 Okay, good. Thank you very much to the organizers for being here. In particular, I guess, Andre and I met last millennium in Trieste many years ago when we were both just coming out of, well, we're still students, but we're just students. And of course, very happy to continue the many discussions we've had in such nice places. And it's wonderful to see so many other old friends and to make some new ones here in with such a beautiful backdrop. So thank you very much. All right. So today I'd like to tell you about another superconducting system, one that's fascinated us for many years. And I'd like to, and when I say I believe in the collective nature of science, so I mean the scientific community. And so today I'd like to tell you about some adventures that Pavel, oh, it's working. Good. I think I have to stand here. Pavel Vokov, Piers Coleman, Pavel Vokov, who's across the Atlantic, Piers Coleman, who's upstairs, and I have had thinking about superconductivity in dilute, quantum, critical metals. Now you'll have to excuse me because there's something that doesn't quite work with this Mac. And so, oh, here we go. Okay. So I have to be coordinated, which is not my strong point. Okay, so I'll try. So what's the big question? The big question is, in these systems, how do we get isotropic superconducting pairing without retardation to overcome Coulomb repulsion? Okay. And what we will argue is that we have interactions between the electrons that are mediated by energy fluctuations of the polar medium. So we're going to argue that the electrons don't interact with the zero point fluctuations directly. But like variants and dark matter, they interact via the energy tensor. And I'd like to tell you why we've come up with these details. So let's begin at the beginning. The details of what I'll tell you about can be found there. So let me start at the beginning. What is a polar metal? Okay. Well, you might say there's sort of contradictions in terms. In a metal, we have screening in a system with polarization. We have a microscopic dipole moment. So what does it mean to have a polar metal? Well, yes, we have screening of dipole moments. But many years ago at Bell Labs, Anderson and Blount realized that you could have the inversion symmetry breaking transition into a phase with a polar space group, even if you don't have to have a macroscopic polarization. So the point is you basically have everything to do with a ferroelectric transition except the polarization. And that's what we have in a polar metal. Now, this idea sort of laid dormant for many, many years. But now more recently, we've found, and again, when I say we, I mean the greater community, we found that there are many intrinsic and engineered polar metals that exist. And in particular, in recent years, in the search for vile semi metals, as we've heard this conference, particularly, for example, tungsten dieteluride, bilayer and bulk, we find that there are a lot of polar semi metals. And in particular, we can tune the polar transition with chemical substitution or strain, as we've seen in many talks. And the question that we can ask is, do we have exotic phases? Marco, it's working great. Thank you. Okay, can we get exotic phases? Right? We know that close to a quantum critical point, we've heard in many talks here that we have the possibility of exotic phases. What about close to a quantum, a polar quantum critical point? So the question is, can metals, near polar quantum critical points, host strongly correlated phases? And we know from looking from, again, work, many of you hear, that for example, the nature of those strongly correlated phases depends a lot on the nature of the quantum critical point, whether it's pneumatic, whether it's anti-fair magnetic, ferromagnetic, whatever. Now, what is the challenge near a polar quantum critical point? Challenge is how do we get strong electronic coupling to a critical q equals zero polar mode? I remind you that the standard way of getting strong coupling and getting strong correlations near a quantum critical point is Yukawa coupling. You couple the critical mode to the electron density. But here the critical mode breaks inversion symmetry, so you can't do that unless you have something like spin orbit coupling. So how do you couple it? And this was a question that Pavel and I asked recently. And what we realized was that without spin orbit coupling, the only way to get a Yukawa type approach was to actually have multi-bands. That's how you get around inversion symmetry breaking. You have two bands with different parodies, and you can do that. But then what we were after was, could we get a non-fermic liquid? Well, of course, you have the strongest coupling when you have interaction of the critical mode with dapless particle whole excitations. So we looked at a number of different band crossings. We looked at 3D. Then we looked at 2D. And we looked also at 3D vial points. What we did find was we could get non-fermic liquid behavior in 2D systems with vial nodes. Okay, so it is possible to get strongly correlated phases. Of course, it's like four priors fine tuning. There are some experimental systems where this might be appropriate. Okay, but that's not the story I want to tell you about today. That's just the backdrop. So now the natural question is, can polar quantum criticality drive superconductivity? And we have a poster child for this. In fact, we only have really one, namely superconductivity in dilute quantum critical polar metals. There's been for a very long time the mystery of doped strontium titanate. Now strontium titanate is a perovskite. It's a close cousin of barium titanate, which was the darling perovskite in the 60s, partly because it's a ferroelectric and at that time, particularly at places like Bell Labs, people were very interested in making memory. Okay, and at Bell Labs, there was an, as a, I was lucky enough to be a summer student at Bell Labs. And that's when I first learned about strontium titanate. Barium titanate was studied quite a bit. Actually, the best samples were by John Rameca, Joe Rameca, and I heard about Rameca crystals actually the other day from Malta. He was an incredible guy. So Joe Rameca made these wonderful barium titanate crystals, and they were working really hard to make ferroelectric memory. Okay, it lost out to magnetic bubble memory, but they had wonderful, wonderful samples. Now strontium titanate was the black sheep in the family. Strontium titanate is isostructural to barium titanate. It's very similar in many ways, but it doesn't go ferroelectric. Okay, so from the point of view of memory, it was completely useless. It was just there. People studied it, but whatever. But recently it's come into its own. Okay, and of course I like the rebellious one in the family. So I like strontium titanate. So in any case, but let me tell you a little bit of the story here, because we're on the last day. And as Andre said, we need philosophy. We need stories. I need to keep you engaged. Okay, so let's talk about strontium titanate. This is a review recently by Cameron Venia and friends. I certainly couldn't put all the references to the papers, but let me just show you what the story is. We have a TC and we have as a function of density for strontium titanate that's doped with oxygen vacancies. We have this lower bum and here we have a dome. Look familiar. This is the first perovskite superconductor. Okay, so what's the big issue? Very nice. Well, this is an incredibly dilute superconductor. TF, for example, for these densities is about 13. TD is that the by temperature is 400. That means that TF is much, much less than TD. And we have slow electrons and fast phonons. Now, remember that for BCS, we need slow, but we want fast electrons and slow phonons. Remember, normally we have T divide is less than TF. But here we have the opposite way round. So we cannot use the conventional BCS theory in this. However, we have two gap over TC, which is very close to the BCS value. And we can't use our LO phonon. And the point is we seem to have a good Fermi liquid in the system in the normal state. So what is going on? Okay, so let's talk about the challenges in this system. How do we overcome Coulomb refulsion? We have no retardation and we have a good old-fashioned S wave superconductor. We can't play the angular momentum game, okay, like in helium. The other thing is that even though the optical phonon doesn't couple to the electron density, and here I'm going to assume no spin orbit coupling, we'll get back to that shortly. In the experiments of the Cambridge group, Gil Lonswitch and colleagues, they find that there is a very, very, there is an enhancement of TC as you approach the quantum critical point. So even though the electrons cannot couple directly to the critical boson, they definitely know about it and you can enhance TC with that. So how does this work? Because remember the critical mode is a transverse optical phonon. It doesn't couple to the electron density. Another way of saying it is it breaks inversion symmetry. You can't couple to the electron density. So what's going on? So we have negligible direct coupling between the electron and the soft mode unless there's spin orbit coupling. And this is titanium, okay, it's not high Z. So let me try and put some context to all of this. And I'm thankful to the review that our colleagues Maria, Jonathan and Raphael have made on this, because of course I couldn't put all the references since 1967. Now, particularly for those of you who are students in the audience, I want to tell you a very interesting tale and perhaps the take home message here is to beware of theorists. And I am a theorist. But I worked in a lab and I have a huge respect for experimentalists because I do not have magic fingers. So superconductivity in in strontium titanate. How did it happen? So the story was that Marvin Cohen, who's still at Berkeley, who is still at Berkeley then, he was an expert in semiconductors. And in the early 60s, you know, there was a huge excitement about superconductivity. And people were really interested in applying VCS and all of this. Well, Marvin's approach was look, semiconductors have been really well studied, because of course, for application, for applications. Would it be possible to study superconductivity in semiconductors? So he had a lot of it. So of course, one of the problems in semiconductor semiconductor, the idea was you would dope it and see if it could go superconducting. Well, one of the problems with doping dilute is the fact that the Coulomb repulsion will be very strong. Okay, the Coulomb repulsion will be long range, which is a q equals zero instability. So what Marvin thought about was he said, wait a minute, I know a lot about semiconductors. And in particular, he had done quite a bit of work on gallium arsenide. Gallium arsenide is a multi-valley semiconductor. So what Marvin suggested was he said, let's consider superconductivity, but not in the usual case, but by invoking large wave vectors from valley to valley. And if you do that, then you can somehow get around the Coulomb interaction. So he wrote down a theory. And then, and this is one of the few cases where it was predicted, then he went downstairs to the experimental lab, and he calculated a TC as a function of density. He caught a nice dome, and then he predicted it, and they did the measurement. And this is what they got. You see TC as a function of density, this is niobium doped. And with all due respect, Andy, you said, if theory and experiment match beautifully, then it must be right. Beware of theorists will include be showing you fits to domes. Because this fit beautifully, but what happened? Well, Len Matys at Bell Labs did the band structure calculation for strontium titan. There's only one ballad. So this is like the Cheshire cat with the smile. It was one of the few superconductors that was actually predicted, beautiful analytic work, predicted very good fit with experiment. But unfortunately, the main assumption is not correct. So we still have this. Marvin Cohen's a pretty good theory. And in fact, I should tell you that Marvin Cohen as Sebastian, who we know from Rutgers and works in density functional theory, can attest some of the best suit of potentials for density functional theory were developed by Marvin Cohen. And when I tell my density functional theory friends, oh, I'm working on strontium titanate superconductivity, they say, wait, Marvin solved it. What's the problem? Okay, so let's continue. So there are many possibilities to extend conventional BCS and the flavor that I'm going to talk about, and Dimitri will talk about also later, is a two phone on process. The notion is you can't couple to one polar phone on because of inversion symmetry breaking, but you can couple to two. Okay, we're going to talk about that in a minute. Quantum criticality is important. Strontium titanate is very close to its quantum critical point. The good news about that for us is that the dielectric constant diverges. That means Coulomb interaction is weak. We like that. Yes. Sure. So you made two comments where she say the phonons are fast compared to the electrons. And then the second thing is you're talking about critical phonons, those on the surface sound contradictory. Could you just clarify? Sorry. So the point is the Debye frequency is small compare is this Debye frequency is large compared to the Debye frequency, which usually refers to the acoustic phonons is very large compared to the Fermi energy. Yes. And now what I'm going to talk about with the critical phonons is the transverse optical mode. Okay. Sorry for the confusion. Thank you. Yes. Same issue with more conventional. If you start with the model of interaction. Yes. Yes. Okay. So Andre is asking me a hard question. So I have to repeat it slowly. What Andre is asking me is if I consider and I think Nikolai is going to ask something similar, if I consider a problem with just electrons where I include the phonons only in the dielectric constant. Okay. Maybe. Oh, cool. I'm interaction. Is it. Yes. Yes. Yes. Yes. I'm only focusing on low energies. Yes. And I know you're going to make I thought you were going to ask me about Coulomb repulsion. Okay. Yes. Go ahead. We're going to talk about that. It's much less than one. And the reason it's much less than one is because in the because of the dielectric constant. Okay. Yeah. Okay. So yes, I will get there. Okay. So quantum criticality is important. Okay. You can have multi band effects. You can have spin orbit coupling and a popular approach to this problem is to involve some sort of rush by coupling. And then you can have a direct coupling between the polar mode and the electron density. Okay. And I have more to say about that. Okay. And again, this is a wonderful review, which I recommend. Okay. So let's talk about spin orbit coupling. I myself have trouble understanding why spin orbit coupling should be so important in a system with titanium. There's a large spin orbit gap. But the point is that what we're interested in is how that gap changes as we distort the crystal. Okay. And this is a question that with our density functional colleagues we've been looking at. Okay. But the another point is can we determine this experimentally? Okay. And with Abhishek and Pavel both wonderful postdocs at Rutgers, we've looked into this. Suppose you did just very briefly suppose you did have a rush by interaction. Okay. And let's look in the nearly polar phase. Well, in that case, oh, I have to use this. In that case, we would have dynamical spin splitting of the bands. And if we turned on a magnetic field, then we would have collective modes associated with the development of spin currents in the polar metal. So just to make a long story short, we would have collective modes as a function of field. But we would also have our soft mode. And those two would hybridize. Okay. And the gap will be a function of that spin orbit coupling. And we worked out the experimental numbers in terms of fields and all of that. And that can actually be measured. Okay. And I should also say that we're doing DFT studies with our Rutgers colleagues, both to get the strength of the spin orbit coupling and also to get the strength of the two phone on coupling that I'll be talking about. So what are the guiding observations that Pierce Pavel and I have have played into hours? Okay. The first thing is that we have very strong ionic screening. So we have very weak coulomb interactions between the electrons. Okay. So the dielectric constant is very large near the polar quantum critical point. And we'll see that that means that RS is very small, even though it's dilute. Then the other point is that we have a critical mode that is inversion symmetry breaking. So there's no linear coupling to the charge density. And here we're assuming everything I'm going to say we're assuming negligible spin orbit coupling. So the electrons do not directly interact with the zero point fluctuations. Okay. And all right. So the model for our coupling is looks like this. It's a model of coupling the local energy density, oh, sorry, the local energy density here, the local electron density with the energy density of the polarization. Okay. Again, the electrons don't couple directly to the zero point fluctuations. This is a little bit like variants don't couple directly to dark matter, but they do know about dark matter through the stress energy tensor. And that's what we're arguing here. Okay. And what you can see is that this will suppress, having the electron density there will suppress the fluctuations of the critical phonons. And similarly, the critical phonons will suppress the nearby electrons. And so you can get a reduction of the chemical potential. And we see that fluctuations of the critical phonon energy density near the electrons will result in attractive potential. These are pictures I'll give you a better sense of how this works in a minute. So what is the action? Well, we have to have an electronic part. Okay. Here we have the electronic part. Then we have the energy fluctuation part. And we have the electrostatic interactions and the polar fluctuations. So let's start simply. Let's take g equals zero, no coupling. If we take your g equals zero, we integrate over the longitudinal modes. And what we have here is we have our dielectric constant. Okay. And the dielectric constant is that we're going to put in is going to be only ionic. Okay. All right. And once again, we see that the quantum critical transverse modes are completely decoupled from the electronic degrees of freedom. Okay. And once again, this is why we invoke energy fluctuations. Now to answer Kulia's question, weak versus strong coupling. Normally, when we have a very dilute system, we have strong coupling. So when we look at weak versus strong coupling, we compare the coulomb to the kinetic energy. RS is then one over kfab. And since kf goes as n to the one third, you might think that this might be very large. But as has been emphasized by Dmitry Masloff, here in dilute quantum critical metals, what's interesting about them is the fact that because of the Bohr radius, the Bohr radius includes the dielectric constant. So it's quite large. RS is very, very small. And so RS is much, much less than one. And so we're weakly interacting. And that's because we're close to the quantum critical point. Okay. So we can use weak coupling theory, even though Sri likes to talk about as the tyranny of weak coupling theory. I like weak coupling theory because it's where I can do, I know how to calculate. But Sri knows about other things. Okay. So it's a lowest order. We're in 3D. Yes. Calculate any unsuccessful question. If you start with the diagram that you're talking about, with the quantum and then calculate any vertex correction, just to check. Yes. Normally, the closeness of the critical point will be used in the elimination of the correction. That's right. That's right. What happens here? Well, to be honest, we haven't done that. We've only done the weak coupling calculation. So weak coupling in the sense that overall factor is small, but you don't know whether it's still the way in which you're getting less vertex correction. I think that's probably the right, I would say. That's probably a fair statement. We assume that because RS was so small, we just had to do the leading order. Okay. But we have not calculated the vertex corrections. What I can say is that, for example, I talk about, this is clearly a two-phone on interaction. And you might ask, what am I talking about? Also energy fluctuations in 3D, the corrections to the two-phone on interactions, which come from quartic terms, which is related to your question, are logarithmic. So maybe that's an answer in two dimensions, which I will discuss. They're not logarithmic. Yes. Yes. So I think that's an answer to your question. That's all I can say. That's it. In 2D, it's not. The energy fluctuations have an anomalous dimension. And in fact, phonons are not well-defined, but we can still talk about energy fluctuations. And as we'll see, they actually save us. We still have a family look. So that's the best I can answer. Since this is less relevant in the RG sense of the standard you call a coupling, you would expect the vertex correction. Yeah, that's what I, that's, that was basically the argument that, thank you, when we, in this interaction is irrelevant in the scaling sense. And so we don't expect the higher order to be important, but we haven't actually calculated it. Okay. So link with prior work. This two-phone on exchange, first of all, Abhishek and Dimitri, were able to solve a rather outstanding puzzle in Strontium titanate using this two-phone on the exchange. We have a row that goes like T-square. Recall that Malta mentioned a T-squared yesterday. We have a row that goes as T-squared. Now you might say, what's the big deal? The problem is the Fermi temperature is something like 10 degrees, and T-squared is going just all the way up. And so why is it still T-squared, even though it's in the classical regime? And so Abhishek and Dimitri and his collaborators, their collaborators, actually showed that with this, oh, I have to do it here, with this two-phone on exchange, you can explain that. And that got us thinking about the superconductivity. We are not the first to suggest the superconductivity. It was proposed many years ago by Nagai, actually based on Raman data at Bell Labs by John Raman. Okay. Two-phone on Dirk van der Merle has also suggested it. So we're sort of fleshing out some of the arguments that they had. So the question is, what happens when we apply this to dilute quantum critical polar metals? So now let's talk about the coupling to the energy fluctuations when G is finite. The first thing we get when G is finite is we get a hardening of the polar mode. Okay. So what we get is that omega T as a function of n is the original transverse optical mode, plus another term that's proportional to the density. So omega T is now proportional to n to the one-half. Okay. So this suppresses the polar state, but that means we suppress the polar state by charge doping. This has been known for a long time. This is neutron data, omega versus Q you see. And note that G over A0 cubed is about 6.6. Okay. That's the value of the electron phonon coupling. Okay. And the reason we'll talk about that is because you'll see that there's going to be a unified picture. This is consistent with observation. And of course, we're taking G to be positive because of links with experiment. Okay. So the first thing we learn is that the doping shifts the quantum critical point. Okay. And if we want to look at density depends, we have to take this into account. Now, the energy fluctuation coupling cannot be integrated exactly. However, we know from our past experience, and as Shreve just mentioned, that this interaction is irrelevant in the scaling sense. Our Fermi liquid is stable. And so we're going to fearlessly go ahead doing perturbative treatment. Okay. All right. So what's our effective electron-electron interaction? Our effective electron-electron interaction at the quantum critical point goes as 1 over x to the fourth. Okay. Here, omega goes like q. The phonon propagator goes as 1 over q squared. So here it goes as 1 over x squared, 1 over x fourth, because we have two phonon propagators. So at the quantum critical point, and here I'm taking x to be a four vector. Okay. So I'm including space and time. At the quantum critical point, I have a 1 over x fourth behavior. All right. Well, we're going to get, yes. Yeah, but I'm just being very heuristic at the moment. Okay. I'm just trying to give the lay of the land, because I think it's too late in the week to be showing lots of calculation. But you're, but it's 1 over x to the fourth. Okay. There's a question from the chat, which is, is there a limit on the value g can attend? That was the choice of word. Is there a limit? Well, we're going by experiment. So I obey nature's limits. So we're going by fitting to experiment. Okay. I mean, at some point, that's all I can. So what happens away from the quantum critical point? Away from the quantum critical point, I have to think about length scales and energy scales. So suppose I have my quantum critical correlation volume. Okay. In the volume, I have the, as goes as 1 over x four to the fourth. And again, x, I'm being very heuristic here. X is a four vector. Outside, I have some sort of exponential behavior. Okay. Now, what are the scales in this problem? Well, in time, the correlation in time is defined by 1 over this omega t. Okay. And that's again, including the doping dependence. And in space will be the speed of sound over omega t. All right. So those are my scales. Okay. And the next question is, what do the electron sample? Well, the important scales for the electrons are one over KF and one over EF. And remember, both have density dependence. One has density of pensions of n to the minus one third, and the other has density dependents of n to the minus two thirds. So in order to find out what the electron sample, we have to compare these scales. So again, we're comparing the scales here with the scales here. So let's go ahead and just do that. All right. So let's take the first case. If 1 over KF and 1 over EF are very small compared to these correlation lengths, then we're quantum critical in space and time. The electrons don't know about the boundaries. And we're just quantum critical. Okay. But remember that KF goes as n to the one third. So this is a very high density regime and we're not there. So let's think about the next possibility. Oh, here if he goes as 1 over X to the fourth, the next possibility is what if 1 over KF is 1 over KF is less than the spatial correlation, but 1 over EF is larger than the temporal correlation. Well, then we have quantum critical in space, because in space, the electrons think we're critical, but we are local in time. And this is low density. And this is sort of Newtonian, it's instantaneous in time, but it's power law in space. Finally, what happens if the correlation length is less than 1 over KF and 1 over EF. That means that our correlation volume is small compared to what the electrons see. So then we're local in space and time. Okay. And that's what we have. So just in summary, what we have is first, we're quantum critical in space and time, but that's for high density. At low density, we're quantum critical in space, but not in time. And then at very low density, we're local in space and time. Okay. So let's just summarize this in a different way. Let's think of the density dependence of the effective interaction. So we can look at log of E versus log of N. Now we have three important energy scales. We have omega T, which goes as N to the half and is density, of course. We have CSKF, which goes as N to the one third. And we have EF that goes as N to the two thirds. And so we have the different regimes. Here we have completely local. Here we have local and time, but parallel in space. And here we didn't discuss this case, but we would have local in space and parallel in time. So in the first regime here, there's no Q dependence, no omega dependence. Second regime, we have no omega dependence. And third regime, we have no Q dependence. So this first regime here is one that Dimitri Kisilov and Misha Feigelman have worked on. And Dima will be talking about it shortly, okay, within a very similar approach. What we are focusing on is this regime. Okay. All right. And there's no Q dependence. And part of the reason we focused on this regime is in this regime, the superconductivity is definitely bulk. Okay. So how do we do our calculations? Well, when we, we talked about in space, but of course, we do our calculations in momentum. And when we integrate this, we're going to get a log. Okay. But now what happens when we are away from the quantum critical point? Well, just again, very horristically, when we're away from the quantum critical point, we're going to have these cutoffs. Okay. So if we integrate over the Fermi surface, then we actually have a complicated formula. But let's just, let's just look at the physical aspects of this. What we have is we have a log omega t is the cutoff just for the transverse optical mode. So it's like a Debye frequency for the transverse optical mode. Okay. And then we have the max of omega t CSKF and EF just as we discussed before. Now the crucial thing to realize here is that large momenta contribute, we have a log divergence, large momenta contribute. So we're going to be looking at momentum across the Boolean zone. Remember about what Marvin Cohen did with the Intervalli scattering here, we're going to say that we can have the sum of momenta that is q equals zero, but we can take momenta from across the Boolean zone. So now let's get, yes. Yeah. And I actually have the full formula if you want that. So what do you think E and k prime still form the formula? Yes. Yes. So this effect is to do with the scattering in the correct direction? Yes. Yes. Because, well, I'm doing, okay, so what I'm saying is that a large momenta are important. That will come. Yes. Yes. No, no, no, I meant large, no, no, I meant of order kf. Yeah, that's what I mean that you can have. Thank you, Nikolai. So I can have k1 plus k2, but they have to add up to be small. So in that sense, it's very similar to what Marvin was talking about. Thank you. Okay. So superconductivity. So for low carrier density, we'll see that the attractive interaction can overcome Coulomb repulsion, but let's just look at the attractive part of the effective coupling. So the attractive part of the effective coupling goes basically as kf log 1 over kf. Okay. That's a dome. Okay. That's very familiar for those of us who played with BCS equations. So it's kf 1 over kf and we'll have a maximum when kf is of order the cutoff divided by 2cs. Okay. So without just looking at this, and it's that log that's crucial. That's why I kept hammering about the log. Okay. The log is crucial and the log has an argument that's density dependent. And kf goes as n to the 1 third. So what do we get in terms of the maximum? Where's our maximum in terms of density? Well, here, and this is why it's important that we have, that we are looking at k vectors across the brio zone. We can look at the dispersion. Of course, we just consider a very simple quadratic dispersion. But in strontium titanate, it's very steep and goes like this. So when we put in, when we calculate the cutoff to the optical phonon, we don't just take the speed of sound from here. We actually do a Debye average over the whole brio zone. Okay. And you'll see why that's important because nmax then is going to be cs average over cs. Okay. And so the maximum, because we have a phonon dispersion that flattens near the brio edge, we have the average value of cs is much less than cs. And nmax of az cubed is going to be much less than 1. So we're well below half filling. Okay. And the other independent, the other point that I want to make is that the maximum density is independent of the coupling constant. Okay. Thank you. All right. So now we can actually look at the superconducting coupling. And here in this region, the Coulomb repulsion is just going to be a constant times the density of states. So it's just going to move things around. But as long what we need for this theory is we need to be close to the quantum critical point for the dielectric constant to be large. And we have to be low density so that the log wins. Okay. And when we do that, we get TC, we can use the formula by Gorkov and Malik Bakroudarov. Is that right? Bakroudarov. Gorkov and Malik Bakroudarov. Okay. I'm still getting it. Bakroudarov, which was summarized more recently by Gorkov. Okay. And so TC has a dome behavior as a function of carrier density. So when we actually, yes, do you want me to go back? No, that's okay. This is what happens when I can't say a Russian name. Okay. Yeah, that's what we're going to do. Oh, I have to, for that, I have to go back to this kind of problem. Yeah, I think it does. Actually, I can go to the end. I have the full, can we do it after I have the full formula, which I can do. And the G is in the log. It goes in the line as one of those columns. Perhaps that's something to check. We haven't, the honest truth is we haven't done that. Okay. But G, I wanted to just get across the main aspect of this formula, but I can show you the thing. And I think it's definitely there. We have very similar formulas, that's surprising. Okay. So our maximum, just to say, so what do we have here? Okay, we have strontium ruthenate. Again, it's not strontium ruthenate. I was listening to your talk. Strontium titanate. There's a strontium there and an oxygen strontium titanate. What do we have here? Well, the red points are data. The black points are what is our fit to that. We use G over A0 cube to be 0.7. That's important because that coupling constant is very similar. That's why it has to be in the log. The coupling constant is very similar to what was used by Dimitri and Abhishek and also by in the other one. We'll talk about that in a minute. We can push this. I'm running out of time, so I'm just going to say our calculations are only good until here. This is where the dynamic effects that Andre was talking about at the end of his talk on Monday have to be included. So EF is less than CSKF below that line. Above, we have to include the electronic contributions to the dielectric constant and look at the frequency dependence. As we learn from Andre, the coolant repulsion will get stronger as we increase the frequency dependence. The attraction will go down, and so you're going to have the interaction changes a function of frequency, and so you can play some games with re-dartation. We haven't done that yet. Did I get that right, Andre? Good. See, I listened to you, too. So a couple of things. Harold Huang mentioned a measure in STM measurements that we have a good BCS, superconductor. It's not, I mean, it looks like BCS. We get that. We also are able to, putting in our numbers, get the kind of experimental values that the Cambridge group gets. In terms of predictions, it's always very nice to post-ict what's been seen, but what can we make a prediction? Well, one of the things we can do is say that in 2D, this should be more enhanced. And so we said, well, if you could make a 2D doped strontium titanic film, this is what we would expect. So what are the distinguishing features I want time for questions? What are the distinguishing features of the energy fluctuation mechanism? And you're going to hear about more of it from Dima shortly. First is that it's a unified approach to a couple of different issues. One is that rho goes like T squared is the dominant coupling to the electron, to the energy fluctuations. I should say there's still some mystery at low temperatures, but for a very large temporary regime. The second is that we have suppression of the polar state with doping. And in all of these cases, we have a G over A0 cubed that goes from 0.5 to 0.7. And so it's a unified approach to very different properties of this material. Now, we have a scaling of TC with the photon frequency and sensitivity to carrier density and low doping concentrations. And remember that in this, our normal state is a Fermi liquid. There's nothing particularly fancy about the normal state. And so in summary, we have dilute quantum critical polar metals. I've done my best to tell you our story about two phonon coupling. It works for intermediate densities. Dima's going to tell you about what happens at low densities. And we believe that there are plenty of polar metals that if we suppress TC, then perhaps we can get some new low density superconductors. So thank you very much. Thank you. Questions? Yes. Do you want to see the full, this is, I think Dima will show it too, but this is what the full formula looks like. And you see that G is going to enter because G always enters an omega T. Yeah, very good. So that's, that should have been my answer to you right away that you see we have, this is a crossover function. This is, remember what I did was, this is the function that gives us all those maxes. One thing we learned from our dear friend, Elehu, is when you have a complicated function, always when you have crossover presented as different maximum cutoffs. So this is what's giving us our cutoff, but you see right away that we have a G. Yeah, I was asking many questions, so I will try to be quick. No, you've been asking questions for years. Yeah, we might as well continue. You mentioned that, and you used in fact, it's Gorkov-Melik-Burgudarov formula. Yeah, question about this formula and whether you need to use it. And I'll try to ask the question. The story with Gorkov-Melik-Burgudarov is that you have a pre-factor of Fermi energy, and anything that goes at energies larger than Fermi energy in fact can be incorporated in normalization of interaction to scattering amplitude. That's right. Which you don't do, you deal with whatever interaction you have. Yeah, very simple. So then question is this, don't you restrict yourself to paving very near the Fermi surface, putting EF as an upper cutoff in a situation when EF is small, and you may have a paving extending far away from the Fermi surface, which will give you larger TC because then you will involve fermions not necessary on the Fermi surface. It's a question. Okay. Well, I guess you also have the same question. I mean, okay. Yes, I know I under. And I wonder if you will get higher. Because again, the story with Gorkov-Melik-Burgudarov is that everything that comes from the atmosphere can be incorporated in normalization. Okay. So then you can view this as a minimum TC. You're viewing this as a minimum TC. I mean, okay. That's actually an interesting point, particularly because one of the things we've been thinking about, which is related to Andy's talk, is everything here was in the nearly polar phase. But you can also go into the polar phase. And then instead of p, you'll have p plus average p. And when you have p plus average p, that means your TC will be a function of strain. Okay. And so what it means is you could really boost your TC in the system in your kind of in cliffs thing. Suzanne Stemer at Cintor-Berber has some work on this, which shows TC to be higher. But I actually think you could really bring it up. And that would might be a test for this kind of mechanism. But I should say that we were just happy to get a TC that seemed reasonable. But I see what you're saying. Do you have an answer to that? Okay. Let me read some statements from the chat, and then we should wrap up. The first Lara Barbatos says the other scenario based on two phonon processes can also explain the same features, resistivity, shift of omega, et cetera. Question mark. In other words, do we have a real way to decide which proposal works better? That's one. And then here's a bunch of comments, which I'll invite him to say. Yeah, just a moment. So Lara is what is, I didn't quite understand what Laura was saying. Lara, do you want to speak? All I'm saying is it's a unified approach to- Lara cannot. Okay. Oh, she's giving an exam. It's a unified approach to these different phenomena, but we're still working on, I don't, we haven't solved everything yet. There's still work to be done. Okay. Marco has his hand up. Maybe I can verbalize what I wrote in the chat. How about after Marco, and then I'll come back to you. He's a direct reply to what was just discussed, and you didn't read it out. Go ahead. So you are selectively taking one part of the story, which is, of course, what certain news channels do, but you don't do that. So let me say it. Okay. Oh, well. So the point is that, as I think Prémy said, is that it's a little bit like in Stapmec. In Stapmec, when you're above the upper critical dimension, you can just do perturbation theory. But once you go below the upper critical dimensions, things start to normalize. And so it's correct that above the critical dimension, which we saw here, two phonon processes are just perturbative. But once you go at or below, they renormalize, and it's more correct to refer to them as energy fluctuations, because the exponents that come in are the same as the energy fluctuation exponents. Pierce, but I believe that Andre was asking why we... I wasn't replying to Andre. I was replying to Lara. Oh, I see. Yes. Okay. Okay. I didn't get a chance to reply to Lara, but that's okay. Yeah. I actually have a slide exactly about that. So let me just show you. Basically, when Pavel did it in 2D, let me get to it, when Pavel did it in 2D, we had to, in fact, in 2D, the interaction, the Fermi liquid is marginal, which could be a problem. But thankfully, the quartic interactions save us. And so the energy fluctuations have an anomalous dimensionality. And what we've done is just the first step. So thank you. Let's ask Marco to turn on his... Sure. Hi, Premine. Nice to see you. Hi, Marco. I'm sorry that I couldn't make to come to in person to Trieste, but nice. I'm sorry, too. Thank you for your nice talk. So let me ask this question. We are in a situation in which we have a rather weak and strongly suppressed Coulomb repulsion. We have a rather small kinetic energy, and still we have a rather effective attraction mechanism lurking around. So the distinctive aptitude for me would be to worry about phase separation or weekly frustrated phase separation. So of course, superconductivity is a good alternative to that because already 30 years ago, we studied a condolates model, which was introduced by Pierce and Nathan Andre, and we found precisely that usually the system likes to be superconducting, but if you kill by hand superconductivity, there is a large region in the phase diagram which becomes phase separated. Now I was wondering what happens here. So you have again superconductivity, which protects the system against phase separation, or this is something we should still have and find somewhere? At low density, we probably have some phase separation. And in fact, that's one of the reasons that we only, let me go back to the original plots. At very low densities, it's believed that the super, at very low densities here, it's believed that we have filamentary superconductivity. There's no, as I understand it, to date, there's no bulk evidence for superconductivity. The only evidence is transport, and so that would play into what you are saying. However, my experimental friends, particularly Dylan Cumberland, suggest that above about here, 10 to the 18 or so, we have bulk superconductivity, and they don't see any evidence for phase separation there. So that's one of the reasons that we focused on this region here and didn't pay as much attention to the lower density state. Still, I should say that philosophically, it's very interesting to ask what happened with such low densities, and Dima is going to talk about that shortly. So let me ask then the question. If you, suppose you kill by hand superconductivity, then is the compressibility positive or negative in this case? Did you check that? Suppose in this region, in this low region, no in the blue one. Oh, the blue one. I don't know if that experiment has been done. I understand what you're asking, but I don't believe that experiment has been done. I think the experimentalists are so thrilled to get to superconductivity that they don't want to kill it, but I can ask. Okay. Thank you. Sorry that I can't answer. We have 15 minutes for coffee, but before we go, let's thank Remy one more time. So just for the online audience and for everyone here, it's 11 o'clock. We'll meet back at 11.15 for the next talk, which is online.