 So let me start by thanking the organizers for working. Perfect. Thank you. Yeah. All right. Sorry for the technical problems. So today I'm going to talk about some recent work on pathological state in mod insulators. And this is motivated by recent experiments on semiconductor mori systems. But the theory I'm going to talk about is hopefully in more general. So yeah, let me first acknowledge my collaborators. The theory work is done with two post-docs at MIT, Yang Zhang and the truth of debacle. And we are fortunate to collaborate with experiments Cornell, GSM, and the King Fine Mac. So yeah, let's start with some basics about mod insulators. As we all know that it's a prime example of a strongly correlated electron system. When there's a strong on-site repulsion at the filling of one electron per site, the band theory would say the system has to be metal, but repulsion makes it into an insulator. At every site there's a single electron and the spins of electron can do interesting things. But as far as chart degrees are concerned, this is a very simple state. And the mod insulator can give rise to many interesting phenomena. When, for example, you could dope it, it can turn into a superconductor at a higher temperature, there may be a pseudo gap state, and all this is motivated by the study of mod insulators strongly motivated by high-temperature supernet as a culprit. And a very different kind of electronic states is a topological insulator. And, for example, one of the earliest examples of a topological insulator is this Hodain model, honeycomb lattice, and Hodain introduced complex hopings that microscopically corresponds to loop current states, honeycomb lattice. And he showed that this state actually supports a quantum Hall effect without any external magnetic field. And this quantum model states has one-way moving edge states on the boundary, despite the bulk has an insulating gap. So this kind of a topological state is fundamentally different from an atomic insulator. And one way to see this is that hydroelectrons on the edge cannot be localized. So this is very different from the case of an atomic insulator, from the case of a mod insulator where electrons are localized at sites, both in the bulk and on the boundary. Okay, so topological states cannot be fully localized everywhere. And this quantum model state was experimentally observed about 10 years ago, in my nanodoped topological insulators, and you see this foundation of the Hall conductance without any external magnetic field. So I'd like to highlight the fundamental distinction between these two different states, the mod insulator and topological insulators. And I think the distinction between the two really represent a fundamental dictony between the particle and the wave aspects of electrons. The mod insulator is understood from the point of view of particles, from the point of view of real space, while topological insulators are naturally described in terms of the waves, coherent block waves in a periodic medium. And also another fundamental distinction between mod insulator and quantum norms for insulators is that in a mod insulator, the magnetic order is usually antiferromagnetic. So there's no global breaking of time rule of symmetry. We perform time rule, so followed by a translation, the state is restored. While in the case of the quantum norms for state, the chirality of the edge states requires a global breaking of time rule of symmetry. And in the case of nanodoped TIs, the system is ferromagnetic ordered. Okay, the magnetic impurities in the synfilm etopologian order ferromagnetically, and the sign of the magnetization, whether it's pointing in the plus or minus z direction, that determines the sign of the all conductance, whether it's plus or minus e square of h. So all these differences makes one wonder, one may ask that not modern topological states are not only distinct, but is there conceivably any direct pass between the two phases? Okay, so what I mean here is the following. Imagine in the phase diagram of all possible phases, is there a direct transition, a continuous direct transition between a mod and quantum norms for state without additional fine tuning? Okay, so we know, for example, this kind of question, you can know we know that in the case of a topological banding disorder, such a direct transition, continuous transition, is well understood, is described by a massive diaphragm with a mass changing sign. And we also know that, as far as I know, there is no such direct transition, at least I don't know any, between a fractional quantum pulse state and a ordinary banding disorder without any fine tuning. So the question now is that imagine I have a hemorrhagic invariant system with an odd number of electrons per unit cell, we know there are one type of states, which is a mod-insuiting states, and we know that in principle, it can also realize a quantum nonthal state as the same feeling. So the question is, can we tune the system in some way to achieve a continuous direct transition between the two, by only tuning one parameter? Okay. So I was motivated to think about this problem by the recent experiments from the Cornell group, and the experiment is done on two layers of the transition method that charge night, monolayers, MOT2 and WSE2. By itself, these are just ordinary semiconductors, nothing fundamentally so interesting, but putting them together, you can use electrostatic gating to achieve a feeling of one whole per unit cell, per ordinary unit cell, and remarkably, by tuning the electric field between the two layers, a transition from a mod-insular to a quantum nonthal state was observed. Okay. Both layers are non-maniac semiconductors, and yet it can arise to a quantum nonthal state spontaneously, like camera symmetry. So what I'm going to describe is a theory for a continuous transition between the two phases, and this phase transition is described by a universal effective field theory. And instead of dealing with square lattice, and we know that these TMD systems have three, four rotation symmetry, it has a triangle lattice symmetry. Okay. So first, let me give you some basic information about these TMD systems. A monolayer TMD, such as MX2, these are large-gap semiconductors, and near the, for example, connection between the span edge, the dispersion is simply parabolic, and we're going to deal with, for example, whole-doped WSE2. So the only interesting, the relevant states are near the band edge, and the band edge is located at both k and k prime. They are degenerates related by time-rosal symmetry. Because of the spin-optic coupling, there's a spin splitting. So at a given valley, the spin is polarized, and the two valleys have opposite spin correlation due to time-rosal. So together, they form a spin-full system. Combining the two valleys k and k prime, they form a 2D hole gas with a relatively large effective mass on the order of electron mass. And this large effective mass leads to strong interaction effects, even in a monolayer MOSE2, within a crystal state was observed without any external magnetic field. And that's because RS is large at this density range on the order of 10 to the 11th. And this experiment actually measured the X-tone energy spectrum in the system as a function of the carrier density. So when the carrier density was introduced, X-tone would interact with the electrons and to form the so-called trials. And as a result, the X-tone energy shows a blue shift. And this is shown by the red line here. And this is well understood. But interestingly, at low density, you see that an additional feature appears, a second X-tone peak appears. And it turned out by detailed analysis in this work from the ETH group, they show that this additional X-tone peak actually has come from the Bragg scattering of the X-tones from the electron and Wigner crystals. And so this is evidence of the Wigner crystal states. And the physics of the TMD is getting more interesting when you consider stacking two layers of TMD together. And when the two layers have a lattice mismatch or has a finite twist angle, you introduce a long wavelength periodic structure that's called the Moray Suplatus. And it was first proposed by Allen's group in 2018 that these systems may be an ideal validation of the Haber model. And the idea is actually very simple. Again, think about this, for example, the TMD bilayers WS2MOS-SE2. And this can be because of the fundamental work-function difference between the two layers. Low energy carriers, in this case, only live on the WSE2 layer. And the role of the MOS-SE2 layer is basically to introduce a periodic modulation of the structure. And this leads to a periodic modulation of the valence band edge. And this can be modeled as a Suplatus potential for the carriers in the WSE2 layer. So what we have here is really the simplest example of Moray band structure described by a free left hole in a periodic potential. And when the Moray period is large, the characteristic kinetic energy at the Bronson boundary is small. It can be made arbitrarily small if you go to small twist angles in the case of the homo bilayers. And in this example, the Moray period reaches on the order of 10 nanometers. So the kinetic energy scale is very small. And in this regime, when the kinetic energy is small compared to the potential energy, then the system really behaves just like a periodic array of quantum dots. Basically, you think about the periodic potential. Each potential minimum provides a harmonic trap for the carriers. And so the system behaves as an array of quantum dots, weakly connected by tunneling between these potential minimas. And this naturally leads to a tight binding description. And once you include the repulsive interactions, it leads to a hybrid model. So following Allen's proposal, the experimental groups at the Cornell and the Berkeley, they actually studied this WS2, WS2 semiconductor hetero bilayer. And indeed, they observed characteristic physics of the hybrid model. So shown on the left are the resistance imagining transport as a function of the feeding factor. You see at the feeding factor of two, when the first Moray band is complete field, you get a banding disorder shown as a resistance peak. But even more prominent is what's happening. Because of the strong hyper repulsion, the insulating state appears. And that insulating state is a modeling disorder. And the Berkeley group observed not only the modeling disorder, the banding disorder, but also insulin states at fractional feeding fractions, including one-third and two-thirds. And again, these were quickly understood as generalized WS2 crystals. Basically, you take the limit of the kinetic energy goes to zero, you have charges on the triangle lattice, and the ones you include the on-site repulsion as well as first neighbor repulsion, then the crystal states will form at one-third and two-thirds feeding. At these feelings, you can avoid nearest neighbor repulsion. And that's why these WS2 states are favorite. And by now, many more WSFs have been observed at other feelings, and all these can be understood by just the extended hyper model on the triangle lattice in the strong coupling regime. Yes, please. Right, so RS here is on the order of, so it depends. So in the case of these experiments on MOS E2 in the electron connection band, the mass is heavier. The mass reaches on the order 1.5 electron mass. And in this case, RS I think is on the order of 30, 20 to 30. In the case of the the WSE2, the effective mass is smaller. So in these systems, RS I think is only of order 10, if I remember correctly. So the fact that you see a Wigner crystal at such RS is again telling you that this is not just a Wigner crystal in free space. Okay, this is actually a Wigner crystals pinged by the periodic model potential. And these are all incompressible states. If you have a Wigner crystal without any pinning effect, it's actually a compressible state. If you change the density, the Wigner crystal period just changes continuously. While in all these measurements, there's capacity measurements clearly show that these are there's an incompressible gap at these feelings. So these are incompressible Wigner crystals pinged, strongly pinged by the lattice. So it's sometimes called generalized Wigner crystals. Next slide. Yes, next slide. Okay, good. So another important feature of multi physics is that once modeling certain states is formed, the charges are localized, but the spins are active degrees of freedom and there are undifferent magnetic impactions between these spins that we expect from the hyper model. And indeed, so one can actually probe the magnetic response of the system using circular dichroism by looking at the difference of the reflection, optical reflection using left and right circular prize light as a function of magnetic field. You see a zero field, time rule symmetry tells you that the left and right circular prize give you the same signal. But once applied magnetic field, small magnetic field like 0.2 Tesla, you see that the x-tone energy peak shifted depending on left or right circular prize. And if you plot the x-tone energy splitting as a function of magnetic field, you see this interesting contrast. So this red line is without any carriers, without any carriers. And in this case, you see that the x-tone limit splitting is very, very small. And on the other hand, once you go into the modeling state with one whole per Moray unit cell, you see the x-tone limit splitting is much, much larger and it rises steeply with applied magnetic field and then eventually saturates. So this behavior is what we expect from magnetic field, the induced spin polarization as a function of the external magnetic field. And one can even look at the slope of this magnetic field at around zero field, which defines some kind of a spin susceptibility. You plot this as a function of temperature, you see that nicely follows the Curie-Weiss behavior and the Curie-Weiss constant is negative depending on there's undifferent magnetic impactions in the system. Okay, good. Okay, so so far everything seems to be well understood from the point of view of a triangle lattice Haber model. But we, from very early on, from more than two years ago, we realized actually the system can be more interesting. Yes, Eris? Ah, excellent. So it's not and it's just because there's a spin value locking in the TMDs. So the in-plane magnetic field actually doesn't do anything in these k-value systems because you cannot, you know, couple the two values by in-plane magnetic field. An interesting question is about gamma-value systems. So we also worked on the gamma-value systems and there I think the magnetic response would be should be isotropic. So you could, you could, yeah, but that effect, so for example, right, if you look at the case the red line without any doping, without any doping, then this external Z-man splitting basically comes from the conduction of valence band edge split on the magnetic field and the amount of that splitting depends on the free electron defect depends on the optimal magnetic moment. One can include all that, but the point is that that is all very, very small compared to the magnetic response in the modeling system state. Yeah, yes. No, it's a good question. So it turned out in the simplest model there isn't any because even though the two spin-off and spin-down states are associated with different values k and k prime, the dispersion at both k and k prime are just parabolic, right? So within the simplest description there isn't, there's an emergent spin-rotation symmetry. One can go beyond that and it turned out that this is a small effect. So there could be some sort of some sort of effective magnetic flux, so to speak, within each valley or opposite flux is within each valley, but that effect is very small. Yeah, okay. Or another thing that you know, there could be, you can think about barrier curvature effects within each valley, but those barrier curvature are very small because the parent state is a large gap bending sort of, it's a bending gap of 1.5 eV. Okay, good. So right, so from early on, we realize that actually if we look at this more potential more carefully, so if you have a potential with a symmetry of a triangle lattice to the leading order, you can express it as a sum of three cosine terms where this reciprocal aspect of g points in three equivalent symmetry-related directions. But in addition, there is a phase factor of phi and this phase factor of phi, the inside the cosine, that determines the landscape of the potential. And when phi is to zero, for example, you find the potential minimum forms a perfect triangle lattice, okay, but when phi is actually pi over three, for example, you find that there are two degenerate minimums at the different locations within the unit cell and this corresponds to a perfect honeycomb lattice. So for generic phase parameters phi, it's actually in between. And when I say different locations in the unit cell, it actually corresponds to different high symmetry stacking regions. For example, the mm region is where the metal atoms of the two layers are vertically aligned. In the xm region, the x item of one layer is aligned with the metal atom of the other layer and the same for mx, right? So depending on the range of primary phi, so in the range of between zero and pi over six, the potential, there's only one type of potential minimum and that form a triangle lattice. And this is indeed the case for MOSC2, WSE2, studied in Allen's original paper, but for WSE2, WS2, which is the recent experiments is about, we find that in addition to a primary energy minimum, there's a secondary potential minimum and so including both minimums would require a two-band model and this is actually a two-band model on the biased honeycomb lattice, honeycomb lattice with an asymmetric potential. And this is the band structure from our large-scale DFP calculation by Yang Zhang. And you can see that, for example, these two bands, the first and second Moray band, this is well-described by a honeycomb lattice type binding model with potential difference between A and B sub-hattice. So why this is important? So you might say that if we are at a fitting factor below two, only the first band is involved and the first band primarily lives on the triangle lattice potential minimum. And why do I care about this? Well, there's a big difference once the Hubbard interaction U is introduced. So the two types of mod insulators, the single-band Hubbard model really describes the so-called mod Hubbard insulator. And in this case, this applies when the second potential minimum is far away in energy when the Hubbard repulsion U is smaller than the energy difference between the two potential minimums, which I call delta. So for U equal less than delta, if you add an additional charge on top of the mod insulating states, it's preferable to create double occupancies. And this is all captured by a single-band Hubbard model. But when the repulsion U is very large, U is bigger than the charge transfer energy delta, then above the filling, half filling, an extra charge will actually occupy another sub-hattice, a different sub-hattice. And in this case, it leads to a charge transfer insulator. And this charge transfer insulator actually is the case for high-temperature signatures. So basically in between the lower and upper Hubbard bands, the lower Hubbard band is band associated with holes. The upper Hubbard band is a band associated with doubloons. But actually the chemical potential sits in between the lower Hubbard band and a different third band, which is called charge transfer band. And this third band actually comes from the second potential minimum, a different atom, so to speak. And in this case, the insulating gap is set by the charge transfer energy delta instead of the Hubbard U. So we believe that this charge transfer insulator picture applies to many semiconductor Moray sub-hattices. And for a while, our work is completed and all. But finally, the direct evidence of the charge transfer insulator came. So this is Twisted WSE2. And in these systems, when you apply a large electric field, charges are all localized in one layer. So essentially it's a one-layer system. Yes, please. That's exactly what this slide is about. Yes. Okay. Oh, the question is what is the magnetic order in the charge transfer insulator? And this is exactly what I'm talking about now. Yeah. Okay. So under a large electric field, charge is still living in one layer. So it's still a one-layer system. And so as I said, in this system, you can change the filling factor continuously from zero all the way to one, where you see a modern insulating state, and then further all the way to filling factor two. Now, in the single band description, okay, for a multiple insulators, at the filling factor of two, you expect double occupancy at every site. And such a system will have no magnetic moments. On the other hand, in a honeycomb lattice model for a charge transfer insulator, at filling factor one, you have one electron at every primary potential minimum at A site sites. And then if you go to filling factor two, because of large cover U, instead of creating double occupancy on the A sites, you have one electron at every A site and one electron at every B site. And in that case, you still have local magnetic moments at the filling factor two. And this is exactly what's observed, okay, that at the filling factor of two, this measurement of the magnetic circuit dichroism, which tracks the validation as functional field, you see again, the pure wise behavior of the main accessibility of high temperature. And then the stability start to drop at low temperature, indicating an anti-permanent infection. Okay, and again, that's what you expect for honeycomb lattice is a bi-bi-bi lattice, and the electron are closer to each other, because it uses both A and B sites. So you expect a much stronger AFM infection at a filling factor two compared to the case of a filling factor one. Okay, so all this is telling us that at filling factor one, electrons occupy the A sites, leading to a charge transfer in through and doping in the further, the dope charge occupy the B sites. Okay, now with this charge transfer picture, we can look at this MOT to WSE2 system. This is again, where the mod and the polynomial state was observed. So in this system, turn out both layers are actually involved. Okay, so this is because when we look at the intrinsic potential difference between the MOT2 layer and WSE2 layer, this delta essentially is a charge transfer energy. Okay, it's an energy cost associated with transferring charge, not within the same layer, but between the two layers, between the two layers. And our DFT calculation says if you just look at the work function difference for the two layers, there is a charge transfer energy, which is 130 mEB. This is much smaller than the case of WSE2, WSE2. Okay, so then this suggests when you apply electric field, one can actually tune strongly, strongly tune this charge transfer energy delta, and this is a way to control the charge transfer energy in the system. Okay, this is something that is hard to achieve in something like cuprids, for example. But in the 2D materials community, the charge transfer energy between the two layers can be tuned by the electric field. So now we can look at the physics of such a charge transfer system. And here are the experimental data. Again, let us first look at the case, sorry, sorry, when all the electrons are in one layer, in the MO layer. And so here the, there's two axes. One axis tunes the fitting factor and the other axis tunes the electric field. So when charges are all in one layer, this is near the top of this figure. And as you change the fitting factor, you go from the no carriers to one electron per unit cell to two electron per unit cell. And at the fitting factor of two, we expect the bandings are when the first moribands is complete field. And, you know, this is the starting point of our discussion. And if you maintain a fitting factor of two, and you increase the electric field between the two layers, you find that the resistance first decreases and then increases again. Okay, there seems to be a transition between two types of similar states. And this is made more clear by the capacitance measurement of the mg gap at fitting factor two. You see that the gap closes and then reopens. Okay, so at the banding sort of fitting, there's a transition between two types of banding filters. So we studied again using the large field DFT calculation of the moribands. And so basically once you put the two layers together, because of the lattice mismatch, a moray structure forms with a wavelength of about 4.6 nanometers. And so because the moray period is relatively small, the bandwidth is much larger. So this is far away from the flat bandwidth gene, very different from magic angle graphene. And here there's nothing relies on a special magic angle. So in the MOT2, for example, what we first do is to just look at the effect of the lattice corrugation on each layers. Okay, we put the two layers together, optimize the lattice structure, and then take the two layers apart, maintaining the corrugated structure. So this is a way to help us understand what's going on. So we find that in each layer, because of lattice corrugation, you see a whole set of mini bands, mini bands separated by superlattice gaps. And the bandwidth is on the order of 40 millilitron volt. And the band gap, superlattice gap is on the order of about 10 millilitron volt, or less. So here all I want you to pay attention to is that the red and the blue curves are the mini bands from different values. Okay, again, the two values are independent of each other. So let's just focus on one value, the red curve. You see that in the MOT2 layer, the band top is at a kappa point, and band minimum is at a gamma point. Kappa and gamma are the locations in the mini-brone zone. And on the other hand, for the WSE2 layer, now the band top is actually at the gamma point. Okay, and remember that the two layers have actually intrinsic potential difference. So at a small displacement field, so all the electrons live in the MOT2 layer, okay, with the band edge at gamma. And the unoccupied states is in the WSE2 layer with the band top, actually at also, sorry, the band minimum is at kappa prime. And here on unoccupied states, the band top is at kappa prime at the same location. So then when we change the displacement field, we reduce the charge transfer energy between the two layers. And as you can see now, you can create a band inversion at the mini-brone zone corner. So there's, for bands from one given value, band inversion occurs at a single point, gamma point in the mini-brone zone. For bands from the other value, it's time also conjugate. So band inversion occurs at another point, kappa prime in the mini-brone zone. So this is what the DFT calculation shows. And without going into the technical details, in addition to this band inversion, there's a small hybridization between the two layers. And this hybridization has a P wave form factor that creates a topological known trivial gap. So after band inversion, there's a trend number of opposite sign in the two values, plus one, minus one, the two values. And this means that at the fitting factor of two, there's this transition, there's a transition tuned by electric field from a trivial insertor into a time reversal invariant topological insertor, which is known as a quantum spin-haul state. So this not only explains the experiment data, the gap closes and reopens, but also it predicts that after band inversion, there should be helical edge states, and this can be detected by non-local transport, for example, non-local transport. Okay, so the more interesting phenomena occurs at the fitting factor of an equal one. So how much time do I have? Okay, that sounds good. Yeah, all right. So at the fitting factor of an equal one, what experiments observe is a foundation is on this line here. You see from a highly resistive state, a modding switching state, into a quantum non-force state, where the haul resistance appears spontaneously without any external manifold, and it becomes quantized at a low temperature. Okay, so to my knowledge, this is the first and so far the only observation of a transition between a mod and a quantum non-force state. Also, notice that there is no hysteresis with respect to the applied electric field. This said, at least it's consistent with a continuous phase transmission. So in order to understand the physics at fitting factor one, band theory is not going to give us the answer. Band theory will predict half of the system is a metal. So we have to think about the interaction effects. And in order to deal with interactions in the Moray bands, we try to downfold the system into a effective type binding model. And so basically, so we constructed a type binding model for the charges in the MOT2 layer. There's the red band. So I've done the particle transformation. So instead of talking about holes, let me just talk about electrons. There's electron in the MOT2 layer. There's the first Moray band has a distortion like this. And then there's a Moray band from the WSE2 layer. You see that the bending minimum is offset from gamma and k. So together, the effective type binding model once again become a honeycomb lattice. So the A-sites form a triangle lattice. It's associated with the MO layer. And the B-sites form another set of triangle lattice. It's associated with the WSE2 layer. And there's a hoppings within each layer. And there's an interlayer hopping TAB. And in the B-layer, there's a spin-off coupling responsible for the fact that the band edge is no longer at the gamma, but at the k point. So these are all the ingredients of the type binding model. And then on top of this, we add the interaction effects. And the predominant interaction effect is just the harbor u on-side repulsion. And this picture summarizes everything I'm going to describe. When the interaction is weak at half-fitting, we get a metal. When harbor u is very, very large and when the charge transfer energy is also very large, then we basically get a charge transfer insulator, a charge transfer insulator, the electrons on the A-sites at half-fitting form a anti-firmament model insulator with a harbor u large energy gap. And as we reduce the charge transfer energy delta, this charge transfer band moves towards the Fermi level. As the charge transfer energy is sufficiently reduced, the energy gap between the charge transfer band and lower harbor band become inverted. So I'm going to show that in this inverted regime, this inverted charge transfer insulator is actually a quantum non-sponsor. Yes, it is 110 degrees. That's what I meant by that. This is exactly the picture. So now, this is a strongly correlated system. You may ask, you know, is there any reliable way to do an analysis? And I want to argue that in this system, there is a controlled limit where we have a symbolic exact theory. So let me first start from this this uninverted regime. In this charge transfer insulator, the magnetic order, as Nicolae just said, is this 120 degree state. And if you look at the holes in this AFM insulator, it has a dispersion. The detail of dispersion is unimportant. All I need to pay attention to is that the quasi particle hole band in this 120 degree AFM state is synchronized and the band edge is at a single point, at a gamma point. And then on the other hand, the charge transfer band, which is associated with the B sites, because the B sites is unoccupied, essentially. So it has a spin degeneracy. It's also located at a gamma point in the magnetic prion zone. Yes, because in the AFM order state, the question is why this spin polarized, why the holes in the dispersion is spin polarized in the 120 degree AFM state? Because of magnetic order, the spin degeneracy is split. Well, in this example, in this case, in this AFM, in this 120 degree non-colonial state, there isn't. So if you look at collinear case, it will be different. In fact, that's important. The non-colonial magnetic order is the starting point of this analysis. Okay, so now I'm going to focus, because band enverging only involves lower energy degrees of freedom. So all the action is going to take place near the gamma point of the magnetic prion zone. So this allows me to go into a continuum description, focusing just on the cosy particle bands near k equal to zero. So there's one spin polarized band on the A sites, and there are spin degenerate bands associated with the B sites. One is the whole light, the electron light, and then there's a hybridization with a p-wave form factor. Again, that is dictated by the magnetic order. The magnetic order still has a three-fold rotation of symmetry. Now, so this is single particle physics all captured by a three-by-three matrix in the continuum limit. And then on top of that, we should add interaction effects. And it turns out that the only relevant interaction is this dense interaction, the hybrid interaction on the B sites, on the charge transfer band. And if you look at the band structure, you find there's a two-fold degeneracy associated with the B sites. This two-fold degeneracy at k equal to zero remain protected when we turn on the hybridization between the A and B layers. And that has to do with an exact symmetry. I'm not going into the details. But here is the evolution of the cosy particle bands as I reduce the charge transfer energy, delta. Okay, for now, let me keep the interaction g to be zero so I can analyze the band structure by diagonalizing the three-by-three matrix. And you see that as delta is reduced, this band from the B sites comes down in energy. And then it swap places with the whole band on the A sites. So after band inversion, the band ordering goes from a two-fold on top of one-fold to a one-fold on top of a two-fold. So band ordering changes. And after band inversion, the Fermi level will be exactly at a quadratic band touching points. In the finite range of charge transfer, inverted charge transfer energy range, there is a quadratic band touching that appears at the Fermi level. And it's known that this beautiful work by Farakim Kielsen collaborators more than 10 years ago, that this quadratic band touching is inherently unstable to interactions. Okay, so in our context, this quadratic band touching comes from the spin degeneracy on the essentially comes from spin degeneracy on the B sites. So the hardware polishing give you an energy splitting and give you a spontaneous energy splitting. And this was exactly what leads to a gap opening at the quadratic band touching points. And again, once the band gap opens, there's a quantized barrier curvature associated with this gap quadratic band touching points, and that leads to a turn number. Okay, so basically after band inversion, when the Fermi level sits at the quadratic band touching point, when I include a weak repulsive induction on the B sites, it opens up a magnetic gap. And this magnetic gap at the quadratic band touching points leads to a turn influencing states. It's a quantum noncephalic gap. And what's important here, like highlight is that, so this magnetic opening on the B sites corresponds to spin pointing along the Z direction. So in this inverted regime, the A sites spins are predominantly still 120 degree AFM order states, but on the B sites, the spins are pointing in the Z direction. So it leads to now a non-coplanar spin structure. So the important thing is actually the change of magnetism from in-plane AFM into a canted AFM, a non-coplanar AFM state. This change of magnetism occurs at the same time as gap closing and reopening at the same time with the change of topology. So in that sense, charging the spin degrees are strongly coupled. So everything so far I've described is based on field theory, invoking only low-end degree of freedom. Everything is universal. And we also checked by numerical calculations, self-consistent Hartree-Fock and DMRG. And both calculations indeed support this scenario. And here on the left, I show you the Hartree-Fock phase diagram as a function of the hyper-U induction strands and the charge transfer energy delta. And you see that for fixed induction strands, you, for example, as I change the charge from the energy, we can go from the modeling thing state to the quantum-month-house state and finally to a metallic state. And that's consistent with experiments. And I also want to say that our theory is very different from an alternative scenario. So for example, in the case of magic angle graphene, the quantum-month-house state turning disorder was also observed. And there it was believed, generally believed, that the physics is coming from the spontaneous full-value correlation, okay, which corresponds to basis ferromagnetism. In our picture, we start from a 120-degree ordered AFM state. And even after bending regime, the magnetic order is still primarily AFM. So, and this, this can be experimentally tested. So in our theory, the modeling state before bending regime should have a zero spin polarization. So if we do the minus of the dichroism, you should not see any signal at the zero field. In the quantum-month-house state, there's now a finite SC radiation coming from the MOT2 layer. But importantly, this SC radiation is far from 100%. So especially near the transition. So immediately after transition into the quantum-month-house state, the SC radiation is finite, it's actually small. And it should further increases with the displacement field that drives it some deeper and deeper into the quantum-month-house region. So that's a prediction. So let me just use, yeah, two minutes to summarize. So the work, as I said, is motivated by the TMD systems. But I think the message is more general. And so we know already for quite some time now that if we deal with systems at even integer fitting, there are different types of bending filters. There's a conventional trivial bending through and there's time rule also invariant topological bending filters. And the two phases are connected by a bending versing transition, where the direct fermion changes sign. So what I just described is now a new understanding for a strongly intact system at all integer fitting. At all integer fitting, interaction can create a model in certain states with a multiple gap or with a charge transfer gap. These are interaction-induced gaps. These are many-body gaps. But by reducing the charge transfer energy, we can actually invert this many-body gap. And this provides a route to topological states in strongly correlated systems. And these two different theories share a common feature that I'm dealing with a continuous transition between topological distinct phases, very different from the flat bend scenario. As I said, even to this day, I don't know if we can understand the fractional common pole states from the point of view of a continuous phase transition. But I think that in the case of model integers, we have such an answer now. So there are many other materials which are charge transfer insulators. And there's a discussion in the literature that charge transfer energy maybe is negative. So it'd be generally interesting to look for topological states in systems with negative charge transfer gap. And also, because we started with AFM model insulators, AFM model insulators are much more widespread than the ferromagnetic insulating states. And also, the AFM transition temperature is intrinsically higher. So this raises the possibility that maybe we can start with a high-temperature AFM model insulator, and we tune it in some way. We can induce a high-temperature on the non-small state. So back on the slide I showed earlier. So I can say now that the modern QH states can be actually connected, and they are actually just steps away from each other. So this is a picture I found online. I turn out that there's a shortest bridge, international bridge connecting two different countries, the two islands, one is in Canada, the other United States, and this bridge is only 30 feet long. So it captures the essence of what I'm saying. Thank you. So questions. I understood. So the P-weight form factor hybridization, that just came from the 120 degree. That comes both from the 120 degree order, also is related to a single-punical hybridization between the two layers. So yeah, I skip the technical details. Just one very short question. Was the level of quantum whole state in the whole connectivity was actually observed in this intermediate state, the gap state? I mean, just to make sure I understand the question. So the data, yeah, this is the quantized whole effect as zero field. Yeah, I see. It's quantized. That's the motivation for this work. And actually, back to your first question about the wise P-weight form factor. There's actually one way to understand this is that, yeah, so let me just show you this slide. Yeah, so one can look at the symmetry eigenvalues of these bands. It turns out that they're in total three bands here. And it turns out all these three bands have different C3 eigenvalues. And the difference in the C3 eigenvalue is such that the match of them between the red band to one of the blue band has the form factor of P plus IP and to the other is P minus IP. Yeah, so it's sort of dictated by the C3 eigenvalue. And here when I say C3 eigenvalue, I have to rotate both space coordinates and these things. Yeah, because in this 120 degree order state, the spring and the lattice are locked together. So on the transition, so can I think of it in that you find the mode to QH so that the transition corresponds to fair magnetic order on the b-sides, which polarize in 120 degree structure on a-sides. Yes, yes. So after b-sides developer FM are ordering the z direction, we expect generally the a-sides also can't, in terms of symmetry they should. But so, and this is by increasing field, the electric field, you can increase that polarization. Yes, yes, absolutely, yeah. So you can drive to non-trivial states of the a-sides. As well, yes, as well, yes, yes, agreed, yes. Do you understand directly that you account for the 120 degrees order just in mean field to have the single particle description, right? Yeah. Okay, and then after that, when you have the degenerate quadratic touching of the bands and interaction, I think it should be different from Balear graphene without twist, I mean, graphene without twist. Right, so yeah, this is an excellent question. Yeah, so there is actually some difference, right? Yeah, there's an important difference. Yeah, difference in the following. So you see, so in biographene the curve and touching, they are in total four degeneracy, right? There's K and K prime values and they're the spins, they're spinning. So that leads to many competing instabilities. And so, you know, I remember like about 10 or 15 years ago, there was a lot of discussion, you know, which instability of wings and et cetera, et cetera, right? So I don't know if it's settled today or not, but a nice thing is here in this system, right? There's only two-fold degeneracy, spin degeneracy, there's no additional value degeneracy. So there's only one type of relevant interactions and that's why so yeah, the unstable state is unique. If I understood correctly, the quadratic band touching being at near the Fermi energy is important in your story. So in the work that you referenced, there's some symmetry that pins the quadratic band touching to the Fermi level. In your work, is there some symmetry that pins it or is it accidental? So there are two questions. One is why the quadratic band touching the two-fold degeneracy is there and that requires a symmetry, right? So in this work by this Yao Hong and Kai Sun and collaborators, they were dealing with the spinless model and the two-fold degeneracy is protected by lattice rotation symmetry and the spinless time-rosal symmetry. In my case, this two-fold degeneracy is protected by a different symmetry. The lattice has a three-four rotation symmetry and moreover, I'm not dealing with the spinless case. The physical electrons really carry spin. But it turns out in this AFM 120 degree order states the spins lie in the x-y plane. So you can define a time-rosal symmetry combined with a spin rotation by 120 degree around z-axis. So we have right as t and sz the product. So this magnetic order breaks time-rosal, so time-rosal is lost. But you can combine time-rosal with a pi rotation of spin along z-axis, right? Because time-rosal flips the spin and the sz rotation restores it. So the combination of time-rosal and sz is an anti-unitry operator and it actually squares to plus one because t squares to to plus one and the spin rotation square also to minus one, right? Because you know spin rotation by if you rotate a spin by two pi actually is minus one. So combining the two the product time-rosal and pi rotation of spins give you an anti-unitry symmetry which is square to minus one. So this actually behaves in the same way as a spinless time-rosal symmetry. So combining these two symmetries it protects a two-fold generosity. Yeah. The second question is why it's pink. It's never pink. So what happened that the question really is that why the grab and touching appears at the Fermi level, right? So this is guaranteed by the fitting factor because I'm dealing with the case of one electron per unit cell. The fitting factor dictates that this grab and touching has to be at the Fermi level. And in this range, because if you go to the deeply inverted regime you find the manufacturer is like this. So then to maintain a fitting factor one you can have spontaneous electron whole pocket, right? So that's why in the deeply inverted regime despite the grab and touching is still there you're not it's not at a Fermi level. But in a finite range of the bending version the manufacturer is such that there's no negative curvatures. So the fitting factor one it implies that grab and touching has to be at the Fermi level. Yes. Okay. So I'm reading online questions. Could you once more review your evidence that the transition is continuous? Yeah. Do you know what the universality class is? Yeah. Do you inherit the universality of the underlying magnetic ordering transition? So this is the actual question. So yeah. So okay. So let me yeah. So the transition is continuous is because there's only so here what I'm showing is the beneficiary in the absence of infections, right? So everything is continuous obviously here because I'm just tuning the single particle of the structure. The question is what happens in the presence of infections. So this infection is before bending version there's an energy gap. So weak interaction is not relevant. After bending version there's grab and touching appearing at the Fermi level and there's only one type of instability and that is due to this interaction. Okay. So that's why. So in other words this interaction changes from irrelevant to relevant as I tune the bending version. So what about universality class? Yeah. I think it's new. It's really new type of a system situation. The closest analogy I can think of is the KT transition. So if you think about the 1D spin is a system and you know the own club scattering, right? It's irrelevant in certain range of latino primers and they become relevant once the latino primary becomes in the right range. So the situation is here that the hyper interaction G is irrelevant before bending version but changes to be rather an upstanding version. So there's an online audience raising their hands. Marco really? Yes. Hi. I'm Marco really from Rome. Yes. Hi. Can you hear me? Yes. Hi. Hi. I have a question because from the cooperates we learned the lesson that usually when strong correlation tends to suppress and reduce the kinetic energy but still there are phononic or magnetic interactions lurking around. The system often gets prone to phase separation and then long range Coulomb repulsion leads to the standard story of frustrated phase separation with the striped charge density wave formation and so on. Now I was wondering whether there is any fundamental reason why moving slightly away from N equal 1 in this case of more a system. The system should be protected against phase separation or you still think that there is some Maxwell construction splitting the system into regions with N equal 1 and regions with N 1 plus something and then you're valid in N equal 1 regions. What do you? Excellent question. Yeah. So let me clarify. You know what you describe is basically doping a modern sort of away from the factor. Here I'm talking about the case of inverting a modern sort of the fitting factor is always at one. Yeah. This is clear. So my question was related to what would happen if you move just slightly away from it. Right. So yeah, actually there are some very interesting experimental data that I don't know maybe still not published but but it's been talked about in public doping the system away from N equal 1. These two fascinating phenomena. Basically there's seems to be a ferrimanism looking around and there's heavy from illiquid. So there's a lot of interesting physics when doped away from fitting factor 1. Yeah, that still needs to be understood. Okay. Exactly. The transition you have three bands touching, is it right? Yeah. Two of them are linear and the one is quadratic. That's right. Yes. So our interactions are irrelevant. It's an excellent question. So this is exactly what we spend some time on. So yeah, turn out at this band touching point. As I said, there's a one linear band touching and there's a quadratic band touching. We found that interaction is irrelevant. One way to think about this is that you know, when you say interaction is relevant or marginal relevant, it always involves some diverging susceptibility in some channel. So we checked for this in anointing susceptibility and we found no diverging. Yeah. I think it's because the linear dispersing does not give you a finite range of state. Right. So we know that interaction is irrelevant at the point. Now there is a quadratic band touching, but there is a quadratic band that band. It only has band bottom touches the Fermi level. Right. So there's no field from EC. Yeah. So we checked. There's no diverging susceptibility of any kind. So yeah. One last question. Can you say a few words about what the density of states is near that touching? Because near a linear touching, you would expect a vinyl point and when you have quadratic touching bands, you have power law vinyl point. So it's a sum. Right. For the linear dispersing band, you give your densest data like this. And then for the quadratic band touching, you give your densest data like this. For quadratic band. So you take the sum of the two. Yeah. Yeah. So there's no doubt. Let's go to this. And it's probably up. And let's thanks now for you. So we just have short discussions for the questions of all the morning talks, but we are running a little bit behind the schedule. So what do you think? Just couple of short questions or we leave this to the lunch. Oh, yeah. So there might be questions to Professor Falco's talk or or Eric Berg's talk. So we could spend a couple of minutes here. Are there questions to the earlier talks this morning? Any questions online? Not yet. Okay. Yes. Yes. Oh, okay. Yeah. So this is to Liam. So when you when you showed the susceptibility, it had a QT law behavior at high temperatures, but then it deviated from QT law. And I