 Good morning to everybody. So we are starting our Thursday morning session. And I'm very happy to introduce Eliseldov, the first talk of today. It will be online talk. And the topic of the talk is Imaging the Local Bent Apology and Chernazide in Magic Angle Griffin. Ellie, it's your time. Thank you. Ellie, can you hear me? Ellie, we cannot hear you. Yes, sorry. I was muted. Yes, so thank you, Natalia, and thank you to the organizers for giving me the opportunity to present our recent work. I'll be talking about imaging topology and the Chernazide in Magic Angle Griffin. The picture that you see here is an image of orbital magnetization in Magic Angle Griffin. That will be the main topic of our discussion today. This work was done in close collaboration with Barcelona, a group of Dima Efetov, Peter and Giorgio, and our team at Weizmann, Matti, Samir, Aviram and Renil, Yurin Alex, and our theory friends, Juen, Kishav, Binghai, Adi, and Erez, and the HBN crystals come from NIMS. So let me start from the big picture, and in particular topology. So topology is a concept that emerged in the recent years as a central theme in condensed matter physics. And in particular, if your bands are topological, then they're described by a chair number. And we think of it as a global topological property. But still, topology remains quite an abstract concept. And I want to pose the following question. Can we image in some way or visualize topology on a local scale? It doesn't sound a valid or meaningful question because, as I said, we think of topology as a global property and of chair number as a global topological invariant. And therefore, the question of local topology sounds not very meaningful. So let's ask ourselves what is the source of topology, and this is Berry curvature. So if we integrate Berry curvature over the Fermi surface of a full band, it must be an integer equals C, and this is our chair number. So for example, in graphene, the source of Berry curvature is the Dirac crossing of the bands. And if we open a gap due to interactions or breaking of symmetry, the singular Berry curvature at the Dirac point spreads into the bands. And for example, at the bottom of the k-valley, the integral of the Berry curvature or Berry flux equals to pi. And in the k-prime-valley, it will be minus pi. In contrast to chair number, which is defined only for a full band, the Berry curvature is defined for any chemical potential. And therefore, we can ask ourselves a more general, even more fundamental question whether the Berry curvature can be visualized in some way. But again, this question is not clear because Berry curvature is defined in the k-space. And it's not clear what's the meaning of the Berry curvature in the real space. But you have to remember that Berry curvature in the k-space acts as a magnetic field inducing orbital magnetization very similar to a regular magnetic field. So now, orbital magnetization is a measurable quantity. And it is well-defined locally. So actually, orbital magnetization is the answer to our questions. Because if we can image orbital magnetization, we can address all these questions of local topology and local Berry curvature. The problem is that orbital magnetization is quite weak. And so far, it has evaded direct experimental visualization and image. Another important point is that the orbital magnetization due to Berry curvature has a close analogy to orbital magnetization induced in the quantum whole state by real fields. In this case, whereas here it is by magnetic field in the momentum space. So if you think of a quantum whole state, then in the bulk of the sample, you think of closed electron cyclotron orbits, which contribute a diamagnetic moment. So this is in the bulk. And along the edges in the incompressible states, the electrons follow what we call skipping orbits, which contribute a paramagnetic moment. And these currents are a result of in-plane electric field due to confining potential that gives rise to these chiral channels that flow in the opposite direction to the currents in the bulk. Similarly, the Berry curvature induces two components of orbital magnetization. One is the self-rotation of the wave function, which we call M self-rotation, MSR, which is analogous to this cyclotron orbits in some sense. And the other is the anomalous velocity of the wave functional wave packet, again induced by a transverse electric field, which causes the drift of the center of the mass of the wave function, which we call churn magnetization, because it is dominant in the incompressible states, in the churn gaps in contrast to the self-rotation, which is dominant in the metallic state. So by this analogy to the quantum whole, we generally expect the MSR and MC to be of opposite sign. So now we want to actually try to image the orbital magnetization and to understand some insights on the topology. Our tool is Quid-on-Tip. We take a quartz tube of about 1 millimeter and pull it to a sharp pipette down to a few tens of nanometers. And then we use one of the superconductors to do three-step deposition. In the first step, we coat one side of the pipette, then the other. And then the third step, we coat the ring at the apex. So as a result, where the leads overlap the ring, we have stronger superconductivity. And in the gap between the leads, we have weaker superconductivity, which form the weak links of a squid. So now this is how it looks in scanning electron microscope. This is an indium squid. So you see one lead, another lead, the gap between the two, and you see the hole at the apex. So now we have an extremely sensitive device residing at the apex of a sharp needle, which is ideally suited for scanning probe microscopy. These devices, these squids can be as small as 40 nanometers. They operate in fields of over a Tesla. Actually, for Molarine, we can go up to five Tesla. They have very low flux noise, a very good field sensitivity. And in particular, they have extremely good spin sensitivity. Namely, if you ask yourself, how many spins I can detect if I put them at the apex of the tip? And the answer is 0.3 Bohr magneton per square root hertz. Namely, we have enough sensitivity to image the magnetic field of a single electron spin. It turns out that these devices are also extremely sensitive thermometers. And we can study dissipation in quantum systems with a microkelvin imaging capability. But today, I will not discuss any of the thermal imaging. I will focus only on magnetic imaging. In order not to crash into the sample, we attach this squid on tip to a quartz tuning fork, which has a sharp resonance at about 30 kilohertz. And we monitor the change in this resonant frequency as we approach the surface, which is essentially a force sensing, which allows us, like you do in atomic force microscopy, which allows us to scan at a height of few tens of nanometers above the surface. So today, I'll be describing orbital magnetization in magic angle graphene. These samples come from Barcelona. So this is a sample in the form of a hole bar. And again, it is a magic angle encapsulated in HPN. The angle is about 1.8 degree. And these are the transport measurements at 300 male kelvin showing the RX-X and RX-Y. And actually they display all the beauty and the fascinating things that you have in magic angle graphene, like correlated insulating states at integer fillings, land off fans, churn insulators, these lines here, superconductivity, the suppressed resistance goes to zero in this region. And in addition, this particular sample also showed magnetism in this region of filling factor one, namely one electron per moire unit cell. Orbital magnetism has been observed in several types of devices, both in magic angle graphene and in trial area graphene by a number of groups. Usually the orbital magnetization transport is visible near filling factor three. In this case, we see it pronouncedly in near filling factor one. So in transport measurements, if you measure RX-Y as a function of field in this window, this is what you see. The RX-Y shows a pronounced hysteresis around zero magnetic field at close to filling factor one, which is a clear signature of orbital ferromagnetism. You can also scan, sweep the filling factor at the fixed magnetic field. And also here, you see a clear hysteresis as a function of filling factor now at the given low magnetic field. It is less intuitive to understand that it arises from orbital magnetization, but in fact, it is equivalent to fields as sweeping. We will be mainly focusing on sweeping the filling factor rather than magnetic field. So these were our global transport measurements, conventional ones. So now we want to focus on our local imaging. So this is the schematic setup. And this is our squid and tip. This is the sample of the magic angle graphene with top and bottom HBN with graphite back gate. And we ground the sample. So all what I will be showing today is there is no transport current flowing in the sample. We just apply a DC voltage to the back gate, which can control the carrier density in the sample. And we will be focusing in the vicinity of filling factor one. And in addition, we apply a small AC back gate voltage which modulates the carrier concentration of the filling factor by 0.08 in the particular dataset that I will be showing you. And what we measure is the local AC component of the magnetic field. Namely what we measure is by how much the local magnetic field changes as we add and remove 0.08 electrons per moire unit cell. This is our signal. We are at 300 millikelvin and applied field is very small, 50 milli Tesla. And this is what we measure. So this is the position along the sample in microns. And this is the AC magnetic field that we measure locally in units of nano Tesla. So these are very weak signals. And we're in this particular case near a filling factor one. We can take this local magnetic field and do magnetization reconstruction, essentially inversion of the BIOS of our law and translate it into local magnetization. So this is the local differential magnetization in units of more magneton per electron. Namely, if we add one electron to the system by how much the local magnetization increases, you can think of it as if you add one electron per, let's say more ray unit cell, this will be the change in the magnetization per unit cell. And so you see both positive, red and negative, blue magnetization with a scale of about 20 more magneton per electron. So this is very large magnetization, much larger than the spin magnetization that arises from spin. And essentially we will be ignoring spin focusing only on this orbital magnetization. So this image was acquired at this filling factor after initiating the system at large, at sorry, at low filling factor, let's say zero and then sweeping the filling factor up to this value and taking the image. We can do the opposite. We can initiate the system at large filling factor and then go down to the same one and again take an image upon sweeping the filling factor down and then we can just take numerical difference between the two. So just one minus the other divided by two and this shows the difference here. So the first thing you see that in large part of the sample magnetization is completely reversible regardless of the history. And in other parts of the sample the magnetization is hysteretic. And if you look carefully at this pattern looks identical to this pattern which means that this entire region has flipped its magnetization as a whole between this and these images. So this is at filling factor of close to one but we can do it at various filling factors. And here's the movie of the orbital magnetization as the function of filling factor from 0.7 to 1.17. And we start from low filling factor you see mainly red patches. So these are red means paramagnetic namely this is favorable energetically whereas blue is diamagnetic namely M dot B term is unfavorable. So we start from a few patches which are favorable magnetically and we increase the filling factor you see that the magnetization grows but also there are quite a lot of blue patches appearing and overall the magnetization grows and so does the hysteresis. And now we're at filling factor one where we reach maximum magnetization and if we continue to increasing the filling factor both the reversible and irreversible magnetization decrease. So we can take this data and make a three dimensional rendering of it. So these are the two spatial coordinates and this is the filling factor through one. So this is upon sweeping the filling factor up this is upon sweeping down and this is the difference. And we're looking at the different cuts of these three dimensional presentation. You see that the orbital magnetization is highly non-trivial having both paramagnetic and diamagnetic regions and sharp transitions between them and quite complicated dynamics. So in order to make some deeper understanding of the orbital magnetization we first do some band structure calculations. So we assume complete degeneracy lifting. So we are talking about four bands, two K bands and two K prime bands and we ignore our spin. And this is the band structure calculation of valley K for example, of this conduction valley. And I'm sure you are familiar with the general structure of the band structure in magic angle graphene. And it has a sharp maximum the bands have the dispersion has a sharp maximum near a gamma point in energy of a few milli electron volts. We can calculate the barrier curvature in this single particle picture. And we see that essentially most of the barrier curvature is concentrated at the top of the band near the gamma point. And we can also calculate the orbital magnetization arising from this barrier curvature and not surprisingly it is also concentrated where the barrier which is concentrated near the gamma point. And the numbers are comparable to the experiment of the order of 30 more magneton per electron near this gamma point. So now we can plot the calculated this differential magnetization which is what is essentially the self rotation part of the orbital magnetization as a function of filling factor. So at low feelings we are here at the bottom of the band. So the self rotation magnetization is very small and when we approach the top of the band, it shoots up and the self rotation magnetization is zero in the gap. And then if we go to K prime value, we get the same thing just with negative sign. The second part of the orbital magnetization is the churn magnetization which is essentially zero in the bands and it has a maximum negative sign in the gap and in fact in the gap the churn magnetization derivative of the magnetization with respect to the chemical potential is just given by the churn number. So the total magnetization should increase gradually has a maximum at the top of the band and then dive into negative region in the gap and so on. So this is what we expect from band structure calculation. And if we look at our data, this is orbital magnetization, the same that is calculated, the total orbital magnetization that is calculated here as a function of position and filling factor and we see that it looks very different from this simple picture. But in fact, after you stare enough on it and analyze it, you can make a lot of sense. So let's start from this patch. So we start from relatively low feeling factors. So we're somewhere here and we expect to have a sharp red followed by sharp blue orbital magnetization and this is in fact, what we see a sharp red followed by a sharp blue, which means that we have now shifted our Fermi level into the gap. So we understand where we are here. So now I want to understand what's going on as we continue to increasing the filling factor. For this, rather than looking at this differential magnetization, we should look at the total magnetization, the integral of this. So the integral of the self rotation magnetization increases gradually as we move towards the top of the band has this overshoot here and remains constant in the gap because there are no density of states there whereas the turn magnetization is zero in the metallic state and decreases or increases in magnitude linearly in the gap as a function of chemical potential and becomes largely negative. So now, so the total magnetization should increase, become positive and then in the gap region it should decrease and if the gap is large enough, the total magnetization will become negative which means that it will become highly unfavorable, energetically. This is the point where the magnetization should flip side and this is this point here. So now the system has two options. Either we can continue to fill the next band, the K prime band, which means that we will become negative and unfavorable energetically or we can recondense all the electrons from the K valley to the K prime valley, flip all this, recondense all the electrons between all the valleys and therefore flip all these curves, flip their sign through a first order transition. Ellie, there is a question from under the table. Yes, sure. Ellie, can you elaborate on your electronic structure? Is there other evidence that the band you label as K reach a maximum at gamma at approximate one, at near approximate one? You are talking about this band structure calculation. So this is just a straightforward single particle band structure calculation. If this is the question. Yes, if I can, this was a question, if you can hear me. Yes, I can hear you very well. Very good, sorry. I'm just staying in my room just out of precaution. So you have bands at K and K prime. These are, they have direct points of charge neutrality. Then you start feeding in these bands. And the question is- We assume full degeneracy lifting due to interactions. The degeneracy lifting is by hand. The fact that you see orbital magnetization, it means that the time reversal symmetry is broken. If all the bands are degenerate, orbital magnetization is zero by definition essentially. Because the orbital magnetization of K and K prime values cancel each other. Absolutely. But if you start with the band, with the direct point at K and start feeding in the band. Yes. Many things happen before the band reaches maximum at gamma point. You probably go into one point, et cetera. Yes. I'm asking simple, maybe just to not to interrupt your talk, a question is, is there any other experimental evidence that around N equal to one? I'm talking about N equal to one. The band, one of the band reaches maximum at gamma. Okay. There is a gap at new equals one. Okay. This is, we see it locally. So the fact that you see negative orbital magnetization actually means that here you are in the gap. Okay. So this was the previous picture. And the fact that you have, so people also observe quantum anomalous whole state, quantized, which also tells you that you must be in the gap for quantization. Absolutely. So the fact, yes. Yeah. This question was about gamma, about point gamma. How you know that it's physics related to closeness to gamma point. Just from experimental perspective. Experimental perspective, again, we know that we are in a gap at filling factor one. So you can think of complicated scenarios where you will be at filling factor one, we'll open some gap due to some interaction without having one full band and empty three bands. But, you know, we start from the simplest scenario and try to make sense. Okay. Yeah. So this is the conventional simple picture of the general soliciting due to interaction where at filling factor one, you have one full band, completely spin and valley polarized and three empty bands. Okay. Okay. Yeah, continue. Good. So now we're exactly at this point and this black line, you see that all the magnetization changes discontinuously in contrast to here where the magnetization changed sign, but in a continuous way. Here along this line, black line, the magnetization has flipped its sign discontinuously. So this is the first order of transition which we interpret as recondensation of carriers from K value, let's say, to K prime. And then we continue and then the magnetization becomes positive again and we continue now filling the K, which is now empty, the K value whereas the K prime is full. So we understand this slice. But now what about this slice, this patch? It seems to have a completely opposite behavior. So for this, we have to change slightly the way we think about the bands and rather than classifying them in terms of valleys and for each valley, you have a conduction and valence band. We want to think in terms of churn bases where the valleys are classified by the churn number and the churn number is dictated by the product of the valley and sub lattice polarization AB. So instead of drawing schematically our bands like this we should draw them like this where all four bands are equivalent and given by the product of valley and sub lattice polarization. And the sub lattice polarization is dictated by staggered substrate potential due to HBN. So if our substrate potential is negative, namely we open a negative gap if you want, then the A sub lattice has a lower energy and B sub lattice has a higher energy. So these will be our bands. So in this case, this full band at filling factor one will have a churn number of plus one. And if our substrate potential is positive all this flip B sub lattice now is favorable. So it has a lower energy and A sub lattice has a higher energy and this full band now has a churn number of minus one. The key point here is that by looking at the churn magnetization or the magnetization in the gap the churn magnetization is directly given by the churn number. So all we need is just to read out the colors. Red color tells us that we have a churn number plus one here and blue color tells us that the churn number is minus one. This is completely general, independent of the band structure, independent of the model. This is just a result of strata formula. You can just think of it that your local sigma xy here is negative or positive doesn't matter. So this is just imaging the local sigma xy which is dictated by the churn number. And if the signs are opposite it means that your local churn number is opposite. So now we have a unique tool to determine the local topology directly. We've no essential model independently if you want. So, and now we can color our regions by we color them by the self rotation magnetization here which has an opposite sign to the churn magnetization in the gap. And now that we can do this analysis in 2D and this is what we find. So we find that the local churn number is position dependent. So you form some churn mosaic with positive and negative churn numbers on a scale of about a micron. And there are regions that show no orbital magnetization meaning that either the local churn number is zero or you don't have a gap. So you have a semi-metal with overlapping bands. This is quite different from what we usually think. So what we find is that topology rather than being global property or churn number rather than being topological invariant it becomes a local property. So this is changes our way we think of churn insulate and in general in topology in particular. And this has been realized by several recent papers theory papers and it has quite far reaching implications. One of them is that one of the main ones is that now you will have topological edge states between the differential numbers. Namely you have states in the gap and you will have edge currents which are quite significant. In particular, if you are in the gap the currents that flow on the edges are topological currents and we can image them directly. So essentially you can either describe your magnetic field due to magnetization or due to currents and we can just invert the local magnetic field into local currents. And these are the currents that actually flow in the system. So now I'll show you a movie. The units here are micro amps per micron is a very large currents larger than the typical currents that you apply in transport measurements. So this is how the currents involve as a function of filling factor. This is the absolute value. This is the magnitude of the current. So some of these current loops flow clockwise. Some of them form counterclockwise. We know exactly just I didn't know how to plot it. So what is shown here is the magnitude the absolute value of the current. It reaches maximum near filling factor one. And as we increase the filling factor in the case I'll show you in a second. But what is important to realize that these are topological currents that couple to transport current. So when you apply transport current to this state you have to think how the currents flow which are very different from what we naively think. And again, this is because you have gap regions and you have edge states which are combined. So as we increase the filling factor the these currents decay but they remain essentially at any filling factor quite far from filling factor one. So now we understand these red patches as that having a chair number of minus one and we denote them as K A and we want to understand what are these blue patches. So we know that they must have a chair number of plus one but there are two possibilities for that. Either we flip the sub lattice polarization or we flip the valid K to K prime. So let's start from this option K prime A. So this option has two penalties. So this blue patch has one penalty of having unfavorable M dot B. And the second is K, K prime domain wall. So we have two prices to pay and there is no energy gain because we don't know of any mechanism that favors valley K over K prime or vice versa. So we can rule out this option. The KB option again has two penalties. It has a penalty of unfavorable magnetization and in this case AB domain wall. But this option does has an energy gain, which is delta. So we conclude that these patches are essentially stabilized by the sub lattice potential and they have domain walls between A and B sub lattices. So the edge states or the band structure across these two patches should look something like this. So the neighboring patches rather than flipping their valley as we would naively think in the conduction band if we think in the, not in the churn basis, instead of flipping their valleys actually they flip conduction and valence bands. So a K A band, which is conduction band in the, on this red patch becomes the valence band in the opposite patch and the opposite for the KB. So again, this has strong implications which means that we will have now topological edge states and non-trivial edge states at these interfaces. Yes, and there will be chiral, of course. So now let me finally discuss this first order transition. So in order to understand this transition we have to consider two energy contributions. One is the just M dot B term. So M is the orbital magnetization local and B is the applied field. And this magnetic energy scales as the domain side of patch size square, just proportional to the area of this domain. The other is the domain wall energy which scales with the size of the, linearly with the size of the domain. And recently the domain wall energy has been calculated to be on the order of 0.05 millilectron volts per nanometer. So we can just plug in all the numbers that we know from experiment. And what we find is that the minimal domain size should be of about five microns. Namely, if we have domains that are smaller than about five microns, the domain wall energy is the dominant one. Whereas if the domains are large, then you can gain from the bulk orbital magnetization. So essentially this blue patch, the system prefers to have unfavorable M dot B term in order to avoid the large domain wall energy. And therefore this gives rise to the fact that when this domain flips, it is unfavorable. It is more favorable to flip when this patch flips. It is more favorable to flip a large domain on the order of five microns in order to avoid additional KK prime domain walls. So this is an analysis of this. If there's a question. Yes, please. So from Sonya Haddad, what governs the space distribution of the chain mosaic? Are there H states at the system boundaries or local H currents cancel each other? Okay, so this goes to the previous slide that I've shown the currents. So think of this red patch and blue patch. So around red patch you will have, let's say clockwise currents. And around blue you have anti-clockwise current, but where the red midges blue, they add up, they don't cancel out. So actually the H states or the H chiral currents, they add up, they don't cancel out each other. So I don't know if this answers the question. I think so. Okay, this is in contrast to when you think of orbits, let's say closed orbits of electrons in quantum Hall, they all have the same chirality. So they cancel each other and you have H states only along the edges. But here because the chair numbers are opposite, actually they add up. Of course you will have also along the edge. If your patch reaches the edge, you will have a current on the edge, but they do not cancel out in the bulk on the contrary, they add up on the edges. Okay, so we are back to this question. So this is the statistics of where this first order transition occurs for each pixel in our sample. So the chair number are dense on the order of microns, but now when we analyze at which filling factor each pixel flips its orbital magnetization, we see a very different pattern. So for example, this large domain flips as a whole at the filling factor of one. Or I don't know, two. And this large domain flips at the filling factor of 1.1. So when it flips, all the patches, both blue and red, flip their orientation in order to avoid causing additional domain walls. So this is what you see here. And as you can see this typical scale of these two-dimensional elements, the scale of this large domain flipping is on the order of five microns consistent with this very naive calculation. So with this, I think I should summarize. So there are several points to take home. One is that there is, we have a first tool of imaging orbital magnetization and orbital magnetization is just a representation of your local topology on your local beryl curvatures. So now we can open this whole range of understanding the local topology in a wide variety of topological materials. The other is that specifically in magic angiography and we find that rather than having a global topological invariant chair number in a gap state, actually the sample is broken into chair mosaic on the scale of about a micron. And this is a new type of a disorder. It is a topological disorder driven by sub-letters polarization and substrate potential due to HBN. And it is essentially unavoidable. Okay, and because you have a more relatives coming from HBN and more relatives coming from the bilayer graphene and you will always have regions which are mostly aligned to K. This, the HBN is mostly favors A sub-letters and regions where it mostly favors B sub-letters. So it's a new type of disorder that we should be aware of and think seriously about it. The last point is that we observe a first order transition in which the electrons recondense from valley K to K prime giving rise to hysteresis and formation of domain walls. So let me leave you with this movie of orbital magnetization and thank you for your attention. Thank you very much for very interesting talk questions. Just... Thanks for the talk. The very curvature is very sensitive also to hetero strength. Can you quantify if you have them there or not? Okay, so our length, our characteristic scale that we can measure are tens of nanometers and we don't know the local atomic structure. So we can do guesses. We can think why the chair number differs from point to point. This is what I've presented is just a simple interpretation in terms of HBN alignment without strain. And in fact, this particular sample by all our above bifurcation and by transport measurement is not aligned. So it even complicates the matters even further and I'm sure a strain is an important factor. Twist angle disorder is an important factor. But eventually the system, if you are gaped and we know that the bands are topological the system has to decide whether locally it has a chair number plus one or minus one. And it is not at all trivial that it should choose the chair number of let's say plus one through the entire sample on the contrary. There are very good reasons why it shouldn't do so. And I think this is the main message here. Why is the orbital magnetization not periodic? Because in Moira you have periodic lattice. Yes, so again, our scale at which we image is much of the Moira unit cell is about 12 nanometers so 13 nanometers and our scale. So we average over a much larger scale. But even if we would have, okay, so maybe I should show this. So again, in this picture that I've presented that the local chair number is determined whether your A sub lattice, carbon sub lattice is closer to boron or nitrogen. And obviously when you think of these both HBN and the Moira unit cell or lattice there will be regions where you're mostly closer to the B sub lattice, to the boron or to the nitrogen. And actually recently McDonnell has done this calculation taking a aligned HBN with commensurate angle. And so in this case, you have super Moira. You have super structure on a much larger scale. And this is what you get. So these are again, perfectly periodic crystals. And these are the chair numbers that you expect locally on the scale of about a micron, okay? But you're absolutely right. They should be completely periodic. But if you take in the constraint and twist angle variations this picture will become highly non-periodic and this is what we think we see. Do you also get orbital magnetization away from magic angle? We haven't. So generally we see strong orbital magnetization in any system that we have looked at so far. But specifically we haven't looked at magic angle at the twisted bilagraphy and away from magic angle. Oh, we have one more question to the left. So the question is from Rakesh Dora. Is your sample a different phase of the matter not in chair insulator? Because here the chair number is distributed specially. Can you repeat the question you're asking? What is the chair number? The question is, so if what you're finding is a different phase of matter than chair insulator and the reasoning is just chair number is distributed specially. Okay, so this is just, this is a matter of semantics. How do you call this thing? So we call it chair mosaic. So locally within each patch you are chair insulator but the different patches are insulating states with different chair number. And of course the interface, the boundary between them you will close the gap, right? And you will have edge states. What is interesting if you look at transport, okay? So this is the churn gap. This is what we sort of usually define as a churn gap. And in transport, you see it very clearly at high magnetic fields and it has a chair number of minus three, okay? Instead of plus one. So this is something that one has to understand how you go from one to another. We concentrated on very low fields where you don't, where you have globally chair mosaic. So eventually when you increase the magnetic field the magnetic field will force all the domains to have the same chair number, okay? So, but still you will have domains. So instead of having domains between plus and minus chair number due to, so instead of having domains between K A and K prime and K B, let's say you will have domains between K A and K prime B both having the same chair number dictated by your large magnetic field because it will force the same chair number in order to have favorable orbital magnetization but you will still have AB domain walls and one has to think how all this evolves at high magnetic fields, so this is work for future. Thank you very much. We have to go on. Let's thank Ellie once again. Thank you very much for your attention.