 Thank you very much for the nice introduction and for the invitation. Probably most of you are aware that there is currently an ongoing revolution in condensed matter by characterizing states, by characterizing solids, by topological invariance, or topological quantities and parameters. And I'll give you an example here. For example, in the field of two-dimensional topological insulators. So imagine this is the bryon zone. The bryon zone is periodic. So you can wrap it topologically to a donut. And on each point on this donut, there is a k-point. On each k-point, there is a Bloch wave function. And the Bloch wave function is not unique. It has a gauge invariance, or has a gauge freedom. And this gauge freedom is basically from 0 to 2 pi. And basically, you can go now, you call that a fiber bundle. And you can make now a tour in this two-dimensional bryon zone. And you can ask yourself, what is this fiber bundle doing? And then you see, you can either have a ribbon here. The face is always the same. Or for example, the face can be twisted. Say for example, from 0 to 2 pi, or from pi to 2 pi. And this, if the electronic structure behaves like this, we call this object trivial. And if it looks like that, we call it non-trivial. And so we have basically trivial insulators and non-trivial insulators. So here, this would be the topology of the Bloch wave function. Here would be trivial. And this would be the untrivial one. So you walk in this bryon zone and you have a facelift. And you have a different classification of these two different Bloch wave functions. One has this topological classification, z2 equals 0 and z2 equals 1. And basically, if you have now a topological insulator, which is non-trivial inside, or you know your vacuum is trivial outside, the only way that a wave function can go from inside to outside is by ripping off your Möbius band and connect it again on this lease finally to these two dimensional surface states. So you cannot avoid them. They are topologically protected. And the interesting aspect of this topological meta is with these topological quantities, there are also response quantities involved. And these response quantities are basically dissipation less edge current, edge states, which have dissipation less edge current. And these are basically to one spin hall effect. And this is not only two for two dimensional topological insulators, it's also two for three dimensional topological insulators. You see basically an insulator, but here you see the surface state, and this is the raccoon. And as I said, the electronic structure causes non-topological invariance, non-trivial topological invariance. G are connected to the barrier phase. The barrier phase is then connected to these topologically invariance. And these topologically invariance tells you whether the topological meta is trivial or non-trivial. And in this way, for example, you can have also the anomalous hall effect. You could use spin currents. And these spin currents are able, for example, to exert a torque on magnetic nanostructures. And you can switch the magnetization by these anomalous hall currents. And basically, what I explored so far in this lecture is basically the momentum space. Of course, we have a real space. We have a magnetization space. We have a time domain. And basically, you can imagine that in this 10 dimensional space, you can have an enormous amount of invariance, which are mostly unexplored. And I give you one other example of an invariant, which is in this magnetization space. And that is basically the chiral magnetics germinals. So think about the ferromagnet, or the magnetic moments of all atoms aligned in the same direction. They are collinearly aligned. And these are these red things here. And then on this ferromagnetic background, we call it also the vacuum state, you have basically precipitate two-dimensional localized orbitals, or in three-dimension strings, two-dimension strings of finite size, which behave like particles. So if you zoom in, they look like this. So this is the ferromagnetic background. And then you have basically a two-dimensional particle here of finite size. So it could be between 5 to 200 nanometers, where you have a winding magnetic structure. And this structure is stable, because the ferromagnetic background is topologically trivial, but this curbion itself is topologically non-trivial. And the non-triviality, you see as following, you have the magnetization at each at a certain spin pointing in a certain direction. And then you can make a projection, a stereographic projection, from this curbion in two-dimension to a sphere. And the important thing is to cover the entire sphere. So for example, this spin is the north pole. This spin here denotes the south pole. And you cover the entire sphere. And there's a topological number involved, this topological number q, a topological charge, which you can calculate by the curvature, basically of the magnetization on the sphere, which tells you how often you wrap this sphere, how often this magnetic structure wraps the sphere. And it turns out that topological charge plus minus 1 is the one which has the lowest energy among all non-trivial topological charges. This also exists in nature. Here I give you an STM image. You see, for example, here a ferromagnetic state. You see some spiral magnetic structure. You see this biomagnetic structure in this image here. So your magnetization is spiraling. And then you apply a magnetic field. And you see in this magnetic field, the skewmeans are precipitating on basic covering at the entire sample. And this happens typically at interfaces. For example, here iron iridium, manganese tungsten, collidium iron iridium. But it happens also in three dimensions. For example, for the P-20 alloys, iron germanium, manganese silicon, iron cobalt silicon. When you see the same experiment in using Lorentz microscopy, you basically have here a spiral state. And if you apply a small magnetic field, you raise the magnetic field, you see the skewmeans are precipitating from the skewmean lattice. The same you can do instead of changing the magnetic field. You can also change the temperature and keep the field. And you see, basically, that the skewmeans are emerging. And you have different techniques to probe these skewmeans. So if you think about topological arguments, all these topological arguments work on smooth functions. So basically, the people work in continuum field. And you have here an energy expression, we call it a micromagnetic model, of these magnetic structures which I have shown you. And so basically, this is what we call the exchange interaction, this is the magnetic anisotropy, and this is the magnetic v field. But you can get only these two-dimensional localized states, these two-dimensional solitons. If you have here this interaction, which has a gradient term, we call it the chiral instability, or the chiral zimity praying. And this interaction is called the Jalochinsky-Maurier interaction. On this Jalochinsky-Maurier interaction, and that is typical for many of these things, you can only have, if you have spin orbit interaction, and you need a solid, which has a broken inversion symmetry. For example, the interface or a bike broken inversion symmetry. Of course, we have our DFT model, and we have to relate our DFT model to this micromagnetic model. And it is very convenient to use spin lattice model in between. So basically, we do a Miley scale modeling. You go from DFT to spin lattice model, and from the spin lattice model to the micromagnetic model. And our motivation is to get all these parameters, which entering here, the exchange, spin stiffness, spiralization, the magnetic anisotropy, Heisenberg exchange interaction, the Jalochinsky vectors, by total energy calculations. And we have implemented this in our Flare Code. Usually what we do is we take spirals, magnetic spirals, insert magnetic spirals in our code, and then we relate the total energy of the q vector of the spiral to these properties here. For example, the spin stiffness is given as the second derivative of the total energy with respect to this q vector. So it would be the curvature here. When the spiralization basically is the first derivative, so it would be the slope here. And that is to get the constants of this micromagnetic model in a similar way you can relate these constants to the Heisenberg model, the J's and the D's. They are related to each other. Sometimes you want to probe the electronic structure of your Skiumion directly. And for this, you have to calculate basically the entire Skiumion in a unit cell. And for this, we developed a code which is highly parallelizable. And you see here, this is the KKR nanocode. Basically, you see the code scales linear, as a big linear scaling. Here, I'll show you for the Luginq calculation with 250,000 atoms and scaling over 30 racks. So in general, what this spin orbit in direction does for you is remember, if you have a time inversion symmetry and space inversion symmetry, then basically you know that you have a degeneracy. At each K point, the band structures occupy twice, once with spin up and once with spin down electrons. But if you have spin orbit in direction and you have at the same time a broken space inversion symmetry, then the Kramers degeneracy still holds. But at a given K point, the band's degeneracy is lifted and spin up and spin down for a certain K point has a different eigenvalue. And therefore, the electron is fooled. It thinks locally there is a magnetic field, which splits the general states. And therefore, you have an effective spin orbit field. And this spin orbit field acts like a local magnetic field and changes your properties. And many of these spin orbit coupling properties are then listed here. Basically, you have a lot of spin orbit coupling phenomena related to magnetism and the lifting of degeneracies for the orbital moment, topological orbital magnetic moments, magnetic anisotropies, the Jarder-Chinsky-Muria interaction. Then you have effects at non-magnetic samples, such as rush by effect and Dresselhaus effect. The topological insulators and wild semi-metals are a consequence of the spin orbit interaction. So you have spin relaxation phenomena. You have the anomalous Hall effect, the spin Hall effect, spin orbit torque, spin orbit Hall effect, quantum spin Hall effect, quantum anomalous Hall effect. So there are enormous amounts of transport interaction, transport effects, exchange interactions, all related to this relatively small quantity, which is the spin orbit interaction. I mean, the same as a very small quantity, you mean immediately you need a lot of k-points, you need precise calculations. And due to the spin orbit interaction, of course, you lift all the degeneracies. That means you need a bigger Hamiltonian. Hamiltonian is not anymore real. It's complex, and so on. So the calculations become quickly time-consuming. And this aspect which I mentioned is part of a bigger field of magnetic materials and spin electronics. So basically, the magnetic materials enters the area of energy. You need permanent, for many things, you need permanent magnets. The area of storage, for example, hard disk drives. And now it's also part, the M-RAM is not also part of the devices. So it goes into the L2 cache of the mobile phone. And magnetism has also the future in magnetic caloric materials. This depends very much on the energy price and in the area of the internet of things, in particular due to magnetic sensors. So basically, behind there is a big field, but I'm concentrating here on the spin orbit interaction. And I would like to give you one example where the proper treatment of the spin orbit interaction is important. And this is, for example, the band structure of topological insulators. So basically, most of the calculation and the field of topological insulators are done by using LDA. And basically, do LDA and then spin orbit interaction. And the better calculation is already doing LDA, then do GW on top of it, and then you do two types of LDA calculations. One, LDA calculation without spin orbit interaction, and then you build up your GW. And you do an LDA calculation with spin orbit interaction. The difference between this band splitting between the LDA, two different LDA calculations gives you a difference, and this you add on top of the GW calculation. These are called GW plus R. And there's a difference if you implement it completely correctly. So you do an LDA calculation with spin orbit interaction and add the GW on top with spin orbit interaction. And this I call spin orbit SOC at times W SOC. And this is about 10 times more time consuming. And here I show you an example of the differences. Here at the example of Wismuth-Tellerite, you basically see here an LDA calculation, this green one. And you see the GW calculation. Both of them is LDA plus SOC and GW plus SOC. But if you do full GW plus spin orbit interaction, you see the change between GW plus SOC and GW SOC. W SOC is basically of the same order. So these corrections are very large and have to be included correctly. And here you see basically a calculation for Bismuth-Tellerite. You basically see an LDA calculation with this very famous state, DERA point. And here you see a GW calculation with the DERA point. First of all, you see in LDA, Bismuth-Tellerite is an indirect bandgap semiconductor. And you see in GW, it's a direct bandgap semiconductor. When you see the DERA point, it looks totally different. How does it compare to experiment? So we look at only this insert. And then you see here a variety of photoemission results. They all agree to each other. They are slightly different, because the doping of the crystal is different, the Fermi energy is a little bit different. The resolution of the experiment is a little bit different, but in totally they agree. And here is the comparison between LDA and GW. And you see this DERA cone agrees much closer to the GW calculation than to this LDA calculation. Let me come to a second example. This is a Scyrmian design. So if you want, Scyrmians are nice, but if you want to have Scyrmians in a framework of a technology, then these Scyrmians have to have certain properties. Yeah, for example, they should be in thin films, maybe in the order of three layers, four layers, not too thin, not too thick. They should be not too small, but also not too large, in the order of five to 10 nanometers. They should be existing above room temperature and close to zero field. They should fit the field of spintronics. That means you need to know how to inject them, to transport them, to detect them, to manipulate them at reasonable fields and currents. You have to read out them, you have to read out. You have a read out mechanism. It should be fast, moving and energy efficient. It should be also possible for available for logic operations. And people like to work with metallic systems, so it should be part of the metallic mechanism. Coming back to my micro-magnetic models, I have a parameter A, I have a parameter D, and I have a parameter K. So basically I can modify my materials and the electronic structure such that I vary this AD and K such that the Scyrmians, which come out, fulfill all these properties. I have to say we are not there yet. But we are coming close to it. And the same on a more microscopic level, we can use the parameters JD and K of the spin lattice model. But here I give you an example. This is Manganese-Tangsten. Manganese-Tangsten, the experimentalist, tell you that you have this wavy pattern here and this wavy pattern might be a result of a spin spiral. This gives you the idea that you have maybe a Geologinski-Mollier interaction, which is very strong. The Geologinski-Mollier interaction wants to have a chiral symmetry breaking. It doesn't want to have a ferromagnetic state. So as you can imagine, you have here a Geologinski-Mollier interaction. So then mathematics tells you if you have a Geologinski-Mollier interaction, then there exists a magnetic B field such that you will form a skirmjall. So therefore, your only thing of what you have to do is now to recalculate the system. You calculate your A, D and K. And then you see, you put it into a phase diagram. You do more on the color and get the phase diagram. And then at the end of the day, you know also your B field, which you have to apply to get a skirmjall. And that's basically what we did. Basically, here you see a spin spiral calculation with outspin orbit interaction. So basically you expect here a minimum. You don't have a minimum here because what turns out is that these J's here have positive or negative sign and your exchange leads you to a frustrated magnetic structure, which is a spiral in itself. But then the Geologinski-Mollier interaction selects one of these spirals, here the left rotating one. And this left rotating one has then a lower energy. And then by minimizing and taking derivatives and things like that, you get all the parameters. You put it into a Monte Carlo code. You get the phase diagram. And here you have a phase diagram of the magnetic field versus temperature. You see here you have a spin spiral phase, which you saw in experiment. You have basically here a skirmjall lattice phase. You have here the isolated skirmjall phase. And here you have the saturated state, which is the ferromagnetic state. Here you see for example the typical skirmjall lattice. And these skirmjalls here are pretty small because they are small because you have here a frustration. And this frustration determines a length scale. So we have finally small skirmjalls. Yeah. And these small skirmjalls are basically, you can achieve them, but you need magnetic fields in the order of 30 teslas. But this is not exactly what you want to have in a device. So you have to get rid of these teslas. And therefore we have the following suggestion. What we do is an interlayer exchange structure. So you have the tungsten. You in slide a certain cobalt layer here, which gives you, which is ferromagnetic, which polarizes the other tungsten material. And this polarized tungsten material exerts an internal magnetic field on this skirmjall active magnetic manganese layer. And the question is now, how do we tune the thickness of the cobalt layer and the thickness of this number of tungsten layer such that we are becoming in the right ballpark? And this we have done, but they're observing if the cobalt layer becomes thicker, we realize the magnetization becomes in plane. This is very bad. We need out of plane magnetization. And therefore we have to slide in another platinum layer here of certain thickness such to stabilize the magnetization to be out of plane. So we have a couple of parameters. We have the thickness of the platinum layer, the thickness of the cobalt layer, the thickness of the interlayer spacing. So this is already a three-dimensional problem. And for these three-dimensional problem, we have to sort out all the parameters and then get into the phase diagram. And here is selected two systems. So they say six layers of tungsten, four layers of cobalt and one layer of platinum, or seven layers of tungsten and four layers of cobalt. And then you see, you move finally along this line where you come into this curing phase when you're becoming, if you change your cobalt thickness, you move it also in this isolated curing phase. This is one example. The other example so far, you have in these micro magnetic equations, you have the exchange, you have the Jaleczynski-Moria in action. So maybe you want to do something at the exchange and the Jaleczynski-Moria in action simultaneously. So if you look at this exchange, there are some materials like iron here where if there's small relaxation, small changes, you can vary the exchange in the action already back quite a bit. And this we make use to, so we have a one substrate iridium which has a strong spin orbit in action. So iron will polarize this iridium and has broken inversion symmetry. This will produce the Jaleczynski-Moria in action. Here, palladium has a large susceptibility, iron will polarize palladium and manipulates the exchange in the action. On the disc, two interfaces now, on varying these two interfaces, we can modify the exchange interactions. Here, the example of palladium, two layers of palladium, FCC palladium, HCP palladium, changes this effective parameter chain. And this we can continue. We cannot only take thin films, we can also take multi-layers and we can also work with rhodium and different concentrations. And then you come into a ballpark here where you can design your material such that you have different exchange in the action. And at the same time, you get different Jaleczynski-Moria in the action which I don't show you. So you have possibilities to tune the exchange and you have possibilities to tune the Jaleczynski-Moria in the action. And this way, you can construct, again, a phase diagram and you see at which fields you get which magnetic phase, the spin spiral state, the squirmyon lattice phase, or the ferromagnetic phase. And the last point I would like to mention is the detection. So if you want to detect the squirmyon, electric detection would be the best. For this, you have to calculate, somehow, the conductance. For calculating the conductance, you need the electronic structure. So you put your squirmyon as a super big impurity into a Green's function code embedded in the ferromagnetic state. When calculated, basically the electronic structure of the winding magnetic state compared to the ferromagnetic state and the chairman is waving, so I go very fast. And therefore, you can develop a number if you call the tunneling spin-mixing-magneto-resistance ratio which is the difference in first approximation to the electronic structure in the vacuum of the ferromagnetic state of the squirmyon state divided by the ferromagnetic state times 100 to get to percent. And then you see, basically, here you can vary this tunneling spin-mixing-magneto-resistance by 20% depending whether you have a squirmyon or not. And this is totally sufficient to detect the squirmyon. Now, with this, I'm coming to an end. So the field of spin-orbitronics, the field of quantum material is a very rich one. We look at spin textures of a neuro-inspired computing, ultra-fast and interferomagnetic spin-tronics. A very big field which is coming up is three-dimensional nanoscale magnetic textures and dynamics. In America, they build new synchrotrons to detect it on a scale of three-dimensional 20 nanometer structures, so magnetization solitons in three dimensions. We look at emergent complex phases in complex 10-dimensional topologies. And, of course, we have to find the appropriate materials for that. And this, of course, we would like to do using materials discovery and functionality discovery. And therefore, for us, the MAX project is extremely important because the phase space is huge, the number of k-points and the set-ups, the number of atoms, the sizes of the systems are huge. And therefore, for us, it's very important that our codes run on supercomputers, and it's also important that we get a systematic screening of the solids. And let me thank a few people. First, the FLIR group, which is composed of Daniel Wortmann, Gustav P. Meyer, Gregor Mehalichik, Juliana Alexieva, who are here. And then the Kiki Anano group, which is led by Rudolf Zeller, Uman Kovacic, Marcel Bornemann, Paul Beimerster, and Jack Pleiter. And we profited a lot from the Vani 90 code and these developments, and we are funded by the Magic Sky and the MAX project. Thank you very much.