 Okay, thank you and good morning everyone. So what I want to do in today's lecture is switch from talking about classical systems to talking about quantum systems and specifically I'm going to be talking about what are called quantum dimer models and of course I'll explain what's meant by that idea. The overall theme will be that we can build on a lot of the ideas that I set out in connection with classical dimer models and we can see that they have consequences which are extremely interesting in these quantum problems. So this was my overall plan for the talk and the central ideas are these emergent degrees of freedom and fractionalized excitations and also some topological consequences of the emergent degrees of freedom in the sense of having these different sectors in both the height model and in dimer models. What I'm going to discuss today is spin liquids in the context of quantum dimer models and I originally advertised that I would also talk about the Ketayev model but I think it's highly unlikely that I'll have any time to start on that topic. So before we get into details I thought it would be useful to spend a little bit of time talking about what the idea of a spin liquid means and just starting at the most basic level it's pretty obvious that the term was invented to suggest an analogy between states of magnetic systems, spin systems and the familiar states of matter. So we talk about gases, liquids and solids and of course in a gas we think of the atoms or molecules as being pretty much uncorrelated. In a liquid we have strong correlations on the scale of the nearest neighbor separation between the particles but they're only short range and in a solid of course we have long range crystalline order. So some of these analogies carry over rather well if we think about paramagnet at high temperatures in either a quantum system or a classical system then we have uncorrelated spins which we can think of as being like the uncorrelated atoms in a gas and if we go to a state with broken symmetry and long range order in a magnet for instance the nail state then it's pretty clear that the periodic spin order is analogous to the crystalline order. But on the other hand the analogy between liquids and spin liquids is really not such a good one despite the fact that we use the term a lot and in fact I'd argue that spin liquids are much more interesting than ordinary liquids and the central points will be that they have topological order and they can have fractionalized excitations in a quantum version of the sort of fractionalization that I already introduced for you in the classical context. Okay so at least one of the points there is that although there are strong correlations there's no long range order so how should we define a quantum spin liquid is it enough to say that we're looking at the ground state of some spin Hamiltonian or some other model for our quantum spin liquid in which all of the symmetries of the Hamiltonian are intact because in an ordinary liquid that's more or less the state of matter that we're dealing with, translational symmetry is unbroken if you're not in the solid phase. So certainly this is a requirement that we want to make, it's a minimal requirement for recognizing a quantum spin liquid but it's useful to realize that it's not really a sufficient requirement, there are some things that meet this condition that you probably wouldn't want to think of as spin liquids and I've got probably the simplest example of that category sketched on the slide here and what I'm imagining is a bilayer Heisenberg antiferromagnet so I have Heisenberg antiferromagnet on a square lattice and I'm just taking two copies of it to give me this bilayer system and so I have two exchange couplings between nearest neighbor spins, within each layer I have a coupling J and between the layers I have a coupling J prime. So if the interlayer coupling is weak then it's pretty easy to imagine that in the ground state in each layer independently you'd have Nail order and the effect of J prime is just to orient the Nail order in the two layers. If we go to the opposite limit where the interlayer coupling is much stronger than the coupling within the layer it's also pretty easy to see what happens if you have the spins at the corresponding sites in the two layers strongly coupled by an antiferromagnetic interaction then what that will do is in the ground state put the pair of spins into a singlet but then once they're in a singlet state this coupling of the nearest neighbors within each layer won't have any effect. So then the system is in a singlet state a product of singlets over the pairs of sites from each layer and therefore it meets this requirement but you wouldn't want to call it a spin liquid because actually it's something that you could deform by making changes in the Hamiltonian into something that you would just call an ordinary insulator. So if you think of the spins as coming from electrons and if you think of energy levels at one of the double sites in the two layers then if you have a pair of energy levels which would evolve into a valence band and the conduction band when you couple the sites together then in a band insulator you'd have these valence band states filled and the conduction band states empty and in a sense this is a spin singlet but it's obviously a rather ordinary uncorrelated state. So we want a definition for spin liquids which at least separates out those boring possibilities and one way of ensuring that something more interesting happens is to say that in any case you're only going to consider systems that have half odd integer spin per unit cell and clearly this bilayer model since it has two spinner halves in each unit cell is excluded in that way. So any any questions about any of this introductory material? Okay so the next thing that I'd like to talk about is yes sorry what it's okay so it's quantum phase transition so it's in two space and one time dimension but it maps on to a classical transition in three space dimensions and that classical transition is the transition in the classical Heisenberg model between the paramagnetic and the ordered phase. So yeah well it depends on the type of spin liquid so I'm going to talk later on about models with power law and short range correlations. So the most simple spin liquids have exponential decay of correlations and a gap for excitations but they're different from trivial paramagnets in having topological order and it's how that topological order connects to the things I've been saying about diamond models that's one of the themes of this lecture. Okay so what I want to talk about next is a theorem which is really in mathematical physics but which I think tells us something quite interesting about what we should be looking for in spin liquids and the version I'm going to give to you is at least in the first instance for one dimensional systems and it's been extended to higher dimensions and I'll give you a kind of poor man's version of the extension to higher dimensions but of course I don't want to get too tied up in the technicalities but what we're thinking about is systems which have a chance of being spin liquids in the stronger sense that we just settled on that's to say they have half odd integer spin per unit cell and then more specifically I'm going to focus on systems that have conserved at least Z component to the spin although you could have full SU2 symmetry and I'm going to talk about things in the context of this XXZ Hamiltonian which some of you will know is anyway exactly solvable so you could say why am I focusing on this when we want to discuss much more general possibilities the point is just to keep the notation short and hopefully you'll be able to see that the approach where we're taking would generalize to systems that are not integrable in in a straightforward way so what the theorem tells you as I'll explain is that in systems like this the gap between the ground state and the next state the first excited state must vanish as we go to the thermodynamic limit and then afterwards we need to discuss various scenarios that can give us a vanishing gap but first of all let's see how at least in outline the proof goes so the idea is we imagine that we knew the exact ground state of the system and we generate a second state from it by applying this operator so it's the exponential of some coefficient times the SZ operator at each site in the chain and in a long chain the coefficient changes slowly as you go from one site to the next so the exponential of I times SZ is affecting a rotation of the spins about the Z axis at the corresponding site and what this is doing is twisting the state that was the exact ground state but it's twisting it in a way that's slow in space if the chain is long and then we're going to do two things I mean really the aim is to use this as a variational approximation for the first excited state and doing that will think about the energy difference between the ground state and the first excited state and then the idea is going to be that since this is a slow twist the energy difference should be small and we should be able to show that it's small but of course a variational approach would only make sense for the first excited state if this twisted state is orthogonal to the ground state because otherwise you get a difference in expectation values that was small for trivial reasons so first of all we want to show the state that I get with this twist is orthogonal to the ground state and we're going to do that by thinking about the effect of translations so let's take T to be the translation operator by one site along the chain and then if we think of the effect of T on this variational state then well we get this state by acting with this twist operator U on the ground state so we've got T times U and then we've got the ground state and we can say straight away that the ground state should be an eigenstate of the translation operator and the point is if the ground state weren't translation invariant then the ground state subs that have to be a ground state subspace with more than one state in it and in that case we'd have zero energy gap anyway between these two copies of the ground state which is the thing that we're trying to show so this must be true if the ground state is unique so we can insert we can insert in here T inverse times T and T operates on the ground state and gives us the ground state again so we've got T U T inverse and then we can show that for the system we're talking about this is minus one and if that's true then the ground state and this variational state must be orthogonal so the fact that this is minus one that's something rather more detailed which I'll sketch now but the crucial thing is the result so if we think about the U T inverse then U is this exponential and we can think of the action of T in the exponential so it's the exponential of I times 2 pi and then if we translate on the label on SZ inside the sum we have a sum over N and N over L and then the effect of doing the translation inside the sum is that what was SN becomes SN plus one and then if we've got a chain with periodic boundary conditions when I translate the last site I get back to the beginning of the chain so let me take this sum to go from one to L minus one and treat the last term separately so this is S of L plus one which then becomes S1 and then I want to get this back into something like what we started with for U and a correction term so this is so if I increase the N here to N plus one I'd have just U again so I can say this is U and then I have a correction term which subtracts off the one that I had to add in here so that's minus and then I have sum over N of one on L times SZ and from the last term I have e to the 2 pi I S1 Z now this is just S total Z and if SZ total is equal to zero then this exponential is just one and if SZ total is non-zero then we have a pair of degenerate states with the positive and negative values of SZ total as the eigenvalue that distinguishes them and in that case we'd have two degenerate states and therefore a gapless system so we can assume if we're looking to see whether there must be a gap that we only need to consider the case where SZ total is zero and so so this is one so what we're left with is that the effect of translating U is to give us back U multiplied by e to the 2 pi I S1 Z and this is minus one if if the spin is half integer which is the condition that we want to focus on in order to have a chance of an interesting spin liquid okay so we can generate a state with a slow twist which is orthogonal to the ground state and then we want to think about the expectation of the Hamiltonian in that state and get an estimate of the energy gap so what we want as the second step of this argument is to think about the expectation minus the ground state energy and we can write this in terms of U so psi we get by acting with U on the ground state and this bra we get with U dagger and sorry and we have the Hamiltonian in there and then the ground state energy we can get just by subtracting off the Hamiltonian so in order to understand this energy difference between the variational approximation for the excited state in the ground state we need to think about the effect that this twist has on the Hamiltonian and if we look at the three terms that I've got in the Hamiltonian there for the XXC chain then the last term with the coefficient delta is unaffected if we do rotations around their Z so the terms that we have to think about adjust the in plane couplings and if we think about the effect of the twist operator on a given site then well if we twist through some angle theta the raising operator on a particular site so this is like U dagger S plus U then it just in fact multiplies it by a phase e to the i theta times S plus so what you find is that the coupling terms here get multiplied by these phases which tell you the twist between two adjacent sites but this twist is slow so what you get is a cause of 2 pi over L from this term and then just unity from this term multiplying some sum over the length of the chain of S plus n S minus n plus 1 plus the conjugate term expectation in the ground state so now if we think about how things scale with n sorry with the length of the chain then the difference of this cosine from 1 goes like 1 over the length of the chain squared and in this sum there are of order L terms so all together combining the two factors we get something that goes like 1 over L so the the argument was firstly that this slowly twisted state is orthogonal to the ground state and then because it has a slow twist the difference in energy from the ground state energy is small and it goes to zero as the length of the chain is sent to infinity so in other words under this condition that we have spin a half in the unit cell and rotational symmetry we have a vanishing gap and the interesting thing then will be to think about the different scenarios that can give us a vanishing gap so does anyone have any questions about the arguments that I was using that yeah but you're doing something at every site I mean what what gives you the minus one is is obviously this and the fact that the spin is half integer I mean so so if you started out you might think that by doing a 2 pi twist as you go along the length of the chain you were going to generate something that was rather similar in other words had overlap with the ground state and what we find from this argument is that you'd have to do a 2 pi twist down the length of the chain to get something that had overlap with the ground state okay so then it turns out that there are several different scenarios that you can imagine which would give rise to a vanishing gap in the thermodynamic limit and two of them are familiar and from the perspective that we're taking today are the uninteresting ones that we want to exclude so one possibility would be that you had some kind of quantumizing model and this is actually what would apply in the XXC chain that I was talking about if delta was large and that you simply had long range order and broken symmetry with some order parameters so in a case like that you would have one broken symmetry state that had all of the spins up and another one with all of the spins down of course in reality dressed by some quantum fluctuations and then you could make states which are symmetric or anti-symmetric combinations of those two and those would be the ground state in the first excited state and the gap between them would go to zero as you took the thermodynamic limit and that's one way in which this theorem might be satisfied another possibility which is also familiar but not what we want to talk about today is what happens in fact at the model in the model I was talking about if delta is equal to one so that we go to the Heisenberg model and then you have gapless excitations so in the one-dimensional Heisenberg model you have a branch of spin-on excitations with energy or frequency varying as a function of wave vector and going to zero at one point in the Brian zone but suppose we exclude both of those possibilities which in a sense are rather familiar less exotic than a spin liquid then what this theorem tells us is that there must still be a degeneracy and the other way in which a degeneracy can arise is for topological reasons and I'll spell out in more detail how that works as we go on but at this stage the thing that's useful to have in mind is the idea that we had in the case of dimer models that you had different sectors to the configuration space and those are going to become different quantum states with zero or small energy gaps between them okay any any questions on that okay so that was trying to set the scene for the sorts of things that we're looking for in in a reasonably general way but now what I want to do is focus in on what are called quantum dimer models as a context for exploring these ideas in more detail sorry before I do that there's one more thing I should say so an objection to the presentation I've given so far is that it was strictly about one dimensional systems and really we want to talk about higher dimensional systems so there are some rather sophisticated extensions of the Liebschild's matter theorem to higher dimensional systems but you can make a kind of cut price version for for yourself by saying that we're not going to think about a strictly one dimensional system we'll think about a system consisting of M chains like the single one that I was discussing and then we say we have M sites in the unit cell of some effective one-dimensional model and provided I take this M to be odd I am still complying with the condition that I have half odd integer spin in every unit cell of the system and then the proof that I've given would go through but this sum over sites would give you not just a factor of L but also a factor of the width of the system so the bound on the gap would not be one over L but it would be M over L and then if you're not squeamish about the order in which you take the thermodynamic limit you can take L to infinity first and M to infinity afterwards and then you've arrived at the result but there are better ways of doing it than that but they're too technical for this lecture yes it's not really symmetry protected so this is topological order in the same sense as in the fractional quantum Hall effect well I'm most interested in the applications in higher dimensions actually I think what I mean there will become much more clear as I go on okay so what I want to get to now what are called quantum diamond models but I want a bit of lead in as motivation and so really the issue is how should we try and think about the possible states of quantum spin systems and the best way to think about them depends quite a lot on what the state actually is that you're attempting to describe so if you have a system that breaks symmetry and for example has nail order then the classical picture of nail order as a sequence of spins with alternating orientations is a very good place to start and this picture of course has an immediate translation into quantum language because we can use it to write down a wave function which is a product state over the sites and on alternate sites the spins up or down and then you can take that picture and dress it with quantum fluctuations and do spin wave theory and so on but if we're trying to describe a state that's not at all like this then maybe that's not a great starting point and what might be a better starting point well if you've just got two sites then with antiferromagnetic interactions we know that they form a singlet so perhaps it's better to think about a wave function in terms of what pairs of sites are doing and we can describe the fact that a particular pair of sites are in the singlet by putting this kind of object which looks like the diamonds I was talking about over that neighboring pair of sites so building on that and going back to the 70s Anderson suggested that maybe we should think about the ground state of a quantum spin system that has some interesting quantum disordered phase as being some sort of linear superposition of possible arrangements of singlets which are represented by the diamonds that I've drawn here now once you start down that track there are an enormous number of questions which you want to think about and which people have thought very hard about so this is just a list of what some of them are so for instance right at the beginning these different possible components in the wave function that I've drawn are not really orthogonal to each other if these diamonds represent singlets because if you form singlets between a pair of sites in one way it's not orthogonal to the other way of forming them so for example if I take a square like that and make two singlets here it's not completely orthogonal to forming the singlets in the other fashion okay so you have a basis which is not uh orthonormal uh it's not even clear whether it's a complete basis in the sector with total spin zero for the system uh if it is a complete basis then even the nail state had better be representable in terms of this and you can ask what you have to choose for the expansion coefficients in order to get the nail state and so on and so on um so that's an interesting direction but sometimes it's actually productive to cut off a lot of these questions and formulate in a clean way a problem that sidesteps them so the idea of quantum diamond models is to do exactly that uh what you decide to do is to study quantum mechanics on a Hilbert space that's uh got the kind of correlations that we were talking about in the classical context uh built in and uh in other words we're going to declare that each of these uh dimer configurations close-pack dimer configurations that we were talking about in the classical context yesterday is an orthogonal state in uh my Hilbert space and so the wave functions that I'll be dealing with are linear superpositions of these basis states with uh some expansion coefficients that I'm calling a of c here so um what we're doing is quantum mechanics on this space uh that has a microscopic number of states in it as I was explaining in the classical context and that also has uh these interesting correlations built in which again uh I've been talking about in the classical context okay so that's the idea and then what you need to do is pick some Hamiltonian which will give you uh dynamics uh in this Hilbert space and hopefully it doesn't matter exactly what choice you make but it better be something that gets you between the different states in Hilbert space and uh usually uh we want to pick a Hamiltonian that's uh local and uh so the simplest thing is to uh choose something which uh makes quantum transitions between different arrangements of the dimers so for instance if you're doing all this on the square lattice uh you might simply have uh a term in Hamiltonian that uh gives you transitions between two arrangements of parallel dimers and if you're doing it on a triangular lattice then you can have three uh corresponding terms that uh work on parallel dimers uh with the different possible orientations so that's the philosophy of what we're going to be doing and uh of course next we want to uh try and investigate the the consequences so obviously if I lose some of my audience here uh it's going to be a bit sad for the rest of the lecture so I should check does anyone have any questions even the most basic questions okay so so that's the idea and now we want to get more specific and uh think about a concrete Hamiltonian with the minimal ingredients to implement this idea and um this Hamiltonian is what's known as the Roxard-Kibbleson Hamiltonian and it's written down at the bottom of the slide now no doubt some of you have seen it many times but presumably a few of you are seeing it for the first time and this is standard notation and it's very compact and elegant uh but it does take a bit of unpacking so let me try and unpack it for the people who haven't seen it before so um firstly there's a sum over terms uh and then the terms are represented in these pictures um so what these pictures are telling you is what happens when this term in Hamiltonian acts on a particular plaquette in the lattice and the sum here is overall the plaquettes and uh if you were being careful about things you'd want to remember that these terms should be in tensor products with operators that are unit operators on all the other plaquettes in the lattice and uh let me try and explain what one of these terms does when it acts on a dimer configuration so um okay so you remember that the uh the wave functions in the system we're writing as super positions of these classical uh dimer configurations and uh so if I think about one term uh which I've labeled by c up there then I can get more explicit and say that uh in this term on a particular plaquette I have uh two dimers which are parallel like that and then over here there's all sorts of other stuffs happening which I'm not specifying and uh now I'll think about what happens if I act with that on this term in the Hamiltonian so uh in the notation I'm acting with this term so uh as I said this is really tensor product with the unit operator on other plaquettes so the stuff out here is unaffected and uh what this does is recognize that this plaquette is in this state and then replace it with a plaquette in that state so uh the effect of this term acting on that component in the wave function is to give us uh a different component in the wave function in which the dimers have been reoriented and uh to be sure that you understand how this notation works suppose we go back and consider the same term in the Hamiltonian but acting on a different component in the wave function say one that has the dimers on this plaquette arranged somehow differently so now this bit of the wave function is orthogonal to this and so when I take the scalar product here I simply get zero so uh this term in Hamiltonian acting on this bit of the wave function would just give me zero so um um does anyone have any questions about the notation uh so I I think the question is well the question is under these rotations can you overlap and I I guess you mean uh put differently can this Hamiltonian take you out of the space of allowed dimer coverings that I talked about classically and and the answer is no because um um you know each of these terms if if on a given plaquette I have two dimers horizontal then that means that what's going on in the rest of the dimer configuration is consistent with that and it can be replaced with two dimers vertical uh without having uh dimers overlapping on any sites okay so so that's the notation and then just to pick out the details of the Hamiltonian we have two terms that you can think of as a kinetic energy with coefficient minus t and a potential energy with coefficient v so the potential energy counts how many nearest neighbor pairs of dimers we have which are parallel either like this or like that as distinct from uh more complicated arrangements on plaquettes and uh if v is positive it penalizes those uh parallel arrangements of dimers and t is a kinetic energy in the sense that it hops us uh between uh different dimer arrangements just as uh kinetic energies in lattice models hop the particles around and uh what you usually do is uh take a negative coefficient for uh hopping so that the uh lowest energy state is something uniformly uh arranged across the configurations so it's natural to have a minus sign and t be positive here okay so so the idea is that we're going to do quantum mechanics on the space of dimer coverings and uh a sensible Hamiltonian is this rocks archival someone and uh it has two parameters in it but of course uh what matters is uh dimensionless combinations so uh the ratio of the parameters t and v is the dimensionless combination and what we once think about is the phases of this model as we change uh the ratio t over v and also as we go from one lattice to another because as i explained to you uh there are differences uh even at the classical level uh important differences between dimer models on uh bipartite or non-bipartite lattices now uh the beautiful thing that rocks are and kibbleson realize is that there's one point in this parameter space this one-dimensional parameter space where we can write down the ground state exactly and that's the one where uh these coefficients are both equal to each other so if we put the coefficient t and v equal to each other then uh we can uh factorize this Hamiltonian in the way that's written here and hopefully you can quickly convince yourself that if you multiply out all of the terms then you simply get back to the previous Hamiltonian with v set equal to t okay now uh why does that help to factorize the Hamiltonian like that well um as i'll explain in a moment the terms in this sum are projection operators and that means that we've written the Hamiltonian as a sum over projection operators with a coefficient t that as i said we want to take positive because uh that way the kinetic energy term has the conventional sign so if we have a Hamiltonian that's a sum of projection operators with positive coefficients then uh the uh eigenvalues of that Hamiltonian must either be something positive or zero so if we can find a zero energy eigenfunction of this Hamiltonian it must be a ground state and then the the statement is which uh i'll try and unpack in a moment that if we simply take a uniform superposition that's say with equal coefficients of all of the dimer arrangements within a given sector in the sense of flux sectors that i discuss yesterday then that will be uh an eigenstate with zero energy of the Hamiltonian at the Roxard-Kithlson point and and therefore uh must be a ground state so um let's think about why uh that should work and again we need to think about how um terms in the Hamiltonian act on uh states so um if we start with a term in the Hamiltonian which on a particular plaquette has uh two uh dimers parallel uh and horizontal then what this uh operator and Hamiltonian will do is uh multiply it by a minus one and what this operator and Hamiltonian will do is simply give us zero on the other hand if we had a configuration where the two dimers were uh parallel and vertical on this plaquette then uh this operator would give us zero and this other one would give us plus one so if both those configurations uh appear in the wave function with the same stuff outside uh and with equal weight for the two dimer orientations when i act with the combination of these terms i'll get something with a plus one and something with a minus one uh and so i'll get complete cancellation uh and so uh i'll uh get zero as the result of the operation of this term on uh those two components in the wave function and then if i think about other components well if uh the dimer configuration on this plaquette were a more complicated one with the dimers not parallel but going off in some other direction then both terms uh here will give me zero anyway so um a uniform superposition will uh give me uh an eigenstate with uh eigenvalue zero uh and that must be one of the ground states of the Hamiltonian and it's reasonable to restrict ourselves to uh a superposition of configurations within uh a given sector of the uh dimer arrangements uh because we can expect that the effect of the Hamiltonian will be to move us uh between uh all the different configurations within the sector but uh because as we said uh yesterday local moves can't uh local moves of the dimers can't get you from one sector to another uh we expect the Hamiltonian to be block diagonal in the different sectors um and now because all of the dimer models that we talked about yesterday uh have uh a number of different sectors uh we have now uh a degeneracy in these ground states which is exactly the degeneracy uh that um we expect from the uh uh Leibschild's matter theorem uh but now arising not because of some uh global broken symmetry nor because of uh some branch of gapless excitations uh instead it's coming about for this topological reason um any questions on uh things so far right so so the question is uh maybe uh how do we know whether this ground state breaks symmetry or not um well uh because the uh state that we're looking at is a uniform amplitude uh superposition of states from the uh classical dimer configurations the question of whether it breaks symmetry is the same as the question that we were investigating in the classical context of whether the dimers have some kind of crystalline order or whether they're fluctuating so for example in a two-dimensional system that would be the question of whether the cosine term is relevant or not and uh we at least uh saw that in some circumstances is not relevant and so uh the dimer configurations uh um uh a liquid like uh okay uh so i think that would be a good point to take five minutes break although the uh general spirit of the discussion will continue in exactly the same way after the break so let's take a break now hello again everyone um so we got to the point of saying that uh when we study our quantum dimer model Hamiltonian at this special point in the parameter space we have a very simple situation where we can connect the ground state wave function to the dimer configurations that we thought about classically and um so the next thing I want to do is uh see what follows from the quantum problem uh from this fact about the ground state wave function and then of course it's natural to ask what happens if we're away from this point and then uh in the final part of the lecture I'll talk about excitations in quantum dimer models okay so what do we learn from this idea that we can uh take the uh ground state wave function to be an equal amplitude superposition of uh dimer coverings uh well the the first one is the one that I stress before the break that the uh ground state has a degeneracy and the degeneracy is simply the number of distinct uh dimer model sectors so in the simplest case of a non-bipartite lattice I explained to you uh that we have a z2 uh distinction between uh different sectors for uh each uh direction uh around uh a torus for a system with periodic boundary conditions so that means that uh in uh two dimensions we'd have uh two sorry we'd have uh two squared two to the power d uh state so uh four ground states and in in three dimensions eight uh and for the bipartite lattice systems we have much higher uh degeneracies um the other consequence is that uh at least some correlators in uh these quantum ground states we can get just from a classical average so if we're working out some uh quantum correlator what we imagine doing is taking some combination of operators and working out its expectation value uh in this ground state so the correlators that we can work out in a simple way are ones that are diagonal in the uh dimer configurations and uh those are just uh averages over the uh classical uh configurations in the sector so for example if we talk about correlators between dimers maybe the uh probability to have a dimer in one orientation on one bond and uh at the same time uh a dimer in a given orientation on another bond uh those are the sorts of correlators that we were working out in Coulomb phases when we were discussing uh flux correlations and uh on uh bipartite lattices uh we saw that you have these characteristic uh power law correlations on the other hand uh if we go to non-bipartite lattices I told you that the results from uh calculations and simulations are that uh dimers uh have just exponentially decaying uh correlations um and so uh the uh dimer quantum liquid on non-bipartite lattices really is highly disordered um we can also consider extending the model uh using the idea of monomers that we talked about in the classical context and this really is an extension because we're adding some new states to the Hilbert space but uh we could for example choose to introduce a pair of monomers replacing one of the dimers in the system and um we might want to give the monomers their own dynamics but in the very simplest version we could take the monomers to be static and have the dynamics of the dimers given uh by the Roxard-Kivelsson Hamiltonian and then what we find is that we have a zero energy state a ground state wherever we put the pair of monomers so uh I talked about the possibility of having uh deconfined excitations and I talked about entropic potential in classical problems as a function of separation between a pair of monomers and here we've got the quantum analog and uh if the energy is completely independent of the separation between the monomers then clearly there's no force whatsoever between the monomers uh and we have uh deconfined excitations so to see why that might be interesting it's worth revisiting the pictures that we use to motivate uh this dimer construction so if the motivation even if we didn't follow it very precisely was that this is equivalent to uh a singlet state then in a quantum system if we go in and excite the singlet we'd uh make a triplet on the bond and if we think in terms of total spin quantum numbers then obviously going from a singlet to a triplet is changing the total spin of the system by one and actually that's always going to be the smallest change that we can make in the total spin of the system because of course total spin uh varies in steps of one and that's true whether or not the spins at that is sites are integer or half integer so in some situations when you make a spin one excitation uh that is an elementary excitation of the system that then propagates so if you're in a nail state and you make a spin one excitation that's a spin wave which propagates uh as a single particle um but uh if we uh carry over this idea of of monomers what we're saying is that uh we've um made an excitation breaking up this uh singlet and produced two monomers and if uh the quantum mechanics of the system allows these to separate then uh what started out when they went next to each other as a spin one excitation has broken up into uh two spin a half excitations so we've got fractionalization and uh deconfined uh excitations okay so uh what i want to do next is talk about phase diagrams uh as we uh vary the one parameter in this model uh which is to say uh the ratio of v over t and i want to uh talk about different limits and see what we can understand in each of the limits and then put everything together uh into a phase diagram uh adding in uh information that's come from simulations so um at the rocks archivism point where we can understand the ground state in a precise way we had v equals t and if we go to values of v that uh larger than t then we can write the Hamiltonian as the rocks archivism Hamiltonian plus uh something with a positive coefficient times uh these potential terms so um the uh ground state of the rocks archivism uh Hamiltonian had energy zero and uh since this additional term comes uh well it's it's diagonal in the diamond configurations and it comes with a positive coefficient if we can find ground states which uh ground states of the rocks archivism part of the Hamiltonian which are annihilated by these additional positive terms then there'll be ground states when v is bigger than t and uh if we arrange all of the dimers in the system in a staggered fashion in the way that i've drawn here then we have no plaquettes with parallel dimers and so these configurations are indeed annihilated by the uh extra terms uh but they're also since their uh states from a given sector uh their ground states of the rocks archivism Hamiltonian and so their zero energy eigen states and uh in terms of the fluxes that we talked about in the classical dimer language this is uh a maximum flux state it's carrying uh the largest possible flux uh in the uh vertical direction so did that make sense okay so that's actually told us the answer for one whole half of parameter space uh on one side of the rocks archivism point and then uh well on the other side it's not quite so simple but if we go to the extreme limit where the coefficient uh of the potential term is large and negative uh to find ground states what we should do is uh choose arrangements which uh favor these terms and since these terms pick out parallel pairs of dimers on plaquettes we should choose uh configurations like this um and uh you see here that uh this is uh a state which is uh broken uh a symmetry and uh in fact the size of the unit cell has been uh doubled in the vertical direction um and you can also imagine other more elaborate symmetry breaking states so for example you could double the unit cell size in both directions and have uh something where you had uh around uh a fraction of the plaquettes uh resonance between two dimers in the horizontal direction and two dimers in the vertical direction uh and the uh and then uh weaker correlations on the the other plaquettes okay so um in one half of the phase diagram we know what happens we know what happens at the rocks archivism point and we at least know uh in the limit when uh v goes to minus infinity uh what the behavior is uh but to get the uh full phase diagram uh you need to do uh numerics and um here the advantage of quantum dimer models is that they can be simulated using quantum Monte Carlo uh without the uh so-called sign problem which makes the simulation of most frustrated spin systems uh very difficult uh at low temperatures and for large system sizes so when you do that uh for the square lattice what you find is well as as we said when v is larger than t uh you have this staggered arrangement of dimers as the uh ground state configuration when v goes off to minus infinity you have a columnar state and uh the belief now is that this columnar state goes all the way up to the uh rocks archivism point so that's actually rather disappointing because um the rocks archivism point was very interesting in having this liquid like arrangement of dimers and having uh de-confined monomers but if you go into the ordered states either side then uh the um monomers in fact will be confined you'll have a linear potential as a function of separation for reasons that I set out yesterday and so uh this interesting uh spin liquid like uh point is uh something that's that's unstable so uh historically at that point people got disappointed and thought about other things uh until quite a few years later uh it was realized by uh Roderick Merzner and Shivaji Sundi the things are quite different on the triangular lattice and uh on the triangular lattice it's true that uh for v bigger than t you have this staggered phase and if you take v off to minus infinity you go back to the columnar phase uh but the arcade point is just the boundary of uh a finite region of uh liquid like dimer configurations and because the triangular lattice is one of these non-bipartite lattices then uh the um conserved fluxes are these z2 fluxes and so we can think of this as a z2 uh and then the jargon is resonating valence bond uh state so uh these singlet states are somewhat sometimes called uh valence bonds and resonating means that we have a quantum superposition of uh many uh arrangements of the uh resonating valence bonds yes uh so uh an intermediate piquette phase is a possibility and the belief uh first when people did uh Monte Carlo simulations was that uh there was a columnar phase and then a piquette phase here uh and then the staggered phase and uh I I think the the current uh understanding is that the columnar phase goes right up to the arcade point and that uh the simulations which suggested uh a piquette phase here were uh on system sizes that were too small but I haven't followed the details of of that work I think it's pretty hard to do it in a controlled way I mean I agree that uh it's it's the obvious thing to to do and I can't tell you straight off exactly what the difficulties are but I think it it doesn't give you a magic cure okay so those are uh examples on two two dimensional lattices of course we can go to three dimensions and uh for example we can stick with bipartite lattices and uh go from a two dimensional lattice that has coordination number four to a three dimensional lattice with coordination number four and one of them is the diamond lattice and uh what we find in that case is that uh again as on the triangular lattice the uh rocks archival some point is the endpoint of a whole liquid phase uh but now because we're on the diamond lattice we have uh fluxes that are conserved uh not just modulo two uh so instead we have uh this u1 phase um now just presenting these lattices uh and the different phase diagrams it seems uh perhaps a collection of arbitrary facts but uh actually there's some context which helps you rationalize those results and the context comes from uh the fact that uh people starting out in high energy physics had studied uh gauge theories on the lattice in a lot of detail and there's uh ideas about the phases which are possible in different types of of gauge theories and in in different dimensions um and uh so the the summary is that uh on bipartite lattices uh in two dimensions uh the rocks archival some point is uh an exceptional point and away from it uh we have a phase in which monomers are confined uh but if we go to three dimensions then we can have uh a finite region of spin liquid phase and on non-bipartite lattices where we have these uh z2 gauge theories uh we can have uh deconfinement in either two or three dimensions and that fits with what's known about gauge theories so uh z2 gauge theories have deconfined phases in both two and three dimensions but uh u1 phases uh you have to go to uh three space uh dimensions to have a deconfined phase so uh i mean that's calling in uh a whole body of of understanding from uh that is gauge theory which i don't have time to to develop further but it gives you some rationalization of these results okay any any questions so the the story so far is that uh we've set out to study these quantum dimer models and we've uh understood uh something about their phase diagrams and in particular we're interested in these liquid light phases um but so far we've been focusing on ground states so what we'd like to do next and and this will be the remainder of the lecture uh is to think about what we can understand uh about excitations uh in these models and you can talk about excitations in in different categories uh so i have actually been talking about one type of excitation here when i talked about the possibility of breaking up a dimer and producing monomer pairs and uh then uh the story was that if uh they're deconfined then uh a spin one excitation can fractionalize into uh two separate spin uh spin half pairs and uh these uh in the jug and uh called spinons and um if we ask about their energy since we have to extend the Hilbert space of the dimer model to even discuss this possibility uh the energy is a free parameter we uh uh liberty to decide what the uh energy of a state like this is compared to that one but since the change that we're making to the system is local it's natural to think that there's uh an energy gap to to create these but but in any case there's a limit to what we can say beyond the fact that they may be deconfined uh within the dimer model language uh so the more interesting category is the uh class of excitations that are within the Hilbert space of the original dimer problem in other words the uh higher lying levels above the ground state of the uh of the um quantum dimer model um so i mean in general these are wave functions which are superpositions of the uh different uh uh classical dimer coverings and we we specify them by uh saying what the expansion coefficients are and if we're studying these excitations at least to begin with exactly at the uh rogts archivism point then uh the excitations have better be orthogonal to the ground state and the ground state uh was uh an equal amplitude superposition of all these classical configurations so uh the uh expansion coefficients all had the same sign which we can take to be say positive and uh if the excited states are going to be um orthogonal to the ground state then some of these expansion coefficients have better be uh negative um and that's going to be uh a guiding principle in writing down what will actually be variational approximations variational pictures uh for the wave functions of excitations and it's going to turn out that we can think of two types of excitation which uh go under the name uh vizon and uh emergent photon so um the vizon will be a a particle like excitation in the sense that we can write down a variational wave function which has something happening just at a point and uh because we're uh affecting uh the wave function of the system just in the vicinity of the point uh it's natural that it should cost a finite energy uh that it should be uh a gap to excitation the the other type of excitation this emergent photon uh yes uh i mean uh i mean a point in real space uh and i i think when i get on to the next slide that will become much more clear uh there's a cartoon which uh explains what i mean uh but before i go on to the next slide i just want to say something about this idea of an emergent photon so the point i want to make is that that's something whose existence you might already have been beginning to expect on the basis of the things that have been discussed uh yesterday and and today so for these dimer systems on bipartite lattices um i'd say yesterday at the classical level that uh in the coarse grain theory we can uh describe configurations in terms of uh an emergent gauge field uh which uh was just like the uh gauge field that you have in magnetism um and what we've done here with these uh quantum dimer models is uh quantized that uh situation and uh so in other words we're moving towards uh some quantum theory that uh at least has uh some similarity to electromagnetism and of course uh in electromagnetism we know that we have uh photons uh so we ought to be looking for photons here at the same time it's pretty remarkable if you get photons out of uh this theory because we're starting with something terribly discrete and uh photons of course are gapless excitations it's rather uh rather special to be able to get gapless excitations out of these discrete arrangements of dimers and and that's something i'll talk about in in detail in a moment okay but but first of all let me be more specific about the idea of uh a vison excitation so the the point was we wanted to have excited states with uh some variation in the sign of the uh contribution for different super positions so if we're going to write down uh variational wave function then uh we're basically going to specify some rule for deciding the signs of different configurations and um what you do is uh think of a trial wave function where you combine uh the dimer configurations uh with equal magnitude but uh a sign that varies uh according to accounting and the counting is just the number of dimers that cross a line that goes from uh the place where you've put the excitation out to either the edge of the system if you have open boundaries or if you have closed boundaries you can only introduce these excitations in pairs and this uh string would have to end on the uh other the other vison and um what we want to do now to understand what the uh wave function for the excitation is like is uh get a bit of a feeling for how the sign uh of in the super position changes when we go from one configuration to another so going from one configuration to another basically means rearranging dimers on some loop in the system and uh then we've got two types of loop that we should think about one would be uh a loop that uh cuts this string uh an even number of times and the other would be a loop that uh encircles the excitation itself uh and therefore cuts the string a not number of times and obviously in the simplest case just cuts it once so um if we shuffle dimers around some loop that cuts the string once or an odd number of times then clearly uh we uh change the number of dimers crossing the line uh from one parity to the opposite parity and we change the sign of the contribution in uh this sum but if we move dimers around a loop that cuts the string twice or an even number of times then uh one of the properties of uh these z2 uh dimer uh models that I talked about yesterday is that uh the parity of n is conserved and so uh the uh contribution to this sum uh has a unchanged sign so in other words the presence of these minus signs is only relevant close to the uh place where the uh excitation is located and uh because we're just disturbing the wave function of the system locally uh we can imagine that it'll cost a finite energy there'll be a gap for this kind of excitation so is that that okay okay so what I want to do now really is the the final topic is to uh try and describe these emergent photons and um again the way that we think about them will be in terms of uh a trial wave function and there's a prescription for writing the trial wave function which is given on on this slide here and I'll try and explain it in a reasonable amount of detail to give you a intuitive picture so the the first thing that we need as a tool is to have an operator which tells us whether uh there's a dimer present in a certain orientation on the bond leaving a particular site so um if the r there is the site and if the labeled tau is equivalent to say x then this operator sigma and tau of r will be one if we have a dimer like that and uh zero if we have a dimer in the other orientation and uh what we're going to do is take a ground state wave function with its superposition of many um different dimer configurations and uh we multiply it by the Fourier transform of this dimer counting operator and uh that's going to be our our trial wave function for for the photon so what we want to do is understand what the effect is going to be of multiplying the ground state wave function by a factor like this so uh let's uh think of uh things in a bit more detail so let me suppose that the wave vector q which appears uh as a parameter in the trial wave function is in that direction and let me suppose that this quantity tau uh is in the x direction and um I think of the action of this um Fourier transform uh on uh different contributions to the ground state so um in particular because the phase factor e to the i q r will change with position I want to think about the wavelength uh two pi over q and if I take some reasonably ordinary dimer configuration then there'll be dimers in it in this preferred orientation all over the place and when this uh factor sigma of q acts on the wave function it'll pick out this dimer and give you a contribution with one phase it'll pick out this dimer and give you a contribution with another phase and so on and so forth and all those different contributions added up with random phases will more or less cancel each other so lots of components in the ground state wave function will be killed off when we multiply by sigma of q um but there'll be some special ones which have waves in the dimer orientation which maybe uh at this point have lots of dimers in the preferred orientation and then one wavelength away have lots of dimers in the preferred orientation but in between have dimers in different orientations and then when I act on a configuration like that with uh sigma of q uh things will add up coherently so what uh this operator does is extract for me from all the components in the ground state wave function the ones that have waves in the dimer orientation uh with uh the wave vector that I've selected okay is that that's yeah right so that gives us a trial wave function and now what we'd like to do is uh work out the energy of the state and um this will be uh a variational calculation of the uh energy of the photon so it'll be an upper bound on the energy of photon although actually it turns out to be a good upper bound so in the usual way um we want to take the expectation of the Hamiltonian uh relative to the ground state energy in the state that we've constructed here and uh since this state won't be normalized because we killed off lots of components uh we'll need to divide by the normalization so uh we need to think about this ratio um and uh one thing that's quite helpful is that uh what appears in the uh numerator uh can be written as a double commutator so let me just uh point out how that works uh so uh it's easiest to go backwards so in the double commutator we have sigma and then Hamiltonian commutator with sigma and so if I expand that out I've got sigma h sigma minus sigma h minus and if I combine terms then I've got sigma h sigma from here and uh from there uh with uh plus signs both times and uh then I've got uh minus um sigma squared h minus h sigma squared and now if I think of putting that back inside the expectation then the terms with h will be next to the ground state and so they'll generate me my e zero and the terms with sigma h sigma will uh be the expectation in the trial state that I need and there's this factor of two but it's been included uh in the denominator here okay so um to get somewhere we need to have an estimate for the uh numerator and for the denominator and uh we can get both of those and in fact we can get them using uh things that we uh or partly using things that uh we understood about the classical dimer configurations uh yes um okay so that's uh a summary of of the task to to understand these two functions and if we think uh first of all about the numerator um what we're really trying to do is to discover whether there could possibly be gapless modes which would then be candidates for photons uh and uh if we uh are looking for a gapless mode then we're looking for a wave vector where the numerator vanishes uh and the denominator doesn't um so the uh numerator as we've just said can be written in terms of this double commutator and so it would vanish if uh the inside of the two commutators vanished and uh so what we want to do is ask ourselves whether this can ever happen um and uh if we act on a state with the Hamiltonian first and then with uh sigma it will uh switch the orientation of a pair of dimers uh and uh having switched the orientation then uh sigma will will give us zero uh if the new orientation is in a different direction to the direction tau so uh in order that these commute what we need is that uh the operator sigma should give us zero and uh sigma is uh counting uh a phase factor for each dimer that's in uh a given orientation so uh what we need to get zero is uh a cancellation between the phase factor uh associated with this dimer and the phase factor associated with the dimer above uh so what we need is that uh e to the i k r1 plus e to the i uh sorry q r1 plus e to the i q r2 should be equal to zero and um r1 and r2 are the coordinates of these two sites uh where the dimers are located and uh that will be true since uh this separation is a lattice spacing if q is uh at the uh zone corner um and for that to work with uh all orientations of of dimers uh you need uh all components of q uh to be pi or um you can have an additional component uh to the wave vector that's uh in the direction parallel to the preferred orientation tau uh because uh the separation to these uh dimers is in the uh transverse uh coordinate so um what we've seen is that there'll be lines uh in um reciprocal space where this uh numerator uh vanishes and uh we'd have gapless modes uh so long as the uh denominator is non-zero so what we want to do is think about the denominator and that turns out to be more or less the uh flux correlator so uh this is the correlator between uh two dimer counting operators and uh it's written in Fourier space so the uh dimers in the classical discussion uh translate into fluxes and uh i explained with a detailed calculation yesterday that the correlator between the fluxes uh has this form that follows uh from the flux being uh divergence free uh there's just one wrinkle which is that here the singular behavior is uh near the origin um but that gets translated to uh singular behavior near the zone corner for a rather simple reason uh which is to do with how you map fluxes uh from uh dimer configuration so we're talking about bipartite lattices and so if this is on sub lattice A then this will also be sub lattice A and this will be sub lattice B and we had an orientation convention for the bonds from A to B so the orientations are like that so if we ask about uh dimers being present on neighboring bonds then that translates to an opposite sign for the flux and uh because of that minus sign uh these correlations uh near the origin in terms of fluxes get translated to uh correlations near the zone corner uh in terms of of dimers um but the upshot is that uh the the numerator vanishes in uh some directions but is finite in others uh so if uh k is parallel to this uh direction tour then uh s of q vanishes but if k is perpendicular to direction tour then uh the second term is zero and s is just unity uh so if we put the numerator and denominator together then um approaching uh the zone corner from uh directions which uh not a long tour uh what you find is that the uh denominator remains finite and the numerator goes to zero and in fact it goes quadratically to zero so uh what we found is that we've got gapless excitations in this quantum dimer model on uh a bipartite lattice uh and we were using information from uh classical uh dimer configuration so it was information that applies uh at the rocks archival some point where we know the ground state wave function exactly uh and of course uh dispersion relation which is quadratic in wave vector is not like the real photons that we're familiar with uh but it turns out that if you go away from this special point which you can do whilst remaining in the quantum disordered phase if you're in three dimensions for example on the diamond lattice then uh this quadratic dispersion is converted into a linear dispersion so i've got one more layer to add to this uh which takes uh two or three slides um are there any questions on this rather detailed calculation that i've tried to present here yes yes yes um the point really is that the uh rk point is uh on a phase boundary between the quantum disordered phase the liquid phase and uh an ordinary ordered phase and um it's quite common to have soft modes at phase transitions and so on so it's that sort of reason that gives you an enormously soft dispersion and uh once we go into the bulk of the phase then uh yeah we have an emergent uh larynx symmetry at long distances okay so the the last points that i want to make are to do with uh using um a continuum field theory uh to um summarize these results and uh to give some justification for this this claim and and to rationalize what's happening so um the the the first point is that we have these modes with uh a quadratic dispersion uh at the rk point and um we can ask ourselves can we imagine writing down uh a continuum theory which would capture that and um the uh action in imaginary time which you're led to write down is is this one that i i've got here and and let me uh try and explain why that makes sense so uh first of all we we have something uh with a double derivative in time uh which which is reasonably natural and then in in frequency that will convert into uh omega squared but if um we have a dispersion omega going like k squared then this omega squared term has to go along with a k to the fourth and so uh going back to imaginary time and space uh k to the fourth means we want uh four gradients so perhaps uh del squared h uh all squared so if we if we do have an action like this then uh correlators in wave vector and frequency have have this form and going from uh imaginary time to real time we'll get poles uh where omega is proportional to k squared um we can also use the information that we have that um the equal time correlations that's to say the ground state expectation is given by classical dimo model properties so to get equal time correlations we integrate over frequency and if we integrate this which is a Lorentzian in frequency then we'll end up with uh dependence on wave vector that goes like one over k squared and um that's exactly the uh form for fluctuations that we had for the height model and uh so it's what we expect since uh the uh physics of the ground state wave function at the rk point was uh supposed to be uh the the physics of uh the classical dimo model coverings which are represented by the height model okay so so the story is that this makes sense as an effective action that will capture uh the uh results of the variational calculation and the other facts that we know about the uh ground state wave function at the rk point but now if we look at it as an effective action it seems pretty fine tuned because we've got this uh del squared all squared uh and there's no symmetry reason that would prevent us having uh a grad h term as well so a reasonable guess is that if we're not necessarily at the rk point then the form for the action should have the two terms that we had before but also uh something with uh a lower gradient grad h uh squared instead of del squared h or squared and uh then uh because as i i talked about in detail in the language of classical height models uh this uh h field uh microscopically is discrete we can add in a cosine term and and now uh because the roksak give us some point was the place where this term was uh excluded we can expect that its coefficient should go like uh one minus uh v over t uh so now we can ask ourselves what happens within the uh approach of this effective action if we go away from the roksak give us some point and in other words turn on this grad h or squared term and uh depending on which direction we go away from the uh rk point the coefficient will either be negative or positive so if we uh make v larger than t this is a negative coefficient and uh the ground state will be something which has the maximum possible gradient to h uh and that's precisely uh in height model language uh the uh staggered configuration of dimers uh that i said must be the ground state of the quantum dimer model when uh v was bigger than t um if we go in the other direction then uh we have this gradient term with a positive coefficient and uh now we're thinking about a dimer model in uh at the moment two space and one time uh dimension and that means that this cosine term in uh two plus one dimensions is always relevant and and that pins the dimer configuration since some kind of crystalline phase uh so uh that's the columnar phase that we had on the other side of the uh rk point so we have an understanding of uh why we have uh just an isolated uh u1 uh critical point uh at the rk point between two ordered phases in two plus one dimensions okay does that sound at least halfway plausible any questions yeah um well i think there are there are difficulties there with the continuum limit and so i i think you really want to stay on the lattice for the c2 theories yeah he uh so that was a classical spin Hamiltonian that took me to the height model but also from dimer models i can go direct to a height model so i can think about the height model just in relation to the dimer models what the mapping between height models and uh yeah it is uh though i see that i'm uh already overrunning my time uh so uh i think i won't do it now uh so there's really just one more point to make uh and and that is what happens if we go up one dimension um and uh really everything falls into place in the way you would want uh so um we want the equivalent of this but in three space dimensions so we already had the idea that we should write uh our effective theories in terms of uh a divergence free field which we can represent with uh a vector potential so we should have an effective action in terms of a vector potential and um so we have something with uh four gradients here and something with two gradients and uh two time derivatives here and um again we can expect the coefficient to the two gradient term uh to vanish at the rocks archivism point and uh if uh this uh two gradient term has a negative coefficient and again we favor a state with maximum possible flux through the system but now if we're in three plus one dimensions when this coefficient is uh positive then the higher gradient term is irrelevant and uh the time derivative of the vector potential if you're in the Coulomb gauge it's just the electric field and uh this of course the curl of the vector potential is just the magnetic field so what we've wound up with is the familiar uh electromagnetism uh of course an effective electromagnetism where the microscopic degrees of freedom are the dimers in the uh dimer model so I think it really is remarkable that you can start with something that's uh so discrete and then at long distances uh get a theory uh which is the same as the one that we would have for continuum fields in in electromagnetism and of course with this theory uh we know that there should be uh an emergent photon uh excitation okay so this is really a summary uh of what I've tried to say today uh so I've been talking about quantum dimer models and with this idea of studying quantum mechanics on a Hilbert space that's provided to us by uh the classical dimer configurations which has all the constraints that we were discussing in the first two lectures we get uh a lot of interesting new physics and in particular the uh classical uh distinct sectors of the height models and the dimer models go over into uh topological order in the quantum system and uh we can have uh excitations uh that are de-confined uh in the uh z2 models in both uh two and three space dimensions and in the u1 model uh in uh three space dimensions okay uh thank you very much