 Second, I want to thank everyone, every brave person who has enough courage to stay here to the very end, to hear to me in person, thank you a lot for that. And I'll talk about nearly the same thing that Bremy told us an hour ago, maybe. It is about the superconductivity in dilute polar metals, especially in strontium titling. Well, I suppose that Bremy told everything about the philosophy, about the stories and I only have some specific numbers and specific effects. And so let's go. Yes, this is my work with Michael Fegelman, and I will mostly talk about it. So I will briefly go through the main properties that would be important for us, because a lot of things was told about the strontium titling ideal in this country. Well, there is two main, I suppose, interesting things. First, it is peroelectric, but it is close to the peroelectric transition as we were talking. And as a result, it has a tiny gap in the spectrum of transverse optical components. And as a result, it has a large electric constant and the column interaction is suppressed. It is like over the 20,000. This is the first part, the second part. It is an insulator, but you can dope it either with niobium or reducing oxygen. And it became a metal and even superconducting metal at very low concentration of three careers. The Fermi energy is like one milleopetite, and it is even less than the gap itself. So naturally, when you start to think about this, you notice that first of all, okay, there are really small Fermi energies, really dilute regime, small density of states, that sounds not good for the critical temperature and superconductivity. But then you think about the second thing that, okay, we have a really, really small gap in optical phonon spectra. And as Aviclain persuaded us, one has to think about the interaction of phonon or optical phonons with electrons. And the next you think, okay, I have to think about the optical phonons. The longitudinal phonon opening was considered by work of, I suppose, in 2016, and there was no luck. So you're on the left with the transverse optical phonons. Okay, so what is the simple kind of interaction that you can write? This is basically the square interaction. So we consider such a vertex where the electron interacts with two phonons, and so we calculated the first term expansion of free energy. And so we obtained the effective potential of two electron interaction via exchange of two optical phonons, transverse optical phonons. And so the main result is described here at the most dilute, oh, I suppose it works, at the most dilute regime, we obtained a static attracting potential between two electrons. And so due to the giant's electric constant, it is even bigger than the Lombre-Pulsion. So firstly, this mechanism was proposed by Nga in 1974, and Wander Marl group was thought about it in, I suppose, in 2019, yes. But there was no analytical calculations. The fun fact is that that's our papers with Premichandra and Pyrrhus Kolman and Pavel Volkov, they appeared on the archive with the difference of like four days, I suppose, something like that. Yes, and basically we have done the same thing, this is the same diagram. But, so this is, I suppose, basically what I have talked about right now, this is how we can calculate this potential. This is quite understandable, and what one may want to look at is this is an action, and this is basically the first term, it is just a simple optical phonon action. And the second term is the interaction I could introduce to this square by the phonons. Well, so what's next? Next we can recall some old results by Berkman-Medik-Berkudarov, and they calculated in this very, very dilute regime, the critical temperature, and how it depends on the interaction. And again, the main result, this is just some numerical factors, and the main result is that here instead of the bifurcancy, the pyrrhus thermoenergy. So, then we calculate the zero momentum, static potential, I will explain why. I'm really sorry. What one have to keep in mind while working with this formula that it is calculated in the static limit. So, this gives us some restrictions so we can do anything with the dynamics. So, we restrict ourselves to only the static potential, and for now we think about the zero momentum case because, as I will tell you later, we restrict ourselves to the most dilute regime. So, one obtain such result for the potential, and so, once again, eta is some numerical factor that came from the cutoff, and omega t appears in the logarithm, and this is the main, this is where the eta appears in the formulas. So, now we just insert the result into the critical temperature and we obtain the plot on the left. So, as I said, we restrict ourselves to the most dilute regime, so from here to here I suppose, we're trying to describe the oxygen doped strontium titanate. So, once again why firstly because the dynamics about everything else I will tell you. Yes, and so we get the plot on the left, and it looks like we get the plot fits, especially if one will consider the pretty bigger bars from the right, from the right plot. From the binias group or from Francis, it appears on the archive like two weeks ago I suppose. Yes, so, nevertheless, we finally describe the critical temperature dependence on the electron density. And we estimate our only fitting parameter that is sitting in, oh, I'm sorry, in here in J. The lambda is basically dimensionless G. So we have the one fitting parameter theory. And so it is, it has some reasonable number like the order for one. This is the first result. Second part. And at the same time, there are a couple of figments, which are not there and they done the theory work. They tried to explain the anomalous high temperature desk where resistivity and strontium titanate another feature of this material. And they explained it via the simple perma liquid theory, but with, again, with these two optical phonons interaction. And so they again had one, one fitting parameter this lambda and their lambda is 0.9 that is pretty close to what we have. This is the second part. There is, I suppose, Remy mentioned this work. There was a work about from underlaying at all. And so they applied hydrostatic pressure to the, to the sample of strontium titanate and they observed mainly two things first of all the critical temperatures. The critical temperature drops. The second part, the gap in the spectrum of optical phonons prices. So, all that we had to do is just looking at these plots at pressure dependence to the our optical gap and again recollect everything and look at the plots. Again, take a glance at the left plot. First of all, it's the critical temperature at five kilobars drops like several times and the second, I think I want you to not notice that the density of electron density is that corresponds to 0.2% of the niobium, it is something like 10 to the power of 20 I suppose. So, so these plots. So these plots are done in suppose somewhere here. And we're sitting in the very, very dilute regime but nevertheless, we are trying to just make some predictions to see if we will get this effect. We get, we get it. So we also have the drop like several times three times I suppose five kilobars and this is a plot from critical temperature dependence on the pressure for two different critical concentrations but again, these critical concentrations that we're restricted to they are much lower than what those experimental group. So this is still the main effect I suppose the same. This was third part, no fourth part. There is a anomalous isotope effect in strontium titanate. This is basically discussed in the old experimental paper from 1999. In some more recent paper. But the main idea is that you can as isotope substitutes oxygen with it's more heavier isotope and surprisingly in strontium titanate critical temperature goes up and goes up like by the half. This is not what you want to expect in some regular superconductor and we think that we describe this effect also because the second, the second fun fact is that when you isotope substitutes oxygen. The system goes closer to the ferroelectric transition so the optical gap became smaller. And one hand have to think about that and eventually this plot the black points corresponds to the just normal strontium titanate with reduced oxygen so it had three carriers of course but still and the red points they correspond to the strontium titanate with isotope substituted sorry it corresponds to the isotope substituted samples and they're substituted in such a way that red points are sitting in the directly in the ferroelectric transition. So they are corresponds to the case when the gap in the spectrum of optical phonons is closed. It is literally zero so what can we do with this we can again recalculate everything but taking accounts not on the zero momentum case but taking account k dependence and that's the gap is zero so okay yeah I hope it perfectly thank you so one can recalculate again the schedule and amplitude and he will again get the nearly the same logarithm but instead of omega t one having the logarithm k-firm. So this is I suppose the corresponds to the case I was too shy to be honest to steal your plot from your paper so we will just draw it later so it corresponds to the case when the k-firm is the dominant term in all this. So what and then again we can just put it inside the Gorkov-Melik-Berkudar formula and obtain the critical temperature so we get these relations and what can we see there we have a factor of 2.5 at the really the lowest density regime and we have the factor 1.8 at the higher density and the higher density is like 1.2 and 10 to the power of 18 and what do we have here and here the left points corresponds to the case when we have like 4 multiplied by 10 to the power of 18 so it is a little bit bigger than this and the factor there is 1.5 and we have here 1.8 and so looking at the dependence on the concentration one can understand that if we will go a little bit further we will have exactly this 1.5 factor so I suppose that all the effect with the isotope substitution lies also in this fact so this is some theoretical predictions as how critical temperature depends on the on the gap itself you can see that it drops like with factor of 1.8 well now why don't we go to the higher densities because one can calculate this dynamic potential this is the logarithm that you have seen and this is first two terms in the expansion over the momentum over the energy and you can see the natural scales that we have we have the logarithm that is not a function as we all know we have momentum divided by omega t square and we have the energy divided by omega t square so this is where one have to think about because the characteristic momentum is the fermi momentum characteristic energy once again when the density of the carriers goes up then these terms became bigger and one cannot neglect them and one have to think about that if we still can easily take into account the momentum and that was done by premiums Volko this is okay but if one start to think about the dynamics then everything crack apart because we don't have a theory for superconductivity with dynamics because once again we use the formula and they are all written in the static limit this is the first part the second part why don't we go to the higher densities because in the there is a non-trivial three-band structure and there is non-trivial effective mass dependence on concentrations and also one have to think about how different ways of doping affect the parameters of the systems of the system so we restrict ourselves we restrict ourselves to this region to this really small tiny region at the lowest density limits where we have only the superconducting that corresponds to the oxygen reduction and there is no multi-band regimes because the second band starts feeling somewhere there I suppose it's two multiplied by 10 to the power of 18 and so that we can use our simple theory but with this simple theory we can describe again three different physical effects well so here okay so again the conclusion we describe successfully the temperature dependence on the career concentration we describe the isotope effect we describe the pressure effect once again the fitting parameter was found to be really really close to what was found by the completely another experiments with another temperature range and so etc and so what everyone have to do next one have to recalculate the Kodara theory to the dynamic region when we can go to the higher densities okay thank you questions I heard so this is the very last part of your talk you showed this face diagram with yeah with two dots okay and you said that basically take a second bent into consideration and then what you can obtain this he sees that goes down no no no this is why we do not go to the higher densities because we restrict ourselves to this region where Tc goes just up then somewhere there the second bend starts feeling what prevents you to calculate the same with two phonon mediated bending with two bends well once again nothing except the fact that it was my bachelor's work and when we have done everything that we were so fun we were so happy so that was all well one of course have to again recalculate all that at the higher densities thinking about the multi-band structure other questions see anything in the chat I suppose premium talk to all the questions also you understand you correctly the dimensionless coupling constant was 0.9 1.1 yes under those conditions for example why don't you worry about the generalization of the single particle properties because when you solve the gap function equation you have coupling rate but the moment the coupling constant is 1 they normalize the propagator as well maybe I got the idea you mean the coupling constant electron and electron interaction you already have electrons coupled to phonon yes yes yes they give you coupling and you say coupling is not weak then why don't you renormalize single particle properties to begin with the propagator itself is part of the BCS game the effective message I understand I understand this is again this is again the questions about the renormalizing everything because we have one question yes I understand well again the simple answer is that because we tried this and it worked the fact is that of course the second our thought was that okay there is a lot of things that we do not understand here and we will have eventually think about all those things but the fact is that the renormalizing they are among these things to be done we discussed it but well let's thank the final speaker maybe we should also thank the organizers once more for putting this great thing together thank you yeah sure so I don't want to say much there was wonderful answer to the last question it works so I hope it works and it works this time so from all of us Tigran is here pierces upstairs I'm here how young was here Alex Kamini was here and unfortunately Oscar Wafik and media could not come but anyways they all participated in the organization so I hope you enjoyed it and communicate with each other don't forget about posters that were put online so if you want to talk to these people please do so most of them put their email addresses into chat so you can get them and have a nice trip back or stay here for a couple of days and enjoy the remaining of sunshine and I hope you enjoyed this part of the world and again thanks very much for coming from all of us thank you I'd like to second what Andres just said can you hear me yeah we can hear you I'm never really sure from up here and I wanted to probably I'm the organizer who did the least organizing particularly a great pity that Masloff couldn't be here because he put a huge amount of work into organizing this event but I'd like to call for a round of applause for all the really active organizers who really made this happen and I hope we'll all manage to come to Trieste again in the not too distant future so cheers and thank you everyone thank you I think it's lunchtime Manjare