 And we go for the second lecture from Ashwin, please. Okay, so I guess I have the bad fortune of having to follow up on Nima's talk, but hopefully I'll try to keep you engaged, maybe not as energetically as he did. But let me just remind you to begin with what we learned last time and where we are planning to go today. So we were trying to figure out how best to describe this microscopic model. It's a model that has a Z2 symmetry and we saw that there were two phases. There was a disordered phase, symmetry was respected, and an ordered phase where it's spontaneously broken. And we also saw that there is a dual description of this theory. You can either talk about these particles, which are essentially the spin flips. So I said, let's go to the basis of sigma x, which is what you want to maximize when you're at large g. And then as you lower g, you begin to introduce some of the oppositely directed spins. So that's actually, it's convenient to call these two basis states, which point along the x direction. Let me call this one zero. So that's like the vacuum at very large g. We have no particles as it were. And then the oppositely directed state when the spin flips over, it pays some energy cost for this g term, but then it's going to pick up some energy gain from the second term. So as you lower g, those events become more and more likely. And eventually those particles, they move around these ones that live in a sea of zeros. They move around and they condense and they end up giving you the ordered state. So what I want to do actually after the class yesterday, there were a few questions that were, I thought, very good questions, where this mechanism was kind of, there were more questions about this mechanism. So today, before I actually get into the meat of the lecture today, I want to give you a simple intuition for this process of condensation. Okay, so we'll begin with that. But that's sort of completing some unfinished business of the last class. But really what I want to do today is to sort of show you a different duality. We already saw this duality between two bisonic descriptions. This was a particle domain wall duality. I'll show you a different duality from bisonic fields to fermions. And that will allow us to describe this critical point, the transition point, very simply. And I'll also talk about thinking about this as a model of intrinsic fermions, what's really changing for the fermions when they go from the left to the right. And there we'll get a lesson in locality, why locality is really critical. When you look at these kind of models and interpret their physics. So that's the plan for today. At the end of it, I want to derive some lessons from the one-dimensional case. The lessons will be very carefully constructed so that we'll be able to generalize all of this to higher dimensions. So there's a little bit of hindsight involved over there. But I'd like to distill all that we learned into some lessons that we're going to apply next time to the two-plus one-dimensional situation. So let me just begin by giving you a little more intuition as to how this symmetry breaking proceeds. If I imagine beginning at very large g and lowering the coupling constant. So the way in which you get a broken symmetry state in this language, a very simple model of it is the following. At every site, you can imagine that there is a single state that is repeated. So I have a product over all sites. And if I'm in the limit of g very large, I have just the zero state. But imagine that I spontaneously develop some amount of the one state on every site. And I'm going to suggestively label this amplitude phi. We're going to say it's very close to the phi field we had in this phi to the fourth theory that we wrote down last time. Let me try to normalize this. So the question is, within this simple approximation, of course the real physical states are a lot more complicated than this. If you think about the state, it has no entanglement between the different sites. It's simply a product state over all the sites independently. It has no entanglement. It's a very simple class of states. But I can think of this as a variational state. I'm going to test how good these variational states are. And I have a parameter phi which I can vary to find the best state. So let's try to do that. Let's phrase this as a variational problem. I'm going to try to minimize my Hamiltonian. Take this energy and ask what's the best phi I can choose within this very restricted set of states. And the point at which you actually develop a finite value of phi is where the symmetry breaks. So if you were to evaluate what is the... So this is essentially, if I think about small phi. So this is really the order parameter. The sigma z operator mixes the 0 and the 1. So it picks up an expectation value of phi. So as soon as I develop non-zero phi in this set of states, it means that I've broken the symmetry. You spontaneously choose a sign for this phi at the positive or negative. That tells you whether your spin is pointing up or down. So let's try to evaluate what the energetics of this system is. So if I look at the average energy of the Hamiltonian, there are two terms. So the first term is this product, sigma z, sigma z product. And within the simple ansatz, there's no correlation between the different sites. It just looks like a product of the terms on a single site. I'm going to do the expansion for small phi. And in the lecture notes, I have it for a more general case, but let's just do this here. So of course, there's an advantage to having some non-zero phi because it gives you some interaction between the neighboring spins. That's why you want to eventually develop some of this phi. But there's a cost. The cost is that the expectation value of your sigma x operator is going to go down. You're putting in these excited states. You're inserting them into your ground state. There's going to be an energy cost for that. And this is going to go as some number over here. I'm not going to try to calculate that at this point, but there's some number, one half or something like that. And there's a reduction of the overall sigma x of phi. And then there are higher order terms in phi. Of course, only the even powers of phi appear over there. So there's something about a phi to the 4 and so on. So it's good to keep track of all of them. But for my purposes, this is sufficient. So what you find is that you have some term which is... So if I were to plot this function, what it looks like is... So if I'm at a very large value of g, this coefficient is positive. If I look at the energy as a function of phi, it's of course quadratic at small values of phi. It looks like that. The best that you can do is to sit at the bottom of this potential and your phi is 0. So for large g, for an extended range of g, until you get to the point where this ga is equal to 1, it actually does not pay to get some expectation value for this phi. But on the other hand, if I start reducing this value of g, there's a point at which it's a g star in this approximation, which is 1 over a, where this thing turns quadratic. You can calculate the quadratic term and you'll see it's positive. So it looks like that. And eventually, if you go below this value of phi, below this value of g, you get some potential that looks like this. So this is for g less than 1 over a. And the minimum configuration is no longer at 0, rather it's at one of these points, plus or minus phi naught, let's say. The precise value is determined by... calculating also this higher order terms. And it's at this point within this approximation that you get spontaneous breaking of symmetry. So that's not a very accurate approximation. If you actually figure out what point this is, it's not at j equal to 1. It kind of overestimates the ordered state. But nevertheless, it gives you the right qualitative picture that there is a transition where you get spontaneous symmetry breaking. So just to make contact with what Nima said in his talk, he mentioned tensors. So it turns out that there's a way to make this approximation more and more accurate. So he said we had a variational state that looked like this, where the coefficients of these two were just numbers. And the problem with that kind of approximation is that there's no entanglement between the different sides of your chain. These are literally different states that you just plug in on the different sides. So you can do better. So right now what we did was we wrote on every side, we had this, where u and v were numbers. So instead, let me think about these as now matrices. I think about generating a state. I take the product over all the sides. And I generate a state, a variational state if you like. By taking these products, I'll get, for example, for the term that is simply the product of zeros, I'll have all of these use, a string of use present. I think of multiplying the use as matrices. And then I finally take the trace. So this is sort of a matrix generalization of the simple mean field variational state. And you can work out some simple examples and see that this does give you quantum entanglement. So this is really how people try to simulate these models. They write on a variational state for the ground state. And you make these matrices bigger and bigger and you get a better and better approximation to the ground state of your system. So this is sort of a systematic way. At least in one dimension, it's believed to work extremely well, especially when there's an energy gap. So as long as you stay away from this critical point, you can get a very good approximation of the ground state by writing down these types of variational states. So the first non-trivial generalization is to take two by two matrices. But in general, the use and views are going to be some chi by chi matrices. And the bigger this chi is, the better the approximation to the exact ground state. So chi is one is just this simple mean field theory. If you use two by two matrices and then you optimize, you take the energy and you optimize these two matrices, you get the first correction to mean field theory. And then you keep going, make these larger and larger, and there are theorems that tell you how big you need to go to get a very close approximation to the ground state. And in fact, you don't have to go very large. So it's a good exercise to just start with two by two matrices and see that this gives you some state that has quantum entanglement. That's actually what I do in the notes. For example, pick u is sigma x and v is sigma y. And you'll see the state has got some non-zero quantum entanglement. So that's a nice exercise. You just take two sides. Pick u is sigma x, v is sigma y. These are just some other poly matrices, not related to the spins. And you can show that this gives you just a EPR. So if you follow this prescription for a pair of sides with those UNV matrices, you get a correlated state. Okay, so that's sort of an aside. Before we talk about the main topic, which is to go towards writing down this Bose Fermi duality. Any questions about this basic phase diagram? Okay, so now let's introduce these fermions. And again, the details you can find in the notes. But the idea is that you can introduce these fermions that live on the sides by taking products of this spin flip operator. This creates, goes from a zero state to a one state and vice versa, and taking the product with a domain wall creation operator. Okay, so physically this fermion operator is it flips a spin. It attaches a domain wall. So you can also write down another operator that flips the spin but in a different fashion, which is using the sigma y. And you can verify that these are actually myerano fermion operators. These are real fermions. So for example, they are their own complex conjugate. So that's a reality condition. And they also satisfy that the square is unity. It's simply products of poly matrices. The squares are just unity. But they're anti-commute with one another. So you should take chi on a side, chi bar on the same side. That's the easy one. The anti-commute, they also anti-commute on different sides. Ah, tilde, yeah, sorry. I'm trying to avoid bar because it looks like it's some sort of complex conjugate. It's not, it's just a different fermion. So these are myerano fermions. You can construct a complex fermion from them. If you wish, you can construct C dagger on every side, which is just chi plus i times tilde. And the number of these complex fermions, you know, if you go back and see what it means, the state zero is no fermion. The state one is one fermion. Okay, so on the single side level, there's a sort of a direct mapping to just the zero and one basis states. But when you look at this on a chain with many sides, there's an additional factor, phase factor that comes in. The strength, which is going to make this fermion operator different from just the operator that looks at one side and, you know, goes between these two. Okay, so there's a bosonic operator that only changes one side. Only cares about one side goes between zero and one. That's this operator sigma z. The fermion operator also does that, but it's sensitive to all the occupation numbers on the different sides. Okay, so that's what gives you all the weird properties of fermions. They're really not local operators. When you think about the fact that they anticommute rather than commute on different sides. Okay, so those are the fermions. It's good to verify all of these anticommutation relations. Physically, it will tell you why binding a charge to a defect, that's really the domain wall, why this combination is a fermion. And we'll see that that carries over to higher dimensions. In two dimensions, we're going to look at defects that are vortices. We're going to bind a charge to the vortex and you'll see that that's a fermion as well. And same thing happens in three dimensions as well. If you have monopoles, you bind charge, you get fermions. Okay, so that's a very general kind of, you know, outcome and it's kind of good to see it in the simple context. Okay, so the whole point of introducing these fermions is to go back and rewrite the Hamiltonian in terms of the fermions and you'll see that something pretty amazing happens. We'll get an extremely simple looking Hamiltonian that we can solve exactly. Okay, so the first thing we need to do is to rewrite all the operators. So let's look at the second operator over here, sigma x. So what is this in terms of fermions? Okay, so sigma x requires me to take the product of these two operators, the z and the y component. The product will give me sigma x and these things will cancel out. So let me get the sign right. So that's the sigma x operator, the sigma z operator, the product that I'm interested in which will go into the Hamiltonian. Okay, so this is going to involve the two sides and this is just i tilde. Okay, so again that's a simple exercise. You get the chi tilde because you're going to get a little bit of the string now. You're looking at two different sides so you get a bit of the string. It changes one of the sigma z's into a sigma y and so I'm giving you one of these chi tilts. Okay, so those are the two terms that are going to appear and the interesting thing is that both of them are actually quadratic in the fermion operators. So these are just canonical fermions and my Hamiltonian is just a quadratic function of these fermions. In other words, you can solve it. It's exactly solvable and we'll see what this solution turns out to be. Okay, so let me take h. I think I have minus signs for both of these. Okay, so what's little j? Little j is this. J plus one, yes. Okay, so let's write this out pictorially. So as a picture, I have on every side I have a pair of fermions. Okay, so let me draw them like this. I have a chi tilde on side j, let's say, and a chi and this is the next site over, chi tilde, j plus one, chi. Okay, and the two terms are doing two different things. The term with the g in it is simply pairing these maron fermions on a single site. So this is the g term. There's some directionality that's assumed when I write this term. You go from chi tilde j to chi j. That's where I've drawn the chi tilde first and then I have the other term. Let me draw it as a double line. So that's like the j term. Okay, this is the z, z term. Okay, that actually couples the different sites together. Okay, so that's what this Hamiltonian looks like. Okay, so we know that the g equal to one is the critical point. Okay, so let's go and look at what this Hamiltonian looks like. Okay, when I set g equal to one, well, then both of these couplings are exactly the same. Okay, it makes no distinction of pairing fermions within a site or pairing them between the sites. Okay, so we also saw that g equal to one corresponds to the self-dual point. And actually, self-duality in terms of these fermions is extremely simple. It simply means that you group these fermions in different ways. Okay, so this was one grouping of fermions within the sites. But if you wish, you could do an unphysical grouping which is to take the pair like that. And if I rewrite my theory in terms of those pairs, that's actually the dual theory. Okay, so everything looks really simple in terms of these fermions. And in particular, the critical point at g equal to one, let me call that hc. It looks just like that. Okay, so now you can sort of exploit the fact that everything's invariant. It kind of makes sense to not label things by the sites anymore. But let me introduce these new fermions. Okay, so this is chi tilde. Okay, so I have these itters that are on this lattice with twice as many indices because I'm going to take chi tilde and chi and put them into correspondence with these itters. And at the critical point, I get a very simple Hamiltonian. It's just i sum over all these indices. Let me call them. Okay, so I have the sum. It's now a double sum. It's on twice as many sites, but it's simply a product of these itters. Okay, so we're going to use this. Right now it's a lattice model. We're going to focus on low energy physics. And we're going to derive a field theory that describes this transition. Okay, so we're going to assume that these fields are slowly varying. We're talking about low energies. And we're going to just retain the parts of the fields that are going to give me low energy excitations. Okay, so if you go ahead and solve this, you can actually solve this completely. It turns out the dispersion is, you can solve it by using Fourier transform. The dispersion is the sign of the momentum. Okay, so the momentum runs between zero and pi. It turns out the sign of that is the energy. So the low energy modes are near zero, zero wave vector, which is essentially nearly constant fields, or wave vector pi, where the fields oscillate, they change sign from side to side. Okay, so let me just, you know, make the following ansatz. You can see, we'll see in a minute that it actually corresponds to the right low energy fields. Okay, so I have a field that varies slowly in space. When I use this notation, it's varying slowly. And there's another one that actually varies rapidly, but I take out the rapid variation. Okay, so this is also slow. And this accounts for the fact that there are fields which change sign from side to side, but they're also low energy fields. Okay, so I can pull out that dependence, minus one to the r. It changes sign whether r is even or odd integer times this slowly varying field. Okay, so the field theory will be developed in terms of these slowly varying fields, eta one and eta two. Okay, so you can kind of see, you can anticipate where the two component structure of your Dirac theory is going to come from. Okay, it's really these two fields. Okay, so now let's write out the Hamiltonian. Write out this product. So it's eta r. Let me say this r happens to be an even integer. Just to make things clear, it doesn't really matter. You can keep that factor of minus one to the r. Actually, let me keep it. Okay, but there's an additional minus sign over here because you're looking at it at r plus one. Okay, so now you make the product and then you sum over all the values of r. Of course, there are four terms. There is the Dirac terms. Okay, so let's write out the Dirac terms first. Okay, so that's the Dirac term. There's a relative minus sign again because this is a rapidly varying field, a chain sign between r and r plus one. And then you have the cross terms. Okay, there's an eta one times eta two, both of which are slowly varying and their product is multiplying something that's rapidly varying. Okay, so when you do the sum on r, there's no Fourier component in this product that can cancel minus one to the r. Okay, that really oscillates on the scale of a lattice. So all the cross terms will vanish when you do the sum. Okay, so that's really just saying that there are two slowly varying fields and there are such different momenta that they don't mix. Okay, so that's the result after doing the sum. You only have these two terms. You do not have the cross term. Okay, we'll see later when the cross term comes. But at this point, I just have this term. Okay, so we said that these are slowly varying fields so I can make a tilde expansion. I have eta one at r plus one. I can approximate this as eta one at r plus a derivative. Okay, and if you want this to be a length scale, this derivative should be the half the, essentially the lattice spacing. But there's a derivative that tells you how much this field changed. Okay, so let's use that. The square of this eta one square is just one. That'll cancel out the other one. So at the end of the day, the Hamiltonian at the critical point takes on a very familiar form. Okay, so eventually this j will become some velocity. Call it vf. I'll pass to an integral rather than a sum. I can do that because they are slowly varying fields. I put in a length scale that'll give me a velocity rather than an energy scale. And this eta one. Okay, so I can make this look even more familiar by writing down a two-component field. And then this over here looks like eta transpose sigma z. These are now fields. Okay, so this really just looks like the Dirac equation. Actually, these are Majorana fermions. You have a real doublet. And there's some Pauli matrix, which is like the velocity matrix of your Dirac theory. If you were to trade this eta transpose for an eta bar, this would become a gamma matrix. Okay, so you can go through all those manipulations. I won't do that. But essentially, this has the spectrum of a non-Kirl, Majorana fermion. So the energy is a function of momentum. Looks like that. Okay, there's a right mover. D by dr, which sets the momentum. If the momentum is positive, you get positive energy. And there's a left mover, which does the opposite. Okay, so there are a pair of branches. So this is on Kiral, Majorana fermion. And the very important fact is that it's massless. There's no mass term over here. Everything moves at the speed of light, which happens to be some velocity v in our theory. Okay, so the theory of this critical point in terms of the Majorana fermions is very simple. It's just a gapless, a massless Majorana fermion. And, okay, so let me just draw that over here. Okay, it just looks like that. Okay, so there are two descriptions we now have for this theory. One is the fight of the fourth theory with the quadratic term switched off in terms of the boson strongly interacting, or I just have a free theory in terms of the Majorana fermions. Okay, so the next step, of course, is to ask what happens if I move away from exactly the critical point. Okay, if I move away from this point, g equal to one, how does this change? And that will allow me to read off critical exponents. The exponent, for example, that's controlled by changing this coefficient g. Okay, so example, how does the, if I move away from this point, I get a finite length scale. The correlation functions are not completely long-range. They have some exponential decay set by a length scale. And one of the critical exponents is how that length scale is set by the deviation from this g equal to one point. Okay, so we'd like to figure out the anomalous dimensions or the dimensions of various operators at this critical point. Okay, so let's try to write down the field theoretic description of the term that moves me away from this g equal to one point if I were to reinsert the g over here. Okay, so what happens when I have the g back in there? Okay, let's imagine this g is very small, a small deviation from unity. And what happens then is that these two bonds are no longer equal. Okay, so this bond is slightly different from this one because g is not exactly one. Okay, so the additional term, okay, so you can sort of guess it. It's something that depends on g minus one. Okay, of course it's got to vanish when g is exactly one. And it oscillates, okay, it's either above or below the average value depending on whether you look at this bond or this one. Okay, so it's going to oscillate from the new sides from side to side and it's going to give me this sort of term. Okay, and we can write down what this means in the field theory language. We go back to our operators eta one and eta two, the slow fields. And I put them in and then I do the sum and I see which term survives, right? And of course the one that now survives is just the cross term. That's the only one that can sort of cancel off the overall factor. So this essentially becomes up to some signs. If I go back to my continuum language, it becomes this term, the product of my pair of slow fields. Okay, but this is really just the mass term for my run of fermions. Okay, so you can write it. So the term eta one, eta two, you can write it in terms of this two component notation as eta bar, eta, where this eta bar is eta transpose times sigma one. Okay, so you're going to augment this, the first term in your Hamiltonian by another term, which is the mass eta bar times eta one. And the mass is actually proportional to the deviation from g equal to one. Of course your g to one is zero, but you also know exactly how the mass depends on the difference between g equal to, between one and the value of g that you have. So over here you have some gap spectrum. So this spectrum looks like p squared plus m squared. And over here as well you have a gap spectrum with the opposite sign of the mass. But if you just look at the energy spectrum, you don't know the sign. It's simply gapped and the mass, at least for small values of deviation is proportional to g minus one. Yeah, question. So when you say, okay, let me repeat the questions and I have to understand this question. So you say at the critical point there's a chirality. So what do you mean by that? Yeah, so we can directly go back and see what exactly these fields correspond to. So I think I erased it, but the fermion fields themselves are simply the product of the spin operator and the domain walls. So there is some combination and I'll sort of describe this in a little more detail in a minute. There's a certain sense in which these fermion operators are actually ideal operators to describe this phase diagram. And so we'll do that in a minute. We'll kind of try to unpack what this fermion operator means in terms of this phase diagram. So I think you're asking for a physical picture of what these fermion excitations are and we'll see that in a minute. Any other questions? So this is actually the fact that you understand what the mass term does as you deviate from the critical coupling already gives you one of the critical exponents. It gives you the exponent nu, what's usually called nu, and it tells you this nu is equal to one. So it's sort of trivial to read off at least some of the exponents. At least some of the exponents are trivial to read off in this free theory, which would be very hard to do within this bosonic description. So one of the questions is how do I understand these fermion excitations, how they correspond to things that are more readily visualized, either in terms of spin flips or in terms of the domain walls. So what are these fermions? We set this chi, for example, as sigma z times the domain wall creation operator music. So let's just look at this operator and some limits. For example, let me look at it on this part of the phase diagram where I'm deep inside the ordered phase. So deep in the ordered phase, we set that the sigma z is some number. There's some expectation value to the sigma z operator. So when I look at this fermion operator, it's a product of these two, but essentially the first one is just a number. I can replace it by its expectation value and what happens is that the fermion begins to look more and more like simply a domain wall operator. So on this side of the phase diagram, the chi just looks like a domain wall and in the disordered phase, we set that there's this duality. So on the one hand, sigma z was ordering over here and on this side, you could think of it in a sense like the domain walls were condensing. So there are very few domain walls over here. All the spins are either up or down. As I increase this g, I start getting domain walls, eventually they condense. So there's an exactly dual picture. So in the disordered phase, essentially my fermion operator kind of dissolves into just the sigma z, into just the spin flips. So this operator has been designed such that in the two extreme limits, either on this side or on that side, it matches up with the operator that produces the excitations. So for example, if I had this vacuum, if I wanted to create a single z2 spin flip, I'd operate with the sigma z operator and that's exactly what the fermion does in the limit that you approach deep into this disordered phase. And similarly on this side, the fermion becomes just the domain wall which is the simple excitation in your phase. And of course, at the critical point, things are complicated because neither of these two is a good description. It's only the bound state of them that ends up being continuous to be a good description. Some string operator that creates a domain wall attached to a spin flip. So that I guess is the best I can do in terms of giving you some intuition for what ends up being these excitations. That's a very good question. So let's examine. I was going to do this a little later, but let me examine it right now. What corresponds to the symmetry that we started with? What does it mean in terms of the fermions? And this is going to help us when we think about what the fermion phases are. Symmetry, in terms of the spins, we said there's a generator which is this product of sigma x. So what this is doing is it's going down the chain and it's counting how many of these, so if I have some state of 0s and 1s, it's essentially counting the number of 1s. Let me call that number of 1s, but it's just the parity of this. That's what this operator in words does. Every time there's a 1 that gives you a minus sign and that product will give you the value of the symmetry generator. So if you see what that means in terms of fermions, we said that every time there's a 1, that's like having a fermion. So this G is nothing but the fermion parity. So the symmetry generator corresponds to the fermion parity. So that's a little bit strange because we think of fermion parity as always being a good symmetry of any fermionic system. But how do you break fermion parity? You can imagine breaking the symmetry G, for example, by adding a magnetic field along the z direction. But what this is saying is that if you do that and you map that system back to fermions, the fermion parity is no longer going to be conserved. So let's try to do that. Let's try to break. So there's a term which is, for example, just sigma z on every side. Now we have broken the z2 symmetry. If I conjugate with G, this term changes sign. So what does this correspond to in terms of the fermions? Well, that's going to be an extremely non-local object. But for example, the term which is sigma z at the last side, let me call it n, this corresponds to simply the Myron of fermion chi at the side n. Other terms, for example, sigma z at the first side will be chi at the first side and then product of minus 1 to the nj where j is on all the sides greater than 1 all the way up to n. So each of these is going to give me something with a single fermion sitting there times something that's measuring parity of fermions for some other set of sides. But this is a particularly simple one. There's no string attached because it's at the very last side. It tells you once you break the z2 symmetry you're allowed to add just single fermion operators to your theory. So that would be a disaster if these were real. This was a model with real fermions. This was a model with real electrons. We're certainly not allowed to add a term in the Hamiltonian which is just a single fermion term. And the reason you're not allowed to do that one way to think about it is that fermions, the fermion operator is not a local operator. It doesn't commute on different sides the fermion operator does not commute and we're told that a field theory or a Hamiltonian of a physical system has to be local. It should only involve local operators and you seem to be violating that when you simply add a single fermion operator. The resolution of course is that these fermions are not the physical degrees of freedom of our system. We started with spins and the demand is that the physical degrees of freedom obey locality. What we wrote down is a perfectly local Hamiltonian, both those terms and the one we added are perfectly local in terms of the spin degrees of freedom or the boson. These kites which we used as an aid to solve the problem are not physical degrees of freedom. And therefore we have no reason to demand that the Hamiltonian is local in terms of that. So this locality for these fermions but it's okay since they are not physical degrees of freedom. It's only when you go back to the spins that you have this very strict demand that the local operators of which your theory is constructed also appear in a local fashion in your Hamiltonian. So this is important because if you're really talking about electrons there's just no way to break the symmetry. If you're talking about the state of electrons in a model like this but the fermion fields themselves are physical let me ignore the fact that the electrons are coupled to a gauge field just think of them as fermions that do not have this gauge coupling then those are your local degrees of freedom those are the physical degrees of freedom this fermion parity should always be conserved. I don't know any connection here to supersymmetry. Oh, sorry. So the question is G looks like a written index so is there any connection to supersymmetry? So I mean the most simple interpretation of G is that it's just the fermion parity besides with written index and supersymmetric theory but here it's just the fermion parity so it's a basic kind of ingredient in characterizing any fermionic system even if you don't have supersymmetry and for physical fermions this is always conserved because you're not allowed to add just the fermion operator in your Hamiltonian. So you can label all your sectors by the values of the fermion parity and you're guaranteed that they're not going to mix by any physical term but of course here these fermions are really derived quantities and all hell can break loose if you sort of lose that original z2 symmetry. And it becomes much less useful once you break the z2 symmetry it becomes much less useful to utilize this fermion language and charges essentially yeah, that's right. That's a good question so we're going to try to describe this similar kind of point and say 2 plus 1 dimensions it turns out that the defect if you want it to be a particle you need to have a u1 symmetry so if you just have a z2 symmetry the defects are domain walls so then the defects are lines it's not that convenient to work with lines let's think of a symmetry where the defects are points so a u1 symmetry will have vortices you can attach the charge to the vortex so you can describe this critical point in terms of fermions so now you have two descriptions one description in terms of phi to the fourth theory phi is now a complex scalar and then you have another description in terms of Dirac fermions which are these composite objects which are massless the thing that's different in 2 plus 1 dimension is that's not a free theory that theory is actually coupled to a churn simons gauge field so it's an it's an interacting theory so on both sides you have interacting theories you have a phi to the fourth theory and you have you know fermions and so now it's a question of taste which is your preferred variables here we have a clear choice because these were really free that's the beauty of 1D that everything becomes simple and it's just this another question so another point that I can mention since it came up is what happens to this theory when I add this particular term add this magnetic field along the z direction so one very interesting thing that happens is that if I look at domain walls we said before that domain walls the only cause to the domain wall was where you reverse the spin so if I had a pair of domain walls the energy cost of this configuration doesn't depend on how far apart these are the energy cost is just a penalty here and a penalty there so now if I apply a magnetic field along the z direction it definitely prefers all the spins to be up rather than down so this pair of domain walls will have an energy cost that depends on their separation so if I have a field a field along the z direction the energy cost of this pair of domain walls will be the field times the separation let me call it delta r between these domain walls okay so there's a potential connecting the pair of domain walls which increases linearly with separation so that should remind you of quark confinement these domain walls are no longer free particles in fact they get bound up okay so people who do experiments on these kind of materials they can work in the limit where there's no magnetic field they can see that there are these domain wall excitations that move around you put on a weak magnetic field they're still kind of free they kind of go beyond a certain distance they realize that they have the string you know connecting them and they kind of bounce back and then you get the sequence of bound states you know very much like you know meson resonances and they can see that in neutron scattering where they can excite these sort of resonances okay and there's a very interesting prediction that if you apply a magnetic field right at the critical point a very weak magnetic field these bound states this sort of sequence of excitations they form a representation of a very large group called the E8 group okay so there's a lot of beautiful mathematical physics even in this very simple model once you turn on a magnetic field along the Z axis okay now of course all of that physics cannot be exposed using free fermions okay because the minute you turn on the Z Z axis magnetic field you have to deal with terms like this sitting in your Hamiltonian extremely non-local terms that involve operators on you know pretty much all the sites in your system so it's no longer a tractable theory as soon as you lose the fermion parity symmetry okay so that's why all these weird and interesting things can happen but you can still analyze them because it's very close to a conformal field theory and these are perturbations of that conformal field theory okay so so just to go to my last point which is really what is the distinction between these fermion phases which are on either side of this critical point okay so if I just look at it it looks like they're gapped on either side there's some mass term to my my run of fermion but we said that it's unlikely that we get a phase transition without something changing in the phase itself okay so something must have changed what is the character that changed across this transition and we'll see that that is really a topological distinction between these two phases actually how much time do I have I kind of lost track of the time two minutes that's all I need okay so the distinction between these two sides you know we can ask what does it mean in terms of this diagram that we wrote if you're on the g greater than unity limit it's kind of continuously connected to this picture where you have these bonds that are connecting the pairs of myaranas they combine into a pair of states that simply get a gap okay so the g greater than one is simply a gap state and we'll call it a trivial state because it essentially just involves the physics on single sides okay we're really not communicating between the different sides so there's nothing very physical that can happen nothing many body-ish that can happen in that limit okay but on the other hand if I go to the opposite limit where I switch off g and I only have this term it kind of looks the same you know you pair up myaranofermions you know this pairs with the next one except you see that this one is left unpaired at the ends of this chain you know there's another one out here where the myaranofermion has no neighbor to pair with okay so this is chi L chi N let's say and this is chi 1 chi tilde 1 okay so g less than 1 it all looks gapped in the bulk okay but there are some zero modes at the ends there's a single myranos zero mode at either end there's a Hamiltonian which you know couples all of them except for the first and last myranomode chi tilde 1 and chi N there's no coupling between them in fact there is a very tiny coupling that arises from some exponential sort of coupling between them something that's exponentially small in the length of this chain because there's an energy gap in the bulk there's a little bit of tunneling you can get to the other side and come back there is some coupling but this will go to zero as a function of the number of sites in your system which is exponentially decaying in the length of the system okay so apart from that if I just said that equal to zero every eigenstate of my Hamiltonian will have a two fold degeneracy okay it really doesn't care what I do with this pair of myranofermions I can put the pair you can think of this as a complex fermion that can either be zero or one and the energy couldn't care less which you pick whether you pick zero or one so this is if you like a topological degeneracy if you actually had electrons you had an electron chain with this particular Hamiltonian then you would find that if you had an open chain with some boundaries there's a two fold degeneracy of my chain of electrons okay so there's a and you can ask where this degeneracy lives is there a mode that I can point to where the degeneracy lives well the degeneracy lives on either side of this chain okay it's equally split between the left and the right end so each one of these should be sort of having a square root of two degeneracy in order to make up this complete degeneracy okay so it's as though you have Hilbert spaces, local Hilbert space okay because this degeneracy, the minimum degeneracy that you can imagine is actually split between the opposite ends of the chain okay so people think this is a very good way of storing quantum information if you put this in a state of either 0 or 1 the environment cannot really read this state of your system if it wants to read the state of the system it has to apply some operation on this end and then currently apply it at this end as well and then read out this information so this is a protected qubit okay so it's protected against certain kinds of errors these fermions are really just electrons okay so it's important that they're not these toy fermions which rely on the Zeta conservation but really electrons for which the fermion parity is a good symmetry so people have actually constructed this in the lab after sort of much effort we now have these chains these superconducting chains which are seen, which are have been observed to have this two fold degeneracy okay so maybe that's a good place to stop and take any further questions