 Okay, so the next lecture will be by Professor Vishwanath from Harvard University and he will start his lectures on dualities and topological phases of matter. Yeah, so can you all hear me? Yes, so yeah, so I should thank the organizers. So not only did they invite me, they also put up with all my changes of plan and they got me here from the airport. So if you're wondering why I look like I just got off a flight, I did just get off a flight. So yeah, so anyway, so I'll be talking about dualities that are very much, very popular these days and dualities is a technique. So I'd like to not talk solely about a technique, I'd like to talk about some physics application as well. So I'll talk a little bit about topological phases of matter and how we can use dualities to sort of characterize them and look at phase transitions between different topological phases and so on. So that's the title of this sort of lectures. So I should confess that I'm not a string theorist, I'm not even a high energy theorist, I'm actually a condensed matter physicist. So everything I tell you will be kind of informed from the condensed matter point of view. But one of the amazing things I feel is that a lot of the things, problems that we have been interested in ultimately seem to connect to things that people in high energy physics are very interested in, there's like this real unity of ideas, but I think it's also good to see the same set of physics ideas from different viewpoints. And so one of the differences in my set of lectures, we'll always talk about systems that have a cutoff, there's always a short distance cutoff. So when you do integrals, you'll find that these integrals often converge. So that may be quite shocking for some of you that you may do a calculation and it actually gives you a completely sensible looking answer without any need for renormalization or anything like that. But those same ideas come back in a different form and we'll try to see how the kind of theories that I'll talk about in some limits can be described by field theories. And you'll see that that's a very powerful technique, but it's not the only technique. There are other techniques as well that we'll use to characterize phases of matter and phase transitions and sometimes there'll be quantum field theories. Sometimes we'll look at exact solutions. We'll look at a bunch of different things and we'll see exactly how field theories sometimes emerge. Okay, so that's sort of the philosophy of what I wanna talk about. But in terms of topics, I want to emphasize that the overarching goal or the organizing principle is to really think about different phases of matter. So you have many particles or many degrees of freedom and we're interested in quantum phases. So I'll be thinking about zero temperature and we're thinking of large numbers of particles. So these are some degrees of freedom in my system. It could be the number of particles that are interacting or spins. We'll often look at spin models to make them very simple. But we're gonna take the limit of a very large number of constituents. Very much like in quantum field theory, you actually have an infinite number of degrees of freedom. Because you say that you have a continuum. At every point in the continuum, there's a degree of freedom. And it turns out that when you take these sort of limits, you get new properties. There are new properties that you cannot quite associate with a small number of particles, one or two particles. This is really a property of the collective system. We call that emergence. And one of the things that really emerges in the limit of many degrees of freedom is that you get phases of matter. You get situations where there are certain parameters and the properties of your system change, but they only change a little bit. They change in some quantitative fashion. But then there are phase transitions where we all know about examples of phase transitions, at least classical phase transitions. Where the properties change in a singular fashion and then you get a new phase. And these differ from one another by having not just being quantitatively different, but actually being qualitatively different. So phases have qualitatively different properties. There's no smooth way of going from one to the other. And you need a phase transition to intervene, a phase transition where singularities occur if you want to go from one phase to another. And we are still trying to figure out what is the set of qualitatively different properties. So this is actually very easy to tell. If someone tells you that this is a property that cannot be smoothly changed, you immediately recognize it as such. An example is symmetry breaking. You can either have a symmetry or it can be spontaneously broken. Those are two qualitatively different states for a system to be in. And that is one way to distinguish phases, an absolute way to distinguish between a pair of phases. So symmetry breaking is one, and it's kind of obvious. You either have a symmetry or you don't. If you change parameters a little bit, if you want to go from one to the other, you have to have a phase transition. And many of the phase transitions we know are of that kind. So if you think about a magnet, the spins align in a certain direction. They break the rotation symmetry. And that happens at a certain temperature, curie temperature. And that's an example of being able to distinguish phases in terms of symmetry. And in fact, for a long time, it was thought that was the only way that sort of exhausts the set of qualitatively different properties. So if you had a symmetry distinction, those were two different phases. If you had no symmetry distinction, those are actually the same phase. They may look different. For example, a liquid and a gas seem very different. But if you think about it, they really do not have any qualitative property that's different. Liquid and a gas have different densities, but that's a quantitative aspect. And in fact, if you think about the phase diagram of a fluid, usually there's a phase transition to go between a liquid and a gas. But if you change the pressure and temperature sufficiently, you can actually avoid the phase transition. You can take your system on a path that never has a phase transition, converts from a liquid to a gas. Because there's really no qualitative distinction. You cannot do the same for going from a liquid to a crystal. A crystal really has a different symmetry from a liquid. There's no way to avoid the phase transition going from liquid water to ice. You have to encounter a phase transition and that's really because they have qualitatively different properties. Okay, so people thought for a long time it was really symmetry. It was really just a symmetry distinction. And there was a theory to account for that, the Landau's theory, where you could sort of capture all the different ways in which symmetry could be broken. And it was thought that that is really all there is to distinguishing phases of matter. But more recently, we have realized that there are other ways in which phases can be different. For example, you can have topological distinctions. And you can have the interplay between symmetry and topology. So you, at one extreme, you have no symmetry. And you still have distinctions between phases. We'll give some examples of that. And sometimes you have to combine the topological properties and the symmetry properties together. And that gives you more ways of dividing up the states of matter. Okay, so this is really one of the overarching themes. How do we distinguish different quantum phases of matter? And in fact, this is something that I feel we have still not completely understood. We understand it fairly well in low dimensions and one dimension, one spatial dimension. Maybe in two spatial dimensions as well, there's been a lot of progress. But in three spatial dimensions, I think there are still a lot of open questions, what are the different phases of matter for interacting particles? You can add various caveats, like, does this system have, can it be represented by a continuum theory? Does it really look like a continuum space or not? Does it only have a lattice realization? There are different ways of dividing the question. But we are still far from an answer, really trying to figure out what are all the different ways qualitatively different distinctions between states of matter. Okay, so that's why we have background. Okay, so what does this have to do with dualities? So it turns out that when you try to describe a phase, you have to describe them in some convenient way. So we need some way to describe these phases. And this may become more concrete as I go on today. I'll give you an explicit example. So we need to pick some set of variables to describe a phase. And what you do in dualities is to really just change your point of view. You trade one set of variables for a different set. And sometimes this can actually be useful. You end up going from a problem where the system is not actually very well described by one set of variables. Usually that means if you write down, for example, a field theory in terms of those variables. The fields are very strongly coupled. You cannot make quantitative statements. But then you can pass to a different set of variables where the fields may actually be weakly coupled. And in fact, we'll see an example today where the fields are essentially free. And then you can solve the problem completely. You can solve it exactly. And that's actually the ideal situation where you find a set of variables that perfectly describes your system. And you can really make a lot of progress. So that's one of the things I want to emphasize that a lot of progress in this, if you want to describe one phase or describe a phase transition between two phases. The key to succeeding is to be able to choose the correct set of variables. The correct set of degrees of freedom that efficiently capture these phases. Okay, so now let me sort of switch and talk about a very concrete example where all of this physics can be explicitly described. And of course, that only occurs in one dimension, in one spatial dimension. So you may think that there's something pathological about one spatial dimension that we can make so much progress, which is probably true. But it still turns out to be extremely useful. And we'll see that if you view the physics of one spatial dimension correctly, you can derive a lot of lessons for higher dimensions, which actually turn out to be extremely useful. Okay, so while I was preparing these lectures, I realized, going back to one dimension, how much of the physics we struggled with in, say, two plus one dimensions, how much of that physics is exactly contained in the one dimensional model? Yes, maybe it's a little bit of hindsight, but I'll try to guide you through that and talk about model system. So this is going to be a one plus one dimensional system. And what we're going to describe over here is something where there are two phases, okay, so we're going to actually have an interesting phase diagram. And there's going to be a phase transition in the system. And we'll talk about how to describe the phases and this phase transition. And in the course of this, we'll come up with a duality. Let me call it duality one, which is going to be a duality between a pair of bisonic fields. So this will be nice, it'll get us some mileage. And then finally, I'll describe another duality, which will be between a Bose and a Fermi field. And it'll turn out the fermionic fields are actually completely free. So not only will this duality give us a new set of variables to describe the problem, these are exactly the right variables to talk about this physics. And because it's completely free, you can solve it. And we'll try to go back and see what made this such a special description. Really, why was it such a good description to begin with? And what will turn out is that in terms of these fields, it's very easy to describe both phases. So there'll be two phases in the phase diagram. This set of variables will be kind of uniquely positioned to describe both those phases. And the fourth thing that will come out is in these new variables, there is actually going to be a topological change in the topology for these fermionic fields, and we'll sort of characterize that. And that's actually, if you think about the fermions as being electrons, rather than some mathematical trick you use to solve the problem, that corresponds to a very active area in condensed matter physics. The search for these MyRuna is zero modes, okay? So all of that will be contained in the simple model that we'll talk about. So the model is actually just the Ising model. Yeah, I'll use a quantum, so we'll work in one plus one dimensions. So if you like, there's a mapping of this model to a statistical mechanics model in two dimensions, where it will just be the classical Ising model. So a lot of this physics will be, those of you have covered that in some statistical physics course. This duality over here will be the Kramer's Vanilla duality. This one over here will be the exact solution to the Ising model. The reason people could solve it exactly was because there's a set of fields in terms of which this is just a free problem. That's the only problem we can really solve. And people like Onsaga, through some tour de force of mathematical physics, figured out that you can make this transformation. They wouldn't have phrased it in these words. But now in retrospect, we know that they really found this duality and they went and solved the free problem. Okay, but it'll turn out that when you do the quantum model, it's much easier to see all of this physics, so we'll work with this quantum model. Okay, so before I actually get started with the meat of this lecture, are there any questions or comments or observations? I didn't realize it's gonna be such a big classroom with so many people. So I was thinking of this as being more interactive. Maybe I should just pause every now and then. How many of you have seen the solution of this quantumizing model? Just to make sure it's not everyone. Okay, there's one person. Okay, so let me start with describing this problem. And I want to make sure that everyone is on the same page. So literally, we are gonna begin with a system which is extremely simple to describe. You have n sites, and on each side you have the absolute simplest quantum mechanical system you could think of, which is just a two level system. So there are two states, let's call them up and down. And there's an operator, there's a poly matrix that I can use to conveniently label these basis states. I can also have a different poly operator, sigma x, that'll give me transitions between these states. So, and we know that these poly operators anti-commute. That's pretty much all we'll need in terms of the algebra of each of these sites. And you want to build up the system made up of n sites. Okay, so really all of the physics, new physics occurs when you take a large number of sites and go into infinity. So there are two ingredients, the second ingredient actually which people don't usually emphasize, why is it that many body physics are so much more rich than quantum mechanics? In a quantum mechanics can also have a large Hilbert space. But the reason you get new physics when you have many particles or so on, is really because of locality. There's a notion of what constitutes a physical operator, or what constitutes a physical Hamiltonian. And the assumption is that the degrees of freedom, they're arranged in some way, and they only sort of interact with their near neighbors. And of course, that's really what gives you the notion of dimensionality. There's a notion of what is close and what is far. And if you want to have some physics that's really different and different dimensions, there must be some notion of locality. It could be a very strict notion that you only literally interact with your nearest neighbors or some finite range. It could be a more relaxed definition where you have, say, power law interactions, things that fall off with distance. But it cannot be that you interact equally with all the sides in the system. When you do that, you're really not describing the kind of physical systems that we're going to be interested in, where all of this new physics emerges. Okay, so I'll try to emphasize locality as in when we get some new properties, how that ties in with locality. Okay, so that's the Hilbert space on each site. And if I have many of these sites, I have two to the n states, a basis states for my Hilbert space, which are just products of what each of these spins are doing on the n sites. Okay, so that's the space of my system. So now I want to think about some Hamiltonian that's going to give me some interesting physics, and we want to introduce some symmetry. So let's introduce an icing symmetry. I'm going to demand that the physics is invariant under flipping my spin from up to down. So I have a generator of my symmetry, which is just the operator that flips my spins. And we're going to demand that things like the Hamiltonian commute with this operator. And as you can see, this operator is a Z2, generates a symmetry group, which is just Z2, because its square is just the identity. Okay, so can we write down a Hamiltonian, which is consistent with these symmetries? So of course, one operator that commutes with G is sigma x itself. So I can add terms. Yes, let me add a term like sigma x. And of course, I want to sum over all the sides of my system. I pick some parameter for that, G. Let me not worry too much about the parameters, I'll put that at the end. But this gives you a somewhat boring model. All you do is apply a field along the x direction. You increase the strength of this field, nothing really much happens to your system. The spins are just polarized along that direction. You really want some competition to make this more interesting. Okay, so you can add another term which also commutes with this G. So imagine that we wanted to add this poly operator, sigma z, which does not commute with sigma x. So it gives you some interesting quantum fluctuations. You cannot add sigma z by itself, of course, because that would not commute with the symmetry generator. It would break the symmetry. But if you had a pair of these sigma z's, you know, product it together, the pair, because each of them individually anti commutes with the sigma x, the pair will commute. So you're allowed to add a term like sigma zi, sigma zi plus one, let's say, nearest neighbors interact. Okay, so that looks like an interaction term, that's good as well. And this model taken together is symmetric by construction. And we'll do something interesting, as we'll see in a few minutes, okay? So let's take that model. There's a competition between these two terms. So I runs from one to, let's say, n minus one. Here it runs from one to n. Okay, I'm doing open boundary conditions. You can do closed boundary conditions as well. We're going to eventually think of large number of sites. So for many things that I say the boundary conditions will not be important, they will be important for one or two things, but I'll kind of highlight that when we come across that. Okay, there's an overall scale to give these units of energy, but that scale is really not going to matter much when you're at zero temperature. You're really just looking at the ground state. You want to find the eigenvector of this Hamiltonian, which has the smallest eigenvalue, okay? So what's the ground state? Okay, and as you vary this parameter g, how does that ground state evolve? Okay, and because this is a large system, n is going to infinity, there could be a phase transition in the nature of your ground state. That vector could change in some qualitative way, and we'll kind of quantify exactly what we mean by that. We have a symmetry, so we can ask about symmetry breaking in the ground state, okay? So, you mean you want sigma i, sigma i plus two? Is that the, is that your question? This one, it's like an interaction potential. Yeah, you could say that. We'll see actually, maybe a better way of thinking about it is. This is a term that actually hops particles, so there are particles that are going to have, carry the charge of this symmetry. So this is something that measures the charge. And this is an operator that either creates or destroys charge. Because this is z2 charge, you can kind of think of it as creating one and moving it to the next site. So roughly speaking, this is like the kinetic energy of some particle. And this is like some potential energy for that particle. But this is intrinsically going to be an interacting model because the Hilbert space on every site is not like that of a free particle. You can either have zero or one particle, but not the rest. So you've already introduced some interactions in that process. But we'll see there are many different perspectives on this. And one of the things that brings out is what you call your particle is really just a point of view. So here, there is some natural thing to call your particle. But there's going to be a dual description, something else is called the particle. Where this will look like the thing that measures the number of particles. And this will actually hop the particle or move it, be the kinetic energy of the particle. But yeah, we'll get to that. We'll define what exactly we mean by the particle and see what these terms do. Okay, so maybe that's a good point to, how do we think about this? So we said that this operator sigma x, it's kind of measuring, so it's kind of like the number operator. If you had like a U1 charge, which is actually an integer, the analog of this would be some exponential of that number operator. But here we just have Z2 charge, it's either sort of one or zero. So this is Z2, I think, and this is measuring the Z2 charge. So the product over all the sites measures the total Z2 charge. Okay, and the operator sigma Z, and this if I write it out in a basis where sigma X is like the usual Z axis, okay? So in a sigma X basis, the sigma Z operator, the roles of these two will be flipped. And the sigma Z operator is one that actually raises, changes the spin of the site, changes it by one plus one or minus one. Okay, so let me define a new basis, which is the sigma X basis. I mean, denote it like this, okay? So sigma X acting on this is just plus, and sigma X acting on that one is just minus. And so now sigma Z in this basis will flip the two spins around. Okay, and so it acts like either a raising or lowering operator, depending on which spin you're looking at, okay? So if you like this sigma Z operator, creates Z2 charge. So if this had not been Z2, you would have a distinct operator, want to create and want to destroy the charge. But because this is just defined model of two, the two processes are equivalent, okay? Creating and destroying charge are the same. So we have just a single operator, sigma Z that does both of them. Okay, so with that background, we can kind of try to describe what happens in this model as we change parameters, okay? As we change this parameter G, let's analyze this model, okay? So let's draw the phase diagram, okay? So if G is very large, I can essentially ignore the first term and this Hamiltonian, I just have the second term, where there's a field that's polarizing all of my spins. So the state of my system is very simple. It's like all the spins are along the x direction. The wave function looks something like that. All of the spins along the x direction and the limit of G going to infinity, that's the ground state, okay? Now as I lower G a little bit, what happens is that I start getting quantum fluctuations from this term over here. So what this does is it takes the spin that's along x and flips it around along the minus x direction and it begins to start to change this vector. So you can ask, what is the excitation? So what is the excitation of your system? Again, let's be in the large G limit. So what can happen is that you can take one of your spins and flip them around, okay? There's one spin that's bad, that's doing something that's energetically unfavorable, so the excitation is just a spin flip. It's something you create some Z2 charge, okay? So when you have a limit of large G, let's call that the vacuum. There's no charge anywhere, all of the spins are along the same direction. But an excitation corresponds to flipping one of the spins, okay? So that's like you inserted one unit of charge somewhere in your system. But that's going to cost you, okay? It's going to cost you energy and the amount of energy it costs you is at least in the limit of very large G. It's simply twice gg, okay? So you took a term that was, sigma x was exactly one. You switched it around and made it minus one. So you pay energy cost of two times g times its coefficient. That's the energy cost to these excitations. And of course, this is a very large energy cost of g is large. You don't see much of this affecting the ground state. So if you were to draw a spectrum, okay? So this is again g, I'm not sure if people can see this. Let me go here, okay? So there's a ground state, let's call that zero. And there's an excited state which is an additional twice gg up there. And as you change, as you lower this g, what happens is that this term can move that particle around. It can hop it from one side to another. It ends up lowering the lowest energy state over here. And eventually at some point, the excitation energy to make this kind of excitation, the spin flip excitation goes to zero, okay? So it comes down and intersects your ground state. And at that point, it's actually favorable to insert Z2 charges into your ground state. And we call that a condensation. You condense these spin flips. You can think of it like Bose-Einstein condensation. You have these excitations, their density increases, and eventually they both condense. And there's a certain critical value of g at which this happens. That's something we'd really like to determine. What is this critical g at which these excitations come down? And they kind of become degenerate with your ground state. And they kind of change the nature of this vacuum. And if you go beyond this, it's like these Z2 charges have condensed. You can easily make these kind of excitations. And what you expect is that you have an expectation value for the sigma z in the ground state. If you go beyond this particular point where these spin flips have condensed. So that's what you would expect. And in fact, you can verify that if you were to analyze this phase diagram from the opposite side, from the opposite limit where this g is zero. Let's say we start from that side. I don't have this term. I have only this term over here. The only interaction is the interaction between spins. And you want neighboring spins to be parallel to one another. Both of them are plus one, or both of them are minus one, you gain energy. So if you analyze it at that limit, it's easy to find the ground states. The ground state is of course the spins polarized along the z direction. All up or all down. And these turn out to be degenerate ground states if you're right at that point. And obviously, if you picked one of these ground states, let's say I picked this one. Evaluated average of sigma z, it's going to be non-zero. So what happened in this process is that you have a Hamiltonian that's symmetric. So the Hamiltonian commutes with g, but the ground state need not necessarily be symmetric. Over here, the ground state is symmetric. You act g on the ground state. Nothing happens if you're already along the sigma x direction. So the symmetry is preserved in the ground state, but when you get to the left hand side, g times the ground state is not equal to itself. It becomes a different state. For example, if I took g times all up, it becomes all down. I act with this sigma x operator that flips this z axis spin. So the ground state actually changes when you act with symmetry operator. So the symmetry is spontaneously broken and you end up with a pair of ground states where the symmetry is not realized in the ground state. So you have to be a little more careful making this analysis. This is like a pathological point, g is exactly zero. What you would like to make sure is that turning on a small amount of g will still preserve this character of the system that you have broken the symmetry in the ground state. And it turns out you have to be a little careful when you do that. You only get symmetry breaking at finite g if you're in the thermodynamic limit. You really need an infinite number of spins to get symmetry breaking. But what you see is that these two states will actually mix. There's a tunneling between the all up state to the all down state that's mediated by this particular operator. But that tunneling matrix element is extremely small. You need to bring down one sigma x for each of these spins to flip them over. It's going to be exponentially small in the number of spins. It's this parameter g raised to the power of n, where n is the number of sides in your system. And g is a very small number. So that's an exponentially small matrix element. And that matrix element goes to zero in the thermodynamic limit. So ultimately, if you go to large system sizes, you're back to this kind of picture where you have states that actually break the symmetry. So there's more detail about that in my notes, which I'll put up. But essentially, the ways to argue that this picture is not just at this point. It extends through this entire range of parameters. So you have symmetry breaking over here. And you found a qualitative distinction between these two phases, which is whether average of sigma z is zero or not in the two phases. Over here it is zero, whereas it is not. So one way to describe this transition is really by this Bose condensation kind of idea. Bose condensation of these z2 charges. And you can try to write down a field theory for this transition. So the sigma z operator is the one that condenses. So let me sort of sketch. So when I say a field theory, what I mean is something that does not concern itself with the minute details of what's happening at the scale of one ladder spacing. So instead of thinking about the physics set on a lattice, let me think about fields, which are a function of space and time. But now they are a function of continuous space rather than discrete points in space. And this will be a valid description if things do not vary very rapidly on the scale of one ladder set. So if the physical quantities really don't change very much on the scale of a few lattice sites, I can pass to a continuum description. I don't really tell you exactly where the lattice site is sitting. Replace it with this continuum field. And we can assume that this is sort of like morally the same as sigma z. Something that carries z2 charge is z2 odd and should not have an expectation value in the ground state that the ground state is symmetric. So you'd expect that if symmetry is preserved, this average of phi should be 0. And the phase transition corresponds to giving this phi some finite value. And of course, this is just a z2 charge. So you should think of phi as being a real scalar field. And in fact, you can do this all a little more rigorously. I won't try to do it right now. You can literally go from a lattice model to a field theory. So two things are happening. One is the operators over here, they only take on values plus or minus one. You want to trade that for an operator that is actually taking on a continuous set of values. So phi is just a real number. So that's one kind of softening that you have to do. The other is you need to forget about the lattice structure. And there's a systematic way of doing this, trading this for a continuous field. And then going near this transition point where it turns out that the length scales involved in your system diverge, they become very big. And then you can rigorously claim that all of your physical quantities, at least at low energies, are varying very slowly. And then you can literally go to this field theory that accurately captures the physics within a certain range of low energies and things that don't vary rapidly on the lattice scale. So in terms of that, you can try to write down a field theory. And as usual, you can just use symmetry to guide you. You have a real scalar field. There's a symmetry that changes the sign of phi. So phi goes to minus phi is the z to symmetry. So your Lagrangian better not have terms that are odd in phi. So you can write on a theory which is d mu phi squared. And then some terms that are even powers of phi. So that's the kind of field theory that you would guess. I would describe this transition. The transition kind of corresponds from r being greater than zero. That gives you phi is equal to zero. R less than zero. So phi is not equal to zero. And that when r is negative, if you want to minimize, let's say this is the Euclidean Lagrangian. You want to minimize this, you actually give an expectation value to phi. That's set by minimizing this potential. And that corresponds to r being less than zero. So I should put this in quotes. Strictly speaking, you have to worry about quantum corrections. The actual value of r could be more negative. But basically by tuning r from positive negative values, you can induce, you can trigger this transition. So this is the field theory. So the field theory of the transition itself, again within quotes, the critical theory is delta mu phi squared plus just phi to the fourth. Yes, I've tuned this to zero. I'm just at the transition. So now this is certainly true. And people have studied this theory in great detail. It's called the Wilson Fisher fixed point. It exists in different dimensions. It's very simple in three plus one dimensions. It's essentially a free theory in three plus one dimensions. As you go down from that, it becomes more and more interacting. You can calculate properties in two plus one dimensions, which matches well with even simple calculations give you good results. But now we are down at one plus one dimension. It's a long way from where it's free. So this is a possible description. It's internally consistent. But these are not the best variables to describe the transition. They're very strongly interacting. And you really have to struggle if you want to get predictions starting from this viewpoint. But it'll turn out that there's a different way of thinking about this problem in terms of fermions that we'll get to shortly, where it'll be trivial to determine most of these physical properties. And those are really the right set of variables to talk about this problem. How much time do I have? 15 minutes, okay, good. Okay, so before I get to fermions, let me talk about the first duality over here, which is to describe this physics again in terms of another bisonic field. But we'll see that in this process, we're going to be going from changing our coupling constant. So G was the coupling constant that we had. We're going to change that coupling constant to one over G. A very typical kind of duality, what happens in dualities, what you call strong coupling becomes weak coupling and vice versa. But it'll turn out the new degrees of freedom are also bisonic. And in fact, there's an exact, this is actually a self duality. The new degrees of freedom have essentially the same properties as the old one. So let's do that. So roughly speaking, this R is going to be G minus G star. Okay, so the more direct connection is R is essentially G minus G star. Or small values of, you know, small deviations. So the lambda itself is not directly, for a particular model, you can try to calculate what lambda is. But it's not, it will only vary smoothly as you go through this transition. Okay, so lambda is lambda G star plus something else, which will be an irrelevant change on the value of lambda. The thing that really changes across the transition is G because it goes through this instability, is R because it goes through the instability point. Yeah, but you can, you know, try to calculate these things microscopically. Of course, as you get near the transition, it gets harder and harder. Things, it's easier to calculate if you're far away, but there's some simple description. Okay, so what is this dual description that we're talking about? So we said that the excitations over here were these spin flips. Okay, so I took one of these spins that are pointing along the x-direction. You know, the ground state was all spins pointing along the x-direction. I picked one of the spins and I flipped it. Those were the excitations. Okay, so now let's ask the analogous question over here. In the symmetry broken phase, what are the excitations? Again, that'll give us a hint how to, you know, change variables and find a new way of describing this, you know, the system. So, yeah, conveniently I have this over here. So this is G much, the strong coupling limits. So let's say I'm G much smaller than one. I have a ground state. Let's say I pick the ground state, which looks like this. And I actually have a pair of them. Let's say I pick the ground state, it looks like this. And I want to find the excitation. Okay, what is the lowest energy excitation I can write down for very small G? Okay, so any suggestions there? Good, okay, that's the answer I wanted to hear. Okay, but turns out that's not the right answer, as you'll see in a second. Okay, let's take the spin and flip it, flip one spin. Okay, so the cost of this excitation, well, you take two of these bonds and you mess them up, right? So let me say I'm on the limit where I can ignore G. I just look at this term over here. So now there are two bonds that were happy before that had energy minus G each, and they both become plus G now. So I get an energy contribution from each of these, which is plus two times J, sorry, it's J other than G. So the energy cost of this excitation, roughly speaking, is four times G. Okay, but the question is, can you do better than that? Okay, good, yeah. So if you did just the first spin, you only have two J. Okay, so that's, but let's say first or last, I guess, right? That's right, yeah. So the way to create a lower energy excitation is to kind of dissolve this bad bond by going and flipping all the remaining spins. At each stage, you're kind of transferring where this bond is. You keep doing that, and then you pop it out of the system on that side. Okay, so anywhere inside the system, you can create an excitation which has just energy of two times J. And this is a domain wall. It interpolates between the two vacua of the system, if you like. All the spins being up and the spins being down. And that is actually the lowest energy excitation in your system. Okay, so the previous theory, if you think about it, the theory we developed over here was based in terms of spin flips. So the spin-flip operator is essentially sigma z, and we derived a theory based on that. If you want a dual theory, you need to derive a theory based on domain walls. That's really the dual description of all of this physics. Don't talk in terms of spin flips, talk in terms of domain walls. So before we had the spin-flip operator, which was sigma z, this is what created a spin-flip. Now we want to create a domain wall. We want to write down the operator that will create a domain wall for us. So let's call that operator in analogy, let's call it mu z. And it's located at some particular site. Actually, it's more natural to think about it as located on the bonds. Because the domain wall lives exactly midway between these two sites. Let me label it as the midpoint of a site. So I'll use i plus half to denote the location of this domain wall. The thing about the domain wall is that it's not a local object. At least to create a domain wall, the operator that creates a domain wall is not something local. You pick all of the spins, say to the right of this, and you flip them over, right? So you've got to have a sigma x for each site to the right. That's some choice that you have, make it to the right or left. Let's take the right, you go and flip the spins everywhere on the right. So it's a string operator involving sigma x all the way to the end of this system. That's the operator that creates a domain wall. So whenever you have a particle, so the spin flip is a particle. There's an operator that creates it, creates or destroys it. But there's also an operator that measures it. That tells you whether it's there or not. For the spin flip, the operator that measures it is just sigma x. And of course, this has to have some non-trivial commutation relation with the operator that creates or destroys it because the number changes. Okay, so we want an operator that, we have the operator that creates a domain wall. We want something that will measure it. Tell us if there's a domain wall or not at a particular midpoint between two sites. Okay, so let's call that operator mu, mu x by analogy. See that, I'd like to write this operator in terms of the original spins. Okay, so what's the operator that measures whether there is a domain wall or not? Any suggestion? Yeah, so the product of sigma z's are neighboring sites. Okay, so if I want to know whether there's a domain wall sitting at this particular point, I just take the product of sigma z's on either side. If that product is one, the spins are parallel, there's no domain wall. That's the convention of mu, mu x is plus one, there's no domain wall. But if they're opposite, this product is minus one, then there is a domain wall. So I've used mu x and mu z to make them look like poly operators. In fact, you can verify they are poly operators. Okay, so they will anti-commute. You'll see that there's just one operator that overlaps, which is the site i plus one, and there's a x and a z operator which will anti-commute. Then these operators square to one. You take mu z squared is just one, mu x squared is just one. So these are actually poly operators as well. Okay, so I'll leave that as an exercise to verify. Also, we have to verify that you have as many operators here in this dual description as you had initially. And up to maybe one operator, you can describe everything in terms of domain walls. If I give a configuration of spins, if I just tell you where the domain walls are, you can reconstruct the spin. Okay, I have to tell you what the first spin is. But after that, every time I tell you there's a domain wall, you flip the spin over. There's no domain wall you continue with your spin. You can reconstruct the configuration of spins. So the domain walls give you a complete information, maybe model a one-spin, but otherwise complete information of what the spin state is. So this is a completely faithful representation of your original problem. You don't lose information, and you can rewrite the Hamiltonian in terms of these new variables. So let's try to do that. That'll be the description of this problem in terms of the new domain wall variables. Okay, so let's see what the Hamiltonian looks like. So that's the nice thing about one dimension. You can do all of this completely explicitly. You can't really do this in higher dimensions, but we'll kind of use this as some, a little bit sloppy about my summations. There's a summation, so there's an operator which is sigma z times sigma z. Okay, so that's great. We already have this mu x operator. That's exactly that, just measures the number of domain walls. And you pay a penalty every time you have a domain wall. Okay, so we have this mu x. Okay, so that's the first term. And then we have this term over here that was measuring the spin flips before. Okay, so what is that term in terms of domain walls? Well, we already talked about this term, so it's got to have something to do with this one over here. The one that creates the domain walls. So I want to isolate just one of these sigma x operators. Okay, I have this whole string. Yeah, I can do that by taking mu z at one particular site and taking the product with mu z on the side to the right. Okay, so that'll produce another string. But the overlap of the strings will be all of these remaining sigma x operators, which will square to one. Okay, so the way to get the second term is plus g, mu z, i, let me call it minus half. Okay, and to make this look a little more like my original problem, let me take out g as a common factor. I'll get one over g here. Okay, so you have the Hamiltonian, the original Hamiltonian. I just rewrote it. So you're still describing the exact same problem, but it's written in terms of your new variables. And what you notice is the Hamiltonian actually has the same form. There's a term which involves the product of z poly matrices, same thing over here. And then you have an x poly matrix with a coefficient. The only thing that changed, okay, the overall factor changed, we said that isn't very important. The only real thing that changed is that g got flipped over to one over g. Okay, so there's a dual description where, okay. So whatever description you had before in terms of spin flips, you take g to one over g, you have the same description, but every time you see the word spin flip, you convert it into domain wall. So one of the things that tells you is if this is really a phase diagram, there's a symmetry broken phase and a symmetric phase. The phase transition has to be at the point where these are equal to one another, right? There's only one special, if you say there's only one special point in this phase diagram, it has to map to itself under this duality, okay? So it tells you that the g star, it better be unity. You know, if this phase diagram has this particular form, okay? So the transition occurs at g equal to one, where there's a perfect symmetry between these two descriptions. Yeah, and this transition, you can think of it in two ways. You can think of it as starting with the vacuum to spin flips. There are no spin flips in your ground state. And then the spin flips become more numerous and they condense. That's one picture. The completely equivalent way of talking about it is, you start on this side, where you have the vacuum to domain walls. You have very few domain walls, because for example, your spins are all up. And then because of quantum fluctuations, they become more and more numerous. You get more and more domain walls. Eventually, those domain walls condense. And that's the right-hand side, which you can think of sort of as a condensate. So it's as though this side is average of mu z is not equal to z. Got to be a little careful with this statement, because these operators are not local, in case we emphasized locality. But let me ignore that for a second. These domain walls condense. And that's a completely equivalent way of describing this transition. Okay, so there are two ways of describing the physics. And it gives us something, it tells us where the critical point is. But it doesn't solve the problem completely, because since it looks the same, it's equally hard in either set of variables. So you want to have a set of variables which will interpolate on one side, which will look like this pen flip. The easy thing to describe is the vacuum. You really don't have any excitations. So this is the vacuum over here. As you get over here, you get more and more particles. So they start to interact, and this makes things hard. Use the dual description, this part is easy. It's the vacuum for domain walls. Then they get more and more numerous, and then they begin to interact. So we'll see that the fermion operators that we'll introduce are actually going to be a product of these two operators. It's going to be Sigma Z times Mu Z. It's like a bound state of a spin flip and a domain wall. And those will be exactly right to interpolate between these two sides, and give us the properties of the critical point. But I guess that's for next time.