 So I'd like to start by thanking the organizers for putting together this workshop in the middle of pandemic. So obviously it was a very difficult task and I appreciate that courage and organization. Okay, so today I'd like to talk about this subject of fracton phases or fractonic phases. And I'd like to think about realizing this phase in the real material. So, you know, we discussed a lot about quantum spin liquid and as you noticed, quantum spin liquid is a mature field as a theoretical subject. So we now understand pretty well we have an exact solution and we even talk about candidate material but this is kind of a new so-called long-range entangled phase. So I'd like to start by introducing these phases like what they are and why they are interesting. Okay, so we have this expanding landscape of topological phases of matter. So we have this so-called symmetric protected phases and these are mostly short-range entangled state. The most famous and the oldest one may be the Haldane phase in a spin-1 Heisenberg chain but then we have this very popular area of a bentopology and these are all examples of symmetric protected phases. And then there is this long-range entangled state like fractional quantum-hole state and the quantum spin-liquid state that you heard about a lot. So now I'd like to introduce what we call a fracton phases and this is a new kid in town. They usually get a lot of attention, especially by theorists. Okay, so it turns out these phases depends not just on topology but it also depends on the geometry of the system. So you have to worry about both and it turns out these phases as an enormous ground-state degeneracy and because of that, a lot of quantum information people wanna think about this as an error-correcting code. So because there's so much redundancy in the ground-state so the chances are that you're not gonna make a mistake very easily. And now because of the fact that these phases depend both on topology and geometry of the lattice, then you have to worry about how you can, you may be able to write down a continuum theory for this because now you have to worry about both a long distance and short distance physics at the same time. So they have some development like that. Also, some people think about these phases as a good example of quantum glasses. So these are some of the motivation. So I don't have time to describe the full details of these properties of these phases but I like to just describe some key properties of such a phases. So if you create some kind of quasi-particles, it turns out that these quasi-particles have restricted mobility. For example, you can create a particle that can only move along one direction in a three-dimensional space. And you can create a point like particles but you can always, you can only create like as a cluster like here, you can only create four particles at the same time. And if you want to move them then basically you have to move this membrane like object at the same time. Or sometimes you can create those particles at the vertex of some kind of fractal structure. So it sounds very exotic but actually there are exactly solvable theoretical models for this. And all these, most of these theoretical constructions are coming from a quantum information literature. And these are so-called committing project models and if you're familiar with the Torico model, that's the committing project model. So one of the most famous model is so-called XQ model. The model is very simple. Just think about a three-dimensional cubic lattice. And then for each X, Y, Z plane, you think about this nearest neighbor born in that plane then you have a product of a sigma Z operators, the poly operators on the plane. All these poly operators, they live on the link of the cubic lattice. So you can do that for X, Y and Z plane. So they are three possible force spin interaction term. But then you have this interaction on the cube. You place a sigma X operator on the edge of this cube. So, but notice that this involves the product of 12 sigma X operator. So this is the monster Hamiltonian. The reason why it's called XQ is very simple. There are three X and there's a cube. This is why it's called XQ model. It's not very creative name, but that's what it is. And the point is that all of these terms commute to each other. So that's why it's called committing project model. The ground state is pretty simple. Ground state is given by the constraint that the eigenvalue of every operator is just plus one. Then obviously that minimize the grounds and energy. It's pretty simple. Okay. So it turns out, even though it looks very simple, it turns out this model actually contains a lot of interesting information. For example, if you start from a ground state and if you apply a sigma Z operator along this membrane like object, what happens is that your cube operator at the corner basically gives the eigenvalue minus one. And that's the way that you create an agitation, if you like. And if you try to move them, but basically you have to move all these four agitations at the same time. You can also create a agitation by applying a sigma X operator along some direction. Then again, then you are basically, or you always create a string like agitation like this. So it's pretty hard to create a just single agitation that would move freely in free space. If you make a dipole operator like that, then it turns out you can actually turn the corners. Those guys basically, this guy can only go in one direction. Okay. So why, okay, it looks cool. So what, right? So the reason why you get a lot of attention is because these kinds of phases have this massive ground state degeneracy. So if you have some open boundary condition on a cubic lattice, if you count the number of degenerate ground state, then log of the ground state degeneracy is basically proportional to the perimeter of the system. So what that means is that if I increase the system size, then I'm increasing the number of degenerate ground state. So that suggests that I can in principle store massive amount of information in this ground state manifold. And it's ever increasing if you increase the system size. So this is very, very attractive as a quantum memory. So that's why quantum information people are very interested in this. So there have been a lot of development to write down a quantum field theory. So this is not of great interest of this audience, but it got a lot of attention from high energy theory, literature, for example, Latin cyber against collaborators are actually working on this subject. And this relation to elasticity theory, basically dislocation disclinations in elasticity theory have something to do with this agitation that I talked about. But then there's also connection to rank to tensor gauge theory. And this is something that I like to use to construct a more realistic model, right? Okay, so let's talk about this higher rank gauge theory. This is nothing but the generalization of electromagnetic theory that we already know. So the usual gauge theory or electromagnetism is just a rank one new one gauge theory. What that literally means is that your electric field is a backup. So why do we have a Gauss law? Then if you violate the Gauss law, then you have a Poisson equation. And there is a corresponding gauge transformation. Now you can generalize this introducing rank two tensor. So this is going to be a two index generalization of the electric field. Then you can think about what kind of Gauss law I can possibly write. And there are two ways of writing that. Basically I can use a two gradient or a single gradient. Then if you think about a possible Poisson equation, either I can introduce a scalar charge or the vector charge, depending on how you do it. So there are two kinds of tensor gauge theory or rank two gauge theories. And then using this idea, you can solve this thing and then you can write down the corresponding gauge transformation. But at this point, it's the trivial generalization. Yeah, go ahead. That's right. So in the usual construction is symmetric and the trace list, but it doesn't necessarily have to be the case. So at this point is a pre-general. Yeah, okay. So what are the properties or the requirement for such a theory? So by the time you look at the say, rank two gauge theory like this from these Gauss law, the new Gauss law, case you that there are more than one conservation law. So usually we require a charge conservation, but in the scalar charge version of rank two theory, you also have to conserve the dipole moment. So you're conserving both total charge and the dipole moment. So typically the charge configuration will be something like a quadrupole, for example, like this. But then allow charge configurations, something like this. So total charge is zero, for example, to begin with. Then whenever you make a deformation, the total charge should remain zero. The dipole moment should also not change, right? So there are only certain charge configurations to allow. It's not, you know, on the other hand, in the usual case, you just have to conserve total charge. So you can have a finite dipole moment, et cetera, et cetera. Similarly, if you think about a vector version of this, then again, you have a charge conservation. Then it turns out that there's an additional contribution, conservation law. So if you think about the charges of momentum, then this is like angular momentum conservation in analogy. So if this is what happens, then this vector charge can only move along the charge vector direction. Again, it gives you a mobility restriction. So at this point, there's some similarity between fracton phases I've told you about and this higher-ranked gauge theory. But in order to make a more precise connection, you have to work harder. It turns out that these are the U1 gauge theories and I have to introduce some kind of Higgs field, if you like, then you break U1 symmetry to Z2. It turns out that that theory can be connected to a fracton phase. So I'm not going to talk about that. I'm actually more interested in realizing phases like in this real material. So in order to do that, I like to borrow two ideas from previous studies. So I like to review what happens in the case of U1 gauge theory. We have a very good example of immersion U1 gauge theory. That's called quantum spin ice. And this is a very story example of how the usual vector charge, so usual electromagnetism can be immersed. And then there's this very nice paper by Nick Shannon and his collaborators that they constructed a classical version of this rank two U1 gauge theory. So I'm going to piggyback on the theory and I'm going to construct a quantum theory for this. So this is basically the plan. Okay, so I'm going to briefly review this. It's going to be very short. Okay, so the usual story is that we start from the classical spin ice. It's an aging interaction on the famous particle ice, is a connoisseur in tetrahedron. And you can rewrite this model by defining a cluster spin. So you sum over all the aging spin on the tetrahedron. Then you scale them, you sum over all tetrahedron, then you realize that every side is counted twice. So you correct it by multiplying factor half. And these two models are equivalent up to constant. Now, by the time you write this way, then it's clear what the ground state is. Ground state is given by all the spin configurations where the cluster sum is zero. And this defines the manifold, the manifold of two-in-two-hour configuration. This is the famous ice manifold. Now you know to make a connection to electromagnetism, so it was proposed that if you define the spin operators, for example, at the link of the dual-diamond lattice, then you can identify this aging spin operator by the electric field. Then you can show that essentially, if you write down a divergence of this field, and that becomes basically the same as the cluster spin, sum of G component of the spin on the tetrahedron, so the constraint on the ground state manifold basically becomes the Gauss law. So this is the electrostatic mapping from the classical spin ice manifold electrostatics. Now we wanna go to the dynamics. So in terms of dynamics, in order to do so, I have to add a quantum fluctuation. So you add some transverse quantum fluctuation term like this as plus and minus, now I can flip my spin. So now if you assume that this transverse interaction is much, much smaller, and the aging interaction, I can do a degenerate perturbation theory. And it turns out that the lowest term I can generate is basically six-spin ring-change term around the hexagon in the particle lines. So what it does is that if you start from one of the degenerate classical ice configuration, and if you apply this ring-change term, that you flip, flip, flip, you spin, and then you generate another configuration, this configuration is not exactly the same as the initial configuration, but the new configuration is a member of that massive degenerate classical manifold. So you always remain in that degenerate space after applying this ring-change term. So that's why you never leave your greenhouse manifold. And that is very important. And then if you do so, then you can have this mapping, namely that the g-component maps to electric field, the creation operator maps to e to the i vector potential. Then this famous commutation relation between g-component to spin and the ranging operator becomes canonical commutation relation with the vector potential and electric field. So this is a one-to-one mapping. And by the time I have that, then I have a quantum electrodynamics. In fact, this ring-change term maps to cosine core of A on the lattice. So the lattice action, lattice Hamiltonian looks like that. So I have a electric field square, and then there's a cosine core of A. And if this theory has a t-compined phase, then I can expand my cosine, it becomes column A square. So then I naturally get E square plus B square Hamiltonian. And that is precisely the usual electrodynamics that we know of. So this is the way that the U1 rank one gauge theory can be immersed in the real material. Okay, so having said that, now I'm gonna move to the more complex object, the new kid in the town. Good, okay. So in order to realize this, we are going to use what we call breathing particle lattice. In fact, a material like this exists, namely that the size of the tetrahedron on the A and B sub-lattices of the dual diamond line, they are different. So naturally, the exchange interaction on the A tetrahedron is bigger than the exchange interaction on the B tetrahedron. Okay, so that's basically the situation. So the particle lattice looks like this, there are A sub-lattice tetrahedron, B sub-lattice tetrahedron. And then each sub-lattice is basically FCC lattice. And since I'm lazy and I don't want to draw all this tetrahedron all the time, I'm gonna replace my tetrahedron by a dot. So the screen dots are basically up tetrahedron, the yellow dots down tetrahedron. So there are A B sub-lattice structure in this particle lattice. So that's the diagram that I like to use. Okay, so now just think about the most general nearest neighbor spring exchange interaction on the particle lattice. Just write down every possible term that's allowed by symmetry. Then you have a Heidenberg interaction, Telotovsky-Maurier interaction. You have a famous Kitaev interaction. And also what people call a gamma interaction, the symmetric-on-isometric exchange. And these are some constant shift. So this is the most general top if you only require the nearest neighbor interaction. So we are going to use this very general model. Okay, and the assumption I like to make is the interactions on the B tetrahedron is much smaller than those on the A tetrahedron. So there's a separation of energy scale. And I also like to assume the Heidenberg dominance of other unisotropic interactions. So these are very, very reasonable assumptions. Okay, so but instead of working with this very complicated spin model, it turns out it's much more useful to write down this model using normal mode of spin operators or those spin. So what happens is that, yeah, that's correct. Yeah, so at this point, right, right, exactly. So at this point, let's think about a classical model first, okay? So I'm going to start with a classical model. So these are not operators. These are some vectors, okay? So in this case, instead of dealing with that complicated spin model, it turns out it's much more useful to write down a model using the normal mode. So basically the fundamental unit is nearly tetrahedron. So there are four spins at the vertex of the tetrahedron. And using the fact that the local symmetry of the tetrahedron is the TD, I can classify the normal mode of collective motion of these four spins using the illusible representation of the tetrahedron group. So there are five illusible representation. As you know, A means single mode, E means doubly degenerate, P means triply degenerate. So there are one, two, three, three, three modes, normal mode, if you like. Question? Okay, right. And there are interactions like normal mode like that are in the A tetrahedron and B tetrahedron. And you may argue that how can you separate degrees and so like that because each vertex is shared by A tetrahedron, B tetrahedron. You just count twice. You correct that again by multiplying half factor then you're perfectly fine, okay? Great, great. So for example, you can work out the mapping from that microscope model to this normal mode analysis and all of this A coefficient, these are basically the mass term or the interaction coefficient is basically represent energy cost to excite this mode. All of that can be re-expressed as some combination of those exchange constants that I wrote down. So this is the most general model that you could write. Okay. So again, I'm gonna assume that Heisenberg dominance over all other small interactions. And if you do so, then you immediately realize that the heaviest mode is this guy. This guy has the largest energy scale. So I'm gonna get rid of this heaviest mode and I'm gonna set those modes to be zero. So I'm gonna get rid of this mode because this mode basically holds the largest energy possible in my model. Okay, so it turns out that you can do so if Heisenberg interactions and these nano-magnets, okay. And that basically imposes certain constraint and that's going to be important, especially I'm getting rid of this mode for the beta trident. And remember that my beta trident spins are all connected to the 8th trident. So these guys, of course, part of the beta trident but they also belong to the 8th trident. So by the time you remove one mode here, it's gonna give me some constraint on the behavior of normal mode of 8th trident. So it gives you some constraint and that constraint is very important. So it turns out, it looks very complicated but what actually happens is that the normal mode in this 4th trident, they're all interrelated. And that's why you have to, in the continuum, it becomes some kind of gradient operator. So it looks complicated, but actually it turns out I can nicely write down some kind of Gauss law using basically a rank to tensor electric field. So it looks like this. So for example, there's a symmetric traceless part. There's a trace-full part. Then there's an anti-symmetric part. So it consists of three matrices but it's just the rank to tensor. And if you use this representation, then you have a Gauss law. And so that's, you immediately see that all the entries of the electric field is basically a normal mode variable. But then basically I have a nice Gauss law. So there's some, so you can already see the analogy to the rank two theory that I talked about. Okay, so, okay, right. So, but then in order to make further progress, I like to successfully remove a higher mode and I like to keep on the low energy mode. So it turns out it's useful to keep only the so-called E-mode and T1 minus mode. So E-mode again, W degenerate, T-mode is triplet degenerate. So if you only keep this mode. That's a question in the chat. What's the question? Are you contracting the divergence with one of the indices of P, I, J? Yes. So that the zero is a vector. That's correct, yeah. So it's going to be a vector charge theory. Okay, good. Okay. So if you make this simple vacation, then it turns out that we can just keep traceless symmetric part of the rank two electric field. And in that case, there is particularly simple. And then the Hamiltonian just becomes essentially E-square with this Gauss law constraint. So we have a mapping from that classical spin model to a rank two cancel gauge theory. But this is a classical theory. And what we really want to have is a quantum theory or fracton. So how you construct this? Oh, by the way, this choosing low energy manifold sounds like a fine tuning, but actually it's not. It turns out, for example, in this case, you can just set all this K-gamma interaction to be zero. If you only keep Heisenberg and Jalowski-Moria, then you just automatically arrive at it. So, you know, it's pretty natural construction. Now you know how to go from, maybe I will skip this. Okay, so you know how to go from classical model to from classical model to quantum model, you have to work much harder. And you may immediately encounter a lot of difficulties when you try to do so. The reason why it does that is that remember that these electric fields that I introduced, these are actually spin. They are not committing variables, right? So it has all the difficulties with dealing with the spin Hamilton, okay? So if you try to quantize this theory directly, then it just immediately becomes a non-committative field theory. And I don't know how to solve that, okay? So that's why it's not easy to solve this model. So instead of directly attacking this model, we decide to look at a different limit. Here, we are choosing actually different low-energy limit. The details are not important. We are just choosing these two modes, the low-energy limit. If you do so, then among three possible components of the rank to tensor, I just pick out the diagonal component from the symmetric part, but I allow a finite trace. It turns out that this model is much easier to solve. And so, but still a rank to gauge theory. And it has the same Gaussian constraint. So for example, you have basically the same kind of a conservation law that I talked about, okay? So let's see how it goes. Again, it doesn't really require a lot of fine tuning. You can get there by making some choices of interaction in the original model. Okay, so let's talk about a quantum theory. So we are gonna work with the theory. I'm gonna define this new electric field as a sum of a symmetric piece and the part that has a finite trace. And then they satisfy this Gaussian constraint. Okay. So now I want to rewrite this model. Remember that I'm basically keeping two normal mode. It's what we call a E mode and A2 mode. And they have the same mass, okay? So then we should not forget that there is actually interaction in the b-tetrider. And that's going to be quite important. And so that is still hanging around there. And by the time you write the model like this and rewrite this variable in terms of this electric field operator, it turns out it's actually very simple. Remember that I only have a diagonal component here. So it's just a diagonal component square, some of the diagonal components square. If you regard those guys as the three components of the single vector, then it just becomes a roto model, basically some electric field square. So this model itself is pretty simple. And in fact, since these are spin variable, they satisfy nice SU2 commutation on this. This model is simple, but all the difficulties are in this constraint as usual. All the difficulties are in the constraint, this model itself is pretty simple, okay? Now, just like ordinary quantum mechanics, the E square is like S square and G component of electric field is like S chat. So the quantum numbers are like this. So we can actually figure out the ground state of each tetrahedron first. And for each tetrahedron, the ground state is bifold degenerate with this quantum number. It's basically a state of SU2 manifold for each tetrahedron. But point is that I have to arrange spins in such a way that I satisfy the constraint. And if you do so, if you count all the degenerate grounds there by imposing this Gaussian constraint out of this manifold, then you find that we actually have a massive degeneracy. And that massive degeneracy is what I'm going to describe. So again, another important information is that if you relax the Gaussian constraint, what that means is that you are now allowing the excitation to show up. Then again, these are basically vector charges, the X, Y, Z component. It's a rank two KG series with our vector charges. And what do you call these charges or electric charges? It turns out they are actually located at the b-ketrahedron center. So my electric fields are defined at the a-ketrahedron site, but my charge is actually defined at the b-ketrahedron center. And this is basically a Poisson's equation written in this coordinate. And these vector charges, now we are thinking about quantum theories that they satisfy some SU2 algebra again. So these are spin-off charges basically. And also, if you apply a raising and lowering apparatus, then you can either create or destroy those charges at the b-ketrahedron center. So again, my rank three electric fields reside at the a-ketrahedron center, but my charges are located at the b-ketrahedron center. That's the important thing to remember. Okay. So now, I should not again forget the fact that there are regular interactions that I have in my b-ketrahedron. And this b-ketrahedron model for the part of the Hamiltonian, I can actually completely rewrite it in terms of variables in the a-ketrahedron. Because again, it just spin belongs to also a-ketrahedron as well as b-ketrahedron. So again, without giving you details, it turns out that if you rewrite it in terms of variables in the a-ketrahedron, that gives me some kind of raising and lowering apparatus. And this raising and lowering apparatus basically gives me a quantum dynamics. If I start from some spin configuration or charge configuration, then if I apply this perturbation, then I can move my charges, then I can move my spins. So before that, before I introduce this, they are just degenerate spin configuration essentially. But I can move from one state to one configuration to the other by using this perturbation thing because then I can create or destroy my charges. And I like to describe to you physically how that actually happens. So let's focus on this talk. Basically, I'm creating and destroying charges at nearby a tetrahedron site. So I apply e-plus here, I apply e-minus here, for example. So that's what I'm going to do. Great. So imagine that I start from some spin configuration or charge configuration, then I apply just single raising operator on the a-ketrahedron site. Then you can show that if I apply a raising operator on the a-ketrahedron site, I immediately create a full charge configuration like this. Basically some kind of quadrupole charge configuration like this at the nearby p-ketrahedron location. Okay. So in terms of three-dimensional picture, it looks like this. You create a extra spin up or spin down configuration. And this is not an individual spin, but basically some kind of collective spin of made of this a-ketrahedron configuration. So now if you apply e-plus and e-minus at a nearby a-ketrahedron site, then you see that e-plus generate this quadrupole configuration, e-minus create another quadrupole configuration. And interestingly, the charges created at the center, they basically annihilate and disappear. Then I can keep doing this by applying more operators like this. Basically some high order perturbation theory that you keep canceling charges in the bulk. You keep doing that. Then essentially what happens is that all the charges that push to the boundary and the bulk charges all disappearing. So you are basically creating a cluster of charges at the edge of the membrane like object and these charges are moving towards the boundary of your system. And if you keep doing this, and if when you reach the boundary of a system, now if you impose a periodic boundary condition, then you can show that with a periodic boundary condition, now you can remove all the boundary charges. Then I go back to grounds and metaphor. So again, just like the example of quantum spin ice, if I start from some spin configuration at the beginning, I apply all this stuff, I move all the extra charges created to the boundary. Then when you annihilate them by boundary condition, I go back to that grounds and metaphor. It's not exactly the same initial state, but the resulting state will be part of this metaphor. Okay. And one case, so one case show that this way, I can basically reach every degenerate spin configuration, every degenerate state in my grounds and metaphor. Okay. So you can get basically every, and this is pretty similar to, example of like quantum whole effect where the way that we generated degenerate grounds that we usually create quality particle and quality whole pair, that you move them along the boundary, you annihilate them, you go to another ground state. So this is kind of a glorified version of that, but it's a much more complex process. Okay. Great. So ground state degeneracy, we couldn't find the analytical formula. I think that we need some algebraic number theories to find this expression. We just couldn't find it. So we decided that we are gonna torture the computer. So we just count it by proof force, basically counting all the degenerate state, basically make the computer to do all the work. So this is my unit if you like. So what I call LX, LY, LJ, one on one means that this cube actually already contains many tetrahedrons. So these are, remember, green does the A tetrahedron, yellow does the B tetrahedron. So there are already a lot of spin. So for example, in this case, I would call that volume one, perimeter three, this kind of cube computation. If you count the ground state, then it's actually 85. Now you keep doing this, then now you increase the system size in this unit by 211, then it becomes already over 1,000. Okay. So now you keep doing this, you see that you have a massive number of degenerated ground state. Then we try to find the rules here. Interestingly, for example, if you look at the system size where the, you know, either the perimeter is the same or volume is the same, the number is different. So obviously it's not just a function of volume or a perimeter. But one thing we notice is that for a fixed perimeter, the larger volume actually gives a smaller ground state. In contrast to your user orientation. For a fixed volume, the other hand, the larger perimeter gives you a larger ground state. So this much, we can sort of qualitatively understand why that happens. For example, if you just think about a case where your system size is almost one-dimensional, elongated only one direction, then actually it's pretty simple. Then the ground state degeneracy monotonic could increase the length of the system. In all other cases, what happens is something like this. So ground state degeneracy does not monotonic increase the volume or perimeter. And we, again, as I said, for a given volume, if you have a larger parameters then ground state degeneracy is larger. And that we can understand this way. Remember that I was basically using some kind of membrane-like operator to construct a boundary agitation. And I moved them to the boundary and I analyzed and created another degenerate ground state. So obviously, if I can construct more and more of those membrane operators, then I can create more and more agitation, right? So more and more degenerate ground state. So this is, yes. So it violates. It's not a semi-permanent ratio, let's say. And I don't know, okay? So I don't know the result in the thermodynamic level. So I don't know, I don't, yeah. Yeah, I think it's self-extensive. It is self-extensive. Self-extensive. Self-extensive. Yeah, sure. The interaction coming from the beta triad, yeah. So that basically gives you the dynamics. Yeah, the spin flip top. There's a spin flip top, the E plus E minus top. That's like S plus S minus top in the, yeah. So that's coming later. Yeah, so that's actually, yeah. So, okay, I'll talk about this briefly, yeah. Okay, great. So, okay, so what I was trying to say is that if I can construct more and more membrane-like objects like this, then what that means is that I can have more and more degenerative ground strength. So I can count the number of membrane operator like this. It turns out that for FCC lattice, there are two LI number of planes in each direction. So the total number of planes is this. So obviously because of this, you have a larger ground-state degeneracy for a given volume. Now, the reason why you have a smaller ground-state degeneracy for a given parameter is because it turns out total number of constraint increase the volume. So there's a competition effect. So there's a competition between parameter and volume effect. And that basically limits the number of possible ground-state degeneracy, yeah. Okay, so now I, okay. Okay, again, this is answer to Chandler's question. So the ground-state degeneracy is actually non-extensive with the volume, but it depends on the geometry. It depends on the geometry. So notice that depending on the geometry, ground-state degeneracy is different. So to us, it's very strange, but for quantum information people, this is very useful. Okay, now, tonally, okay. So now I only talked about electric field, but then you can ask me, where are the magnetic field? Because in the fully quantum theory, I should also have a magnetic field, not just electric field. And these are basically the totaling process that you talked about. So how final fluctuations may actually generate the magnetic field in the form, okay. So we look at that. And interestingly, we don't generate this in the finite or the perturbation field. So again, I don't have time to describe all the details. It just, it's the property of this particular model. I don't think that this is generic, but this particular model, because of the geometrical way that charges are created, we find that if you do the finite or the vacuum fluctuation, that's what basically what you're talking about. We basically, the system does not come back to the ground state. In fact, you can do this if your system size is very small. In fact, this is like, we only talked about electric field square at this point for the Hamiltonian. We are asking how, whether you can generate a B-square term. It turns out the one over the magnetic permeability, you want that to be finite, to have a finite speed of light, right. So here, according to, if you do a perturbation theory, then in order to generate a magnetic field, it turns out you have to go all the way to the boundary. You have to apply a perturbation theory many, many times until you reach the boundary. That's the property of this phase, okay. But because of this, this magnetic permeability depends on your system size, essentially, L-square, right. This guy, paternity parameter to the L-square. So in the thermodynamic limit, your magnetic, one over magnetic permeability goes to zero. So therefore, for any finite size system, your speed of light is finite. But when the system size becomes larger and larger, your photon becomes slow and slower. And in the end, photons are basically localized in the thermodynamic limit. And what it means practically, is that it takes a very long time to turn it from one ground to the other. And that's basically the connection to this quantum glass-like behavior people talk about in the fractured phases. No, no, it's not Lorentz in there. Actually, the photon goes like K-square, the length two-tier. It's not linear, it's not Lorentz in there. Okay. So there's some connection to quantum glassiness. Actually, people like Claude Chavong and Raphoul Nankishio and they talk about this in general. The question is, you do get a maximum education, but then the question is, what the coefficients are? Right? Exactly. I mean, again, you get this P-square term. It's just that the coefficient goes to zero in the thermodynamic limit. So if you only care about finite size system, there is fine. I mean, your photon is slow, but it's there. You just have an awfully slow photon, that's it. No. Yeah, so actually, I mean, what I mean by maximum education is the generalization of the ordinary maximum education. There is a gauge invariant. Okay, there's a gauge invariant. Yeah, there's a gauge invariant associated with the right truth there. So again, everything makes sense. If your system size is not infinity. Good, I think I'm going to the end of that. It's okay, so that's all I wanted to say. And this is the summary. So again, I claim that there's a model that only involves a two-screen exchange interaction on this green particle lattice. And there are gap charge excitation that can only move with a cluster. And there's a serve extensive ground set. You just see that depends on the large geometry. In this particular model, photons are localized in the thermodynamic element. But I would say this is far more realistic than your SQ model where you have a two-spin interaction. So that's all I wanted to say. Thank you very much. So you did this on a part floor lattice, but all of the numerical things that you did were essentially one D or two D. So essentially, are you saying that if I just have a two-dimensional subspace of the parachlorolattice, where I just such a user tetrahedra from the parachlorolattice form a two-dimensional Kagome lattice, let's say you would have all these properties in 2D? No, no, no. Actually, the theory is constructed for the three-dimensional parachlorolattice. It's just that when I counted the ground set, degeneracy, I can only do that. Yeah, I can only do that for the finite size system. Okay, and those instances happen to be two dimensions. But then that leads to the next question, is there a two-dimensional model which has these properties? So these fractional models are inherently three-dimensional. I see. And actually, some of us believe that the two-dimensional fractional model may be confined. What that means is that you may not get this space. It's just like some of the two-dimensional spin-legged phases are unstable. So it's possible that some of, most of the 2D fractional models we can construct is most likely unstable. So that's what we believe, but I don't think there's a proof. Then the last question is just a physical intuition. So you started off by saying that you wanted long-range entanglement. So the long-range entanglement is contained where at the end of this, I mean, what I think the C was sort of the reclassical calculation. So the entanglement, where does the entanglement appear if I wanted to? Yeah, so in fact, there are some studies, some people actually computed entanglement entropy for the ideal fractional model. In that case, you can show that indeed, the entanglement entropy is quite large. Actually, it depends on your system size as well. So it depends, so even for this model that you showed, so if I divide them, I can divide it. Yeah, the entanglement entropy depends on the geometry of the lines. So it's a very weird property as well. Yes, so I'm still a little bit intrigued by this idea of C going to zero. And I'm wondering whether there's a way to take some limits in order to have it depend on the order of limits you take. For example, if you go to T equals zero first, will you get a different entropy? If you go to the thermodynamic limit and then you go to the T equals zero, or whether there was some sort of ratio there that you could play with, can you take some limits? So we haven't tried that. So it may be a good idea. But what we tried is that, I could add more lattice side on top of this breathing particle lattice. I mean, there's no such a lattice, but I can add artificially add some lattice side. Then I can make it work. Then I can make the velocity like to be finite, the thermodynamic limit. So that's why I'm saying this property actually depends, even this property depends on the geometry of the lattice. So, yeah. Yes. The zero in the thermodynamic limit, when, yeah. I don't know what's the corresponding quantity in the elastic theory actually. I'm not an expert of that. But there is an object that corresponds to be in the, yeah, I think that there is a mapping. Yeah. Yeah. I think it's related to that. Yeah. So there is this article written by Leo Lajowski. He's an expert on elastic theory. I think he has this mapping between rank to gauge theory and the topological defects in the elastic medium. But actually, I'm not the right guy to answer the question. Partway through your explanation, you talked about the low line states that you're interested in on, I think it was the eight tetrahedra as being equivalent to a spin two degree of freedom. So I wondered if you could write the problem that you're dealing with in terms of the spin two degrees of freedom coupled by the B-tetrahedra. I mean, there's no technique. I imagine you could get some kind of spin two Hamiltonian spins on the eight tetrahedra and then... I think so. I think there is a way to write a model like that using the usual skin variables. I believe so, yeah. It's just that there's some constraint on that spin model that's related to this Gauss law. So it was just more convenient to write this way. But I think that there are ways to write down in some more conventional degrees of freedom. Okay, last question from the chat. Did you find an approximate formula for the ground state degeneracy? No. Perhaps I missed the answer. Yeah, no, I think that somebody has to find out but maybe it's not me. I was just not able to do so. Yeah, we tried very hard, but it was hard. We just don't have a formula. It would be nice. So follow up, no, but how does it stay? So the only thing, I basically describe everything I know. So it's a self-extensive and it depends on both parameter and volume and that there are two competing effects. Basically, there's a number of constraints that scales with the volume and number of operators that regenerate a more degenerate ground state. They scales like parameter. So they have two competing effects. And that's the reason why the dependence is not monotonic. So that's why it's hard to find that formula. For the usual model, the most interesting is actually the model is probably what people call Haas model, Haas code. And that one, he has an exact expression for the ground state degeneracy. And in that case, he knows the answer because actually he constructed his model starting from algebraic geometry construction. So in some sense, he already knew the answer when you write down the model. So in that case, he know, but in this case, we don't know the answer. Let's thank the speaker again. Here you go. And because of time, let's meet at four 10. So 20 minutes coffee break.