 So, yeah, thank you for the introduction and I would like to thank the organizers. I feel extremely honored to be here. So I'll be talking about many body formulation of topological phases of matter. In particular, I'll be talking about symmetry-protected topological phases, such as topological insulators and the other variants. In particular, I will be focusing on the thermionic ones, which has extra complications. So we had to work a little bit harder. So I'd like to thank my collaborators, my student Hassan, who is graduating next spring and my former postdoc Ken. So I've been working on this subject for a few years by now and we have a series of publications. So you may have heard some version of my talk, or at least closely related to the topic which I'm going to talk about today. But today's focus will be time reversal symmetry, which, so, yeah, like I said, I've been working on this project of constructing many body topological invariants. But time reversal symmetry was almost the last symmetry I wanted to crack, which was the most difficult, but I thought it is most interesting, I think. So here's a quick overview. So as the title suggests it, I would be interested in many body formulations of topological phases. And I want to contrast this formulation with the single particle formulations of topological phases of matter. For example, for two-dimensional time reversal symmetric topological phases, yeah, not just for two-dimensional time reversal symmetric topological instilators, but many topological instilators of various kinds, we often rely on the band picture. So we have the bands and we have a gap and at each band we have a block web functions and the topological phases or topological invariants can be described in terms of the topology of this block web functions, which is parameterized by the single particle momentum. And for example, for 2D topological instilator, we have a formula for the topological instilator, which may be looking like this. So detail the form of the formula is not important now, but it suffices to say that it is written in the language of single particle web function. So I want to say this had the tremendous success in the sense that it predicted topological instilators and it got detected later and so on and so forth. Nevertheless, question remains, so if we have an electron-electro interactions, which may be weak, but which may be rather strong, and in that case, we want to talk about topological phases of matter, so we want to go beyond this formulation. So that is the main goal of this talk. So our formula for topological invariant, for example, for the case of topological instilator, looks like this and it looks rather complicated at this moment, unfortunately, but I just want to mention that in this formula, it's written entirely in terms of this many-body ground state and the reduced density matrix constructed out of the ground state. So there's no trace of single particle formulation in this formula. Everything is written in the many-body language. So the purpose of my talk is to explain what is in this formula and where it comes from. So this is one example, but there's other cases we can talk about and I'll discuss that. So let me say a few words about symmetry-protected topological phases and by that, the obvious example is topological instilator, but by that, I mean phase which is pretty gap and the more importantly, ground state is unique. So if you draw this tree diagram of phases of matter, we are interested in basically phases which cannot be detected by other parameters. So we need some sort of topological invariant to characterize them and we are interested in gap and we are also interested in particular kind of gap phases which is symmetry-protected topological phases. So just one more slide for SPT phase. So in SPT phase, we make topological distinction of quantum ground state in the presence of symmetry. So if we don't have symmetry, SPT phases may be adiabatically connected to topological trivial phases such as the atomic instilators, but if you impose some symmetry such as time reversal symmetry or some other symmetries, then your phase diagram will be disconnected into pieces like this. So then phase of your interest may not be connected to a trivial phase anymore in the presence of symmetry. So here we are making a distinction between different ground states in the presence of symmetry. So we are assuming symmetry is preserved. So there's no other parameter to distinguish different ground states and instead we need something topological invariant to distinguish different phases of different ground states. So I will explain the many body invariants first for the simplest example, which will be the Harden-Spain chain. So this is the Bosonic system. Whereas for fermionic systems such as topological instilators, I need a little bit more techniques or technology. So I will first describe basic strategy for the Bosonic cases, then I will move on to fermionic cases. So the key ingredient is something we call a partial transpose operation. I first defined this partial transpose operation for Bosonic systems and as it turns out inventing this operation for fermionic system is the kind of main obstacle or main difficult part, but for Bosonic part it's rather easy. So I consider Hilbert space, which has some tensor decomposition. So there are two parts in your Hilbert space. You just write out your density matrix in this Hilbert space using some basis, say E1, E2, and then you write out the matrix element. So in this partial transpose, you take a transpose, but not for the whole density matrix, but you take the transport only for let's say the second Hilbert space. So it's maybe a little bit difficult to see here, but if you look at this equation carefully, I exchange column and low only for the second Hilbert space without touching the first Hilbert space. So J and L, they are swapped, but I and K are intact. So that's the partial transpose. So this is going to be my main tool to construct a topological invariant, but I'm not the first one who uses this operation. So this operation has been used in quantum information community to describe a quantum entanglement. So partial transpose has been used to construct some sort of entanglement measure, which I will not describe in detail, but there's some use in this history. But as it turns out, the partial transpose is also useful to construct a many-body topological invariant of symmetry-protected topological phases in the presence of time reversal symmetry. So the connection to time reversal symmetry maybe understood from the fact that taking transpose is basically like taking a complex conjugation, which is basically taking time reversal. So I want to go through some examples, which is the ordain chain. So it's the canonical example of symmetry-protected phases. It can be protected by various types of symmetries. For example, spin rotation protects the hardened phases being deformed into topological attribute state, but here I will be focusing on time reversal symmetry. So I will impose time reversal symmetry. And then a hardened phase is an example of SPT phase. Famously, the ground state of the hardened phase can be picturized by the collection of singlets or dimers or maybe bell pairs. So basically in this construction, we split a spin 1 into two virtual spin 1 halves, and these virtual spin 1 halves, they form a bell pairs. I always forget, is it inter or inter? This is the inter-site bell pairs. So this is the kind of picture for the ground state of the hardened phase. So I said something about quantum entanglement before. So to demonstrate the usefulness of the partial transpose operation to detect quantum entanglement, I first want to focus just a single bell pair and ask if there's entanglement. Well, we know that there should be some entanglement, but how do we quantify the entanglement? So using partial transpose, we can construct some sort of number or measure which can quantify quantum entanglement. So I have a bell pair, which is a part of the hydrogen thing, if you wish, and then I construct the density matrix. So following the definition I presented before, I take the partial transpose, which means in this case, we exchange these number 1, 0, or 0, 1 only for something sitting on the second entry. So here, partial transpose doesn't change this part of the density matrix, whereas for the last two terms, taking partial transpose changes these two terms into this. So we get the new matrix, partially transposed density matrix. So what people have noticed before was something strange. There's something interesting about partial transpose density matrix. So if you look at the eigenvalues of this partial transpose density matrix, it's actually not a valid density matrix in the sense that some eigenvalues are negative. In this case, there are four eigenvalues, but there's one eigenvalue which is now negative after taking the partial transpose. So it's some sort of condensed metaphysics language, some sort of response theory. So we take a partial transpose. If there is some quantum entanglement, your state will be battery changed, whereas if you have a classical state like this, taking partial transpose doesn't affect it at all. So here, partial transpose, the effect of partial transpose is so strong such that we now get the negative eigenvalues. So basically counting the negative eigenvalues which appear after the partial transpose gives some sort of some sense that how much quantum entanglement you have in the bare pair or in quantum state in general. This is called a PPT criterion. For our purpose, this story about quantum entanglement is related, but not exactly, because we are going to talking about topology rather than entanglement. So let me now move on to discuss topological invariant of hardane chain. And for that, we follow this work by Paulman and Turner. So they constructed topological invariant of hardane chain using tensor network, but here I phrase their invariant using the partial transpose operation. So their construction goes as follows. So I take the hardane chain which is living here, but then I take out the interval I. So then I first trace over all degrees of freedom outside of this interval. Then I get the reduced density matrix for this interval I, so I call it the law sub I. Then I partition this interval into two subintervals, I1 and I2. And with this bipartitioning, I take the partial transpose for the, let's say for the, for I1. So then I get the partially transpose reduced density matrix. So that's step three. And then I form this quantity, which is the trace of original reduced density matrix and partially transpose reduced density matrix. And then I will look at the phase of this quantity, and the claim is that the phase of this quantity is quantized as far as this I, I1, I2 are sufficiently large as compared to the correlation lengths of the hardane chain. And then the value will be plus one for topological attribute ground state, whereas for hardane ground state, it's minus one. So that's the construction. To make some, to give you some more intuition, we can also write this invariant using tensor network representation, where I represent my ground state using these smaller boxes. For the ground state, I have one column of boxes, which has one dangling physical spring index. There are two other indices, which are just budgetary there to encode the quantum correlation of this state. So using this, this topological invariant can be represented in this way, where you can sort of see there are four units of this, because each density matrix has ground state bra and ground state ket, and they have two such, two kinds of density matrices. So I have four columns, four rows, sorry. And then this trace and the transpose basically connect to these bonds. So this invariant, if you look at this diagram long enough, as it turns out, you can sort of deform this network, assuming it's a topological phase. And then this tensor network can be deformed into a, which is some interesting space time, which looks like this. So the tensor network, we want to view it as a space time pass integral, discrete version of space time pass integral. But this space time now is this peculiar space time. In the discrete notation, it looks like this. But this can be deformed into this space time, which is called the real projective brain. So it can be viewed as a two-dimensional plane with a small hole and this circumference, opposite points on the circumference of these points, they're opposite to identify. So that's what is called the projective brain. So that's the topological invariant of the hard engine. So it's a pass integral, so you can interpret the topological invariant as a pass integral on unoriented space time, which is a real projective brain. So why we are interested in this quantity to give you some more intuition, I want to draw an analogy to the topological invariant of the quantum hole system, which is basically just hole conductivity. So the idea is that we want to look at, we want to detect the phases of matter by introducing some sort of background, which I denoted x and a here, and then I just trace over or integrate over all degrees of freedom, which defines effective action for this x and a. And in this notation, I mean x is some space time, which can be torus or maybe a real projective brain, for example, and a is a background gauge field configuration. For quantum hole effect, sorry, so then for topological phases, this pass integral gives you something pure, imaginary sitting on this exponential. And for the quantum hole effect, this phase part is given by the celebrated the chance I monster with quantized coefficient, I forgot to put I here, but this is a phase quantity. So drawing the analogy to this quantum hole example, what you are doing for the hard end phase is that for hard end phase, there's no even gauge field because charge is not conserved. But we are considering a background, which is the unoriented space time. And for the unoriented space time phase part of the partition function is quantized, either zero or pi. And for hard end phase, it's it's pi. So okay, so I think I'm spending too much time for the Bosonic case. But now what I want to do is to basically follow this strategy for other fermionic other cases, including topological insulators, and also KitaF chain, their fermionic topological phases of matter. So KitaF chain is a, I would say, fermionic analog of hard end spin chain. It's a one-dimensional chain where fermions are hopping on the lattice, and there's a hopping as well as this pairing term. And in this simplest model, the phase diagram looks like this. So they're topological phase, which exists when this chemical potential is smaller than the hopping and the pairing, whereas if the chemical potential dominates a hopping and the pairing, we get a topological and trivial ground state, okay? So skipping some details aside, of course, this is a system which hosts a Myrona fermions, and there's experimental interest, of course. But this is the very close analogy of hard end spin chain in the sense that ground state can be also written like this. So here, we take the physical electron, which we can split into real and imaginary part, and then in the topologically non-trivial phase of the Myrona fermion chain, KitaF chain, these real and imaginary part Myrona fermions, they form an inter-site kind of pairs, fermion version of pairs. Okay, so the question is, can we detect topology of the Myrona fermion chain by using basically the same technique as we use in the hard end spin chain, okay? So that is the question I want to ask. So this interesting complication for the Myrona fermion chain in the presence of time reversal symmetry, I want to impose time reversal symmetry. So then with time reversal symmetry, previous study suggests that different phases, topological phases in the Myrona fermion chain are classified by Z8 quantum number. So classification for the hard end chain was Z2, but now we should detect Z8 distinct topological phases in this business. So anyway, we want to follow as closely as possible the strategy we use to construct the topological invariant of the hard end chain, but then there's this issue, what is then the partial transpose operation which we used before? So partial transpose operation is very transparent for bisonic systems, but for fermionic systems, the problem arises because fermionic field of space doesn't nicely factorize in the sense that if you have a two fermionic operators, let's say here and there, even when they are far apart, they still know each other because of fermionic sign, fermionic statistics. So this actually complicates the construction of partial transpose operation, which constructing varied fermion partial transpose is rather complicated. So this is the first thing we tried to construct, and once we construct the fermionic version of partial transpose, we can then use partial transpose to construct the many body topological invariant for fermionic phases with time reversal symmetry. And as it turns out, people have looked at the definition of partial transpose for fermionic systems, but we found that the existing definition was rather useless for our purpose. The reason we noticed that was the following. So we looked at the entanglement property of the fermion chain. So by previous definition, I mean people previously looked at the fermion version of partial transpose essentially by taking the Jordan-Wigner transformation. So they take the fermion chain, but they convert it to bosonic system. Once you rewrite your fermion system as a spin chain system, you can use the definition, you can take the definition of bosonic system to define bosonic partial transpose. But as it turns out, it doesn't detect the presence of Myrana dimers, which I used to picturize the ground state of Myrana fermion chain. So the protein here is the entanglement measure, which is going to entanglement negativity. I will not go into the detail of this quantity, but it's something which is supposed to detect quantum correlation. This quantum correlation using the Jordan-Wigner version of partial transpose says entanglement in topological non-trivial phase of ketaph chain is actually zero, close to zero. If you are deep inside the topological phase, which is sort of very strange. So that's why we noticed that it is actually important to construct the partial transfer operation for fermions from scratch. So that's what we did, and this slide exists here just to say we constructed it. Details are, I think, I want to skip some details, but we can construct fermion version of partial transpose. And for fermionic systems, it is convenient to use expansion to define your density matrix. I mean any operator can be expanded as a polynomial of fermion operators. And then when you take a partial transpose, we add some phase to the matrix element which didn't exist for bosonic systems, which is important. But I think I'm going to skip this detail of the definition of the partial transpose, but I just want to say we really did it. So now using this partial transpose, less will be rather similar to the bosonic case. So this is basically the same as what we did for bosonic case. So we have a now k-tive chain living on this line. And then for the same procedure, I construct the reduced density matrix. And then I take the partial transpose for the part of this interval i. And then I consider this quantity, or the phase of this quantity. So as before, this is supposed to be a pass integral of fermionic systems living on an oriented space. And this really works. So here is the numerics of this quantity. So here, this is a phase, and this is the amplitude of this quantity, z. And let's just focus on the amplitude, sorry, the phase part. And it's plotted in the unit of pi over 4, which is 2 pi divided by 8. Okay, so once again, there are two phases, topologically trivial and the topologically non-trivial phases in the k-tive chain. So I will be focusing on this case. So in topologically trivial phase, this phase is basically just 0. But as you go into topologically non-trivial phase, this phase of this quantity is quantized in the unit of 2 pi divided by 8. So it's 1. So that means this quantity can detect z8 classification of the k-tive chain. Okay, so now I want to, for the last, I just want to present one more example, which is the topological insulator. So topological insulator is, there's some difference. Now it's a two-dimensional system as opposed to one-dimensional system. Another difference is that k-tive chain doesn't preserve a particular number. It still preserve even the parity of the particular number. But in topological insulator, a particular number you want is strictly conserved. So there are no two symmetries to talk about, time reversal and charge you one, okay? So then we have to construct topological invariant taking into account these two symmetries, okay? So, but other than that, construction is rather similar to the case of k-tive chain. So now I have a two-dimensional system which is wrapped into a cylinder, okay? And as before, I focus on a sub-regions of this cylinder. So now I have a region R1 and R3. And the other part, I just call them R2, okay? And then I first trace over this region R2 to get a reduced density matrix, which is living on region R1 and R3, okay? In addition to the partial trace, I have to decorate this reduced density matrix by adding this factor which is the, which twists the phase of the wave function along y-direction only for region R2, okay? So this is something I call partial U1 twist. So it's a kind of twisted version of reduced density matrix living on region R1 and R3. But then I will take the partial transpose for region R1, okay? And then I combine these quantities to form a, this quantity z. So there's similarity before. So the previous example was simpler because it's basically raw partial transpose raw and that was it. Here because of the extra U1 symmetry, I need to add this phase factor. And what it does is to produce, so as before, taking partial transpose, I basically simulate the passing to grow on an oriented surface. But because of the U1 symmetry, I have to introduce background U1 gauge field on top of this background spacetime manifold. The spacetime manifold is a crime-bottle times periodic S1. But this extra U1 gives you additional background, appropriate background gauge field configurations, okay? So these are the final things. So the topological invariant, if you compute it numerically, it's the phase part is either plus or minus one. So for topological insert, it's minus one. Okay, so I just put my summary here. Thank you for your attention.