 of posters, instructions on how to run them up and all that. So we are very happy to have, I think, the record of posters at the STP school. But this also implies that we trust that you guys are very disciplined and really help us arrange them properly. So it's important that we are able to put two posters for each side of each wall. So if two posters are A0, they do not fit by roughly 20 centimeters. So the thing that you have to do, you have to fix your poster in the middle end, and you have 10 centimeters outside. If your poster is A1, it's no problem at all. If your poster is A0, by the way, in this way, this is really problematic. So search for a partner that has an A1 or A2 poster. Otherwise, it's very hard. And even these rules, we should be able to put all the A2 posters up. Please be collaborative. And if you have already fixed your posters upstairs and it's not according to these rules, please do that over the line so that people have space for their own posters. OK, with that, we start with the second lecture of the morning. So good morning, everyone. I'd like to continue in this lecture the theme that I started yesterday talking about systems which have macroscopically many degrees of freedom, but constraints amongst them. So as you remember, I started out using the largest part of the lecture yesterday to talk about this example of the triangular lattice anti-ferromagnet. And we saw how, well, firstly, degeneracy could arise in a reasonably realistic interacting system. In fact, as a result of frustration. And then we saw how you could get a long-distance description of the system, which I think is certainly much simpler by mapping configurations onto a height field. And that mapping can be done exactly at a microscopic level. But having got this more convenient degree of freedom to describe the system, that's to say, the height field in place of the spins, we could then start making approximations and go to a coarse-grained continuum description of the system. And then we have to check a few things, such as the relevance of cosine terms that might pin the height to integer values, but provided just the leading grad H squared term is relevant. Then we have a rather simple description, which allows us, for example, to understand power law correlations, and also allows us to go further. Although the height mapping was developed initially for ground states of the system, therefore, to describe zero-temperature properties, if we go to non-zero temperature, then we introduce defects into the system. But if the temperature is low on the scale of the exchange interaction, then these defects are dilute. And you have large regions of the system between the defects, which can be described by the height model. And we can capture the entropic interaction between defects by doing calculations in the height model. And then, depending on the stiffness parameter, you may either find that these defects, which you can think of as vortices in the same sense as in the XY model and the KT transition, if the stiffness is small enough, then the vortices are unbound. And that gives you exponentially decaying correlations at long distances, which is the case in the spin-half model at finite temperature. So then I started talking about dimer models, and I'll continue with that during this lecture. And then at the end of this lecture and in the lecture this afternoon, I'll talk about a class of magnetic materials that goes under the name spin ice, which is probably the best realization that we have in a concrete material of the physics that I'm talking about. So the endpoint of the lecture yesterday is this slide. And the overall idea was that if we think in the dimer language, then we can use dimers to specify the strength of flux on links of the lattice. And this flux is divergence-free at the microscopic level. And so again, it's a suitable quantity to use when we construct a coarse-grained description of the system. And we can think of a divergence-free flux, of course, as coming from the curl of vector potential. And if we want to use that idea in two dimensions and use a three-dimensional curl, then we can say that we have three-component vector potential, which is in the z direction and with a size given by a height field, which is just, in fact, the height field that we were talking about before, because then the flux strength is the gradient of the height field. And we get back to this effective description on the basis that if we have large values of the flux, then we have low entropy. And so in these effective theories, we should penalize those microscopic configurations. And in three dimensions, we have something similar. We can construct an effective weight for configurations in terms of an integral over the flux density in the configuration. And the main difference between the two-dimensional theory and the three-dimensional theory is that in two dimensions, the fact that the height field is microscopically discrete can, if the stiffness is large enough, give you a crystalline phase where the height locks uniformly to an integer value. But in three dimensions, there are no small perturbations of this theory that are relevant in the RG sense. So what I want to do next is talk about calculating correlation functions starting from this point. Are there any questions which I should answer before I get going? Yeah. Well, firstly, you say if the ground state is the zero flux state, I mean, I think I'd like to emphasize that in these problems, we have many configurations which all have the same energy. So in the dimer problems, we have microscopically many allowed dimer configurations. And if we do a statistical average, we should take account of all of those. And it's only after we've done some coarse-graining that we say that the configurations that in a coarse-grained sense have lower values of this flux B are the ones with more microscopic entropy. So by that argument, what we want to do is suppress configurations with large values of B. And it's B itself rather than its gradient that matters. I mean, that's why it's not gradient of B. I mean, I had some pictures which are on the previous set of slides. But for example, if I have a dimer configuration which is like this, then it translates to a large flux. And what's more, there's no way of locally reorganizing these dimers. So it's got a small entropy. Whereas a configuration which is like this has lots of possible local reorganizations. And what's more, this turns out to be high flux. And this is a low flux. So it's for that sort of reason that you penalize flux itself rather than the gradient. OK, so what I want to start with is a calculation of the correlation functions of the flux here. And so obviously, that's likely to be somewhat analogous to the calculation that I did just of spin-spin correlation functions. If it were completely the same, I wouldn't simply repeat it. And the difference is that when we talked about mapping from the height field back to spins in the triangular lattice antiferromagnet, the argument was that the spins are given by a periodic function of the height field. And in the leading approximation, you just take the lowest Fourier component, which was this cosine. And for that reason, what we had to average was the exponential of i times a constant times the height field. What I want to talk about calculating averages of now is components of this B field. So what we want to calculate is an average of one component of the B field at the origin and a possibly different component of the B field at some point r with the average meaning of functional integral over B with this effective Hamiltonian as a weight. And the effective Hamiltonian is some microscopic stiffness times the flux density. So the first point to make is that the fact that this B field is constrained to have zero divergence is absolutely essential in having the possibility of a non-trivial answer here. Because if you did a calculation like this, but where B was completely unconstrained, then the B's at different points in the functional integral would be independent of each other. And this would just be some delta function on r. So what we want to do is impose the constraint that if B is zero, and of course, there's more than one way of doing that. One way would be to express B in terms of the vector potential and switch the integration variables to the vector potential. And another possibility, which is the one that I'll follow, is to put in some penalty in the weight for having a non-zero divergence and take the limit at the end of the calculation of infinite value for this parameter, which suppresses things. So if we do Fourier transforms, we can talk about Fourier components and we have the inverse transform. So if we use Fourier transforms, we diagonalize the effective Hamiltonian. And what we have then is a sum over the Fourier modes and the spin components or field components. And then as the weight, this term, we just have unity in the space of components. And here we have something to impose this constraint, so plus mu. And if I think of the action of divergence on B, then of course, the gradient just brings down a q. So what I have is, if I'm thinking about field components i and j, what I have is qi, qj. And it's useful to be clear about what we've got here and to see what we've really got. It's convenient to take out the length squared of this vector and I can write it as mu times q squared and then qi, qj over q squared. And that's in fact a projection operator. So I can think of this projection operator, qi, qj over q squared. And to convince yourself that it's a projection operator while projection operators square to themselves and it's one line of algebra to see that's the case. OK, so then we want to calculate the firstly in 0 if the Fourier components are different from each other. And then what we've got from doing the Gaussian integral is 1 over k times the inverse of that kernel to the weight. So we need to take the inverse of 1 plus mu q squared and then we take the ijth entry. And thinking of things in terms of a projection operator, I think makes it easy to see what the inverse of this is. What you get is a piece which is just 1 minus the projection operator and another piece which involves this Lagrange multiplier that's going to keep the divergence 0 and that's 1 plus mu q squared times p. And then the point is that when we take the limit and fix the divergence of b to be 0, then because mu is on the bottom, this term goes away. And what we've got is something that's proportional to delta ij minus qi qj on q squared. So what you see from that is that there's some kind of non-analytic behavior at q equals 0 because depending on which direction you approach q equals 0, once you've picked on components i and j, you get different values. Well, that's the correlator in reciprocal space. But what we'd like to do is understand the implications in real space. And so if we go back in the Fourier transform, then we have Fourier transform of this correlator and then there's also 1 over k. And all the interest comes from the behavior of the Fourier transform of this term and a convenient way of doing the integral if I turn this into 1 over 2 pi cubed integral over q. A convenient way of doing this is to think of the integral of eta iq dot r and take derivatives with respect to components of r in order to bring down these factors. So we have 1 over k and d2 by drid drj of, and then as far as this term is concerned, it's minus 1 over q squared times e to the iq dot r. I'm missing a view. Well, I took mu to infinity, which killed this term. And so what I'm dealing with is this term. And what I'm saying is, in fact, this first contribution is not the important one. The interesting long distance behavior comes from this term because of its singularity. And so the Fourier transform of 1 over q squared gives me just 1 over r. So what I have finally is d2 by drij of 1 over r. And if I evaluate the derivatives, then I get 3 rij minus r squared over r to the fifth. And if I'm careful and put back the prefactors, then it goes like 1 over k. So if we stop and look at this result, we can compare it with what we had in the triangular lattice ising model. And there are some similarities, but also some differences. So the first point is that whereas in the ising model, we had a power law of 1 half, which we saw later on would vary continuously with the sniffness in the height model. Here we've got a fixed power. It's basically 1 over r cubed, regardless of the value of the stiffness. And we've also got an interesting angular dependence, which is basically the angular dependence that you have in the dipolar interaction. And I'll come back later to some consequences of that. So any questions about the calculation? Sorry, should I write the final result higher up so that it's easier to see? So the result was 3 rij minus r squared over r to the fifth for the correlator that we had originally. OK, so in other words, we've done a parallel calculation for the three-dimensional system to the one that we had of spin correlations in the two-dimensional system. And what I want to do now is go back to dimer models and talk about other aspects of their properties. And so the first aspect that I want to talk about is the analog for dimer models of the possibility in the triangular lattice antiferromagnet of introducing excitations out of the ground state. Now, in the Ising model, we started with a statistical mechanics problem with some energy scale, and we got our constrained set of states by going to the ground states. In dimer problems, you don't initially have any kind of energy scale. You just have this way of constructing allowed configurations with dimers on links and exactly one dimer touching each site. But we can go in the same direction if we allow ourselves to break up a few dimers into pairs of monomers. And then we consider configurations in the system with monomers arranged in particular places. And presumably the interesting situation is going to be when the concentration of monomers is low. And so let's focus on what happens if we have a single monomer in a dimer configuration. So our basis for talking about dimer configurations was this mapping from dimer configurations onto fluxes, which involved, first of all, taking a convention for the orientation of links from, say, sublattice A towards sublattice B, and then associating a flux with each link with a value and a sign that depends on whether there's a dimer present or not. So in a situation where we have a monomer here, that means that we have four links around a site with no dimer on. And that means that they all carry flux in the same direction relative to the site. And so we see in the right-hand picture that the monomer acts as a source of flux. And if I place the monomer on the opposite sublattice, then the links would have been directed in the other direction. That's to say towards the site where the monomer is. And in that case, we'd have had a sink of flux. So then we can ask about the entropic interactions that we have between monomers. So that's to say we can think of placing a pair of monomers in the system at two fixed sites and ask about the entropy of the configurations of the dimer background as a function of the separation between the two monomers. Yes, question. No, there's something that we introduced. So before you decide that you'd like to study monomers, then you might just stick with a rule that you can only consider dimer coverings of the lattice, which have exactly one dimer touching every site. And then there's no possibility of monomers. So it's a conscious choice to break that rule and to investigate the consequences of breaking it. OK, so then we ask what the entropic interaction is between a pair of monomers. And that means something very concrete. We put monomers down at a given separation and we count the number of dimer configurations that there are for that separation. And then we vary the separation and see how the number of dimer configurations varies with separation. So that gives us an entropy as a function of separation. And that's what we're computing. And the idea is that this continuum effective theory can tell us what that entropy is. And I haven't actually put any explicit calculations because the results of the calculations that you do are familiar to you from electromagnetism. So the point is that if we put a pair of dimers, one on the A-sublattice and one on the B-sublattice, they're a source and a sink of this flux. So if you like, you can think of them as positively and negatively charged electric charges. And when we do this calculation of the entropy, what we're asking for is in the electrostatic analogy, we're asking for the energy of the electric field between the two charges. And in other words, we're asking for the interaction potential in electrostatics. So in three dimensions, you know that the interaction potential goes like 1 over r. And we start with some attractive interaction for opposite charges when they're close together. And so we have some value of r that corresponds to the lattice spacing. And the crucial thing is that if we separate the charges to infinity, we only have to pay a finite amount of electrostatic energy or in the analogy entropy. And so in three dimensions, these charge pairs are unbound. In two-dimensional electrostatics, because the electric field due to a point charge falls off in two dimensions like 1 over r rather than 1 over r squared, if we integrate that up, we get a potential that varies logarithmically instead of as 1 over r with separation. And so we have a logarithmic entropic binding potential. And as I explained yesterday, we should then see how that plays off with the increased entropy from different positions to put the charges as you separate them. And if the coefficient here is small enough, then charges are unbound. And on the other hand, if it's large, then charges can be bound in pairs. So any questions about that? Yeah. I'm sorry, I didn't get the question. Yeah. Yeah, this is quite a long way from quantization. I mean, we're really, well, we're strictly in three dimensions rather than 3 plus 1 dimensions. So I think it's a formal analogy. I mean, I can talk more afterwards. So there's one caveat to that idea that in three dimensions or in two dimensions, if the stiffness is small enough, monomers are unbound from each other. And the caveat is that it's crucial that the background of dimers in between the monomers is in a liquid-like state and not in a crystalline state. So I mean, I explained that if the stiffness is large enough, then that can lead the height field to lock. And that corresponds to the dimers ordering in some kind of crystalline pattern. And if we start with a crystalline pattern and then break up one of these dimers into two monomers and shuffle the dimers around to separate the monomers, then what happens when we do that is that we leave a clear string behind along the line on which we've separated the two monomers. And because we can identify this visually when we look at the configuration, we can expect that there'll be an entropy penalty associated with this string, which is proportional to the length. And so instead of having an entropic potential, that grows logarithmically with separation in two dimensions or goes like 1 over r in three dimensions, if we have a crystalline background, we can expect a potential that increases linearly and therefore binds pairs of monomers rather strongly. Well, this is a limitation of drawing cartoons. But suppose you have a situation in two dimensions where the stiffness of the height field is just bigger than the critical value so that the height field is relevant, but only relevant at long scale, sorry, so that the discreteness of the height field is relevant on long scales, then the configurations that you'll get will not be as rigid as this, but they'll have some average order but with fluctuations around the order. So there'll be a finite entropy density but also an order parameter and it's how that finite entropy density is affected by the pair of monomers that I'm talking about. Thanks. Okay, you can also go back to the triangular lattice problem and think of the mapping from spin configurations to dimers and check how this idea of monomers fits in with what we were talking about there and the answer turns out to be to some extent it fits but to some extent it's a slightly different idea. So in the context of the triangular lattice, the idea you remember was that if we have a pair of spins in a triangle which are parallel to each other, then in the mapping to a dimer configuration, we place a dimer on the corresponding bond of the hexagonal lattice. So now we should see what that rule means if we have a triangle that's excited out of the ground state so that all three spins are parallel. Well, clearly what that means is that on the site of the hexagonal lattice which is at the center of this triangle, there are dimers arranged on all of the three hexagonal lattice bonds that go outwards from that site. So the spin excitations correspond to having sites of the hexagonal lattice not with one dimer touching them but with three dimers touching them. And you can see that this will give you sources and sinks of flux but in a slightly more elaborate way than the monomers that I was talking about. So in the context of the hexagonal lattice dimer model, we can introduce monomers but these don't have any correspondence back in the Ising model language. OK, so there's one final point which I want to make about this type of dimer model and it's the analog or the extension of the point that I made about sectors of states in the triangular lattice antiferromagnet. So the point there was that if we consider ground state configurations that satisfy say periodic boundary conditions on a tourist, the fact that the spins satisfy periodic boundary conditions doesn't necessarily mean that the height variables will satisfy periodic boundary conditions. And as I explained yesterday, the boundary conditions on the height fields are that they have to match modulo 6 in both directions around the tourist. So you can have configurations with a tilt to them. And there are analogous ideas that apply in these dimer models. So the first point I want to make is that if we take a dimer configuration and ask about the flux crossing a line through the system or a line in two dimensions or a surface if we're thinking about a three dimensional model, then that flux is conserved in the sense that it doesn't matter where you locate the line. We'll calculate the same flux across the red dotted line as we will across the green dotted line. And you can convince yourself that that's true. Across this green line, we've got three units of flux going down and three units of flux going up. So those cancel. And then on these two, we've got four units of flux in the upward direction altogether. And if we look at things across the red line, then again, we've got four units of flux upwards here. And again, we've got a cancellation but a different cancellation on the two leftmost links. That was just one example. But you can convince yourself more generally that nothing that we do with local rearrangements of dimers can change the flux crossing one of these hypersurfaces. So for example, you can start with this top picture and think about rearranging the dimers on this green loop. And the loop is chosen so that on alternate steps, it crosses a dimer and then crosses an empty bond and then crosses a dimer and so on. And that means that you can rearrange the dimers on the loop moving the dimers from the occupied to the empty bonds. And if you do that with this loop, then you'll switch from this bond that goes across the cut being occupied and similarly with this one being occupied. But if you check things out, you'll find that this link is in the convention for directing the links from one sub lattice to the other. This link has one direction relative to the cut and the other link has the opposite direction. And so the contributions to the flux from these two dimers cancels. And that remains true whether they're both occupied or both unoccupied. And alternatively, if you find a loop which crosses two bonds that are directed in the same sense, then on these bipartite lattices, you always find that in the initial configuration, one of the links is empty and the other is occupied by a dimer. And that remains true if I shuffle the dimers around the loop. So whatever rearrangements I make of dimers around local loops, I can't change the flux cutting any of these hyper-surfaces around the system. So what that means is that we can separate the configurations of dimer models according to the values of the flux. And in d-dimensions, we'd have fluxes around all d-directions on the hyper-tourists. And you can't get from one sector to another just by making local rearrangements of the dimers. So any questions? I mean, I've got a bit more to explain on this slide. But any questions so far? So if you want to get between sectors, then one way of doing things is to break a dimer up into a pair of monomers. And then because they're initially on different sublattices, they correspond to charges with opposite signs. And if you then make dimer rearrangements locally, you can eventually move one of the charges right around the torus. And if you then bring the two monomers together, you can recombine them into a dimer. And in the process of doing that, what you've done is arrange for an extra flux line to thread around the torus. And that gets you between one sector and another. So one thing that you can get from this height model description is an understanding of what the fluctuations ought to be in the fluxes winding around the torus if you average over all these different sectors. And the result is easy to work out and useful to see. So do it, calculate it now. So the point is that if we want to describe configurations of the flux in this coarse-grained way in the presence of some average flux winding around the torus, then we can split B into something which has no net flux and then the average. And if I have a flux winding around the torus, which I write as 5, then the flux density will indeed dimensions be like that. And so if I take this decomposition and put it into the statistical weight that we're using, then I get, obviously, a contribution involving delta B squared and then something involving pi squared. And the coefficient here, since this is a constant, it'll be the square of this 1 over l to the d minus 1 times the volume of the system from the integral. So that will give us 2 minus d as the pre-factor. And then you might think that you'd have a cross-term. But if you have net flux 0, then that averages to 0. So the point is that this gives you a statistical weight for fluctuations in the flux between sectors. And what you see is that pi squared average varies with system size like delta D minus 2. So in two dimensions, this is an order 1 quantity. And when it's an order 1 quantity, the fact that, microscopically, it's actually discrete valued becomes important. And then we get back to the relevance and the cosine terms that I talked about in detail yesterday. But if we're in three dimensions, then the fluctuations in the flux grow with system size. And then the fact that we're fluctuating through integer values rather than fluctuating through a set of real values becomes unimportant in a large system. And that's how you see that the discreteness of the flux at a microscopic level is not important in three-dimensional systems. So any questions? OK, so the next topic that I want to talk about is what difference it makes, what type of lattice you define your dimo model on. And in the examples I've talked about so far, the lattice was bipartite. So a lot of my pictures were drawn for a square lattice. And the lattice that I arrived at starting from the triangular lattice sizing model was hexagonal one. And both of those are bipartite. And the fact that they were bipartite was crucial in the way that I went from dimo configurations to a definition of fluxes that I could coarse grain. So you would expect things to change a lot when you go to a non-bipartite lattice. And indeed, they do. So when we go to a non-bipartite lattice and ask what's conserved or whether there's anything that's conserved when we shuffle dimers around loops, it turns out that there is something simpler that's conserved. And that's just the parity of the number of dimers cutting a particular line. So for example, suppose you start with this configuration and then you pick out the dimers lying under this blue line. And again, I've selected a path on the lattice where the links alternately go over a dimer and over an empty bond. And that means that I can make a rearrangement of the dimers on this blue line, switching the dimers to the empty bonds and leaving the occupied bonds empty. And clearly, if I do that, then I'll change the number of dimers cutting this red dash line by 2. And so I leave the parity unchanged. And you can think about other examples. So here, I've chosen things differently. But again, if I move the dimers around, I'll leave the parity of the number of dimers crossing the dash line unchanged. So what this means is that the set of configurations is still separated on a tourist into disconnected sectors. But now the number of sectors is much smaller. So in the bipartite case, the sectors were labeled by these total fluxes, which had integer values ranging up to something of order the size of the cross-section of the system. Here, the sectors are labeled, but just by the parity of the number of dimers crossing a line in each of the directions encircling the tourists. So in d dimensions, we have 2 to the d sectors. Or if you took a surface or a hypersurface to embed the model on with some more general genus, then you'd have 2 to the genus disconnected sectors. So you can also ask the other questions that we've talked about in connection with dimer models for these non-byte-partite lattices. And in particular, you can ask about what the entropic interaction is between monomers. And what you find, well, firstly, dimer correlations on these non-byte-partite lattices are generically short-ranged rather than giving you these power law correlations. So they decay exponentially. And in turn, that means that you just have short-ranged entropic interactions between monomers provided you don't pick some model where the dimers are ordered into some crystalline state. So any questions on that? Right. So you have to do a proper calculation, of course, to get that. And in two dimensions, like a lot of other classical statistical mechanics problems in two dimensions, these dimer models are generically exactly solvable. And there are Grasman variable techniques for calculating the entropy and the correlation functions and so on. So you have to do a concrete calculation. In three dimensions, you can't do exact calculations. But of course, you can do Monte Carlo simulations. And you can measure the correlation functions in simulations. OK, so that's probably a good point to stop, because I want to move on next to an entirely new topic, which is about these magnetic materials that realize Coulomb phases. So maybe I can stop there. Thank you.