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So for our next story we have Professor Sid Parameswaran who is a condensed,
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massive theoretical physicist, and Sid is going to tell us about axioms in the solid state.
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Thank you, Julia, and thank you all for being here. It's really nice to be back for one of these events in person.
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So I should confess that I made the stop this week. I have an 18 month old son and I tried it on him a couple of days ago.
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He promptly fell asleep, which I wasn't complaining about, but I'm hoping for a different outcome today.
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So in Joe's stock, what he mentioned was that he talked to you about some of the exciting physics of actions and action like particles.
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And so what I want to do in this stock is actually illustrate how some of those ideas apply in a setting,
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perhaps where we're more familiar with in everyday physics is sort of an introductory courses,
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which is Maxwell electrodynamics, but with an ingredient, which is what do these axioms mean for thinking about electrodynamics?
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So that's what I want to do today. So I want to start with a pair of propositions and axioms from this paper by Frank.
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Well, check. And you know, he famously coined the phrase Axion in its original context.
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And quickly, Jordan mentioned this sort of running joke on axioms.
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The word it sounds like a fancy particle was actually a form of laundry soap because it cleaned up problems in the standard model.
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And so that was why the name came from. So the first quote expresses a sentiment that many of us in the centre can probably sympathise with,
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which is that whether or not axioms have any physical reality, that study can be a useful intellectual exercise.
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So that's great. That keeps us employed. The second point, he said, and that's the centre of the stock,
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is that it might be is not beyond the realm of possibility that fields whose properties partially mimic those of Axion fields,
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can be realised in condensed matter systems. So it's the second proposition that's going to be my primary concern in the stock.
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And the first is something though I'll return to at the end. So to start off with, let's just say let's set up rules of the game.
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So what Joe talked about in high energy physics and what John will talk about is the Axion field theta as
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a fundamental field that describes Axion dynamics in quantum chroma dynamics and its original setting.
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This is a challenge to with I stated goal of finding Axion like physics and solids because we know the standard model of solids.
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The standard model of solids is just electrons and ions governed by Maxwell's equations, interacting by Coulomb forces.
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And so there's no QCD here, there's no strong force.
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And so very clearly I can't change this description when I'm talking about solids, the sort of ultraviolet physics is fixed.
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And so I want to think about axioms. And so my question is, how can I do that in a solid.
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So of course, if you've been on Saturday Theory before and seen a condensed matter session, you sort of know the answer.
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That instead of looking at the physics of the high energy scales,
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we look at low energy scales below at some very low scale that we can probe and experiments
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and ask for new physics that are emergent when many particles cooperate on those scales.
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So the goal of this talk is to find a way to obtain.
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So what I've written down is the Maxwell Lagrangian. So this is just ordinary electromagnetism, so nothing exotic of it.
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So this is very similar to what Joe talked about when he talked about Axion like particles.
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So the Maxwell Lagrangian with the term that comes from Axion electrodynamics and the disclaimer is that for most of the stock,
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I'm just going to think of this theta as an angle rather than a field.
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It's just going to be a constant. But I want to talk about what that constant means if we add that to our equations.
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Okay, so that's what I'm going to do today. So. As a warm up.
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I want to start off by reviewing a little bit about Maxwell's equation.
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So, you know, Maxwell's equations are sort of things we know about and love in vacuum,
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but I'm going to be interested because of my solid state physicists I care about matter.
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So I'm interested in Maxwell's equations in media.
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So this is part of the dreaded second year course where you have to deal with all kinds of terrible things for those.
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So trigger warning, I'm going to take you back to those days. So, you know, so those of you need to leave the room, you might want to do that now.
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So matter is a source of energy.
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We know that charges and currents and solids associated B and B and two of the Maxwell's equations.
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Oops. Then get there are two of the Maxwell's equations don't really care about the fact that they're sources of energy.
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So those are the fact that there are no magnetic monopoles, although we'll come back to that later.
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And the second is that there are that the electric and magnetic fields are linked.
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But do equations do care? Gauss's law and you know, the curl of B the currents generate fields.
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So Faraday's laws, they care about fields.
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And so to go any further, we have to actually model what we mean by charge and what the RO and g density and currents, what they mean.
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Because depending on whether we think of the system as a metal or an insulator, the behaviour will be different.
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So today I'm going to just talk about insulators where charges on electrons are fixed by positive and negative charges can't move very far apart.
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They're tied to each other and similarly currents don't roam freely, they're tied to sites of individual atoms.
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So if I draw a picture of a solid in a field, an insulator in a field,
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then it's just a bunch of little dipoles because every positive charge is pretty close to every negative charge.
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And if I looked at the current loops inside, they're all being fixed positions in space.
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And if I dropped you into this insulator and I walked around and I could measure electrical magnetic fields,
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you'd see wildly fluctuating electric and magnetic fields, because every time you went past a positive charge, things would jump up and down.
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So of course, this is very hard for us to deal with and compute, but what you can do is blow your eyes a little bit.
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That's very easy for me. That's all I need to do, you know? And what I do then.
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And there's a mathematical, mathematical statement behind that blurring, which is, of course, screening.
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All that means as we average over distances that are small compared to the size of the solid, but they're large compared to atomic distances.
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And if I do that, I see something nice that happens. If I average, then I'll cancel everywhere except right at the boundary of the system.
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So you see that there's positive and negative charges. Cancel accepted boundaries over here.
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And so that means that traditionally I can rewrite these charges in terms of physical things that I call polarisations and Magnetisation.
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So I've got but the density is related to a thing that I call the polarisation, which is just measuring these little dipole moments inside.
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And the current is related to magnetisation and these are average, so they only really matter at boundaries.
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And so what you can then do is say, well, let's play around with this and put those in the equations, do some rearranging,
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and I can introduce new fields which have these sort of unedifying terms like
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the displacement field and the rather confusingly magnetic field strength,
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as opposed to magnetic field density, terrible historical terminology, but we're stuck with that.
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But they call D and H, and the nice thing about these is that these look these equations look like they satisfy the old Maxwell equations.
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They have no divergences and they have very simple equations of motion.
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So we're very happy because we can work with those and solve them. Okay.
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So I want to make another specialisation, which is that if I turned off the fields outside,
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if I turned off electric fields and I turned off magnetic fields, the solids go back to being not having polarisation or not having magnetisation.
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So I'm not describing you a refrigerator magnet or something that could have a frozen polarisation.
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I'm describing things that only are polarised or magnetised when I stick them in a magnetic field.
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So there's a simple way to understand that for polar polarisation in particular, if I have a bunch of neutral atoms, the think of a single atom,
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the cloud of one atom of the electron is sort of perfectly cancelled when I look
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outside by the positive charges because there's a cemetery in the problem.
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But if I put on an electric field, I distort that and the system develops a dipole moment.
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But since it wasn't there, in the absence of a field, I can see that the direction was set by the electric field.
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You can see that that polarisation is just linearly proportional to the field.
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And when I do this, these displays and the same thing would be true.
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I'm not going to run this argument, but it'll be similar if I did it for Magnetisation.
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So what that means is that these coarse grained fields that I introduced are actually linearly proportional to the electric field,
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the external electric field, the externally imposed electric and magnetic fields.
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Now, of course, this can only be true on average distances, because microscopically that doesn't make any sense.
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The real electric and magnetic fields fluctuate, but as long as I'm willing to give up a little bit of information, these things must be true.
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Now, it turns out that this can be captured in a very nice way.
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I can just go into Maxwell's equipment, make the Lagrangian I wrote down,
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and just add two terms epsilon and a dielectric constant in a permeability like so.
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But otherwise nothing has changed. What these do will change the speed of light.
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So which was one in purest units before, will now become square root of Epsilon Times new.
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But that's it. They've changed some properties, but it's all absorbed in those two constants.
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So our lesson is that every insulator is effectively a new vacuum for electrodynamics.
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That's what insulators are vacuum. So what we want to ask is kind of an insulator provide a vacuum where that
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electrodynamics has a hidden missing term that we the data B term that we'd like to get.
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So what does that mean for an insulator to do that? So the insulators we normally encounter, as I've said, have this.
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And what I want to do is find a way for an insulator to get this additional piece that will check and others wrote down, which is Theta E Derby.
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So if I want to ask what that does, may be a good way to do it is to write down how Maxwell's equations change when I put that in.
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So there's a bit of painful work to do that. But what I've done is write down the I leave them as an exercise for the reader.
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So homework for you, you put that in and what you find is that you get a mess.
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It looks not particularly edifying. What if I write it like this?
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You just get a bunch of extra pieces in Maxwell's equations, but if I stare at them, I see that actually they can be got off in a very nice way.
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This if I pull out a piece here, it just looks like if I change my polarisation to redefine it,
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to include this little piece B And redefine my magnetisation, I go back to the old Maxwell equations.
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So what I would say is that the stern tells me that a magnetic field,
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in addition to introducing a magnetisation to my material, also somehow magically generates a polarisation.
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And similarly, an electric field, apart from changing the polarisation of the medium, also introduces the magnetisation.
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So what we've got is somehow a material that's responding in a crossed way.
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It's sort of E is triggering B and B is triggering.
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And maybe that's not so surprising because I've sort of gone off diagonal over here because I have an E metre.
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So intuitively it makes sense. And so what I want to ask is how does this effect, which is called a magneto electric polarise ability,
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possibly appear in a solid and it's actually not easy to get that in a classical solid.
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So to understand where that comes from, we have to go into the quantum theory of solids.
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So again, you know, Q groans here for some fraction of you.
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So electrons in solids, they describe like electrons everywhere.
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In the absence of relativity, apply the Schrodinger equation. But the SchrÃ¶dinger equation that they satisfy is very special because it's periodic.
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If I shift the coordinate by a spacing of in the spacing between atoms, what if everything looks exactly the same?
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Because if I just run by one unit cell over the solid looks identical.
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And so Felix Bloch, shortly after the advent of quantum mechanics,
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Felix Bloch actually pointed out that the eigen states of those equations are very special.
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They can always be chosen to be a plane wave and a periodic function that allows us to solve these problems.
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So this is the foundation of thinking of the quantum theory of solids. And so I want to take two features of the solution.
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So rather than, you know, go to the maths of that,
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I just want to give a representative picture of how you might see the solutions of these equations on a solid.
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Now we traditionally label the solutions by some number, some label K and index N,
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and what I'm drawing here is the different allowed solutions of this problem labelled by the quantum numbers that I have.
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And the two things you want to look at from the solution are the first is that if I look at the solutions and I just look at their energies,
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there are some energies where I have solutions,
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their energies where I don't have solutions, the energies we don't have solutions imaginatively called gaps and insulators correspond to the case.
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When you have electrons, you have an electron in each energy level in this region, and then you stop right here.
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So to put in the next electron, you have to pay an energy. Set by the Gap.
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That's the quantum mechanical statement. The charges are bound in insulators because you have to pay energy to move them around.
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They're not mobile. They're bound in these states. And so this is important because if I if I ask about questions of energy is higher than this gap,
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then my picture of insulators as effective vacuum will break down because I've got charges moving around.
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That's what controls when you say low energy,
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you have to say low compared to what it's load compared to the gap the creating expectations that insulator.
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The second feature is actually something that we've sort of known,
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but its implications were unexplored for about the first 50 or 60 years of the theory of solids, which is the following.
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So in it, in free space, we can we have a free particle.
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We can label states by momentum because momentum is a good quantum number.
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A solid isn't free space, but it's not complete chaos randomness either.
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It's got a regular array of sites. And so in that limit, the momentum becomes actually a periodic variable known as the crystal momentum.
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In this picture, what that means is that these green dashed lines should be identified.
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And really I'm describing things that live on a circle. And so what that means when I think of things that live on a circle are that the
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allowed momenta form circles in one dimension or tauri in higher dimensions.
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So these are the sort of allowed states. So somehow the states that I have, if I think about their labels, the labels live on a Taurus.
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And that's sort of an important fact about solids.
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And this the stories is known as the Brill One Zone, and that's where all the action takes place in solids.
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So why would I care about labels that live on a Taurus? Well.
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That feature has particularly important topological consequences.
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So something we're taught in first quantum mechanics course you take, you know,
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all introductory courses tell some lies and this is one of those lies that usually doesn't have any consequences.
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And the lie is that you don't need you don't ever care about the phase of a wave function.
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That's almost true. The one case you care about, the phase of the wave function is if it changes and it changes in a way around a loop,
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then you come back to the place where you started. So this is sort of, you know, one of my favourite authors, Terry Pratchett,
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has a quote that coming back the way you start it is not the same as never having left.
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Well, this is an example of that. So if I wind around this loop and come back, something has changed.
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And so that change was actually something that Michael Berry pointed out, known as Berry's phase.
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And so people had not appreciated that solids would be such a rich place for Berry's phase,
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partly because the wave functions fundamentally have this periodic direction that I can wrap around.
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And so if I think of these block states and think about how they move as I change,
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these are parameters of these states, these labels, then there are all these phases that they can pick up,
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and that non-trivial winding, surprisingly, can lead to new forces on electrons, and those forces mimic electric and magnetic fields.
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And in very mathematically identical ways, they can sort of be shown to be exactly equivalent in some ways.
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And in particular, such forces can give rise to a theta term.
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So I've written a rather unedifying equation over here. So this is something like a vector potential, but it's sort of souped up.
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It has additional indices because it's got matrices that every place where used to have a number for the vector potential of electromagnetism.
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And it depends on some very complicated way on these microscopic wave functions.
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But it turns out that once you've repackage this, this looks like a very familiar form for topologies.
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So Jim Simons was a mathematician who discovered these many, many years ago.
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He went on to become the chair of the maths department at Stony Brook, and his hobby was investing.
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And he left academia to found Renaissance Technologies, which is a sort of hedge fund.
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That's one of the more successful ones. Fun.
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Funnily enough, in a nice twist, his foundation now funds an enormous amount of research into areas like topological solids, including my own postdoc.
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So, you know, very grateful for people who have gone on to do things outside of physics.
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Help for us. So coming back to our story, the microscopic details are really quite gory.
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So if I looked at one of my textbooks that I look up to these things and I said,
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Let's see if I can come up with an elementary solution, it turned out that this particular time is the very last chapter.
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It's almost the very last equation of this textbook. So I'm not going to go into that.
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So, you know, nobody wants to see how the topological sausage is made. We just want the product.
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So what I'm going to try and do is take the spirit of an effective field theorist and just ask.
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What are the consequences of solids in the effective field theory?
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So let me just keep this term in there, but still think of this thing as a solid state physicists and ask What are the experimental consequences?
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And what could I observe in experiments given that this term is sitting around?
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And so what I want to do in this rest of the talk is give you illustrations of what the consequences of this are.
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Okay. The rules of the game. Clear. Very good.
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Maybe it's actually even though we usually wait for questions at the end, it's good to maybe pause here because I want to change gears and go back.
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So any questions on this aspect so far? Very good.
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Everything was clear. So everyone's going to get first class results on the exam that I'm handing out shortly.
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So the first thing I want to do is make some very general remarks.
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And again, in the spirit of thinking about theoretical physicists, we know we don't really like to go into details if we can avoid it.
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So I want to ask very general things based on two guiding principles, which are topology and symmetry.
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Most of you are familiar with symmetry. Both seem that topology is perhaps a new entrant into how we think about physical problems.
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So the first thing to remember, and this is something actually conveniently enough, Joe has this on the board over here.
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In quantum mechanics. We actually don't care about engines themselves.
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We only care about their effect on the equations of motion. And we care about how they enter quantum mechanics.
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And really, Lagrangian is like the action term only into quantum mechanics in an exponential.
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So that means that if I stuck that in that term, I wrote down always sits in this e to the I time stuff.
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So why is that important? That's important because there's a calculation I can't do on the board here.
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But if you take my word for it, the structure of electromagnetic fields requires that that integral is actually something that I know.
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It's just e to the eye. Peter Times And what data was this number I gave you?
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That forces theta and theta plus two pi to always lead to the same physics.
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And so if two things lead to the same physics, we just think of them as being indistinguishable.
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So we demand that data and theta plus two pi always have the same consequences.
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So they must be the same. So theta lives on a circle. So this goes back to Joe's original assertion that axioms are angles.
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So this is the proof that it's an angle and not just an arbitrary parameter in a theory.
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Because angles live on a circle. The second point is to ask about how symmetry is might constrain this angle.
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So notice at this point I have not said anything about what the nature of the media that I'm sitting in.
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It's right. I've just told you that it produced this data term. That's it.
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Now I'm going to ask about two symmetries that are perfectly reasonable to ask about solids.
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The first is the solid has no magnetism sitting around, so it's not magnetic.
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There are magnetic solids. I'm not talking about them. I'm just thinking about non-magnetic systems.
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The second thing is I'm going to ask that the solid has a centre of inversion.
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That's a fancy way of saying the solid has some kind of internal mirror reflection symmetry.
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It looks if I send X to minus X, it stays unchanged, so all the atoms will line up exactly where they are.
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If I just happen to flip all my axes, so if I use right hand rule or left hand rule, the solid doesn't care.
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That's the second step I'm going to do. Now.
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Why do I care about these two cemeteries? Well, they do something rather nice.
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So. Under the first time reversal symmetry.
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What I'm doing is taking space and time and leaving space unchanged and flipping the sign of time.
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Electric fields don't care about this, but magnetic fields because they care about currents.
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We'll start currents circulate in the opposite direction because you run time the other way.
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And so magnetic fields flip sign under this operation. This is why it is very important that I said the solid is non-magnetic because if it
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has frozen in magnetic fields then it would not look the same on the time reversal.
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And that's one symmetry. The second symmetry is inversion symmetry, which is the exact opposite.
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It sends x to minus x, but leaves T unchanged. And since if you think about a solid, it generates a think about an electric field.
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It has a spatial coordinate. So if I flip the sign of space, it goes the other way, right?
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So. Magnetic fields. It's a bit more subtle way you can see that magnetic fields don't care is you can imagine
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taking a current loop and putting it in a mirror and it will still circle it the same way.
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On the other side, the magnetic field will point in the same direction and it's mirror image.
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So magnetic fields don't care. And so why do what these two terms do, though,
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is that notice that they don't constrain ordinary Maxwell electrodynamics at all because Max Electrodynamics has e squared and they have B squared.
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E squared doesn't care whether E goes to minus C or dot.
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Now that is B squared,
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but an E not beta really doesn't like this because it flip sine east is unchanged and b flip sign B stays unchanged in e flip sign.
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So if I enforce these symmetries there, if any, if I ask the theory to be invariant under these symmetries,
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the equations of the theory have a change in sign.
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Every time I see an equation with theta,
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I have to put in a minus theta and demanding invariance under these symmetries requires theta and minus theta to be the same.
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So this is an elementary thing. If I had if theta were not an angle, it could only be one solution to this equation.
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That would be zero. But actually if they doesn't angle this equation as two solutions.
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And that's because PI works perfectly well.
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You see that if I take pi to minus pi, it looks like I've changed the sign,
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but I've already agreed that I'm minus phi and you're just going halfway around the circle.
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And it doesn't matter whether you go halfway around this way or halfway around back way.
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And so there are exactly two solutions consistent with these Symmetries Zero and PI.
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All other values are forbidden. If I have either of these symmetries, I can only have two solutions to these equations.
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So a remarkable thing I've understood is that having had the stamps,
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I could I claim that the solid has this term fine comes out but if I demand either inversion or time reversal symmetry, I can fix this term.
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It doesn't matter any. None of the microscopic details of the solid matter.
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This term has to be zero or it has to be five. That's it. So what I.
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So that's a nice fact. So for the rest of the talk, I'm going to assume these symmetries and so they're equal.
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Zero. I understand what it does because that's just Maxwell and we've gone through that.
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So I've got to ask, what happens if I fix theta equals pi? And that's what I want to do.
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So what does this mean in a solid what is this non-zero data angle mean?
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So we saw that naively when I did the wrote those equations down earlier,
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I said I could rearrange them a little bit and see that a magnetic field induces an electric polarisation.
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Electric field induces a magnetisation. That's a very appealing picture.
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But we do know that what we really care about when we solve things, our equations of motion, we don't care about all of these labels of things.
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We care about how the equations of motion change. And so if I do that, I'm actually going to be in for a little bit of a surprise.
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So let me there's actually quite a straightforward calculation. So let me take you through it step by step.
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These are the equations that I wrote down earlier. These are the modified Maxwell equations.
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So you've got this expression of this expression. Right. Remember, there's only two of the Maxwell equations.
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We'll come back to that in a second. Two of them were not modified by there being matter in the system.
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All I'm going to do now is just take the red bits.
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I've got a divergence outside a curve over here, so I have to dust off my vector calculus identities.
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Full confession. I used Google to figure this out. I've forgotten half of them, so I have to go back and get the signs right.
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I put that in and I rearrange and what I've just done is keep the conventional natural equations in blue on the left and the changes on the right.
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And, you know, this is basically a product rule. I have a divergence.
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I have two things. Either theta could vary or be could vary.
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And so I just have to combine the two changes. Okay.
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So, so far, so good. This is just expanding and rearranging.
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But now I have to remember that there were two Maxwell equations that have never not entered the game, but they're waiting in the wings.
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And those are the two that didn't care that they were sources. They were pristine, unchanged.
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They will always be true. So let me see what they do.
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What they do is actually cancel two terms over here, arrange them in such a way.
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These two equations precisely cancel these two bits here.
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And actually, now I have a surprise. If I look at what's left notice of this new term with theta as a constant in both space and time on the right.
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Either I have a gradient of theta or I have a time derivative of data.
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So what I've realised after doing all this work of getting this term in a solid is
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that the equations of motion apparently don't seem to care if this term is there.
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They just don't care if it's a constant.
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Which is kind of it's a bit deflating because I've done all this work and I've just found out that in the end, nothing seems to change.
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But I can engineer a situation where these terms change.
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And it's something that's, again, familiar from thinking about the physics of solids is to think about boundaries between media and vacuum.
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And at a boundary, the vacuum has theta equals pi.
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We know that because there was no either metre.
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But imagine that I had a solid with equals by I can ask what happens at the interface between these two regions.
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So this is sort of a standard thing that we do a lot of Vietnam.
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If you remember your secondary in them, you spend a lot rather in order the amount of time stuck in the rod cam doing boundary value problems.
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That's what we're going to do now. So let's look at such an interface that I'm going to talk a little bit about the physics of such an interface.
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So I'm going to look at so I've got a region up here with it,
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equal SPI a region down here with it equal zero being a so I'm going to imagine this goes on
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forever so I don't have to worry about things fringing on the boundaries and stuff like that.
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And what I'm going to do is I don't I'm interested in asking.
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So normally when I think about a solid,
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what I do is put on an electric field perpendicular to the interface and ask how it does polarisation, but that's an ordinary solid.
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So what I'm going to do is something that you probably don't normally do in a solid,
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which is put on a magnetic field perpendicular to the interface and ask what happens inside.
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So I've just translated for the particular coordinate system of the equation we had earlier, so that the divergence of the electric field, remember,
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this is just like Gauss's law talks about divergence of electric fields,
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and this says that what's on the right side is a source of electric fields, which is a charge.
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And so it says that there's a charge that is proportional to the change in theta as I move in the Z direction, but I have an interface.
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So that means that theta is constant everywhere but jumps at that boundary just so that boundary jumps.
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So what this equation is told me is that there's actually a surface charge density that's triggered.
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So I put on a sort of bizarre I put on a magnetic field and suddenly there's a surface charge density that's produced at that boundary.
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But this is just a consequence of the forces of acting inside that solid,
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that things start moving around so as to produce a charge density on that surface.
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Now I force the electric field outside to be zero,
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which means that the charge here somehow when it create an electric field somewhere and the logical
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consequences that there's an electric field created inside the material parallel to the magnetic field.
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So what I've produced is an electric field parallel to B,
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and what's remarkable is that that electric field has a strength that's related to that of the magnetic field by the fine structure constant.
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So I've done, you know, I've done something remarkable because I've just taken Maxwell's equations with this extra time thrown in.
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And I've told you that there is in principle a way to measure fine structure constant by doing an electrodynamics measurement,
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which is quite, quite remarkable if you think about where that came from.
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I can do the same thing the other way, and that will actually lead us to something quite neat.
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So what I'm going to do is now take an interface between three equals zero and theta equals pi.
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And let me just set it up before I do that and I'm going to put an electric field now parallel to the interface.
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Now, again, if I take this equation, kernels of magnetic fields,
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whatever appears on the right hand side should be a current because currents are sources of magnetic fields.
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So what it tells me is that there's a current that points in that that set by this that has this gradient and Z times e y.
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And if you unpack all of this, you find that there's actually a current that points into the board at this interface.
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So it's sort of pointing into the plane perpendicular to the electric field.
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And that current looks set up a magnetic field parallel to it.
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So just the exact reverse of that effect. And again, this proportionality constant is linked to the fine structure constant.
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So again, something remarkable, but actually this effect,
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I'll come back to the first effect towards the end, but this effect is actually something familiar.
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So on the interface between the theta equals zero and theta equals pi region,
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I put on an electric field and the electric field generates a current that's perpendicular to it.
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Now, this is actually something that is a very, very, very old effect.
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It was discovered by Edwin Hall during a speech this studies in 1879.
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This is something known as the hall effect. We don't think of it in the setting.
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The hall effect usually emerges when we think about.
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An electron could be in some looking at a surface, but I've got a magnetic field out of the surface and I'm trying to push a current through this.
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But of course, if I have moving charges, there's a Lorenz force in moving charges.
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So the current push a car into one direction, the magnetic field makes it veer away from the direction I want to push it in.
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So if I want to maintain that current, I'm going to have to apply additional force to keep it going in a straight line.
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Otherwise it keeps going off in the other direction. So it's like having a car with a wobbly sharing wheel.
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I have to actually put a bit of force to even keep it going in a straight line if my wheels aren't aligned.
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So you can just think of that, that analogy.
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And so there would have to be an electric field perpendicular to the current in order to keep a steady current flowing.
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So this something that's well known. This is the whole effect.
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But what we've heard is that remember that if I go back to the previous slide over here, these coefficients were all fixed.
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There's a fine structure constant here. The theta had to jump from zero to pi, so that jump is quantised.
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If I have a system with the symmetries, there's no intermediate value of data.
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It has to jump from zero to PI because on the surface it can't have any other value, so it has to jump abruptly.
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And so what must be true is that that coefficient, there's no freedom in that coefficient.
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Alpha was fixed, the jump in theta was fixed.
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And so if I translate that into more conventional units, so the whole conduct conductivity is usually called sigma using initial.
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So the subscripts x and Y to say that it's a response of a current that's perpendicular to a magnetic field.
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So one is an X and the other is in Y.
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And it turns out that once you put in all the constants, you get one half a but a minus here because in this particular case, the minus.
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But I could rearrange that to get a plus sign in the geometry times squared over H and E square over h is a combination of fundamental
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constants that has the units of resistance and in this case the same as resist units of inverse resistance or conductance.
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And so that happens to be the same units as conductivity in two dimensions, which is where we are.
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So I've got this remarkable result that there's a quantised response that is sort of a universal thing, and there's no freedom in this response.
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There's just a half time something. So I've got something that's very, very rigid and fixed.
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But in fact, this is really, really, really surprising. The fact that you get this half here is really surprising for a profound reason.
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It turns out that if you give me a purely two dimensional system, well, the whole conductance has to be an integer.
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Times is quite over.
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H Unless you have very, very strong electron electron interactions which we don't have over here, we're imagining the solid is very ordinary.
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It doesn't have anything exotic going on. So if you have an integer times it's quite over.
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Well, any perturbation I do to the surface I can always think of as something like gluing stuff onto the surface.
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Let me glue whatever I can give you a complete freedom to do whatever you like to the surface.
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You can glue any two dimensional system you like, but don't modify anything on either side.
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You can't put a three dimensional system on there. You can do anything in two dimensions.
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Whatever you do, you will actually not be able to change the oddness of this.
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So let's see how that works.
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I'm proposing doing a sequence of things where I start off with my S quite over to H, so maybe I glue on something which has is quite over.
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Sorry, glue on something that has squared over equal.
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I get three squared over two h. Let's suppose I glue on something which at minus two is quite a rate that's allowed.
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I get minus B squared over two h. So notice I'm changing this number drastically.
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But what I can't do is remove that two in the denominator. It's always a half integer.
380
00:32:45,150 --> 00:32:48,900
So this half mess is very strange. It can't be removed on the surface.
381
00:32:49,260 --> 00:32:54,180
And so what we've actually found is something interesting. The surface is actually illegal as a two dimensional system.
382
00:32:54,510 --> 00:33:00,360
No two dimensional system with the same laws of physics that we're put in with weak interactions could ever do this.
383
00:33:00,960 --> 00:33:07,440
So this is okay because the only I defined my surface as one between something that has theta equal zero interact
384
00:33:07,440 --> 00:33:13,800
with spi and such a surface fundamental means the third the third dimension in order to define its existence.
385
00:33:14,130 --> 00:33:18,000
That's okay because it's embedded in three dimensions. So it's a very peculiar property.
386
00:33:20,040 --> 00:33:26,009
And so I should point out that actually understanding I just wrote this down the sigma x y
387
00:33:26,010 --> 00:33:30,270
equals this squared age actually understanding this is one at least three Nobel Prizes to date.
388
00:33:30,270 --> 00:33:35,100
And so, you know, it's it's a deep piece of physics, but it's sort of incidental to the story here.
389
00:33:35,970 --> 00:33:45,270
Okay. So I've done all of this, but I've told you that these consequences of materials which have this data electron action, electrodynamics.
390
00:33:45,510 --> 00:33:50,280
What I haven't convinced you is that there are materials that do this. So you've got to take that on faith so far.
391
00:33:50,670 --> 00:33:57,210
Let me give you some examples of materials that do this, focusing again on the two symmetries I talked about time, reversal, inversion, symmetry.
392
00:33:58,940 --> 00:34:02,030
So there's one little subtlety that comes in.
393
00:34:02,660 --> 00:34:11,210
If I have a solid that has the state equal spy and let's assume that there's never magnetism anywhere, the solid is always time reversal invariant.
394
00:34:12,770 --> 00:34:19,400
Well, it turns out that an insulator can have a hole conductance only if time reversal is broken.
395
00:34:19,490 --> 00:34:23,180
It's forbidden for an insulator that preserves preserves time reversal.
396
00:34:23,810 --> 00:34:27,620
That's a complication, because that seems to contradict what I just wrote down.
397
00:34:28,250 --> 00:34:32,120
Right. So the resolution of this is that somewhere my assumption should break down.
398
00:34:32,450 --> 00:34:38,270
And the assumption that I had to get this piece of physics on the surface was that things were insulating everywhere.
399
00:34:38,740 --> 00:34:41,130
The only way to resolve the tension between these ideas,
400
00:34:41,150 --> 00:34:45,680
the fact that if things were insulating everywhere, the surface would have to have a whole conductance.
401
00:34:45,980 --> 00:34:49,970
But if things were time reversal invariant everywhere, the surface couldn't have a hall conductance.
402
00:34:50,240 --> 00:34:54,410
Something has to give. And what has to give is that the insulating everywhere piece?
403
00:34:54,890 --> 00:34:59,420
So it's actually a remarkable fact that if I demand that the symmetry of nature,
404
00:34:59,420 --> 00:35:03,050
time reversal is present everywhere in the system and the surface and the bulk throughout,
405
00:35:03,560 --> 00:35:08,660
then the surface between equals pi and a theta equal zero insulator has to be a metal.
406
00:35:09,380 --> 00:35:13,520
And the reason that's okay is because metals actually have mobile charges that can get
407
00:35:13,520 --> 00:35:17,000
rid of that surface charge and those surface currents and sort of cancel the effect.
408
00:35:17,790 --> 00:35:24,029
So we've actually discovered I'm taking you through the discovery of actually a profound fact that a time reversal, symmetric system,
409
00:35:24,030 --> 00:35:30,780
which I said equals PHI, is actually in a different state of matter from our vacuum and from an ordinary insulator.
410
00:35:31,050 --> 00:35:33,150
It's something known as a topological insulator,
411
00:35:33,330 --> 00:35:40,170
and it's special because it's interface with our normal vacuum will be a perfect metal as long as time reversal is preserved.
412
00:35:40,680 --> 00:35:46,620
So it turns out that this observation was actually predicting the people predicted this effect comments about 15 years ago.
413
00:35:46,830 --> 00:35:52,979
And one of the people who did this, actually, the person who worked the most individualistic prediction was a graduate student who is working alone.
414
00:35:52,980 --> 00:35:57,750
Rahul Roy, who some years after he made this discovery, spent several years in Oxford as a postdoc.
415
00:35:57,750 --> 00:35:59,850
So sort of a nice Oxford connection there.
416
00:36:00,570 --> 00:36:07,110
So I've just told you about all these special surface effects, but now I've kind of poured cold water on that and said, Oh, the surface is a metal.
417
00:36:07,110 --> 00:36:12,690
So this effect, I told you, is not there. Well, it turns out, though, that that metal itself is very special.
418
00:36:13,350 --> 00:36:20,640
That metal has to remember somehow that if it broke time reversal, symmetry would have to have this half integer conductance.
419
00:36:21,030 --> 00:36:25,800
And it turns out that there's a very special, special consequence of that.
420
00:36:26,310 --> 00:36:31,350
So if you could measure and this is something that experiments next door in the cloud in lab do,
421
00:36:31,800 --> 00:36:37,379
you can measure the surface dispersion of metals very well. And if you measure that, there's a certain thing you can count in another.
422
00:36:37,380 --> 00:36:39,810
If you can see these, it says one, two, three, four and five.
423
00:36:40,140 --> 00:36:45,600
What it's counting is the number of times the dispersion crosses this energy, which is the Fermi energy.
424
00:36:46,020 --> 00:36:52,650
And it turns out that you can you cannot cross it an odd number of times except in this topological special metal.
425
00:36:52,890 --> 00:36:56,010
And so that measurement tells you that into it is what that equals.
426
00:36:56,010 --> 00:37:02,160
Pi exists in this sort of convoluted way, but it's actually evidence that we have an insulator where theta equals pi inside.
427
00:37:02,550 --> 00:37:09,150
Sort of a remarkable fact. But of course, there's another cemetery and that's a bit more amenable to what I want to do today.
428
00:37:09,570 --> 00:37:13,800
So if there April Spy is enforced by inversion symmetry, there's never a problem.
429
00:37:14,130 --> 00:37:18,140
That's because I can't set up the start experiment with inversion symmetry.
430
00:37:18,150 --> 00:37:24,990
So if I gave you a surface that below is one material, above is another material and I send x to minus x,
431
00:37:24,990 --> 00:37:28,470
I've swapped them around because now one material is on top and the other is below.
432
00:37:28,860 --> 00:37:34,769
So a surface can never preserve inversion symmetry. So that whole set of logic breaks down at the surface.
433
00:37:34,770 --> 00:37:39,030
And the surface can always have this property where something can rearrange and be insulating.
434
00:37:39,420 --> 00:37:42,510
And it turns out that other symmetries can do this in various ways.
435
00:37:42,780 --> 00:37:46,919
But for purists, the term axial insulator is reserved for inversion,
436
00:37:46,920 --> 00:37:52,020
symmetric or similar systems where you have data close by in the bulk and it's quantised on the surface.
437
00:37:52,230 --> 00:37:56,129
Break symmetry and it's no longer this perfect metal. So how would you observe that?
438
00:37:56,130 --> 00:38:00,090
Because you no longer have this ability to look to count things on the surface.
439
00:38:00,900 --> 00:38:04,920
So it turns out that these Hoffmann to your whole effects can be done in a clever way.
440
00:38:05,340 --> 00:38:09,540
So there's a effect that is studied in optics, which is called the Faraday effect,
441
00:38:09,750 --> 00:38:13,530
which is that the plane of polarisation, of light rotates in certain solids.
442
00:38:13,890 --> 00:38:17,580
And it turns out a prediction of this material is that there's a quantised Faraday effect.
443
00:38:17,820 --> 00:38:19,620
And so that's something that one can work out.
444
00:38:19,890 --> 00:38:27,180
The only problem with that is that it's quantised if you have a single surface and you can isolate it, but it's very hard to do that with real solids.
445
00:38:27,180 --> 00:38:32,280
Theorists can talk about a single surface, but if you go to a lab, you're going to have to surface this for any finite object.
446
00:38:32,430 --> 00:38:35,250
And, you know, you have to do experiments on finite objects.
447
00:38:35,880 --> 00:38:46,080
So it turns out that this is an ongoing challenge to understand where to actually come confirm that this piece of electrodynamics works out.
448
00:38:46,350 --> 00:38:51,090
So people are still trying to do this in the lab, and they do various ingenious ways of disentangling these effects.
449
00:38:51,940 --> 00:38:56,700
Okay, so to close, I just want to flag one last piece of exotica,
450
00:38:56,700 --> 00:39:03,180
but I think that's sort of interesting because it sort of suggests how solid state materials offer prospects to explore very nice new physics.
451
00:39:03,540 --> 00:39:07,409
So that's going back to this first of the two interface effects I talked about.
452
00:39:07,410 --> 00:39:13,260
So the second one I spent a lot of time on, which is saying that if I put on electric fields, I generate currents.
453
00:39:13,710 --> 00:39:17,040
But the first one was that if I put on magnetic fields, I generate charges.
454
00:39:17,400 --> 00:39:23,430
So imagine I have a sphere with theta equals zero embedded inside a region with that equals by tiny little sphere.
455
00:39:23,940 --> 00:39:28,070
And inside that sphere, I drop in a point source of magnetic field.
456
00:39:28,110 --> 00:39:31,709
The thing that's forbidden by the first by one of Maxwell's equations. But never mind.
457
00:39:31,710 --> 00:39:35,010
For now, let's stick in a source of magnetic field.
458
00:39:35,460 --> 00:39:40,440
If I did that. Well, what I've told you is that, you know, I've got magnetic field everywhere.
459
00:39:40,440 --> 00:39:46,139
I'm going to interface between data equals PI, so actually create electric fields parallel to the magnetic field.
460
00:39:46,140 --> 00:39:49,080
Sorry. Create electric fields everywhere.
461
00:39:49,620 --> 00:39:55,500
If I look outside and I'm only listening to theta equals by region, I don't ask what's going on under the hood.
462
00:39:55,980 --> 00:39:59,070
Then this looks as though I have an electric field of a point charge.
463
00:39:59,610 --> 00:40:01,530
So it looks like if I put in a magnetic monopole,
464
00:40:01,530 --> 00:40:09,390
it triggers an electric field that looks exactly like that of a point charge with a strength related to the magnetic field by Alpha.
465
00:40:10,300 --> 00:40:14,260
So it looks like a magnetic charge triggers an electric charge.
466
00:40:14,770 --> 00:40:18,459
Now, this argument was a bit convoluted, but it turns out that you can shrink.
467
00:40:18,460 --> 00:40:22,870
This data equals zero region all the way to nothing and the argument would still be there.
468
00:40:23,140 --> 00:40:27,219
So monopole moving inside this material would actually carry an electric charge.
469
00:40:27,220 --> 00:40:32,170
And it's something known as the die on. So this is something called the Witten effect, and it's a rather beautiful piece of physics.
470
00:40:32,860 --> 00:40:38,079
So you'd like to be able to observe this, but there's a problem with that, which is that we don't know about magnetic monopoles.
471
00:40:38,080 --> 00:40:38,889
So in a Stanford,
472
00:40:38,890 --> 00:40:44,260
they have an experiment that's designed to search for a magnetic monopole that's just been sitting there looking for a monopole to swing by.
473
00:40:44,620 --> 00:40:49,430
And in 40 years, they've seen one signal that's very controversial. So we don't think that we can find them.
474
00:40:49,450 --> 00:40:51,760
They're not they're not hanging around our universe very easily.
475
00:40:52,240 --> 00:40:58,450
But I think maybe if I've convinced you of one thing, it is that, you know, solid state materials, each of them are in some sense their own universe.
476
00:40:58,450 --> 00:41:02,979
They can generate interesting new back here. And here's an interesting Oxford connection.
477
00:41:02,980 --> 00:41:09,520
So about 15 years ago, a group of people who have either past or very present Oxford Connections predicted that
478
00:41:09,520 --> 00:41:13,600
there are certain magnetic materials that could emulate the physics of magnetic monopoles.
479
00:41:13,900 --> 00:41:19,900
So there's a long and interesting story there. But the neat piece of physics there is that the electric and magnetic fields that are
480
00:41:19,900 --> 00:41:24,490
involved in these monopole problems are sort of emergent electric and magnetic fields.
481
00:41:24,910 --> 00:41:29,020
But very recently people have pointed out that those electric and magnetic fields,
482
00:41:29,020 --> 00:41:35,410
even though they're sort of not the standard ones we think of in max electrodynamics, can actually have their own theta terms.
483
00:41:35,650 --> 00:41:41,229
And so there have been recent papers that predict the this wouldn't effect and having observable
484
00:41:41,230 --> 00:41:46,209
consequences of actin electrodynamics in sort of materials where you can actually do measurements.
485
00:41:46,210 --> 00:41:50,930
So that sort of fictitious sort of special emergent monopoles can have consequences.
486
00:41:50,930 --> 00:41:54,069
And so we can test the sort of rather exotic prediction of field theory in
487
00:41:54,070 --> 00:41:57,430
an experiment that you could realistically imagine sitting on this tabletop.
488
00:41:58,620 --> 00:42:02,160
So let me. I'm sorry.
489
00:42:02,170 --> 00:42:05,670
Going the wrong direction. Let me go back to my sort of two propositions.
490
00:42:05,700 --> 00:42:10,830
The first, I think I hope I've convinced you this is the the the sort of second statement that will check made was that,
491
00:42:11,070 --> 00:42:15,800
you know, it seems clear that we can emulate the physics of axioms in the solid state material.
492
00:42:15,810 --> 00:42:21,390
So just to summarise, you know, we start with the idea that if you think about insulating matter,
493
00:42:21,570 --> 00:42:24,150
then you can view it as a new vacuum for electromagnetism.
494
00:42:24,540 --> 00:42:32,370
And we know now several insulators where this conspiracy of quantum theory allows you to have Axion electrodynamics as the effective description.
495
00:42:32,820 --> 00:42:35,070
And there are many active experimental searches going on.
496
00:42:35,070 --> 00:42:39,450
I gave you a samples of some of them to look for consequences of these Axion electrodynamics.
497
00:42:39,870 --> 00:42:44,640
And I should say that everything I talked about today is treating theta as an angle and not a field.
498
00:42:44,910 --> 00:42:51,450
But there are situations where theta can become a field where this physics comes in, when you have an insulator with some additional complexity,
499
00:42:51,450 --> 00:42:58,409
where there's some magnetism or some other charges moving in very constrained ways and they can generate dynamics for these Axion field.
500
00:42:58,410 --> 00:43:02,610
So you can actually there are also prospects for observing dynamical axioms in the solid state.
501
00:43:04,620 --> 00:43:10,320
So the second point, which is whether or not actions have any physical reality, that study can be useful intellectual exercise.
502
00:43:10,560 --> 00:43:14,220
I have very little to say, but let me close with words of one of my heroes, Duncan Holden.
503
00:43:14,610 --> 00:43:18,890
It's very difficult to know whether something is useful or not, but one can know that it's interesting.
504
00:43:18,900 --> 00:43:20,760
So I hope you'll agree with that and let me thank you.