 Our first speaker in the second section will be Lorenzo Piroli and he will talk about universal aspects of low-temperature transport from GHD. And Lorenzo, please. Okay, so let me first also thank the organizers and the host for this invitation and for this very nice conference. Right, so today we'll talk about some work mainly with Bruno and also with Martin and Pasquale. We should also be online today. So differently from other works that we've seen, especially the first day where we had a glimpse of quite advanced applications of GHD, most of the things that I will discuss today can be understood as early applications of the theory. So these works are more concerned with elementary aspects and foundational aspects of GHD. Right, so since this wasn't really addressed during day one and since this is now day two, I thought it would be appropriate to start with a bit of historical motivation of GHD. So we all know that GHD was introduced in these two very nice and influential papers by different groups and I think it's fair to say that one of the main original motivations for GHD was to try and extend the theory of quantum quenches in homogeneous settings. So let me very quickly say something about quantum quenches. So what is a quantum quench? Well, imagine you have a many-body system and extended many-body system and imagine you prepare your system in some initial state, which for example can be the ground state of some local Hamiltonian that was not here by H of G, where G is some set of internal parameters. And then imagine that at time t equal to zero, you bring your system out of equilibrium by a sudden change of the Hamiltonian parameter. So then what we want to do is to study the dynamics of the system by probing local correlations. So this is a problem that is interesting for many reasons and already in the idealized situation where you take an infinite and homogeneous system, so no disorder, no dropping potential, already in this case this is an extremely difficult problem. So the physical intuition suggests that if you really are in a idealized situation, what you should have is a local equilibration towards something stationary. And now a lot of a decade of literature has shown through a huge amount of numerical evidence that if you take generic systems, what you will have in fact is local thermalization. So the idea is that if your system is very, very large and you focus on a small part of it, then the rest will act as an effective path. And so if you wait long enough after a quench, you will have thermalization. Now it is known that something different happens when you have a local conservation loss and particularly so for integrable systems. Right, so now a working definition of integrable systems is many body systems that display an extensive number of local conservation laws. So what we have is a Hamiltonian written here in one dimension and a set of operators that are written in the same form, so a sum over space of some localized densities. And then all these operators are in involution with the Hamiltonian and are thus conserved quantities. Now the nice thing about this integrable systems is that they are generally interacting models but also exactly solvable in the sense that you can compute analytically a spectrum beyond the so-called bed yarn sets. But from the physical point of view what is really special and nice is that these integrable systems admit a quasi-particle description which is the basis of the thermodynamics and the generalized hydrodynamics. So essentially you can think of them to many respects as a generalization of ideal Bose and Fermi gases. So for example the energy levels can always be written as a sum over single quasi-particle energies where each quasi-particle is clearly tried by a quasi-momentum which again is a generalization of the concept of momentum for ideal Bose and Fermi gases. Now the crucial thing is that in thermodynamics you can associate with each eigenstate. I will define a quasi-momentum distribution and so for thermal state for example you have something which is very similar to what you have in Fermi gases. Now in as a result of 10 years of investigations a quench problem has been essentially solved for integrable systems at least conceptually. So in particular what happens after a quench is that your local properties still equilibrate towards something stationary but instead of having a thermal Gibbs ensemble to describe the physics you have a generalized Gibbs ensemble which is written by also taking into account all higher conservation laws of your system. And the most convenient way to describe the GG is by providing the distribution function of the quasi-particle, quasi-venta. And nowadays there are techniques to extract this, to extract the correlation functions from these distribution functions. And finally in several cases of interest now it is also possible to compute explicitly this Rob Landa. Arguably the most important turning point was the introduction of the so-called quench action approach by J.S. Ko and Abbey Nessler. And in any case so this was already sorted out four years ago and you can read more about this in these very nice reviews appeared in 2016. Right so as a consequence when we think of integrable systems out of equilibrium we should always think of local relaxation where the local stationary state can be something very weird, something highly different from a thermal state. And there are many examples in your literature where the quasi-momenta distribution functions for the quasi-particle look nothing like a thermal state. For example can it be a double peaked or a for example can it be a double peaked or have pet tails. Okay so this was four years ago and then at that point a very natural question pop up started to pop up which is what happens if we take inhomogene systems. And some referees were obsessed with this question because this is really what you want to know if you want to compare with experiments. And once again in the spirit of going to idealized situations the simplest thing you can do if you want to study inhomogene systems is to prepare a bipartite initial states. So essentially you do inhomogeneous quenches. Okay so imagine again you have a many body system for example it's pin chain in one dimension you cut your system into two and then you prepare the left and right side sites to be thermal states at different temperatures. You can actually prepare them as any other say equilibrium state if you want. Right so in this setting then you attempt equal to zero what you do is you let evolve the whole system and inhomotone and then you simply ask what happens at large times. Okay so this was the kind of problem that was started to be investigated at the point but this problem has actually a very long history and starting from early 90s people already provided very nice results especially for free theories and in conformal systems as I will also mention later. And several people who worked on this problem are also online today. Okay so but before a GHD this problem was not cannot be tackled for interacting theories besides brute force numerical calculations and so GHD finally was invented in the in the effort to try to solve this problem. So now let me briefly recall what is the GHD solution in this case. So based on all the experience that the community gain from the study of homogeneous quantum questions it was postulated that in this setting at very large times one could associate to each point in spacetime a quasi stationary state which is described by an appropriate quasi particle quasi momentum distribution function. And then the intuition is that since a defining feature of renewable systems is the presence of quasi particles we should expect that the stationary state only depends on the ratio between the distance of the origin and the time t that has passed after joining the two halves together. So these are essentially the conceptual ingredient that you need in order to write down some equations and then once you accept these postulates you can simply arrive solution by assuming two further ingredients. So the first is the continuity equation which is very natural and this actually can prove and then you need a formula for the expectation value of concert quantities and the currents and we we know about the story of the expectation value for the currents by the nice talk by Takato at day one. So mixing all these ingredients it was shown in these two papers how to actually arrive a solution of to this problem which is written in a very concise form here which is essentially a continuity equation for once again the quasi momentum diffusion functions of the quasi particles. Okay so what is really important is that when you have these idealized settings the claim is very strong namely that these are exact solutions. So if you had a infinitely powerful computer and you run it for infinite times then you should get exactly the predictions for GHD. So the status is the same as traditional thermodynamic calculations that people did it for interval systems and indeed already in the first paper by the Italian group they compared against EMRG calculations finding very nice agreements. So these are two plots for the profiles of current of energy and energy after joining together the two halves you see that here you have x over t on on the first axis and and then you have something that you could expect so if you go to very large distances on the left or on the right essentially nothing happens because information has not propagated yet and in the middle you have something in trivial. Right so the important point is that GHD not only gives you access to how conserved quantities such as energy spread but also how the profiles of any local observable look like after joining together two halves and so early on we realized that these profiles that are emerging from from this kind of bipartition protocols can be very very strange and so one natural thing to do at the very beginning was to try to provide some phenomenology of this kind of transport profiles that emerge in this kind of problems. Okay so there are many questions that we asked and then one can ask but the one that I will focus on here is about a low temperature regime of this kind of bipartition protocols. So there are three main points that I will discuss in the rest of this short talk. So the first one is to tell you about our test of GHD against previous CFD predictions and then how GHD led us to discovery of new universal effects beyond the CFD prediction and finally I will also discuss some spin charge separation effects that emerge naturally when applying GHD to these kinds of bipartition protocols in multi-species integral models. Right so let me focus on one of our favorite toy models which is the X-axis in Heisenberg chain. So here we have the Hamiltonian these are standard Pauli matrices and this model is nice and because if you upon varying delta and h essentially model goes through at zero temperature different phases so here I will restrict the regime of delta and h for which the model is gapless. Okay so now imagine you want to do some phenomenology here you want to produce some plots what you need to do is to take that very concise equation flesh it out and solve it in a miracle. So this is essentially a all the equations you need to provide plots so it doesn't matter so much what are the different terms here but you see that there are complicated equations that in general you can all only hope to solve numerically but the point is that if you are interested in this mole temperature regime then there are a lot of simplifications to the point that you can get another result and this is what we did at the beginning with Bruno so let me tell you the results right so the first result is something that he was actually expected as I will mention in the next slide but these are the profiles for energy and current of energy after joining together two activity chains at different and small temperatures so you see that the profiles here are extremely simple and they display a three step form where again so on the far left and on the far light right sorry we have the thermal predictions but there is a third plateau emerging in the middle of the light cone where we can make a prediction on the expectation value of energy and current of energy so the energy for example it's simply proportional to the sum of the square temperature whereas the current of of the energy is simply proportional to the difference of the squared and temperature so here we didn't discover anything new because this prediction was given a long time ago in a series of very insightful and remarkable works by Denis Bernat and Benjamin Guillon where essentially they're studying what happens when you join together two CFTs and if you think of this problem this way then it's clear that the situation has to be like this because in a CFT you only have one type of velocity and so the only thing that the system can do when it evolves by joining together different thermal states is to display a light cone spreading with this single velocity here the prediction was more general was for every career CFTs in our case we simply had C equal to 1 because a x-exit chain at small temperature is described by a CFT with central charge equal to 1 right but then we could go further because as I mentioned CFT gives you gives access to arbitrary local thermobots and so for example you could look at what happens if you focus on the magnetization and you see that there is something very strange happening at the light cone edges so this is essentially a snapshot taking at a given time in this cartoon so what we did we computed this form of the light cone analytically and then we did the same for different observables and different models for example the Lieb-Lieb-Lieber model in one dimension and essentially in all these cases we found the very same function which has a simple form and a difference here and this of course was a hint of something universal in the sense of an usual sense of a randomized review and so indeed in a subsequent work with Pasquale what we were able to show is that this kind of behavior at the light cone could be predicted based on a universal non-linear light liquid description so this is a theory that was introduced now more than 10 years ago and essentially allows you to predict some non-perturbative effects in linear light liquids based on taking into account the most dominant irrelevant term beyond the ammo tone so in our case we have the xxz chain which falls into the university class of the lightning and liquid so we could apply this logic and what happens if you if you apply this logic to this kind of my partition protocol is that you have to modify your calculations or letting your liquid calculations by allowing for a curvature of your dispersion relation so this pop-ups as a phenomenological parameter m star so the calculation is extremely simple within this non-linear liquid approach and what we find is the exact same result that we derive with a very long the dense calculations and importantly this m star pops up as a non-perturbative effect in the final result okay so this was an example of how ghd allowed us to explore a little bit of phenomenology of low temperature transport and arrive at a universal result that was not noticed before so now let me give you the last piece of phenomenology that we discovered in this early attempt and this pertains the study of more complicated interval systems that are made of more than one particle species so the vast majority of beta-ansits models are solved by an elementary beta-ansits where essentially you only have one type of excitations in xxz you can think of these kind of quasi-particle as magnonic excitations now these can form bound state but there are only one type of excitations but there are more complicated models where you actually have different species and the example which is most relevant one for experimental realization is arguably the younger than model which is essentially a one-dimensional model of spin-fold fermions which is written here so the Hamiltonian simply tells you that this is a model of spin-fold fermions that interact with a point-wise interaction and so it was natural to also consider these kinds of more complicated models and see what happened at small temperatures and so the surprise was that here we in the guidance regime we see something different in a way we do not see a three-step profile for the energy and the energy current but a five-step profile so we have the emergence of two velocity two natural velocities and these are exactly those that correspond to the excitation of the spin and the charge so in this sense this kind of bi-partition protocols show you how if you go to low enough temperature you can see some kind of separation effect even though this is not exactly the spin charge separation effect that you see at exactly zero temperature because in that case you can find observable that are only sensitive to one or to either the charge or the spin whereas here for example the energy we see both traces of the charge and the spin velocity but still it is a separation effect because if you play the same game in a model without different types of excitations you will only see one velocity okay so as a last slide let me give you some outlook so ghd has been exceptionally rapidly it's been an exceptionally rapidly developing field over the past three years and many excellent works have been appeared and most of the authors are actually online now but so it was now so as a series of these works is now established that ghd can be applied to more general settings that are relevant to experimental setups so for example you can include trapping potential in homogeneities and phasing and it was particularly exciting to see that last year also an actual experimental confirmation of ghd appeared in prl by these authors where essentially they studied one-dimensional bosogastis trapped onto atom chips and so a natural question based on these work is whether these low temperature features that we have observe survive in the presence of these more realistic experimental settings and so one thing that we're now looking at which is particularly looks particularly interesting is whether it's kind of spin chart separation effects at small but finite temperatures can survive when we put in traps or find a number of coordinates okay so with this I have finished and I thank you for your attention thank you very much Lorenzo so I think we have time for some questions so if you have a question just unmute yourself and ask excuse me can I ask a question about the bipartite system can you hear me please go ahead what would happen if you got three parts like a red one blue one and red one do you think this things will interfere or just spread independently so say it again you want three particles pieces or you have three settlements ah yeah right so so in that case the problem is that you first you should define how to scale these intervals in the in the hydrodynamic limit because here you have these intervals are are infinite in one in one direction and the other so it doesn't matter how you scale you will always have the symmetry if you have three three segments and you don't scale correctly the middle one say then if you scale too much then this will disappear so instead if you if you allow for scaling this third segment I guess that at each interface we'll see the exact same physics that you see here so you essentially can treat your three part tight problem as a collection of bipartite problems okay thanks sorry I have a question as well can you hear me please go ahead more okay no it's about the effect of a correction to ghd to the behavior of the bummer because is there some limit because I think that you the extension of the boundary depends on the temperature or the difference of temperature right so you're so you're talking about yeah yes when you see the the like okay behavior which is universal I mean so I'm wondering whether there is a limit when you you expect some some effect due to the correction to ghd to the first order ghd right now so here we're always in the hydrodynamic limit so so the point is that these light cones they're they they they do not disappear at the hydrodynamic distance so you see that they're they're they're they're proportional to the temperature whereas these plateaus are proportional to temperatures square so there's no way in in senate temperatures zero in such a way that these problems disappear so I guess that if you as long as you're in throw in the in the hydrodynamic limit you can neglect anything that is not ghd and the only corrections to this and the hydrodynamic limit comes from the contributions coming from ghd that you neglected so here we're still in the in the in the in the game where we really send t to infinity t to infinity x infinity and and so here there is no correction to ghd if you want to if you want to take finite x finite t then of course and if you want to have something quantitative you need to add these corrections to ghd by including quantum corrections you know yeah yeah I meant for example if you can imagine some scaling with the difference of temperature and the and space and time no I mean right no in that case if you if you start to take finite systems I'm not sure at what point the quantum correction will be more relevant to uh more relevant than than the ghd correction thanks could I ask a question yes please yeah so I'm looking at the slot to the left and I see this this peak uh up and down so did you compute this analytically and uh what yes it's a screen current or is it the uh right so okay so on the left here you have a profile for the magnetization so here it's a zoom out because these peaks are are very very high so they're proportional to the temperature whereas the three step profiles uh you have a separation which is proportional to t square but yeah so here we could take the ghd equation compute the limit t goes to infinity and and essentially derive this another formula here you see so the reason why here we also have this is well this this here will disappear as the temperature goes to zero the important part is this here you see that this this temperatures are smaller than this because I computed similar things but for heat transport in cft and then these peaks in that case appeared due to shorchin derivatives in the uh one thing that I didn't mention very similar no no indeed like the claim is that these are are universal so if you take any observer we generally will be there there is a caveat so here you don't see these these peaks when you compute the energy or the energy density uh and the reason is that if you look at the formula the next one I don't know if you see it essentially these coefficients here there are non-universal of are simply zero in that case so our claim is that whatever you do whatever observable you compute in an interval model which falls on in the universality class of uh linear letting the liquids you will always uh find this kind of broadening of your icon at small temperatures unless you're considering the energy or the energy current okay because in our case we started exactly in the heat transport because that's where you have the conformal anomaly uh so that's and the peak looks very similar but I mean the set up could be different and not just when I saw the peak then it corresponds exactly or it looks very similar to what we found exactly by a shorchin derivative it's that great but okay maybe you can discuss later yeah I can show it to you I'm just curious to to to see where this came from thanks but in this case you can just check the picture that Lorenzo showed for the energy density for example there is no peak so I think yeah that's why I'm interested to to to understand the results was uh in our case was and you can compute it you think of a transformation and it just pops out without any no expansion or nothing just just one step thing essentially okay but thanks I think we are probably running late if there is a very quick question otherwise we need to pass to the next talk