 Thank you very much for your kind invitation and I'm going to talk about classification gap on state phase using quantum spin system. So first, what kind of classification problem it is. So we consider a classification problem of Hamiltonians with a spectrum gap between the lowest eigenvalue and the rest of the spectrum. And the classification criterion is as follows roughly, two gap Hamiltonians are equivalent if there exists a smooth path of gap Hamiltonians connecting them. That is a classification criterion and that kind of classification problem we would like to consider. Recalling that a state, a ground state with a spectrum gap shows exponential decay of correlation function. We can interpret a gap to state as kind of normal phase very roughly. And in this sense, this is a classification of normal phases and looking for the connected component of normal phases. That's a kind of classification problem I would like to consider in this talk. And for that is a kind of a problem I'm interested in. And in this talk, we consider this problem in operator algebraic framework of quantum statistical mechanics. So that is our mathematical framework. And the advantage of operator algebraic framework is that we can treat infinite system directly. And the reason why we would like to the reason why being able to consider infinite system directly can be seen as advantages is because in our classification problem, there are object characterizing phases that can get meaning only when we consider infinite system. In other words, this object become something trivial when we illustrate them to finite system. So in this sense, I feel some considering infinite system is meaning for this classification problem. So I would like to do so in this talk. So first, I would like to introduce this mathematical framework. So with these small letters, I wrote a corresponding thing in finite dimensional quantum mechanical system. So I introduce a sister dynamical system, which is a generalization of finite quantum mechanics. So in finite dimensional quantum mechanics, physical observable was given by mn, which I denote n by n matrix algebra. So in finite quantum mechanics, most simple case n by n matrix gives physical observable. And I would write in this and this sister dynamical framework, this physical observable are depressed by general sister algebra. So mn is depressed by sister algebra, which is a Banner-Huster algebra with no satisfying this sister norm condition. And our original n by n matrix, it is one kind of sister algebra with respect to this uniform norm. So this is a generalization of our original setting of n by n matrix in finite dimensional quantum mechanics. So also now I would like to talk about physical states and recall that in physical finite dimensional quantum mechanics, physical states are given by this formula with density matrix rule here and by this formula. And note that this map, a procedure of taking expectation value, a expectation value of physical observable with respect to the state, this procedure has for this map, this one in the following condition that it is linear and positive. Linear means a linear, but positive means if we put A star A here, then this is positive. So this is a physical state in finite dimensional quantum mechanics. And I would like to, we would like to generalize to a general sister algebra, namely, we just take this abstract property and define that such a map is a state precisely, a state omega over sister algebra is a linear functional over sister algebra A with norm 1 satisfying this positivity condition. But that is a definition of state in sister dynamical system. And note that recall that in physical state, we say state is pure if this law is given by a one-dimensional projection. But in other words, this situation can be characterized that the state cannot be done as a convex combination of two different states. And we don't have the projection or density matrix infinite system. So we say that some state is pure if this latter condition is satisfied, namely, a state omega is pure if it cannot be written as a convex combination of two different states. So that is a definition of state and its pureness. Now, as maybe you know, there is a famous about this state on sister algebra, there is a famous theorem that is called GNS representation, which basically tells us that there for each state on a sister algebra, we can associate some representation on here with space of the sister algebra essentially unique way. More precisely, let us try to look at this theorem. So for each state omega on a sister algebra, A, the GNS representation pi omega on a here with space, and there is also some special unit vector in this here with space satisfying this condition. So this is telling us that expectation value of A with respect to omega can be written like this in the by these objects. And also this is a this it also satisfies this condition memory vector of this form is dense in this here with space. And it says that this is unique up to unitary equivalence in the sense if there is another triple like this, then this they can be related to each other by some unit. And so in other words, it is essentially unique way we can associate such a triple. And here I wrote the detail just to say that it is some concrete definition, but this detail doesn't matter to for my talk. And what I wanted to I want you to remember is that just that if for each state in on a sister algebra, we can associate some kind of representation essentially unique way. I'm sorry, can I ask a question? Yes. Is this valid for any state or for pure states? This is for any state. Okay. Okay. So when state is pure, it has much structure. It gets more structured. This holds for general state. Okay. So that is state and or assist algebra and we notice GNS representation. And now a time evolution I would like to consider in which is given by Hamiltonian of course in a finite dimensional quantum mechanics by this Heisenberg experiment revolution, but in this system framework, it is given by sister dynamics, by which we mean a strongly continuous one parameter group of automorphism, which is defined right here. So it is a group homomorphism from out to automorphism group on sister algebra. So satisfying some proper continuity, namely for each element a, this map is continuous with respect to the law of the sister. So anyway, so instead of talking about Hamiltonian, we consider this a abstract property of the group action. So this is time evolution. Now this talk is about the ground state. I was talking about Hamiltonian whose we have a gap between the lowest eigenvalue and this of the spectrum, which means that I'm interested in ground state. So in finite dimensional mechanics, the ground state was defined in terms of Hamiltonian, of course, which means that a state on environmental matrices is a ground state of this Hamiltonian, means this density matrix has a support under this as project spectral projection of Hamiltonian corresponding to the lowest eigenvalue. So this was a ground state definition of a infinite quantum mechanical system. But my program now is that we do not have Hamiltonian in this framework, but we do have time evolution. So in this framework, for that reason, we talk about the ground state is defined in terms of dynamics instead of Hamiltonian. So here is a definition. So let a be a sister algebra and a sister dynamics on it. And let delta be the generator of this sister dynamics. And a state omega over a is called a tau ground state. If this condition, this inequality holds for any element in the domain of the generator of the dynamics. So this is the definition of ground state in this operator algebra framework, which may look a bit weird compared to this condition. But actually, if you consider this condition in our original setting where a dynamics is given by Hamiltonian, then you can easily show that this condition of original condition of ground state is equivalent to this new condition. Therefore, what I want to say is this may look strange, but actually, this is just a generalization of our usual definition of ground state. And it is taken as a ground definition of ground state in this operator algebra framework. Just a question, where does this generator live in what space? This is a generator on the sister algebra A. The domain is the one which has this limit. So I see. So you didn't, because I thought maybe you were representing your dynamics on the GMS. No, I'm not doing that. Okay. Okay. Okay. Thank you. So that is a sister dynamic ecosystem, but the basic situation. And now I would like, as I said, our problem is about kept ground state phases. And to so what I have to specify what I mean by kept ground state phases in a, what I mean by kept ground state in this framework. And in order to do that, I need the following propositions here. And so let me explain this proposition. So let a omega be a tau ground state for some sister dynamics tau on some sister algebra. And let h by omega be a GMS to the canonical representation of a sister algebra associated to our state omega. And then the proposition tells us that the register unique positive operator h omega on this hillbill space is satisfying this condition. So it's basically implementing the dynamics tau in this hillbill space. So this proposition tell us that there is some, we can associate some positive positive operator uniquely to each ground state. And I will call such a unique positive operator h omega the Hamiltonian associated with this state omega. So this is a proposition. And with this proposition, I can now define what I mean by kept ground state in this talk. And I will define it as follows. We say that sister dynamics has kept ground state is the following two conditions holds. So first condition is the following. So because h omega satisfy this condition, for example, putting one to here and then it punish. And we see that omega omega is invariant under this time evolution, which means in other words that omega omega is an eigenvalue eigenvector of this Hamiltonian corresponding to the eigenvalue zero. And this first condition is that he is telling us that it is non degenerated for pure state. So for any tau ground state omega zero is a non degenerate eigenvalue of this Hamiltonian h omega. That is our first condition. And second condition is that about the gap. So it says that the existing constant gamma positive gamma satisfying this condition for any pure tau ground state omega. And so here, sorry, this h omega, sigma h omega is a spectrum of our Hamiltonian associated to our ground state omega. And it's here we are considering the spectrum other than this lowest eigenvalue zero. And it's saying that there is a gap between the lowest eigenvalue and the rest of the spectrum in this sense. So this is a definition of a gap to a ground state. Can I ask a question? Yes. I don't understand the first part of the definition in physical terms. Can you explain like, why do you need this condition? What does this mean in physical terms? Yes. So very roughly, state being able to representation corresponds to macroscopic status of the physical system. And so, sorry, we can define for each element in for each unit vector in this hubris space, we can consider this kind of functional. This also defines some positive linear function, which is normalized. So this defines a state on the system algebra. And state given like this way, we regard it being close to our original state omega. Such a state given like this form is represented by, you can interpret that they are physically macroscopically, the difference is macroscopically the same. So, and here I'm considering this Hamiltonian in this GNS setting. And I'm considering the Hamiltonian's eigenvalue zero. And if there exists other vector, which is in the kernel of this H omega, it means this is another ground state. So what I'm saying here is that there is no other ground state, which is macroscopically the same as our ground state. That is a condition what the first condition is telling. But can you give example of some spin chain, which relates this condition I, because I'm familiar with the spin chains a little bit. So, we can consider using chain where all the, there is all up spin and all down spin ground state. And they are macroscopically different, right? Yeah. And so, but there is no other ground state, which is can be represented inside of this GNS representation. Okay, so that would be okay. That would pass your definition. But can you give an example of some spin chain, which would not pass your definition? Yeah, so I don't really want to give you this. Yeah, so there is xxz model, where we know there exists a kink, right? And this kink, this kink and that kink, they are macroscopically the same. The difference is only local around here. So, I agree that it is, does not cover all the ground state definition, gap ground state definition, in use of physical sense, in the sense that I would like to eliminate the possibility of degeneracy of ground state, which are macroscopically the same. So, yeah, anyway, the answer to the data question is xxz with this kink should not satisfy this condition. Okay. Okay, thanks. So, in this talk, we are interested in sister dynamics on quantum spin system. So far, I talked only about sister dynamics in general setting, but I would like to actually, we would like to consider is a quantum spin system in this talk. And of course, the dynamics have to be constructed from physical model. So it should be given by something called interaction. And that is what I would like to explain next. So probably it's not that necessary, I don't know. So, let me start. So, throughout my talk, that we are fixed term natural number D and let nu be a spatial dimension. And a new dimensional quantum spin system is a sister which is given as an infinite tensor product of D by D matrices, like this. And we can do, so we can, if it takes a product of D by D matrices along new dimensional lattice. And for each gamma in the subset gamma subset of new dimensional lattice, we can do the same thing, namely with a tensor D by D matrices along this gamma and obtain some sister by algebra A gamma. And this A gamma is naturally regarded as a sub algebra of our new dimensional quantum spin system. So, and of course, we probably it's not necessary, but we say that we observe going here is localized in this gamma or has a support in gamma or so. And a physical model are given by interaction. And in this framework, when I say interaction, it is stated a little bit formally, namely an interaction is a map five from the set of finite subset of new dimensional lattice to our new dimensional quantum spin system and satisfy this condition. And namely, of course, for each x finite subset of new dimensional lattice x, we assign some interaction time five x corresponding to interaction between spins inside of x. And so therefore it should be self-adjoint and it should be localized in x in the sense it belongs to this sub algebra. And out of this interaction, you'd like to construct some some dynamics. And that's not too free to do so. We need to require some physically reasonable locality condition for us to define the dynamics out of interaction. And so we require some additional condition for example, we require finite uniformly bounded interaction in this sense. And from this interaction, we saw from such interaction with three double locality, we construct a dashista dynamics in order to do so. First, we define local Hamiltonian and amnesia that's the summation of all the interaction term inside of each finite set lambda. And out of that, we define hyzenberg dynamics and take the thermodynamic limit and then we learn the thermodynamic limit. And if our interaction is nice one, like uniformly bounded and finite range, or other some sort of locality conditions that is satisfied, then this limit is not to exist. And that defines the sister dynamics on new dimensional quantum spin system. And that is the sister dynamics we are interested in. Um, sorry, I'm having a little trouble understanding the physical definition of this interaction. If you give me like a subset, like just a couple of points near each other, what is the meaning of phi of x? And am I supposed to think of that as like some interacting Hamiltonian that relates all of like that interacts between exactly those points in that subset? Uh, it's, what do you mean exactly? Sorry. What is the meaning of phi of x? Phi of x is like interaction between spins inside of x. So, so why if, why would it be a finite range if I take an x and let's say the diameter is big? I mean, why would I expect it to be zero? Isn't it still going to interact with some? I'm writing here just an example of some locality condition. So even if it's not finite range, it should decay some sort of proper way, right? And this is just an example and you can generalize it to accommodate a little bit more long range interaction. But as this is easiest one, I wrote it. But in general, it's not necessary. I see. Okay. Thank you. So, now, so as I said, we are interested in this sister dynamic star phi given by an interaction phi with gap and this is a kind of interactions. And we are interested in gap to cross state of such a dynamics. And thinking about such a system, we recall that there is some nice property, namely the grand state satisfy exponential decay of correlation function, which is known as a theorem here in this setting. So let phi be a uniformly bounded finite range interaction with gap to cross states. Then in the state per city shown that the correlation functions of any pure tau phi grand state decays exponentially fast. Namely, if x and y be some finite region, this joint region and just say a is localized in x and b is localized in y. And we consider the correlation function and this theorem tells us this correlation function decays exponentially fast with respect to the distance between x and y. That is some consensus, I guess, in physics, but it is it can be is proven in this paper as a theorem as well. So any such property of a gap to cross state. So exponential decay of correlation function make us feel that this ground gap to cross states are kind of normal phase. And that kind of normal phase we would like to classify now. That kind of classification we are interested in. So classification of gap to cross state phases. So in this classification problem, we are interested in a uniformly bounded finite range interaction with gap cross states that are kind of set we are interested in. And as I said, because of the explanation the correlation function of pure cap to cross state physically, we be more or less regard this P as a normal phase. Sorry, I have a question. So in your definition of the set P, the gap can be arbitrarily small, right? Yeah, then then it's possible that then if then you can you can you can get a bunch of phi with smaller and smaller gap, then it might it might look like some guy placed ground state. Is it possible? Actually, it's included in the classification criteria, which will come next. That's that. So namely, basically, we consider this kind of classification. So we introduce the classification that two interactions are we say they are equivalent. And if we can connect them smoothly via a path inside of P, and I'm writing here smoothly, very roughly, but actually this contains more condition indeed, as you said, that we say that two interactions are equivalent if we manage to connect them in a path inside of here, while the gap on the path should be a strictly bounded from below by some strictly positive number. So I'm requiring some path existence of smooth path of interactions connecting this phi 1 and phi 2. Sorry, it's very bad notation, but t equal to 1 is phi 1 and t equal to 2, t equal to 1 is phi 2, sorry. But anyway, it's in a path of interaction connecting phi 1 and phi 2. And the mean in this smoothness, which I didn't really write it down, contains the condition that so each phi t, which has some gap by definition, and I'm requiring this infimum of gamma t to be strictly positive along the path. So in this sense, we don't regard two interactions, which is connected by passive interaction in this set, but not satisfying this one being equivalent. Actually about that, won't that condition always be satisfied? I don't know that's apology, I'm assuming you're going to get to it in a moment. But even before the topology, if zero one is compact, so it's infimum is always, is always greater than zero, right? So I couldn't understand. Isn't the infimum over a continuous path into a space? The continuity of gamma t is quite a hard analytical problem. So that will be great, that is the case. But for the moment, I don't know. So we assure this. I guess I misunderstood. Then smoothness in this case is not stronger than continuous? It continuous, smoothness means that, you know, this story, it's all because I didn't write the conditions precisely. So each finite subset in new dimensional lattice, we have this matrix valued path, right? Smoothness means this map is smooth. Okay. Okay. Thank you. So I have to say that's not all the things I hide it from you. But there is more condition because of analytical technical reason we have to require. And some of them like pointed out here, it's something essentially important. And if you want to know the detail with look at my paper with the moon, and to know that why such a condition, what is the condition and why such a condition is required with look at this paper. That's interesting. So I mean, so I've been to a few like condensed matter physics talks where they usually have the same similar condition, but they're actually talking about specific, like Hamiltonians. So they connect the actual Hamiltonians on your system. And can you indicate just intuitively why you're replacing that picture with this one where you're you're not just looking at Hamiltonians, but you're actually looking at these more fundamental maps, these maps fee. It's same. But as I said, in the beginning, Hamiltonia is not well defined in the infinite system. So that's all. Okay. Arthur, if I can comment, Arthur, this fight is a term in the Hamiltonian. Yes. Yes. So it's like the Hamiltonian can think as a naively as a sum of all these fights or translations. Yeah. So it's, I think it's identical to the definition that you might have seen in physics talk. Yes. Okay. Thank you. Thank you. So that is a classification we would like to consider and verify roughly to interaction which are equivalent. But by the way, we write like this when two interactions are equivalent. And we went to interaction, which are equivalent can be moved to each other without experiencing his transitions. So that is a classification we would like to consider. And the question here is what are the invariant of this classification? So that is a kind of question I would like to know, but how we can do it is a purely not very much clear. And there's some theorem to help us for such kind of analysis, which is called automatic equivalence in the field. So let me explain about it. So for each interaction phi I denote by g phi is a set of all tau five grand state. And the automatic equivalence is a theorem here, which says the following. Suppose that phi 0 and phi 1 in this set P, which would like to classify the interactions. Suppose that phi 0 and phi 1 are equivalent, then the theorem tells us that there exists an automorphism which interpolates the grand state of the first interaction g phi 0 and grand state of the second and interaction phi 1, which interpolate these interactions exist. And that is an automatic equivalence, but actually it's not that there exists some automorphism interpolating these two sets, but actually this alpha automorphism connecting them has a very nice property, namely it belongs to a nice class of automorphism, which I wrote q odd here. So let me define, explain what I mean by q odd. By q odd, I mean that automorphism given by time-dependent interactions. So let me explain, but first time-dependent interaction is just like we did before, we considered before. It's a pass of interaction parameterized by t. So it is a and so that phi b are such a continuous pass of interactions. And out of interaction for each time t, we can do the same procedure defining two defined local Hamiltonian and we can consider the solution of this differential equation. And before it was a this was a not a time independent, this was time independent before. So as a result, we obtained some high level dynamics like this. But now we are considering a time dependent ones. So in general, it may not be that far, but what we are doing here is the same as we did to construct sister dynamic system. And so anyway, we consider this solution and take the sum of dynamic limit. And as again, if our interaction pass of interaction is local enough, then this limit exists and define a continuous pass of automorphism. And what I mean by this q odd is the automorphism given in this manner with some of t and some some pass of phi. So that is q odd. In our world, it is automorphism given by time dependent interaction as an analog of a sister dynamic system, sister dynamics given by interactions. So what this theorem is saying that the pi zero and pi one equivalent, then the corresponding ground state can be interpreted by some automorphism, which belongs to this class. And so this class become very important and its property become important for us. So I'd like to take a little bit more time to and I'll see what kind of automorphism it is. Yeah, can you give us an idea? Is this an important? Yeah, why is this on three well theorem on the previous page? What's the difficulty in showing this? I'm not sure I understand what's the big deal. Can you give some idea? What's a big deal? Why is this not obvious? So it seems like you are almost guaranteed if you define alpha wave equation. What can go wrong? What do you have to find against when you prove this theorem? Well, it's natural, but it's maybe not to prove something which look natural doesn't really mean we don't need to do anything. No, no, of course, I'm sure you have to do a lot of things, but I was just wondering what's the main difficulty? What is the main problem here in proving something like that? Difficulty? Well, the idea is quite simple because of the gap we can construct some differential equation connecting ground state and yeah. So actually, I don't know how to answer. Slava, if I may try to intervene, I suspect the difficulty is that you want to actually construct a solution time dependent showing your equation, but if the Hamiltonian is gap that all time you have the adiabatic theorem that tells you that okay, if you go infinitely slowly, you stay in the ground state manifold, but here you want to get to your new ground state by a finite time evolution. So you have to kill all the non adiabatic effects, which I think can be done because the gap exists at all time, but you need a bit of fine tuning probably of the time dependent Hamiltonian so that you don't get any virtual excitation above the gap at any time. I don't know if this helps. So yeah, that's something that I did not indeed appreciate. So this path, yeah, Yoshiko, can you say this path 5xt that appears on this slide, is it something that has to be chosen very carefully or is it something you just choose the same path which comes from the definition of the fact that phi 0 is equivalent to phi 1? No, okay. The path giving this automorphism is different from the path given by this one. Using this existence of gap, we cook up, so let's say this is given by phi t, then we cook up some other path of interaction depending on introducing the information of gap or whatever. I see. Yeah, okay, this is something that, thanks, but no, that's was a very, yeah, now I see the point. Thank you very much. Thanks. Okay, so now I'd like to explain what is a, but I have only 10 minutes left, so I'm not sure. So let me skip this part, but with some reason, so maybe I did only this part. So for if our interaction is on site, then namely, if 5xt is 0, if x contains more than 1 point, then corresponding automorphism becomes infinite transfer product form, which does not create any entanglement at all between sites, different sites. But in general, it's not that extreme situation. It's not on-site interaction, in general. This is our class. It has more tail. As a result, we may a little bit more create entanglement, but for some observation, we can conclude that it does construct entanglement, but maybe not short range, not long range more. So there is some indication we can interpret as an alpha in this class, not creating long range entanglement. And as a result, from based on such observation, I would like to define short range entanglement and long range entanglement as follows. Namely, a state omega which can be written as this form, where omega 0 is just infinite transfer product state, which does not have any entanglement between different sites times, and for which is automorphism, our class qult, which does not create entanglement. And so we say that a state which can be written in this form have a short range entanglement, and we say that otherwise it has a long range entanglement. So that was remark about this property of qult. And we recall that we are interested in, our question was what is the invariant of our classification, and let us come back to this question. And the automorphic equivalence was telling that if two inductions are equivalent, then they interpolate the corresponding ground state of via automorphism in this class qult. And from this theorem, we can conclude that we can see that if some object is stable under this alpha, under any alpha in this class of automorphism, then from this theorem, we can conclude that it is an invariant of our classification. So what I want to say is to find invariant, what we should do is to find some object stable under this automorphism in qult. And that is a hint. And that is what we would like, we should do apparently, but how we can do and what to do, always physicists tell us that in this case, physicists talk about string-like excitation, which is string-like excitation like anion. And in fact, in this Kittai quantum double model, Kittai derived anionic string-like excitation. And the physicists always tell us that these anions or string-like excitations are invariant of that classification and classification. And Kittai also talked about superselection sector in his papers. And because of that, Peter Narkins showed that Kittai quantum double models, this string-like excitation can be understood as a kind of superselection sector. But the difference between Kittai and Narkins, maybe he is considering it in operator algebraic framework, which we are working on. So based on Narkins' work, we think that a superselection sector in operator algebraic sense should be an invariant. So what I would like to think now is the superselection sector in operator algebraic sense. So let me define it here. So let h pi 0 be an irreducible representation of no-dimensional quantum spin system. We say a representation pi of no-dimensional lattice satisfies superselection criterion for pi 0 if this condition holds for any cone and lambda in no-dimensional quantum spin system. So here, cone means that, so let's say this is axis and any element which has an angle with a theta between this axis, the set of such point is called cone in this context. So we see a representation pi of no-dimensional quantum spin system satisfies superselection sector if this condition holds for any such cone. Let us look at what this condition. Here, we are considering the complement of this cone and we are restricting our representation pi and pi 0 to our complement. And the requirement is the unitary equivalence. U dot E dot means unitary equivalence, namely there is some unitary in the human space which satisfies this condition. So this is the definition of superselection criterion and we say that such kind of representation or representation of superselection sector for pi 0. And this is the definition of superselection sector in operator algebraic framework and based on Narkin's observation about a quantum double model, we expected such a sink should be invariant of our classification and from our previous observation based on a automorphic equivalence, what we have to do is that these superselection sectors are invariant stable under this structure of superselection sectors are stable under our automorphism in this class. That is what we have to do to show that superselection sector superselection structures are invariant of our classification. And this theorem written here says that is a case. Okay, so let me read it. Let h pi 0 be an irreducible representation under this alpha via an automorphism and from this class and suppose that a representation pi satisfies a superselection criterion for pi 0. Then pi alpha satisfies the supersection sector criterion for pi 0 alpha, which means the following. So when alpha maps this representation pi 0 to pi 0 alpha, it maps the corresponding superselection sectors to corresponding superselection sectors. So there is bijection given by this alpha between the superselection sector pi 0 or pi 0 and pi 0 alpha. In other words, this superselection sectors are structure of superselection sectors are invariant of the classification. I'm sorry to interrupt you again, but can you explain why cone is a natural set to consider in this context? Why cone and not some other set? Like why not cylinder or whatever? Why is it a cone? Because yeah, why is it a cone? Yes. In quantum spin system, you know, this automorphism when it acts on some some regions and there's time. And so there is, you know, leaving some bound. So there is a speed of a speed of transportation. And memory when we, so for example, when we map. No, there is, I mean, if you involve time, I understand there's a cone, but here there's no time. You said cone lambda in that new and you're just spatial coordinates. Yes. Where's time in this definition? I don't see time. Yes. But you see, we are considering this kind of map. And this is given by time, time independent interaction, which we can regard as time evolution. And this cone shape, analytically, technically, fit very well with this analysis of this cure. Because it has some finite time, finite velocity for the time evolution. So for example, this cone lives on a spatial lattice, where is velocity? Yeah, maybe I didn't understand what the cone means. So yeah. Yeah, what cone itself is not super important. You can put up a cone if you want. But the reason why technically convenient to consider this cone is because if you apply this alpha to some observable localized in this cone area, then you can approximate it with some automorphism localized in a little bit larger cone. Can you draw the coordinates on this drawing? Where is time? Where is x? So I didn't understand. So you draw the cones, but I don't understand. Here it is just spatial. Yeah, but why are you calling this speed of propagation if it's just spatial? Again, how about we forget about time? That was my mistake. Okay. Okay. Okay. Suppose we are in 1D. Suppose they are in 1D, then what is this cone? Is it just half axis? Yeah. Okay. Okay. Sounds good. Thanks. Okay. So my conclusion is that the super selection sector is is some invariant of our classification, but and and furthermore using this fact that this super selection sectors are stable under our our queue out. We can also show that short range entanglement implies a trivial sector theory namely trivial sector theory you can regard as a long existence of trivial and not trivial union. And they are corollary vectors. And furthermore, sorry, I think I need to conclude now. So let me conclude that first this super selection sector is invariant of our classification that one thing. And although I don't have a time to explain, but this super selection criterion a super selection sector, we can interpret that any one is a kind of a super selection sector because we can derive some mathematical object called graded sister tensor category out of the super selection sector, which actually maybe you know, because many a paper about a QFT do this kind of thing a lot. And so, but what I want to emphasize is that our setting is different from a QFT, but nevertheless, we can still derive some a building structure out of super selection sectors. And sorry, it's a bit out of time. So I would like to stop here. So yeah, thank you very much for your attention. Thank you, Yoshika. So you've been asking questions. So let's see if you have an audience has more questions, please. I was a little curious about that last comment you made because you had introduced this definition in terms of these cones. But initially the motivation was wanted to capture Katayev's notion of anions in your definition of super selection sectors. And I'm assuming that very last thing you said towards the end of the talk was addressing that. So is that right? Yes. Basically, we can say this. So, graded sister tensor category in a QFT setting is given roughly like this. So super selection sector is object and morphism are intertwined, which means the bounded operators which satisfy this condition for each super selection sector both sigma. And in this framework, they can extend the super selection sector is originally just a representation of, at least in quantum spin system, it is just a representation of a sister algebra. But it can be, so there's no composition in the beginning, but it can be extended to endomorphism on larger sister algebra so that you can consider a defined composition. That's kind of things down here. And furthermore, you can define direct sum and you can also define sub-object and you can define grading. And so all the structure called a graded sister tensor category, it can be derived out of this super selection sector based on DHR theory. And the setting of a QFT is a little bit different from our setting. And so technically we need to work a little bit different thing, but it never is a basic basic recipe works for quantum spin system and we can derive such a this sort of thing out of our gap to ground state cases. I see. Thank you. I have a question. So these theorems that you and other people proved, so can we consider given these theorems that in some dimensions, perhaps in one dimension or in two dimension, I don't know, the gap phases have been classified fully or is this problem not yet fully solved? It's not fully solved at all. But can you, yeah, it was not clear, it was not clear to me like how far we are from solving this problem in some particular dimensions, like what is the, what is the main difficulty towards the full classification? Is there some, I mean, I'm not sure if you can summarize it, but if you can say something, I'd be curious to know. Maybe it depends on who to whom you ask, but for me, it looks very difficult. We are far because our classification problem is based on the gap of Hamiltonian with gap. Yeah, sure, sure. I only can see the gap. This is really hard analytical problem. And so basically, if you want to show that this and this with the same index can be connected to each other, you have to construct some path of gap Hamiltonian, which means that, and but in general, to show existence of gap is really hard problem. And so, yeah, that I understand that I know the fact that the practice is very hard to show that there's a gap. But, but leaving this problem aside, you know, if we knew how to show for any Hamiltonian gap to not, but do we know like how many classes are there or something like that? Yeah. Do we have the full list of invariants, for example, perhaps it follows from your theorems, but I'm not sure. Yeah, I don't know. So basically, you're asking if this is the only invariant that is what you're asking? Yeah, something like that. Yeah. In this general direction, could you say something about this? Personally, no, sorry. But if you consider, of course, people talk about modular tensor category. And I often see that people are claiming modular tensor category plus conformal charge are completely impervious. But I never understand why. I don't know if this is a related question. But which which is easier to compute in this case? Because I'm not really, you know, I don't know how to compute either of these, to whether to check whether, you know, there exists such an alpha or whether to check that the two resulting braided C star tensor categories are monohydrally equivalent. How can you say something about how easy it is to check either of those conditions? What do you mean how hard in which situation? I guess, well, I'm not really sure. But I mean, maybe if you're looking at a particular example, you have a system in mind, is it easier to construct such an alpha? Suppose these things are related, you just need to prove it. Is it easier to generally construct such an alpha that exhibits that exhibits this equivalence? Or is it easier to prove that the resulting C star tensor categories are monohydrally equivalent? Right, because I mean, like in modular, I believe it's in modular tensor categories, you have like these, you know, these things that you can just compute, right, kind of like analogous to the algebra brackets. And I thought that, you know, maybe I'm thinking of the wrong word, but they're usually like invariant to compute. But I don't know how easy it is to do it in this setting. If you are talking about this setting, this monohydrally equivalence is given by this alpha using this alpha. But it's not an if and only if, right? Your statement goes in one direction. If omega p2 equals omega p1 composite alpha, then the C star tensor categories are monohydrally equivalent. So it tells you, if I can prove, sorry, if I can prove that they're not monohydrally equivalent, then I know that there exists no such alpha, right? So maybe my question is, is it easier to prove that these categories are not monohydrally equivalent, or is it easier just to prove that no such alpha exists by some other means? Well, I don't know. I think to derive this C star tensor categories highly in concrete example is highly non-trivial issue, I think. So yeah, both sounds sad for me, sorry. Okay, thank you. Okay, other questions? Well, actually, a sort of basic physics question. You mentioned that the axiomatic quantum field theory setting and the setting of spin systems have to be handled differently from the mathematical viewpoint. What is the phenomenon that causes this difference? Because, okay, we may naively think that is it because your systems are defined on a lattice and quantum field theory is usually defined on a continuous space? Or is it something else? What makes these two settings different when you try to prove these theorems? To prove theorems, in essence, this QFT is quite a clean area with nice assumption which is reasonable only in a QFT setting. But our physical quantum lattice system, which where dynamics is given by these interactions, it's kind of messy. And so there is a nice causal cone in a QFT, but we do have, as I said before, we do have some time revolution, but there is always error going out of this cone, which caused some mess for us. Okay. Maybe I'll ask a question. Could you please show back this slide where you had a theorem about Z2 lattice? Yes, this one. Yes, perfect. So let me ask what happens if we replace this dimension two by another dimension? First of all, what's the analog of this theorem in dimension one and what happens in higher dimensions? I think things should work well as well, actually, for a general dimension. It's just that I wrote this paper with this dimension because it's easier to write it down. But basically, I think everything goes fine with higher dimension. The only problem I think is that this braiding, which we can define everywhere, but in higher dimension it becomes trivial because braiding two times in higher dimension. Yes, exactly. So braiding becomes trivial. Is there any higher non-trivial thing remains? Sorry? Does there remain some kind of higher non-trivial thing? If braiding becomes trivial, but is there some kind of higher structure like what people discuss in topological field theories and just trying to ask what do you expect in higher dimensions, says analog of this statement? I don't know. Actually, just looking at that, I'm not sure if we can derive something. But okay, if you go to one dimension, what happens in one dimension? In one dimension. Just one dimensional, let's say, spin chain. What would be the theorem about it? In one dimension. Yes, so if you take spin chain in one dimension. There is no non-trivial sector theory. So, publishing is trivial. Okay. Because in one dimension, the system is almost like a tensor product of left infinite chain, right infinite chain. And so the graph structure is like this tensor product form, which means that super selection sector is trivial. Okay. And maybe also I'll ask like in this way. So suppose I bound the range of interaction by some finite number. Let's say I consider just near-spin interaction or next to the nearest. So then the space of fines is just some finite dimensional space, right? And in this finite dimensional space, we consider points which are equivalent by path with your condition. So if I fix some very low number for dimension new and the range of interaction, let's say one or two, I have just finite dimensional space to study. Can be my complete classification of their sectors that exist in this space? Not that I'm aware of. I don't know. Not even like in dimension one or two, and let's say nearest interaction. It still looks to me like some finite dimensional problem to study. Yeah. No, not, I don't know. Sorry. Maybe. Okay. All right. Anything else? Or if not, maybe you'll be ripping up, Slava? Yeah, I'm, I have no further question. Thanks, Yashiv again for your nice talk. Thank you. Thanks for joining us from so far away. Thank you very much.