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So I come back, we have now proven that our algorithm, the Monte Carlo method, really works.

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We start from a randomly generated chaos of magnetic moments that are oriented along random orientations.

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And then after just taking one after another and randomly reorienting them and then accepting the change

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or not accepting the change, we can turn this into a nice ordered Ferrer magnet.

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So, of course, so far, the result was pretty much trivial, and we have just tested our method if

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it really, really works.

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So the thing is, our energy term is still pretty simple, and in reality, there are many, many more

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terms here.

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And as our troops and dipole dipole interactions that all lead to interesting phenomena and especially

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a hot topic in today's research and physics are non linear spin textures like domain volts, for example.

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And here I want to show you how you can generate those.

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So these are often induced by the so-called Lashinsky Maria into action.

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Or you could also call it as a metric exchange interact.

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And this corresponds to the last term here.

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And you see, this is such a triple product out of some vecteur de IGY and then the two neighboring

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vectors I and J.

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So it's a bit similar to the exchange interaction, but it does not favor parallel orientations.

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But due to the cross product here, it favors perpendicular orientations.

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So you see there is some kind of fight between this term and this term.

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We have here a parallel orientation that is favourite, and here we have a perpendicular orientation

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that is favored.

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And you can imagine that this can give rise to some interesting phenomena like nonlinear spin textures

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and these defectors.

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They are, of course, a bit difficult to explain now in this cause, and this is not really what this

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course is about.

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I will just tell you that these vectors are determined basically by the symmetry of the sample or more

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precisely by a broken inversion symmetry.

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So it sounds difficult.

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But for example, if you take two materials and put them on top of each other, then you have some broken

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inversion symmetry.

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And then at the interface of these two layers, you will have these challenges.

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Shinseki, Maria Vectors De.

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And in our example, we will consider a pretty simple case where these de I.J will always point along

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the bond direction connecting two of these atoms.

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So for example, if we consider these two, then the bond will go from here to here so that the I.J

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will be pointing along this direction.

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So, for example, we could call it the X direction.

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And so as we see here, we only have x minus x y minus Y.

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So we only must consider the terms we must calculate.

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The terms were the points along the X direction or along the Y direction.

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So here it is, gives us a term D and then unit vector along X times this and unit vector along y times

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this.

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And if you calculate the cross product, then you will end up with such a term.

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This is what we have to implement now in our new energy function.

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So since it's pretty similar to the exchange, I will copy this one here and actually let me switch

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off the magnetic field for now because we will now focus not on the magnetic fields, but on the DMI.

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So I call this energy DMI and energy DMI contribution.

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And here we call energy DMI my contribution.

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And now here we must just call the results.

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So since it's a bit difficult, I will write.

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The result is 0.5 times the times and then the neighbors right plus left plus up plus down.

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And we will now specify, right?

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Left.

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Up and down.

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And of course, we must also specify the strength, which we could say is 0.3.

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So typically, it's very, very small, much smaller than James.

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So in is one than this is probably something like zero zero something.

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But yeah, let's just consider a bit of a larger value here so that we see something in our result.

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So now for the individual neighbors here.

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So basically, we start from one moment am I and then consider all the foreign neighbors?

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So we have now four terms here and the term with the right neighbors.

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So we could say along the positive x direction would be magnetization and then x y zero.

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And then we have here the Y coordinate.

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So this would be index one times the neighbor and the Z component.

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So I copy this.

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Take the neighbor to the right.

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So here plus one and the z component to.

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And then we subtract the other term.

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So basically, I take this one and just right here, two comma one.

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So no one thing that we learned here for the exchange interaction was we need this modulo with the length

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so that we don't run into problems with the indices.

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So whenever we have X plus one or X minus one, I write modulo length.

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And so this is already our first term.

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Now, along the left direction, it's pretty similar.

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We have here just x minus one and X minus one.

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And now the sign is reversed because the vector node doesn't point from here to here, but from here

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to here.

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So and since the vector should always point along to bond, it's now opposite.

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And so this means we have to change the sign here minus.

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And plus.

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And now we only need the other one.

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So this one we have this time we have magnetization the Z component x y zero times magnetization x component.

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So does this zero.

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And then along the Y direction.

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So this would be y plus one modulo rulings and then minus with the reversed components.

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So zero and two and then the down direction was exactly the same, just with the reversed signs.

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And of course, not two plus one, but minus one.

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OK, and that's it.

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Does.

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Those are the four neighbors.

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We add them up, multiply them by D and zero point five as in the formula here and now, we must just

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include this in yeah, in the step exchange.

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No notice to exchange, but the step T.

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So here I already wrote of day three.

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We include the energy corresponding to the geologists De Maria into action, so it would be here.

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And since this considers neighbors, we need to factor two this time again.

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So energy, DMI contribution and here as well.

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OK, that looks good.

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And here we don't really need to change much.

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Just update two and three at interaction with magnetic fields and DMI.

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And I noticed one thing that I missed actually previously, but it didn't really matter because it's

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just optional just for tracking.

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I didn't update here the energy.

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So of course, here I must write

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that this is energy exchange, energy, magnetic.

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This is what I missed previously, but it didn't matter because we never used this energy list for anything.

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We only need the energy change in this one I updated and energy the my.

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All right.

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So I think I will play now a bit with the parametres and show you then a good solution, so see you

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in a second.

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So it turns out the first result already was very good.

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So I left the step number at five million and I left the devalue at zero point three.

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And this is what we got.

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So we get a so-called spin spiral.

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So it's a spin or two magnetic moment really spirals once around a circle when you go from left to right.

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And this is a typical signature of the so-called gelatin scheme area interaction.

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So in total, the magnetization is zero here.

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So if you add up all of the arrows, they would compensate and would give zero.

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But still, you have some order in your system.

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So it's nothing compared to the chaotic starting configuration.

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It's really well ordered.

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But it's the so-called spin spiral where you have this compensated magnetization in total.

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All right.

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So this was really all I wanted to show you.

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This is once again the energy.

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And really, we have shown that you can include several more energy terms here.

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We have no exchange interaction, interaction with magnetic fields, DMI and now you could go ahead

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and add more and more terms to make the system more and more realistic.

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And you could also change these parameters, for example, J.

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D, to really resemble actual physical samples and materials.

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And for example, there exist different materials that have different ratios of DOJ constants that give

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different lengths of these periods of these spin spirals.

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What you can also do is experiment with the step, size and experiment with the change of the temperature.

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And you could, of course, also increase the sample size by increasing the value of length.

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But this will then lead to new problems because then it will take more steps to converge because you

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have the more moments and then you would have to simulate for even a longer time.

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So the whole purpose here was to show you how you can implement a Monte Carlo method and of course,

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in general to learn more about Python and physics.

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And we've we have really accomplished a working algorithm, and we have been able to solve a different,

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difficult problem.

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We have been able to show that this chaotic configuration turns it to the smell ordered configuration

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just by providing this energy term here.

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And this was something we could have never solved analytically, at least not straightforwardly, because

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they are just too many variables involved in the system.

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So it really works.

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But now to get further and to get good results and interesting results and to include more interactions

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here is really a science on its own.

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And there are really experts in this field who only do Monte Carlo simulations so it can get difficult,

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pretty fast.

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So here, just as an example, I have increased the size of the system.

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So here now we have 40 times, 40 magnetic moments.

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And you see, when I now relax the system, it still looks pretty much ordered and we see the non-coding

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clarity.

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But we do not see this nice spin spiral that is parallel to the edge of the sample.

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So here we really see some domain walls that go in a disordered way through the sample.

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And this is really what happens when you make the system larger than it gets more and more difficult

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to really converge to the actual energetic minimum.

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So in this case, you would have to increase the number of steps for the simulation or for the relaxation,

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and you should play with the temperature and really from here and it gets difficult and very demanding,

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very fast.

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So in my research, I also did some of these Monte Carlo simulations, and some of those simulations

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took almost three days on the very, very fast computer.

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So you see, it gets really, really difficult quickly.