 OK, so hello everybody, my name is Steve Bramwell from University College in the centre of London in England. It's a pleasure to be here and thanks to the organisers for putting on this interesting conference. So I'm going to talk about Coulomb gases in magnetism, but the very specific aspect of Coulomb gases in spin models and how they can be realised in magnetic materials. And I'm going to be mixing theory and experiment. I'll keep it quite simple, just try and give some ideas that may be worth taking away. And I'll also give a bit of a personal take on the subject, although I obviously will review the subject to some extent as well. So spin model magnets are of course rather an old fashioned thing, they've been around for a long time. And as probably most people know it goes back to easing in 1925, the easing model of spins that just point up or down a two state spins with an exchange constant. And depending on the sign of the interaction, the ground state is either a ferromagnet or an anti ferromagnet. So the interactions are near neighbor and can either align or anti align the spins. Now, obviously, despite being a simple model, it's famous for the fact that in 1949 Onsaga provided an exact solution which is far from simple or trivial. Which is one of the sort of great moments of statistical mechanics in the 20th century. Much later on in the sort of 1970s, people started to do neutron scattering on real magnets. This is a measurement of Hirakawa and Aikida from 1971 on real material where they measure the order parameter. So that's actually this is an anti ferromagnet. So that's a staggered magnetization. They measure the order parameter of this material with neutron scattering. And the line there is Onsaga's exact solution. So the line goes through all the points. Now, I've always been rather delighted by this figure. You can justify the reason that this model applies to this magnet and I won't go into that in detail. It raises a few questions actually because this first of all, this is a three dimensional magnet, whereas this is a two dimensional model. But the main point I want to make is that in terms of model building, spin models offer tractable many body systems and experimental realizations thereof. And that's the short file recipe for discovery when you've got theory and experiment in such close harmony. So my particular personal interest in spin model magnets is I'm particularly like where you can get analogies between systems that are formally unrelated. And my own work over the years has been related to two major areas there. First of all, I've been interested in 2DXY magnets which map to the two dimensional Coulomb gas as has already been mentioned yesterday. We found all sorts of interesting connections like for example with order parameter statistics and statistics of turbulent flow. And this is added to a debate on long Gaussian statistics that goes as far as galaxy density distributions and that sort of thing. A bit closer to home as it were, this is a plot of a data collapse, experimental data collapse between a helium film and a magnetic film near the costless, thawless transition. So you get a lot of analogies in these systems between formally unrelated systems. My second major area that I've worked on is spin ice which is something I invented a long time ago in Mark Harris. And this maps to a three dimensional Coulomb gas. It's actually a magnetic analog of proton order in water ice. But it's a little bit more than that as well. There's been a variety of spin-offs, if you forgive the pun of spin ice. So first of all to so-called artificial spin ice which is micro magnetic arrays designed to mimic the spin model. And secondly to magnetic monopoles in spin ice that were introduced by Castle Overhead Talon by Rich Kinn. And this is the subject of that neutron scattering pattern that I'll show a few times in this talk. So I want to, that's just a little bit of background on my personal interests. So I'm going to talk about Coulomb gases and obviously I maybe don't have to say this but Coulomb gases are important. If you just look at this fish for example, it lives in an electrolyte, it lives in a Coulomb gas. But also the processes of life that go on in there are processes of electrochemistry largely. And they're the Coulomb interaction. And occasionally it's long range nature do play an important role. So there's a lot of Coulomb gas physics in that picture. So I'm going to stick to sort of pose some rather simple questions in this talk and indeed an next talk. I'll elaborate on them. So the first question is how can we create a Coulomb gas out of spins? We've already heard from yesterday that you can do that. But I wanted to try and sort of answer the question a bit more generally today. What kind of Coulomb gas do we get when we create a Coulomb gas out of spins? We'll see that there's a number of caveats that one has to think about but those are interesting in themselves. And finally the most interesting question perhaps is what kind of new insights do we get? So what's the point of all this? I mean is it worthwhile going to all this effort to make Coulomb gases out of spins? And I'll argue a little bit today but especially in the next talk that it is really worthwhile if you're interested in Coulomb gases to think about the spin realisations of Coulomb gases and the sort of things that give you different perspective on the problem which I think is interesting. So that's a sort of preamble and now my plan for today's lecture. So first of all I'm going to address that question how to create a Coulomb gas out of spins. I'm going to give just two examples. They don't necessarily exhaust the possibilities but the first is the well-known two-dimensional XY model that maps a two-dimensional Coulomb gas. And the second is spin ice that maps a three-dimensional Coulomb gas. And then I'm going to talk about magnetic monopoles in spin ice and finally draw some conclusions. So that's the plan. So the first question is how to create a Coulomb gas out of spins. Well let's start with a simple observation. Coulomb's law is long range interaction as we already heard. So it's algebraic one over r. Coulomb of course derive your text experimentally in 1785 just before the French Revolution. Good time to keep your head down and do science. But Maxwell's equations are local of course. So the energy for example is an integral over the electric field, the local electric field. So the question is can you simulate a Coulomb gas locally? Now a very elegant discussion and answer to this question has been given in the work of MAGS and co-workers. So particularly MAGS and Rosetto from 2002. And these workers were interested in simulating, in coming up with methods of simulation, practical methods of simulation of the Coulomb gas of electrolytes really. And as we heard yesterday, they start with the observation that a naive Monte Carlo approach would give a computer time that scaled as n squared, the number of particles in the system. So it's very inefficient. Avalt's summation does a bit better than that. But they were able to present an order in Monte Carlo algorithm. I'll just briefly say something about it without going into any particular detail. So they start with the observation that the energy goes as the square of the integral over the square of the field. But what they show is that you actually need to sample all solutions of Gauss's law. So this is Gauss's law, of course. And the general solution of Gauss's law is that the electric field is a gradient of a scalar potential plus the curl of effect potential. In terms of rather more simple terms, you can imagine the electric field as being Helmholtz decomposed into a longitudinal part with divergences which is the sort of physical part in electrostatics. And then a transverse part which is closed loops of field, and then a harmonic part which neither has divergence nor curl. So just in terms of pictures, that's a longitudinal part, field aligns beginning and ending on charges. What I'll call the transverse part is closed loops of flux. Then the harmonic part could be, for example, just a uniform field spreading to infinity. And what they show is that if you have these fields decoupled energetically and statistically, then you can use this as an algorithm to simulate the Coulomb gas. So they come up, for example, with, they sort of have a charge on a lattice site. They allow it to hop. It drags all of these fields with it. And then you have some moves that allow relaxation not only of the Poisson part of the field, the physical part, but also these other fields as well. So these are auxiliary fictitious fields in terms of the Coulomb gas problem, in terms of the electrical problem. But when you do that, you are able then to have an efficient order n Monte Carlo algorithm for simulating the Coulomb gas. So I won't go into that in any greater detail, but I can refer to all this interesting work on that subject there. So my interest in raising all this is what the relevance it has to spin systems. So it gives an idea of how to create Coulomb gas out of spins. So the spins essentially become the lattice electric fields in the problem. So we need to map spin fields, which are generally sort of discreet vector fields on a lattice in some way to lattice electric fields. And the components of the fields, the longitudinal and transverse components, need to be largely decoupled. And I'm going to show how this works in these two classes of spin model. First, the 2DXY model giving the 2D Coulomb gas with its logarithmic potential. And secondly, spin ice giving the 3D Coulomb gas with its familiar 1 over R potential. So just back to the plan then, that gives an idea on how to create a Coulomb gas out of spins. And I'm now going to talk about these models in detail. So first of all, the 2DXY model. Well, as most people probably know, the 2DXY model is just spins that rotate within a plane, also called the plane rotator model. And they interact via the cosine of their angular differences. So I could adopt product interaction between the vector spins. And for many purposes, you can do a sort of continuum approximation, where you introduce a spin angular difference field theta and take the gradient of that. There are two excitations in the problem, the spin waves, which are wave-like excitations, long, slow deviations of spins in which there isn't much difference between the angles of neighbouring spins. And then vortices, where you get these sort of rotational textures in the spin field. And of course, the main observation that came from Berezynski and Kostlitz and Thalys in the 1970s, I'll call them BKT, is that the spin waves and vortices are largely decoupled in this problem. So, for example, if you integrate this angular difference field round or the gradient of it around a plaquette, then you get plus or minus zero or two pi. And you can decompose this into two components, one for the spin waves and one for the vortices. And that starts off a sort of laser groundwork for an analytical approach to the model, which is extremely successful, mainly due to Kostlitz and Thalys. And what comes out of that is that the vortices form a symmetric two-dimensional Coulomb gas on a spin wave background. A symmetry is related essentially to the time reversal symmetry of magnetism. So, it has to be symmetric. And it's most famous for its pair and binding transitions. So, at low temperatures, the charges are bound into pairs. And at high temperatures, you can get free vortices or charges. And there's a famous BKT transition that separates these two phases. By the time BKT published, this was already established as a property of the Coulomb gas, at least to some extent, starting with the work of Salzberg and Prager back in 1963. So, this pairing transition, which is a property of a logarithmic potential. So, just in terms of the pictures of charges, you get pairs at low temperature, then a transition and then free charges as they unbind. And, of course, this, as has already been mentioned, gives a very good model of vortex interactions in two-dimensional superfluids, for example. And you get a really excellent agreement between theory and experiment on those systems, just picturing the vortices as a Coulomb gas. So, in terms of the electric field representation of what I was saying about earlier, so this is something I, me and my collaborators, particularly Peter Holesworth, have been interested in in the last few years, just because it draws together a lot of information about different models. And actually, we're very much inspired by Beautiful Paper by Valiton Beck from 1994. We don't really go much further than that, but we just made this connection to some of the things I was talking about earlier with the MAG's algorithm. This work was done by Michael Faulkner, a PhD student, who's now looking for postdocs, incidentally. So, you can already start to see the electric field description emerging just if you make the electric field a difference, angular difference field on the lattice. The spin difference field, the angular difference field, goes to the lattice-elect, maps to the lattice-electric field. And this is the actual mapping. It's very simple. And so, you can take any old spin configuration and create a lattice-electric field configuration that corresponds to it. And once again, this lattice-electric field has, is a general field. So, where you Helmholtz decompose it, it has all three components that I already mentioned. So, the longitudinal divergence-fool component, the transverse divergence-free component, and the harmonic component. And not surprisingly, it turns out that this is the vortex field. This is a field coming from the vortices. And this is a spinwave field. And I'll say more about the harmonic component in the next slide. And so, if you look at the actual spin models involved here, those of you who know the XY model will realise there's a kind of hierarchy of spin models. First of all, there's the full cosine XY model, which is the one that's sort of relevant to real magnets, as it were. And then you can make a harmonic approximation to that. And if you can actually, what we call the harmonic XY model, has a potential that sort of goes like that. It's a piecewise parabolic potential. And then there's the more famous villain model, which is basically a parabolic potential, plus vortices with any integer charge. It's not really a spin model, in the sense you can't simulate it with spins. It's a bit more subtle than that. And what we can show is that the villain model actually maps exactly to the Mags Rosetto two-dimensional Coulomb gas in the Gronchononic ensemble. This is an exact mapping. And so it answers the question immediately why this system with short-range interactions is able to model a Coulomb gas with long-range interactions. It's for the same reason that Mags and Rosetto's algorithm works in electrical systems. So these fields are perfectly decoupled, both energetically and statistically. As I mentioned, the villain model isn't really a spin model. The harmonic XY model is the nearest you can get to it while being a spin model. And this also maps exactly but with a temperature-dependent vacuum permittivity. And similarly, the XY model does something rather similar with some corrections. Okay, so one interesting thing that just came out of this. So this is really just another way of looking at an old and very familiar model. So maybe it's surprising something new would come out of it, but we did get something out of it which is what we call topological sector fluctuations and ergodicity breaking at the BKT transition. So the vortices are charged as a topological defect in the electric field, meaning that you can't remove them by stretching or bending the field lines. You have to sort of cut and paste if you want to remove the charges. Now the harmonic fields include a polarisation term and you can show that they include an integer value winding term. So this is done on a torus. So it's really done with periodic boundary conditions. We don't have the curvature of a torus, but we have the topology of a torus. And the harmonic fields, those without any curl or divergence, you can picture just wrapping around the torus like that. And they define to distinct topological sectors. They're excited with sort of integer values. And you can actually simulate this using the simulation technique already mentioned. And you can see the above, the costalous transition, you get these jumps where this field is excited. So this is when two vortices, for example, travel around the torus and join up again, leaving behind a field like that. So these are what we call topological sector fluctuations. Now with local dynamics, it turns out that the suppression of these below the costalous-sparaginski transition is very strict. They're fully suppressed. It's actually no gidicity breaking. Without going into too much detail, you can see it here, for example. So this is a sort of susceptibility for fluctuations of these fields. And it's a ratio between a susceptibility which allows global moves and one which has local moves only. And you can see it goes very much to zero below the transition, and it goes to unity above... Sorry, this has died. No, it hasn't. Above the transition, there's very big fluctuations in between. Now in the actual spin models, these fluctuations correspond to global spin twists. So the fields look like this. They're just uniform, the electric fields. And the spins look like that. So they kind of twist round as you go down. This is all quite new. I mention it because it's of general relevance to Coulomb gases. There appears to be some experimental evidence of this. I don't ask me to explain why, but this paper, if you're interested. And this would be in the phase field of the order parameter of the superconducting state. OK, so that was a quick look at the 2DXY model and just some sort of general lessons to learn from that. First of all, spin models provide lattice fields that map to the electric fields of a Coulomb gas. The Coulomb gas is symmetric and in the grand canonical ensemble. We can expect spin models rather generally to realise a general Coulomb gas in which you have these auxiliary field components, the transversal harmonic components. They exist and in any old spin model that they won't necessarily decouple perfectly. However, those auxiliary fields are interesting in their own right and lead to, for example, these topological sector fluctuations. And later on, I'll give other examples of what these auxiliary fields do. So just back to the plan. So that's the 2DXY model and now onto the second example, a spin ice, which is the three-dimensional Coulomb gas. And that's going to be the main part of my talk both today and tomorrow. So spin ice is a spin model realisation of a three-dimensional Coulomb gas. But how you actually get there is quite subtle. I want to give a brief history of spin ice so just to explain how we came up with this sort of rather strange concept. So if we go back to Nernst himself and the third law, I'll take that to mean just for simplicity, I'll take that to mean entropy zero at zero temperature. Of course, following Nernst's proposal of the third law, there was a lot of debate and a lot of experiments. Some of the most beautiful experiments came from William Joke who measured the residual entropy, the zero point entropy of real materials. And one particularly interesting one was water ice where he measured with very high precision and most extraordinary measurement as one of the most beautiful calorimetric measurements ever made, the entropy of water ice and discovered it was non-zero in the approach to zero temperature. And this was famously explained by Pauling in 1933 in terms of proton disorder. So if this is a fragment of the ice structure, you have oxygens which are the big ones and hydrogens which are the small ones. And so you have an ice rule which means you have two hydrogens close to each oxygen and two further away. And so this satisfies chemical bonding requirements. So two short covalent bonds and two longer so-called hydrogen bonds, these sort of electrostatic bonds. Now it turns out that when you go through the system applying the ice rule, you have a choice side by side but it's not a completely free choice. Nevertheless, you build up a number of equivalent states that grows exponentially with the size of the system. And when you take the logarithm of that, that gives you the entropy and Pauling calculated this on the basis of the ice rules and remarkably got a number, allog three halves per mole of water molecules that agreed with an experimental error with William Jokes experimental measurement. And hence it was largely accepted immediately. Now there's long been a sort of an interest in creating a magnetic analogue of proton disorder in ice. A significant effort was made by Anderson in 1956 who considered an antiferomagnet on this pyroclor lattice. This is a lattice of corner-linked tetrahedra in which spins can point up or down and it's frustrated because on one given tetrahedra the best you can do to satisfy antiferomagnetic interaction is two spins up and two spins down and you can see at least roughly speaking that this could map to too close, too far to an ice rule and indeed it does in a slightly complicated way actually but one has to just look closely at it to follow it. When I came into the field in around about the early 90s one thing that struck me was that people had been looking for this for years and never found it. There are no experimental examples of Anderson's model in the purest sense, at least not in the sense in an ideal sense at any rate. I think when you see something like that the most likely explanation is always that the model is lacking something that makes it maybe unphysical in some sense and far be it for me to say Anderson's model was unphysical but you'll see that in a sense it is. Now the problem that struck me and others and my colleagues was that the real issue here is that the local point symmetry on this lattice is not just up down like that, it's a trigonal point symmetry that goes along the long axes of the cubic structure and so the local point symmetry if you're going to get easing like spins in real material they're going to point in and out of this tetrahedron and you can't really do anything about that and instantly in real materials you always have some anisotropy that will reflect the point symmetry or nearly always you have to work very hard to avoid it. And so the idea that we had was that up down classical easing model was incompatible with the point symmetry so we studied models where the spins just pointed in or out and the first discovery was it kills the zero point entropy completely so when you have an anti-ferromagnetic model with in-out spins you get an ordered state, all spins in or all spins out alternating on different tetrahedra so Andersen's model falls to bits. However, not long after that we made a sort of what we thought was a very surprising discovery we were thinking about an allergist ferromagnet particularly in the context of these materials called homeium titanate and dysprosium titanate here they're nice sort of crystalline solids only the homeium and dysprosium is a magnetic and they're very big magnetic moments compound spins with huge magnetic moments and we sort of just wrote down our simple model for that and discovered to our enormous surprise that with ferromagnetic interactions the ground state of a tetrahedron is two spins in, two spins out and this maps this goes straight back to the ice model so rather remarkably by changing the symmetry of Andersen's model and changing the sign of the interaction you do generate an ice model and now the reason that in these materials that the spins point along these axes is a crystal field term that glits the free iron state such that the ground state is a doublet the maximum value of the spin it's a quasi-classical spin that really does point along those directions either in or out and it can flip between the two and so the water ice mapping is generated by ferromagnetic interactions plus the crystal field and so quite literally if you imagine putting a proton in every spin in a configuration of this ferromagnetic model then you get a configuration of protons in water ice there's a one-to-one mapping and you can carry over the statistical mechanics of water ice in its entirety almost at any rate to describe a spin ice and so this is why we called it spin ice for obvious reasons very direct mapping from spin model to the proton model and the ice rule there is therefore two spins pointing in, two spins pointing out of each tetrahedron okay so over the forthcoming years a lot of experimental verifications of all of this and one of the most beautiful was the Pauling entropy measured by Ramirez and colleagues and so they did a similar calorimetric experiment to the ones that had originally been done on ice and they measure the Pauling entropy in the ground state Over the years we and others have improved the neutron scattering treatment of these experimental systems such as this is about the best we've got so far this is a neutron scattering pattern straightforward neutron scattering no smoothing or adornment it's really a remarkable pattern those of you who know neutron scattering it measures spin correlation function in reciprocal space and you can by using polarised neutrons you can get all the components of this tensor now if you know X-ray or neutron scattering you'll know that if you have order periodic order then you get Bragg peaks we don't have any of those if you have say spin glass which has a kind of very subtle order but you just get a blob like a potato so we don't have that either what we have is something in between which has some considerable structure most notably including these narrow necks now we call these pinch points now come back to those in a moment now this is a simulation the one on the right that's experimental data that's a simulation just of the Pauling ice model and you can see it's pretty similar there's one or two discrepancies in fine detail that turn out to be interesting as I'll show later so just a word on these pinch points so these are rather fundamental to ice type systems so if we think about the ice rule two spins in, two spins out and if we grab these spins as link variables on a continuous flux that I'll call the magnetisation then the ground state because you can see this is a sort of conservation law as much flux flows into a tetrahedron as flows out and so the ground state then the ice rule state it consists of closed loops of flux the smallest of which is six membered rings larger loops even span the entire lattice and this particular correlation function or neutron scattering pattern reflect that very closely so you can show this in a very simple argument if you say the free energy is entirely entropic going as a square of the magnetisation an integral over that and then you simply impose the constraint that the magnetisation is divergence free then you can calculate the correlation function and you discover it gives these pinch points this wasn't originally discovered for spin ice but actually there are analogous arguments for ferroelectrics and for ice itself indeed and a lot of set of very nice papers by youngblood and axe in the 1970s discovered that but obviously to see it in a magnet is rather remarkable but back to the sort of electric field description so the electric field there for other obvious reasons the magnetisation of a spin field is the one that's going to map to the electric field directly again it has all its three components so in the ground state you only have the transverse component so the ice rule gives you that but the longitudinal component comes from defects in the ice rules H3O plus or OH minus defects so basically if you take a proton off it off it's water molecule and take it to another water molecule you get H3O plus an OH minus and you can separate these defects and in spin language this gives you three spins in and one out, three out and one in and these are divergences then in the effective electric field so that kind of charges in that sense and so again this transverse field is the ice rule configurations and the harmonic field again see next slide so just to complete the sort of overall picture of spin Coulon gases we'd expect topological sector fluctuations in spin ice as well I'll just briefly mention that this is mainly work of Ludovic Jobert and so system spanning loops transformed to harmonic fields so you have two types of loops in the ice rule states you have closed loops and system spanning loops rather remarkably you can show theoretically that there's an experimental signature of this there's a crossover in the magnetic susceptibility from the Curie law example C over T to the Curie law again but with the factor 2 you can actually measure this experimentally as well so this is the crossover of the Curie amplitude doesn't work perfectly we understand this but I won't go into that in fine detail and essentially entirely because of the long range interactions you have to use very perfectly spherical crystals in these experiments because of this very big de-magnetising fields ok so just to sort of complete that comparison with electrical Coulon gases and also the 2DXY model so comparison of with spin ice is in the XY model it's a spin difference field maps electric field here it's the spin field itself that maps electric field in spin ice and in both cases a Helmholtz decompose components nearly but not quite decoupled I'll come back to the not quite a bit later on because it's important in itself so back to the plan for today I think I'm more or less on time so I've gone through sorry that's not coming out very clearly ok so I'm on to magnetic models magnetic monopoles in spin ice now so magnetic monopoles in spin ice really come from two papers one by Rishkin in 2005 and the other by probably a slightly better known paper by Castle and Overmass and Sond in 2008 and the reason I wanted to explain sort of how these were discovered because it's a good example of why spin models are kind of useful so early on although we we sort of realised collectively in the field realised that we had a sort of Coulomb gas in spin ice we not ever thought about the effect of dipole-dipole interactions in the system because we thought that they would just make a mess of the whole thing because they're complicated and unisotropic and long range and we thought that simple picture that I've given would be totally ruined by adding dipole-dipole interactions into the mix but people who were interested in the material side of things rather than the model magnet side of things wanted a proper Hamiltonian for spin ice rather than the phenomenological spin ice Hamiltonian that Harrison I invented so this is the Hamiltonian they came up with and it's called the dipole-spin ice model coming from Shastri and Sidarton and Jingran colleagues and you can see it's a really ugly beast it's got a magnetic dipole-dipole interaction and then exchange interactions about third nearest neighbour very ugly thing and the question immediately rose was how could you square this Hamiltonian with the very simple observations that spin ice really worked for for the real materials and that was a big question in the field for quite a while and it was solved first by Jingran and then by Isaacow and colleagues and so it turns out that the dipole-dipole interaction has the property that it's self-screened in the ground state so you can map all of this to very good approximation onto an effective near-neighbour ferromagnetic exchange interaction hence that's why spin ice works at least in the ground state however the dipole-dipole interaction survives in excited states now then there were two lines of reasoning that led to the idea of magnetic dipoles in spin ice so one coming from the water ice community so Ryskin is a water ice physicist many years earlier in 1984 he'd written down a pseudo-spin Hamiltonian to describe electrostatic interactions in water ice itself and lo and behold it was the same Hamiltonian as this one with electric and magnetic interactions replacing each other now the other way of doing it was to go straight from this Hamiltonian and this is what Castelover, Mercer and Sondi did they found a way of retaining the long-range part of the dipole interaction whilst getting rid of one or two quadripole-connect corrections that occur at relatively short range and what they discovered and Ryskin discovered as well was that you actually have a Coulomb interaction between the H3O plus and OH minus type defects and that's what we call magnetic monopoles now the most important thing about magnetic monopoles is really shown in this picture here just to try and understand it so if you look at a tetrahedron where the ice rule is obeyed so there's as much flux going into it as there is coming out of it then in magnetostatic terms the divergence of M, the magnetisation is zero and hence the divergence of the H field of magnetism is also zero because there's no the divergence of B is zero of course because of Maxwell's law and so there's no charge density in the ground state whereas if you add some defects in so three spins in, one out three out, one in for example then we now have divergences in these fields and so you can immediately imagine that these are going to interact coulombically and indeed they do and you can show that as Ryskin did very simply actually by integrating a line of spins between the defects now these lines of spins that exist between the defects we sometimes call Dirac strings rather grandiose-ly because you know that a dipolar string monopoles at the end but the important thing is that the so there's for example a very short string going between two of these monopoles now but the important thing is these strings don't pair the monopoles in any way so you can draw strings in different ways the monopoles kind of exist on a string network but they're not paired and that's why you can call them monopoles because they're not paired yet they interact via the magnetic coulomb law with a well-defined charge and within a factor of two the charge is what you'd get if you took a magnetic dipole and naively stored it in half twice actually twice that value and that's all and this is mu naught the vacuum permeability so what these authors showed then was that real magnetic coulomb interactions exist on top of entropic ones real in the sense they come from the true dipole-dipole interaction in this Hamiltonian so in a sense we've combined an effective interaction with a real interaction fundamental one and the picture to have in mind is lots of little north and south poles interacting coulombically and moving around the system okay so I want to just mention the monopole conduction because this is an important puts an important caveat on the picture so this goes back to Rishkin's original treatment so in water ice if you wanted to conduct protons which it does it's a very good proton conductor you have to move the protons through the system so one way of doing that is to move them along the bonds oh I was about to say that so monopoles are dragged by real magnetic fields in the same way protons are dragged by real electric fields so if you want the system to conduct you have to push the protons along the bonds like this and that starts to separate charge so you've got positive charge down here and negative up there but you've also polarised the system now the similar thing in spin ice is shown here so you flip a spin you create two monopoles then you flip further spins restoring the ice rooms locally and you've left behind this string of a lined spin so again you've magnetised the system in this case you've polarised it or magnetised it so again that's another picture of the same thing so as monopoles move through spin ice or protons move through water ice they polarise the system so the monopole hopping drags the electric field as it were with it and polarises the spin system now in the context of water ice Jackard treated this theoretically by defining a configuration vector so this is just a vector that describes a spin order and a finite vector cos entropy now for spin ice Ryskin showed that the configuration vector is just the magnetisation divided by the charge and hence the entropy grows sorry the entropy is diminished by the square of the magnet the local magnetisation of the system so there's an entropic cost of monopole conduction and the thermodynamic equation of motion then is the rate of change of magnetisation with time is proportional to the field but then there's this reaction field that eventually brings the system to equilibrium so if I want to discuss this with respect to the magsresetto coulomb gas that I introduced earlier basically in that you hop a charge and then you relax the local fields but in water ice it turns out that the there's another kind of defect in the system so-called bonding defects where you have two protons on each bond or no protons these relax the polarisation and they allow a conducting steady state they allow you to pass protons from one side of the system to the other and it's actually quite a bit of magic that this happens it's really difficult to see intuitively but they do actually succeed in doing that but in classical spin ice we don't have that luxury and in spin ice all relaxation is mediated by monopole hopping the transverse relaxation is relaxed by monopoles only so all the fields in the system are relaxed by monopoles only and they're not entirely decoupled in this sense and that's really the origin of the of the jackard reaction field so I'll just say a quick word here about quantum spin ice I'm not going to do any justice to this it's a very topical field and a huge number of papers and very interesting ones it's less connected with experiment though than is classical spin ice I'm just going to mention one paper by Shannon and Benton and colleagues which is particularly nice one in my opinion so what you can do is these transverse fields so these closed loops of spins you can actually introduce some dynamics in those so you can introduce a tunneling sort of matrix element in those now in principle this could happen experimentally it doesn't in these particular materials that we study but in principle it could happen it could come from transverse terms in the spin operators but when you do that you can argue that the transverse M then transverse magnetisation then behaves like a lattice gauge field and along with the monopoles you then get a complete analogue of lattice, electromagnetism, quantum electromagnetism and you get photon excitations this is a dispersion and you can there's another picture of the dispersion of the photon excitations and how they relate to the field correlations in reciprocal space and in fact there's a sort of netexperimental sort of signature of this is when you have these processes going on it washes out the pinch points that you see in the classical pattern okay so I've more or less finished for today and I want to draw some conclusions so the three questions I posed at the start then were how can we create a coolant gas out of spins what kind of coolant gas do we get what kind of new insights do we get so in terms of the first question then the answer is really that you need to map the spin fields to electric fields and decouple the longitudinal and transverse electric fields and 2DXY model and spin ice although at first sight rather different actually very similar in that regard what kind of coolant gas do we get so we get a symmetric coolant gas it has to be symmetric because of time reversal symmetry of the spins and the kind of general coolant gas so it is a coolant gas but it includes all solutions to Gauss's law in general so there are longitudinal and transverse fields that will always be present and retain some coupling and in spin ice the monopole coolant gas is very much conditioned by the so-called Jackard field this reaction field coming from the polarization of the monopole vacuum as it were so finally done what kind of new insights do we get well I mean just by thinking in these ways you can see that you're always going to get topological sector fluctuations of the harmonic components magnetic monopoles in spin ice are in sense new insight by comparing water ice and spin ice we've already learned something about the for example the pseudo spin Hamiltonian of water ice so it was a worthwhile exercise already in that sense and emergent electromagnetism in quantum spin ice but most of the new insights I'm going to be talking about tomorrow so tomorrow having sort of set it up I'm going to focus on spin ice and show that it really is a model lattice coolant gas but I'm also going to show that it has emergent chemical kinetics of these monopoles you can write down chemical kinetic equations coupled equations and you can get non-equilibrium effects and all of that sort of thing and I'll argue that it's basically a model system for studying charge generation from a vacuum and I'll show both theoretical results and experimental results to try and convince you of that so that's my talk for today and thanks for your attention