 You had the mattress typically there made a kind of 10-minute break, you can decide, see looking at the place. Thank you, and good afternoon everyone. So I'll be talking about topological lattice model from gauging. Okay, so I know that this is your fifth hour of lecture today and you're a little bit tired and this title sounds a little bit abstract. I'll try to make it followable as possible. I'll try to do examples, very, very explicit examples. Lattice model, which are exactly soluble, meaning that if you take down the nose and you want to go home and reproduce all the calculation, you can do that. Okay, and so the title of course contains two parts, one is topological models and the other is gauging. So, well, we can have a full lecture on either topic, but for the lecture today or for the three lectures that I'm going to have, I want to put them together because these are very tightly connected subjects and understanding topological model from the perspective of gauging is one that I find particularly useful in my years of study. Okay, so I will try to explain to you through very simple examples how these two are related. So I want to say something more about gauging. So gauging is something that is not very easy to explain because, well, first of all, the name is very confusing. The name gauge, when you read about it, you don't know what it means. And that is because of a lot of historical reasons why people started this subject and how it evolved over the years. But as can as matter people, sometimes we don't know about all the histories of where it comes from, like gravity and all that. And actually, in a lot of cases, we don't need to go so much back into the history in order to understand it. So that's something I'll try to convey through these lectures that there's actually very simple ways to understand what gauging or what gauge theory means in a can as matter setting. Okay, in can as matter setting, I mean, we have local degrees of freedoms being both on the electrons and they couple to each other in certain way. They might hop around, they have interactions, and they usually sit on a lattice. And then the goal is, we ask, what is the ground state of the system? What are the excitations of the system? Things like that. Okay, completely in a can as matter setting. Okay. So the first reason is that the word is confusing, that gauging. So actually, whenever I see gauge or gauging, I just replace it by local or more explicitly local symmetry or local constraint. Okay, so that is something I find very useful that whenever I see the word gauge or gauging, I just, in my mind, I replace it by local symmetry. Okay, and then you will see that that's exactly what we're talking about. We're talking about, we'll be talking about lattice model with local symmetry. And somehow, the fact that the model has local symmetry can give rise to a lot of exotic phenomena as can as matter phases. Okay. So the second reason that gauging is not a very easy subject is because the way that it's usually taught, the way I first try to learn about it is in a quantum field theory class and the teacher started writing down Lagrangian and started writing down fermions coupled to the gauge field and the gauge field transforming certain way. The fermion transforming certain way and there's Lorentz invariance and there's covariant derivative, all of that. Okay, so very soon into the lecture, the teacher was writing down like g nu nu and a lot of indices, upper indices, lower indices. So I will not do that. Okay, I'll try to keep things very, very simple and even very naive to the extent that we can do exactly several models and try to explain what a gauge theory is. Okay, so for those of you who already know a little bit or already know too much about the gauge theory, my version might look different or might look too naive and I will comment upon how the version I'm talking about is related to the usual version that people talk about. Things like quantum QED, that kind of thing. But that will be further back into the lecture because I want to just start from very, very simple examples and show you what we mean by gauge theory and why that's something that's non-trivial. Okay, so the first example that I'm going to talk about without explaining why I want to talk about it is the model called Tori code. How many of you have heard about Tori code? Oh, wow, good. Okay. And that's great because I'm going to go over it. Well, for the other half who hasn't seen Tori code, I'll still go over it in every detail and we will review all the properties of Tori code and try to see how to reinterpret it or how to interpret it in terms of a gauge theory. Okay, and then see how Tori code can emerge by gauging something, by taking another model and apply the procedure called gauging and then reproduce Tori code. Okay, so we'll see if we can get through that today. So the first part is Tori code Z2 gauge theory. Z2 gauge theory. Tori code is a very popular model. It is taught in mathematics classes, it is taught in quantum information classes because it's exactly solvable so everyone can play around with it and it includes some very, very non-trivial properties. So usually the simplest version of Tori code that I like to talk about is one that sits on a two-dimensional square lattice. Okay, so on this two-dimensional regular square lattice we have one qubit or one spin one-half degree of freedom on every edge. Okay, so all the edges contain a qubit. I'm just drawing a few of them. So with a qubit of course I'll label it as tau, I'll label the qubit as tau and as a qubit of course it has a two-dimensional polyoperators, tau x and tau z. Okay, so these are the operators that act on a spin. So of course tau x and tau z and the combination gives you tau y. All right, so this is a hubris space and the Hamiltonian is a sum of terms and some of the terms are centered around a vertex V while the other terms are centered around the placet which we call P. And Hamiltonian contains a bunch of terms. It's a sum over the kinds of term that's centered around vertex which I will call A V and for the second type of term they're centered around placets and I'll call them P P. Oh sorry, I forgot to say that whenever you think I'm not making sense or you have any question please just raise it to me. I want to make sure that everyone's following. Okay, so the Hamiltonian contains the A V terms, the vertex terms and placet terms and this is the vertex term. The vertex term involves four qubits that's around the vertex and it's a tensor product of tau z operators around the vertex. Tau z, tau z, tau z, tau z and that's it. Okay, so we do a tensor product of four tau z and that's our A V term. And secondly we have this B P term and this B P term is around the placet and the tensor product of tau x term around the placet. So A V term is tensor product of four tau z and B P term is tensor product of four tau x. And that's it. The Hamiltonian sums over all kinds of terms centered around all the vertices and all the placets and our goal is to find the ground state and properties of exactitation. Of course this is a strongly interacting model. It involves spins and the spins they enter into these four body interactions so on the surface it looks very, very complicated. But what's nice about this model is that if you look at it, all the Hamiltonian terms, the A V and the B P term, they all commute with each other. So you can check that explicitly. So of course all the A terms commute with each other because they only involve tau z operator and tau z wherever, whether they overlap or not, they always commute with each other. And similarly, all the B terms commute with each other because it involves only the tau x operator and tau x operator always commute with each other. The only tricky part, the only thing that we need to check explicitly is whether the A term commute with B term. So if we have an A term here and a B term here and they don't overlap, then we don't have to worry about it because the spin operators, when they don't overlap, they always commute. The only thing that's tricky is if we have a vertex here and a plocket here, right? Then they overlap at certain locations but rest assured that these terms they still do commute because you can see that they overlap at exactly two locations at this qubit and this qubit, right? So the tau z from the A term, they anti-commute with the tau x term from the B term but they anti-commute at two locations. So the minus sign, you have two minus sign and add it together. These two terms commute with each other. Does that make sense? So these terms, usually we don't have that asynchronous model but this is a special model. This is a model where the terms, even though they overlap with each other, they commute with each other meaning that they can have a complete set of eigenstates, including ground states, right? Whenever operators commute, they have a complete set of common eigenstates. So that means if we want to look for the ground state, we just look for the ground state of each of the Hamiltonian terms. We can individually minimize energy for each of the terms and if we successfully minimize energy for all of the terms, we get the ground state of the Hamiltonian. So this is the nice thing about exactly so. Hamiltonian terms commute and we can start looking for the lowest energy states by just looking at the local energy requirement without going for the global energy. Okay, so we can try to see how do we minimize energy for the A term and how do we minimize energy for the B term, right? And in the end, what we want is the ground state which minimize energy for the A term and minimize energy for the B term altogether. Well, the A term is a tensor product of 4 sigma z. And as a polyoperator, what kind of eigenvalue does it have? What kind of eigenvalue does the A term have? Yes? Plus minus 1, yes, because it squares into 1. It squares into identity. The possible eigenvalues are plus minus 1, okay? And same for the B term. The B term is a tensor product of tau x and as a polyoperator, it can only have eigenvalue plus minus 1. And because I've chosen to put a minus sign in front of all these terms, so the minimal energy state is the eigenstate with eigenvalue 1 for all the A operator and eigenvalue 1 for all the B operator, right? That is something we're looking for. We're looking for a state that has eigenvalue 1 for all of these operators, okay? Okay, we can try to see what that means. For example, if we want to have an wave function which has eigenvalue 1 for the A operator, that actually has a very nice physical interpretation. A operator is in terms of these tau z operators, right? And the tau z, it has two eigenstates. Tau z is equal to 1 on the 0 state, and tau z is equal to minus 1 on the 1 state. This is the two states of the spin 1 half. And to have a picture of what the ground state looks like, we're going to assign some physical meaning to this 0 and 1 state or to the tau z equal 1 and minus 1 state, okay? We're going to say that if the spin is in the state 0, then the particular edge is not occupied by a string. Well, on the other hand, if the spin is in the state 1, then it is occupied by a string, okay? You can imagine that in the state 1, there's a color string. Let me find another color. So let's say we have blue color. If it's in a state 1, now we have a blue color running along the edge. Well, if we're in a state 0, then there's no string running along the edge. This corresponds to no string. This corresponds to... And this is very intuitive because the qubits, they live on the edges, so it's very straightforward to interpret the two different states of the qubit as the edge being occupied or not being occupied by a string. Of course, this is just another way to say that there are two states of the qubit, but then it gives a very nice interpretation of what this A term in the Hamiltonian was. So the A term is a tensor product of all the tau z operators. And remember that if we want to look for the ground state, we want the A term to be 1. We want the A term to have an eigenvalue 1, but the A term involves a bunch of tau z's, which means that we can only have an even number of tau z in this term to be minus 1. We can have all of them to be 1. We can have all of them but 2 to be 1 and the other 2 to be minus 1. That's fine. Or we can have all 4 of them be minus 1. That's also OK. In order to satisfy the A term, the allowed configurations are like this. We can have a configuration where everything is not occupied. We can have a configuration where 2 of the strings are occupied in blue. Or we can have the string going in different directions. We can have them go right through the vertex. Or we can even have the configuration where all 4 edges are occupied by the string. And of course, there are rotationally symmetric configurations of these ones. I'm not drawing them all. But all of these kinds of configurations at a vertex are allowed by the A v term. You can see that this gives us a very nice way to understand the requirement on the wave function imposed by A. Because if we have no string, if we have 2 string or if we have 4 string, the common property of all these configurations is that a string goes into a vertex and has to come out. For example here, it goes into a vertex and it has to come out. This one goes in and come out vertically. And this one, you can think of it as one going horizontal and one going vertical, or 2 of them turning corners. But either way, a string cannot end. If we want to satisfy all the A terms, then at every vertex, the string cannot end. The string must go on. And the way for a string to go on is to form loops. For example, we can have configurations like this. We can have small loops going around a placket. This kind of configuration satisfies all the A terms. Or we can have bigger loops. We can have bigger loops that covers more area. And that also satisfies all the A terms because the string never ends and forms a loop. Or we can have even more complicated shape of loops that twist around and wiggle. But in the end, as long as everything forms a loop, now we satisfy the A term. So we can have small loop, big loop, or multiple loops, but that's okay. As long as we have loops, we're fine with the A term. So the A term is saying that I want loops. So any kind of loop is fine, but I only want loops. I don't want the strings to end anywhere. If the strings end anywhere, that's bad. That costs extra energy for the A terms. Okay, so that's what the A term says. Yes, right. So you can imagine that we have one loop like this and another loop like that. And that's fine. Yeah, they just cross each other. Oh, if they share a bound, if they, like, one loop like this and a small loop like that, if they share a bound, this bound gets removed, right? So because you've... So to go from no string to string, you flip the spring. But if you flip it again, you go back to no string. This is a Z2 spring. So whenever two loops touch, they just cancel each other and become a bigger loop, which still satisfy the A term. Yeah, good question. Okay, so all these configurations are allowed, and they play a role in the grand segue function. Okay. All right, so next thing we want to look at what the B term is doing, right? If we only have the A term, if we only have the A term, all these configurations have minimal energy, meaning that they're all degenerate, right? If there's no B term, then there's no way to distinguish all these configurations from each other in terms of energy, and they generate a huge ground-state degeneracy. And the huge ground-state degeneracy will be removed once we add the B term. So now we can see what the B term does. The B term is a tensor product of 4 tau x around the block head, right? And if we apply the B term, what does it do? If we apply the B term to somewhere where there's no string, what does it do? It creates a small loop, right? It generates a small loop like that. And if we keep applying, if we apply to another loop, if we apply to another block head, then it makes the loop bigger. So if we want to look for the ground-state wave function, where BP acting on the ground-state wave function should be equal to the wave function because we want the wave function to have eigenvalue 1 for the B operator, right? That's the way to minimize energy for the B part. We want it to be an eigenstate of B with eigenvalue 1. So we want to look for a wave function that satisfies this condition. And the way to do that while still satisfying the A condition is that we make a big superposition of all possible loop configuration. Okay, so let me try to draw the picture. So a ground-state wave function, if I want to draw the picture for it, let's say this is a two-dimensional system. We can have a configuration where there's no string. All the spins are in the state 0, right? And then we can have a state where there are small loops like that. And we can have a term where there's a small loop somewhere else. And we can have terms where there's a small loop, well, or there's a bigger loop like that. Or we can have multiple loops and so on and so forth. And we have to make a superposition of all the loop configurations altogether. Right? Remember we mentioned that all the loop configuration, they satisfy the A term, and they all have minimal energy for the A term, but they will degenerate if we don't have the B term. Now with the B term, the B term splits the degeneracy, but the B term maps one loop configuration to another by flipping some plaquettes and generating a loop or moving a loop or enlarging a loop. But it doesn't break the loop configuration. It just maps from one loop configuration to another loop configuration. So if we want to satisfy all the B terms, the way to do it is to make a superposition of all possible loop configuration and that will be our grand state wave function. Does that make sense? And you can explicitly check that this wave function will be invariant under BP because what BP does is to create a small loop like that. It just maps this part of the wave function to that part or maps this part to that part. But in the end, the total wave function is invariant. This is why we say that the grand state wave function of the torical Hamiltonian is a condensation of loops. So pictorical is very easy to imagine what is going on. But of course this is a highly non-trivial wave function. Usually we don't think about wave function this way. We think about spins polarizing certain directions. But this is a highly entangled grand state wave function, although in terms of this loop configuration, it's easy to picture what is happening in the wave function. So it's a superposition and it's an equal wave superposition of all the loop configurations. We don't have any phase factor or any weight in front of the configurations. All the weights are equal. Sorry, of course I didn't put it in the normalization of the wave function. It should be properly normalized, but I'll just ignore it here. All right. Okay, so this is the wave function. But this is not all about wave function. If we imagine the situation on a more topological manifold, the story can actually become different. That is, if we imagine the system to be living on a torus, or another way to say it is that we consider periodic boundary condition between left and right and also up and down. Imagine that we glue together the system between left and right and up and down. So make it into a torus. If we make it into a torus, that is, we're connecting left and right and up and down, there are actually configurations in the Hilbert space that also satisfy all the A-term and all the B-term, but doesn't look like this at all. Okay. So imagine a configuration where we just have one loop going across horizontally. So this is periodic boundary condition, so nothing is broken, even though it looks like it's broken, but actually it's connected back to this side. So everything is still closed loop. So it's still satisfied A-term. Yes. For the minus one of BP, yes. So you just change the sign structure here. For example, if you want this placate term to have an eigenvalue BP, then you just change plus sign to minus sign. And all the corresponding pairs, you make them having opposite signs, then BP will have eigenvalue minus one. So on the torus, this is a legitimate loop configuration, right? But it doesn't look like anything that we have drawn before. And you can imagine that there are others. For example, we can have one where we have a string going vertically. Or we can have one where we have both string in the horizontal way and string in the vertical way. These kind of configurations, there are still loop configurations, but we cannot start from the configuration where we have nothing. We flip a little placate and generate that. If we flip a little placate, we can only generate the loop. That's what we call contractable. It's like a bubble and we can shrink it back to nothing. But these kind of loops, it's a non-trivial loop on the torus, and you can move it up and down. You can move it up and down, but there's no way to shrink it back to nothing. Which means it cannot be generated by applying the BP operator. On any particular loop configuration. So these are basically different universe for the ground states of the torus code on the torus. And you can see that there are four different universes. This is one, this is number two, this is number three, this is number four. And they give rise to four different sectors, four different dimensions of the ground state hubris space. This is the first dimension. The second dimension, we can start from this particular configuration, and then add all the small loops. We can do a silver position with configuration with one horizontal string and a small placate. We can do everything that we did for this, no string configuration, and we make a big silver position of everything generating that way, so that it again satisfies all the BP terms while satisfying the AV terms, giving rise to a degenerate ground state of torus code on torus. Similarly, we can do the same thing for the configuration where we have a vertical string, right? You can start from the thing with just one single vertical string, and we can add little placates, or we can add even little placates on top of the string so that we move the string a little bit, but it still goes across the system, and we make a big silver position of all kinds of configurations like that, that becomes another, the third ground state wave function. And of course, the fourth one is that we start from this point and apply all the little placates, and we generate a big silver position, and that is the fourth dimension of the ground state hubris space. Oh yeah, why not two? Good question. So let's see what happens with two. If we have two, start to putting little placates and try to remove part of it. We can put in this placate, and then remove this part, and this part, so then we open the ribbon in the middle, okay? And then we can keep doing that. We can put in another placate like this, and move the ribbon down, and put another placate here, move the ribbon down, and in the end you can finally see that by putting in little placates, we can actually remove two string into no string at all, which means that two string is actually something that's already here, even though I didn't draw it. So there's actually already a configuration starting from the no string configuration that we can generate the two string configuration. And this one can actually be generated by just applying BP-Cloquettes one by one just along the strip. It actually only matters in the case of Turricode, whether there's an even or odd number of big loops in a certain direction. If there's no loop, it's the same, it's equivalent to the case where there are two loops, okay? And of course the case where there's one loop is still a different thing. Yes. Okay, diagonal one is more like this. So, for example, we can have one that goes this way, right? But this one you can deform it. I should use blue, sorry. We can move it around and deform it such that it just becomes this. Okay, they're equivalent to each other. It only matters how many times you wrap around the torus in the x and y direction. Sorry, I can't hear you. Oh yeah, sorry, it was confusing. And when I, yes, when I drew these diagrams first, I was talking about open boundary conditions, but then I moved on to talk about periodic boundary condition and putting this extra term, which is confusing, sorry. What I meant is that all the previous configurations I drew still applies to periodic boundary conditions. So you can imagine that putting all these configurations on periodic boundary condition and then add this extra one. Yeah, sorry. So this is periodic boundary condition. Right. So, so Tori code on the torus has ground state degeneracy four. Knowing this, knowing this, now I'm going to ask you a question. What is the ground state degeneracy if I put Tori code on the sphere? It might sound like a bad question because I talk about Tori code on the square lattice and it's not obvious how to put a square lattice on the sphere. You probably need to involve some pentagons or triangles and things like that. But let's forget about the lattice for a moment because, you see, when I do this discussion, I didn't even put the lattice in, right? Actually, this kind of story holds on whatever lattice. You can just imagine a manifold where there are loops floating around. So just thinking in terms of the picture of loops and of these kind of big loops or small loops and try to argue what the ground state degeneracy should be if I put the model on the sphere. Okay. So a sphere would look something like this. And what do you think the answer is? What? Yes. Because there are no big loops on the sphere, right? I can draw whatever loop like this and it can shrink to a point. Even if I take the equator and the equator can shrink to either the south pole or the north pole. So a sphere doesn't have a noncontractable loop, meaning that if we put toracode on the sphere, the ground state degeneracy is just one. So you all got it. Un-degenerate. And actually, this kind of thinking applies to all kinds of manifolds. Yes. Okay. Why we don't have more, right? Okay. Good. Good question. So for this kind of Hamiltonian, it's actually easy to argue why we exactly have four. Okay. So the way to do it is to think about how these terms, they put constraint on the Hilbert space. So originally, we start from a bunch of qubits, right? And each qubit has a two-dimensional Hilbert space. And we have two spins per unit cell. So that's four-dimensional Hilbert space per unit cell. So the dimension of the Hilbert space is two to the n of, sorry, four to the n of vertices. Now let me do it over there. So dimension of total Hilbert space is four to the number of vertex, the number of vertices, or the number of unit cells. And then we can see how this big Hilbert space gets cut down if we try to satisfy the A-term and the B-term, right? The A-term and the B-term, they're polyoperators and it's going to one, meaning that half of the Hilbert space has eigenvalue one and half of the Hilbert space has eigenvalue minus one. So if we try to satisfy one of the A-term, we cut the Hilbert space into two, right? So if we satisfy another A-term, we cut the Hilbert space into two. And if we try to satisfy the B-term, we again cut the Hilbert space into two and that just works similarly for all of the terms because they all commute. They all commute, meaning that cutting the first one doesn't affect the way you cut the second one, you just cut the Hilber space into two every time you enforce eigenvalue being one for each of the operators. So we just count how many operators there are in the Hamiltonian and then we'll get the ground-state degeneracy, right? So let's imagine we leave a torus with periodic boundary condition and let's see how many Hamiltonian terms we have. Well, we have one vertex term per unicell and one plocket term per unicell. One vertex term per vertex and one plocket term per vertex. So that means we want to Hamiltonian terms per unicell, which means we want to divide this number by four to the number of unicell. Okay, not quite, right? Because not all the A-terms are independent from each other and not all the B-terms are independent from each other. There are certain global constraints among them such that after cutting a certain number of times the final one A or final one B is already fixed by all the previous ones and you don't just have the freedom to cut anymore. We can look for the constraint. And the constraint is such that if you multiply all the A-terms together you get identity, right? Because every edge participates in two of the A-terms. This edge participates in this vertex term and this vertex term and the horizontal edge participates in the left vertex term and the right vertex term. So every edge participates twice. So you take tau z and you square and you get identity and you get identity everywhere. Okay, so there's a constraint. There are actually exactly two constraints, that the product of all the A-V terms is identity and the product of all the B-P terms is identity. Which is saying that we divide it by too much, right? So we should divide this Hamiltonian constraint by a factor of 4. So now this is very simple algebra and you see that this is exactly equal to 4. So we have a four-dimensional degenerate Hilbert space in the ground state. This is a great question, thank you. And actually I'm going to follow your question because this is not the end. We found that there's a full ground state degeneracy, right? But how do you know that this is some degeneracy that we want to care about? And when I say whether we want to care about it, I mean whether this is a ground state degeneracy that's stable to local perturbation. Because in connective matter systems, connective matter systems are very dirty. There are lots of things that can happen that can deviate from the exact models like this and in general we should allow the possibility of having all kinds of local perturbations to the Hamiltonian. So the question now becomes we do have four-fold ground state degeneracy, but what if we add some extra terms like tau z, tau x, whatever to the Hamiltonian, making it not exactly soluble, but what if that can allow us to remove the ground state degeneracy? And actually of course this is not the case because otherwise I won't be talking about this model here. The ground state degeneracy for tau z, this four-fold ground state degeneracy is stable to all local perturbations. And the way to see that is to realize how can we map between different ground states? And that actually can show us that in order to map from one ground state to another, we need to do something global and very, very long range and which becomes something that's not possible to do just by doing local perturbations. And you can immediately see what's going on if you want to map from one ground state to the other. You need to draw a big loop, right? If you want to map from this one to this one, you need to draw an uncontractable loop in the horizontal direction. If you want to map from this one to this one, you need to draw a loop in the vertical direction. And that is something that's global that you need to do consistently along the loop to make sure that it does close up into a loop. And this is something that local perturbations simply cannot do. And local perturbations cannot even tell whether there's a big loop in the horizontal direction or in the vertical direction. So more formally, we can write down what is called a logical operator. Of course, this is borrowing terminology from common information, but this is a very useful terminology here. So a logical operator is one that allows us to map between different ground states. For example, one of the logical operators is one that draws a big string in the horizontal direction. The color doesn't look like red, a different color. So what we do is to apply tau x operator all along the string. We apply tau x operator all along the string. We'll be able to map between this wave function and this wave function. And another logical operator is when we do it vertically. We apply a tau x operator along a vertical string and draw a vertical loop. And that gives us a second one. And the fact that these operations are very big, they have to cover the whole loop, tells us that the ground state degeneracy is stable to local perturbations. So this is how much we want to know about the ground state wave function of tarot code. So it's a civil position of all possible loop configurations and their degeneracy. If you have big loops on the manifold, if you're living on a tarot, or if you're living on even more complicated manifold with non-trivial genius, then you will have ground state degeneracy. And you can map between the ground states by drawing big loops, by drawing big circles on the manifold. I want to talk about excitations. The ground state is boring. It's just there. At zero temperature, we have some degeneracy, but that's not the most exciting thing. The most exciting thing is the low energy excitations of the system. Now I need another square lattice. So bear with me when I draw another square lattice. Imagine how we can make excitations in the ground state. Suppose that the system is already in the ground state, satisfying all the A and B operators. And we want to take it out of the ground state so we can do something to it. And the simplest thing you can do for a spin model is to apply the tau z and tau x operator. Let's think about what happens if we apply a tau x operator at a certain location in the system. What happens if I apply the tau x operator? It's not the ground state anymore, because the tau x operator changes the wave function and it changes the eigenvalue of some of these Hamiltonian terms. It doesn't change the eigenvalue of the Hamiltonian terms if the Hamiltonian terms are far away. If the Hamiltonian terms are here, it doesn't care about it. But if the Hamiltonian terms overlap with the operator that I apply, then the eigenvalue might be changed. And in particular, this vertex operator, this A, V, and this vertex operator, A, V, they anticommute with the tau x operator I apply. So they get excited. Their eigenvalue change from 1 to minus 1. Meaning that now I have a higher energy for the Hamiltonian. Just by applying this tau x operator, I change the eigenvalue from 1 to minus 1. I change the eigenvalue of this one from 1 to minus 1. So all together, I cross the energy of 4 for this Hamiltonian. I can keep doing that. I can keep applying the tau x operator. Just do it when that is basically done. And what happens is that, well, I flip back this A, V term. This A, V term, I anticommute with it again. So its eigenvalue got back to 1. But another one got flipped. And this A, V got flipped. So I do tau x and tau x again. I create two excitations. And you can see that I create two A excitations. And I can move them further apart by just keep applying these tau x operator. For example, I can even make them turn. I can do tau x again here, such that I move this excitation to this point. I can keep moving, move it too. So if I start from a point, create two excitations and keep moving them. What happens if I close it up into a circle? If I apply tau x in a fourth circle, what happens? Yes, it goes back to ground state. Because I keep moving them, moving them, moving them. And finally, this A, V overlaps with this A, V. So they annihilate with each other. So this A, V, this excitation of A, V is actually what we call a fractional excitation. We create them in pairs. And once we create them in pairs, we can move one of them anywhere. We can move the other anywhere. And finally, if we bring them back, the two annihilate with each other. And we go back to the ground state. Now we're actually going to make some connection to gauge theory. Because this A, V, this violation of A, V is something that I'm going to call a charge excitation. Of course, charge is something that we talk about in gauge theory. We have electromagnetism. And then we have charge. So here, of course, I'm just introducing the terminology in a very brute force way. But I'm going to show you later why this is a legitimate way to call it. Why we think of it as charge. Why it is actually related to the actual charge, the electromagnetic charge that we usually talk about. So this is one. So applying a tau x string is one way to make fractional excitation. In particular, it's the charge excitation we make. So the question is, if I apply tau x, does it affect BP? It doesn't because it commutes with BP. It commutes with BP. So let me just write a few lines. So we start from ground state. And we apply tau x somewhere. And you can ask, what happens to BP? And because they commute, so this is just tau x BP psi, which is tau x psi. Meaning that the wave function after the application of tau x is still an eigenstate of BP with eigenvalue one. On the other hand, if we ask the question of the A, V, which overlaps with the tau x. And because they anti-commute, so there's a minus sign coming out. Which means that the wave function after the application of tau x is an eigenvalue minus one eigenstate of A, V. Meaning that A, V is excited. Yes, good question. That's the second step. Now we're going to play with tau z. It should be the other side of the story. So let's apply tau z. Find tau z using the same logic. It doesn't do anything to the A term because the A terms are made up of tau z and they all commute with tau z. But the B term, the B term, they're made up of tau x. So they might anti-commute with the tau z if they have overlap. So here, if we apply the tau z here, it's going to anti-commute with the B operator in this placate and this placate. Making two excitations. So again, it's a pair of excitations. And now we can move them around. We can move them around by applying tau z on the subsequent edges like that. And we can even make them turn such that we can move the two excitations apart. And very similar to the tau x case, we create a pair of excitations, move them around and finally we can bring them back. If we apply the operation to a full circle, we annihilate the two excitations so we go back to the ground state. So we call the violation of BP. We'll give it a name of flux excitation. And we'll see later why this is a good way of calling it. So the charge and flux excitation are the two basic type of excitations of tariff code. We can actually get all the fraction excitation by starting from them and doing some composition. Let's give them some notations. Let's call this charge excitation. Let's denote it by E. Well, that's the usual notation we give to charge. Let's call the flux excitation M. We can ask what kind of quasi-particles, what kind of excitations are these things? If we really think of them as point charges, and we can ask, are they bosons? Are they fermions? And do they have statistics with each other? And that's something that would make sense in a gauge theory. Even though I haven't explained to you why this is called charge, why this is called flux, but if that connection is true, then we would be able to talk about whether, for example, the charge is a bosonic charge or fermionic charge. And we can talk about the flux excitation in a two-dimensional case. We can talk about the statistics of the flux excitation as well. And we can talk about something which is what happens if we bring a charge around the flux? Remember that this is what happens when we do the Harnoff-Bohm effect experiment. Now we have a flux loop, sorry, we have some flux going through a hole, and we put some electrons, and the electrons go around the hole, go around the flux. And what happens in the Harnoff-Bohm effect is that the electrons go around the flux, even though on the outside of the hole, the magnetic field is zero, because magnetic field is only non-zero inside the hole. But the electron, it does feel the existence of the magnetic flux in such a way that the total flux through the middle will change the phase factor of the ground state wave function. If you learn about the Harnoff-Bohm effect, that's the famous experiment, which shows that electrons are actually quantum mechanical particles, that its wave function has a phase factor, and the phase factor can be changed by coupling to an electromagnetic field. It's coupled in such a way that even though the magnetic field itself is zero, the phase factor can be changed by putting some flux through the trajectory of the electron. So this is what we expect to happen if we call something a charge, and we call another thing a flux, that the charge going around the flux is going to accumulate some phase factor. So in the Harnoff-Bohm effect, of course, it's a real electric charge, a real magnetic flux, and the electron going around the flux will accumulate a charge, will accumulate a phase factor of e to the i phi times the charge, electric charge and over h bar or something. The unit here might not be accurate, I apologize, this might be some constants here. But the point is that the phase factor times the electric charge, this is the real electric charge of the particle, changes the phase factor of the wave function. And that is exactly what we expect to happen here, that we have some pseudo charge, and we have some pseudo flux, we give them the name, and we expect that they have certain pseudo statistics, pseudo phase factor induced when one of them goes around the other. So let's see how that happens before we take a break. So this is the two-dimensional system, the lattice model that we were talking about, and let's create some flux, let's create some flux out of the vacuum. So the way we create the flux, of course, is by applying these tau z strings, we still have one flux at this end and one flux at the other end. We do tau z, tau z, tau z all along the string, and we have one n particle here and one n particle here. Remember that these n particles they literally correspond to a BP operator with eigenvalue minus one. So this is like cutting a hole in the system and putting some flux through the hole. And now what we want to do is to bring a charge and let it go around the hole. Well, the way we do that is by stretching out some tau x string. And with this tau x string, we're creating two charges at the end. And we want to bring one around the flux and finally annihilate them. And when I draw this circle, what I mean is that I literally apply this tau x string all along the circle. And the thing we want to compare is between the case where there is a flux with the case where there is not a flux. We want to see how existence of this string changes what happens when we bring a charge around this particular point. Well, the only thing that can happen is when they intersect. So imagine that it's zooming to this point where there's a cross at this point. This is part of the square lattice that we draw over there. So at this point, what happens is that when we created the m particle, what we did is we applied a tau z operator on this particular qubit. And then when we bring the e-charge around, we applied another tau x operator at this location. And now there's a difference. If we first generate the flux, bring the charge around, compared to the case where we bring the charge around and then generate the flux, that corresponds to a commutation between these two operators. What is the commutation relation between the two operators, between the two strings? How do they commute with each other? Just because that they overlap at this particular location, one of them acts with tau x and the other acts with tau z. So the different order of doing things, whether we create the flux first, bring the charge around, or bring the charge around, and then create the flux, that differs by a minus 1. And that is exactly the chronoform effect in this particular case. And of course in this case, the flux cannot be arbitrary value. It is only minus 1 corresponding to e to the i. So that's why we usually call this flux a pi flux. We've learned about super connectivity. That's the same pi flux we're talking about. So this is the chronoform effect. This is the mutual statistics between the charge excitation and the flux excitation. Now there's also something called the self-statistics, which is concerned about whether the charge and the flux are bosons or fermions or some other kind of anions. That can also be found, and that can be found in a way that I will not explain too much, but just tell you how we can find it. The way to find the self-statistics is again, on the ground state, we want to use the string to do something, and the way we use the string to do something is to draw a figure of 8. We draw a figure of 8 either with the tau x string and tau z string and see how that changes the ground state. And the answer is that it does not, and you can simply do the computation yourself. You can do a figure of 8, tau x string or tau z string, but because all the operators are just tau x and tau z, so they have no commutation with itself, so if we do this exercise with either the E excitation or the M excitation, we're going to find that this is exactly equal to the ground state itself, meaning that the topological spring of the E particle and M particle are just one, so they are bosons. So I hope you can... These are the two conclusions that we reach by just looking at the lattice model, studies excitation and the string operator, how they commute with each other, how they commute with itself, and these are the two conclusions I want you to keep in mind. And later we're going to see how it actually is related to the idea of gauging. One is that the E particle going around the M particle has a phase factor of minus one, or correspondingly the Hanoff form phase factor is pi, and the other one is that the charge and the flux, they're both bosons. Okay. So I guess we can take a 10 minute break and come back and then I'll show you how this can be understood in terms of starting from a much, much simpler model and then gauge it into this gauge theory of toricode. Welcome back. Okay, I got several very good questions during the break, so I just want to address them with all of you together. I think they're very important points before we move on. So one of the questions is why the fact that we map from one ground state to the other using these big string operator implies that local perturbation cannot remove the degeneracy. Okay, so that's a very good question. And the reason is that let's say we have one ground state, which is let's say energy, let's shift the energy to zero. Okay, let's just shift the energy to zero. Or some energy, just energy minimum. And we have another ground state, which is also energy minimum. And we know that in order to map from one ground state to the other, the thing we need to do is to apply a string of toracs. And we need to apply a number of toracs that's as large as the system size in one direction. This is one. Okay? So you can ask, well, what happens if we just add the term of toracs to the Hamiltonian? And take the original Hamiltonian, which is exactly solvable. Now we add to it with some small coefficient epsilon, the toracs term at every location. For every spin, we add a toracs term, and basically we're like adding a magnetic field to the spin model. Okay, of course, but this is a small magnetic field. And you can ask, what happens to the degenerate ground space if we add this perturbation? But once we add this perturbation, the Hamiltonian is not exactly solvable anymore because this tau x term, it doesn't commute with the AV term. AV term involves a bunch of tau z, so they don't commute, so we cannot solve it exactly. But we can argue about things using perturbation theory. And this is degenerate perturbation because these states are degenerate. And the way the degenerate perturbation theory goes is that we can try to map from one ground state to the other using the term in the perturbation. For example, we can put one tau x in between psi 1 and psi 2 and try to see how it gives correction to the ground state energy. And it doesn't because a single tau x cannot connect psi 1 to psi 2, so this term is actually zero. And similarly, if you put two tau x, it's still zero because there's no way you can connect psi 1 to psi 2 unless you put all the tau x along the string together and use it to do the perturbation theory. So there is indeed some correction to the energy, but that term comes in the form of the product of all the tau x between psi 1 and psi 2 along the string. So there are all of them. But remember that each of the tau x term comes with a coefficient of epsilon in the perturbation Hamiltonian, so there's a factor of epsilon to the else power. So the whole thing goes as epsilon to the else power. So there is indeed some changing energy. There's a tunneling between the ground states that's induced by the perturbation to the Hamiltonian, but it's only on the order of epsilon to the else power, meaning that if we have a very, very big system, if we take the thermodynamic limit, then the energy splitting is exponentially small, which we can safely ignore. But if we don't have a very large system, if we only have a small system, then the energy difference can be big. Now we can actually actually split the degeneracy. That's what happens if you have finite size effect in your numerics. You might not actually see the fourfold ground state degeneracy. You might actually only see energy splitting in all the ground states. Only if you take the thermodynamic limit, make your system size big do you see that the four states become closer and closer in energy and they finally become degenerate. But their approach should be very fast because we're guaranteed that the energy splitting should get small in an exponential way. So that is the good news. Of course for numerics, it still might be hard to see the degeneracy. The second question is regarding the cell statistics and the figure of eight that I drew here, so I want to say a bit more about that because I did it in a very quick way. Okay, so I claim, of course I can't explain very clearly, but the way we want to look for the cell statistics of these excitations is by drawing the string operator in a figure of eight and then see how it changes the ground state wave function. For example, if we want to do it for the E particle, what we can do is to draw a string operator like that and it goes around and around in a figure of eight and crossing itself at this point and then finally fuse back. And you can do homework and see that if we just do tau x along the figure of eight, it doesn't change the ground state wave function. On the other hand, if we do for the M excitation, that is if we do it on these kind of edges with tau z operator, imagine that we apply tau z whenever the blue dotted line crosses the edges of the square lattice, that is for calculation of the topological spin of the M particle and homework again, you can show that this doesn't change the ground state wave function, it's again one. But there is an interesting case here which is the composite of E and M particle. The composite of E and M particle which we usually write it as E cross M equals psi and this is a fermion. This actually has to do with the fact that E and M particle has a mutual operating statistics of minus one so that this psi particle is a fermion and you can literally check it yourself because psi is the combination of E and M so the string operator of psi is just by applying tau x here and then tau z on some neighboring edges. So the way we can calculate topological spin for the psi particle is by drawing the string of tau x and tau z together. Sorry, this is tau z. We do tau x and then tau z on the edges that's sticking out. So like this, tau x and tau z and then tau x and tau z and tau z and then moving above we do tau x and tau z. So we go like that or we complete the figure of eight and you can calculate it very explicitly that this gives rise to a minus sign if you apply the figure of eight for the doubled string operator on the ground state wave function implying that this psi particle is actually a fermion. This is the fun thing you can do when you go home today. All right. Okay, any questions? So now we can move on and see why we are calling it a gauge theory and why we can think of the excitations as charge and flux. We're going to actually retreat back to a very, very simple model, the model that you're learning the first day of the work in this matter class which is called the transverse field ising model. How many of you have seen the transverse field ising model? Okay, a larger proportion, good. Again, I'm going to do the transverse field ising model on the square lattice just to match the discussion we had on taric code. Imagine that we have a square lattice and we have ising spin sitting in the middle of each poquette. We have ising spin sitting at the middle of each poquette which I'm going to label as sigma. Okay, so from now on, I'll be very careful with notations. So tau corresponds to the gauge field and sigma corresponds to what we call the matter field. So again, this is a spin one-half degrees of freedom and we have sigma x and sigma z. Sorry, I think it's better if I put the sigma spin at the vertices. Sorry about that. Let me put the sigma spin at the vertices. The same thing, I'm shifting things around, but I put the sigma spins at the vertices and we have sigma x and sigma z. And the transverse field ising model is one where we have the interaction between the spins given by sigma x, sigma x on nearest neighbor sites together with another term in the z direction summed over all lattice site. This is transverse field. I forgot to add some coefficient. This is with coupling J and let me just set this one to one. Okay. And you probably, if you learn about transverse field ising model, you know that there are two phases. One is the symmetry breaking phase and the other is symmetric phase. But whatever, the most important thing about transverse field ising model is that there's a symmetry. And that symmetry is a global symmetry. And the symmetry is simply implemented by a unitary operation of the tensor product of all the sigma z operators. You can see that this transverse field is symmetric under the symmetry and also this ising coupling term is symmetric under the symmetry because even though sigma x anti-commute with the symmetry, we always have two of them. So it's a symmetric term. And we all know that there are two phases when J is much, much smaller than one and this transverse field dominates. So we have a symmetric phase and if J is much, much larger than one, then we have a symmetry breaking phase. So the symmetric phase, in the simplest case, you can imagine where J is just equal to zero. If J is equal to zero, we only have transverse field and then the eigenstates is simply all the spins polarized in the z direction. All the spins in the state of zero. That gives us the lowest energy and that becomes the ground state of the whole system. So it's a unique ground state. It's a pencil product wave function. Very simple, no entanglement at all. The simplest wave function you can think about. In the case of J much larger than one, they're actually, or we just said J to infinity and ignored the transverse field term, then they're actually two ground states. One is all the spins in the plus direction or all the spins in the minus direction in the positive or negative sigma x direction. So of course all the spins want to point in the same direction to minimize energy here, but they can point either in the plus x direction or minus x direction. So each of these ground states violates the symmetry. That's why we call it a symmetry breaking phase. I suppose this is a story that you're more or less familiar with. So what's the relation between this very simple model and the toricode model we were talking about? That is we're actually going to take the transverse field ising model and do something called couple to gauge field. Couple the transverse field ising model z2 gauge field, such that it actually turns into the toricode. So here the word gauge appear and as I said at the beginning of the class that whenever you see gauge you replace it in your mind by the word local. So what we are trying to do, when we say we want to gauge the symmetry or gauge the model, meaning that we want to turn the global symmetry into a local symmetry. Of course right now this is a global symmetry. We need to apply sigma z on all the lattice sites in the transverse field ising model in order for the Hamiltonian to be invariant. If we only apply it to a point like here, if we only apply sigma z at this particular point, then some of the term is not going to be invariant. And those kind of terms involve the coupling term here, the coupling term here and the coupling term here in terms of sigma x and sigma x. All these four terms are going to get a minus sign if we only apply symmetry locally at sigma z. So without doing anything special this model, this transverse field model is not locally symmetric. It has a global symmetry but doesn't have a local symmetry. And what we want to do is to promote the global symmetry into a local symmetry and turns out that something magical happens once we do that. And the thing we do, of course, we want to do something highly non-trivial in order to achieve that purpose. And the thing we do is to put in the gauge field, is that we're putting these degrees of freedom in the toracal, these tau fields. And we couple the tau field and the sigma field in a certain way such that the total model become locally symmetric. That is we're actually going to have symmetry acting locally, acting near each lattice site, and the whole model is symmetric under each of the local symmetry transformation. So let's see how that goes. So what we're going to do is to put the tau degrees of freedom on all the edges, connecting nearest neighbor sigma spins, just at the usual location where we did for the toracal model. I'm not sure if the color is clear. So the green dots are for the matter field and the white dots are for the gauge field. But the matter field lives at the lattice site and the gauge field lives on the edges. Thank you. Okay, it's helpful. This one's good. It's to take the matter field, take the gauge field, and couple them. And of course we know what the problem is. The problem is these sigma x, sigma x terms. These sigma x, sigma x terms, they're not locally symmetric. So we want to modify them by inserting some tau operator, such that on the local symmetry, the sigma's transform and the tau also transform and altogether their transformation cancel. So let me show you how that goes. So let's define what the local symmetry looks like. Local symmetry acts around the vertex and there's one matter field at each vertex and there are four gauge fields around the vertex. And we'll define the local symmetry operation as sigma z, which is originally how the symmetry works. We'll act on the matter field and then tau z. Tau z on all the gauge field around it. You can see this is like putting the matter field at the center of the AV terms, putting them together. Of course, now we have made it a local symmetry. This is still a z2 local symmetry, meaning that it squares into identity. So if we started from a global symmetry that squares into identity, we want to still keep that property when we promote it into a local symmetry. So the local symmetry still squares into identity. The weird thing about this local symmetry is that each gauge field transforms under two local symmetry. Each gauge field transforms under local symmetry at around this vertex and around that vertex. But that's what gauge fields do. That's why we want to introduce the gauge field. And the gauge field transforms under gauge symmetry at different spatial locations such that they can mediate the gauge interaction and we can make the model gauge symmetric. Without doing that, we cannot in general make a model gauge symmetric or locally symmetric. So this is local symmetry. Now we have defined what the local symmetry looks like. Our next step is to make the model locally symmetric. We just take the model and we want to write down all the terms in a locally symmetric way. What do we have to do about this term in order to make it locally symmetric? Do we have to do anything? No, it's already locally symmetric. It commutes with all the local symmetry actions, so we just keep it. We do nothing and in that term we just copy them into the gauge model. But we have to do something here because this term, originally, it's not symmetric under these kind of local actions because one of the sigma x is going to anticommute with the sigma z and it's not invariant. So what can we do in order to make this term locally symmetric? Yes, exactly. We can insert a tau x in between such that the tau x sitting here is going to anticommute with the tau z in the local symmetry action and altogether they commute. So sigma x i tau x i j meaning that this tau field sits in between the nearest neighbor, sigma and we sum over the i j terms. So this is the way that we can make globally symmetric terms locally symmetric. And actually it's not just restricted to nearest neighbor coupling terms. Let's say we add a coupling term that's diagonal on the square lattice. Sigma x, sigma x. This is again a symmetric term, right? Symmetric under global symmetry. And we want to make it locally symmetric. We want to make it symmetric under all the local symmetry actions at the vertices. What can we do? Maybe it's easier if I make it horizontal. Let's not make a term. We have two sigma x coupling that way. How can we make it locally symmetric? Yes, exactly. We take two tau fields. And you can check that it's locally symmetric under symmetry at this vertex, at this vertex, and at this vertex under all of the local symmetry actions. So the claim is that when you introduce a gauge field like that and define your local symmetry like this, any global symmetric terms, any term that's symmetric under the original global symmetry can be made locally symmetric by coupling into them some of the gauge field along the way. I cannot prove that. I can more or less prove that because this longer distance coupling term, they can be obtained by composing nearest neighbor coupling terms. We just have sigma x, sigma x, tensor sigma x, sigma x, and the middle one cancels and becomes a longer-term coupling term. So if we can make the local ones locally symmetric, then we can make a longer distance when locally symmetric as well. They don't have. They're independent operators. So I didn't get your question. So this is the local symmetry. Maybe that's the thing that was not clear. This is the local symmetry action at each vertex, around each vertex. It involves one sigma z and four tau z. The tensor part of the five gives you the local symmetry action. So you can check that if you only have this term, it's not going to commute with all of these kind of local symmetries. Only if you make it into sigma x, tau x, sigma x does it commute with everything. So the question is, for example, if we have sigma x, sigma x coupled diagonally, we can choose different paths to put the tau, right? We can put tau along this path or we can put tau along this path. That's a good question. I cannot address it now, but later we can see that this is actually equivalent if it enforces the zero flux condition. This is something I'm going to come to, I don't know, in five minutes, hopefully. But that's a good point. Okay, so the claim is that as long as we have a globally symmetric Hamiltonian, by putting in these gauge fields we can make them locally symmetric on the local symmetry action that looks like this. Yes. Yeah, right. Yes, good question. Okay. So the reason is that if you do that, then something magical happens. For example, there can be emergent topological order. Of course, the original reason is that people find that nature works in this way, that the standard model is made up of gauge theories for whatever reason, we don't know. Nature is all the couplings, strong weak electromagnetic couplings that are generated by gauge fields. So it's not why we want to do it, it's just that. In condensed matter, it's more about coupling to a gauge field so that emergent topological order can come out. And this actually becomes a very useful way to identify what is the order in the symmetric model itself. For example, if we are in a symmetric phase or in a symmetric breaking phase and you gauge it, you're actually going to get different result, which is something I'm going to talk about tomorrow. I don't think I'll have time today. But we can very explicitly see that starting from different phases under global symmetry we end up with different gauge theory phases. So where am I? So now we made the matter field all the terms in the matter field Hamiltonian locally symmetric. This is good. But this is not the end of the story because now we have introduced a lot of gauge field but we still have the same number of Hamiltonian terms which means we'll have a lot of degeneracy in the system. So we need to actually talk about what happens to the gauge field themselves. If the gauge field will just let them be, then there's too much degeneracy going on. What happens to the gauge field is that we want to impose a condition that blocks around a placette is zero. So this is a condition that's consistent with the local symmetry. So we want to add to this actually the BP terms. And the BP terms here is pure gauge term. So it's a tensile bottom of four tau x around the placette. And we're going to interpret that as the zero flux condition. Of course in the continuum if you take the continuum limit it just means that there are no magnetic field. Zero magnetic field because magnetic field costs energy so we don't want magnetic field. In the discrete case we just impose the zero flux condition on every placette. And that's it. This is our gauged Hamiltonian. This is the original one. This is the gauge one. The gauge one involves all the original terms made locally symmetric and also the flux term. Oh yeah, sorry. Thank you. That is a good point because that's something I'm going to use. So there's a J here. Okay. And now it's the moment of truth. We're going to see that why taking this transfer field ising model coupled to the gauge field into a gauge version of the model is related to the tariff code. Yes. If we don't impose the BP term it will be too much degeneracy. So this degeneracy can be removed by local terms like BP. So if you have a lot of degeneracy that's removable by local perturbation it's generally bad. So I want to put in something that removed the degeneracy in the gauge field but also retain the local symmetry and BP turns out to be the simplest term and that's that. Okay. Yes. This is a Z2 symmetry so it's a Z2 charge so it's either zero or one. There's no integer labeled charge. It's either even or odd. If it's odd then it's coupled. Yes. And we're going to see. We're going to see. So that question becomes how do we solve this Hamiltonian, right? This in principle is doable at least in the limit where j is very large or j is very small because if j is very large or j is very small ignore either of the term this thing becomes commuting Hamiltonian again because the BP term they commute with themselves and they commute with the local symmetry constraint and they also commute with these things. No, sorry. The BP term also commute with any of these gauged terms that come from the original Hamiltonian. So let's see one of the limit where j is equal to zero when j is equal to zero the original Hamiltonian is very simple. The original Hamiltonian is just transverse field so we know that the ground state doesn't break the symmetry there's a unique ground state which is pointing everything pointing in the z direction and let's see what happens to the gauge Hamiltonian if we set j equal to zero. So if we set j equal to zero the gauge Hamiltonian involves the transverse field and then also the BP term. It still looks a bit complicated because there are all these gauge constraints there's gauge field, there's matter field but what if we can get rid of the matter field and indeed under this gauge constraint a single sigma z is equivalent to the product of four tau z. So we can just replace the sigma z by the product of tau z which is actually the AV term that we had before and of course we have used the condition that we kept the local symmetry we enforce the local symmetry such that we can integrate out the matter field in a sense. We get rid of the matter field so that we get a pure gauge theory with only the AV term and the BP term which is exactly a tau r code Hamiltonian. Right because we have this local symmetry and we enforce the local symmetry to be unbreakable. So this is just the constraint on the Hilbert space that we consider. Everything has to be eigen value one under all these local symmetry and it's possible because all the local symmetry can meet with each other. That's the sector of Hilbert space that we'll be looking at we don't want to consider sectors where gauge symmetry is broken. In that case the sigma z becomes equivalent to the product of tau z and we can just do the replacement. So just as a conclusion we see that we started from the symmetric phase transverse field ising model with global z2 symmetry and what we did is to gauge it or more especially we coupled to z2 gauge field such that it becomes locally symmetric. That is tau r code with topological and one last thing I want to point out before we end the lecture today is why the charge is both why the charge excitation is a both sound in tau r code. You can see where the charge excitation comes from right charge excitation which is violating the product of tau z around the vertex is equivalent to violating a single sigma z at the lattice point because we enforce the gauge symmetry so they're equivalent so the charge excitation if you interpret that in terms of matter field that's just violating a single sigma z that's just flipping a spin and flipping a spin is a bosonic excitation there's nothing fermionic to it so we don't really have a fermion excitation so all the charge the symmetry charge because flipping a spin changes the symmetry charge changes the symmetry charge from 1 to minus 1 or minus 1 to 1 and that is a bosonic excitation and that's exactly what you see when you use the string operator for the e-excitation and do this figure of 8 you get 1 and you should get 1 otherwise we'll be in trouble. I'm going to continue the discussion by putting J very large and ignore this term and we're going to see what happens when we're in that limit. Okay, thank you.