So for our next story we have Professor Sid Parameswaran who is a condensed, massive theoretical physicist, and Sid is going to tell us about axioms in the solid state. Thank you, Julia, and thank you all for being here. It's really nice to be back for one of these events in person. 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.
He promptly fell asleep, which I wasn't complaining about, but I'm hoping for a different outcome today. 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.
And so what I want to do in this stock is actually illustrate how some of those ideas apply in a setting, perhaps where we're more familiar with in everyday physics is sort of an introductory courses, which is Maxwell electrodynamics, but with an ingredient, which is what do these axioms mean for thinking about electrodynamics? 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.
Well, check. And you know, he famously coined the phrase Axion in its original context. And quickly, Jordan mentioned this sort of running joke on axioms. The word it sounds like a fancy particle was actually a form of laundry soap because it cleaned up problems in the standard model.
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, which is that whether or not axioms have any physical reality, that study can be a useful intellectual exercise. So that's great. That keeps us employed. The second point, he said, and that's the centre of the stock, is that it might be is not beyond the realm of possibility that fields whose properties partially mimic those of Axion fields,
can be realised in condensed matter systems. So it's the second proposition that's going to be my primary concern in the stock. 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. So what Joe talked about in high energy physics and what John will talk about is the Axion field theta as a fundamental field that describes Axion dynamics in quantum chroma dynamics and its original setting.
This is a challenge to with I stated goal of finding Axion like physics and solids because we know the standard model of solids. The standard model of solids is just electrons and ions governed by Maxwell's equations, interacting by Coulomb forces. And so there's no QCD here, there's no strong force. And so very clearly I can't change this description when I'm talking about solids, the sort of ultraviolet physics is fixed.
And so I want to think about axioms. And so my question is, how can I do that in a solid. So of course, if you've been on Saturday Theory before and seen a condensed matter session, you sort of know the answer. That instead of looking at the physics of the high energy scales, we look at low energy scales below at some very low scale that we can probe and experiments and ask for new physics that are emergent when many particles cooperate on those scales.
So the goal of this talk is to find a way to obtain. So what I've written down is the Maxwell Lagrangian. So this is just ordinary electromagnetism, so nothing exotic of it. So this is very similar to what Joe talked about when he talked about Axion like particles. So the Maxwell Lagrangian with the term that comes from Axion electrodynamics and the disclaimer is that for most of the stock, I'm just going to think of this theta as an angle rather than a field.
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. Okay, so that's what I'm going to do today. So. As a warm up. I want to start off by reviewing a little bit about Maxwell's equation. So, you know, Maxwell's equations are sort of things we know about and love in vacuum, but I'm going to be interested because of my solid state physicists I care about matter. So I'm interested in Maxwell's equations in media.
So this is part of the dreaded second year course where you have to deal with all kinds of terrible things for those. 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. So matter is a source of energy. We know that charges and currents and solids associated B and B and two of the Maxwell's equations.
Oops. Then get there are two of the Maxwell's equations don't really care about the fact that they're sources of energy. So those are the fact that there are no magnetic monopoles, although we'll come back to that later. And the second is that there are that the electric and magnetic fields are linked. But do equations do care? Gauss's law and you know, the curl of B the currents generate fields. So Faraday's laws, they care about fields.
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. Because depending on whether we think of the system as a metal or an insulator, the behaviour will be different. 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.
They're tied to each other and similarly currents don't roam freely, they're tied to sites of individual atoms. So if I draw a picture of a solid in a field, an insulator in a field, then it's just a bunch of little dipoles because every positive charge is pretty close to every negative charge. And if I looked at the current loops inside, they're all being fixed positions in space.
And if I dropped you into this insulator and I walked around and I could measure electrical magnetic fields, you'd see wildly fluctuating electric and magnetic fields, because every time you went past a positive charge, things would jump up and down. 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. That's very easy for me. That's all I need to do, you know? And what I do then.
And there's a mathematical, mathematical statement behind that blurring, which is, of course, screening. 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. 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. So you see that there's positive and negative charges. Cancel accepted boundaries over here.
And so that means that traditionally I can rewrite these charges in terms of physical things that I call polarisations and Magnetisation. 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. And the current is related to magnetisation and these are average, so they only really matter at boundaries.
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, and I can introduce new fields which have these sort of unedifying terms like the displacement field and the rather confusingly magnetic field strength, as opposed to magnetic field density, terrible historical terminology, but we're stuck with that.
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. They have no divergences and they have very simple equations of motion. So we're very happy because we can work with those and solve them. Okay.
So I want to make another specialisation, which is that if I turned off the fields outside, 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. So I'm not describing you a refrigerator magnet or something that could have a frozen polarisation. I'm describing things that only are polarised or magnetised when I stick them in a magnetic field.
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, the cloud of one atom of the electron is sort of perfectly cancelled when I look outside by the positive charges because there's a cemetery in the problem. But if I put on an electric field, I distort that and the system develops a dipole moment.
But since it wasn't there, in the absence of a field, I can see that the direction was set by the electric field. You can see that that polarisation is just linearly proportional to the field. And when I do this, these displays and the same thing would be true. I'm not going to run this argument, but it'll be similar if I did it for Magnetisation.
So what that means is that these coarse grained fields that I introduced are actually linearly proportional to the electric field, the external electric field, the externally imposed electric and magnetic fields. Now, of course, this can only be true on average distances, because microscopically that doesn't make any sense. 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.
Now, it turns out that this can be captured in a very nice way. I can just go into Maxwell's equipment, make the Lagrangian I wrote down, and just add two terms epsilon and a dielectric constant in a permeability like so. But otherwise nothing has changed. What these do will change the speed of light. So which was one in purest units before, will now become square root of Epsilon Times new. But that's it. They've changed some properties, but it's all absorbed in those two constants.
So our lesson is that every insulator is effectively a new vacuum for electrodynamics. That's what insulators are vacuum. So what we want to ask is kind of an insulator provide a vacuum where that electrodynamics has a hidden missing term that we the data B term that we'd like to get. So what does that mean for an insulator to do that? So the insulators we normally encounter, as I've said, have this.
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. 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. 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. So homework for you, you put that in and what you find is that you get a mess.
It looks not particularly edifying. What if I write it like this? 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. This if I pull out a piece here, it just looks like if I change my polarisation to redefine it, to include this little piece B And redefine my magnetisation, I go back to the old Maxwell equations.
So what I would say is that the stern tells me that a magnetic field, in addition to introducing a magnetisation to my material, also somehow magically generates a polarisation. And similarly, an electric field, apart from changing the polarisation of the medium, also introduces the magnetisation. So what we've got is somehow a material that's responding in a crossed way. It's sort of E is triggering B and B is triggering.
And maybe that's not so surprising because I've sort of gone off diagonal over here because I have an E metre. 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, possibly appear in a solid and it's actually not easy to get that in a classical solid. So to understand where that comes from, we have to go into the quantum theory of solids. So again, you know, Q groans here for some fraction of you.
So electrons in solids, they describe like electrons everywhere. In the absence of relativity, apply the Schrodinger equation. But the Schrödinger equation that they satisfy is very special because it's periodic. If I shift the coordinate by a spacing of in the spacing between atoms, what if everything looks exactly the same? Because if I just run by one unit cell over the solid looks identical.
And so Felix Bloch, shortly after the advent of quantum mechanics, Felix Bloch actually pointed out that the eigen states of those equations are very special. They can always be chosen to be a plane wave and a periodic function that allows us to solve these problems. So this is the foundation of thinking of the quantum theory of solids. And so I want to take two features of the solution.
So rather than, you know, go to the maths of that, I just want to give a representative picture of how you might see the solutions of these equations on a solid. Now we traditionally label the solutions by some number, some label K and index N, and what I'm drawing here is the different allowed solutions of this problem labelled by the quantum numbers that I have.
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, there are some energies where I have solutions, their energies where I don't have solutions, the energies we don't have solutions imaginatively called gaps and insulators correspond to the case. When you have electrons, you have an electron in each energy level in this region, and then you stop right here.
So to put in the next electron, you have to pay an energy. Set by the Gap. That's the quantum mechanical statement. The charges are bound in insulators because you have to pay energy to move them around. 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, then my picture of insulators as effective vacuum will break down because I've got charges moving around.
That's what controls when you say low energy, you have to say low compared to what it's load compared to the gap the creating expectations that insulator. The second feature is actually something that we've sort of known, but its implications were unexplored for about the first 50 or 60 years of the theory of solids, which is the following. So in it, in free space, we can we have a free particle. We can label states by momentum because momentum is a good quantum number.
A solid isn't free space, but it's not complete chaos randomness either. 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. In this picture, what that means is that these green dashed lines should be identified.
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 allowed momenta form circles in one dimension or tauri in higher dimensions. 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. And that's sort of an important fact about solids.
And this the stories is known as the Brill One Zone, and that's where all the action takes place in solids. So why would I care about labels that live on a Taurus? Well. That feature has particularly important topological consequences. So something we're taught in first quantum mechanics course you take, you know, all introductory courses tell some lies and this is one of those lies that usually doesn't have any consequences.
And the lie is that you don't need you don't ever care about the phase of a wave function. 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, then you come back to the place where you started. So this is sort of, you know, one of my favourite authors, Terry Pratchett, has a quote that coming back the way you start it is not the same as never having left.
Well, this is an example of that. So if I wind around this loop and come back, something has changed. And so that change was actually something that Michael Berry pointed out, known as Berry's phase. And so people had not appreciated that solids would be such a rich place for Berry's phase, partly because the wave functions fundamentally have this periodic direction that I can wrap around.
And so if I think of these block states and think about how they move as I change, these are parameters of these states, these labels, then there are all these phases that they can pick up, and that non-trivial winding, surprisingly, can lead to new forces on electrons, and those forces mimic electric and magnetic fields. And in very mathematically identical ways, they can sort of be shown to be exactly equivalent in some ways. And in particular, such forces can give rise to a theta term.
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. It has additional indices because it's got matrices that every place where used to have a number for the vector potential of electromagnetism. And it depends on some very complicated way on these microscopic wave functions. But it turns out that once you've repackage this, this looks like a very familiar form for topologies.
So Jim Simons was a mathematician who discovered these many, many years ago. He went on to become the chair of the maths department at Stony Brook, and his hobby was investing. And he left academia to found Renaissance Technologies, which is a sort of hedge fund. That's one of the more successful ones. Fun. Funnily enough, in a nice twist, his foundation now funds an enormous amount of research into areas like topological solids, including my own postdoc.
So, you know, very grateful for people who have gone on to do things outside of physics. Help for us. So coming back to our story, the microscopic details are really quite gory. So if I looked at one of my textbooks that I look up to these things and I said, Let's see if I can come up with an elementary solution, it turned out that this particular time is the very last chapter. It's almost the very last equation of this textbook. So I'm not going to go into that.
So, you know, nobody wants to see how the topological sausage is made. We just want the product. So what I'm going to try and do is take the spirit of an effective field theorist and just ask. What are the consequences of solids in the effective field theory? 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? And what could I observe in experiments given that this term is sitting around?
And so what I want to do in this rest of the talk is give you illustrations of what the consequences of this are. Okay. The rules of the game. Clear. Very good. 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. So any questions on this aspect so far? Very good. Everything was clear. So everyone's going to get first class results on the exam that I'm handing out shortly.
So the first thing I want to do is make some very general remarks. 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. So I want to ask very general things based on two guiding principles, which are topology and symmetry. Most of you are familiar with symmetry. Both seem that topology is perhaps a new entrant into how we think about physical problems.
So the first thing to remember, and this is something actually conveniently enough, Joe has this on the board over here. In quantum mechanics. We actually don't care about engines themselves. We only care about their effect on the equations of motion. And we care about how they enter quantum mechanics. And really, Lagrangian is like the action term only into quantum mechanics in an exponential.
So that means that if I stuck that in that term, I wrote down always sits in this e to the I time stuff. So why is that important? That's important because there's a calculation I can't do on the board here. But if you take my word for it, the structure of electromagnetic fields requires that that integral is actually something that I know. It's just e to the eye. Peter Times And what data was this number I gave you? That forces theta and theta plus two pi to always lead to the same physics.
And so if two things lead to the same physics, we just think of them as being indistinguishable. So we demand that data and theta plus two pi always have the same consequences. 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. So this is the proof that it's an angle and not just an arbitrary parameter in a theory. Because angles live on a circle. The second point is to ask about how symmetry is might constrain this angle.
So notice at this point I have not said anything about what the nature of the media that I'm sitting in. It's right. I've just told you that it produced this data term. That's it. Now I'm going to ask about two symmetries that are perfectly reasonable to ask about solids. The first is the solid has no magnetism sitting around, so it's not magnetic. There are magnetic solids. I'm not talking about them. I'm just thinking about non-magnetic systems.
The second thing is I'm going to ask that the solid has a centre of inversion. That's a fancy way of saying the solid has some kind of internal mirror reflection symmetry. It looks if I send X to minus X, it stays unchanged, so all the atoms will line up exactly where they are. 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. That's the second step I'm going to do. Now.
Why do I care about these two cemeteries? Well, they do something rather nice. So. Under the first time reversal symmetry. What I'm doing is taking space and time and leaving space unchanged and flipping the sign of time. Electric fields don't care about this, but magnetic fields because they care about currents. We'll start currents circulate in the opposite direction because you run time the other way.
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 has frozen in magnetic fields then it would not look the same on the time reversal. And that's one symmetry. The second symmetry is inversion symmetry, which is the exact opposite. 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.
It has a spatial coordinate. So if I flip the sign of space, it goes the other way, right? So. Magnetic fields. It's a bit more subtle way you can see that magnetic fields don't care is you can imagine taking a current loop and putting it in a mirror and it will still circle it the same way. On the other side, the magnetic field will point in the same direction and it's mirror image.
So magnetic fields don't care. And so why do what these two terms do, though, is that notice that they don't constrain ordinary Maxwell electrodynamics at all because Max Electrodynamics has e squared and they have B squared. E squared doesn't care whether E goes to minus C or dot. Now that is B squared, 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.
So if I enforce these symmetries there, if any, if I ask the theory to be invariant under these symmetries, the equations of the theory have a change in sign. Every time I see an equation with theta, I have to put in a minus theta and demanding invariance under these symmetries requires theta and minus theta to be the same. So this is an elementary thing. If I had if theta were not an angle, it could only be one solution to this equation.
That would be zero. But actually if they doesn't angle this equation as two solutions. And that's because PI works perfectly well. You see that if I take pi to minus pi, it looks like I've changed the sign, but I've already agreed that I'm minus phi and you're just going halfway around the circle. And it doesn't matter whether you go halfway around this way or halfway around back way. And so there are exactly two solutions consistent with these Symmetries Zero and PI.
All other values are forbidden. If I have either of these symmetries, I can only have two solutions to these equations. So a remarkable thing I've understood is that having had the stamps, 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. It doesn't matter any. None of the microscopic details of the solid matter. This term has to be zero or it has to be five. That's it. So what I.
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. Zero. I understand what it does because that's just Maxwell and we've gone through that. So I've got to ask, what happens if I fix theta equals pi? And that's what I want to do. So what does this mean in a solid what is this non-zero data angle mean?
So we saw that naively when I did the wrote those equations down earlier, I said I could rearrange them a little bit and see that a magnetic field induces an electric polarisation. Electric field induces a magnetisation. That's a very appealing picture. 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.
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. So let me there's actually quite a straightforward calculation. So let me take you through it step by step. These are the equations that I wrote down earlier. These are the modified Maxwell equations. So you've got this expression of this expression. Right. Remember, there's only two of the Maxwell equations.
We'll come back to that in a second. Two of them were not modified by there being matter in the system. All I'm going to do now is just take the red bits. I've got a divergence outside a curve over here, so I have to dust off my vector calculus identities. 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.
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. And, you know, this is basically a product rule. I have a divergence. I have two things. Either theta could vary or be could vary. And so I just have to combine the two changes. Okay. So, so far, so good. This is just expanding and rearranging.
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. And those are the two that didn't care that they were sources. They were pristine, unchanged. They will always be true. So let me see what they do. What they do is actually cancel two terms over here, arrange them in such a way. These two equations precisely cancel these two bits here.
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. Either I have a gradient of theta or I have a time derivative of data. So what I've realised after doing all this work of getting this term in a solid is that the equations of motion apparently don't seem to care if this term is there. They just don't care if it's a constant.
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. But I can engineer a situation where these terms change. And it's something that's, again, familiar from thinking about the physics of solids is to think about boundaries between media and vacuum. And at a boundary, the vacuum has theta equals pi. We know that because there was no either metre.
But imagine that I had a solid with equals by I can ask what happens at the interface between these two regions. So this is sort of a standard thing that we do a lot of Vietnam. 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. 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.
So I'm going to look at so I've got a region up here with it, equal SPI a region down here with it equal zero being a so I'm going to imagine this goes on forever so I don't have to worry about things fringing on the boundaries and stuff like that. And what I'm going to do is I don't I'm interested in asking. So normally when I think about a solid, 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.
So what I'm going to do is something that you probably don't normally do in a solid, which is put on a magnetic field perpendicular to the interface and ask what happens inside. 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, this is just like Gauss's law talks about divergence of electric fields, and this says that what's on the right side is a source of electric fields, which is a charge.
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. So that means that theta is constant everywhere but jumps at that boundary just so that boundary jumps. So what this equation is told me is that there's actually a surface charge density that's triggered. 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.
But this is just a consequence of the forces of acting inside that solid, that things start moving around so as to produce a charge density on that surface. Now I force the electric field outside to be zero, which means that the charge here somehow when it create an electric field somewhere and the logical consequences that there's an electric field created inside the material parallel to the magnetic field.
So what I've produced is an electric field parallel to B, 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. 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.
And I've told you that there is in principle a way to measure fine structure constant by doing an electrodynamics measurement, which is quite, quite remarkable if you think about where that came from. I can do the same thing the other way, and that will actually lead us to something quite neat. So what I'm going to do is now take an interface between three equals zero and theta equals pi.
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. Now, again, if I take this equation, kernels of magnetic fields, whatever appears on the right hand side should be a current because currents are sources of magnetic fields.
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. And if you unpack all of this, you find that there's actually a current that points into the board at this interface. So it's sort of pointing into the plane perpendicular to the electric field. And that current looks set up a magnetic field parallel to it.
So just the exact reverse of that effect. And again, this proportionality constant is linked to the fine structure constant. So again, something remarkable, but actually this effect, I'll come back to the first effect towards the end, but this effect is actually something familiar. So on the interface between the theta equals zero and theta equals pi region, I put on an electric field and the electric field generates a current that's perpendicular to it.
Now, this is actually something that is a very, very, very old effect. It was discovered by Edwin Hall during a speech this studies in 1879. This is something known as the hall effect. We don't think of it in the setting. The hall effect usually emerges when we think about. 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.
But of course, if I have moving charges, there's a Lorenz force in moving charges. 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. 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. Otherwise it keeps going off in the other direction. So it's like having a car with a wobbly sharing wheel.
I have to actually put a bit of force to even keep it going in a straight line if my wheels aren't aligned. So you can just think of that, that analogy. And so there would have to be an electric field perpendicular to the current in order to keep a steady current flowing. So this something that's well known. This is the whole effect. But what we've heard is that remember that if I go back to the previous slide over here, these coefficients were all fixed.
There's a fine structure constant here. The theta had to jump from zero to pi, so that jump is quantised. If I have a system with the symmetries, there's no intermediate value of data. 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. And so what must be true is that that coefficient, there's no freedom in that coefficient. Alpha was fixed, the jump in theta was fixed.
And so if I translate that into more conventional units, so the whole conduct conductivity is usually called sigma using initial. So the subscripts x and Y to say that it's a response of a current that's perpendicular to a magnetic field. So one is an X and the other is in Y. 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.
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 constants that has the units of resistance and in this case the same as resist units of inverse resistance or conductance. And so that happens to be the same units as conductivity in two dimensions, which is where we are.
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. There's just a half time something. So I've got something that's very, very rigid and fixed. But in fact, this is really, really, really surprising. The fact that you get this half here is really surprising for a profound reason. It turns out that if you give me a purely two dimensional system, well, the whole conductance has to be an integer.
Times is quite over. 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. It doesn't have anything exotic going on. So if you have an integer times it's quite over. Well, any perturbation I do to the surface I can always think of as something like gluing stuff onto the surface. Let me glue whatever I can give you a complete freedom to do whatever you like to the surface.
You can glue any two dimensional system you like, but don't modify anything on either side. You can't put a three dimensional system on there. You can do anything in two dimensions. Whatever you do, you will actually not be able to change the oddness of this. So let's see how that works. 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. Sorry, glue on something that has squared over equal.
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. I get minus B squared over two h. So notice I'm changing this number drastically. But what I can't do is remove that two in the denominator. It's always a half integer. So this half mess is very strange. It can't be removed on the surface. And so what we've actually found is something interesting. The surface is actually illegal as a two dimensional system.
No two dimensional system with the same laws of physics that we're put in with weak interactions could ever do this. So this is okay because the only I defined my surface as one between something that has theta equal zero interact with spi and such a surface fundamental means the third the third dimension in order to define its existence. That's okay because it's embedded in three dimensions. So it's a very peculiar property.
And so I should point out that actually understanding I just wrote this down the sigma x y equals this squared age actually understanding this is one at least three Nobel Prizes to date. And so, you know, it's it's a deep piece of physics, but it's sort of incidental to the story here. 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.
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. Let me give you some examples of materials that do this, focusing again on the two symmetries I talked about time, reversal, inversion, symmetry. So there's one little subtlety that comes in. 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.
Well, it turns out that an insulator can have a hole conductance only if time reversal is broken. It's forbidden for an insulator that preserves preserves time reversal. That's a complication, because that seems to contradict what I just wrote down. Right. So the resolution of this is that somewhere my assumption should break down. And the assumption that I had to get this piece of physics on the surface was that things were insulating everywhere.
The only way to resolve the tension between these ideas, the fact that if things were insulating everywhere, the surface would have to have a whole conductance. But if things were time reversal invariant everywhere, the surface couldn't have a hall conductance. Something has to give. And what has to give is that the insulating everywhere piece?
So it's actually a remarkable fact that if I demand that the symmetry of nature, time reversal is present everywhere in the system and the surface and the bulk throughout, then the surface between equals pi and a theta equal zero insulator has to be a metal. And the reason that's okay is because metals actually have mobile charges that can get rid of that surface charge and those surface currents and sort of cancel the effect.
So we've actually discovered I'm taking you through the discovery of actually a profound fact that a time reversal, symmetric system, which I said equals PHI, is actually in a different state of matter from our vacuum and from an ordinary insulator. It's something known as a topological insulator, and it's special because it's interface with our normal vacuum will be a perfect metal as long as time reversal is preserved.
So it turns out that this observation was actually predicting the people predicted this effect comments about 15 years ago. 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. Rahul Roy, who some years after he made this discovery, spent several years in Oxford as a postdoc. So sort of a nice Oxford connection there.
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. So this effect, I told you, is not there. Well, it turns out, though, that that metal itself is very special. That metal has to remember somehow that if it broke time reversal, symmetry would have to have this half integer conductance. And it turns out that there's a very special, special consequence of that.
So if you could measure and this is something that experiments next door in the cloud in lab do, 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. If you can see these, it says one, two, three, four and five. What it's counting is the number of times the dispersion crosses this energy, which is the Fermi energy.
And it turns out that you can you cannot cross it an odd number of times except in this topological special metal. And so that measurement tells you that into it is what that equals. Pi exists in this sort of convoluted way, but it's actually evidence that we have an insulator where theta equals pi inside. 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.
So if there April Spy is enforced by inversion symmetry, there's never a problem. That's because I can't set up the start experiment with inversion symmetry. So if I gave you a surface that below is one material, above is another material and I send x to minus x, I've swapped them around because now one material is on top and the other is below. So a surface can never preserve inversion symmetry. So that whole set of logic breaks down at the surface.
And the surface can always have this property where something can rearrange and be insulating. And it turns out that other symmetries can do this in various ways. But for purists, the term axial insulator is reserved for inversion, symmetric or similar systems where you have data close by in the bulk and it's quantised on the surface. Break symmetry and it's no longer this perfect metal. So how would you observe that?
Because you no longer have this ability to look to count things on the surface. So it turns out that these Hoffmann to your whole effects can be done in a clever way. So there's a effect that is studied in optics, which is called the Faraday effect, which is that the plane of polarisation, of light rotates in certain solids. And it turns out a prediction of this material is that there's a quantised Faraday effect. And so that's something that one can work out.
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. 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. And, you know, you have to do experiments on finite objects. So it turns out that this is an ongoing challenge to understand where to actually come confirm that this piece of electrodynamics works out.
So people are still trying to do this in the lab, and they do various ingenious ways of disentangling these effects. Okay, so to close, I just want to flag one last piece of exotica, 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. So that's going back to this first of the two interface effects I talked about.
So the second one I spent a lot of time on, which is saying that if I put on electric fields, I generate currents. But the first one was that if I put on magnetic fields, I generate charges. So imagine I have a sphere with theta equals zero embedded inside a region with that equals by tiny little sphere. And inside that sphere, I drop in a point source of magnetic field. The thing that's forbidden by the first by one of Maxwell's equations. But never mind.
For now, let's stick in a source of magnetic field. If I did that. Well, what I've told you is that, you know, I've got magnetic field everywhere. I'm going to interface between data equals PI, so actually create electric fields parallel to the magnetic field. Sorry. Create electric fields everywhere. 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. Then this looks as though I have an electric field of a point charge.
So it looks like if I put in a magnetic monopole, it triggers an electric field that looks exactly like that of a point charge with a strength related to the magnetic field by Alpha. So it looks like a magnetic charge triggers an electric charge. Now, this argument was a bit convoluted, but it turns out that you can shrink. This data equals zero region all the way to nothing and the argument would still be there. So monopole moving inside this material would actually carry an electric charge.
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. 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. So in a Stanford, 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.
And in 40 years, they've seen one signal that's very controversial. So we don't think that we can find them. They're not they're not hanging around our universe very easily. 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. They can generate interesting new back here. And here's an interesting Oxford connection.
So about 15 years ago, a group of people who have either past or very present Oxford Connections predicted that there are certain magnetic materials that could emulate the physics of magnetic monopoles. 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 involved in these monopole problems are sort of emergent electric and magnetic fields.
But very recently people have pointed out that those electric and magnetic fields, even though they're sort of not the standard ones we think of in max electrodynamics, can actually have their own theta terms. And so there have been recent papers that predict the this wouldn't effect and having observable consequences of actin electrodynamics in sort of materials where you can actually do measurements. So that sort of fictitious sort of special emergent monopoles can have consequences.
And so we can test the sort of rather exotic prediction of field theory in an experiment that you could realistically imagine sitting on this tabletop. So let me. I'm sorry. Going the wrong direction. Let me go back to my sort of two propositions. 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, you know, it seems clear that we can emulate the physics of axioms in the solid state material.
So just to summarise, you know, we start with the idea that if you think about insulating matter, then you can view it as a new vacuum for electromagnetism. And we know now several insulators where this conspiracy of quantum theory allows you to have Axion electrodynamics as the effective description. And there are many active experimental searches going on. I gave you a samples of some of them to look for consequences of these Axion electrodynamics.
And I should say that everything I talked about today is treating theta as an angle and not a field. But there are situations where theta can become a field where this physics comes in, when you have an insulator with some additional complexity, where there's some magnetism or some other charges moving in very constrained ways and they can generate dynamics for these Axion field. So you can actually there are also prospects for observing dynamical axioms in the solid state.
So the second point, which is whether or not actions have any physical reality, that study can be useful intellectual exercise. I have very little to say, but let me close with words of one of my heroes, Duncan Holden. It's very difficult to know whether something is useful or not, but one can know that it's interesting. So I hope you'll agree with that and let me thank you.