 Thank you. Welcome back. So this is the third and last lecture of this topic. So I'll be talking about gauging symmetry protected topological states and also gauging just some simple states with so-called subsystem symmetry in order to get what we now call fractal order. Not exactly topological order. I would prefer to not call it exactly as topological order just as fractal order. But I'll explain what I'm talking about with very explicit examples. But before that, I think I need to clarify something because I got a lot of question yesterday regarding the idea of spontaneous symmetry breaking. And I totally agree that this is a very important notion, but it's also a very confusing notion. So let me say something before we actually get to talk about the main subject of today. Continuous symmetry breaking. So first let's deal with global symmetry. So for example, like in the Ising model we had yesterday, where the coupling is sigma x sigma x ij. The whole system has a z2 symmetry, which is the tensor product of all the sigma z, right? So what does it mean to spontaneously break the symmetry? Well, the usual explanation is that if you try to find the ground state of this Hamiltonian, you will find there are two ground states. One of them is all the spin pointing in the plus x direction. And the other is all the spin points in the minus x direction. And usually we will just say that because each ground state breaks the symmetry, and actually each of the map to the other, under the symmetry transformation. So we'll say it's symmetry breaking. And this is a statement which is OK if you really understand what's going on. But not OK if this is used as a complete justification for symmetry breaking, because we're dealing with a quantum system. If we're dealing with a classical system, we can talk about the number of ground states. But as a quantum system, we can only talk about the dimension of the ground space, meaning that the whole ground space is now actually two-dimensional. And any superposition of these two states are ground states of the Hamiltonian. And in particular, we can make superposition of these two states. We can just do an equaway superposition, or we can sum them up with a minus sign. And we would end up with some wave function which actually are invariant under the global symmetry transformation. If you apply the z2 symmetry onto either of these two states, you either does nothing, or it adds a minus sign to the whole wave function. So then we can just, as well, choose these two states to be the basis states of our ground space. Then in what sense is the system symmetry breaking? Both of these states, they are well-defined eigenstates of the symmetry transformation. But of course, spontaneous symmetry breaking is still a legitimate idea, even though we're doing quantum mechanics. And the reason is that, well, the reason we're simply stated is that in a many-body system, we would prefer to think about states like this, but not states like this. What's wrong with states like this? States like this are very big superposition of totally different wave functions. So these two wave functions, where this means either pointing in the plus direction or in the minus direction, they are like two universe. And when we make a superposition of them, it's like saying that we want the Schrodinger's cat to be both alive and dead. We know that macroscopically, it's usually not very stable to have this kind of macroscopic superposition. And usually, as long as you have some perturbation in the system, the big superposition will decay into either one of them. But let's try to make the statement more precise. What do we mean when we say we like these kind of states, but not these kind of states? Because we know that wave function can always be written as superposition of something, even though it's a tensor product states, we can write it as superposition of something. So what is the criteria for saying that which wave function we want to look at when we try to determine whether the system is undergoing spontaneous symmetry breaking? The criteria here is correlation, correlation function. So this is a product state. And for product state, if you calculate something called the connected correlation function, it's going to be 0. Connected correlation function, can you see it down here? I should move up there. Connected correlation function is the expectation value of 2 operator 0102 minus the expectation value of 01 times the expectation value of 02. What we do is that we put 01 here and 02 there and separate them by some distance. And then we try to make the distance become larger and see how this correlation function changes with distance. Using this kind of function, we can distinguish these kind of states from this kind of states because you can see that with some choice of 0102, these two big superpositions, they can actually have very long range correlation. Well, for these product states, product states can never have a long range correlation. For product states, as long as you have two non-overlapping operators, the connected correlation function is exactly going to be 0. And more generically, we might have a Hamiltonian that's not exactly solvable, and we're going to have some wave function, ground state wave function that's not exactly tensile product states. But then we can still look at the connected correlation function. We can require that the basis state we look at to be short range correlated, meaning that the correlation function decays with distance in an exponential way. And that is when we say, OK, this is the wave function we want to look at. Of course, product states are the limiting cases where the correlation function just go to 0 beyond certain finite distance. But usually there's a tail of decay, but the decaying tail is very fast. It goes exponentially with distance. So the way to tell whether there's spontaneous symmetry breaking is to, the first way is to look at short range correlated, ground state, and see if it breaks symmetry. So when we require short range correlation, these kind of big civil positions, they're not allowed. We can only have these kind of wave function. And we find that each of them has to spontaneously break symmetry. And the other way to do it without saying that we prefer certain ground states more than others is to use the idea of an other parameter. An other parameter is something that should have a zero expectation value if symmetry is not broken. So this is something that has a non-zero symmetry charge, for example. In this case, usually the kind of other parameter we can choose is the operator like sigma x, which is odd under symmetry. If the whole system is symmetric, if nothing's being broken, then the expectation value of that operator is going to be 0. So we can either do sigma x, or we can do sigma y, or we can do some more complicated operators. But as long as it transforms non-trivially under the symmetry, then we can consider it as an other parameter. So with other parameter, again, we can calculate correlation function. We can actually calculate, sorry. So what I'm writing is that when we calculate correlation function, we can actually calculate it. We can actually calculate on these symmetric states. And in a symmetry breaking situation, it's actually going to give a very long tail. For example, we can calculate on this wave function, or we can calculate on this wave function. And you see that in both cases, there's going to be a non-zero expectation value for this correlation function, even when we take the distance between r and r prime to be very big. So we can do either one. But they both involve, in general, calculation of some correlation function and taking the distance to infinity, and then see in a thermodynamic limit whether there's real spontaneous symmetry breaking going on or not. So this is, again, emphasizing that spontaneous symmetry breaking is a notion that's only valid in the thermodynamic limit. And if you have a finite system, it's actually quite not very easy to tell if the system is spontaneous symmetry breaking or not. For example, when we do numerical calculations, if we take a Hamiltonian like that plus b sigma z and do numerics, and we try to look for the ground state and try to tell if the system is spontaneous symmetry breaking, what will happen in a finite system if you look for the ground state? The ground state is always going to be something like this, not exactly like this, because there's a non-zero b. But it's going to be something like this, which is actually invariant under the symmetry. Because as long as there's some tunneling between the two sectors, in a finite system, this state is going to have lower energy than this state. So in a finite system, you're doing your marks, you're always going to see there's a unique ground state and a unique ground state satisfy the symmetry. And the only way to know whether the system is actually spontaneous symmetry breaking or not is to take the system size larger and larger and then do either one of those things. And only when the system size is large enough so that you can see the trend of correlation function of all the parameter, so on and so forth, can you tell whether the symmetry breaking is actually happening? OK, so this is regarding global symmetry. Any questions? Yes. Oh, yes, yes. Some local operator which transforms non-trivial under symmetry, which are usually charged under symmetry. OK, yes. Yeah, exponential. Yeah, but there are different ways of being slower. And there are polynomially slower polynomial decaying correlation function or it can be just not decaying at all. It's going to infinity. So in that case, the superposition between plus state and minus state, that's infinite correlation length. But no, no, no, sorry. What I'm trying to say is that the correlation length actually doesn't, the correlation function doesn't decay. But if we have a critical wave function, like a gapless wave function, that's going to have usually a polynomial decaying wave. Are there some other questions here? Oh, they're just local. And 0102, so just this statement is that for any 0102, we want this to be short-range entangled, or it's short-range correlated. Now we require that we are looking at a ground state which is short-range correlated for any 0102. OK, so this is for global symmetry. And then yesterday, because we were talking about gauging, someone was asking, what is this idea of spontaneous symmetry breaking of gauge symmetry? Or actually, what is this idea of there's no spontaneous symmetry breaking of gauge symmetry? So there's actually a paper which says, this is by Alice Huar. And the paper is about impossibility of spontaneously breaking local symmetries. So I can give you the reference. 1975 paper. And the title is just impossibility of spontaneously breaking local symmetries. OK, so the story was that people were studying some model with gauge symmetry, and the kind of gauge symmetry that we have seen in the previous lectures, where you have some local action of symmetry, and then you have independent symmetry operators everywhere. And one of the confusion at the time was whether the ground state wave function can develop non-zero expectation value for some other parameter, meaning that an operator, which is actually the vector potential, whether the ground state wave function can have a non-zero expectation value for the vector potential. And the author of this paper is trying to say, that's not possible. And if you get the true ground state, the true ground state is always going to have a zero expectation value. That symmetry is not broken in the ground state, in at least in a spontaneous way. If you explicitly break the gauge symmetry, that's what was caught in this paper called gauge fixing. If you gauge fix, meaning that you spontaneously add term to the Hamiltonian, which breaks the gauge symmetry, then, of course, the symmetry is already broken. There's nothing against having a non-zero expectation value of some other parameter in the ground state. But unless you do that, then that's not going to happen. And having explained spontaneous symmetry breaking for global symmetry, I think it should be straightforward to, at least intuitively, to understand why it cannot happen for local symmetry. Because the fact that a global symmetry can be spontaneously broken is tightly related to the fact that we have different ground states. They're globally different. And the way to connect one to the other is by applying this global operation of flipping all those things, which is a highly non-trivial thing to do. And spontaneous symmetry breaking only happens in the thermodynamic limit. Only when the system size is very, very large, only when flipping from one guangxi to the other is a very hard thing to do do we have real spontaneous symmetry breaking. Imagine what would happen for a local symmetry. If you locally break the symmetry, you can imagine that you do something locally to flip the system around. Originally, for example, you have a spin pointing in the plus direction. You just flip it to be in the minus direction. And you can make a simple position between these two configurations, which are different in a local way. Then you restore the symmetry. So these kind of local symmetry breaking can always be restored using some local perturbation to the system. And that is exactly why local symmetry cannot be spontaneously broken. Spontaneous symmetry breaking is a notion that only applies when you have very, very big symmetry operators which map the system in a global way. So actually, if you, of course, this is all related to the Higgs model that we talked about. Remember that we started from the symmetry breaking phase of the Ising model. And we gauged it by inserting some tags into the Ising coupling term, adding some blockhead term, and then also enforcing, and then also adding the gauge symmetry term into the Hamiltonian, which is sigma z tau z tau z tau z tau z. OK. OK. So you can actually just play with this very, very simple toy model and see what happens in terms of symmetry breaking. For this model, we know that it has spontaneous symmetry breaking, that there are twofold degeneracy in the ground states and one map to the other under the symmetry transformation. For this Hamiltonian, you remember, we explicitly argued that there's a unique ground state. There's a unique ground state, and all the other excited states have a gap away from the ground state, meaning that there's no breaking of symmetry. So this Hamiltonian, of course, still has the global symmetry. It still is invariant under the global symmetry of the Ising model, but the spontaneous symmetry is gone. The spontaneous symmetry breaking is gone. It is somehow absorbed into the fluctuation of the gauge field. And once the gauge field is here, the gauge field interacts with the spontaneous symmetry breaking state with the condensate and then get rid of it. So no spontaneous symmetry breaking for global symmetry, no spontaneous symmetry breaking for local symmetry, no symmetry breaking at all for this Hamiltonian, and there's a unique ground state. So this is somewhat related to the usual Higgs mechanism that we were talking about when we say that, for example, in a superconductor, the charge conservation symmetry is breaking down to Z2. So usually when a U1 symmetry, a continuum symmetry, is broken down to some discrete symmetry, there will be go-stone walls. There will be some gapless walls because of the symmetry breaking of a continuous symmetry. So once you couple it to a gauge field, and because of the coupling, the go-stone wall is gone. And the gauge field comes in and fluctuates together with the condensate. Not only the go-stone wall is gone, the photon wall of the gauge field is gone, that's because they combine together and both of them disappear and the whole system becomes gapped. Because that is a more advanced version of the Higgs mechanism. This is a very toy model of what is happening. In our case, there's no continuum symmetry, there are no gapless walls, but the physics is actually very similar. Okay? Okay, I spent a lot of time on this. Yes? No. No, HG does not. HG has unique ground state on any manifold. So, the one that does have ground state degeneracies, which when we gauge the symmetric limit of the Ising model, which we end up with Tori code. So, let's get to the topic of today about gauging, so-called symmetry protected topological states and end up with something called twisted gauge theory. Okay, you might not have heard either of them, but no worries, I'll give you a very, very explicit example about what I'm for for this topic. And there's actually a very nice reference for this part of discussion. And for some other part of discussion, so just write them down. So, this is a paper by Michael Levin and Zhen Chenggu in 2012, PRB. And the title is, Braiding Statistics Approach to Symmetry Protected Topological States. And I highly recommend that you read this paper because it's very nicely written and a lot of details are explained are contained in this paper. And another one I maybe you will find useful is Electronode by Kitayev and Chris Laumann back in 2009. And this was never published, it's only on archive, archive number I'm writing down here. And this paper is called Topological Faces and Common Computation. This paper contains a very detailed explanation of toricode, of the excitation of toricode and more generally about topological phases. And in particular, it talks about the phase diagram of toricode, which I don't have much time to talk about in this lecture, but I got a lot of questions about it. So if you want to learn about the phase diagram, about the Higgs transition, about the confinement transition of toricode, you can look at this paper. Of course, this lecture note actually contains a lot of other useful materials, like the discussion of Mariana chain of the two-dimensional hexagonal lattice model, which are all very useful. Okay, so for the discussion of Symmetry Protect Topological States, engaging them, this is the paper. The first one is the paper that talks about them. Okay, so I took the pain to draw a triangular lattice. This is as good as I can do. Somehow I cannot use square lattice anymore. I'll tell you why in a little bit, but the fundamental physics is still very similar. We're going to have matter field living at the vertices of the triangular lattice and we're going to have gauge field living on the links. So the green dots are the matter fields, the sigma degrees of freedom, and we're going to have the tail, the gauge field living on the edges. So I'm actually going to talk about two different Hamiltonians, two different systems in parallel, just for comparison. The first one is, well, the first one is the one we have been talking about the whole time, the symmetric phase of the Ising model. So the Hamiltonian is simply sigma z i summed over i. Maybe I should write v, okay, let me write v. Sigma z v at the vertex and sum over the vertex. This is the symmetric limit of the Ising model. The second Hamiltonian looks very similar, but with a little bit twist added to it. There's still a sigma z part, but then it's multiplied with some phase factor. And the phase factor, first let me write it down. It's one minus sigma x w sigma x w prime over two multiplied over triangles v w z. W prime. So if this is v, this green dot is v, then we find the triangle, which contains v and two other vertices, one is called w, the other is called w prime, okay. So there are two extra matter fields at the w and w prime points. And the phase factor that we multiply onto the sigma z is for each of such triangles, okay. So there are six triangles like this, right. So v is involved in six triangles and for each of the triangle, we add a phase factor like that. And the phase factor is related to the difference between sigma x for w and sigma x for w prime. If they're the same, there are no phase factor. If they're opposite in the x direction, there will be a phase factor of i. Is the notation clear? Complicated. Hamiltonian now involves seven. Seven matter field degrees of freedom at the same time, right, looks terrible. And I don't think without insight, anybody would be able to solve the Hamiltonian. But the thing is that we do have insight coming from. Somewhere, such that we know that there's a, this is again exactly solvable Hamiltonian, meaning that all the Hamiltonian terms commute. So let me label this one as dv. And you can explicitly check that dv and dv prime commute with each other. Amazingly, even though they look so ugly, they still commute with each other so that you can, there's a unique ground state to all these Hamiltonian terms. And the second reason why we're talking h0 and h1 together is that they're both symmetric under the global z2 symmetry, right? But obviously the first one is symmetric. And the second one is also symmetric because it involves the product of two sigma x at the same time. So if you apply the symmetry, this term doesn't change, this term doesn't change, now Hamiltonian is totally invariant. So these two Hamiltonian, they're invariant on a global symmetry, and we can ask, does the ground state spontaneously break the symmetry? So for the first one, it does not, for the second one, it does not either. So I'm actually going to tell you what the ground state wave function is like. But the thing is, these two Hamiltonian both have unique ground state, which doesn't spontaneously break the global symmetry. However, these two ground state wave functions are very, very different. And there's no way to map from one to the other in a way such that we preserve the global symmetry. Okay, so when the global symmetry is preserved, these two Hamiltonian actually belong to different phases. This is what we call the symmetry-protected topological phase. That is, the H zero is in the symmetry-protected trivial phase, and the second one is in a non-trivial symmetry-protected topological phase. Of course, this might sound not very familiar, but I'm sure that you probably have all heard about topological insulator. How many of you have heard about topological insulator? Okay, good. Topological insulator is exactly a symmetry-protected topological phase in Fermion system with charge conservation and time reversal symmetry. The different system of degree of freedom, the different set of symmetry, here's the unitary Z2 symmetry in topological insulator case, it's time reversal and charge conservation symmetry, but the idea is the same. As long as you keep the symmetry, there's no way to deform these non-trivial symmetry-protected topological states to the trivial one. And in the topological insulator case, you can never deform a topological insulator into a atomic limit of a mod insulator, for example, while preserving the whole symmetry of the system. Okay, so let me tell you what the ground-state wave function looks like. It's actually not that terrible. So let's first think about what the ground-state wave function looks like for H zero. H zero is simple enough. And we can immediately write down what the ground-state wave function is. Ground-state wave function is a product state. All the spins are polarized in the Z direction. So we have a tensor product between all the zero states. So this is simple enough, but it's too simple. We want to write it in a more complicated way. Let's expand it in the basis of plus-minus states of the eigenstates of sigma x, because we know that zero state can be written as a superposition of plus and minus. I'm not keeping track of normalization, right? So we can write it like that. And then we expand everything. So we get configuration like all plus, and we get a configuration where we have all plus, but one minus. And we have configuration where we just have plus-minus, plus-minus. And then we have a superposition of all possible spin configuration in the x direction. Yes, yes, yes. Yes, eigenstates, yes, sorry. I should make that clear. So sigma z acting on zero is zero. Sigma z acting on one is minus one. And sigma x acting on plus is plus. Sigma x acting on minus is minus, minus. That's my notation. So this gives us another way to interpret the symmetric wave function of h zero, right? Because if we only look at one term, this is a configuration that breaks the symmetry. Usually we call it a domain configuration because it's something like a magnetic domain that breaks the symmetry of the system. And we have a domain configuration where all spins are plus, and we have domain configuration where all the spins are minus. And now we also have domain configuration of all kinds. So the symmetric wave function is actually a superposition over all domain configurations. So we can start from one domain configuration, allow the spin to fluctuate, and then we make a superposition of all possible ways the spin can fluctuate, and in the end the fluctuation will restore the symmetry and get us back to the symmetric ground state. That's the way we can interpret the symmetric ground state as a way starting from the symmetry breaking ground state. Of course, this is just making a story too complicated, right? We had a product state, we want to write it in an expansion and as a superposition of exponentially many terms, but why do we want to do that? And the reason we want to do that is now we would have a very easy way to write a wave function of h1. So this is a sum over all domain configuration. It turns out, I won't be able to show you, I'll just claim that the ground state wave function of psi one can also be written as the superposition of all domain configurations with a twist. And a twist is the phase factor of minus one to the number of domain walls. Okay, what is a domain wall? Well, if all the spins are in the plus direction, there's no domain wall, right? If all the spins are in the plus direction, but this one is in the minus state, if only this spin is in a different state than all the others, then there will be a domain wall around it. You can draw a domain wall between this particular state, between this particular spin and all the others in the system. If we have another spin, which is in the minus one state over here, then we have another domain wall, okay? So we can count the number of domain walls. If we only have this, we have one domain wall. If we also have this, we have two domain wall. And the coefficient in front of the configuration is minus one to the number of domain wall, so we care about the parity of the number of domain wall. And of course, here I'm only drawing very, very small domain wall. We can have bigger domain wall. We can have a whole patch of spins in the plus direction while the others are in the minus direction. So we have a bigger domain wall like that. We can always count. Yes, it can be odd. For example, if only this dot is plus and all the other spins are minus, then there's only one loop of domain wall, right? Then that means it's one. And that kind of states gets a minus sign. What, sorry? The domain number of loops for the domain wall. So it's domain walls. So how many loop of walls there are in the configuration? Yes, a loop type because this is, we're talking about closed system. Domain walls have to be closed. And now I can explain why I have to take the pain to draw the triangular lattice, not the square lattice. Because in a square lattice, we might have domain wall configuration that looks like this. We have two square that touch each other, right? And then there can be ambiguity whether there are two domain walls or whether there are just one domain wall. Because it depends on the detail at the crossing whether the crossing is like this or whether the crossing is like this, right? So this will give you different solutions. So we'll try to avoid a square lattice but rather go to a triangular lattice. And the nice thing about triangular lattice is that the domain wall live on the dual lattice of hexagonal lattice. And hexagonal is a trivalent lattice meaning that you always only have three links going into each other. So there will not be a confusion as to whether they're crossing or not or whether they go this way or not way. Very important that we do this. But of course, it's not like we can no longer talk about square lattice anymore. If we do want to talk about square lattice, we need to just be very careful and define what happens at the crossing. You can just take the square lattice and make it look like that at each vertex and just split the vertices into trivalent and then we're fine. So that's the two-way function. One is just the superposition, equal weight superposition of all the domain configurations and the other is the equal weight superposition of all the configurations up to a phase factor depending on the even and oddness of the number of domain wall. Okay. And you can see that Z2 symmetry is indeed preserved. Okay. Because in this case, of course, it's easy to see why Z2 symmetry is preserved because Z2 symmetry flips all the domains. Flips all the spins. So it maps from one configuration to another configuration, which is like all minus. It always maps between different configurations which are involved in this big superposition so the big superposition doesn't change. On the other hand, for this side one, it is also invariant under the Z2 symmetry action because the Z2 symmetry action maps between different domain configuration. But for those two domain configuration, they have the same domain wall, right? It doesn't matter whether the inside is plus or whether the outside, sorry, whether the inside is plus or minus. As long as inside and outside are different, you have the same domain wall configuration. So Z2 symmetry action doesn't change the number of domain walls and doesn't change the plus minus sign in front of the superposition. So the total wave function is still invariant under the global symmetry, meaning that there's no spontaneous symmetry breaking in the system. Nice. This is saying that we have two different Hamiltonian invariant under global symmetry and they each have a unique symmetric ground states. Now let's try to gauge them. So we're just doing the same thing we have been doing in the past two lectures. We have some matter field at the vertices and if we try to gauge the global symmetry, the first thing we want to do is to put in some gauge field. And we know that we want the gauge field to live on the edges. So we add a gauge degree of freedom tau and tau again, this has been one half. It's a qubit degree of freedom with operator tau X and tau Z and we add one per each edge in the system. So I'm sure you can just already write down the gauge Hamiltonian for H zero in a very straightforward way. I've been talking about that all the time. So to gauge H zero, the first thing we need to do, well, first thing we need to do is to talk about gauge symmetry. What is the local symmetry action that we want to preserve while we promote the Hamiltonian into a gauge version? It will be the same as what we have been doing all the time except that we are doing it on the triangular lattice. So there will be six gauge fields associated with a single matter field. So there will be a sigma Z and then six tau Z. Tau Z, tau Z, tau Z. So it's a bigger term, but it's not any more complicated. This is the local symmetry action involving both matter field and gauge field. So now we want to write the Hamiltonian in a way such that it is invariant on the global symmetry. Sorry, under this local symmetry. But for H zero, we don't need to do anything because each of the term is already gauge symmetric. We just keep them there. And then we remember that we need to add some dynamics to the gauge field because otherwise the degeneracy is too big. So we add the BP terms. And the only non-trivial thing here is that the BP's, the placets, they are triangles. They're not squares anymore. And we just, for each of the triangle, we add a term like tau X, tau X, tau X. Finally, we can put the gauge symmetry term as a Hamiltonian term into the gauge Hamiltonian just saying that we consider gauge symmetry as some dynamical constraint but not as a hard constraint on the Hilbert space. This will not change anything for the ground state or for the low energy dynamics which are the things that we care about. So we just put sigma Z and all the tau Z. Tensile product of them into the Hamiltonian. Okay, so I will not write that in detail. Okay, this is for H zero and about for H one. Now we do the same exercise, right? For H one, we take this big ugly term and try to make it gauge invariant. It's actually not that bad because this part is already gauge invariant. We just need to make this part gauge invariant but this part is just, even though it's complicated, it only involves this Ising coupling term. And we know how to make Ising coupling term gauge symmetric, right? We just insert a tau X in the middle. So we simply have this thing and then a product over the triangles that involves V, I to the one minus sigma X w tau X w w prime, sigma X w prime divided over two. So these triangles are the ones that includes V and we're done, right? So the first term, we make a gauge symmetric just like that. And then we just copy whatever we did upstairs. We have flux term, the placket term and then the gauge symmetry term. So we get two different Hamiltonians with the same gauge symmetry, okay? Because the matter field is doing something different. Remember, we summarize what we learned from gauging the symmetric phase of the Ising model the other day and we said a few things, right? First of all, the gauge charge comes from the symmetry charge, right? So, well, in both models, in both cases, there's spin models and the symmetry is this, the two symmetry and the symmetry charge can be simply generated by applying a sigma X flipping a spin, right? So this is telling us that in both cases, no fermions are involved, there are no fermions at all. So we would expect the symmetry charge to simply be a bosonic symmetry charge. So we should, so for either Hamiltonian, if we find them to be topological, we should find that they have one species of particle which is bosonic, which corresponds to the original symmetry charge. And of course, we expect it to satisfy this kind of fusion rule that two symmetry charge should fuse to something not fractional because this is a Z2 charge. So there's only an even-outness to the symmetry charge. Okay? And secondly, we know that this is Z2 symmetry. So we should have Z2 charge braiding around the Z2 flux, giving rise to a phase factor of, of what? Giving rise to a phase factor of, and it can only be pi because this is a Z2 symmetry, right? We don't have other options. So this is the harm of Paul's effect between E and M. If you have an E here, oh, sorry, if you have an M here and you bring an E around it, that's going to generate a phase factor of minus one or E to the I. So these things are all fixed. And the only thing that can be different is in terms of the magnetic flux, in terms of M. So M is the flux of the Z2 symmetry, okay? Meaning that if the charge hops around, whether it sees a zero flux or a pi flux. But a zero flux and pi flux, the only thing that the charge can see, so this flux for the Z2 symmetry, again fuels itself into a non-fractional thing, right? Because two pi is equivalent to zero. So if you have a pi flux, fuse it with a pi flux, you get back to having nothing. So we do expect this. What's left for the M particle to be different in these two models is their topological speed. So M zero, which is the flux in the first model. Well, now when we know, because this is exactly the model that we studied the other day, except that it is now a triangular lattice, right? And we can just safely remove the matter field because we can just consider the states which are the ground state of this term. And then we can get rid of the matter field in the center of the gauge symmetry term. And then we have a pure gauge theory without a matter field. And that's exactly the taricode Hamiltonian on the triangular lattice. There's nothing more to it. And it will have the same universal property. It would have the same ground state degeneracy. We'll have same quasi-particle species and their statistics. And they just follow exactly from our discussion the other day. And actually, even though we started our discussion the other day on the square lattice, as we moved on, I forgot about the lattice. Remember, I drew the picture without specifying which kind of lattice I was on. Just because for the taricode, actually we don't really care for all the universal properties. So even though we move on to the triangular lattice, all the properties that we talked about the other day still applies. Okay. So the M flux quasi-particle is bosonic for H zero. For the flux in the second case, turns out to be semi-ionic, meaning that it has a topological spin of pi over two, meaning that if you do a figure of eight for the string operator, you get a phase factor of I. Okay. I can roughly sketch how this works. I won't be able to do the algebra very carefully. If you do want to check the algebra, go to this paper, contains all the details and you can follow through. You can see how things work. But right now I'm just going to roughly sketch. Okay. Why? What we expect should actually happen for these two particular models. Okay. So find the statistics of the quasi-particles. How can we determine the statistics of the quasi-particles? The way we can determine that is by, well, drawing the string operator, right? And if we do a figure of eight, we get the cell statistics. And if we do this kind of configuration and we do it in different order, their computation relation will give us the braiding statistics. So the key becomes finding the string operator for the different species of quasi-particles. And what is string operator? String operator is an operator that creates excitations at the two ends and only at the two ends. Okay. So we need to find some operator that as it pass through the ground state, it doesn't create an excitation in the middle, it only creates excitation at the end. Also, let's see how that works for taricotin, for H zero. For H zero, first of all, we can create flux excitation. You can create one flux here and one flux here. The fluxes, they live in the triangles. So let's try to create a flux excitation. The flux excitation, they're just applying sigma z to this line. Sorry, tau z, tau z, tau z, tau z, to all the edges that cut through the blue line. So doing that, we anti-commute with only the triangle placate operator here and the triangle placate operator there. And that is a string operator, right? Because it only creates excitations at the end. And this corresponds to the string operator of M zero. And then how do we create charge excitations? Well, charge excitations are created if we hop charges using the icing coupling from one point to the other. So one charge is here. Let's say we want to create another charge here. And the way to do it is to apply sigma x and tau x, tau x, tau x, tau x all along the way. And finally, sigma x. So this is the string operator for the charge. So they look exactly the same as what we had in the square lattice sort of code. And then you calculate the self statistics is one. You calculate mutual statistics, these two strings, when they cross each other, they anti-commute, giving rise to the pi statistics, the AB phase factor of a charge with flux, just as we expected. Okay, how about for H one? We now have H one, what changes? What has to change? What does it? This string operator for the charge, that's still okay. All we need to worry about is whether the middle part of the string operator would commute with all the Hamiltonian terms in the middle. Right, because the difference is that originally we have this kind of term, now we have this kind of term. This kind of term, right? So they look different, and in particular, they involve this kind of garbage in the Hamiltonian. So we need to be careful and check whether it still commutes with the string operator. And it does because there's only a tau x here. And the string operator is composed of tau x. And so it just doesn't see the change in the Hamiltonian at all, right? So what the string operator does is still to anti-commute with a vertex term at the end creating two charges, okay? So this string operator is still valid. It is still the string operator of the E quasi-particle, which means that all the properties which involve E itself, still valid. It still fuels itself into trivial, still has a bosonic statistic and so forth. How about the one for m? How about the blue one? One involves a bunch of tau z along a line. But that's trouble, right? Because tau z would see this tau x in this part of the Hamiltonian. So it knows about the change in the dynamics. So that's why I say that the flux knows about what the matter field is doing. The charge is just the symmetry charge, but the flux is somehow gets involved with the dynamics of the matter field and knows the matter field is doing something on trivial. And so this string operator does not work anymore because as it passes along, it also raises the energy of this term. So it's not a string operator anymore. It's not just creating an expectation at the end. It has to be modified. And it has to be modified in a way such that there's some extra phase factors involved in the sigma x spaces on these edges, on the edges that's neighboring this string. So it becomes fatter, it becomes more complicated, but what happens on this edge is phase factor in tau x spaces. And by properly choosing the phase factor on these edges in tau x spaces, we can make sure that the string operator again can mute with all the Hamiltonian terms in the middle of the string. Of course, I'm not going to write down the explicit form of the phase factor. It looks complicated and not very useful if I write it down again in this paper. But then you can see what this phase factor does or it doesn't do. For example, the commutation relation between the green line and the blue line, they're still the same. Because the way we change it is by adding some phase factor in tau x spaces, which can mute with all the tau x in the green string. So the commutation relation between them is still the same, which is minus one, which is exactly what it should be because we expect a pi phase factor for the Harnoff-Bohm effect between charge and flux. And the only way things can be different is when we do a figure of eight. For this more complicated string operator, that is not going to end up being one. And it turns out it ended up being i. So topological spin of i corresponding to a phase factor of pi over two. Okay. So of course, I did a lot of hand waving here. I can't show you the algebra in a very explicit way, but I hope you can trust me and see that this is one example where I want to show you that by gauging, we can tell the difference between different symmetric phases. Yes. The worth, sorry. The n particle is not bosonic, it's semionic with a topological spin of i. A fermionic is topological spin of minus one. So semionic is like half fermion. I think that's where the name comes from. Half or half boson, depending on how you have it. And not everything is consistent. Here only i is consistent. If you put some other random phase factor, it's not going to be consistent. Or the wave function. It's just that i is some special. Yeah. Yeah, I cannot say that topological spin is just that i, but something related. Something. Yeah, but it's only consistent in this way. Otherwise, it's not like if I put a e to the i pi over. There are no other general statement regarding other phase factors here. Sorry? Is it that such thing as written with the. You mean other phase factors? Yeah, yeah. No, I think they only discussed this one. Otherwise, it doesn't commute. It's not exactly solvable. It's not too symmetric. And there's not much more you can talk about. Okay, I think let's take a 10 minute break. And when we come back, I'll talk about the fractal. Okay, so before we move on, so just to clarify what I just said, yes, before the break. So I wrote down these two Hamiltonian, they have symmetry protected topological order. And here I gauge them. And I guess some gauge theory. So if some of these terminology makes sense to you, then these are short range, short ranging tangled phases. These are long ranging tangled phases. These are considered to have intrinsic topological order. These are not, these phases, they don't really have topological order. We shouldn't say that they have topological order, but they have symmetry protected topological order. In the sense that if you enforce symmetry, then there are different phases. That's the only non-trivial meaning of the term symmetry protected topological phase. If you enforce symmetry, these two are in different phase. But they don't have the usual sense of topological order where we have ground state degeneracy on different manifolds, fractional statistics, and all that. All of those things only happen after you gauge it. After you gauge it, you get some intrinsic topological order which have, well, both of them have fourfold ground state degeneracy on the torus, four different species of quasi-particles, grading statistics, cell statistics, et cetera. And if you have ever heard of the name, this is in the toricode phase of normal z to gauge theory. This is a twisted gauge theory and the topological order is usually called double seminal topological order. Okay, so for the last half hour or so, I do want to tell you about something that's much more recent called the fractal model. And the fractal model is something that came out of nowhere. It came out of a study by quantum information people who were trying to ask the question of how do we build a quantum hard drive? So now we have hard drive today for classical computers. But one day if we have a quantum computer, how do we build a quantum hard drive? So people spend a lot of time on that. I think that problem has been studied for maybe 20 years or so. There hasn't been a very successful answer to that question yet. So if you're aspiring graduate student, you can try to think about it, how to build a reliable quantum hard drive. But in the process, in the process of trying solving not very hard question, which hasn't been solved yet, people came up with a bunch of other models. And in the quantum information community, people always come up with exactly solved models because that's mathematically clean and you don't need physical intuition to tell what they're going to do. So they come up with a bunch of exactly solved models. And even though those exactly solved models still cannot make a good quantum memory, they caught the interest of the connect matter community who is asking, what the hell is going on with all those models? Because all those models, they have some very weird property which look like topological order. For example, they have ground-state degeneracy on non-trivial manifold. They have fractional excitations and the fractional excitations seem to be braiding with each other in certain way. They somehow look like topological order. But they also look more exotic than just topological order. For example, these are usually three-dimensional system which have a ground-state degeneracy that increase with system size. So it goes like exponential in linear system size, which is something people have never seen before in a topological order system. And also, there are fractional excitations that does not quite move. Some of the fractional excitations just don't move. They're pinned at a point and they cannot move by themselves. Sometimes they can move in pairs, sometimes they move in a cluster, but by themselves, they cannot move. Sometimes the fractional excitations only move along a line. And those are all very strange because we've never seen that in a usual topological order. In usual topological order, we know we have fractional excitations but those fractional excitations, I can bring them wherever I want. I can just draw the string operator and wherever it terminates, I'll be able to bring this fractional excitation to that location. Right, so the Canis-Matter community started to ask the question of what the hell is those models and what kind of phases do they belong to? And what kind of physical mechanism is responsible for generating such kind of exotic behavior? So what I'm going to talk about next is one way to try to answer this question is that some of these fractal models, they can be understood in terms of gauging system, not with global symmetry, but with something called subsystem symmetry. Subsystem symmetry is a symmetry that doesn't act everywhere. It just acts somewhere. For example, if you have a three-dimensional system, we can have symmetry acting on the XY plane. We can have one symmetry generator acting per each XY plane. So here is symmetry generator, another symmetry generator, another symmetry generator. All of them are Z2 symmetry generators, so we have lots of symmetries. And if we do something called gauging, very similar to what we do there, but also following a slightly different procedure, we'll be able to get some of the fractal models where the fractional excitations don't quite move. Okay, so let me get two examples. So one of the most famous fractal model is called the X-cube model, a three-dimensional lattice model with qubits on the edges of a qubit lattice. So I'm drawing one cube here. And in this cube, there will be 12 edges and hence 12 qubits. Let me call them tau. I call them tau because later they're going to become the gauge field. The X-cube model has a, exactly so of a Hamiltonian, which is kind of complicated, but not too bad. The first term is a cube term, which is a tensor product of tau X over all the edges around the cube. So there will be 12 of them. I miss one, I miss more than one. Okay, so 12 tau X tensor product together gives you one of the Hamiltonian terms. And we sum over all the cubes. Just summing over all the cubes. And the second type of term lives at the vertices. At each vertex, we can take out a cross shape like that. And there will be four degrees of freedom associated with this cross. And we take a tensor product of all the tau Z at this cross. So this is a three dimensional system, so there's three ways we can take out crosses and we do them all. So for each of the cross, we do a tensor product over four tau Z, and we sum over all the crosses. So this is, of course, again, a very strongly interacting spin Hamiltonian involving talk about interactions, which is not nice. But the nice thing about this Hamiltonian is that, again, it is exactly solvable, meaning that all the Hamiltonian terms can mute, which you can check, because you can, the only non-trivial thing is whether these kind of cross term commute with the cube term, right? And you can check that whenever they overlap, they always overlap in two places. You have a cube, and you have one of the cross at the corner here, then they always overlap in two places. All of the terms commute, yes. All right, so this is not the three-dimensional taricode. The three-dimensional taricode is, it involves vertex operator, which is tau Z, tau Z, the tensor product of six tau Z, six of them, and then the placate term, which are tau X, tau X involving all different placates. Well, this one might look like this one, but this one is definitely not this one. Yeah, and they're very, very different in terms of their universal properties. Okay. Okay, then we can ask, well, it's exactly solvable, so let's ask for some of its properties. If we have the system in L by L by L lattice with periodic boundary condition, what is the ground state degeneracy? We can ask that. Turns out that the log of the ground state degeneracy is three L minus six, three L minus six, maybe three, sorry. I do not remember exactly what is the number, but it's some finite number. But the point is that it is something that increase with the system size and the total ground state degeneracy actually increase exponentially with the linear size of the system, which is very fast. And secondly, we can ask, what are the fractional excitations? How can we create fractional excitations and move them? Well, we can create fractional excitations and move them just by applying tau X and tau Z operators. That's what we did for the Tori code. The Tori code applied tau X, created a pair of flux, applied tau Z, created a pair of charge. And here we can try to do the same thing and see what happens. So let's apply a tau X here. I apply a tau X here. I doesn't care about the cube term, but it does care about the cross terms. So the cross terms actually going to be excited, but at each cross, I'm sorry, at each vertex, two of the cross terms are going to be excited. So if I do it that way, this term and this term are going to be excited. So, there will be a red excitation and a blue excitation. I label this as red, this as blue, this as green. So create a red and blue here and also a red and blue here, but we can keep moving, right? We can keep applying tau X, tau X, tau X until we have moved the red and blue to this end. So this is very similar to what we did for Tori code. We created a pair of excitation and we moved them apart. But then the difference is that we cannot make it turn. If we try to make a turn, for example upward, by applying tau X here, we're getting into trouble because a vertical tau X is going to excite the red one, pencil the blue excitation. Well, at the same time, it creates an extra green excitation. So if we do that, well, the red one will follow, the red one will go up, but we will have a blue excitation and the green excitation sitting at the turning point. We can keep doing that, tau X, tau X. Well, moving the red excitation and the green excitation upward. So this is very different from the excitation we see in a two-dimensional top logic order, which can definitely make turns while still keeping their type, right? Here, if we try to make a turn, we always deposit something at a corner, meaning that we cannot make turns. These fractional excitations, they're quasi-particles, but they can only move in a straight line. We have three different kinds of so-called one-dimensional particle. One is a combination of red and blue. The other is a combination of green and blue. And finally, we have the combination of red and green. Three of them. Three different kinds of one-dimensional excitation, one moving in the x direction, the other moving in the y direction. And finally, the third one moving in the z direction. One moving in each direction, okay? Okay, so what happens if we do tau z? Suppose that we have a bunch of cubes, just drawing one surface of the cubes, and I do a tau z. If I do a tau z to a vertical edge, this is going to violate the cube terms, right? It's going to violate four of the cube terms around this edge. So it's going to be one, one, two, three, four. There are four cube excitations surrounding this edge. Is that clear what I'm drawing? No, I can do a better job. So there are cubes. So these two cubes, they become excited, and also these two cubes, right? All four cubes become excited if I apply the tau z. So you can see that by a single tau z, I create a square of excitation. So if I take a top-down view, if I take a top-down view, I would have four excitation if I do something in the middle. And then I can keep doing it. I can keep doing it and separate the excitations, but I always have four, and those four is going to sit at the corner of a rectangle. So of course, now these three are gone, and they move apart, but they separate at the corner of a rectangle. And there's no way, there's no way that I can just move one of them. I cannot just take one of the cube excitation and move it by one step that I cannot do. All I can do is to maybe move a pair of them along a line, and if I try to move one of them, I create three more, right? So these cube excitations by themselves, they cannot move, and they are called the fractal excitations. That's the fractal in the name. Fractal excitations. And these are the one-dimensional excitations. Yes. Oh, okay, let's see. If we try to move it, where do you want to apply the tau z operator? You can try to apply here, right? So this one gets canceled, but then we have three more excitations. So that's not called moving one excitation, it's replacing one excitation with three. No matter how you do it, you cannot just take one excitation and move it because these excitations they're not generated as the end of string operator. They're generated at the corner of a rectangle operator somehow. Okay, so how can that happen, right? How can that, how can something only move in one dimension and how can something only move as the corner of a rectangle? Well, there are different ways to understand this, but the way I will try to explain is of course in terms of gauging something. For the symmetry case, we know that the gauge charge come from symmetry charge. So the motion of the gauge charge is just the same as the symmetry charge and symmetry charge can hop anywhere. So the gauge charge as a fractional excitation can hop anywhere. So if we try to explain this x cube as the result of gauging something, well, we probably start from some system where the symmetry charge doesn't have the full possibility of motion. And that's what happens when we have system with subsystem symmetry. Okay, suppose that again, we're on cubic lattice the matter field is at each vertex. It's again a sigma spin, sigma z, sigma x, the matter field at each vertex. The on gauge Hamiltonian is very simple. It's very, very simple. It's just the transverse field at each vertex. So simple as Hamiltonian can think about with a product ground state. The only non-trivial thing is how we define symmetry. What kind of symmetry do we require? Well, it is invariant on the global Z2 symmetry. It is also invariant under subsystem symmetry that only acts along a particular plane. So we're going to have symmetry on a plane, for example, where the z coordinate is fixed to be z naught. So this is the tensor product of all sigma z v, where v belongs to this plane. For example, we can take this horizontal plane, which would involve one, two, three, four of the spin that I draw here, but of course it also goes infinitely and involves all the other spins in this plane. And similarly, we can have symmetry on xy, on the x plane, or symmetry on the y plane. Very similar definition. Now consider what happens when I define symmetry in this way. Let's consider the case where I just have one symmetry. Okay, let's for a moment forget about these. Let's just say we have symmetry on a particular plane in the z direction. Then the symmetry charge sitting on the plane cannot hop off the plane. If I want to preserve symmetry, the symmetry charge can only hop within the plane. It cannot hop to another plane because the symmetry charge for that symmetry will change and starts violating the symmetry that I want to impose. So if I impose one planar symmetry, one planar symmetry, then the symmetry charge will be constrained to move in a two-dimensional plane only. If it tries to hop off the plane, you violate the symmetry, that's not allowed. Well, compared to the case of global symmetry, if you have global symmetry, the symmetry charge is allowed to hop anywhere. The same universe and it hops anywhere it preserves the symmetry. Now what happens if we have two sets of planar symmetries? If we have one planar symmetry, another planar symmetry and they intersect along a line. Well, suppose that some symmetry charge, like this one, is shared by this plane and also the horizontal plane. And if we try to enforce symmetry in both planes, then this charge can only move along the intersection line. It has to be in both planes at the same time and that means it can only move either to the left or to the right, but it cannot get off the line. And now finally, if we have all three planar symmetries and if this point sits at the intersection point of all three planes, it cannot move anywhere. By itself, it cannot move anywhere because it wants to be on three planes at the same time and there's only one point that's satisfied this condition. So if we start from this model and somehow able to gauge it, we might be able to get some phenomena like that. When the symmetry charge in this case becomes the gauge charge in those fractal models, those gauge charge are going to have restricted motion. Yes. Yeah, symmetry, here I'm talking about symmetry living on a plane can also live on line. Right, like in two-dimension, it can live on line. Even in three-dimension, it can live on line. Okay, right, so now the question is how do we gauge it, right? This is, we have a different symmetry. How do we want to gauge it? Now we have a loss of symmetry generator and we need to decide what's the proper procedure for gauging it. Well, the first thing you want to answer when you try to gauge something is what is the gauge field? The gauge field is probably, again, a Z2 variable because this is a Z2 symmetry, right? But then the second question is where do you want to put it? Where do you want to put the gauge field? So in all the previous examples, we put the gauge field on the edges. Why do we put the gauge field on the edges? Because we know that the kind of term that's not locally symmetric, which is something that we want to solve when we try to make the symmetry local, is this kind of Ising term. This is the problem we need to solve. It's not these kind of terms. These kind of terms, they are locally symmetric, but we need to solve the problem of Ising coupling. And because of that, we put a gauge field in between and solve the problem of Ising coupling and make it gauge symmetric. Here, if you want to decide where we want to put the gauge field, we need to ask, what is the counterpart here? What is the Ising coupling that we can put in the system right now? Can we do sigma x, sigma x? Is this a coupling that's invariant under all the symmetries? Let's say we have all three symmetries. We have symmetry in x direction plane, y direction plane, and z direction planes. Is this term invariant under all symmetries? No, because if you apply symmetry in the horizontal plane, you only touch it at one point, right? The other two directions are actually fine, but if you apply symmetry in the z direction, it touches the coupling at one point. Therefore, it's not symmetric under that z two symmetry. So the minimum term, the minimum Ising coupling term that we can add is actually a full body term involving four spins around the same square. And that always works. You can check that no matter whether you do symmetry in xy plane or yz plane or zx plane, it is always symmetric because it touches the interaction at two points. And because each of such Ising coupling term is associated with one square, one placette, that is where we want to put the gauge field. Put the gauge field at the center of each placette. So for each cube, we have six tiles. One, two, three, four, five, six. We have six of them, okay? And then the story just falls, okay? What do we need to do? We need to take the original Hamiltonian, try to gauge it. The matter field term is already gauging variance, so we just copy them. Then we need to add the local gauge symmetry term. What is the local gauge symmetry term? The local gauge symmetry term involves the matter field and the gauge field around it. The gauge symmetry term is also going to be the matter field and the gauge field around it. How many gauge fields are there around it? You can see the picture, this is the picture here. This is the matter field. On all the little squares, there is a tau and there are 12 tiles around it. So the gauge symmetry term is the product between one sigma z and 12 tau z, 12 of them. Come over the vertex. Finally, we need to add something that looks like a flux term. What is a flux term? Why do we want to add a flux term? A flux term is the minimum pure gauge field term and commute with the gauge symmetry, right? The term in terms of tau x, but still commute with the gauge symmetry. And it turns out just to be a full body tau x term around a cylinder. I don't know how, let me try to draw it. Let's see if I can draw it. Tau, tau. And another one is tau, tau, tau, tau. Sorry, tau x, tau x, tau x, tau x. What do I want to do? Tau x, tau x, tau x, tau x. So four tau x around the side surface of the cube, that one commute with the gauge symmetry and it's a minimum term that we can add for the pure tau x. Oh yeah, yeah, yeah, sorry. So this is, that's not helping. Sorry. How do I do it? Okay, maybe I can do this. So this is consider an open cylinder with boundary at the red edges and open cylinder with edge with boundary at the red edges. And we have tau x on the side surface of the cylinder. Okay, so it's side surface of the cylinder, one way, the other way, and the third way. And you put four tau x on the side surface. All right. And I have one more minute because I'm almost done. So this is exactly the X-cube model. If you get rid of the matter field just like we did for Torico, you say that, okay, let's be in the ground state of the sigma z term. So I get rid of the sigma z here. This becomes exactly the same as the X-cube model that I drew there if you realize that in three-dimension cubicle lattice, surfaces are due to edges. So just replace surface by edge. You get from one cubicle lattice to another cubicle lattice. But here degrees of freedom are on the edges. Here degrees of freedom are on the surfaces. They're actually the same. It's just drawing the lattice in different ways. And then gauge charge. This is the gauge symmetry term. Violating this term gives you the gauge charge. And we know that the gauge charge is a fractal excitation which cannot move because the gauge charge transforms under symmetry in all three directions. So it cannot move, it's just pinned at the point. Questions are welcome.