 At occasion, I met many people who would become lifelong friends, one of whom was Andre Trubacott, others included Lev Joffy and Anatoly Larkin. So anyway, it's a great pleasure that I introduced Andre today, who's going to tell us about the twists and turns of superconductivity from repulsive interactions. And Andre has promised to provide 30 minutes for discussion and questions. And we will definitely cut him off at the end of 90 minutes. So please bear that in mind. Okay, thanks. It's interesting that Pierce mentioned it because my memory also goes back this year. Pierce and Preve were here and there were the leaders. Every day after the conference, you just waited for orders, where we go tonight. And it was wonderful. You don't need to think about anything. You were told when to go swimming, when to go for food. It was all taken care of. Okay, look, it's tutorial. So the talk will be of two parts. One part will be really for people who are not actively working on superconductivity and so others know what to do with their laptops during this talk. This part. And then the second part, I will tell you something that is more modern and hopefully it will be some exciting about what's going on. So it goes supposed to work, right? Yes. So let's very start with something very simple. Superconductivity is the resistance states of interacting electrons. Resistivity goes to zero at TC. It all started more than a century ago. And normally, colloquium-style talk will tell the story how it started and how by sleeping you can get a wonderful discovery. But the question is, let me start it in a very simple way. What do we need for superconductivity? And if we start with conventional Drude theory, which predicts that the resistivity should remain finite at t equal to zero, the theory is based on on flow. On flow means that current is proportional to electric field, dissipative current, coefficient is proportional to lifetime of electrons. You immediately divide electric field by current and you get the resistivity and everything is fine. Good. But I guess it's even before all the stories started with BCS, etc. It was pretty well known that if by any chance the system had a macroscopic condensate, the macroscopic number of particles in the same quantum mechanical state, this condensate has amplitudes, this condensate has a phase, and there is additional current associated with the gradient of this phase. And this current will be not accompanied by any dissipation and as such would exist at zero electric field. And of course, a nonzero current at zero electric field means that resistivity is zero. Good. So, in this respect, in the simple language, really simple language, once we have macroscopic condensate, we have superconductivity. Good. For bosons, appearance of condensate is absolutely natural, there's boson-steined condensation and so on. But of course, fermions are fermions and two fermions can simply occupy the same state as Pauli principle. However, trying to figure out which one is which. However, if two fermions form the bound states and bound state become boson, boson condense, and therefore all this standard logic means that we need to pair fermions into bound state. Great. Remember, my talk is about superconductivity out of repulsion. So, in order to have bound state, you need attraction between fermions. Obviously, you cannot get bound state on repulsive interaction. You need attraction. So, the question is how to get an attraction. And of course, BCS is a big click here, because besides doing something on general trends, namely that bound state exists for even for arbitrary weak attraction, there was a specific prediction for mechanism that two electrons attract each other by changing something else, a genetic quant of lattice vibrations which are phonons. They're a zillions way to tell how it actually happens. One electron comes in, creates disturbance of the lattice, and other electrons come to the same area, read the disturbance of the lattice, and through disturbance of the lattice, the two electrons talk to each other, and this gives you effective, attractive interaction. And this has been definitely confirmed for many, many ordinary superconductors. I guess McMillan-Rovall is often cited as work that really establishes that phonon mechanism is the mechanism of superconductivity for conventional cases. But as we heard from Kole-Prakofi talk last week, the problem is still ongoing and it's not fully solved. And B-Polaronics superconductivity is still active field of research here. Okay, this is electron phonon case, and it should work. Yes. And the new era goes back to one year or two years before Pierce and I met here for the first time. It's 1986 and Cooperates when the TC went up almost overnight. And again, I will put several talks of this conference just in the context of what I'm going. These three talks are on subjects more generic than just simply Cooperates, but at least in the first, Leonid definitely mentioned Cooperates and I think Lara with Lara Ben-Farter will also mention Cooperates. So Cooperates, then next breakthrough in 2008. Iron tinctites, not the highest TC, but definitely highest amount of nature and science paper in the field of superconductivity. And then, of course, twisted bilayer graphene that was last week and will be this week pretty much subject of discussions at this workshop. And besides this, number of other materials have a fermion materials, rutanates, titanates, nickelates, Kagame materials, for which some topological superconductivity has been at least proposed and also observed in one of the many materials irradiates, including Ketayev materials, uranium-pilurium too, which also shows time reversal symmetry breaking. So this is not a complete list. The question is this, is high TC really relevant for what is called new non-phononic superconductivity? The answer is definitely no. There is one slide which I borrowed some years ago from Geert Bloomberg. This is MGB2, wonderful superconductor, nice phonon superconductor, STC is 39 Kelvin. Let's not forget that the highest TC superconductors are hydrites under pressure and hydrogens still fit STC larger than 200 Kelvin. And the record, I guess, is really now close to room temperature, depending on which room you are. But then the question is what really is relevant. And with this, I will say phrase that not everyone agrees with, that in corporate sirens, nickties, routinase, titanase have a firm material, organic superconductors, I am willing to put twisted bilayer graphene there, but I know that this is not solved problem. Electron-phonal interaction is not responsible for the pairing. By one reason or another, in each case, you need to set the reason, either by symmetry, it doesn't give right symmetry of the order of state or because it's weak. And I guess one good example here is really iron-based materials in which at the beginning of this iron-based era, people like Igor Masin and his collaborators calculate as the best case scenario for electron-phonal superconductivity found one Kelvin. Since then, they move the temperature up, but it's still not 60 Kelvin. It's still order of magnitude, at least order of magnitude. Let me take this as a premise that suppose that electron-phonal interaction by one reason or another is not the mechanism for superconductivity. It doesn't mean that it's not relevant to physics. It may be relevant to physics. Most likely, waterfalls called waterfalls in the cooperates observed in photoemission are due to opponents, but it's not agreed. Then, obviously, the question is simple. We need to take this out. Can I eliminate it? More and I don't want to leave the meeting. Okay. I better do nothing. Which one? Okay. Good. Thanks. Okay. Good. Right. So if not phonons, then we have actually one choice. Electron-electron interaction. And electron-electron interaction is repulsive. So here is the story. Let's suppose that phonons are out and we consider electron-electron interaction how to get superconductivity out of repulsion. And that's exactly what I'm going to talk. And Sri will be talking about how to get pair density wave superconductivity with non-zero finite momentum again, also out of repulsion. So the question is how one can get bound fermion pair out of repulsive interaction. Story is old. Story started in early 60th of previous century and it started with two works, one done by Anderson with Marel, another landow with Lev Pitayevsky, who unfortunately passed away a couple of days ago. But the story, if I can guess, the story has two parts, essentially. First, it's all about spherically symmetric system. That fermions can form bound pairs with any angular momentum, not necessary with angular momentum zero, which we normally call S wave. So BCS is the story about S wave superconductivity when fermions bound in a state with zero angular momentum. The first statement from both groups worked independently was that it's not necessary. It can be any angular momentum. Fine. But the second statement is much more important. That pairing problem decoupled between different momentum components, which means that interaction can be completely repulsive in all channels except for one. And you still get paving because there is just complete decomposition of different channels. Again, this all for spherically symmetric case. Very good. There is one more here. If you look at what happens at large angular momentum, let's not ask what's the number 40 C you will get there. But just for simple reasoning, the largest angular momentum, the largest distances from which this components come from. And then another piece of information that at large and at large distances, column interactions occasionally gets over screen. There is free del acylation because of the sharpness of the Fermi surface. And because of free del acylation, yes, interaction is repulsive. Most of them are repulsive, but there are some regions where it's, yes, I was told that if I keep it, yes. There are some regions where the interaction gets over screened. And the question is, can it be that this over screened, that some component of angular momentum pick up interaction in this over screened regions. And this brings conlatangere story from 1965, which by itself is quite interesting story because there are two papers, one by conlatangere and then detailed papers of the same by latangere only without con, by reasons which I have no idea about. And they did something very serious and very serious. It was not just second order calculations. No, they did quite accurate analysis of non-analytic correction to regular interaction. And basically they said, yes, that the effect of free del acylations, which remember as a power law, compared to general screening of interactions that give you exponential decaying angular momentum component, expansion decaying with the number of m. What they found is the statement that if you go to large angular momentum, and particularly if you take odd angular momentum, 25, 27, 29, something like this, you definitely get attraction. Definitely means that interaction will be negative full square. There was a story about the temperature that they put in, which was 10 to minus 40. But it was interesting story. It was just a really story about how important to be honest. At the time they were doing their calculations. Again, by the reasons which I don't fully understand, I never get a definite answer. It was well believed that in helium-3 pairing should be d-wave. So angular momentum equal to 2. So despite having formulas for any value of l, both in short paper and in long paper, they only put the number for l equal to 2. Get temperature 10 to minus 40. Sorry, 10 to minus 70. Yes, e to minus 40. Everyone laughed and said, great. Best of luck of finding Tc of 10 to minus 17. If they put l equal to 1 into their formula, they get 1 mK, immediately. And this was before the discovery of superfluidity in helium-3. Why they didn't do it? I have no idea. But well, they didn't extend their formulas to where they didn't work, supposedly. Okay. Anyway, what is interesting is it was not done in short paper. It was also not done in long paper. Yes. Yes, yes, yes, yes. Another story is what is the pre-factor. But if you assume the pre-factor is just Fermi energy, you get 10 to minus 3 of Fermi energy. Fermi energy is Kelvin, so you get me together. Okay, but long story short, I guess this was really the fourth example of how one can potentially get superconductivity out of repulsive interaction. And the recipe here is that, yes, it's repulsive, but you can always find at least one attractive component. The story was sort of forgotten. It developed, but it was forgotten. And then it resurfaced again, and you will see how it's related to what's now called spin fluctuation mechanism of the pairing, because it turns out that it's all the same. Yeah. Okay. There is some somewhat shorter version, a simpler version of conlatangir, which was put by Fe and Lauser a couple of years later, already after the discovery in helium-3. Suppose that we have Hubbard interaction. Let's not play the game with screen coolant interaction. Suppose we start with Hubbard. Then the idea is this, you start with Hubbard, and to first order, just Hubbard you give you repulsion in S wave channel, m equal to zero, and give you nothing in all other channels. So you act on top of zero. Then you go to second order, and the question is what you get on top of zero, we get attraction or repulsion. It turns out that in this case, you get attraction in all components with m not equal to zero, the largest is p-wave. Yes. So basic message here is that if you start with Hubbard, if you do calculations in three dimensions, later it turns out that if you do it in two dimensions, you get the same, but it'll work a little bit harder. Then you find the final answer is that for attraction, you get S wave for repulsion, you get p-wave. This is all for isotropic k-square over 2m dispersion, so it's not latest systems that we are interested. Nevertheless, it's good to know that you can get it. Good. So what I will do is the rest. I will look at the latest systems, and let me first start with some sort of negative state. There is no theorem that superconductivity must be there. You well may have a latest system in which there will be no superconductivity, no matter what, down to equal to zero. So there is no, because different channels do not factorize any longer. They're normally only finite number of different one-dimensional or two-dimensional representation of the corresponding group. For example, for square lattice d4h, you have four one-dimensional representations, so only four different components factorize as or not. And so you're only searching whether it in one of the four, you can get attraction. You may or may not have it. But you will see that the number of studies of in spirit of con-lattinger give you at least right symmetry of the order parameter in the system that people were talking about after 1986. Question? Yeah. In a crystal, a very low filling, doesn't it become effectively like a continuum system? Oh yes, absolutely, absolutely, absolutely. Yes. But you said there's no theorem. But then the question is this. If you are, of course, if you are extremely low filling, you get TC, which can be very small, then you're going to latest system. You start putting in latest effects and you are not guaranteed that latest effect will not kill you what you get at very low filling. So in other words, there's no guarantee that what you see at very low filling will survive, say close at quarter filling or close to half filling something like this. And we know examples. In the square latest model, if you start at very low filling, you don't get DX square minus Y square. You get actually either DXY or something else. And then that component disappears. And then DX square minus Y square appears already at some finite filling. So it's interpreted. Thank you. Yeah. No, there is no question. The question was either rigorous statement that if you have repulsive new Hubbard model, then S wave superconductivity is not possible. Okay, good. Let me, it will be, if Pierce will allow me to go for two and a half hours, then it will be last slide in my talk about this. But so far, let me quickly give you the answer. If you want to combine strong Hubbard you with electron phonon interactions answers, there is a possibility to get S wave superconductivity. If it's only Hubbard you, I don't know. I think there's you cannot get. Yeah. Okay, so what I will try to do, I will try to say let's assume that we screen coulomb interaction and we do it on a lake, but even on the latest, there is analogy of free data oscillation. So I don't want to go to large distances. I want to go to large angular momentum because there is no such thing as angular momentum on the latest. I just want to see what you get out of there. And this stuff is supposed to work. Yes. So cooperates. Nobody wants to talk about cooperates because it's a very complicated story. So let me go to either this region or the same region on the other side, when cooperates are supposed to be nice metals with latenter type for the surface, and I will do some very simple so really for the next 10 minutes it will be quick analysis for same analysis for several petitions. So I want to take this for my surface and do something simple. Introduce dimensionless coupling. Let's assume that you start with Hubbard U dimensionless coupling is U time density of states without any more at the corner density of state is the larger simply by Fermil fermiology. So I want to introduce patches at the corner call it one and two obviously when I change K to minus K the same patch. So I want to play some very simple game instead of splitting interaction and angular momentum blah blah blah I just want to introduce two quantities interaction inside the corner. Let's call it G one and interaction between the corners are between patches call it G two solve for the pairing two by two problem what I want to do I want to see what is the pairing what's the sign of the pairing couple by construction here you need coupling negative to get attraction positive coupling means repulsion. So I have two channels you have two couplings solve two by two meaning that you solve for superconductivity in this corner and that corner meaning two gaps so linearized equation for two gaps there's are two couplings that you get G one and G two are both positive so one coupling no question is positive absolutely it's repulsion it's S way but the either is well it's a different between these two guys and again if you do Hubbard U then to first order you get nothing you get in the second channel you get zero then you again play on top of zero great let's play it up top and zero let's do calculation to order U square I just put one diagram because everything else cancel out also one need to check this and there is an answer that came out I think already pretty much in plain louder paper if you read carefully what they did that interaction at large momentum transfer becomes even more repulsive than interaction at small momentum transfer will try to find out physical explanation for this in 10 minutes but right now it's just a logical statement you do calculations you find that yes repulsive interaction becomes even more repulsive but the one that finite momentum becomes more repulsive that's the one at small momentum but look here this guy becomes more repulsive than this one so mdb becomes negative negative means attraction so for more repulsion you get actually attractive interaction in some particular channel what is this channel quite obvious you solve for eigen functions you find that the gap changes sign between one and two so it's sign changing gap between patches combined with the fact that k and minus k are the same point obviously have four points where the gap must go through zero on a firm surface it's a d wave and again without discussing cooperates very much this is the work for which buck the price was given arpe status of superconducting gap which clearly shows doesn't show it's arpe that the gap changes sign but it shows quite clearly that the gap goes to zero along that very good so iron based very quickly the same story pretty much it's a zoo sorry first of all face diagram same players magnetism at zero doping superconductivity at finite doping dope either whole by whole by electron there is also isovalent doping there is a variant of the system which is superconducting already at without doping so variety is there look at the terminology it's a zoo many different well many mean five different orbitals and of course a zoo of states but as everyone who worked in this business knows that if you zoom into what happens near the firm surface you get nice whole pockets you get nice electron pockets whole pockets are in the center electron pockets are in the corner so you can play a simple oversimplified of course game when you just put whole pockets whole firm surfaces in the center and electrons in the corner and then you get the same stories and cooperates only when instead of patches I now have firm surfaces so I get the same story I can write down interaction within pocket and interaction between pockets same story again as it was before patch the word patch is replaced by pockets so it's really same story one coupling is definitely positive another well depends on which interaction is larger you do the same convalescent analysis and again you find out that the second order in you you get more repulsion between pockets than within pocket and because of more repulsion one of the coupling becomes negative and you get again sign changing solution which will be simply what's called s plus minus why it's s because you make a circle around each pocket nothing happens with the gap so it's s but it's not conventional s it has the changes signed between when momentum goes changes by pi so it's s plus minus of sign changing whatever question yes aren't you promulgating an urban legend here the urban legend of s plus minus if you take many of these systems you have some without pockets of holes some without pockets of electrons and in your own papers you know that the ratio of two delta over tc seems to be the same for all of them which wouldn't be the case if you use a different mechanism for each particular configuration of electron a whole pocket so is this not a very and it's a very nice story but it's very unlikely i would not say this because as you know in every election there is a majority that decides and there are extremes on both sides the examples that you showed are extremes yes you can go to the extreme when there are no electron pockets you can go into extreme when there are no whole pockets but in this extreme nobody actually measured to delta over tc to the same way as it's measured for conventional materials so in the extreme i'll be a little bit more serious with dancing in the extremes it's entirely possible that for example in one extreme as louis telfer keeps existing you can get d-wave instead of s-wave and we know the reason but if you go into systems which have both electrons and whole pockets then let me do show you two pieces of evidence that it's s plus minus and again we can discuss evidence but let me let me show it again i'm talking about systems with whole and electron pockets there are extremes very high electron doping or very high hole doping whether it's only whole and electrons but in this extremes tc is one kelvin and i'm not gonna be about that materials with 60 kelvin those cases are on the plot in your paper also and they have the same high value of around seven of two delta over tc we can look at it later but it's a paper with your your co-author and it has pure hole pockets and pure electron pockets okay yes i i know how to defend uh in uh let me put in one place in a situation when you have say only electron pockets there is no laser arpec in this situation you cannot reach electron pockets with laser arpec so i don't think that you can really accurately say what is the delta over tc but so those cases shouldn't be imploded in your paper let me check carefully what is the paper okay all right let me check okay very good okay so now piece of evidence one is simple how to prove that the system is s-wave not d-wave well do arpec and look at the firm surface this is the original arpec from i over group this is kaminsky group this is laser arpec from uh chikshin unfortunately also passed away recently uh and uh clearly it shows this is really very accurate um arpec data which includes also small another firm surface that i didn't talk about uh and it shows that the gap doesn't change as you make a circle around the firm surface this is s-wave i cannot imagine that this is not s-wave from this data but this is not to say much s-wave can be anything it can be can be due to phonon how to prove that it's sign changing s-wave this is a more sophisticated analysis and there were several evidences let me show you one it's the same story as with cooperates in 1965 1995 everyone was talking about this wonderful resonance peak in cooperates uh that pumps out at four on the MEV in YBCO and the same happens here i mean that if the gap changes sign between at some momentum different and you ask what superconductivity does to spin response at this momentum because of sign changes there is a game of coherence factors and you get a resonance peak below two delta this what seory says uh dachs kalapina and tom mayer did calculations yeah you remember with caution of the calculations and this experiment from um i've doubled your group in germany and you clearly see that this all is the momentum connecting center or pole in electron pockets exactly right momentum you judge by yourself but to me there's clearly a sharp peak emerges below superconducting temperature as you expect for sign changing gap there are evidence from s e m um but uh let's not talk about question on the web yes i missed um can we also consider from far nude i'm sorry and far nude says can we also consider that the rkk y interaction may induce some bound state of electrons in principle yes in principle yes yeah yes in principle yes and in fact i can elaborate a little bit more on this there was a ruchman lee recent papers and essentially the same but again it will not be ordinary s wave okay let me give you the third example before i move to the second part of the talk and just do something it's a little bit more sophisticated i could possibly twisted bilayer graphene although of course the judgment is there it's a story about uh in this case just simple graphene which has dirac points and k and k prime and suppose we managed to dop graphene up to a point where chemical potential goes into one whole point and one whole point is when Fermi surfaces that start at small triangles around k and k prime at some point merge and end up with one large Fermi surface in fact there was there were experiments that are just experiments when you put calcium um and potassium on top of top and below um singular graphene and claim was that yes you can actually even reach this but uh simple argument that's why i present want to present this is the same story about large density of states in some patches of the Fermi surface looks the same story as before but with one interesting twist because of symmetry there are three points three non-equivalent points where you have one whole singularity is a density of states other three are related by k to minus k symmetry and number of interactions is two because of the symmetry so you still have the same points they have one interaction inside the patch another interaction between the patches but interaction between patch one and three is the same as between one and two and between two and three by symmetry so we have three by three problem because three different corners two interactions what you get you get immediately when you solve this equation you get of course one channel is s wave no question about this it's repulsive it will stay repulsive forever but there are two other degenerate channels for which interaction is the difference between these two couplings story is the same as before we start with zero if we start with hub and interaction we need to go to second order to see what happens and predictably what happens is that this guy becomes negative so uh right it was the additional one i said this one yeah and you you gain you get more repulsion at large momentum transfers at a small momentum transfer same story but there is a twist the two are degenerate and degenerate by symmetry as a result if you look what happens you will get two different orders which develops simultaneously in d wave formulation both have four nodes in d wave formulation one is analog of d square minus y square another it's d x y they're degenerate by symmetry and then you ask when they develop you need to write down of course gingburg-landau for this and bryphal coefficient in gingburg-landau but then you ask okay wonderful when they both appear what happens well they do appear with the factor of i or minus i so you immediately get chiral superconductor chiral because it's combination of d x square minus y square and d x y and the superconductor breaks time interval from symmetry and you get it instantly just because you have to the degenerate channel yeah okay it's a it's questions that goes a little bit further than what i'm talking about so i in the it's not me when people like uh ganzales and others extended this to twisted bilayer graphene i think they were talking about harborton mori uh then in graphene they know there are two values and if you have k from one valley minus k is always from the other valley but i don't think that this matters much for this kind of mechanism so i think you'll still get and you know you you have one whole points in graphene so you can then the question is do you have six or twelve one whole point there are two possibilities and two two different scenarios depending on how transformation is from direct points to one whole and then on top of this of course strong coupling effect that may tell you that the dispersion may be completely different but even dispersion that has minimum a different gamma point it still has one hope so one hopes are there so short answer is that it is harbored in mori and it's maybe oversimplification you don't need to do it and two values must be included but two values will not change the story here yeah okay good now now the second part of the talk that this is great but it's too good to be true because in everything i did i said let's take harbort interaction let's assume that harbort interaction doesn't give you nothing in channels which are known as way so you have a wonderful playground you start with zero and then on top of zero second order can give you either positive or negative if you give you something negative then you immediately get a great but in reality of course we never deal with harbort interaction we always deal with screen cooling interaction so it's completely natural to expect that if you that your starting point this interaction with small momentum transfer this interaction with large momentum transfer and you don't it's not rocket science they will say that this guy should be larger they always get larger initial interaction at small momentum transfer so which means that you start with repulsion and once you start with repulsion then immediately it's a question about perturbation theory which is in this respect con lateral mechanism is if we want to talk not about isisotropic system with infinitely large angular momentum if we want to take real lattice system you start with repulsive interaction you go to second order and second order somehow has to overcome the repulsion which cannot do if it's weak coupling so somehow you need to go to stronger coupling an abandoned idea of weak coupling theory and to a certain extent this is a picture for those who remember what happened in the world in 1963 right and let me try to briefly describe three approaches one will be ghost under realization group analysis and I think it's nice approach it's still weak coupling but it tells you that in some situation you can go beyond second order and in fact sum up infinite series of graphs for the ability for con-lattinger effect it's basically con-lattinger taken to incident order second one is also con-lattinger taken to incident order in a different way this is what is ghost under the umbrella of electron both in theories this is what happens when you consider interaction near some density wave instability whether it's spin charge density wave circulated currents pneumatic something like this it's different line of arguments here and the third one again if I have time it will be very quick about what happens can you get s-way out of repulsion the answer is you can get s-way out of repulsion but it will be citation when you will get a gap which changes sign as a functional frequency and it all goes into superconductivity with embedded dynamical vertices in so let me tell about RG approach very quickly it works well for best for iron based materials let me quickly repeat same story you get one interaction which I used different notations here so notations will be back I just didn't change it so let me quickly get it here so the question is what's this G3 G4 is what I previously called G1 G2 and they will be called G1 G2 in all other slides so the story was like this can we go beyond second or and still get still keep coupling small the answer is yes we can there is BCSC or superconductivity for this I mean that if there is some some logarithmic contribution from from the renormalization then we can go in order of coupling then coupling square times log coupling cube times log square etc and at the same time neglect simply coupling square coupling cube etc standard argument for BCSC superconductivity normally collateral normalizations are just normalization the powers of the coupling so we cannot do it but if they're by some reasons also logarithmic then we can try to sum them in all orders and this is what goes into what is called parquetron normalization group approach when you look at iron based materials and you find out that you do normalization in a particle whole channel so what's connected to something you take in direction of particle particle channel and say no it cannot be just simply pairing in direction you you need fully irreducible interaction so you need to collect all the normalization from other channel this would connect to dust at second order you take normalization from particle whole channel and here they are also logarithmic why because one is whole pocket one an electron pocket and it's a story which goes into excitonic oscillator story that if you have whole and electron pockets that particle whole bubble behaves in the same way as particle particle bubble so details aside if you have logarithms in the particle whole channel and have logarithms in the particle particle channel then treat both equally don't tell that one renormalization is important one renormalization of particle whole channel only has to be taken to normalize interaction in the particle particle channel taken direction the particle particle channel normalize interaction the particle whole channel so it's all mutual and you get two interactions now back to normal notations which i uh keep here g1 g2 are interaction in the particle particle channel there is also g3 and g4 two different interaction in the particle whole channel uh great so put all of them together you get a set of renormalization group equations because all of them are logarithmic and this means that i just sum infinite series of diagrams and instead of summing this infinite series of diagrams i know i can write down differential equations what's the result the result is interesting i'm interesting and these two guys g1 and g2 this is a bad guy this is against pairing this is a good guy this poor pairing i want to make good guy more repulsive than the bad guy this was this equation that i'm not making this they just follow up from this equation that you start with repulsion you keep going and now you can do this because couplings are small but couplings times logarithms are of order of one and you see that good guy goes up bad guy goes down you get a scale it's interesting system by itself develops a scale below which it has attraction so it develops if you like upper cut off for any attractive interaction and there is a reason why this this happens because there is an interaction which is in the particle whole channel which wants the system to become anti-pharmagnetic and in the process of becoming anti-pharmagnetic it wants to enhance all interactions at large momentum transfer including component of the pairing interaction and once this guy grows up it pushes up good component of the pairing interaction because this is the one with large momentum transfer while the one with small momentum transfer here goes down while it's not necessary which means that what's the story this is intelligent way to say that spin fluctuations promote superconductivity that just means the present so basically you get interaction which is favoring something else in this case favoring anti-pharmagnetism and it pushes up good component of the pairing interaction how this to compete it's a different story sometimes red line is pairing interaction you see it changes sign positive here means attraction and sometimes it just loses to tendency to magnetism sometimes an eventual wins over tendency of magnetism and becomes the leading instability these are details but basic message is here that I want to go to the message here right so spin fluctuations enhance the tendency interactions that favor spin fluctuations enhance the tendency superconductivity superconducting fluctuations at large momentum transfer intensifies the tendency towards making which actually goes like this that as long as both are not ordered they support each other once one is ordered then it to try immediately to kick out everything else and to fight against any other order which in good physical terms means that competition is good but monopoly is bad as long as the two compete great they promote each other as long as one is ordered they try to kick out everything else now sorry there was a question no no no it's only sorry the question was whether this rg equation depends on the weather hole and thermosurface has the same diameter no the ratio of masses goes into overall factor in the coupling and they just incorporate it into the coupling so you can have one 10 times larger than the other as long as both are circled in the same equation just very normal ready this would be same story sorry sorry sorry the question was about nematicity how the nematicity plays here it will be the same story but you need to extend it a little bit what you need is two different orbitals dxz and dyz then you write down different couplings and you get the same story that repulsive interaction between hole and electron pockets will favor nematic order nematic order this q equal to zero order however that order changes sign between hole and electron pockets like s plus minus superconductivity it's the only nematic order that develops due to repulsive interaction and for this we have this wonderful experiment from japan that shows that yes nematic order has opposite sign on hole and electron pockets so this is well established and then of course again the same interaction that favors plus minus and nematic channel also favors plus minus in the superconducting channel so the game is exactly the same okay great so this is good monopoly is bad we are great here i want to say a few words about yeah um let me go to next slide yeah i already went ahead of myself yeah i will go to next slide and we'll answer this question yeah oh the question was i use the word spin flotation which i should not because i didn't explain why i'm using this word that was essential as a question right yeah i know that what this interaction does at the end of the day i just need to explain why uh i say that spin flotation promote super superconductivity uh-huh it yeah it very much depends uh what's the window you have suppose you have a large window for rg you can start with weak coupling so g all g's are much smaller than one you go to the regime when g times log is a further of one but if rg develops already substantially up to the scale then log is large log is large means the g is still small so then i can neglect all higher powers of g etc if i don't have this window then you know what the answer will be then everything depends on numbers that's all assuming somehow that there is a window for rg okay let's do the last part permanent both models here i need to answer the question from three because i want to give you one example it doesn't have to be one but this one example is spin flotation scenario what it means we said and i said itself that interaction at large momentum transfer enhances becomes more repulsive this is how i get attraction great this conletting job but i want to go a little bit further take this guy and ask this i have pairing interaction dressed pairing interaction randomized by conletting as usual for super conductivity we need anti-symmetrized interaction we need vertical this is how we split our interaction into spin singlet and spin triplet channel so i can take this guy and ask which component of the renormalized interaction becomes attractive is it in spin channel or in charge channel this i can do because i can take this interaction again and i need to deal with anti-symmetrized interaction for pairing and anti-symmetrized interaction have both spin and charge component and the answer is spin so already at this level at the level of conletting we can say that yes it's spin component of the interaction that enhances and basically spin enhancement of the effective interaction and spin channel promotes you attraction in this particular case d-wave channel so knowing this i can say okay let me now forget about rg rg was a good way to justify calculations now it closed my eyes i don't know how to justify anything i just take ladder series in spin channel sum up this ladder series some use the word rpa with this which is forbidden word in some circles uh and uh then you obtain effective interaction mediated by pi pi spin flutation pretty much the same that you get energy but here you do it without saying the word i want to stay with weak coupling i really want to bring the system close to in this particular case pi pi if our magnetic instability and i want to see what happens with superconductivity and then sometimes the answer is immediate and what you want you took component of interaction which is good for superconductivity and enhance it artificially by saying that the system comes out close to transition into state which is the state with the same large momentum transfer in this respect everything is good we start we cheated a little bit but we definitely get attraction no question it's just by construction interaction is attractive in the d-wave channel but there is a caddy if we want to do this and bring the system close to the critical point meaning to the order in a particle whole channel yes you substantially enhance attractive pairing interaction but there is a competitor the same interaction that mediates pairing also if you view it in a particle whole channel this is the interaction which destroys formulaic and now you have a different story attraction is guaranteed to you but you have a competitor the in rg there was no competitor because we assume that the coupling is weak and we didn't include any self-energy effect here we must include them and this is what was presented last week a number of talks is what will be presented this week it's a story about strange metals and strange metals to me means essentially non-fermil liquid and large cells and again attraction is guaranteed but you have a competitor and the question is competitor wants something different competitor doesn't want superconductivity competitor wants non-fermil liquid so the question is who wins tendency towards pairing or tendency towards non-fermil liquid they come from the same interaction this is just nice flight to show this this is just to say that they're competing why because if you make fermions and coherent you destroy cooper logarithm if you make system fermions ordered and superconducting you eliminate scattering at small frequencies and therefore electron recover nice fermi liquid behavior so superconductivity works against or acts against non-fermil liquid non-fermil liquid acts against superconductivity if you look what happens then this all goes into the story about how to treat pairing at a quantum critical point in the metal and long story short yes sure it's the question was that in the particular spin fluctuation case this is hot sport story which the first approximation is true you can do equally the same story near enigmatic transitions and will be full thermos surface but in both cases there is a competition between non-fermil liquid and and tendency towards pairing and you have much weaker superconducting tendency right so in some sense hot spot determined what happens or if you don't want this go to a situation of stronger coupling when the whole furnace surface becomes hot okay i just want to show you one specific features here that are when you deal with this quantum critical models dynamics plays essential role because first of all destruction of fermi liquid come from frequency dependence on the self-image and at the same time effective both fluctuations of the order parameter becomes massless as the critical point and if you look at the end of the day what you need you need effective interaction averaged over the Fermi surface which turns out to be singular functional frequency with some exponent and three are the question about this exponent let me not go into details of all the stories with different problems and a lot of people working on them with different exponents i want just to show you one thing here that what is a competition between non-fermil liquid and superconductivity in terms of equation it's basically said two equations like this one for the pairing vertex when pairing vertex is non-zero system is superconducting another is for fermionic self-energy when this guy gives you non-fermil liquid then you get non-fermil and very long short story long story short you look at the equation for five sigmas and denominators the larger and sigma the less tendency towards pairing look at equation for sigma phi than denominator so the larger as phi the smaller is self-energy it's competition on a trivial level they compete because they open the term stand and denominator so when you look at this by the way superconducting gaps the one we like to associate with superconductivity essentially the ratio of phi and sigma and if you go to bcs case which in this case means exponent equal to zero and pairing interaction is just a constant at low frequencies you get nice features that nothing depends on frequency self-energy zero and you get a nice bcs but if you want to go to any value of the coupling here you immediately get non-fermil liquid self-energy and non-fermil liquid self-energy means that your pairing emergence out of non-fermil liquid if it emerged very good you ask computer first of all to see whether pairing emerges who wins yes well wait a second wait a second pairing vertex is obtained using fully dressed green functions uh there were talks about syk so this version uh same version of syk in some sense what it means it means that uh you deal with absence of quasi particles but still presence of the Fermi surface the Fermi surface is sharp self-energy is large but it's only a function of frequency at zero frequency self-energy it's zero and i have a locus of point where g minus one is zero at zero frequency so Fermi surface is sharp there's no quasi particle what it means that there's no quasi particle it means that uh this guy in denominator is larger than omega so you start destroying what we normally call cooper logarith yes it's there so all all strange metal feature all absence of quasi particles is built into this formula is built here it's built in syk type models which give the same equations at the end of the day i hope i give i'm not sure that i answer this question but at least the answer is this all non quasi particle features are here but it's a situation without quasi particles but with sharp Fermi surface which i think pretty much everyone has okay uh let me quickly move to the last few slides that i want to show so first of all we want to check who wins non fermi liquid or superconductivity for this we ask a good friend called computer and the good friend just calculates what is superconducting tc if it's zero then non fermi liquid wins it turns out no it's not zero for any value of the coupling gamma tc is finite don't look that it goes to infinity when gamma goes to zero simply because it's unconstrained bcs but it's important that for any value of the exponent it's the same it's it's finite means superconductivity wins in the competition with non fermi liquid great you may say this is the end of the story superconductivity wins forget about non fermi liquid this is what you get and moreover you calculate solve non-linear gap equation this your gap function i really plotted delta not phi here and it behaves completely normally in a sense that it's finite at zero frequency as you expect in bcs and then you know in bcs it's a constant up to some scale and then you put it to zero and here it's just short soft cutoff instead of sharp cutoff the gap just goes down and disappears at large frequency yes right um in this case the question was uh in what parameter range superconductivity wins notice that there is only one parameter here exponent gamma that's only parameter everything else is universal i will show you one slide when i artificially add extra parameter but so far we don't have any freedom it's it's surprising to just like an s y k well in s y k there's a ratio of um number of fermions and number of bosons which you can use as a parameter here there is no even no saturation so in this respect it's completely universal problem and if something wins it wins if something loses it loses there is no parameter you can very accept for this exponent yeah yeah that's exactly what i'm saying yes i'm coming to this i'm coming to this yeah sorry no it comes out from here the question was do i need another scale for sigma no this equation for sigma put phi is equal to zero you get a standard equation for sigma which give you non-fermic liquids oh you need something oh yes you need to there's will be omega naught here right some scale this g so if you i measure everything in units of g then everything becomes universal so yeah g times number determines me yes yes yes so in this if i measure everything in units of g coefficient here is just a number universal number okay uh pierce do i have zero right oh really great then i can talk for 10 minutes yet sure i said okay guys i don't really want to keep your lunches i know lunch is an agenda i will be done pretty soon thank you i just want to make one point here so i hope i can mean these are equations and again keep repeating that there is another model very similar called syk at the end of the day the equation for syk is pretty much the same as this equation and conclusions are the same okay uh this i already said this i already said great everything looks great and now in the last 10 minutes or five minutes i want to say that something is interesting here we face here fundamentally non bcs state and i will tell you what is fundamental yeah uh the mental means this uh let's depart for the normal state let's try to calculate pairing susceptibility and see what we get from pairing susceptibility if we do it in bcs we know how to do it everything is frequency independence so you need to put temperature otherwise you get singularities so what you do you write gap equations find delta is the same here you put extra and then you calculate susceptibility as the reaction to what you put so phi divided by phi naught it's susceptibility we know how to solve this equation it's standard cooper logarithms we sum up series of cooper logarithms and we get wonderful denominator i'm telling you that bcs theory of superconductivity you get wonderful denominator divide phi by phi naught we know what happens there is a critical temperature exponentially small when you go below this temperature and go for example to zero temperature your pairing susceptibility is way negative and when you calculate when you obtain negative susceptibility you know what it means it means instability it means basically that the pole in this susceptibility is in the upper half plane rather frequent rather than in the lower half plane very good let's do it in our case same absolutely the same but we depart from non-fermilic with normal state and we want to look what kind of pairing susceptibility we get look at the equation we get two components here this guy is nothing but fermionic self-energy at small frequencies it's larger than omega because we have non-fermilic with sigma is parametrically larger than omega in other components interaction which is singular and if you combine these two powers you find that scaling dimension in the kernel is one like in bcs so you do get logarithms in the perturbation theory means that if i start putting phi naught instead of phi and start doing integration i will get logarithms like in bcs i this all works at frequencies again smaller than g which i put at the upper cutoff i can do it a little bit accurately but doesn't matter so let's do it there is one difference there is no weak coupling limit the coupling is a constant because every again everything is expressed in terms of g and the rest is numbers and numbers have numbers so i sum up logarithms and i find that yeah same but there is a factor one half one six etc in front of higher powers here and you sum this logarithms you get power low not one divided by one minus log which means simple thing you sum up logarithms you get nothing this is not a cooper pairing problem if you sum up logarithms you get pairing susceptibility phi divided by phi naught which remains positive even at zero temperature at all frequencies you don't see any trace of instability but you know computer tells you there is a finite dc there is an instability so where it comes from well plus this guy again this guy tells you well that let's look at this equation at small frequency kernel is marginal means dimension one which means simple thing that we need to search for the solution in the power law logarithms extreme of this power law but let's search for the solution of this equation in the power law without phi naught then we add phi naught and what we find is that we search for a solution in the power because kernel has dimension of one same power goes on the other side of the equation all we need to do is to match the two sides and obtain this exponent alpha and you get it if you do it like summation of logarithms you get real real alpha I already showed you you get solution in the form of the real power but now substitute and see what you get substitute here is what you get you get complex exponents instead of real exponents you get complex exponents it's by itself it's interesting there is nothing complex here in this equation it has completely real coefficients yet solution give you two complex conjugated exponents and what it means it means that at small frequencies this function is the sum of two complex power and complex power is cosine of logarithms so it's a salating solution this was done by three and others in condensed matter and that's included yeah and this was done by other group who was working in high energy part and doing holographics hyperconductivity and York Schmarin gave a talk at the same time last week and was talking about equivalence between two approaches what it means from our perspective that there are oscillations it means that perturbation theory starting from a constant will never match what I get at low frequencies because again I have a question with positive coefficients when I do perturbation theory I can only increase the value of five I start with a constant at something positive next time again something positive so from this perspective it's only increases from other perspective the function has zeros you cannot match this to function which means that you need to integrate break a symmetry you need to introduce order parameter in order to break low frequency behavior and and find the match and this exactly what happens when you see this there's indication that something that the system is unstable you introduce non-zero phi non-zero order parameter which means superconductivity you will be able to solve non-linear equation but to me this means that this is really fundamentally non-BCS pairing not because of attraction versus repulsion in this case but because of competition so there are another phenomenon that competes and forces completely different reasoning for the pain so this is part number two that everything goes well superconductivity wins competition in a Fermi liquid but Fermi liquid strikes in a sense that it's really not BCS pairing and I keep repeating this this is a phenomenon called holographic superconductivity which uses holographic dual between gauge and gravity and York gave a talk so who didn't listen to his talk it's all recorded essentially about analogy between what we are doing and what people doing holographic superconductivity and gravity do it's the same equation so at the end of the day the same thing okay Premia asks about extra parameter I don't have one but I have hands so I can introduce extra parameter by hand like an syk model this extra parameter is introduced by varying number of fermions and number of both and both go to infinity but their number their ratio is the number so what I can do here I can say that suppose by hand we make interaction in a particle particle channel smaller Shri will say no I don't do it by hand we do it by going from original SU2 to matrix SUN which would be absolutely right that's why I put a factor of n so we can do it accurately but the next result is this what I want to do I want to make tendency towards non-farm liquid stronger I want to take interaction in a particle particle channel weaker and ask a question does it mean that we have a threshold because obviously when I do go between real and imaginary exponents it looks like a threshold problem and yes it is a threshold problem you can solve this and we did it for s for this model and with laura class and we did the same for syk model it's the same story yes making parameter n larger for fixed gamma you go from a state with complex exponent to the state with real exponent and this is naked non-farm liquid ground state so the answer is that if you allow me to add parameter I can get situation when non-farm liquid wins over superconductivity it will be here and I can have a situation when superconductivity wins over non-farm liquid it will be here but I can only do this because it's completely different panic mechanism which is threshold here in this region I have a real exponents which means padding susceptibility is always positive and not oscillating here I have complex exponents it means that I cannot match low and high frequency behavior I need to break symmetry and introduce order parameter to match and it's a lot of interesting math including some of with matches with dense on talk here last week this is the tip of the artwork because the actual phase diagram contains two different superconducting phases and transition between them but if you give me two extra hours or three extra hours I will tell about this in the last five or even less than five minutes I want to talk about the very last part this is related to the question that Kolya asked at the beginning of my talk can it be can we I always talking about non-S wave whether it's Rg whether it's electron boson problem it was always started yes conventional S wave is always repulsive you want to do something extra whether it's s plus minus g plus d1 plus i d2 or simple u8 something non-S wave question is can we get S wave from repulsive interaction and the first glance answer is no we cannot but at the same time at the same time all textbooks will tell us that yes of course take electron phonon interaction and you get S wave superconductivity but the same textbooks are saying that there is also cool of interaction and one of the order of textbook sitting on the first row and he put it absolutely right in the book so the question is BCS story is attraction due to phonons but on top of this we have animal of the room we have elephant in the room sorry and this is electron-electron Coulomb repulsion which is stronger than electron phonon interaction and there is a myth which translated into literature that yes you renormalize Coulomb interaction over some window frequencies get smaller and at some point it gets smaller than electron phonon interaction you get an attraction this myth there is no such thing as attraction and total interaction is always repulsive no matter what because you start with Coulomb interaction you start screening it and phonons effectively change dielectric constant bake it frequency dependent but you will never change attraction repulsion and attraction so total interaction is repulsive and there is a question how to get S wave superconductivity out of repulsive interaction then it's a story that really goes back to Gurevich Larkin fears of story to Tanaka and again it's correctly done in a Pierce Coleman book and a story goes like this in a certain approximation certain approximation you can just say well what happens due to electron phonon interaction you take repulsive interaction and it becomes still repulsive but smaller and smaller frequency this is what I wanted to show here this interaction is repulsive it's positive but as a function of frequency it's smaller at smaller frequency than at larger frequency that's still completely completely repulsive interaction you want to solve for superconductivity for this repulsive interaction and the answer again known and then forgotten and then recovered again is that you can get a solution but this solution will be in something similar to what I told you before only instead of momentum it will be frequency it will be solution in which the gap at small frequencies and the gap at large frequencies have open the time so what basically you do by looking at this interaction you want to take a gap which on average will cancel a constant and then on top of the constant there will be deep at small frequencies and this what is electron phonon attraction so you need to get that on average will go to zero when you integrate over all frequencies and by doing this it cancels the large constant if it can so anyway there's a solution you get plus minus same that we talk about with respect to s plus minus or d but now as a function of frequency and I want to finish with the word there's an element of topology here because part of our workshop is in topology topology here is in a trivial way but still it's interesting that the gap function that changes sign on Matsubara frequency is a topological gap function in a sense that each zero is the center of dynamical vortex you just take imaginary frequency real frequency Matsubara axis is here you get zero here and you also get of course another zero here just make a circular on this point you will find that the face of the gap will change by two pi when you go along the circle so there is a dynamical vortex sitting here it has consequence of what happens on the real axis but it sits there and so in this respect the vortex this is vortex this also vortex they cannot just appear they can either come from infinity or appear due to annihilation due to unbinding of vortex anti vortex pair what is anti vortex anti vortex is a pole of gap not a zero in the upper half plane the gap must be an elite it doesn't have any pole so as long as the vortex is here it cannot disappear because there is again it either go to infinity or it has to do something to accumulate anti vortex and there is no anti vortexes anywhere here so then the question is how transition happened first of all what is transition transition is this I need to quickly go back and say yeah it looks like plus minus solution but there's all of these problems as I try to say these are threshold problems if you take coupling lambda the same and stop and just start shooting this interaction up up up as the end of the day you'll get a negative constant sorry positive constant there will be no superconductivity at all so you need a particular deep of the interaction in order to be able to get this plus minus solution and for fixed coupling lambda if you change the ratio of these two frequencies or you keep the ratio of the frequency the same and change coupling lambda doesn't matter either way you will eventually come to a situation when the system is no longer sustained with plus minus solution and it becomes normal state and the question last question that I want to address is how the transition happens by all numerics and also analytics the position of zero goes to zero as a transition so this point and this point merge with each other very good and then you ask a question uh what happens when you want to go from the other side and see how uh superconductivity emerges long story short it emerges exactly like in BKT theory of supercondu- off two-dimensional superconductivity or whatever breaking u1 when these two guys come close they approach zero frequency and at the same time there are two anti-vortices that live in the lower half plane come to the same point and at the transition there is a binding of the vortex and anti-vortex which if I review rewind it differently as the system emerges in the superconducting state it emerges because of unbinding of vortex anti-vortex piers at zero frequency and then after unbinding anti-vortices pulse go into lower half plane while zeros move along with subare axis so it's really vortex anti-vortex unbinding and what is nice here is that you can write down formulas which I probably don't bother writing here that really shows where the poles are where the zeros are and you can really see this in real time okay so this is the element of topology here okay I think I'm done and let me flash the conclusions and conclusions basically is going back to first part that if you ask what gives eventually superconductivity in case of repulsive interaction it started with conlatinger story it evolved but we still qualitatively use the same wording namely we start with repulsive interaction we want to screen this repulsive interaction from contribution from particle whole channel and it turns out that when we screen them with we most cases we find the channel which becomes attractive it either becomes attractive instantly if we start with hybrid you or it becomes attractive after we do this screening and screening has to overcome the threshold and I gave you example rg shows how this threshold emerges and then all this non-bcs pairing also threshold phenomena it also shows how threshold emerges and final example also has a threshold so in all cases the whole example that I give you there is a threshold for the pairing either it develops or it doesn't develop but it always develops because we do something with the interaction either we create attractive component in momentum space or we make interaction frequency dependent and allow this time changing solution as a function of frequency and I guess with this I'm done thanks that's very simple question of the old generation so the last part of your talk so you seem to say that all this macmillan picture where there is lambda of attraction and then to much of logarithm of coulomb repulsion reduced by to much of logarithm that this is just me and this is not but full so is there any region where this old macmillan picture is still waiting so you are talking probably about strong enough in theory no I'm I need to say more accurately what I'm saying the results if you look at the results at the end of the day the results are all correct it's a question about interpretation and interpretation goes this way if you want to interpret the result as repulsion translates into attraction I will be very vigorously opposed to this interpretation because repulsion is repulsion however if you want to say that this interaction I roughly put it like this and I approximated just like two different interactions as a result two different gaps let's call data one per data two here in the right equation data one is something data one plus something data two and do the same for data two if interaction is repulsive all the time but if I eliminate this guy from the equations means I do rg I integrate over this energies and I write down effective interaction in for data one only this is where you get all this macmillan stuff and this is where you can interpret the result as effective attraction there will be solution for data one and data one will be equal to data one and here it will be minus coulomb interaction you divided by one plus new times log plus electron polynomial this is your macmillan stuff but only after you integrate this part and if you don't this is your interaction so in terms of interaction it's always repulsive in terms of splitting in the two parts and integrating one part you get effective interactions that you can interpret as attractive the slime things that you need to start with you need to deal you don't split into two you need to solve the full equation yes yes but you ask is there a limit when this interpretation is when this result result is right yes of course when there is a big logarithms and that we coupling then the result is right but again only as only after you integrate out high energies all I'm opposing is the wording that goes into textbooks that somehow you convert repulsion and attraction you don't convert repulsion and attraction interaction remains repulsive at all frequency the other question yeah first of all very nice talk thank you are there experimental consequences of having complex exponents yes yes there are if you look at but there are in terms of the easiest experiment experimental consequences if you plot density of states as a function of frequency yes there is a gap and then it starts doing like this and the larger are the more complex are the exponents the larger are this look this amplitude of this location the end result is a crazy situation with a bunch of data functions instead of continuum this is one result second is that exponent here is not one half one half is BCS result the rest anomalous exponent close to the threshold yeah but the easiest is of course this variation that you see in density of states any other questions yes in order of which the hands went up yeah it's probably somewhat relating to the previous discussion it's not kind of the question so you don't have phonons whatsoever just coulomb interactions themselves in the lower frequency subspace they can lead to pairing you don't need phonons for that just if you simply call homogeneous electron gas do rpa you will immediately see equal large enough rs it's already sv superconducting which means phonons are not required for the frequency dependence which you are saying yeah this is uh yeah this is what you are saying is related to this plus morning mechanism of the pairing essentially no no I mean that you start getting I agree with you so uh I have to be more careful the thing if you you're saying that even if you start with harbor and start tracing it you start picking up frequency dependence and if you are lucky you pick up frequency dependence when the gap is I don't know how generic is this but I agree with you you pick up frequency dependence and it's not guaranteed that you will get a minimum at small frequencies sorry that you get a minimum it's right right but I guess I guess but again still the interaction is the largest at zero frequency and then it goes down as a function of frequency yeah but not small omega not q right but then you go down and it drops to the complete screen answer it's very small now it's and but very large at small at small frequencies it's still very large at small frequencies and very small at low frequencies oh this what you're saying it's unscreened for finite frequencies okay okay okay yeah finite frequencies cannot be screened completely okay yeah good hi thanks for the nice talk I had a general question about the spin fluctuation picture which is that as written down it seems as if the spins that are fluctuating come from the same degrees of freedom as the electrons that could prepare yes in in other materials like the iron nictides there are different degrees of freedom and if I can searchfully think about the crossover when the magnetism is not derived from itinerant fermions but some species of fermions which are doing a transit crossover from itinerant magnetism to local magnetism does anything break down in this the kind of equations that you write down the spin fluctuation mechanism as you go towards the local moment regime of the magnetism what breaks down is this um this all it's a good question I should have mentioned this uh part of the story here is where this interaction one over omega to power of gamma appears it appears because your bosons are lambda over them how their lambda over them depends on whether it's q equal to zero instability or finite instability but in all case they're lambda over them lambda over them means that even if you start with some original say spin fluctuation coming from different electrons and you have a sharp spectrum with poles in the bosonic susceptibility if you couple them to electrons this approach assumes that if you couple them to electron that you go to frequencies at which land out dumping will be larger than any bare dynamic that you put in and assumption here is that the action happens at this frequency the temperature instability temperatures that you get and the boundary scale for self and non-fermal liquid and self energy scale g that I put in is the smaller than the one at which it's actually within the range where land out dumping is stronger than the bare dynamics of the bosons that you start with if it's not then the whole theory has to be done differently then you have to start and do everything with bare dynamics of the boson then of course it will be different theory yeah and he says you understand correctly your last reasoning if we have a discontinuity kink in the effective interaction so we may have a kink oscillation in the sporia transformation which we can have the same normal that occurred in free del oscillation this may do the same traction no I don't think that it's a little bit stretch of interpretation it is similar in some sense but I would not relate this kind of kink to free del oscillation because still free del oscillation in real space right and this is just frequency domain so yes you can think about analog of free del oscillation frequencies I mean but this will bring us too far probably here's another this chunk who's in taiwan your fermion boson approach seems to assume the sdw fluctuations arising from the fluctuations of the long-range anti ferromagnetic order in kink breaks however one popular view is anderson's rpb scenario but the short range anti ferromagnetic rpb may get further condensed and leaves the kink going superconductivity when the charge motion is so here can you comment on this yes of course uh no it's another good point of course look uh it goes like that uh there's a scale g and the scale of bandwidth if the scale of g is smaller or comparable to bandwidths takes this if the scale of g is much larger than the bandwidths go to a different scenario I can we actually we thought we were working about crossover from one to another it exists but I didn't talk about yeah so short answer is yes you are right there is another limit when the approach at least initial approach is totally different but then you have to judge for yourself uh what is the cooperate story whether coupling and bandwidths is uh which one is much larger charge transfer gap is 1.7 electron volt which is the scale for the interaction bandwidths is 8 t t is 0.4 electron volts so 8 t is about three so bandwidths three electron volt interaction is close to two electron volt the rest is just to put the numbers in yeah so this goes back to the age of debate between Slater yeah more than Anderson yeah about local moments versus you know I say yeah yeah yeah