 Okay, let me start by thanking the organizer for inviting me here. It's a great pleasure to have the possibility of opportunity to present here the results which we achieved in my group. So the talk is about understanding stone correlation effects in spin orbit, T2G materials. As first talk, I was thinking to the board two slides to introduce the topic of stone correlation. Eric already did that, but let me remind you some main aspects. Stone correlated system are those for which the single electron picture breaks down qualitatively. Consum eigenvalues do not provide a realistic electronic structure, and the stone correlation mostly arise from the local part of the Coulomb interaction. So typically there are compounds involving atoms, we partially feel that the f or some time p shares. And here I will focus on materials which involve atom in these rows and particularly rotates and perhaps if I assign row dates. The method which became the state of the art approach for this system is the LDA plus DMFT method, which starts from Consum eigenvalues to build Hamiltonians to build models which are augmented by a local Coulomb interaction, so generalized Habard models. Models are then mapped onto some self-consistent quantum impurity model, which remains very hard to be solved and which is typically solved by quantum Monte Carlo in realistic cases. And as was already mentioned, if you want to know more on these methods and the various aspects, we have a website with many, many lecture notes. Now the difficulty is that with quantum Monte Carlo we can typically solve only certain limited type of models, so with a limited number of degrees of freedom and with specific interaction. And so this is different from DFT where we can solve the full problem in one go. And at least if we want to stay in the situation in which we solve the problem exactly with quantum Monte Carlo. But a lot of progress has been made in the last year so that the complexity of these solvers, so the problems that can be accessed, increase thematically, at least for the point of view of the many-body physics. And for example, in my group we built general codes which can now deal with realistic Coulomb vertex, like the one that you obtained from Abinishow for constrained LDA and even with spin-orbit interaction. And of course we did this by exploiting the supercomputer center in Eulich, you see here behind the Ukraine which has now been dismissed for a new supercomputer. So what I will show you today are results which can now be obtained where we include general Coulomb interaction and spin-orbit coupling which were not possible till a few years ago. The system I want to address are routinates and materials which are in the same, which involve atoms in the same row. So routinates have 4D, T2G4 configuration, nominally, and they are particularly difficult to describe because if you look to the electronic structure it looks simple, but many, many energy scales are very similar. Not only the Coulomb interaction and the bandwidth, but also Hoping and Christophe field and spin-orbit interaction, and as you will see, terms in the Coulomb vertex that are typically neglected. So you really need a general solver and you need the state-of-the-art quantum Monte Carlo for doing that. They are very interesting, so they are particularly interesting and they are still hotly studied because, well, from this side of the phase diagram we have Stonson 2-routinium-O4, which is a metal which becomes superconductor at a very low temperature, and the nature of this superconductivity, it's not understood to date. If we don't replace Stonson with calcium, we remain in this layer, a perovskite structure, but distortions start to take place, and on this side we have a system which is metallic at high temperature, where it has this LPBCA phase, and then becomes insulating and even magnetic with exotic properties at very low temperature. So we start to discuss this system, and in particular the puzzle of this Fermi surface. The Fermi surface is particularly interesting because it gives us information on what site and interaction are crucial at the Fermi level, and those are the electrons which are essential to understand the superconductivity, which remains still a puzzle, as pointed out by Adi Mackenzie in this nice review of 2017. So what is the electronic structure of Stonson 2-routinium-O4? Well, if you do LDA calculation, you obtain a picture like this, with the band crossing the Fermi level, which are the T2G band. So one is almost two-dimensional, it's the XY band, and two are almost one-dimensional, the XZ and YZ's band. They are split by a crystal field splitting, and although I say one-dimensional, two-dimensional, the openings are relatively long range. Now, if you calculate the band structure of such a system, you expect something like this, straight lines for the quasi-one-dimensional bands and around Fermi surface for the two-dimensional band. And indeed, more or less, this is what you find in LDA, these are calculations. This one I took it from the paper of Masin and Singh in 97. The Fermi surface was measured by several groups, here I'm comparing to the work of Dabashelli with Alpes. And you can see that the LDA Fermi surface and the Alpes Fermi surface are quite similar, at least on a qualitative scale. But if you look to the details, differences start to arise. In particular, here, we see that the LDA has the generacy which don't exist in experiment. And the relative sheet size, it's not correct. So one started to wonder what is missing. And more or less in the same year, 2000, it was pointed out that this system is actually strongly correlated. There's high mass renormalization of at least a factor of three. And together with Igor Masin, by trying to understand NMR experiments, which are crucial for superconductivity, we understood that spin-orbit effects are strong. Of course, a lot of works went on since then, but now that we had the means of starting this problem with LDA plus MSD, we looked at this problem with this approach. So the models where the solving is something like this, a tie-binding-like model with open-constructed ab initio for the T2G bands, local coulomb interaction, and spin-orbit interaction, whose largest term is the local term. So the result is what we obtain is collected in this picture. Up there, you see the LDA thermosurface and the LDA plus spin-orbit thermosurface. And as one expects, the generative problem is solved because spin-orbit now splits the generate bands. So adding spin-orbit improves the agreement. However, if we now add correlation, we see LDA plus MSD, LDA plus spin-orbit, plus the MFT, the agreement doesn't really improve, or better, it improves some sheets, but it gets worse for some other thermosurface sheets. So the surprising result of a very tough calculation was that the correlation do not improve the agreement, at least not remarkably, so it doesn't solve the problem, what we were instead expecting. So to understand this result, we should look at what the self-energy is actually doing to the thermosurface, and it's simple. If you have a thermo-liquid, the self-energy is modifying the thermosurface by changing the on-site parameters of our Hamiltonian. So changing the crystal field and changing the spin-orbit coupling in this case. And what happens is that both the crystal field and the spin-orbit coupling are enhanced for the system by Coulomb repulsion, more or less they double the value with respect to LDA. And these two effects together both make the smaller the internal sheet and larger the external sheet so that one gets better and the other gets worse. And there is no way of tuning U and J parameters to solve this issue. So something is missing. And one might think that there are no local effects which are not taken into account in the dynamic Hamilton theory, but this in particular because, sorry, because here one could think, okay, the larger discrepancies at some k-point, specific k-points, but we found that there is a simpler explanation which is also a general mechanism. The Coulomb interaction that we should use in this model, well, typically we use spherical Coulomb interaction like in the atomic limit. But actually when you take into account linear function and screenings, the Coulomb interaction is not spherically symmetric, and in fact if you look to CRPA results they show a tetragonal contribution, for example, in this system. And this should be taken into account. Typically it's difficult because you need a generalized quantum Monte Carlo solver and you need to take into account the double counting correction a proper way. But we solved these issues, so we have this code and we have a solution for the double counting. So we could now include these terms and now we recover perfect agreement. Why? Well, what is happening is that this term in the Coulomb tensor, so the difference between the U on the X and Y and the Z orbital is positive. This is the CRPA value and reduces the crystal field announcement. So this is the missing force which retues the parameter so that we can get the crystal field at the end, which is in this case very close to the LDA initial value without enhancement. And this is what gives the final result. So in short, instead of this large enhancement we have a smaller enhancement, delta prime, which is for realistic parameter basically zero. This is what gives this perfect agreement. And this is a generic phenomenon which will occur in many other multi-orbital systems at the Fermi server where symmetry is crucial. Now very recently I saw that a paper in Cormat which re-analyzed the problem and confirmed basically our result with a more refined higher solution for termitional experiments. Of course, if the spin orbit is important, if the Coulomb interaction is important, this should show up also away from the Fermi surface and that's why we try to look to conductivity experiment, optical conductivity. And here again there were a lot of data and even theoretical description based on LDA plus NFD which reproduced quite well experimental data, except that perhaps the height of the do-the-peak was always a bit too small with respect to experiment unless you make the Coulomb interaction unrealistically small. But these two work made a coherent picture where the contribution of single orbital or single bands could be identified separately. Now the question is if spin orbit is important as we found out does this picture stay? Can we talk about the resilient quasi-particle as we do in this paper or the thermal regime? So we re-analyzed this problem not only in this system but also in the table layer materials and what we found is the following. Well we can reproduce of course as they did while the experiments qualitatively were even without spin orbit but the spin orbit interaction is much closer to the data for what concerned the height of the do-the-peak. And why is it? Because in this specific case it reduces the imaginary part of the self-energy and therefore enhances the peak at zero frequency. This is still within the orbital picture but then we decided to analyze the components of the optical tensor divided in the intra-orbital, inter-orbital and the rest. So this would be interband if there was only bands orbitals and bands not equivalent in this system. And what we found is that while the inter-orbital term is still large there is also a sizable rest component which changes the picture of the conductivity. And this transfer of weight is very important and recently we found it becomes even more important in the single-layer rodate where the singular band picture breaks down completely. Why is this very important? Well if we think about superconductivity as was pointed out in this paper right in 2014 if the spin orbit is strong and how we describe the Cooper pairs we cannot talk about spin and orbital parts separately we now have to make a different classification which perhaps will allow to solve the open questions which we have about pairing in this system. If spin orbit is important what about when we increase the distortion so the effect of correlation becomes even larger what happens when the system becomes an insulator? What happens when we go to calcium 2 which is an insulator at low temperature? So we are on this side of the phase diagram and as I was telling you there are different phase LPBCA metallic SPBCA insulator paramagnetic insulator and very low temperature magnetic. So this was a very intriguing problem because initially it was proposed that the metal insulator transition was orbital selective in nature with the XY band the larger band became staying metallic and there are two bands becoming insulating but in 2010 we have shown that this is not the case so we don't find any orbital selective transition the metallic phase all four bands are metallic and the transition is more associated to the effects of structural distortion from the LPBCA to SPBCA change instructor so in the ordered phase sorry in the insulating phase also we find orbital ordering of the type two electron on the XY and one electron in the XZ and YZ orbit this was somewhat confirmed also by ARPES which saw three bands in the metallic phase but soon after we published that paper a new result came out which a new theoretical description came out from Guangquan Yu in Stuttgart and he actually used the LDA plus you approach to analyze this problem and the result were quite different from ours namely found that the spin orbit coupling is now the dominant interaction which gives rise to the metal insulator transition now as we know the LDA plus you approaches not the best approach to determine the onset of the metal insulator transition because it doesn't describe properly the metallic phase but nevertheless nevertheless he had a point because the spin orbit interaction was indeed neglected in our calculation and it could be that it plays an important role so we now reanalyze this problem by using the LDA plus MFT approach and what we found is summarized here in the metallic phase we still have a metal with three bands that are for the old metallic in the insulating phase this is without spin orbit and this with spin orbit you see very little difference so the spin orbit does very little to the onset of the transition if it does something it slightly reduces the gap a phenomenon which is a bit exotic but you can understand it already in the atomic limit by looking at the effects at the multipliers so this is exactly the opposite of what was found in LDA plus you so what was wrong in LDA plus you well first qualitatively after it fork gives in this specific case exactly the wrong train so it enhances the gap but also the wrong coulomb interaction was used which is this U minus J to J U minus 3 J form which is correct for real orbital but not for spherical harmonics and unfortunately this enhances the effect of spin orbit I put a warning that it's always important to use the correct coulomb vertex because you make approximation which can be crucial in some cases if our model is correct then it should be possible to use our results to describe also the magnetic phase this is then the real test of our results so what about the magnetic phase so here the debate when we started to work on this was two opposite picture one picture proposed by Guignac Alunic was that the spin orbit dominates the ground state is a J total equal to zero state and the magnetism arises via the brown-black mechanism the second picture was proposing that instead there are effective moments with spin one and the spin orbit is perturbative in nature what do our results say well with spin orbit we find that the ground state is still close to X, Y orbital order with two electron in the X, Y orbital it's slightly modified but very close still to this configuration so starting from this we calculated the Hamiltonian and we found that we are closest to a spin one with the perturbative spin orbit coupling this is the minimal effective Hamiltonian which we can calculate where this is the super exchange coupling and this is the tensor the zero field tensor due to the spin orbit interaction and with this tensor we can we find the easy axis in the right direction we can describe the magnetic structure but also then we went and obtained the spin wave spectra which we now compare to experimental measurement from this paper here and as you see we can reproduce them quite well so this is the the well A so it's the full calculation we can even reproduce this gap which is called pseudo gold goldstone mode perhaps the next speaker will come back to this point so it doesn't go to zero because of this anisotope special anisotope of the zero field field tensor so this strongly support our picture there is still one debated paper which is this one which proposes which found experimentally a mode which very ascribed to to amplitude mode the Newtonian values to interpret is quite close parameter wise to ours and preliminary result indicates that we can put also this mode in this picture but I don't have the final result yet while I expect the next speaker to talk about this point later in his talk what we certainly can exclude that we are so close to this quantum critical points because the parameter that we have however we look them already in the atomic limit show that we are close to a spin one state with a perturbative spin orbit effects okay so I guess I'm coming to the conclusion so I have the last slide of the talk here I collect the result which I presented I've shown you that I worked done for this series where as I said many many different interaction which are they have the same order of magnitude so it's not easy to simplify the problem by neglecting some of these parameters because they are all important in some way or in some regime we have seen that the Fermi surface for example is crucial to include the spin orbit coupling but also the diagonal symmetry Coulomb terms which are typically neglected and effect which is likely to be important in many other materials with multi orbiters we have found that the spin orbit interaction is also key to interpret correctly the conductivity data according to what has been done so far for the insulating phase we confirm our old result that it is the LPBCA to SPBCA transition that is the main mechanism driving the metal insulator transition and the spin orbit plays a little role in that it shrinks a little bit the gap but it's crucial for magnetism although it can be treated in a perturbative way and this is all let me thank the people that did their work Gini Gorello, who developed at least the first version of the continuous time interaction expansion code that we were using which contributed to the spin orbit extension and did most of the spin orbit calculation and as myself are starting with the conductivity calculation and then I would like to thank you for your attention and you have more information on our website for the school for the students that are present