 If we have not found space, do not worry. We will have plenty of spaces on the windows on the entire terrace, so there's space for everybody. And we have special things for that. We will have to do it however, at the start of the session. But we will cross the queue and put a sign. And it's locked on the boards. Don't panic. But instead, the people that are already on the boards, but outside of these rules, please don't drive with them, because this allows you to fit. OK, so good afternoon, everyone. So in the lecture this morning and the lecture yesterday, I tried to explain an overall idea to you, which is that in these systems with microscopic numbers of degrees of freedom, but constraints between them, we can have emergent degrees of freedom at long distances, which are quite different from the microscopic degrees of freedom that we started with. So in the models that I've talked about, we had microscopic degrees of freedom, which were the spins or the dimers. And we ended up with these long distance degrees of freedom, which we could think of as fluxes, which satisfy a divergent zero constraint. So that's like electromagnetism. And in the jargon, you talk about that as being a U1 gauge theory. And then when I discuss differences with dimer models defined on non-bipartite lattices, we had the alternative possibility of Z2 gauge theory. So what I want to talk about this afternoon is a set of real physical systems, magnetic materials, which are the best realizations of the general physics that I've been describing. And these materials have become known as spin ice for reasons that I'm going to talk about as I go through. So the overall context is given by this slide that I used at the beginning. So I was talking about frustrated units of spins with anti-ferromagnetic interactions between all the spins in the cluster. And we could think of clusters of various sizes, but now I want to specialize focusing on tetrahedra. So the number of spins in the cluster Q is equal to 4. And I want to think specifically about ising spins so the number of components becomes 1. And if we're just considering a single cluster, as in these pictures, if we're thinking about ising spins, then according to this way of thinking about ground state energies, minimizing the total magnetization of the cluster, the ground states are clearly ones in which two spins are up and two spins are down. And so if we count the number of ground states for a single tetrahedron, it's 4 choose 2. In other words, there are six ground states. And the question that we're going to be thinking about is how that generalizes when we connect these clusters together, in fact, in a corner-sharing fashion, and also, of course, how the ground state constraint in each cluster generates long-range correlations of the types that I've been talking about in terms of a Coulomb phase. So not all of the details of the lattice will be important, but this is a picture of the so-called pyrochlor lattice built out of corner-sharing tetrahedra. And perhaps the one simplifying fact to keep in mind is that it's a cubic lattice in the end, but obviously a cubic lattice with several sites in the basis. So the actual materials are ones that have magnetic ions on these sites that form corner-sharing tetrahedra. And the magnetic ions are rare earth ions, which have a large angular momentum when they're isolated. But in these materials, they're not isolated. They feel a crystal field, and that splits the angular momentum levels. And the lowest levels are ones that correspond to the magnetic moments being orientated either out of or into the tetrahedron along the high-symmetry direction. And the effect of the interactions between the spins, which are actually a combination of exchange interactions and dipolar interactions, is to give you lowest energy configurations, which have two spins oriented into the tetrahedron and two spins oriented out. So if we represent in and out by up and down for ising spins, then we have a Hamiltonian, which is equivalent to the nearest neighbor ising anti-ferromagnet on this lattice. So are there any questions about that basic starting point? OK, so now we want to connect these tetrahedra up to form a big lattice. And there's a kind of amusing aside, but it also has quite a useful point to it, which I'll get on to in a moment. But the aside is to do with why this material got called spin ice. And the point is that there's rather good analogy, precise analogy with problems which people had thought about in physical chemistry and in particular Linus Pauling thought about. So these are problems involving ice in its solid phase. And of course, you have oxygen to minus ions and protons. And the whole ice crystal is held together by hydrogen bonds. And if you think in detail about the structure, then one oxygen ion has four neighbors. And the protons on these bonds, which, sorry, I should have said the open circles represent the oxygens and the black points represent the protons. And if you think of water as H2O, then two of the protons on the bonds associated with a particular oxygen should be close to that oxygen. And two of them should be close to neighboring oxygens. And this too close to far rule is just the same as the two in, two out for a tetrahedron in the magnetic material. So there's a complete analogy between spin configurations in the ground states of this magnetic system and proton arrangements in water ice. So one of the points that Pauling made, which we can take over, involves an estimate of the number of ground state configurations that there are in a problem like this. And I guess the first point to make is that there is, as with the other problems that I've been talking about, a macroscopic degeneracy in these ground states. And the simplest way of seeing that is by seeing that you can construct things which are analogs of the loops of flippable dimers that I talked about when I was talking about dimer models. So you can think of going through the lattice forming loops, which alternately visit spins that are pointing into a tetrahedron and pointing out. So for instance, we could draw a loop that goes through this site and then goes through that site. And we can always continue in the next tetrahedron and the one after and the one after because we can always find an exit point with a spin in the direction that we want. And presumably some of those loops will end up being closed in typical spin configurations. And then once we've formed a loop like that, we can reverse all of the spins on the loop because if in each tetrahedron the loop includes one spin that's going in and one that's going out, if we reverse both of those spins, we preserve this ground state rule. And if we reverse spins on independent loops, then we're generating a macroscopic amount of entropy. So what Pauling suggested was a way of estimating this entropy. And it's very much in the spirit of mean field theory. So you might think that it wouldn't be a terribly good estimate, but as I'll show you in a moment, it turns out to be extremely good. And it's quite nice to see how things play out. So the way the estimate goes is in two steps. So first of all, we think about a single tetrahedron and we've got four spins, each independently with two states. So altogether we've got 16 states for the tetrahedron. Of course, some of these states will be excited states. The ones that are ground states are just six out of 16. So this is the fraction that are ground states. So now we take that information from thinking about a single tetrahedron and go to the whole lattice. And we say that the best simple estimate that we can make of the number of ground states in the whole lattice is to take the total number of spin configurations and then reduce it by this factor for each tetrahedron in the system. And so to complete the calculation, we just need to put the dependence on system size in full. So the number of states is 2 to the power of the number of spins. And the number of tetrahedra, well, there are four spins in each tetrahedron, but each spin is shared between two tetrahedra. So the number of tetrahedra is half the number of spins. So this power of the number of tetrahedra we can replace with half the number of spins. And then if we combine these two factors in a single fraction and cancel things down, we have an estimate for the total number of states, which is 3 halves to the power half the number of spins as being our estimate of the ground state entropy. Are there any questions on how that logic goes? So then the beautiful thing is that actually you can do experiments which see almost precisely that. And so the experiments involve measuring the entropy as a function of temperature of one of these materials. Now, of course, experimentally you don't measure entropy directly. In fact, all you can do is measure entropy differences between states at two different temperatures. And the way that you do that is by measuring the heat capacity as a function of temperature and then saying the entropy changes the heat divided by the temperature and integrating it. Of course, even talking about the heat capacity, we have to be careful because our interest here is clearly the magnetic degrees of freedom. But if we measure the heat capacity, then there'll be contributions, at least in principle, from the lattice vibrations and so on. So sometimes you can subtract the contribution from the lattice vibrations by making a similar compound but without magnetic degrees of freedom and subtracting the two heat capacities and so on. So you do that kind of thing and have a heat capacity as a function of temperature that's just from the magnetic degrees of freedom. And then by integrating it, you can get entropy differences between two different states of the material. But what we want to talk about is absolute entropy. So we need some reference state with known entropy. And here, things are nice and simple because if we go to high temperatures, then the system will be completely thermally disordered. And so the entropy should be kb log 2 per spin or r log 2 per mole. So what's done in this experiment is to take that value as the high temperature value and you see from this temperature scale that all of the interactions in this material are rather weak there on the scale of a few Kelvin. So we only have to go to 10 Kelvin to see the full entropy of the high temperature state. And you measure how much entropy comes out as you cool down. And now because you've used the high temperature state to get an absolute value of entropy, you know that if you cooled to a unique ground state or a ground state with a finite degeneracy, you'd have zero entropy. And so you expect to end up down here. And what you actually measure from the heat capacity is some residual low temperature entropy. And the picture is that that's because of precisely the degeneracies that I've been talking about. So any questions there? OK, so that's kind of reassuring. It means at least at the zero-tholder, the physics that I was talking about yesterday and this morning is present in this material. But of course that was really just our starting point. And so what we'd like to do is go further and see whether some of the more sophisticated things to do with this Coulomb phase are also there as well. Yes. So well, it's very much in the spirit of a mean field calculation. I mean, it's outrageous really to say that we can simply take the fraction of ground states that we compute when we think about an isolated tetrahedron and use that fraction when we're calculating the ground state of a whole lattice. So the surprise is not that there's a discrepancy, but that the match is actually as good as it is. But I mean, of course, there could be other reasons why there's a discrepancy. And I don't actually know whether it's because of failures in the mean field calculation or because of differences between the real material and the model. Yeah, so the question was, can you also get this entropy difference from the coarse-grain model? And that's a very useful question. And the answer is no. And the point is that when we coarse-grain, we're somehow integrating out short-distance degrees of freedom. We're averaging over short loops of spins that can be flipped. And the entropy adds up contributions from all scales. And unless we have that short-distance physics under control, then we can't calculate this number, which is not exactly a non-universal number, but it's very specific to the details of the lattice and so on. OK, so to get at the Coulomb physics, we really want to be thinking about long-distance properties. And so locally, we know that the constraint written in terms of these Ising variables is that in each tetrahedron, we have two spins up, two spins down. So the total Ising magnetization of the tetrahedra is 0. And you can actually see in a rather elementary way that that must lead to some long-range correlations. Because imagine that you worked out the total magnetization of the spins in the lowest plane of this picture, which I'm trying to indicate with a pointer. And you compared it with the magnetization of the spins in the plane above. Now, because together they make up the tetrahedra in the lowest layer, you know that whatever the magnetization is in the lowest plane, you must have opposite that magnetization in the plane above. And if we go up through the stack, then we have some alternating pattern of magnetization. So there's some kind of long-range correlation there. And this is a calculation of correlations as you would see in the Fourier transform. And what the Fourier transform shows, obviously, is some sharp features. And sharp features in reciprocal space must be related in real space to some long-range correlations. So this idea in real space matches qualitatively with this picture in reciprocal space. And what we'd like to do is see how things work out in a more detailed fashion. So there's a mapping, which I'm going to talk about at various levels, to precisely the three dimensional Coulomb phase description that I had this morning. And the simplest way of presenting things is in terms of the pictures that I've drawn here. So the rule for constructing ground states is that in each tetrahedron, we have two spins pointing in and two pointing out. And I obviously haven't drawn all of the spins in this picture, but at least I've drawn arrangements for the spins that I have represented, which are consistent with this rule. So now at the simplest level, in your mind, you can imagine connecting these spins together in a way that gives you flux lines and in a way that ensures that the flux lines never end. They just are either infinitely long or close into closed loops. In other words, these flux lines are divergence less. And the point is that if every tetrahedron contains two spins that are pointing in and two that are pointing out, then we can take one of the ingoing spins and pair it up with an outgoing spin and say that that's a segment of a flux line and we'll always be able to continue this flux line into the next tetrahedron and so on until either we reach the edge of the sample or it closes on itself. So there's a direct mapping from the spin configurations to flux lines and that's exactly the sort of thing that we had when we were talking about Coulomb phases. Of course, you can do that in a more precise way and that's what I'm explaining in detail on this slide. And there are some specific points to take in, but I think it's worth concentrating on how the details actually work. So the first point that you have to absorb to see how the details work is that we need to have in mind two lattices. So on the one hand I've drawn in blue the pyrochlor lattice made out of corner-sharing tetrahedra. And at the same time we can introduce the second lattice which is sometimes called a parent lattice and the sites of the parent lattice are at the centers of each of the tetrahedra and the bonds of the parent lattice these arrows that are drawn in red here. So there was something similar which happened when I talked about the triangular lattice ising model and the dimo representation and maybe it would help the visualization if I explain it in that case. So in the case of the triangular lattice ising model, if I draw a bit of the lattice, we might have something like that. And then I said that for the mapping to the dimo model, the dimos were located on the bonds of a honeycomb lattice. The analogy is that these sites of the honeycomb lattice that I've drawn in yellow are at the centers of the triangles and in the same way the sites of the lattice, the parent lattice with the red bonds are at the centers of the tetrahedra. Now actually if you think about the geometry of things, you realize that this parent lattice with the red bonds is in fact a diamond lattice. And the diamond lattice is bipartite just as the hexagonal lattice here is bipartite. So the great feature of bipartite lattice is that they give us an easy way of introducing an orientation convention on the links. We can take sublattice a and b and direct the links from sublattice a to sublattice b. And that's what the red arrows are showing in this picture. The site of the parent lattice at the center of the picture is on sublattice a. And the orientation convention is that the arrows are going to the other sublattice. OK, so that's setting up some machinery. And these oriented links we represent with unit vectors with a corresponding orientation in real space. Now the mapping from spin configurations to fluxes says that on one of these links we have unit flux in the direction of the link if the corresponding ising spin is plus 1 and minus unit flux if the corresponding ising spin is minus 1. And then the ground state constraint, which says that two spins are up and two spins are down in each tetrahedron, means that we have two units of flux coming into the tetrahedron and two units going out. And so this divergence-free condition is exactly the same as the ground state condition in the ising language. So in other words, the way this is working is exactly analogous to the mapping to a height model that I had for the triangular lattice antiferromagnet. But now, of course, we're thinking about a three-dimensional problem. So are there any questions about how any of that worked? Yeah, well, I think that's the sublattice. That's the point about the parent lattice, the diamond lattice, having two sublattices. OK, so now we do the hand-waving bit. And we say that we're interested in long-distance physics, and so we're going to coarse-grain a bit. And that will be averaging over these short-length loops of spins, averaging over their orientations. And by the same sorts of arguments that I used before, we expect that the configurations with the most entropy will be ones with lowest local values for this flux density. And so we get a probability distribution which I could write either in terms of the vex potential or in terms of the b, the flux density, in the way that I have been doing before. And so the picture that we get from that is, well, if we coarse-grain, we have some ensemble of flux loops. And then by the calculation which I started with in the lecture in the morning, we end up with this correlator between components of this emergent field. But now this emergent field also corresponds to the spin correlations themselves because there's a direct connection between the local value of the flux and the local orientation of the spins. And so we can expect that we can see these correlations in the spin correlations themselves. Now, if we're going to think about the details of how that translation from the fluxes to the spins, if we're going to think in detail about how that works, we need to go into things reasonably carefully. And I suppose the basic point is that as far as the emergent flux is concerned, the long wavelength correlations are revealed at small wave vectors. But when we think about the translation to spins, it turns out that we're not looking at zero wave vector for reasons that I'm going to try and explain. OK, so I said that the pyroclore lattice is a cubic lattice, but it has several sites in the basis. In fact, we have the four sites in the tetrahedron as the sites in the basis. And so we can think of transforming between the spin orientations on those four sites, or if we go into reciprocal space, the Fourier components of the spin orientations on those four sites, and the emergent gauge field that appears in this long wavelength description. And there are four spins and three components to the emergent gauge field. We expect there should be some other degree of freedom, and the other degree of freedom turns out to be the local average to the magnetization or its Fourier component. And of course, in principle, there should be some transformation between these different degrees of freedom, and it turns out that it's given by this 4 by 4 matrix. And I want to try and give you a feel for how you read off some of the entries in that matrix. And hopefully, if I can give you a feel for some of it, then the rest will seem plausible. So it seems to me this is something that is best explained in real time. So let me, first of all, draw a picture of a tetrahedron. So this is meant to be a three-dimensional drawing, and this is the edge which is behind. And I'll number things in a way that's consistent with the picture. So these are sites 2 and 3 and 1 and 4. And let me suppose that I have a particular ground state spin configuration in this tetrahedron with, say, the spin at site 2 and at site 1 pointing into the tetrahedron, and the spins at sites 4 and 3 pointing out of the tetrahedron. And then I want to relate this to some axes in the space where the crystal lives. And so these can be the cubic axes. And I'll take x, y, and z like that. And now I want to think what this spin configuration corresponds to in terms of components of this emergent field. So on average, what this represents is a field that's in this direction. So in other words, we have an x component to the field. And it's true that up here there's a component in the z direction. But on average, it's canceled by the minus z component here and similar things on the other sites. So what I'm saying if I invert that relation is the bx, it's, well, if I have plus 1 and minus 1 to represent spins which are either into or out of the tetrahedron, then the fact that spins 1 and 2 are pointing into the tetrahedron means that they contribute positively to bx. And the fact that 3 and 4 are out of the tetrahedron, that also means that they contribute positively to bx. But if they're pointing out of the tetrahedron and I used s equals plus 1 to record the fact that these two spins are pointing in, then those would have a minus sign. So absorbing that, I'd say that the x components of flux is related to the spin configurations in that way. And in a similar but rather trivial way, we'd say that the total magnetization in terms of these ising degrees of freedom for the tetrahedron is just the sum of the four spins. So now we can go back to what I've got written here. And so the claim is that there's a transformation between the spin orientations at these four sites and some degrees of freedom that we can use when we talk about things in a coarse-grained way. And for instance, the magnetization is the sum of these four spins. That's why the first row of this matrix is 1, 1, 1, 1. And the x components of the field is the sum of the first two spins and then subtract off the spins 3 and 4. And that explains this second line of the matrix. So there's another step to see what we learned from that. But can I check whether that came across as at least halfway plausible? No, no. The half is just because I wanted to write down an orthogonal matrix. OK, so now the point is that we have a theory which is expressed in terms of these degrees of freedom. But when we go out and do an experiment, then we're probing these degrees of freedom. And we're doing all of this in reciprocal space in terms of wave vectors rather than in real space. And when we take account of this transformation, then these minus signs look like phase factors which you get from e to the i, the scattering wave vector, times the position vector of the different sites. And so if we look at small q structure in the flux, well, small q, you might think that would be near the origin of reciprocal space. But because of these minus signs, it gets transferred to the vicinity of Bragg points in the scattering with non-zero wave vector. So is that all right? So now we're ready to look at some experimental data. So what you can do is take one of these materials and do elastic neutron scattering from it. And so that basically gives you Fourier transform of the spin configuration. And what you see is the experimental version of the theoretical pattern that I had earlier on in one of my slides in red and blue. And the point is that you have sharp features in reciprocal space which tell you about these long-range correlations in real space. And in fact, these precisely Fourier transforms of the power law correlations. I mean, in fact, I said that you had correlators which were of the form qi, qj over q squared. And that is a function of angle as you go around q equals 0, which now is transferred to somewhat finite wave vector through the transformations that I described. And the fact that the intensity goes from high to low to high to low as you go around here is just the behavior of this function as you go around the origin in q space. OK, so the message is, yeah, good. OK, so the horizontal and vertical axes, two directions in reciprocal space, and you're also taking a particular slice in the third direction. And the intensity is the scattering cross-section. Yeah, thank you for that question. OK, but the basic message is that you have these material spin axes which are a very good realization of this Coulomb phase physics. And on the one hand, from the heat capacity and entropy measurements I was talking about, we see the ground state degeneracies. And on the other hand, from measuring correlations, we see these long-range correlations that follow from the local constraints. OK, so now we should go on to this issue of excitations, which again will follow rather closely in parallel with what I was talking about for the triangular lattice antiferromagnet. So what I want to think about doing is starting from a ground state. So you can check for yourselves. I think that this is a ground state. You can see that each of these tetrahedra that I've drawn has two spins pointing in and two spins pointing out. And then you can say we're going to make an excitation and we'll do that by picking on one of these spins, for example, the one that I'm indicating here and reversing its orientation. So when we do that, we have three spins pointing in to this tetrahedron and three spins pointing out of that tetrahedron. And in that sense, this lower blue tetrahedron becomes a source of the emergent flux that I've been talking about and the one above becomes a sink of the emergent flux. So you can probably imagine what's going to happen next. We've made a single spin flip and generated two tetrahedra that are not in their ground state and they can be fractionalized. They can be separated without paying any extra energy price because if I flip, for example, this spin over here, then I return the blue tetrahedron to its ground state but I excite the green tetrahedron and make it a source of this flux and of course I can carry on flipping spins to separate these two excitations further and further. So these excitations are like the monomers that I was talking about in dimer models in their sources and sinks of the flux and so in terms of this emergent gauge field, they're like monopole charges. And so one question we can ask is what about the entropic interaction between a pair of oppositely charged monopoles and now we can use the ideas that I was talking about this morning. So this is like three-dimensional electromagnetism and the entropic potential is the same as what you'd have from Coulomb's law. So we have one over our potential and that means that if we create a monopole, anti-monopole pair in neighboring tetrahedra, then we start with an interaction which instantly would be attractive if they're oppositely charged and with R being the distance between two neighboring tetrahedra and the effective potential that you'd pay to separate them to infinity is finite and so these excitations are de-confined. So if we think about the interactions between these monopoles, then according to the long-distance description of the ground states that I've been talking about, we have this one over our potential and that's purely entropic in origin. So if we're going to put something into an ordinary Boltzmann factor, it's beta times the potential that goes like one over R. In other words, there isn't really any temperature in here, it's just entropy. And if we had a material with just nearest neighbor interactions, that would be the end of the story. But of course in any magnet, there are dipolar interactions and if the exchange interactions are large, then those dipolar interactions are not very important but in this material as you saw from the heat capacity measurements, the whole energy scales are rather low, a few Kelvin and also since the magnetic moments are rather large, the dipole interactions are rather large. So actually dipolar interactions are quite important in these materials. So once you realize that, you feel that that's rather unfortunate and it probably spoils this picture because these different ground states that I've been talking about, the macroscopically degenerate ground states are macroscopically degenerate really because of an accident of the nearest neighbor model and once we include further neighbor interactions, these dipolar interactions, then that degeneracy is going to be lifted and at least in principle, this whole story collapses. And that's true but fortunately only to an extent. So you can find out the effect of these dipolar interactions on spin ice. So this is a phase diagram as a function of temperature and the strength of the dipolar interactions compared with the nearest neighbor exchange and the details don't matter very much but the overall point is that at a rather low temperature this material would order in principle but in reality what happens is the things, the dynamic slows down as you go to low temperatures and it would take much longer than experimental time scales for the system to arrive at the ordered state. So what you probe experimentally is this highly correlated state that you would understand even in the nearest neighbor model. So in other words, the lifting of the nearest neighbor model ground state degeneracy from dipolar interactions turns out to be less important than you might have thought. So it's a bug but one that doesn't ruin the whole story. But then the beautiful thing is it turns out to be a bug that's also a very nice feature and the nice feature comes when you think about the interactions between these fractionalized excitations which are monopoles of the emergent gauge field and the point is that they turn out also to be magnetic monopoles and to understand how that goes the nicest way of viewing things is to say that although these spins are really microscopic dipole moments we can think of them in a fictitious way as being made up out of magnetic charges plus and minus with some finite separation with the size of the charge and the separation chosen to restore the correct microscopic dipole moment. So in other words what we're doing is replacing something which really should be an exact dipole with a pair of charges with finite separation and what that does is introduce higher multiple contributions but reproduces the dipole contribution exactly. So where does that get us? Well if we think of all the spins in the lattice like this and ask what happens in a ground state of spin ice then since in a ground state there are two spins pointing in and two out of every tetrahedron that means in this representation if we choose the distance between the two charges to match the distance between the centers of the two tetrahedra there'll be two positive charges and two negative charges at the center of each tetrahedron in other words these charges will cancel and that means in fact with this approximation to the dipolar interactions will restore the degeneracy of the ground states that we also had in the nearest neighbor model. But now if we think about making excitations the effect of making an excitation is to spoil the cancellation that we had at the center of each tetrahedra between the magnetic charges and so we'll have a net charge of one sign at the centers of some of the tetrahedra and a net charge of the other sign at the centers of the other tetrahedra. So what this means is that in addition to the entropic interaction that comes from the emergent gauge field because of the dipolar interactions we've also got a real magnetic interaction between these monopoles. So it's a pretty remarkable situation. So there are situations which are probably familiar to you where you have long range interactions, Coulomb interactions which then get screened. So you may know that in a plasma you say at finite temperature you can have Dubai screening and so on. So it's easy to convert long range interactions into short range ones. What we're doing here is exactly the opposite. We're taking not a completely short range interaction but a dipolar interaction which falls off like one over R cubed and we're converting it into an interaction between monopoles which falls off like one over R and so we're promoting more rapidly decaying interaction into a more slowly decaying interaction. It's a kind of anti-screening and it comes because of all the correlations in these Coulomb phase ground states. So when that came up as a theoretical idea of course the question became is there any experimental evidence that really shows that there are these Coulomb interactions at work in the material and the first suggestion and I think probably still the best one is this and it's a comparison with an experiment that had been done before the theoretical ideas were developed which in fact used a magnetic field to control the density of these monopoles. So just to give you an idea how that works if I draw a tetrahedron like that and take a ground state with two in, two out like that and I apply a magnetic field in this direction then if the field is strong enough it'll make it favorable to flip this spin so that it points in. In other words it'll control the density of monopoles. So by varying magnetic field and temperature you're able to go between a low density and a high density phase of monopoles and that's a bit like the liquid gas transition and when you work out the theory if you go through a transition like that in this situation just in a model with short range interactions then it turns out to be continuous but if you include the long range interactions these magnetic monopole interactions then it's converted into a first order transition and experimentally it had already been found to be a first order transition so this albeit slightly indirectly really does show you that you have these long range interactions between monopoles. So of course this is very famous work by two of the organizers of this school. Okay, so I think that finishes what I want to show today. I've tried to explain how spin ice gives you first of all a realization of this Coulomb phase physics in three dimensions and we have very clear experimental signatures of these parallel correlations via the neutron scattering experiments and we have classically fractionalized excitations in this three dimensional system. So questions, sorry is only, yeah okay so the question was when you calculate these correlations from the emerging gauge field description are you only allowing for spin configurations with div B equal zero and the answer is yes and so then the question would be what's the justification? Well if you go to low temperatures then the density of these monopoles should get very low. I mean in fact there's a Boltzmann factor because it costs a finite multiple of the exchange energy to generate these monopoles. There'll be a Boltzmann factor which will suppress the density as you go to the low temperatures. Yes, but the experiment still had a sufficiently low temperature that the density of monopoles is very low. So I mean the basic point is that a non-zero density for the monopoles sets correlation length and this Coulomb phase physics is still good out to the correlation length and the correlation length corresponds to the density of monopoles and that density can be made very low if you go to low temperatures. I mean then there are questions about equilibration and so on. Yeah, so if you have a pair of interactions which are close to each other, sorry, a pair of excitations, which pair of monopoles which are close to each other in one part of the sample and then another pair over here, then you'll have an interaction between the two pairs which you can understand by adding together the entropic contributions and the magnetic monopole contributions. Well, we should distinguish here between the entropic contributions which are described by the emergent B field that I talked about and also the, and on the other hand the magnetic contributions which come from the fact that the spins have dipolar interactions. But I mean in both cases you expect to understand what the interactions are by thinking about the corresponding version of electromagnetism. I mean the fictitious electromagnetism for the emergent gauge field and real electromagnetism for the monopoles. So you'll get an interaction between a pair of dipoles but it's what you would expect from adding up the contributions from the different charges. Yes, but yeah, I mean you also have to think about as they move you have to think about the effect that their motion has on the spin background. So I mean the point is that the magnetic field couples to the dipoles so to the spins via dipoles so in the presence of a magnetic field you affect the energy of the spins and so you can promote the reorientation of the spins and that's the same thing as exerting a force directly on the dipole. Sorry, on the monopole. Yes. Yes. Well when you go to the limit it's actually not very interesting because if every tetrahedron has a monopole in it that simply corresponds to a state where the spins are doing what the magnetic field tells them. So in this picture I have that orientation for the spin and if I just repeat that then I have some unique state for the system which I can describe as a lattice full of monopoles but it's a unique state so there's none of the interesting statistical physics to talk about. Well you wouldn't really want to be talking about spin waves in this system because it's very strongly ising well often when you talk about spin waves you're talking about the dynamics and the dynamics here is a separate story but it's rather slow relaxation in any case. But if you're talking about the statistical mechanics then as you reduce the field from the one that saturates the system then you can reverse chains of spins and you can think of those as being like flux lines inserted into the lattice and there is some interesting statistical physics associated with that transition from the saturation magnetization.