 Okay. So welcome everyone to this Quantum Encounter seminar at IHS. And today we are very happy to welcome Peter Holtzworth. So Peter is a condensed metaphysicist. He's a theoretician, but he's also involved with experiments. And Peter got his PhD in Oxford in 1985. And he's a professor of physics at Ecole Normale Supérieure in Lyon. And over the course of his long career, he worked on a great many things on magnetism, in particular in frustrated systems, and on critical phenomena, both in and out of equilibrium on analogies between critical phenomena in magnets and in turbulence, and many other things. And in recent years, he's been particularly interested in the subject of spin ice and its analogies with electromagnetism. And that's, I guess, what he's going to tell us about today. So please, Peter, I'm very happy to have you. So it's a pleasure to be here. So as you said, over many years, I've been working on these condensed meta systems that show this emergent behavior so that this complex, frustrated magnetic system can, to an excellent degree, be written in terms of simple notions from field theory, from lattice field theory, from the physics of Coulomb interactions, both in the classical regime and in quantum regime. So this is what I wanted to tell you about today. So here's my program. I hope that this is relevant for this audience. I wanted to start with a prologue that reminds us about both field theoretic and Coulomb representations of charged systems. And then tell you something about this condensed meta system, with its magnetically charged quasi particles that have been called magnetic monopoles. So I don't know if anybody in the audience works on real monopoles or if this is going to be divisive or if there's going to be some discussion about the legitimacy of calling these objects, magnetic monopoles that would be interested to have some feedback. And then so then there's several points I want to talk about energetics and perhaps the most interesting point for a theoretical audience is the description of the magnetic correlations in terms of emergent correlations in an emergent field and some notions of fractionalization of the magnetic moments that appears. And then some analogues of Dirac strings in these systems. And finally, a discussion of work slightly outside my own field of research, but there's been much research on quantum equivalence and emergent quantum electrodynamics at the end. Okay, so please don't hesitate to ask questions. I'll try and I'll try and remain pedagogical and if I burn up too much time I'll jump a few chapters. So before going on there are many people. I'm going to try and talk about work that's been that I've been doing over several years. So there are many people involved on the left. There's my colleagues and on the right. There's some students. Maybe I won't go through this entire list, but I should really maybe underline the role played by Ludwig Jobert in Bordeaux that used to be on the right, but now he's on the left. And and Stephen Bramwell and Roderick Mosner from University College and from Dresden. And in green I've highlighted Flavien and Joffre, who are actual students at the moment. And maybe I'll see a few words on what they're doing towards the end of the seminar. Okay, so let me just start with this gentle reminder. So here's an electrostatic problem with two charges, a neutral system, a positive charge and a negative charge. And there are two equivalent ways of discussing this problem. So if you do, if you come to a conduct matter physics or physical chemistry, most probably you'll discuss Coulomb interactions between charged particles or quasi particles. But if you have a more theoretical background, then you might be used to a more field theoretic description where you have an energy of the fields that the integral of the square of the electric field integrated over the material together with the constraints of Gauss's law that tell you where the singularities in the fields are that represent the charges. Okay, so these two points of view are connected because this field energy contains both the energy of correlations of interactions, but it also includes the self-energy, the electric field that's required to build up those charges independently of their neighbors. Okay, and so the Coulomb interaction is actually the difference between the total energy and the sum of the self-energy. So the sum of the electric fields that would be created if individually one after the other, you ignored the interactions but you just summed up the energy scale. And if for a neutral system the Coulomb energy is often negative, it's because the total field energy in a correlated system, the total energy density of fields is reduced because of the correlations in the system. Okay, and so in this field theoretic representation then you have Gauss's law and the electrostatic and electrostatic problem is usually solved by taking the simplest solution which identifies E as the gradient of the scalar potential and the electrostatic problem is usually given by an electric field from a scalar potential only, but of course this is just one of an infinite number of possible solutions as you can add to this any electric field that's the curl of a vector potential, so divergence zero. And so the total energy has both a part from which the charge is constructed and a circulating part with no divergences and the electrostatic case would correspond to a minimum of this rotational part and of course then you get electrodynamics if you include a rotational part that is coupled to the vector potential of a conjugate magnetic field. Okay, and two problems that deviate away to two sets of work that deviate away from this electrostatic minimum, first one that I could recommend to you that I found extremely interesting if you if you consider an electrostatic problem but you allow yourself the luxury of an arbitrary divergence free field then it's very easy trivial to find a local solution of Gauss's law and so if you have a system where there's no electrodynamic coupling and you allow yourself this extra term you can locally solve Gauss's law and then afterwards you find that the partition function that deals with the charges and the auxiliary field they cancel and this kind of starting point allows you to address a Coulomb problem with only short range interactions of course then the price that you pay for that is having to deal with two fields both the field you're interested in and a Gaussian auxiliary field but it gives you the luxury of having short range interactions and this can be read you can read about this in this paper by Max and Rosetto from 2002. Okay and related but not quite the same as that is the problem that I'm going to try and talk about today is classical spin ice in which you have a situation where the energy of magnetic quasi particles comes from a longitudinal part comes from a part that comes from a scalar potential but there is a rotational part that's present that doesn't contribute to the energy so there's a large field element of this divergence free that contributes zero to the energy and this will be this will be spin ice and that's essentially it except that the problematic is magnetic rather than electric so everywhere there's an electric field there'll be a magnetic field. Okay so let me try and talk about such a system so such a system emerges from so-called spin ice materials and generically the two generic materials that are much discussed are homeon and dysposium titanate so this is a rather exotic material homeon is rare earth homeon titanate titanium rare earth materials homeon and dysposium have a large magnetic moment made up of orbital and spin components 10 ball magnetons per spin and these magnetic moments are localized and they sit on the apexes of corner sharing tetrahedra the so-called pyroclore lattice and these magnetic moments are discrete objects to an excellent approximation they can point either out of the tetrahedron or into the tetrahedron and these are discovered more almost 25 years ago by bramwell and harris and collaborator so this is the first point is that you have discrete degrees of freedom which will renormalize at larger length scales to give you emergent fields that look like look as if they have continuous degrees of freedom of an electrostatic or a magnetostatic field and the the pyroclore lattice then it's corner sharing the centers of the tetrahedra this kind of dual lattice the center of the tetrahedron forms a diamond lattice which is bipartite itself can be split into two sublattices a and b okay peter is this a an antiferous there's an antiferous magnetic um which actually actually it's a it's a frustrated pheromagnet pheromagnet yeah okay so it's pro frustrated in the sense that the the predominant effective coupling is pheromagnetic so the spins would like to be parallel but they are immensely strong crystal fields which make them point along these local that direction so they can't all uh map they can't they can't line parallel so the best they can do this is the pheromagnetic best they can do pointing in or out with pheromagnetic coupling but actually that all that maps onto an effective uh antiferomagnetic system you can define an an equivalent antiferomagnetic system with antiferomagnetic coupling of icing spins up and down on a tetrahedron and two in two out would correspond to two up two down of the effective spins and that antiferomagnetic was invented by uh by anderson in the 1950s uh to discuss the ice problem actually so um these spin ice materials they're called ice materials because the phase space of icing configurations either spin in or spin out maps exactly onto the phase space of uh protons in water molecules in an ice crystal okay so so if you place a water an oxygen ion in the center of each tetrahedron the two in two out would correspond to two protons close and two protons far away this is exactly the configuration of water in the cubic phase of ice and um uh this physics was explored extensively in the 1930s uh by Gauki and Stout and Pauling Gauki and Stout were the Nobel Prize for their experiments on water where they measured the configurational entropy through the measure of the specific heat and calculated the the configurational entropy and they find that you never at any temperature regain the configurational entropy of proton ordering at low temperature it's missing entropy okay and this is this famous Pauling entropy of proton disorder and the icing spins share the same phase space so if you do this Gauki and Stout experiment this is a real experimental days of data from Disposium Titanate there's no ordering phase transition and there's a kind of a shop key peak and if you integrate uh C over T underneath this curve rather than getting R log 2 as you might expect for icing degrees of freedom you get R log 2 minus exactly this Pauling uh entropy so there's no there's no phase transition in any field and also I should see that this ice physics um this was uh um one of the motivating elements of much of the work on vertex models from the Baxter school of exactly solvable models in the 60s and 70s so much of what I'm going to say actually has some connection with vertex models on a diamond lattice okay and so let me talk about uh some of these models now so I want to talk about two specific models that are very close but but but but different even so so the first is a nearest neighbor model so this is exactly Slava's question so you have an effective nearest neighbor coupling J effective with with with a coupling between spins and it's ferromagnetic and now this on each tetrahedron the minimum of energy is two spins in and two spins out when you tile this on the pyroclore lattice you can identify loops of spins that can be turned so if I flip all the spins in this loop I re-establish I go from one ice state with two in two out to another ice state with two in two out so it's a loop model it's essentially a loop model and by construction these loops are exactly degenerate so this model it violates the third law of thermodynamics and it has its exact ground state entropy as its ground state entropy uh this uh powering entropy now this model it describes spin ice at the 80 percent level okay in order in order to go from 80 to 97 percent then you have to add dipolar interactions to it because this exchange coupling this is the order one kelvin and the dipole interactions for objects of 10 bar magnetons it's also of about 10 kelvin and this would be the strong crystal fields that make the the objects icing like along these so this model uh gives remarkably good numerical uh comparison between disposing and homeon titanate and and simulations and also it has a remarkable property that it has very important long range interactions but when you put the tile system in one of the powling states with two in two out everywhere the long range part of the interaction is almost screen it's not perfectly screen but it's almost screen so what happens is if this nearest neighbour model is exactly degenerate among the powering state this dipolar model has a very small band of states so if these energy scales are two kelvin then the band of states is 150 mili kelvin so in to allow to an excellent approximation you can forget it and you can presume that this model okay also has a degenerate uh set of of states okay and so uh uh sorry were there a question question no okay and so uh uh this brings me immediately to this mapping to this magnetostatic problem because anybody wearing glasses should maybe take their glasses off and then you can see that this two in two out configuration it's a local constraint and if you imagine these as lattice field elements okay then you can write this local constraint as a lattice constraint of the in a in a style divergence of m equal to zero okay so it's like a loose paramagnetic system but with a local constraint that everywhere that divergence of this emergent field is equal to zero okay so it's like the the field of magnetic moments in its low energy stick you can write it you can write these magnetic moments are sitting on the back of the curl of a vector potential and so it looked like the magnetic moments a lattice field elements with uh diver that are divergence free okay can i ask a question peter yes please so this constraint the divergence is zero it would naively also allow one in three out so is this some uh well no uh that's a very good question but you have to ask it again in 15 minutes okay one in three out you know even even even at i i h e s i believe that three minus one is not zero right so so um that's a that's a state that will bring a topological defect a magnetic monopole okay yeah no sorry sorry i will i will say something really stupid so no no it's an excellent question it's a it's a it's an excellent question that will come back to bite you okay okay okay so so this is the statement that this configuration of states that has no long range order it will have magnetic correlations that look like this emergent field okay and these magnetic correlations that look like its emergent field this is what they look like this is a pinch so-called pinch point neutron scattering maps in reciprocal space so this is a neutron scattering map from a single crystal of homium titanate and this divergence m equals zero corresponds in reciprocal space to skew to q scale m of q is equal to zero okay so q here it's not a real momentum it's a lattice momentum so that has uh that that um has repetitions over a b b one zone and so when you combine the lattice the trans discrete translational symmetry with the neutron scattering constraints this divergence m equals zero corresponds to this rather elegant pattern with these sharp pinch points and the sharp pinch points that give you the sign of that an indication of this divergence zero okay so maybe this mapping to an emergent field it can be made more clear if you make a further abstraction so i'm going to allow myself the luxury of defining a third model so instead of having point dipoles on the corners of the tetrahedron i'm going to extend the point dipoles as needles or dumbbells okay so these infinitely thin needles they carry a dumbbell of north and south poles and the dumbbells of north and south poles they meet at the centers of the tetrahedron okay so then there's a there's a thorny problem of a of a divergent self-energy where these objects meet here but that shouldn't be any problem in for this community so apart from that i think you'll be able to see that by construction then that any configuration with two spins in and two spins out will be magnetically neutral because there'll be two blue dots and two red dots and also that all configurations with two spins in and two spins out will be iso energetic okay so this small band of energy that opens up when you include the dipole interactions okay this is essentially the difference between having point dipoles and having needles extended to the centers of the tetrahedron okay so now instead of having two in and two out i take three in and one out okay so this is slaver's question okay so there's a net charge of two blue dots here and there's a net charge of two red dots here and so these tetrahedra are carrying magnetic charge okay so this is this is this is a topological defect in this emergent field and and as it's a magnetic problem it looks extremely like a magnetic monopole it's classical but apart from that it shares many of the many of the properties of Dirac's monopoles as i'll try and show as we go on and then if you go to all in or all out so there's four blue dots here or four red dots this is like a double monopole okay so this is so then this is a mapping from this dipolar spin ice model to the dumbbell model and from now on everything that i talk about will be related to a dumbbell model okay and you can see that if you go from a two in two out configuration to a three in one out configuration you change the magnetic configuration delta m by two m and i should say that small m is the magnetic moment of the dipole or of the the needle and i flip this dumbbell through a distance a where so a is the distance between the the diamond site centers okay and so this monopole it carries a charge of plus or minus two m over a and this double monopole carries a charge of four m over a and this has the right units because this is the dipole divided by a length so this is the magnetic charge okay and now there's another important point for spin ice physics here that maybe as far as in emergencies goes it's a complication but a very important point for spin ice because you have underlying dipole interactions okay these are real magnetic moments okay so this is real magnetic charge so you've got to think of these these needles as carrying real magnetic flux and so the the the standard magnetostatics of that magnetic problem it's all riding on the back of this emergent field so the emergent field has riding on the back of it real magnetic flux and so this is real magnetic charge and these objects then will interact with a real coulomb interaction and and so you can see that if you start off here with two in two out everywhere and you flip this spin for example so you now you have a pair of three in one out and three out one in separated by a nearest neighbor okay now i can flip a second spin i can flip this spin i re-establish the ice rules two in two out on this tetrahedron and i move the magnetic moment monopole from here to here okay so in the absence of dipole interactions or in this nearest neighbor model there's no energetic interaction between these topological defects but because there's real magnetic flux then you have this real coulomb interaction between these quasi-particle excitations okay so this then it opens the door to both coulomb that is quasi-particle and field theoretic descriptions of spin ice so what i'm going to try and do now is discuss both the energetics in which i'm going to treat this Hamiltonian as essentially a Hamiltonian of coulomb interactions between monopoles above this set of equivalent ground states okay so the number of monopoles as you can see here this is not conserved yet i can create and destroy monopoles so there's a chemical potential for the creation and the destruction of monopoles and a chemical potential for the creation and destruction of double monopoles okay and i can calculate this by going back to my dipolar spin ice and this chemical potentially some function of the exchange part and the magnetic moment and the dipolar interaction so i can calculate these things and also because there's a constraint there the chemical potential the chemical potential goes like q squared the chemical potential of the double monopoles is exactly four times the chemical potential of the single monopoles and that will be an important point as i as i go through so this will allow us to attack energetics like specific heat in this low temperature phase the apparition of phase transitions as you change the value of the chemical potential and then i want to switch to a lattice field description to to enable us to get a hand on the spin correlation some of this we've already seen because we've already seen these pinch point patterns okay so as we'll see this this needle it will represent an element of a lattice field and each needle it will be your strength m over a where m is magnetic magnetic moment and a is the distance between sides just peter just you said this so the the the cost of creating one of these monopoles is what a few kelvin yes so it's a few kelvin so the exchange so the energetics both the energetics and the chemical potential are a few kelvin so two two three kelvin in each case okay and that's going to be important in what follows and the you said the typical interaction energy of two monopoles at one lattice spacing is also yeah that's right that's right so you can see this this is so this is specific heat data for homeon and disposing tightly so the the blue crosses are experimental data okay and you can see that there's a shot keep peak and it's around two kelvin so this corresponds this is exactly the the the temperature where you go from a high temperature high density fluid to a low temperature low density fluid okay and so here i've got experiment i've got numerical simulation of the dumbell model which only has coulomb interactions and i've got the bi-huckel theory and so this is our best shot there are no fitting parameters here and so you can see that first of all the experimental data is extremely well represented by simulations of this dumbell model with only uh coulomb inter with only coulomb interactions and this chemical chemical potential cost and then given this model i can make and improve mean mean field theory like the bi-huckel theory and i can get an excellent description of this data so you you might ask actually if you start off with dipolar spin ice if it gives an excellent numerical description of everything what why bother uh doing all this monopole this but this dipolar spin ice that anybody who's worked on dipoles are confirmed that they're messy doing doing analytical calculations is extremely complex and this allows you to separate to give you some fractionalization that allows you to identify quasi-particles and make simple theory so here you have simulation experiment and theory together so give you some idea of how successful that is another idea of how successful one uh uh this coulomb interaction description is of this dipolar model here's some numerical data of a phase diagram so that you should concentrate here on the gray so this is simulations of dipolar spin ice Hamiltonian and here you've got varying this ratio of exchange to dipolar interactions and you've got monopole fluid which is like spin ice where it's predominantly two in two out and you've got a phase here there's a phase transition first order phase transition to an all-in-all-out phase where spins are pointing all in or all out on uh uh alternative tetrahedra so now actually going to your question the system moves here from a predominantly ferromagnetic fluid to a predominantly anti-ferromagnetic or ordered magnetic system and these are where experimental systems sit on this phase diagram okay but I can model this because I have coulomb interaction so if I look at the zero temperature ground state energy of my system you remember that I've written I can write I've written the Hamiltonian as essentially the coulomb energy plus the chemical potential cost of creating these particles and if I change the ground state configuration it corresponds to a change of sign of the coulomb energy minus the the energy cost of creating the monopole so this is the coulomb energy gain of having monopoles and this is the energy cost of creating them in this phase this is a positive number um in this phase this is a this is greater than zero so at zero temperature the system doesn't want to have any monopoles when it changes sign at low temperature it wants to have as many monopoles as you can that is a monopole a double monopole uh crystal and the ionic energy of this crystal I can calculate it if I can remember from my condensed matter physics courses um there's something called the Madeleine constant this this ionic energy of the crystal this is the nearest neighbor energy the energy of a nearest neighbor pair times this Madeleine constant okay so then when the chemical potential if I vary the chemical potential to go below this threshold across this barrier and I'll go from a monopole fluid which is a monopole vacuum at zero temperature to a monopole crystal okay and this monopole crystal will be exactly the zinc blend structure with positive charge and negative charge alternatively on the ice on the on the interpenetrating uh uh sublattice is the diamond lattice so so north poles and south poles I'm sorry I'm not sure it's important but where is this first or transition to the monopole crystal on the orbit diagram it's here it's here so so this this is dipolar simulation so this is varying these parameters and this corresponds in the monopole language to changing the chemical potential so as I move along this actually this axis I change the chemical potential okay okay and you should bear in mind that I'm a stickler for tradition okay so traditionally there's a minus sign here so that means that if it costs energy to create particles the chemical potential is negative so there's a double minus sign this is an energy cost yeah okay what's the vertical what's the vertical axis of that diagram is important okay this is temperature this is this is temperature essentially okay okay so this so you can think this is as temperature temperature and chemical potential okay and and I can and this is the threshold okay so now if I translate this Coulomb language into dipolar spin ice language I get this value for this parameter here point minus 0.918 and it's almost exactly where dipolar spin ice where this phase transition occurs okay so I mean if you believe the first bit then this shouldn't be a surprise it's just telling you that this dipolar spin ice model it can be approximated to have to really excellent approximation by the Coulomb interactions of these induced monopoles and here you go from a monopole vacuum to this monopole double crystal can I ask a question yes please so in your previous slide I guess you're talking about the quantum critical point and that's why there's no entropic contribution to the free energy here ah okay okay okay okay there's no quantum mechanics there's no quantum mechanics for the more is there a temperature critical point then yeah I mean okay so someone this is a logarithmic scale okay so going down here there'll eventually log there'll eventually be some quantum mechanics and loss of the godicity kicking in I mean it's a very good question that's related to the end of my seminar so here you should think you should try and force yourself to continue thinking classically here even though it's at low temperature okay but I would remind you that I started off about talking about these Galician stout experiments on ice okay so the proton degrees of freedom in ice they never freeze and they never become quantum mechanical experimentally okay so this question was more simple minded than that I the question that why I'm discussing stability of phases in terms of the energy and not free energy that's all okay okay so all right so think classical statistical mechanics then you have a classical system um you have a classical system and you ask what its energy equals at zero temperature okay and so in in this phase it's a monopole vacuum and in this phase it's a double monopole crystal okay and the condition for changing it's when you gain energy by having monopoles okay as I saw at the beginning as I showed try to show at the beginning the Coulomb energy of a correlated charge neutral system is always negative but this energy cost this is a big number so this is bigger than this and so the system it doesn't want any monopoles here normally I would expect a minus ts right there yeah normally you would normally you would expect a minus ts and if we were here for a week I could discuss that but that's what you see in custom install is for example it's exactly that the change of that factor and here you see so you say temperature is too small the entropy that's right this is a purely this is an energy energy phase transition okay and and exactly actually it's a good question because there's a multi-critical point here this is the first order transition and this is a second order transition and across the second order line entropy is playing a big role and across the first order line it's spending it's playing a much smaller role okay so um I don't know how I'm doing for time here I've used lots of time already so um well Peter don't worry you would like to understand your story so okay okay so let me let me talk let me talk a little bit more so this double monocle crystal it's also degenerate with a single monopole crystal instead of having all in all in all out I could have been had three in one out three out one in I could have played the same game with single monopoles rather than double monopoles and as I said there's a degeneracy so you have the same condition here for single monopole crystal and it's the same constraint okay and so uh actually at this point here it there's a multi-phase region where there's a double and single monopoles they're equally equi probable when you move away it's the double monopole that wins but you could also have a single monopole and in fact I'm going to probably going to blow myself away with too many details but um this system has an incredibly rich phase diagram this is temperature chemical potential and now I can add a staggered field a staggered chemical potential that's conjugate to this crystalline order parameter and on this phase diagram I've got the double monopole crystal I've got the spin ice phase but I've also got these wings of single monopole crystal okay and these are planes of first order transitions this is a line of second order transition and this is a multi-critical point and any classic people in the audience they'll recognize this is very similar to a s equal to bling bloom cappell model uh where there are multi-stage ordering it can go from so uh zero zero monopoles single charge crystal double charge crystal okay and on all that uh can be confirmed with x with with numerical simulation and so maybe I should move on from energetics um try and talk a little bit about spin correlations in terms of um this lattice field picture so um this Coulomb fluid it satisfies the magnetic Gauss's law that is that there's no there's no revolution these objects carry induced magnetic charge but Maxwell's equation divergence b equal to zero it's still valid okay so this when you have magnetic in the absence of magnetic monopoles divergence m would be equal to zero in the presence of magnetic monopoles divergence of b is equal to zero so divergence of m is equal to divergence of h the magnetic intensity which is equal to minus row this density of magnetic charge okay so uh that's a standard electrostatic magnetostatics actually but now let me uh apply it to my emergent system so rather than rather than the differential form of Gauss's law I need to think about the integral form of Gauss's law so let me take a tetrahedron and I integrate over a surface containing four needles okay so this is the theoretical construction all the magnetic flux is contained in infinitesimally cross sections of the needles okay so this integral it's really just the sum over the magnet magnetization density inside the four needles okay so then this shows you how these how these needles they become elements of an of a lattice field okay m i j so this is magnetization density times this infinitesimal cross section that's air that's that's so this is magnetic moment per volume times an area so this gives an object that has units of m over a it has exactly the units of charge and it's a lattice field element okay so this element that has three spins in and three spins out this corresponds to a vertex a set of four elements that they're m over a a with three times minus one and one okay so now have a sum over j which is just the the integral the discrete integral it gives me the sum over j of m i j that's four that's three times minus one plus one this is minus q where q is equal to two m over a this is exactly what I told you when I introduced the dumbbell model okay so all this means then that I can do lattice field theory for this object where these are the these are the lattice field elements and in the presence of magnetic monopoles of magnetic monopoles these magnetic field elements they're going to fractionalize into two fluids this is what we've called magnetic moment fragmentation and the there's an interaction between these two fluids because for each element each one of these field elements it has fixed amplitude m over a so let me give you an example we started talking about configurations that have two in two out okay divergence three so this means the set m i j uh it has two in two out this is entirely divergence three what I've called m root okay but if I take an element that has three in one out here's the three in one out I can cut this into two parts one it has four times minus a half this gives me minus two this is the charge and a second part which is minus a half minus a half minus a half plus three halves and this has divergence zero and this is a divergence free part okay so this is a sleight of hand this is the theoretician hiding turns onto the table it's not like the magnetic moments really fractionalized into two fluids they just appear to fractionalize okay and if I go to a double monopole that's all in okay so they're all in they'll have four terms that are part of this what I call the ec now it's m m that gives me the magnetic monopoles okay so this cutting into this description in terms of lattice fields allows me to predict that the the ensemble of magnetic moments can be written in terms of a lattice field with a part that comes from a scalar potential giving me the magnetic monopoles and a part that comes from a vector potential and this is bound to be there because each element is a fixed length so this is like the leftover once I've paid the price of having these magnetic monopoles okay let me give you a simple example or what I tried to be a simple example take two post charges next to each other nearest neighbors so these both have three spins in and one out so here's the out spin and here's the out spin and I can separate this in this representation this is like an old-fashioned Michelin map the number of chevrons gives you the strength of the field so one chevron is m over 2a so one chevron is like a half here okay so this is the three in one out three in one out and this splits into a longitudinal part giving me the two monopoles two in charge and a leftover that's divergence free what is the intuitive reason why this pure gradient corresponds to all arrows pointing in this something that I didn't then didn't get well I I I I I don't know if I go right back to the beginning sorry I don't know how to do this I mean that's what it looks like isn't it that's a charge okay so you look at the field is emanating from this charge and it's emanating and it's it's homogeneously flowing out of the of the of the of the of the of the of the of the singularity okay okay so so so so now this is the lattice version sorry I've come so far this is the lattice version of this okay you've got a tetrahedron the tetrahedron has a considering it's discrete it has a very high symmetry and so you expect the lattice field elements coming out of this monopole to be home to be distributed in some ice discrete isotropic manner and that's this okay so you can cut this into this plus this and this is this plus this okay okay thanks okay and so you can do this and so you can do that actually for any monopole fluid configuration although it's rather involved a messy okay because it involves solving Laplace's equation for finding what m m is at each site and then subtracting it from the the total configuration to give you m root and that's complicated and I'm not going to discuss that here but there's one case where it's it can be done trivially and even though it can be done trivially it's very interesting so let me go to this single monopole crystal this crystal of alternate sites of three in one out and three out one in okay so this has both charge ordering and residual entropy if you look at this figure if I flip the spins around this loop okay then I'll still satisfy three in one out three out one in on alternate sites and the red spin is the minority spin okay so if I flip the minority spin I'll jump from here to here from here to here from here to here if I go around here so you can see there's an extensive entropy even though there's this magnetic order of the crystal okay and so you can do this you're a weird theoretician so we can choose any value of the chemical potentials and we can force the system to be in this monopole crystal okay and so this is what we did in this in this paper and so alternate sites have have three in one out three out one in if I scatter numerical neutrons of the numerical spin configurations then I find Bragg peaks corresponding to all in all out but those Bragg peaks are contained exactly a quarter of the total intensity that is that they contain exactly a half of the spectral weight and if I zoom in from the Bragg peaks to look at diffuse scattering I see down there a whole scattering pattern of diffuse scattering that looks like the pinch point pattern that I had for the two into out configuration so this is a mixed state it has anti-pherominated order described by the monopole crystal and it has cool on phase ferromagnetic correlations that correspond to the circulation of loops like this so this is a superposition of a loop model and a long range ordered model and maybe there's too much information here but actually if anybody is interested in hardcore dimers over the last 15 years has been an incredible amount of work on hardcore dimers so if you put hardcore dimers onto a diamond lattice you find that these they're just the hard cores give you dipole like correlations and you can map this dimer configuration onto an emergent field so you can see you can say there's a field of one along the dimer and a field of minus one over z minus one between dimers okay and you can see that this pattern here corresponds exactly to this fragmented part here so actually you've got a long range ordered part and a field part that moves the that maps exactly onto the phase space of hardcore dimers on a diamond lattice sorry i'm confused before things were happening in 3g and now dimers are into yeah sorry sorry i i was too stupid to i was too stupid to represent these in three dimensions so this is just an example for a square lattice but excuse me that was not very clear so this is an example of dimers on square lattice with emergent field but if you put dimers on the diamond lattice you get the same emergent field and it's this emergent field okay okay so the correlations of this emergent field are the correlations of dimers on a diamond on a diamond lattice okay so i've been speaking for a long time so maybe uh maybe maybe uh you should tell me when to wrap up i can i can jump some stuff here but uh i i could say i could jump very quickly the experiments maybe because until now this has been just a set of theoreticians having fun right but as we were doing this like a present from heaven uh my collaborators from granobal they discovered the material homeum iridate and so homeum forms spin ice and the iridium it orders antiferromagnetically at a much higher temperature and it gives you strong internal fields that look exactly like this staggered chemical potential and miraculously this system forms an all-in-all-out magnetic order where the magnetic moment is is exactly half the total moment of the iridium iris okay there are no single crystals but if you look at the diffuse scattering of a powder then it corresponds to Monte Carlo simulations of these Coulomb phase patterns so this is exactly a uh exactly a a a a reincarnation of this monopole crystal with uh fractionalization that are discussed and then we recently i've been working with these people on the system material disposing iridate and we measure actually uh we measure this residual dimer entropy at low temperature okay um how much yeah how much time i i can speak for as long as you wish but i don't want to uh i don't want to um i don't want to to maybe we should pause for some questions if there are any questions and then i i mean i'm gonna be maybe yeah maybe we can choose make some choice from the things that you yeah depending also what the questions are going to be so yeah i think yeah would it be okay with you yeah fine fine i should just say that the next chapter was going to be on Dirac streams i don't know that's of any interest anybody yeah so i my question was it's also like maybe a suggestion to one direction of this what we could choose to continue is that okay you talked about this um um field but okay it was never up to now quite clear if there is some regime where this m field could really be maybe at some in some range of temperatures or some range of distances could be imagined as a continuous field if there's some continuous approximation uh yes yes yes that's something that for me would be interesting to know more about okay maybe other people have other preferences i don't know so people should just pick up i should say you can make you can make a continuum uh yeah you can make a continuum field theory of course you've got to incorporate these the these constraints and there are papers on that by moester and my collaborators and i've tended to i've tended to concentrate on the discrete on the discrete nature but people people making this this dimer mapping for example they go on to discuss continuum descriptions of the die of the of the dimer dimer correlations so actually the identity the identification of this uh discrete lattice but this lattice field theory it's just the first step in uh in a continuum description i would say um so yes it can be done so the fact that there are these long range correlations uh described by this nice formula one over our cube it means that in fact that some long distances you can just use this that's exactly right that's exactly that's exactly that's exactly right and if you you know if you look at correlations in this system they come out to be one over our cube and uh and uh and the thing behaves essentially like a continuum system okay well i don't know since the audience is not giving us any input maybe i don't know how about like well you started a bit late because there was a technical glitch so maybe like 10 15 minutes okay just um maybe i'll maybe i shall skip the this the Dirac strings but um you know Dirac in 1931 uh proposed that you could create these magnetic particles monopoles and as i understand it he proposed that Maxwell's equation could be saved by by connecting the charges by an invisible Dirac string so here's a charge that's radiating flux okay and so if you just take the green part the divergence of b is non zero but it's it's um the the the monopoles are corrected connected by a string that that contains the missing flux so if you take the green flux plus the red flux you get db equals equals zero okay so this is not the Dirac system it's not a vacuum these are induced charges and so naturally you have db it's equal to zero but you have something very similar uh that happens so if you start off with a reference state for example and then you create a pair of monopoles and you separate them by flipping this set of spins so this set of spins here is flipped to separate these two and then you subtract the initial configuration from the final configuration you get a string and this is essentially a Dirac string and i think this is how people in the 1930s thought about electromagnetism and i think that the Dirac idea was that a monopole is induced from the vacuum just to spin ice monopoles are induced from a magnetic medium of course this is completely uh completely classic and you have to need to add quantum fluctuations but it's essentially the same and and uh and uh maybe i'll skip this you can see evidence of these strings if you start off from an ordered phase and then from an ordered phase you flip a line of spins and it becomes visible in the background of ordered spins and you can see this with neutron scattering and these these people here Mosner and collaborators they claim to have seen this in this in this scattering experiment my last word would be on quantum fluctuations for nearest neighbor spinites so for the moment it's been completely classical uh but now what you do is in these models you allow transfer spin fluctuations and this means that once you allow these fluctuations the Hamiltonian is no longer diagonal in Sz Sz and so this this transverse fluctuations allow quantum dynamics of loop flipping so the low energy sector includes quantum dynamics of loop flipping and so you introduce quantum fluctuations in the form of loop flipping and this gives you quantum spin liquid uh phases and these quantum spin liquid phases they can be interpreted essentially as quantum electrodynamics so what happens now is that if my magnetic field in the absence of monopoles if my magnetic field could be the thought of as the curl of a vector potential the transverse part gives you a second vector potential that's conjugate to the first and it's an emergent electric field so now you build up a ground quantum ground state which is a coherent superposition of these two in two out states and there are many different coherent super positions and so you introduce the band of states from the degenerate powering states and the excitation from one superposition to another superposition gives you a photon like excitation so the ground state is non-magnetic it's spin zero but you go from one spin zero state from one state to another by a magnetic excitation and it looks essentially like a photon and if you want to read more I can really recommend this paper by Benton and collaborators it's not necessary the first paper on the subject but it's the most pedagogical and it's a really terrific read okay and this modifies this structure because now instead of having a flat band of states you have this wider band of states and there's a hunt for doing experiments on these materials but finding materials that are experimental examples okay so I should stop I should say there are two things I didn't talk about one is dynamics and we made contact with one of Onsager's very famous papers from the 1930s on non-ohmic conduction in low density charge fluid so-called veneffect and I didn't talk about castellane transition which is a topological transition within the emergent field that's what my students one of my students is working on and this is extremely like costless foulness confinement deconfinement transition and so I think I've been speaking for long enough and I'll just leave my conclusions saying that I hope I've given you an insight that this is a garden for emergent physics where you can observe spin fractionalization Coulomb fluids Helmholtz decomposition derax frames quantum electrodynamics front and face transitions nonlinear dynamics so breath essentially everything so thank you for your attention well thanks a lot let's thank Peter which is very nice talk and are there any questions I probably bombarded you with information I think I'm sorry about that so let me ask you so spin fractionalization so this means that this was happening when you were taking this m and you were decomposing it into pieces that's right that's right okay so you know all this is just manipulation of a vector field right you can make out you can make a Helmholtz like the composition of any vector field okay but it turns out you know in examples of fractionalization that I'm familiar with where it's related to quasi-particles and then there are like some quasi-particles you can really separate the two things here well is it accompanied by the ability to to separate the the because the magnetic the quasi-particles here not the quasi-particles the excitations are molecules and it's not like we can really separate them into two different types of well I mean yeah yes and no okay so this is not spin charge separation yeah I guess but it's something it's something rather similar because you start off with this with a bunch of dipoles and the excitations the monopoles for a start so it looks like the most naive level you think you've broken the dipole into two monopoles okay but so then the field and that's a fact then the field-theoretic consequence of that is that because you've got monopoles the monopoles are described by a field and this field decomposes into these two parts so now if you've got a low density of magnetic charges right this part the monopolar part it separates essentially completely independently from the circulating dipole apart the amplitude of the dipolar part is essentially one is essentially this because the the amplitude except very close to the monopole the amplitudes here of this part are very small so it looks like you've got a set of magnetic charges and a set of magnetic moments so in that sense actually I think it's pretty it's it's not dissimilar okay and and also and also you know often we call these parts transverse and longitudinal because the magic of taking a Fourier transform is that the monopolar part is always parallel to the the transformation back to Q and the dipolar part is always perpendicular uh transverse so this this is m of q longitudinal and this is m of q transverse okay so so uh these this set of magnetic moments really looks in reciprocal space like it decomposes into two fluids so I hear what you're seeing but but actually I I think it's a pretty spectacular example of fractionalization actually okay are there any other questions um yeah jofa I do have a question actually it's on the sorry okay um it's on the the the model uh the the exact model of the spin ice model um dipoles messable the diameter right down it's uh in terms of interaction between the spins uh this this one exactly this one yes and uh just after the demo model you are rewrite it in terms of um quasi particle imagine quasi particle yeah so going from here to here exactly yes uh and um the my question uh is how do you do that and more precisely why it's not exactly the same uh I should say it's a kind of magic no so um so you might like a skeptic might say that this is just a multiple expansion of this Hamiltonian okay um but it's more it's it's it's it's more subtle than the than this first of all it relies very heavily on this pyro pyro claw uh symmetry right um it's it's a fact that on this pyro claw symmetry the the ensemble of two in two out states they're essentially quasi degenerate so essentially self-screened so now if you put this dipole Hamiltonian on a square lattice for example this uh uh dumbbell model approximation it's nothing like as good right because if you may if you make a multiple if you make a multiple expansion of this object on the pyro claw lattice essentially all you get to an excellent approximation is this so it's very lattice dependence because this pyro claw and diamond lattice they're very high symmetry in three dimensions okay so I'm not quite uh uh uh answers your question um um but there was also some magic that you wanted to put the dumbbell model so that the the length of the dumbbells was exactly such that they all convert to the center that was a bit a slight of hand I didn't understand why that's that that's just like a fun and uh to read to know more about that then you need to read some of Mosner's papers so so what you can do is you start with a point dipole and you replace it by a dumbbell of length x okay and then you you increase it continuously until x is equal to a yeah right so when x is equal to a you have a dumbbell model that only has Coulomb interactions yeah right so if you start off with a square lattice and you do that you change the energetics considerably extending from when x goes to zero to a but on the pyro claw lattice it's really small it's like a separation of energy scale and that's because of the cancelling effects specific to the pyro claw lattice okay this is this is these papers by uh Mosner and Sondi there's a paper by Mosner and Sondi called projective equivalents why the why spin ice obeys the ice rules and essentially when you do this on the pyro claw lattice there's virtually zero change okay uh but if you do it on the square lattice is a huge change the the dipole the dipole Hamiltonian on the square lattice and the and the dumbbell model on a square lattice is completely different because that was a slightly found did that help I don't know maybe another one one more thing that one more thing I could say for people who are interested in particle physics you might notice that the inner inside these dipole these monopoles there are three little objects okay so a monopole is made up of one blue spot and three red spots and it's I think we've out we've exploded our time today but there are lattice symmetries where you can see also the fractionalization of a monopole into smaller objects that also happens in some some geometries so the monopole is always the fundamental object for dynamics whenever something moves it's a monopole that moves but you can observe a fractionalization of those charges so this is a an elementary example of fracture of charge fractionalization well actually I I would like to continue is going to have a few other questions but if Peter you are available but maybe I am available I would like to read the recording then yeah