 We've had the introduction, we've had a few talks in this meeting about twisted biometric graphene. So just very quickly, let me tell you, we are taking talking about two layers of graphene twisted stacked on top of one another. And as it was sort of discovered in the in 2011, or 10 by number of groups, particularly McDonald and collaborators at a magic angle of about one degree. The more a superstructure of these two layers of graphene gives rise to interference effect basically that that gives us these very flat bands to flat bands near the chemical potential and near the charge neutrality. So we have two bands, we call them the upper and lower or conduction or valence flat bands. And when you include the relaxation effects as it's been shown by a number of people, the remote bands kind of pull away from these flat bands to so energetically they can be quite isolated. And, and we have graphene so of course we have valley degrees of freedom, and we have a spin as well, which gives us basically the electronic system with eight flavors if you wish to bands to spins to valleys degrees of freedom. And my pointer is dying. So hopefully, I can do this. Oh, no, it's saying something. The meeting is being recorded. Okay, that's fine. All right. So, we're basically putting electrons into this manifold with eight flavors, and Pablo Jerry Herrera that made these samples for the first time in a controlled way, discovered that when you partially fill these bands. You get a correlated insulating phase. So in particular, the correlated insulators are more stable when you have two electrons in this manifold of eight, or you have six electrons in this manifold of eight. And doping away from this new equals minus two or plus two that's how we keep our counting the convention in this talk is going to be eight electrons going from minus four to plus four. You dope away and you get superconductivity. So of course this diagram looks a lot like the superconductors we've heard about, for example in this meeting, and trying to understand both the nature of the correlated state, the insulator, the normal state that's correlated as I'll show you today and superconductivity is kind of the mystery. And of course, having very large density of states associated with flat bands. You might just wonder, this could be just a conventional superconductor small amount of electron phone or interaction maybe enough to give us superconductivity. So whether the role of electronic correlations is important is something will be nice to experimentally establish the role of top a lot topology of these bands we are we are borrowing basically states from graphene. So graphene has this very curvature, this very phase associated with the direct point and what that does to the band structure and the nature of pairing what kind of pairing interaction can we get are among the key questions. So today I'm going to tell you how we use the scanning telling microscope to look at the system. And I'll show you it's actually quite nice. This is probably the first electronic system where we can tune the material in our experimental setting from the to the non interacting limit by filling and empty in these flat bands and watching what happens to the spectroscopic properties. I'm going to focus quite a bit on superconductivity and what we see in terms of the spectroscopic property of the superconducting phase. And I'm going to show you something about topological phases. And if there's time I'll tell you where we are headed in the lab. So the way these samples are made is basically by stack and tearing. You can take a piece of graphene which have scotch tape on a piece of silicon. There are many different methods of ripping graphene apart into two pieces, and you use basically these polymers glued to basically a glass slide to pick up pieces of graphene and stack them on top of one another. And we use boron nitride as a substrate underneath to isolate the sample from gates and our transport colleagues also encapsulate their samples with boron nitride on top. What we are of course going to vacuum tunneling experiments with the SDM. We only have boron nitride on one side and gate only on one side of the Sam. Now, we have worked very hard to get very clean samples question. That actually matters. And actually, if you look at the question was is boron nitride aligned and does it matter. And the question of what the boron nitride does is very important. Because boron nitride can break the graphene a B sub ladder symmetry, and I could get gap the direct points, which are actually here in these flat bands as well and that will come up a little bit later. But that's actually one of the powers of the SDM actually if you look at this image, you can see a lot of different features on a very fine scale is not this scale is finer than that. Actually is the graph is the is the carbon atoms which you cannot really see in this projector, but they're there. Then there is a finer scale that sort of you can see here, not that the bright spots but within the bright spots. That's actually a more that's created because of the misalignment of the boron nitride underneath with the two graphene sheets that are almost aligned with one another one degree. So that's one more a superstructure. And then you have this other more a superstructure that comes from this one degree angle misalignment of the two graphene layers. And what structurally that that that means is that within the regions that are very bright, which we call a stacking is your graphene are almost aligned with one another. The two hexagonal lattice is right on top of one another, and they go out of register and they go into the register and so on as you go out into this superstructure of the more. So unit cell of the more is involves 10,000 atoms, which is makes this a very difficult problem to solve the band structure up so you have these continual models that you've heard about. Okay, so why do you see these bright spots. Actually, if you go to the McDonald's continuum model, you see that these bright spots are associated with the enhanced density of states of the graph of the flat bands. Electrons for the flat bands basically live in these a sites, which will become kind of relevant in parts of this call. So we have a gate underneath and and we can tune basically the electronic structure to fill and empty the flat bands. And when you do spectroscopy when the flat bands are completely full that you see two very sharp peaks near the chemical potential you notice that they're below the chemical potential. And this is totally gives you a sense of the bandwidth involved with these flat bands is about 12 million electron volts. And each one of them, whether they are filled or empty to roughly have the same kind of shape. And of course all the action is happens when you partially fill these states. The kind of data I'll show you in this talk looked a lot like this, where we, on one axis we have the tip sample tunneling. Okay, the two peaks that you saw here are these two flat lines for example here up here about 40, 40 gate voltage, which when you have the flat bands completely occupied below the chemical potential, and then you can tune them through the chemical potential, and get all the way to minus four, where you could completely depleted them. Okay, so there's a lot going on in this slide, but there is kind of two key features I want you to notice. The first one is that the way you read this slide is we go engage voltage down to go up here, and go down again and so forth. As soon as we partially fill these flat bands those sharp peaks in the spectroscopic properties of these states become extremely broad. And this is a feature that we first notice. You see it's very broad here is very broad here as well here and all the way here when you start exiting the flat band is to begin to be sharp again. The broadness of the, this broadness of the spectroscopic property of the flat band is a signature of strong interactions. And it's telling us about very strong charge fluctuations that are taking place in these flat bands, which makes the quasi particles extremely broad, when you when you partially fill these things, which is quite interesting. If you look a little bit carefully you notice that there are finer features that are kind of going on in this data. In fact, if you look here you see that there is sort of repeating features, which I'll bring out by taking this is the IDV. And I can take the IDV and normalize it by I over V which is measured for every curve. And what that does is basically normalizes the fact that the tip actually the tip height can be slightly different between each of those spectra because the spectroscopic property of the system is changing. Now what you notice here is is this cascade of features in the data, and you can kind of see that the cascade of features involve some spectral property that comes from near the chemical potential goes out, and then we sets again goes out we sets again and it means every basically quarter filling within these bands. So the system knows that it has these flavors for for up here and for down here. When you sweep the, the, the, the, the carrier density. These are very high temperatures. This is not when the system is an insulator or a superconductor this can be observed up to, you know, 20 Kelvin or so. And this, you get a question. They want to say, yes, yeah, I would say the broadening it's like 40 millivolts energy scale so I don't think it's very sensitive to temperature. The broadening is, I think it goes on to like 1020 Kelvin, the question was the what the, what to what temperature does the broadening can be observed I would say 1020 Kelvin. The cascade or yeah independent you do yourself a change in the density of state independently of the cascade. I would say we haven't measured it carefully enough but my, my intuition is, you know, about 20 Kelvin or so, all these features begin to show up together. Now one thing you notice is that this this data is a little bit different. This actually data taken not with the tip over a sites is the tip over the ad sites and when you put the tip over the ad sites. You become very sensitive to not the flat bands but the remote bands that's where they have the largest density of state. And you see out here at higher energies which where the remote bands are these are the remote bands. You see that the edge of the remote band shows these sort of cascades of features, sort of in line with what you saw in the last slides. We just simply interpret the edge of this band as some single particle state for tunneling into that edge of that band, plus what the chemical potential of the system is doing what we find is that the chemical potential shows this resetting behavior, every quarter filling in the data. The correct measurements of the chemical potential was done also with an SCT by Shahal Elani's group, and they also observe actually it's eerily the same, the numbers even match between these two very different experiments done on the same system which is nice. So this resetting is something that one can try to have different theories of we have a very simple idea of how to understand this resetting. And our simple idea is that, well, you know if you look at the SDM images that they kind of look like electrons are living on the AA sites, it kind of looks like a quantum dot. And you can just think about the quantum dot, of course there's hopping between the dots, I don't know how to solve that problem. So let's ignore the hopping, we think it's very small because the flat bands are very flat. And essentially just think about, you know, a number of quantum dots with with basically eight flavors. And, and all you're doing is you're asking what's the what's the energetics of adding and removing electrons to these dots. And if you go through that exercise, you basically have of course the energy of the levels if you just say there are two levels e and minus e not. There is an onsite pool of repulsion you within each dot of course you can make this more modern more sophisticated, but it's very simple model you can sort of convince yourself that there will be of course jumps in the chemical potential. Every time you go through filling each of the flavors, and, and these jumps in the chemical potential, and the associated addition and removal spectral features that you see in a calculation of the simple model is one which tells you roughly the size of the long repulsion in the system. These features can be interpreted kind of like an addition removal spectra, as you go through each of these filling states within the simple, you know, infinite view, Hubbard model. What's interesting about this is that this exercise tells you that the energy of the interaction is about 23 million electron volts so the size of the cool on repulsion, you get from the simple model is larger than the bandwidth of the system. Okay, by by a lot. Okay, and, of course, you know you need a theory of these features. And that's remains to be fully done. And one idea that's being explored right now in trying to understand our kind of data and put it together with of course the fact that the system is conducting and showing extended electronic state is to think about the fact that maybe you can think of the system as having two flavors of electrons, localize electrons on the AA sites and electrons that are conducting on the on the right, which are have a band structure very similar to just buy their graphene, which is, you know, ignoring twisting and this is a work of under Burnaby and song, which are building this heavy fermion model of the system which is I think quite interesting as well to try to see whether you can recover some of the spectroscopic feature from such a model. So just before I get to superconductivity let me quickly tell you that of course these bands are topological, and the way this was first exhibited experimentally came to light was when people made samples that accidentally were aligned with boron nitride. And when you align these graphene samples with boron nitride, the direct points which are very hard to see in this band structure if you zoom into them, basically they get gapped out by the princess of the breaking of the symmetry. And it was a discovery of anomalous Hall effect and quantize anomalous Hall effect as well in samples it one in one particular feeling of new of three elect of seven electrons in the system, showing basically chair number of one. What we discovered was that, even if the samples are not aligned with boron nitride. And if you cool the samples down to mili Kelvin, you, we find that at some values in presence of small fields, the system exhibit these gaps that come in the spectrum and it looks like this so this is the same kind of spectra I've been showing you before the colors represents that the div the tunneling density of state, and this dark blue region here is a presence of a gap that opens up the chemical potential. And this turns on at some particular value of density you see it's not new equals one or two is that some value of magnetic field. And what we find is these gaps can basically be moved around they move around as you change the value of the magnetic field in the density and magnetic field, you know, knobs that you can turn. And essentially you can show that this is a signature of a formation of a chair and insulating state, and the chair and insulating state with a quantize hall conductance, of course has a gap in the bulk. But this quantize hall conductance, the quantize Hall state, and the gap basically are such that if you change the density and the magnetic field. You have to, you basically are changing where the gap occurs in density, and following essentially how this gap moves as a function of magnetic field of been flux per unit cell as and as a function of the filling factor, you can actually back out the number associated with those gaps. So essentially what this plot is is that every, at every density, you find the gap at every magnetic field and you change the field and follow where the gap is, and from the, this is called the strata formula. And from the slope of this line, you can basically extract the turn number associated with the state. This is something we found in STM and it's also been seen in transport studies and other STM groups, and also with the STT experiments which also measure the gap in using the, you know, compressibility measurement as well. So this system, these, these, these phases are caused by interaction. They're not caused by, you know, a, you know, a single particle effect alignment with the BN and so on. This is purely from the interaction in the system. And it needs a little bit of magnetic field to get stabilized question. The size of the gap. And that's a good question is very hard actually to measure the gap precisely because when you were getting an insulating phase you actually have a charging effect that adds to the, to the size of the gap you measure in a tunneling experiment. So, but I think that's a good question we have to look at the compressibility measurement because they're not so sensitive. They can actually measure the gap directly determine dynamic gap. This is a tunneling gap. Okay, so these bands are also topological. All right, so now let's talk about superconductivity. So, the superconductivity is most robust in this system around the new equals minus two insulating state. I think we will answer the questions from the chat at the end, or you want to read it. Okay, so the question is, what mimics the magnetic fluxes in Tuesday's ballet graphing on for a nitride to give rise to the quantized animals. So this is, this is actually magnetic field we are applying to actually measure the strata formula so we apply the magnetic field and basically look how the gap changes so treading flux actually is happened experimentally it's not a fictitious Okay, so this is the same kind of data I was showing you before but now zoomed in close to the chemical potential and also measured at 200 millikelvin electron temperature. And what you see here is low conductivity near the charging charity point. Okay. This is not insulating the sample is not insulating at charging charity, and minus two actually the sample has a insulating state I'll show you that a little bit more carefully the moment. And then we see a gap that develops and low temperatures near new equals minus two. This actually new equals minus two gap closes and reopens another gap opens between minus two and minus three. And this is where our transport colleagues have found superconductivity in these materials. The first thing is how do you distinguish two gap phases. And this is of course, it would be nice to do transport experiments simultaneously, but our devices are actually relatively big. And even if you measure transport experiment and show zero resistant you're not really sure it's really coming from the area under your tip, it could be percolating somewhere else. So we realize we could use the tip to do Andrea reflection experiment. And this experiment turned out to be more complex than we first thought, which I'll talk about a little bit today. But just to just to remind you in a conventional as wave superconductor. If you are in the tunneling mode, you of course measure the BCS tunneling gap. And make a contact between the normal metal and a superconductor, you have a conductance which is bias dependent. And at the energies below the energy of the gap you have the Andrea reflection process where electron from the normal metal is retro reflected and a Cooper pair goes across. And this of course is a situation where your conductivity is enhanced, if you wish, half of this junction is superconducting. So when you go below this gap energy, the conductance will be doubled in the ideal case of an Andrea reflection from a S wave superconductor. So we went along looking for such a signal. And, and actually they were under reflection experiments also on the wave superconductors will discuss that in a moment. We found that, indeed, at low temperatures below about one Kelvin in the region between minus two and minus three, I'll show you the gate dependence more carefully in a second. You see this enhanced conductance of about 30% or so, from a contact resistance, where this is about five kilo ohms, or, or, or so in the in the resistance channel. And so this is a signature of superconductivity. Okay, so this is this is nice. So we can put this side by side. So we can take an area of the sample performed tunneling spectroscopy. Find the gaps correlated insulator gap, but you looks like a superconducting gap. And then we can say goodbye to this area because we're going to crush the tip into it, because it is no longer recoverable for STM measurement and record this point contact And what you notice here is purple is low conductance and, and, and orange is high conductance, you notice that you can cut it's very nice you can see oh that's the that's the new equals minus four this is the ordinary insulator. This sample is a sort of semi metallic a charger chalice at plus two and minus two you have an insulating phase. But then here if you look carefully, this is the Andrea signal that we are measuring between between these two phases, this gap, this state is gapped, but it has an Andrea signal when you crash the tip into the sand. So very good. So we know we have a superconduct. So now let's look at the spectroscopic property of the superconduct. So, this is actually a nice plot. This actually shows a very important feature which is the new equals minus two insulating gap closes before what appears to be like a superconducting gap opens. Okay, this is actually very different than what we see in the group rates for example. So here are represented from two different devices when you're in the middle of the largest superconducting region of the sample. And you see that the tunneling spectral has this coherence peak. It has this very V shaped looking curve sometimes it hits zero and sometimes it doesn't. This is actually very characteristic of things we have seen in the cooperates. You know V shaped gap and it not always touches zero. And it doesn't look like your ordinary one Kelvin aluminum superconductor, which we can easily measure in this instrument and see the BCS density of states. So now you can start playing around and try to fit this curve with, you know, S wave, if you take a simple S wave, you have to broaden the data beyond. You know what's reasonable to fit such as such spectra, like, you know, to Kelvin or something which is not reasonable. If you just assume a nodal superconductor with very, you know, small browning, you can fit this experimental results. Okay, so this is all very. One thing I should say is that the two delta over KTC if you if you interpret this as a superconducting gap is something like 25. Okay. It comes the interesting parts. We take the sample we warm it up above one Kelvin. I showed you that the Andre signal got went away. But if I look at the tunneling measurement with the tip far away, it still shows a pseudo gap. Okay, so that it shows a pseudo gap. So this is very interesting. So this is the thing down here. This is the data at four Kelvin. And you can you can this is answering your question. For example, you can see the cascades are very clear in the sample at four Kelvin already. And the size of it is very hard to interpret experimentally because of course there is term of broadening as you warm up to four Kelvin. But what you can do, let me go forward. I'm missing a slide, which I want to show you. Sorry, has disappeared. Okay, maybe I get it up to do in the question session. Oh, here it is. Of course, in this experiment suppressed superconductivity with a very small magnetic field, when you apply the field perpendicular to the sample. So with just about half a Tesla or even 100 minute Tesla, the superconductivity in both under reflection and transport studies go away. But this pseudo gap is still there. And it's actually rather sharp. Okay, so if you compare the zero Tesla and half a Tesla and one Tesla data. And I find that this gap in the spectroscopic properties is it sensitive is filling in as you as you turn on the magnetic field, but it actually retains most of this characteristic, even when the sample is not showing the real zero resistance state. So you know, you could ask whether this is due to some pairing without face coherence, or whether this is due to some other ordering phenomenon that needs to turn on before superconductivity is possible. Right. So now, let me talk about the fact that if you do these two different experiments, you also seem to observe sort of two energy scales in this problem. So in the tunneling experiment, the gaps we see are about two delta over KTC of 25. But when we do our Andrea reflection, we somehow see a smaller energy scale in the experiment associated with this enhanced conductance. Okay. And this is of the order of four to six. I don't know quite how to interpret these things. But the fact that there are these two energy scales may be important in the in this problem. And here's a sort of a plot of what it looks like. This is quite sample dependent. But this is a plot of what it looks like for one particular sample. So the blue is the tunneling gap. The red is the energy scale you get from the Andrea reflection, which is smaller. And I've also plotted here the correlated insulator gap for you, which is turning on and off and your new equals minus two. So what's interesting about this is that this is actually quite different than the cooperates. This, the gap that we call the pseudo gap in the cooperates when we go to under dope samples would continue to rise as you approach the mod insulator. And here this gap seems to die before some other gap turns on, which is associated with our insulating states. So this could be this would be important in understanding the nature of this correlated insulator and its connection to the superconducting So the energy scale being two different. That's different from the corporate but what is similar to the cooperates is that there seems to be some experiment for example in Andrea reflection in the cooperates this was done in the side way tunneling side way and reflection in the high TC cooperates. There is that turns on at TC, and it has some energy scale. And there is another energy scale which is a sort of the pseudo gap which I just told you goes up and up. As he goes higher so the fact that there are two energy scales, and sometimes in under dope samples in tunneling spectroscopy this is a sample which is TC is like 60 Kelvin down here, you can actually see two energy scales in the tunneling spectrum. And, and, you know, there is a question of whether you associate this energy scale associated with superconductivity. This is the pseudo gap which is remains to be still fully understood in the cooperates. So there are these connections. Now, let me also mention that, in addition to twisted by layer graphene, there is an experiment on twisted trial air graphene which is also a superconductor. And this is a work by step and much Pergis group very similar experiments than the one I showed you. What they also observe is that there is a region where you have a correlated insulator and then there is a region where you have superconductivity curiously in the region that think they have superconductivity. There is either you shape curve, or a V shape curve in there in the spectrum this is a paper. That's online. And now I think they have repeated this experiment, also doing Andrea reflection just like I showed you. And I think it's only in the V shape area that they see an Andrea signal, we have to look at that very carefully. So, since this work has been published has been a couple of people who've been interested in trying to understand, understand our experiment. I think maybe you heard from Leonid glasman who was here about, you know, Andrea reflection in a nodal superconductor and there is also previous to that there is work by center which is online analyzing kind of all the different evidence from all different experiments concerning the pairing symmetry question in the bio air graphene. And one thing that we kind of, we kind of didn't think about was in detail about how would you how does the, how does the process of under reflection works in a nodal superconductor. So, if you think about a nodal superconductor, of course the the phase of the order parameter is changing as you as you go around in all directions. So, if you have a process by which you think about, not, not tunneling but under reflecting from a point in the system from a single point and compute basically the under reflection signal, adding up all the phases from all the direction. Of course you end up at zero voltage you end up with actually zero and reflection signal. So, plus and minuses of the order parameter cancels. It's a mystery to try to sort out the central and limit that. So if you compute from a point, you get this, the signal should look like this the IDV versus V should basically go to zero. This is central calculation and this is Leonid and Felix and their postdoc calculation. And, you know, they're the way maybe they thought about understanding our experiment is sort of like a fine tuning. So, some components, maybe you have a S wave, that's an isotropic and you know, but the reality of our experiment is that this V shaped tunneling gap is very insensitive to, to gate voltage, basically. So if you change the density of the sample, you still see this V shaped curve, and it doesn't change very much. In central's calculation, they also consider the possibility that you have a Andrea reflection process in which you conserve momentum. So when you, Andrea reflects from one point, you don't think about momentum conservation. So they consider well maybe this experiment somehow conserves momentum. Okay, and you might ask how would that be possible hold that thought for a moment. And if you, if you do that calculation, you do in fact expect to see an enhancement in the low energy conductance with for the Andrea process if you conserve momentum. So this got me thinking, me and my students thinking about, what did we actually do in the lab. So this was a kind of a fun exercise so here's what the signal is when the tip is very far away. So, so it incorporates essentially Andrea reflection was mostly done in AB plane sideways into the internal tunneling. Well, okay, so this is why you know, I don't actually like this experiment. This was the last resort to find that we actually have superconductivity in the sample because you don't know what's happening. And you'll, I think you'll, you'll appreciate maybe the answer to your question is coming. So think about what we are doing in tunneling we are a giga ohm a giga ohm is like five angstroms away from the sample. Okay, and now you asked you bring the tip close to the sample what is really happening. Well, you know people's romantic view of our experiment is this you have this very beautiful sample. You have this beautiful tip, and you know, this is maybe what's happening you bring it and touch it from a point and this is. But this is actually what the real experiment looks like. This is what the curves look like in when you're blind. You're not imaging anymore. So when they're tunneling they get a curve like this, but then they start bringing the tip closer and closer and these are in the micro Siemens. Okay. So at first, they move the, move the sample a good, you know, you know, one nanometer 10 angstroms. They still get a V shape, they still get a V shape, they have to go a good bit closer. What is happening here in the experiment. Probably what's happening is something like this, and actually maybe even more violent, because if you actually read off the numbers in our experiment. It's, if you're around 20 kilo ohm, which is roughly when you have a quantum of conductance, we actually still get a V shape curve. Only after we get up to like, you know, five kilo ohms or so, where you have multiple channels of conductance in the perfect world of transmission one, which may not be true here, I doubt it's true. That's when you get this very enhanced conductivity and their signal. And now it's not completely clear what's happening. Maybe what we are doing is we have brought the tip so close that we are actually sideways conducting. We are basically doing under reflection in the side of the sample effectively. Now, there are a lot of questions that are not resolved. Is the tip doing that, or is the tip in this case is actually a potential impurity for the D wave superconductor creating an Andrea state. We still need to work all of this out, but I am encouraged by, by this data because after their theory I went to the lab and they said let's look at this curve. This looks more like their theory in terms of where the regime of the experiment would be a point contact. And maybe actually this is all consistent. Okay, I think I'm running out of time. Five minutes. Okay, I'm almost there anyway. So let me show you another curious experiment. The question was asked because does the alignment the boron nitrite matter. It matters a lot. So I showed you experiments that people have done where they see this churn insulator in actually kind of almost call them all state. We also see that state actually here it is. There is a sample which actually has perfect commensuration in this region between the boron nitrite and the, and the, the bilayer graphene, such that the, the more is are just they line up and make this very beautiful structure on the on the AA science. This is the same kind of data again. So this spectroscopy you notice there is this giant gap. And now we can do this point contact measurement and we can say oh look this is an insulator. Okay, I charge with charity, and that's what you expect to see open up gap at the direct points. There is a there seems to be somewhat an insulator here around new because plus two in this sample. And this is the new equals plus three insulator actually if you follow this with a magnetic field, you can measure his turn number. His turn number is one, as he was discovered with the transport studies. But look down here. This is a Millic elements. There is still event whole singularity associated with the with the flat bands. This sample shows nothing. It shows no pseudo gap physics. It shows no superconductivity. And this is consistent with the anecdotal evidence from transport colleagues which when they get an online sample, they haven't seen the superconductor yet. But this is actually a sample where you can go back and forth and say oh this is actually a line. We really know it's a line. It has a transport signature of new equals plus three. And it, it, it is not super. What is the conclusion of this, the conclusion must be that maybe the pseudo gap phase, you know, first of all, it's very sensitive to this breaking of symmetry the superconductivity see to see symmetry. And it looks like the pseudo gap phases also. It just disappears as a question. Okay. Good. So I'm pretty much done. So, well, I can actually if I have a moment then I can speculate about what we're working on. So let me just review this, the V shape gap, very large gap to TC ratio. There's also a very curious effect, which is that if you if you integrate this Andrea signal and and watch this address signal as a function of temperature hard vanishes I showed you that it goes away at at TC. It vanishes linearly, which is very strange. And this strange behavior has also been observed in strontium ruthonate, and, and actually some heavy Fermi on superconductors previously in under reflection experiments as well. There was some interpretation associated with that which I won't get into here. But, but it had to do with nodal superconduct. So let me tell you what we are working on now. So looking ahead. We want to understand the nature of the insulating state. And how do we gonna how we're going to do that. And it looks like a working with Andre and also Mike Salato has comparable work and collaboration with us you'll see where some of these ideas come from. And that basically computed different types of correlated insulating states, and they've computed is local density of states in real space. And these states have real, you know, some of them have very sharp signatures, if you can map their way function in real space on the atomic scale. And the motivation partially for this, these works came from actually a recent experiment we've done where we look, we went back to monolayer graphene. And there is this been this mystery about what happens in monolayer graphene, when you have filled the zero land on level. So remember monolayer graphene has four flavors of electron to spin and to valley, two of them. And the question is what kind of a state do you get from transport studies we know we get these broken symmetry states every time we go filling a quarter of such a manifold. The first one is kind of obvious you pop you populate, for example, you know, a spin or a valley state is a spin polarized for magnet. The one at half filling is not obvious what you could get. Theoretically, there are basically two candidates, you can get a canted anti ferromagnet. So every so you have the graphene, you have two electrons on two sites on AB sub lattice the two electrons can be anti ferromagnetic. They could make a, what we call a calculus state or some people would call it valence bond state. Okay. So, we just learning how to do these experiments on magic and a graphene, we kind of have realized how to do experiments without tip, the turbine the sample. And what you see here is the land on level spectroscopy of just graphene in a magnetic field. And what this is here is the zero land on level, and you can see it's broken up into pieces. And that's breaking up into the different broken symmetry states of the zero land on. And the question is, you know, which of these states do you have. And what we see is that the system actually makes this balance bond crystal, it makes this calculate state. And what we see that is just by imaging on the atomic scale. You can actually see the signature of this bond state on the on the sub lattice of graphene. And essentially from this experiment, we kind of realized that from this kind of data the four year transform of this kind of data, the real and imaginary part of this, we can actually get a lot of information about the nature of value polarization in the sample, the nature of valley coherence in the sample. So when you make different kinds of coherent states. When you have the phase of the wave function being different between these two valleys, actually the local density of state is different. So let me just put that up as something that is a technique we have developed, and it has a lot of applications you can actually watch these states form. So I'm on the magnetic field, and I see the chairman is getting nervous here, and it has a lot of application to this problem to be able to tease out. Basically what's the nature of the states by actually looking at the wave function and their foreign analysis. And there's a lot of information there. The problem is, as we start working on this we realize we need data, which is a little bit ridiculous. You need data which has atomic resolution of maybe 10s of more sites. Where you can actually get enough information to rule out different states from one another. So that's, that's a kind of very hard data to get because you absolutely pristine samples and be able to get really, really high resolution on a very large scale. So let me stop there and just put up my summary slide and see if there are any questions. Thank you. Thank you very much. In the transport measurement they use the HBN encapsulated samples right. There's HBN on both sides. So does it matter in the tunneling experiment you have HBN on one side so it probably matters. Yeah. Yeah, it's very sensitive. So I think we have heard experiments from transport colleagues where they think one of the BN is aligned, but the other one is not. I mean, so I think our experiment is simpler. Just vacuum. Thanks. So maybe you said that they miss it about the pseudo gap of these are conventional superconducting behavior. So you show that with the magnetic field the pseudo gap survives to very large field. But what does it do in temperature? Does it survive above the sea? What is the critical, the T star? So where does it close? It's maybe six Kelvin or something five or six Kelvin. Yeah. Okay, so it's really like in a sense cooperating. Okay, thanks. Let's talk about the first part of your talk about cascade with apologies that I asked similar question before, but that was about experiment. Now I want to ask about the model. So suppose I start with charge neutrality and then move into either direction positive new or magnetic. I don't understand the models that you have. There are components of initially 440 generate peak that start moving one after the other and the relative scale is you. Or delta E naught. Those are the only energy numbers in the problem. And then you reach new equal to either plus four or minus four. That's why they have to recombine. And again, because bandwidth in your case is zero. Essentially, they have to recombine back at zero energy. Because again, only four components move for other components don't. Yes. And at the end of the day bandwidth between the two is because you neglect hoping bandwidth is zero. When you started you said that originally have before. Yeah. I think, I think you're talking about the spectral weight of these features. I'm asking about positions. You started by saying that you have very narrow band of 12 family. Yes, and there are two four fold generate peaks. And then components of one of that start moving components of the other probably don't. Well, I think what happens is this is that actually, we can go through this in detail, each of the lines in this calculation corresponds to different type of excitation you can make. And add an electron, remove an electron and add an electron and pay no you add an electron and pay you remove an electron gaining you said all of those. You just have to go through all the different possibilities. So that's what all these lines are. Right. But at some points the distance between the slides is you and then the question to the I see the picture. So they combined back into fourfolded to fourfold the generated peak close to each other. That's right. So there is a movement to one side and then there is the movement back. Maybe you're talking about how they go out and they come back in. Yes. Yes, exactly. And then again question about experiment. Do we see anything like this in experiments that moves move one direction and then comes back. It's all a blur. Just to be completely honest right, but the fact. The thing that this model gives you is that the chemical potential is doing these jumps, but the chemical potential has this very weird cat cusp like behavior, which this model doesn't recover. Now, be outdid an exact diagonalization with like some number of sites with a little bit of tea. And he we could with mother's loving I we could just say, oh, it does kind of look like a feature that's not jumping but it's, but it's, you know, kind of smoothly changing and so on. I haven't looked at the detail but some of the features are resembling one, for example, the plus and minus features at this point, they may go out a little bit further. I'm not sure, but also. Yeah, I think it's a. It's hard to tell. Because of the time. How close is proximity effect. I also I have a question about the double gap structure. So first of all, with the Andrea reflection. So I guess you do have some fit to BTK but from the data you showed us at the end you have very kind of dirty conductance at the hot when you bring the tip closer. So how much can actually trust this, this a four to six that over to see. In terms does the line ship really fit the BTK formula or do you have to do. All I want to say is the energy scale for Andrea reflection signal is small. That's true. And that's very consistently happening. And how well do the two gaps extrapolate to similar to seas. Yeah, one of the gap doesn't care about TC the tunneling gap doesn't care about go straight like that, because you had some kind of look like a square root but Okay, that's that's the gap fitting to the the BTK model for for the Andrea reflection that one looks a little bit square would like. I asked about the last part of your talk where you mentioned the kick list they disappearing. Did you look into that reaching that disappears and if you see any changes because that specific point and critical point right. Disappearing as a function of density. I don't, it was too quick for me to see that's why I'm asking. So there are two transitions that we see one is as a function of field. So as you the calculus state actually appears as you turn on the field above a critical point. And that critical point is determined by the AB sub lattice potential created by the boron nitride substrate. The system either would like to make a valley polarized state, or you can make a calculator state when that potential is weak, it will say I'll make a, I'll make a calculator, and you can tilt the balance with the magnetic field. So we haven't studied that critical point very carefully but but there is a very nice critical point and we have checked as we twist the boron nitride angle relative to graphene. The point this point actually shifts in magnetic field very precisely. Yeah, it is an interesting thing to think about, but we haven't studied carefully what happens at that transition. Thank you. Okay, all right. I know there are more questions but questions to Ali probably Okay, just to be fair. Okay, so yeah, so thank you very much. I like to think a little more time for the next question. Next speaker is Francisco.