 Good morning. We are sailing into the second day of our workshop, just two announcements. First, there will be a small reception on Thursday after the poster session at the open terrace here in the main building, the precise time to be determined. And second, this is more important, if you are presenting a poster, there will be a piece of paper outside this room. Please put down your name so that we know how many of you, so that we can count the space for posters. Right, and with that, we start our first session, and the chair of the session is Michael Arkyselov, ICTP. Good morning, everyone. We start session of Magnetism 1. There are two talks, and the first talk is given by Gabe Epley from ETH. Now, please. Okay, it's a great pleasure to be here, and I really want to thank the organizers for inviting me. It's, of course, an even bigger honor to be here, because of Pierce Coleman's birthday. It's too bad that he won't be here, but even though he's not here, I just maybe want to, you know, because there probably will be too many after dinner speeches on Friday. I just say a few words about Pierce. Actually, Pierce, I've actually known for a very, very long time. This is a picture, as Premie was saying yesterday, she kept referring to the last millennium. This really is the last millennium. This is in 1987. We were actually both at a meeting in Bangalore, and those of you who are old enough can remember some of the people who are circled. You'll notice here, for example, is Phil Anderson, the Chandra Varma. Back here is a young man, and if you actually blow up that young man, that is the person we're honoring today. That's Pierce in 1987. Next to him is this year, which I had thought that Phil Anderson was his thesis advisor, but maybe this was a surrogate thesis advisor when Phil was inventing the RVV theory. So this is what Pierce looked like then. Actually, since that time, actually first met Pierce in the early 80s when I was at Bell Labs, and he used to come down from Princeton because Phil Anderson was also at Bell Labs, and we had many chats. Actually, these many chats led us to actually a very, very productive collaboration over the next decade or so dealing with the issue of quantum criticality, and in particular, trying to understand essentially what happened, the different kinds of quantum critical points you could get with an antifuramagnet. In fact, here's a sort of a conventional quantum critical point where you cross over between an antifuramatically ordered state and a paramagnetic state as you might think you do in chromium. In some of the heavy fermion systems, however, you would form some kind of a strange unbound condo state which would essentially exist just at the quantum critical point and give rise to all of the funny transport properties. This figure essentially became our icon for guiding our research for something like a decade. We had a lot of fun trying to study this problem which I think is still with us today. Now, of course, there have been many other social occasions. I just want to finish here by saying happy birthday to Piers. If I do get a chance, I'll make a few more personal remarks on Friday, but I think the great thing about Piers and that comes through I think in this picture is sort of his unbridled enthusiasm at both social occasions, but also at sort of doing serious science and basically the idea is that even though science is serious, it also has to be fun. The whole creativity, I think Piers' particular creativity, derives from essentially this spirit of having fun. And I'm hoping that actually I'm hoping that everybody here at this workshop is having fun thinking about the new developments in physics going forward, the way we were having fun together with Piers thinking about quantum criticality. Okay, so let me get to the hard stuff now. And this is not so hard. Let's just talk about some new science that we've been doing over the last year or two on magnetism and try to relate sort of all these concepts, interesting concepts of topology that have been so fruitful for electrons to magnetic systems. And the collaborators here are especially Alexander Bolatski, who is basically various institutions. And I won't list them all, but actually they're all sort of listed here. And then his various students and postdocs. Also, I'll show some work that we were doing or I did with Angelos McLeese in London, the LCN, and also with collaborators from both the LCN and EPFL. Something you already know, I don't need to show this, graphene. So of course, everybody has heard this many times before, is graphene is this interesting substance where at the Fermi level you can have essentially Dirac particles and the dispersion is given by this famous comb, which exists at the k points in the Brion zone. So that's Fermion, so not very new, not very exciting. This is also not very new and not very exciting, but people tend to forget is that, of course, in the single particle picture, the dispersion for phonons and electrons is essentially the same. You have for phonons also this type of Dirac-like phenomenon, also at the k points because you solve exactly the same equations when you look at the motion of the masses as you do when you look at the motion of electrons. Here, actually the red lines are just simply the calculations for an isolated sheet, and then these black points are real inelastic actually scattering data on a piece of graphite. So people tend to forget that the phonons and the electrons look the same. You can, of course, also create any kind of honeycomb lattice find these Dirac-like cones. Basically the reason they arise is that I have a number of a lattice, and then if I have proper symmetry protection in this number of a lattice, I can get essentially a bank crossing at the k points. You can even do it for artificial materials, such as quantum dots, where you can look at essentially exotongue polar condensates, for example, on such honeycomb lattices. Now, last but not least, and this has been sort of neglected, sort of one of the most popular bosons, especially with the people in this room, who have been coming to Piers's meetings over the years, is Magnon's magnetic excitations, and again the, if you linearize the equations of motion here, you again get this, these spectra or these eigen frequencies as a function of momentum, and again at the k points you get your Dirac cones. And so basically this raises a question is that there are fermionic systems, but actually they're just electrons, for example, in graphene and it's analogs, but those are fermions with long-range interactions, and there's an interesting question of if when you start to deal with interaction effects, the similarities between the bosonic systems and the fermionic systems go on. So from a spectroscopic point of view, I've just shown you there's really no difference between the fermionic systems and the bosonic systems you get essentially for essentially these linearized excitations, there's no difference, and it's really fair to ask is there any difference when you when you put in interactions, is there any difference when you think about the statistics of the quasi-particles that you're interested in. And so the rest of the talk I'm going to, I'm going to deal with this, deal with this question. So I'm going to talk about the nonlinear effects, so linear looks boring and the same, what about the interaction effects, and what about the surface states, edge states? Okay, so the key thing of course is in, the key thing that gives rise to these cones is the fact that you have a number of alattes. Once you have a number of alattes, of course the block function is no longer just simply a scalar, but it's an interesting vector which then obeys in particular simple cases the Dirac equation or more complex cases some kind of a while equation. So here's a number of alattes, and when you think about magnons, of course you can convert the Magnon problem into any of these other linearized excitation problems using the Holstein-Premikoff transformation. So you start off here with the usual Hamiltonian, and then you essentially write the spin-raising and lowering operators in terms of basically bosonic annihilation and creation operators. And then the crucial thing here that there's z actually is s minus a dagger a, which of course produces a nonlinearity, which you throw out to first order. But if you keep only the linear order terms, then you have a Hamiltonian, effective Hamiltonian in these degrees of freedom, which of course exist on the two sublattices, so if you wish there's an a and a b for the two sublattices. And this Hamiltonian consists of essentially just j times s, and then this term here, which encapsulates the momentum dependence, the interesting momentum dependence, which gives you those Dirac bones at the k points. And then there's a diagonal term here, which is 3. That's actually not going to be unimportant. So this is just a standard thing, and this is how you convert your eigenvalue problem for the spin case into an eigenvalue problem, which looks very similar to that for fermions or in fact for phonons. So there's a 3 that's important. There's a diagonal piece. And that gives you, excuse me, yes, yes, I'm expanding around the fermi-dentagrastate. So that's what you get. So this is why you get the same thing that you got before. The 3, the 3 there, which is the diagonal term, which effectively comes about because you can have multiple occupancy if you wish of the sites rather than single occupancy. It just shifts the energy of the spectrum, and you just get, again, this famous form as if it were graphene, electrons and graphene. Now let's think about the interaction effect. So there's a rather famous result for the, for electron-electron interactions for the, for graphene, which I just summarized here, is there are, in fact, self-energy corrections due to the Coulomb interaction. And that's, the calculation is actually just shown here. It just comes about from this diagram. You get essentially a creation of electron-hole pairs mediated by the Coulomb interaction. At the end of the day, though, you get what amounts to an integral basically of this, of this form here. And of course, that form gives you a logarithmic correction to the velocity. So in fact, the velocity with electron-electron interactions does, is actually not constant as you go towards the Dirac point. It actually diverges logarithmically. Effectively, you're just dealing with a 2D transform of a Coulomb potential. And what this means is that you don't have a Dirac cone, you actually have a Dirac hourglass with the interaction effects. And this Dirac hourglass was seen already seven years ago by Andre Geim. He actually measured the velocity, which you can do by changing the chemical potential with a gate. And sure enough, it more or less agreed with this logarithmically diverging effect that was predicted. Okay, so we know, of course, that with the magnons also, there are these corrections. Remember, I said that SZ actually was involved actually not an A dagger, but an A dagger A. So there's a number operator. So when you reinsert that into the original Hamiltonian, you, of course, generate terms which are not just quadratic in the AMB sublattice operators, but also actually come in to higher order. And so we have to start thinking about the correction terms here. Their correction terms, of course, has been raising and it's been lowering operators as well. And so you can start to expand the Hamiltonian in these operators. And you get, of course, all manner of terms, particularly get all these fourth order terms, which essentially couple the two sublattices together. Notice that they never appear by themselves. So you never get an A to the fourth type term to this order because, of course, we're dealing with only the case of nearest neighbor interactions. And the nearest neighbor interactions are strictly between the two sublattices, not within a particular sublattice. So you have all of these extra interaction terms. And, of course, here we have well-known processes from just many body physics to try to correct the self-energies. And, of course, you have a heart tree diagram, which, as you'll remember, just simply produces a shift of the energy. And then you have these rainbow diagrams, which we'll talk about a little bit more in a moment. And so we go ahead and calculate this as the heart tree correction. This is just this shift of the self-energy. It scales essentially like the population squared of the magnons, mainly the zone center magnons. And the correction term is exactly this. That's the interaction. That's the spin. And that's pi. You can see, of course, that as s gets smaller, the corrections get smaller. Got bigger. This goes as one over s to the third. Okay. So that's what we've got. Now, what about this diagram? So this diagram, again, is quite simple to write down. It's simply self-energy correction here. Again, scales essentially like this population factor times a matrix element divided by this energy denominator where energy has to be conserved, going through this in here. And in other words, one can then consider here that, of course, you can calculate both the real and imaginary parts of this. The imaginary part gives you decay rates, and the real part gives you a spectrum renormalization. You should remember that if I have a Bravais lattice, such as a square lattice or a cube, Dyson showed essentially that the correction, this correction here, again, was simply a constant. And so let's look at these two. Let's just try to understand how the calculation actually works out. So here are the two integrals that one can consider just to think about what's going on. So this is the real integral, which has matrix element effects plus this energy conservation, energy conserving delta function. And this is just this density of states, which just takes account of the delta function without thinking too deeply about the matrix elements. Of course, you know that from ferromagnets of matrix elements generically are weakly momentum dependent if at all. So if you want to build intuition, really, you can build intuition just by looking at this particular integral. And so one can then go ahead and calculate these two functions. The solid lines are just simply the density of states. The dashed lines have the matrix elements in it. You can see that there's actually relatively little difference. To get even more sort of into this understanding of what's going on here, because you can see that when you calculate those integrals, there are actually quite a few singularities in momentum space. These are van Hove type singularities, which are gotten essentially from kinematic constraints. So all of the singular behavior here, you see both in the density of states and in the matrix element weighted density of states, which is the self energy. You see them in both terms. And what you can see is there are singularities at these important symmetry points in the zone. And those singularities appear in both functions, both the density of states and the self energy. So basically you're dealing with van Hove singularities. And these van Hove singularities come about in a way that's shown diagrammatically here at left. So you have to be a little bit careful here. So there's various colors here, which correspond to different momentum points in momentum space. And then what I'm showing you here is coded to the same color as the contours along which that delta function enforces energy conservation for those particular values of the initial Magnon energy momentum. And what you see here is that as you, depending on whether you look for, let's say, the up to the up down band, remember there are two bands because I have two sub lattices. Those are the AMD operators. In this case, these contours evolve a particular way as I move through the zone, as I change the colors here. The most interesting one is going to D plus D. So that if I move, for example, from this point in K space, this is the contour where I find the spectral weight, which matters for the self energy. Then if I just move a little bit over here, that contour comes across. And then the purple one at the K point jumps here. And when, of course, there are these jumps in these equal energy contours is where I have my von Hover singularities. So I have singular behavior. In particular, I have singular behavior at the K point at the end of the day. And so one can ask, here again, it's just drawn in more detail what these rainbow terms do. And I just showed you now just not just the imaginary part, which was shown before, but also the real part. Again, I see these singularities in the real part. And I see this curious sort of double singularities near the K point. So there's this minimum and then these two maxima around it. So that's a rather curious structure, very different from what I get for Coulomb interactions for the electrons. And so if I really blow up the region near the K point to look at the many body renormalization effects, what I see for the Coulomb interactions and fermions, I see this hourglass shape, logarithmic, whereas actually I see something very different for the magnons. I actually still see, of course, a shift in the Dirac point. I see linear behavior at the going through the Dirac point. And then I see these odd looking singularities coming from the van Hove effects, which I just described a minute ago. So there seem to be very different many body effects in the case of the fermions on graphene versus the magnons on the same honeycomb lattice, even though I started off, of course, with the lowest order, the same excitation eigen frequencies. So a question is, does this have anything to do with reality? Well, it does. It turns out that about 50 or 60 years ago, these materials are very popular. These are transition metal trihalides, which consist essentially of a honeycomb layers of transition metals are indicated by the gray spheres, which are sandwiched essentially between layers of the halides, for instance, bromine or chlorine, which are denoted in purple. So I have a nice Fonderwals magnetic analog of graphene. And the renormalization of the spin waves in these materials was a topic 50 years ago. This is a paper from 1971, when I can make the usual remark that most of you were not born then. And I was not even, I was still wearing short pants. So this paper actually, when we started, when I last checked, actually was cited only five times. It's actually a very, very pretty paper. And what these guys did is they did neutron scattering measurements on a very large crystal of this material. And they did the usual thing. These are just neutron data. They don't look any different today than they looked then. You fix, for example, the momentum, scan energy, and you can see the eigenfrequencies, you can see peaks just like people do nowadays. So nothing has changed. The mass of the neutron hasn't changed. And neither has its spin, which is crucial for these experiments. And they actually had this diagram in the paper. This is direct crossing here. And this is the k-point. And what they found was when they measured the eigenfrequencies near that k-point, they found very curious behavior. What they thought was very curious is here is the k-point indicated by the red line. And they saw a renormalization and, of course, the error bars, because these days are pretty large, but what you see is there's a fairly substantial renormalization of the spin wave energy, which is quite momentum dependent near the k-point. And then it looks like there's a maximum just a little bit away from the k-point and then it goes down again and then there are large error bars here near the gamma point. Okay, so that was not explained at the time and just parked. And when we picked this up again and did our theory, we then actually tried to overlay the theory, which has, of course, exact coefficients out in front. And here I've just re-plotted the data from 1971. And this is the theory with all these van Hove effects. And sure enough, the van Hove singularity actually counts precisely extremely well for these old data. So this is the renormalization, the renormalized dispersion relation near that point. So actually the interesting thing here is you could say that the renormalization of essentially a Dirac particle due to many body effects was first measured 50 years, in this case 40 years, before Guy measured it with his collaborators for fermions and graphene. To say the other interesting thing is the citation index is a very interesting guide to what you should be looking for, new physics, and you should be looking at papers with very few citations and exhume them 50 years later. This is actually a very beautiful paper. Yeah, yeah. No, I think they didn't, they just simply didn't have the ability to do those, no, they tried going beyond. But they, if, Silverglitt, they did try to do calculations, but I think they were just not able to do the integrals. But they knew you should be doing those integrals. Of course, Holstein-Premikoff thing existed when you and I were wearing short pants, when you're very small. So it's an amusing sort of history of science thing, but it's also amusing just to read this whole paper because there's a lot of physics there, which these guys actually understood. But basically that resolved that old mystery. Okay, now what happens when these bows on them? So this is the sort of the next order in this theory. What happens, of course, when the amplitudes become even larger? Well, then you enter what we sort of like to call a nonlinear regime. I'm just going to breeze through this because here, of course, in one dimension, you enter the regime of solitary waves. In two dimensions, you enter the regime of sort of large-scale discommensurations. You enter an interesting regime of nonlinear dynamics. And one can ask this question for the bosonic systems. In this case, actually, we tried to answer this question for graphene, which is being stretched. Okay, and it turns out that graphene, which is being stretched, if you do molecular dynamics on it, actually has, in addition to the linear excitations, has these nonlinear solitonic-type excitations, which look like that, essentially, which look like very, very large waves. And with very, very large displacements, and these large displacement waves, solitary waves, actually show up when you essentially stretch, essentially, when you stretch the material and you look for the elementary excitations just in the molecular dynamics calculations. And actually, if you look, you find out that there are very long-lived, actually, large-amplitude excitations, which are not simple phonons. This is now for phonons. We suspect something similar might happen for the magnons. So we're expecting that there might actually be nonlinear phenomena in these especially low-dimensional magnetic systems, which will be analogous to this. Now, the reason we were working on this problem with phonons was that we were interested in the amazing ability of graphene, actually, to carry water bubbles on its surface. And this was related to the problem, actually, of using graphene in biological sensing systems and water filters. But the main point here is that, in addition to the spin waves, the spin wave approximation does not account for all of the qualitative features that one would expect. And we're certainly hoping people will look for the nonlinear excitations in the magnetic van der Waals solids. Okay, so what might these nonlinear excitations be good for? Well, as I was saying before, these nonlinear excitations are very good for carrying water. We suspect that perhaps in the case of the magnetic systems, they may be good for carrying charge around. I won't go into this because I'm running out of time. In fact, there's another interesting thing about these large amplitude ripples, rather than the low amplitude phonons, is that what we discovered a number of years ago is that if you have large amplitude ripples, you can actually destroy charged NC wave states in graphene, which are present when you have very flat pieces of graphite. You could get beautiful charged NC waves. Whereas if you crumple the graphite, in other words, if there are these large amplitude excitations, actually you can get, you actually destroy the charge density waves, which implies again that if these ripples are moving around, these nonlinear, the soliton type excitations, which arise when you go to nonlinear order, are strongly coupled to the electronic degrees of freedom. Actually, we think that might give rise to some interesting new physics over the next few years. This just shows the same picture in reciprocal space, and this shows essentially the two classes of samples. This is two types of doped graphene. They're all doped in the same way, but the samples which have charged NC waves are the ones where the fluctuations, the surfaces are very flat. There are no ripples around, whereas the ones which don't have charged NC waves have surfaces with much greater roughness. It just shows the height spectrum of the material, which you just measured with STM in this case. If you have a very rough surface, there's no charged NC wave, lots of ripples, lots of these nonlinear excitations, very flat, you get this. I think there's going to be some interesting electron phonon and electron magnon physics coming up soon in this field. The other thing, of course, that's happened recently as people have rediscovered, not just themselves, have rediscovered these transition metal trihalides, and they find very curious ferromagnetic behavior and also layered anti-ferromagnetic behavior in single flakes of chromium triiodide. These materials have come back. They have very, very large optical rotations, so they're very amenable to doing physics. Again, the transition temperatures also depend on the number of layers that you have, so there's some detailed physics to do with the couplings between the layers as well. Just wanted to introduce this topic a bit. Let me just conclude with one particular and very simple demonstration of where fermions and bosons are different, going back to that three that I mentioned at the beginning, different between the graphene and these layered honeycomb ferromagnets. This is, again, the honeycomb lattice, and I have two sublattices. Now, if I think about assembling this into three-dimensional structure, what I get is, of course, the possibility. These are the different stacking sequences that I can have, A, B, A, B, C, and so on. What you'll notice in particular is that if I think about that last layer, that last layer could have one sublattice, for instance, in this case, the blue ones in the last layer could be decoupled from everything underneath. So I have a surface, I have one isolated sublattice at the surface, B sublattice. So if I think about the, now, the consequences of that when I have bosons rather than fermions is, of course, that if I consider this, this Hamiltonian here, of course, I have my usual in contrast to what happens for the fermions. Of course, I have these off-diagonal terms as always for the fermions, but in addition to that, I have this famous S plus, this famous term derived from SZ, SZ. So what that means is that I do have, actually, a B sublattice term living by itself at the surface, which is not present in the fermionic case. So I have a B sublattice term, and of course, that B sublattice term now, this diagonal term is missing in the case of the fermions, but is present in the case of the bosons. That's that three. And of course, that three, combined with a weak coupling down below, actually gives rise to a dispersive surface state, which is non-dispersive in the case of graphene, but is dispersive in the case of the magnons. So I could get, actually, surface magnons, which are qualitatively different than what I get for the surface fermions in the case of terminated graphene-like structure. And of course, you can do this all by, you can check this for yourself by doing sort of a classic slab calculation. And again, the key point here is that I have this three, which is not zero. And as I say, when you do this slab calculation, you get what I just showed you, and I think I'm about to run out of time. Of course, this, what the slab does for you also, it tells you how this k point wanders around in three-dimensional space as a function of kz. So it twists around, and of course, that twist as it goes around is related to the chirality of the interaction so what this means, among other things, is that if you had, for example, a Jozinski-Maria interaction, you would break the degeneracies between basically right-handed and left-handed polarized k point trajectories, and that in turn would actually also gap out the Dirac spectrum at the k point. Okay, so let me conclude. I've shown you sort of the excitement now of moving from fermions to bosons. Bosonic Dirac materials actually are not exotic at all. I would almost claim that they're much more common than the fermionic variety. There's so many things. There's phonons, polaritons, magnons, and particularly we've recently rediscovered these transition metal crihalides, which are fattened walls materials, so you could isolate them and do all kinds of physics with one, two, three, and four, and so on, and layers. Combine them also with other founder walls materials and heterostructures, if you like. There's a lot of new physics, including dispersive surface states, which did not exist for the simple graphene model. I've also shown you that there are corrections to the self-energy of a very different kind that you can discover in these materials, a very different kind than you can discover for the electron systems. The last thing is that solitons, actually not only do you have sort of interesting corrections to the linear theory, you also have nonlinear excitations which show up at large aptitudes, and these actually have some, if you wish, practical use. They can carry water, but I'd like you also to think about how they might be able to carry charge excitations along. Thank you very much.