 with a twist and strain. Very much looking forward to your talk. Okay, so good morning and thanks the organizers for the invitation. The ability to isolate atomically thin layers has made it possible for us to engineer band structures to change the Fermi energy in the system. And it has opened an enormous vista of possibilities for creating correlated states, by for example, creating flat bands as we discussed here and bringing the Fermi energy into the flat band and then watch new the emergence of correlated states that have possibly interesting topology. So what I'm gonna tell you about is I'm gonna start with the experimentalist understanding of flat bands. And I'm gonna put that in a context called Landau levels. And then I'm gonna give a very quick introduction to graphene. And then I'm gonna discuss two cases of engineering flat bands. One is in twisted by layer graphene. And there I will ask a question. We know that we have superconductivity and twisted by layer graphene at the magic angle. But I'm gonna ask the question is the flat band also topological and can we say, can we observe that experimentally? And then the second part of the talk, I'm gonna introduce another way of creating flat band by using strain if I get there. Okay, so let's get started. Okay, so let's get started with the most important people here. So the pictures here are of postdocs and students or former postdocs who've done all the work at Rutgers in red here. We have our theoretical collaborators. This is the team of Andre Geim who provide graphene on an open-dice cell night where we did the work on strain and the Japanese group that provided the boron nitride. So let me get started. Of course, the reason that we are also interested in flat bands is that you have the kinetic energy is quenched and when you're able to bring your Fermi energy into the flat band, you have enhanced correlation effects and you give rise to a whole slew of possible correlated states. So the most interesting one for this discussion here is superconductivity. So we all know very well that conventional superconductor, the BCS theory, the pairing of the paired electrons live in a very thin sliver on the Fermi C and the TC as we've seen already many times is exponentially suppressed here in the pairing energy and density of states. So the traditional way of increasing TC is to increase the density of state. For example, usually by chemical doping for in graphite, you do your calcium intercalation and you can bring TC up to a few Kelvin. Sorry about this. And in transition metal dichalconegide, you can use ionic liquid doping and people have observed the TC, superconductivity with TC as high as 15 Kelvin. So is there another, an alternative to that? And that was proposed in the 90s by this mostly Finnish group. And what they show that alternative approach to increasing TC is to create flat bands and then because of diverging, you get a diverging density of state of the Fermi energy. And what is very interesting about it that the pairing spans the entire Fermi C in this case. And what Volovic showed is that and his collaborator that in this case, TC is not exponentially suppressed, but it's linear in the pairing energy times the area of the flat band or volume of the flat band within the Brillouin zone. And these are pretty very interesting and intriguing predictions. In 2007, in fact, Volovic predicted that you could get a room temperature superconductivity in graphite due to in the presence of screw dislocations. And of course, there's a big question whether that is possible. So however, the caveat about all this, and we've heard it just now from Deban John and yesterday from Andre Bernovic, is that even though the pairing can happen at very high temperatures, unless the superfluid stiffness is finite, it's completely useless. And if you have a single band, isolated band in the Ginsburg-Landau sense, of course, the superfluid stiffness is going to go to zero because the mass diverges. So in 2015, this Finnish group found the way around this and they pointed out that if you have a non-trivial geometry of the wave function, then you actually have a lower band on the, which is determined a geometrical bound of the superfluid stiffness. And they found that this geometrical bound was a lower bound, which is proportional to the churn number here. However, subsequent words show first that, in fact, you don't need a finite churn number, all you need is a finite bearing curvature. And more recently, all you need is the non-trivial, you need a quantum geometry. So this condition has been relaxed quite dramatically. So in fact, so the bottom line is not only do we need flat bands, but we want some kind of non-trivial topology. Experimentally, what we can measure is the churn number. We cannot measure the metric or anything else. So we're going to be looking to whether those flat bands have a finite churn number. So now let me do a little bit of a detour for the students on graphene. And you've heard the lecture on Monday in the tutorial. So this is the crystal structure of graphene. It has two sublattices, A and B. The ingredients for the band structure are very simple here. Two-dimensional honeycomb structure and identical atoms on the two lattices. This gives rise at the low energy to this very simple Hamiltonian here. You have two Dirac cones. The low energy excitation are described by two Dirac cones, which are related, which each of them is chiral. And they're correlated by time reversal symmetry. And that gives a density of state, which is linear and vanishes here at the Dirac point. Experimentally, the fact, and this is where charge neutrality sits. So experimentally, the fact the density of state vanishes makes it possible with relatively small gate voltage to walk around here and to change the Fermi energy almost as well in this kind of a geometry about one volt in gate voltage by 10 to the 11th carrier per centimeter square, or it changes the Fermi energy by 30 millivolts. Now, if you add a magnetic field, you had to have a gauge field here. And that gives rise to lambda levels, which in this case go like the square root of the magnetic field times the lambda level index. And what's interesting here, and most interesting for the discussion today, is the n equals 0 lambda level, which actually, if you look here, it's made up of both electrons and holes. If you park your Fermi energy in between the zeros and an equal one level, you find that there's no quantum hole effect, your number is 0, and it's going to stay 0 unless you remove this degeneracy either by spin or by value. OK, now experimentally, what we do with we measure, in this case, a scanning tunneling microscopy. In scanning tunneling microscopy, we measure the tunneling current, which here, this integral here goes from the Fermi energy to the bias voltage that we apply. Rho here is the density of state, and this exponential factor comes from the matrix element. And so that allows us to do two kinds of measurements. One is what we call topography. So we keep the current constant, the bias constant, and we raster the sample so we can see where the atoms are sitting. Another measurement that we can do with STM is take the derivative of the density, dI dV, and that extract the density of state here as a function of energy, which is controlled with the bias voltage. So this is what you see, what it looks like in graphene. This is graphene on graphite. And when you apply magnetic field, you see the Landau levels pop up, and there's a lot of information that you can extract, in particular the Fermi velocity from this sequence here, quasi-particle lifetimes from the line width, coupling to the substrate, et cetera. So there's a lot of information that you can extract from here. And so basically, Landau levels are bequins-decentral flat band with non-trivial topology. So for example, if you park your Fermi energy here, you can see the quantum whole effect, which basically measured directly the chair number. Basically, this is the number of edge states that you have in your system. If you park your Fermi energy in one of these Landau levels, you can see fractional quantum whole effect. That is the emergence of interactive state. However, the problem is that, you know, we have to apply magnetic field here. So the question is, can we create flat bands without breaking time reversal symmetry? And there are, I know, of three ways of doing it. One is the magic angle twisted graphene. So you take two graphene layers, superpose them, and you have a certain angle where the band flattens here. That is kind of a resonance effect. That is when the interlayer and intralayer hopping are comparable to each other. This is not a trivial flat band that you would get by just putting the system in a periodic potential. So I'm gonna be discussing this. The other way of doing it is apply periodic strain that creates a pseudo magnetic field, which doesn't break time reversal symmetry, but act exactly like a magnetic field in the sense that it gives you these pseudo-lander levels, flat bands, and by bringing the Fermi energy into one of them, you can hope to create non-privile correlated states. The third way, which I'm not gonna discuss, but Andre Bernovich discussed it yesterday, is by partite lapis and their line graph partners. And these are a few examples. And these have naturally symmetry protected spiral flat bands. In this case here, for example, the LibBand. And these are very interesting systems that have not yet been really fully explored experimentally. Okay, so let me go to Twisted by Lea Graphene. So you've seen this probably many, many times. You bring the green layer on top of the red and twist it and you're gonna create a super period here, which is inversely proportional to the twist angle. But something that not many people may be realized is that when you shift it, when you shift the two layers, this is what happens. You see the pattern remains the same. The only thing that happens, you change the origin. Okay, so the pattern, the translation, the pattern shift, but nothing very interesting happening. You will see that you have more nice right present. The story is totally different. So if the two lattices have different lattice mismatch, of course, this period is a little bit more complicated and then there is a lower bound on the size of your Moiré cell. So Moiré patterns have been, macroscopic Moiré patterns have been known for hundreds of years in the textile industry. At the nano level, microscope atomic Moiré patterns have only been discovered with the invention of the STM. So that's when they were first observed, when people, the first thing that people looked at to when they build the first STM was to look at graphite and they saw these Moiré patterns. But what they did not realize, and then you see here, as you decrease the twist angle, the period gets larger and larger. But what people didn't realize for quite a long time is that not only that it generate pretty patterns, but this periodic Moiré also changes the band structure. So as you decrease the twist angle, you see the band structure changes. You see the appearance of van Hof singularities that get closer and closer together as you decrease the twist angle. And this is here 1.8 degrees and then you go to 1.1 degrees and the two merge. Another thing that you see that at the Fermi energy, you open a small gap here, which is kind of a hint that you might have correlated states. So here is the gap between the van Hof singularities. And you see here that it merges at a finite twist angle, which is quite a surprising at the time. And another thing that you see, you can measure the Fermi velocity as a function of twist angle by measuring from Landau level spectroscopy. And you see that it tends to decrease and maybe tends to zero right at the magic angle. So that is another signature of the flat man. The illustration there. Yes. For activity, is it possible to measure the... On the previous slide, how accurately is it possible to measure near one degree? How accurately? It's very accurately because when you look at the Fourier transform, we can measure it within a fraction of a degree. No, I meant... That you mean the twist angle, right? Very accurately, right? So you go to the Fourier transform, you compare the Bragg peaks from the atomic lattice to the Bragg peaks of the twisted of the Moray pattern, and you can see exactly that. And then it's going to depend on how large the area that you look at, but we are pretty accurate and it's much more accurate than we can actually control the twist. Andrei? What's the meaning of this? You can add in one whole piece to your experiment. This k-o in your work is in electric control. So they repeat for... We see them merging, but at this point here, you see the Fermi energy is here and there's a little gap right at the Fermi energy. So the minimum distance is our resolution, which in this case was not very great. It was like five millivolts. But I'm looking at the lower plot. You want to use two millivolts, what? I'm looking at the lower plot when you put... I guess this is the distance between one whole points as a function of the angle. Right, right, right. So I'm asking about the lowest points that you have because the vertical scale is in electrical... The lowest point that... Okay, the arrow bars are pretty large. This gives you an idea of the arrow bars. They go down as you go here. I took them out, but it is... I don't know, 0.1 degree or so. You mean... It's an energy. It's about five millivolts. Okay. Any other question? Okay. So in order to understand what's going on, let's go to reciprocal space. And here we have the diracons of the two layers, red and green. And you see that they're the distance between adjacent red and green diracons in one of the cave valleys is proportional to twist angle that gives rise to this. This spans the mini-brillow and zone here. And because each one of them is descended from a particular cave valley, they have opposite chiralities. And now we have too many brillow and zone with opposite chiral, eight orbitals per more so two from valley, two from layer, two from spin. So now let's go back and look at this Hamiltonian. This is Hamiltonian in one particular cave valley. And we're looking at these two diracons. And you see that if there is tunneling between the two layers, which means that you have off diagonal term here, then the intersection point between the two cones goes down and creates a saddle point. And by keep perp here is the interlayer tunneling. And it creates, so as we know, a saddle point in the band structure gives rise to one of singularities in density of state, a logarithmic divergence basically in the density of states. And at the magic angle, and you see as you decrease the twist angle, eventually you get to an angle, which is the ratio of the interlayer to intra-layer tunneling, which is about at one degree. And this is just an illustration of how the flat band is created. This is work from the Cretaceous group. And you see as you decrease the twist angle, the band closes here. And eventually when you go up to 1.1, you get along a perfectly flat band. The problem is that if you look at this, there's no gap to the remote bands. So there's nothing interesting gonna happen. You need a gap to the remote bands for something interesting to happen. And here is where a nature is kind to us. And let me show you what happens to open the gap there. So here is again, the Moiré cell, the bright regions here correspond to what we call AA stacking, where every atom in the top layer eclipses an atom in the bottom layer. The darker regions correspond to AB or BA stacking, where in one case, the atoms in the A sub lattice eclipse atoms in the B sub lattice in the bottom layer and vice versa for BA. Now there is something, we can plot something that we call registry index, which is the extent of which the top layer obstructs, the top layer atom obstructs the bottom layer atom. And if you have a rigid lattice, that the AA regions and AB regions have roughly the same size. However, if we look at the energetics of stacking, you see that the AA stacking is much, much more expensive. It's like 12, it's about 20 millivolt per atom, more expensive than AB stacking. So the system really doesn't like to be in the AA stacking. And because this is an atomically thin layer, it can really readjust to reduce the size of the AA layers. So this is how, for example, at 0.3 degrees, the registry index looks like this. So the AA regions have shrunk to basically 10% of the Mora cell. Experimentally, we see that as well. This is 0.28 degree. The AA regions are the bright ones here, ABBA, and they're separated by domain walls. Now in order for this to happen, you need a lot of built-in strain in your system. And how does that happen? It happens actually in a very interesting way. Every atom here, I don't know if you can see the arrows, is kind of shifted a little bit in the Mora cell. And they form kind of vortices that are, in one sense, you have a vortex lattice that is sitting on this Morae. It actually has opposite vorticity in the top and bottom layer. Very interesting phenomenon. Okay, and this reconstruction, this is what gives rise to a gap. So now, because of the reconstruction, we have a nice gap between the flat band here and the remote bands. And that's what allows us to do all this, interesting flat band physics. So bottom line in magic angle twisted by layer graphene, we have an isolated flat band. It's eight-fold degeneracy. The degeneracy protected by C2P symmetry inversion times time reversal. And the band filling is controlled by the electrostatic gating. So by gating the sample, we can move the Fermi energy within this system. And there's a way of, for us, this is the way we counted. Other people count some were different, but I think this is a good way of counting the filling. So N naught is one carrier per Morae cell. So N equals to zero is charge neutrality. Electron sector is positive N over N naught and holes is negative N over N naught. And minus four here would be an empty band, plus four would be a full band. Okay, so since, you know, back in 2010 when we did our experiment on suspended graphene layers, we didn't have much control over the twist angle. What we did have was CVD graphene that had all the possible twist angle in the sample. So you could just walk on our sample and look at different angles. But there were several important technical developments in the intervening years. First, Coridine et al and Jim Hones group discovered that boron nitride is a very good substrate. If you don't align it with that, they didn't know that with the graphene and this enabled the observational fraction of quantum whole effect without actually suspending the sample, which we had to do. The second development was the ability to perfectly control the twist angle. This was developed by Emmanuel Tutu and using these techniques, a group of Pablo Jariah-Hilaero and MIT were able to create this device graph twisted by layer graphene sitting here. They had a whole bar here and that, you know, they observed superconductivity but it created an avalanche of both theoretical and experimental paper. And this is, I'm listing that just the highlights, churn insulator, superconductivity, pseudo gap phase, nematic charge order, orbital magnetism, blanket dissipation, fractional churn insulator, all this observed in this one system. And this is a subset of the many, many papers that have done that. So I'm gonna ask now the question, are these, is this flat band also has some non-trivial topology? I can't measure the quantum metric but I can measure churn number. Okay, for that, there's one more theoretical thing here. Let's go back to the Hamiltonian here. So this is the Hamiltonian in the layer base. So top, in red we have layer one, in green we have layer two and there's something very interesting that happens when you change basis, when you go from layer basis to sublattice basis. And if you do that, what emerges is a kind of, it's a gauge field and you see here A and plus A, this is only in the K valley. So on the A sublattice, you have the gauge field has a positive side and the B sublattice is a negative side. Bottom line, it looks like a magnetic field except you haven't broken time reversal symmetry, right? So it looks like a magnetic field, it should act like a magnetic field. If you take the curl of that thing, you can estimate, it's kind of 120 Tesla, this is a rough estimate. And if you have a magnetic field and of course the first thing that you know, you're gonna get sudo-landau level. These are sudo-landau levels because you haven't broken time reversal symmetry. And this is in the K valley. So you see sublattice A in red, sublattice B in blue. And if you look in the moiré cell, you're gonna have a magnetic field pointing up on sublattice A, magnetic field pointing down on sublattice B. This looks like almost a Haldane Hamiltonian. Should the one hand still come from three or? It comes from going, it's inherent in this structure here. You just do a change of basis to the sublattice basis. No, the strain is not included here. No, there's a U here. This is the, of course the interlayer tunneling, that is when you do the transformation, it's buried in here. And the zero here is kind of because it's tunneling between the A B sublattice and that can be neglected for the N equals zero level. Yes? Do you think it's gonna come back to this thing? No, no, not at all. You're gonna see in a moment what it looks like. You know, it's kind of, okay. Now, if I add to that the K prime valley, of course it's time reversal symmetric, then it's gonna look like this. So you have the K and K prime valley. And now we have this flat bed, which is just the equivalent of the N equals zero lambda level, N equals zero pseudo lambda level, eight-fold degeneracy protected by C2T symmetry. So unless you break one of these symmetries, you're not gonna see anything that has to do with topology. Okay, so here we have again, the pseudo magnetic field induced by more A potential. We can describe these, the wave vector in two ways. One is what I call the churn basis. So you put on the left everything that have churn number plus one or field up on the right, everything that has churn number minus one. And then on the left we have K valley on sub lattice A, K prime valley sub lattice V and vice versa. You can also, now, whether you use this basis or, so for example, when I'm having a filling of three, I have all these states filled except for one, with minus one here, for example. And, but this is not gonna make any difference unless you really break the symmetry because you're gonna fill these seven states in different regions of the sample, they're gonna be all over. So you're not gonna recover the symmetry unless you break some of the symmetry. And I'm gonna get into that into a moment. The other possibility is look at a sub lattice basis. So you separate A on the left and B on the right. So in A we have K and K prime valley, churn number plus one, minus one, and vice versa here. In a moment you see why I'm talking about this. Now, if you look in real space, in the real more ASL, and that goes back to your question here, that this pseudo magnetic field creates this chiral current that counter propagating the diamagnetic current loops on the AB and BA regions alternating here with opposite sign pseudo magnetic field. So depending on how you fill your states, you might discover this orbital magnetism. Okay, so let's see how we can break the symmetry. So first let's see how we can break the C to Z symmetry, the inversion symmetry. We put the sample on a staggered potential. In this case, we have to perfectly align it with the boron nitride substrate. And let's fill it to a filling of, so now what happened, the A sub lattice is lower in energy than the B sub lattice. So when I start throwing in electrons, they're gonna first go in here and only once this is filled, they're gonna go up here. So for example, when I'm at the filling of three, this is what it's gonna look like. The last electron that say goes here, we have a hole that has, it has both, well, it's both valley polarized, spin polarized and sub lattice polarized. Okay, yes? The other thing of the one of the sub lattice. Yes. The alignment with the boron nitride is the whole thing is aligned. So which depending on, okay, boron nitride has two atoms, right? So the boron, depending on the sub lattice on the boron is gonna stay lower than the sub lattice on the nitrogen. So they have different energies. So that's how I break it. So if you go back to the graphene Hamiltonian, the reason we had this, the rock cones with no gap that the gap was the, was that we had exactly the same atoms on the sub lattice, the same potential on the sub lattice. The moment you break that, you open a gap. My question is really how do you use to go only at the one of the layer of the... You only alight a lower one. The higher one is already automatically aligned. I'm gonna get to that in detail if I get there. Okay. Okay, so experimentally people did this, two groups, Goldabert-Gorbans group and Andrea Young's group and Andrea Young's groups by training the sample with a magnetic field. So, you know, a quantum hole effect, anomalous quantum hole effect and with hysteresis here, which is a signature of orbital magnetism. Now, the other way to break the symmetry is to break the time reversal symmetry by applying an external magnetic field. This time you're not breaking the sub lattice symmetry, just the time reversal symmetry. And again, for C equal to three, this is gonna look like this. Now I'm looking in the churn basis, not in the sub lattice basis. And again, I have one hole here. So the system for the case of three is gonna be valley polarized. And sub lattice polarized, it should show an anomalous quantum hole effect, which it does. So here is the straight up formula. So filling of two, we go to precisely quantized state at about three Tesla, filling of three, it's precisely quantized at seven Tesla or so. So both depending on how you break your symmetry, you can reveal the underlying topology. It's not there unless you break some symmetry. It's not visible unless you break some symmetry. Okay, so let me now focus on this part here, breaking the symmetry with the boron nitride. Now, aligning two graphene layers. Experimentally, we know how to do it very, very precisely. However, we have no idea how to align the boron nitride. So you have a magic angle to survive that graphene, you put it on your boron nitride, you cannot control the angle, very, very difficult. And why am I saying that? In order to achieve this state, you had to be aligned to within a fraction of a percent of the angles. And let me show you why. So the conditions for that, you have to have commensurability between the boron nitride moire pattern and the twist TBG moire pattern, okay? And here is the condition for commensurability. We want to have the TBG at the magic angle. And so NPQ are three co-prime numbers. So the anomalous quantum whole expected in a non-trivial case is graphene, graphene is 1.1, graphene boron nitride is 0.55, or you can have graphene, graphene 1.1, graphene boron nitride is 0, and that's it. And there's the very tight bounds on that. So let me show you why. So this is my first layer of graphene, I'm bringing up the second layer of graphene and twisting it to 1.1 degree, okay? And now I'm going to add the boron nitride underneath and I'm going to twist it to 0.55 degrees. And you see, you create a pattern. Now, no matter how hard I try, I can't hit the exact twist angle. And I'm going to a shift. Let's say experimentally, I have no control over the shift. And you've seen that in TBG, the shift doesn't do much. But in TBG on HBN, it completely, it's not, it's completely different. Every shift is a little bit different. It's very critical to have both the twist angle right and the alignment right, unless you're at the perfect angle. But what does perfect angle mean? So this is a calculation that was done by this group here. Two groups here, this was Centiel's group and Allen McDonald's group. And Allen McDonald's show that if you are perfectly aligned, you have a sure number of either plus one or minus one that spans the entire sample. If you're just a tiny bit off. And this is, red is sure number plus one, green is minus one. Blue is minus one, green is zero. And this is the displacement from perfect alignment. And you see that to create regions that are not, you're not uniform here. And unless this red percolates throughout the sample, you're not going to be able to see anomalous quantum whole effect. So let's see how tight those bounds are. Yes? Yes. So here we have that you have a lattice mismatch with the boron nitride. Now when there are, I don't haven't tried that. At what point? I haven't tried that. I don't know. But certainly when you have a lattice mismatch, it's a mess. Okay, so how tight is it? Now that Andrea Young's sample was here. And you see, this is the region. You know, P graphene graphene versus P graphene boron nitride. This is a region that we have to be in order to see anomalous quantum whole effect. You have to align it within the precision of a fraction of a percent, impossible to do it. So we were wondering what's going on. Either these guys are just unbelievably lucky or something else may be going on, okay? So it turns out that something else is going on. And this is what's going on. So we prepared a whole bunch of samples and we looked at them with the STM. This is 0.46 for TBG, 0.45 for GBN and so on and so forth. This is the Fourier transform. And you see immediately that we have two groups here. On the left, we have commensurate because you see that the black peaks, the vectors are beautifully aligned. And in this case here, they're perfectly aligned. On the left, on the right-hand side, they're much further from being commensurate and indeed they are not commensurate, okay? And you can measure the built-in strain and it is not negligible because the lattices do readjust to each other. And if I'm looking at 1.03 TBG, 0.3 boronitride, this is what the topography looks like, perfectly aligned in the reciprocal lattice here, the Fourier transform. You see you have beautiful second order peaks and the black vectors here are perfectly aligned. So we have its commensurate. However, if you look at a number, it's really very, very far from the alignment that was predicted for rigid lattices. It's really like 50% away. And still we get perfect commensuration. So we get relaxation and Volodya Falco talked about this. There is a lot of, when you have these 2D layers, there's a lot of relaxation going on. You cannot assume rigid lattices. And then you have relaxation but you also have strain buildup. Now, what does that do? So the lattice is gonna try to align, it's gonna try. But as it aligns, it's gonna build up strain and that's gonna grow and grow. Eventually it's gonna give because it competes with the Van der Waals energy. So that will create domains. And here is an example of this sample here. And here we have three commensurate domains. I'm gonna focus on one. It depends on how far you are from the twist angle. This one is about 300 nanometers. So one can model this by a Franko Cantorova model. I mean, we roughly estimate that we need to do better. We would be good to have some theoretical help on that. Okay, so this is our commensurate domain. And here we have a tensile domain wall that it gives and then we have a hole here. It becomes incommensurate and then the organizing starts over again. So now we are looking at then we're taking density of state maps across this domain and we're taking them at different energy, zero millivolts, 40 millivolts, 60 millivolts. This is the domain. So you see the domain here lights up. We have a gap in the bulk, the domain lights up. There is a state within the mid gap state in the domain. So we walk away from the domain into the domain and plot on the vertical, these are the spectra as a functional position and you see zero millivolts here, 40, 60. We are filling of three and you see that we have a mid gap state in the domain wall. And no, this is just preliminary work and we trained a system and we look at, we trained a system with 500 millitas law and we observed that there is a well-defined chirality here and that chirality turn is change a sign when we train it with an opposite magnetic field. Okay, five minutes, okay. Okay. So, right. So it looks like first of all, nature helps align these systems and perhaps it's these domains that give rise to the anomalous quantum whole effect. We don't know yet, but it looks pretty promising and we still have to do a lot of work to understand this. Okay, I have three more slides and I'm gonna summarize. So it's scorecard for magic angle twisted by layer graphene. It turns out that in our group, only one third of the samples are superconducting. Pablo tells me that they're now better than that, but the point is that it's not so easy to get the samples to be superconducting. And one of the reasons, probably the main reasons is that there's twist angle disorder. No matter what you do, the twist angle changes across your sample and you can see that with various technique and unless the magic angle percolates throughout the whole sample, a transport measurement is not gonna give you superconductivity. You of course can see locally, but not globally. The other problem is the PC is never larger than four Kelvin. So unlike what, you know, what Volovic predicted, it looks like we're never gonna get above four Kelvin in this system. And one can speculate why. So going back to Volovic's very simple formula here, you see what enters here is the volume of the flat band in the Brillouin zone. And for magic angle graphene, it's only 0.04%. It's tiny. So this factor here kills DC. So unless we can maybe increase this, we can probably not get go much higher. Now, if you look at the formula here, there is a recipe for increasing it, the volume in reciprocal space, because if you make the twist angle larger, the magic twist angle larger, then the volume in reciprocal space is gonna increase. How do you make it larger by increasing the hopping between the layer? And that can be done with pressure, right? So that was done by the Columbia group. So they applied pressure and gigapascal, huge amount of pressure, and indeed they increased the magic angle from 1.1 to 1.27, and they increased DC from an initial, I think two Kelvin to three Kelvin. Not there yet. So it looks like this route is not that promising. So one has to find other systems of flat bands where perhaps the volume, first of all, that are less finicky and where the volume is larger. And since I'm at the end of my talk, I'm not gonna tell you what the system is, I'm just gonna give you a hint. And the hint is twisted by layer graphene, we have a periodically modulated interlayer hopping. So what we thought about, okay, instead of a periodically modulated interlayer hopping, you have a periodically modulator intra-layer hopping. And in this case, what happens, so that you can do that by periodic strain, and that gives you pseudo-lander level, which I won't get into, but we can discuss it later. So here we can control the strain period, no big deal. So that controls the mini-brillouin zone area, controls the volume of the brillouin zone in reciprocal of the flat band reciprocal space. And you can also control its strength by the amount of strain that you put in there, and that allows you to control the flatness of the band. And since I'm at the end of my time, let me go to the end, jump here to the end. Okay, let me jump to the summary. Okay, so summary of what I told you. So basically I only had time to talk about this part. So if you have a partially filled flat band, you can get a whole bunch of different correlated phases, TBG on HBN aligned, the lattice relaxation favors commensurate moray pattern to a certain extent, unless you're too far away from the commensurate angle, you form domains of commensurate lattices, you have the main walls that host mid-gap states, which may be responsible for the quantum anomalous whole effect. And thank you for your attention. Thanks for the last talk, you have time for questions? Just a simple question, since you are not sure whether domain walls host mid-gap states, the quantum anomalous solid effects, have you been able to between the samples, like at the play with the tweening such that somehow you remove these domain walls and see whether transport change. So the domain walls are created by the self-alignment. And unless you are at the perfect commensurate angle, which is 1.1 for TBG 0.55 for TBG HBN, at that point you should be able to create, get rid of the domain walls. You don't wanna get rid of the domain walls, but well, I mean. You're still gonna have the edge of the sample, which is gonna give you, you know, if all these ideas are correct, which they probably are, then it is gonna give you edge states at the edge of the sample. You're not gonna chase them away. So the edge is defined either by domain wall or just by the physical edge of the sample, wherever, whichever is smaller, my opinion. Okay, go ahead. How is the domain wall distributed in the sample? Is it uniformly distributed? I know that, oh, that's a whole different story. That's very, very interesting. So the domain walls, they like to be aligned across some crystallographic axis now, but if they're forced to be along another axis, they go like this, they go along the right axis and then they jump and then, and they become much wider and we have both tensile and shear domains. And we are working on writing this up, but this is, you know, a whole different talk, yeah. Thanks. So yeah, about the last part of the string effect, two questions. One is, what is the energy scale in the experiment they didn't get a chance to talk about compared to matriarchal graphene? The energy scale of what? For example, the flatband and the next band, remote band, what does the energy count? Well, I can show you that. So it will depend on, you know, pseudo-lander levels, right? If you have a periodic, pseudo-lander levels, so it's gonna depend on the pseudo-magnetic field, which depends on the strain, you can tune that. So, you know, we have about 100 millivolts between what, in this particular sample, if I remember correctly, 100 millivolts, the pseudo-magnetic field here's about 120 volt. I mean, here we have pseudo-magnetic field which is about eight of all. So this is graphene on pillars, eight test lines. Another quick question is, have you tried a one-dimensional periodic strain field? You mean tensile strain? One-dimensional, so you can create a pseudo-magnetic field which is periodic varying, but it's in one direction, not in the other direction. I see, so just have one array of pillars. Yeah, yeah. We have done that and we looked at it. There's very interesting quantum whole physics there and there's very, very interesting things going on there. We couldn't understand, we couldn't get anybody to help us understand it, so we never published it, but I'll be happy to talk to you about this. I think the end-of-scale maybe even more favorable in that case. I see, I see, okay. Yeah, if I remember correctly, they see the anomalous quantum hole in the plus three filling, but they don't see it at the minus three filling if I remember correctly. That's right, they only see it at plus three, but now somebody, they only see it so at plus one. Now, there is a strong electron hole asymmetry here, like for example, superconductivity is strongest in the whole side. It looks like this anomalous quantum hole effect, if you break the sublattice symmetry, then the samples don't show superconductivity. There's some evidence that they do, but it's probably different domains. So it looks, and that might help rule out certain models, but it looks like the breaking inversion symmetry is inimical to superconductivity. Yeah, my question was related to that. Do you have this domain wall kind of a picture in the whole side where they don't see the phenomenon? Does it help explain that as well? You mean no domain walls? Yeah, I'd like to see that in the other side. Nobody's looked at this before, so a lot more work needs to be done to correlate those things. So for example, we haven't done transport on these. Okay, and one last question is that, do you have some quantitative measures on the twist angle disorder thing? Like you say that there are disorders. The twist angle disorder is huge. I showed you like in STM, like you have maybe the best sample is at best, you have five or six more cells that are the right angle. Really, the disorder is huge. And there's no way of controlling that, because of the strain and because of how you put, by the way, if you do CVD graphing, they're much, much more ordered. If you can, the problem with CVD graphing that you can't control the twist angle, but those are the samples that we looked at back in 2010, they were much, much more ordered. Good, so maybe one final question. Okay, thanks. There's an online question. What is the interplay between the domain walls and the stability of the superconducting phase? Well, so as I just said, there's no, if you have a, if you're aligned with boronitri, there's no superconducting phase. It's somehow that the two don't like each other. Now, we don't know why, but that's what stimulates, the whole idea of this, the pairing happening different valleys came from that. So if they happen in different valleys, if you break the valley symmetry, then you don't have superconducting pairing, but we don't know. I mean, and there's no superconductivity in these samples. Good. So I would like to thank you again for your burners talk into discussion. I would say that we have been at 11.15 for about an hour today. Yeah, the quick question. So when you resolve different shunt then, like you have the flood rate of the sea density.