 Okay, so welcome back to the second part of this afternoon session, the third talk will be given by Roderick Mössner from Breslin on the fine structure constant in QSI. Okay, so hello everyone, I'd like to thank the organizers very much for putting together this meeting, inviting me here and going through all the trouble of making sure things actually work, which in hybrid meetings isn't so easy. So today I'm going to explain to you what the title means and the conclusion of the talk literally is the fine structure constant. First author of this paper happens to be at the school that I'm organizing at the moment, which is why I'm also delocalized that cell who's sitting here with wearing a mask. So Chris Laumann is lecturing up there as well. So several of us are here. Okay, outline is very simple. I'm going to say a few words about quantum spin ice, then how QED emerges there, calculating the fine structure constant. And if I've got time in the end, I'm going to say something about not just the photons in the system, but also the gauge charge excitations. And I'm going to try to unpack this as things go along. Okay, so the title is the fine structure constant. Of course, all you know, the fine structure constant is one of 137 in QED famously, and there are many ways of defining it. And famously, Pauli is said to have said that when I die, my first question to the devil will be what is the meaning of the fine structure constant. So even at the end of this talk, you will not know what the meaning of the fine structure constant is, but I'm going to try to tell you what the value is of the fine structure constant in an emergent system. Sorry. He may not have said this at all. The thing that he may or may not have said I read is the devil. Okay. Okay, so then several definitions of this are one perhaps, which is very physical is the ratio of the Compton wavelengths to the bore radius. So, you know, energy terms, you know, the Compton wavelength being the wavelength of a photon, which corresponds to the rest mass and electron. And similarly, the bore radius tells you something about the binding energy of a nucleus. And it turns out then that the maximal stable nucleus you can expect to have is roughly one over alpha. So atomic number of 137. And the subject of this talk is that we don't just have QED that lives around this, but we also have systems where we have QED as an emergent low energy theory. And so that will also have an alpha and the question then is, well, what is its alpha? And is it different? And if it is different, what are the consequences? So before I go there, I'd like to give a materials motivation. And this, if you're not interested in materials, you can simply switch off for the next few slides. So rare earth pyroclore magnet. So a lot of frustrated magnetism takes place on the pyroclore lattice, which is shown here instead of corner shearing tetrahedra. And the typically for the strongly frustrated systems we're looking at on the so-called A site, which is one of the pyroclore lattices here, there is a magnetic rare earth ion. And because of the coupling in tetrahedra, this is highly frustrated, as I'll show in a second. And there are many, many samples now, and many compounds, which people are looking at. So all of this started in the well late 90s, when it was recognized by Harrison Bramwell, that homium dysphysium titanate were really very special. They found a residual entropy at low temperatures, about which I'll also say a couple of words in a few slides. But things have moved on since then. So these task of spin ices are strongly anisotropic easing systems. But now people are looking at less strongly easing systems, so systems where transverse degrees of free and transverse terms in Hamiltonian play an important role. So here's an incomplete list of systems which have been proposed here. And I'm going to give you a few, I'll tell you a few words about one of them that we've been studying just as a motivation for why quantum spin ice may be moving from the realm of a purely theoretical idea to a more material idea. So if you actually look for a candidate quantum spin ice material, so there are several challenges and several levels. So first is you actually need some material, which is somewhat promising. You then need to identify a likely Hamiltonian for this. You then need to establish the phase diagram of that, which is a completely separate problem. And then you would ideally like to know where the Hamiltonian parameters of that material that you're guessing based on whichever consideration actually plays your system. And then you hope that your system is actually going to be placed in an interesting spin-liquid phase. So in Zerium-Circunate, the example that we've looked at, there's some, so we know what the local degrees of freedom look like. You can then write down the simplest nearest neighbor Hamiltonian. And there's a priori, no particular reason for assuming that the further neighbor terms are completely negligible, but at least you can symmetry classify the nearest neighbor terms. And then you can try to fix the pre-factors by essentially fitting simultaneously to as many different experiments as possible. And one of the very nice developments in the field separately from the availability of these nice samples, which have been coming online for a while, is that there are lots of experiments available. In particular, there are these four d-neutron scattering experiments now, where you can have four spatial resolution and inelastic scattering as well. And then you can fit those. You can then combine this with fits to magnetization curves and so on, so susceptibility and finite fields. And then you can try to extract some model parameters. But you're always limited by the fact that we don't actually know in a controlled way to analyze three-dimensional, strongly correlated magnetic Hamiltonians. And so you still end up doing some mean field theory or some approximate treatments, which may or may not be reliable. Okay, but if you do this for zirium circumnates, what I just said, then you end up with a mean field phase diagram and the best guesses based on essentially best fits to these experiments all have this thing living in a quantum spin ice phase. So the statement would be that if the terms in the Hamiltonian that you have left off do not lead to instabilities, and if you can cool the system to low enough temperature that the dynamics becomes quantum coherent, then you should have a quantum spin ice. This is work in progress. And I think it's imaginable that in the next four or five years, there will actually be a system where with some confidence we can say that it's a quantum spin ice. Okay, so this is the material side of things about which I won't say anything more for the rest of the talk. And now I'm going to come to the core of the talk, which is given that we believe we know the UV degrees of freedom. And we believe that we know what the Hamiltonians doing looks like. But nonetheless, we've got this hard many body problem. What we try to do instead is we try to write down the effective low energy degree of freedom. So there we've got a different set of pre factors, like point structure constant speed of light and so on. And we try to fix those from studying the short distance degrees of freedom. And then we say, okay, for low energy, long wavelength analysis, then the effective low energy degree of low energy system is going to be what we are studying. Okay, so let me walk you through this. So we're starting say with classical spin ice, we're getting about the quantum dynamics, just to explain this degeneracy. So classical spin ice has easing spins living on the sides of the pyrochlor lattice. There's an effectively anti ferromagnetic exchange between them. And if you ask what are the ground states, you find that the ground states are those states when each state for even two spins point in and two spins point out. And then there's an excitation gap. So you can flip a spin, but that costs you finite energy, the size of the easing exchange. And if you do this, you have these excitations. So okay, so if you do this, you have excitations which have the following property. So if you have two spins pointing in two spins pointing out, you can think of the spins essentially as a flux field. And you see that on each tetrahedron, the total amount of flux coming in equals the total amount of flux going out. And this is like say Kirchhoff's current law, if there's something conserved, it's divergence vanishes. And so that's what you have here. So the divergence of the spin field on the lattice vanishes. And if you've got an excitation, it no longer vanishes. And so just like say Gauss's law where you have divergence of E is row, can basically say the divergence of the spin field defines a charge for you, right. And so the excitations are charged in this sense. So in the same way that in that electric charges are charged according to Gauss's law, here similarly, if you have tetrahedron violating the ice rule, you end up with something which is charged according to this emergent Gauss's law. Okay, so that's the important starting point. And then the emergent electrodynamics. So there's several ways of seeing this. I think the cleanest way of seeing this is okay, we believe these are the degrees of freedom, we write down the simplest theory, which incorporates all the symmetries and which incorporates this physics of the divergence free field. And then we say this is the low energy theory. But here I'm trying to give you some pictures to motivate why this is a sensible thing to do. And this is starting with the ice rule states. So you can see here on this cartoon of spin ice and each tetrahedron, which I've just plotted as a cross here, there are two spins pointing in and two spins pointing out. But if you flip a spin, then this is the same as creating a charge anti charge pair. And the important thing is that these charges are de confined. But it's to say you can flip further spins and then rather than creating more charges, you actually move the charges if you do it in the right way. So this is shown here. Okay, and since there's a degeneracy, there's actually no unique path, no unique assignment of arrows, which you have to flip to have these two charges here. So I first showed you this zigzag path. So what's shown here is a path which is much more straight, but there are other paths which wiggle to different degrees, which are all consistent with the two charges of these locations. And so then we do statistical mechanics. We average over all of these. And if we average over all of these, we get a density of flipped spins. And that density of flipped spins actually looks just like an electric field in three dimensions. So you start off with Gauss's law. You know, div s is the charge. And then if you do all of this averaging, you have two charges like this, the spin density distribution is precisely that that you get for two charges separated by this distance. So this effectively at long distances, dipolar field, and at short distances, the Coulomb field one over r squared, I'm coming out of the side of these charges. So this is how classical spin ice is related to this Coulomb physics. And then the basic idea. Okay, sorry. There's no fast forwarding that I'm aware of, which can save you the pain. So this is the basic idea behind this. So the central ingredient is this high degenerate of the classical grand state manifold, which is reflected in the residual entropy that was mentioned earlier on. And then you can ask, okay, I've got this huge degeneracy. I'm basically doing, I'm basically doing statistical mechanics and this. And then you can ask if I just decide that I only allow these low energy states, and I add quantum dynamics, I've got a problem which is a lot like the problem of the Landau level, where you have a completely general Landau level, and then you add interactions say between electrons living in them, it's very similar. You've got a completely general set of classical spin ice states. And then you add quantum dynamics to them. And you can see the quantum dynamics can't be a single spin flip. There's a single spin flip creates a pair of charges and takes you out of the spin ice manifold. But rather you have to flip a loop of spins arranged head to tail, because that doesn't create any defects in this divergence free background. Okay, so this is what's known as the ring exchange operator. You simply take these spins, you reverse them. And so this is the simplest model, simplest microscopic model for quantum spin ice is the easing exchange, which gives you the spin ice manifold, and the ring exchange terms which give you quantum dynamics. And this kind of treatment is controlled when the easing exchange is much stronger than the ring exchange. But the problem is if it's much stronger than the energy scales don't work out very well. So what we're looking for in practice is that the easing exchange dominates and the ring exchange isn't all that much lower so that you can see it's affected at relatively reasonable temperatures. Okay, and then the classical spin ice is these energy scales just work out in a way that you basically don't get to see the ring exchange physics at all. Okay, so that's that. So we start with a microscopic Hamiltonian. And then we assert, if you don't believe me, read any of these papers, we assert that the low energy action actually is a quantum Maxwell action with an electric field and a magnetic field, which is just exactly the same form as Maxwell, Maxwellian physics. And so then we expect the low energy degrees of freedom to be effectively phonon photons, just like normal photons, emergent photons. And then there are charges of the magnetic field and charges of the emergent electric field. So that's the basic, these are the basic degrees of freedom of the long wavelength theory of quantum spin ice. And what we're trying to do now is make a connection between this Hamiltonian and the pre factors of that Hamiltonian. Okay, so we're going to use exact diagonalization techniques. And as you know, three dimensional quantum spin models are not very friendly towards exact diagonalization. But I'll tell you how we tackle this. But basically the idea is, we've got these different pre factors that we need to fix. And we actually only need to fix two different quantities. So essentially, the strength of the color energy and the speed of light. And if you combine those two, you get the fine structure constant of quantum spin ice. The special thing about the fine structure constant of quantum spin ice, it's the one dimensionless quantity which characterizes the theory, the other two, the other things like the speed of light actually dimension for quantities. And they change, for instance, if you just double the lattice constant, but if you double the lattice constant, say the, or do something like this apply pressure to your system, the fine structure constant remains unchanged, more or less. Okay, numerical plan of attack. I mean, you know the world record for exact diagonalization on spin systems is 50. But we don't need to do a complete exact diagonalization because we are restricting ourselves to the spin ice manifold. Even though that's got an extensive entropy, it's considerably less than that of free spins. And we were able to do or we rather self was able to do a complete exact diagonalization using up to 96 spins. So that that turns out is already is still small. I mean, the cube root of 96 is not a spectacularly large number. But nonetheless, I'll show you in the next few slides how the analysis proceeded to extract the various, the various quantities that we're looking for. Okay. So what we basically did is to get independent estimates for these quantities is to vary between different topological sectors and also vary the shape of the unit cell. So we basically got lots of data points. And then the hope is that the, you get in some effective averaging, which averages out the noise, which may be due to some fortuitous resonances of at finite sizes. And so that you can then fit to a large data set and obtain reliable estimates. So I think the central insight in all of this was that if you want to know what the strength of the Coulomb interaction is, so sort of the properties of the electric field, you do not in fact need charges, right? Because if you need charges, then you need to go outside the quantum spin ice manifold and your Hilbert space explodes. But you use the fact that they're different topological sectors, which in simple parlance means you can take a system with periodic boundary conditions and a non-zero electric flux. So this is shown here. You've got this pair of charges. You just take them around the cylinder, unilate them, and then you've put in a flux line. So if you now do the same course grading I showed you earlier, then what you'll find is that everything averages to zero, but there's, except for there's a non-zero flux in one direction. Okay. And then you can vary this flux. You can put a flux, you can have a counter flux going the other direction, which cancels it, but you can also have a flux in one of the other dimensions and so on. So you basically get a three-dimensional flux vector, which tells you in which direction the average electric field points. Okay. And so this triplet of fluxes is then one of the things that we, so okay. So then given this triplet of fluxes, you can then ask what is the ground state energy, right? And so since the electric field, the electric field lines have an energy, which is proportional to e squared over epsilon. Looking at those vector, which are permeated with a non-zero electric field allows you to get the essentially the, this e squared over epsilon without putting in effect, without putting in actual charges into the system. Okay. So here's the equation, which was used to extract the answer. And so this is what we got. So there's all these different values of the fluxes and so on plotted on the x-axis and effectively an energy on the y-axis and we use, and the color code tells you the density of the points is otherwise you, so most of the points actually do lie on this line and the width of this is given by rare outliers. And if we do this, we get the following results. So as I said, we get a dimension for quantity for this. The dimensions that are involved here is a lattice constant and the strength of the ring exchange. Questions? Okay. So that's that. Then for the photon, yeah. The thing is you actually need a number, right? So this is not universal. The number could have been anything. So if you had a reliable way of extracting the difference in energy between the ground state and the state with a non-zero flux, then you would have the answer, right? It's just I'm not aware of how you could do this. The only things that I'm aware of are always things which work up to the, up to the pre-factor. And since you have to solve a full quantum problem, I don't know any shortcuts either, but yeah, that doesn't mean they don't exist. It just means I don't know them. Okay. Other questions? Yeah. So in this case, it's negative. So in this case, it's such that the ground state wave function is no less, but I don't think that matters all that much. Right. I think it doesn't matter all that much. Okay. Was there another question on this? Yeah. As I, can you read it because I don't want to touch this and then not be able to go back to the current settings? Oh, can you show it there? Brian Skinner, go ahead. Yeah. Please give more details. Yeah. Okay. So the other thing we wanted is the speed of light. And so there, quite obviously, if you want to measure the speed of light, you take omega goes to c times q and then you fit the straight line. But the problem is, even if you've got a linear size of eight or something, you really don't have enough points to be deep in this, in this linear regime, as I'll show next. So what we basically did is take the full dispersion, say that we know which lattice this takes place on. We take the lattice, the plus sign of that, to get the dispersion as a function, not just in the linearized version, but in the, for the full Bria zone, since this is rotationally averaged, we get a curve which broadens. And then we plot all the data points we get from these different samples and then fit this line. And what I was saying is basically, we don't have any data points in the strictly linear regime, we just have it where it sort of starts deviating. But again, we have the points lying pretty narrowly on this line, expected from the lattice, the plus sign, then here's the speed of light. Okay. So these were the two ingredients. So now, if you're not interested in the technical stuff, just forget the last few slides, these are the answers. So this is what I showed you and the combination between the two of them is that the fine structure constant of quantum spin is 0.08. So you may say in contradiction to the title I gave you, 0.08 is not a large number. And so I'm going to explain to you now why it's actually a large number. So okay, so just before we get there, so the speed of light in quantum spin ice for reasonable parameters, whatever that means, is about 10 meters a second. So Usain Bolt is faster than this. But it's really a human speed from that perspective. And it's very, very different from the speed of light in free space. And this has a lot of implications say, in principle, you could shine light onto spin ice and turn it into emergent light inside. But it really doesn't work very well, because the phase space factors don't agree. Okay, but one of the things that I really like you to take away from this is that the fine structure constant quantum spin ice is an order of magnitude bigger than the fine structure constant of the photons that are hitting us all the time. And an order of magnitude actually is a lot. So relatively speaking, it has a high fine structure constant. Okay, let me just say you can also perturb the model. They're too natural, too simple perturbations, we could think of one as something that's known as a rock cyclism potential. Again, you don't need to digest particularly what that means, but basically counts the number of resonances which are possible and assigns an energy to them. And another one is just a further neighbor exchange. That's the first thing that we tried and it worked, which is why we use that one. So if you do this, so you get a perturbation parameter, which is the size of these perturbations called u and zeta here. And if you vary mu from minus a half to one, so at the so-called rock cyclism point is it's known that the quantum electrodynamics breaks down and the fine structure constant vanishes in the square root behavior. And then you see that you can increase the value of the fine structure constant by putting in a negative rock cyclism potential. If you put in the nearest neighbor exchange, the further neighbor exchange, it turns out that you can again change the fine structure constant, you can make it quite considerably bigger by more than a factor of two. And then when you hit point two, the system actually becomes unstable and goes into a different phase. And there's some arguments and it's not clear to what extent they apply outside for quantum spin ice. But in QED, there's some arguments which make you believe that actually QED itself becomes unstable at a fine structure constant of point two around there. So in that sense, point one is a fine structure constant is very large because it's only a factor of two within what you can actually do. But no one really knows what happens in detail at this very strong coupling. So but what I'm trying to say, not only relative to alpha QED is our quantum spin ice strongly coupled, but it's also a PSP strongly coupled compared to the maximum possible value. So was it a first order transition or? Yeah, okay. That I don't know. So whether you can think of it as an instability brought about by some collectivists, I don't know. How much time do I have left? That much? Okay. Wonderful. Questions? Okay. So then I'd just like to take one brief detour on a separate piece of work which relates to this, however, which is the question. So this is the basics of quantum spin ice, the photons, the speed of light, and so on. So this thing is strongly coupled. So I'm going to say a few more words about this. But I'm going to return to the question of what actually happens to the charges because I'd like to advertise. There's also a question, is there some, yeah, some other estimates we can do. So here's an example of something where you can do a very simple calculation and get what apparently looks like compared to exact diagonalization, a reliable result. And so I'd like to motivate this the following way. So I said, you can do some finite size ED as long as you restrict yourself to the ground state, spin ice ground states. If you put in charges, the scaling gets worse. It doesn't get catastrophically worse. If you only put in two charges, it's just like L, because the charges can only have L cube locations and so on. So it doesn't just blow up completely immediately, but you certainly can't do 96 sites. And what we then try to do is do even smaller ED and come up with an effective analytical theory, which allows us to combine to sort of get a guess a little bit like with the lattice Laplacian I showed you, so to get a qualitative guess for what the answer should look like. And so the question, the reason that it's worth looking at the quantity I'm showing here is that two monopole density of states is because if you flip a single spin, which is what say a neutron may do, you actually create a pair of monopoles, a pair of charges. And these charges then, you know, they're various form factors, but the density of states is what you would see in principle in in elastic neutron scattering. So the two, the two particle density of states is a very natural quantity to look at if you're interested in looking at a fractionalized model. Okay, so just to clear up this detour, neutrons coupled to spins, but the spins fractionalized into these charges. So you can never create a single charge by itself the way you can create a single magnum by itself. So you always get two. So the natural thing to look at is actually a two particle quantity in here, the density of states is just the simplest thing to start with. And so there's a very cute way of looking at this, which is, those of you who've worked on MBL are probably familiar with thinking of a spin system of actually defining a hypercube, where basically each dimension of the cube corresponds to one spin. And if the spin points down, you're sitting on the left hyperface, if it points up, you're sitting on the right hyperface, so that flipping a spin always corresponds to going from one hyperface to the other. Okay, and so here we have this hybrid view of things. So we have these charges, which the location of which determines where the defect tetrahedra are, but we still have the spins, the dynamics of which this is the fundamental quantum dynamics. And the spins are more naturally thought of as living on this hypercube and the charges are more naturally thought of as living on the real cube. So if you look at the hypercube, it turns out that, and I'm not expecting you to adjust this in real time, that you can have closed loops of length three on this hypercube. And then the next shortest closed loop that you can have on the hypercube is actually length eight, you have to go around the parallelogram two times. So then we do what theoretical physicists are good at. We say we ignore everything above length six, so we just pretend that we have got a lattice, which has got closed loops of length three, but no longer closed loops. And so this turns this hypercube into a cactus known as a Housimi cactus. And the Housimi cactus is soluble. So you then can basically, I've got a hopping problem Housimi cactus, I can get a spectrum, I can then translate the spectrum and effectively, so okay, I get a spectrum and the spectrum is the solid line here. So this asymmetric line shape, a steep singularity here, and then this slower come down here. And then we compare this to a 32 site two to charge simulation, where we also got the two particle density of states out and they're basically overlap. There's a lot of uncertainty in this, say there may be processes which are present neither on the Housimi cactus, nor on the 32 site ED, which would make this just wrong. So we don't know, but the agreement here is actually quite good. So in other words, if you do low energy, low temperature, neutron scattering, the in zero's order, the thing that you should be looking for is a line shape like this. That's all I'm saying. If you find it, it doesn't mean you've got quantum spin ice. If you don't find it, it doesn't mean you don't have quantum spin ice, but this is definitely kind of thing to look for. Okay, sorry. Yeah, so it's basically convolutions of things which have, what are they called? What are the singularities of the band bottom called funho singularities? Yeah, so it's basically about just that physics. And it's depending on, yeah, and then it depends the dimensionality you're sitting in, but this is what you get. And if you change the sign of the hopping, it just gets flipped. No. Okay, so I don't know. I sort of suspect that they're not perfectly protected because, you know, the Housimi cactus is, lives in a different dimension than the actual physical system. But I also suspect that it's the, you know, the weakness of these other, the other perturbations, which preserves the sharpness of the onset would be my guess. Okay. All right. Okay, so let me, so this is the end of the results section. I'm going to say a few more things about the interpretation. But before I do this, I'd like to thank Sal, the Chris Lowman, who's the Leonardo building and Sid Moran Pudi, who's at MIT. And the last bit I did with Masa Ute Gawa, who's Tokyo. But I'd like to finish with a following observations. So famously, right, the integer quantum all effect paper was submitted. I think I can't remember what the submitted title was, but the ref repointed out that it's actually what's actually being measured is not E squared over H, but it's actually the fine structure constant. And one of the miracles of their topological physics is that you can take a dirty, relatively dirty semiconductor sandwich and extract the fine structure constant of the vacuum of QED to an accuracy of like 10 digits. So I haven't much they've got these days. Okay. And one of the, you know, ironies of life is this was so good that the standard standard of units was changed. And I think now with a change of the standards of units, the original title of Clitzing's paper was correct. And this one's incorrect, but I don't remember the details of that. Okay. But the thing was, you know, topological physics tells you, you can measure take a dirty semiconductor and get the fine structure constant spin. Okay. So exactly the opposite happens here, right. In quantum spin ice, we have a tunable fine structure constant. So we still have topological physics, because this is a spin liquid, that's an emergent gauge field. So that's what you can use to define a topological phase. The point now is we have quantum, you know, alpha quantum spin ice, which is large and tunable. So it's exactly the opposite of the quantum Hall effect version. So we go to a different vacuum and the different vacuum has completely different properties from our vacuum. And the miracle still remains that our vacuum, you know, there's that, you know, the permittivity and whatever of gallium arsenide don't matter. And you can still extract alpha from the quantum Hall effect. Okay. So just to summarize, so in quantum spin ice, we've got emergent QED with photons, electric and magnetic charges. There's promising experimental progress. We are moving towards qualitative and quantitative descriptions, I think more so than I would have hoped a few years ago. So we've got effective theories, finite size numerics and relatively simple analytical models. And between those, we're basically getting a grip on sort of the universal long wavelength physics. But in experiment, you actually most of the time do not measure the universal long wavelength physics. And for that motivates the study of sort of less, less universal, but arguably more characteristic and more, yeah, more characteristic features, which are seen at intermediate time and length scales. Thanks very much for your attention. I guess you probably remember that the limeshift causes a fifth power of a fine structure constant. So the question is, is there a solid state analog of the lamb experiment? So you could propose, which would measure this right. So in principles, it looks like just about every phenomenon QED in principle exists here as well. So spin ice is special in that there's a perfect symmetry between the charges. So rather than having a hydrogen atom, you're going to have positronium. And experimentally, the issue is that having the two charges orbit each other this way involves the collective motion of a large number of spins, because that's what's involved in moving the electric field lines along and doing this coherently is currently not experimentally realized. So that's right. So is there any corrections to crystal field levels? I mean, so crystal field levels for what I'm saying here are infinite temperature. Okay. Yeah, yeah. So there will be ground state correction of the energy. But if you talk about defects, there's an even stronger effect, which is if you take one spin out of spin ice. Okay. So if you take this spin away, you also have charges except for the only half the size. And so then you have basically a dipole, a charge, anti charge pair of half the size present, and you can ionize this to get the stationary charge, immobile charge, a defect charge there. And then everything that you said about something orbiting it, all of that's possible. And yeah, I mean, yeah, I think, I think as far as actually doing experiments, I concern that that isn't possible at the moment, but there is, in principle, something you could measure having this defect charge surrounded by an opposite charge which is bound to it and asking, what is its spectrum? I mean, so the likelihood that you will see corrections to this, of course, grows with alpha considerably, right? But I suspect that, you know, so if I were to ask what are the most striking features that you expect to see at look for things like triangle radiation, rather than, you know, the lamp shift, because, you know, there's nothing sacred about the speed of light here. And in principle, you can kick one of these things to move faster than the speed of light. So, you know, I think that's a lower order effect. And it's perhaps a more spectacular effect. And that's, that's why I start looking for, for strong coupling QED. What about magnetic flux sectors? Are they very magnetic flux sectors? Are they very high up in the... Okay, so just to use language. So in, in gauge theoretic length, so what we called magnetic monopoles in, in like 10 years ago, in this language are electric charges. That's what you talked about, right? You talked about electric flux sectors. What about... That's right. So these were the electric flux sectors. The magnetic flux is actually a lot lower. So this is, again, different from QED. I mean, I'm not going to get into, like, what magnetic monopoles do in normal QED, but yeah, the magnetic, so the magnetic flux sector has a characteristic energy scale, which is that of the ring exchange, and not that of the easing exchange. So it's actually down there. And so... Are you talking about the defect or you're talking about the state on the torus? On the torus. So it's basically the energy scale in center 2D, easing gauge 3D would be the energy scale of the vison. Yes. And that's typically, you know, in these kind of systems where you have a very high energy constraint giving you the constrained Hilbert space, the vison energy is typically a lot lower. That's the same here. So can you find it in your exact diagonalization? Check, check that your alpha is... I mean, one goes as alpha, right? The other goes as one alpha, something like that. Yeah, so the problem is that the, is separating out the photons from the purely magnetic degrees of freedom because they live at the same energy scale. So basically, it's much more likely that the photons enter a continuum determined by these magnetic excitations. And this is another reason why things become very difficult to analyze at these low temperatures, but these degrees of freedom are there. Photon continuum. The photons enter a continuum. So this is, you know, the separation I've shown you isn't strictly right in the sense that the param, so there's no reason why this thing at the zone corner should be below that. That's it. So the speed of light is so small, you should see it as a dominating effect at some temperature in the heat capacity. Is this feasible? And if I integrate the total entropy that is in this photon system, has it anything to do with the residual entropy of the classical system? Okay, so the answer to the first question is, as a matter of principle, there really is nothing stopping you from seeing that contribution to the specific heat. And if your measurement is accurate enough, you should see that there are two or not three photon polarizations. I think all of this should go through and there have been things approaching claims that you can do something like that. I think not everyone's convinced, but I think that it's not unlikely that this is going to work. Then the total entropy of the total polling entropy is going to be everything below the electric charges. To the extent that the photon branch is not separated from the magnetic excitations, I do not see how you can do a quantitative, you know, a half log three halves for the photons either. I think it must be split between the two. Perhaps there's a clever way of arguing how things should come down at low frequencies. We haven't thought about that. I think that might be possible. But I think just a straight integral would be the spectral weight here and not just the spectral weight there. Thanks a lot. So if I understand correctly, in real materials, in the material part, the models which describe them, they differ from the generic model for which you did your calculation. So that means that for each of these materials you have your own fine structure. That's correct. Is there any kind of upper limit for this alpha? Yes, so we expect that the fact that it became unstable at point two may not be a complete coincidence, but we don't have a particularly strong argument for this. I mean, you saw that even with relatively strong perturbations, it only changed effectively with a factor of two both ways. But this is the first determination and no one's looked at it systematically. Well, with these two perturbations and in principle, you could look at other perturbations and see what happens. It's just that's a lot of CPU time and it would be nice to do this systematically, but there's no way of doing ED systematically on families of models. There are a few questions online. So the first question is by Brian Skinner, who says, at a conceptual level, I don't understand how you are able to extract e squared over epsilon from your numerics instead of just epsilon. Since we are looking at states with electric field, but no charges, I would think that the proportionality between the squared electric field and the energy density is just excellent. So the size of the flux that you can put in is given by the size of the charge. So basically, it's true the energy is electric field squared over epsilon times electric field squared like in the capacitor, but the size of e is fixed by the size of the flux going through is fixed by the size of the charge. So it's like having a capacitor with epsilon times electric field squared and putting one charge in one capacitor plate and the other charge in a capacitor plate. It's that quantization which allows you to extract e squared over epsilon rather than just epsilon. Another question is what modifications do we need to make to realize the vacuum value of alpha? Okay, so the vacuum value being that of ideal quantum spin ice, anything. It's a non-universal quantity and unless you do something special, fine-tuned or clever, anything will change alpha. And last question is how you choose your model or system that it is icing or not other orientation direction that xy or hyzenberg. Okay, so the, okay, it's the short answer is point group symmetries of the pyrrole lattice. Slightly longer answer is you have each side has a local three-fold rotational axis. So you can either have an easing axis, easy axis or easy plane. That's the difference between easing and xy. And then it's an issue of, okay, so that's the next level. And then the next layer down, it's an issue of, you've got a certain amount of angular momentum. And if you think of an Anderson super exchange type calculation, whether or not you've got a matrix element between safety up in the down state depends on how much angular momentum you can actually move in one of these virtual hops. And then the classical spin Isis, which are the most strongly easing, there isn't a lowest order matrix element flipping things. So this is why it's very strongly easing. If you go to these others, you actually also have xy magnets, right? So it's not like you either get this or that, it just depends on details. Some of them are easing, some of them are xy. So both are possible. But there are reasons which are understood why the classical spin Isis are so strongly easing. Okay, I don't see other questions right now. So let's thank the speaker. And in principle, we still have a question session for all the afternoon speakers. So if there are questions, no, I guess we went already over time. So thank you.