 So the last lecture for today, in the first lecture by Professor Yom Baikin from University of Toronto, he'll be talking about correlations and topology and quantum materials, and we continue. Thank you very much for the introduction. So I have a very general title here today, and there was many because I didn't know what I wanted to talk about, so if I give a very general title, then basically I can decide later what I want to talk about. But I sort of decide today. Okay, so in the last four lectures or so, you learned how powerful the Landau theory is. The topic of my talk today, I used to tell you that actually there are situations where the Landau theory is not very useful. And the question is then what can you do about it? So this is the topic of topological phases of interacting electrons in our quantum materials. So roughly my outline looks like this. So I'm gonna give you a very general introduction to topological phases of matter. So there are several different definitions of this. So I'd like to just give you a quick overview of this idea of topological phase of matter. Then I'm gonna use a quantum material with a strong spin-off coupling as a platform to discover some of these topological phases that are discussed in the literature. And I have two broad examples here. The first example is material called the paracolor ide. And I'm gonna tell you that theoretically we expect to see a number of inducing topological phase of matter. Then I will switch the gear and I will talk about recent activities on quantum spin liquid and guitar materials. Here, I will mostly focus on theoretical ideas. Later in the week on Wednesday, Professor Hideo Takagi will give you much more sort of broader perspective on quantum spin liquid phases in quantum material. And if I have time, I'd like to make a connection to a so-called topological superconductivity. Okay, so topological phase is a new paradigm. So unlike the Landau theory that you heard about, this topological phase is really only like 30 years old because really the beginning of the topological phase is a quantum only fact. So we have in condensed matter, we have more than one standard model, different from the high energy physics. And one of the standard model is this Landau theory that you heard about. Essentially here, the Landau wanted to a classify possible broken symmetry phase of matter. And then you have to introduce this idea of order parameter. And one nice thing about this concept was that you can actually measure this order parameter by experiment. So for example, if you use an x-ray, you can understand the crystal structure basically by measuring the charge density modulation. If you use a neutron, you can measure the magnetic structure and by looking at a spin density modulation. And all these things like charge density modulation and spin density modulation, you can define an order parameter. In fact, you can define Landau order parameter for those objects. And that's the way that we classify different kinds of broken symmetry phases. So that's the way we classify crystal structure, magnetic order, and even superconductivity. So that's essentially what you heard in the last four lectures. And there's another standard model that's called Landau for Milky theory for interacting electrons in matter. And there's also proposed by Landau. So Landau at least proposed to standard model that we use in everyday life. Okay. So now if you wanna go beyond the Landau paradigm, then what is out there? And so one interesting concept is this topological phase is a matter. And really the beginning of topological phase is the integer quantum whole state. So you can think about this as a grandfather of topological insulator. So this is basically the experimental setup that you are already familiar with. So you take a piece of two-dimensional electron system, then you apply a current, then you measure the longitudinal and the transverse voltage drop. And that way you can measure either the whole resistivity and longitudinal resistivity. Then when you look at, for example, the sigma xy, the off-diagonal component of the conductivity tensor, and this quantized in the unit of e square of h. So this is the famous quantization of whole conductivity of integer quantum whole state. And notice that when I change my magnetic field, you basically go from one kind of quantum whole state to another kind of quantum whole state. And basically there is no auto parameter related to any of those phases of matter. So basically you cannot use a lambda auto parameter to define such a phase. And this phase of matter has a very interesting property. So the bulk is cal. On the other hand, if you look at the boundary, then because of the fact that tiny bit of symmetry is broken, essentially the electrons at the boundary or the edge can only move in one direction. So for example, at this boundary, the electron should move on the right-hand side. This way and the other guy should move with the other way. So essentially the motion of the edge state is a chiral. And because of that, this chiral edge state cannot back-scatter. And the interesting thing about this picture is that imagine that I'm starting from a usual parabolic band structure without a magnetic field. And we know that if I apply an orbital magnetic field, then I end up with a lambda level. So this is basically drawing our energy level as a function of spatial coordinate here, in that momentum. So this represents the boundary of the sample. This is the center of the sample. And in the center of the sample, you have a bulk, usual bulk lambda level. But when you go to a boundary of the system, there's an edge confining potential. So your energy levels are all going up like this. And by doing so, you create an edge state. Essentially your lambda level meets the foamy energy. And that's the way that you are creating a boundary state. And notice that if you create a boundary state like this, for example, those edge states are moving into the page of the Vigra. And those edges are coming out from the page of the Vigra. So that way, you are basically physically separating right-moving particles and left-moving particles. So if you just focus on one boundary, then the boundary edge state is essentially chiral. Of course, you have both right-movers and left-movers, but they are physically separated. So if you just focus on the boundary, and then it becomes chiral. So what that means is that if you just focus on the boundary, and those guys cannot be realized in the standalone 1D system. So in the usual 1D system, of course, you have both right-movers and left-movers. Here, you can separate them physically. Okay, so people realize that you don't have to characterize such a phase of matter. You cannot use a land order problem. You have to use something else. So imagine that I'm starting with a semi-classical dynamic of electrons. So this is what you find in Ashkirk's moment. So basically, the time derivative of a momentum is a force, so it's given by the Lorentz force. The velocity is given by the K gradient of the band structure. So if you use the Bloch electoral function in the solid, and this is basically the semi-classical equation of motion you discover in your Ashkirk's moment textbook. And we understand now that this equation of motion is not complete, so we have to actually add additional term here. And this additional term has some similarity to this Lorentz force in the real space. So you can think of that as a Lorentz force in the momentum space. Essentially, this emergent magnetic field, if you like B, is given by the K-space curl of what we call a Bayer gauge field. So this is just given by overlap of K gradient of this part of the Bloch wave function, and sandwiched that with another brass state. So this acts like a gauge field or the magnetic vector potential in momentum space, and that's the way that you can generate this what people call a very magnetic field. And that acts like an effective magnetic field in K-space. So you can think of that as a Lorentz force in K-space. So it turns out, if you do the integral over entire brilliant zone of the magnetic field, then that gives me a net flux, momentum space, a magnetic flux. And it turns out this number is quantized in terms of integer number. So if you sum up for all occupied state, then the total turn number of occupied state is an integer topological invariant, and one can show that this quantity is directly related to the off-diagrammatic segment of the conductivity tensor. And that's the way that we now understand why sigma x, y is quantized in integer quantized state. Notice that here, the center object is this topological invariant. So generally, if we think about topological spatial matter, and this spatial matter cannot be characterized by some local order parameter, like a magnetization and magnet. Also, another way to think about this phase is that if you try to deform this phase ideologically by using some kind of inter-transformation or local perturbation, then you cannot go to, say, simple phase, simple metals and band insulators. Just by doing simple operation, you have to go through the phase transition. And you have to, perhaps many of these phases are characterized by some topological properties or non-local properties. So it turns out that there is a way to characterize these phases by thinking about phase transition between different phases of matter. So for simple discussion, let's focus on a gap phases of matter. What that means is that I'm thinking about phases where there's a bulk gap. So imagine that I'm trying to connect some arbitrary topological phase with a simple phase of matter. So what I mean by simple phase of matter is that these phases can be fully characterized by a Landau's local order parameter. And then I'm trying to go from this phase to the other by using, say, successive local inter-transformation. So basically what I'm saying is that you are changing your Hamiltonian ideologically. So the claim is that you cannot go there without closing the bulk gap. So in order to go to this kind of phase of matter, you have to close the gap somewhere and go through the phase transition and only then you can reach the other phase. So that's one possible way to characterize such a phase. And the examples of such a phase of matter are basically quantum whole state that I talked about. And also the quantum spin liquid belongs to this category. And in this case, symmetry doesn't play any role. And that's why these type of topological phases are called intrinsic topological phases of matter. Okay. So a lot of us learn about this quantum whole state either in graduate courses or maybe in even in undergraduate courses these days, or at least you heard about this. So I only told, so far I only told you about the integer quantum whole state but there's a more interactive version of it. There's fractional quantum whole state. So for example, when you fill out, say, one third of the lowest Landau level of two-dimension electron gas. So that's what I mean by nu equal one third. And you can form a correlated version of the quantum whole state. And people are more familiar with the idea that for example, when I measure the optargram matrix, optargram or conductivity, then it's quantized in terms of like some fractional number like one third. But you may not be familiar with the idea that if you actually put the system on a non-trivial manifold like a torus instead of a sphere, then you can actually change the number of the number of ground state. For example, if you put the system on a torus, then you can show that there are three degenerate ground state, if you put the system on a sphere, surface of the sphere, then there's only one ground state. And obviously ordinary phase of matter wouldn't do that because if I have a piece of copper and I put them on a surface of the torus or the surface of the sphere, the ground state must be unique. Okay, I mean, of course, you wouldn't do this experiment. This is like a Kedankin experiment. And in this case, if you go to the boundary, then that edge state is again a one-dimensional chiral electron liquid. But in this case, different from the integer case, these electrons are strongly interacting each other. So they form a so-called Latinger liquid. But you can think of that as a strongly interacting one-dimensional electron gas moving only in one direction. Also, such a phase also supports this so-called non-trivial excitation. So it supports this charge one-third, low-point-cold particle, and these are the elementary excitation. So these are the known facts. Okay, so I'm not gonna talk about quantum avoid effect. So I'd like to switch my gear and explain why quantum spin liquid phases may have similar properties like what the fractional quantum state has. Okay, so quantum spin liquid is another example of increasing topological phase. So what do you mean by quantum spin liquid? Generally, is that we are talking about quantum paramagnet. So essentially, if you take the expectation value of the total spin out of respect to ground state, there's zero. So there is no magnetic order. So your spin rotation symmetry is not broken. And also generally, we say, we are talking about coronary insulator with no broken translational symmetry, and that's the reason why it's called liquid. So there was this old idea by Philip Anderson. Basically, his idea was, since we want to have this quantum paramagnet, why don't we start from basically spin zero object to begin with? Imagine that I have some kind of balance bond between two spin, two nearby spin. So if they're interacting in terms of anti-ferromagnetic exchange coupling, it's very natural to have up-down spin configuration. But by making this spin-singular configuration, I'm basically starting with the spin zero object. So I'm gonna put this spin zero object on the lattice in some arbitrary fashion. And if I just take a one configuration like this, then obviously this state satisfies this criterion, the expectation value of S is zero. But I don't satisfy the other criterion, that I don't want to break translational symmetry. So if I take only one of them, then obviously this will break translational symmetry. But imagine that now I'm thinking about all possible configurations of this balance bond configuration. And imagine that I take a linear superposition of all possible configuration like this. Then by definition, by doing so, I will be recovering a translational symmetry. So I recover a translational symmetry. But again, since everything made those spin zero objects, I will naturally get spin zero state. So this is one way to construct a spin liquid. So here, this is sometimes called regulating balance bond state because this balance bond can fluctuate in time and quantum mechanically. And victorially, this is just a sum of all possible balance bond configuration with some amplitude for each balance bond configuration. So, but there's a cartoon picture, but there's a way to make this discussion a little bit more precise. So, and it turns out that it's very useful to start from a superconductor. So you may wonder why I want to talk why I want to start from a superconductor when I try to describe an insulator. The idea is as follows. So this is the famous B.C.S. Hamiltonian. So here, this is kind of hopping term. This is the pairing term. And now imagine that, okay. So then we know that if I do the field transform, I can write the B.C.S. Hamiltonian like this. We understand the ground state. This is the famous B.C.S. wave function written in a momentum space. And this is famous U.K. and V.K. factor. And this is the quasi particle dispersion relation. But now you can transform this Hamiltonian as follows. So you can easily commute yourself that you can factor out this U.K. factor outside the product. So I can rewrite this way. Then using the remarkable property of the exponential function, I can exponentiate this guy like that. The reason why you can do that is because C operators are fermionic operators. And this G.K. is nothing but the ratio between V.K. and U.K. And then you do the field transform. Then, okay, so before I do that. So the product of K, you can put that as a sum of a K in a momentum space like this. Then you do a field transform. Then you arrive at this wave function. Namely that you are pairing the electron sitting at the position R and R prime. And they have spin projection up and down. And this G, R minus R prime, is nothing but the pair wave function. And this pair wave function is a funeral transform of this V.K. to a U.K. And this is in fact the real space form of the B.C.S. wave function. So in the textbook, usually you see this form, but this form is basically the equivalent form. So if you think about this species wave function like this, then it's clear that this wave function is not gonna conserve the number of electrons because when I expand this exponential function, I'll be generating one pair, two pair, three pair, et cetera, et cetera. This is a two particle state, four particle state, six particle state. So the linear superposition of all those, okay. So how can I construct the spin liquid out of this picture? So imagine that I'm starting with a B.C.S. superconductor. And I like to think about a situation where the average number of electron per side is one, okay. So remember that in superconductor, there's a large charge fluctuation. So the particles don't have a definite charge state. So charge is not a good quantum number in superconductor. But I can still define an average number of electron, average charge. So average number of electron per side is one. Yeah, sorry, sorry. So now I'm assuming the average number of electron per side is one. But the charge of the electron is now well defined in the superconductor. So let's start with this real space form of the B.C.S. wave function. Then I'm gonna take the following limit. Then I'm gonna say, I'm gonna freeze out the charge fluctuation in the superconductor in the following sense. So if I have upspin electrons and downspin electrons sitting at the same position, then the coulomb energy cost will be infinity. So what that means is that if I take this limit, I cannot have a more than one particle per side. But since I already started from average number of electron per side to be one, if I take this limit, I have no option than placing exactly one particle per side. Do you agree with that? Does it make sense? So average number of particle per side is one. That's the situation I began with. Now I don't want to have a situation where I have upspin and downspin sitting at the same position. Because the energy cost for that configuration is infinity. So then I'm forced to have the situation where I have exactly one particle per side. Do you agree? So if I do that, that's an insulator, right? Because the electrons cannot move. So by starting from a superconductor like this, by taking this limit, infinity repulsion limit, I end up with an insulator. Now the question is, what's the wave function of such a state? So I started from a vicious wave function, then p sub z means I'm doing this projection to exactly one particle per side situation. So it turns out that if you do so, then the resulting wave function is just some of our valence bond configurations where the amplitude of each valence bond is precisely given by product of a copper per wave function. So you can think of that as an explicit construction of wave function of the superconductor. Sorry, the spin naked. So another way to see the topological nature of this state is as follows. So now imagine that I take a superconductor, say two-dimensional superconductor, and I put them on the surface of the cylinder. So I wrap around the cylinder. Then imagine that I thread Hc divided by two vortex in the center of the cylinder. So I have a vortex threading in the center of the cylinder now, right? So then because of the fact that I now have a vortex in the center of cylinder, when I go around this cylinder along the circumference direction, then my electron basically pick up the phase vector pi. And because of that, if I look at now the corresponding Kupapov wave function, this wave function is periodic in the x direction, but it's anti-periodic in the y direction because I pick up a phase vector pi. So you get some modified wave function. And typically if you do this in a superconducting state and such a state has a higher energy than the ground state, meaning the state without a flux has a lower energy, this guy will have a higher energy. It turns out under certain conditions, now if you freeze out the charge fluctuations in both cases, and one case show that if I construct a two-state like this, then the energy of this state becomes degenerate in the thermal dynamic limit. On the other hand, if you compute the overlap of this two-state, the overlap goes to zero. So what that means is that these two-state generate this way, they are degenerate ground state. Energy is the same, but the overlap is zero. Does it make sense? I have two-state on the cylinder, I end up with two-state on the cylinder, energies are the same, but the overlap is zero. So these are two distinct ground state. Does it make sense to you? So this is the explicit construction of two-degenerate ground state on a cylinder. So if I think about a spin-like state, obtained this way and that way, they are degenerate, but they are distinct state. They are distinct ground state because the overlap is zero. So if you construct this ground state like this, then notice that if I do any local measurement, then no local measurement can distinguish these phases of matter. Because the change is essentially global. I hope it makes sense. You can please ask questions if you have any. Yeah. So it turns out that the energy difference goes like e to the minus length of the cylinder. And in fact, that's the energy cost to pull these vortex out of the cylinder, turns out. And obviously if the cylinder is infinitely long, it takes infinity, then your vortex cannot escape. So that's why the energy cost, energy becomes degenerate. And another way of saying the same thing, is that now if I look at the content of the wave function, here I have some of the valence bond covering. It's just that amplitude is now different. In this case, we have to use a periodic boundary condition for both directions here. Remember that another direction, for another direction it has to be anti-purearity. So another way to see the same structure is as follows. Imagine that when I think about a superconductor, I take a short coherence length limit. So what that means is that I'm thinking about a very, very small size copper pair. Remember that coherence lengths in superconductor is essentially the size of the copper pair. So if I take a very, very short coherence length limit, what that means is my copper pair size is now like one large spacing. Does it make sense? Yeah, one large spacing. And in this case, that frozen copper pair is represented by this, the short line here. And it turns out if you take a symmetric and anti-symmetric combination of those wave functions that I constructed, and then you can show that in such a state, if I start from some valence bond configuration like this, and say, imagine that I stick to some particular column of the lattice on the cylinder, that if I count the number of valence bond that cross this line, and if I start with the even number of dimers or valence bond like this, and you can easily convince yourself that if I wanna move this valence bond, that for example in this case, in this case, sorry, initially you cut two bonds, but when you move diamonds, then it becomes zero. In this case, initially I had making mistake. In this case, I cross three diamonds like this, but I end with one. My point is you can only change it here. You can change the number of valence bond crossing this line only by an even number. So you cannot go from this sector to the other sector. So in that sense, they represent two degenerate ground state. So there's another way to see that there are two degenerate ground state on the cylinder. Hope it makes sense, does it make sense? Yeah, and if you repeat this argument, if you put this system on a torus, then how many ground state do we have? Four, right? Because on the torus, there are two holes that I can put a flux, right? Does it make sense? Does it make sense at all? Any question? Does it make sense? If you have a torus, if you have a torus, in the cylinder it's clear that I can thread the flux, right? In the center of the cylinder, does it make sense? Okay, so that's why there are two degenerate ground state. If you have a torus, then you can thread the flux inside the torus or in the hole of the flux, right? There are two different ways of threading the flux, right? Each time I generate two degenerate ground state. So on the torus, you will end up with four degenerate ground state, right? Now, if I put this system on a, say, some strange manifold of, say, imagine the number of holes is N, then you end up with two to the N number of degenerate ground state, yeah? You cannot do this experiment, but you can do a Kedanki experiment. Yeah, question? Yeah, so here I'm already taking a thermodynamic level. Yeah, so that's the manifold without boundary. The torus has no boundary, yeah? By that side, so again, I'm taking a, if you like, I'm taking a thermodynamic, thermodynamic means number of spins, number of spins on the manifold is basically infinity. Yeah, so here when I say they are degenerate, I'm already talking about a thermodynamic level, yeah? So another interesting property of this phase is, is that the elementary excitations carry non-trivial quantum numbers. So let me start with the usual elementary excitations in superconductor. So in superconductor, probably the cosy particles are the elementary cosy particles. And remember that I started with a situation where the average number of particles per side is one. Remember that? Yeah. So in order to have that, I should have a particle horse nutry. Remember that Pogolbo-Kozhe particle is a linear superposition of the electron and particle state, remember that? And if I have average number of particles per side is one, then I have an equal superposition of particle state and whole state for the Pogolbo-Kozhe particle. That's what I mean by, that's what I mean by particle symmetry. So in that sense, Pogolbo-Kozhe particle has zero average charge, but they carry spin quantum number. Yeah, I hope it makes sense. So the equal superposition of particle state and whole state, therefore average charge is zero. But remember that in superconductor, you cannot really assign a charge quantum number to your Pogolbo-Kozhe particle because charge is not a good quantum number. But now, if I do the projection, meaning, again, I remove all the doubly occupied side on the lattice, then I become, basically my state becomes an insulator and insulator charges will define. Now, charge is a good quantum number. So you can ask what happens to this Pogolbo-Kozhe particle after the projection? So since I started the zero average charge, it makes perfect sense to think that the charge of this particle becomes essentially zero. But now, it's exactly zero because I'm in an insulator. I'm not in a superconductor anymore, yeah? So that way, I can get rid of the charge, but I still keep the spin quantum number. So such a particle will have spin half, but no charge. So that's the way that you can create a charge in neutral object with a spin half quantum number. And these particles are called spin off. So you can think of that as a fractionalization of electrons. But what really happens is that you just screen out all the charge degree of freedom from the Pogolbo-Kozhe particle. So you end up with a non-trivial excitation, non-trivial excitation. So you see that, you see the analogy between quantum whole state and the spin liquid. I can have a non-trivial ground state degeneracy or a non-trivial manifold. I can also have non-trivial excitation that carry only some fraction of the quantum numbers of the electrons, right? So some properties spin liquid are very, very similar to quantum whole state. So in the literature, there's the different kinds of topological phases discussed. So this is what we call symmetrically protected topological phase of matter. So again, the argument is very similar. So I wanna think about a possible phase transition between some simple phases of matter, say, simple phase means metals and band insulators to some topological phase of matter. But now I wanna put one more condition that when I'm thinking about changing from one phase to the other, so I'm thinking about some performing a unitary transformation starting from this phase to the other. But if I insist that there's certain symmetry in this process, there's some symmetries in place so that when I do a unitary transformation, I have to obey the symmetry. So symmetry cannot be changed when I try to do a unitary transformation from this phase to the other. So if you insist that this transformation also test satisfy certain symmetries, then basically there is no transformation you can connect to phases of matter. But if you're willing to break the symmetry, so in this plane, so to speak, the symmetry is preserved, but out of that plane, symmetry is not preserved. So I can definitely go from this phase to the other by going through some path or unitary transformation that breaks symmetry, then it's okay, then you can connect these two phases of matter. And famous example of that is a topological band insulator that you are familiar with. And here the symmetry is nothing but a tiny bit of symmetry. So if you think about this as a band insulator, this guy is a topological insulator. And if you try to connect them by preserving the time of symmetry, then you cannot connect these two phases of matter. But if you're willing to break time of symmetry along the way, then you can in principle connect these two phases of matter. It's okay, you can ideologically change band insulator to topological insulator once you break time of symmetry along the way. So in that sense, this topological phase is symmetry protected. And this is somewhat different from the previous case. In the previous case, you don't require any symmetry at all. So in practice, okay. So in practice, this is the typical way to distinguish a trivial band insulator and topological band insulator. So this is the, say, the balance band and conduction band. Then this band structure represents the surface band structure. So these are the bulk band. These are the surface band. And if you look at what we call tiny muscle invariant momentum. So we're at this momentum position under tiny muscle symmetry transformation, the momentum goes to minus K up to reciprocal R specter. And at that momentum positions, we know that you should have at least double degeneracy because of tiny muscle symmetry. So there are two degenerates that are here, two degenerates that are here. They are so-called Kramer's partners. And at some other momentum position, these electronics that are also degenerate, meaning they should be also, so each band is doubly degenerate here. Now you have to connect this point to the other point by some band structure. And there are two different ways that you could connect. For example, here, when I go from this side to the other, you can split this Kramer's degenerate state, but then you can combine the same pair here, but you could switch the partner. For example, you can use one of them here, the other guy here, then you can make a degenerate state there. So there are two different ways to connect the electronic state, boundary state. But notice that in this case, I necessarily have to cross even number of band crossing here. Here I can have, say, odd number of band crossing. So if I move around the chemical potential, here I can get rid of the surface state, but here, no matter what I do, I can not get rid of the boundary state. And this kind of situation corresponds to so-called topological band insulator, and that situation corresponds to trivial band insulator. And this is distinguished by looking at the surface, a band structure, but you can also look at the bulk state, especially when your system has an inversion symmetry, you can label each block state at tiny muscle inversion momentum in terms of parity eigenvalue, so because parity is a good symmetry. So if you take a four-independent tiny muscle inversion momentum position, then think about the block wave function, and there should be an eigenstate of the parity operator, so there's the parity eigenvalue, there's plus minus one. So it turns out that if you multiply all the parity eigenvalues here, if the answer is minus one, then it corresponds to this situation. If the product or parity is plus one, then it corresponds to that situation. So you can make a connection between some kind of bulk topological invariant and the surface state, boundary state, and you can generalize it to three-dimension. So here's a demonstration that if you're willing to break symmetry, then you can connect two different phases of matter. So the question is how can you connect band insulator and topological insulator? So imagine that I start with some Hamiltonian here, this gamma, it's some four-by-four matrices, it's called a gamma matrices, and they anti-commute. So if you have a Hamiltonian like this, so K i is the momentum, let's think about a three-dimensional system so that I can have K x, K y, and K z. So if M equals zero, then I just end up with a Dirac-Pomion Hamiltonian. But depending on the sign of the mass term I put here, I can go from band insulator to topological insulator. So it turns out that by changing the mass of the, sign of the mass of the Dirac-Pomion, I can go from band insulator to topological insulator. But notice that the transition happens when M equals zero, that's essentially a gap response. So that's what I mean by saying that if I wanna connect band insulator and topological insulator, by insisting that time of symmetry is always satisfied, then there's no other way. You have to go through the M equals zero point, mass equals zero point. On the other hand, if you are willing to break time of symmetry along the way, so here for example by multiplying these matrices, one can show that this combination breaks time of symmetry. So if I add this term here, then this entire Hamiltonian breaks time of symmetry. But since all these matrices are anti-committing, the spectrum of this Hamilton is always gap and I can basically go from M equals negative M state to positive M state without closing the gap. But you can do that only when I'm breaking a time of symmetry. So this is an explicit demonstration of that idea of symmetry protected topological phase of matter. Okay. So, okay, so this is the idea, but then the question is, where can you possibly find any of these phases of matter? So I'm gonna use this idea of paramaterial strong spin-off coupling. So just to motivate the discussion, this is essentially the situation that we are thinking about. So I have, there's some band structure. There's an interaction between electrons. They imagine that I also have some kind of atomic spin-off coupling. So, for example, when there is no spin-off coupling, when I change the interaction strength, say with respect to the bandwidth or the hopping strength T, we know that you can have a transition from the simple matter of a band insulator to a mod insulator. This is the correlated insulator. If you have a very weak interaction and if I only have a spin-off coupling, then we learn from various example that I can go from simple matter band insulator to say topological insulator or the semi-metal. So many of the, say, topological insulators discovered in nature basically belongs to this category. I just increase the spin-off coupling that I end up with some kind of band inversion and then get into a topological insulator. So now the question is, what happens if I have both interaction and spin-off coupling? So then, so you end up with this kind of phase diagram and you realize that there's a very large area here where both interaction and spin-off coupling may be large. And this is the place where we may expect to see an interesting phase of matter in the sense that you have some interesting band topology coming from spin-off coupling, but at the same time interaction with the electron is very, very strong, so that you may expect that you may discover different kinds of phases other than just ordinary topological insulators. So that's basically the idea. So it has been proposed by many people that perhaps four to five detentions of matter oxide is a good place to stop. And so a lot of interest in now in a 40 or five detentions of matter oxide system. So here's one famous example. So many of these materials have this idiom ion. So imagine that I have idiom four plus ion. Then this guy has five electrons in the 5D orbital. So if you remember your freshman chemistry, of course, if there is no crystal, then these D orbital states all degenerate. You remember that there are 10 state. There are five D orbitals, then that's spin projection, so there are basically a 10 state there. Then, but when these idiom ions are surrounded by oxygen octahedral, then your five D orbital may split into T2G and EG level. And the reason why it does that is very simple. Just because of the steady coulomb interaction, so these orbitals are basically in the lower energy state. But now if you think about the angular momentum operator, of course, for the orbital angular momentum is two. But if you only think about lower T2G manifold, then by looking at the matrix element of the angular momentum operator, you can easily see that if I only take those guys, then they look exactly like an angular momentum of the P orbital at equal one state, but the sign is opposite, right? Sign is opposite. So they actually, if you project this operator to this manifold, they actually act like angular momentum one operator, but there's an additional minus sign, yeah? So now if I turn on the spin over coupling, then this angular momentum minus one combined with spin out quantum number, I can generate a total angular momentum state, half and three half. And they basically split because of the spin over coupling. And since there are five electrons here, four of them goes to lower energy state and the remaining one goes to the upper level state. So this guy now carries a pseudo angular pseudo total angular momentum half, but essentially it's half here. And this is the wave function, essentially a linear combination of three T to G orbital. So you just end up with half field pseudo spin half system, and you can do that by using a strong spin over coupling. So here's the interesting picture that say, this is an atomic picture, but imagine that I go to a lattice, then this atomic level becomes some kind of band. So the entire, the five with the orbital manifold is split into two pieces. So it's just like J effective three half manifold and J effective half manifold. This manifold is totally filled, but this manifold is only half here. So this situation is very, very similar to say, if you're familiar with it, it's like a Q-plane system where you have half field band. Effective in the effective one band picture. So now important thing is that here, now if I apply a Hobart U, so normally if you have a total, the entire band like this, then the bandwidth can be very, very large. So it's very hard to make the system to be an insulator, but now notice that the bandwidth around the formula is relatively narrow. So by applying a small Hobart U, I can split this back to the low Hobart band and upper Hobart band. So if I can do that, then even with a small strength of the Hobart U, I can get into an insulator. I can get into an insulator. So there's one of the reasons why, for example, nobody asked me this question, but if I go back to this phase diagram, notice that the slope is actually downwards like this. I emphasize that when spin-off coupling is large, interaction effect is larger. So one reason for that is that using spin-off coupling, effectively you can make the bandwidth near the formula level narrow, and that way the interaction effect can be amplified. So that's one reason why I get this. The reason why this slope is like this is because if you have a strong interaction, your band is already very narrow. So even with a small lambda, you can make a big change. So either way you win. Either side you win. That's why you generate a large area like this, okay? So as an example, I'd like to now discuss this special class of material called parachloridate. So this is the 3G material, has this, having this chemical formula. So this is a rarest ion. So R2 is into O7. So the rarest ion and idiom, they are sitting at two inter-painting parachloridates. So parachloridates is nothing but this connoisseur tetrahedron structure. There are two of those. They are inter-painting each other. It's a complex structure, but each one of them sit on an independent parachloridates. So you can put various different kinds of rarest ions on the R side or H side, and this is roughly the phase diagram. So by changing the radius of these ions, essentially what you're doing is, you are changing the bandwidth of the conduction electron system. So perhaps, how about you, for idiom ions are about the same, but because of the fact that your bandwidth is changing, you can go from a more insulating state to a more metallic state. For example, when you put a presidium in here, it turns out that it remains metallic down to very low temperature. On the other hand, if you put, say, Europe in here, then there's a matter of insulator transition at parate temperature, and yeah, and it turns out that this magnetic order is what we call all-in-all-out magnetic order. So each tetrahedron, if I look at the basically moments at corners of the tetrahedron, for this tetrahedron, all moments are out. For this guy, all moments are in. So this is what we call all-out, all-in magnetic order. Some kind of antiferous magnet. But notice that with this magnetic order, the total magnetization is zero. Yeah, total magnetization is zero. Okay. Yeah, so you can start with this picture. Then you can construct some very simple tight binding model out of this JFK to half degree of freedom. Then you can put, how about you? Then imagine that I study a phase diagram. So this is the schematic phase diagram you can obtain by doing that kind of exercise. So here is the non-interacting Hamiltonian you can take. So here, my operator C represents the electron in the JFK to half basis with up and down spin. So because of a spin-over coupling, you can have a spin-preserving opening that represented by T1 term. But you can also have a spin-flip-hopping. And it turns out that this term is extremely important. And in fact, if you ignore the interaction, just this line is non-interacting limit. And this limit of the phase diagram is already interesting. If you change the ratio between, say, spin-flip-hopping and spin-preserving-hopping, then you can get either topological insulator or you can get some kind of semi-metallic state. It's called quadratic betting semi-metallic. So you can have a transition between, say, metallic state, semi-metallic state, to topological insulator. So even without interaction, question? So could you, could you speak up? Which phase diagram? This one? Oh, so non-metallic means the temperature dependence, the resistive, yeah. The temperature dependence of resistivity is non-metallic. So what that means is that usually for metal, a resistivity should go down in temperature. It doesn't do that. But nonetheless, it's not an insulator. It's a finite temperature. So you can say something is insulator or metallic only in the zero-temperature limit. So in that sense, it's a non-metallic. I think this experimentally is, try to be a little bit more precise. Yeah, okay. So you can have interesting band topology even without interaction. You can go from topological insulator to a semi-metallic. Now when you increase interaction, it turns out that with some intermediate ratio between T2 and T1, you can indeed go into all-in-all magnetic order. So this simple model can actually explain the experimentally-discorded magnetic structure. But what is really interesting is that this part of the correct magnetic order is actually in this side of the phase diagram. Basically you have to start from a semi-metallic band structure. Then you go into an antiferous magnet. And it turns out you cannot just directly go from this semi-metallic state to antiferous magnetic state. You actually have to go through some other intermediate phase. And that phase is the famous vile semi-metallic phase. And this is what I'm gonna explain. And unfortunately, the real material is not here. It's here, or maybe fortunately, because this part is also interesting, but this part of the phase diagram is not realized yet. But it would be interesting if there's a way to get there by changing some parameters of the model. But most of the materials are here in this part of the phase diagram. So here is what happens. So this is some result using a very simple minded Hartley-Falk calculation. So without interaction, in this part of the phase diagram, in this part of the phase diagram, I have what we call a quadratic band-touching semi-metallic. So remember that on the particle lines, there are four sides per unit cell. So there should be eight bands, including spin. So without magnetic order, each band is doubly degenerate. And then I end up with this band structure. So if you look at the joint center, this is the chemical potential. You end up with this quadratic band-touching. And if you go to the other side here, then the band structure looks like this. So why there's a difference like that? So group theory tells you that with this, with the cubic symmetry of the crystal, if I'm sitting at the gamma point, the joint center, then group theory can only tell you the band degeneracy. So it turns out that band degeneracy has to be some combination of two for two. So here band degeneracy is like two for two, right? But here band degeneracy is four to two, right? But group theory doesn't tell you, which one should be realized. It only tells you what multiples are possible. So both configurations are possible. But now if I have this situation, four to two, for the half field situation, I end up with a band insulator. Well here, since I have two for two, I should have a semi-metal. This is essentially what happens. And there's some kind of band inversion when you go from here to here. And interestingly, you don't get a boring band insulator. You actually get a topological insulator. But now, if you get into a magnetically order state, then your tiny versus smith is broken. So there's now no reason to have two degenerate bands, so bands split. But it split in such a way that I generate a band crossing along gamma and L line, like that. And in fact, it shows only one crossing point here. It turns out that actually there's another crossing point. And these two crossing point, they all move to the L point. And at some point, they annihilate. They open up the band gap. You end up with a boring magnetic insulator. So that's the way that the band structure evolves. So essentially what happens is that if you start from this cardiac band touching, and when you split this band, the simple picture is that each cardiac band touching split into two pieces like this. So it's very easy to generate a two crossing point. And that's the way that this band crossing is generated. Notice that this is the crossing of two non-degenerate bands. And that's very important. In fact, in the experiment, this quadratic band touching has been observed in this material. This is the most metallic system. There was an obvious experiment basically conformed that at high temperature. If you look at the band structure, and in fact, the band structure you observe is consistent with this one. Yeah, this guy. So there's a very simple way to understand this entire picture. And that's using what we call a latino model. So this is nothing but expanding your band structure around the zone center. It's a K-dapiparabation theory. So you just expand it up to quadratic order. And remembering that this is a cubic crystal, I have some linear combination of cubic harmonics. And that's essentially the latino model. And if you use that, then you can easily explain why there could be a quadratic band touching. And in fact, when you apply a time-inverse estimated working perturbation, you just split that quadratic band touching, you generate a wild permeance. Okay, so one of the reasons why this state is interesting is that you can actually detect it by doing all sorts of experiments. So as I said, this wild permeant, so actually they occur always in pairs. And they are related, for example, in this case, these two wild points are related by inversion. So you already create those points by pair, and it's characterized by so-called this chirality. So if you compute this triple product, triple scalar product of this combination for that point and that point, then they carry plus one and minus one. And once you get into this phase, this phase of matter is extremely stable because especially in three-dimension, you can associate each velocity to one power matrix. The reason why you wanna use the power matrix to describe this band crossing is because this represents the crossing of two non-digital band, so really there's only two states to take into account. So for each velocity component, say Vx, I can associate that with the parametric sigma x, Vy, sigma y, Vg, sigma z. But by the time you do that, then there's no other anti-committing two-byte matrix you can add. So it tells you that there's no, I cannot get it out, there's no mass term I can add to this Hamiltonian. So that's why once you create such a state, it's extremely stable. So this kind of state has a very interesting property. So for example, imagine that I'm thinking about some kind of a projection of these file points to a surface-brilliant zone. And so, and then let's imagine that I'm thinking about in a bulk-brilliant zone, I'm thinking about some cross-section like this. So this is a two-dimensional cross-section momentum space. And if you compute the turn number that I talked about at the beginning of the talk, then it turns out that they actually form an integer quantum whole state. So if you only take one 2D cut of the three-dimensional-brilliant zone like that, say between these two points, and they, this hypothetical 2D system extract the integer quantum whole state. So what that means is that if I go to the boundary of the one, boundary of 2D system like that, but in the boundary-brilliant zone, I should get at least one gapless point because we know that for 2D quantum whole state, there should be an edge state. The edge state is gapless. So there should be at least one momentum position where the gap finishes, right? And you can do that for the next layer, next layer. But by the time you leave this thing, meaning you go outside this space between these two points, it turns out that this state is trivial. It's not an integer quantum whole state anymore. So those guys, they don't have any edge state. So notice that if I do this, then I generate a continuous set of boundary state like this, but this guy should end here. It cannot continue. So that's why you end up with this, what people call a forming arc state. It looks very strange, but that's basically the origin. It can be explained by integer quantum whole fact, essentially. Yeah, so here, the only thing you care about is the chun number. So chun number, basically some bands can carry a chun number without magnetic field. You don't need a magnetic field to get a finite chun number. And you just need the, basically, this is the property of the wave function, right? You remember that? Property of wave function. So the wave function defined on the 2D plane in the momentum space. If you compute the chun number, that turns out to be integer number, what? It's the same, same as the nuclear one state. So if you only look at the one surface like this, then of course you get only one forming arc. It turns out that if you look at say, but you can have another boundary and the other side. So if you like, the missing half is actually in the other side, the other side of the surface state. And notice that this is reminiscent of the situation where when you have integer quantum whole state, remember that? You can have a right moves, a left moves, they are physically separated, yeah? Physically separated. And that's the reason why you could realize some anomalous 1D state. This is the same. If you combine these two, you just get an ordinary closed zero energy contour. But because of the fact that bulk is of a very strange state, you can separate this guy into two pieces like that. So there's certain similarity in physics, okay? So even though this vile semi-metallic phase was first proposed in a particle relay, it was not observed here so far. But it was observed in other materials. And most of the materials that we know where the vile semi-metallic is observed is actually a semiconductor system where you actually break inversion, not time of the symmetry. So here, this is an example where you go into a magnetically ordered state so that you actually break time of the sort but no inversion. But in a semiconductor system, there is no interaction really. So you cannot realize a magnetic order, right? So the only way to get that is to break the inversion symmetry. So it turns out that there are two different ways to generate a vile fermion, either breaking, either you break time of the sort. You can also break inversion. And most of the examples say publishing, nature and science. So inversion breaking a vile semi-metallic. You keep seeing them, right? In the archive. I think there was a time that every week there was a paper, yeah? But most of them are breaking inversion, not time of the sort. So for a long time, we thought that particle relay is an example where you could see time of the sort breaking vile semi-metallic by going into a magnetic order state. It didn't happen here but there's a recent example of that. This is a Manganese III tin. Here, Manganese ions are seen on a calcumelaris. They make a 120 degree order. So you get into a magnetic order state. So it turns out that this is an itinerant magnet. So this is a magnetic order. And it turns out that this material actually realizes the magnetic vile semi-metallic. So basically you have a magnetic order so you break time of the sort and then you get into this vile semi-metallic state. Okay. So another interesting property of this vile semi-metallic is that in principle, you can have what we call an anomalous hole effect. So again, if you use this idea that if I think about 2D plane for each cut, they act like an integer column or state. So each one of them should contribute whole conductivity of e square of h, right? Each 2D plane should contribute like e square of h, whole conductivity. Now, you can say that the number of 2D plane like that should be proportional to two times k, the distance between two vile point. Make sense? So each 2D plane carries, maybe I should show this picture. You know? Each 2D plane carries e square of h, whole conductivity. Yeah? Yeah, I'm almost done. Yeah, yeah. So each plane contributes e square of h, whole conductivity. Then I have many of them. The number of 2D plane that contribute to whole conductivity will be proportional to the distance between two vile points, right? Make sense? So the total whole conductivity has to be proportional to the distance between two vile points. Yeah. So I think I'm gonna, yeah. I don't think I have time. Let me see, maybe I should skip this. Yeah. So, in principle, you can have a very large whole conductivity, but in a cubic system like a particle, there are actually many other pairs of vile points and they actually contribute some negative contribution. So cubic symmetry enforces that if you sum them up, the answer is zero. So unfortunately, if you have a strictly cubic system, then you don't get an enormous whole effect. In fact, in the real system, there are 24 vile points. And if you sum them up, then whole conductivity, enormous whole conductivity is zero. But you can imagine that you break cubic symmetry, for example, by making a film or you apply a strain. So then all these contribution of vile point, they don't cancel. And in principle, then you can generate a finite enormous whole effect. And in fact, that has been done. So I just wanna show the experiment. So that is a very recent experiment done in New York, Tokyo. Basically, they make an epitaxial thin films of pristine eidimoxide. And they found that at very high temperature, like 20 Kelvin, I think, they start to see a large animal's whole signal. And independent of detail, the reason why you could get that is basically because you're breaking a cubic symmetry. So I just want to say that effect like this, it actually has been seen by making a film. Okay, so I think I wanna end my talk today like that. Okay, thank you very much.