 Hello again, everyone. So this is an afternoon session. So, well, so I'll be giving some discussion about topological states of matter. And I should apologize from those senior people who are, who are familiar with the subject so it's going to be a very basic introduction to the subject. So I'll be talking about, I will start talking about one dimensional systems exhibiting some non trivial topological states. In particular, I'll talk about one these superconductors, one the topological superconductors, or the so called my runner chains, and then I'll switch to two dimensions, which is more interesting. And I'll talk about the same, perhaps the simplest two dimensional system exhibiting non trivial topology. Those are the cold so called churn insulators. And I'll talk about the, the first churn insulator that has been studied by holding in the 80s, all things a model to the model on a complex. And this discussion will allow me to also make links to topological insulators, which I will very briefly mentioned into the, and in particular, I'll talk about came early construction in the model for topological insulators. The next topics will be covered real quickly. And the last point here, which will be directly affected by the discussion of churn insulators is the topological order into the. And my main idea here will be to draw similarities and also contrast logically ordered systems with those that do not exhibit disorder, but are still topological such as topological insulators or churn insulators. I'm going to start with the plan. And I really want this to be informal so if there are any questions, please just go ahead and interrupt me and ask the questions, but this is supposed to be a tutorial for students basically. Yeah, I'll be careful to follow the zoom link as well. All right. The topological systems in one dimension. The topic appeared in the beginning. Well, I should say the subject has been in the center of discussion, starting from late, starting from 70s. And in fact, most of the things I will be talking about concerning topological order were known in 70s. But they were known mostly to high energy physics community. And then the subject resurfaced in a condensed matter, because such systems became possible to realize in labs. And most of the interesting properties of these systems have been started quite recently. So, in one day, the interest is coming mostly from the possibility of engineering my run a chains that can be used in topological quantum computation and Microsoft is investing huge amount of money to make these computers. And to realize the systems. So, and this is the most motivation for me to briefly cover this topic here. So, the observation by P type in 2001, which was heavily based on earlier studies of many other researchers as well. And a nice, nice Hamiltonian that exhibits this topological properties is the following so consider a simple one dimensional chain a regular lattice of spinless electrons that live on this one dimensional chain. So, fermions can hop from one side to another side. So that's the hopping parameter P, and we have a way of pairing. The pairing is has the P wave symmetry, because these are spinless fermions. Okay, these are spinless fermions, and they can in real speed they can get paired if they reside on the nearest neighboring sites, I and I plus one. So it just explicit pairing term incorporated into the Hamiltonian is a quadratic Hamiltonian and fermions have certain chemical potential and I, where and I here is the particle number operator. And the second quantized form of the Hamiltonian simply tight binding chain with pairing pairing is has P wave symmetry, if you do to get transformation here, you will get basic operator corresponding to such translation is the momentum of your order parameter in case space will be proportional to the momentum. And the chemical potential needs. Okay, so it's on. Of course, simplicity assume that this phase of the order parameter delta is just a real number. It has no phase because it's in this discussion is going to be relevant. Okay, so an interesting mathematical so it's a quadratic Hamiltonian everybody can solve this straightforwardly, but there's an interesting way of studying this Hamiltonian, namely, if one rewrites it by introducing some terms of freedom. My run a fermions, gamma, I, one. I, two. And if I is written is in this way in terms of my run a permanent on CI dagger is written as gamma, I, one minus. I, two. That. Yes. This one. Thank you. Is this better. I don't think I can do that, but let me try to put it here, maybe. I'll try this better. Okay, then. New operators gamma operators are my run a fermions because they're anti commuting the original real, the original complex fermions. I commit with each other and therefore this one's are also anti commuting. So gamma I one is equal to CI dagger plus CI and gamma I two is equal to I CI dagger minus CI. Given in terms of complex fermions but the interesting property of this gamma fermions is that they're real. You take gamma and emission conjugated is equal to gamma itself. So, these are clearly our mission operators. So, why I have these two additional indices one and two it just simplify the. It simplifies the picture a lot. If we just think about this complex fermions see CI dagger as a box consisting of to my run a fermion. This one is electron or fermion, complex fermion, consisting on to my run a fermion. This one is called gamma one. This one is gamma two. So these are my run a fermions. And in terms of this my run a fermion the whole Hamiltonian has a very simple form. And similarly, if we take this, let's call this my run a chain chain Hamiltonian. So the same chain Hamiltonian in the simplified case when the chemical potential is equal to zero, and the hoping parameter t is equal to pairing parameter delta. And this one is especially simple. You can simply check that it is equal to minus it. And I go to run from one to n minus one, gamma I to gamma plus one. So it's a bio linear combination of my run a firm. So these are important here. I runs from one to n minus one. And we see that if initially we had a chain of fermions, numerated by index I. So this was complex for me on one complex for me on to so on complex for me on and so complex for me on one. So I see so to my run a fermions one is a gamma one one. This one is gamma one to a complex for me on to essentially is gamma 21. This one is gamma to two and so on. The last one is gamma and one and gamma and two. Notice that the sum runs from I to, sorry, from I to one to n minus one. So, and also when I is equal to one, gamma one one is not entering into the Hamiltonian. So this is an interesting observation. Which suggests that something, not all the my run a fermions are entering into this Hamiltonian in the same way to understand what's happening. It's convenient to do this transformation once again by introducing some set of new formula degrees of freedom. So complex fermions like real like electrons by defining them as follows. So let's if I represent an elation operator for complex fermions. It's going to be gamma I plus one one plus I gamma. I two. Elation operator is going to be one house gamma I plus one one minus I gamma I to essentially what I did by this. Yeah, the question is, is it necessary to write the Hamiltonian in terms of my run a fermions and whether the open boundary is necessary. The boundary is important here and this is the fact of the boundary that I'm going to consider in a little bit. The boundary conditions are open. And whether it's necessary to write it in terms of my run a fermions. It simplifies a lot. So what I'm going to like in very simple terms what I'm going to argue today was show today was actually known from 60s. It was written by a paper by a lead Mazzis and others who did not switch to my run a fermions and they miss very beautiful and extremely simple physics here so essentially what I'm telling you known from 60s. But not in terms of my run a fermions which basically boosted the field toward this topological quantum computation and attracted many investments. So physically speaking, it is. The physics can be understood without my run a furnace although it's going to be extremely hard and not so useful. So in that sense my run a simplified a picture a lot. I'm just switching to this new set of for me on it. For me on degrees of freedom, which basically is nothing but composing what I'm doing here I'm composing new set of complex fermions out of these two my runners. These two my runners binding them together into one for me on so this is going to be at one. This is going to be up to and so on. Remain and the first one will remain on pair. All the rest are getting pair these two are paired together these two are paired together. This is paired with the next one and so on, but the very first my run and very last my run are not going to get the pair. And let's see what we get in terms of this new operators what do we get for the for the Hamiltonian. Hamiltonian becomes extremely nice and simple. It is going to be 2t I remind you that delta is equal to be here. So that's the case. I consider for simplicity although many things. And you will remain the same if we start varying me on t me on delta. A little bit. So the Hamiltonian is going to be 2t some with respect to I went from one to n minus one. And I dug up. So it's diagonal. Just like a term corresponding to the chemical potential for this new fermions. So the energy of creating an electron f electron is 2t. And that's it. How about this first my run and the last my run. We have these two operators, the very first one and the very last one. What we can do we still out of these two operators, we can cook up one complex fermion. Let's call it FM, which is one how gamma and two plus I gamma 11. And this is a real thermion, which, which can exist in the Hamiltonian, depending on the, on the parity of fermions that we have. If we introduce a new operator corresponding to the period of permeance, let's call it an M, which is equal to zero, or even parity of permeance and one for odd parity. When M is equal to zero, we will not, we are not going to have anything on the boundaries, but when M is equal to one, we are going to have one additional fermion complex fermion, but which is no local which is split between this boundary and the other boundary. This is what exactly this math is telling us, you don't have one single fermion somewhere in the, like complex fermion somewhere in the chain, but it is split between two ends of the, of the lattice. Okay, so that's this is what has been observed. Well, it's just simple analytical derivation. And the idea was to engineer such a system and there has been numerous proposals put forward out to engineer this type of Hamiltonians in experimental setting, and it's still ongoing. The engineering is still ongoing. One of the interesting properties of this chain is that if the parity is odd, you do have this to my run a fermions localized at the boundaries of the chain, and whatever you do with your system break symmetries, introduce some moderate, okay break symmetries torture the system in a moderate way such that it's not being cut or something extremely drastic happening to it. This to my run us will remain there. It's very robust, the effect is very robust, and the robustness of this effect of this effect is so so attractive. You don't have to have some special additional conditions to have this my run us they they will be there, along as you're not doing extremely drastic things to your Hamiltonian to your system. The situation is different for this third class of topological states I'm going to mention briefly where like the logical insulator where symmetry of the system is important. So, here, basically all symmetries you have are more or less broken. We don't have to worry about preserving some symmetries in topological insulators assist the story somewhat different. You have edge states, but those are states are protected by certain symmetries and if you are breaking those symmetries ages are changing. There's one example where you don't have to preserve any symmetries that they did this my run us are still there, and you can encode the information and you can work with this my run us. And basically it's from the point of your quantum computation. It's very attractive. Yeah, thank you. Yes, please. Is it possible to break each scene to more than to operate. Yeah. Okay, you're talking about some anionic degrees of freedom, probably, if you're doing. If you are doing linear transformation like this one. All you can get is going to be a familiar. My run is also for me. If you're, but you can, you can rewrite a fermionic degrees of freedom in terms of some functionalized excitations, each of them is no longer a fermion if this is not a linear transformation, some, some other complicated transformation you can get any statistics you want you can get some anion degrees of freedom and so on, and try to work with that. But the question is going to be those anions that you're interested in. Are you going to stabilize those in the edges of somewhere in the bulk of the system or not. So that's a very good point. I'll touch upon that later. So, yeah, yeah, I don't have, I mean, I wasn't planning on talking about that in details but experiment is ongoing so it's not okay it's a mathematical trick just this Hamiltonian itself, but you can realize a very similar Hamiltonian experimental if you want to introduce some spin orbit coupling and proximity to the superconductors, take a wire. Okay, so this is the part of the wire just hoping for put it on a superconductor proximitize it with the superconductor will which will induce pairing of fermions inside the chain, and that can be done with the help of some additional spin orbit couplings. At that Hamiltonian is believed to host such such H states. Now experimentally how to measure this actually there are techniques. It's not very easy to understand whether. Okay, it's easy to understand experimentally see the edge states, but whether this edge states are my run us or not. You cannot like it's hard to prove it, only to prove it you have to have, at least, I think for chains like this. You will have to break them together to see that the braiding property of the edge state is in this way is what you expect from my run us, not from just simple electrons. Because I can say okay yeah you have a permanent degree of freedom at the boundary but how I know it's my run a not a complex for me. You will have to break them and study their properties that many body properties, a way function made out of let's say for my run affirm. To start studying this experimentally experiments are going Charlie Kane Copenhagen is doing is one of the leaders in the field doing that experiment. And as I said Microsoft is investing a lot into this business. So it's ongoing. But very strictly speaking, it seems like it's still not theoretically but experimentally still being debated how to observe my run. Okay, yeah, adults also yeah they're experimental efforts to realize this my run us also adult in Netherlands. Yes. Yeah, it's fine tuned now. Okay, yeah, as long as new is smaller than 2p I believe I don't take it for granted but I think it as long as me smaller than 2p the physics is the same. The only thing is that if these two my runners are exactly localized at the very last on their first sites. If you go away from this zero delta for the people and you still have a localized degrees of freedom I run a degrees of freedom at the boundaries, but the way function is going to be exponentially decaying here is just like a delta function is one here or zero otherwise. In that case you will have some exponential decaying tales. Okay, so this is one area where topology. Okay, this is one one interesting field, which with promising applications with ongoing experimental efforts in the United States and Europe and elsewhere and China. This is one dimensional from theoretical point of view a lot is clear already already at this level. So what else, what are the other interesting effects one can observe them as was mentioned for example how to observe the grease of freedom localized in the system or on the boundary of the system that are different from electron to my run a firm and so those are not fermions but have some more exotic properties like an unique ones, how to get anions to get anions in one day is hard but in to be, you can do that. And this is what I'm going to talk about. I'm going to switch to the second part and talk about churn insulators. Yeah, I hope everyone here is familiar with graphene. Those are the type of churn insulator I would like to discuss is the classical one suggested by started by Haldane is defined on the honeycomb lettuce. I'm going to put a link on offer honeycomb lettuce here. Honeycomb lettuce has a human cell, which has two sites in it. You can take, for example, this dotted there dashed area as a unit cell of the honeycomb lettuce. And you will cover the whole lettuce again. So this unit cell has two sites. Let's call it site A and site B. And because of this structure. Let me draw it again. This is the unit cell with two sites and being, and I can introduce a lot of specters. Let's call this one, let's say you want to be two. E three. Everything in this lattice can be. All all coordinates of each other site can be found using this. A lot of specters you want to turn the three. There are some is equal to zero they're not independent. So those are the vectors connecting nearest neighboring sites. You can also talk about, let's say, that there's new one. This one. Me or two, which is 32. This is me or two. And you three. This one connecting next nearest neighboring sites. And can be chosen to be minus E three plus E one. This is minus E two plus E three. New three is minus E one plus E two. So the vectors allow us to write down the Hamiltonian of the graphene of electron, let's say, living in a honeycomb lettuce. So the nearest neighbor tight binding binding Hamiltonian, which is minus the time some over. That's nearest neighboring sites, I J. See, I did the C J. But this is the tight binding model describing a particle stopping between nearest neighboring sites I and J. So you can write this down as minus the time some over our are in the radius vector, let's say, describing position of the site. Some with respect to alpha running, wearing three values 123 which correspond to the emphasis of this three vectors you want it to be three. So this is going to be CRI. So this is one way of writing the hopping between nearest neighboring sites, nearest neighboring sites belong to different sub lattices and B. So operators one correspond to sub lattice a one corresponding to sub lattice B. And this is the Hamiltonian with nearest neighboring cops. If you diagonalize if you can diagonalize this I'm going to know this way. By Fourier transforming it. I'm not going to write Fourier transformation I'll just write the form of the Hamiltonian into representation. It's going to be minus T times some over this quick momentum. So it's convenient to write it in a matrix form so I'm introducing a spinner, because of this two sub lattices. I can introduce a spinner. A dagger, CKB dagger a creation operator corresponding to different sub lattices are combined into one spinner here. Then the Hamiltonian is two by two metrics. Exponent minus I K one K one exponent minus I K E two plus exponent minus I K E three zero. The same thing with mission conjugation. So this is the metrics times the annihilation operator, corresponding to sub lattice a and annihilation operator corresponding to sub lattice. So that's the Hamiltonian after Fourier transformation. Let's call this, this part here, HK, which is the two by two metrics. The finalization is straightforward. We need this to diagonalize this one, and the eigenvalues of HK can be found. Straight forwardly. So, so it's a two by two metrics will have two eigen energies to eigenvalues. If you do it exactly, you will get plus minus. Mod p t is the hopping parameter here. To go sign. Three K x a a here is the lapis constant. This in between, between nearest neighboring sites and continuing from here. So that's not extremely informative, but what this eigen energies having common is that there are certain values of the momentum K, where the gap is closing between two energy branches, and in the vicinity of these two points that this person is linear. So these are the points so called K and K prime points, which are defined as, for example, as plus minus four five over three in the brilliant zone. So these are the, these are the points. And if you calculate the dispersion nearby, it's capital K here. So if you go away from this single points where the gap is closing by amount of Delta K, you will see that dispersion is linear. So the leading term is linear here in Delta K. So basically, if you look at the dispersion, it looks like this along K K prime line is K, this is K prime along this line, you have this to direct points. And then the middle of the Berlin zoning gamma point dispersion is quadratic. Okay, so this is graphene stuff which is interesting. But to get some non trivial, let's say, age mode. They don't exist in graphene. Okay. One can do the following. Let us take not only hops between nearest neighboring sites but also let us include next nearest neighboring hops, hops between next nearest neighboring sites like that. And also, in the presence of, so we do that in the presence of some flux field. So let's suppose that when particle hops between two nearest neighboring sites, its wave function acquires some phase exponent. If I let's call it H by H. So, so the hopping parameter here is complex when particle hops from here to here, the hopping parameter is complex with the phase exponent if I H. So that Hamiltonian can be written down as graphene simultaneous, like as we described here. Each graphene. Plus, the next nearest neighboring, let's call it next nearest neighboring terms in the presence of this complex phase five H. That model is already very interesting because very interesting properties. Hopefully this is clear, I, it will be, it will help me to avoid writing huge formulas but the idea is this. So let's take next nearest neighboring sites connect them with each other and the in front of the hopping parameter we put exponent if I H. And make the whole Hamiltonian Hermitian. And that case. Hamiltonian, we can do still. Retransformation analogous to what we did here. Let me write down the term corresponding to this two by two metrics in the presence of the complex field five. And what this five means is essentially it's, it's a magnet, it's a magnetic field, you can think of it as a magnetic field that that is spreading this 120 degree triangles everywhere. So let's say if these two sites belong to the same sublattice a, the magnetic field is let's say suppose it's perpendicular to the blackboard pointing inside. If these two sites belong to sublattice B, there is similar magnetic field pointing outside of the blackboard. So this one has plus sign going in this one has minus sign going out, but at the overall magnetic field threading the, the unit cell of the honeycomb lapis is zero, but within the unit cell itself there is a modulating magnetic field. So that's the Hamiltonian. So that's corresponding to the so called how they churn insulated. So in this two by two representation that Hamiltonian will look like this. In the momentum space. So, if next nearest neighboring hops are represented by P prime along the hops along these lines are the prime times exponent by each. P prime is real. Then the two by two metrics is minus the prime. Okay, so it's still a two by two metrics, which we can write in terms of Pauli matrices. So it's, it has this one. Let's write it as a zero K times identity method so this is diagonal a zero case and function of K I will specify in a moment. Minus D times age of K. H X of K times Pauling X component about Sigma X Pauling metrics. Minus D times H Y of K times Pauling not X Y, the second Pauling metrics, minus the prime times function, H zero of K times Sigma Z. So I see that the it's a two by two metrics in the momentum space representation, which can be written as a dot product or some vector with components age zero H X H Y and H Z with the power the matrices. So H zero itself as some form. I'm going to write it down twice cosine by age by age is the phase that went to do the complex space or sign K mu, K mu one plus cosine. K mu two plus cosine. K mu three. So, there is also H X H Y and H Z. So I don't want to specify all of those. I'm going to write an audience paper of AD, AD four, I believe. What is important about what is interesting about the simultaneous is that it has several phases in it. So first of all, in order to introduce next nearest neighboring cops in graphene. We do that without this complex phase five, right, five is equal to zero. And the simple way of writing that Hamiltonian in the graphene simultaneous in the presence of next nearest neighboring cops. It has t times the two by two metrics are just erased corresponding to graphene. In case space, plus the prime which is next nearest neighboring cop. In the stopping parameters times tk squared. Where tk again is a two by two metrics with entry zero, it's four and minus I K ex plus exponent minus I K, E Y. Zero. So this two by two metrics respond to the hops between nearest neighboring sites next nearest neighboring copying appears in the Hamiltonian as a square of this amount of square of this two by two methods because all by obviously between two next nearest neighboring slides is equivalent to hop from this side to this side and then from this side to this side, and hence this t plus t square structure of the graphene simultaneous. In the presence of the flux five field is not that straightforward because in principle. It is still hold it still holds, but one has to be a little bit more careful, but by looking into the flux is threading these triangles when particle goes from here to here. It is equivalent to this hop only if a particle acquires certain phases so let's say when it helps from here to here with the with the hopping parameter p prime which has a face. If I age that that face should be also accumulated when it goes to this way around that trajectory, but this is one little subtlety, but which is, which is, which can be taken into account. So I'll skip that technical part and just write down the phase of the face diagram of the whole dance model. So, okay, so bit late in time so let me just specifically say what what is interesting about this model here. So first of all, this model has this hz term, which corresponds to the muscle the Hamiltonian right hz is always giving you the spectrum which is massive. Depending on the depending on on the whether you have the space here or not this hz term can. You can have different signs basically corresponding to k point of the building zone and k prime point of the building zone, which means that the mass in the vicinity of k point here. So hz term opens up a gap, both in k point and k prime point, but in the presence of the fire fire field here, the function hz has different signs. I should have written down the expression for the. Yeah, let me write down the expression for hz, which is the most important function here. So hz prime times hz was let's say q is equal to minus three square root. E prime line of five eight. And three square root three. E prime. By age. So k is equal to k point and the one equal to the prime point. We see that the difference in masses of the spectrum corresponding to K and K prime points of the building zone. So this difference is essential. And in the presence of that. It ensures that there are there's some interesting topology and some some interesting age states appearing in the propagating along the edges of the sample was two dimensional sample. So one can complicate this. So let's add Hamiltonian a little bit more by adding to the Hamiltonian original Hamiltonian that we discussed staggered chemical potential, but as I assume that we're adding a chemical potential to sub lattice a. Okay. And a different chemical potential to sub lattice B. So this is the difference between chemical potentials corresponding to sub lattice a and sub lattice B and this difference let's call it M delta mu a chemical potential corresponding to sub lattice a minus mu B is equal to M. So if you add a chemical potential that one can introduce, then this. So this in the presence of this staggered chemical potential this function HZ will change just a little bit, you will add M to both terms here. So this one becomes in the vicinity of Q point it's M minus square minus this expression here it's M plus this expression. Otherwise it's the same. So having that one can draw a face diagram, separating two different phases. So one phase when is when both masses have the same sign. Okay, you can have such a situation is M is M is sufficient to large can the both masses can be positive. This one and this one can be positive. In that case, the, the exception spectrum of the model is pretty pretty much the same as the one in the absence of the five field. So you have two masses one is bigger another one is smaller but both are positive. So this is the so called topologically trivial phase. That's the outside of this region here. This region here is when both masses have different signs. For example, if you set them equal to zero they're clearly so this is this error represents M this error here represents by age. When M is, let's say if M is zero. You always have different signs of both mass terms. So therefore, the phases along this line is in this topologically non trivial region character right by the so called churn invariant, which can be written as one of times. So this invariant tells you whether C is equal to zero or one, you will immediately just by looking at C, you can immediately conclude whether these two gaps have different signs or the same sign. If C equal to zero. They both have the same sign if the C if C is equal to one or minus one. They have different signs. And in that case you have different phase basically the topologically different phase, which is inside this box. So here basically are the lines when M minus three root three T prime sign is equal to zero. Basically when the war or this is equal to zero. So in this region, C is equal to one. C is equal to minus one. And these two and these phases, this topological phases are very different from the phase outside of this region. The phase inside the region has an edge state, you can study the spectrum again, and see that there's an edge state. So the qualitative description of this H3 state can be as follows. The gap spectrum. Suppose you have a finite gap. And let us look at the vicinity of the K point only for getting about K prime point let's concentrate in the vicinity of K point. So how did the dispersion in the vicinity of this K point plus minus square root K squared plus some master M squared and M is on this expression, the total muscle the system in the vicinity of the K point. Okay, so this is the dispersion. Can we. Okay, it has a gap. There's no way I can close it. Obviously no because if M is finite K prime K squared is positive. So there's no way to close it. But what if, what if this K, it's a two dimensional vector right. So it's K x squared plus K y squared. What if I assume that let's say, this is my boundary along y axis. If I assume that let's say K x is imaginary. Then it will be minus i kappa. So it becomes if K x is i kappa x, that will become minus i kappa f squared plus K y squared. In that case, K y can be equal to zero and a couple x minus couple x square can compensate for this gap. And the gap is closing, but obviously, obviously this can happen only if I have a one dimensional dispersion basically. So this one is imaginary and constant. The wave is propagating along the boundary of the sample basically in y direction only so this is a free way to in one direction and exponentially became way in x direction. So that kind of mode can close the gap, essentially one dimensional more than a to the nearby the boundary of the two dimensional sample can close the gap, and that is the signature of the edge state. The point here. The difference between these two phases is the following. Yes, you can always do this trick near a K point and near K prime point. If you are outside of this region in this black region. If you do this trick, basically, you will get two modes that will cancel out each other and you don't have them any edge mode. So this is the case in graphene gap graphene graphene in the presence of this target potential, and so on. But if you are inside this region. They don't cancel each other but they actually are propagating along the same direction that happens because the two masses near K point and K prime point are different, they have different science that happens because of that reason. Basically, for that reason you have a propagating edge mode here. So that's kind of qualitative description how this boundary mode appears in the channel and later. Okay, so this Hamiltonian is an interesting one has an edge state. And also in the same time it. This edge state appears because of the presence of this target magnetic field. Okay that is threading this 120 degree triangles, although these two different triangles of different fluxes that if you look at the overall flux through the unit cell it's equal to zero. In the unit cell there is a modulated magnetic field that gives rise to this interesting edge physics, and has the name of the paper by a Haldane quantum whole effect without one of our levels so there's no one of our levels because there's no overall magnetic field, but there is modulated magnetic field that gives rise to the edge states. Any questions. Yeah, there's a question if I can elaborate more on the dispersion relation of the Hamiltonian. I would prefer not to do this because of the time constraints but it's very simple it's written in many textbooks on Haldane's paper as well. I would be happy to forward some literature on that some later point questions from the audience, yes. Can you say it again. Can you speak a bit louder I guess. Yes, and the difference. T and the prime are entering into here and here. And then what. Right, in the graphene model, it enters a little okay. This structure is the same right all the difference only comes about because of this function HZ of K in graphene HZ of K such that the gap near K point and the gap near K prime point if you introduce let's say for example this staggered potential they are the same, both are positive both are equal to M. So that's the situation in graphene. But if you're adding this complex phase of this magnetic staggered magnetic field. You're generating minus here and plus here, and the overall sign of this term. So which is the gap near K point and this term which is the gap near K prime point they both can have different signs. The game is about the sign of this, the most term here basically. If the gaps have different signs you are living in this day dashed region. There the chair number is one and you do have this boundary age modes, because of this physics here. If you're outside of this dashed region then no edge modes, the physics is precisely like in gap graphene. Yes. And I think that the negative mass. Yes, yes, that's exactly. That's a good question. It's a good interpretation but but one has to be careful with symmetry transformation that you will have to redefine your some symmetry transformations if you do that. qualitatively yes, but but then you will have to redefine some symmetry transformations. So, I have about some time left, also an hour. Then in that case, let me, my aim is to get to the logical order so very quickly say what happens if you, for example, take two copies of this audience model. So one, if you have one how dense model. Here, let me draw the unit cell of the honeycomb lettuce once again so these are next nearest neighboring cops. These are nearest neighboring cops. Okay, so this is a lettuce a this side belongs to some lettuce be. And this is the. We have five h here, five h threading this triangle and five h trading this one. And if you choose a unit so like this, where this is a bloodless a, and this belongs to the bloodless be. Then these triangles are threaded by minus five h minus five h this one and minus by each this one. The rest is the same. So the total, total flux through the unit cell that let's say through the hexagon is equal to zero. All right, but if you have one copy one whole bench chain, how lens model then the time reversal symmetry is broken because of the magnetic field because if you're reversing the field. It's precisely like changing this minus two plus and this plus two minus. Now the gaps, flipping the science and one situation is time reversal opposite of the other situation. Okay, flipping the magnet direction of the magnetic field is equivalent to doing time reversal symmetry transformation. So one single copy of how dense model is breaking time reversal symmetry. But if you are taking two how then models how dense models, for example, one would feel being it. Let's say one copy of how dense model were copy number one, where if you take this 120 degree triangle. Let's ABA. You have five age reading it. Copy number two to take the same triangle. ABA. You have minus five age. There will be no time reversal symmetry breaking because this five and minus five overall symmetry overall field is going to be equal to zero. But in each of these two copies we had boundary modes propagating boundary months right. So let's say here you have, for example, a mod that propagates clockwise. You will have a mod that is opposite to it that will propagate counterclockwise. So, even though the magnetic field will be absent will cancel out you will still have two counter propagating boundary modes, both are Bosonic, or you can say, for me on it but in one day bosons or for months are the same basically they have two directional propagating Bosonic mode in two directions clockwise and counterclockwise. If, let's say, electrons living in one the first how they model have let's say spin up and the electrons living in the second model have pins pointing down so this will be kind of like when to counter propagating modes with spins they both survive and both are there. Okay. And how to engineer such situation. Again, in my lab proposed an interesting way of doing that, that obviously the Hamiltonian now will have the Hamiltonian which was a matrix in in the Hamiltonian. So you have two sub lattice degrees of freedom to valley degrees of freedom. And now if you also add to spin degrees of freedom, you will have eight by eight metrics now if you add also spin degrees of freedom to your fermions and the Hamiltonian can be written down. Now, if the girl analog of graphene in that situation if you have some spin degrees of freedom. We'll be eight by eight metrics. So age zero. So this is analog of nearest neighboring graphene says the bear one. You can write it as some velocity. Okay, if you're going to write it, expand the Hamiltonian and the vicinity of K and K prime points, your Hamiltonian will have the form of four by four metrics. Sigma X, how Z, the X. Sigma Y. Okay. You can write it like that. Now, this additional metrics style. It's still a power limits which is acting in the space of K K prime point so this distinguishes between two values to direct points K and K prime. This is distinguishing between two sub lattices. So it's the is this is the four by four problem that we kind of built it already. Now if one is adding also spin degree of freedom. So there is a way of doing this we will to have this physics of two different copies of all day model all day model we have to have to ensure that the gap of whatever it is the gap of the layer one. The gap corresponding to layer one electrons living in which f spins up the gap is opposite to the gap of spin down particles living in layer two. In that case, the master should have should be like this so we have to have Sigma Z let's say for layer for the upper layer and minus Sigma Z for the bottom layer. And that can be achieved by writing it. Sigma Z cross. Sigma Z. Right, as he is already Paul the matrix corresponding to spin degrees of freedom so these are real spins. So this is just an identity and the Hamiltonian itself can be written down as. In. Is the term that couples these two layers with each other that can be written as some sort of spin orbit coupling term, which is basically Sigma Z. Sigma Z cross. As the firm. Okay, so basically this different must terms for the form for species with different spins. The presence of the value degrees of freedom this distinguished with the between each other via this additional policy power matrix. Okay. So I'll try to skip. Responding to this. So this model, as I said, it preserves time reversal symmetry I can write down the transformation that reverses times with time symmetry. But this is not that important the important qualitatively one can see that the time reversal is preserved because magnetic fields are opposite to each other basically. So the time reversal would flip these two layers with each other, bringing the system to the same system we started with basically so time reversal is preserved. But there are two H states on this two H states are giving rise to the spin whole effect. Right. So, let's say in this, if you have your sample has a boundary spin up particles will move clockwise well spin down particles will move counterclockwise, they won't cancel each other because these are two different currents, but yeah you will have two boundary modes basically in that system. In this in this came a lesbian orbit coupled system. What is so it's a very interesting system itself. The property of the system is the following if you're breaking the time reversal symmetry let's say by introducing an additional external magnetic field. Then this edge structure will be gone. Depending on the strength of the magnetic field or and depending on the details how you break time reversal symmetry. Things can happen, but definitely one of the modes will be gone or something will happen to it so you don't have exactly this age structure anymore. So these are interesting time reversal invariant systems like unlike churn insulators for example where time reversal symmetry was gone from the beginning you have to have this magnetic field. Here you don't have to have it, but the the edge modes are fragile. Any weak perturbation some some additional magnetic field or breaking. The question from the zoom, is it applicable to coupled ladders or at least two leg leather. So this physics I'm talking about is essentially two dimensional. What I discussed about my run a furnace in the beginning it that part is applicable to ladders or multi leg letters where you can have very interesting edge stage with different statistical properties. On a, you do have to have a clearly two dimensional situation in order to have this two dimensional, or at least you have to have a wide stripes, let's say, but you cannot regard the wide stripe as a ladder right. This direction has to be sufficiently large to have two dimensional physics in it to possess the two dimensional physics in it. Okay. So the last part, which I spent 10 minutes talking about is about the logical order. Now, that, in addition to this interesting states that we discussed already, there are states that do have, let's say boundary edge modes that are extremely robust to with respect to any perturbations you can apply from the outside. You can add disorder you can add magnetic field break time reversal symmetry of the Hamiltonian by different means. If the system is topologically ordered and possesses this edge modes, it will possess it, no matter what you do, what you do to it. Hamiltonians that types of that type of quantum systems is hard to engineer and even study analytically and numerically, but in principle, they exist. So these are states where fractionalized statistics is realized so basically you you build a system out of, let's say bosons or fermions or electrons, interacting with each other with some strongly interacting system of bosons or fermions. But if you cool down the temperature down to zero, let's say, and the ground state you stabilize low energy excitations that do not have the same statistics that the original constituent particles had. Let's say you build you start with bosons. You pull them together they interact into the let's say if you have, for example, everybody knows that if you have spin one half XY magnet. XY, basically magnet can be mapped on to hardcore bosons. So you have a system of bosons, basically interacting with each other strongly. And eventually, if you at low zero temperature if you stabilize a ground state that has statistics different from bosonic one so your low energy excitations are not bosons not electron but anions. This can be achieved only if you have basically topological order in the system. You cannot change the statistics of your low energy excitations without an emergent gauge field and the emergence of the gauge field makes the systems very robust with respect to the external perturbations. So, one example of a topologically order state is for example is if you take this hold and churn churn insulator and put it in and introduce a fluctuating gauge field to it in high energy literature one starts gauging model. Basically gauging a model is means that couples this electrons that live on the honeycomb lettuce on the hold and hold this model and make these electrons to interact with some external fluctuating gauge field. So that that that that that's in the literature that's called gauging. So gauging procedure gives very interesting properties. So, let me just explain a couple of these properties real quickly. So suppose you have this audience model. Let's start with churn insulator. Churn insulator and introduce a fluctuating gauge field. So suppose when particle helps from one side to another side. There is a side number one side number two. The hopping parameter between one and two is multiplied also by exponent. So A12 A12 is some phase, which is fluctuating. Okay, it's time dependent, and also position dependent depends on these two endpoints. And it has some internal dynamics which are not going to talk about so it can be Maxwell like I call it from magnetic like it can be some other type gauge field. So let's put it aside for a moment, but let us assume there is a fluctuating gauge so that when particle rotates around a closed loop. All these phases are being multiplied. And in principle, depending on the structure of the ground state, they can stabilize some some some sort of so called Wilson loop so if you're calculating an order parameter which corresponds to a product of creation and relation operators along the closed loop, calculate that expectation value, it may have a phase in it and that phase will be the phase that is straight in this closed loop basically. So this type of states are inherently topologically ordered. And let me just discuss that because example for them for just a short time now. I'm going to introduce a gauge you into a dance model on a lattice is simple right you write in front of each shopping parameter this type of one term and assume that it has its dynamical it has some internal dynamics in it. Okay, we can do that. If you go to the continuum limit representation. I simply. So this was a dirac electron with a mass in it right so they're a cooperator we all know. It can be written as ID slash means that it's multiplied by Sigma matrix in condensed matter of gamma matrix in the high energy physics. So the gauging means this minimal substitution, you're adding this gauge will multiply by Sigma matrix also here and plus the master. Okay, so the master, as I said in the whole dance model it you have different masses corresponding to different zero points. But this is your zero cooperator anyways, and this is the fluctuating gauge field that you have. The whole dance model, as you noticed it is quadratic and harmonic degrees of freedom. So basically which means that there are cooperator is multiplied by two for me on spiners. For me on for me on a degrees of freedom in the in this model can be integrated out easily is quadratica more than it can always integrate out using Gaussian integration. But what you will get out after the integration is a determinant of this operator, which is not that simple to calculate. This is called the effective model, effective low energy theory, which comes about after integration over for me on a degrees of freedom. So, let me just not talking about coefficients at the moment which are irrelevant. What we get after integration over for me on a degrees of freedom is logarithm of determinant of this operator here, which is some beast one has to deal with it somehow. It's an internal gauge field, which is fluctuating. So it's time dependent calculation of determinants, which has explicit time dependence on it. It's really hard. It's not possible, because you can expand in gauge field, assume that this like the gauge field itself is weak right you can treat this as a, a, as a small parameter and expand this action in terms of powers of a. So then, in perturbation theory, this determinant becomes. Let me put a coefficient here. Which basically deals with the number of harmonic degrees of freedom living on a lapis. So, and in front of it. Please. Oh, ID plus M. Okay, so this is the leading term where a gauge field is gone. The term linear in the gauge field and then plus the term that is quadratic in the gauge field. So, so this is a standard expression for the expansion of this logarithm of determinant. So this this term is the term corresponds in this term we already know what it gives us this is just basically turn insulator with no gauge field at all. This is some table table diagram which doesn't do much linear in a green's function basically times the gauge field. This one is the most important term that defines the fact of low energy theory, and this is called nothing but polarization operator right, we have one has to calculate a polarization operator, and that defines the low energy effective So it just a convolution of two greens functions as you see from here. So what is the polarization operator of electrons in a chance turn insulator. That is the question. Let's take all this turn insulator and integrate out thermals calculate polarization bubble. If we do this. It gives us an interesting expression. I will write down and then stop there. Just look about applications later. So polarization operator. Let's, let's call it. Now, in case space and in the presence of the mass M. Can be written as epsilon new new role. So it has to indices new and new that correspond to different indices here a new and a new. Basically, you can calculate diagonal. Polarization operator or off diagonal one so distinguished between various values of me a new. So let's concentrate on the so called odd term which is linearly proportional to the momentum. And a function, which is all done for, which isn't, which defines basically the odd entry in the polarization operator under permutation of new and new. Switch new new places of new and new. This is anti symmetric tensor, which requires minus sign in front of it. The rest is not changing. So this is the term that is under permutation of to indices new new. So I'll write it as odd here is squared M. The odd term. If you do the calculation basically calculate the polarization operator in how they model exact calculation gives you some function of M and P, which is arc sign, but you can expand it. When P goes to zero long wavelength limit or small momentum. And it becomes one over four pi over M. So that's the sign of the master epsilon new new role. Plus terms that are quadratic in the momentum. Okay, so terms that are quadratic in the momentum are familiar from in them, for example, Maxwell terms are quadrat quadratic in they contain by linear, they are by linear in derivatives so Maxwell term is of this type. But the leading term that is linearly proportional to the momentum. It's an off diagonal term. It has to be different from new in order this to be not zero, but this off diagonal term has this linearly scaling in momentum term, which is called turn silence. I'm done with the calculation of parts so let me just say a few words about the interpretations and what what this means basically. We see that in terms of turn insulator we're generating a term in the polarization operator which knows about the sign of the master. Multiplied by momentum in the first power. When P when the momentum is small. The momentum term P squared is subdominant so what we are generating here after gauging this Haldane's model and integrating code for months. We are generating this turn silence term first, and then Maxwell term is sub leading and higher order terms are sub leading. So which means that for some reasons, one is able to couple turns turn insulator to gauge field or gauge the turn insulator. Then the effect of low energy physics is going to be dominated by the turn silence term, which this is a subject of a separate discussion, but depending on the coefficient in front of it. Which is just right coefficient for turn insulator model, for example, one can stabilize anionic excitations, and so called topological order. So the edge states in this turn insulator in the presence of the gauge field will be still there. So those edge states cannot be broken and they don't disappear if you do some some drastic things with this model for example introduce some additional terms that breaks that break time reversal symmetry or so some of the time reversal is already inherently broken here. Basically, there's very little one can do in order to get rid of this edge states that are present in the in the gauged turn insulator model. These are topologically order states, low energy excitations. I didn't talk about this but our kind of vortices of the turn silence theory, which are anionic, depending on the coefficient, you can get different opinions but those are basically anionic quasi particles, not electrons not bosons but some some sort of onions, which are very interesting again from the point of view of topological quantum computation or just from purely academical point of view. So I think I should start here and any questions. Yes, please. I'm going to add disorder to this. Okay, so if you're adding disorder then it becomes a metro comparison of different scales, you know, if disorder is, let's see, if disorder if the scale corresponding to the disorder is smaller to the energy scale that's it is smaller than the other energy scales in your Hamiltonian, then you are safe. The more later the situation is completely different you add some infinitesimally small time reversal symmetry breaking for example, they already changes the edge structure right here is no. Of course if you add like a lot of disorder that just basically localizes everything all your possible expectations that will change the total ground state. So what I'm talking about is adding amount of disorder which corresponding energy scale is smaller or comparable to other inherent energy scales in the system was are not that that is not going to change much. I'm not sure I caught your question, but if you're going to add some kind of magnetic field structure where if there is a point on the lettuce, it sees magnetic field only at the beat of lettuce and then it can act as an opposite I'm not sure I caught your question but if you're asking if you want to get rid of the real spin of electrons. You need to introduce some additional don't call it a spin but you will need to distinguish between these two all then layers. In the earlier scenario, how can we can take that same model by adding spin. I know if you add a spin is the different model. Okay, in the original Haldane's paper he said that internal magnetic moments in the lettuce can give rise to staggered magnetic field. This is somewhat similar to what you're saying, but in fact in all materials you don't have this this type of situation. So it's somewhat. Other questions, yes please. This is so if you don't have. But when you magnetic field is that the training so they don't say you get an extra is inside the gap. Then we define it by putting field. I wonder if the coupling to the charge. So what do you call a charge is basically deficient in front of the coupling to the gauge field or for something else. Yeah, yeah, right. Right, right. Right now. I see, I see, I see. See, if you. Yeah, yeah, I should have. I should have identified this a bit. So I constructed this Hamiltonian by postulating some externally fluctuating gauge field right. But in fact, if you would like to stabilize a system, like in nature like find a material that has this property. This gauge to should not be added from the outside, it should be internally generated so called emergent, it should be an emergent gauge field. How do you get an emergent gauge field in a system of bosons for example right start with bosons that interact with each other somehow like nearest neighboring terms maybe next nearest neighbor in terms. Let's say spin health operators interact in between nearest neighbors and next nearest neighbor some magnetic exchange model. How do you get an emergent gauge field there you don't introduce it from the outside it should emerge inside. Well, it's simple you do some non local transformation represent your spin operators or bosons via some non local combination on a lattice of some can be fermions for example you can always represent your spin rising or lowering or residing coincide I as a fermionic operator and spoon and I E. So now ease or some new charge some argument or I minus rj and be aware and so this is a fermionization procedure. This is a fermionization that allows to represent spins residing on the two dimensional lattice in terms of complex fermions sees it for me on a creation on a relation operator and is a number of fermions residing coincide J, and this is just the argument is an angle function between our i and rj to to reduce vectors. So this transformation allows it's a non local transformation that allows to express let's say bosonic operators in terms of fermionic one and some sort of fixed gauge. Now this is already a non local transformation you can write it down as an emergent gauge field. With some sort of churn churn assignments dynamics. Okay, so that has a different charge. It's not the charge of the electron, but that is some some sort of let's say a topological invariant or some, some number here that takes that takes care of the statistics of your particles you want to have in your system. For example, if you want to have electron here. This has to be equal to one. Basically, you can represent your bosonic particle as a fermionic one. And some sort of flux attached to it. It's like a flux attachment procedure. If you want to represent if if you have you want to start with fermions, you can take a fermionic particle represented as yet another fermions but then you will have to attach two fluxes in order to have the same statistics. Then this number should go to to be true. So your effective charge determines the statistics of the particles you want to have in your system this is mathematically. And it tells you what the coupling to the, the, to the despolarization operator hence to the transformer storm is going to be for that determines the coupling you want. And this gauge field should be not imposed from the outside but it's internally generated so I did not talk much about this but it has to be an emergent gauge field to stabilize onionic excitations. So your question is now. Yeah, so you have this one continuously tuning this one. And you want to tune it to zero. Like mathematical perspective. So you're saying right. Let's say if that charge is tuned to be equal to zero you don't have any gauge field at all the gauge fuel is gone. Right. So these terms are gone. You have started with now. I see. So at small values. Okay, I, it's a good question. It's kind of a little bit artificial because coefficient itself stabilizes the statistics and that type of transition you won't express. Okay, transitioning statistics in physics is you cannot expect it right. You don't have continuously. You don't have quasi particles that continuously change change their statistics. You have particles characterized by statistics right and not a transition where particle changes changes changes statistics. Like, all the time, and then eventually become a permanent for example, don't have such situations but mathematically, it's a very interesting question. Yeah, you have such a transition like that one can possibly study it. Other questions. Okay, there's a question. Can you have a topological order without an emergent gauge field. The answer is no. The final correction is that in some literature you can see for example this one dimensional my run a chain is referred to as a topological order because see it's still those my run edge states are very robust. Even if you introduce some additional perturbation to the system. So in one dimensional situation which is very simple, very, very special in 2D, you do have a gauge field, you do, you have to have a gauge field in order to stabilize. You do have to have an emergent gauge field in order to have topological order. There's question what happens if the gauge field fluctuation is large. Again, the gauge field fluctuation is determined by terms here and here. So, a small moment, the, the leading term is proportional to the momentum, which is the so called turn Simon's term. The sub leading one is the Maxwell term, and then you can have other terms as well that are higher that contain higher derivatives than the Maxwell term has. So Maxwell has quadratic in derivatives you can have more. Now the fluctuation large means that the next terms Maxwell or the term which is next to the Maxwell is even stronger. Then you, you have to study those models. For example, if you just take Maxwell term plus turn Simon's term is called Maxwell turn Simon's action which is by itself is extremely interesting one in this action, some news mysteries are happening by the way. The Maxwell term plus turn Simon's term. You know in Maxwell action, you have a photon field, right, which is musless, the photon propagator is musless, but you're adding turn Simon's to it. Even though the propagator still is gapless. So still is diverges let's say it's zero momentum, but, but the photon field is gap. It's a gene way of generating a photon gap in Maxwell turn Simon's theory which is just, which is very different from Higgs mechanism. It's a topologically generated master which is very interesting because a fantastic physics is happening in this type of gauge theories, which have up which has applications in condensed matter in in those topologically ordered systems. Yes. How it comes, how an emerging field comes into the picture. Let's say, for example, you have a boson, right. I said, start with the system of bosons for example. Now each boson. I'm going to answer your question technically not not qualitatively. So technically, each boson you can represent as a, as a gauge filled couple to not okay some flux filled couple to fermion. If you write the theory down in terms of fermions, we already have naturally emergent gauge filled there. If by some reason this fermions become low energy excitations in your system, then the gauge will remain there forever. So that's a way of engineering generating emergent gauge field. Other questions. Why don't you generate the key? Why don't you generate key? Because of the small gauge field approximation or because of the small key? No, no, no, it's just because of the topological reasons you you do generate. Even part, not this one. You do generate it, but it's, it is subliving it is here, like Maxwell like. So I just disregarded it. Small wave, let's say, absolutely. It's a Maxwell type. So, to be honest, I have to take into account Maxwell, Chen Simons and even other terms. But yeah, for the moment I just wanted to pay attention to the Chen Simons one. Yes. Very, very good question. Well, physics is topological is very rich, but it is based on topological order with discord symmetry. So the top of the symmetry group in topological in Torico is Z2 symmetry. The order is Z2 topological order. Here I mostly talked about you see a gauge field A which is a U1 gauge field. So I talked about continuous symmetry and topological order with inherent U1 symmetry. Or you can do SU2 symmetry if you want to. Hello, Tigran, can you hear me? Yes, yes. I just introduced you. All right. Thank you. You didn't hear that, but I'm sure that you said some very nice words. Can you see my screen? No, no. All right. About now? Yes, yes. On my screen, do you also see the zoom panel?