role this boost operator could play in generalized hard dynamics. And yes, that would be very nice to be clarified. Okay, I think, yeah, I'm more or less done. So thank you for your attention. Okay, so thanks. I don't know. I can clap for myself. Good. So we have five minutes for questions. Are there any questions? To ask questions, just unmute yourself and ask it. Okay. Can I just, will you hear me? Please. And I want to ask Takato, at the end of the story, the presence of a single subconcept current can prove the symmetry of the matrix. At the end is just one end, you know. I mean, the B matrix, I mean, B matrix is always symmetric for any component. The B matrix, maybe they see, I don't remember the notation. You can go back to the slide. Yeah, you can stop me. The creation of the current and the Q. Where? I'll set the point to say that we are using the presence of a subconcept current to prove the symmetry of a correction function. Yes, that's indeed. Yes. So is there the B matrix? Yes, this is defined by a single subconcept charger. There is a charge, there is a current, and the current is conserved, but it proves the symmetry of the whole B matrix for whatever kind of charges you want to put in. I didn't get this point. Why you need only one subconcept? It's not, it doesn't have to be. I mean, it happens to be that it's just, there is always a unique subconcept current. I mean, the point is that from just the existence of only a subconcept current, it doesn't have to be unique, just some subconcept current. And from that, you can generate, I mean, do this kind of... I understand why a single one is sufficient to prove the symmetry of the whole B matrix on all the other currents and charges? No, I think I said, like I said, the B matrix is symmetric for... Let me just... I think there are two separate points, Alvize. So he's saying that B is always symmetric, if I understand it. Yeah, yeah, yeah. And then you also need the second point, which is this subconcept current. It's a separate issue, but B is always symmetric. Okay, okay, okay, sorry, I didn't understand. Yeah, we do need two inputs. I mean, not failing to have one of them is really, I think, invalidate hardware. So we have to have both of them. Okay, okay, okay, thank you. Hi, you hear me? Yes, I do, yes. I just wanted to ask a simple, probably trivial question, but I was wondering if there is anything non-trivial when you have more than one particle species, so more than one effective velocity? Well, I don't think so. I mean, for example, in the X-X-Z-P half chain, you can also generalize this argument very trivially. The fact that many case models, I mean, accept some very special models like, how about the model? The fact that those models have a subconcept current does not change. So argument goes through really. I mean, this, for example, this identity has nothing to do with the particle species. So having more particle species, that fact will probably enter into some, yeah, like for example, somewhere like this, but it does not change anything. You can do a similar argument for any every single component. So yeah, this has nothing to do with a number of species, I would say. Thanks. Okay, hi everyone and hi Takato. I would like to just add that you didn't mention, but there is also a new microscopy proof, which I put to the archive like three weeks ago or something like this. I know, I'm aware of that, yes, yes. Okay, but you didn't mention, so it... Sorry, yeah. Does it, it is kind of a new proof for the effect of those tests? Well, I thought just thought that you established that the current operators, I mean, you basically found a way to construct current operators. Yeah, but it also, it leads to the proof, yes. Okay, okay, nice. Sorry to mention. Okay, thanks. But also, it's good for Balazs to mention this, but for going deeper, I think it's better to do it in the discussion session later. Okay, a very quick question, otherwise we have to move on. Okay. Okay, so let's thanks Takato again.