 is Gil Lonserich who was actually a nose pierce from when he was an undergraduate. So if my Coleman number is one, I think Gil, you might have a negative Coleman number. But anyway, so Gil has known Pierce since he was an undergraduate and he's going to tell us about normal state and electron pairing in fur magnetic and for electric systems. So Gil. Good morning and it's a real pleasure to be here on this special occasion for Pierce. I have known Pierce since his early days in the University of Cambridge and for a fleeting moment for maybe a year or two, I managed to keep up with him as his tutorial supervisor. At least I think I kept up with him. Then ever since the roles have been reversed, the pupil and the tutors are inverted. This is Pierce, you may recognize it, he hasn't changed very much. When he was inducted to the fellowship of the Trinity College as a research fellow in 1984 and behind him is his college's famous great hall which holds the portraits around the perimeter of former famous fellows of the college including Newton, Maxwell, JJ Thompson and Rutherford. JJ Thompson and Rutherford. And the hall is also known for being the inspiration for the idea of how to visualize the Rutherford model of the atom and the phrase that was invented was fly in the cathedral. And the fly is a nucleus and the cathedral is the atom as a whole. And one night during the formal dinner, the fellows and their guests were treated to a demonstration of this idea when they were startled to see high up in the hall, a beautiful model glider navigating the hall over long periods of time. So it was thought that this was supposed to be a representation of the fly in the cathedral in real life. And at first it was thought by most people that this was that an experimentalist was behind this demonstration, no one came forward. But then it came to light that for peers the construction of glider was a great passion. So we all had our suspicion of who was behind this beautiful demonstration. So today I want to talk about some problems that Piers and I have discussed over the years are just two problems that overlap with our discussions to make this manageable. And they concerns the problem of quantum phase transitions in particular on the border of sure magnetism for our example and on the border of fair electricity. And in particular I'd like to discuss the possible interpretation of recent findings published in these articles or presented in these articles by Montessucena, Stephen Rowley, and their co-workers particularly Matt Cokes, Seb Hines, Haines and Carsten Enderwein and others. So the first problem and also I want to mention that the very, very fruitful discussions that I've had with Premi on these problems and she is the first author in a review concerning quantum criticality in ferroelectric materials. So I want to talk about, as I said, two problems, selected problems. The first is the existence of a ubiquitous peak in the order parameter susceptibility both found both on the border of ferromagnetism and ferroelectricity. So one can think about the problem in this way and in a conjugate applied magnetic field, the order parameter initially appears to grow as a result of thermal agitation. And this is surprising for the site and it's sometimes described under the phrase order by disorder. And here, for example, are a few cases among relatively simple d-metals on the border of ferromagnetism where the stoner enhancement factor which is closely related to the Wilson ratio varies from about 10 to 100. Although the peaks are very weak, they're completely reproducible and been seen many times. And what we see is the peak position shifting down towards zero as you approach the ferromagnetic critical point. But this phenomena has been seen with even greater precision accuracy on the border of ferroelectrics. In particular, there's recent data in strontium titanate which is right at the edge of being a ferroelectric with a tiny negative pressure that goes into a ferroelectric state. So what is plotted now is not the susceptibility but the inverse susceptibility. So the peak is now a minimum. And what we see is that the minimum position as the pressure decreases and one goes enters the border of the ferroelectric state, the minimum shifts down. It appears to disappear at the ferroelectric point on critical point itself. All the evidence is that the ferroelectric transition in this case is second order down to the lowest temperature. Now, curiously, although the peak and the susceptibility in the ferromagnetic has been known for many years, a consistent and agreed upon interpretation has not been found, at least not a quantitative interpretation. These peaks has not yet been found, although there are many proposals that are very interesting. But in this case, I think we can actually say where this phenomenon arises, that is minimum of the inverse susceptibility or peak and the susceptibility itself. And to try to describe this, we begin with a discussion of what these soft modes are, the soft critical modes, which are paramagnons in the border of ferromagnetism for ferromagnetic metals, but in this case correspond to relative displacements of the charged titanium and oxygen atom. And these displacements give rise to the lowest transverse optical mode, the polar transverse optical mode. The gap of the mode can be tuned either with pressure, strain, or isotopic substitution or chemical substitution to essentially map out a phase diagram a bit like this. And strontium titanate finds itself very close to this edge right here at ambient pressure and without doping the stoichiometric state. So if we look at the standard description of a problem like this, we would start by noticing that the dispersion relation for the soft mode becomes linear in Q as the gap goes to zero. So why did the critical point? So the dynamo co-exponent in this case is one so that in a cubic case, the effective dimension for a quantum critical phenomena is the marginal dimensions for this case. And this means that both scaling arguments, and mean field, the self-consistent mean field theory based on the side phi to the four free energy functional or action lead to the same results apart from a logarithm correction. And in particular, the mean field description leads to a prediction that both the Grunheim's ratio and the susceptibility, the dielectric susceptibility or dielectric concept, diverged low temperatures one over the temperature squared. Also predicted is that the ferroelectric QA temperature squared very linearly with a distance, the pressure, the distance from the quantum critical point. And in fact, these findings, these predictions agree with a lot of measurements present. However, this picture only holds right at the quantum critical point. And if we go out in pressure, for example, further out, we lose this quantum critical behavior indicated here in green, and we get into activated behavior where the inverses of the ability is slowly varying monotonically increasing function of temperature, which is predicted both by the self-consistent mean field theory and the famous Barrett model, which has been traditionally used to describe the quantum perilelectric state. However, this picture breaks down as I indicated, instead of this monotonically increasing dependence, we get a drop, a minimum as shown here, and a characteristic and a distinctive feature of this. In fact, this is the most thorough data yet collected on any of these types of problems. And the most distinctive feature is that the minimum, the temperature of the minimum, T min, the square of that temperature is a linear function of the pressure difference from the critical pressure. So it goes to zero at the critical pressure in the same way that the inverse susceptibility, the T equals zero inverse susceptibility. So therefore, these two quantities are proportional to each other. Now, this behavior is precisely what is predicted by the model that Premi discussed on Monday, which involves the coupling of the order parameter field, which in this case is the electric polarization, and the volume strain field. So we have a free line action that involves the contribution from the order parameter from the lattice displacement U, and then the cross term involved with this coupling, what this is called an electrostrictive coupling. And this, all the parameters of model are actually known independently of the experiments I just presented. And within these model parameters, one finds that this theoretical description here does in fact predict that the minimum temperature squared should go linearly with the pressure difference from the critical pressure. And these are basically three slopes corresponding to different values of the transverse optical frequency gap at zero pressure. And the correct slope that explains the data can be obtained with a gap which is within a factor of two of that measured by a neutron scattering experiment. So the way then to describe the quantum per-electric state is that above a certain temperature of order T min, something like the Barrett formula applies. But this breaks down below T min. And there we must think of the quantum per-electric state as this hybridized state, hybridized the hybridization being between the polarization field, the order parameter field, and the volume strain. And that this crossover line can actually be understood quantitatively as I explained in previous slides. So we think that this problem is on its way to being understood for the ferroelectrics. And one then may wonder whether we could now go back and see if a similar effect could play an important role on the border of ferromagnetism as well. But there the problem is much more complicated and other effects are competitive. The second problem I wanted to discuss in this short talk is also connected with a peak. But in this case, with a peak of the superconducting transition temperature is a function of the proximity to the quantum critical point. So we know that in the case of magnetic quantum critical points, there are many theoretical studies that suggest that the superconducting order in temperature due to the rising from the exchange of spin fluctuations which would be per magnum or anti per magnum that this type of transition temperature tends to rise, generally speaking as one approaches the magnetic quantum critical point. And there are now many, many examples of this phenomenon experimentally. Most of them are involved the border of anti-ferromagnetism. But there are a few intriguing examples involving the border of ferromagnetism. And I mentioned two cases here. A quasi two-dimensional ferromagnet counts the Rubinane and a highly anisotropic uniaxial material with uniaxial magnetic anisotropy, uranium and germanium too. And there are a number of other examples in the same class now that also exhibit this phenomenon. So that has been a evolving story going back to the beginning of early discussion Piers and I had in the 1980s already. But the corresponding problem on the border of ferroelectricity is still being studied. And there is some evidence that something similar is happening. So for example, this is data from Stephen Rowley in his group which shows that the superconducting transition temperature in a charge carrier doped strontium titrate that the TC tends to rise as we approach the effective ferroelectric critical point where defined in the doped system by the position where the optical gap goes to zero because we cannot measure the dielectric constant at that point. But we can still measure the optical gap. There is also some new data from Benia's group and others that suggests that by using chemical and isotopic substitution that suggests the TC actually drops on this side on the ferroelectric side. So although we don't have a complete mapping at the moment within a single system, there is evidence that as in the case of the magnetic system, the transition temperature tends to be enhanced around the ferroelectric critical point. So now we might, what I want to do in the remaining part of this talk is just consider whether this phenomena can be understood or thought about in terms of the exchange of these soft modes that is the carriers doped in the system interact with the soft modes that we've been talking about the polar transverse optical mode and that the exchange of these soft modes could somehow play the role equivalent to that of paramagnons on the border of magnetism. However, it turns out the problem is in this respect actually more complicated than that suggests. And in some ways it's simple but different from what we may have naively expected. So if we use the dielectric function theory for the effective interaction, then we have to express dielectric function in terms of the effect due to these transverse polar modes. So that deals one resident shown here where the frequency squared in the denominator is the frequency of the soft mode. So when we reach the quantum critical point delta goes to zero. However, minimally we have to also and importantly include a contribution from the doped carriers themselves. So there are two resonances minimally, one due to the transverse optical mode, one due to the conduction electron which should be represented by the linear function but there is a simple analytic expression that represents the linear function exactly in the limit of low omega over q and low q over omega. And it's shown here but it's useful because of a similarity to the resonance produced by the polar mode and the polar frequency squared is replaced by the Fermi velocity q squared. So there is a restoring, is if the Fermi system has a restoring force and of course that restoring force is due to the poly principle. So this simple expression is exactly reproduces the linear function and the two limits I indicated and it's good enough for our purposes. But of course the story does not end there. We now have to work on the pairing interaction or the total interaction by taking the inverse of dielectric function and that can be written in terms of two resonances and a constant term. So this term here is due to the direct repulsion between the electron and these two resonances potentially attractive when the frequency is lower than the omega plus omega minus and the omega plus and omega minus are the longitudinal frequencies as opposed to the transverse frequency that appear here. So this is a rewriting. There's nothing new here that isn't already here. However, what is being exchanged is not equivalent to the problem of what is being exchanged or in fact longitudinal fluctuations. And the effect of delta however is not absent but is subtly inside of these functions omega plus and omega minus which can be written out explicitly but are very complicated. So it's not so easy to see a pairing rising from the exchange of transverse optical modes in this case. It is really actually a pairing due to the exchange of longitudinal optical modes, two of them which are hybridized but which are affected by the gap delta. And so that interaction in a sense identifies the problem uniquely. And so at this point it's a question of how to solve it. You cannot use simple BCS type analyses. Ideally you should use the full Eliashberg theory where you cannot assume that the frequency of the bosons is small compared with the Fermi energy for one thing. You can also try them to simplify the numerics by using the so-called KMK approximation for the Eliashberg equation. But in fact both have been done and it's still unpublished paper by Enderman and Litterwood. I've used the full Eliashberg equation. We use the KMK approximation. But what happens is more or less the same, namely that the model predicts that the transition temperature for the density relevant to the experiment, the superconductive transition temperature tends to rise as we approach the ferroelectric critical point. What is also predicted is that TC should have a peak as a function of carrier density and the peak position is roughly in the right place around 10 to the 20 carriers per cc. Now it was interesting about this is that these are completely independent of free adjustable parameters. Now maybe the first time over since the 50s that a parameter free calculation for this problem had actually been carried through. Although I think that kind of did something very similar in the 1980s, a long time ago, but then partially forgotten. But it's an important point is that although it predicts, the model predicts qualitatively what it's seen and in particular, we see that the collapse of the gap of the transverse optical mode does have consequences and tends to enhance TC on the border of ferroelectricity. But what cannot be done in this type of calculation is to predict the actual value of the transition temperature at the maximum because it turns out that value depends on the cutoff in the high frequency cutoff in the Liashberg theory. So this problem is still being debated. So we can say if we examine what the model is doing, the drop in TC at low density is a simple origin. And of course it's just due to the fact that the single particle density of states for the carriers is dropping off at low density. But the drop off at higher density was considered more puzzling for many years. It is origin is physically very transparent. It is simply that as carriers are added to the system, the ion becomes green. So this high supercontinuity in this region is a consequence of the fact that the interaction is essentially an interaction mediated by ions which are charged. Whereas the normal supercontinuity at much higher density is described in terms of neutral atoms that have only a weak polarization and therefore a weak effective interaction. So this is a very interesting phenomena which was already foreseen many years ago by Larkin and can be called ionic supercontinuity. So in the summary slide, I just want to mention one point, the last point which is the following, that it may be at the, of course the anti-peak of the transition. The density is still small, 10 to the 20 carriers per cc. And to get a transition temperature of water half a Kelvin or higher, one needs an extraordinarily strong effective interaction between the carriers. In fact, as strong as one may invoke for example high-temperature superconductors. But high Tc is not obtained because of the screening effect at a higher density. And so the problem in this area is to see if it is possible to prevent the screening from clicking in too early as a function of density. And this might indeed occur in the case of ferroelectrics and anti-ferroelectric materials. And it's possible the materials like BKBO are of this kind, ionic superconductors, and they have through conducting transition temperatures approaching 40 Kelvin. So now I'll just simply end where I started and went to wish theorists a very happy birthday. Thank you.