 Okay, thanks so much. All right, are you able to see my pointer? Yes. All right, so thank you very much. Sorry for not being able to do this in person as much as I would like to. And so this is a tutorial on the graphene bilayers. Of course, our main motivation, as we will see, are the twisted graphene bilayers and the heterostructures inspired by that. But in fact, what I will show you today, and this is again mostly a tutorial for students, is more general than just twisted bilayer graphene. And so the goal of this tutorial for the students is two-fold. First, to show a pedagogical derivation of the effective continuum model for the graphene bilayers. And we will do so by systematically expanding in real space gradients of both the slow fermion fields and the atomic displacement fields. All of this should become clear as we go along in the lecture. And we will therefore arrive at the result, which allows for an arbitrary inhomogeneous smooth lattice deformation, of course, including a twist. And then we can check our results with more microscopic calculations and find out how far in the gradient expansion we have to go in order for the match to be accurate. Okay, so this is the first part. And in the second part, we'll discuss the topology of the narrow bands. I will show you how to construct a smooth gauge throughout the more everyone's own, which makes this topology explicit. And maybe if there's time towards the end, I will discuss a little bit about the many body effects. My understanding is that there will be several talks later this week, which will delve deep into the many body physics of these narrow bands. And so hopefully this will provide a background for understanding some of those experiments. So let me start with the motivation. So imagine we take two graphene layers, which are rigid for this illustration. And then we twist the top relative to the bottom by the angle theta shown over here. So we will then create a new pattern, the so-called Moray pattern. And if this angle theta is small, the period of this new pattern is going to be long. And so the famous Dirac electrons in each of the graphene layers will then experience this new long wavelength potential. They will bragg scatter off of this potential, reconstruct their spectrum and form mini bands. And so now we can imagine that we fill or empty these mini bands using the field effect transistor method. And then study this new system. That's the basic idea. And so let's try to get some intuition for what to expect, and we will sharpen this as we go along in this tutorial. So let's say that the large Brillouin zone of the monolayer graphene from the bottom layer is marked by the blue large hexagon here. And that the twisted layer on top of it corresponds to what will have the large Brillouin zone, which for a moment we're going to consider to be completely decoupled from the other layer, to be this large red hexagon. So these are our two Brillouin zones. And the corners of these Brillouin zones host the famous Dirac points. And then this angle theta is just the real space twist angle that I showed you on the previous slide. So now imagine still for decoupled layers that we were to plot the dispersion along the line that joins the Dirac point k1 and the Dirac point k2. Well, then the dispersion would look something like this. We will get one Dirac cone, which will intersect the other one. And this is all shown in one valley, and there is a there are two clear energy scales in this problem. So one of them has to do with the energy scale of this cross point. And that of course has to be set by the separation between the two Dirac points. That itself is set by the twist angle multiplied by the slope. The slope is nothing else but the Fermi velocity. So that's the first energy scale in the problem. Now if we introduce the interlayer tunneling, which couples the two layers, then the crossing will turn into an avoided crossing. And that itself will split by a scale set by the interlayer tunneling. And I will introduce in just a second what the W0 and W1 are in more microscopic picture. But think of them as the typical tunneling strength through the AA stacked regions in this Moray pattern and the AB stacked regions. And so now these two energy scales will determine whether we are going to end up with flat bands or not. So how will this work? So now imagine that we hold the twist angle fixed for just a second. That fixes the energy scale associated with the crossing. And we were to increase the interlayer tunneling, let's say by pressure. Then these bands will start pushing against each other more and more as we increase the pressure. And at some point the level of repulsion will become comparable to the bandwidth. And at that point we may expect to obtain something like narrow bands. So the dimensionless parameter to which everything is so sensitive in this particular case is indeed the ratio of the interlayer tunneling to the energy scale associated with this crossing. As I said the denominator here is controlled by the twist angle and the numerator is controlled by pressure. Now needless to say this field has experienced a minor revolution in 2018 when at the March meeting Pablo Jarrilo Jarrero introduced their results on the so-called magic angle twisted bilayer graphene where the ratio which I mentioned is close to being one. Namely the bands are almost completely, well they have been flattened out as much as possible. They are almost completely flat. And so now what they're showing in this first graph is the two terminal conductance in the system formed by twisted bilayer graphene at the magic angle versus the filling of these two narrow bands. So there are eight electrons which we can add into these bands and start from completely empty to completely filled. So there are eight electrons which fill these bands so why is that? Eight electrons per more unit cell. So why is that? Well the sketch which I showed you is for a particular value and for a particular spin and there are two bands which means that two electrons will fill one value with a specific spin or we have two spins so that means we have four plus two valleys that makes it eight. So the charge neutrality point would be here emptying the bands would correspond to minus four filling factor and filling the bands would correspond to plus four filling factor. And so when the MIT group empties the bands which is shown over here they see a band insulator that's the minus four filling and when they completely fill the bands they also see normally a band insulator that's over here in this blue film. The charge neutrality point is in between and so remarkably they notice that near the filling factor minus two they saw an insulating like behavior in transport and next to that insulator they saw a superconnection similarly near the filling factor two and there were some signatures of correlated states filling factor three and maybe even minus three. These results were quickly reproduced and extended by other experimental groups so the figure here shows the results from Andrea Young and Coridine's collaboration where the device was made at a twist angle which was away from the magic so remember the magic value here was something like 1.1 degrees somewhere between 1.05 and 1.16. It's extremely sensitive to the twist angle. The band with grows very quickly if you go away from the magic angle and so in particular if we were as they do as they take the device at 1.27 degrees which is already sufficiently far from the magic angle that the gray trace which is an ambient pressure does not show signatures of correlated states except for maybe a small dip in the conductance near the filling factor two but this is nominally a not strongly correlated band no longer it's not yet made narrow they see an isolation of that band from the remote band so that's a filling factor four here and filling factor minus four but other than that there's only a dip at the charge neutrality point where we expect direct cones anyway. But when they apply an external pressure as schematically shown here they are increasing the interlayer tunneling and as they do that they push the bands apart stronger and they're able to flatten out the band sufficiently so that they see signatures of the correlated insulating states at the filling factor two and minus two as well as plus three so all filling and they also saw superconductivity near minus two and then of course there are the data from Demetriy Effetov's group which followed shortly thereafter where in this specific device they saw correlated insulators at essentially every integer filling factors factor and superconductivity almost everywhere in between so the to be more precise and go beyond just this hand wavy schematic for narrowing the bands let us introduce the commonly used continuum model which is in the form of a four by four differential Hamiltonian. The first two by two part describes the massless direct particles moving with the Fermi velocity PF P is the momentum operator and then the sigma matrix here is a two component vector sigma x and sigma y which is rotated about the z-axis in the Pauli matrix space by a small angle theta so this is the rotated top layer effective Hamiltonian in the vicinity of the k point and then the other two by two block corresponds to the bottom layer which is also the gapless Dirac Hamiltonian in this case not rotated so there's no Pauli so there's no substitute theta on the second Pauli matrix and these two layers and so these Pauli matrices they act in the sub lattice space remember honeycomb lattice is not a Bravais lattice it is a lattice with a basis it's two interpenetrating triangle lattices and so the Pauli matrix acts in the two sub lattices of the honeycomb lattice underlying carbon carbon honeycomb lattice and then the two layers are now coupled through the interlayer tunneling matrix T the information about the more periodic potential sits in this matrix T and it can be written as follows it is parameterized by two interlayer tunneling strengths which I introduced before w0 which corresponds to the interlayer tunneling through the AA regions in the Morin pattern and then w1 which corresponds to the interlayer tunneling through the AA regions these small wave vectors q0 q1 and q2 are sketched in this little plot and they are set by the twist angle between the two layers and it is this term together with this one which introduce the Morin potential that scatters the Dirac particles reconstructs them into the bands and I should say that if there are any questions feel free to interrupt me and please just unmute yourself and ask a question all right now this Hamiltonian will have a perfect particle of symmetry if we were to ignore this twist angle theta in this Pauli matrix this is actually a rather weak approximation in a sense that we are not committing too much of a sin by doing this as was shown by Andre Bernovich's group and Leon Balances group remember it is not this term in the Hamiltonian which introduces the Morin periodicity and the Morin physics it is mostly coming from the interlayer tunneling term through these q's so if you were to drop this angle theta here you are actually not getting rid of the entire physics of twisted bilayers and the Morin physics you are just ignoring the rotation of this Pauli matrix which if you do that will give you a perfect particle of symmetry in this problem now experimentally as you might have noticed there isn't a perfect particle of symmetry in these layers for example the pressurized data which I showed you from Corey Dean and Andrea Young's group's collaboration saw an insulator at filling factor 3 but not yet at minus 3 so one should go beyond just assuming particle whole symmetry in these Hamiltonians and one of the goals and one of the byproducts of our gradient expansion that I will show you in this tutorial is that we can capture systematically the particle whole asymmetry terms there's a very nice perspective on this Hamiltonian as opposed to deriving this in momentum space deriving some of its key features in the real space from the effective field theory perspective by Leon Balances which actually inspired what I will show you later today and so if we were to take this right so if we were to take this effective Hamiltonian that I just showed you and computed the energy spectra numerically then it would it will look something like what I'm showing over here the mini Brillouin zone the Moray Brillouin zone is sketched over here the cut we're going to take starts at k prime so this is the corner of the Moray Brillouin zone it takes us to the other corner to gamma and then through these end point back to gamma eventually back to k prime so that's that's the that's the x axis cut the y axis the energy and in this little movie that I will show we will change the twist angle we will start at a large twist angle where the bandwidth is not flattened yet by the interlayer tunnel and then we will decrease this angle towards and through the magic value to see the formation of the narrow bands and then of course as we change the twist angle we are changing the period of the Moray lattice in the real space we're increasing it as we decrease the angle and therefore we are simultaneously changing the size of the Moray Brillouin zone so as we increase the period in real space we decrease the Moray Brillouin zone but instead of showing you the results of this calculation for by adjusting the size of this Brillouin zone what I will do is I will continuously rescale it so that the x axis is the same and the only thing it will be changing is the y axis but you should remember that the Brillouin zones themselves all are also shrinking as we decrease this twist angle so we started three three degrees and hopefully this movie is running on your side as well and as we approach roughly the 1.1 degrees we see that the bands start narrowing down and roughly at 1.1 we see that they become very very flat they also separated from the remote bands here that's because in this particular calculation from the continue model we chose the interlayer tunneling through the AA regions so W0 to be smaller than the interlayer tunneling through the AB regions which is W1 and now if we continue decreasing the angle notice that the bands get broader again okay and so from the picture that I introduced is clear why this should happen the bandwidth starts out too small at these small angles or smaller angles and the interlayer tunneling repels them simply too strongly and so you sort of over shoot and create and the bands are no longer all that narrow past the magic end um okay so any questions okay so now yes oska we have a question yes please thank you can i ask about the validity of the model in the entire Brillouin zone because if i put w equals zero it looks just like two cones at the k and k prime points which for graphene is only a low energy kind of model right you mean if you said w zero equal to zero w one equal to zero yeah both of them equal to zero right if you set everything to zero then these t's are all zero and you have two decoupled monolayers yeah so if we were to talk about just simple graphene then that is a model of these two cones right that's right and that's valid only at low moment for single layer graphene that's right so the question is how low right so uh more than that also uh you seem to also be able to get the dispersion for the entire Brillouin zone oh but this is a this is the moray Brillouin zone so this is the Brillouin zone which has already been folded many many times um and became very small right so that Brillouin zone has a size which is set by the large is basically yeah so you can see it from this picture um sorry you can see it from this picture um if this angle is one degree then um uh this side is extremely small and so the Brillouin zone for which i'm plotting the dispersion is not the large hexagon but the small hexagon by the way in this picture this is uh this is not to scale the angle is not as large as it is in this picture in reality near the magic angle it's much smaller than this so so you're really only expect you're i'm only showing you the dispersion over here for few of the lowest bands does that answer your question uh yes and also uh does it also fall within um a small fraction of is the spectrum sufficiently linear there um yes so so so i will discuss uh high order corrections to this um in the non-linearities as we go on it's actually a very good question it's a rather subtle question so um it depends on whether you want to talk about the magic angle precisely or whether you want you're allowing to go a little bit off the magic angle um it turns out that the non-linearities should be included if you want to describe the magic angle accurately for the reasons which i will explain in just a second but if you are off the magic angle so that the linear terms dominate um then uh then this will be fairly accurate thank you you're welcome are there any other questions there's something on yeah very quick question about the movie that you showed is there only one critical angle or if you keep increasing the angle that you see re-entrant behavior and against the narrow bend you're right andrey so thank you for asking that question so in the model which i just showed you uh where we are rigidly rotating the two layers although um it's sort of a mixed in match because uh the model which i just showed you sets w zero and w one to be different that is justified by um saying that in reality the AA regions uh will shrink and the AB regions will expand i think there was a talk on the dichocogenites uh where the speaker mentioned something similar in that case there's a competition between the adhesion energy and the elastic energy within the each layer in any case in this particular case this is included only by setting the w zero and w one to be different from each other other than that it's a rigid twist so in such a model there will be indeed a re-entrance and you will get additional magic angles however um in reality by the time you go to the next uh magic angle the reconstruction is fairly strong um meaning that there are terms beyond this simple model which i just showed you um and uh the magic angle gets basically that the narrowing of the bend gets avoided once you go to next uh angle because of this reconstruction so it's really just the first one where the reconstruction although present is not too strong um for it to be there at least that's that's um my current understanding of uh the results in literature does that answer your question osuka we have a question uh online question here uh so could you please comment on the substrate induced disorder not the twisted angle induced one in the tbc and if there is experimental data on the strength of the disorder okay there is experimental data on the strength of the disorder so um in a monolayer graphene you can't quite approach the dirac point very precisely um because of the so-called puddles uh now where do these puddles come from electron hole puddles so where do they come from well it's rather simple you have um an energy potential and the energy potential will shift the dirac cone up and down uh in real space and so some regions uh in space will be let's say electron uh dope some others would be hold up already fairly early data uh by elizeldo where they were looking for um the uh the the magnetic response uh using a scanning squid um found out that the twist angle uh disorder uh causes much stronger uh effects on the band structure than this puddling type uh potential i think there was at least order of magnitude uh if not two uh difference in the strength of those terms um so so yes uh there is that that term is present but at least currently it is uh believed to be subdominant um uh uh at least away from the charging trity point in these narrow bands uh to the twist angle in homogeneity i don't know if elie is in the audience but uh he has information about this okay so let me continue so um these questions are sort of a good segue to what i really want to do in this tutorial um and so now although the main experimental findings where indeed reproduced by a number of experimental groups uh there is a nagging lack of reproducibility in the finer details of the physical characteristics of devices and this is true even if the devices were manufactured within the same lab and even within the same device this is likely due to the spatial inhomogeneity in the twist angle as i mentioned and an intentional strain uh produced during the device fabrication or more generally due to lattice deformations which vary over distances long compared to the microscopic spacing between the neighboring carbon atoms um and what i really mean by this is something like hetero strain where the strain is opposite in the two different layers so it is being recognized that the twist angle is not the only parameter controlling the physics of the specific device uh and this fact motivates a development of a theory whose input would be more than just the twist angle theta the Fermi velocity and the two interlayer tunneling constants through uh the AA so w zero and through AB regions w one um and this was the case in the model which you just saw and so instead um we would like to build a theory whose input would be a smooth and possibly inhomogeneous configuration of the atomic displacement and we imagine that such a configuration is in principle extracted from either STM topography or maybe from BRAC interferometry so that um we um we have one fewer um variable to to worry about unknown variable to worry about so imagine we extract this from an experiment we would like to input this into some continuum theory uh and then make predictions and compare those predictions uh directly to the experiment um and so that's what i want to do here in this lecture uh i want to show our derivation of an effective theory for these graphing bilayers where we systematically expand in the real space gradients of the slow fermion fields um and the atomic displacements um and we're going to allow for an arbitrary uh inhomogeneous lattice deformation including twist um the two papers on this have been posted on the archive um a couple weeks ago uh and they were done in collaboration with Jiankang who used to be a postdoc at the magnet lab and uh he's now a faculty at Shanghai Tech University so most of this uh first most of this lecture will follow the first part the first paper uh over here which introduces the general uh systematic method for doing such expansion so what will follow is going to be somewhat technical but uh since this is a tutorial and i think that students would appreciate seeing technical details um i opted for going that way as opposed to just describing the results okay so uh let me go through the slide slowly so uh imagine for a moment that we have a true position of the carbon atom marked by this coordinate x and the two subscripts here j and s label the layer so j could be top or bottom and s is the sub lattice uh so this could be a and uh or b so imagine that this is the true position of the carbon atoms uh in a specific layer for specific sub lattice and we can reference that position by an undistorted honeycomb lattice um whose lattice vectors are are just simply the bravet lattice vectors plus the basis vector tau s okay so the a1 and a2 are the primitive lattice vectors for the triangle lattice okay n1 and n2 are integers and tau for the b sub lattice is shown by this little vector over here that's the basis vector and for the a sub lattice so tau a is equal to zero there's no shift over here now this rs um it's really just a label that we use uh to describe um the true position of the atom which happens to be somewhere else at x uh j s and so the difference between the actual position uh the difference between this reference position um and the actual position is of course the displacement field u which also will depend on which layer we are sitting so j either top or bottom or s which is um the sub lattice okay um and now uh the atoms let's say in the top layer are therefore given by this map x j equal to top s equal to a sub lattice uh which take this reference position uh to the true position and this true position um may be deformed within a plane and it may also have a deformation out of the plane so the side view here is allowing for um a corrugation in the z direction so therefore we will split this uh displacement field into a parallel component which is in plane and a perpendicular component which is out of plane and um we assume we have this map uh that this is either obtained from some numerical calculation which relaxes the two sheets sitting on top of each other um with a twist or this is extracted from an experiment um now uh not only that we assume that we have this map we assume that this map is one to one so we have the inverse of this map as well um that's that's this dashed arrow um and so then uh we have some hopping model this hopping model will depend on the distance between the two atoms let's say between these two different layers so here uh the blue would be the bottom layer and the red would be the top layer um and in fact this hopping may depend not only on the distance between these two atoms it may also depend on the orientation of this long black vector compared to this short vector um relative to the nearest neighbor of this atom and similarly to this one as is true in the microscopic uh vanier based models for the interlayer tunnel now this um parametrization is so-called is referred to as Lagrangian coordinates in other words that everything is um referenced back to these undistorted uh honeycomb lattice coordinates r s and the displacement field is also given in terms of those coordinates there's an alternative uh uh way to describe this uh so-called Eulerian coordinates where um we imagine we invert this map uh we describe r in terms of x and then we replace the r in here by its x dependence in the displacement field um and then um the true position of the atom would be given by the reference position plus the displacement fields now in terms of the true positions of the atoms um now in this particular case because the graph we have additional simplification because the graphene sheets um uh do not make overhangs we can go to so-called monge gauge where the uh displacement uh can be entirely parametrized uh by the in plane true positions of the atoms x parallel okay uh this position of course will have an out-of-plane component uh in general and that's this u-perb uh in addition to the in-plane part so this is just to set up uh the the language and the transition from one set of coordinate system to the next set of coordinate system will become very natural um uh in our uh in our derivation of the narrowband hematonic so any questions about this part yeah also we have some question here wait so if i understand correctly in the Eulerian coordinate each point is mapped to the it's uh real coordinate like where exactly it's in the yes phase than the lattice that's right so that's right so this x is the true position of the atom in the layer j so either top or bottom with sublattice s a or b and it is referenced by an undistorted position uh this is just you can think of this as just an index um which i'm showing over here does that answer your question yeah yeah so maybe maybe my question is uh if two two atoms which are close in position might not be close in like ours is that true correct you're absolutely right uh in particular right now we made no assumptions about these displacements being small so if we have a rigid twist for example for one layer then the atom can actually move uh significant distance away from its original position rs and will become close to another atom so let's say the bottom layer did not get rotated at all but only the top layer got rotated um then um you know uh in the top layer uh the uh x could be quite far away from the original r oh thanks you're welcome so um again this is this is kind of important there's no assumption being made here about the smallness of these displacement fields in particular because we would like to be able to include strain um and rigid rotations uh but we will make an assumption that these fields are smooth in other words that their gradients are small okay um so uh just as a quick example were there any other questions not at the moment you can okay so as an example let's consider a rigid twist um so for rigid twist let's just say that the top layer is rotated in this case the true position of the atoms in the top layer will be given by um a rigid rotation of uh the positions our reference positions which of course all in plane um in addition to that there will be an out-of-plane uh shift just so that uh you know they're not sitting on the bottom layer and in this particular simple example uh the the u-perp would be a constant there would be no corrugation um okay so so that first equation should be fairly clear the true position is given by the reference positions and now we would like to figure out well um how big are these uh displacements in plane so we invert this equation um and so if we only take the in-plane component of the uh x we can drop the u-perp on the right-hand side uh invert this matrix it's simple because it's just a rotation matrix this gives us the in-plane positions rs um in terms of the uh true positions uh x uh parallel um and then uh we can just simply uh i think there's a minus sign here yeah sorry this should be a minus sign over here so then so we just go to a definition of the um then we go to the definition of the relationship between x and r we have that relationship we've just figured out in this first equality over here i substitute this um i substitute this r into this equation okay um and uh uh and i'm able to obtain therefore what the displacement field in terms of x uh is and so if we work uh in a small angle approximation if the if the twist is small then uh the displacement field will be just simply set by the angle and then um the out-of-plane unit vectors he had crossed into uh the true positions uh x so that would give us u uh of x um okay um so to illustrate our main idea uh we are going to make a simplifying assumption although in the papers we um go beyond uh these simple models um and uh and work it out for more complicated configuration dependent hopping terms which depend on the relative angle between the nearest neighbor and the vector connecting the two uh atoms between between between which we hop but here we're going to assume that the hopping amplitude t between the two atoms depends only on the separation between the two carbon atoms and this is indeed the case um uh in the slater coaster type models so the interlayer hopping and the interlayer hopping are not exactly the same but other than that uh if you consider two atoms between the two different layers um there is no angular dependence uh on the strength of the amplitude for that hopping uh in such models it only depends on the magnitude of the vector that connects the two um the two atoms um instead in general t will also depend on the orientation of the vector connecting the two atoms relative to the nearest neighbor uh sites of these atoms so it will depend on for example the angle laser pointer here it will depend on the angle that um uh is made between the small black arrow and this long black arrow and it will also depend on this angle uh we're gonna start with considering a microscopic model where such a dependence is ignored and moreover in general the on-site terms will acquire configuration dependence um and uh i'm not gonna treat this in this tutorial uh because we don't have enough space to do that but we treat this uh these more intricate cases in the papers is there a question um so uh let's start with the uh a microscopic Hamiltonian uh for uh the two layers uh we assume as I said the hopping only depends on the distance between the two atoms or on the vector uh between the two the true positions of the two atoms so again s and s prime here sum over the sublattices j and j prime sum over the layers and then r s and r s prime um well uh they sum over all the sites uh within um our reference sites within the two layers so the hopping which is a physical process will depend on the true position difference between the two atoms but we can label our creation and annihilation operators uh the fermionic creation and annihilation operators um uh by uh simply our reference positions they don't have to be uh labeled by the true positions in fact this is much more convenient uh to do it this way um so this is rather generic um the only thing we're going to assume and this is true in general uh is that this hopping function uh is exponentially decaying as the atoms become further further apart from each other so this t is short range and we're going to try to take advantage of the fact that it is short range um now uh because our tibanding Hamiltonian is Hermitian it has to be true that this hopping function uh has satisfies the following property if you complex conjugate it and you change the sign of the vector inside it has to be the same um and then because uh we are assuming no spin or decoupling and we preserve the spin less time reversal symmetry um it has to be true that uh this can be chosen to be real um and so as a example which is often used in literature um here is uh uh one uh functional form for this interlay for this hopping function um uh it's it's uh basically a slater coaster like parametrization for the hopping um for the pi and the sigma orbital hopping it's exponentially small um and the decay length delta here is set by um uh this number uh this is fitted to some dft uh at twist angles which are sufficiently large that the dft is feasible um so we don't have too many atoms per unit cell then a naught over here um is the uh is the basis a vector that connects the two different sublattices um and then the d-naught is the interlayer distance uh for the top and the bottom okay um and as you see this depends yes please oh we have a question here online question um does only the term uj pup i think it's slides before um uj pup include the lattice relaxation effect um we can include it in both but the example which i showed was just for the top but uh we can include it in the both top and the bottom no problem okay so here's an example of a microscopic model which is often used in literature um which describes the tunneling between the two layers and it is indeed of the form that uh i discussed uh it only depends on the difference between the true positions uh of the atom uh vectors uh x okay um so our next step so this is a lattice model we would like to start with this lattice model and arrive at a continuum theory um theory which contains only gradients so we take this and then we introduce an identity in the form of an integral of delta functions and these delta functions are uh uh describing some continuum variable r and these are our reference lattice positions rs and we integrate over r so this is clearly identity we integrate over r prime that's also clearly an identity and this of course references the um the other uh set of atoms to which we hop um and so um because of that all the x's that we had previously in our hopping function uh we just write them out in terms of rs and uj rs uh but because we have these delta functions we can replace the rs with the continuous variable r here and our prime over here and same with the fermion fields so so far i haven't done anything and now i notice that the sum over the uh lattice vectors which are undistorted remember this is the reference uh honeycomb lattice that sum over these delta functions uh can be written um through so-called dirac comb identity as a sum over the uh microscopic uh reciprocal lattice vectors g um which are the undistorted brillon zone large brillon zone um of the um of the monolith graphene so this g runs over uh this set where m1 and m2 are uh integers this is just the triangle lattice uh reciprocal lattice vectors um triangle lattice corresponding to the atomic carbon carbon uh lattice and so this is just an identity um and so uh we will then use this as identity uh to first interchange the integration and summation uh over rs and our integration over r prime um and then rewrite all of these delta functions as sum over all uh these g vectors we're going to get uh one sum for the first delta function the second sum for the second delta function okay so um and uh now we know that the physically important states come from the vicinity of the dirac points and therefore we take these microscopic fermion fields which live on the lattice uh originally and then we decompose them into uh slowly varying envelope fermion fields um which from the valley capital k uh we call them little psi and from the valley capital k prime or minus k we call them little phi um and then uh we multiply them by the fast spatially varying functions uh from these two valleys as follows so we take this fermion field here um and it is approximately expanded um through this so this first part is the fast uh envelope and this psi is the slow envelope field and we like to write a theory for those slow envelope fields is there a question okay um oh yeah okay what is the justification for this decomposition um so in the undistorted case we know that the low lying modes live near the k and k prime dirac points of the underlying uh carbon lattice and so we would like to write the theory uh in the vicinity uh i mean we'd like to write low energy theory which is valid in the vicinity of those points um that's the justification maybe uh why why do you separate the slow varying function the fast varying function because um i would like to do a gradient expansion on the slow varying part um and stop this gradient expansion um at you know appropriate order so they can recover the lower energy physics accurate um thanks yeah you're welcome okay were there any other questions no not no okay so these uh so we know the canonical commutation relations for the um lattice fields c um and from those uh using this normalization we can also work out what are the commutation relations for the slow varying fields and they are written out over here the rest are uh equal to zero so uh so now we have our program we can set it we have it all set up we're gonna replace the sums over the delta functions with the sums over these uh plane waves we're gonna get two sums because we have two sums over delta functions we're gonna get two sums over g's um and then we're gonna replace our uh lattice fermion field c with these uh slow varying fields times the past envelope now um you can already sort of see uh quickly what's going to happen we're gonna get four types of terms for the fermions we're gonna get terms which are all intra valley so they will only contain psi dagger psi uh phi dagger phi um and then they're gonna cross terms the cross terms correspond to the intra valley scattering um now i'm not going to discuss this too much in here but you can show that the intra valley scattering terms um contain a wave vector which is not a reciprocal lattice vector g they contain a wave vector 2k um it is 3k that will be equivalent to g and because everything else is slow uh there really isn't anything to compensate for that difference um in the large wave vector between the two valleys you would have to go to very high order perturbation theory or you have to have very strongly uh spatially varying displacement fields to be able to compensate for that difference in the vector um and so the internal valley scattering terms are indeed um very small uh and so from what i will discuss uh from what we'll follow i will drop those internal valley scattering terms and we'll only focus on the dominant intra valley scattering terms so we're gonna focus on one specific valley let's say it's the valley k and so this first line uh that's exactly what i uh just promised we replaced the sum over rs and rs prime um and the delta function through the plane wave sums um and uh and then i also uh replaced the fermion fields with the slow envelopes and they all have this large well they have this fast envelope card uh sitting in front of them so i haven't done anything i just made a substitution um okay now uh this uh this hopping function t is short-range uh as i promised um and what i would like to do is i would like to take advantage of the fact that it is short-ranged um now notice that it is short-ranged in the x variables so at this point um i would uh very much i think it would be very advantageous clearly to go from the Lagrangian formulation into the Eulerian formulation we would like to change variables and call this object which is sitting inside of this microscopic hopping function called an object x um and then of course this will be called x prime and then we have to rewrite everything not through r but through uh x and x prime and that's what the Eulerian formulation does for you okay so the i think as far as the position uh is concerned that's fairly straightforward you just invert the expressions which i introduced previously um and that introduces our capital U um but we also need to do this on a fermion field and now um remember our fermion fields were written originally in terms of r and we would like our new fermion fields which are now written in terms of x uh to be uh obviously just as slow as the previous ones were which is fine that's not difficult but they would we would like their uh canonical commutation relations to also contain a delta function in these x variables and in order for for us to do that we have to introduce uh the square root of the Jacobian um that follows from uh the transformation of delta functions under the change of variables so now we know how to express the slow variables from valley k uh in the Lagrangian coordinate system uh to the Eulerian coordinate system x so we can substitute psi uh for psi x uh as we did over here and then everywhere of course we're going to pick up Jacobians for the transformation because we're changing from r to uh x there will be a Jacobian from this measure Jacobian from this measure but we're going to pick up one over squared of a Jacobian from the fermion field so that gives us these factors over here this one and that one and the rest of it is just um a straightforward replacement of the uh x and u of r in terms of uh uh sorry uh r and u of r in terms of x and u of x so uh so although this this is somewhat busy it's fairly straightforward uh we're just substituting in terms of our new variables now why we why would we want to do this because now our hopping function t depends only on the difference between x uh and x prime it also contains this corrugation piece but remember uh there's just a small corrugation on top of a uniform uh on top of a flat uh sheet so so so these u parallels are not large those we can treat as small vectors um and since this function is short range uh the x parallel and x parallel prime will be forced to be near each other if we are gonna try to do this double integral over x and x prime okay so let me just pause here and see if there are any questions uh about this yeah we have a question one second hi uh you did talk about the uh inter-valley scattering being forbidden because there was a 2k momentum transfer now um presumably the uh uh the u can also have some uh momentum which is not small so for example if i turned on a lattice deformation that had a momentum of k is it possible that uh inter-valley scattering would be possible in that case and how should i see it in equations yes you would see it so so uh so first of all in the situation which we are considered that cannot happen because we are considering only smooth deformation from uh the original positions and therefore by definition the gradients of you are therefore small which means that you cannot contain large wave vector that will be able to compensate for the difference between um k and k prime however so so in that case uh if you were to do the substitutions which i just introduced uh previously um you would discover that uh there would be um there would be a term here uh and also when you transform into the x variables which will vary as a 2k and the only way to compensate for this would be from you as you as you uh already noticed but you does not have uh those wave vectors um if it does they're extremely small now if you had a configuration so so so for the configurations which i'm considering and which are um also obtained from uh models which relax between the two layers um you is does not contain large gradients okay so there this is this is justified but if you did have a situation where there was a rapid variation of the atomic positions on the length scale much shorter than the Moray length scale uh so that you did contain a large wave vector then you would not be justified or i would not be justified in dropping the um intermally scattering terms okay but in the cases that we are studying this is justified and i will show you the comparison between the original microscopic brute force calculation with deformed positions and the continuum model and the match between these two uh hopefully uh soon okay does it answer yes thanks all right you're welcome all right um okay so so we're almost through this um now uh with these substitutions uh we now have the follow we have the top equation and we would like to take advantage of as i said of the fact that the hopping is a short ranged function so notice that what happens uh so let's for a moment uh ignore the out of plane corrugation although we can introduce it it's not difficult let's just focus for a moment on the difference between the x parallel and x parallel prime and clearly we would like to go into center of the mass coordinates uh little x and the relative coordinates a little y because the interlayer i'm sorry because the microscopic hopping function depends primarily on y again the the dependence on this uh out of plane corrugation can be handled easily uh the the key is the dependence on the difference between x parallel and x parallel prime so if we switch to center of mass and relative coordinates then we will be able to take advantage of the fact that the hopping function will be a function of the relative coordinate and it is a short ranged function of that relative coordinate so if it multiplies a slowly varying function of the relative coordinate then we can take the slowly varying function and the gradient expanded near the small value of y that's the key idea okay and so that's what we're gonna do uh we are going to uh introduce uh the center of mass coordinate and the relative coordinate uh substitute um and then what do we notice is that the factor which was sitting over here okay which contained g and g prime can be written as follows the the center of the mass coordinate x is premultiplied by g minus g prime and then of course the relative coordinate is premultiplied by this plus now notice that everything else including this fast factor only contains y so all the fast factors will contain y and not x so everything else that this this factor will multiply the center of the mass part will multiply will be slow in x okay the fermion fields are small so remember the x parallel will then just becomes become little x plus half of y okay but y is small and anyway x and anyway psi is a slow varying uh function okay um these u's they're all slowly varying they're sitting over here they're sitting over here everything else depends on u's or derivatives of u's so everything else is slow in terms of the center of the mass coordinate and so when we do the integration over the center of the mass coordinate we'll notice that we have this factor which if g is different from g prime is very fast because that comes from the underlying carbon carbon uh lattice uh getting free a transform um into the reciprocal lattice uh vectors so g minus g prime for g not equal to g prime is fast there's nothing to compensate for that um in the rest of uh this expression and so uh uh up to uh negatively small terms we are therefore allowed to replace g to equal to g prime so this double sum will therefore collapse to a single sum everything else uh will be very very small um so we do that uh that makes things a lot simpler um and then uh all of our fields let me illustrate this on a displacement field all of our fields which used to be at this coordinate x parallel they now are at the center of the mass coordinate plus uh half the relative coordinate and because we have this hopping function t which is short ranged in y this slow field can therefore be expanded in gradients and a polynomial in y this polynomial in y will never be large because the hopping functions will compensate they're exponentially small for large y and so now uh we can do this not only for the displacement fields uh the atomic displacement field but we can also do this for uh the fermion fields uh same idea these are also slow envelope uh functions um and uh and that's it uh we can just truncate this gradient expansion at the particular order uh and uh compare it to the uh explicitly computed um that type binding uh model okay so that's that's the basic idea um as i said yeah this oscillates strongly unless g is equal to g prime and all other factors are slow functions of x so if you do that and if you expand the fermions to leading order in a gradient expansion uh which is in this big bracket the theory that you get for fermions looks like this um the hopping function uh is now just the function of the relative coordinate uh the Jacobians transform uh over here they only become functions of the center of the mass coordinates also a slow function of the center of the mass coordinates um we will obtain a difference between the displacement fields uh uh here in this exponential so if we are in the same layer in the same sub lattice so um j let's say is top j prime is top s is equal to s prime then this part will be zero but if we consider interlayer tunneling so j is different from j prime so j is let's say top layer j prime is the bottom layer then we will pick up a term which um uh remember at least for the rigid twist uh is linear in uh in x and this will correspond to the interlayer tunneling functions t which i introduced before and then uh in addition to that uh we also have gradients of the displacement fields uh which uh which sit over here uh uh and uh they are multiplied by y which is forced to be small by this interlayer uh but by this atomic hopping function t so now you can expand this in polynomials uh in um in y um and you obtain higher and higher order terms in the gradient expansion for for u uh and for psi okay so uh uh any questions about this nope okay um okay so yeah these symbols they were just uh taken from the paper they are just square roots of these Jacobians which are themselves also small and so now uh you know it's it's up to us how far we want to stop this gradient expansion um and the good thing to do would be then just to check um okay so um how does this continuum theory that I introduced previously how does it get modified um if we stop at second order gradient expansion so there are several terms um so first of all um if there is a lattice relaxation we are going to pick up the so-called pseudo uh magnetic uh vector potential terms which may be different for the two different layers uh which we marked by the script a here these terms they so these are the in the interlayer part of the Hamiltonian these terms are linear in gradients of the displacement fields um and they do not introduce any gradients into the slow fermion fields so they're actually of the same order as the Hamiltonian uh that is commonly used although they are often dropped in such a Hamiltonian now they will necessarily be introduced if the lattice deforms by expanding the ab regions and shrinking the a a regions even if the deformation is three fold symmetric so even if it's not really an externally applied strain term the pseudo vector potential terms will be present and they will be of similar order of magnitude as the terms which are being kept so there's actually no justification for dropping these pseudo magnetic uh vector potential terms uh in the presence of uh uh lattice relaxation now of course in addition to those terms there will be high order gradient terms of fermions so second derivative or cross term between the first derivative of the fermions and the first derivative of the displacement field um in addition to uh terms which um uh people would recognize from the ab bilayer um and again these are just similar these are also cross terms they contain one derivative in the displacement field and one derivative in the fermions um um there are other high order terms sorry there are there are other second order gradient terms which are not written in this expression but we checked uh numerically for the configuration that corresponds to relaxed uh rigidly rotated structure and those were negligibly small even though there are second order terms so the second order uh you can get away with with with these terms now for the interlayer tunneling terms um there's a contact interlayer tunneling term which is parametrized by this um sabbatis matrix s and s prime which is the analog of the t i introduced before this is where the information about the moire potential sits but in addition to that there are also gradient terms gradient interlayer tunneling terms uh sometimes they are called non-local they're not really non-local um there as you saw this is all through the gradient expansion so they necessarily have to be uh local but they do include gradient terms and so um uh so these contact interlayer tunneling terms are of the same order of magnitude as the first order gradient terms in the interlayer and then these first order gradient interlayer terms are then in turn of the same order of magnitude as second order gradient interlayer terms and that's the way the pattern uh goes um uh now we can ask um why would we want to go to high order terms uh high order expansion apart from of course being able to introduce um the spatial inhomogeneity due to the deformation um there's a deeper reason for this actually so um the results which I showed you originally um where computed uh where we obtained the narrow bands were computed by including only the first order gradient expansion terms in other words we only kept uh uh let's say this term over here um and then we kept the contact terms in the interlayer terms now the order of magnitude of these terms can be estimated by simply multiplying the Fermi velocity times the distance in momentum space between the moire brillant zone corner uh corners so k and k prime in the moire brillant zone so the order of magnitude of this first term is about 200 milliolectron volts um the order of magnitude of this term is about 100 milliolectron volts so uh the the terms which we would be dropping in this high order expansion um are about eight percent of those so normally one would not care except for the magic angle phenomenon because when we achieve magic angle with first order gradient terms the bandwidth is anomalously small it is not 100 or 200 m e the bandwidth is less than 10 m e which is precisely the order of magnitude of the terms which are being dropped they're eight percent of about 100 to 200 m e and so in fact if we want to be accurate um in our gradient expansion at the magic angle uh to the order of the bandwidth of the narrow band then these high order terms need to be included if we go away from the magic angle where the bandwidth is large uh it is indeed set by the first order gradient terms again it is of the order of 100 to 200 m e then these high order gradient terms are indeed just eight percent correction to this so um let me show you is that a question okay um so let me show you a result of a calculation uh for the rigid twist in this so-called Slater-Coster model um so the figure on the left shows the energy dispersion um for two different calculations so the red dots correspond to a tight binding model for originally twisted uh layers uh this includes everything it includes the interlayer tunneling that uh somewhere worried about sorry uh intervalley tunneling that somewhere worried about um it includes um uh all order in gradient expansion equation uh it's the it's the it's the lattice model um and then you might not be able to see it very well but it essentially perfectly matches the calculation uh using the continuum theory which I showed you uh where we do the calculation separately at valley k and then by spinless time reversal obtain the spectrum for valley k prime and as you can tell they're on top of each other um now uh you can actually for this rigid twist calculation you can get away with um uh including uh so this is the green curve including only the first order gradient terms in the intra valley and the contact terms in the intra valley but as you see without the lattice relaxation so just for the rigid twist the narrow bands in the full calculation as well as the uh continuum calculation so in here on the right we are only showing one valley so that that's why uh this red portion of the tight binding calculation which corresponds to the other valley is not being compensated but here on the left we show both values so they match so but but you see because there is no lattice relaxation there is no separation of the narrow bands from the remote bands while experimentally you saw that there is such a separation so um how will this uh calculation reproduce uh the tight binding model when we do introduce lattice relaxation so uh we do this similarly to what was mentioned last week uh we have the so-called generalized stacking fault uh energy functional for different stackings um that in the case of the graphene bilayer favors the AA sites in addition to that we uh we have the intra layer elastic terms so if it wasn't for the intra layer elastic terms then um if you were to twist uh the AB regions would expand as much as possible at the expense of the AA regions and then you would form domain walls between the AB and the AA regions now the price you pay for those domain walls is of course the elastic energy um in within each layer and so there's a length scale that is associated with those two energy scales and that length scale happens to be not too far from the more period at the first magic angle um it is longer than that though and so um as a result what happens is that there is lattice relaxation it is not uh entirely as strong as it is for tiny little twist angle like 0.03 um where you get very strong domain walls between the AB and DA regions um but in this case there is there is a deformation which is a lot weaker than that nevertheless the AA regions shrink and the AB regions grow and the displacement field can be computed using such a calculation as a correction to the rigid twist now this is a two-dimensional in this particular case periodic vector field and so by Helmholtz theorem we can decompose it uh into a irrotational part uh and the solenoidal part so this is the pure curl and this is the pure gradient it turns out that this um a pure gradient term is extremely small it's negligible and the uh relaxed configuration is dominated by the solenoidal part the pure curl and so we can just parameterize it by a single function uh epsilon u which whose contour plots i'm showing over here um and the corresponding arrows then uh describe this vector field delta u but what these arrows really represent is the fact that the AA regions um are getting smaller and the AB regions are growing now uh so with this calculation we can therefore input not just the rigid twist uh but we can input this displacement field um and and ask how good of a comparison will this make um and so this is again for the simple Slater-Coster type tight binding model now you see that the narrow bands are separated from the remote bands by gaps that's seen experimentally this is computed for um a commensurate twist angle um these are the uh commensurate values m and n 30 30 31 and 32 for those um who want to know more about this i can tell you but in any case it would correspond to twist angle 1.05 the lattice relaxes according to this model um and again the red is the full tight binding calculation with the 10 or 11 000 atoms per unit cell which are now relaxed based on this input from delta u and the blue and the green which you can almost not see overlay with that perfectly um uh if we stop at the second order gradient expansion in the intra-layer term and a first order gradient expansion in the intra layer term um okay so uh any questions about this yeah we have one and by the way you have about seven minutes left okay um sorry i didn't get something about so you separated the the fermion with the everything with the past field and and there's low field variation it's uh sci-fi because you said that the physics relevant was relevant near the bellies k and k prime but the flat band expands also over gamma essentially so oh i'm sorry i'm sorry for the confusion no again the um there are two different k and k prime points so there's a large k and large k prime of the monolayer graph nothing to do with more okay and then then there is the more everyone's on which is this tiny little bronze on so so this k and k prime i'm sorry they should have subscribes a little m okay sorry for the confusion are there any other questions so um okay can i go yes okay um okay and so i promise that i'll say a little bit about the topology of the narrow band so let me just go through this uh next few slides so there are different ways uh to to study this so um to me the cleanest way to see the topology of the narrow bands um is by constructing the so-called hybrid vanier uh basis this is a technique an idea that was introduced by uh these gentlemen back in the day not for twisted bibliography but in general for specifying the for studying the topology of the narrow bands topology of bands so what do we do so imagine you have a set of bands which are isolated from the rest of the spectrum um in our case it's very clear what they are we saw them in the previous slides um there would be these bands over here um and now i would like to study a position operator projected onto those bands and in particular i like to study its eigen values and its eigen functions now of course if i didn't have the projector then the position operator is easily diagonalized with delta functions it's it's it's just as sharply defined as it possibly can be um but the projection makes it uh impossible of course uh because we don't have the full hill full Hilbert space at our disposal now we would also like to be able to do this with block states on bands so um so therefore instead of studying just the position operator let's say the x component of the vector r or the y component of the vector r we put it in an exponential and dot it into a tiny little wave vector delta q which is our discretization of the brillon zone so this delta q is two pi over the system size where we have the periodic boundary conditions and the advantage of studying this object as opposed to just the position operator is that it fits the periodic boundary conditions nicely um and so now we take this operator um and we would like to try to diagonalize it uh unfortunately uh i'm running out of time so i'm not going to show you how to diagonalize it but it's not very difficult what you have to do is you have to come up with an appropriate linear combination of the block states separated by this delta q and of course they will connect back through the edge of the brillon zone and therefore it's called the wilson loop so we have to we have to we have to mix those states in such a way that this object will be diagonalized okay um so um so that's just what i said um you will get a set of uh recursion relations for these coefficients alpha it's fairly simple to solve those recursion relations and you will find an object which is a product of these overlap integrals of the block states along each of these links as you go along one of the non-contractable uh circles of the brillon zone um and then you just have to find the eigenvalues of that object in our case since we only have two bands within a given valley for a particular spin these landers will be two by two matrices so when we take this product we will obtain uh some other two by two matrix um and then we'll have to find the eigenvalues of the two by two matrix um and they will be related to the expected position of the eigenstates of this uh operator um and they are related to it in this particular fashion in our case because the twofold rotation symmetry about the z axis followed by time reversal is a symmetry of our effective Hamiltonian uh so that's the form it will take so we can think about the um eigenvalues of this as telling us something about the expected position uh of uh of this operator of this eigenstate and because we still have a translation symmetry uh corresponding to this uh wave vector this wave vector will be a good quantum number so we can therefore study the expected position of our states as a function of uh k as we for example could in the Landau gauge for Landau levels and what we find is that the eigenstates look like this now uh oops um the the four plots on the left uh they just correspond to one of these eigenstates it's resolved to be at the layer uh at the top layer sub lattice a top layer sub lattice b bottom layer sub lattice a bottom layer sub lattice b um and as you see it is exponentially localized in one direction and it is blocked extended in the other direction um and the good quantum number that I mentioned um which we can vary is chosen to be zero in this particular case and so uh the the four plots on the right then correspond to the similar hybrid binary states um except that we change this good quantum number uh from being zero to being halfway through the Brillouin zone and now you see that the distribution which is peaked on these AA sites that's what the black dots are um we used to have an average sort of in between these two columns now shifts to the left so this is very similar to what happens with the Landau levels in Landau gauge as you change the good quantum number k the Landau level Gaussian drifts either to the left or to the right depending on the orientation of the magnetic field now here that happens with the states which we constructed by appropriately making a linear combination of the two narrow bands um so uh what do the uh expected positions therefore look like um so we study the so-called Wilson loops uh for uh different ratios of w zero over w one the so-called chiral limit and now here it goes towards something more realistic in either case they wind in such a way that one of the linear combinations will drift to the left and the other will drift to the right so they behave as if they were churn plus and churn minus states okay um and that's independent of the value of the ratio in each case they wind the same way but the actual shape of these so-called Wilson loops uh does change uh with the ratio of w zero over w one so you can already get a feel for the topology of these narrow bands just from this picture you are able to decompose the narrow bands into uh hybrid binary states which act as if they are um churn plus band uh hybrid binary states and uh churn minus hybrid binary states um so or we can go to block bases and so I promise that we will construct a smooth gauge um so once we have these states um we can um uh we can we can construct them in such a way that if we shift their unit cell remember they are localized in one direction so the n corresponds to that 1d unit cell index so if we shift this by one it is equivalent to shifting the k by by one that's just what the winding means and we can also make sure that under the two-fold rotation out of the plane followed by time inverse of symmetry the churn plus and the churn minus state gets swapped so once we have these hybrid binary states we can just go back um and construct uh uh block states uh out of them simply by making a one-d linear combination of them so now these phi's correspond to churn plus block state churn minus block state q and k there are two momenta in the moray-brylon zone uh along the g1 and the g2 uh axis and now we can write our kinetic energy Hamiltonian uh in this basis now if we do this it's of course within a valley and per spin it's going to be some two-by-two matrix since this is a two-by-two matrix we can write it as a linear combination of the three-poly matrices in identity so that's what this thing says with some coefficients prefactors which will be functions of q and k now because our states are constructed to transform um simply under the c2t symmetry we can show that one of the poly matrices in this particular case sigma 3 will have a vanishing prefactor so we really only have an identity sigma 1 and sigma 2 but identity is easy it just shifts the energy overall it doesn't tell us anything about the topology of the possible Dirac touchings between the bands and so the only thing we now therefore have to focus on is the prefactor of sigma 1 and sigma 2 and those are shown over here can you wrap it up this is the last thing i wanted to show okay good all right okay so if we were to look at the contours of zeros for the prefactor of sigma 1 it's the red contour and and this red contour notice they are not periodic with respect to k they cannot be made periodic with respect to k but and and at the same time we have zeros of the coefficient of sigma 2 that's that's that's this line that is blue line and so now if you go around this Dirac point this is now all in the more everyone's zone okay so if you go around this Dirac point you go from plus plus to minus plus in a counterclockwise fashion similarly if you go from plus plus to minus plus it's also in the counterclockwise fashion so what you have is you have two Dirac nodes which have the same winding number and you know that this is impossible to construct if you had um uh exponentially localized uh uh vanier states which respect the symmetries of the problem um and so this is one way to see the topology um anyway so i'm not gonna tell you much about the many body physics which i promised but um i just want to point out that um now that we are are able to decompose the Hilbert space into turn plus and turn minus you can imagine that strong coupling physics due to the Coulomb interaction will favor a population of let's say turn plus over the turn minus due to spontaneous symmetry breaking and indeed something like that is seen experimentally with this famous observation of interaction induced churn insulators um okay so let me just stop here and thank you for your attention we have passed over five minutes but let's see if we have some original question last one yes so hi thanks for the nice talk um so you you you have presented this model uh these uh including these variations in in space in this large-scale variation with the idea to explain the uh lack of reproducibility in uh twist bibliography in in different samples um my question is um so i'm having the impression i'm not sure if right or wrong that um twist trilateral graphene is a bit more reproducible so my question is whether you uh see something in the model that uh if applied to the twist trilateral could give an idea that why this is more reproducibly i don't know if if you can't say something about that we have not studied the trial layer yet um now my understanding is that experimentally it is the i'm not sure about the reproducibility uh but what i hear is that the yield for making twisted trial layer is higher than the yield for making twisted bilayers which work but um i have seen results where the twisted trial layer has um actually moray on top of moray reconstruction as seen by stm from the columbia group um and um that there is some variation of that in space as well um i think other groups do not report that so um the idea here as you noticed um is that we would like to combine this starting with the bilayers which are easier but of course this can be extended to multi-layers uh maybe an explicit observation of the structure with the electronic properties and then there will be one fewer uh unknown uh in uh in our starting hematonic we have a couple of others but okay those who can stay can stay let's have more questions from a student does it matter what is the point of rotation between these two layers um to first order it doesn't um but you can include that effect as well uh it's just a you add a constant to the u uh in one of the layers so for example uh you can you can also study the ab bilay using the setup it's uh it's easier right you just you know exactly what your use are uh it's just uniform uh shift and then do the gradient expansion by the way what I sort of forgot to mention is that this technique gives you electron phonon couplings automatically because it gives you the couplings between the gradients of the displacement field and and the fermions yeah i have a very small speculative question is that uh this slow and fast moving fields that you are talking about and the heavy fermion and uh fast fermion that andre bernabé talks about are they related or they are completely different um right so um so imagine that we have the narrow bands already uh let me just go back this i may have to click too far back um okay here um so what andre is talks about um is taking this narrow band and then decomposing these narrow bands um into states which are exponentially localized and centered on the aa sites um which will uh uh account for something like 96 percent of the weight of the wave function within these narrow bands plus um dispersing states uh which touch quadratically at the gamma so uh these states are exponentially localized on the more length scales you see so so they are built out of the slow envelope field so uh there's no contradiction here you could you could take the result of this model and then apply the procedure um of andre and jida song and obtain uh the decomposition into um the exponentially localized states at the aa sites uh plus the um dispersing uh i think we call them conduction uh states with parabolic touching um so so it's it's it's it's not it's not very very different it's it's an additional step on top of what i just showed does it answer your question yeah chef thank you welcome so uh i had a question related to the topology so you early on you had a model in which you had exponentially localized vania functions but without going to uh valley polarized model now i suppose you have a model which is valley polarized and you need to go to hybrid vania basis could you comment on right right right right right yeah so models and adding interactions on top of them right so our um work that you're referring to um um had some intervally mixing it is not essential uh so you see this uh obstruction uh story uh is rather subtle so there's a theorem which states that if you have a set of isolated bands whose total turn number is equal to zero you can always construct exponentially localized vania states for this okay now the narrow bands which i showed you uh within a valley so you don't have to you don't have to valley mix okay within a valley these set of bands have zero turn number they can be decomposed into a turn plus and turn minus but overall turn number for this band composite is zero so you could do what was done um basically concurrently uh by uh koshino and uh liang fu um where they started with the continuum model where the um where there's no valley mixing okay now you can as i said by this theorem you can always construct exponentially localized vania instance now the question is do you pay any price for this due to the topology of the narrow bands which is there and there is a price you pay for that the price however is not exponential localization a priority the price you pay is simple representation of in this particular case the c2t symmetry so you have a unitary transformation the unitary transformation takes you from block states to vania states those vania states are guaranteed to be exponentially localized because the total turn number of that band composite is zero but now if you take one of these exponentially localized vania states and you act with the c2t on that state you're not just going to get another vania state what's going to happen is that you get a linear combination of vania states in its vicinity that linear combination falls off exponentially fast so uh you can recover numerically if you wish c2t exponentially fast if you are willing to check it to further and further coefficients so now what we did in that work um had almost valley polarized uh vania states um Cauchino and Fuss uh collaboration had a fully valley polarized vania state they had this three peak structure you can project the Coulomb interaction onto those states the inter-valley mixing is still going to be very small in either case um and you discover that such models do not favor anti-ferromagnets as Hubbard models would do but they favor generalized ferromagnets and an odd feeling they also follow they also favor um charge density wave type states like like a strike now um in such models it is not easy although not impossible to answer the question does your state break c2t because the representation of c2t uh is non-local um so but they still give you an intuition for what sort of states to search for if you choose a different basis so if you now go to hybrid vania bases or if you go to block bases you sort of know what to look for at this point which is what people did after our paper um and sure enough this type of states that they found are generalized ferromagnets they are inter-valley coherent states all of that was in the paper you're referring okay any addition stripes were also found okay now the advantage of that is that you can say something about uh the symmetries of the state without having to work through the non-local representation of c2t um the disadvantage is that these interactions are rather convoluted in uh both the hybrid vania bases and in the block bases um so i view these methods as complementary um okay good um so let's uh given the time of interest let's stop here and then let's thank the oscar gate all right thank you thank you there were some questions on the on zoom sonia hadat does only the term uj perp include lattice relaxation effect no uh the uj parallel will also include it and in fact the model that we studied um had a uj perp constant i don't know if sonia is still there but and and you can hear me but you answer that question oh okay oh you ask me okay thank you sister okay you're welcome