 We're recording in progress. Going to turn this. Did you mute? Okay, so we have. So it's mine. I thought it's smooth. So the first speaker is a plea from ETH and the search for flatbends and pack of metals. Right. Okay, so it's a, it's a big pleasure to be here and I want to thank the organizers again for having this marvelous meeting. That seems to have started many, many years ago, and the topics seem to have moved on to, you know, starting I guess with heavy odds of ITC and now, now this time is dedicated more to top logical effects. And I am going to talk about essentially a search for a key ingredient of interesting topological effects in real, real material. Let me just, let's just see how this works. Why is this not advancing? Oh, there we are. Okay. It's funny that it doesn't. Let me just acknowledge that my collaborators in advance is always important, especially if you run out at the end. So I've highlighted here at the beginning of the list, the junior people, the postdocs who've really done most of the work on making this happen at the variety of institutions, both in Switzerland, including, of course, ETH and EPFL, but also University of Geneva. And the work that I'll describe is, is covered in these two preprints and there'll be another one shortly dealing with the end of the talk if I manage to get there. So just by way of outline, I'm just going to tell you probably something you already know, just sort of briefly reintroduce the Kagame lattice and what can happen on the Kagame lattice with electrons and spins and why is it so interesting for the correlated electron community. I'll then just briefly touch on what's being done with real materials, insulators and metals in the search for flat bands. And then I'll talk about the material, which is the main topic of my lecture today, and then deal with electronic structure and magnetic excitations thereafter. Okay, so frustration, you all know, triangles are interesting because you can, if you have a third spin, I'll say in the Ising model, all coupled equally to each other. That one is frustrated, doesn't quite know what to do if you set the other two spins in a particular orientations. And of course, these, this gives rise to, you know, sort of non-trivial geometric frustration on bigger lattices of triangles. So the triangles are the basic motif. And of course, you can decorate these triangles with spins. And actually, the other thing that you can do with the triangular lattice is you can thin it down. You can dilute it. And one thing actually that I'm sure you all know is that if you just remove, let's say, if you remove one third of the spins to make a hexagonal lattice, actually, you're no longer frustrated. You can generate, you can create a lattice where all spins actually are happy. So playing around with the triangular lattice and its derivatives is interesting. So in the case of going from the triangular lattice to the hexagonal lattice, the dilution actually helps relieve the frustration. And, but there are other, and so this is, of course, a simplest case of a vacancy lattice derived from the triangular lattice. You can, of course, also have more interesting vacancy lattice. He's then simply hexagonal, which we know so well from graphene these days. You can have a two by two vacancy lattice. And this actually gives rise to the Kagome lattice. That actually maintains some of the triangle. So there's frustration that is actually continues to exist. And this fitting out exercise in this case, you know, leaves you essentially with, with a problem for the simple ising model. But actually, you can relieve the frustration, of course, to some degree, if you, if you have a continuous spin degrees of freedom on any of these lattices. But what happens with the Kagome is, is, is, is especially interesting with this vacancy lattice. You have this opportunity here to create, let's say, some ordered structure, be it anti ferromagnetic or ferromagnetic where you actually maintain local degrees of freedom, which can fluctuate at no cost to their environment. Okay, so, so what this effectively means is that in real space, you have these, these localized degrees of freedom, which, which essentially can rotate in the presence of everything else without any cost. And that of course leads to these famous flat modes that are the topic of this particular, particular talk and I think also suspect the topic of many other talks at this meeting. Just, if you want to read about this, this is of course a very old story, something that probably Chandra and I wrote, you know, 25 years ago. So moving into case space, how does this happen? So if I look at the ferromagnetic spin waves on the honeycomb lattice, of course, those are have a very close relationship to the electrons in graphene. You have this famous dispersion relation with these touching points. These are going to type these Dirac nodes at the, at the K points of the, of the, of the honeycomb lattice. So there's this, of course, is has been much discussed and has interesting course implications also, you know, when you think about edge states and everything else, whether it spins or electrons. If you, so there's two modes here, essentially that touch. Now, if you, if you go to this is the root three by root three, super lattice of the, of the ordinary triangle lattice. Now, if you go to the Kagamine, there's actually an extra mode that gets added in case space that corresponds is flat in case space corresponding to these localized degrees of freedom that I mentioned to you a few minutes ago. And so you still get actually inherit essentially these Dirac, these, these, the, the, the inherit the hexagonal modes, including the Dirac points, which are marked by the green arrows. You inherit those, but you add, in addition to those Dirac modes, you also add a flat mode on top. And these are just spin wave calculations, but as you know, these are completely essentially just related by a simple square root to what you see for electrons. And this topic of what might happen in such lattices when you add correlations really came up in a serious way a little bit more than the decade ago in the paper from the group of Jarvan Wen at MIT. And they, they started to try to tune these dispersion surfaces that I just showed you using first and second neighbor poppings. And they also added spin orbit coupling to make the physics a bit more interesting. And they also added ferromagnetism. But basically with this tuning, well, these modes here up here, you see the ordinary Dirac type modes, or if you wish, the graphene type modes, but they of course acquire gaps, which, which are, which basically become a function or a function of these these additional parameters that are thrown in. There's also down here, there's this, this originally flat mode, whose curvature and distance from these others is can be essentially fixed by these first and next year's neighbor parameters that I just illustrated in the last slide. And what we have here really is a metallic ferromagnet with a spin orbit interaction. And it's of course interesting at that time was thought to be interesting because maybe this could be a residency for high temperature fractional quantum pulse states in zero field. And so long story short, here's a particular tuning that they use, they got sort of they introduced flat bands in this simple lattice, Kagame lattice for the U. So these are potential hosts for exotic quantum pulse states. And in some sense, this is not in some sense in every sense is the motivation for this kind of work is exactly the same as a motivation for all these wonderful studies of twisted violet graphene that you've heard about for the last five, six years, especially from the MIT and Columbia groups. So what about experiments to actually see these flat modes actually to spectroscopically make sure that they're really there and that all of this theory based on very simple models is somehow relevant for actual physical measurements. It turns out actually that it's rather more difficult there. I mean, on the insulator side, essentially for the simple ferromagnetic Kagame systems, the best data are available for this horrible organometallic thing that copper does have ideal this copper VDC I can't pronounce that benzene dicharboxylate that I just did. Those are the things that essentially are stuck between these layers. You see the benzene rings there. And then there are these Kagame layers which are actually quite perfect for copper. Now the unfortunate thing with this is that it's not possible to get really large crystals. And so all of the data that we have on this system, the dynamics of looking or looking in detail at the flat modes is on powders. Now that's an insulating realization. That can just look at the magnetism. There's of course metallic realizations. And the most popular one with the sort of people doing experiments has been this iron iron 3-10-2. It's misspelled there. And what this is is Kagame bilayers with interspersed with Kagame bilayers of iron and 10 interspersed with simple 10 triangular layers and then another Kagame bilayer on top and so on and so forth that gets repeated. So this is a proposed metallic realization of this material. It's actually not quite as simple as that in the sense that the Kagame layers, these bilayers actually are actually distorted and they're distorted in such a way so that actually you lack inversion symmetry in the points of inversion symmetry even in the plane. And the other drawback of this stuff is that the crystals are rather small, but you can get reasonably good things in the sort of fractional, a fraction of a millimetre range. This stuff has been studied for a very long time. Actually it's been studied in the last 50 years. It's curious, it has a very high paramagnetic transition temperature. It's not 630K. So it's a very strongly coupled sort of stoner type system one thinks it has a curiosity that attracted most of the attention, which is it has a spin orientation transition sort of at around the 120 Kelvin where the moments go from fine essentially perpendicular to the plane to in the plane. And it turns out that there's a recently we will establish there's a nice critical endpoint associated with that transition. This is not the topic of this talk, it's really thinking about the dynamics trying to see if we can see the flat modes that are needed to get all this interesting many body physics. So what about the electron? So we actually have performed many, many of the people from Oleg Yazev's team in our collaboration have gone and calculated using DFT, all kinds of parameters. I'll get to some of those later. And what you see is actually is a band structure like this. It looks like any other band structure of a complicated material. There's actually in this, any DFT calculation that you look at bears no resemblance at all to the simple dispersions that I showed you at the beginning, even the dispersions that when he expanded the range to be to second year's neighbors. So it's a fully three-dimensional metal. So that's a little bit sad. It's got an absolutely baroque Fermi surface. So again, nothing at all like what you were told to believe. And and then people, of course, did photo mission. And what I'm trying to convey to you here is that actually the photo mission is doesn't actually show very much of that spaghetti. The photo mission also has a drawback, especially as photo mission is usually performed, is that not only do you see the spaghetti from the bulk, but you also see all kinds of fill from the surface states, particularly at the energies where people like to do photo mission, which is, you know, near, you know, between 50 and 100 EV. And so there's some ghost-like features. And a lot of time was spent in the early days, meaning a few years ago on looking near the K points. Of course, these are the same K points that were inherited from the hexagonal lattice and triangular lattice. And there's this feeling that, well, there are probably Dirac points there. Now, what the calculations actually do show is that those so-called Dirac cones are first of all, they're not predicted by DFT for the bulk, but they're actually derived from the superposition of surface and bulk states. And so that's sort of the, you know, that's happy to engage in more discussion about that, but that's basically no Dirac points here. The other thing is there's no, nobody can see any flat bands there either. New discovery, which there'll be, I think, a talk later in the meeting about is that, although there are no Dirac points, there are tons and tons of well-knowns, which are very close to the Fermi surface. So the other thing, a thing to say is experimentally, you know, there are these, although there are these features which have fooled people for a long time, there's no sign of well-resolved quasi-particles near the Fermi surface and no sign of resolved flat bands anywhere, but there are curious features in the 3D band structure which ARPES does see. And they're seeing it actually in a very ghost-like way. So what I'm showing you here is essentially a photo mission. You of course all know that with photo mission you only conserve momentum in the plane, but of course you could always fix the momentum along the Z direction for the initial state by looking at the energy, the outgoing electron energy. So you have a way actually to perform KZ scans by varying the energy KZ where Z is perpendicular to the surface, perpendicular to the basal planes of this material. You scan the photon energy into the soft X-ray regime. So this is now hundreds of EV. And so what I'm showing you out left is a section of the Fermi surface actually in the KYKZ plane. And what you see is that there are actually curvatures, there are band curvatures there. There's also what's noticeable is that if you look in detail and there's some optimism near the gamma and Z points, is there looks like to be some essentially zone center Fermi surface which is actually fully modulated along Z. And you can make this out in this cut along this dash, vertical dash line. You can see here that there are alternating sort of dark and bright regions exactly at the Fermi surface basically centered on the gamma and Z points. So that was just a very, very small hint. This is hardly definitive, but it said that perhaps to look for, we should try to finally see if we can actually image directly real quasi-particles near the Fermi surface by going into a different regime with the instrumentation that's available for our pitch. And what we decided to do to try to see a little bit more in detail, particularly what goes on near the gamma points, is we went to do laser arpes with a very low initial photon energy, just 6EV. And of course what that 6EV means is that your typical kinetic energy for the electrons, you have to subtract the work function with the electrons from the photon energy is going to be measured in single digit EV. And the great thing about that is that these low energy, the very low energies, the electron mean-free path becomes much longer. It's basically a border of 100 instruments or beyond. In contrast to what that mean-free path was in this regime where I showed you the photomission data, again which misled people about these so-called direct cones. And so we're now dealing with the mean-free path for the electrons exiting the sample, a good more than ordered magnitude longer than we were dealing with in this typical soft sort of UV range that people work in. The other thing that actually the laser brings, it brings a much smaller spot size than you get out of the normal cyclotron source. But these two facts are very important. So the first thing, the three things that you get, one thing is you get much more bulk sensitivity so you don't have to spend all your time trying to sort surface from bulk. The other thing that gives you specificity to particular locations in the sample. And of course last but not least, it gives you much better energy resolution so you really can start to look at quasi-particle lifetimes and mean-free paths. So at the same time, of course, one needs to do some DFT to try to understand those pockets. DFT does in fact give these pockets, DFT for the full three-dimensional material. These pockets don't exist in any kind of simple model calculation for the bilayers or the monolayers. And the most important thing I know this, unfortunately this screen is much smaller than I thought it would be. The key point that we want to look at is here near the gamma point. And what you'll notice is there are indeed electron pockets for the electron-like Fermi surfaces centered on the gamma point in a very small range of q-space and a small range of energy transfers which is actually perfectly matched to the 16e incident of energy we are using for this particular photo-vision experiment. Now the other interesting thing that the calculation, of course, reflects is the fact that there is no, that there is actually a missing inversion symmetry in the plane for these real materials. Again, if you squint at this, and again I'll show you a little bit more data later in a minute, is depending, in fact, on how you, depending on essentially the orientation, let's say, of your experiment relative to the crystal, you'll find that actually these Fermi surfaces are not inversion symmetric going to the gamma point. Okay, so that's, and the last but not least is what goes on near the gamma point is very, very sensitive to you in the DFT plus you calculations which were performed here. And so if somehow you can get your hands on these gamma points in a clean way, you can actually decide, determine what you might be. This band minimum really moves very quite strongly in view of 1.3. For instance, a band minimum is something like minus 0.1 EV below the Fermi surface and if you view as a half then it's substantially lower than 1.3. So if you can with high resolution measure that band minimum you can also establish a key parameter for the many body physics here which is basically this U term in the DFT plus you calculation. Okay, so now finally I'll show you some data but to the data do not disappoint these are now subtly when you do this experiment and sure enough you actually resolve these features at the around the gamma points and you see a nice seemingly sort of circular thing which has three fold not six fold symmetry. So it doesn't invert so if they rotate by 180 degrees that pattern is not the same. The other thing that you see is you see evidence that there are actually three bands there and we just label them alpha, beta and gamma. And the minimum is minus 0.1 EV which means now you could actually compare it directly with the DFT plus you calculation and then isolate this U term. Now if you move the beam on the sample actually you rotate this pattern. Okay, you rotate it if you like by 180 degrees or if you wish it 60 degrees which of course is equivalent to 180 because it's of course three fold symmetric pattern. So what you can see here at the top you can see that this pattern here is rotated so flipped through the origin. Also if you look at the energy Q dispersion curve it's simply also just simply flipped around the origin. So this pattern here is really crucial is not symmetric going through between the M and M between the M prime and M point. So depending on where you are in the sample you get different looking band structure but it's always symmetry related in the sense that there are two band structures that you see which are simply rotated relative to each other by 180 degrees. So the twinning matters, yeah. Yeah, and now there are and sure enough I'll tell you in a minute. Yeah, that's the next topic but let me just do the material science housekeeping first and then we can do the physics. So just on the material science side you can actually map out this crystal and essentially what's happening here is the crystal is twin and you have you can actually map out the band structure in different locations and actually have my postdoc, Sadie Kihana wrote a fancy machine learning algorithm to go through all the data and identify which to which of the two rotations the different bits of the sample belong and so here's a nice map and now getting on to the physics just one other piece of housekeeping these data are in fact consistent with synchrotron data at the other gamma point if you add up the data from the different from the different rotations okay so the synchrotron data is just sort of the circle if you then simply add the two types of domains that we have you get something which looks rather similar to the synchrotron data but as Andre just pointed out the resolution is much better you can start to do physics with those streaks so let's talk about the physics you could do so this shows first of all the comparison in detail with DFT DFT actually this is a DFT calculation where we've done a little bit of averaging along the Z direction simply because of course even here there's still a finite escape depth for the electrons but what you see here next to those data basically going from the M prime to the M point is three major bands crossing the Fermi level three major ones and then amazingly actually up here there is a flat band apart just about 20 or 30 millivolts above the Fermi level and so this actually the DFT really replicates that structure with one big proviso namely that there are three bands crossing the Fermi level not two so there seems to be there's a spare band that's come in into the data that one, the beta band it turns out that beta band is the sharpest one of the lot and in fact what I'm showing you here is just cuts through the actual data just showing you how sharp in fact that beta band that beta band really is and what you're seeing here at the at the left is two polarizations and that turns out actually whether you see that band or not depends on the polarization how you see that band but what's found is you get fairly sharp what we call these alpha bands but then if you tune into this vertical polarization you actually get you can actually see what we call a gamma band which is here and on top of that you see an extremely sharp beta band and in fact if you just analyze a mean free path that you get from that it's an axis of 100 angstrom it's something like 150 angstroms which is actually comparable to what you get as a mean free path from the just the very simple fluid analysis of the bulk resistance this is around 4 kelvin actually it's around 4 kelvin it's down the domain size is just measured in several 20 microns type so the scale bar is 40 microns so you just park actually the data that I'm showing most of the data would park here yeah so depending on the polarization so here here you see basically you have to look at the contour plot you see two things and I'm just cutting very very close to the front edge here and so what you see here is you do actually see two things as if this thing looks like a shoulder but it's actually another peak and then you see this extremely sharp object rising from that yeah on the other side you see alpha band is a fairly sharp peak but not quite as sharp as one so beta is the sharpest one of the one beta is also the one that makes the least amount of sense because alpha and gamma just sort of exist everywhere beta just stops just just sits there okay so this is this is interesting actually just for the record I'm not going to go through this since it's a little bit too detailed is these flat bands somehow these these I didn't mean these flat bands these the small fermi surface looks relevant for other measurements for instance the haspen alpin there's actually for instance you can calculate from our measurements you can calculate the dhva frequencies for this for the small gamma band and actually get values which are which are close to those actually they're actually attributed to something completely different they're attributed to this to these Dirac points which DFT shows are just not there so we have a feeling that we actually have resolved why you do see this the haspen alpin even though and it's just by a coincidence that whatever calculation was done for the the haspen alpin coincides with the incorrect band structure for the material but this is this is a detail but the fact is these bands matter in the bulk ok now there's another interesting thing about this this stuff so we said that there are three bands when in fact I really expected only two based on the DFT but if you warm up from six kelvin or 70 kelvin that beta band just disappears so the beta seems to be a low temperature a property of a low temperature state doesn't the field get worse? ah that's a synonymous I'll talk about broadening now of course of course it could just be broadened out but the fact is you cannot identify it anymore yeah let me let me now show you what what else we learned here so we actually have plotted up here for you I've plotted the the widths of these peaks which is of course just the inverse scattering length as a function of the distance from the Fermi surface and so you'll notice this is here in inverse angstroms so that delta k there's a half width and it turns out that and what I'm plotting for you here are the data sets where the different colors correspond either to different bands beta band or alpha band and different temperatures so the hotter the alpha band which we can still identify here at the high temperature and then at the low temperature at 6th Kelvin it moves down to there this is the beta band which we can only resolve at the low temperature and what's very interesting is these straight lines here correspond simply to the really silliest marginal Fermi liquid ansatz where you say that essentially this width here this half width is precisely equal to essentially the is precisely limited by the wavelength of these quasi particles as measured relative to the Fermi surface that's a slope that slope is exactly one so that's a funny thing so in other words these things to here over a phase shift which is precisely defined by the distance to the surface okay so there's something interesting going on here and we suspect basically that there's some kind of strong correlation physics here which may now come from the combined effects of an empty flat band which we did not see and then perhaps you which is quite sizable 1.3 EV and of course let's not leave out spin orbit coupling so the physics of generating this band here this extra band may be physics that we're all familiar with from other systems including at the fermion systems where you could actually generate some kind of a sharp quasi particle resonance which is induced on account of partial immobilization of carriers via hybridization with this flat band so this is so we look for a flat band I don't think we found it we did find it in DFT calculations if you look in detail at the photomission data which we believe now is bulk and very well resolved it seems that there's action interesting many body physics effect which we ought to be looking at so just summarize arcus plus DFT no flat bands northera coins and simple cagamate monon bilayer calculations but near the gamma point strong correlation physics perhaps connected with flat bands just above the last two minutes let me just talk about the spin sector of this material so couldn't get anywhere with the simple models for this cagamate lattice when we looked at the electronic spectrum and photomission the question is do those simple models make any sense even for the spin sector rather than the ordinary particle sector and here I actually have to affirm you that on the spin sector things are amazingly simple so we've done relatively elastic x-ray scattering measurements of the magnoms here and indeed actually we do find a strong optic mode and that optic mode is indeed coming is indeed what you want for this dispersion surface remember what I said at the beginning dispersion surface essentially the cagamate thing has three parts to it it has essentially what I expect for the honeycomb lattice for this Dirac points plus a flat K independent thing stuck on the top and one can actually see that flat K independent thing and the data extremely simple much simpler than photomission data are life continues to be simple there's actually only one coupling constant that we can identify that coupling constant just comes from the nearest neighbor it's 25 and a half and actually that gives you even though now ignoring the two dimensionality a mean field transition temperature for the material of 800 K which is not terribly far from the 640 K that's being observed so we have the simple light in the spin sector rather complicated life in the pausing particle sector so summarized interesting flat band physics without without highly localized orbitals or spins on the cagamate lattice at first sight iron 3102 looks like a great realization but there's really no evidence for the flat electronic bands as one would have hoped from the initial you know jargon when type calculations however it's still worth thinking about this stuff I'm going to cover this gamma point centered small primary surface where there's clearly some strong interaction physics going on and there is also DFT evidence for a small sort of flat band near that near those parabolic bands that are defining those small primary surface at the same time at first sight second sight but at first sight actually the spin waves look extremely simple let me just leave you with this the stuff is both simple and complex and I think it has some legs left for people like yourselves to do some theory on excellent other questions about probably question about the second talk that you are going to give if I you phrased magnetic results in terms of spins just spin Hamiltonian if instead I will use fermionic length and takes point of view that you said that there are bands near the Fermi level and presumably flat band above if I take this and calculate particle whole bubbles and then from them calculate spins susceptibility do you think did anyone do this and second that probably flat band will have some effect on Sfqonomica that tiny flat band I don't know yeah what I can tell you is that this story on the spins is much more when you start looking at the details of that it's much more complicated than that the physics of this magnetic response function the poles of the magnetic response function are correctly given by the stupid theory with one parameter but everything else the real parts of the pole are given but everything else is completely different but in general this is a metal it's a very good metal and the life is extremely different I mean I can just maybe give you a preview because I skipped over those slides this shows the this is the essentially the magnetic susceptibility as a function of q and omega for the simple Kagami lattice of course with rakes you only have access to the zone center but the fact is that the selection rule for normal normal system with localized moments which says actually the optic mode should not be visible for example in the cubos to zero limit that selection rule is hugely violated in this stuff and the other thing of course if you start looking at the we've then of course done an analysis data the eigen frequencies as I say the real part of the eigen frequencies is actually given correctly by the stupid theory but the imaginary part is much more interesting and also the scattering amplitudes are not at all given by the do not obey the normal extension rules that you have for an insulator so life is much more interesting you showed some polarization dependent data on our piss is it clear from that what orbitals are involved in the beta band that is puzzling well from the polarizations of course you can tell that but as I say this beta band doesn't seem to show up in the bank calculation so what orbital is it which orbital it obviously has to be the same orbitals that you see in the gamma and the gamma point the gamma band we do know a lot about from the from the DFT because it does show up in the DFT that one does show up in the DFT so the our hypothesis is basically that the beta band is derived from a hybridization of the gamma band you see it's out there hybridization of the gamma band and the flat band and then the alpha band for some strange reason doesn't really hybridize maybe for some symmetry reason with that flat band above it but this at this point this is just conjecture and one should be able to you know one needs to go and ask very detail questions of the DFT calculation which we have not yet asked you mentioned something about marginals from a liquid telling that delta K precisely equals to K minus K if so I just wanted to you don't mind what is delta K? delta K is the half width delta K is the half width of a Laurentian that's used to fit the data I see, so it's Laurentian feet and it fits yeah those are these fits these are the these are the fits here okay right and the half width of this thing is exactly this thing it's exactly that but what that means is if you have a confining it's as if you said that it's a funny coincidence it may just be a coincidence but basically if you say I have to fit one oscillation of a quasi particle within that mean free path it's a numerological thing I'm not sure I'm surprised to know that Laurentian fits at all quasi particles with some lifetime but a half width doesn't it doesn't depend on whether I'm telling you about Laurentian or Gaussian right? yeah but it is it does you know the way these things have tails on them so you certainly need there are tails on these things and so and there's asymmetry okay I don't see other questions so let's thank the speaker again so the point is basically that those beta bands are generated dynamically at low temperature this is an exercise for the theorists it's validator we're through that plane