Welcome back, everyone. This is the 21st lecture of the condensed matter course.
When we left off last time, we were talking about magnetism and in particular, we were talking about feral magnetism.
Feral. Magnetism and everything that we said about magnetism was really focussed on understanding what happens at zero temperature,
all of the moments aligning with each other. But generally you can ask the question what happens to a power magnet as you raise the temperature?
And we have a little bit of an intuition about what happens in power magnetism at Finite, which will be the subject of much of today's lecture.
You know, at zero temperature, all the moments align.
But then when you turn on the temperature, there's going to be some thermal fluctuations and the moments won't be completely aligned anymore.
And at some point we might not have our magnetism at all.
So in general, what we want to know is we would like to know as a function of temperature what is the magnetisation,
some sort of equation of state, and maybe will also allow for an externally applied magnetic field.
So this is what we're interested in finding out. Now, I should start with a bit of a caveat.
In the last lecture, we spent a lot of time thinking about the idea of magnetic domains which are created due to long range dipolar interactions.
And even though the local physics is to align all the moments in a ferromagnetic low temperature,
there will be some global physics which will make some regions point up in some regions point down to minimise the global magnetic energy.
So we're not going to think about that kind of physics today. We're only going to think sort of locally.
So for us, a ferromagnetic at zero temperature would have all their moments aligned, at least for for the purpose of the day.
And then we can put back into long range dipolar interactions later if we're interested.
So we're going to use a very simple model and try to understand what happens to the magnetisation and find a temperature.
We use the spin one half easing model, one half easing by spin,
one half is what we mean is that the spin on each side will be some sigma i pointing the
z direction where sigma I equals plus or minus one half and we'll write the Hamiltonian,
which we've seen before. There will be a coupling to the external magnetic field.
B Some overall sides sigma.
I And here I've put the external magnetic field in the Z direction and then we have
the coupling between neighbour spins minus one half J Some of our neighbouring spins.
I am J Sigma I Sigma J So again the bracket notation means that I enjoy our neighbours and here we're concerned with our ferromagnetic.
So we said J greater than zero. All right.
So this is the the easing model, a fairly simple models write down.
But as I mentioned before, it's actually quite difficult to solve.
It is solvable in one dimension, in two dimensions. It's solvable without the magnetic field.
And beyond that, it's not solvable at all. So that means we're going to have to resort to some sort of approximation method to get some
sort of understanding of what happens to the magnetisation easing model of finite temperature.
Now there's lots of approximation methods one can use,
but the first resort of theories or first resort of scoundrels is what is known as mean field theory.
I mean field theory, which we are going to use to try to estimate what happens to this Hamiltonian at finite temperature.
So mean field theory can be approximately defined as whenever you approximate some operator or a quantity some operator.
Or quantity with an expectation with an average or a mean.
And this is used very, very commonly, not only in condensed matter physics, but also outside of condensed matter physics.
This is sort of a very general statement. There are many things that you can start approximating as an expectation are an average.
So we're going to use a particular variety of mean field theory known as molecular field theory.
Very or also it's known as vice minefield theory.
Vice mean field theory. This is the same guy, Vice, who was discovered the idea of domains in magnets as well and discussed them last time.
And I have to admit, I don't know if it's I before or before either.
So apologise about that. My spelling is not very good.
So the idea of molecular field theory, it has basically two very simple steps that you can always follow.
Step one is that you should treat one piece of the system.
Exactly. You isolate one little piece of the system. It might be a single unit cell or it might be just a single site.
And you to treat that. Exactly. So treat one piece of system.
Exactly. Treat one piece of system.
Exactly. Everything else should be averaged.
All else is averaged.
So we've actually seen something like this before. When we study the Einstein model of the solid, we treated one atom.
Exactly. We treat it as quantum mechanics correctly, but we average the potential from all the other atoms.
We made it into some rough average and it just became this approximate potential, well, that one atom is living in.
So even though it is a very crude approximation in that one atom we treated very honestly,
it was the effect of all the neighbours that we approximated. The second step of the field theory.
Step two is self consistency. Self consistency.
Which means all pieces. All pieces should be the same, should look the same in the end, should look the same at the end.
And and so it shouldn't matter which part of the system we focus on to treat.
Exactly. At the end of the day, the whole system should look homogeneous again.
It shouldn't be anything special about the one piece we're going to look at. So for us, we will treat the easing from the magnet.
And we're going to focus in on just one side, a one unit cell. In this case, these ferromagnetic unit cells, a single site.
So the so we focus on, okay, so step one,
look at one site look at one site site I for the easing ferromagnetic I'll write down the Hamiltonian for site I only
so Hamiltonian for site I is going to be B Times Sigma I That's the coupling to the externally applied magnetic field.
Then there's going to be the coupling to all of its neighbours, which I'll write this in this way.
J is neighbour of I. I Sigma J time sigma i.
So I've written out all the terms in the Hamiltonian that contain the spin sigma i.
Right. Now, you may have thought here for a second that I may have dropped the factor of a half this written off top, but I haven't.
Remember that the factor of a half of top is there, because in that sum, strictly speaking,
there is signal one, sigma two, but there is also sigma two, sigma one. In this sum, if sigma I is sigma one, it does not occur the other way around.
It only occurs once a sigma two single one would just be represented once in this Hamiltonian.
So you don't need the factor of half when you write it this way. All right, so this looks a little complicated.
Let's see if we can simplify it a little bit. Will right.
Will define define an effective magnetic field.
Be effective, be effective via the following equation.
G b b effective equals g b the actual magnetic field, the external plus or minus the influence of the neighbours.
J time sum over small j is a neighbour of ii of I sigma j So what we're doing is we're wrapping up
the interaction with the neighbours and we're calling it part of the effective magnetic field.
In terms of this quantity, we can rewrite the Hamiltonian for site I as just give you the be effective time Sigma II, which looks extremely simple.
It just looks like a single spin sigma I coupled to an external effective magnetic field be effective
and we know how to solve that we solve that before it is a single spin in a magnetic field.
Should be easy, right? Well, okay, that's a little deceptive because we've sort of lied here.
To be effective is not just a number here. It's not a constant.
It actually has some degrees of freedom hidden in it. You should think of it as an operator.
It has all these sigma JS in it, and that complicated life is going to make everything really complicated.
So we can't really treat this simply as a spin coupled to a constant magnetic field.
So that's where my field theory comes in handy.
The rule of mean field theory is that you should approximate everything except this one spin in terms of an average.
So step one,
step one is approximate be effective by some average of the effective average would be effective by taking this average instead of being an operator,
it's now just a number. So the Hamiltonian becomes one, the Hamiltonian becomes H, II is going to be expectation of B effective,
which is now just a number couple of the Sigma I and that really is just a single spin coupled to a constant magnetic field.
So we know how to solve this. We have done it before. We did it. We've done it last year.
We did it in the last lecture. So I'll just write down the first step.
You write down the partition function, it's either minus beta G, new B expectation B effective times plus one half plus.
E to the plus give you the beta gbps expectation.
B Effective times one half.
How this looks familiar that we can differentiate this partition function in order to get expectation of Sigma I and the
expectation sigma I then becomes minus one half hyperbolic tangent of beta g movie expectation and be effective times one half.
Okay. So now we have an expression.
For Sigma I for the expectation of Sigma I in terms of the external the effect of magnetic field the temperature and Gobi.
So far, so good. Happy with this? All right, good.
So now the last step is we have to figure out with this expectation and be effective is so expectation of right this way, be expectation.
Be effective is just going to be, uh, be the external magnetic field minus j some over j is is neighbour of i.
Of I. Times expectation of Sigma J.
Okay so I just took the average of of be effective here.
And the the final step of the minefield theory is that at the end of the day, all sites should look the same.
So the average of sigma on site I should be the same as the average.
So this is step to self consistency. Average of sigma on site.
I should be the same as the average of sigma on site j. There is nothing special about site II versus idea, and we'll just call it Sigma.
Okay, so we're just setting the average of Sigma the same on every every site.
So if we do that, then our equation here with the tension, the hyperbolic tangent can be written as sigma equals minus one half hyperbolic tangent.
And now I'll plug in our expression for be effective.
Where? Well, actually. All right.
Well, okay. Uh. Up here.
This some here, I suppose. This some here. I can be the Times Sigma.
Where Z is the number of neighbours. Sorry. Number of neighbours.
Since we set Sigma expectations of Sigma J to be just this number Sigma and there's G terms in the sum Z terms in sum.
Exactly what I just did. Sorry, was a little messy. Okay.
All right. So this ends up coming out of minus beta jay z over two times sigma plus beta, JMU B B over two.
So this is just a rewriting of this of the equation up there with the with the hyperbolic tangent.
I plugged in the effective and I plugged in j time z four where I had j times expectation j times
z times sigma is where I had j times sum over neighbours times expectation of Sigma J and yes.
All right, all right. All right. So this is our our final result.
And it is some sort of self-consistent equation for the magnitude of sigma, which is the expectation of the spin on site ie.
And so in principle, we now have our solution for what sigma is as a function of the temperature,
the coupling to the neighbours, the number of neighbours, external magnetic field.
It is a complicated equation, but at least in principle we can solve it.
The moment per side would then be minus g, b, times sigma.
Where are the sign? So this is the moment per per spin.
This minus sign comes from the fact that electrons have negative charge, so the magnetic moment points in the opposite direction of the spin.
Okay. So in principle, we've now solved our problem.
However, this is sort of a complicated, transcendental equation, and it's not so obvious what its solution is with some equation four sigma.
So we have to figure out how to solve it. And let's simplify to a an easy case first.
So case. Case one hour. Just to set the case to be equals zero.
External magnetic field is zero, in which case simplifying this equation.
Moving a sign inside we get sigma is one half hyperbolic tangent of beta jay z sigma over two.
Good. It's still a hard equation to solve because it's still a transcendental equation for Sigma.
But there is a nice graphical way to solve this equation,
which is to plot y equals sigma and y equals one half tense beta jay z sigma over two on the same plot.
And where they intersect is the solution of this equation.
That's what we're going to do. That's how we're going to solve this equation. They're also with me and we're happy.
Yeah. Okay, good. So. So here we go.
First we have to decide, you know, put it over here now.
Okay. This right here. Why? Sigma over here.
Temporary sigma. Sigma over here.
Then we'll plot the first curve, which is y equals sigma like this.
Y equals sigma. That is the first curve that we want to plot.
And then the second curve is this hyperbolic tangent which asymptotes at one half and at minus one half.
So we'll asymptote here and it will asymptote down here.
But there's two possibilities as to what this plot looks like with the hyperbolic and hyperbolic.
Hyperbolic tangent with this hyperbolic tangent looks like. One possibility is what occurs for a low temperature.
Low temperature, what happens outside this new high temperature? First high temperature.
High temperature is the first possibility. So in high temperature Beira is small and so the tense is a very gentle function.
So it looks kind of like this. Okay.
So this is a high temperature. High temperature.
Can people happy with that? That's a hyperbolic tangent curve and it asymptotes to this dotted line.
So what we see is these two curves, the hyperbolic tangent and the Y equal sigma curve intersect only at one point at sigma equals zero.
So the only solution only solution is sigma equals zero.
And that tells us that you do not have fair magnetism at high temperature.
You in fact get R and equals magnetisation equals zero or aura magnetism rather than power.
Magnetism. Power magnetism. Not very magnetism at high temperature.
At high temperature.
And this is actually quite typical that any firm magnet, if you raise it in temperature at some point, the pro magnetism will go away.
And you have this sort of intuition that what that's coming from is that the moments are fluctuating
thermally around so much that they eventually stop pointing in any particular direction,
and they don't point in any if they're not pointing in any particular direction is not a fair magnet anymore.
And in fact, it will be a power magnet. The show in a few moments.
The other possibility is that we're at low temperature.
So think about that possibility, low temperature, in which case the hyperbolic tangent is a much steeper curve like this.
Because the argument of the hyperbolic tangent is larger because beta is larger at low temperature.
And so in fact there's three solutions. So this is the low temperature of low temperature.
So we have a solution here. We'll call that sigma, not a T with a solution at zero.
And then there's a solution here which we'll call minus sigma, not of T.
So we get 3 to 3 solutions, three solutions which are sigma equals zero, and then plus or minus sigma not of t which are non-zero.
The fact that we have non-zero solutions for Sigma is telling us that we have a foul magnet and
naturally the foul magnet can have it spins pointing up or it can have its pins pointing down.
So these are the ferromagnetic solutions.
And this is not surprising because we know that if you go down to zero temperature, this curve will be extremely steep and will intersect right here.
As a matter of fact, when you get complete polarisation that sigma,
the expectation of sigma is one half, meaning all the spins point in exactly the same direction.
Okay, so that would be what you get at zero temperature.
At higher temperature, the intersection moves down to a slightly lower point and you get slightly smaller
sigma not of t so you get slightly smaller magnetisation but not zero magnetisation.
Now we might wonder what about this guy? Here we have another solution which tells us we have an expectation of Sigma being zero.
And this one actually is a little puzzling. It turns out it's a not a physical solution.
Not physical. And you'll actually show that for one of your homework assignments.
So it turns out what your show for your homework is that this self-consistent
equation that we solve is actually the same thing as extra missing a free energy.
And as you know, an extra mile is something you can either get a maximum or a minimum.
So we're getting one maximum and two minimum of the free energies.
The minimum are the free energies are the things that are realised physically and the maximum of the free energy is sort of mistake.
It comes out of the equations, but it's not physical and it is much clearer when you actually write down the free energy
which you'll do for homework and you'll see why it is that we don't keep that solution.
So this one we sort of ignore and we only keep track of these ferromagnetic solutions.
So we'd like to know at what temperature do we go from being a power magnet to being a feral magnet?
Just write that down. At what t at what t do we get?
There are magnetism. Well, there's actually a name for that.
D.C. is known as for less than D.C. You get feral magnetism from AG,
and D.C. is known as the Curie Temperature Kerry Temp or the critical temperature critical temp.
Although if you have feral magnetism. So we'd like to know when do we get at what is TSI or when do we get this magnetism at below what temperature?
Well, the way to see that again,
looking graphically is to realise that the transition between having one solution and having three solutions occurs when the
hyperbolic tangent the function is exactly tangent to the y equals sigma line where it aligns up just exactly with the same slope.
And the reason that that's the important point where,
where the where you have the transition from power magnetism to magnetism is because of the
hyperbolic tangent where any less steep you have only one solution and if any more steep,
you have three solutions. So we have to look for the for the situation where the hyperbolic tangent has exactly the same slope as y equals sigma.
Okay. So t from finding slope of tangent of tangent.
Equals one. So if we just expand the Y equals equation here, the slope of which we're setting equal to one is better.
Jay Z over four or just rearranging a little bit.
We get KB T equals Jay Z over four, which is the critical value of the temperature.
Is everyone still with me? Fairly. Fairly happy now?
Yes, maybe. Okay. Maybe. All right.
So let me just pause for a second to sort of.
Okay. Well, it's time for chocolate, actually. So I suffer from one chocolate bar.
What's the. What is the critical importance of. Think outside the box music is really good.
He said that, as we say, he's married to Beyoncé, but it's close enough.
Yeah, so. Oh, yeah, very good.
Yeah. Okay, now Jay-Z is beside me managing Beyoncé.
The importance of Jay-Z here is that it is. In fact, it is telling you that Beyonce's middle name.
You. Not her last name. No, no, it's a sister's name.
Yeah, right. No, no. Does anyone know Beyonce's baby's name?
Yeah. Someone got it over there. Yes. Yeah. Okay.
I have one for you also. All right.
So those are the important of Jay-Z here is actually is the total amount of energy of coupling of the spin to its neighbours.
And so we have the neighbours here and j is the the coupling strength from the spin to each of its neighbours.
So when the temperature gets above the total amount of coupling strength to its neighbours,
the fluctuation in the thermal fluctuations of that spin so overwhelmed the amount that it wants to stay in the same direction as its neighbours,
which is what is keeping it a power magnet. It wants to have the same direction as its neighbours and when the energy of
when the thermal energy is is greater than the energy holding it in place,
you lose power, magnetism. Okay, that's the intuition behind this picture here.
All right. All right.
Anyway, so at least graphically, we can take this picture and we can solve for our sigma as a function of t just have to figure out where,
where those two curves cross as we change t and we'll get a picture that looks kind of like this.
Here is temperature. Here's sigma at zero.
Temperature sigma is actually a half because of all the spins are aligned and then at TSI it goes to zero and above.
Above T is exactly zero, but then it comes down kind of like this at a finite temperature as you changed.
So this is it actually gives you the equation of state of how much magnetisation you
get as a function of temperature and you can get it from this new field approximation.
It is not exact, but it is fairly good. Okay. All right.
So the last step is that I promised you that up in this region here, we have a power magnet, so we should.
We actually convince ourselves that it really is a power magnet. And one way to do that is to calculate susceptibility.
So things that we know is up in this region. If you do not apply a magnetic field sigma zero.
If you apply a small magnetic field in Sigma is probably going be close to zero.
So what we can do is we can take our original self-consistent equation and we can expand it for a small argument of the hyperbolic tangent,
because everything inside there is by definition small C the sigma is smaller, b is small, everything is small.
So we can write sigma equals.
As you just writing this all out, I'm going to observe a couple of minus signs here.
Beta C sigma over two minus beta.
Jim, you b over to the magnetic field.
Okay. And then we just rearrange this.
So we solve for sigma as a functional magnetic field, we get sigma axial right this way.
One minus G over for beta.
Equals minus data. I give you B over four times a magnetic field.
And this notice this quantity here is actually itsy bitsy.
So with a little bit more rearranging, we get sigma equals minus one quarter grubby magnetic field B divided by k minus kb.
T c. And then we want to actually convert this into some sort of susceptibility.
So the magnetic moment is minus G, movie sigma and magnetisation.
So this is the moment. Precisely. The Magnetisation is the moment perceived times rho the density of spin's density of spins,
and then the susceptibility will be new, not by definition the m d b at zero.
Which is just plugging in a couple lines up here.
We then get one quarter row g b squared mu,
not over KB times one over one minus t over t and this whole quantity here is actually the query susceptibility query for a spin one half.
In other words, it is a susceptibility we would have calculated with just a free spin, one a half.
In fact, it is exactly what we calculated when we calculated the magnetic susceptibility of a spin one half in the last lecture.
But then this whole thing gets multiplied by one over one minus t over t.
So the susceptibility of our power magnet above the AC is a susceptibility of just a system of free spin,
one halves which aren't coupled to each other.
And the effect of the coupling in the easing model, at least with the mean field theory, is to multiply it by this factor of one minus t over t.
Okay, now you may notice that at the critical temperature or the curie temperature, this factor diverges and that actually is physical, it is real.
And the reason it's it's there is because we have to remember that that susceptibility tells you
how much magnetisation you get when you turn on a little bit of magnetic field by definition.
But when you go below the critical temperature, you get magnetisation even for zero magnetic fields,
susceptibility must have gotten infinite somewhere along the line. So that is what this term is doing there.
All right. So this is the general scheme of, ah, of minefield theory and how it works.
Minefield theory, where was it? A minefield theory and how it works where we generally approximate some operator by an average.
In this technique, a minefield can be used much, much more generally than what we did here.
For homework, you'll do minefield theory calculation for an ANTIFERROMAGNETIC.
Well, you could do it for a very magnet. You could do it for spins that aren't spin one half.
The fact that we got a hyperbolic tangent in this in this whole story came from the fact that we were thinking about a spin one half.
So it had only spin up and spin down. If we're thinking about spin one, we would have spin as the equals, one as equals, zero as equals minus one.
We get something more complicated than a hyperbolic tangent. We get a so-called broken function.
But the general themes of how you how you approach it treat one piece of the system exactly.
And then insist on self consistency that remains the same.
And the technique is used outside of magnetism in condensed matter.
It is used for understanding liquid crystals, it is used for understanding fluid dynamics, is used in astrophysics, is used in atmospheric physics.
Mean field theory is just a very general technique which is used for many, many things, and it is good to internalise the ideas of it.
All right. We are going to use mine field theory for one more thing, which is to understand itinerant, feral magnetism.
Itinerant, which is the next subject, an itinerant magnetism.
And the word the word itinerant means wandering, basically, or sometimes it means lost.
But what it really means is that the magnetic moment moments are carried are carried by mobile electrons.
By mobile electrons.
Now since we started talking about magnetism a couple lectures ago, we thought about putting magnetic moments on sides,
and the magnetic moments can move and point in different directions.
But we did not consider the possibility that the moments could actually hop from one side to another side.
But in real magnets, like in iron, for example,
is quite typical that the same things that are giving you magnetisation these electrons are actually moving around to in bands.
And yet they managed to give us a magnetism.
So as with other types of magnetism, itinerant magnetism is driven by interactions.
It comes from the interaction between electrons from from interactions.
So somehow we have to introduce interactions and treat them at least within some Enfield approximation in order to understand itinerant magnetism.
Now of course, once you start introducing interactions between electrons, things get complicated very quickly.
So we need a very simplified model which will represent the interaction between electrons and still give us the right, the right physics.
So the model we're going to use is the so-called Hubbard model.
Hubbard wrote this down in 1963 to try to understand itinerant magnetism.
And since he wrote this down, the Hubbard model has taken on a huge life of its own.
It is potentially it is probably the most important model of interacting electrons in all of physics.
There have been, probably, without exaggeration, 100,000 papers written on the subject of the Hubbard model.
It's that important to the field. And the reason it's so important is because it is exceptionally simple, and yet it is.
Physics is quite complex. So by the way, Hubbard treated it in mean field theory is what we're going to do here.
It is fairly simple. You get a good result, which helps you understand itinerant feral magnetism, which is what we're going to do in the model.
It's pretty straightforward. What we do is we start by writing a Hamiltonian that is type binding, type binding type binding electrons with hopping.
With some hopping t and this part of the Hamiltonian a tight binding electrons with some hopping.
We have solved this before. We we looked at this earlier in the term.
We got these cosine shaped bands. You can throw up the cosine shaped bands.
If it's a Hatfield, then you get a metal if it's a full band again.
INSULATOR But never in just this model here with just type binding electrons with hopping which you ever get from magnetism,
you would never get a situation if you only have this type binding structure
where you would ever have a different number of up moments from down moments,
they would always come out. Even so, we have to add something to it, which is going to change the story.
And what we're adding to it is the interaction term.
Some overall size I parameter you use greater than zero times the number of electrons on site.
I would spin up times the number of electrons on site I will spin down.
Right. This is number of electrons on site I with spin up down to down.
Oh, okay.
So what this is doing is it is just telling you that you have an energy penalty of you whenever two electrons sit on the same atom, on the same side.
And we've actually discussed this Hamiltonian before, we didn't use the word hybrid model,
but I'll remind you, when we talked about Mott Insulators, Smart Insulators,
we discussed a situation where the interaction between electrons was so strong that you could only put a single electron on a site,
and then you can get a situation. If there is a single electron on a side,
then everything gets frozen in a traffic jam because no one can hop to the neighbouring side because there's already someone sitting there.
And even though you have something that looks like it should be a half filled band, you still have an insulator known as an insulator.
So this is exactly the same physics. We're just going to allow ourselves to turn on and off that interaction.
You make it as strong as we want or as weak as we want. Okay, so not too complicated.
It looks like a very simple model, but again, it's one of these models. It looks simple, but isn't it solvable in one dimension?
And absolutely nothing is known about it, despite 100,000 papers written on it in higher dimensions.
Okay, maybe not absolutely nothing. But. But. But not as much as we would like, which is why people keep writing papers about it.
Okay, so how do you treat this? Well, first thing we're going to do is we're going to write this interaction term as in the following form.
We'll write it as you over four and up plus and down minus two over four and up of squared, minus and up, minus n down squared.
So that is just an identity that is easy. We haven't done anything different here.
What we're going to do now is we going to make the minefield, approximation minefield,
which a minefield always somehow or other means take something that average
where you're not allowed to do something that is incorrect by taking an average.
And what we're going to do, which is not quite correct, is we're going to take an average of these terms.
So we're going to take average of analysing down and square it, minus you over for average of and up, minus and down and square it.
Okay. So that is the mean field approximation or it's one type of minefield approximation we're going to use and we see what it gives us.
Well, first of all, this term here, average of up and down, that is just a total density, total number of electrons, total density.
And that is fixed, probably fixed at the beginning of the day by the conditions that you have.
So that is just going to give that first term is going to give us a constant.
The interesting thing is this term here that end up minus and down is actually the moment for a site divided
by the more magnetite and then it gets squared minus you over for a moment divided by the town square.
Right, because the number of spin ups and downs is the magnetic moment on on the side.
Okay. So we can rewrite this interaction term as some constant which we're not interested in, minus you over four.
And then I'm going to rewrite this moment as magnetisation times the volume of the unit cell over new B squared.
So Y equals volume of unit cell.
And the reason I'm rewriting it that way is because I would like to have this thing
is a magnetisation and magnetisation has dimensions of moment per unit volume.
So I have to put in a the volume of unit cell to have the dimensions come out.
Right. Okay. And the thing to notice about this is that the energy, the interaction energy gets smaller as the magnetisation gets larger.
Why is that? Well, the intuition for that actually comes from an exclusion principle if all the spins were aligned.
In other words, if the magnetisation were maximum, then there would be no interaction energy whatsoever.
Because you can never have a situation where two electrons sat on the same side and you
only have to pay this energy penalty when there are two electrons on the same side.
So if you polarise all your spins in the same direction, you get the minimum possible interaction energy.
And that is why it is that if you increase the magnetisation, you drop the interaction energy.
Is that clear? Good. All right. So from this picture, it makes me think then, okay, once you turn on the interaction, all the spins want to align.
But it is not so easy because aligning the spins cost you kinetic energy.
And you have to you have to sort of let these two things fight it out with each other.
So let's remind ourselves what happens to the kinetic energy as you change the magnetisation of a system.
And now we studied this before. So if you recall poly power magnetism, which we did in the first week.
Pauli Paramagnetic. We actually calculated, although maybe not in the same language.
You'll recall we had the density of states of spin up versus energy kind of looks like this.
And then we had the density of states for spin down versus energy and it looks exactly the same.
And if there is no magnetisation, you fill them both up to the same level.
Does it sound familiar from poly power magnetism? Now, if you want to make this person look identical, if you want to have a nonzero magnetisation,
what you have to do is you have to take some of the spin downs and make them spin ups.
So to do that, let's take a sliver of the spin downs and we have to move them up to higher energy up here.
So it cost you energy. So these were supposed to be aligned with that.
So it costs you energy to have a non-zero magnetisation kinetic energy to have a non-zero magnetisation.
All right. So we save some interaction energy, but it costs kinetic energy, so we have to trade those off against each other.
So in order to figure out how those are battle against each other,
we have to figure out how much energy it actually costs you to have some non-zero magnetisation.
So let's do that. So first we'll go back to the poly paramagnetic calculation.
So we're going to set you equal to zero for a second.
So now we're back to non interacting electrons and I'm going to write the free energy as a function of magnetism,
as a function magnetisation per unit volume, and that will then be minus B.
I think everyone is probably will accept that,
that there should be a term like that in the free energy minus B that wants the MAGNETISATION to align with the with the external magnetic field.
But then the second term is less obvious,
but it takes the form m squared mu not over two times the poly susceptibility, poly paramagnetic susceptibility.
And I'll just remind you, we calculated this in the in the first week of the term with Chi Poly.
Was high poly is new, not new B squared times the density of states at the Fermi Energy.
Does that kind of sound familiar? Okay. Now, why is it that I wrote down this complicated term here, and why do I think this is correct?
Well, first of all, this is sort of a tail or expansion in the magnetisation.
So this is linear and magnetisation is quadratic in magnetisation.
In principle, there would be higher times as well, which we're going to ignore because we're interested in small magnetisation in general.
But why would I give it this term? Mu not over to Chi Pauli?
Well, it turns out that it actually has to have this form and to see that it has to have this form.
We know that the free energy should be should be minimised.
So the steam has to equal zero. And that tells me that minus B plus m mu not over our chi poly has two equal zero.
And if you just rearrange that, that tells me that the magnetisation is chi poly over mu not times the magnetic field, which is what defines Chi Poly.
The definition of the poly susceptibility for our free electron gas is the magnetisation should be chi, poly times, the magnetic field.
So that tells me the free energy has to have that form that I just wrote down. Okay.
So a little bit of a trick you could go through the agony of of trying to actually calculate the magnetisation the
energy is a function of magnetisation you get the same result you would discover it has to come out this way.
Okay, so, so let's turn off setting, set the external field, set B equal to zero and turn, turn you on you not equal to zero.
And then the free energy we get is going to be a combination of the two terms that we just calculated.
So free energy volume is then, well, okay, we have this term here.
I turned off B, so I just get m squared nu not over to chi poly but then also are just about to scroll off the top.
There is minus u over four times while k v squared over new b squared.
With this squared of times I'm going to get rid of a v magnetisation squared was v squared,
but now it's going to be v because I'm writing the free energy per unit volume, not the free energy in this case.
Okay. So I dropped a factor of a volume, so it's not like V squares is v.
All right.
So what this is, I guess we can factor out the M, so we have m squared and then new not over to chi poly minus u over for the over mu b squared.
And this tells us there will be higher terms, two terms which are quadratic and magnetisation and so forth, which you're not interested in.
But the key thing to realise here is if this thing in the bracket is less than zero, well magnetisation wants to be non-zero.
Whereas if the thing in the brackets is greater than zero, then the magnetisation wants to be zero.
Okay, so the criterion. So our criterion for magnetism.
For Farrow, which is known as the stoner criterion.
Stoner and Criterion.
Is that this new not over too high poly minus u volume with you would sell over four more magnets on squared should be less than zero.
So when you turn on the interaction strength strong enough, then the interaction strength wins and it wants to have this the spins polarise,
whereas if the interaction is not strong enough,
then in fact it is more beneficial to have an equal number of spin up since spin downs because you say kinetic energy that way.
Actually, we can rewrite this in a just putting in the definition of chi poly.
We can write this as our volume of the universal cell times. You times get f should be greater than two.
And equivalent. Okay. So that is what we get out of this minefield approximation for figuring out when we have firm magnetism.
But one of the things that's interesting about the Hubbard model is you can also get
Hubbard ANTIFERROMAGNETIC Hubbard ANTIFERROMAGNETIC ISM and Thai Pharaoh magnetism.
And this occurs when we have you as large, you large and one electron parasite.
So this is in the Mott INSULATOR regime. Mott INSULATOR.
So if you is large, it prevents you from having too electrons parasite or mostly prevents that.
And then since it's one electron parasite, you have this traffic jam of electrons where you have exactly one electron on every side.
And the way to convince yourself that this thing is going to be antiferromagnetic is to try the two possibilities.
So case one is consider a ferromagnetic. Case, one is considered ferromagnetic.
Ferromagnetic. So we have a bunch of spins.
Electrons each with their spin sitting on each side. They're all aligned.
And the energy of this situation is actually zero.
The energy of the ground state is is exactly zero.
And the reason zero is, well, you never pay any interaction energy, because due to the Pauli principle, no electrons sit on the same site ever.
So you don't ever pay the price. You don't ever pay the new energy.
And also, the electrons can't hop from one side to the other at all because they're in this traffic jam so they can't hop.
So the total energy of hopping is also zero. Now the second possibility is that you have an antiferromagnetic case to antiferromagnetic.
And this is a little more complicated because we have spin up, spin down, spin up, spin down and so forth.
And here, well an electron can hop to the neighbouring side but it has to, it has to pay a price to do so.
So you get spin up, spin down, you have zero on this side and then you have up,
down on this site, which costs you energy you and then up on this side.
So this is an excited state, which is a valid, excited state. It costs you some energy.
You you might be large, but at least virtually you can make such a thing.
And so in second order perturbation theory, during this sort of hopping process, you get a shift in the energy,
which is some over all possible excitations excitation hopping Hamiltonian ground
state squared over energy zero minus energy excited state and this is less than zero.
So in fact, the change in energy is less than zero.
So Antiferromagnetic energy for Antiferromagnetic is less than energy referring magnet.
When you have one electron parasite in this in this might insulator limit.
Now, this might seem a little complicated, but let me rephrase what we just did.
If you have a ferromagnetic so in the case of a ferromagnetic, each electron is basically sitting in an infinitely tall well.
You can't hop to the left, you can have to the right. That is absolutely forbidden by the exclusion principle.
So you should think of it as an electron sitting in an infinitely tall well.
Whereas for an antiferromagnetic you have an electron in a finite well.
So the height of the well is only you. It can hop to the neighbouring side.
It costs an energy you, but it can do it. It is not forbidden by Pauli and the energy of the anti.
For a magnet with a finite barrier is lower than the energy of the Fuhrer magnet with the infinite barrier.
So therefore Antiferromagnetic is favoured. So this is sort of one of the things that makes the anti for our magnetism a will.
So it makes the hybrid model quite interesting that it can give you anti-foreign magnetism.
It can give you foreign magnetism.
It can give you lots of other physics as well, including superconductivity, charge, density waves and all sorts of other things,
which is why there have been a hundred thousand papers written on this hybrid model.
You're not responsible for all of that, I promise. You're supposed to know something about itinerant power magnetism.
But that's all. Okay. So sit still.
We're going to go over on this lecture because we actually have come to the end of the course.
So I just want to wrap everything up. We're going to cancel tomorrow's lecture.
This is this is actually on schedule, which is it's actually planned to be 21 lectures.
But I leave an extra lecture in case of disasters or running behind.
And we didn't run behind. We did everything exactly as planned. So we are actually finished the course more or less.
A couple of organisational comments before before the closing remarks.
First of all, obviously no lecture tomorrow. That is the first comment.
The second comment is in no one or not, no one, but very few people actually use the message board heavily this this term that that's okay.
I don't know why it is people didn't like it that that's fine.
But you know, this message board is still up and running.
And I think as it gets closer to exams, people will get more enthusiastic about using it and asking questions.
That is fine. That is what it's meant to be, therefore. So don't hesitate to use it.
Ask questions. Your heart's content. I will try to answer as many of them as I can as you revise for your exams.
The only thing that I want to ask you to do is let me know what you want to have for the revision lectures.
We have two revision lectures in the fourth week of 20 term and I am always completely at a loss as to what the most useful thing to talk about is.
If you think I should try to just give an overview to do that,
if there are particular questions from that from previous year's exams that you want to go over, let me know.
I'll try to assemble something that people will find useful.
All right. So just a couple of closing ideas for the course.
When we started seven and a half weeks ago, I promised that that by the end of the course,
we'd actually know something more about the world around us. And I think we genuinely do.
We know things like why diamonds transparent and and why ions a magnet and why solids expand at higher temperature and many, many more things.
And some of them very esoteric and some of them less esoteric things like quantum why quantum effects are important
for heat capacity or how a transistor works and and how you determine the crystal structure of a material in X.
For those of you who go on in science,
I hope this forms a good foundation for for future learning and discovering more things along the same lines or even in other fields.
For those of you do, don't go on to science. That is fine too.
I hope that at least in a good and an entertaining exploration of the natural world,
and I hope it leaves you with a better appreciation for the subject at the very least.
I do realise that there is an awful lot of material in this course and I apologise about that.
It is more or less a fixed amount of material fixed by the exams, the exam committee and the syllabus.
So we don't really have that much freedom to remove material.
So I realise that a lot of people view this as drinking from a firehose.
It's just so much information coming, coming at you. At the same time it's hard to absorb.
But ironically, the the biggest problem with the course is that there is actually not enough in it.
And I don't say that because I feel like you need to learn more for your exams.
I say that because condensed matter is such an enormous subject that we have hardly scratched the surface.
It seems such a shame to leave it here, but this is all we have time for.
We could easily do a year of lectures or more on material physics or quantum effects or statistical effects or device physics,
or any one of a dozen types of things that we just barely mentioned at all.
And as well as many, many things that we didn't even get to superconductors, superfluids, fractionalised, particles, so forth and so on.
I hope I really hope many of you have the opportunity to go on and learn about those things in the future,
and hopefully it will be as entertaining or even more entertaining as what you've learned so far.
Finally, I just want to thank you all for having come to lectures and having looked enthusiastic throughout the term.
Thank you very much.