 I'm recording in progress. You have to follow through. Okay, I will give an introduction to the theory of crystal biologi and the new things we have talked about superconductivity and as there have been many talks this summer in the subject I have included some new topics, which hopefully will be of interest to some of you. Nice. I hope from problem to problem. I don't know why this is not working. Okay, finally. I don't remember how these things are started so I will note that onto that I will just emphasize that there are people with many backgrounds in this audience, the connections or the differences between crystal by layer graphene and other more ostensibly study the cooperates and the mid ties and so on. And in many cases, as I mentioned, in, in this slide, the hover model has been the starting point to the study of correlations and superconductivity in these systems. The correlates states and superconductivity in carbon compounds were there before the experiments that might be from Pablo. So, people have found superconductivity in graph and what it's called graphite intercalation compounds which you can think of as heavily doped graphene layers. People have found the famous renormalization of the Fermi velocity due to the fact that it's a direct system and because of that long range coom interactions are not screened and people have found correlated behavior in the quantum whole regime in graphene and also superconductivity in doped fully relatively high critical temperature, but that was it. Definitely superconductivity have never been found in graph I don't mean they were in post but not very credible to say the least of superconductivity in graphite. They have been repeated very recently, but there was no clear evidence for strongly correlated behavior of superconductivity in graph. The idea that twisted by layer graphene were interesting started with tight binding calculations in about 2010. But before that, Joao López Rosantos, Muno Pérez and Antonio Castorneto have made a generalization of the direct model people used to study the low energy properties of single layer graphene to twisted graphene is a continuum model, a KP model. That model in 2010 was extensively studied by Alan McDonnell and Rafi Bistriza in a famous paper in the NASP where in agreement with the tight binding calculations, I just mentioned they discovered that there were very flat bands at certain twist angles and there was so flat that even the Fermi velocity, the effective Fermi velocity of these bands went all the way to zero. And then shortly later we cooked up a more simplified model where the bands were not approximately flat, were infinitely flat, which is now called the tight end. And experimentally, as I just mentioned it in the previous talk, such features, features related to new fan hop singularities and such things were already measured with a state with an STM by Ivan Ray and collaborators. Okay, the model itself is very interesting. I'm going to review this briefly, this continuum model put forward to study the properties of twisted by layer graphene as more twist angle. This is a scheme of what's going on. You have the large brilliance zones associated to the two twisted layers and the small brilliance zones are associated to the more lattice. And what you have to know, and you assume that tunneling concerns momentum tunneling from one layer to the other, and you have to notice that if you are in one layer, there are three ways in which a plane wave can tunnel into the second layer, which are not equivalent also the three K points in one layer are equivalent the way the, the plane wave which starts in one layer shows up in the other layer can have, it's a superposition, let me put it that way, of three plane waves, then these, these three plane waves are reflected back in the, in the first layer, and they become even more plane waves, and the whole thing complicates so there are multiple interference with processes, and at the end of the day for reasons not fully understood in my opinion, you have these flat bands for certain So, at the end of the day these wave functions have a lot of internal structure. And this is an interesting issue that was pointed out The point is not going to appear in this paper. You have flat bands and typically flat bands in a strong in a narrow band system they are associated to highly localized atomic orbiters, let's say F orbiters in uranium or something like that. That is not the case, and then they look the same no matter where in the band you are. That's not the case in twisted by layer graph, it happens that there is a big difference whether you are at the gamma point or at the K point of the mini brilliance zone. The wave function of the associated states are totally different at the K point these wave functions are piled at the regions as I mentioned before, while at the gamma point the wave functions are totally different they spread out more or less uniformly over the entire real space units. This is again another picture of the charge density of different points in the same flat band and emphasize this thing doesn't happen in a typical flat band strongly coordinated system. And things become more complicated as started by Leon Valens and others when you will be on the first magic, but I will concentrate on the first magic. As I said, in other systems the flat bands become because they're based on highly localized orbiters which are far away from each other and more or whatever they become flutter because of interactions. And they are normalized because they're the electrons and they need to drag excitation so the medium and the effective mass increases and advanced become flutter. None of these things is included in the continuum model I just outlined for you. In the non interacting model, all the parameters you plug in are in the scale of 100 mbs or larger, that's the interlayer tunneling and the interlayer hopping. And at the end of the day you get bands whose width can be a few mbs that's very unusual and has no counterpart in a typical strongly correlated system. And because of that because of the complexity of the model, this is just a cartoon with theories have tried to analyze it from many points of view, the Harvard light picture will requires first the knowledge of the Vanier functions, which is a quite challenging task. People have used the radar model and extension and exploit it in order to show similarities with the endow levels which is the other clear case of flat bands in condensed matter. And then into dimensions the density of state diverges that, of course, enhances the role of interactions. And then the singularities and then it is from how singularities also play. I can expect it to play a major role on what happens in the face that. First let me mention that the question of defining the Vanier functions something we've had to work it out. Here in Trieste is quite more challenging than what it looks. First, because you cannot do it starting from the continuum model a continuum model assumes that this value is independent of the other. Each valley is topological in the sense that it's like a level it has a final turn number it has a very curvature, which doesn't average to zero. And because of that the Vanier functions you define and need to have a power or takes they are not exponentially localized. And moreover, at the end of the day, you can calculate them for instance using the type of calculations as I mentioned, you get Vanier functions which are very unusual. This picture spinner shape that they're called. They are defined in an effective honeycomb lattice of the size of the more unit cell, but the peak note at their origins but at the center of the unit cell, and they seem to overlap Vanier function by construction cannot overlap, but they are defined in the same space again and common that's not the typical Vanier function you get in other flat bands, which you can characterize very easily in terms of atomic orbit. There's a question of how useful these Vanier functions are the question of the obstruction. You have, as I said, you define wave functions starting from individual valleys you have to pay the price of power lot as they're not exponentially localized. They are Vanier functions I have to warn you, they are not uniquely determined is you have to optimize them but you're never sure that you are choosing the best Vanier function appropriate for the problem at hand, but nevertheless, they have a well defined for and they are normalizable the power and it doesn't imply that they are not normalizable so they are actually not that useless as some people may believe you can take them and calculate umka processes and things like that related to the more they will not. Now let's go to the interactions, the very large size of the unit cell this large number 10,000 atoms in the unit cell and allows you to characterize the first approximation, the role or the relevance of the interactions. And the leading interaction is going to be the Coulomb interaction as I mentioned this charge accumulation at the center of the unit cell. Even if you have a gate which cancels the average charge you still have an inhomogeneous charge of zero average, but which is not negligible at all that defines for you an electrostatic self energy, which scales like the square root of the number of atoms in the unit cell or the radius or like the size of the model unit cell, all other interactions they scale like one over one over the number of the area of the unit cell. So, first approximation, the interaction you have to look first is the Coulomb interaction the long range Coulomb interaction rather than for instance in the atomic hover. And now, let's do the simplest approximation you can imagine which is the mean field heart rate calculation. The crucial point is that as you increase or to change the charge of the system you are creating this accumulation of charge at the center of the units. As Ali also mentioned the energy scale associated to this accumulation of charge is much larger than the bandwidth of the central bands themselves. And this charge, you create this potential this electrostatic potential you create changes in different ways, the states at gamma and at k because they have associated charge distribution which are changes from place to play. So, because of that, I will show you later, you're changing the not only, you're not shifting you're not only shifting the band you're changing the shape of the band itself, because the states are gamma do not feel this potential that they are uniformly distributed in the circle and this potential has zero average, while the states at k are strongly pushed up or down by this potential you create that charge up the system. And that, that defines for you this change in the shape of the bands implies like an effective interaction if you try to define effective interactions in terms of local picture. This is what we call in the way back for the fish for sister hopping. Namely, as the interactions change the very tight binding parameters which determine the banks, the shape of the bands themselves. For example, so what I'm trying to tell you this for instance down here this red bands are the bands are the neutrality point where there's no charge accumulation, but as soon as you put some charge in the system. The points of the bands are pushed up or down depending on whether you want the electron on the whole side, while the gamma point remains more or less an alter. There is a big change, and at the end of the day the band with this control, not by the initial model the model I described earlier, but by the interactions themselves the total bangui is more related to the interaction than to the initial parameters of the model. And then you have the complex possible fermi energies with the many pockets per value. Even if you start with flat bands interactions will make them dispersive so even in the kind of model you have this phenomenon and that's interesting because nowhere I know, I mean, in condensed matter I know of bands become much wider because of the interactions as I said, the typical effect of interactions is to make but not never why that yet. That's a good point. I couldn't rule out happening at other angles, at the magic angle I think it will not be the case because it's very clear that you accumulate the magic angle by definitions of a flat. And what they do is they go in the opposite direction they become wider, but I agree that that's a question to restart it maybe at other angles, it will happen. I cannot roll it up. There's a good question. I also mentioned this nice paper by Andre Bernévy, and where he discusses the flat bands as hybrids, the hybrids of itinerant electrons and a flat band, a kind of condolite band. So if you look at, there is like a flat band which is moving up and down. And then in addition to your gamma you have dispersive bands so so I, I like the model actually has a point on it and it is based on this difference between wave functions at different levels of the bivalve and so on of the flat. These are all the theory works which basically show the same results. There's no question that this change in the shape of the bands is there. And more interestingly, experimentally this fact that the bands that the bandwidth is never a few MVVs is always larger than that has been confirmed in many STM experiment. So you are in incompressibility experiments. And this is the kind of effect I am trying to, this is a cartoon of the effect I was discussing before this kind of effective hopping which is in news interested by layer graphing. For what I had it. Oh yeah, I always associated assistant hoping to horse and he's my colleague and he's a kind of told me like that, like in many other situations. Russian physicists have anticipated this phenomenon way before. Okay, let me make a short in discussion of a simplified model of a superconductor, which arises from repulsive interactions that was introduced by Lee and food. We have collaborated with them in extending the simplest model you can think you have an electron in a chronic spinless electrons in the whole honeycomb Larry the only the simplest interaction. You can cook up is a repulsive interaction between nearest neighbors, you put a strong two lattice bias, and you dope it slightly away from half feeling and it becomes a superconductor and you can do it just hand waving. And that was shown by Valentina Leon. Again, according to Misha told me that some results in this model were anticipated by Russian physicists before. And what, what our contribution was to study the superconductivity of this model in a more conventional way a la con la pincer. Discuss extensively and clearly yesterday, and what we found is that you have a kind of F way, the superconductivity is a kind of F way like, where you have values to different valleys, the gap parameter is more or less constant within each opposite science in the two banks, that's why it's called as well. And it's, so these are the data we calculated the thing is that you start with an interaction between nearest neighbors, but the dramatic picture outline here creates new interactions between electrons on the same sub lattice. You start with a parallel interaction range, but that interaction is attractive and this, this attraction overcomes the repulsive interaction, as you started with, and that's what leads to superconductivity. And there are similarities to, to twisted by layer because in the sense, you're playing with the intrinsic richness of the unit series in the model you have two atoms in the model you have in the properties of those two atoms by applying a sub lattice potential, and you can do it at a much larger scale in twisted by layer graphing and we think that there are similarities, they're not negligible similarities between this model and twisted by layer. Okay, one can go beyond hard tree calculation one can do hard tree for problem, and then one has a lot of possible polarized face or gaps face, there are many faces, which are very close in energy. This is our calculation, different polarized cases were facing the fail by a few me be per unit cell, but that is very consistent with the cascade I mentioned with the general properties known about the insulating faces in graphite. Let me mention here this nice and this animation put forward. This is an animation which shows how things change how the advanced change when you change the chemical potential. This is the first central one now the chemical potential reaches the first central one. And as you will see, the band shapes are changing. This is the situation where there is a gap because there is a substrate that doesn't matter very much. Now you get into this gap. Now you get to the conduction band. Again, things change again because the change in the number of times in the central ones, eventually, the chemical potential leaves the central bands. And that's the end of it. Nothing changes anymore. Everything looks like a real one. Okay, so again, this is the possible. These hard to fall resource give you a possible explanation of the cascade a number of polarized faces and so on, which may be out of safety and twisted by layer graphing. So let me summarize this part of the talk by saying that the relatively simple and straightforward hard to fall calculation give you a lot of information about what's going on in twisted by layer. So what is left is superconductivity how similar the superconductivity in graphene is to other strongly correlated system. And let me emphasize that superconductivity is very prevalent in twisted by layer graphene. And there are cases where you see superconducting faces but you don't see insulating faces, while that's definitely not in the case in a strongly correlated system. Okay, so a possible explanation of superconductivity we had extensively studied is a similar to the cooperates superconductivity due to intervening low energy most associated to broken symmetry faces. But I will emphasize now another approach based on the phone seminar phone latin your paper. Two of yesterday, which is a very early papers discussing how superconductivity can arise from the passive interaction. We will emphasize this particular diagram out of the four diagrams considered by the code and that is the reason is that there are four types of these diagrams because there are four flavors in graphene, and the others are not a general. So, so this diagram has, it's like the word of an expansion in field theory in high energy people use one of an expansions and any the number of watts which is three. So, and it's also we are at the same level, let me put it that way. And as they said, for Conan Lantiger studied the worst possible scenario for con Latina superconductivity, the electron gas in three dimensions. And then they also mentioned they didn't include that here but if you include these the diagram these charge fluctuation you should include also longitudinal acoustic mode because longitudinal acoustic mode coupled to charge fluctuation. And these are other well theory was by no means completely related to this topic I will not mention what we have done and this is another type of study due to this is a rather treating a dancing to remote but I will not discuss it here. The point I want to make is that this is a kind of excitation you expect in in twisted by laser in a film, you have plasma you have electron hole pairs and you have fun and they are all overlapping one on top of each other. This is not what you have in a typical to the electron that they are the phonons are very much the couple from the electron whole because they basically occupy different energy. And you can see it in this nice calculation of the dielectric function by Cyprian Lewandowski and then you leave it off. You have the continuum and the plasmons close together. This is our version of the same problem this is without phonons and this is with phonons. These faint lines are the longitudinal acoustic phones which are coupled to to the electron hole continuum. And so at the end of the day, you have a kind of more complicated response function which includes all these electron electron electron hole excitation sorry, and these phonons together. And you can estimate what is the effect that the normalization of the phonons due to the electron hole pairs and for twisted by laser graph it is by no means negligible. So the opposite is also true. The longitudinal phonons will influence the screening of the long range interaction. So this is the kind of diagram we studied it's very simple it's called Latvian linearize from Latvian, we just put electrons that day and one value and at the opposite value we couple them via the full screen interaction not we don't stay at second order like on Latvian. We realize that screening is very important these problems will go to infinite order. So by then I emphasize we're looking to this diagram. And we include the phonons as I just so this is just to show you the full gap equation we need to solve is by suggest. Yeah, absolutely. That's what I wanted to emphasize here is by no means a trivial calculation, because the very interaction contains a single momentum, but the screen interaction is a full matrix because you have a club processes. And then you have to add with q and n with q plus G where it is a reciprocal value. So it's a very complex calculation where you have to include form factors, which depend on the full wave function of the, of the crystal. I mean I had a more primitive comment. How do you justify this approach when small parameters the opposite. Could you say that again. Well, you can do the same for example, we are according to Bodhisattva should a good theory should be valid beyond its limits of applicability. I will come back to that. Absolutely. The reason we stay at zero frequency is because we can't do anything. We're aware that it would be interesting to go beyond zero frequency with our computer computational means we can. I will tell you what limitations that impact. Okay. These are results. So you do get superconductivity you get superconductivity comparable with critical temperatures comparable to to the experimental ones, also I said from the 12 ones that no respectable theories should trust the calculations of the critical temperature and I think the current point, but at least the range is all right the trends are all right because you change the hunger or change the feeling and everything Similarly with the density of state that are similar resolved but without the phone or by Mike Saleta at all. And they are great but they get a very low critical temperatures, but these are the main messages so far. The nature of the pain interaction and we have studied it in detail, we can exclude phone or we can simplify the matrix elements or simplify the matter, the way functions will include and basically both aspects are crucial complex matrix elements from cloud processes and all that. And at the phone. And these are the kind of order parameter we find the red, the black lines are the Fermi surface. This is a case where there are different pockets per valley. So, but the interesting thing is that we get something like in the Lianfu model, we get an order parameter which is constant, basically constant at the surface of each bar. And these are more calculations and these are some what we have included or what we don't have included we haven't included acoustic and optical forums. They will typically increase the tendency towards failing, however, so depending on whether one has S wave or F wave forums will be good, definitely they will increase DC for S wave painting, and what they're doing for F wave painting is unclear. We have not considered polarized phases. Also the method is basically the same for polarized phases. The only difference is that this factor of four in the bubbles in the screening you have to replace it by two. So, we expect superconductivity anyway. And we don't have retardation effects and that means that if our TC where was comparable to the phone on frequencies for instance that the calculation would not be reliable but that's not the case. I mean, it is at least a factor of 10 between the DCs we have estimated and the phone on frequencies. So we can expect that in this regime of not very strongly coupled superconductivity we have said. As I said, we didn't study polarized phases also in principle is something we have planned to do. And now let me emphasize that even the F wave superconductivity which is likely to occur, it will quite robust against elastic scattering, because only short range scattering scattering which takes you from one valley to the other will be harmful for superconductivity. Long range scattering which is the common scattering mechanism you have in very clean graphene. Only couples stays within the same valley and as the gap is more or less constant within the valley. That's okay. We have studied all the situations, but I want to move fast and they're totally different for instance a famous cases to twist a graphene by layers here you don't have this flexible bands, I have described to you is there more than the bands which are more rigid bands we have not completely finished it now the calculation for superconductivity for this case, but definitely the hard to follow calculations are totally different from the hard to follow calculation for single for two twisted monolith. But I will not go into that we have looked at twisted trial layers were definitely the situation is very similar to a twisted by layer pointed out by this one and collaborators. And we have superconductivity very similar to the superconductivity one has for twisted by. This is our other theoretical was some of them are actually very close to the one I have presented based on the same diagram. So it tends to reference our opinion that long range interactions play a crucial role in the superconductivity of twisted by layer. What's this. How can I. Okay. Okay, now I will move to a non more assistant. I don't know how much time do I have left. So, so how much time we have left. So this is a non more a system twisted by layer and we have looked to superconductivity in the same way these were other calculations on the topic, we have done this contract in just like calculation. This is the bands. This is the kind of that and we have you will have also included an intervalid coupling defined by you, and this is a kind of the effective interaction one has. And it's similar to what Andrew mentioned yesterday is weak. It's always a positive but it's weaker for small Q, and then for last Q. In fact, we do get superconductivity but at the very low energies, actually a very low critical temperatures, and with a very complex phase right up notice here, this is, this is the order parameter but it shadows, the Fermi surface, notice this crossing of superconductivity when the Fermi energy is very close to a fan hole singularity, which is consistent with the previous theoretical results I mentioned to you by Andrew. Okay, we actually will have other direct spin orbit coupling because they can take a group, if I'm not special groups and amazingly we found that the critical temperature grows with spin orbit coupling and then it's not so it will depend on this. Now let me move quickly to the results I had shown about under a bus catering as measured by STM throws in a twisted by laser graphene. I miss a theory paper I mentioned my center. I apologize about that these are two recent papers one of them was presented here last week by Lenny's last month. So what we have done is to study this type of superconductivity this F wave superconductivity, which we think phenomenologically it is not unlikely that it is the kind of superconductivity which is present in in in in crystal. So we first started a toy model, which is just graphene plus how they superconducting up which means that superconducting up has opposite science in. And notice that this is this are calculation for answer notice these H models, these are Majorana H most in in in only in in an armchair edge, and that is not quite the same as what has been studied in this paper by. And they are protected is under a stage in the middle of the gap are protected by the symmetries of the armchair edge and they are not present in the six. And if you go to a more sophisticated model of twisted by laser graphene you have even more mid gap or more a mid gap and the best days and some of them seem to grow, and they are precisely at the edges of the nano so this is an equivalent calculation but with parameters with a larger model you need to learn so on is a scale model. So the last thing I want to speak is about what happens when you put a tip on this system. So we modified a little bit the model outland by last month or by Ali. So we have a tip and instead of having a single complex channel going out we'll have to challenge one per value. But in one body, the gap is the superconducting up is positive and the other body the superconducting up is negative. And then let me go briefly I'll show you what it is. If you have close contact if you are in the BTK limit very good contact this is what you get at the red curve is under a mass scattering the blue line. This two grass is a normal back scattering. This is a way this is a way and as Ali mentioned, and then the black man found under a mass scattering is totally surprised in the, in the way case. So we have to tell also that the BTK model assumed that the Fermi velocity is the same on both electrodes. That's definitely not the case if you're tunneling into twisted by later I think because you go from a white band to a narrow band. What happens is that you suppress. Should I answer now or let me finish with some finishing because on this question is very much related to what I'm going to tell you right now. So what happens is that the mismatch in Fermi velocity is changes a lot at red mass scattering in the S wave. But there's something else you can do which has also been mentioned by Ali. The tip is a short range scatterer itself. So you can mix the values. So you can jump from one valley to the other and that changes a lot what happens in the S wave system. It doesn't change much what happens in the S wave case. And that makes sense that associated to Anderson theorem that elastic scattering doesn't screw up very much. But it changes a lot and basically what is happening is that the tip is inducing underage states inside the gap and these underage states break all the symmetries and then you can backscatter an electron come backscatter as a whole even if you have a S wave superconduct. So you are in the tunneling regime. Things are similar but you still have. I mean, this is a, I mean, then you, you, you miss the, the, the, the underage for backscattering in the S wave case. And are you also you don't have also a lot of under scattering but you still have a resonance due to the presence of these. You still have weakened as the tip is supposed to be removed from the sample, the entire valley tunneling is weaker here but you still have a resonance in the case of F wave tunneling. And you can also include spin orbit coupling because that will break the equivalence between the two. Valid but the model allows for it, and you still have a lot of structure inside the gap in the F wave case. And, and I see this is what I wanted to tell you, first of the largest interaction is a cool of interaction. Then the narrow bands are very fragile I didn't mention how strains or the substrate modifies the narrow bands but they also do a lot of damage to the bus. And then you can look at superconductivity in a diagrammatic fashion mediated by electron, four pairs plasmons and digital phonons. And you can look at this excitation low energy excitations and they can mediate superconductivity. This is it this is mostly the group and this is our institution and notice that the facade of our lab or our institution is a purpose a giant model structure, triangular model structure. Thank you very much. Thank you. There were a couple of questions online. I would like to read the two questions here. So this question is from Pierce Coleman. So interested by a graph in there is an essentially perfect values in the legacy is an additional orbital quantum number of the flat pan. So we have a perfect as you for and the input back as you asymmetry has anyone quantified how well the one of an expansion actually works. For example by actually calculating the diagrams that are normally normally one of us smaller. This is a very good question. And to my knowledge, no, that has not been done in our case. Again, is for lack of computing capacity. So online. Do you expect a re-entrance of the superconducting phase at high magnetic in. We haven't looked into that. That's a good question again. The principle that is doable within our limitation. I appreciate the question because we can look at it. Okay, so any questions from the audience. Nice talk. I just had a question. The diagram you should to see as a function of doping. It's basically superconducting everywhere right. Okay, I think it's on your last slide. Oh, yeah. Yeah, like this probably just because in a mean field theory, you only put one possible broken symmetry state, you define it or you don't. But I agree with you. This may not happen is like in the hover mode that you may have other phases which killed the superconductive, but in principle, if those other phases are not there. Then you get superconductive in an image again that if as the painting is mediated by electron hole excitation. If you're near these other phases, but you're not quite in a broken symmetry state that will show up in the screening function we are using there will be quasi soft mode so that is included in the calculation what is not included is the situation will have a real broken symmetry. Okay. I'm sure that I understood correctly in the calculation you did or the gap using the essentially the plasma and the phonons. The symmetry the sign changes between different family pockets so that would be an again without the notes, while the calculation you show at the end to spend Andre if assume that you have a gap varying inside each pocket. So not all gaps for the ballet. Yeah, yeah, yeah. Okay, so so the question will be then how can you imagine that somehow there's a long range media that the pairing can be modified to get the real F wave. See material the gap. Okay. The second good question first, our first calculations included only the long range interaction which is in travel and from those resources we couldn't tell whether one has S wave or F way. But the passive interaction will favor F wave, because there will be some. How are you turn, which may be couples the two values and that will immediately change this degeneracy between S and F wave towards F way. If on the other hand what matters is a coupling via optical phonons arcade for this time, then you will get the opposite. In the case of the by later we included everything, including how about you for the inter valley processes and there we didn't get exactly F way, but something similar. I mean within each valley. In each valley the order parameter is not constant anymore. So it's a more complex super, super complicated. It's not. If we have to work it out if it's shown there, and it has a change of sign. We didn't. I didn't show the color code but there is a change of sign. At least one, maybe more. Yeah, no, no, this is a very challenging calculation look at this case is a melee Kelvin, and in addition we have to deal with the large bill once we don't have to deal with narrow small more every once this is much more challenging and complicated. In other cases. Okay, so from the audience. This will be the equivalent but it's plus in one valley and minus in the other valley but the two valleys are associated are connected by a rotation. So this is a way, but it is the counterpart of plus minus in the nicknames I think. Exactly. Well, break is about 20 minutes so we are going to come back here by