 So we'll continue the discussion of topological lattice model from gauging. And I want to clarify a few things that I talked about by the end of last lecture yesterday. So I got a lot of questions about that, so I think I didn't make things very clear. So let me start with some clarification. First of all, if you remember yesterday, what we did is to introduce a Tory code and then show how the Tory code can be obtained by gauging, doing something called gauging to the transverse field ising model. And in particular, to the symmetric phase. And what we did is to take the exactly solvable limit of the symmetric phase and then gauge the global Z2 symmetry such that we make it a local Z2 symmetry and turn it into a Tory code by the end of last lecture. So this is the transverse field ising model. The transverse field ising model with a global Z2 symmetry. But then we want to make it into a gauge model with local symmetry that acts on a correction of what we call the matter field and the gauge field altogether like that. And we modify the Hamiltonian such that each Hamiltonian term is now explicitly invariant under all the local symmetry transformations. And the way we did it is, of course, by addressing these ising coupling terms by this gauge field term. And then we said, OK, so these two terms were just inherited from the original transverse field ising model. But there will be too much degeneracy in the model because we didn't say anything about how the gauge field would behave, what kind of quantum dynamics they would have. So we want to add some extra term just for the gauge field because the gauge field is just wandering around, don't know what they should do. So we said, OK, we want to add the BP term. And the reason we want to add the BP term is because we're looking for terms for the gauge field itself. So it's a pure gauge term. But we still want to maintain a gauge symmetry. The gauge symmetry, the local symmetry is something that we want. This is the whole point of doing this process of gauging. So we want to add some terms while maintaining the gauge symmetry. And you can see that if you want to maintain a gauge symmetry, we cannot just add one-body tau x terms. What we have to do is to add a whole loop like that. Remember that the BP terms, each of them is a loop of tau x. And that turns out to be the minimum term that we can add for the pure gauge field to introduce some dynamics for the gauge field. To relieve the huge ground set degeneracy if we don't include these kind of local terms. That's the first thing. So we don't just add BP term because we want the BP terms. It turns out they are the simplest term. The simplest term that we can add while preserving the local gauge symmetry. And secondly, another thing which I said in a very confusing way is I said that if we enforce the gauge symmetry and by enforcing gauge symmetry, what I mean explicitly is that if we only look at a sector where all the gauge symmetry terms are equal to 1, so we force them to be 1. If we only look at a sector where all the gauge symmetry terms are equal to 1, now we can have the equivalence between sigma z and the 4 tau z. And that is something we used to reduce this Hamiltonian explicitly to the Torre code Hamiltonian and show their connection. But of course, enforcing all the gauge symmetry to be 1 is different from saying that the model has the symmetry. Usually if the system has symmetry, then the hubris space might transform non-triviality still under the symmetry. For example, if it's a z2 symmetry, then the eigenvalue can be 1 or minus 1. That's solely possible. So here we are saying that if we forget about all the minus 1 sectors, if we only look at a plus 1 sector, then we have this equivalence and then we can do the reduction and the model becomes exactly equivalent to the Torre code Hamiltonian. But of course, we can look at other sectors. We can just look sector by sector. We can look at other sectors where, for example, some of the local symmetry transformation is equal to minus 1. Now other sectors where some of them are minus 1. But that's just fine because that just means that in those sectors, the sigma z would be equal to the product of tau z times minus 1. So there's still a 1 to 1 relation between what we consider as a gauge charge excitation and the original symmetry charge excitation. It's just the labeling is opposite. So as long as you fix a particular sector where the UV takes either 1 or minus 1 value, then there is a correspondence between symmetry charge and gauge charge. OK. So of course, I did that in order to show you that this gauge model of transfer to your Ising model is just Torre code. That is the whole point that we start from transfer to your Ising model, gauge it, and then we get Torre code. But actually, I don't really need to enforce the gauge symmetry to be 1 or minus 1. I don't need to enforce that into be a particular value in order for this model to be in the same phase as Torre code. So of course, right now, it doesn't look like the original Torre code we were talking about because it involves sigma field, involves tau field. So there are more degrees of freedom involved. So it looks kind of complicated. But it is totally still possible that this gauge Hamiltonian has the same kind of topological order as Torre code. That is, it shares all of the universal properties that we talked about yesterday, which are like four full ground safety generously on the torres, the gauge charge excitation, the gauge flux excitation, each of them being both sounds. But they mutually have a minus 1 braiding statistics. Everything like that is actually the same between this gauge model and the simpler version of Torre code that we talked about at the beginning of the lecture yesterday. And we can actually work that out. So let's see. So this Hg, let me just write everything out. So OK, so we were looking at the limit where the transverse field ising model is exactly solvable symmetric point, meaning that we set this J term. We set this J to be 0. Can you all hear me? This is better? OK. Maybe I should. I was OK. Isn't this much louder? I'll just leave it like that. OK. All right, OK. Right, OK. So I was saying that, OK. So let's try to see how gauging the symmetric limit of the transverse field ising model gives us something that's still in the same phase as Torre code. OK, so we're going to look at the limit where J is equal to 0, so J is equal to 0. And the gauge Hamiltonian looks something like minus i sigma zi minus sum over p bp. And now, instead of enforcing all the local symmetry terms to be 1, we can just add them as another Hamiltonian term. And we add them as, well, they're local terms, so we just think of them as something in the Hamiltonian that we can add. And it involves the sigma z and the 4lz. And that's it. OK, so the Hamiltonian contains three kinds of terms. One is the transverse field on the matter field, the plocket term, and the vertex term, which is a little bit more complicated than the vertex term in the Torre code because it also involves the matter field. OK. So I claim that the ground state property and the low energy excitation universal property are the same between this model and the Torre code model. And the easy way to see this is that if we are looking at low energy, we can, first of all, set this term to be 1. We can, first of all, say, OK, we want to minimize energy for this sigma z term, meaning that we want to set sigma z equal to 1 everywhere so we actually ping down the matter field. The matter field is just pointing in the positive z direction everywhere. And by doing that, we get rid of the sigma z in the middle for this term. And again, we get Torre code. Oh, why do I choose this to be minus? Well, choose to be minus because I'm just saying that I want the ground state to have eigenvalue 1. Yes, so this particular choice of a minus sign in front of all this UV term now corresponds to saying that I want ground state to have UV equal to 1. Of course, I can choose different signs here. It's just saying that at different lattice side, I might have different gauge constraint. Right. OK, but now there's a very, very important point that is in the ground state of this term, where I said sigma z equal to 1, I literally get rid of the sigma z in the middle of the vertex term so that I only have the placate term and the vertex term. Right. And that is literally the Torre code. And it's going to have all the same universal properties as the Torre code. But on top of that, I can add some other small terms. Like we talked about yesterday, we can add tau x term with some small coefficient. We can add tau z term with some other small coefficient without affecting too much the universal property. We argued yesterday why small perturbation doesn't break the 4-4 ground state degeneracy of the Torre code on the torres. And actually, small local perturbation doesn't change any of the universal properties we talk about, like quasi-particle types, statistics, topological spin, braiding, whatever. OK. But on the other hand, you see that when we add terms like tau x, we're explicitly breaking the gauge symmetry. OK, this used to be the local symmetry that we want to impose. We said that the whole process of gauging is to promote the global symmetry into a local symmetry. But now by doing this, by turning this into a local Hamiltonian, we see that we can actually, we are allowed to add terms that explicitly break the local symmetry constraint without affecting the physics too much. OK, so this is something that's more of a condensed matter point of view about gauge theory. That is, gauge symmetry is a way to motivate everything that we do here. And we start from things with a global symmetry, promote it to a local symmetry, and we write it down such that everything's gauging variant. But after that's done, it's not that important, especially for the case where we get a gapped Hamiltonian. Once we get a gapped Hamiltonian, we actually don't care so much about gauge symmetry. And gauge symmetry is, in some sense, an emerged symmetry. And what's actually emerging is the topological order that's emerging. So ground safety generacies, the fractional expectations, there are statistics that's emerging out of this whole process. So we do take the gauge symmetry seriously when we try to say how we want to gauge a model. We started from a model and want to gauge it. And during this step, we want to make sure that we enforce the gauge symmetry, because otherwise we don't have any clue how to do the gauging procedure. But once we did the gauging procedure and we get a gauge model, we actually can relax the constraint a little bit. But not by too much. We cannot put this term to be too big. If it's too big, then something bad can happen. But as long as this is a very small perturbation, all the physics survive in the gauge model. Topological order at the topological age? Yes, exactly. I'm saying that. Yes. Yes, yes, exactly. So what gauging we get there is something that doesn't mean that it's big. Yes. What are they doing? Right, right, right. So that's a good question. The question is about now the topological order is not protected by the gauge symmetry, whether they're still related in a certain way. Yes, exactly. So I would say that topological order is not protected in a way that we don't need to enforce explicitly operators like that. We can break operators like that. But there's a sense of renormalized gauge symmetry in merging. So as long as we stay in the same phase, you can imagine that it's not exactly this operator that's preserved, but there is some dressed version, some expanded version, which would look much more ugly than this one. It may become bigger. It may even have a tail, decaying tail. But there's some version of emergent symmetry transformation in the model. And that literally is if you see that this is related to the hopping to the string operator of a flux. The string operator for the flux is the tau z, tau z, tau z, tau z. So this gauge symmetry is something like creating a pair of flux and bringing them around. And that always exists whenever we're in the topological phase. So you can say that maybe on a larger scale, you can always do this process of creating a pair of flux and annihilating them around a circle. And if we're in a ground state, that's always an invariant operation. As long as we don't have excitation in the system, in a ground state, then we can always create a pair of flux and bring them around and finally annihilate them. And now will be the local symmetry transformation in this. So as long as we're in another way of saying it is that as long as we have the same topological order, we always have this kind of symmetry. But that symmetry would not look exactly like this. It will look different. In principle, yes. If you give me an actually Hamiltonian like this, I wouldn't know how to write down exactly the operator. Because the operator will look bigger and it will probably not even be exactly finite size. It will decay. It's local in some sense because it will be an operator that has a profile that looks like this with a tail decay. But then I don't know how to write it in an explicit way. Yes. Well, yeah, exactly. So now we still keep the matter field here and we involve the matter field in the UV. So the high energy excited states will be different. But now we care about ground state and we care about low energy excitations. And now we'll be the same. And we care about fractional excitations on top of the low energy on the ground state. So those will be the same. And actually explicitly, we can see that the string operator almost still look exactly the same with some caveat. So the string operator say for a flux, if we want to create a pair of flux along this line, all we do is still to apply tau z, tau z, and tau z along the whole way. Now if we want to generate, if we want to hop the charges, let's say we want to hop the charge along a line from this point to this point. So originally, in the transfer field icing model, the way to create a charge is to do something that anti-camille with the symmetry. We want to do a sigma x. That anti-camille with the symmetry and locally generate a charge. So we do sigma x here. But of course, we cannot just do one sigma x because we want to preserve the global symmetry. So we need to hop the charge onto the next site. We need to do another sigma x here. But now we gauge it. After we gauge it, we know that this is not gauging variant. What we need to do is to add a tau x in between. So this becomes a single charge hopping step. We keep doing it. So we do sigma x, tau x, sigma x. And then we do sigma x, tau x, sigma x again. But the two sigma x cancel each other like this. So we just have a string of tau x in between two sigma x. And we can keep doing that so that we actually stretch a line of tau x until we have moved the sigma x to the other end. So this is the string operator for the charge hopping in this model. And you can see that even though at the end it's just by the matter field, in between it's exactly the same as the previous string operator we have for the pure gauge theory. And because of that, if you calculate self-statistics by doing the figure of 8, or if you calculate the mutual statistics between this string operator and that string operator by doing some commutation relation, it's going to be exactly the same. So of course, the dressing at the two ends, it doesn't really matter. OK, so let's try to summarize a little bit before we move on just to see what we learn from this case. We take the transfer field icing model, putting the gauge field such that we can make it locally symmetric. And the first thing I want to point out here is that look at the way we changed the term that was in the original Hamiltonian. So this transfer field, it doesn't change at all. This icing coupling term, we dress it with some gauge field in the x basis. But that is all we're going to do. If we have other terms like xx coupling at a longer distance, we talked about yesterday, all we do is to putting more tau x along the way, connecting the two. So if we only look at Hamiltonian terms involving the matter field, it corresponds to the same dynamics because the operator algebra is still the same. So the gauge field, we put them in there, but they just go along for the right. They just connect the sigma x operator whenever they need to. And they only come in the tau x form. They never come in the tau z form. The tau z form comes in these kind of terms. So if we only look at the dynamical terms for the matter field, it's exactly the same as the original model, which means that whatever the matter field was originally doing, it is still doing the same thing after we couple it to the gauge field. If it's symmetry breaking, it's symmetry breaking. If it's symmetric, it's symmetric. If it's gapless, it will be still gapless. If it's some non-trivial insulator or superconductor, it's still a non-trivial insulator or superconductor. So dynamics for matter field is the same, at least in the ground state and the low energy sector, where this term is said to be 1. So at least at low energy, when we put don't put in extra flux, when the background flux pattern is just trivial, and we don't have flux anywhere, the matter field is just doing whatever it used to be doing. Of course, the non-trivial thing is how the gauge field get involved. Now there's a gauge field, and the gauge field is doing something, and I want to understand what the gauge field is doing. So first of all, as we saw yesterday, the gauge charge, the gauge charge is just something that we inherit from the ungauge model. The gauge charge is simply the symmetry charge before gauging. So gauge charge comes from symmetry charge. And this immediately tells us a lot. For example, originally we have a Z2 symmetry, so we know that the Z2 charge is either 0 or 1. And if we have 2 charge, that's equivalent to 0, which means that the gauge charge in the gauge model, if you fuse two of them, that goes into the identity channel. So if we have two of the gauge charge, and if we fuse them, that goes back to just what I write as 1, meaning that the trivial fractional excitation channel. So the fusion channel of the gauge charges come from the symmetry charge. And secondly, the statistics of the gauge charge, the topological spin of the gauge charge is also inherited from the symmetry charge. And in this case, we know we started from a spin model, so the local gauge charge excitations are all bosonic. So these are bosonic excitations. Of course, there can be other possibilities. We can have a model where we have a fermion degrees of freedom, and the fermion degrees of freedom might have a Z2 global symmetry. So a fermion model always have a Z2 global symmetry, which is the fermion parity symmetry. And we can gauge that global Z2 symmetry. In that case, the gauge charge will be a fermion. Yes. All right, good question. So in QED, which I'm actually going to talk about later, so to say, what the connection is to things I'm talking about here, but just to answer your question, in QED, we get photon because that's a continuous gauge field. And here, this is a discrete Z2 gauge field, so actually the photon is gapped out, so we don't have a gapless sector. And you can see that all the excitations are gapped. I think we were writing something here yesterday. I was saying that the e-excitation, each of them cause energy 2, and the m-excitation also cause energy 2, so all the excitations are gapped. Of course, also in this case, the matter field is gapped. If the matter field is gapless, then it will be gapless. OK. And the third thing is that there's going to be an aharonov-Bohm effect, aharonov-Bohm phase factor between the gauge charge and the gauge flux. And yesterday, we saw that come from the braiding statistic between E and M, which is the commutation relation of the string operator of tau x and tau z. And that is set just by the symmetry, because this is a Z2 symmetry, so the gauge charge going around the gauge flux can only generate a phase factor of minus 1. If we do have U1 symmetry, if we have real electromagnetism, where electrons go around some flux, not a phase factor can be anything, between anything of 0 and 2 pi. So this factor is determined just by symmetry group, the original global symmetry group. So fusion rule comes from symmetry. And the cell statistics comes from the original model, from the statistics of symmetry charge. So you can see that even without doing this exercise of taking the model and actually change the Hamiltonian and then solve for the gauge Hamiltonian and solve for the Bromsey degeneracy excitation braiding statistics, we can already tell a lot about the gauge theory. We know that the gauge charge has two fields like this. We know the gauge charge has to be bosonic. We know the gauge charge and gauge flux has to have braiding statistics like that. So that is all just set by symmetry alone. So in a lot of case, it's actually quite hard to do this exercise. For example, when j is not equal to 0, then the model is not exactly solvable. And you can do this exercise and writing down this gauge Hamiltonian, but it will not be very straightforward to analyze the Hamiltonian. Or in some more complicated models, it might not be very straightforward to analyze the gauge Hamiltonian. But even without doing that, we know we should get at least 1, 2, 3. This is just guaranteed when we do this procedure of gauging. The final fourth one is the only tricky place. Now, we do need to analyze the gauge Hamiltonian in order to figure out what the flux is doing. So the flux excitation, unlike the gauge, unlike the charge excitation, it's something that didn't really exist in the original, in the original global symmetry model, in the transverse field model. It is something that we add into the model by hand. And of course, we were saying that if there's no flux excitation, the matter field is doing whatever it does. But if there's a non-trivial flux pattern, for example, through each of the placets, we can put a pi flux. We can set up the flux configuration such that it is non-trivial, then that might change the dynamics of the matter field. Just imagine that this tau x, if it takes a non-trivial value, then it might change the dynamics of the matter field. And exactly because the gauge field, the gauge flux, is something that's involved with the dynamics of the matter field. So the gauge flux somehow knows about the dynamics of the gauge field. Unlike the charge excitation, the charge excitation doesn't really know the dynamics of the matter field. But the gauge flux, it does. And it responds to that. And it responds to that in a way by changing its own statistics. So of course, in this case, we see that it's a simplest case. It's the simplest model that we can think about. So the gauge flux is doing something very, very trivial. It's a bosonic quasi-particle. And bosonic is the most trivial thing you can think about. But once we go to some slightly more non-trivial state, which maybe I have time to talk about today, then the flux excitation is actually going to respond to that by becoming a non-bosonic excitation. For example, if the state is not a simple transverse ionizing model, it's not the symmetric phase of the transverse ionizing model. It becomes something called the symmetry-protected topological phase, still with Z2 global symmetry. Now, one, two, three still holds, especially two and three still holds. But four is going to change where the gauge flux is going to respond to the symmetry-protected order by turning into a semi-ionic excitation. It's going to change itself's statistics. That's what gauge flux is, too. Gauge flux, they encode the non-trivial order of the original system in terms of its own statistics. And that's why gauging becomes a useful thing to do, because if we start from some model, which we don't know really what is going on, and then you can try to detect for the order by coupling to the gauge field, and then extract and see what the flux excitation is doing. I'm not saying that's an easy thing to do. Of course, ideally, or logically, that is a possible path to investigate the original model. OK? All right. Good. So this is just trying to summarize what we learned yesterday, and now we're ready to move on. We're not moving on very far away. We're just moving on to the limit where j is much, much larger than 1. So yesterday, we were only looking at the limit where j is much, much smaller than 1, and we actually literally set j equal to 0. And now we're going to look at the case where j is much, much larger than 1. So we're going to ignore the transfer field ising term. Sorry, the transfer field term in the model. And we all know what happens to the transfer field ising model, the model goes into a symmetry breaking phase. And we're going to see what happens to the gauge theory. OK, so the Hamiltonian for the transfer field ising model is simply sigma x, sigma x, which is job, the sigma z term. And the gauge Hamiltonian, well, we know how to write it down. We just take this Hamiltonian and then drop this term. And I'm putting the gauge symmetry as another local term into the Hamiltonian. So we have sigma x i tau x ij sigma x j minus the BP terms and minus the UV terms sigma z tau z tau z tau z tau z sum over vertex. OK, so we get another Hamiltonian. And we get another Hamiltonian where all the terms commute with each other. So it's still nice. So this is exactly so for model. So in principle, we should be able to figure out what is going on in this gauge model. So what I'm going to claim and I'm going to show you is that this model doesn't have topological order. Yes. Oh, yeah. You're saying that why I'm not dropping all the other terms for the gauge field as well. OK, sorry. And a better way to say this is that maybe I add a B coefficient here. And now I'm sending B to 0. I'm just saying that I want to be in the central breaking phase of transfer field icing model then I couple to the gauge field. I do want some non-trivial gauge field involved and then couple it. OK, so I'm dropping this term, but I still do want to keep the gauge field term. So I'm actually increasing the coefficient in front of the gauge field while I do that. So I should have put a j here and a b here and a j here and a b here. And now what I'm doing is to set B to 0. So I didn't worry too much about these coefficients because mostly I'm talking about ground states. And these j and b, they're going to give some energy scale to the excitations. But as long as that's finite, I didn't care too much. Yes. Yes, yes. So we literally just take the transfer field icing model where we gauge everything. And then we say, OK, we were originally in the central breaking phase, so we can ignore this term. But all the other terms are just here. I just copied from here. I did nothing. I just copied from this side to the other side. Oh, the last term. Oh, the last term. Yeah, the last term, you can always do that. You can always just put it in as an extra local Hamiltonian term. Or a better way to put it is that you can deal with it in two ways. Either you set them all to 0. You say that, OK, I only look at the gauge invariant sector. And the other way is to include them into the dynamics, such that they don't become hard constraint. They're soft constraint that can be violated, but the ground state doesn't violate them. So the ground state properties are all the same. Yeah, so once I involve them in the Hamiltonian, it's not a constraint anymore. The hubris space does contain terms that violate this AV term. But at low energy, I don't violate them. OK, so the goal is to show that this is not topological. Not topological in the sense that there's no ground state degeneracy on the torus. There are no non-trivial fractional excitations. No braiding statistics. Accentations are just local excitations, and they move around, but they don't braid with each other. And we call it not topological. OK, so this is actually possible to do, and even possible to do, as I said here, in class. First of all, you notice that this is exactly a solved model because all the terms still explicitly commute with each other. You can check that. That's how we came up with the Hamiltonian in the first place. They commute with each other. So in principle, with some effort, we should be able to solve for the ground state and low energy excitations, at least. And OK, so the way you can see that is first by noticing that the BP term is actually right now redundant. So the BP term involves tau x, tau x, tau x, and tau x. Four of them. But it's simply a composition of the first term, which is the Ising coupling term around these four edges. Because the first term is like sigma x, tau x, sigma x. And if I do that around the four edges of a single plocket, I cancel the sigma x part and get a BP term. Which is saying that if I do get a ground state wave function, which has eigenvalue 1 for all the sigma x, tau x, sigma x term, then I automatically have eigenvalue 1 for the BP term. So BP term is kind of redundant. We need to only look at the first term and forget about the BP term in this case. Forget about it. Of course, this is only true if we're looking at the ground state where everything is eigenvalue 1 for the first kind of term. Now we have a square lattice with matter field and vertices and gauge field on the links. They're coupled. Still not very simple Hamiltonian because everything's coupled and everything will be entangled. But this is actually a Hamiltonian that's very popular in quantum information. And quantum information people study this Hamiltonian a lot. And this is actually called a cluster state Hamiltonian. So the kind of term that we have in a Hamiltonian is sigma x tau x sigma x. And then sigma z tau z tau z tau z tau z. We have two different kinds of terms. So we can do the same exercise as we did yesterday. We can count what is the total hubris space dimension for the system. And then we can count how many poly operators do we have in the community Hamiltonian, where each of them cut the hubris space in half. Finally, we need to look for possible global constraints between all these poly operators because yesterday we see that that's how it give rise to the 4-fold degeneracy in the tarot code. There was some global constraint between all the AV and BP term in the tarot code. That's why we have a 4-fold degeneracy for the tarot code. So we can do that here. We see that we have some matter field. We have some gauge field. And you can see that these kind of term they're centered around the gauge field. We have one term per gauge field. They just live on the links. And they're centered at the gauge field. So one term per gauge field. And this kind of term is centered around the matter field and one term per matter field. So we literally have the same number of Hamiltonian terms and the same number of degrees of freedom. So the total hubris space dimension, if we keep dividing it by 2 per each of the Hamiltonian term, we get just down to 1. And the only tricky part that we need to worry about is whether there's going to be global constraints among all these kind of terms. If there are no global constraints, then we have a unique ground state. If we do have global constraint, we might have more than one ground state. So if you stare at it for a long enough time, I hope that you can convince yourself that there are no global constraints among all these kind of Hamiltonian terms. Remember that yesterday when, if you remember, the AV term, there are also these kind of term, but without the sigma z in the middle. And the AV term, if we multiply everything together, it goes into identity. But here, because there's a sigma z term in the middle, so if we multiply all these kind of terms together over the whole lattice, we get a tensor product of all the sigma z, which is not identity. So this is, you can do the same thing as yesterday, but that doesn't give you a global constraint on the local terms. So this is just me trying to argue my way around, just trying to say that this model, if you do the exercise and later solve it for the ground state, you're going to see, now the ground state is not going to be degenerate, no matter what kind of manifold you put it on. You need ground state, no fractional excitation, no statistics. It's actually just a totally trivial, topologically trivial model. The wave function is going to be a little bit complicated because everything's interacting, so the wave function is still entangled, but it's something that we call short range entangled because we can build up the ground state by starting from a product state and doing some local unitary transformation. So it's very, very different from the Toricota Motonian, which we get by gauging the symmetric limit of the transverse field ising model, which does have ground state degeneracy, does have fractional excitation, does have statistics, and the ground state wave function, you cannot just get it by starting from product state and doing simple local unitary. Okay, so what is the punchline here? What am I doing? I take the symmetry breaking phase, couple it to the gauge field. And the coupling to the gauge field procedure, of course, it's generic. It's something that we can do for whatever model that has global symmetry without any relevance to what the phase is for the original global symmetric model. And then I see the gauge model is not topological. Something trivial. How does that make sense? And that makes sense if you have heard of the term Higgs. I assume everyone has heard of about Higgs, the Higgs particle, the Higgs mechanism, but probably in a very different context. You heard about Higgs about how the Higgs particle was discovered at LHC and how it gives mass to all the fundamental particles. And maybe you have also heard about it in terms of superconductors. Superconductor also undergoes so-called, it also is an example of Higgs mechanism. And so Higgs mechanism with respect to the electromagnetic field. So there are two places where Higgs is usually mentioned in the standard model. In a standard model, the Higgs mechanism involves the electrical weak interaction. So in a standard model, there's a SU2 gauge group and a U1 gauge group. And these are the gauge group that are responsible for the weak interaction and the electromagnetic interaction. And it turns out that this gauge group is breaking down to just U1. It started from a much larger gauge group and gets reduced to U1. And this U1 turns out to be the electromagnetic gauge field, giving rise to photon electromagnetism. On the other hand, we have superconductor, where originally in superconductor, we know how to get a superconductor. We start from metal, which has U1 symmetry, which can couple to external electromagnetic field. But once it becomes superconducting, the gauge group becomes Z2. So we say that the electromagnetic field in a superconductor is Higgs. The gauge group started from U1 gets reduced to Z2. So Higgs, if you want to have a very simple, very naive way of understanding what is Higgs, it's a reduction of gauge group. The reducing gauge group from a bigger group to a smaller group. This is my naive interpretation. Of course, there's much more to Higgs than this. But this is like the minimum version of understanding what is called Higgs. And the reason why we can reduce gauge group is because there's some symmetry breaking going on. In superconductor, we know that it's the copper condensation. The electrons, they form copper pairs, and the copper pairs condense so that we don't have charged conservation anymore. We only have conservation up to two. We can only talk about the even and oddness of quasi-particles in a superconductor so that the global symmetry is reduced from U1 to Z2, and that happens spontaneously. So superconductors basically undergoing a spontaneous symmetry breaking from U1 to Z2, and the gauge field gets Higgs along the way. And the standard model is the same. Some spontaneous symmetry breaking happens and reduce the gauge group from this big one to the smaller one. That's also exactly what's happening in our toy model of transverse field icing model coupled to a Z2 gauge field. Of course, the original gauge field we started with is Z2. It's already the smallest group you can think about. Unless we have spontaneous symmetry breaking, it breaks into nothing. It doesn't get reduced to a smaller group. The Z2 group just becomes not a group, just becomes nothing. So originally we started from the taric... Once we gauge it, on the symmetric side we have the taric code, which is the de-confined Z2 gauge theory. But if the matter field undergoes spontaneous symmetry breaking from Z2 symmetry to no symmetry, then the gauge group in the gauge theory also reduced to nothing, which is why the gauge theory has no topological order at all. The gauge group is now nothing. The word limit? Yeah, the trivial group, yes. Exactly, a group with one element. Right, so given that, given that we know what happens when things get hick, it shouldn't be a surprise that when we do this exercise and we take the symmetry breaking limit, couple it to gauge field, and analyze what happens with the gauge Hamiltonian, we found that it's not topological. This is what we should find. Otherwise we'll be in trouble. And this exercise of talking about cluster state, counting the number of Hamiltonian terms, this is just to confirm what we should be able to find. So lesson learned is that if we start from the symmetric phase of the model, we get some truly topological order with ground state degeneracy, quasi-particle. If we start from the symmetry breaking phase, we get a non-topological model. Of course, if we start from a bigger symmetry, if we get, say for example, if we started from Z4 symmetry, then it is possible to partially hick the gauge field. We can partially spontaneously break maybe a Z2 subgroup. Of the total group, such that we retain some gauge structure in the gauge theory, and we can still have a Z2 topological order in the end. But if we are in a symmetric phase, we should get a Z4 topological order. Maybe this is a good time for a break. And when we come back, I'll talk about how this notion of gauging, how the gauging I've been showing you is related to the kind of gauge theory, gauge field you have all heard about. Especially everyone, when we take electromagnetism, even classical electromagnetism Maxwell's equation, we have heard about gauge field. I'm going to talk about how they're related. How that version of gauge field can be reduced to whatever I'm showing here. Okay, I'll see you in 10 minutes. So this is regarding posters. You have time until noon to submit your abstract, if you have not done so. By the way, many people have already submitted during the application procedure, and you don't need to send again because the secretary is receiving multiple copies of the same thing. So if you have done it already, don't fret. And by the end of today, I'll tell you until noon to submit the abstract of the poster. This afternoon, I will tell you which posters are accepted which not. We'll tell you by, yes. Yeah, this was the third point that I was going to, so yes, the next slide answers that. So, okay, so this afternoon, I will tell you by email who has been accepted and who has not. Tomorrow, I will give you an instruction on how to print your poster in case you have to print the poster. Now, yes. Oh, I cannot hear that. What did you say? You can send an email to the secretary. The address is in case you have not seen the webpage of the school at all in the last two weeks is ASMR35 at ICTP.IT. Okay. Tomorrow, I will give you an instruction on how to print the poster in case you need to print. By the way, how many of you already have it printed? Okay. Those of you who have printed, please do not ask to reprint it because you found that there was a minus sign that should have been a plus or things like that because it costs us 10 euros per poster, okay? So, we are going to pay for those who have not printed it, but if everybody has to reprint it, I will ask, you know, I will ask a contribution to the budget. So, it costs us something to print the poster, right? So, tomorrow, I'll give you more instruction about this. So, you're finishing at 11, and then there will not be a coffee break because the lunch break is just extended until 2.30, and then at 2.30, we have another lecture, and then there is the colloquium by Professor Dalibar, who's going to be here and a lot more people will join us. That's all. I'm not going to take more of your time. Thanks. Okay, now it's up. So, now I'm going to tell you how the gauge field I'm talking about here is actually related to the things you probably have heard about, about gauge theory starting from very early years in terms of electromagnetism, Maxwell's equation, how those two are actually the same thing. It might not look like that at all when I wrote things down over there. Maxwell's equation. Let me remind myself what Maxwell's equation looks like. Of course, Maxwell's equation is classical physics. When Maxwell wrote down his equations, it's not quantum mechanical yet. It's a classical theory of how electricity and magnetism turn into each other, and electric field turns into magnetic field, into electric field, and how the charge and current respond to that. And that is a set of four equations. Curl of v is equal to zero, sorry, divergence of v is equal to zero. Curl of e is equal to minus partial b partial t. And then the divergence of e is not necessarily zero. It's related to charge density. So in the integrated form of Maxwell's equation, we know that this can be interpreted as you integrate the electric flux through a surface, and that gives you how much charge is enclosed in that surface. Of course, I'm writing down the differential form of Maxwell's equation. I'm turning everything into an integrated form. And finally, we have the curl of b has two contributions. One is the real current, and the other is the time derivative of electric field. So this is Maxwell's equation in terms of e field and b field. e field and b field, there's something measurable, they're real. But what people notice is that we can introduce some fictitious field, the a field and the phi field, the vector potential, and the electrostatic potential to simplify the equations. And in particular, if we set the magnetic field to be the curl of a, then we automatically satisfy this equation, because the divergence of a curl is zero. So if we choose b to be equal to curl of a, we don't need to solve the first equation at all. And second thing is that we can choose the e field to be minus gradient phi minus partial a partial t. And if we choose the e field to be such a combination between these fictitious fields, now we don't need to solve the second equation. The second equation is automatically satisfied. So this seems to be simplifying things because we introduce some a field and phi field, and of course we can plug them into the following two equations to get some non-trivial relation between a and phi, and also their relation between the charge density and current density. Now we can try to solve for the whole dynamics of the system. If we have only vacuum, we will find some plane waves or some other form of waves up to some boundary condition. I'm sure everyone did that exercise a long time ago, maybe. But the thing is, when you use the a field and the phi field to try to solve for the Maxwell's equation, it gets some redundancy. And in particular, if a field transforms as a plus gradient f where f is any function of space time. Let me just write space. Three, three, three, space. Phi is if f is any function of space, and then phi goes to phi minus partial f. I guess I need to put in the time coordinate anyway. So the time derivative of f and f is again a function of space time. Those two transformations, those transformations of a and phi for any kind of f give you the same e and b. You can plug in this transformation and see that it doesn't affect e and b at all. So in classical electromagnetism, the a field and the phi field are considered as something that's fictitious, as something that we just introduced for the purpose of mathematical convenience. But once we solve the equation in terms of a and phi and we try to get back to e and b, we need to remember that not all different solutions of a and phi are actually physically different because they might give rise to the same e and b and in particular, any two different solutions related in this way, they're physically the same. And so gauge symmetry, in that case, just represent a redundancy in our attempt to change variable in the Maxwell's equation. It's something that's fake, that's something that we should get rid of but it simplifies the equation so we do it anyway but we need to get rid of it in the end. People go to quantum in actual magnesium called QED, quantum in actual dynamics. A big realization when we go quantum is that the a and phi field are not fake. They're real. There's something that actually exists and there's something that's actually fundamental. And the b and e field, on the other hand, becomes something that can be derived from the a and phi field. And the reason people think that way is exactly because of the Ahronov-Bolm effect that we talked about yesterday. Can put a current carrying solenoid where there's a current going through, generating some magnetic field but the magnetic field is confined within the very, very thin solenoid. You have a configuration like that and the magnetic field leaking of magnetic field out of the solenoid is very, very small. However, if you put some electrons and electrons might go along some paths around the solenoid but at a distance that's very large away from the hole such that the magnetic field in the region outside is literally just zero. For the electrons going around the solenoid still feels the existence of the electromagnetic field by changing the phase factor of its wave function. We know what happens when the wave function changes phase factor. The interference pattern that we get at the final screen will shift its location. The experiment is done like originally you don't put any current in the solenoid so there's no flux through the hole and you get some interference pattern and then you turn on the current and increase the flux through the hole and you see that the pattern will shift. That's because along the same trajectory the electron will accumulate different phase shift along the trajectory such that the interference maximum is moved from here to here and the minimum is moved from here to here. This is just saying that the A field and the phi field there are actually real things in quantum mechanics. In the quantum mechanical formulation of how electrons or how fundamental particles coupled to electromagnetic field we cannot use E and B to write the equation. We actually need to apply into the Hamiltonian the Lagrangian or whatever you want to write in order to describe the system. So the gauge symmetry is still here the gauge symmetry for example where A goes to A plus gradient F not still there and that is still a redundant symmetry so it's a quite tricky situation in quantum mechanics that we want to think of A and phi as real but somehow they have this weird redundancy of socialism. It's just how nature is that's how the whole center model is formulated that we have some field but they have some redundancy but because of that there are a lot of amazing things happening. So we learn to get used to that and live with that and dual theory while remembering that there's this kind of redundancy involved in the theory. So you might already see now there is some similarity between this transformation and the transformation I'm talking about over there even though they still look very different right now. So here gradient of F is what we call the local gauge transformation or just the gauge transformation and F can be anything F can be any function and in particular it can be a delta function localized in space if F is localized in space then only and the A field that's right next to it gets transformed everywhere else it stays the same so there can be symmetry transformation that's localized around the particular point in space and that's what happening here here we have the UV operator which is acting around a particular point in space and on the other hand in the quantum theory of electrodynamics we not only have the gauge field transform but at the same time we have the matter field which is the electrons moving in space suppose that this is the electron field you can think of it as the electron wave function you can think of it as the creation operator of the electron at every spatial location that will transform under the same local symmetry transformation and as you can imagine the way that the the matter field will transform is by accumulating certain phase factor and that phase factor is exactly this F so we have the gauge field transforming we have the matter field transforming and then we put them together and they couple in certain way so how is that related to the UV over there let me try to walk my way back so the first thing we want to do is to put things on a lattice because the torque code we did it on the lattice so let's try to put the electromagnetism also on a lattice I shouldn't have erased this so we have let's say again square lattice where the matter field is at the lattice point where the matter field here are the electron matter field and and the creation annihilation operators given by C dagger and C and almost the same thing as phi over there just using the more common lattice notation for electron creation annihilation so the electrons if they hop on this two-dimensional lattice they if it's an insulator or a metal if it's not superconducting then it has a global symmetry so if we do the transformation of e to the i alpha C dagger and C goes to e to the minus i alpha C and we use a constant alpha everywhere this is a global symmetry for electron hopping in two-dimension because all the Hamiltonian terms we are going to have is C dagger C plus C C dagger something like that so if we have C and C dagger transforming the opposite way with certain phase factor the whole Hamiltonian is invariant and this is the global u1 symmetry okay but now we're not satisfied with just being globally symmetric what we want is local symmetry we want to promote this coefficient alpha to be a local thing meaning that we want alpha to have spatial dependence so we want local symmetry such that C dagger goes to e to the i f x C dagger and C goes to e to the i minus i fx C dagger and in particular I can choose this f function to be a delta function at a particular lattice site so I'm doing transformation only at that lattice site so this is how the matter field transforms but well if we want to make the symmetry local we need to pay the price of introducing gauge field well the gauge field as you can imagine they now live on the edges I live on the links of the two dimensional lattice where on each link we have a gauge field of A and now here's where the quantum mechanics of the gauge field comes in in quantum mechanics this A field is not just a number field it's not like for each spatial location I have a vector or a number or sets of numbers this A field is an operator a non-trivial commutation relation with another operator which is the E field so in quantum electromagnetism ENA become a pair of conjugate variables conjugate fields conjugate electromagnetic field so of course ENA are both vectors so let's label the directions by A and B and their location spatial location as X and X prime and the commutation relation is given by I delta AB delta X X prime so on the lattice we'll have a quantum degree of freedom on each link both variables are given by ENA and on the lattice we'll choose the direction of ENA just to be along the direction over the direction of the edge so on each edge there's a fixed direction so this is a quantum mechanical degree of freedom and ENA don't commute and if you look at the commutation relation it's pretty much like a rotor degrees of freedom a harmonic oscillator where conjugate variables have commutation relation like that that's just what quantum mechanics do in quantum mechanics ENA are not mutually independent variables they have non-trivial commutation relation meaning that they cannot be measured exactly at the same time and in particular this E field once we put it on the lattice then we can see what is the variable for E it turns out that E is an integer variable and A is a phase factor so E is an integer variable taking value from 0 plus minus 1 plus minus 2 and all the way up and A is a phase factor taking value between 0 and 2 pi it makes sense why E is an integer variable because well because of this relation this is the so called Gauss's law in Maxwell's equation it is saying that if you integrate the electric field flux through a surface it tells you how much charge is inside so on the lattice what we do is that we integrate the electric field on these four edges 1, 2, 3, 4 we add them together and we should get how much charge there is at this particular lattice site but charge is quantized you know that charge is quantized and if charge is quantized we better have all the E field to be integer otherwise we wouldn't always get integer charge on the surface the charge quantization tells us that E is integer and quantum mechanically we know that if we have a variable where one set of operator has integer eigenvalue then the conjugate one better be a phase factor taking value from 0 to 2 pi yes oh half integer, oh yes if it's half integer then half integer is pretty much integer with one half in front reskill everything yes maybe the phase factor go from 0 to 4 pi ok so the electromagnetic field in quantum mechanics is described by pretty much rotor degrees of freedom right it's characterized by two sets of variables one is integer and the other is a phase factor and now we can see how the symmetry transformation on the gauge field can actually be generated well we know what the gauge transformation does the gauge transformation takes A and send it to A plus gradient F and then and of course the C field C dagger goes to E to the I F and if we want to change A now we have a way to do it with a unitary operation because A and E has non trivial commutation so if we make a unitary operator by doing like something like E to the I something like a what's a good notation like beta E and use it to conjugate A and because A and E doesn't commute this is actually going to generate some transformation on A so this commutation relation allow us to write this transformation on the A field together with this transformation on the matter field in the following form so it becomes E to the I gradient E minus rho where rho is the charge density the number of charge on lattice side where the charge operator at the lattice side which is a C dagger C so this is now looking more and more like what we have over there so it has two parts one it involves E to the I F rho which involves the symmetry charge on the other side and the second part is E to the I gradient F E to the I F gradient E and actually we can exchange the gradient and move the gradient to F so it becomes E to the I gradient F E it should be just gradient FE I think I may have made some mistakes here but it should just be a gradient FE so let me write it this is gradient FE and there is F rho yes 1, 2, 3, 4 here so you can use this to conjugate the A and C variables so let's say this is the unitary at a particular vertex and you use this operator to conjugate the A and you see that A will shift like that just because E and A has this commutation relation and also it takes a little bit algebra and also if you use it to conjugate the C operator, C dagger operator it also transform in the way that you want to pull into two parts one is a rotation generated by the symmetry charge on each lattice side and the other is the rotation of the gauge field generated by the E field generated by the electric field on the vector potential part of the gauge field yes this is the unitary that I want to apply in order to generate two transformations and I'll apply it in the way that I conjugate the variable like in quantum mechanics that's how we do transformational operators ok and by restricting to the lattice I can just choose F to be a delta function delta at a particular location such that this is just rotation on a particular lattice side and this one only involves the E field on the edges that goes out from this particular point everywhere else the transformation will be trivial it's only at the side and also on the edges so it's almost looking like what we have over there and then the final step in order to make the connection is to realize that this is a U1 gauge field and we have a Z2 gauge field over here and when we go from a U1 gauge field to a Z2 gauge field a lot of things can change for example the charge is not labeled by integer anymore it's only labeled by even and odd or 0 and 1 so the E field also takes value in 0 and 1 ok and that actually corresponds to the tau z operator if we take the exponential of E so tau z is the exponential of E and takes value in plus minus 1 as we would want for a poly operator on the other hand the field also needs to get a Z2 character to it and the way we reduce the A field from a U1 to Z2 is by defining this tau x operator as E to the I integration of A along this edge and we require this integration to be either 0 or pi it has to be either 1 or the value and again tau x contains value of plus minus 1 which is exactly what we want for a poly operator and what's nice about this is that you can check that tau x and tau z defined in this way the anti-commute with each other which is exactly the anti-commute due to the commutation relation between the E field and the A field an anti-commutation relation is exactly what we want for the Z2 gauge field and now you can see that this UV operator if we choose the F to be a delta function at a particular spatial location it just reduced to what we had over there because at the lattice side what we do is we take e to the i pi the charge on the lattice side here it has to be pi because it's a Z2 charge and we can have the charge to be either 0 or 1 so this is the sigma z part which is e to the i pi rho and then the tau z part is just this tau z part is this and this is just that one if we set f to be pi delta x at this particular point and then 0 everywhere else so tau z is equal to e to the i pi so see that when I say okay this is the local symmetry I want I didn't just create something totally crazy and this comes from the comes from electromagnetism it comes especially from the quantum version of electromagnetism and how the gauge will couple to the matter field and on the lattice with discrete gauge group this is exactly how it should work yes when you're on the same edge they don't commute for the same degree of freedom they don't commute so basically we're introducing some bosonic degree of freedom some rotor degree of freedom to describe the gauge field of course again a fundamental difference between what's going on with Maxwell's equation or the standard model is that the gauge symmetry is something inherent it's something it's just what it is it has to be like that but from from what we're doing here we see that the gauge symmetry is something emergent something that comes with the topological order something that we can impose but we don't have to be literally serious about it we can relax a little bit but then as long as we just do it it always emerges at long distance as a global property of the topological phase so I have a little bit of time so maybe I can we'll talk a little bit about this one where we have Fermion's Hopping on a lattice and then we want to couple it to a gauge field this is probably the the most familiar case where you would encounter coupling a matter field to a gauge field something people talk about all the time that we have because this is real literally have in a material where electrons hop between the lattice side and then we have real electromagnetic field that couples to the to the Fermion's Hopping there so let me talk a little bit about how to do this coupling following exactly the same procedure now we showed which work for the transverse field Ising model we have again two-dimensional lattice and we have Fermion's Hopping so on each lattice side we have C dagger C and we can create or annihilate Fermion's and the Fermion's can help meaning that we have some term C dagger C C dagger between each bound and and then how do we couple it to gauge field coupling it to gauge field meaning that we want to put some vector potential along each edge so there are actually different situations that people talk about in the simplest case actually a common discussion is where we don't consider the gauge field as quantum mechanical gauge field we just consider some classical gauge field and that's totally legitimate especially in common condensed matter setting where we just add some electric field or magnetic field to our system and we don't consider the quantum dynamics associated with that we just consider that the field is there we add it by hand it's so strong that its dynamics is not going to be affected by the electron in the system so if we take that limit if we think that the electromagnetic field just exists in the background and we don't need to consider the back and forth interaction not interaction we don't consider the back action on the electromagnetic field by the matter field in the system then we just treat the electromagnetic field as something fixed which means we fix the A into a certain pattern or we fix the flux within each placette to be certain value and then the electron will hop according to that in a way this A field will change the hopping by changing c dagger c to c dagger e to the i a exactly the same as we would do the coupling for the icing one so for the icing one we're putting between a tau z term where is it? sorry we're putting in between a tau x term and this tau x term is exactly the exponential of the A field of course integrated along the edge and that contribute an extra face factor to your hopping and if you have non-trivial flux pattern then this change of face factor is something that's actually going to change your dynamics so this is one this is just treating the electromagnet field as the background on the other hand you might consider a situation where the electromagnet field is actually dynamical it's quantum mechanical so we actually need to think about them as a quantum mechanical degrees of freedom whose dynamics might be affected by the matter field in the system and in that case we literally need to consider them as a set of conjugate variables by E and A and we need to do something like this okay we need to take the hopping term this is the hopping term we take the hopping term modify it such that it's gauge symmetric at the flux term the flux term is just saying that I integrate A around the fourth circle or I can do the the unitary version of it I can integrate A and take the exponential of it and multiply over the whole circle that should give me identity and then I also have gauge symmetry I put that together and I have a full quantum mechanical description of quantum mechanical matter field coupled to quantum mechanical electromagnetic field okay so one final thing if we have a superconductor we don't have just an insulator if we have another insulator or a metal if we have a superconductor which involves terms like delta c dagger c dagger delta star dc right so this is the kind of term you will see in a bcs bcs Hamiltonian where you have a mean field kubo pair terms for kubo pair creation and annihilation if we have these kind of terms kind of term they break the u1 symmetry if we are trying to do this kind of symmetry transformation that kind of term is not symmetric anymore unless this r5 is equal to pi so the u1 symmetry gets reduced to a z2 symmetry when we go from a metal to a superconductor something I mentioned just now so that when we have a superconductor Hamiltonian we cannot couple to the usual actual magnetic field we need to couple to the z2 gauge field like for the Ising model so now in this case we don't have ena as the gauge field instead we have tau x and tau z as the gauge field but we can still couple them to that tau z gauge field so the thing we need to do is to insert between here a tau x and insert between here a tau x and we do the same procedure we take the Hamiltonian modify the Hamiltonian at the flux term at the gauge constraint and then you can ask what is that phase difference superconductor actually give rise to different gauge theory there's still z2 gauge theory but with fermionic charge this time because we know the charge electrons or at least of course the particles so they are fermionic and they should have high statistics with the flux but other than that how the flux should respond how the flux should behave what kind of statistic does the flux have is related to what kind of superconductor we started from and if you are interested in this this is what Kitai wrote about the 16 fold way big paper in exactly several models than beyond huge 130 page paper so I will go into that if you are interested you can read it so this is for today and tomorrow I plan to talk about how to gauge a symmetry protected topological phase again with z2 symmetry so here we had a spin model with global z2 symmetry but it's a pretty trivial one everywhere the spinning is just polarized but tomorrow we are going to talk about non-trivial symmetric phase under z2 symmetry and how gauging net can give rise to a different topological order a different z2 gauge theory and finally I want to go a little bit beyond talk about how to gauge system with subsystem symmetry something called subsystem symmetry into something that people got very interested in called fractal order so these are pretty new stuff but they are actually very simple in terms of exactly the model and the algebra that's something I hope to cover tomorrow