 I guess we have to move to the last speaker of this morning, Nathan Goldman. Nathan, I think you can share your screen now, and you are unmuted, I think. So. Okay, can you see my screen well? Yes. So, okay. And the talk is about donabillion block oscillation and high order topology. Nathan, go ahead. That's right. Thank you very much, Taro. And thanks a lot, Christoph and Taro for giving me the opportunity to present this work here, which was performed in Brussels in collaboration with Marco Diliberto, was really the main actor in this project, and Gian Domenico Palumbo. And so this work entitled nonabillion block oscillations in high order topological insulators, describes a rather exotic and rather intriguing type of oscillation that takes place in those systems. So all the things that I'm going to present today can be found in this publication, which appeared last year. Okay, so since this is about block oscillations, let me first start with a little pedagogical introduction. So let's just start with the problem of a quantum particle in say a 2D square lattice, okay? So this is really a generic 2D square lattice with some hopping J, JK, which connects sides J and K. And let's assume that we have some generic band structure associated with this lattice. The only non-generic thing is that we're going to assume that there is some gap in the structure. And well, the idea will be here, it's very standard, create a wave packet in the lowest band of this energy spectrum, and add a force along some direction. And so the main question that I'm going to address in the next couple of minutes will be, what do I expect in this rather generic situation? So first of all, I can propose a very, very naive picture based on a 1D picture, if you want, which would be, well, what I would be expecting would be essentially block oscillations. So the main idea is to say, well, K dot equals F, right? Where K is the quasi momentum. So this is essentially the Ehrenfest theorem, meaning that the wave packet will essentially perform cycles in K space, right? Because K space is periodic, so it will move in this brilliant zone then come back and so on. And then what the semi-classical equations of motion tell me is that the mean velocity X dot is essentially given by the band velocity dE over dK where E is the dispersion. And as we can see from this picture, the velocity is first positive, then negative, positive, then negative, which leads to a periodic motion in real space of period, which I'm going to call TB, given by two pi over A, which is the size of the brilliant zone divided by F. So the strengths of the force. Okay, so this is really the very standard, very simple block oscillation picture that you would get in one D. So as we all know, this is no longer valid in 2D or in higher dimensions. So a better picture when the dimension is larger than one is to realize that the motion is actually set by the geometric properties of the band. So K dot equals F, this is still there, still the Ehrenfest theorem. But in the equation for the velocity, we actually have an additional term here. So we have the band velocity as before, which describes block oscillations, but then we have this additional term, F cross omega, where F is again the supplied force, and omega is the very curvature of the band. So it's this quantity, which you probably all know, which is responsible for the very phase. And as a reminder, this anomalous velocity was already identified in the 50s by a carplus and litinger. And so the main lesson here is to realize that after one block period, so after one cycle in case space, because of this term, this anomalous velocity term, the wave packet can actually exhibit a net transverse drift. Okay, so transverse to the applied force, which is really the origin of the anomalous whole effect in condensed matter. Okay, so so far so good. Now, what is this talk about? Well, I'm going to start with the results of a little numerical experiment. Okay, so I'm just going to tell you what we did and what we observed, and then we will try to elucidate that. So we have performed an experiment using a 2D lattice, which I'm going to define in a couple of minutes. It will be a rather exotic 2D lattice, but let's put this aside for now. So we perform an experiment where we put a wave packet, we create a wave packet in a 2D lattice in the lowest band, and we apply a force. And what do we observe? We observe enunciating whole drift. So what does this mean? That means that the wave packet starts somewhere here in space. Okay, I apply a force. Then the wave packet will follow the force and then it will be deviated. It will reach a point here, which will be the maximal extent if you want of this trajectory. And then it will come back exactly to the same place. Okay, so it's a little bit like a combination of a blood association, so a periodic motion in real space with an additional transverse to our whole drift. Okay, now I'm going to tell you what is really special about this effect that we have identified here in this experiment. The most intriguing part is that the oceation period, okay, so this is an oceation. So the period of this oceation is not arbitrary. It is exactly given by twice the period of the bare blood oceations. So the blood oceations that you would get based on the size of the brilliant zone. Okay, so the period of those oceations is given by this two times the bare blood period. And the second thing which is quite intriguing is that this effect takes place only for two special directions of the force. Okay, which I'm going to elucidate in a couple of minutes. And if we respect those conditions, namely if we indeed apply the force along one of these special directions, then this effect is extremely robust. Okay, we basically always see it and meaning that there seems to be hints of topological protection behind this effect. Okay, so these are the results of this numerical experiment. And now I'm going to give you all the details and all the topological and geometric structures which are hidden behind this effect. Okay, so what is the model? The model is the so-called Benal-Kazar-Bernevig and Hughes model introduced in 2017. Some people like to call it the 2D SSH. What it basically means is that you have alternating hopping parameters on this square lattice, J1, J2, J1, J2 along one direction, but also along the other direction, J1, J2, J1, J2, J1, J2, et cetera. And additionally, we also have a pi flux within each plaquette, okay? So you might see these little arrows here. These arrows mean that we actually reverse the sign of the hopping amplitude, which generates a pi flux in each plaquette. So this system is extremely famous nowadays because it is a very simple model for those so-called higher-order topological insulators which are famous for their quadruple moments in the bulk and corner states. And here, as you see, I'm putting this under parenthesis because we're not going to talk about those topological properties here today. So we're going to focus here on the spectrum as a first step. So the spectrum of this module is represented here. So it displays four bands, okay? So that's important. There are four sites here in the unit cells. So there are four bands, but they are two-fold degenerates. So here there are two bands which are exactly degenerate, okay? So here you see there is a slight offset, but this is just for the illustration. Here we have two bands which are exactly degenerate and here as well for the higher bands. And these degenerate bands are separated by a gap. And as we will see now in the following, this degeneracy of the bands is really the crucial element for having what we call a multiplied block period, namely, oscillations in some classical dynamics, if you want, which are associated with a multiplied block period, okay? So I'm going to elicitate this in the following. So first to really understand the effect that I'm going to analyze today, we have to go back to the basics regarding the oscillations in degenerate bands, okay? And this is explained already in this review of 2010 by Xiao, Niu and I. So to remind you how this works, I'm going to propose the following thing. I'm going to create a wave packet, which is a superposition of my states here in the lowest band, in the lowest degenerate band at some quasi-momentum k, okay? And so in the following, I'm going to describe such an object in the following manner. So I will be looking at this time-evolving state here, which is given by a superposition of my states u1 of k and u2 of k in those lowest bands with some coefficients eta1 of t, eta2 of t, which will be time-evolving, okay? So here u1 and u2 are really the eigenstates in those two lowest bands, which are completely degenerate. And in the following, please keep this in mind that I will call eta, this vector, which collects eta1 and eta2. And so this quantity eta will really describe the internal band dynamics, if you want, when I perturb my system, for example, when I apply some external force, okay? Okay, so, well, we know actually, well, what are the dynamics for such a situation? So the result is the following. So under an applied force F, well, we still have the Ehrenfest theorem, which tells us that k dot equals F. And this eta vector, we know that it's going to evolve in the following manner. So you see that here we have the initial eta. So these are our initial populations, if you want, in the lowest two bands. Here there is some dynamical phase, which I'm not going to discuss today, which is trivial. But importantly, we have this object w, which is actually a two time two matrix, which is going to act on this vector. And this matrix w is what we call a Wilson line, okay? Wilson line is a path ordered integral of A along the path, which is depicted by my wave packet in k space of this quantity A. And A is the Bary connection. In this case of degenerate band, it's actually the non-Abelian Bary connection, which is written in terms of my eigenstates u1 and u2 in the following way. So here you see there is a partial derivative with respect to kx or ky. And here you have u alpha and u beta, where alpha and beta equals one and two, okay? So this object here is a two time two matrix. This is the non-Abelian Bary connection. And from this, we can build the Wilson line, which is going to dictate the time evolution of those eta. Torinata in five minutes, five minutes. Still five minutes? Okay, I'm going to run. Okay. Yeah, don't worry. So these objects were actually observed experimentally in cold atoms in 2016 in the group of Emmanuel Bloch. Okay, so if you consider now a close path in k space, which is what we're going to call a Bloch-Oceation, well, you see that the itas will evolve in the following way. So here you see that there is this operator, which is now a close path integral of A, which is what we call the Wilson loop. Okay, the Wilson loop is this object which dictates time evolution. And so if w, the Wilson loop is equal to the identity, that means that the state comes back to itself after one Bloch period, so after one cycle in k space. So this is really the standard Bloch-Oceation picture. However, you can find situations where w is not equal to the identity, but where w to some power n, where n is some integer is equal to the identity. Meaning that in those cases, the system actually recovers its initial state after n cycles or n Bloch periods. And this is what Heuler and Alexandra Dinata have extensively studied a few years ago and coined topological Bloch-Oceations. Okay, so here, let me just go very rapidly through this then. So these quantized Wilson loops, which are such that w to the power n equals one. There are actually non-accidental objects. They are actually related to the topological and symmetries. So topological properties and symmetries in the system. In particular, n here can be shown to be a topological invariant protected by some crystalline symmetries. And so just to give you an idea in the experiment in Emmanuel Bloch's group, they were looking at a honeycomb lattice, which has some C3 rotational symmetry. And you can show that this topological invariant, n here, is then equal to three. So you need three Bloch cycles. So to recover your initial state here, which correspond to this path in momentum space. And so what we have identified is that in the BBH model, you also have those quantized Wilson loops along two paths, C and C bar, which are generated by forces aligned along the two diagonals. Okay, so if you follow these two paths, W to the power of four is equal to the identity. And we can show that this is related to mirror symmetries and C for symmetries in the system. So we have connected this to a winding number, which is protected by the crystalline symmetries in the BBH model. So let me just highlight this expression here. So we find that W along the path C is can be expressed as e to the i to pi over four, W times sigma one, where W is a winding number. So a topological invariant, which is protected by crystalline symmetries, such that you indeed see that after four loops, you recover your state. And if you're looking at the band populations, so captured by eta one square and eta two square, you see that those populations are completely recovered after two loops. Okay, so these are the dynamics in internal state space. And as we saw, they are characterized by a quantized Wilson loop associated with a topological invariant. But as I said a couple of minutes ago, we observed this oscillating whole drift. So to explain this, we have to go in real space. And we know how to write down the semi-classical equations of motion in real space in this non-Abelian situation. The equations are written here. So you see that the velocity along x is given by this band velocity. And then there is this additional anomalous term, which now involves those omega xy, which are the components of the non-Abelian bary curvature. Okay, so you have those two anomalous contributions here, both for x dot and y dot. And in the BBH model, we have time reversal symmetry, which imposes that omega 11 is equal to minus omega 22. But importantly, and this is rather exotic, we have non-commuting mirror symmetries, which actually lead to non-zero bary curvatures, which is something which is typically not the case in previous studies of non-Abelian bluff negotiations. So I know I only have a very limited number of minutes left, but let me show you here on one slide, what are the- Less than limited. Okay, I'm less than limited. One minute, one minute. So this is, okay, so let me elucidate then what we see here. So let me take some initial states. So we prepared this wave packet at the gamma point and in the lowest band one, and we act with the force along the diagonal path here, right, past C. In that case, we can find for this particular situation that the anomalous velocity is simply given by this difference of band populations, time omega x, y 11. And from the structures of the bary curvature distribution, and from the fact that there is this perfect band inversion after cycle in case space. Okay, so since I have to go very fast, I will let you check this in the paper. You can indeed find that the behavior in real space will be these exact or perfect oscillations which involves a net drift along the direction perpendicular to the force. And so this is really a direct manifestations of the quantized Wilson loop and the existence of non-Abelian bary curvature. Okay, so this is for a general initial state. So I will skip this. We also saw additional structures which I'm not going to describe here. And so here are my conclusions. So we have identified a novel form of topological velocityations which involves annussiation, annussiating whole drift that was never identified before. This phenomenon does belong to the family of topological velocityations because it builds on the existence of quantized Wilson loops associated with topological invariance and crystalline symmetries. And so this phenomenon that we have revealed here really involves a very rich interplay of quantized Wilson loops and non-Abelian bary curvature and something which is of course, very interesting for experiments. This effect is very simple effect that we have identified here can actually be used as a probe for non-Abelian geometric and topological band properties. I thank you, Saru. Thank you very much. Thank you. I would like to use this time, there's many time for questions, if any. Probably I can ask myself because I'm curious about the last part. So do you have an explicit proposal for experiments that rebuild an un-Abelian character? Or you say it's just the fact that you get this fractional, I mean that just winding over multiple periods is enough. Right, exactly. So the fact that you find that the association is exactly given by an integer at times the bare block period which is associated with the structure of your brilliant zone. This fact alone is already a manifestation of the quantization of the Wilson loop. I'm just mentioning because in fractional pumping where you have, you know, related things one can imagine to just to have a combination of paths that shows that pump charge does depend on the way you're ordering the path because it's not sure or something. So I was wondering if something similar one could event here or not. Yes, I see what you mean. So here there is no notion of non-computing operations. Yeah, exactly. Yeah. You do this experiment and the only way you can explain it is through those non-Abelian structures. You need the non-Abelian curvature to see this whole drift. Which, and I remind you that here there is time reversal symmetry. So the only way we can actually have this is through the non-commutativity of mirror symmetries and therefore the existence of a non-Abelian very curvature. And so I don't remember the end of my sentence but that's essentially it. Yes, you see, you made this experiment you see this associating the whole drift and that's a witness, a smoking gun of the entire non-Abelian structure both associated with the Wilson loop and the non-Abelian very curvature. Thank you. Yes. More questions? No? Ah, yes. On that, go ahead. Hi, Odette. Hi, Nathan. Good to see you. It was a great opportunity to say hi again and thank you very much for the very nice talk. Yeah, it was pretty fast, right? Yes. I have a question regarding the non-Abelian contribution. So effectively in the model that you're using it's what the purists would call a pure quadrupole guy that doesn't have any other contribution from lower polarization creatures. There are these mixed guys that have non-quantized quadrupole and some polarization contributions. What do you expect to see there? Okay, so I guess the question that I would ask you would be what are the symmetries in those systems? So here the effect that I'm showing here really relies on those special mirror symmetries which are non-commuting. So I think that what you have in mind now is a system which do not present those symmetries. No, it has other ones with some non-symorphic ones. Yes, yes. So I mean, yes, I would have to dig into the model but I don't have a generic answer to that. Okay, maybe I'll holler and we chat. Sure, but indeed here I really take this opportunity to emphasize the fact that the topological invariance that we have in our quantized Wilson loops are not the same as the topological invariance which are associated with those quadruples which are associated with those Zac phase and the Vanier bands and so on. So here they are very different of a very different origin. So losing the quadrupole, I think shouldn't be too dramatic with respect to that. However, we do need the mirror symmetries to define our topological invariance. That's really crucial. I hope this answers the question somehow. Thanks. So no more question. Thanks a lot, Nathan. I also would like to thank all the speakers of this morning. So we have a lunch break and I remind you of the colloquium by Frank Wilczek connected to this conference but it will be on a different Zoom ID. So I will see you at the colloquium and then for the afternoon session. So have a nice lunch, rest and see you later. Bye-bye.