 This talk will be given by Massimo Capone. It's a orbital selective mode physics and the phase diagram of ion-based superconductors from CISA. Thank you. Thanks to the organizers for the opportunity to talk. It's always nice to talk at a series of conferences that have really grown up with. Thank you for sticking around so late. So, okay, the title is basically the one that has been already introduced. I start with the list of coworkers. Sorry. So, there is a long list of coworkers involved in various sub-projects, but I mainly would like to list the main ones. Look at the Medici, who has a space AE, sorry. There is a little mistake here. And Laura van Farilow, who is at CISA now, and look at GeoNet, who used to be at CISA in my group until a while ago. So, I can skip this thing. And essentially, so what I will discuss mainly today is the origin of these orbital selective correlations that have been mentioned yesterday by Lara and the stock, and are now becoming a big point in the field of ion-based superconductors, and actually in general, of materials with multi-orbital correlations. So, okay, this is something that you basically all know about, but it's just to point out the fact that when you look at the phase time of the cuprates and you say, well, all the physics of the cuprate is basically about doping. A multi-insulator doping is a key word. So, organizing things as a function of doping is what makes you really realize that these materials are highly correlated. You start from this multi-insulator, and then you have a series of phases which are sort of less and less strange, less and less correlated, until you eventually end up in a Fermi liquid. If you take a snapshot, at some point, then asking the question about the role of correlation will be a little harder unless, of course, you take the snapshot in the antiferromagnetic insulator. This is actually an example where the thing of taking snapshots has been really important. This is a cluster of materials, the alkali doped metal fullerites that we worked a lot about in Fieste in these years, where now we have this nice phase diagram as a function of basically the volume per C60 where you have a superconducting dome with some maximum at some distance from a multi-insulator, which occurs when you put your fullerine buckyballs sufficiently far apart that the hopping becomes small and you end up in the SMOT regime. So you have now, you have this phase diagram that you understand quite nicely as something where electron-phonal interaction drives superconductivity, but it has to deal, and as a matter of fact, it exploits the fact that there are strong correlations. Actually, before 2000, essentially we only had materials in this region, in the region where, for example, the plot of critical temperatures, a function of the lattice spacing was quite boring. This is the result you find in a BCS theory, and everyone was concluding that these materials were just BCS superconductors and there was no correlation physics whatsoever. So you know, taking a snapshot in the wrong place may be misleading. So this is the point I want to make, is that now if you want to look at the iron-base superconductors, we should really look at the big picture, not take a snapshot in the wrong place. Okay, this is just an introduction, iron-base superconductors that is probably useless now. We have all this plethora of materials sharing this structure with layers of iron and something else. And we have this kind of phase diagrams. This is just a bunch of relatively old phase diagrams which all share an important thing. Superconductivity appears doping a spin density wave metal. So clearly if you look superficially, this phase diagrams you would like to compare with all of these where in all cases, you start from something which is magnetic, you dope it and you end up with a dome of superconductivity. But as we see in a second, then you all know there are some important differences. So I'm asking the question whether we really have to compare this phase diagram and I will give you a sort of twisty answer. So again, looking things very broadly, one has good reasons to think that the iron-base superconductors are somehow similar to the cuprates, so they are correlated. So for example, I mean, mainly say, most people believe that the mechanism of superconductivity is magnetic. Superconductivity appears doping this magnetic state and the metal is not a really good metal. It shows incoherence in several properties. But on the other hand, there are aspects which look point in a different direction. First of all, for most the fact that the parent compounds are not mod insulators. They are metals. This is a big difference. And there are some more subtle things. It's not so easy to identify Haber bands and features which are typical of mod insulators. And the density functional theory doesn't fail as badly as it does in the cuprates and gives you a band structure which at least has something to do with what we observe in experiments. Of course, there can be a sort of simple answer to this dichotomy. That these materials are just like in the middle between these two words. They are like intermediately correlated, whatever that means. Actually, our answer will be a little different. Will be that these materials lie in the middle in the sense that we have some electrons which are really correlated and some which are not. And this is the orbital selectivity we'll be talking about. Okay, this is just history. Clearly, given this intermediate situation, these are all very old papers of the first generation. Let's say a part of the community polarized in the weak coupling direction. Another part of the community polarized in the strong coupling direction. And some started to realize that there were some anomalies which are specific to this kind of materials. But besides this polarization, there was at least something everyone agreed on which is the key point in the study of these materials and which is clearly different from the cube rates which is the fact that the electronic structure has a multi orbital character. We have that many bands cross the Fermi surface and as a matter of fact, basically all the 5D orbitals are needed for a reasonable description of the fermiology. This is because the crystal field splitting is small and you end up with this very intriguing and complicated fermiology. I'm sorry. We start from parent compounds which happen to have six electrons in these 5D orbitals. So in principle, they might well have a multi-insulating ground state even if they don't. And we have to understand why this doesn't happen. Of course, the simple answer is that the upper view is small but we'll see that the answer is actually a little more interesting, I would say. The fact that we have many orbitals and this situation brings about two important things. One is that we certainly have to keep into account the Hohens exchange coupling and the other is that since we have different orbitals, we might have a different behavior between different orbitals which is the orbital selectivity I am mentioning. So now I jump into the experimental world and I show you a diagram where we collected various experimental estimates for the effective mass enhancement in the family of the so-called one-to-two family which is basically barium, iron-2, arsenic-2 doped either with Hohens and with electrons. So here lies the undoped compound. Here I'm doping with electrons. Here I'm doping with Hohens. So here there is many experimental data. This collection is actually a few years old. It's four years old. But anyway, I mean, these trends are confirmed by every experiment. Essentially, what we have to see here is that, one, the black dots here are specific heat, somerville coefficient and this does clearly a very nice and simple thing. It just grows when we go towards hold doping. So when we decrease the number of electrons in our bands, it does steadily and it reaches very large values of normalization. Blue one is an estimate from optics, again, of the effective mass while when you see many symbols, these come from experiments which have some kind of selectivity. For example, quantum oscillations or ARPES which really measure different pockets. And you see that essentially the picture is that, you see the black dots of the specific heat grow. The blue lines from optics are sort of slipping while all this selective data essentially range in the window determined by these two guys. Anyway, overall clearly you see that the degree of correlation if anything increases in this direction. All this data can be summarized by us assuming that in this system, there is a coexistence of different fluids with different effective masses. If it is like that, essentially you see that. If there is a bunch of different effective masses, if you measure the specific heat, which is basically some over bands, these will be dominated by the largest masses you have. While if you look at the optics, clearly the light guys, the one with the smallest effective mass will dominate the conduction. So everything is perfectly consistent. So you have a bunch of data if you can measure them and you have optics and specific heat that sets the boundaries. So basically the whole story about all this data is summarized by the fact that you have an increase of the correlations of the effective masses going in this direction and this increase is selective in some degree of freedom. Some part of the system has a much larger effective mass than some other. Then there are other experiments and of course let me also mention this much more recent experiment that has been already mentioned. In a few talks, the evidence of orbital selective padding in iron ceruleum where essentially one sees this hierarchy of quasi particle renormalization weights which you can picture at this level as sort of the inverse of the effective mass if we neglect the momentum dependence. So this is something that will happen actually in the kind of theories I will show, of course in the real world it's a little more complicated. So we have this picture which comes from experiments and you want to understand why this comes out in theory. So essentially as I already told you if you want to describe the system in terms of a Hubbard-like description, you're bound to consider also the Hohenskapling and the Hohenskapling is what we studied. It gives rise to the fact that the system wants to maximize the total spin and then there's a second choice wants to maximize the total angular momentum. This is how the interaction looks like actually for t2g orbitals and this is how it looks like if we write it down in the basis of orbitals. Okay, so essentially what we do now is to solve a Hubbard-like model with this kind of interactions which are dictated by the fact that I have a multi-orbital system. And I'm asking myself what happens to the mod transition coming to my first question so why these guys are not mod insulators? Sorry, so now actually one realizes immediately that the story is quite complicated. So let me start from a situation where you have the system and you have total half-filling. So for our case of five orbitals, this would mean five electrons in the bands. You look at z, z is this quasi-particle weight which you can picture now as basically the inverse of the effective mass enhancement factor. So when z will go to zero, we will have a mod insulator. When z equals one, you have an interacting system. Black line is j equals zero and if you increase j, you obtain this data. So these data are telling you something quite simple. If you increase j, it's easier to localize the electrons. z drops faster. So you need a smaller u to obtain mod localization. This is quite natural and it's also what you could have expected because mod physics is basically all about the fact that the electrons have a hard time moving because they have to fulfill the constraint given by the Harvard U. Now I'm adding the Hoon's coupling which is a further constraint. So the motion in principle, I can expect it to just become harder and harder and this is what happens here. Okay, fair enough. So this would tell us that it's easier to have mod insulators if you have the Hoon's coupling. So it doesn't look like what we are expecting looking at the iron-based superconductors. But actually the story becomes different if you look at any filling which is integer but not the number of electrons not equal to the number of sides. Like you have six electrons on five orbitals or no, no, two electrons in three orbitals and anything like that. So a situation which of course can give rise to mod physics. You can localize six electrons on one atom. That's perfectly possible but the picture is different. So now black again is u equals zero and the series of color is the same. Red, green, blah, blah, blah. So you see that if I increase j first for small u's I have the same physics. It's actually easier to localize but that the system changes its mind and you have this change of behavior and you see for example for this value of j you see mod localization becomes very, very, very, very, very hard. So the critical u becomes much larger. So now we have something for the first point to understand the iron-based superconductors. We have six electrons in five bends so we are in this situation and so it makes perfect sense that these guys are not mod insulators even if the hardware due is sizeable. But why does this happen? Of course because you know my naive argument would suggest that mod localization would always be easier. Actually if you do the simplest thing you can do the atomic estimate of the mod gap so basically just compute the energy cost of exciting one electron from one side to the other. So energy of n plus one minus plus n minus one minus the energy for the uniform situation. You can do this for the, for our problem and you find that while at half filling the atomic gap grows so it's again easier to mod localize out of half filling the mod gap decreases. So basically your system has some constraints but it just cannot open a mod gap. Essentially the idea here is that you have that even though you increase your interactions you still, since the own scapuling is very large you still select states which have different number of electrons so you can still have opening processes even when the interactions are very large. So you end up building a phase diagram in the density u space. So here this is the hardware due. This is the density. This is total half filling. Six electrons, seven, eight, blah, blah. So the black line is the mod insulator. So this is the black lines are telling what I just told. Here you just need a small value of u to enter the mod insulator. Here you need a big one. And you're left with a wide window where the system, if you have the same value of u which would drive the system mod inserting here it's still a metal and actually happens to be a quite a correlated metal as you can see here from this color plot that tells you when the color is light the z is small, the system is quite correlated. So this different behavior of the critical use opens up a wide window of interactions where the system remains a metal even though it's far from the mod transition. Now this is the same plot as before just cut from five to nine. What I want to draw your attention to is this other important quantities are the charge inter orbital correlations. So I'm looking at the correlation between the densities of two electrons in different orbitals. And basically the point is that these guys are frozen, which means black when I enter in this region. So this means that when I enter in this state which is dominated by the Hohns coupling and I have this resilient metal which doesn't want to die I have that the fluctuations between different orbitals are frozen. This makes a lot of physical sense because if I want to have the high spin what do I do? I just have, for example, up spin in all the sites which means that every attempt to have an orbital fluctuation would result in decreasing the spin so it will be unfavorable. So a direct consequence of this metal which wants to maximize the spin is a freezing of the inter orbital charge correlations which will be extremely important because this tells us that the behavior of different orbitals can be decoupled which is the crucial thing we were looking at now. So now this was just, sorry I never said that this was just a five orbital hubber model so no realistic bent structure, pure model. Now I enter in the world of the iron base superconductors and I'm plotting UZ for different pattern compounds. Iron selenium, barium one to two and this is some 1111, five different groups. And you see that overall the plot look all like what I have discussed before. First you have this region and then you have this rapid drop and then you see this flat region where the Z doesn't drop if I still increase U. This happens in all the cases with some differences which of course boil down to the nature of individual materials. But the thing you see is that when you enter in this region you also see a significant spread between the Z's of the different orbitals. This is plotted in an orbital selective way. So when I enter in this region where the Z are flat since the orbital fluctuations are frozen I have that an orbital selective behavior and even tiny differences in the original electronic structure can turn into significant differences in the renormalized electronic structure and in a significant difference between the Z's. Okay, so here I'm showing you that in the pattern compounds we find this picture plus the orbital selectivity. This is again the orbital decoupling. These are the spins. This is just spin-spin correlations in different channels. They all grow because the own scoupling wants to do that. And these are the orbital correlations. They essentially all go to zero except those which are basically irrelevant because the electrons here do not play a role in the response. Now I can go... As I told you, what I really care about is the doping dependence. So now I've shown you that N equals 6 is this physics. Now let's look at the doping dependence. So now I go back to my original plot. This is essentially the original plot just a little cleaned up from the experiments. And this is the theory. So you see, the theory is telling you that essentially qualitatively you have exactly the same behavior. So if you go in the electron doping region, everything is boring. The effective masses are small and basically orbital independent. But if you reduce the number of electrons, they can become large and orbital selective. So the experimental picture is confirmed by our theory. But now, when I looked at the experiment and just stated that something must be selective, now we know that this something is orbitals. And it's due to the Hohns coupling and to the Haber-Dieu. Now I can dig a little deep into this. So here, this is basically the same plot as before. Before it was the effective mass. It's now a plot disease. This is useful because now you see that 6 is the actual materials. 5.5 is potassium one to two, which is the maximum hold-doping you can actually do. But now in theory, I can take the liberty to go all the way down to n equals five. And I can clearly see that my Z's, which do not vanish here, do not vanish here in perfect agreement with experiments, would vanish if I would go all the way to n equals five to the Mott insulator, which makes a lot of sense because I told you that full half-filling, it's an easy situation for Mott physics. So that's what I should find. Now, so this is interesting because it's already telling me that what controls the overall degree of correlation is the distance from a Mott insulator, which doesn't exist in the iron-based superconductors, but it exists in the world of parameters. Now I can take a little, I do a little exercise. I can plot also the filling of individual orbitals. Okay, so clearly here overall the density is six, but it splits somehow between the orbitals. So I play this game, I obtain this thing, and you can see already that the orbitals which have, for example, a smaller Z like the X, Y here, are also those which have a smaller filling. So now I can plot the Z of the individual orbital as a function of its individual filling, okay? And as you see, all this thing gets sort of much simplified with respect to the original picture. You basically see that every individual orbital has a Z which is proportional to its own distance from a filling. So it's like every orbital behaves like a single band doped Mott insulator. Of course with some different in the slope which comes from the band structure parameters here. So clearly here you see that the influence of the N equals five Mott insulator. It's very interesting because it really simplifies the whole picture into a collection of single band Haber models. Now I can go a little further with comparison with experiments. And now essentially we focus on the guys which are more interesting for us. Which are those who have 5.5 electrons like potassium. You can change this with rubidium or cesium which have a different ionic radius and give rise to different bandwidths. Essentially this is a comparison between the estimate coming from some of the coefficient in experiments and in theory. So blue is experiment, green is theory without changing any parameters. So we do all the calculation with the same Haber view and the same Hoon's coupling, no fitting parameter. Even though we might even think that here since these materials are more correlated the Haber view might be a little larger. But we are being really over prudentic. We just keep the same values of you. And you see that starting from the black dots which is density functional theory which obviously fails except for this uncorrelated region here. It fails badly. You see that our calculation even using simple slave spin mean field not the best thing you can do in the world. And without adjusting parameters just reproduces very nicely the whole behavior. So this is really telling us very strongly that this picture is really what explains the degree of correlation of the different compounds at least in this family. Where now I think I convinced you why I want to focus so much on doping because clearly this gives me a solid control parameter that really controls the degree of correlation. Now I can use of course this picture also for other materials which are outside this single family. Okay. So, okay. Essentially this concludes the part where I try to convince you that the correlations in the iron base superconductors are orbital selective. And this part is explained by a multi orbital hubber model with a hood scapuling. Basically all these results do not really depend on using slave spins, dynamical mean field theory or other single side mean field methods. Now, okay. Yeah, 10 minutes. Okay. Now I can play a little game. I can try to compare the cube rates with the iron base superconductors. So what I, we took here are some data by Emmanuel a while back where I'm plotting, sorry the colors are quite horrible, the quasi particle weight now resolved in momentum space in the patches that Emmanuel discussed before when we introduced the dynamical cluster approximation. So basically now he has tiled down the one zone in two patches and this is the Z for the different patches and here is the density for the different patches. We have few data because of course here are many things happened, pseudo gap and so on. So we are in the safe region. But now you look at this green and yellow data. They follow quite nicely in the whole zoology of the different orbitals of the iron base superconductors. So this is at least suggesting that basically what is realized in the iron base superconductors as an orbital selective degree of correlation is mirrored in the cube rates or at least in the dynamical cluster approximation treatment of the cube rates as a momentum space selectivity. Now, if I go back to my face diagram of the iron base superconductors I can now do a very, very simple game. I can plot everything. This is basically, these are the two superconducting domes of the one to two. Here it's basically invisible for a reason but it's barely visible from my distance. There would be the spin density wave thing. So this is really experiments and I just plot everything as a function of the average orbital doping. So basically this is respect to five. So this is just six minus five divided by 2.2. So this is one to two. And this would be my putative Mott insulator. Essentially what is my story is that if I look at my models then at doping zero in the iron base I would have the Mott insulator. Then I enter in this selective Mott region and then eventually I will end up in the Fermi liquid. And superconductivity shows up right in the crossover region between the selective Mott physics and the Fermi liquid. If I go into the cube rate language this is still the Mott insulator. The selective Mott region is the pseudo gap which is basically a region where some regions of the Ruan zone are gapped. They have the pseudo gap and some others are not. And then you eventually end up in the Fermi liquid. So there is a clear correspondence between the phase diagrams and superconductivity appears right in the same place essentially if you for a moment neglect the spin density wave of course. In this picture the spin density wave is something that happens here that sort of I can be bold and call it an accident meaning something that happens because in these materials actually six electrons is not a random doping. It's a commensurate doping where you have the chance to have broken symmetry phases and that's why the spin density wave is realized but it doesn't really impact on the degree of correlation of my system. So this is now this is our way to compare the phase diagram of the cube rates and their own base superconductor. So coming back to my first slide I don't want to compare them just setting a correspondence between the spin density wave and the Mott insulator. Now the role of the Mott insulator would be played by this putative Mott insulator but still I have a sequence of phases which really mirrors what happens in the cube rates. Okay so far I only were taught besides the speculations about the metallic state with no broken symmetry whatsoever and I have not discussed any relation between the degree of correlation that I think we reproduce nicely in the experiments and the broken symmetry phases that actually happen in the real compounds. So here I will briefly say two things which are let's say much less exhaustive than what I'm discussed so far. So now I will discuss nematicity. Essentially what we do here which will play with Arion selenium which you heard already is like the place where we want to study nematic effects and their relations with superconductivity and we add a nematic perturbation. In this paper we consider three different options which I don't discuss in details but this is orbital ferroordering so a change of populations these are other options that have been discussed in the literature as I am changing bond order and the d-wave bond order this way. My bottom line is that what so here these are the z's these are my favorite plot of the z's the drop at some point. When these z's drops we don't find any spontaneous symmetry breaking in this direction so what we do is we add a field and we really do the basic thing we compute the response to the field and we don't find a divergence so the system doesn't want to spontaneously break this symmetry this is the susceptibility and you see they don't do anything particularly exciting but what we observed is that if we add this small perturbation this triggers a differentiation of the z's which goes perfectly in the picture I showed you before if you add some perturbation you will slightly change the populations of the orbitals and these will change their degree of correlation according to our scheme before so if I add this perturbation I trigger a differentiation of the z's and if I compute something which I can sort of call enemaic susceptibility which is like computing a sort of derivative of the z's with respect to the field I find an enhancement exactly when I enter the crossover region so the bottom line but to be honest the effect numerically is small and it doesn't allow us to for example to be compared with numbers that have been extracted by the analysis of Shamos Davis experiments that has been discussed before still so essentially so my list of results is that there is no spontaneous symmetry breaking there is this effect on the z's but the effect is small actually more recently Chimiao Xi and co-workers have found that if you take mixed perturbation Chimiao Xi has a combination of D plus S bond enemaic ordering you can even get some much more larger differentiation between the orbitals which goes in the direction of the experiments I mean I cannot add much on this because I didn't actually do the calculation now I spend my last second to give you some work in progress about superconductivity where I take a similar point of view as I did before I add a built-in perturbation so now I don't want to hope that pairing comes out from my Haber model I just add a pairing interaction which I will treat very simply in mean field in a BCS way now the game I play so these are plots of the gap as a function of the Haber view where I'm playing two different games one is to use just and so I do BCS using dressed greens function where the dressing comes from dynamical mean field theory so I introduce all these who's physics and essentially if you look at your data for different values of J you see basically the for example let's look here this pink data here are those where I just use a Fermi liquid approximation to my DMFT self energy so I just extract the Z and the real part of the self energy at zero frequency and you see what I obtain is that if I start from a given superconducting gap and I can cap you this is very rapidly killed but if I really include the real self energy that comes out from the DMFT which I remind you is momentum independent but it's orbital dependent of course it's crucial here you see that my gaps survive much, much longer and this effect is much more evident the more I increase the Hoon's coupling so essentially what I'm showing here is that our strongly correlated Hoon's metal which has all these properties that I try to discuss you it's much more it's much more compatible with the superconducting channels we have than a more standard correlated system which is just parameterized by a Z so if you want to go in the other direction the opposite direction if you look an experiment you need a much smaller Z to understand what is serving the experiments then if you use just a Fermi liquid approximation this is exactly what we are doing this is exactly what we are doing so basically this is telling us so this is really somehow working probably but it's really telling us that the coherent part of the Green's function is playing some role and actually the part which really matters is the one on the scale of the Hoon's coupling so of course we are not taking advantage of what happens at the scale of the Hubard U which would be strange it just never happens but we are exploiting a peculiarity of the system here the G that you are taking there is a finite energy scale so this result also depends on somehow do you have a cut-off if you want for pairing or not? Well... So you pair say it up to any energy level or you have a cut-off for pairing so the question would be imagine that you have a cut-off for the pairing so does the result depends on how large is this cut-off with respect to you or Jay? Yes, in the sense that of course if it's negligible with respect to the scale of the Hoon's coupling you don't see anything Yeah, I'm basically done so that's a list of conclusions which is what I said I mean just to be very rapid the Hoon's coupling favors orbital decoupling and orbital selectivity and I mean I would try to convince you that it is a simple mechanism it's not like black magic and that orbital selectivity correlation are the organizing principle of the degree of correlation of the normal state of the iron base superconductors and they are controlled by this hidden mod state which allows me to establish a link with the cube rates then coming to broken symmetry phases we don't find so far that they are the source of these low energy instabilities but they have a non-trivial interplay so at the very least when we studied these weak coupling instabilities in these Hoon's metal we have to carefully take into account the peculiarities of the Hoon's metal beyond the simple picture of an effective Fermi liquid between normalizes Thank you for your attention