 Okay, so good morning, ladies and gentlemen. I'm happy to be here. This is not the first time of my stay in ICTP, maybe the 30s or 35th one, but it is the first time when we came all together here to celebrate the birthday of our friend Boris Selshuler. And I want to start from the picture of 1987. It was also some advanced school under the supervision of Abrikosov in Dushanbe. And I want to, the part of discussions of the character and properties of Boris to show very important his part. It was our adventure. We bought 200 kilos of melons and 300 kilos of water melons. And since 1987, I never ate so good. You see, the seriousness of these people, they are just before the military operation. So Boris, through all his life, conserved this interest for good food. And you see now with what interest and pleasure he is studying the menu in easy in Torino. Then he continued his gastronomic studies and then you see, this is in Tuscany, how he has new ideas about how, et cetera, after such tastings. Okay, I think that it is enough for introduction and I would like to pass to the problem which a little bit younger that Boris, but not too much. In 1968, the Aslamazov Larkin and simultaneously Maki, they discovered two effects, two different effects because each group lost the another one. But totally they took an ex essence of phenomenon which was not in the center of interest of superconducting community during more than 50 years because, no, yes, more than 50 years. They discovered a phenomenon which is called superconducting fluctuations. Actually starting from Camillingones, they in clean materials and in low temperature materials superconducting transition is defined very well and the problem of its definition didn't rise at all. But when people after the explanation of BCS of the nature of superconductivity and the prediction and the theory of second order, second kind superconductors by a pre cause of people studied to look for high critical parameters and they started to study low dimensional system and dirty systems. So already in films, superconducting transition was smeared enough, but it was not dramatic. Much more dramatic today where you see for superconducting transition in high temperature superconductors or close to superconductor insulated transition. So it now to say that TC is the half of transition is just ridiculous. So the, goal of fluctuation theory today is practical sense is to give to experimentalist possibility to define such parameters as TC, semi-microscopical parameters of superconductors as TC, HC2 and tau phi. And actually my talk will be devoted not to so new fields like topological metals, et cetera, but to this old story. But in some sense it will be the final cut because we propose the general solution for any physical characteristic of superconductor which can be for dirty superconductors enough that we can be calculated in all face diagram temperature magnetic field. So I want to recall you what are superconducting fluctuations. So close what this is a practically handbook story. So let's suppose that we are close to TC in normal phase and I approach normal phase from above. So Natura non-facid celtus told Romans. So nothing happens abruptly. So system feels in advance that superconductor transition is close. So the long electron correlations of the Cooper pair type takes place in the system. How long? So from uncertainty principle, I can say that this is nothing works. I cannot. Okay, you can see that the characteristic time from uncertainty principle is h bar divided on some energy. What energy I can put in denominator to get characteristic lifetime of fluctuation Cooper pair. This should be something. I know that this Cooper pair will become eternal below TC. So in TC, this time should be infinite. So the only physical combination of energies which I can put in denominator is T minus TC. So the characteristics just from uncertainty principle I can get Gilsburg-Landau time. The microscopic theory provides us by the coefficient pi over eight. Okay, so I will present for future this in the form of dependence on relative temperature epsilon. So it is TC minus one divided epsilon where epsilon is reduced temperature. Okay, now I will speak today about, oh, thank you very much. I will speak about the dirty systems for simplicity. I can do this for clean also. So what will be the characteristic size of this Cooper pair? I have two electrons. They started, let's say from the same point and the time which they have is Tau-Ginsburg-Landau. So in diffusion motion, the distance between them during Tau-Ginsburg-Landau will be square root of diffusion coefficient for electron Tau-Ginsburg-Landau and we get in this way, the characteristic size of Cooper pair. This is Ginsburg-Landau lengths which is more or less CBCS divided square root of reduced temperature. So these are large objects and these objects have, we cannot treat them as a Boltzmann particles. These are waves. So we should consider the superconductor at the age of its transition to normal metal from side of normal metal as the some box full of long waves. How I can, so this is just consideration from the Einstein and Geisenberg relations. Now I will pass to, now I want to use the fact that these Cooper pairs are bosons. So I could write for them boson distribution for their distribution over momenta and energy is inverse. The characteristic energy is inverse Ginsburg-Landau time. So this is T minus TC plus kinetic motion. So I can use Rayleigh-Gins approximation so I can expand this exponent and get that the concentration is defined by T over energy. Then for concentration distribution for to get concentration of these Cooper pairs I will integrate over momentum space but of course I have to cut off this integration at the limits of applicability. Which limits of applicability I should use? Of course the motion of the center of mass of Cooper pair because I have some energy, I have some energy T minus TC. This is a let's say energy of condensation plus I have the kinetic energy of the motion of center of mass. What it is, it is P square over some effective mass which I do not know but evidently that momentum cannot be more than psi Ginsburg-Landau which I found because they are soft and this if you want is super fluidity, Criterium-Landau for super fluidity. So I cut off my integration and I find nice relation. I do not know what is N from such consideration but because I do not know what is M but I know what is the relation N over M. This is temperature and psi Ginsburg-Landau in power two minus D with D is dimensionality. So I'm not able to define concentration from such I'm not able to say what is the mass of Cooper pair but I know N over M. But N over M is a combination which I need for conductivity for Longivian formula for diamagnetic excitability, et cetera. So I have succeeded too uncertainly to cancel out and from such hand waving consideration I can reproduce a lot of formulas. The exact consideration for two dimensional case gives me that N over M is T over P log epsilon. So the concentration of Cooper pairs close to Tc diverges. This is what happens close to Tc in two dimensional case with super fluid density. This is, for instance, Tc zero, this is a mean field result. Fluctuations decrease Tc and already in the first order in accounting hardly approximation you see that fluctuations decrease critical temperature with which respect to imaginary mean field result for Ginsburg number, so there's a strength of fluctuation, Tc and large logarithm of these Ginsburg number. So this is the noticeable decrease of Tc and I can in two dimensional case find Tc from Nelson-Jump and it will be TbkT, Berezynski-Kostrelitz-Towler's transition. What is interesting that the difference between this temperature and this only in first order calculated temperature shifted down by fluctuations differs only for Ginsburg numbers. So this means that if I will perform all procedure of renormalization group, I will take into account all orders of fluctuations, I will change first order approximation only for Ginsburg which is in logarithmic approximation made negligible. So practically just first step of perturbation theory makes me to coincide the renormalized by fluctuation Tc with temperature of Berezynski-Towler's-Kostrelitz. Now I will evaluate several, so which effects we have, I still close to Tc, so all this is still introduction. So having Tau-Ginsburg-Landau-Ksi and this N over M I immediately can obtain all fluctuation phenomena I can evaluate them. For instance, Aslamazov-Larkin, what does mean? I mean you have the opening of new channel of charge transfer. You have electronic system, but the fact, electron scatters on impurities, but cooper pairs, when electrons form this fluctuation cooper pairs, they do not feel impurities. Impurities are integrated in their effective mass. So their motion is ballistic and you can use just Drude formula with instead of scattering time you use the lifetime what means to be scattered or to die is the same because next the further birth of cooper pair then further cooper pair will one with zero momentum. So it will start its life from the very beginning. So if I substitute here N over M which I found two E square and Tau-Ginsburg-Landau I find that Aslamazov-Larkin, so conductivity due to cooper pairs is proportional to reduce temperature in power D over two minus two. And we get without coefficient famous Aslamazov-Larkin formula. Next effect which is beyond Ginsburg-Landau is can be explained in terms of weak localization and Boris and Khmelitsky idea about the important role of self-intersecting trajectories. Imagine that you have some electron moves along some self-intersecting trajectory and another electron moves in the opposite direction. They can form cooper pair and this cooper pair will be anomalous different from just to parallel. So we can calculate corresponding contribution and conductivity in the spirit of equalization. I can say that I have to calculate probabilities of formation of all such cooper pairs until they are coherent. So up to tau-phi, the maximal time from minimum time is scattering time. So this probability is determined by well-known integral of weak localization theory and then I have to take into account that they interact via close to transition. So I should multiply this probability on effective interaction which is electron-electron interaction and here cooper logarithm. So in result, I will reproduce the result of Maki and then Thomson calculation on fluctuation conductivity. Then next, the fact that I form some cooper pairs means that I steal electrons from the Fermi level. So increase of conductivity due to the opening of new channel of the charge transfer plus this strange cooper pairs and Maki Thomson version on the self-intercepting trajectories, all of them increase conductivity but at the same time, we observe decreasing of drude conductivity because you have less, now you have less electrons on their working places. So just using the Einstein expression for conductivity and I allowed to vary due to fluctuations, density of states and diffusion coefficient, I see that I can expect one renormalization due to renormalization of the density of states on the Fermi level, so evidently decrease. Plus something may be related with the renormalization of the character of diffusion. What previously was not discussed. So easy to evaluate renormalization of conductivity due to density of states. You just take into account that instead of the effective concentration of cooper pairs is a free concentration minus two concentration of cooper pairs. You substitute this density which I just found and you find that this correction will be proportional to logarithm. So it is less singular and actually it is necessary, it does not show up itself only if the pair transport by some reasons is suppressed or forbidden. Like for instance, in the transversal conductivity in layered systems. So renormalization of density of states show up itself, close to TC. The goal of my talk will show you what happens with quantum fluctuations. Actually, I'm so detailed in description what happens close to TC to show you the construct when we will go to AC2 of zero. So at zero temperatures, the hierarchy of these contributions completely is different, you will see. So the last contribution, sorry, renormalization of the diffusion coefficient can be calculated on the diagrammatically and it is not singular at all. It is square root of epsilon. So close to TC does not play any role. Far from TC it is necessary to adjust asymptotics. So I want to summarize what we know. We know that Tau-Ginsburg, vicinity of TC, Tau-Ginsburg-Landau is H over some energy. This energy is T minus TC, epsilon is reduced temperature corresponding time is Ginsburg-Landau time and corresponding length is this one, is Ginsburg-Landau length. Now I move to Hc2. So temperature is zero and I approach Hc2 from above. So the question will be established in the following way. How is forming a precoce of lattice when I approach transition from above? Actually, Lev Aslamazov gave this problem to his PhD student in 1979. I was also PhD student, but it was not resolved. It was not me. I had another problem. Nevertheless, when I saw it, I recalled that it was established. So formation of a precoce of lattice, how it happens? I propose again it is possible to write corresponding propagates of course I invented these explanations after calculation. So I won't repeat the same way. So I approach Hc2 from above. Should be some characteristic time of fluctuations. Superconductive fluctuations happens, quantum, but happens. What is the characteristic time? Again, Ginsburg says me that it is H bar divided some energy and this energy should turn zero in transition. What energy I have in my hands? Cyclotron frequency for Cooper pairs. Only what? So if I will use as this quantum fluctuations energy, the difference between omega cyclotron and the current H minus the same value of Hc2, I will have the analogy with T minus Tc for quantum phase transition. And the answer will be delta BCS H bar. This is a reduced magnetic field which characterizes me. There's a parameter which is very and I pass transition due to this parameter. So I see that corresponding quantum fluctuations time will be H bar delta BCS minus one. H, please look, it is always very similar. The role of T minus epsilon plays H bar. Look, I'm sure that you remember what is delta minus one. So look, I am a little bit above Hc2. Below Hc2 is a precoce of latiss. And delta minus one is a period of rotation of Cooper pair in vertex. So I see that I found some time much longer than one rotation. Fluctuation Cooper pair should form, if it is formed at zero temperature in magnetic field, it is obliged to rotate. But rotation is very short. I see that correlation remains much longer than one rotation. So I arrive to the idea that this is what happens below Hc2. So I believe that the superconducting fluctuations from above, they are not, of course, now there is no place for this box with long ways. You have clusters of vertices. You can take, be an experimentalist, you can take good camera and put exposition. Exposition, you can, you have two possibilities to make exposition shorter than quantum fluctuations time or longer. If you will have longer, you will see nothing because all will smear out. If you have shorter position time, you will see these clusters, which are, let's say, precursor effects for future precoce of latiss. You approach Hc2, these clusters grow, and fluctuations become slower and slower. In the point of Hc2, cluster will be infinite and it will be eternal. So this scenario, as I see, how fluctuations happens above Tc. Again, I can have any hands, how quantum fluctuations and size, so size of this, the size of these clusters, I can by hands evaluate a lot of things. The same contribution of paraconductivity, the Nernst effect, diamagnetic effect, et cetera, and compare them with the results of microscopic fluctuations. So I see the diagram in the following way. Here, you see well-known diagram of superconductor. This is a second order. This is a Meissner phase. This is a precoce of latiss. Here happens who knows what vertex liquid. From above, here I have some Rayleigh-Jeans picture of long Ginsberg-Landau waves. Here, I have this cluster of fluctuating Cooper pairs quantum fluctuations, which lead to formation of a big cause of latiss. And here, when I move along Hc2 of temperature, at some point happens, so you have long ways with characteristic size Xc Ginsberg-Landau here. Here, I do not know Xc Ginsberg-Landau or not, but some size, which shrinks. Simultaneously, you have growing, decreasing magnetic lengths the corresponding to rotation of Cooper pairs. At some point, they become of the same order and the long waves break in vortices. So, from the wave picture, you pass to vertex picture. They're interested in the result of our microscopic calculations for different things. We see at what temperature it happens. It happens, more or less, for enumeric relaxation. We see that you cannot speak more about waves at Tc close to 0.6. So, you cannot find this in numbers. This is some numerical result. I will show it. So now, I will pass to microscopic. This is a well-known 10 diagrams which describe contribution to conductivity from fluctuations. So, all first order fluctuation contributions. I show them only to identify. So, to Aslamazov-Larkin contribution correspond this diagram. You see that entrance of electric field and exit of it are connected only by Cooper pair propagators. So, this is a purely superconducting transition. This is Mackay-Thompson diagram. These four diagrams, I will call those diagrams because you see that all of them are related to renormalization of one electron density in presence of fluctuation pair. So, you see that electron meets virtually the electron which moves in time in opposite direction. They form Cooper pair then the first one continues life. So, this is renormalization of densities. And there is a group of diagrams which you renormalize vertex. So, they correspond to renormalization of the diffusion coefficient. Now, please do not pay attention on the, I will show some formula. What is for, I know this is not next slide. So, what is the wave line? Wave line is just the, this is a Cooper ladder. So, interaction of two electrons, one, two, 10, 100 times. And formation, these two particular functions which corresponds to the life of Cooper pair. And it is written here in the most general way. So, for any temperature and for any magnetic field and dynamical, so T is reduced temperature, H is reduced magnetic field. And this function E, M corresponds to M's Landau level, any temperature, any magnetic field and frequency. Corresponding Cooperon can be written in terms of, okay, this is just account for impurities. This psi function has some nice, probably this is a psi function is a logarithmic derivative of Euler gamma function. It has poles at zero and minus integer numbers. And this is important function of the theory. This is the answer. So, the answer you can see, I just want to say that this part is a slommas of Larkin contribution. This part is Mackey-Thompson contribution and this part corresponds to density of states in normalization, this part corresponds of normalization diffusion coefficient. What is interesting, I show this terrible formula for two reasons. First reason is to show you that exists some solution which can be used, I will show what it is next slide. But what is important? Important that this works in all phase diagram above line H2. Are important that these four contributions we discussed in details close to TC. But this expression allows me to get for instance, so here you see some interesting already results. Our belief that fluctuations always increase conductivity, so always red, is correct close to TC. This is superconductor, this is TC. But exist some line where total fluctuation correction change sign. Here it becomes negative and very negative close to quantum phase transition at H2. This is the surface which shows me full behavior of fluctuation conductivity as a function of temperature and magnetic field. I especially turned the usual presentation of superconducting phase diagram to show you what happens close to H2. Look, here fluctuation correction is very negative. And what is fantastic that a Slamazov-Larkin contribution exactly equal to anomalous Maki Thompson and equal with opposite sign to those contributions. All of them die as a T square in the region of quantum fluctuations where when T over TC is much less than H minus H2 over H2. So when the temperature is so weak that is less than the reduced magnetic field here. And all story now is managed by renormalization of diffusion coefficient which gives you quantum corrections log one over H. This result was first obtained, this result, not this discussion, but this result was obtained by Galitski and Larkin. Here you can see the fittings with this formula of some experimental results of Baturian and company. And now I have still five minutes. So I will shortly show you where else were applied these ideas. We with Quartzov-Galitski and Serbin, we did this fluctoscopy of Nernst effect which was in the focus of attention after the experiments of ONG which attributed some magic properties to Nernst effect in pseudo gap phase but then group of Kamranbenya from Paris demonstrated that the effect of fluctuations that the Nernst effect in low temperature superconductor was critical temperature 038, not 400 times, but 5,000 times more than in normal phase. So we have succeeded to calculate again for any magnetic field in temperatures, the Nernst effect and to find all the synthetics and the position of maximum which is very important because today people are interested how to find for new materials critical field analyzing properties of superconductor of FTC. This is a group of Tefer and the same Kamranbenya. Then our attention, very recently it was published one week ago, Attila Riga-Monte and Alessandro LaShafi told us that they observed peak of quantum fluctuations in nuclear magnetic relaxation rate in some superconductor or this yttrium so it was critical temperature 15 K. So we performed here so we can start from the Karin Gallo for relaxation of nuclear magnetic relaxation rate and which is proportional to the square of density of states. So the first evident effect is that density of states decreases so you should have a gap fluctuation gap in W but above TC was observed peak and this peak was debated many years ago we with Mohedran Deria proposed the following explanations that the same localization diagram imagine that here you have the nuclear so electron moves along self-intercepting trajectory comes here reflects from this nuclear change it momentum of spin flip and then interacts with itself in past and this corresponds to Macky Thompson diagram but not those one which was for conductivity you had two loops and interactions here you have one loop like in weak localization so the sign is the opposite. In result you have competition of two contributions one of these strange self-cooper pair and corresponding positive relaxation and opening of gap in NMR relaxation rate due to density of states. In result you have when tau phi is short along so this effect is this process is effective and you indeed can have the positive contribution but when you study this in all along the line H2 of T so you see that there is some point even for tau phi infinity there is some point where this effect is just due to magnetic lengths which kills you it so this is and the last picture which I will show you is beautiful application for tunnel spectroscopy so you have a tip, you have a tip and you have fluctuating superconductor so when electron enters here tunnel in your superconductor so there are two possibilities one possibility is to enter in normal region and you have depleted density of states or other possibility if electron was lucky it during this energetic relaxation length meets some superconducting region. This electron would like to enter in it but it's his chemical potential still is different so it can extract from vacuum the electron of corresponding chemical potential with the opposite momentum form Cooper pair and enter in the superconducting region. The whole moves along its trajectory back, yes and result that you have the large scale gap and then some small t minus tc square t minus tc width zero by singularity which allows you to extract tau phi in principle you can see the surfaces which we calculated for zero magnetic field and here you can see the experimental curves of Benjamin Cecipe which are very similar but they do not see this zero bicep anomaly and that's all I want to show that Boris because I maybe demonstrated only his interest to bind of melons, et cetera but here his academic, let's say with his academic, how to define it degrees it, yeah and finally, nevertheless we do the same but we changed place from Dushanbe finally we finished at Paris but all is the same Boris, happy birthday to you.