 Thank you. So yes, so this is a second lecture I've given and I'm going to continue here on the topic of spin ice, which is a realization of a three-dimensional coulomb gas in a spin model and also in real materials. Let me just remind you if you weren't there yesterday or if you've forgotten the key facts about these magnetic monopoles or magnetic charges. First of all, the vacuum that they spring from is a transverse or divergence-free magnetisation field. So closed loops of spins that are a kind of soup of closed loops of spins with overall disorder, but strong correlations. Monopoles are thermally excited sources and sinks of this flux, so they're breaks in the loops and they live on a kind of network of strings but they're not paired in any way. Monopole diffusion arises from spin flips. In the real materials that occurs actually from a sort of local quantum mechanical flipping process but there's no coherence beyond that, so it's like a quasi-classical spin flip. And so this is very much like an electrolyte with diffusive dynamics. It's a lattice electrolyte, it's symmetric, there are charges of both types and they have equal properties, a part of the charge of course is opposite. It's overall neutral, there's no charge, overall charge imbalance and it's in the grancanonical ensemble, so the charges are thermally excited from the vacuum. A charge number isn't fixed but the chemical potential is a material parameter which is fixed. When a current of monopoles flows in response to say an applied field it magnetises the system locally and costs entropy so it spreads out these, it pulls out these strings and it costs entropy and that's an important caveat. And this is a thermodynamic equation of motion for the monopoles so the current is the rate of change of magnetisation, it's a polarisation current because no monopoles pass through the boundaries of the system and it's proportional to the field minus this term in the magnetisation which ensures equilibrium is reached. M goes to chi times h at equilibrium. So I wanted to continue asking some of the questions asked yesterday in particular what kind of new insights do we get and I'll hopefully illustrate some of these today. So the plan of today's talk then, I want to first of all focus on spin ice as a model lattice electrolyte so this is a sort of background in a sense to what comes later. I first want to look at the screening of the coulomb interaction and electrolyte and also the pairing of charges so in three dimensions there's no costal, it's howless transition in a three-dimensional coulomb gas but there is pairing, some pairing of the charges which is rather interesting in its own ride. I want to show how then then we'll just look at the linear response of the system and show how that agrees with experiments on spin ice materials. I then want to go on to the main part of the talk which is chemical kinetics of magnetic monopoles and far from equilibrium behaviour so I'm going to argue that ordinary chemical kinetics is quite a powerful tool for looking at non-equilibrium and even far from equilibrium behaviour and so if you can map a complex system onto chemical kinetics which is actually what we do here then you've got a way of addressing this far from equilibrium behaviour and I'll then consider spin ice as a model system for charge generation kinetics and I'll give other examples of that in nature and the particular example I'll be interested in here is what's called the veneffect of electrochemistry and famously analysed by onsages I like to call it onsages veneffect and show how this is an effect, a very non-linear and non-equilibrium effect, a pure consequence of the coulomb interaction and an example of how chemical kinetics is strongly modified in the presence of a long range interaction giving some universal results. I'll show how that works for spin ice and then I'll finish off with showing how we can use that to understand experiment on non-linear and far from equilibrium magnetic response in spin ice materials then I'll draw some conclusions. So just on the first topic then the electrolytes as long as the magnetisation of the system is zero on some reasonable length scale we can expect spin ice to behave as a model lattice electrolyte it's a symmetric one it's overall neutral and you can think of it in the grand canonical ensemble and it's a lattice electrolyte but to a first approximation we can model it as a continuum electrolyte with a hardcore repulsion to remove the coulomb singularities and that of course is is given by the lattice constant in the lattice electrolyte. Physically speaking it's like what electrochem is called a weak electrolyte which is a partly dissociated electrolyte so an example of that is water so water dissociates into into ions in water itself the concentration of these ions is exceptionally small as I'm sure you all know but very important in a general weak electrolyte this concentration might actually be large so weak is not a statement about the density of charges it's a statement that the dissociation is incomplete. So let me give you a little bit of background on electrolytes for those who are unfamiliar or may have forgotten this sort of old stuff. So the first important thing about electrolytes is screening and of course the famous Debye-Huckel theory of 1923 very beautiful theory by Debye and Huckel and so the basic idea is rather than solve the very complex sort of many body problem they focus on a single ion let's say a positive one and regard it as surrounded by a smeared out ionic cloud and they have a finite ionic radius so that so that the potential is kind of cut off at the boundary of the ion and then they consider the Poisson Boltzmann equation so they write down the Poisson equation with a with a Boltzmann factor to describe the charge distribution they linearize it and then they're able to solve the properties thermodynamic properties of the electrolyte self consistently and this process is strictly valid for small density or high temperature so the parameter the small parameter is in fact density over temperature and then just on this picture here the potential of course dies away something like that it's the famous screened Coulomb potential going as e to the minus kappa r where kappa is the inverse Debye length so that's kappa sorry kappa to the minus one is a Debye length and of course this is was the first time in physics that this famous potential was was introduced however there's a an important caveat on on on Debye-Huckel theory um which is the issue of pairing in electrolytes so uh Bjerum in 1926 sort of put his finger on the basic physics of what's going on here so not long after Debye-Huckel um the point is that um there's a you can you can form Coulombicly bound pairs in the system so and it turns out that not all the charges are free so there's a dissociation of Coulombicly bound pairs now Bjerum analyse this approximately by considering a radius around an iron which is a so called Bjerum length and this is comes this is a length scale that comes from equating the thermal energy kt with the Coulomb energy so if if an iron gets further away from another one uh with this length from this length scale then this length scale it becomes free it's it's liberated by thermal agitation this is so called Bjerum length and if a negative iron approach the positive one and falls within this length scale it will essentially fall towards the uh positive iron and stick to it to a first approximation um in fact you can do a very simple back of the envelope calculation which was done by Foss in 1934 just a simple statistic on mechanical calculation of the iron pair distribution function in fact it's bimodal so the uh the first peak is due to Bjerum dipoles these things and that's the Bjerum length and the second peak is due to uh free charges as assumed into by hookle theory um in fact this the trough in between these peaks doesn't go to zero if it did these would be straight this would be a strictly a separate chemical species but Bjerum approximated uh this trough as going to zero and treated these as a as a separate chemical species with chemical equilibria uh to the free charges which is an approximation but a very useful one in practical sense the whole story is much more complicated than that so so there are higher order couplings uh leading eventually to phase separation and strong correlations and this is what was worked out only relatively recently by Fisher and one of the member of the audience so how this relates to spin ice is shown here um so in spin ice because it's analogous to water ice you you need to think of a process whereby the vacuum in a sense the water molecules produce a pair of charges which then dissociate so the initial production of charge from the vacuum can be shown like that so you produce an exciton in a sense or a Bjerum pair these then dissociate um so you over cut they overcome the Coulomb potential to do this so this distance here uh which is more than twice sorry more than half of that energy so this energy is more than half of that energy because of the Coulomb interaction is the free monopole chemical potential which in spin ice materials is essentially a material parameter and then this the screening lowers this energy and actually pulls more charges out of the vacuum um so the the uh charge density rises a bit further than just as will be given by the pure chemical potential and uh so this is the Debye-Huckel screening and uh there's an effective chemical potential effective energy scale which is this one here which is the one that dominates uh the properties of the system and controls the density and you can work uh this out with Debye-Huckel theory and calculate the density and so on it's straightforward to apply the Debye-Huckel theory to to the spin ice system and you can calculate the specific heat um and this uh calculate this is a uh calculation has no uh fitting parameters because we already know the chemical potential from the mapping of the spin Hamiltonian to the spin ice model to the monopole model um and so this this shows theory versus experiment and you can see it works it works pretty well that the point are experimental specific heat for a spin ice material uh i won't go into the fine details of this is an experimental issue but you'll see there's a little bit of a discrepancy up here this is actually uh comes from the breakdown of the Debye-Huckel approximation of small density over temperature but rather interestingly in this system Debye-Huckel theory works pretty well at high temperatures uh because uh the temperature is high and it works well at low temperatures because the density is low uh so the this can this was a continuum theory with the hardcore repulsions first done by Castle Overtel but we've recently extended that to include double charge monopoles which is the only way that uh you can fit the data at high uh over the whole temperature range uh you might ask why don't we use a lattice theory there's some very nice work by Fisher and colleagues from 2002 used doing lattice Debye-Huckel theory uh the answer is nobody's yet done it for a non Bravais lattice and we have a diamond lattice here and so at the moment we've been content just to use a continuum theory but you can see that already works pretty well okay so um just to elaborate on this a little bit electric chemists have something they like to talk about called corresponding states which is where you introduce a rescale temperature and a dimensionless density and you can collapse a lot of electrolyte properties onto this kind of phase diagram uh without going into too much detail uh the region of high temperature and low density is the ideal gas region the particles are too far apart to feel their coulomb forces and then uh you have the Debye-Huckel theorem at slightly higher density slightly uh higher temperature then the sort of Bjerynpairing region uh clustering and so on in eventually phase separation where we are in spin ice roughly speaking is in this part of the phase diagram and indeed up to the sort of high end of that so each individual spin ice material because its chemical potential is fixed by by material parameters the dipole coupling and exchange constants um uh each material tracks out a trajectory in this space so the only way we can reach the strong strongly correlated regime is to find a new material that takes us into that regime unfortunately we're just kind of on the edge of that regime and the spin ice materials that are known there are about 10 of them known um at the present so chemical potential is is uh related to as I mentioned exchange terms and it from a phenomenological point of view it's the energy of a pair plus the coulomb energy divided by two okay so that was a quick look at screening and pairing and now a linear response so we can do linear response for the coulomb gas uh you can write down a set of first of all if one can write down a set of drift diffusion equations um this is like one might get for a semiconductor for example um now for an electrical system you'd write down a series of equations like this uh diffusion and that's a diffusion term and a drift term uh this little e would be the screening field the field from the charges uh big e would be an applied field and then you have a you can have a polarization current so if you have electrolyte in a box for example all the current would be a polarization current um you have a continuity equation where I've omitted some chemical kinetic terms here which prove to be important at later on they're not important at the level of linear response and then you have Poisson's equation or Gauss's law uh which applies to the screening field um now the first step in translating this to like magnetic monopoles in spin ice is simply to replace electrical quantities with magnetic ones which I can do in powerpoint just like that so I've now replaced electric field with magnetic field polarization with magnetization and so on but there's one extra term I need to take a count of which is a term I discussed in the last lecture this um entropic term here which is specific to spin ice systems or ice type systems which is so called jackard field a reaction field which accounts for the polarization of the vacuum by a current apart from that they're these equations are the same as the electrical case now that's just to remind you that's a screening field and those are carrier generation recombination terms which I'll come back to later on and and all of the this can be solved analytically uh for the generalized response functions for from a magnetic perspective we're interested in the wave vector and frequency dependent susceptibility because this relates to a lot of experimental properties so I can just illustrate a couple of solutions so for example at q equals zero which is just the bulk ac susceptibility um I won't again I won't go through this in too much detail but we can the response is simply di by like um and we can measure a variety of these uh various quantities that go into this experimentally and from that we can isolate the mobility so we can get for example the density from the specific heat and we can isolate the mobility uh this is an experimental measurement of the mobility versus temperature I won't talk about what's happening at high temperature here but at low temperature you see the mobility goes as one over temperature which implies a constant diffusion constant uh and is consistent with an ansteinstein equation uh implying brown in diffusion of of monopoles uh with an a thermal hop rate um the hop rate doesn't have to be a thermal and in fact at low temperature there's evidence that it it changes regime and becomes thermally activated but it but it works reasonably well similarly we can look at the solution at omega equals zero zero frequency which gives you the wave vector dependent susceptibility and hence the dynamical correlation static correlation functions in reciprocal space that you can measure by neutron scattering so that that thing um this is the solution uh you can see there's a so this is basically a statement of these pinch points here and this says that the width of a pinch point uh is a lorenzion so going across the width there is a lorenzion and the width of the lorenzion is the debilength so you can measure the debilength in that way um that's the experimental data and you can fit it to the theoretical expectation it works reasonably well there are some issues down at low temperature uh actually the this this uh as you can see actually if you carefully compare that with that it doesn't entirely work well in fact it can be improved by a Poisson Boltzmann approach which retains some non-linear terms that are emitted in the linearized susceptibility there okay but the point I want to make by all of that is basically that the um the system spin ice system is pretty well defined as as a lattice electrolyte the one major caveat being the addition of this reaction field term this jackard field which is this extra property that occurs in in water ice as well so I now want to go on to the main part of my talk which is to talk about chemical kinetics and far from equilibrium behaviour so I want to first sort of give an advert for chemical kinetics so chemists use chemical kinetics just standard chemical kinetics I mean as in physics we know it really goes deeper than that but just standard sort of chemical kinetics is nevertheless a very powerful tool uh to describe far from equilibrium systems so this was first understood by Lotker in 1909 who wrote down a simple reaction scheme a so-called autocatalytic reaction which eventually produces a steady state and you can very simply using textbook physical chemistry write down the kinetic equations for that scheme and you discover that it approaches steady state so the density for example of this substance a whatever that might be uh approaches steady state in an oscillatory way so rather than uh approaching exponentially to the steady state it oscillates and this was the start of uh as many of you will know of uh a sort of industry that discovered complicated far from equilibrium behaviour in chemical systems a possibly the most famous example in experiment is the Belusof Shabotinsky reaction where you get these complicated spatial and temporal patterns so the thing oscillates for a very long time both in space and time before it thinks about approaching equilibrium and this can be pretty much understood using standard chemical kinetics just a very complicated chemical uh kinetic scheme with feedback loops and that sort of thing so and it's quite remarkable I think that something as simple as chemical kinetics can understand all of that I mean you need drift diffusion terms or obviously to get the spatial variations here so later on I'm going to show how a kind of emergent chemical kinetics can describe far from equilibrium behaviour in a magnet which is spin ice which is quite surprising and I'm going to show this plot in particular so this is actually the density of monopoles as a function of time after you apply a field and the blue line is spin ice you can see that it starts here and it doesn't it goes up and then it comes back down again to much lower values not actually zero so there's uh again a non-monotonic approach to equilibrium in that case I'll explain where that comes from a bit later on so it's also worth thinking in this context as of spin ice as a model system for charged generation kinetics so I want to make some sort of broad brush stroke analogies with a whole raft of other systems from elsewhere in in physics and science in general so let me do that for a few minutes before going back to the chemical kinetics um so this you can think of as a universal equation because we've got a vacuum producing a pair of chargers that has many applications though as we've already seen it occurs in water arguably the most important chemical reaction there is as far as the process of life are concerned occurs in semiconductors where you produce electron hole pairs uh and at a more exotic level it occurs in QED in a sense when you produce electron positron pairs the Dirac C uh single law of physics is involved here which is is is the Coulomb interaction um and there's a vacuum and in general in all these cases the physics of the Coulomb law is going to be conditioned by the vacuum in some way so in a rather trivial way in water you have the relative permittivity of water so water's quite polarizable as you probably know you can rub a pen and attract the stream of water um similarly you have the concept of effective mass in semiconductors and and at a much more serious level you have vacuum polarisation and and lamb shift and all that sort of thing in in QED uh one way of probing these um sort of processes is to apply a field so one could apply an electric field to that and what's going to happen is you let's say you have a pair of charges produced from the vacuum then the field is going to give a tendency to drag them apart and uh this field assisted dissociation occurs in a variety of scenarios so in electric chemistry it's called the so-called second vene effect and there's a beautiful paper by Onsarg which I'll return to later where you actually produce charges from the vacuum by applying electric field you can imagine ripping apart a hydrogen atom uh with electric field and a famous calculation by Oppenheimer himself in 1928 did exactly that you need a really big electric field similarly the so-called Schringer mechanism of production electron positron pairs at a more practical level you can imagine field emission uh where you eject electrons from a metal surface surface and they interact with their own image charges uh and the so-called Fowler Nordheim mechanism there and in solid state physics you can actually apply electric fields to solids and ionize them and giving rise to increases in conduction so by the so-called pool frankl effect so all of these involve the Coulombic unbinding in a sense and because they all involve the same broad physics uh there are some universalities come out and this is to some this is related to the similarity of the processes as well as the algebraic nature of the Coulomb interaction so what you have then is if you produce a pair from the vacuum and look as a function of distance then the potential you're dealing with looks something like this so the the flat part there or the sloping part there is the potential coming from the field and it adds to the Coulomb potential that binds the charges together so you get a a peak you get a a Coulomb barrier and there are basically two ways over the barrier through the you can go over the top classically and you can do a simple again back of the envelope calculation if you have an attempt rate with the Boltzmann factor trying to get over this barrier the barrier becomes lower the stronger the field you apply and it's a very simple calculation and you can work out a rough approximation to all these effects where the current goes as the exponential of the square root of field divided by temperature and this characteristic form is a property of the Coulomb interaction for example if you put in a different interaction a different algebraic interaction you'd get a different exponent so this is in a sense a transformation of the Coulomb law the second way of getting through the barrier is quantum mechanically tunneling it that's what happens in Fowler-Nordheim tunneling and in the Schwinger mechanism and in both cases you get something like the current going as the exponential of an energy scale divided by the field which comes out of simple quantum mechanics these are approximate expressions but the universality of them is largely related to the universality of the the algebraic nature of the Coulomb law so their characteristics of the Coulomb law so here i'm going to in what remains the talk talk about how all this applies to magnetic monopoles in spin ice so i'm going to argue that it provides a model system to study such charge generation partly because we have a detailed theory classical theory of both the Coulomb law and the vacuum in this case and also we have experimental methods to isolate the pair unbinding which i'll show later on so i'm not going to talk about this in detail how it works in spin ice by talking about onsager's veneffect in spin ice so onsager's veneffect is an electrochemical phenomenon that we're going to translate into magnetic monopole language for spin ice so uh this is really chemical kinetics meeting long range interactions and there was a very beautiful paper by onsager in 1934 a typical work of genius onsager um he solved a very difficult electro diffusion problem um for which he even had to invent some new mathematics he was undeterred in that and then problem he basically was interested in was if you have molecules dissociating to these barren pairs and then dissociating further to free chargers new plane electric field what happens uh now this experimentally it was found there was an increase in charge density and quite a striking increase which then caused an increase in conductivity and a breakdown of ohm's law breakdown of the linear conduction law so there's a field increase of dissociation rate constant and onsager saw it as an electro as a as a diffusion problem in the combined electric and coulomb fields so this is a three-dimensional potential surface of the electric of the coulomb field of pairs so you're regarded as a two-body problem of pairs diffusing in the coulomb potential and the external field and remarkably it's a two-body diffusion problem a difficult problem to solve remarkably he solved it essentially exactly and his solution which i won't go into detail because it's rather complex um but this is a result of it he discovered that the only thing that happens is that the the dissociation rate constant of these pairs and actually before anybody asks you don't the approximate nature of of the bier and pair pair is not important for onsager solution it's just a way of thinking about it but the dissociation rate goes as a function of the field um which is a the special function and the field is is a rescaled a dimensionless quantity that goes as the modulus of the applied field it's a rare example of of of a physical property depending on the modulus of a vector property of a vector field um and from that he was able to derive the increase in charge density and the conductivity and therefore could um check against the experiment now this function of how the the um charge density or conductivity increases with field is shown here a high temperature is already advertised it goes as exponential square root of field that's the that's sort of rather universal form coming from the coulomb law but at low temp at low field it's much more complex and eventually becomes a linear form um and this is what intrigued onsager so much because that appears to be contrary to ordinary statistical mechanics expanding around equilibrium states where you'd expect a quadratic increase um now when onsager checked his results against experiment remarkably the line goes through the point with no fitting parameters um the only difference is that there's this small crossover to um quadratic form so actually there is a quadratic form at low field onsager returns the problem in 30 years later and solve that as well and that comes from the screening so not surprisingly so the screening eventually restores ordinary expectations of statistical mechanics but the screening cloud is basically blown away by the applied field and and then you the system you basically have a two body problem where charges that are created try and escape each other through diffusion and this is where onsager's original solution applies so the screening the screened veneffect correction also fits experimental data extremely well so just to summarize that rather complicated thing the onsager onsager calculated the rates of increase of charge density with field uh sorry the and very low field it's quadratic because of screening even at low fields in dilute systems it's linear and then at high fields it goes over to this universal exponential root field form so we were interested in translating all of this to see if it worked for magnetic monopoles in spin ice um so we had a two so first of all we did this theoretically and then we also later did experiments so um we had a two step process to the theory the first thing to do was to establish that the veneffect works for a lattice electrolyte and so i worked here with um peter holsworth and roderick merciner and the person who did the bulk of the work was voicet kaiser uh here who um managed to to simulate lattice electrolytes at very low density the veneffect had never been simulated before rather surprisingly uh we discovered which uh helped us get this into nature material so first of all we did this just for lattice electrolyte and we we did we showed that the veneffect works pretty much like for a continuum electrolyte in that case but then we went on to study spin ice where you have the additional effect of these strings between the monopoles and this jackard reaction field so i'll show the results uh essentially so this is what you get this is from a simulation this is log of time this is the density this is log of time after you apply electric field so you have the system sitting at equilibrium not many charges in the system small number two to make screening as small as possible you apply a field and you you measure the density in the simulation now the top line there the brown line uh okay so sorry just to mention of course this is proportionally in an electrolyte the conductivity so the top line there is the is the lattice electrolyte so what you see is the charge density in the conductivity started to certain value then they rise and then they reach a conducting steady state which has a higher charge density and conductivity than it did low field and um so this is the breakdown of ohm's law non-ohmic conduction and this is the field dependent increase here so this is on sarga's veneffect now the blue line is what happens in spin ice so you can see that there are two regions here in this pink region shaded region spin ice more or less follows the lattice electrolyte essentially behaves as a lattice electrolyte but it suddenly realizes it remembers that it's a magnet and uh the jackard field kicks in the entropy cost of conduction becomes too great and the system gets returned to a new equilibrium which actually has less charge than it did originally so rather remarkably in spin ice the veneffect manifests itself as an increase to a kind of pseudo steady state that really doesn't last very long it only lasts for one decade of time there and then the density falls to a much lower value than it started off with so it's a non-monotonic approach to equilibrium now we discovered remarkably that straightforward chemical kinetics captures all of this so the green line there is a kinetic model which is really just textbook chemical kinetics with a little twist and log of time log of time yeah yeah so this is density versus time so so the the green line there is is a kinetic model just based on this these chemical equilibria with a fast step there a slow step there and essentially all one has to do is write down the kinetic equation on sarga's kinetic equation for the veneffect for the slow step and combine that with richkins conductivity equation for spin ice which just to remind you is the conductivity in electrolyte supplemented by this jackard field it's really as simple as that and this kinetic model works pretty well as you can see so and rather remarkably you can see that as why that just referring to this chemical kinetic model if if the magnetization small you can see that the system just behaves like an electrolyte so the monopole gas becomes an ideal coulomb gas in the classical empty vacuum at small magnetization or short times so the thing the system thinks it's a coulomb gas and thinks it's an electrical coulomb gas and then it suddenly remembers it's a magnet and goes back to equilibrium that's what happens and so we we have here in some sense a model electrolyte study non linear and non equilibrium response so just from the theory a lot of there's a lot of more interesting experiments one can think of doing so for example if you find ac field the magnetization of the system oscillates in response to the field but the charge density oscillates with twice the frequency it doubles the frequency and the reason for that is as i mentioned earlier the the charge density increase depends on the modulus of the field linearly and so you get all sorts of funny non-equilibrium and and sort of non-linear effects in this system we were also able to say make a few statements about the effect of microscopic dynamics on this sort of universal mesascale physics so for example for a lattice electrolyte we're able to show that the mobility changes with field in a certain way won't go into that so this is a sort of broad relevance to a lot of real systems that are prodded on this graph this is a chemical potential and temperature there are two regions on this graph this is the region where the screened on saga theory is valid and this is the green region is what's accessible roughly to to simulations with the present ordinary techniques and so that we have a lot of relevance here to all sorts of processes going on in electrolytes glasses and solar cells and that sort of thing but so that so that was the theory of on saga's vene effect in spin ice it doesn't in some ways it doesn't go far beyond on saga's theory it's just modified to include the jackard field um i want to finish up for the last few minutes talking about it is this observed experimentally in spin ice material so experiments on non-linear far from equilibrium magnetic response so these are experiments that we published earlier this year were made were basically done by carly paulson elselotel and shawn giblin with crystals from probachron matzahyra and we published it in earlier this year so experimentally we have to deal with a number of rather complex problems so at the temperatures we're interested in which are very low temperatures less than half a kelvin relaxation becomes very slow is modern because monopoles disappear exponentially there are not many monopoles in the system um so but our chemical kinetic approach to the theory means that the system doesn't have to be at equilibrium we can use a non-equilibrium starting point uh so we actually have a special technique which i won't go into uh to quench the temperature and we create uh an excess of pairs uh at this low temperature now these these there are things in spin ice called non-contractable pairs they're actually pairs that recombine very slowly this was shown theoretically by kustl nobertal and it's rather simple because but basically there's a type of pair you can draw that if you flip a spin in between them it creates double charge monopoles rather than annihilating the monopoles so they have to annihilate by going around a loop and hence they last longer so we can take advantage of this by creating a pair population at very low temperatures so what we do experimentally is after a thermal quench we apply we drop the temperature very quickly down to about a bit down to about 50 millikelvin uh we apply a field very quickly magnetic field we measure the magnetic current versus field we take the slope and that gives us one data point uh and then we reheat and repeat a very arduous experiment we keep on going till we've got millions of points so this is the initial current density versus time so we're basically taking the slopes of these curves um so these these are the guys who did the did the experiment and there's some of their equipment um so this i'll just show you the results this is a current versus field plot at 65 millikelvin so this is the monopole current versus the initial monopole current versus the applied field and the red line there is the exponential root field form that is so characteristic of the Coulomb interaction you can see that if you do a free fit to a power law you get an exponent of a half rather nicely so this is the case where we've we've created an excess of pairs and bound them with a field so this is a confirmation of the Coulomb force between magnetic monopoles and personally this this was a uh at the end of a very long um process for me because we uh we thought up this method a long time ago and the first couple of times we tried it uh it didn't work all that well but so this time it's uh it really worked so uh somewhat higher temperatures there's a transition to a kinetic regime where uh the the principles of on sarga's veneffet can be tested um so this is the log of the conductivity versus field at different temperatures you can see the conductivity going up as a function of temperature um because there are more monopoles coming into the system they're thermally activated and at low field it's a flat line so it's omic conduction but then there's this kind of envelope of non-omic conduction and we can fit the on sarga function in this region using the monopole charge as a fitting parameter we know in theory what the monopole charge is this is just a game to see if you can measure it and also a conductivity parameter and there are actually two regions of temperature one is where this all works very well and we get close to the theoretical charge and the theoretical temperature dependence of the conductivity and there's a lower temperature region where it breaks down and we think this is because of the breakdown of these kinetic equilibria that you do need to some kind of equilibrium or some kind of uh uh a sort of return process in the equilibria uh so basically there's a nice regime where we agree pretty well with the veneffet for magnetic monopoles so i'll just focus on that regime for a minute uh so we can compare uh because we've deliberately uh arranged to work at low magnetisation we can compare with the veneffet theory for an electrolyte directly so that's on sarga's theory which i've already described and uh if we plot the conductivity divided by the on sarga function it should be a flat line but we see an exponential approach to the flat line which comes from screening um this screened veneffet theory works very well so the the red line there is our kinetic screened theory and it puts a line through all these points even when the conductivity goes non-monotonically with field similarly put uh this is the current density versus field you can see it it rises very dramatically with field and finally we can even vary the initial concentrations because we're starting out of equilibrium so we can choose a concentration and the kinetic theory again puts a line through the points so the basically the dynamic kinetic model works extremely well in in in describing these results there's one thing that doesn't work all that well which is the activity coefficient is bigger the sorry smaller than we believe it should be for a coulomb gas and we don't really understand that at the moment but there's some non-ideality coming in that uh that creates that problem okay so let me now draw some conclusions uh first of all we've got a magnetic electrolyte with coulomb interactions and the associated chemical kinetics and some of the uh non-equilibrium effects that they go with that um so we feel it's loudest to study on sarga's veen effect which is a non-linear non-equilibrium effect that can be pictured with chemical kinetics uh and is a direct consequence of the algebraic nature of the coulomb field and and we get this non-monotonic approach to equilibrium that is born out experimentally as well um i want to make a couple of broader points so first of all a point about emergence i'm sure we all agree that emergence is an overused word people trying to get grants and things like that but um i think we do have a case of an emergent property so in magnetism we normally start uh with uh we know the crystal structure for example we know we know what spins there are and and we know the sort of interactions uh so from that we can write down a spin Hamiltonian and we obviously hope that we can describe all properties with that for example we might want to try describing the far from equilibrium response of a magnetic system with that but the problem is when you look at this very ugly Hamiltonian uh it's essentially impossible almost impossible to go from there to there and there's no hint as to why you get this funny response even to simulate it would be extremely difficult um going from there to there so so what we do instead is we have this mapping to the ice model we have this transformation to the of the Hamiltonian then to the magnetic monopole model finally we justify the emergent chemical kinetics and on sarga's veneffect and finally we're able to put a line through the points and understand the non far from equilibrium response and uh when you look at this Hamiltonian there's really no clue uh that this kind of Coulombic response is going to appear in this Hamiltonian and that in a sense is is what we mean by emergence i i believe the only way you can do get from there to there is to go through this mapping to to a monopole system and then a quasi chemical system uh so i don't know what sort of light hearted sense you might ask the question how much progress have we made since the days of Maxwell well Maxwell himself actually there's a line in what in his book if we assume the distribution of imaginary substance which we've called magnetic matter the potential uh will be identical with that due to the actual magnetisation of every element of the magnet so he's talking about he's saying you can use a concept of imaginary matter to model uh magnetism through the concept of effective magnetic charge now you can pretty much do this for any magnet of course so there's nothing new there but the difference with spin ice is that this magnetic charge has quantized charge and it has quantized energy and that's so it behaves much more like real matter than Maxwell would have would have imagined at that point and so this is the picture that you warned against thinking of in in textbooks when they cover this but actually it does occur here um and finally since since we're on uh Maxwell uh something about universality Maxwell starts his treatise uh by with a long discussion of the magnetic Coulomb law and how pole strength can be measured in the same units as electric charge and what that means and what pitfalls you may go into as a result of getting too excited about that um nowadays we tend to forget that magnetic properties can be measured in electrical unit but it's interesting here to put our measurements back into electrical units and so we can do that so we can measure the current in nanoamps and the field in volts per centimeter and um just to to emphasize the universality of the Coulomb interaction in this regard here I've plotted experimental data for field emission from a tungsten filament where the electron escapes its image charge you get this straight line against root field and the lower picture is spin ice at 50 millikelvin where magnetic monopoles unbind in a similar fashion and again the straight line against root field is is a property of of the long range Coulomb interaction in both cases so on that strange note I'll thank you for your attention