 Light show, got it, light show from current slide. Okay. Okay, well, thank you for. Okay, thank you. Last talk of the morning session is given by Professor Andrew Bernoulli. Is there a laser pointer? I don't have a pointer. I just, just my mouse. Yes. Oh, okay. This is truly dual conference. Okay, well, thank you for coming back from the quick break. Thank you, Professor Bernoulli, for the invitation to this wonderful center with such an amazing view. I wanted to chat a little bit today about, about flat bands, and the people that have heard this, my TVG talk is not going to be just the TVG talk. We're just going to try to obtain some solvable models of flat bands, even though they're a little bit artificial. But by the, the function that they're artificially can get a lot of, a lot of information, and then we're going to try to go to twisted bilayer graphing and see which ones of those points in these artificial models still work. Okay, this was mostly work done by the people in the first row, Jonah Dmitru, Aaron, Frank and Frank. Let's see. Okay, this doesn't. Okay, so I'm going to skip this slide. We all know why flat bands are interesting. And this is part of the many, many seminal contributions to, to the field between them. Now, the type of flat bands that we'll be looking at are going to be not atomic bands. So we don't want to look at flat bands that are basically made out of orbitals that are very localized, which have no hopping matrix elements between them. We want to look at flat bands that have, that come out of interference where hoppings are very large in between the atoms. So we want to put topological flat bands. And it turns out that there's a lot of them. There's a lot of these topological flat bands. And in fact, there is very natural for flat bands to be topological. As I'll show later. And it turns out these are not just thought experiments in stoichiometric systems. People can actually find flat bands at the Fermi level. This is a paper from Chalkilsky's group at MIT, where he analyzes some of the flat bands in this compound. Actually, this compound was in the same appeared, you know, a week after this paper in our database. But basically he had already the experiment, but you can also see that there's a lot of other bands around the Fermi level. So you don't can't really know what exactly is due to the flat band or not. So we're going to try to get a better separation of these flat bands from other bands. Okay, so what's known in literature about flat bands? Well, quite a lot of things are known literature about flat bands is known first that you cannot get exact churn bands with finite range hopping, okay, or with finite number of bands. And by exact, I don't mean exponentially exact. I mean really exact. I conjecture that this is true for all the stable topological states. But I have no proof of this. Then, for other lattices such as the Kagome lattice, which have flat bands connected to a quadratic touching, there were extended states written down by Bergman, who and balance back in 2008, which attributed to the degeneracy point. But in fact, they can be attributed either to the degeneracy point or to the flat band. Then the Haug group has obtained models for many, many flat bands. You can see here a crazy model with many, many non flat bands and a flat band sitting right at, for example, minus two. And this is how we got into this game by just trying to compute Wilson loops of these flat bands and finding that most of the time these Wilson loops wind, which means the very phase winds, which means you cannot localize the bunny centers, which means these bands are topological. And then, you know, you can, for example, go into the Kagome lattice and ask what happens if I open that to the degeneracy point here by a gap by a, for example, you know, you find that the only way you can open it is towards topological bands. The band will not become flat and will not be flat anymore, but it will be almost flat. And the only way you can open is towards topological band if you keep, for example, inversion symmetry. So there appears there's some connection between flat bands and topology. And this is not always true that obviously flat bands have to be topological. But I'll give you a reason why it's, it's natural. And for this reason, we have to get a, a field for how orbitals influence eigenvalues of the bands in, in the Breon zone. And what I claim, and this is well known, is that basically if you start with orbitals on the lattice, and if you want to understand all the bands within the bandwidth of all those orbitals, then there's a very simple machinery that you can do that to get from the orbitals on lattice to the eigenvalues of the flat of the, the eigenvalues of the bands in the Breon zone. And this is called representation induction and subduction. You induce from the orbitals on the lattice on the, on your high symmetry or positions or non-high symmetry positions, whatever we call positions you have, you induce into the space group, you Fourier transform, then you subduce to the space group and you get all the eigenvalues except for their order of the bands that stem from those orbitals. These are called the elementary band representations. They were done by Zach for two examples in 2001, but now they're tabulated. So let's see how this goes, because if we know this procedure, then we know which bands come from atomic orbitals, which bands do not come from atomic orbitals and hence are topological and we know many other things. So for example, this is an example of this induction procedure with inversion symmetry. I have on a square lattice or 2D lattice, rectangular lattice, I have four points in the unit cell that are inversion variant. One point is the origin. The other point is the mid-bond in the x-direction, y-direction and mid-cell. So I'm just going to give you this example for two points. The high symmetry points, the maximum wick oppositions are these four because they're invariant under a higher group than just identity. And for example, the origin, the 1A position here is invariant under inversion and I can put two orbitals on it, s or p, just a representation of inversion. And then if I fully transform this, then since I don't have any translations, since my origin is there, then in momentum space, these s or p orbitals that I've put there are going to also have the same eigenvalues at all the momentas that are invariant under inversion that the s or p had in real space because I have no face factor. However, here if I put an orbital, if I have an atom or an orbital at the mid-bond, the mid-bond is invariant under not inversion, but inversion times the translation in the x-direction. When you fully transform that, so you can again put s or p orbitals in real space, which have eigenvalue plus or minus one under this symmetry, but when you fully transform this, which is the induction in real space and then Fourier transforming, you get the face factor e to the ikx, which means that your inversion eigenvalues change between the ks equals zero and k equals to pi points. Same thing for all the other positions. And now you can immediately see that all the atomic bands by construction can come from a combination of only these eight solutions, eight possible basis states which are the s and p orbitals on these four sides, on these four maximum weight oppositions. And if I fully transform to momentum space, these are the atomic limits. So for example s orbital at 1a in the center doesn't have a face, so when you go to momentum space it takes eigenvalues one in the entire four inverse symmetric mantas, the p orbital is the eigenvalue minus one, the s at mid-bond in the x direction remember it has a face factor e to the ikx, so it's got eigenvalues one at x equals to zero, which is gamma, which are gamma and y, but it flips the eigenvalues to minus one at x and m because e to the i pi is one. Same for all of the other ones and what you see immediately is that the atomic limits, for example, have an even number of negative eigenvalues or an even number of positive eigenvalues, not an odd number. So you immediately see that, for example, also that there's a topological band here and the topological band is the band that has an odd number of inversion eigenvalues and this is just half of the Foucain index. And if I had a band that had an odd number of inversion eigenvalues I would immediately know this topological and that's the z2 index. So this can be replicated for all the groups and it's something that we'll use constantly during the stock, so I thought I pointed out. Okay, so now let's get back to flat bands and there's a way to generalize basically Liebes method to build flat bands in a way that basically shows you why it's natural that flat bands would come out of topological. So Liebes method is the following, you build the chiral lattice, you take a lattice which has two sublattices n and l tilde, the l sublattice has more sites per unit cell than the l tilde sublattice and if I only have topics between the two of them then I'm going to get a number of zero-modes just because the S matrix is rank deficient. Okay, so then Liebes Hamiltonian obviously has chiral symmetry by for example the sigma z matrix and then it's got nl minus n tilde, nl tilde zero-modes but it turns out that this is not necessary and you can actually add any hoppings between the smaller sublattice, this bk can be anything as long as ak, this hoppings between the l sublattice has for example a k-independent value a with the genesis na and na is larger than the number of sites in the l tilde lattice, so for example for this this is the general condition, you'll still have flat bands without actually having chiral symmetry in the energy spectrum for example, so this is generalized as a simple straightforward generalization of the Lieb lattice but without chiral symmetry in the spectrum and it's irrespective of the matrix Bfk. Okay, so for example this bipartite type lattices also include all the line graph lattices which you know are not bipartite themselves but and the split graph lattices etc, so basically they include I claim all the flat bands that we know without particle symmetry as new things, but for example if we take the Lieb lattice which is bipartite we get this flat band as 0 this is just exactly Lieb's lattice with chiral symmetry we can break chiral symmetry and get it also but for example we can form the line graph lattice of the Lieb lattice which is the checkerboard lattice is some special value of the of the hopping parameters and we get this flat band this can be understood as just taking the Lieb lattice and putting a huge potential on the blue sites and basically integrating them out right so all these methods includes not only bipartite lattices but also line graph lattices etc okay so now comes the punch line why are flat bands likely to be topological you can prove the following theorem so again we said that if I know all the orbitals on all the in all the lattice I know the eigenvalues of the bands at the high symmetry points so we know that in an L plus L tilde lattice I have the orbitals on L plus L tilde so I know that all these bands here all the blends the dispersive bands including the flat bands I know the eigenvalues it's just the eigenvalues or the band representations which we're going to call Br of the L lattice plus that of the L tilde lattice of the orbitals on those lattices and we know that but what you can prove even though you don't have chiral symmetry you can prove that the eigenvalues of the bands that are not flat come in pairs okay so for example the eigenvalue here you don't have chiral symmetry but you can prove that the eigenvalue of this lower blue band at high symmetry points are the same as the eigenvalues of the higher one and you can see them by labeling them in this in on the figure and then you have and and they're the they're the same as the as the eigenvalues of orbitals coming from the smaller from the L tilde sub lattice so then you have this identity of the band representations on L plus L tilde sub lattice is equal to the L tilde plus the flat band plus another L tilde okay so that means that the flat band eigenvalues are the same as the difference between the eigenvalues of the L lattice minus the ones on the L tilde lattice because I have two L tilde lattice so it means that the flat band cannot can naturally be described as a difference of band representations a difference of atomic limits not as a sum of atomic limits and hence most likely is naturally they will come out as topological it doesn't always have to come out as topological for example the L I can I could have the L lattice contain the L tilde lattice by deformations but that's doesn't if that doesn't happen then it will always be topological so it's very natural that flat bands become topological because they're eigenvalue data can be expressed as the difference of the eigenvalue data of two atomic limits rather than the sums okay so for example this is true in many cases you can build you can build lats and check it and this theorem I don't have time to go into it can also tell you when I have when you have the generously points between the flat band and dispersive bands you can do a little advertisement you can take this you can take this geometric point of view and implement it on database and then you can out of those lattices you can then brute force for the flat bands at the Fermi level and what you find is you know and the flat bands that don't have many bands unlike the initial compounds I mentioned around the Fermi level and what you find is that out of the two whatever 100,000 materials about only 300 have flat bands at the Fermi level so those are you know there's not an investment of riches there's only about 300 of them and for example there's some that have kind of neat flat bands like this one of course DFT needs to be rechecked this was a high throughput calculation when you have complex materials but these are the type of materials that are for example not known as to be magnetic this is nm in the database non-magnetic that have flat bands that could potentially host more interesting states there's just flat band ferromagnets yes this is a very very tough question so very very tough question so the the way the definition is given here is within 15 electron volts bandwidth okay but and separation from and within you know you can play with your you can play with your toggles on the website and you can check you can check how much how much leeway you want to give yourself and then if you get to zero compounds then you're then you should give yourself more leeway but that's a very tough definition right because if either it's exactly flat and then we know what we mean or it's not flat and then we don't then then but it's in any experimental experimentally related thing right that's right so realistically here we don't want we want you know less than less than 10 bands around the around within one electron volt around the Fermi level just you know 15 electron volts bandwidth of the flat portion and then you can say how many portions of the Brian zone are covered by the flat bands on the website but yeah that's that's very good question and we're very tough yes some rule of thumb yesterday should be 300 divided by 1200 thousand no there's no rule of thumb yeah I was very surprised I was expecting I actually don't know what I was expecting but I was surprised yes wow this is not the point I want to focus on but yes in this construction is flat flat in the yes right the high throughput is done by looking by checking for lattices that have we you know that have this lead form because you can check geometrically of course you can do the high throughput by just by brute force also and it turns out that 80 percent of them match but no there's no not always there's not always lead construction in chemistry right lead construction really means really right reconstruction is some very strong okay even this lead generalized construction is very strong right you don't get that in chemistry like you're always going to get the approximate things of course it's a very good point yes so when the number of sites is different for me in each sub could I say why when there's two sub lattices with different number of sites why is there a flat band so the flat band is the zero eigenvalue of the matrix that connects the two sub lattices that matrix is not square is rectangular and a rectangular matrix imposes this less constraints right it's got it's got it's rank deficient rectangular matrix so it has less rank than its highest dimension and that that gives you a zero mode for example and you know I can add a constant to the zero mode and make it any but that is basically the fact that's a non a rectangular matrix a non-square matrix always a zero mode thank you very good point okay more questions okay so now let's add interactions okay so what we're going to study is a family of made up models and these models are very artificial they were first introduced but very good they're first introduced by Sebastian which actually possess analytically soluble dcs ground states with negative but what we're going to try to study is the effect of topology on this model so I've said that it's very likely that the bands are topological we're going to study the effect of topology of this on this model so and then try to go to twisted by layer okay so basically this is the projector into the flat bands and it's our quantity of most interest and the Hilbert space will be by the projected operators into the flat band which are over complete and if I look at this if I look if I write them down in orbital space I could just write them down as a projector operators in mental space will be complete and they have the commutation relation this commutation relation up to the projection operator so there's two two ways of looking at it I can look at it either in this orbital projected operators which are over complete or I can look at them into the band projected operators which are complete both of them are good now by being Sebastian's paper top my science paper basically the positive definite how about Hamiltonian and this is a Hamiltonian that's the sz projected sz on each site okay into this into these bands is a projected spin operator on each side now if we expand this what you're going to get is the square of these number operators okay and this square is proportional to the diagonal part of the projection into the flat bands and if you have this uniform pairing condition which is which says that these projected operators matrix diagonal part of the matrix element do not depend on the orbital index alpha then you can basically obtain from this positive semi-definite Haber Hamiltonian this attractive Haber model okay because the off diagonal term is the attractive Haber model and it's positive semi-definite Haber Hamiltonian you can get this negative Haber model and you know the negative Haber model ground state now how weird is this uniform pairing condition in fact it's not weird at all you can obtain it well it is kind of weird for example interested by there it doesn't work and I'll show you what the what the things that what the results are but in some models such as the ones we've built artificial you can obtain it by symmetry so for example if you have a projection then the projected operator diagonal matrix element will be for example on these two orbitals will be the same by symmetry so you can impose this uniform pairing condition here by symmetry so we consider this uniform pairing condition and you consider no quadratic terms okay so this is a very simple Hamiltonian who also has an eta-pairing symmetry this is not because I start with a positive definite projected interactions well this is positive semi-definite here you can't get it too bad yeah well that's right okay that's right because the ground the ground states overall is so this is the mapping the minus just gives you the negative u-hubber model and now you've got pairing okay it's a weird it's a weird of course I said that these models are fine tuned but there are some things the effect of topology that I'm going to show is not fine tuned this is fine tuned the interaction also I said that yes that's right the specific form of the original interaction is two-fold that gives me the hubber model because it has this sz so it has this minus sign and there's positive semi-definite so I know it's ground state so if I find an operator who's killed which is killed if I find a wave function which is killed by the initial interaction I know it's ground state very good okay so it's got an eta symmetry which is basically a derivative of the c and young's eta it's not a momentum pi it's a momentum zero and this is actually just a co-prepare and it commutes with a Hamiltonian okay and it commutes with a projected spin for example it actually can be enlarged the symmetry u2u1 and then you can find for example these particle non-conserving DCS states by just adding by just doing a coherent superposition of eta is a two-particle operator on the vacuum and because this Hamiltonian commutes with eta the Hamiltonian commutes with eta these are all ground states of the Hamiltonian okay so basically you got an array of ground states of ground states that are bay by just taking this eta operator which is just c dagger k c minus k the projected ones into the ground state now for these states are very simple you can compute odlro this is the result mu is the filling but fundamentally what I'm more excited about because I want to go twisted bilayer is that you can compute the excitation spectrum exactly well not exactly but because they're not well the jurists allow that they're exactly solvable overall but what you do is for the single particle excitation spectrum you just take the ground state at some particle numbers now we're working particle number conserving sectors you take the ground state some particle number which is just you know eta this eta operator to the n where that's a two n particle ground state on the vacuum and you hit it with the band operator into the flat band or flat bands you have more of them and what you find out is that the one body spectrum is exactly flat okay so you say oh and I'm getting something trivial is actually flat with a pairing gap that's proportional to the interaction okay so this is the one body spectrum it's exactly the pairing gap proportional to the interaction now this is something that will change in twisted bilayer because the uniform pairing condition doesn't hold so this the fact that this spectrum is flat is a consequence of the uniform pairing condition okay now you go to the two body spectrum and the two body spectrum you know it has to be interesting because the two body state okay I have an eta operator again so I know the two body state is also eta to n plus one times the vacuum so I know I must get a zero energy state here but I know the two I know the one particle continuum only has only has a pairing gap which is this so I know the two body state has to have a zero energy state so I know I have a Richardson bound state okay so this is the two particle continuum I have a scattering matrix which we can write exactly I'll show you what it is okay the triplet excitations are 2d coupled electrons no no pairing turns out but the singlet excitations turn out to be coupled and give you something interesting so direct computation gives you that the scattering matrix is this with a negative sign where this math Cal U is basically these these non math Cal use okay these these use are the project the eigenvectors of the Hamiltonian of the flat bands of the Hamiltonian so they're just you you dagger okay now you see the semi the scattering matrix semi it's a negative semi definite so you can only have bound states okay and it's got a huge null space because it's rectangular okay again so this null space is actually just the particle particle continuum because I saw that the one particle excitation has a constant energy the particle particle continuum must have you know a lot of these particles of of two particles of this constant energy not scattering and the way you see it's got a huge null space is that the number of eigenvalues of U dagger is the same as number of eigenvalues of U dagger U and that's as much more matrix matrix that's a number of orbitals by number of orbitals matrix so that matrix has a huge number of of zero eigenvalues because this number of non zero eigenvalues the same as that of this matrix which is much smaller and this matrix is the effective Hamiltonian for the charge to non zero eigenvalues of this scattering matrix and it turns out it's just a product of the projection operators of in the in the bands of P is the projection operator into the flat band okay so this is how the two body spectrum looks for example you've got this large continuum of basically this two particle excitations unfair pairs and then you got the Cooper pair and this is this is your this is the of course your maximal bound state gap this is this is the other makes ground state in plus two particle sector but what I wanted to focus here is the dispersion of the Cooper pair okay dispersion of the Cooper pair turns out to be important and I'm just giving you the punchline here this this is the Hamiltonian that basically just characterizes the bound states is this product of the projectures in the in the flat band and it turns out that the Cooper pair mass is actually the minimal quantum metric of the flat band and this is an exact statement okay and the minimal quantum metric of the of the of the of the flat bands basically gives you the Cooper pair mass now it turns out that this model as was pointed out actually by other people including in errors paper also has a symmetry that basically relates the charge to spectrum to the charge zero spectrum to the exit on spectrum okay and so they're actually the same so it's a very it's not a BCS model it's a very artificial model that you know by adding next nearest neighbor for example interactions you can get phase separation it's very artificial model but the what's not artificial is that the Cooper pair mass is bounded is related to the quantum geometry of the flat band okay so this is what I want you to remember out of this that the the the stiffness of the collective mode has something to do with the quantum geometry now how much time do I still have I hope a lot you write down the effective kind of maximum like the tutorial I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 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