 So we'd like to start the last talk in the morning. So again, a continuation of more material. The speaker is Lora Krasen at Max Plan Institute at Shizugawa. Thank you. Yeah, first of all, of course, thank you to the organizers for this wonderful meeting and this even more wonderful place. And as we just said, I'm going to talk about unconventional superconductivity in Muare Transition Metal Dicaicogenides, which is work done in collaboration with these guys here from Bochum and Achen, and in particular Nico and then not really advanced or implemented advanced codes that allowed us to go to a higher resolution, which as you'll see will be important for what I'm going to discuss. Transition Metal Dicaicogenides can also be used as or can also be used for stacking and twisting and in that way create new heterostructures and study maybe new physics. Similar to twisted bilayer graphene, which I guess it's fair to say really brought new momentum to this field when correlated insulators and superconductors were discovered at the magic angle, so that now we also consider these systems really as a platform to study correlated physics. And during the workshop, we've already had the flator of talks about this very interesting system. Now, as we also heard, the Muare bands in twisted bilayer graphene are kind of special because they are based on a Dirac spectrum. This leads, for example, to the occurrence of this magic special angle where we see the correlative phenomena, and this can also lead to topological bands, at least in the very symmetric limit. Now, in Muare transition Metal Dicaicogenides, the situation is in that sense a bit simpler because the model, we know better how to model the Muare bands. And in particular, it's been suggested that you can simulate the triangular lapene with the triangular lattice Habat model with these systems. And this is what I want to briefly sketch, how you get this in the first part of the talk. And in particular, we've kind of taken this seriously. So I want to show to you what type of triangular lattice Habat models you can simulate. And then we use that as a motivation for what we'd like to do, namely for studying superconductivity from repulsive interactions, which is something Andre told us yesterday, how to do. And if you don't like the focus on superconductivity so much, let me also say that what we did is basically consider competing orders in a van Hover scenario and how that changes in his new Muare type triangle lattice Habat models. Experimentally, there's been also a lot of experiments on these Muare TMD systems. By the way, I'm going to use TMD instead of transition method like I co-organized throughout the talk because it's too long to say every time. And so I want to just briefly emphasize a few points of these experiments, of these many experiments that will be important later. First of all, let me mention that in these systems, there's no magic angle, but rather a magic continuum. So there is an entire regime of angles where you get correlated physics. And here you can see this as function of angle and displacement field. I think this was for twisted tanks in the selenite. And you see this insulating state appears for a range of angles. Furthermore, what was pointed out in that publication is that there seems to be a correlation between this insulating state appearing at half-filling and the peaks in the density of states shown here on the right. So the van Hover peaks. Furthermore, also away from half-filling, correlated phases have been reported. For example, generalized weakness crystals, which we've also already heard about, and striped phases. And the point I want to emphasize here is that this means that interactions of extended range must be important in these systems. So in the triangular lattice-hubborn model that we're going to get in the end, we cannot get away with just an on-site interaction. And then as I want to talk about pairing or superconductivity, let me also mention, following there has not yet been a clear observation of superconductivity in moiré TNDs. But there was evidence for zero resistance state in twisted tungsten disalinite down here. And here you see the line shapes that the resistance goes to 0. So as this is not yet clear, this of course raises the question is superconductivity exclusive to these heterostructures based on graphene? Or can we also get it in other systems? And then more generally, of course, we are still figuring out if the mechanism is conventional versus electronic. And in this talk, I'm going to take the theoretical point of view and just discuss what's possible in principle. And I'm going to show you that at least in principle, superconductivity is not exclusive to graphene systems. And I'm going to consider an electronic mechanism. So yeah, we're going to discuss superconductivity within the van Hover scenario. And the other point I want to say about superconductivity is that if it is unconventional in these hexagonal systems, we are basically guaranteed to get something interesting. Because if you go beyond S wave, almost all the first irreducible representations of the symmetry are two dimensional. So in other words, this means we would expect thematic or topological superconductivity if it is unconventional or if it's not S wave. OK, so with that, let's go into the modeling for this more transition method like co-generates. We've also already heard about this in a few other talks before, but let me briefly repeat some of the facts. First of all, we're going to have to distinguish HOMO versus heterobilayers because there are different types of TMDs. For example, tungsten-icellar-knight and molybdenum disulfide. And then as so this means that we can get a moiré pattern like here due to a twist angle, but also in the heterobilayers without a twist angle from the lattice constant mismatch. And then as you see, although from the top they form honeycomb lattices, when you look at these materials from the side, they have this basically three-layer structure and you see that inversion symmetry in plane is broken. So this means it will make a difference if you stack them starting at 0 degree like here or at 180 degree like here. And I'm going to call this AA versus AB stacking. So please don't confuse that with AA and AB ranging within the moiré pattern. Next, let's look at the band structure from the layers we start with. I want to consider the semiconducting transition method like heterogenize. So there's going to be a band gap. For example, here we see the band structure for a single layer of tungsten-icellar-knight. And this band gap will, of course, depend on the exact material the size of that gap. But the more we will consider the situation where the valence band maxima are at the k and the k prime point. And then because there's strong spin orbit coupling in these systems, there's something what's called spin valley locking, which means that you have a definite spin projection inside the valley. So I'm now sketching this band structure just schematically. That would mean just at the k and k prime point, that would mean that here this highest valence band at k, let's say, has been pointing up. And then its counterpart at k prime will have this spin pointing down. OK, and with all of these highest valence band, I should say, that we want to use to start building the moiré band structure. So all of that taken together means that there are differences in the setup for our moiré bands. And I want to show you a few examples now. So first of all, here I'm showing the top and bottom moiré, sorry, grillon zone of the single layers with the slide twist. And then here you have the mini brillon zone and the k valley and it's opposite on the other side of the k prime valley. And for the situation of homo bilayer AA stacking, if I sketch this band from before from top and bottom layer, they will be close to each other, as you see here in the mini brillon zone, both with spin pointing up. In contrast, if I consider a hero bilayer in AA stacking, the top and bottom layer will have a different band cap. So one of these bands will move down in energy and only the other one remains close to the firm level. And then the third situation I want to consider is again homo bilayer, so the energy will be the same. But in this AB stacking, which means that here we have to turn one of these grillon zones by 180 degrees. And now you see that k spin up is close to k prime spin down from the other layer. So these two bands have opposite spin projections, the ones that are close by. Then on top of that, we can of course add, let the layers talk to each other. So we have a hybridization in this first case. Whereas in the other two cases, this is not as important here because the other band is far away in energy. And here because interlayer tunneling will be suppressed due to the spin. So the active bands near the top of the valence band are quite different in all these setups. And now on top of that, we want to add the moire potential. I want to start because that's the simplest situation with this hero bilayer AA stacking where we have this quadratic band in the k prime and it's counterpart in the, sorry, in the k-layer valley and it's counterpart in the k prime valley. And here below, I'm showing you the example for times niacilinide on molybdenum disulfide without moire. And you see that it's through what I'm saying, the states from the highest valence band all come from the tungsten layer and the ones from the molybdenum layer are far down in energy. So let's take this parabola parabola and add the moire potential, which we can, similar to what we've heard about in tricep bilagraphy, expand in inverse lattice constants. What this leads to is that you have to fold back this parabola many times because you enlarge your units of re-space immensely. And then you get this spaghetti of bands. And if you zoom here, zoom in here to the top, you see that indeed the highest valence band, now the moire band that has formed is isolated and it has a very small bandwidth. So we can also tune through this band with a gate voltage. And it's exactly this band here that's been suggested to be well fitted by the tight binding model on the triangular letters plus in the actions to get the Habat. The effective model, this triangular letters Habat model has an effective SC2 symmetry that is formed by these combined spin value degrees of freedom from the two K spin up and K prime spin down values. And this is the first simulation that I'd like to look at on the phone. Then the next case I mentioned is the homo bilayer AB stacking where we have the opposite spin projections and the two layers basically on top to each other because of that. So we end up with a very similar situation. We've just doubled our degrees of freedom, which means that now we get an SU4 triangular letters Habat model instead of SU2. And then finally the third situation I'd like to look at is this homo bilayer AA stacking with the displacement field. It turns out without the displacement field you can again model this with an SU2 triangular letters Habat model. However, if you add the displacement field it will break that SU2 symmetry. And this as I'm gonna show to you will allow us to change the band structure and in particular tune the location of our van Hover points in the van Hover scenario that we wanna study. Okay, so with all that, I hope I could convince you it might be interesting to look into the triangular letters Habat model again. And here I'm showing to you the kinetic energy the tight binding part with the chemical potential in a 3D plot along a path in the beam zone and as contour lines in the beam zone. There's a special filling in this energy dispersion for the non-interacting bands. It occurs at three quarter hole filling when we start from the top emptying this band and that's the van Hover filling due to saddle points here in this dispersion which leads to a peak, a singularity in the density of states at this filling. These van Hover points occur at the end points here in the beam zone. And because of that, we expect that ordering tendencies will get the boost because of this enlarged density of states which actually corresponds to a logarithmic singularity. And on top of that at this filling we also have approximate nesting of the Fermi surface. So we have this almost perfect hexagon. So we do expect competing orders and this has been studied a while ago and led to this schematic phase diagram I'm showing here as function of temperature and filling near van Hover filling. We expect a spin density wave with these three nesting vectors and one and two and three and it's flanked by unconventional superconducting states mediated by particle hole fluctuations or it's been density wave fluctuations. And now what we wanted to see is how does this picture change for these moire models that I just showed to you? There is a quick question online. So question is, could you please elaborate more on how we extend the model from the SU2 to SU4 when the stacking is changed? That's the question. Thanks. Yeah, I was very quick on this. So the point is if we go from the one band here to the situation with the two bands here to first approximation, we can neglect interlayer coupling because they have opposite spins. The one from K value will point up and the one from K prime value will point down. So it's basically really just two parabolas which don't talk to each other. Then you can, when you set up the moire model you can perform a gauge transformation to get rid of the twist that makes the difference in the peak positions here. So you really have just for the one valley two parabolas on top of each other. And then in energy they are degenerate with the same two parabolas in the other valley which leads to these four degrees of freedom. So K spin up from top valley, K prime spin down from top valley, K spin up from bottom and K prime spin down from bottom. And then you do exactly the same as I showed here to this, you add the moire potential just that you start with four degenerate bands and not two. And this is how the SU4 symmetry comes about. Did that mean SU5 has to be? No, because they are really degenerate in energy but you're right, this is of course an approximation if you and you in reality that will be broken. And I think due to the first, the largest breaking comes from injections which bring it down to SU2, SU2. Okay, thanks. Sorry. Yeah, so I was saying that we wanna see how this when over scenario for the SU2 symmetric, how about model with just onsite in action changes for our moire models. And more specifically because like taking this simulation part seriously, we can really ask specific question with each of these types of moire materials. In this first case, as I've already motivated in the beginning, we can look at how the situation changes if we include the effect of longer range long repulsions to this SU2 model. In the second case, we can study the effect of having more flavors in our van Gogh scenario. And then in the third model with the displacement field, we can study the effect of the Fermi surface geometry and the location of these van Gogh points on the competing orders. And this is basically the plan for what I wanna discuss next. To study these questions, what we use is the functional renormalization group which you can think of as a discovery tool for these ordering tendencies and the approximation we use in a sense of very minimal approximation. So a truncation which basically amounts to a will scenario. This allows us to calculate the static two-particle correlation functions stressed by the interactions. So if you wanna think of it as the effective interaction in the system, and the important point is that we do that in an unbiased and momentum resolved way. Unbiased because in the equation that we use which is this differential equation I'm showing here, we do not only include one of these ordering channels, but all of them and importantly also their feedback onto each other. And momentum resolved as you see will be very important for the different states we're gonna discuss. The momentum resolution for one of our calculations in the PN zone is shown here. So with that, let me go through these three models I just introduced. First, we have the AA heterobiler. So the SU2 Hubbard model with long-range interactions. We take the values for this model from the previous papers that they've been estimated. And in particular, we take in the account long range into actions up to the third nearest neighbor repulsion. Their ratio has been estimated as well. And I'm plotting this here as function of the distance for these three interactions. However, in the system, we can tune the overall strength of the interaction. For example, by a substrate, which is what we do also in our calculations. So we keep the ratio fixed and vary the strength of the in our case nearest neighbor repulsion. And because of that, we're gonna study this entire range that I'm showing here. And with that, let me give you a summary of what we find here. You have a phase diagram as function of this nearest neighbor repulsion and fulfilling. On the bottom, I'm showing the corresponding perm surfaces and we find that near one whole fulfilling, we get basically the analog of the spin density wave that I mentioned before, just that our SU2 degree of freedom is now a combination of spin and valley. So in that sense, a spin valley density wave. It's signaled by the divergence of the corresponding susceptibility, which has peaks here at the end points where we would expect it because of the nest. Now, as you see, this instability is very unimpressed by the presence of this nearest neighbor repulsion because here we have basically vertical lines. In contrast, the pairing instability next to it, which is signaled by a divergence of the pairing susceptibility, seems to be stabilized by the presence of this nearest neighbor repulsion. And this is what I'd like to discuss next. So let's take a closer look into this pairing state. To this end, we saw the linear gap equation. And I'm showing the two solutions. There are two degenerate solutions that we find here on the right. The upper plot is along the Fermi surface, so this number of labels positions on the Fermi surface, and the lower plot is our data within the entire view also. And you see that these data points for both solutions are fitted well by the second nearest neighbor harmonics of the irreducible representation E2 of the lattice point group C6V, which is a very long term to say that we find G-wave form factors. And I'm plotting the G-wave form factors here in black, which I think you can see there. Now, these G-wave form factors have the same symmetry as the D-wave form factors, which would be the first nearest neighbor lattice harmonics of this era. So why did the system choose these second nearest neighbors over the first nearest neighbors? Choose the G-wave over the D-wave. And the reason for this is quite intuitive because it now does not only need to overcome the onsite repulsion, which we are maybe kind of used to, which is why we expect something beyond S-wave very often, but now we also have to overcome at least the nearest neighbor onsite repulsion, so the pairing is pushed outwards, and this is why we get the G-wave, so the second nearest neighbor instead of the D-wave, which would be first nearest neighbor. Now, of course, you could ask why do you care if they have the same symmetries? I cannot distinguish them. And of course, as I'm posing this question this way, the answer will be, I will be able to distinguish them, but for that, we have to look a bit closer in the superconducting state. Because as I said, there are two degenerate solutions. So the ground state will be formed by a linear combination out of the two of them, like this. And then we'd like to know which linear combination is realized in the ground state. And for that, we have to minimize the free energy, which I'm showing here. The free energy must have this form due to symmetry. We have the usual quadratic and aquatic terms. And because we have these two fields, that the two parts of the gap, we also have this additional quadratic term. And which minimum is realized will be determined by the parameters in this free energy, in particular, by the sign of this gamma. And that one we can calculate using our FRG data as input. And then we find that gamma is positive. So in the ground state, we want this term to be zero, which happens for a so-called hybrid combination. So a G plus or minus IG superconducting state. That this is favored has, this may be also intuitive because this state has a magnitude without zeros. So there's no zero if you go along the Fermi surface and that sends we maximize the condensation energy. At the same time, the argument of the states, which I'm putting here, winds four times if we go once around the Fermi surface. And this might already give you a hint that this is a topological superconducting state. When it is formed, it also spontaneously breaks. Pamper versus symmetry basically by choosing a winding direction. So by choosing G plus for this G minus IG. And we can classify the topological state by borrowing, for example, the topological invariant from Scrumion physics, when we define this pseudo-spin M based on the superconducting gap properties. I just like this type of calculation because we can then show nice plots of this pseudo-spin which I'm showing on the right here. And it might be a bit small, but what you basically see is away from the Fermi surface when the dispersion here is the largest, this pseudo-spin will point up or point down. It points up here in the middle and down here in the corners. But if we are close to the Fermi surface, so if this term, the dispersion term is zero, it will wind in plane according to the superconducting gap properties. And yeah, I'm sorry that the picture is not good enough but you would see that the spin winds four times if you go around the Fermi surface. So it repeats this phase winding, which is why this topological number is indeed four for this G plus IG state, whereas it would be two for the corresponding counterpart of the D wave. So for D plus ID, which means that although they have the same symmetry, they can be distinguished based on their topology. These are different topological states and this does determine physical properties because it determines the number of chiral edge modes. And this is why we expect an enhanced quantized response in spin and thermal holoconductance, which is determined by this number. More questions in the chat? Should I answer the questions in the chat now or later? Well, I don't think there's any question Okay. What do you mean independent on anything else? Oh, yeah, yeah, I think you can calculate this topological number in a different way. I just, this is the one I chose because it gives nice pictures. But it, you know, as it is a topological invariant, it does not depend on details of the system. The D plus ID and D plus ID have the same symmetry. In principle, we could have a superposition of both. Right. And then perhaps the backscattering at the edge is perhaps in here, where net state is still D plus ID. I'm gonna, so the question was, sorry, I should repeat it. The question is, in principle, because they have the same symmetries, these two states can mix, which is correct. And I'm gonna delay the answer for a bit because I have a slide on that. Maybe it's the next one. Yeah, okay, perfect. So I wanna say that this state we found is kind of robust in the sense that we found it for large parameter regime if we change the hopping values or these interactions. But we also wanted to see if it is robust if we go to stronger coupling, which we cannot really do without within our methods. So the thing we can do is to phenomenologically just add such a exchange coupling by hand and see what it does. And if you do that, the D wave will start to mix in. And I'm showing this here on the bottom. If this exchange coupling is zero, we have the G wave and then you see that the mixing occurs and for large values will get a nice D wave form factor. Then we can again use this data and calculate this topological number. And then we see that indeed there is the topological phase transition between these two when this exchange interaction overcomes a critical value. And then we're gonna change from this N equal to four to an N equal to two change. That's state. We did not consider the edge states so far. So back scattering between them. Okay, so this is what I would like to say about these long range interactions. And then go to the next model that I mentioned, the AB homoval layer where we have the SU4 Habat model. So now we have the tight binding part, just the onsite interactions so no long range interactions. And we again add this exchange interactions now in the SU4 action. We also tested the results I'm gonna show to you regarding breaking of this SU4 symmetry to SU2 by SU2 with different types of foods couplets. And here the main ordering tendencies you find here when overfilling are the following. I'm showing the phase diagram again as function of temperature and chemical potential. Now because we do not have the long range interactions we have D wave. So these form factors away from when overfilling and at when overfilling, we now find the change. So instead of the spin density wave earlier we now find a quantum anomalous Hall state. And this is an interaction induced quantum anomalous Hall state. So it occurs due to spontaneous symmetry breaking not because we start with topological bands. Some people also call this an imaginary charge density wave and it would lead to loop currents in the real space lattice, which I'm sketching here. And the reason why this quantum anomalous Hall state appears now is instead of the spin valley density wave is exactly the enlargement of our degrees of freedom because they, as we also, pardon the previous talk they will boost fluctuations of the charge basically via this diagram. Okay, and then finally the third model I wanted to discuss is this AA homo bilayer where I promised we can study the effect of changing the geometry of our primary surface and the location of the Van Hover singing narratives. Now the model looks like this. Now we only have the onsite interaction and the change is actually in the side binding part where we get such a pile space which occurs due to the displacement field. This phase is between zero and pi over three has been estimated previously for realistic displacement fields. And down here I'm showing the energy contours of the kinetic part for different values of this phase tuned by the displacement field and experiment. I'm showing spin up and spin down for zero displacement fields, they are degenerate. And we have the situation I described earlier with this hexagonal Fermi surface at Van Hover filling. These blocks are the Van Hover point. But if we add the displacement field and five becomes non zero, this degeneracy between spin up and spin down will be broken. This leads to these points, the Van Hover points moving along the mk direction. And at five equal to pi over six, they will meet at the k point and produce a high order Van Hover singularity. If we then increase the displacement field or this phase even further, they will start to move inwards along k gamma. So for example, for five equal to pi over three, we end up with this situation. In all cases, as you see, we retain a nesting between the spin up and spin down Fermi surface. And altogether, this of course leads to a rich phase diagram of different instabilities and potentially symmetry broken phases. So let me try to summarize this in the following. Here you see the instabilities we find, the function of this angle and the filling. And I'm sketching again in very small the different Fermi surface. The color code of this plot encodes the type of instability. If it's of density waves type or of pairing type. So red would be density wave and blue would be pairing. And this dashed line follows the location of the Van Hover point. Where you see that within our recoupling approach, we get the instabilities. Close to this Van Hover location, we get the density wave and then particle hole fluctuations of them can lead to pairing in their vicinity. Now let me try to summarize these different states to because we changed the Fermi surface and the nesting vectors, the wave vector of these density waves will also change as with the type of density wave because we break the S2 symmetry and this is what I'm trying to summarize here. So again, angle and filling in the first plot, the color code encodes the wave vector of the density wave and in the right plot, the color code encodes the spin-spin correlations. And to summarize these two, if you take them together, we see that when we started zero in our previous situation, we again have this density wave with the endpoints which leads to three degenerate style phases. So down here, then if we go along, we move and we move towards this high order when holding singularity, we get a hundred degree spiral with the wave vector of K and then if you go to really high displacement fields will end up with a ferromagnetic order. In between these wave vectors evolve kind of smoothly and also go to incommensurate situations. In the vicinity, as I mentioned, we find pairing and stabilities. So again, then we try to summarize the ones we find. Here with this color code now, the broken SU2 symmetry will allow singlet and triplet to mix and the symmetry is such that S and F will mix and P and D will mix which might be to P plus IP or D plus ID again. And the color code now encodes this. Basically, DP is blue and SF is orange fish. And what I wanna emphasize is that you basically only have one color on one side of the van Hover point of van Hover filling. It's either blueish or orangeish. So this means that, and as I said before the density of the details change if we vary this angle. But that does not have to seem to have a very strong effect on the type of symmetry of the pairing instability. So what I'm trying to say is that the pairing state seems to rather depend on the thermosurface. So on the filling, then on the details of the density of a fluctuations. Okay, and that brings me to the end of the talk. Let me quickly summarize what I tried to show to you. I argued that we can use these more RETMDs as simulators for different types of triangular lattice Hubbard model. And we use that as a motivation to study the van Hover scenario in these different cases. For the a heterobiolayer, we studied the effect of longer range glori pylons which led to this G plus IG topological superconducting state with the higher topological number. For the AB homobiolayers, we can approximate an SU4 symmetric Hubbard model. So we can study the effect of more degrees of freedom which led to a quantum anomalous horse state instead of a spin density. And then finally, adding a displacement field to a homobiolayers allows us to tune the location of the van Hover point and the thermosurface geometry which I showed in the very end. So thank you for your attention. Right, the talk is open to question. I have a question from online. So go ahead. Hi, so I'm Marco Greeley from Rome and I have a few questions which are related but they are maybe naive but let me ask them nevertheless. So one thing is that a lot of your scenario depends on this van Hover singularities. And now I was wondering whether disorder could be dangerous because it could be lift and reduce this van Hover singularity. And in whether possibly superconductivity could be a way to oppose this fragility because superconductivity could in some sense exploit the van Hover singularity and protecting it against the disorder. So whether you do investigate this possibility. So this is one question. And another question is related to the functional renormalization group approach whether there is a feedback on the electronic density of state and electronic spectral weight. So the self-energy corrections maybe in the order you are in such a way to see whether this broken symmetry phases or this fluctuations can give rise to again a lifting of this van Hover singularity in a self-consistent way. And then the last question is whether face separation could be another issue in this regard because interactions when there are density of states which are very large could rather easily induce face separation in the system. So these are my questions. Thank you very much. Okay, the first one was about disorder. Yes, it will probably smoothen the van Hover singularity which will reduce the tendency to these instabilities that I showed. I think probably superconductivity is a bit as long as we still have the particle whole fluctuations I would expect it to be a bit less sensitive than the particle whole instability the density rate itself. What was the second question? The FRG we did not include any self-energy effects just because we wanted to look at this first qualitatively this can be done and just would... So there are also people who develop codes for that. It just takes a bit of time to develop these codes. So basically did not study what you asked. In principle, of course if the ordered states is formed it will change of course the band and reconstruct the band structure. For the case of the spin density wave this has been studied back in these publications I showed before and it's been suggested that this leads to a half metal state at higher temperatures and then to an insulated chiral combination of the three stripes at lower temperatures. And now the third question... Is the phase separation. Is there... Did you check the chemical potential goes in the right direction when you change the feeling of your system that the compressibility... So I don't think we can really answer that question in our truncation. The only thing I can say is that we did not see any instability in the compressibility. But I think we would have to go beyond this approach to really answer this. Okay, thank you. Right. Thanks for the nice talk. I wanted to understand the robustness of the topological superconductivity when you apply it to the real material as opposed to the triangular lattice on the real material. So there is some weight at the first level coming from chalcogens as well. And chalcogens live on the other sub lattice of a sagonal motif. Now, unlike the triangular lattice which has very high symmetry, that system will have other symmetries. In particular, it will not have these irreducible two-dimensional irreducible representations because inversion is broken. Now, I wanted to understand how to understand... So this is the role of the topological superconductivity in the sense that in your model, in the triangular lattice model, the two-dimensional radius representation is important. In the real material, when you break that presumably topological superconductivity destabilized, but in the honeycomb lattice, there is also very phases. And those very phases might work in the opposite direction tending to stabilize topological superconductivity. So there's a competition between these two. And I was wondering if there are simulations on honeycomb lattices that might shed some light on how these steps in topological states are stable. I think, so I know at least of other effigy resides for the honeycomb lattice just for normal graphing. And then I didn't really look into this G plus IG state, they also saw this effect that longer range and they actually will tend to favor higher harmonics in the pairing symmetry, which I think it's, this is really very intuitive and quite universal effect. The question is if it wins in a real material, we did not look into that. The other layer of your question was about these other states, which is why I put this up again. I think, so on this scale of a single layer, they seem to be close to this highest valence band that I discussed where we formed, from which we formed the triangular lattice. However, on the scale of this effective band, they are probably far away in energy. So I don't know if they would play a role. The situation changes, of course, when you start adding the displacement here and you start moving all these spaghettis around. And yeah, does that answer the question? Yes, thank you. All right, next question. Yeah, I wanted to ask about the first part of the talk about the G plus IG with the extended the hubbard. So have you studied how sensitive the symmetry of the final state is to the range rather than just the strength? Meaning if I were to just chop off third nearest neighbor or second nearest neighbor, whatever it was, or vary the relative strengths between them a bit? Yeah, thank you. We did study that. So the minimum model basically you can come up with is just adding the nearest neighbor repulsion. We also varied the hoppings a bit. And as I said, the ratio we can fix with the tail and then studied this great range. But if you really chop them off, you the most drastic thing you can do with still keeping the essential ingredients of the centered interaction is this nearest neighbor. How about model that I'm showing here? And then we get the phase diagram that I'm showing on the bottom. You do get the same effect at large values of these or not large but large share values of the nearest neighbor repulsion. So these triangulation of the G-wave instability for small values. And this is actually known from previous studies. You get different types of pairing and stability. So I-wave superconductivity has been discussed which is a bit fragile in our calculations. And F-wave superconductivity on the side where you have pockets because that can compete because the nodes of the F-wave are not on the film sets. But all this richness goes away if you add the nearest neighbor repulsion then it's just G-wave. That's why I'm saying it seems to be kind of robust. I guess I was wondering about the experimental situation. So no superconductivity has been found in any of these systems, right? And you think, what are they doing wrong? You have a suggestion? Well, you shouldn't ask the theorists about that, right? No, I've asked experimentalism. What they are telling me is that we are kind of spoiled from the graphene systems. I don't know if you can- But people are trying this system, right? Right, but so apparently the problem is context and maybe disorder. So maybe we'll see something when the samples get better but yeah, I don't know, of course. So what are the phases that has been seen? They found these anomalous fault states, right? Yeah, that depends on the model. They found very, very different types of- Yeah, the weakness states, the mod insulator at half filling, striped phases. So in that sense- And do you find these also in your calculation? So we are different in a different coupling regime because that's why I said let's add a substrate because our method is not valid for this type of strong coupling situations. Yeah, so in that sense, we did not study this. I get precursor, I think precursors would probably be some type of charge density base, but we didn't go through this. And then so let me emphasize again that at least this drop in resistance was seen for this program itself. Yeah, quick, theoretical question. Is pair density wave an option here or stiffness prevents it all the time? For most situations, we don't see it, but there is an option in this last situation and there's a paper by Hong Yao about it. I'm trying to remember at which angle it occurs. Yeah, I don't want to say something wrong, but there is an option. Probably it's- Oh, no, I know. It's probably at the situation where you have the high order and holding point. And yeah, I think it's degenerate with the zero wave factor state. And they had an argument why you might push the pair density. All right. Just one more question. I have a question about after the six of them from three higher order random singularities and then with further change, they start moving towards the center. Is it possible to make them meet in the center and then what happens? Yeah, this actually happens if you go to five or two pi half. So you might notice that where we have the high order, this, the critical scales as well as the range of the instability are kind of bigger. And this also happens up here. And the reason is that at- Sorry, this is very small, but at five or two pi half, maybe you can see it here. We only have a single and hopefully an ad gamma. All right. Okay. Yeah, just a small question by one that graph. So you mentioned that there's some incommensurate wave factors that develop. Do you think that your model encompasses all the relevant terms or could those be broken down to commensurate situations from some other interactions with a moiré? Yeah. So what we do is really just the instability analysis. So I think they, they, in that sense, so what we find is the incommensurate that could change if you go beyond this. Okay. So if there is no other question, more important than lunch, then I think I'm going to close this session. Thank you so much.