 Okay, and I looked and speak. It was introduction and good afternoon. Can you see my presentation. Fine. Yes. Today's talk about quantum thermal transport in the short term regime on a such a type model, but this walk was done together with. But yes, I want to encourage everyone to ask questions if you feel that it's appropriate time in the middle of the talk, it's okay and I think we'll have enough time. Well, even kind of meta theorists, not to mention the higher crime community would ask a question what such a type model and what it works on the gym. This model and then only style it makes sense to talk about quantum thermal thermal. I quickly introduce such a model and try to do it as accessible as possible and but still again feel free to ask any questions. This model started, well actually appeared initially in nuclear physics in 1970s. But it was reappeared in kind of meta community in 1993. This paper was such a different year, but became really famous in 2015 after the talk of the time where he introduced it in the modern form but for my run of thermals. Why this model is actually very interesting. He wrote it in the morning for form for charged fermions. And for now I forget about chemical potential and talk in general about the purpose of this model. We have and thermals with random on all the interactions and the interaction constants are random but they have certain symmetries but was important in average, this interaction is zero, but the variance of the interaction is non zero. So the interaction is Gaussian. This model is interesting because it can be so it has exact solution. In large and limit. We try to introduce millions functions, and by mean a few analysis, looking into the several points for the action or by analyzing the contributing. And diagrams we can find the system of recognize equations in large and limit. Basically, this system allows several solutions when energy is quite large, then we can neglect the second term, and to get free thermal solutions. Can you hear me, which is okay. Yes, yes, no worries. But there is an opposing limit when we neglect the first term of this in this equation and then we have a non trivial solution for model. The system is called conformal regime, which I explained the reason a bit later, but it was also shown. Some time ago that the system has one, one more regime, which is called Schwarz's energy gene, which lies even lower in the interest. And these two regimes are really interesting part about this model and generate a lot of interest in various communities. So why this model is so interesting. First of all, it's highly strong interactive non thermal liquid system, which has exact solution which is already quite cool. But as I showed in his talk, this model actually is a simple. A simple holographic has simple holographic duality to a black hole in ideas and to the center to dimensional space. And that's why the community explore it because I think her before it, the easiest holograph, at least best known the holography model was super symmetric young meals, which is extremely intimidating I would need the whole page to write. While in contrast here you have nice to look in Hamiltonian and to reproduce. Basically, a lot of properties which you expect from black holes. So you have a toy model to study holography and even content gravities we see later. And also it was discovered soon afterwards that this model is extremely important for quantum case studies, as well as for understanding the physics of strange metals. And even those different intercrossing between these regions were known before, or at least they were suspected. It turned out that this like a model actually sits in the, in the middle of everything. So this model has something to offer to, to many, well, not everyone, everybody, many communities in can estimate and high energy physics. So let's start with conformal regime. As I mentioned to just neglect one term and the question for the function, and then everything can be solved. Exactly. It's easy to see that this is solution by simply plugging them here, or by doing after doing the free transform this equation also will be satisfied. So, what's why this is important that the same greens function. The system is called do to the black hole and what it means. Basically, for many matter in the vicinity of the black hole horizon has the same greens, clearly propagators. It's possible to show that the dynamic properties of this model are coincide with the properties of the black holes. The most known example is that the zero temperature entropy for this model is the Beckenstein Hawking Hawking entropy of the ADS black hole. And then going to quantum chaos. There is such thing as a liponoff exponent, which shows you how fast to close trajectories diverging time and unlike in like classical physics. It's not shown for any quantum anybody systems in terms of state that this exponent cannot be cannot exceed the bounds of KBT over H. And for all, for most all systems, this x point is lower than this bound with exception of two known systems, one of them is black holes, and the other is this like a model. There is a lot of activity in this now, considering a higher dimensional generalization of this like a model. There are exactly chains lattices where you have spiky dots sitting in there on them, or some other non trivial high dimensional realizations. And for now it's the best toy model for strange metals that to probably have. It reproduces a lot of benchmarks of these things and when I say strange metal it's well that's a term used to characterize some cooperates like tides and other compounds. It's common in them that they have learning that resistivity very peculiar compared to normal metals thermal diffusivity and other things. And some of the strange metals are high temperature conductors. So another, well, we can capture all of these properties with some less likely generalizations. And another and some questions, whether you can make these models superconductive. Since the productivity in this studies are non BCS types, but unfortunately for now it's the same as that there are it doesn't give you high temperature conductivity. The temperature is in fact reduced compared to be says but who knows that this door is not comfortable. Let's go back to the conformal regime and its equations. We can. For now we can see that, well, if we look closer to this two questions we can see that there is not one solution, which I showed you before but actually you can do, you can reprimit rise. There is no time with any not only differentiable function and you'll also have another possible solution. So actually there is kind of a manifold of solutions, which has this general form. And for them, they all, well, this set of points remains invariant under such transformations, and it belongs to the group of conformal transformations. Therefore, this regime is called conformal regime. And this is true both for my runner permeance and for complex or otherwise, in other words charged permeance. But of course if you have charged into the system then obviously you want symmetry is also preserved here. So that's one of the applications of the conformal symmetry. It's, you can do this reprimitization and you instantly get final temperature solution for this I came on without doing any additional calculations just. Sorry, I cannot interrupt you. So when you say, when you talk about charge, S Y K, do you mean when you look at the S Y K model for permeance or bosons? Basically, they're usually two types of almost identical types of S Y K series. Some of them are about my runner permeance and other about complex permeance and it means just permeance is electric charge. But it does not appear. I mean, it doesn't seem to appear in the model. You can have here this thermals and you have chemical potential. So basically this model gives you just charge for months. Okay, so there is here that you want global symmetry. I mean, can add some like I want to show it here. Yeah, maybe can go to this slide. You can do this transformation on your permit operator. And so the solution remains in the variant. This one corresponds to the conformal transformation. And this one comes from the you once in this one. Yes. So the model is always you want to. So far, yes, if we talk about charge for months, yeah, if there is charge to talk about. But, so I told you before, we get this regime we made the one important assumptions, we neglected this term in the effective action. While in principle, this term is of course present and it breaks both of these symmetries. So before you could have any fluctuations without additional energies and now each fluctuation will come with some finite cost. Okay, so just just to understand what so what you're saying is that if you're looking at any swk model. It is actually so what you're looking at just a million. It is actually you want invariant, global invariant. However, this is actually only the case, only if you neglect this extra. Yes, exactly. Time dependent part of the kinetic term in the action. Yeah. And this is this term corresponds to this. Yes. So you have goldstone malls in your system now. Which, well, you have two different types of goldstone malls one or five feet field and then for a field. And you can write effective Hamiltonian or sorry effective action. Counting for this fluctuation for fluctuations. Fortunately for us, these two terms factorize, there will be some higher terms about the suppress this one over and and is extremely large number. So they neglected. Constance K and M adjust of the order and over J and therefore we come to the MG scale J over M, where this symmetry breaking becomes important. And this entity is the short so called personal operator defined. But, well, actually both these terms are nothing new. We all know them from other problems. Like for those who familiar with the column block eight. Problem you can see that it's exactly the same term as appear there and one of a key is placed the role of the effective charging energy. So, for charged firmness like a, and this regime, you naturally have the appearance of the column block eight. And the second term is also well known. It's actually well known high energy physics. So it accounts for fluctuations of quanta fluctuations of metric near the ideas black holes. And so it's, well, there's an addition in my opinion is just additional reason to be amazed by this model, since we not only capture classical gravity but some quantum, quantum collections to classical gravity. And for instance, this whole discovery started a lot of an investigation of warm holes by means of the couple of just like a dots. So when we're staying in the short side regime, we need to account for the different fluctuations and average green function to see how it's normalized by the fluctuations. It's easy to show that the answer factorizes so there is one term which comes from average over five fluctuations and one over after we integrate over. In principle, there will be some other kinds of other type of locations, which are also loud, of course, it's like you always have some goldstone malls and some mass malls. I usually sometimes refer to six malls, but they have a very small comparing to this one to the goldstone malls in this case. So they usually looked as they have smallness of while one of them. And the first multiply here is the coolant later which is where we'll start it last, I guess, 25 years, the problem with the coolant block eight for zero temperature from zero temperature it looks like this. And at fire temperature to be something more complicated, but the idea is the same at large key you have explanation of suppression of the correlations and it goes to one when this argument is worth more. While Schwarzson correlator reproduces the conformal regime at small times or what's the same sufficiently large energies and while so when we stay here. But below this energies. The green function is factor normalized in this way. And so actually, there is even one more regime, which appears that even lower expansion small energies, or you can say that at very, very large times. The body until now knows how to deal with it. It also known that this regime should require some debt. And there are very good arguments why this regime should capture genuine quantum gravity regime of the black hole. So here I go to classical black holes, then the account for quantum. This is just near the classical solution and then we come to basically stream model or something like this. And well now people trying to study this regime of the second model numerically, or there is another way to approach it. So from stream community, try to analyze how the properties of the calls in this regime and applied for the S5 basis. So that was an introduction to the model. There are also useful links, which you can check if you're interested. If you find that this model is actually interesting, and was looking further, such if in the year and kid I for that's where all this story started, and fun fact kid I have never published this results as paper it was presented only during the token recovery and so exist as video recording from this talk. So I think it's safe to assume that it's the most cited video in comments matter and high energy communities now. I developed the ideas for that show dualities for charge complex complex as like a models to charge a plate calls. This is a general nice paper to look into the basics. The story about went in chaos, some small selection of papers about strange metals, as like a and superconductivity. And most important papers for the short story. And there are quite a few experimental really the proposals now for this model, even though unfortunately, none of them are realized at the moment. Well, maybe one of the problems that most of them are concerning myron thermions, and we don't have them to begin with, but this one deals with irregular shaped graphene flake in magnetic fields so I think it's quite reasonable and most of further results actually related to this experiment for for charges like a model. I think it's a good time to make a short store and ask if there are some questions in general we can ask for more details or some additional explanations before I go to the second part of the thermal transfer. Then I will further. I introduce this system will have a lucky quantum dot realized by charged fermions. It has some linear size. Therefore it has finite charging energy. It has spectral symmetry, which related to the chemical potential of the model. There are two leads with applied voltage bars, and the leads. There is high temperature gradient. There is the copying of leads to the dots. It's executed by means of friend of tunnel or front of tunneling. Okay, going to most strict formulation of the problem, we'll have this Hamiltonian was a circuit dot. We have a term corresponding to charging energy of the dot, you can think about it as diagonal matrix elements of four permanent directions density density type and non diagonal terms. And then there are three ferments inside the leads and the coupling between the ferments inside the dot and off the inside the leads. Well, by means how we're, how we realize the model. It's very easy to assume that since coupling constants, there's like interaction accounts are random. The tunneling also will be random. There will be no scousions or average of land is zero while variance is fine. And due to this zero average, there will be no direct tunneling in the system, but there will be indirect on process name elastic tunneling and elastic tunnel. And we also assume that we stay in the victim limit because if the tunneling characterized by large enough captain, then the whole movement for the sake model will be shifted. And it will be very painful to fix this. Well, it can be done numerically but the analytical beauty of the model will be supported. The charging energy as we discussed further always is present for this is like charges for case system, but we have also explicitly have termed this charging energy. So the total charging energy is given by the sum of these two contributions. Sorry one precision. When you take the average of the non die. He did the average of other realization only the average of other sites. You can, well, lambda I can be anything with any sign and any sort of varying. But when you take all when you take some of lambda and average it then you get gear zero basically it means that lambdas also Gaussian Gaussian random. So, do you mean that when you the average you some of the I and then divide by the number of sites, or do you average over the realization. Let's say it's rather realization realization. Thank you. Yes. I have a question like I understand the model correct me like the the coupling in the model are also random. This couple, well, inside it's okay we have random interactions J. And the Gaussian. Yes, the Gaussian and we also have Gaussian couplings lambda I. And the okay. Okay. So, let's look into a series of thermally transport. Basically we have radiant of voltage and temperature. There will be electric current and heat current in the system. They're given by this simple formula. And this current depend on transport coefficients, which is electric conductance, they're electric efficient and thermal conductance, which is connected to this large, large game. And of course there is a thermal power, which is just relation ratio between these two. The current is easy to get simply using the Fermi golden rule. We have here density of states inside the lead density of states in the adult. And difference of fermi distribution functions for in the lead and in the adult. So these two formulas, we can explicitly get all the transfer coefficients and the express here in terms of the team matrix, which defines all the transfer properties of the system. So in principle, what we need to do just to find the team metrics and then compute this integral. However, it has two principle deeper and different kinds of contributions. One comes from elastic growth tunnel links and the other from elastic tunnel links. There is of course some high order corrections, but since the return limit, they can be neglected. So usually this term defines all the transport, but as we see later, at some cases it's completely suppressed. And then as an elastic term defines our. So the correlation functions here, correspondingly two point and four point is a key relays. They all factorize in the same speed as I showed above into column part and just like a part, and the same happens for the four point part when we have four point column which takes every quantum mechanical average of our phase fluctuations and four point functions for the SPI key elements. So you have electric conductance, which is the storm of elastic and elastic processes. All of them are shown here. Well, if the temperature is much higher than the energy of the charging, well, the effective charging energy, then the conductance behaves as one over square root of T while actually at low temperatures. So we would want to expect that conductance should be exponential suppressed by the exponent of the column created. But here, the inelastic proton process come to rescue and the conductance actually retain its behavior. We are about here in the formal regime while it's gives this three over two power and now we can consider thermal thermal power, also known as a big efficient. And for large enough temperatures, thermal power of this like a model is constant and it's four pi over three times spectral symmetry parameter. Basically, how far your system from particle whole symmetry point difference cinema power as well as thermally deficient directly proportional to this parameter. So give zero if you stay in the particle whole symmetry point. But for accounting. Well, when you go to lower temperatures below the cool effect cool block eight energy. Of course, all direct tunneling process expansion suppressed, and you need to one would need to consider elastic tunneling process. So they also it turns out that they also financially suppressed. So both of the process of this exponential suppression and it turns out that elastic brought still dominated a bit over the elastic ones. So we have this exponent exponent with some perfectors in both regimes below the energy of the cool block. Also, very straightforward to get a coefficient, since we know thermal power. What's the implication of the result. We have a selective cool block eight, where some transfer coefficients are suppressed and remain final, while the other ones, namely non diagonal ones. Extremely suppressed. It's not a hard case, but extremely unusual. I think that's the first example of the ceremony electric block eight. When GT suppressed, while G and copper remains sexually. We have a little structure. And well, even though there are some reports of so called heat cool block eight where the conductance routine remains unchanged while thermal conductance. But he is the principal knowledge in the thermal power reduction. The conductance in this situation can be approximated as a DJ simply because it will normal has another 10 proportional to square root of thermal power which is extremely small. In this case, so it always plays the same way as electric conductance up to additional power. And again, we have elastic process of high temperatures and dominating elastic process, which give you a power dependence at low temperatures. Some what's, you know, can discuss some applications. These results, for example, there is a so called current formula, which puts connection between thermal power and entropy basically says that thermal power is entropy per particle. This formula is usually approximate and walks in the thermodynamic limit of transport. But what's really interesting in the systems with conformal symmetry. Just, for instance, as like a system. This relation is exact. So, an interesting consequence is kind of go to zero at least very low temperatures and measure back in Stein Hawking entropy directly looking at thermal power. The answer is no, because even though this ratio relation remains perfect high temperatures, when we go with the region below. It's a EC so the region of cool and block eight. This relation is broken and entropy reminds fine. What's interesting, without cool and block eight this rate this relation would be true even into the Schwerzen regime where the conformal symmetry is broken. But due to the cool and block eight term of our goals to zero while entropy remains fine. There is a question. Yeah. So, is there any physical meaning of constant difference at high temperature is convenient or what. Is there any physical meaning of constant difference at high temperature. Is there any thermal power or what maybe you can meet the person. And it's one subject here and I don't know. Yes, I will give the right to talk so maybe you can. You can talk. Yes. Hello. Hello. It is bigger than the time of our high temperature that seems they are constant difference. Okay you're interesting to why it remain while it's called white constant yes. Constant temperature and there is a goes asymptotically to this constant value. And both don't convert the same. You repeat your second question. They are converging different values, but not converging them like zero or something. Hello. Yes. Basically, there are several arguments for kind of meta theory. We just have, well, the power is just entropy per particle. And in this, in the SPA case system, it remains kind of the same. Such temperatures limit. Basically, I remember that the system has conformal symmetry. So it, the race, the current formal works here perfectly well. And it captures the fact that that constant thermal power corresponds to constant and back and stand how can entropy of the black hole. So that's the question. But yeah, it's, it's, it's going to be very, very loosely shown for the dust is like a charges like a that thermal power is four pile or three. And because it's connected to the entropy of course, the SPA case system and of the contract dual system. And of course it has some correction, which behaves like this, but asymptotically it approaches fine value. It means that the ratio of these two coefficient will be constant at very large temperature. So there is always some conversion between electric between heat and electric current. In the same line at this, just want to confirm that what you say is in the previous plot, your result would go to the dotted value as you keep on increasing key. I mean at this point, it seems that the result is different from any kind of looks like really constant. So it approaches here slowly, but also, well, it's also, there is also, let's say some numerical artifacts in play because we can see the finite values of N and some not not too small values of lambda, just to capture physics. In principle, when you, when you above certain temperatures, there is complete or non elastic contributions are completely suppressed completely small. But here you will have some small interplay still between two contributions simply due to this parameters. Okay, just one more. How, how do I mean, what does it mean that as you see that your absolute value is lower, you're more closer to asymptotic value. Does it mean something. No, I think I can just put it here and check maybe. I mean, I just wanted to see that the red curve is slightly still more far away but then yeah it's the, it will reach there so never mind. Thanks. So, well, for us as proportional to actually asymmetry and red line for that line, it should be twice larger than the blue one. So the difference because epsilon also exists in this term as well. And it takes a little bit longer to rate. But yeah, actually, even with these parameters you can consider only elastic process and you get just perfect to perfectly reproduce this analytical law. But when you add the elastic process to account for this regime. It spoils a bit this large temperature synthetics was conformable. Thanks. There's another interesting fact is that will change to look what happens to the Widman's Widman France law. This law is just fighting all metals and some semi metals. Basically it says that the ratio between heat conductance and electric conductance is proportional to temperature and the constant number, which is known as Florence number is five square was the equal in some systems this law can be broken. And in S like you, if we neglect the cool block eight effect. The whole structure of the middle we demand France law remain the same, but the ratio will be slightly different. It's now by square one five. If we reduce the cool block eight effect, then the law will be completely broken, but we still can define the law in this ratio is zero temperature limit of this relation. And it will give you pi square over two exactly in conformal regime and almost by square one two international. And here are some additional links. That's our paper. Some results will of course this work based on previous results in quantum transport. That's charged conductivity in conformal regime. The charge conductivity in the short sound regime with accounts for elastic process for this charge conductivity. The address of elastic process in thermal power for the thermal power, mainly in conformal regime with some discussion about the Schwarz and additional paper which I want to highlight about the comment later on how to evaluate it because it was a long story for more than decade when different rules were getting we're getting different results and this paper kind of settled the whole story about the potential suppression and the right numbers in the form. As conclusions to my talk, I want to highlight again that we found something principle new. We call it thermo electric cool block eight. More no transfer coefficients only diagonal coefficients survive the low temperature regime in this as a key model, while non diagonal goals are completely suppressed as we go to low temperatures. And that happens due to crucial role of elastic tunneling, which say diagonal coefficients, but cannot do it for non diagonal coefficients. Because the four point cooling correlator for diagonal ones. Well, it gives solid order of the contribution from the cool and later in this case is a constant, which completely fine, but when you can see the JT, this leading order gives you exactly zero simply by arguments and you need to go to expansional small collections. And it turns out that there is no advantage in an elastic process of elastic ones. This particular case. You also see that the cool and block eight breaks down telling formula and to change the organization. And with this, I want to thank you and I'm ready for questions. Thank you for the talk. So, are there any questions and there was a lot of questions already. So, are there any questions. So, like, I have a question so I mean, just to roughly understand like at this result like you considered the system but coupling to be random right. It's just random Gaussian, and you can do this average of them. So, despite, well, the whole idea with this like a, it sounds terrible that you have all total interactions and all of them random of them strong. So what to do, but when you take this other average, you can get pretty nice Gaussian dependence. So you have, well, due to the fact that the variance of this coefficients is finite and constant. You still can do all this analytical results. The same idea works for transport. When. Well, it's not a new idea actually. There were some papers so I think, was one of the scholars about random quantum dots where they showed that when you have even when you have zero. If you have zero average tunnel, you can get fine tunneling due to this variance, not zero variance. So you think that I mean this. I mean, if we consider this coupling from any other distribution. The result of the same. Well, well, first of all, if you can see the wall. The coupling is Gaussian. So, the average is even but you can play some bars and say that you have some finite zero tunneling and distribution of coefficients around this, not zero. But then it will give you direct tunneling process which will dominate over all indirect. No, I'm saying that if there is no direct on anything but but I take this coupling from other any other distribution. The result, I mean, the ratio that you said, the law and ratio, they do the change or do you have an idea. Like how sensitive are those results to how you take the system but coupling. Yeah, you mean just simply change my constant. No, it's, it doesn't change. Okay, okay. This well you can put various lambdas. Because basically, you can say approximately the well. So, for example, let's say that cup is proportional to lambda square or lambda power four and J is proportional to lambda square and they just cancel each other. Okay. And, and another question like I did not understand like when you when you have your connecting the system to do. And then you're always working on the small temperature gradient. You mean, okay. My system and some temperature difference. You have. Yeah, I have my quantum dot and I have leads with some temperature difference so there is heat current. It also generates current and two questions. And yeah, and the DT is your what you're always working on the limit where the delta T is small. Well, I am working in linear response theory. What you do here you basically you put this difference to do zero and look. Okay, okay. Yeah, yeah, J is basically I over delta the limit of the typical zero and the same project in the limit of that you go. Yeah, it's a fast one. Okay, thank you. The other question. We have a lot of time. So, since I have no idea about these systems and just trying to understand that the figure that you show your small absolute is related to some asymmetric spectrum spectrum asymmetry. You do not keep your density of so the charge density fixed right. So your new is not really because in the very first slide when you introduce the the introduce the Hamiltonian you said we forget about this new, but that's just because it's a diagonal term or you keep your density fix. Well, we forgot it simply to describe the whole idea of this work here was what's going on there, but in principle if the system has chemical potential, of course it's present, but it turns out that in conformal solution. You can actually add it as exponent to your greens function here and in the short some regime, there will be some corrections, but they all account here, I did this extra slide. That's your conformal greens function. You can see that there is a theta angle here, which gives, which is a way to parameterize this spectrum asymmetry which I talked about. And what is spectrum asymmetry. Well, you have every charge in your dot. Zero zero if you in the particle hosting three points. Yes. But if you away from this point, it's not zero. And then theta has some final values. And then as consequence, epsilon also has some things that is so it's going to some way just how we're reprimit rise. Okay, thank you. Are there any other questions. It's not that.