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Welcome back in the previous lecture, we ended up with this nice result of a fairer mechanism, but

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to be honest, it was pretty difficult and we had to play with the sample size, with the temperature

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with a number of steps.

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And another reason besides the local energy benchmark that leads to a problem in the convergence is

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that the energy minimum is actually not well defined.

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So here it is really a coincidence that all the magnetic moments point roughly along the out of clean

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direction along Z.

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They could point really in any direction, every so so as long as all moments point along the same direction,

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as long as they are parallel.

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This would give the same energy for the system.

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And so all of these states are equal and they are equally likely to occur as the result of the simulation.

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For example, they could all point downwards.

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They could all point along the X direction, they could all point along any direction as long as they

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are parallel.

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So this also hinders a bit the convergence and makes everything more difficult.

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So as an update, what I will do next is I will include another energy term here, and this will be

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the so-called Zeeman energy or the interaction with an external magnetic field B. So there is a prefect

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here, but we don't really care about prefectures.

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We just take the dimensionless units, dimensionless quantities.

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So I will just write mu is equal to one and then we define our magnetic fields, which is in the radio.

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So I write B is equal to and got Perry.

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And then we have here B x, y z.

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And from the beginning, let's consider a B field along the X direction.

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So, for example, like this, it has a strength one and it points along the X direction.

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So now we can define our energy terms in a very similar way, as is done here.

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So I start with the lower one, a right energy for kinetic.

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So for magnetic field.

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So you could give it really any name you want.

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And all right, energy is equal to energy plus energy, kinetic contribution and then the magnetic contribution

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it will bend will then be dysfunction here, and it will be the term as shown here on the right.

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So it will this time not be a sum over pairs of moments, but just a single moment.

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And then the scalar product with the magnetic field itself will just be minus New Times and P Dot Dot.

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And then we arrived the B comma Mac and then all three components x y zero.

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OK, so this is now our new energy and we can run this, and here I will update now.

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Also, the step change sort of metropolis step because I will now include here the new energy term.

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So he will write.

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Of the two, we include the energy corresponding to the interaction with the magnetic fields, and here

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are basically two just a copy of this cell, and we call this new step just step T. Because this will

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be an hour or final solution.

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And here I just have to add 40 energy terms to new one I right here, plus energy magnetic contribution

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and my x y is correct.

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And here I will do the same thing.

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And that's really it.

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So the thing is here we needed to two and now we don't need the two.

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And this is because we have here a sum of pairs.

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So this means AJ and GI give the same result, which is why we needed the factor of two.

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And here we don't have this.

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We just have a sum over I.

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So really every interaction is only counted once, so we don't need the two here.

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It's not.

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Let's go ahead.

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And maybe the magnetic field is a bit strong here.

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Let's make it maybe a bit larger, a bit smaller, zero point one and let's run it and takes, of course,

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a bit of time because I run it four or five million steps.

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As before, and I will get back to you with the solution.

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Just one thing I just noticed is that I did not update this cell.

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So of course, we have to do this as well.

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So let me do this real quickly.

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So this would be here what we did previously, and now we need update to.

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Ad interaction with magnetic fields.

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And here the routine is just called Step T..

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So now finally, I can see and I can rerun everything and I'll get back to you in a few seconds.

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So I've increased the magnetic field here a bit to 0.5 along the X direction, and this is what we get

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a pretty nice furrow magnet along the X direction.

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So all the moments are parallel to each other, basically due to exchange, but also due to the magnetic

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interaction.

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And they prefer now this particular orientation.

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So no, let me scroll back up and let me change this to the magnetic fields along the Z direction and

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let me rerun the code and show you what happens then.

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And this time, the result looks like a fairer magnet once again, but the moments really point along

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the out of plain z direction.

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Okay, so now with the magnetic field, we have a tool to really dictate the global orientation of all

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the magnetic moments.

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And this was just a generalization of the energy term where we have now not only the exchange interaction,

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but also the Zeeman interaction, which is the interaction with the magnetic field.