 Okay, so your next speaker and last speaker of the day is Jacobo Denaubis and he's going to tell us about logarithmic anomalies of spin transport. So hi everyone, it would be great to be really in physically in Trieste, but to enjoy the seaside, but now the only sea you can enjoy is this beautiful wave from Okusai. Okay, so we continue mostly what beer has been telling about anomalous transport in one dimension chains and indeed this is work in collaboration with many people and I would be in the discussion room, I think would be the target for any questions and so recently we also work with Roman and Sarang on something that indeed you would see in the slides. So let me just quickly introduce the problem since you've been hearing about this. Anyway there is a lot of ways to study non-equilibrium but maybe the simplest way is to study linear response of systems and if you want like one of the few known laws of non-equilibrium dynamics is the fixed law, just to tell you how a microscopic system behave at large space and time given any so I mean given, it's universal right, so in a sense any microscopic dynamic you expect to be given by something like this at large scales and all microscopic interaction is hidden in this constant, just the diffusion constants and recently not just in this field but in many fields there's been an effort to understand a bit more like how for example property of chaotic dynamics are related to diffusion constant and so on. So for me or many people in the community, one motivation is really can we understand this law which is non-reversible, so just the emergence of the reversible dynamics from microscopic reversible dynamics. So you just want to study an independent system and see how at large scales you see the emergence of non-reversible behavior and it's not easy to find exactly solvable system that actually give you normal diffusive dynamics for example free system don't give you that, they just show you usually ballistic transport and of course it's also interesting to know when it fails, when you have anomalous behavior. So it turned out in the past years that again quite a nice set of models that can answer a bit of all these questions are interacting integrable spin chains, not only they actually despite integrability they actually show large-scale diffusion and despite irreversibility of course, but they can also show anomalous, so failure of the fixed law and that's why it's again another motivation to study this aspect that is modeled. So what is diffusion constant? So the simplest way to define a computed by Kubo formula and which tell you the basically DC conductivity which is a matrix in case you have many consequences if you have only one, of course they will be just a number but it's a matrix if you have many, it will be just integrated current-current correlator. So if you have a system which also has ballistic transport you should subtract the ballistic part which is called the root weight which is basically the large-style limit of the current-current correlator. So now this expression is always regular, it's always converged and this it's a diffusion constant. So if it's zero it basically tells you that a current-current correlator is just the root of weight, so there is no dissipation current does indicate and that's a free system if you want. But as was shown a few years ago for an interactive system for an integrated system this is it's positive only if you only maybe one element is zero for example when the current is conserved and but not only there are also some charges, so now ij-level charges like could be energy, spin, you want only one charges but also the higher charges. There are some charges for example the u1 charges like the spin or the charge that actually show infinite DC conductivity and this doesn't mean that it's ballistic because we're already subtracted. So it means that this integral over time diverges so the current decays lower than than you expect. And basically then there are lots of numerical evidence although not perfect because it's hard to simulate quantum system for example this is a this is a T-diamagi simulation of the Eisenberg chain so you can take Eisenberg chain spin one half and state of the arts and numerical simulations so this is done in a group of too much pros and already a few years ago. Basically if you take just the Eisenberg so no one is on z direction you see that basically and now you join two states that basically infinite temperature states so now indeed I forgot to say yes we are interested in transport the finite temperature so is this the case of finite temperature it's the infinite temperature no problem to define it on the chain. So I joined two infinite temperature states with slightly unbalanced magnetization and this is a way to study transport it's really like what Kubo would tell you to do to define this concept and basically you see that sprayed in here it's not T to the one half it's something it's something larger instead if you add nasotropy on z direction so now you simulate the next z you see that the spreading goes as T to the one half as diffuses. So now in this case you can say that there is a spin diffusion constant and and this is a fixed value and you would like to know and then indeed it can be computed nowadays but here you can fit and it's despondent and you see that doesn't increase as T to the one half it increases the T to the one sorry it doesn't go to constant but it increases T to the one third so this was the starting of actually lots of numerical simulation on this and indeed not all just despondent was measured but also what is the profile rescale profile of magnetization inside here. Also one can ask what is actually what happened for non-integral change so because so far you've seen an integrable system okay so I mean I'll study non-integral change for example let me take again isomber chain but for spin one so this is also called a lane chain if you want and or any genetic spin S so one way again to study the transport is for example looking at spin the relaxation rates so just a spin auto correlation and you want to see what's despondent that's the spin decay now so this is done by Maxime Dupont-Juan Moore last year and they did simulation and you see that for isomber spin one half you have a nice decay with the exponent three over two so now three over two is basically the same if you use the the fact that auto correlation decay is a one over square root of T times diffusion cost that saves you the diffusion cost that diverges T to the one third so they confirmed this exponent which is basically KPC exponent so then it did the same for spin one and even if it starts very closely to exponent three over two well they seek some kind of very slow drift towards what would be the diffusive value two which just characterized just one over square root of T so this is hard to say they concluded indeed that indeed for integrability super diffusion this exponent is a non-respondent should be protective why for integrability non-integrrability should slowly drift towards diffusive value so this was also done after that me together with Ney Markham and Denjak and Christof Karas we studied this system and in particular any this this kind of chains in the low energy limit are described again by an integrable system which is the non-linear sigma model and there we could prove the versions of the sequence to tick but of course the low energy theory does not describe the physical system so you see that in the physical system there is something different to take so what are the main statements just to cut it short so basically these are statements that apply indeed as Vir was saying to see that to apply to no matter what clontome or classical spin chains because somehow we are looking at properties of very large scales and and provided you are respecting some rules you can always make quantum classical correspondence between the system so in integrable chains either quantum classical now there is quite analytical and numeric evidence that there is always quasi-sense of ballistic and diffusive spreading although in some cases diffusive is anomalous and indeed in rotational invariant integrable chains for example any system with non non-abillion symmetry like SU2 or SU3 or maybe SUn this is remain to be checked the local monetization it can be phenomenologically described at large scale by a Berger or KPC field then basically it's a field was current is a non-linear and there is an exponent parameterizing this non-linearity and it depends on the millton but for Eisenberg it's 2 and it's KPC class so what happened in rotation in rotation invariant non integrable chains well then basically we cut some phenomenological model that seem to describe what numerics observe sets and also a bit confirm what Vir was saying that basically what you should expect that there is still a Berger field that described your local monetization but the environment or the noisy environment where it moves basically change with time and the late time decays so it doesn't kick you the system and the system basically crossover to diffusion but with logarithmic and we try to tell you a bit more so basically just a very crash course on why there is a diffusion in integrable system so this morning well this morning a few hours ago you heard that Takato telling you about GHD and also Axel of course and as Axel was indeed the remarking you is that the existence of GHD is basically due to the fact that in integrable system at any energy scale not just in the low energy limit you can always have stable excitations on top of a background state and the simplest excitation or one particle excitation where basically you take one or this particle and you kick it and this is basically simulating the fact that the system is there is like a particle moving through a background like a finite temperature state at some point kick with the system and get slightly deviated and the deviation in proportion to the scattering shift and this is what dress the velocity so now there is a ballistic motion with some dressed renormalized velocity but it's still ballistic because particles don't decay they just get renormalized then but particle can also actually have two-body scattering and whenever they meet they have two-body scattering and this redistributes the momenta I mean redistribute either the permute to the remain the same of course we are in 1d but this leads to diffusion and this fusion indeed this phenomena is described not by t itself but the fact that you're in a constant final background and therefore is formalized by a dressed version of the scattering shift nevertheless these two-body scattering all take into account all diffusive term and this was our result with Benjamin Dillon and Daniel Bernard and basically this it's the microscopic mechanism for diffusion of course this is very quick it can be said in many more words but so now okay we were very happy in 2018 to have the finally an expression for the diffusion constant and integral system something that actually was was looked for many years and in the 80s 90s there were many different attempts and so we applied that finally to the most beloved system the ability and it did the spin one out of the chain where indeed there were already numerics available from from Tomas and so on and basically we could write the spin diffusion constant as a sum of contribution and this you can read them as some bounce state contribution and eventually some all complicated computation that you can do to compute this diffusion constant either using this kind of a form factors because this eventually reduced to some you know with some form factors or using what Rohan and Sarang did using more like kinetic approach basically give the same answer which is a very simple answer basically tell you what you can read in in textbook that diffusion is given by some effective loss time some if you want mean free path and but you have to sum over all the all the basically modes that you have in your system these modes are like bounce states and you can read them as bounce states or manuals if you want because you see your elementary excitation even to construct finite temperature states you can use these manuals of course now I picture them like this but they are not like this this is only in the limit where basically if I would have a strong anisotropy in z direction but let me just picture like this you can really picture them as some extensive bounce states where they're their size is exponentially in s and as this number s increase basically this bounce this becomes larger and larger and because of the gravity they're all stable okay and and the fact that basically if you are in this model and the isomer model well it's something critical happened that basically this means is the this mean free path if you want to see is proportional to s and this you can read it because basically you are finite density there is no mean free path the only moment where these bounce states move freely is when they meet each other because when they meet each other they jump is they jump by size s so if you want to the mean free path is fixed it's just given by the dimension of the of the of the bounce states but that velocity as a function of s the decay as one over s and you see that basically a sum is something that becomes constant which basically the bush and this basically is a way to extract the also the scale in spawned as a roman sarang data you get t to the one third and recently with indeed with the nine roman sarang we basically showed that if you look at the dynamic properties of these bounce states at large s and you take this large s limit in some proper way basically you have to introduce a cutoff which is the magnetic field and rescale this magnetic field to zero well something magical happened basically you go from a quantum model with bounce states and so on to a classical model which is basically landau litchitz equation and you see it because this is also an integrable model and it has solitons classical soliton solutions and these solitons have a scattering shift and this is indeed this one with the log basically you can take this limit and get this you get the solid so basically what we understood is that or basically confirm what somehow there was an expectation that this large scale bounce states are nothing else than just soliton solution of a classical PDE and therefore it is confirmed the fact that kpz in quantum chain a classical chain has the same nature there are some wide soliton solutions that move through the system and have zero energy and therefore you can produce them as much as you want and basically this leads to a normal structure and okay so let me just go a bit quicker here because basically we use this intuition to compute the basically this lambda kpz so what is lambda kpz basically if you have a burger equation then you would expect there is a parameter for the non-linearity this parameter basically give you the large term the knocks of because k basically the kpz here italian that's the spin-spin correlation a large scale should be described by the universal scaling function which is not gaussian this kpz function and there should be a non-universal parameter which is called lambda kpz that can now can be extracted with and it's a parameter of these emergent gas also or classical solitons basically you have to run some complicated tba to to terminate better answer to get it it's actually an interesting tba because it's now it's continuous it's not any more discreet of course because classical physics is continuous but anyway then you can compare with the amargi solutions and what's the interesting is that this parameter is non-universal it depends on the model but also depends on the temperature so basically if you compute a spin-spin relaxation rate a different temperature you would see a different emergent kpz parameter and which we try to fit with the amargi for example this is spin-spin correlation multiplied by t to the two-thirds right so it goes to a constant and this constant is can be given it's given basically that by this parameter and this is the value it's quite good relaxation to the to this parameter it's low it seems to go as t to the minus one-third and this is some also what is expected in kpz universality classes for for different systems by the way so this is very good but still it means some kind of interpretation or why there is kpz so at least we understood that we don't have to look at quantum mechanics which is hard but we can look at classical and now indeed as Vir was telling you before basically you can just given a classical amethodian or a quantum just take the semi the semi classical mean field limit and read it as a pde and once you have a pde there is this nice gauge invariant reference frame which is called serendipity where basically you don't anymore look at the evolution of the spin but you look at the evolution of two parameters basic energy density and what's called torsion and the energy density can be neglected in it in a dynamic limit and now you basically have an emergent equation that describes your spin on a large scale only in terms of the torsion of the spin and so basically you can reduce all the problem if you just to know what kind of transport you will have it seems that is good enough to just given an amethodian which you write as some continuous spin field theory just distract what is the leading term in the nonlinearity of the of this pde written in terms of torsion but now of course you ask yourself but okay but i can do this for an integrable amethodian but also for a non-integrable one and given some symmetry property for example as you do or i will get the same equation so integrability doesn't play a role whenever i have a rotational invariance i will have the same question well no because it seems that the integrability comes into basically the noise that you add to this equation why do you add the noise you add the noise because you still trace in a way a lot of other degrees of freedom you're doing the other dynamics so you're doing the dynamics and our noise act as the effect of the other degrees of freedom that act that basically scatter with your spin and if you have an integrability basically you'll have ballistic modes that still travel into your system and basically keeps it in you during all the time but if you have a non-integrable you expect that all the other modes decay and actually decay diffusively so basically you can do these answers for the strength of the noise that you put in your in your torsion equation and then say okay but what do i do with with uh some kind of a kip easy equation with noise dependent and actually we were lucky because last year i mean a few months ago indeed the group of peter dussalle and other people they actually started this kind of a non-linear equation with time dependent noise and they claim that if noise is stable the large time dynamics is indeed kpz but if noise decay is diffusively then you're actually in a critical point and you basically go to the fusy class but we logarithmic corrections basically the claim is that even in an integrable chain so if you're integrable and this n is equal to you indeed display kpz scaling but if you are non-integrable you display diffusion scaling indeed your exponent is diffusively but you have logarithmic corrections and this logarithmic correction actually is what basically delay infinitely the emergence of diffusion in your system so we tested and i need to finish but we tested this and basically on a classical chain which is basically the isenberg classical spin chain and uh this is classical now so it's non-integrable it's non-integrable still as s2 invariant so we didn't it's not it's classical so it's great because we could do simulation up to uh up to several decades and indeed we plotted the diffusion constant function time and we see nicely that it increased with some logarithmic scales while if you instead study another classical system that has a different non-linearity power for example this chain or this this is non-linearity power three and this is non-linearity power four you still see that diffusion constant goes to come to to finite value so this was quite surprising because one usually expect that non-integrable chain displays always finite diffusion and actually this is a largely standard model that can go back to the literature in the 80s people were studying this chain numerically and they always find a contradicting result somebody was claiming was super diffusive somebody was claiming diffusive uh eventually at least this seems to say that yes there is a it's diffusion but with some logarithmic and it seems so related to this emergent kpz dynamics so I wanted to show you other checks of this prediction but okay I overestimated the time let me just go to the conclusion so basically anomalous kpz transport in isotropic quantum chain is related at least now we can see it from thermodynamics to the to this emergent site on gas and as steve said it can be also related to cold some kind of goldstone mode so though the correspondence is still could be done even more clever more better and different type of transport can be all reduced to a non-linearity exponent and the presence of absence of ballistic modes and this basically now will give you a table to characterize transport in isotropic models is if there is something else we don't know we conjecture that this is all you can observe but of course the jury is up and open questions what kpz the emergence still needs to be proven from fifth principle this is a very nice problem that's invite people to do but okay maybe the more interesting problem is can we prove the fact that's non-integrable isotropic chain have infinite diffusion that would be great we designed that sorry for overshooting thanks a lot jacobo for this very nice talk so we have we have some time for questions okay jacobo can i ask a question hi hi uh yes so basically you were talking about uh i mean this classical limit of of hasbro chain and we all know that it's kind of london lift is and the the point is when you try to quantize london lift is actually you can do that and you could you could get some quantum states that's a kind of corresponding to the classical soliton if you want or finite gap integration if you want uh yeah but then there actually strain hypothesis doesn't apply i mean you can really see that a thermodynamic limit this density of the better rules they don't satisfy strain hypothesis but uh if you want to take the limit of tba so tb assume strain hypothesis why this tool can somehow be the same yeah it's a problem of limits right here we are first taking a terminal limit and then taking this large wavelength limit and you are you could do instead uh and see and it seems that to you that you are doing first large wavelength and then terminal limit so this is indeed something that's uh discussing with an a it should be possible i mean this gas of classical soliton could be either constructed by indeed taking a tba and now taking this large wavelength limit but also could be constructed by taking these modes that you have in classical theory and taking in the final number of them so far we only know how to do the first one we don't know how to do the second one but eventually indeed there should be there should be a way to to do it in both ways but so far could only see this yes i'm a bit let's say surprised at how well tba already worked since i mean people assume strain hypothesis there so it's kind of yeah yeah so strain hypothesis works right i mean uh at least to get uh i mean it works in so many other contexts but um i agree that is uh it's it's very interesting limit and uh and indeed we this was just the first one the first time where we noticed this right and also kind of the there's another difficulty as you said when you're taking the other way the limit the other way around is that how to kind of compare the classical and quantum thermodynamics because i think it's it's still not it's a very subtle thing to take a thermodynamic in different systems it's kind of the solitons are kind of fermionic or yeah indeed so they're indeed now what we see what we find is that the for example the this the these are of course these the quasi particle and tba are fermions but as you take this limit they become classical so their occupation number actually are not anymore between zero and one but indeed i mean there is a lot to know and you know a lot because you're going working on this and uh so we'll get to this yeah i guess yeah okay thank you and i can go maybe a bit slower in the session yeah indeed sorry i had to get very fast it's quite subtle yeah um any other question it can ask question actually yeah so just uh just to clarify so the so the noise term somehow uh put my video uh goes down in non-integrable models because you don't have all these uh kind of stable modes that provide a bath that provide that gives them so but of course in any model integral or not there is noise so what you're saying is actually there's that noise is another scale then look at the noise produced by this very large scale stuff which then affects this even larger scale kind of spinways that's that's how you divide things yeah exactly this is the correct way to say it because indeed again uh this is a phenomenological theory that only tells you that you're going faster than if you see or not if you're not going faster tells you nothing because uh you're not you're not putting a lot of stuff that uh this doesn't predict anything of course but if you're going faster then it tells you who who indeed it can read you right and indeed this is a noise that is suspected to be on scales that that uh act on and so similarly of course in non-linear fraternity dynamics is the it's the ballistic mode that provides information about this is was a bit why we maybe broadly conjecture that uh this is not integrability on i mean the distinction is it's not just between integrable and integrable if you would have a one extra ballistic mode it should work yes and indeed in non-linear fraternity dynamics you always have one ballistic modes that's why yeah so that's what we thought that the reason why in non-linear fraternity dynamics you always see kpz is because you always have a at least one ballistic modes but maybe now have it can contradict me okay we have one very quick question anyone else uh hi yeah um uh yeah so with regards to um how quickly you saw a lambda kpz converge in time um i'm wondering if you can um also sort of use your you know theory of the kpz equation with noise to sort of get like a finite time convergence of the full profile to the kpz profile function uh-huh um in order to sort of use the d marg data to sort of more to distinguish whether the d marg data is really showing kpz or non kpz and yeah one should use uh one should solve a burger equation at finite time and maybe using the the one loop theory that uh harbert introduced it's i don't know if this can give you finite time results for the but indeed because you see whenever the problems whenever you write this you write an infinite time prediction indeed and uh i need it one has to find some finite time uh it's probably indeed uh of course the full evolution is more the dependent semest but uh there are there are ways to at least get details and this actually was done as far as i know in uh in uh not in this models but in a stochastic classical system there are ways to get uh details of this so indeed good point this should be done right thanks all right so um if people are interested in talking about super diffusion some more uh they can join the gtsy room um and so i suggest we unmute and uh thank and uh yeah thank you so all right so we have a break again uh so for the speakers Jacopo and Via once you're done with your breakout rooms if students don't have questions anymore you can also join the gtsy room if you want and with this uh all right i will see everybody back on wednesday yes bye bye on uh on which uh on which topic like uh i didn't get the like the final claim sorry the what right this uh yeah so indeed um okay so first should be said that uh this is a mostly phenomenological uh way to to describe uh how things go baddies who are in a mask that's a really good chart too intelligent for mosques sure instead of the mask