De l’effet Hall quantique aux matériaux moirés... (9) - Antoine Georges (2025-2026)

Ajouté : 2026-06-07

[00:00:33]Experimentology Columbia experiment experiment.

[00:01:23]Yes, >> thank you very much. So, uh thanks for coming even if we have a heat wave outside you made the the trip. So, thank you very much. So, today we are going to uh see all what is the topological states that we found nowadays in materials. I'm going to focalize especially in graphin based materials.

[00:01:48]First to keep just track of all the evolution in graphine and second because last week uh Nicoleno uh gave a excellent introduction to uh the topological states that we already see in uh TMDS in twisted TMDS mi materials.

[00:02:07]So this talk uh if you read the abstract is going to go in the following way. We are going to start introducing the quantum hall effect and fractional quantum hall effect into the materials and more materials and then we're going to jump into uh newer things as the anomalous quantum hall effect into the materials. So to be completely honest this is a a little bit ambitious uh abstract or um outline. So I'm going to skip part of the fractional quantum hall effect because also next week you are going to have a talk by Wendel Febe who is going to cover uh most of it at least in uh to the electron gases.

[00:02:53]So the objectives of this seminar what I want you to have a look during this uh talk or to understand at the end of the talk is first uh the experimental observations and phenomenology of the quantum hall effect which comes a little bit to complete Antoan's uh curse of uh all the theory about it. Then the modifications of this uh of the quantum hall effect when we have a more super latattis and how this gives rise to no quantum hall effect and at the end of this talk what is important to discuss is whether we need a mu superlatis or not to uh observe all these effects.

[00:03:41]Okay. So I think it's a little bit uh unorthodox here but if you have a question a quick question something that you need to understand in order to continue please just interrupt me okay if I think it's a very very long answer I'm going to just tell you that we can discuss later okay but don't worry just in interrupt me so for me there is no way to talk about uh topological states states at least in an experimental point of view that do not start talking about the first observation of the integer quantum hall effect in the 80s by Bon Klitsson. So here you can see the first uh measurement was uh performing silicon MOSFETs at quite high magnetic field and low temperatures. And the main characteristics of this experimental uh measurements are that when you are in some gate value. So let's say for example here your longitudinal resistance drops to zero and your whole resistance is quantized and it's perfectly quantized in terms of fundamental constants of nature. So how can we see this? There are two ways.

[00:05:01]Anton just uh said this before. When you have a two-dimensional electron gas and you apply a magnetic field, you can describe this as uh classical orbits of u by the Lawrence force where your electrons are going to start to describe uh these orbits are going to be localized and you can see this quantum mechanically as the formation of Lando levels. So these landown levels are going to be uh are going to have localized electrons in space and are going to be highly degenerate.

[00:05:36]Okay. So now the materials where we uh observe uh uh quantum hall effect are many but the most stable or the ones that are uh most studied so far is uh two dimensional electron gases made of gallion arsenite aluminium gallion arsenite and these are constructed by the fact that when you have two semiconductors that with a different h energy gap you're have a band bending and at the interface of this band bending you're going to have a conduction a two dimensional conduction of uh electrons which is called a two dimensional electron gas. These are uh made in molecular BMPaxi machines which are these huge machines with extremely high uh and demanding uh performing conditions where you need uh to be very very clean and uh have extreme uh vacuum. So just to show you how an MB works, I'm going to show you this example. This is a gallium arssonite nano wire. First you're going to look here. Okay. Aside.

[00:06:56]So you can see now that every uh every layer of atoms is deposited one by one. Okay. So this allows you to have a very good crystallinity and this allows you to have a very high mobilities which are necessary for uh this u structures this conventional two dimensional electron gases.

[00:07:20]So in a two dimensional electron gas like gallion arsenide we have a parabolic energy dispersion. These are in fact uh momentum and energy resolve tunnel in a spectroscopy because unfortunately in gallion arsenite as this is a buried structure we cannot perform arpes as the ones that Antoine show before. So you can see that we have a parabolic energy dispersion and as we apply the magnetic field we're going to see that this energy dispersion splits in Lando levels and as you increase and increase the splitting between Lando levels becomes larger and larger. Okay.

[00:08:00]And that of course is going to have consequence in what we are interested most the quantum hall effect. Okay. So let's say that we have the same uh two dimensional electron gas with an magnetic field applied out of plane and we're going to measure two quantities.

[00:08:18]We are going to measure the longitudinal uh voltage or resistance and the whole voltage on resistance. So you need to remember that now you have land levels that are equally spaced in this case. So when we measure the resistance the longitudinal resistance what we are going to see is as an oscillation a very fast oscillation a low magnetic field and then eventually becomes slower and in between two oscillations this longitudinal resistance drop to zero. At the same time the transverse resistance is going to uh describe plateaus and these plateaus are going to be quantized. Okay, in terms of fundamental constants of nature and this is so well contised that in fact this uh this value is the practical electrical resistance standard of the international system of units since the 90s.

[00:09:19]Okay. So now but why is like this? So we saw the bulk edge correspondence uh just before and to explain this we're going to have the Lando level formation that I told you before and now we're going to see how this energy changes as a function of the position in the material. So as Antoan uh said before we are going to have two different topological orders. Outside the sample you're going to have a zero churn number for example and inside you're going to have a let's say churn number number uh churn number of two which means that your fermy energy is placed between the second and third land level.

[00:10:04]So now if we look what happened on the edges because we know that gapless states must exist in the interface between two topological faces we're going to see that we have dispersive modes. These dispersive modes are going to be the responsible of two things. The first one, the drop into zero of the longitudinal resistance because now the conduction between this uh between this electrode and this electrode is a perfect conduction because we cannot back scatter from this edge to the other edge of the sample. This is exactly the same as Antoan just said before. And now we are going to have a voltage drop between the two sides of the sample which is also quantized.

[00:10:52]So okay this is the integer quantum uh whole effect. We already know this. This is the experimental characteristic of the integer quantum hall effect. Now if we keep either increasing the magnetic field or making our sample cleaner we're going to pass to a different regime which is the fractional quantum hall effect. So this is the original sample where the quantum uh fractional quantum hall effect was measured the first time doesn't look like very clean and I'm here talking about making sample cleaner but remember that this is a berry structure so it's not necessarily what you see on the surface what matters so what happens here is that now interactions are going to uh be more important and in in particular interactions uh your kulum energy is going to become more prominent or larger than both things the thermal smearing which means that you need to be at low temperatures and also the Lando level broadening or the disorder of your system. So in order to obtain a fractional quantum whole effect which can be seen here by the quantization of the resistance this time in terms of the same units uh the same uh fundamental constants but with a feeling factor or a share number of one/ird and that has exactly the same characteristics as the integer quantum whole effect. a plateau in the whole resistance and at the same time a drop in the longitudinal resistance. Okay. So what is important to know I'm not going to uh go into the dict of the fractional quantum hall effect but what is important to know here is that uh this states shows up when your kulum interactions become more important that your disorder and your therm broadening.

[00:12:45]So this uh was the first measurement of the fractional quantum hall effect.

[00:12:51]Nowadays you can see that this well nowadays this is quite old paper. So there are a fullology of uh fractional states each one of them with their own interesting things. But in particular there is one uh important uh state or a series of very important states that are called even denominators. quantum uh holes fractional quantum whole states that uh are very interesting because they are expected to have a nonavilian uh statistics and this Wendel is going to cover next week. So we have integer quantum whole effect, fractional quantum hall effect. But if we keep cleaning our sample, if our sample starts to become even uh better than before, we're going to have something that is called an electron solid. Okay, this electron solid was first observed in the '90s at the end of the '90s by the presence of a strong anisotropy in the measurements of the longitudinal uh resistance and the whole resistance as well. So you can see here that if I measure this in this configuration where I'm applying the current like this and measure it here, this is the crystalline orientation this arrow. Okay. So the crystalline orientation of the two is going to stay the same. I'm going to change whether I measure longitudinal or in parallel to this crystalline orientation or perpendicularly to this crystalline orientation. You're going to see that especially around the even denominators the uh longitudinal resistance is a strongly anisotropic.

[00:14:35]Okay. Then the whole resistance in both cases will go back from the fractional states that was living before to a quantized plateau of the integer quantum whole effect. Okay. So this has been associated with the formation of an electron solid which means that the system in order to decrease its energy is going to h arrange the electrons to form a solid. These solids can be either a bubble face which also includes a beginner crystal or a stripe face. In this particular case, this is an stripe face. That's why is so uh anotropic to measure in passing the current in this direction or passing the current in this direction.

[00:15:22]A more upto-date measurement of this can be seen here. This is still gallium arssonite. You can see that now the this is called a re-entrant integer quantum hall effect is fully quantized and it's not just present in a very particular set of uh land level index but contrary to that is present in many land level indexes. So okay let's just uh take a second and and recap. Quantum hall effect it's integer quantum hall effect.

[00:15:56]If you are in a single particle picture, then if you start to increase your interactions, then you're going to have the fractional quantum whole effect with even denominators, etc. And eventually you're going to pass to the electron solid. What does it mean? It means that if you want to study the fractional quantum hall effect, you have a sweet spot. You cannot have a super high mobility. you cannot have uh the highest interaction possible because then you're going to turn out in the electron solid and this was pretty much the 80s and the 90s for condom matter physicists to studing this uh two dimensional electron gases. Then in the 2000s we had graphin that show up. You know very well that graphin is a array of hexagonal uh hexagonal array of carbon atoms.

[00:16:50]uh that the most common way to obtain graphin in a lab is by the scotch tape method in which we just peel off layers of graph of graphite h and of course the most common source of graphin is graphite. So now in a practical point of view or a more experimental point of view, if we take a layer of graphin which as you know has a linear energy dispersion and we make a device out of it and we're going to measure transport. Okay, we're going to pass the current from this size of the device to the other side and we're going to measure the voltage drop.

[00:17:28]As we change the fermy energy through this gate electrode, what we are going to see is an strong increase in the resistance and then a decrease. And this uh peak in the resistance is the charge neutrality point which means the point where the uh conduction and balance band touch each other. And this is one of the uh most important characteristics of uh graphine. We can tune the carriers from electrons to holes. Okay. So we have a knob that we didn't have uh in gallion arssonite. It was not as useful in gallion arssonite because you decrease the mobility of your systems. Okay. But what about the topological state? What about the quantum hall effect? Well, now we're going to compare graphine with its linear energy dispersion to gallium arseni with this parabolic energy dispersion. So if we apply a perpendicular magnetic field, we are not going to have the same uh behavior between these two. Gallion arsenide has lando levels that are equally spaced.

[00:18:32]Graphin has lando levels that get closer to each other. It also has the presence of an equal zero lando level which is a full topic of uh of rich search. It has been for a long time now. And in particular the land of level spacing between the two first land of levels is much larger in graphing. And why is this so important? Because if you take this spacing between the two Lando levels, you are going to be able to apply this uh this new material to metrology. Okay.

[00:19:11]So in the case of uh when you are looking for a very or highly uh quantized uh two dimensional electron gas the limit that you are looking for is where the energy gap is much larger than the temperature of your system. Let's say that we're going to work at 2 kelvin. So here for example uh 200 kel it establish a much larger uh gap. Okay. So in the case of gallion arsenide as you saw before we can measure the uh whole resistance we are going to find a metrologically uh quantized value only within one uh Tesla at very high magnetic field around 10 Tesla. In the case of graphing on the contrary we're going to be now around four Tesla and it's going to be accurate accurately uh quantized all the way even to much higher uh magnetic field. So what this means is that graphine even in its basic uh characteristic that is just land level formation has already surpassed some materials such as gallion arsenite and it's now used as the new quantum hall resistance standard but well that's graphine on uh CVD graphine on silicon carbide so this is not a super clean system if we want to make our device cleaner we need to go uh to bender valet structures. So in vanderal cetror structures, we can mix materials. For example, like here you see different types of materials. The materials are going to be held each other by bender balls forces, which means that we can stack them as we want.

[00:20:56]We can pull them apart. We can restack them. We can do virtually everything we want. The way we do this, I think Dimma already explained, but I'm just going to pass through it quickly, is we take a polymer, we pick up the first layer with this one, pick up the second one, the third one, etc. So, this guarantees that the graphin uh that we have in the middle, it's always clean. It's going to be perfectly clean. Okay? And in fact, if you take one of these uh stacks of 2D materials and you make a cross-section, you can see in TM measurements that there is nothing between the layers.

[00:21:34]Okay, there are only bounds forces holding them uh together. So establishing this method allowed in fact to go from a quantum hall effect on of graphine on silicon oxide that was barely uh visible at high uh magnetic field and that it was uh didn't even show uh the general liftings to a uh system where we can observe already the generous liftings for the same uh magnetic field And to go even further, one where we can observe the integer quantum hall effect to very low magnetic fields with a quite complicated structure I must say. So today the most or the cleanest uh devices were in fact recently uh measured in a new technique that is called bleaching where at low temperatures you are able to uh bleach the defects in boron nitrite and get even cleaner material. So for example here you can see that the magnetic field at which you can obtain quantum integer quantum hall effect is very low. The same happens to the fractional quantum hall effect is very low magnetic field and you can observe electron solids also a very low magnetic field compared to what it was observed before. Okay. And oh this is graphing perfect. So we have the three landmarks that I show you before integer quantum hall effect fractional quantum hall effect electron solid states. Okay but we are interested in more materials. Okay. So amore material is uh the one where you see or you are able to generate a super latattis. These super latattis are going to be the bright spots that you're going to see in this uh animation. Okay, it's going to depend on the angular alignment between the the layers. In this case I'm using the example of graphine on boron nitrate which has a different in latis constant of uh 1.8%.

[00:23:44]So the size of these numerator latises is going to depend on the difference between the two latis constants on the latis constant and the angle between them which means that just changing the angular alignment between the layer as you saw in the animation I'm going to be able to go from something very close to zero as a super latattis to something very large around 14 nanometers. But what's the origin of this? This is just an effect of who is near you. In fact, so you can see in these three spots that we picked that the mor effect is just related to the fact that not every carbon atom is going to have the same environment. However, another carbon atom at certain distance called lambda is going to have the same uh environment. This creates a modulation a periodic modulation of the landscape.

[00:24:43]Okay. So this has been observed in STM measurements for a long time. So we can see that changing the uh latis um the angle sorry between the two latises we can change the size of this m pattern and very importantly I'm unfortunately not going to talk about it but it's something that h it's extremely important for experimentalist theorists and everyone else is the fact that you can see that for very large super latattises here you are not just going to have a mor pattern uh periodic potential that is superimposed but you're also going to have a strain that is developed inside the mor latis. This is because it's more favorable for graphing to stretch and match the latis of uh boron nitrite instead of just staying in its original position. So this strain is going to develop all around.

[00:25:43]It's what we call the comrated state because graphin stretches to commemorate the latis of boron nitrate and it's the responsible of many uh features that we see but that I'm unfortunately don't have the time to talk today. So let's see in transport what's what is the meaning of all this. So if I have graphin on boron nitrite as uh what I started saying I have a linear energy dispersion. Okay.

[00:26:13]So as in my previous example, what is going to happen if I measure the longitudinal resistance of my graphin uh flake is going to be I just have a strong pick in the resistance. Okay. Now if I have a super latattis, if I align graphine and boron nitrate that have a super latattis of size lambda, I'm going to modify the electronic bann structure and this is just band folding. Now I have a periodic potential that is superimposed. So my brill zone is going to be smaller. My bands are going to be folded and I'm going to have new energy gaps that appear in the system. In particular, the central uh energy gap, the one of the charge neutrality point is has two origins. The first one is because I'm breaking inversion symmetry in my system and the second one is because of the strain that is developed in the common rate state. So now if I measure transport there, I'm going to see a new feature. I'm going to see this pick that shows up. Okay, please keep in mind that these are uh measurements at uh room temperature, which also means that this modification is strong enough that I can see it even at room temperature. Okay. So back in the days in uh long time ago at Colombia, we developed a technique where uh using an AFM tip, we are able to change continuously this uh the orientation between graphine and BN. And you can see that as we change this orientation continuously this P moves because the position in energy of the satellite picss is going to go down and up. This is just the bolding being larger or uh smaller.

[00:27:58]But okay, I have a material. What about topological states? So the first thing that mor materials allow us to do with respect to uh topological states is the observation of the hofstatter butterfly.

[00:28:12]I wish I could uh get into the details of the hofar butterfly but this is going to be like a four hours uh talk so I'm not going to go to that. I'm just going to tell you that uh this is a beautiful representation of what happens in energy to a system where we can put uh at least one uh quantum of uh magnetic flux per unit cell. But now our unit cell is much larger. So maybe we can achieve this uh new limit. So to simplify this hopster butterfly, we can replot it uh as the number of charges per um per super latattis as a function uh as well of the uh magnetic flux. And this is just what we call a ber representation. And what it says is that in fact the gaps we can see these gaps just as a linear trajectories in the density field diagram. Okay. And this in fact these gaps can be represented by the doantine equation. This diopantine equation is going to have the number of particles per m super latattis. It's going to have the flux and uh in particular of a particular importance is going to have this factor t and this factor s where t it's in fact our churn number. Okay. Of course in graphing was not the first time that people try to reach this limit of the hofster butterfly. There are measurements that date back to the 90s where they use gallion arsenide etructures and they pattern holes on desk to try to observe the effects of the um of having a super latattis. In this case the super latattis is much larger. It's around 80 nanometers. And with the technology that it was uh available at the time they managed to see some signatures of the hopsta butterfly. However, the disorder was too large and uh they couldn't see anything more clear. Okay. But in graphing uh aligned with boron nitrite, we were so lucky that these systems are easy to build that in fact three groups in the world observe the hops that are butterfly at the same time. Okay. So this uh this is just a an example of how the motor materials were changing the landscape even uh 10 years ago. I want to give you just a quick uh look of these states. It's going to be a little bit more clear in a in a second. But in particular, I want to flash this measurement here because it seems to me insane.

[00:31:03]It's like you're going to spend the rest of your life just trying to find gaps and trying to analyze and understand what is going on here. Although they did a pretty good uh job. Okay. Now imagine you can have this with one mor super latattis but you can also use the technique I told you before to make your first moratis with the bottom bn aligning the bottom bn and then just change uh the alignment with the top bn and now have a second super lattice. Okay, this will allow you to have even larger more m supercells which in terms of what's happening in the hopsar butterfly it makes everything way more complicated.

[00:31:58]So going to try to keep it simple. I'm not going to go there but I just wanted to show you that we can have we can even modify further these states to have even larger more super latattises.

[00:32:11]So to give you a real h example of what this uh hopstar butterfly uh is doing or what is capable of showing us, we're going to go and talk about a material that uh is as interesting as graphing but it has more tuning knobs which is blayer graphine. So now we are going to just put another layer of graphine on top of it in what is called a vernal stack which means that one layer is shifted with respect uh to the other one. Our band structure is now going to be parabolic. And more interesting now we are able to apply a displacement field. So electric field perpendicular to our sample. And with this changing its magnitude we are able to tw to tune a uh gap in this uh system. This gap is only going to depend of two things. The first one is the inter layer hopping and the second one is the difference in potential between the two layers. So let's put this aside. The way we do this experimentally, the way we tune this displacement field experimentally is by having two gates by applying uh two voltages to the system. Okay. So this is a real measurement. Here you can see that the resistance increases, decreases and increases as a function of as I change the two gates which means that I'm opening the energy gap closing it and opening again. We can reply this as a function of displacement field and density. And we can see that for zero density which is charge neutrality point we have this energy gap opening and closing. More important than that which is already quite important is the fact that with this electric field we can then change the position of our wave function. Okay. What does it means? It means that when we have a positive displacement field, we're going to be pushing the electrons towards the top layer or the holes towards the bottom layer. And the same is going to happen for a negative displacement field. I'm going to be pushing either the holes towards the up layer or to the bottom layer. And this is going to be very important because then it means that I can put my electrons closer to the morel latattis or farther from the mattis.

[00:34:38]Okay. So where are we billayer graphine?

[00:34:43]Uh we have an energy gap. We have a parabolic energy dispersion which also means that our interactions should be larger. And in fact they are. When you look at uh what happens when you apply a magnetic field in the lowest Lando level, you can see that now this lowest Lando lever has a combination of orbitals, spins, uh Lando level index etc. But you can also see the presence of very prominent and very clear even denominators, fractional quantum whole states. And even if you have a look here you can see the formation of electron solids which means that we have recreated already in billayer the same things as in gallion arsenite we have integer quantum hall effect fractional quantum hall effect even denominators which we didn't have in monollayer graphine and now we can recreate and play around with the electron solid as well.

[00:35:43]The other thing that was first time that was possible uh to do it was thanks to the qualities of billayer graphing and its properties is in fact the observation of an electron solid in the integer quantum hall effect. So this electron solid is a a big crystal and uh we are able to observe it because graphin it's on a surface. So we can do STM measurements. These are SDM measurements performed in the Jastani group uh recently. So every time you see a reddish spot that's an accumulation of uh charges and you can see that they are distributed uh they are organized. So in fact they can see as uh they change the uh filling factor of these lando levels how the latis constant changes which means how this becomes larger or smaller and how they compete how is the competition between this beginner crystal and the fractional quantum whole states.

[00:36:49]So this is possible only because graphin it's on the surface in gallon arsenite.

[00:36:54]it was not possible to uh observe big crystallization. It was measured but implicitly uh understood.

[00:37:04]So now hop butterflies no we were at that because this we already saw before.

[00:37:10]Let's put a m on top of that. So if we look at the hostter uh butterfly of a billayer graphine in particular at the very high uh flux very high number of flux per M super latattis which means a high magnetic field we are going to see just a full solology of uh states and these states separate or uh have its own topological significance. So what we are seeing here it's what I think extremely beautiful map of all the topological states that we can have. So each one of the uh represents a topological state obtained by the doantine uh equation and they are going to be classified depending on the values of t and the values of s. So remember that the values of t are the equivalent of the churn number. So we are going to have integer churn number which is the integer quantum hall effect. Fractional churn numbers that are here that are the fractional quantum hall effect. Then we have integer s and uh t which are churn insulators. These are the replicas that you saw before in the baner uh map which are the black lines here which means that these are replicas of the main landal fan that occur at different filling of the mores superlattis. You're going to also have two things that are new and uh very exciting which are fractional SNT which is a fractional churn insulator. We're going to see it just uh in a second. And we're going to have symmetry broken churn insulators.

[00:39:03]Both of them are the result of interactions and they are very active subjects of uh research uh currently.

[00:39:11]Okay. So why can we see this here and not in a regular uh billayer graphine with uh um on BN but not a line. The reason for this is that you need a ratio between uh you need certain number of interactions but also you need to compete in this case with the latis potential and in fact this fractional charge insulator can be stabilized only in a uh range of this ratio. After that what you get is a compressible state which is just a solid uh an electron solid state. So if you think about it all these all these fractional churns insulators integer quantum hall effect and all these they are the origin of them is just land of levels flat bands.

[00:40:09]Okay. So now the question is whether can we make flat bands without magnetic field. Are we able to do this? And since 2018 we know that we are able to do this. So in fact when we take two layers of graphine and change the angular alignment as we can see it in this numerical uh simulation the electronic band structure is going to get modified and at a certain angle called the magic angle what we're going to have is a flat band. Okay, this flat band is the responsible or it's believed to be the responsible of the observation of u strongly correlated state superc conductivity etc in twisted blayer graphing and what is interesting here is that the fact that you have a flat band will quench your kinetic energy and now your interactions are going to become much prominent okay and that's why you get all these states of course This was uh once again the first measurement but uh no long after a lot of new superconducting domes uh correlated states etc were observed in the same system.

[00:41:27]However there is a problem with uh twisted blayer graphing. The main problem with twisted billayer graphing is that you can have a beautiful solology of states to study, but you try to make to replicate exactly the same angle with exactly the same characteristics and you might find something that is different. And this is related mostly to angle disorder, a strain, discomrated state that I told you before is also playing a very important role. the relaxation of the atoms.

[00:42:05]But uh then this is also a very active uh research topic at the moment. How is that we can have the same angle but different erh physics? What are the other parameters that are playing an important role here? So okay this is very interesting but we are mostly concerned about topological states and this is not at least not that I know of topological state. However if we take exactly the same system so twisted blayer graphin align at the magic angle and we also align the botto nitrite that is as a substrate we are going to observe quantum anomalous hole effect.

[00:42:48]So why is this important? Remember that I just showed you you can have two mores and they might interfere with each other. They might lift new uh symmetries of the system. This is what's happening here. You have one mor that is created by the twisted by layer graphine and a second mor that is superimposed that is created by the graphin and the boron nitrite. Okay. So let's see those results of quantum hall effect. So when we measure the resistance as a function of the density in a quantum anomalous whole state, what you're going to see is that for zero magnetic field, you're going to have a strong increase of your whole resistance. And this value here is going to be quantized. At the same time, you're going to see a decrease in the longitudinal resistance. So exactly the same as the quantum hall effect, but this time without any magnetic field.

[00:43:45]Okay. So in fact if you go and h change the magnetic field you place yourself sorry you're going to place your fermy energy at this position so your gate voltage and you're going to sweep the magnetic field back and forth you're going to see thisis loops. So this is the loops uh tells you that there is a magnetization an intrinsic magnetization of the system that is switching as you change the magnetic uh field. Okay. So the reason why we believe we observe this quantum anomalous whole state which by the way has only been observed in one sample in the world since 2020. Okay. It's because when we have twisted blayer graphing, we're going to have this uh these topological bands, but you can see that the total churn number is equal to zero.

[00:44:44]So, we don't see any topological effects. adding the boron nitrite and breaking an extra symmetry with the boron nitrite is going to lift one of this uh degeneracies and now your chair number is going to change and that's what uh makes possible the observation of this uh integer quantum whole effect without magnetic field or anomalous uh quantum hall effect. Of course, since this has been observed in just one sample in the world, there are a lot of questions about it. What is this origin? How is the BN really playing a role? What is the temperature, angle, and current dependence of all these? Can it be metologically quantized? Because it's going to be a revolution if uh for metrologic uh measurements, we can do the same kind of measurements as in the quantum whole effect without magnetic field. Okay. So this is very beautiful and uh is one of the most remarkable uh results on twisted bioraphing to my own personal taste.

[00:45:53]However, at that moment then people started to think about can we make this can we make flat bands and these kind of uh systems to observe this phenomena in other materials. Okay. So I already told you about monollayer graphing and billayer graphing. So monollayer you have just one of this billayer you have the second layer that is stacked. Now you can add a third layer, a fourth layer, fifth layer. And what is interesting here is that you can see that the energy dispersion goes roughly without just a first uh approximation roughly as the power law of the number of layers which means that every time you add a layer you're going to have an even flatter band uh near charge neutrality point.

[00:46:48]Okay.

[00:46:50]So now you can imagine if I add a more potential to all this then I'm gonna cut my bands. I'm going to make the same thing I did before this band folding and I'm going to isolate my flat band.

[00:47:08]So is this possible or not? That's it's a very good question. Is this what is happening or not? That's a an excellent question. And the main problem to see uh all the results of robohedral graphim by the way sorry I completely forgot this organization of layers is called robohedral. There are there is another one which is more stable called bernal in which this layer is going to replicate this first layer and in that case you are not going to have the same bands. the B structure is completely different and it's topologically trivial. Okay. So, but let's focalize now in uh M super latattises in robohedral graphing. So whether this really isolates the bands and it's good for observing uh topological state it's still under debate in the sense that when we have the exactly uh same kind of system and we measure their uh transport response we do observe topological behaviors that I'm going to show you in a second but contrary to what you will ect we observe this uh topological behavior here in the m distant regime. Okay. So remember I told you before if you apply a displacement field you can push your wave function either closer to the m or farther from the m. In the case of robohedral graphing, in fact you can observe topological states of uh integer and fractional anomalous quantum whole effect. But this time is when we are far away from the mor. Okay, which is a little bit contraintuitive if I think that well if the mor is doing what I think it's doing, I should see it h in the other side.

[00:49:11]But okay so what is uh also very interesting uh of no sorry so the other thing that is important to keep in mind that this is a mu distant but it's not pushing the wave function to a place where there is no mu because remember this commensurate state I told you before this will propagate to a certain number of layers we don't know how far it's going to propagate but we know that at least to 10 layers we can still see the more h the commenate state. So it's just a more distant but not a more less uh part of the diagram. Okay. So now to explain this we can perform some numerical simulations of uh of the system and we can find that we are going to have in fact a flat band a non-trivial flat band with a bunch of trivial uh bands uh around as Anton said uh before so this flat band might be the responsible of observing this uh fra this integer uh quantum whole uh anomalous quantum hole state. However, it's not necessarily the reason why we observe the fractional uh hold effect.

[00:50:29]So, just to give you a little bit more of a hindsight of uh these states. So, remember that before I told you if you sweep the magnetic field, you can switch the magnetization, the spontaneous magnetization of the system. And now you're going to see that this is quantized to fractions of um fractions of the e square over h. Okay. So what is very important to keep in mind when you look at this um loopsis loops is that remember twisted b layer before it was not that clean. It had a bunch of jumps in the middle and the reason for that is because you have angle disorder in twisted billayer graphine. You have a lot of different angles in the same sample which are going to give you different um effective magnetizations.

[00:51:25]However, robohedral graphin seems to be an extremely clean material where we have a mor super latattis which as it is formed with boron nitrite and not with another graphine it's way more stable and that's why you can see this is stresses loop being very very clean.

[00:51:46]So okay we have recreated uh integer quantum hall effect without magnetic field fractional quantum hall effect without magnetic field maybe there are other surprises to come and uh in fact uh just just to give you an overview. So now we're going to have to change the number of layers. We're going to go from penta layer that we were working before to four layers. So by the way this effects topological effects has been observed in four five six and seven uh layers superconductivity has been observed in three uh four five layers only unless someone published a paper this morning that I'm not aware of but uh I'm going to try to read it when I go back home. So uh what I want to say here is that uh for brhedral uh four layers graphing you can still see a fullology of topological states. Now you have topological states with different share numbers fractions integers etc. But you can also see superc conductivity pockets okay in exactly the same system. So this here it's what I truly consider to be the most beautiful curve I I have ever seen and I have seen before a lot of curves where you can see superc conductivity and quantum anomalous hole effect in the same device in the same uh at the same moment just by changing the concentration of carriers of the material. Why is this so beautiful?

[00:53:32]Because if you ignore the uh whole resistance, you're going to see that both phenomen have the same behavior. No, a longitudinal resistance going to zero.

[00:53:45]But these are completely different phenomena. So these are not related in this uh in this case. I mean they are related but not in the same in the way we believe. So now this is the first observation of both superc conductivity and uh a topological state in the same device which can bring us to the development maybe in few years of new states such as topological superconductivity using uh quantum hall effect to connect superconducting cubits you you name it we have almost everything. So okay it seems that is maybe even more interesting than uh all the topological states we saw with magnetic field before we can have because we can have uh also superc conductivity. Now the other thing that you haven't seen yet is the electron solid. No at the beginning we were talking about we have integer quantum whole effect fractional quantum hall effect. Keep increasing your interactions you're going to have an electron solid. Well, it turns out that you can also have an electron solid without magnetic field. So nowadays with this incredibly uh clean samples, you can see the re-entrance of the integer quantum whole effect at very low temperatures but without any magnetic field. This means that the interactions become higher and higher and then uh your system is going to organize as it did before. Whether this is a bubble or an stripe phase is still uh under debate. But there are very recent developments by the GIA group by the league group in Texas that show that this might be a bubble phase.

[00:55:34]So does matter? What do you think? Is it important for this topological state?

[00:55:44]Is it really important? Turns out that it might not be the only way. Okay. So this example that I'm going to show you uh it's a pentalayer roboedral graphine although I think there is a no. Yeah it's good h in a close proximity with tunandiseleni the material with highest pinor coupling. Okay. So in fact, if you put these two uh in contact, make exactly the same measurements as you did before, you're going to observe in this little pocket a anomalous quantum hall effect, which is as clean as before. The difference is that the churn number is higher now, but it's still a churn number. It's still quantized. It's still the same uh physics. So truth to be told uh mu might be important for mu materials but m is not the only way and it's not uh a necessary condition to observe anomalous quantum hall effect in robohydral graphing. So with that I would like to summarize.

[00:56:54]First thing is we did it we passed from this million dollar machine to tape. So great we can unlock with this motor systems uh new limits of physics that's we we have been doing all these years we can have a extremely rich new physics uh with these motor systems that go from replicating the original topological states integer fractional and electron solid states in graphing to making also the same integer fraction and electron solid states in robbo graphing without any magnetic field. So with that I would like to thank you for your attention.