Spontaneous symmetry breaking from anyon condensation
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Lu, Yuan-Ming
In the context of quantum spin liquids, it is long known that the condensation of fractionalized excitations can inevitably break certain physical symmetries. For example, condensing spinons will usually break spin rotation and time reversal symmetries. We generalize these phenomena to the context of a generic continuous quantum phase transition between symmetry enriched topological orders, driven by anyon condensation. We provide two rules to determine whether a symmetry is enforced to break across an anyon condensation transition or not. Using a dimensional reduction scheme, we establish a mapping between these symmetry-breaking anyon-condensation transitions in two spatial dimensions, and deconfined quantum criticality in one spatial dimension.
 the hotel room last night. I don't know whether you guys had similar experience. They had an outburst of facts last night. OK, so today I want to talk about a topic which is really to actually to quantum phase transition. So I realized most of the previous talks focused on quantum phases, such as topology in Sweden, and some of them topological orders. So today the topic is slightly different. I want to talk about certain universal property of phase transitions. Is this working, actually? I want to talk about certain universality of phase transition between topological phases. So let me first give you some motivation. Why am I interested in this problem? And what precisely is the problem that I want to study here? So by the way, I'm a kind of smatter physicist, so the language I use may be slightly different. So if you have any question, feel free to ask me. I'll also try to explain some mathematical concept. Try my best. Yeah, in here. So one of the cornerstone in quantum as a matter of fact is actually Langdou's theory. So in Langdou's theory, in the time of Langdou, he believed that phases of matter are described by their symmetries. So therefore, if you want to describe phase transition between two different phases, you want to describe how the symmetry of the two phases changes. So these are the so-called Langdou type of phase transition. So take one example is the most convenient example that everyone are familiar with is for example, one-dimensional transfer of your isymodel. So you have isymesping which pointing to z direction. So they can either point to up or down along the z direction. So you have nearest neighbor interaction between spin along the z direction. Oh, actually. So in this talk, I want to use x, y, and z to label the polymetrices on every side. So this term will be z, i, and z, j. So this term is completely classical, of course. Every term here commutes. But you can add some dynamics here by putting in a transverse field. So by tuning this ratio of the two coupling constant, h over j, you can see there is some kind of translation. So if you tune the parameter of h over j, so you can show when h equal to j for example in one dimension, when h equal to j, you go through a translation. When h is small, you are in a so-called ordered phase. So in order phase, basically the zi acquire expectation value. Or more precisely, if you measure correlation function of z and zj, this becomes 0. Even if i and j, the distance goes to infinity. So this is in the ordered phase. So when h is large, you can see these terms are to flip the spin from this z direction. So it's flip up to down. So therefore, when this term dominates, you get into a disorder phase. So in a disorder phase, you don't have expectation value. Equal to 0. And all the correlation functions are short-wrenched. So there will be a transition between the two phases. So in a disorder phase, apparently the system preserved only one symmetry in the system. So only one symmetry here is, let me call it rotation x, to be a product of order spin along x direction. So i and j label different lattice side. So this obviously commutes with the whole Hamiltonian. So in this order phase, where order spin is pointing along x direction, we preserve the symmetry. However, for the order phase, it has a spontaneous breaking of the symmetry. Because if you look at the ground state of order state, in a very simple picture, you have 2D general ground state. One of them is everything pointing up. The other one is everything pointing down. So separately, they are not in very under this symmetry. So in particular, this is which you map this ground state to this ground state. You can further form a superposition of the two ground states, which is everything spin up point. With the superposition, it's everything pointing down. In this case, you can show this symmetry operator acting on these ground state. These two D general ground states give you eigenvalue plus and minus 1 for these two ground states. However, these two ground states are generally a thermodynamic limit. So therefore, we call in this case, a spontaneous symmetry breaking, SSB short for the spontaneous symmetry breaking over there. So that means although the Hamiltonian actually have the symmetry, the ground state of the system break the symmetry, the thermodynamic limit. So that's the kind of phenomenon I'm interested in today. So in the usual Langdahl theory, as I have said, Langdahl believed that faces are a matter described by their symmetry. So if you start from a phase with symmetry g, in order to go into a different phase, the symmetry has to change. So generally, g goes to some other symmetry group H. And in this case, generally, H just can be any subgroup of g. And the phase transition between these two faces with symmetry g and with symmetry h are described by fluctuation of local order parameter. In this case of E-sync model, the local order parameter are nothing but the expectation value of the spin C by Z on different side. So in particular, more generally, the order parameter of the system leaving this coset space of g quotient H. So in the case of Langdahl theory, all the translations between two different Langdahl faces are described by low energy fluctuations of the order parameters in a Langvelder's limit. So that describes Langdahl type phase transition. So in the case of Langdahl, two different faces, they can have different symmetry. But there is one necessary condition is that the group of the other phase has to be a subgroup of the one phase. So there is a subgroup relation between the two different symmetries. And this H generally can be any subgroup of g. So now the question I try to ask is the following. So in Langdahl's theory, Langdahl assumes symmetry is the only thing described in the system. But we already know there are many things defined in the Langdahl theory, such as, for example, a topological insulator, which has been discussed a lot, and also topological orders, which are described, for example, by Michael and Bruno, among them. So the problem I try to focus on is a problem interplayable symmetry and topological order. So the question I try to answer is the following. Imagine initially I start from a topological order. Let me call it C. So this C mathematically is described by so-called unitary modular tensor category. So it contains some object, object A. And it also contains some morphism for these objects, some morphism. So this morphism, for example, includes fusion and braiding of this object A. So if I start from a topological order, C, if I go through a transition to another different topological order, D, imagine initially the topological order C preserved symmetry G. Now my question is, can G be preserved in this new topological order D? So can G be preserved? Or if it's not, what are the possible symmetry group H, which is compatible with this continuous transition between these two topological orders? So I will make this more precise later. So in this case, I will focus on one special type of quantum phase transition between the two topological or C and D. So the type of topological transition is something people call enion condensation. Translation. So here I need to emphasize that although I'm talking about enion condensation, the type of enion that I condense in a system has to have to be both studies, meaning the twist of this enions, which are condense, has to be trivial. So those are the requirements of this enion condensation transition. I'm going to give a more precise definition for enion condensation transition. But let me first use one example, which is a toricode example, which has been mentioned a lot previously to demonstrate what do I mean by enion condensation. So let me remind you of the toricode model. So in toricode model, you have a spin. Polymetrics is leaving on every link. So toricode model contains two terms. So one of the terms is so-called a vertex term. So toricode model contains two terms. The first term is vertex term, which is eniosum over all the vertices. And there are four spins, which has in the neighborhood of these vertex. So you now have a product of all the spins on these four vertices. And you have, let me call the link, let me label the link to be L. L belong to this vertex. And you have a product of a spin sigma x for these four links. So that's the first term, the vertex term. Then secondly, you have a placket term. You have every placket. There are four spins on the boundary of the placket. And you have a product of all the links on these four boundary of the placket. And you have a polymetrics z along this direction. So these are the two cons ring on the toricode model. Once you put this on a torus, you can show there are four four degenerate ground state. And as Bruno has commented, there are four super selection sectors in this case. One of them is the e particle. E particle is the violation of this vertex term. There's also m particle, which is a violation of this placket term. So in the case of unitary modulate tensor category, the object in this topological toricode, c-toricode, you contains four different super selection sectors. So they are generated by the e particle, which is violation of the vertex term. And the m particle violation of the placket term, there is also a fermion, which is a fusion of e and m. So this one you can show there are very fermion statistics. And the braiding of this system described by the following, so if you have two different particles, say e and m, if you braid one with the other, you always get a minus one phase. So any two different objects has a minus one braiding. So that's the data for the unitary modulate tensor category for the toricode case. So now in this case, as Bruno has commented, this model has no dynamics. In other words, if you look at the excitation spectrum in the system, they are basically flat band. You have, for example, an athorus, you have 4D general ground state. On top of the ground state, you have a very flat band. This, for example, has a gap for 2. So this gap of 2, meaning it has a pair of, for example, e particle. They all have the same energy. And similarly, for a pair of m particle, they also have exactly the same energy. But you can actually change the coefficient here, too. Make this flat band go into two different flat bands, for e particle and m particle. But the point is it's completely flat. So now what we want to do is to add some dynamics into this quasi particle. So the simplest way to do this, and is also the most extensively way that has been studied so far, is to add a magnetic field in the model. So what you do is you have a perturbation of the Hamiltonian, delta H toricode. In this case, let me choose to add a magnetic field, which is along z direction. So it's minus H sum over link, zl. So if you still remember, among these talks, so if you have a z spin acting on the ground state of the toricode, the z spin would actually violate, for example, the z spin here, would violate the x. So the vertex term on this vertex and also the vertex among this vertex. So therefore, if you have a single spin z on this term, it's going to create a pair of e particles on this vertex and this vertex. So in other words, and also because this e particle has a z2 fusion rule, equal c equal to 1, creating a pair of e the same as hopping one of the particles from here to here. So therefore, this term basically gives you some kind of kinetic energy for the z particle. Does it change the ground state space? No, actually, as long as this term is small, the ground state is the same. Yeah, you still have these for the general state. For infinite testing, sorry, for a thermodynamic system, they are still exponentially degenerate. But for finite system, there will be degeneracy. And that will be changed. Sorry, there will be splitting and that will be changed by h. So in this case, so basically what I want to say is this h term give you some. I'm asking whether it actually changes the microscopic form of the wave function, which does, right? Yeah, yeah, that's right. OK, yeah. So the many-body wave function will be changing in that case. So in fact, this is also exactly so. Oh, no, not exactly so. I should say it's mapped to another familiar model. So if you have this perturbation, you can map these two transfer field IC model in two plus one dimension. So this is basically a similarity of transfer field in that case. So you do change the many-body wave function. Yeah, thanks. So yeah, what I want to say is this term give the e particle some kinetic energy. So as long as they give you some kinetic energy, we know the e particle starts to disperse, right? Initially, it's a flat band. They are localized. Now they start to disperse. There's some dispersion. So when this h term is large enough, so in particular, it's the order of one, then at some point, this dispersion minimum is going to touch zero. OK, so that's going to drive you a phase transition. This is when the gap closes. And when we say the gap closes, there is a specific gap that closes. That specific gap is the gap for e-acitation. So this is when we say the e star to condense. This means the gap of the e-acitation go to zero. So in this case, we call it is to the e condensation. And generally, you can also push this concept to other topological order. Once you have some excitation spectrum, you can always push them to zero energy, and then they start to condense. OK, now one question you may ask is, what happened to this e condensation? Even if I have the e condensation, what does the neogram say look like? What does the neo wave function look like? So now let me give you a rough picture of what does it look like. And why do we call it e condensation? So in a physics context, I think one concept we are very, very familiar with is boson-Einstein condensation. In the case of boson-Einstein condensation, if you try to write down the ground state, the ground state of the B, C, I should say more precisely, a superfluid is actually a superposition of different boson particle number with some amplitude r phi m times a dagger for the power of n acting on the Falk vacuum. So it's a superposition of different number of boson for bosons. If you have a superposition of all these states, it gives you actually a superfluid ground state, which have the condensation of boson. OK, so therefore, in the case of a usual boson condensation, you have a superposition of many different boson numbers. So here, in the case of e condensation, there is also a similar analogy like that. So in particular, let's think about one very simple limit. When h goes to infinity, when h goes to infinity, we know the ground state is basically all that's been pointing to z direction, right? But I want to write this way function in a slightly different form. So in this case, the new ground state is basically all that's been pointing to z direction, so all that's been pointing up. I want to write this in a slightly different way. I want to write it as a projection on the original ground state up to some normalization. I have a product of all the link 1 plus zl acting on the original Tori code ground state. So we know Tori code ground state is basically a superposition of many spin configuration. But now if I just do this projection, all the spin just pointing up. So now I want to start to expand this term. So I have many of these zl's showing up on these links. So every term zl, we already know it's physical meaning. It means I create a pair of e on top of the ground state. So therefore, this state is going to give you a creation of many, many e particles everywhere in the system. So therefore, in some sense, this is a superposition of the open string configuration. So when I look in the case of Tori code, in the so-called 11 string net condensation picture, the Tori code ground state can be viewed as a superposition of the closed loop configurations. And every closed loop meaning, so I define closed loop by zl equal to, sorry, xl equal to minus 1. Sorry, xl equal to plus 1. If I have xl equal to plus 1, every link I call there is a loop there. Sorry, there is no loop there. If xl is minus 1, there is one loop. So the Tori code ground state has been proven to be a superposition of all the closed loop configuration. But right now, if I add a zl operator acting on the ground state, Tori code ground state is going to create an open string at that state. So therefore, this new ground state will be a superposition of all the possible open string configuration. For example, I can have one open string with this lens. I can have another open string with this lens. I could have two open strings here and here and so on. So in this extreme limit, it's going to be a superposition of all the possible open string configuration. So that's literally what I mean by e-conversation. I have many e-particles just moving freely in this ground state. So I want to give you a picture of what is conversation. So now let me use a more rigorous mathematical definition for this conversation. So the rules of any e-conversation is given by the following. So a mathematical reference for this one. I think this one is first really summarized by Liang Kong in the following reference. I mean, many physicists has worked a lot of example for any e-conversation. But I think the mathematical formation summarizing here. So in order to have any e-conversation, I have to first have a so-called condensate of algebra. I'm going to use Tori code to demonstrate this. So condensate of algebra is basically some object inside the category C. So this object is generally a composite object. It's a superposition bunch of simple object. So in the case of Tori code, for example, a is 1 plus e if I want to condense e. So it contains two objects, 1 and e. And there are some rules for this conversation, algebra. Is that it's so-called connected at all algebra. So at all means commutative and separable, algebra. Yeah, so I'm going to explain what exactly does this word mean. So it's not just algebra actually. There is also some morphism associated with a. So one of them is if you have 2a, there is a morphism which map you to 1a. There is also a morphism which maps 1, the original vacuum in the category 2a. So let me call this i. So these are the morphism associated with this condensate of algebra. So first of all, I want to explain what does commutative and other things mean. Commutative means the object in a should have trivial braiding with itself. So in other words, this one is the same as this. So if your performer braiding should be the same, this is commutative. So it physically means the object you condense must have both some statistics by themselves. And mutually, they should also have both some statistics. I understand. Sorry? Maybe. Why? Yes. Oh, yeah. So that's a very good question. So as we said. Why not? Why not? Yeah. So when they have both some statistics, meaning if I braided the other one around the other, there is no face, right? Noisier face. So as we said, this is like a superposition of all the open strings. You don't want all the open strings actually cancel out each other. For example, you could have something like this. This is also an open string, right? And you could also have something like this with this. So you don't want these two things to cancel out with each other. So if the braiding is not trivial, this amplitude is going to cancel out. So to the? It's not possible to have something like the fermion state for something. Yeah. The finite density? Free fermion state, finite density. Would that be? You mean you want to excite the fermion excitation in a Tori code. But that's not something we call condensation. I don't know. Yeah. Maybe not, but it seems like an equally interesting base transition. That's true. Yeah. OK. You're right. I should make a comment on that. So any kind of condensation is only one type of possible face transition between different faces. So there are some things which we know for sure does not describe by any kind of condensation. So one of them is transition between different SPT faces. So for example, if you have a fermion, I have number 0, try number 1. That transition is definitely not any kind of condensation transition. So also, if you have a transition which changes the chiral central charge of a topological order, meaning the central charge of the edge state, the number of central charge for the right mover, subtract the left mover for topological order, if this transition changes, chiral central charge is not any kind of condensation transition. So therefore, for two faces on both sides of any condensation transition, it must have the same chiral central charge. So there are many transitions beyond any condensation. I'm not claiming this is the only transition. But this one is a transition which somehow is better understood. So therefore, we can draw some nice conclusion on top of that. Of course, those are also very interesting questions. And I think, mathematically, I don't think there are very general interacting framework to address those transitions. So you guys are very welcome to think about those things. So this is for commutative. There is also another property for connected. Connected basically means the vacuum here shows up once. So dimension of mathematically, arm space for 1 and a is 1. So I should say, physically, you should really think about this as a superposition of many states. Every state has some dimension. You have a direct sum of the different state. And that is how you should think about this commutative algebra. And more precisely, physically, what you can think about is if we just focus on finite region of the system, you ask myself, in that finite region, what are the superselection secret that shows up? As I said earlier, so you have all the possible open strings. So therefore, in this finite region, you could have no strings opening in the region. So that give you a vacuum sector. And you could also have one string which terminate at the region that give you an E sector. So you could have one plus E both of them. So that's how you should physically think about it, the condensation algebra. Finally, there is a condition for separable. So separable, the definition is there exists a map. Let me call it tau from splitting map from 8 to this, such that you combine this map tau and the map mu, the fusion map, which give you the identity map. So well, I have to say I have less understanding of this mathematical rule. But I can give you a physical example to illustrate what the separable means and where this word separable comes from. So now let me give you an example which is, OK, let me first work out the Torico example. So yeah. I think there should be another condition there is a forbidden algebra, which I think it means that the composition of AAA is associated. Yes, yes, that's right. You're right. I didn't describe those things. Yeah. So this has to be associative. Yeah. Yeah, what I mean is here, that's correct. That's definitely correct. So this condensation algebra is described by basically an object and the two morphism. And they must satisfy a social activity. So the associator, all I should say, it's really come from the f symbol. The associator for AAA. Yeah, so these maps have to satisfy this one. Actually, there are other conditions you can return down from these sort of diagrams. I'm skipping all those. Yeah. So OK, so let me first use the Torico to illustrate what does this condensation do. And how do I understand this new phase D that I obtained after the condensation? Yeah. OK. Yeah, let me erase this part. So in a following, I'm going to give you some rule which determines what is this other side of the phase transition. Once I have a condenser algebra, A. So mathematically, this D is defined as local A modules. So let me basically give you some physical rule to determine this one. I can also write down the definition for this one. OK, let me do that. So a module in particular, let me define a left module of a left A module in the following way. So I want to have, again, a set of objects M and also a map, which is mu A on the left. So I want this map satisfied the following condition. So I'm going to explain what physically that this means. So this is a map mu A L. So this map tells you basically, so if I have some object originally in a category C, so right now, I said A is condensed, right? So this means in a new phase, A is really the vacuum. So therefore, initially, if I have two objects in C, if they differ by some object inside A, they should be identified. So that's physically what this means. And a local M module means the object that in this module must be local with respect to A. In other words, local A module satisfied the following condition, of M and A I could have from Brady. So this will be M, OK. This will be A and M. So basically, this one is the same as M and M and A. So meeting the object, that's quite after the condensation. In this new category D, they should have a trivial braiding with A. That's a physical meaning for local M module. So in the case of Tori code, let me explain what do I mean. So originally in Tori code, I have four objects. So it's 1E and 1M. So in this case, if I try to condense E, so first of all, I need to identify the module. So left on A module. So we know it has two objects, 1 and E. So in addition to the object C, also have E and M and Epsilon. Epsilon is E cross M. But I know, as I said, any two different objects which differ by something inside the condensate algebra should be equivalent. So therefore, these two things are the same. You should identify them. And then another rule is you need to require local. So you need to look at these objects. Do they have trivial braiding with E? So in this case, both M and Epsilon, as I have said earlier, they have non-trivial braiding. So therefore, a physical degree of code, they are confined. So therefore, after the condensation, you don't have anything non-trivial. So you don't have a trivial category. So these are the trivial categories of the back. So this is the simplest case where it applies to Tori code. But the simplest case doesn't really explain this separable property. So now let me use a slightly more different example, a more complicated example to explain this separable property. So now let me talk about example which is a category, which is something people call E-sync cross E-sync bar. So physically, it corresponds to, you can think about you have a P plus IP, superconductor on the top layer, and P minus IP superconductor on the bottom layer. You consider your pi-flags on both the top layer and bottom layer as your excitation in the system. So therefore, on the top layer, you have an E-sync, on the bottom layer, it's E-sync bar. So you have one psi sigma plus one psi bar sigma bar. So psi is a fermion on the top layer. Sigma is the pi-flags on the top layer. Psi bar is a fermion on the bottom layer. And sigma bar is a pi-flags on the bottom layer. So bottom layer is P minus IP, top layer is P plus IP. So now on one condensable algebra in this case, you can show it's one plus psi psi bar. How do you do that concretely? You can actually construct the exact solvable model for this case. So in some sense, you can think about gauging the, if you have the Z2 fermion parity in this superconductor, you can gauge the Z2 fermion parity. You make the pi-flag become dynamic excitation. Or I should say, physically, couple them to external UN field. If you give dynamics to the UN field, then you get the pi-flags there. Yeah. So but right now, so far, the. You complained yesterday that this is not what you think of. You should think of it dynamically. Yeah. If you couple the dynamical, you engage with. I don't know who made that comment. No, no. For the other one. It must be Ashwin. OK. Yeah. All right. So now I'm condensing this kind of condensable algebra. So you can see I condense psi psi bar. So finnically, this means I want to tunnel psi and psi bar from the two layer. So you should think about p plus, p layer, and p minus, ever start to talk to each other. OK. So once they start to talk to each other, as you will know, they have trivial topology index. So therefore, after the condensation, if you expect, if this is my C, this original category, the new category D, which is C, a low key in this case, should be just Tori code. OK. Tori code is a trivial superconduct hypervisor Z2-Ga ratio. So now I want to really illustrate this. And I want to use for this a separability. Sorry. Yes. So you describe nicely this condensable. Densable algebra. You didn't say how D is actually. D is C, a low. Yeah. D is defined as the local A module. S, A, low, not dot. The category of local A modules. S, A, low, not dot. Yes. OK. Yes. Yeah. So basically, the object which are not confined, and they are identified with respect to A. Yeah. So we want to think about this case. So in this case, again, the first step is to identify, first of all, the local object, right? We said it has to be local A module. So it has to be something with trivial braiding with psi-psi bar. Actually, we immediately know sigma has an untrivial braiding. Sigma braiding with psi give you a pi face, because it's pi flux. Fermi goes around and gives you a pi face. So therefore, sigma is confined. Similarly, sigma bar is confined. Sigma psi bar is also confined. And sigma bar psi is also confined. OK. So those are gone. And now what you are left with is 1 psi psi bar. And sigma sigma bar. I think that's the only thing. Oh, and also there is psi psi bar. But psi psi bar, we know it's just the same as vacuum. So let's forget about psi psi bar. So we know if psi psi bar is the same as a vacuum, psi is the same as psi bar. Because psi is the same as psi psi bar, feels with psi bar. So if psi psi bar is vacuum, psi is the same as psi bar. So taking a module, meaning these two are identified. But now that you are left with this thing, yes? I forgot. Did you ever mention what the fusion rule was about for this? No, not really. OK. Yes. Yes, thank you. So the fusion rule for this one is, OK, these two copies, I think and I think, but they are separate. They are independent. So the fusion rule here is psi cross sigma is sigma. And sigma cross sigma is 1 plus psi. And psi cross psi is 1. Two fermions become a boson. And two pi flux can either give you all the fermions parity or even fermions parity. And a pi flux combined with fermions is also pi flux. Yeah. OK, so now you're left with this object. But here there's some peculiarity for this object. So if you compute the fusion rule for this object, what you can work out is, according to this fusion rule, is you get 1 plus psi psi bar plus psi plus psi bar. OK, so as we have said, after the conversation, psi psi bar is the same as 1, right? After the conversation. So therefore, you have a dimension two space for the vacuum sector. So this means this fusion rule suggests psi psi bar is its own inverse, its own antiparticle. But you have a dimension two space for the fusion channel into 1. This is not allowed in a UMTC. So you want this to split. So what I mean is generally in a UMTC, if you have a bar, the fusion rule for the 1, this channel, the multiplicity has to be 1. So therefore, this is not allowed. You have to split this. So way to split this is to count the dimension of sigma and sigma bar. You figure out the dimension of this one is 2. So they split into two objects, which is nothing but E and M, E plus M, in the Tori code case. So this is how you ended up with the Tori code. So this psi and psi bar become epsilon part of the Tori code. So psi psi bar is split into E and M. So, sorry? What are E and M in your original category C? E and M, you don't have them. It's like a sigma-sigma-y is a combination of E and M. So when you have to write it's a C, a log, you have to enlarge the category by adding some kind of direct summons, some kind of completion to get E and M. You need to allow an object to split after you take the local M module. And there are certain rules for the splitting to happen. You need to satisfy all the rules, and then you get this. This is the. Like item, code, and completion of the category where you add direct summons of an object. If there's a projection of an object, then you want it to have an image or something. That is probably true. I cannot answer that. Yeah. My collaborator probably can't. Yeah. Sorry about that. OK, so this is, I think, what they call separable for that algebra. OK. So now I have explained what is any accusation. So now the next thing I need to explain is the symmetry action for the topological order. So I think, OK, I'm going to also spend a considerable amount of time explaining how does symmetry act in a topological order. So far, I have specified, if I start from category C, if I do any accusation of if I condense A, what is the category D, the new topological I get. But now the question is, if this topological order has symmetry G, does this one preserve G or not? Yeah. Or is there some abstraction for this one to preserve G? So that's the kind of condition we want to derive. And in order to answer that question, I have to first understand, how does G act on a category? So let me also give you some introduction for that. So this subject in the physics has been studied a lot recently in the past five years, something like that. So this kind of phenomenon is something people call symmetry enrich the topological order. So this can be viewed as some generalization of the topological in-suitter concept to topological orders. So this is called a symmetry enriched topological order, or SET order. So mathematically, this problem reduced to understanding what is the G action on a unitary model tensor category, a group pattern. So this has been studied by an Ettinger of Nick Schich, Ostrich. And the reference is this one. It's fusion categories and homotopy theory, something like that. So basically what they study is G extension of fusion and braided tensor category. So this mathematical work has been sort of translated and putting a proper context in a physics situation by the following two papers mostly. Buck, actually, et al. So Kitab also have some published work on this subject. Yeah, so basically I'm not going to give the whole thing. I'm just going to summarize the main result and give you some physical idea of what does that mean. So I want to understand how does symmetry act on the enion in the system. We know in a topological order you have a bunch of enions, you have fusion and braiding. So you want to have some G symmetry action on the enion. You want the symmetry action preserve braiding, fusion, and F symbol. So therefore, there are basically two general things you can, in the end what you work out is that basically two things that can fully characterize how does symmetry act on the enion. I should say fully characterize a universality class of symmetry action on a topological order. The first one is a symmetry action row, which is a map from the symmetry group G to the automorphism group of the category C. So physically what this means is how symmetry permutes enions. So we know we can have different enions in the system. And I can ask myself, how does symmetry permute a different enion? Let me use that tree Tori code as one example. So in the case of Tori code, we know we have E particle, who sees some vertices. We have VAM particle, who sees some here, right? And actually, if you write a model in this way, where the two terms have the same coefficient, you can show the model explicitly have a symmetry. It's a transition symmetry. You can do this kind of diagonal transition. Let me call this TD, TD-Tutor. It's diagonal transition. Diagonal transition, it physically maps the position of E to a position of F. And more precisely, this diagonal transition is written as TD-Tutor equal to TD. TD is pure spatial translation, multiplied by global spin rotation. Because here is spin sigma x on the vertices. Here is spin sigma z. So you need to flip spin x into z. So it's E i pi over 4 sigma y, yl, sum over all the l. So that's a diagonal translation. So you can show this commutative Hamiltonian. And indeed, it switches E and M. So for example, in this case, you will call rho TD-Tutor would map E to M. Yeah, this is E and M duality, yes. It is really not a translation. Sorry? It's not a translation. No. Is it? Is it? No. It's a delay value. I mean, just a. Oh, you mean? In the super selection sector, there is just a. Yeah, I think what? But even there, I mean, it's like changing integer coordinates to integer coordinates plus 1 half. The physical model has a translation. You can show this just commutative Hamiltonian. In a periodic. In a periodic. Yeah, yeah. It doesn't have to be translation. You can also write it on size, symmetry, same micro as a paper and on some exactly sober model. Realizing that symmetry. I don't know what's saying if you take that TD, you don't land on the left, it's a left. Well, the really physical degree of freedom is just on the links. So this one map a link to a link. So it's OK. Yeah. Yeah. OK, so for example, TD action in Torre code. Switches in E and M. So this is the first level of symmetry action in this case. I should say in this case, there is an abstraction. Let me call it O3 abstraction. So abstraction belong to 3-co-psycho of G with coefficient in a billion and young of the system. So I really cannot explain this very well in some simple way. So physically means somehow sometimes certain permutation cannot be realizing a pure 2D lattice model by some unsized symmetry. You have to realize it is somehow on some surface of a three dimensional system or some other system. Anyway, not a pure 2D local Hamiltonian. And now there is a second level. Oh, is this the TD2? Is this transition? Is this diagonal transition? So you can do it a couple of times. Like the identity is so usually. Oh, it's Z. It's a Z-groove. Yeah, that's correct. I should call Z of, yeah. Yeah. Yeah, in this case, it's Z. But somehow secretly you can see it's like a Z2, right? Because when you go back, it becomes just a transition. Oh, what do I mean? If you do it twice, this becomes a lattice side become another lattice side. Looks like just a usual transition, which doesn't do too much. But yeah, in this case anyway. It's actually a symmetry of everything we're talking about, right? I mean, you can think of this as a diagonal square level as we look at the mechanism. Yeah, exactly. Yes, it's just a transition of that lattice. Yes, precisely. So I should say in that language, if you think about the whole thing as just a square lattice like this, this is rewritten as a so-called one-placket model. Shogun has a rewritten model. In that case, it's really just a transition. And he did show in that case the defect of these transitions, which we just desolation, is just a matrona, fermia, has some dimension square root 2. So just a transition means it's trivial, or you say something interesting about just a transition. Just a transition. It's not trivial because it's paramilitary. And I mean, in this case. Right. Yeah, yeah. So who z is acting effectively in this case? Yeah, yeah, that's right. You're not replacing it with a perspective. But somehow. We really should think of g as the z. Yeah, yeah. Yeah, in the physical model, yes. You should think about it as z. Yeah, yes, yes. But yeah, OK. OK, so the second level symmetry action is less intuitive as this one. It's so-called symmetry professionalization. Maybe I should just give a general introduction. So symmetry professionalization is you can also have some illustration in the Tori code model. So now let me give you some idea of what does this mean. So let me first make some assertion. So symmetry professionalization means the following thing. So meaning if you have some symmetry in topological order, the topological is a gap phase preserving the symmetry. Then you can somehow define a local version of symmetry action. Let's say right now I'm focusing on just unitary finite symmetry. So you can, if the system have a global symmetry, you can always have some version of local symmetry. So the version, let me call it UG local version of the symmetry. So let's say it's just at some region. Find a region in the system. So what you can show is the two action on this consider action of symmetry element associated with the element G and H equal to UGH log up to a phase omega GH. And this phase generally depends on what Anya is inside this region. For example, if I have Anya, A, this phase is labeled by Anya, A. So I have to have a more precise definition by this local symmetry action. This local symmetry action meaning the following. So in some region which is inside this boundary by larger than a correlation line system within this inner region, the action of UG log is exactly the same action of symmetry, global symmetry on the system confined to this region. It's restricted to this region. But near the boundary is somehow modified. So I can use. Yes, it commutes with Hamiltonian. Yes. So this is really an assertion. I'm claiming this UG local commutes with Hamiltonian. It's an actual symmetry in the system. I should say that they commute with some subspace, not the Hamiltonian. Hamiltonian restricts to some subspace where you have finite number of Anya's excitation. So you know the Hamiltonian has a whole spectrum where you can have a many, many number of Anya's. Let's say I have a finite number of Anya in the system. And there is no Anya on this region, on this boundary region. Anya only leaves either inside or outside. If I restrict to that subspace of the Hamiltonian, then I can show this actually commute. I should say this is a belief. It's not really a proof. Yeah. No. No. So if you have some Anya here on this boundary region, then you cannot really find such a low-coaction. So at least I don't know how to find that low-coaction. So let me actually use one example. Again, let me use Tori code. So after showing you this example, I'm going to show you a general picture for this so-called symmetry fragmentation. I should say this is actually a projective action of symmetry on the Anya. As you can see, this is a projective representation. So that's what symmetry fragmentation means. Yeah. I will say that there's also some well-known example in physics. For example, in the fractional quantum model system, we know the Caussi-Hou or Caussi-Lachlan-Kerri-Freschen charge. That's a very well-known example for this phenomenon. So let me consider the following symmetry. I have a transition, Ty, along this y-direction. So this is not a diagonal transition. OK, it's just one lattice transition. And let me consider another little weird symmetry. Let me call it Rx. So this Rx is the following. It's a product of all the links, vertical links. All the vertical links. And you make a product of the spin poly-x operator on the vertical links. Just on the vertical links. You can easily show this commuter as a Tori code Hamiltonian. Just on vertical links. Because you have a plaket term. And this plaket term has intersection with this vertical link and this vertical link. You have two of them. So in total, it's still commuter. And you can easily check this commuter as a Hamiltonian. Now you want to ask yourself, OK, how to define a local action of this symmetry? So locally meaning, imagine I restrict myself to this region. This is my local region. I'm, of course, having some extreme version of local region. I have two lines. OK, between the two lines, this is how I define my local region. I want somehow the symmetry act exactly like this symmetry. Like the global version within this finite region. But you know clearly this operator and this operator, a product of all these operators don't come into its Hamiltonian, right? If I just act it locally. Because for example, this plaket is going to anti-commutative this operator. So this plaket is going to be flipped. So therefore, the Hamiltonian doesn't really come into this one. But in this case, you can do a modification. Is that now let me? Yeah, is there any? OK. So let me use a line to label the symmetry action. So I have this line, right? Any line, any spins crossing the line, I will sign an xl on this line. So I have this line denoting my symmetry action within the region. OK, I have all these lines. Now if you know tory code, this line is nothing but the m string in tory code. The product of sigma x on these links is the m string. So now one way you can do this, to make this the actual symmetry commutative Hamiltonian is to close it at the m string. OK, you can choose this way to come here. Close it. And you can close it here. Closing this spin here. So I want to close it here. If I close it here, you can see it's just a product of a bunch of closed strings. Of course, it commutes with the Hamiltonian. So now you succeed it. But now you can see this action. This is really my local action. This is my urx look. We stick to this region, how it's defined. Now you can see this region also preserves ty, right, transition symmetry. So now you can ask, how do they interact with each other? So we know globally, rx and ty definitely commute, right, on the global level, on the whole system they commute. But if I restrict to this subsystem, I can ask myself the same question. What you can show is that if I do a translation acting on this local action of symmetry, it's going to make this go up. This one go up, right? So it's going to, let me draw the new thing. The new thing is going to look like this. This one is translated to this one. Every blue, sorry, every green one is translated to the red one, right? So it just goes up. So now you can see these two things just differ by it. Because every string is, this one is actually the same as this one after the translation. So what is left is really just a string on the boundary. OK. So what are your left withs is nothing but this one. Yeah. So more precisely what I'm trying to write here is, if I write this, it's equal to urx local, multiply a string, the boundary string. And the boundary string is, let me call it a string of m on the boundary, right? I have a string of m on this boundary and on this boundary. OK. So more precisely I have shown ty urx loke, ty inverse urx loke, inverse local symmetry. So I have proved the local symmetry commutative to the Hamiltonian. This one is equal to just the m string. OK. So now imagine you have an e string. You have an e particle sitting here, let's say. I have an e particle sitting here. And some string would create this e particle here. So if I do this total action on this single e particle, I can just use this local version of the symmetry action. If I do this, this e string would intersect with m string once. As I have said, e and m string has natural statistics. So therefore what I have shown is, in this case, if this total action, if just act on a single e particle like this one, you're going to get a minus one phase. So minus one phase is nothing but braiding of e and m particle on the boundary. OK. So using this example, what I have illustrated is, generally this omega a gh somehow can be written as some s matrix, which is a braiding of the enion you act a symmetry on together with some other enion. Let me label this other enion to be omega gh. In this case, omega gh is m. And what these people have proven is that generally for UMTC with braiding non-degeneracy, this phase has to be just some element of s matrix. Has to be true. And this omega gh has to be something a b-link enion. And so therefore what I have shown is this omega gh belong to the a b-link enion. Let me call it in a topological order. OK. In a topological order. So. We can take a few minutes. Yeah. OK. I expect you to spend a lot of time talking about these things. So now I have decided that this actual phase is really coming from some a b-link enion. So this phase must satisfy actually social activity. So if you first have three symmetry action gh and k, if I first do hk and then do g, and then if I do gh and k, so it has to satisfy a social activity. And which translated into this enion is nothing but a co-psycho condition. So you can show omega gh omega ghk equal to omega ghk times omega hk. So this is nothing but a co-psycho condition. So therefore I have shown basically the symmetry fashionization in this case. Defining this case is characterized by basically an element. Omega belong to an element of two co-psycho h2 of the symmetry group g. It was coefficient in a b-link enion here. Yes. So generally, if you also have symmetry permutation rho in this case here, then it's some twisty version of the group homology. But generally the symmetry fashionation classifies in this way. OK. So now I think I'm ready to talk. Yes. So for the toricode, is the billion hundred rubble to be just this? The whole thing is actually the whole thing. Because every particle is a billion enion. Yeah. So it can be the whole thing. Yeah. So now I'm ready to make some assertion here. So OK. So I should say let me apply that to the case of toricode. So if I apply that group homology to the toricode, this equation just means if in the language group homology is omega ty rx ty. So each one of them is not a gauge invariant. But the ratio is gauge invariant. This one is equal to m, because this is m string. It is one of the billion enions in the case of toricode. OK. So with this knowledge, now I'm ready to make the assertion of what our theorem is about. So our theorem basically has a following physical picture. Is that as I have explained earlier, after I do this enion condensation, the ground state is like a superposition of many open strings. So imagine I assume after the condensation, the state still preserves the symmetry. OK. As I have assumed earlier, if the system still preserves the symmetry, they must admit a local action of the symmetry. OK. So meeting locally if I acted it on the symmetry, the system must still preserve the local action of symmetry after the enion condensation, if it preserves the symmetry. So therefore, what I want to show is basically, after the enion condensation, I have a ground state, which is, for example, in the case where I have just one open string in this finite region. I have another case where I don't. I have two open strings in this finite region. I want to show this superposition of many, many different configurations. This is my new ground state after the condensation. I want to show this new ground state still admit some local action of symmetry in this case. OK. But remember, these particles, they become like a vacuum. This is like the vacuum of the new ground state. This is the ground, sorry, actually, the vacuum of the new phase, the ground state of the new Hamiltonian after the phase transition. So since it's ground state, there is no enion there, right? There's no enion there. So you expect the local action has to be trivial. It's not a projective rep, meaning in a condensed phase, because these enions become vacuum. So therefore, you want the symmetry basically to act non-projectively, but linearly, within the subspace of any enion which condenses. So that's physically the requirement. Now let me write down the mathematical answer for this, what way have derived the rules for the symmetry not to be broken after the phase transition. So there are basically two rules. So I'm only giving you a very rough argument. But that argument can be made more precise if I look at the physics on some edge version of this argument. And it can also be, I mean, my category has proved this within the category theory. But now let me state the conclusion. Conclusion is there are two conditions. And we think these are the only universal conditions based on the algebra structure. So actually, as I have said earlier, in the case of Langdahl's theory, the symmetry breaking pattern is purely determined by group theory, right? So you expect some kind of algebra answer to the solution. So therefore, you should expect some kind of algebra formula. In the end, I should obtain for the symmetry not to be broken across the transition. So there are two conditions. One condition is this condensable algebra A. A is preserved by symmetry. I should say A is stable under G action. So meaning A is a collection of any on that condenses. So if the symmetry G action permeates any on, I don't want it to permeate outside this subset A. I want the permutation to remain this A structure, so A is stable under G action. This is natural, right? For example, in the case of a Tori code, if I condense E, this diagonal transition has to be broken because it maps E to M. So it has to be broken. That's a prediction. So the second statement is that the symmetry fracturing has to be trivial. And that can be made more precise by looking at the following short exact sequence. So I should say mathematically, this is equivalent to saying there is a G invariant algebra structure acting on this condensable algebra. So there is a short exact sequence, which is automorphism A to odd of Cg by A, G to 1. So this is some extension of G. So I need to explain that a little bit. Yeah, sure. So imagine you assume the symmetry is not preserved. So it is preserved. Assume the symmetry is preserved. If I start from a series C, if I gauge symmetry G, I got a new category, Cg. I can also condense A. Then I get D, which is Ca log. In this case, I can also gauge G. I will get, let me call it, Dg. But now, in this Ia is some condensable algebra in Cg. I condense Ia. So I get Dg. So the claim is that this can be proved within categories that if basically this A means some equivalent algebra action of G, then this whole diagram commutes. Meaning if you first gauge and then condense, so if you first condense and then gauge, they are equivalent. Yeah. What symbol did you actually write in the middle of that exact sequence? Automorphism sub what? Automorphism Cg. Yeah, so meaning automorphism within this big category after gauging symmetry G. Yeah. What's inside? Ia is the new condensable algebra. It's a bigger condensable algebra after you gauge the symmetry. OK, but Ia, I guess, is defined by this A. Yeah, that's right. That's right. No, Ia is, yes, that's correct. So actually, you don't need to work out what is in the middle. You just need to know how do you extend the symmetry action G. Yeah. So actually, this part is really given by the cosecler. Yeah. So the same is the following. If this short exact sequence split, meaning if there exists a map from G to this middle group, such that pi is identity on G, this mean it's split. If it's split, this mean G can be preserved. If it doesn't emit a splitting, then G has to be broken. OK, so this is also equivalent to say, if splitting means exist sigma, such that it's my identity, it's the same as saying this middle group. Let me call the middle group to be E. This group to be N, meaning this E is a semi-dimensional product of N and G. Whenever it's a semi-dimensional structure, this means it's split. It's equivalent. So in particular, in the case where if this automorphism group N is obedient, so for example, in case of Tori code, A is 1 plus E. So in this case, all of A is Z2. It's basically braiding an M around E generated by that. So if that's obedient, this is the same as saying H2 of G on this obedient group. The previous omega is trivial. It's a trivial element of the two cosets. If this automorphism group is obedient. OK, so the prediction is basically in this case, because this coset code is non-trivial, if I condense E, you have to either break TY and break RX. I could show you literally why this works, but maybe I will stop here. So if you're interested, you can ask me. Thank you. Thank you. Thank you very much.
 the hotel room last night.  I don't know whether you guys had similar experience.  They had an outburst of facts last night.  OK, so today I want to talk about a topic which is really  to actually to quantum phase transition.  So I realized most of the previous talks  focused on quantum phases, such as topology in Sweden,  and some of them topological orders.  So today the topic is slightly different.  I want to talk about certain universal property of phase  transitions.  Is this working, actually?  I want to talk about certain universality of phase  transition between topological phases.  So let me first give you some motivation.  Why am I interested in this problem?  And what precisely is the problem that I want to study here?  So by the way, I'm a kind of smatter physicist,  so the language I use may be slightly different.  So if you have any question, feel free to ask me.  I'll also try to explain some mathematical concept.  Try my best.  Yeah, in here.  So one of the cornerstone in quantum  as a matter of fact is actually Langdou's theory.  So in Langdou's theory, in the time of Langdou,  he believed that phases of matter  are described by their symmetries.  So therefore, if you want to describe phase transition  between two different phases, you  want to describe how the symmetry of the two phases changes.  So these are the so-called Langdou type of phase  transition.  So take one example is the most convenient example  that everyone are familiar with is for example,  one-dimensional transfer of your isymodel.  So you have isymesping which pointing to z direction.  So they can either point to up or down along the z direction.  So you have nearest neighbor interaction between spin  along the z direction.  Oh, actually.  So in this talk, I want to use x, y, and z  to label the polymetrices on every side.  So this term will be z, i, and z, j.  So this term is completely classical, of course.  Every term here commutes.  But you can add some dynamics here  by putting in a transverse field.  So by tuning this ratio of the two coupling constant, h over j,  you can see there is some kind of translation.  So if you tune the parameter of h over j,  so you can show when h equal to j for example in one dimension,  when h equal to j, you go through a translation.  When h is small, you are in a so-called ordered phase.  So in order phase, basically the zi  acquire expectation value.  Or more precisely, if you measure correlation function  of z and zj, this becomes 0.  Even if i and j, the distance goes to infinity.  So this is in the ordered phase.  So when h is large, you can see these terms  are to flip the spin from this z direction.  So it's flip up to down.  So therefore, when this term dominates,  you get into a disorder phase.  So in a disorder phase, you don't have expectation value.  Equal to 0.  And all the correlation functions are short-wrenched.  So there will be a transition between the two phases.  So in a disorder phase, apparently the system  preserved only one symmetry in the system.  So only one symmetry here is, let me call it rotation x,  to be a product of order spin along x direction.  So i and j label different lattice side.  So this obviously commutes with the whole Hamiltonian.  So in this order phase, where order spin is pointing along x  direction, we preserve the symmetry.  However, for the order phase, it has a spontaneous breaking  of the symmetry.  Because if you look at the ground state of order state,  in a very simple picture, you have 2D general ground state.  One of them is everything pointing up.  The other one is everything pointing down.  So separately, they are not in very under this symmetry.  So in particular, this is which you map this ground state  to this ground state.  You can further form a superposition  of the two ground states, which is everything  spin up point.  With the superposition, it's everything pointing down.  In this case, you can show this symmetry operator  acting on these ground state.  These two D general ground states give you  eigenvalue plus and minus 1 for these two ground states.  However, these two ground states are generally  a thermodynamic limit.  So therefore, we call in this case,  a spontaneous symmetry breaking, SSB short for the spontaneous  symmetry breaking over there.  So that means although the Hamiltonian actually  have the symmetry, the ground state of the system  break the symmetry, the thermodynamic limit.  So that's the kind of phenomenon I'm interested in today.  So in the usual Langdahl theory, as I have said,  Langdahl believed that faces are a matter described  by their symmetry.  So if you start from a phase with symmetry g,  in order to go into a different phase,  the symmetry has to change.  So generally, g goes to some other symmetry group H.  And in this case, generally, H just  can be any subgroup of g.  And the phase transition between these two faces  with symmetry g and with symmetry h  are described by fluctuation of local order parameter.  In this case of E-sync model, the local order parameter  are nothing but the expectation value of the spin C  by Z on different side.  So in particular, more generally,  the order parameter of the system leaving  this coset space of g quotient H.  So in the case of Langdahl theory, all the translations  between two different Langdahl faces  are described by low energy fluctuations  of the order parameters in a Langvelder's limit.  So that describes Langdahl type phase transition.  So in the case of Langdahl, two different faces,  they can have different symmetry.  But there is one necessary condition  is that the group of the other phase  has to be a subgroup of the one phase.  So there is a subgroup relation between the two  different symmetries.  And this H generally can be any subgroup of g.  So now the question I try to ask is the following.  So in Langdahl's theory, Langdahl assumes symmetry  is the only thing described in the system.  But we already know there are many things  defined in the Langdahl theory, such as, for example,  a topological insulator, which has been discussed a lot,  and also topological orders, which  are described, for example, by Michael and Bruno, among them.  So the problem I try to focus on is a problem  interplayable symmetry and topological order.  So the question I try to answer is the following.  Imagine initially I start from a topological order.  Let me call it C. So this C mathematically  is described by so-called unitary modular tensor category.  So it contains some object, object A. And it also  contains some morphism for these objects, some morphism.  So this morphism, for example, includes fusion and braiding  of this object A. So if I start from a topological order,  C, if I go through a transition to another different topological  order, D, imagine initially the topological order C  preserved symmetry G. Now my question  is, can G be preserved in this new topological order D?  So can G be preserved?  Or if it's not, what are the possible symmetry group H,  which is compatible with this continuous transition  between these two topological orders?  So I will make this more precise later.  So in this case, I will focus on one special type  of quantum phase transition between the two topological  or C and D. So the type of topological transition  is something people call enion condensation.  Translation.  So here I need to emphasize that although I'm  talking about enion condensation, the type of enion  that I condense in a system has to have  to be both studies, meaning the twist of this enions, which  are condense, has to be trivial.  So those are the requirements of this enion condensation  transition.  I'm going to give a more precise definition  for enion condensation transition.  But let me first use one example,  which is a toricode example, which has been mentioned a lot  previously to demonstrate what do I mean by enion condensation.  So let me remind you of the toricode model.  So in toricode model, you have a spin.  Polymetrics is leaving on every link.  So toricode model contains two terms.  So one of the terms is so-called a vertex term.  So toricode model contains two terms.  The first term is vertex term, which is eniosum  over all the vertices.  And there are four spins, which has  in the neighborhood of these vertex.  So you now have a product of all the spins  on these four vertices.  And you have, let me call the link,  let me label the link to be L. L belong to this vertex.  And you have a product of a spin sigma x for these four links.  So that's the first term, the vertex term.  Then secondly, you have a placket term.  You have every placket.  There are four spins on the boundary of the placket.  And you have a product of all the links  on these four boundary of the placket.  And you have a polymetrics z along this direction.  So these are the two cons ring on the toricode model.  Once you put this on a torus, you  can show there are four four degenerate ground state.  And as Bruno has commented, there  are four super selection sectors in this case.  One of them is the e particle.  E particle is the violation of this vertex term.  There's also m particle, which is a violation of this placket  term.  So in the case of unitary modulate tensor category,  the object in this topological toricode, c-toricode,  you contains four different super selection sectors.  So they are generated by the e particle, which  is violation of the vertex term.  And the m particle violation of the placket term,  there is also a fermion, which is a fusion of e and m.  So this one you can show there are very fermion statistics.  And the braiding of this system described by the following,  so if you have two different particles, say e and m,  if you braid one with the other, you always get a minus one phase.  So any two different objects has a minus one braiding.  So that's the data for the unitary modulate tensor  category for the toricode case.  So now in this case, as Bruno has commented,  this model has no dynamics.  In other words, if you look at the excitation spectrum  in the system, they are basically flat band.  You have, for example, an athorus,  you have 4D general ground state.  On top of the ground state, you have a very flat band.  This, for example, has a gap for 2.  So this gap of 2, meaning it has a pair of, for example,  e particle.  They all have the same energy.  And similarly, for a pair of m particle,  they also have exactly the same energy.  But you can actually change the coefficient here, too.  Make this flat band go into two different flat bands,  for e particle and m particle.  But the point is it's completely flat.  So now what we want to do is to add some dynamics  into this quasi particle.  So the simplest way to do this, and is also the most  extensively way that has been studied so far,  is to add a magnetic field in the model.  So what you do is you have a perturbation  of the Hamiltonian, delta H toricode.  In this case, let me choose to add a magnetic field, which  is along z direction.  So it's minus H sum over link, zl.  So if you still remember, among these talks,  so if you have a z spin acting on the ground state of the  toricode, the z spin would actually violate,  for example, the z spin here, would violate the x.  So the vertex term on this vertex and also the vertex  among this vertex.  So therefore, if you have a single spin z on this term,  it's going to create a pair of e particles on this vertex  and this vertex.  So in other words, and also because this e particle has  a z2 fusion rule, equal c equal to 1, creating a pair of e  the same as hopping one of the particles from here to here.  So therefore, this term basically gives you some kind  of kinetic energy for the z particle.  Does it change the ground state space?  No, actually, as long as this term is small,  the ground state is the same.  Yeah, you still have these for the general state.  For infinite testing, sorry, for a thermodynamic system,  they are still exponentially degenerate.  But for finite system, there will be degeneracy.  And that will be changed.  Sorry, there will be splitting and that will be changed by h.  So in this case, so basically what I want to say is this h term  give you some.  I'm asking whether it actually changes the microscopic form  of the wave function, which does, right?  Yeah, yeah, that's right.  OK, yeah.  So the many-body wave function will be changing in that case.  So in fact, this is also exactly so.  Oh, no, not exactly so.  I should say it's mapped to another familiar model.  So if you have this perturbation,  you can map these two transfer field IC  model in two plus one dimension.  So this is basically a similarity of transfer field  in that case.  So you do change the many-body wave function.  Yeah, thanks.  So yeah, what I want to say is this term  give the e particle some kinetic energy.  So as long as they give you some kinetic energy,  we know the e particle starts to disperse, right?  Initially, it's a flat band.  They are localized.  Now they start to disperse.  There's some dispersion.  So when this h term is large enough,  so in particular, it's the order of one,  then at some point, this dispersion  minimum is going to touch zero.  OK, so that's going to drive you a phase transition.  This is when the gap closes.  And when we say the gap closes, there  is a specific gap that closes.  That specific gap is the gap for e-acitation.  So this is when we say the e star to condense.  This means the gap of the e-acitation go to zero.  So in this case, we call it is to the e condensation.  And generally, you can also push this concept  to other topological order.  Once you have some excitation spectrum,  you can always push them to zero energy,  and then they start to condense.  OK, now one question you may ask is,  what happened to this e condensation?  Even if I have the e condensation,  what does the neogram say look like?  What does the neo wave function look like?  So now let me give you a rough picture of what does it look like.  And why do we call it e condensation?  So in a physics context, I think one concept we are very,  very familiar with is boson-Einstein condensation.  In the case of boson-Einstein condensation,  if you try to write down the ground state,  the ground state of the B, C, I should say more precisely,  a superfluid is actually a superposition  of different boson particle number  with some amplitude r phi m times a dagger  for the power of n acting on the Falk vacuum.  So it's a superposition of different number of boson  for bosons.  If you have a superposition of all these states,  it gives you actually a superfluid ground state,  which have the condensation of boson.  OK, so therefore, in the case of a usual boson condensation,  you have a superposition of many different boson numbers.  So here, in the case of e condensation,  there is also a similar analogy like that.  So in particular, let's think about one very simple limit.  When h goes to infinity, when h goes to infinity,  we know the ground state is basically  all that's been pointing to z direction, right?  But I want to write this way function  in a slightly different form.  So in this case, the new ground state  is basically all that's been pointing to z direction,  so all that's been pointing up.  I want to write this in a slightly different way.  I want to write it as a projection  on the original ground state up to some normalization.  I have a product of all the link 1 plus zl  acting on the original Tori code ground state.  So we know Tori code ground state is basically a superposition  of many spin configuration.  But now if I just do this projection,  all the spin just pointing up.  So now I want to start to expand this term.  So I have many of these zl's showing up on these links.  So every term zl, we already know it's physical meaning.  It means I create a pair of e on top of the ground state.  So therefore, this state is going  to give you a creation of many, many e particles  everywhere in the system.  So therefore, in some sense, this is a superposition  of the open string configuration.  So when I look in the case of Tori code,  in the so-called 11 string net condensation picture,  the Tori code ground state can be viewed  as a superposition of the closed loop configurations.  And every closed loop meaning, so I define closed loop  by zl equal to, sorry, xl equal to minus 1.  Sorry, xl equal to plus 1.  If I have xl equal to plus 1, every link I call there  is a loop there.  Sorry, there is no loop there.  If xl is minus 1, there is one loop.  So the Tori code ground state has been proven  to be a superposition of all the closed loop configuration.  But right now, if I add a zl operator acting on the ground  state, Tori code ground state is going  to create an open string at that state.  So therefore, this new ground state  will be a superposition of all the possible open string  configuration.  For example, I can have one open string with this lens.  I can have another open string with this lens.  I could have two open strings here and here and so on.  So in this extreme limit, it's going  to be a superposition of all the possible open string  configuration.  So that's literally what I mean by e-conversation.  I have many e-particles just moving freely in this ground  state.  So I want to give you a picture of what is conversation.  So now let me use a more rigorous mathematical definition  for this conversation.  So the rules of any e-conversation  is given by the following.  So a mathematical reference for this one.  I think this one is first really summarized by Liang Kong  in the following reference.  I mean, many physicists has worked a lot of example  for any e-conversation.  But I think the mathematical formation summarizing here.  So in order to have any e-conversation,  I have to first have a so-called condensate of algebra.  I'm going to use Tori code to demonstrate this.  So condensate of algebra is basically some object  inside the category C. So this object is generally a composite  object.  It's a superposition bunch of simple object.  So in the case of Tori code, for example,  a is 1 plus e if I want to condense e.  So it contains two objects, 1 and e.  And there are some rules for this conversation, algebra.  Is that it's so-called connected at all algebra.  So at all means commutative and separable, algebra.  Yeah, so I'm going to explain what exactly  does this word mean.  So it's not just algebra actually.  There is also some morphism associated with a.  So one of them is if you have 2a,  there is a morphism which map you to 1a.  There is also a morphism which maps  1, the original vacuum in the category 2a.  So let me call this i.  So these are the morphism associated  with this condensate of algebra.  So first of all, I want to explain  what does commutative and other things mean.  Commutative means the object in a  should have trivial braiding with itself.  So in other words, this one is the same as this.  So if your performer braiding should be the same,  this is commutative.  So it physically means the object you condense  must have both some statistics by themselves.  And mutually, they should also have both some statistics.  I understand.  Sorry?  Maybe.  Why?  Yes.  Oh, yeah.  So that's a very good question.  So as we said.  Why not?  Why not?  Yeah.  So when they have both some statistics,  meaning if I braided the other one around the other,  there is no face, right?  Noisier face.  So as we said, this is like a superposition  of all the open strings.  You don't want all the open strings actually cancel out  each other.  For example, you could have something like this.  This is also an open string, right?  And you could also have something like this with this.  So you don't want these two things to cancel out with each other.  So if the braiding is not trivial,  this amplitude is going to cancel out.  So to the?  It's not possible to have something like the fermion state  for something.  Yeah.  The finite density?  Free fermion state, finite density.  Would that be?  You mean you want to excite the fermion  excitation in a Tori code.  But that's not something we call condensation.  I don't know.  Yeah.  Maybe not, but it seems like an equally interesting  base transition.  That's true.  Yeah.  OK.  You're right.  I should make a comment on that.  So any kind of condensation is only one type of possible face  transition between different faces.  So there are some things which we know for sure  does not describe by any kind of condensation.  So one of them is transition between different SPT faces.  So for example, if you have a fermion,  I have number 0, try number 1.  That transition is definitely not  any kind of condensation transition.  So also, if you have a transition which  changes the chiral central charge of a topological order,  meaning the central charge of the edge state,  the number of central charge for the right mover,  subtract the left mover for topological order,  if this transition changes, chiral central charge  is not any kind of condensation transition.  So therefore, for two faces on both sides  of any condensation transition, it  must have the same chiral central charge.  So there are many transitions beyond any condensation.  I'm not claiming this is the only transition.  But this one is a transition which somehow  is better understood.  So therefore, we can draw some nice conclusion on top of that.  Of course, those are also very interesting questions.  And I think, mathematically, I don't  think there are very general interacting  framework to address those transitions.  So you guys are very welcome to think about those things.  So this is for commutative.  There is also another property for connected.  Connected basically means the vacuum here shows up once.  So dimension of mathematically, arm space for 1 and a is 1.  So I should say, physically, you should really  think about this as a superposition of many states.  Every state has some dimension.  You have a direct sum of the different state.  And that is how you should think about this commutative  algebra.  And more precisely, physically, what you can think about  is if we just focus on finite region of the system,  you ask myself, in that finite region,  what are the superselection secret that shows up?  As I said earlier, so you have all the possible open strings.  So therefore, in this finite region,  you could have no strings opening in the region.  So that give you a vacuum sector.  And you could also have one string which  terminate at the region that give you an E sector.  So you could have one plus E both of them.  So that's how you should physically think about it,  the condensation algebra.  Finally, there is a condition for separable.  So separable, the definition is there exists a map.  Let me call it tau from splitting map from 8 to this,  such that you combine this map tau and the map mu,  the fusion map, which give you the identity map.  So well, I have to say I have less understanding  of this mathematical rule.  But I can give you a physical example  to illustrate what the separable means  and where this word separable comes from.  So now let me give you an example which is, OK,  let me first work out the Torico example.  So yeah.  I think there should be another condition there  is a forbidden algebra, which I think  it means that the composition of AAA is associated.  Yes, yes, that's right.  You're right.  I didn't describe those things.  Yeah.  So this has to be associative.  Yeah.  Yeah, what I mean is here, that's correct.  That's definitely correct.  So this condensation algebra is described  by basically an object and the two morphism.  And they must satisfy a social activity.  So the associator, all I should say,  it's really come from the f symbol.  The associator for AAA.  Yeah, so these maps have to satisfy this one.  Actually, there are other conditions  you can return down from these sort of diagrams.  I'm skipping all those.  Yeah.  So OK, so let me first use the Torico  to illustrate what does this condensation do.  And how do I understand this new phase D  that I obtained after the condensation?  Yeah.  OK.  Yeah, let me erase this part.  So in a following, I'm going to give you  some rule which determines what is  this other side of the phase transition.  Once I have a condenser algebra, A.  So mathematically, this D is defined as local A modules.  So let me basically give you some physical rule  to determine this one.  I can also write down the definition for this one.  OK, let me do that.  So a module in particular, let me define a left module  of a left A module in the following way.  So I want to have, again, a set of objects M and also a map,  which is mu A on the left.  So I want this map satisfied the following condition.  So I'm going to explain what physically that this means.  So this is a map mu A L. So this map tells you basically,  so if I have some object originally in a category C,  so right now, I said A is condensed, right?  So this means in a new phase, A is really the vacuum.  So therefore, initially, if I have two objects in C,  if they differ by some object inside A,  they should be identified.  So that's physically what this means.  And a local M module means the object that in this module  must be local with respect to A. In other words,  local A module satisfied the following condition,  of M and A I could have from Brady.  So this will be M, OK.  This will be A and M. So basically, this one  is the same as M and M and A. So meeting the object,  that's quite after the condensation.  In this new category D, they should have a trivial braiding  with A. That's a physical meaning for local M module.  So in the case of Tori code, let me explain what do I mean.  So originally in Tori code, I have four objects.  So it's 1E and 1M.  So in this case, if I try to condense E,  so first of all, I need to identify the module.  So left on A module.  So we know it has two objects, 1 and E.  So in addition to the object C, also have E and M and Epsilon.  Epsilon is E cross M. But I know, as I said,  any two different objects which differ by something  inside the condensate algebra should be equivalent.  So therefore, these two things are the same.  You should identify them.  And then another rule is you need to require local.  So you need to look at these objects.  Do they have trivial braiding with E?  So in this case, both M and Epsilon, as I have said earlier,  they have non-trivial braiding.  So therefore, a physical degree of code, they are confined.  So therefore, after the condensation,  you don't have anything non-trivial.  So you don't have a trivial category.  So these are the trivial categories of the back.  So this is the simplest case where it applies to Tori code.  But the simplest case doesn't really  explain this separable property.  So now let me use a slightly more different example,  a more complicated example to explain this separable property.  So now let me talk about example which is a category, which  is something people call E-sync cross E-sync bar.  So physically, it corresponds to,  you can think about you have a P plus IP,  superconductor on the top layer, and P minus IP  superconductor on the bottom layer.  You consider your pi-flags on both the top layer and bottom  layer as your excitation in the system.  So therefore, on the top layer, you  have an E-sync, on the bottom layer,  it's E-sync bar.  So you have one psi sigma plus one psi bar sigma bar.  So psi is a fermion on the top layer.  Sigma is the pi-flags on the top layer.  Psi bar is a fermion on the bottom layer.  And sigma bar is a pi-flags on the bottom layer.  So bottom layer is P minus IP, top layer is P plus IP.  So now on one condensable algebra in this case,  you can show it's one plus psi psi bar.  How do you do that concretely?  You can actually construct the exact solvable model  for this case.  So in some sense, you can think about gauging the,  if you have the Z2 fermion parity in this superconductor,  you can gauge the Z2 fermion parity.  You make the pi-flag become dynamic excitation.  Or I should say, physically, couple them to external UN  field.  If you give dynamics to the UN field,  then you get the pi-flags there.  Yeah.  So but right now, so far, the.  You complained yesterday that this is not  what you think of.  You should think of it dynamically.  Yeah.  If you couple the dynamical, you engage with.  I don't know who made that comment.  No, no.  For the other one.  It must be Ashwin.  OK.  Yeah.  All right.  So now I'm condensing this kind of condensable algebra.  So you can see I condense psi psi bar.  So finnically, this means I want to tunnel psi and psi bar  from the two layer.  So you should think about p plus, p layer,  and p minus, ever start to talk to each other.  OK.  So once they start to talk to each other, as you will know,  they have trivial topology index.  So therefore, after the condensation,  if you expect, if this is my C, this original category,  the new category D, which is C, a low key in this case,  should be just Tori code.  OK.  Tori code is a trivial superconduct  hypervisor Z2-Ga ratio.  So now I want to really illustrate this.  And I want to use for this a separability.  Sorry.  Yes.  So you describe nicely this condensable.  Densable algebra.  You didn't say how D is actually.  D is C, a low.  Yeah.  D is defined as the local A module.  S, A, low, not dot.  The category of local A modules.  S, A, low, not dot.  Yes.  OK.  Yes.  Yeah.  So basically, the object which are not confined,  and they are identified with respect to A. Yeah.  So we want to think about this case.  So in this case, again, the first step  is to identify, first of all, the local object, right?  We said it has to be local A module.  So it has to be something with trivial braiding with psi-psi  bar.  Actually, we immediately know sigma  has an untrivial braiding.  Sigma braiding with psi give you a pi face,  because it's pi flux.  Fermi goes around and gives you a pi face.  So therefore, sigma is confined.  Similarly, sigma bar is confined.  Sigma psi bar is also confined.  And sigma bar psi is also confined.  OK.  So those are gone.  And now what you are left with is 1 psi psi bar.  And sigma sigma bar.  I think that's the only thing.  Oh, and also there is psi psi bar.  But psi psi bar, we know it's just the same as vacuum.  So let's forget about psi psi bar.  So we know if psi psi bar is the same as a vacuum,  psi is the same as psi bar.  Because psi is the same as psi psi bar,  feels with psi bar.  So if psi psi bar is vacuum, psi is the same as psi bar.  So taking a module, meaning these two are identified.  But now that you are left with this thing, yes?  I forgot.  Did you ever mention what the fusion rule was about for this?  No, not really.  OK.  Yes.  Yes, thank you.  So the fusion rule for this one is, OK,  these two copies, I think and I think,  but they are separate.  They are independent.  So the fusion rule here is psi cross sigma is sigma.  And sigma cross sigma is 1 plus psi.  And psi cross psi is 1.  Two fermions become a boson.  And two pi flux can either give you  all the fermions parity or even fermions parity.  And a pi flux combined with fermions is also pi flux.  Yeah.  OK, so now you're left with this object.  But here there's some peculiarity for this object.  So if you compute the fusion rule for this object,  what you can work out is, according to this fusion rule,  is you get 1 plus psi psi bar plus psi plus psi bar.  OK, so as we have said, after the conversation,  psi psi bar is the same as 1, right?  After the conversation.  So therefore, you have a dimension two space for the vacuum  sector.  So this means this fusion rule suggests psi psi bar  is its own inverse, its own antiparticle.  But you have a dimension two space for the fusion channel  into 1.  This is not allowed in a UMTC.  So you want this to split.  So what I mean is generally in a UMTC,  if you have a bar, the fusion rule for the 1,  this channel, the multiplicity has to be 1.  So therefore, this is not allowed.  You have to split this.  So way to split this is to count the dimension of sigma  and sigma bar.  You figure out the dimension of this one is 2.  So they split into two objects, which is nothing but E and M,  E plus M, in the Tori code case.  So this is how you ended up with the Tori code.  So this psi and psi bar become epsilon part  of the Tori code.  So psi psi bar is split into E and M.  So, sorry?  What are E and M in your original category C?  E and M, you don't have them.  It's like a sigma-sigma-y is a combination of E and M.  So when you have to write it's a C, a log,  you have to enlarge the category by adding  some kind of direct summons, some kind of completion  to get E and M.  You need to allow an object to split after you take  the local M module.  And there are certain rules for the splitting to happen.  You need to satisfy all the rules, and then you get this.  This is the.  Like item, code, and completion of the category where you add  direct summons of an object.  If there's a projection of an object,  then you want it to have an image or something.  That is probably true.  I cannot answer that.  Yeah.  My collaborator probably can't.  Yeah.  Sorry about that.  OK, so this is, I think, what they call  separable for that algebra.  OK.  So now I have explained what is any accusation.  So now the next thing I need to explain  is the symmetry action for the topological order.  So I think, OK, I'm going to also  spend a considerable amount of time explaining  how does symmetry act in a topological order.  So far, I have specified, if I start from category C,  if I do any accusation of if I condense A,  what is the category D, the new topological I get.  But now the question is, if this topological order  has symmetry G, does this one preserve G or not?  Yeah.  Or is there some abstraction for this one to preserve G?  So that's the kind of condition we want to derive.  And in order to answer that question,  I have to first understand, how does G act on a category?  So let me also give you some introduction for that.  So this subject in the physics has been studied a lot recently  in the past five years, something like that.  So this kind of phenomenon is something  people call symmetry enrich the topological order.  So this can be viewed as some generalization  of the topological in-suitter concept to topological orders.  So this is called a symmetry enriched topological order,  or SET order.  So mathematically, this problem reduced  to understanding what is the G action  on a unitary model tensor category,  a group pattern.  So this has been studied by an Ettinger of Nick Schich,  Ostrich.  And the reference is this one.  It's fusion categories and homotopy theory,  something like that.  So basically what they study is G extension  of fusion and braided tensor category.  So this mathematical work has been sort of translated  and putting a proper context in a physics situation  by the following two papers mostly.  Buck, actually, et al.  So Kitab also have some published work on this subject.  Yeah, so basically I'm not going to give the whole thing.  I'm just going to summarize the main result  and give you some physical idea of what does that mean.  So I want to understand how does symmetry  act on the enion in the system.  We know in a topological order you  have a bunch of enions, you have fusion and braiding.  So you want to have some G symmetry action on the enion.  You want the symmetry action preserve braiding, fusion,  and F symbol.  So therefore, there are basically two general things  you can, in the end what you work out  is that basically two things that can fully characterize  how does symmetry act on the enion.  I should say fully characterize a universality class  of symmetry action on a topological order.  The first one is a symmetry action  row, which is a map from the symmetry group G  to the automorphism group of the category C.  So physically what this means is how symmetry permutes  enions.  So we know we can have different enions in the system.  And I can ask myself, how does symmetry  permute a different enion?  Let me use that tree Tori code as one example.  So in the case of Tori code, we know we have E particle,  who sees some vertices.  We have VAM particle, who sees some here, right?  And actually, if you write a model in this way,  where the two terms have the same coefficient,  you can show the model explicitly have a symmetry.  It's a transition symmetry.  You can do this kind of diagonal transition.  Let me call this TD, TD-Tutor.  It's diagonal transition.  Diagonal transition, it physically  maps the position of E to a position of F.  And more precisely, this diagonal transition  is written as TD-Tutor equal to TD.  TD is pure spatial translation, multiplied  by global spin rotation.  Because here is spin sigma x on the vertices.  Here is spin sigma z.  So you need to flip spin x into z.  So it's E i pi over 4 sigma y, yl, sum over all the l.  So that's a diagonal translation.  So you can show this commutative Hamiltonian.  And indeed, it switches E and M. So for example, in this case,  you will call rho TD-Tutor would map E to M.  Yeah, this is E and M duality, yes.  It is really not a translation.  Sorry?  It's not a translation.  No.  Is it?  Is it?  No.  It's a delay value.  I mean, just a.  Oh, you mean?  In the super selection sector, there is just a.  Yeah, I think what?  But even there, I mean, it's like changing  integer coordinates to integer coordinates plus 1 half.  The physical model has a translation.  You can show this just commutative Hamiltonian.  In a periodic.  In a periodic.  Yeah, yeah.  It doesn't have to be translation.  You can also write it on size, symmetry,  same micro as a paper and on some exactly sober model.  Realizing that symmetry.  I don't know what's saying if you take that TD,  you don't land on the left, it's a left.  Well, the really physical degree of freedom  is just on the links.  So this one map a link to a link.  So it's OK.  Yeah.  Yeah.  OK, so for example, TD action in Torre code.  Switches in E and M. So this is the first level  of symmetry action in this case.  I should say in this case, there is an abstraction.  Let me call it O3 abstraction.  So abstraction belong to 3-co-psycho of G  with coefficient in a billion and young of the system.  So I really cannot explain this very well in some simple way.  So physically means somehow sometimes certain permutation  cannot be realizing a pure 2D lattice model  by some unsized symmetry.  You have to realize it is somehow  on some surface of a three dimensional system  or some other system.  Anyway, not a pure 2D local Hamiltonian.  And now there is a second level.  Oh, is this the TD2?  Is this transition?  Is this diagonal transition?  So you can do it a couple of times.  Like the identity is so usually.  Oh, it's Z. It's a Z-groove.  Yeah, that's correct.  I should call Z of, yeah.  Yeah.  Yeah, in this case, it's Z.  But somehow secretly you can see it's like a Z2, right?  Because when you go back, it becomes just a transition.  Oh, what do I mean?  If you do it twice, this becomes a lattice side  become another lattice side.  Looks like just a usual transition, which doesn't do too much.  But yeah, in this case anyway.  It's actually a symmetry of everything  we're talking about, right?  I mean, you can think of this as a diagonal square  level as we look at the mechanism.  Yeah, exactly.  Yes, it's just a transition of that lattice.  Yes, precisely.  So I should say in that language,  if you think about the whole thing as just a square lattice  like this, this is rewritten as a so-called one-placket model.  Shogun has a rewritten model.  In that case, it's really just a transition.  And he did show in that case the defect of these transitions,  which we just desolation, is just a matrona,  fermia, has some dimension square root 2.  So just a transition means it's trivial,  or you say something interesting about just a transition.  Just a transition.  It's not trivial because it's paramilitary.  And I mean, in this case.  Right.  Yeah, yeah.  So who z is acting effectively in this case?  Yeah, yeah, that's right.  You're not replacing it with a perspective.  But somehow.  We really should think of g as the z.  Yeah, yeah.  Yeah, in the physical model, yes.  You should think about it as z.  Yeah, yes, yes.  But yeah, OK.  OK, so the second level symmetry action  is less intuitive as this one.  It's so-called symmetry professionalization.  Maybe I should just give a general introduction.  So symmetry professionalization is  you can also have some illustration in the Tori code  model.  So now let me give you some idea of what does this mean.  So let me first make some assertion.  So symmetry professionalization means the following thing.  So meaning if you have some symmetry in topological order,  the topological is a gap phase preserving the symmetry.  Then you can somehow define a local version of symmetry action.  Let's say right now I'm focusing on just unitary finite symmetry.  So you can, if the system have a global symmetry,  you can always have some version of local symmetry.  So the version, let me call it UG local version of the symmetry.  So let's say it's just at some region.  Find a region in the system.  So what you can show is the two action on this consider  action of symmetry element associated with the element  G and H equal to UGH log up to a phase omega GH.  And this phase generally depends on what  Anya is inside this region.  For example, if I have Anya, A, this phase is labeled by Anya, A.  So I have to have a more precise definition  by this local symmetry action.  This local symmetry action meaning the following.  So in some region which is inside this boundary  by larger than a correlation line system within this inner region,  the action of UG log is exactly the same action of symmetry,  global symmetry on the system confined to this region.  It's restricted to this region.  But near the boundary is somehow modified.  So I can use.  Yes, it commutes with Hamiltonian.  Yes.  So this is really an assertion.  I'm claiming this UG local commutes with Hamiltonian.  It's an actual symmetry in the system.  I should say that they commute with some subspace,  not the Hamiltonian.  Hamiltonian restricts to some subspace  where you have finite number of Anya's excitation.  So you know the Hamiltonian has a whole spectrum  where you can have a many, many number of Anya's.  Let's say I have a finite number of Anya in the system.  And there is no Anya on this region, on this boundary region.  Anya only leaves either inside or outside.  If I restrict to that subspace of the Hamiltonian,  then I can show this actually commute.  I should say this is a belief.  It's not really a proof.  Yeah.  No.  No.  So if you have some Anya here on this boundary region,  then you cannot really find such a low-coaction.  So at least I don't know how to find that low-coaction.  So let me actually use one example.  Again, let me use Tori code.  So after showing you this example,  I'm going to show you a general picture  for this so-called symmetry fragmentation.  I should say this is actually a projective action  of symmetry on the Anya.  As you can see, this is a projective representation.  So that's what symmetry fragmentation means.  Yeah.  I will say that there's also some well-known example  in physics.  For example, in the fractional quantum model system,  we know the Caussi-Hou or Caussi-Lachlan-Kerri-Freschen  charge.  That's a very well-known example for this phenomenon.  So let me consider the following symmetry.  I have a transition, Ty, along this y-direction.  So this is not a diagonal transition.  OK, it's just one lattice transition.  And let me consider another little weird symmetry.  Let me call it Rx.  So this Rx is the following.  It's a product of all the links, vertical links.  All the vertical links.  And you make a product of the spin poly-x operator  on the vertical links.  Just on the vertical links.  You can easily show this commuter  as a Tori code Hamiltonian.  Just on vertical links.  Because you have a plaket term.  And this plaket term has intersection  with this vertical link and this vertical link.  You have two of them.  So in total, it's still commuter.  And you can easily check this commuter as a Hamiltonian.  Now you want to ask yourself, OK, how  to define a local action of this symmetry?  So locally meaning, imagine I restrict myself to this region.  This is my local region.  I'm, of course, having some extreme version of local region.  I have two lines.  OK, between the two lines, this is how I define my local region.  I want somehow the symmetry act exactly like this symmetry.  Like the global version within this finite region.  But you know clearly this operator and this operator,  a product of all these operators don't  come into its Hamiltonian, right?  If I just act it locally.  Because for example, this plaket is going  to anti-commutative this operator.  So this plaket is going to be flipped.  So therefore, the Hamiltonian doesn't really  come into this one.  But in this case, you can do a modification.  Is that now let me?  Yeah, is there any?  OK.  So let me use a line to label the symmetry action.  So I have this line, right?  Any line, any spins crossing the line,  I will sign an xl on this line.  So I have this line denoting my symmetry action  within the region.  OK, I have all these lines.  Now if you know tory code, this line is nothing but the m  string in tory code.  The product of sigma x on these links is the m string.  So now one way you can do this, to make this the actual symmetry  commutative Hamiltonian is to close it at the m string.  OK, you can choose this way to come here.  Close it.  And you can close it here.  Closing this spin here.  So I want to close it here.  If I close it here, you can see it's just  a product of a bunch of closed strings.  Of course, it commutes with the Hamiltonian.  So now you succeed it.  But now you can see this action.  This is really my local action.  This is my urx look.  We stick to this region, how it's defined.  Now you can see this region also preserves  ty, right, transition symmetry.  So now you can ask, how do they interact with each other?  So we know globally, rx and ty definitely  commute, right, on the global level,  on the whole system they commute.  But if I restrict to this subsystem,  I can ask myself the same question.  What you can show is that if I do a translation acting  on this local action of symmetry,  it's going to make this go up.  This one go up, right?  So it's going to, let me draw the new thing.  The new thing is going to look like this.  This one is translated to this one.  Every blue, sorry, every green one  is translated to the red one, right?  So it just goes up.  So now you can see these two things just differ by it.  Because every string is, this one is actually  the same as this one after the translation.  So what is left is really just a string on the boundary.  OK.  So what are your left withs is nothing but this one.  Yeah.  So more precisely what I'm trying to write here is,  if I write this, it's equal to urx local,  multiply a string, the boundary string.  And the boundary string is, let me call it a string of m  on the boundary, right?  I have a string of m on this boundary and on this boundary.  OK.  So more precisely I have shown ty urx loke, ty inverse urx  loke, inverse local symmetry.  So I have proved the local symmetry commutative  to the Hamiltonian.  This one is equal to just the m string.  OK.  So now imagine you have an e string.  You have an e particle sitting here, let's say.  I have an e particle sitting here.  And some string would create this e particle here.  So if I do this total action on this single e particle,  I can just use this local version of the symmetry action.  If I do this, this e string would  intersect with m string once.  As I have said, e and m string has natural statistics.  So therefore what I have shown is, in this case,  if this total action, if just act on a single e particle  like this one, you're going to get a minus one phase.  So minus one phase is nothing but braiding of e and m  particle on the boundary.  OK.  So using this example, what I have illustrated is,  generally this omega a gh somehow can be written  as some s matrix, which is a braiding of the enion  you act a symmetry on together with some other enion.  Let me label this other enion to be omega gh.  In this case, omega gh is m.  And what these people have proven  is that generally for UMTC with braiding non-degeneracy,  this phase has to be just some element of s matrix.  Has to be true.  And this omega gh has to be something a b-link enion.  And so therefore what I have shown is this omega gh  belong to the a b-link enion.  Let me call it in a topological order.  OK.  In a topological order.  So.  We can take a few minutes.  Yeah.  OK.  I expect you to spend a lot of time talking about these things.  So now I have decided that this actual phase is really  coming from some a b-link enion.  So this phase must satisfy actually social activity.  So if you first have three symmetry action gh and k,  if I first do hk and then do g, and then if I do gh and k,  so it has to satisfy a social activity.  And which translated into this enion  is nothing but a co-psycho condition.  So you can show omega gh omega ghk  equal to omega ghk times omega hk.  So this is nothing but a co-psycho condition.  So therefore I have shown basically  the symmetry fashionization in this case.  Defining this case is characterized  by basically an element.  Omega belong to an element of two co-psycho h2  of the symmetry group g.  It was coefficient in a b-link enion here.  Yes.  So generally, if you also have symmetry permutation rho  in this case here, then it's some twisty version  of the group homology.  But generally the symmetry fashionation classifies  in this way.  OK.  So now I think I'm ready to talk.  Yes.  So for the toricode, is the billion hundred  rubble to be just this?  The whole thing is actually the whole thing.  Because every particle is a billion enion.  Yeah.  So it can be the whole thing.  Yeah.  So now I'm ready to make some assertion here.  So OK.  So I should say let me apply that to the case of toricode.  So if I apply that group homology to the toricode,  this equation just means if in the language group homology is  omega ty rx ty.  So each one of them is not a gauge invariant.  But the ratio is gauge invariant.  This one is equal to m, because this is m string.  It is one of the billion enions in the case of toricode.  OK.  So with this knowledge, now I'm ready  to make the assertion of what our theorem is about.  So our theorem basically has a following physical picture.  Is that as I have explained earlier,  after I do this enion condensation,  the ground state is like a superposition  of many open strings.  So imagine I assume after the condensation,  the state still preserves the symmetry.  OK.  As I have assumed earlier, if the system still preserves  the symmetry, they must admit a local action of the symmetry.  OK.  So meeting locally if I acted it on the symmetry,  the system must still preserve the local action of symmetry  after the enion condensation, if it  preserves the symmetry.  So therefore, what I want to show is basically,  after the enion condensation, I have a ground state, which  is, for example, in the case where I have just one open string  in this finite region.  I have another case where I don't.  I have two open strings in this finite region.  I want to show this superposition of many, many  different configurations.  This is my new ground state after the condensation.  I want to show this new ground state still  admit some local action of symmetry in this case.  OK.  But remember, these particles, they become like a vacuum.  This is like the vacuum of the new ground state.  This is the ground, sorry, actually,  the vacuum of the new phase, the ground state of the new  Hamiltonian after the phase transition.  So since it's ground state, there is no enion there, right?  There's no enion there.  So you expect the local action has to be trivial.  It's not a projective rep, meaning in a condensed phase,  because these enions become vacuum.  So therefore, you want the symmetry basically  to act non-projectively, but linearly,  within the subspace of any enion which condenses.  So that's physically the requirement.  Now let me write down the mathematical answer  for this, what way have derived the rules for the symmetry not  to be broken after the phase transition.  So there are basically two rules.  So I'm only giving you a very rough argument.  But that argument can be made more precise  if I look at the physics on some edge version  of this argument.  And it can also be, I mean, my category  has proved this within the category theory.  But now let me state the conclusion.  Conclusion is there are two conditions.  And we think these are the only universal conditions  based on the algebra structure.  So actually, as I have said earlier,  in the case of Langdahl's theory,  the symmetry breaking pattern is purely  determined by group theory, right?  So you expect some kind of algebra answer to the solution.  So therefore, you should expect some kind of algebra formula.  In the end, I should obtain for the symmetry not  to be broken across the transition.  So there are two conditions.  One condition is this condensable algebra A. A  is preserved by symmetry.  I should say A is stable under G action.  So meaning A is a collection of any on that condenses.  So if the symmetry G action permeates any on,  I don't want it to permeate outside this subset A.  I want the permutation to remain this A structure,  so A is stable under G action.  This is natural, right?  For example, in the case of a Tori code,  if I condense E, this diagonal transition  has to be broken because it maps E to M. So it  has to be broken.  That's a prediction.  So the second statement is that the symmetry fracturing  has to be trivial.  And that can be made more precise by looking  at the following short exact sequence.  So I should say mathematically, this  is equivalent to saying there is a G invariant algebra  structure acting on this condensable algebra.  So there is a short exact sequence,  which is automorphism A to odd of Cg by A, G to 1.  So this is some extension of G. So I need to explain that  a little bit.  Yeah, sure.  So imagine you assume the symmetry is not preserved.  So it is preserved.  Assume the symmetry is preserved.  If I start from a series C, if I gauge symmetry G,  I got a new category, Cg.  I can also condense A. Then I get D, which is Ca log.  In this case, I can also gauge G. I will get, let me call it,  Dg.  But now, in this Ia is some condensable  algebra in Cg.  I condense Ia.  So I get Dg.  So the claim is that this can be proved within categories  that if basically this A means some equivalent algebra  action of G, then this whole diagram commutes.  Meaning if you first gauge and then condense,  so if you first condense and then gauge, they are equivalent.  Yeah.  What symbol did you actually write  in the middle of that exact sequence?  Automorphism sub what?  Automorphism Cg.  Yeah, so meaning automorphism within this big category  after gauging symmetry G.  Yeah.  What's inside?  Ia is the new condensable algebra.  It's a bigger condensable algebra  after you gauge the symmetry.  OK, but Ia, I guess, is defined by this A.  Yeah, that's right.  That's right.  No, Ia is, yes, that's correct.  So actually, you don't need to work out what is in the middle.  You just need to know how do you extend the symmetry action G.  Yeah.  So actually, this part is really given by the cosecler.  Yeah.  So the same is the following.  If this short exact sequence split,  meaning if there exists a map from G to this middle group,  such that pi is identity on G, this mean it's split.  If it's split, this mean G can be preserved.  If it doesn't emit a splitting, then G has to be broken.  OK, so this is also equivalent to say,  if splitting means exist sigma, such that it's my identity,  it's the same as saying this middle group.  Let me call the middle group to be E.  This group to be N, meaning this E is a semi-dimensional product  of N and G. Whenever it's a semi-dimensional structure,  this means it's split.  It's equivalent.  So in particular, in the case where  if this automorphism group N is obedient,  so for example, in case of Tori code, A is 1 plus E.  So in this case, all of A is Z2.  It's basically braiding an M around E generated by that.  So if that's obedient, this is the same as saying H2 of G  on this obedient group.  The previous omega is trivial.  It's a trivial element of the two cosets.  If this automorphism group is obedient.  OK, so the prediction is basically in this case,  because this coset code is non-trivial,  if I condense E, you have to either break TY and break RX.  I could show you literally why this works,  but maybe I will stop here.  So if you're interested, you can ask me.  Thank you.  Thank you.  Thank you very much.
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