 The first stop is by Pauline Mediador, going to tell us about the chiral stop and the magnetic optics effect from magnetic dipole. Okay, thank you. So, um, today I'm going to show you the results of a very simple experiment to realize a new way to create a spin orbit coupling like the Lachinsky-Morilla interaction in a magnetic system. So I will start with some preliminaries. Just remind you about chirality, a chiral system is such that all the mirror symmetries are broken or are absent, and that fact doesn't change in the case that you have time reversal. In the, when you have a magnetic chirality, that means that the magnetic degrees of freedom of your system are chiral. Okay, so chiral states can be like helicoids or any winding texture like conicals. And in magnetism, actually, in magnetic systems, they are very, they can be frequently found, in particular in rare earth metals, with centro-symestrical stylographic lattices. Nowadays, these magnetic systems with chiral structures can be also in general in two dimensions, so it's something that has been happening in the last decade. So why chirality is important? Why we want to have a chiral magnetic system or some chiral structures in a system is because interesting things happen, for instance, if some electrons actually pass through some of these winding textures or chiral textures, they will fill an artificial gate field and they will then realize a very phase and then that can give rise to, for instance, the whole effect. Right, so, and there has been some large topological whole effect from two-dimensional block type chiral spin textures. And in those cases, the responsible for those chiral structures have been the Jerochinsky-Morilla type of interaction. So now we focus on that one. So this particular spin orbit type of interaction is able to stabilize winding textures such as skirmish, chiral domain wall, or helical state, conical state, and the such. What happened then is that the magnetic moments essentially periodically wind around some wave vector and the wavelength of these periodicities is determined by this Jerochinsky-Morilla interaction and the typical exchange interaction of the system. So it's the confidence between the exchange interactions and the Jerochinsky-Morilla interaction which sets the wavelength of these textures. So we want to be able to control Jerochinsky-Morilla interaction, just be able to create one in a controlled manner and to tune it at will, because if we are able to do that, we will be able to control also the sense of rotation of these chiral textures. So the strategy up to now has been combined, the broken invasion symmetry with a strong spin orbit interaction. So this is what has been done. And in magnetic systems in particular what you do is that you sandwich or you couple two magnetic layers through a metallic layer in order to create this spin orbit interaction. So this is the approach we propose. It's a very simple system. It consists of a zigzag chain. The chain is made out of dipoles. This is a picture of the experiment. The zigzag chain can be seen like one dimensional lattice with two sublattices. So we call this sublattice, the sublattice P, because it orders usually collinearly. And the other sublattice we call P. The difference between these two sublattices is that they have perpendicular DC planes of rotation. So dipoles in the C sublattice, they can rotate in the XZ plane, while dipoles in the P sublattice, they can rotate in the YZ plane. So what we do, we place the dipoles here. They are made out of neodymium. So they are hinged in this PTFE plate. And they are able of rotation. This is the radius. They measure about 1.2 centimeters. This is the lattice constant for the two sublattices. This is the saturation magnetization of these neodymium dipoles. And this is the mass. So we place our dipoles, which are now here. It's easier. So we, they stay still. And these ones, the P-dipoles, they are mounted on a translation stage in such a way that they can move. So if you want the lattice constant of this system, delta is fixed, but the vertical lattice constant, which I call L in units of delta, can change. So these, again, are the easy planes of rotations of P sublattice, P sublattice, okay. And because they are dipoles, they couple through a dipolar interaction. The G here, storage all the parameters of the system, mu zero, the permeability, the magnetic intensity of the dipoles, the lattice constant. So this is the energy scale. And here we have the magnetic moments of any dipole at side I, which can be in any of the two sublattices, and this K can be another dipole or the same sublattice. And this is the vector that joins the two dipoles, and this is the distance between the two of them. So in order to describe each of the sublattices, we define the magnetic vector. So it's defined in terms of a theta angle, which can variate. This is this variate like that for the P sublattice and here for the C sublattice. And the phi angle is going to be fixed in order to have the easy plane so phi sub C is going to be zero and phi sub P is going to be pi over two and that set the easy planes then and we have a magnetization vector for each of them. Important experimental details, the lattice has disorder. There is some geometrical disorder in this phi angle actually, which is about 0.005. We take into account this disorder, but there are these additional disorder this friction, and the friction for each of the, for the dipoles for all of them is not exactly the same so that will induce some some imperfections in the dynamic. So, what we do is the following, we use dynamic in the system, just by the 18 this lattice constant, which I call L remember this is in units of Delta so everything is non dimensional. This lattice is kept constant so if the two sub lattices are very far away. So this L for instance is about larger than one. Okay. And what we see is that the collinear sub lattice, the C sub lattice, settled in a ferromagnetic collinear state, while the P sub lattice, it settles in a, in a, is in a parallel state. The parallel state is anti ferromagnetic, but the magnetization vector here the black point is the South Pole, for instance, and the white pole is the is in white. So you can see that there is a modulation here of the position of the dipoles in the piece of lattice. So we approach P to our C, we reach a regime in which the system can have two possible magnetic configuration, these are two meta stable states. And this is the state which I call a F square in which the two sub lattices are in the in the XY plane. But both of them are anti ferromagnetic. So the C sub lattice is collinear anti ferromagnetic, the piece of lattice is parallel anti ferromagnetic. We also find this other phase in which the C sub lattice now is collinear ferromagnetic, and the piece of lattice is also anti ferromagnetic parallel. As we decrease the distance between the, the, the two sub lattices. So this one is this the magnetic state which is going to be selected, which is here. And finally, at very low L these two super lattices settle in an anti ferromagnetic parallel state, pointing into the C direction. The first one is going to be, for instance, plus C, the other minus C, plus C, minus C, et cetera. Here, we can see in the experiment there are some imperfections and this is due to the friction. It's very difficult to stabilize the lattices when these dipoles are very close because the force between them is very, is very big. And we see some imperfections there. So we record the full process so we start receiving the piece of lattice from the C one. And then once we reach L equal one or two 1.5 we start approaching the piece of lattice to where the C sub lattice once again so this define a loop. So this is, for instance, the magnetization along the x direction, which is determined by the C sub lattice. And here we see the distance between the two sub lattices defined by L. So this is how this what happened with the magnetization of the two sub lattices become a part, and then this is what happened as they approach. So here we have this phase, the anti ferromagnetic phase where all of the dipoles are pointing out of the XY plane, they are all pointing into the C direction in an anti ferromagnetic way. Here we have a phase where all of them go into the plane, so it's going to be something like a spin flop transition from here to there. And here we enter this metastable regime where we can find the Faf phase where C sub lattice is in a collinear fashion and the piece of lattice is in an anti ferromagnetic state, or the one which both of them are anti ferromagnetic. These are also phases which are in the XY plane, and then we reach this twisted phase, which I am going to show. We can also do the same with the stagger magnetization along the x direction. Here you can see like around this point is going to be the transition between this out of plane and in plane magnetism. And here we have the magnetization along the C direction for the piece of lattice and the C sub lattice. So this is the way how we induce dynamic in the system, and this is what we are going to study. In order to see more clearly what happened, because in that way we can also avoid friction. We perform a molecular dynamic simulation so we have the equation of motion for every dipole in the system. And this is by the polar angle theta. So, so, change in time of this angle are going to be, this is the initial moment of if each dipole, they are going to be defined by the internal torques and some dumping. Okay. So the theta is extracted from experiment and T here can be seen like why divided by the with and D is the velocity at which we move the piece of lattice from the C sub lattice which is constant and we define that in the experiment. So these are the internal torques, the internal torques are going to be equal to the magnetization of each of each dipole cross the internal magnetic field in the system, and this internal magnetic field is going to be the derivative of the total energy of the system. Okay. And so again, we can see what happened as L increase. Okay. Here we have a very small L this anti ferromagnetic phase where all the, the dipoles are pointing out of the plane in an anti ferromagnetic fashion. Here we have something new from experiments we can have a metastable state where we have these anti ferromagnetic state and we have here an anti ferromagnetic square. Here we have the metastable phase that we see also in experiments. And here we see the twisted face once again, which, as an experiment consists of the C sub lattice in a collinear for magnetic fashion, and the piece of lattice in this, in this twisted configuration. Here we study then the magnetization dynamic of the medics. Okay, it's very noisy at very small distances, because you know the force between the dipoles is very large the torque between them. We see here, the metastable phases also, and we see additional metastable phases on the, the critical distances L for which the transitions happen are very close to experiments. So, now we want to understand what is the origin of these twisted face that we have here. So we just take the bipolar interaction. And it was in the first slide, and we separate the energy between different conclusions. So essentially, we take the dipolar interaction, we include their, the magnetizations of each of the dipoles we massage the questions without any, any approximation at all, and the dipolar is actually separate into this. Here again is the energy scale. We have a term here, which couple dipoles in the same chain. So we can call this the inter chain, a symmetric type of couple coupling, right, energy, or this here. So all these parts here accounts for, we call it UC and UP is the interaction between dipoles that belong to the same sublattice. So this store all the dipolar interaction between the C dipoles with the C dipoles and this store the interaction between all the P dipoles with the P dipoles where this coupling is just a typical dipolar coupling. So we can call this exchange in inter sublattice change. And then we have here another term. This term is again symmetric. And we also have a couple of dipoles among different sublattices. So we can call this an exchange and inter sublattice or inter chain exchange. And this J sub AK, and this J sub AK is even here. And of course it depends on the distance between the two sublattices, of course, between also the distance between the dipoles like this. And finally, we find another term. And we call a use of DM, because it resembles a spin already interaction in the sense that it couples to vectors by a cross product and is multiplied by, and is this is perform a performing adult product between between with another vector. So this is like spin orbit type of interaction. And it's the same as a yellow chain scheme or a type of interaction. So at this be ice to K is given here is a vector, of course, and it points along the Z direction and is given in terms of L and the distance between the dipoles. So, here we have the energy the dipolar energy for this system. Okay, which can be separated in these four terms and we can. So exactly which is the, the formula for, for each of them. So, now we, we see how can we change this coupling. So, I'm using the experimental data we have recorded the pollution of these dipoles during the whole time so we have the angle of all of them so we can compute the energy of the system this is the energy of the experimental system. We have the angle of L, the full energy. Okay, and we see it has a deep and then grows and then stays constant. So, here is the regime where the system is anti for magnetic square here is the regime where the system is out of the plane. This is the most stable phase, and this is the twisted phase. And we find the same thing. These are plots when we compute this energy when the system is receding when I don't know for that I will have to compute over. I haven't done that. So here I separate the different energy. So, in red is the energy between dipoles that belong to the C sub lattice in black is the energy, the symmetric energy between the two sub lattices in blue is the anti symmetric energy between the two sub lattices and P is the symmetric interaction between dipoles of the same sub lattice P. So, they use a P energy of the P sub lattice alone is very small so it's a plot here is one order of magnitude smaller than all the other contributions. So we see that the collinear the the energy of the collinear state goes up state constant and then goes down but remains fine it's large L. So the general change camera interaction actually zero for the when L is zero, and then it gets, it gets maximum, some value about 0125, and then goes up and then goes down for them in magnitude and then it reached zero in the twisted phase. Also, the symmetric interaction between the two chains start being finite, and then it becomes zero so essentially in the twisted space we see that the yellow cheese kemoria energy and the symmetric exchange energy. They become zero and the only contribution comes from the collinear state because this one is very small and same thing happened. In the case of the piece of lattice so it's useful to have this because now we can say we can, we know which regimes, which couplings are important, and also having this we can see up to how many neighbors, interactions are going to be to be relevant. Again, and this is again the yellow cheese kemoria coupling. Okay, now I call instead of L I call it why because we can imagine we have a long system right. So here is the yellow cheese kemoria in terms of x the position along the zigzag lattice and L the distance between the two sub lattices. And we see actually that there is a place where this interaction is this coupling is going to be optimized. Okay, so, and this is going to be in terms of the distance between the dipole so for two dipoles which are x apart. So the optimum distance at which the two sub lattices should be placed in order to reach a maximum anti symmetric interaction is given by this. So, when four nearest neighbors, right dipoles which belong to different sub lattices x equal is equal to zero and this gives a one divided by four, which coincide with the deep in the in the in the yellow cheese kemoria energy. So we can assume we have a very long chain, and we can integrate out along the, the chain, right, and we get an effective to effective coupling, which may be relevant. This is the yellow cheese kemoria interaction. Okay, that one chain is one. And this is the the symmetric interaction, right, and we can see that while the symmetric coupling of course is maximum. When they are the closest the closest the yellow cheese kemoria interaction is not it has an optimum. So it's not. Yes. So we can assume lattice constant between the two sub lattices that will optimize the skill interaction. So we can see that the system is not stable here. This is the energy landscape. This is the angle of the angle of the collinear sub lattice. This, this energy has been they have been separated along the y direction for clarity sake and those are computed for different values of L, and we see that about the critical L that we saw when the system evolved from the anti ferromagnetic square to the meta stable phase. It actually about this L we see that the, the, the barrier is going to be almost flat. Okay, so what's that this barrier between the two meta stable state. This is fed by a block domain wall. In order for the collinear chain to evolve from a ferromagnetic into an anti ferromagnetic state and visa versa block domain wall need to be created. Okay. The block domain wall. The energy to create one of those can be compute. We know that the yellow cheese kemoria interaction is going to be important. They said in the previous slide up to a second neighbor nearest neighbors so we can safely neglect all of the other terms. And we can compute this difference between energy of a pristine system and a system with a block domain wall for the case of states that are at the border at the bulk and near to the border of a chain. And also we need to take into account that there are some frustration. There are some states that are going to be energetic really very costly. For instance, here we come. We are approaching the two chains. This one, even so it can be a weak link won't won't change because it will create a lot of dipolar energy. So taking into account these things very easy to compute the width of the loop. So we can compute this, the width of the status is moving the system, and it's going to be a function of this distance between the two sub lattices. So, because we have a spin, or the type of interaction, we expect to have a spin current in the system. So we can, we can compute the spin current from the Heisenberg equation and adapted to the classical system at hand to give this question here, where you is the total energy of the system. And of course, this one is the internal field created by the dipolar interaction with all the other dipoles. And so, this is a torque. So the torque. Okay, it's essentially the current of the system, the torque on the spin, I is going to be the the flux, okay, of angular momentum transfer to that spin. Okay, that's so I'm having this formula for computing the spin current or magnetic current. We can write the dipolar interaction. Okay, in terms of interaction matrix is a quadratic form. So we can always write it like this, this I here is a matrix which contains all the geometrical aspect of this system. Okay, and writing this you in this manner and using this formula we can compute the spin the magnetic current of the system. And we see that this ABC are X, Y and Z components. And we see that in the, the, the, the current of the system, essentially it's one, this is the data leader operator so it's going to couple magnets, along, along different directions and the responsible for coupling those magnets in this is going to be the yellow team's committee interaction. So, computing this, we find a spin current along the city direction. The spin current is proportional to this yellow team's chemical interaction coupling which we know exactly what these, and it couples spins in the two sub lattices. The components along X of the collinear or C sub lattice and the components along Y of the, of the piece of lattice. So, we have a magnetic current in the system, which is induced by this yellow team's chemical interaction. So in the anti ferromagnetic square phase, this is the planar phase where two of the, the two sub lattices are in the XY plane and the two of them are anti ferromagnetic. We can write this spin current in this manner. Okay. And, and remembering that the in the anti ferromagnetic square phase. Okay, we don't have a coupling, a symmetric coupling between the, the, the C and the P sub lattices, we have that the, the full energy of the anti for magnetic square phase is given by, by this formula here. We can just write in it in a different way, where this J sub AKPP is that is just the change constant in the same chain P, and this J sub AKCC is minus two times that. So in that, we can also write, we can take this. Okay, and write the different, the, the energy of the anti ferromagnetic square phase in this, in this manner, where this term, this cheese AK ID cosine of ice of AK is given by this term here. And using this energy, once again, we can compute the spin current using the previous formula, and we find that the spin current actually can be writing in this way. Because we have this fellow here, that means on this is proportional to the yellow cheese Kimoria vector. That means that the yellow cheese Kimoria vector act in this case as a vector potential or a gauge field, which is associated to the magnetic current in the system. So we can go a little bit further. So, we have them are an effective yellow cheese key Maria interaction right, we can, or can be exact or can be effective wherever this is the little vector of the piece of lattice. Again, this is the yellow cheese Kimoria vector which we know a selfie what it is. So, so we can define a magnetic flux. And this magnetic flux is going to be equal to the yellow cheese key Maria field, which points along the x direction dot surface a so we have here our system. We can define a loop here is which has a length equal to two pi y, for instance, and this loop has an area, which is pi y y x, and points along the x direction. So, this loop lives in the white z plane. Right. And we also have the yellow cheese Kimoria field which point along the x plane, right. So, we have the x direction so we have this magnetic flux, and we can dedicate this magnetic flux by far a law with, with, with, with fine FM. This is a yellow cheese Kimoria family, right, which is going to be the fine, which is equal just derivate in this we have the expression for that. This is the two pi V, V is the speed at which the two sublattices are moving apart, why, and this is multiplied by the effective there just give me interaction at linear order in why. Okay. So this FM is going to be proportional to the integral over a close loop on our electric field, right, so we can just replace what we found here, and we can obtain an electric yellow cheese Kimoria or an electric field and effective electric field due to the key to using the system, which is given by, by this expression. And, and then we can find, we can write the, the, the energy of the system in the anti for magnetic square phase. Now we replace this H in terms of E that we found in the previous slide, and we have an energy of a magnetic system which has an E vector there, an electric field vector. So that means that if we just derive the energy of the system respect to the vector, we identify an electric polarization. So here we see a manifestation of the magnetic electric effect in the system product of this effective yellow cheese Kimoria interaction. Finally, we explore a little bit that we didn't texture. So I will go a bit fast here so here I am changing L, L equal almost two then one on 0.9 and 0.8 and we see this is the magnetization of each of the dipoles in the system. This is actually that it has, it goes up down, this is along the y direction, this is along the z direction. Okay, and so we see that actually this texture evolve. I am moving the two chains apart. We define an ill vector. Now we have this texture has is a field right, which depend on X and L will define the new vector. And we minimize the energy considering also that the yellow cheese Kimoria and the, the symmetric energy are zero in the regime of the wind in texture to find a solution for this texture. The solution is a soliton. So, essentially, what we are finding here is that the yellow cheese Kimoria interaction is able to stabilize solitons or twisted textures in the system. So if you come the solitons, they have conservation law that we have the continuity equation, right, this is the topological charge and the, the, the, and the topological current, right, and this, they fulfill the continuity equation. For the materials we can, we can assume that there are some crystal fields, which are going to induce dynamic of the system so we can study this Hamiltonian density now in terms of our nail vector here, and where we know that this a change is going to be proportional to J in the same sub lattice, and this is a susceptibility, which we know is going to be proportional to the yellow cheese Kimoria interaction. And here we have this K, which is going to be on an isotropic term, which is going to be proportional also to yellow cheese K and to the symmetric exchange on this M can be the transfers. So there's a magnetization vector. So we can. We know that the our texture on the transfer motion, they are conjugate variables. So they satisfy the, the was born bracket. So we can induce, we can find or figure out the dynamic of the system just solving the, the Hamilton equation, ignoring that thing for now, and we arrived to assign Gordon equation which has actually the solution. So it's actually a tone which is a domain wall which is going to have some width, which is going to be given by the ratio between the, the symmetric coupling in the same sub lattice on the anti symmetric and symmetric coupling between sub lattices, and it's going to have the same wave speed, which is also be proportional to, to the symmetric coupling in the same sub lattice and to inter sub lattice couplings. So, I finished with this. So the dynamic of at six aglattice of dipoles is induced by the intrinsic interfacial magnetic torque, which arise due to the interchange dipolar magnetic field. So the magnetic dynamic of all the magnetic also bubbles in the system is propelled by internalized interlier gap variations. We can think of that, like strain, for instance, which tuned intrinsic torque and interfacial that yellow team Kim or Ian the absence of any external driving force and induce an electric field and allows the manifestation of the magnetic electric effect. The fact that this wind in texture in the system is stabilized by the internal gene scheme or interaction. Thank you. So, I don't, I don't use, I don't use periodic boundary condition. I, I solve the molecular dynamics as it is for the same system with 37 dipoles. Maybe you already mentioned what up in the experiment what is the physical dipole. What is the what. Experimental figure. Yeah, what is what is the dipole made up of those specific physical dipoles. The dipole. Yeah, what, yeah. Magnetization. Saturation magnetization of each is about 10 to the six is there. Yeah, like what are those make it is what is that the individual dipoles. So those are, each of them is a dipole. It's a magnet. It's a physical magnet. It's a physical magnet. Yeah, this is a picture. It's a physical magnet, you can buy them. Thanks. Any more questions. You don't have the photo of this market. No, they are like this size. Again, we are theoreticians. We cannot imagine. They look like that essentially the same. No, no, they come like that. No, because this is okay, this is like the, this is like an acrylic plate where, and this is the magnet. This is a piece of graphite that allows the rotation of the magnet. This is painted with a pen like black to be able to when you record it to distinguish one part of the other. What. So this is the radio of this all of them they are cylindrical magnets. This is the length, and this is the lattice constant. So the distance between these on this. And this is the pressure magnetization of the neodymium, which is given on this is the mass, the mass of each of them. Since it's one day, do you expect phase transition. Why do you want to find. You're showing different phases so that's Yeah, but phase transitions are useful if you want to find something of the system that you cannot find any other way. I mean, it may have a phase transition, but I just see a face change on temperature is not important here. Right, because this magnet are massive. So temperature won't do anything to the system you can assume that this is done at zero temperature, right. So probably if there is, it may be a first order one because you have a hysteresis, right. First order. Usually another that's interesting. Any more questions. Thank you.