 So this lecture is not about stuff that we would have discussed 20 years ago, which was so far the lectures, which I presented. But it's now connecting what current research in this area is about. So this is an example of techniques that you have seen in the first two lectures being boosted up in the current research. Now, what is current research about? I talked about briefly in the very first lecture about this phenomenon of many-body localization. That's the claim of physicists that this phenomenon of Anderson localization, which is clearly a one-particle phenomenon, survives the onset of interactions. OK. And as I said, I mean, somehow this is kind of remarkable, because what this entails, in particular, is that you can prepare experiments as they are done in Munich by Bloch's group, and where you have atoms in a cavity, and you put all the atoms to the left half of the cavity, then you release the wall in between the cavity. The cavity has a random potential before those atoms, which in Bloch's experiment in this just laser light. And you will see somehow that most of the atoms will remain in the left half of the cavity. So something in which statistical mechanics actually would, by its very foundational hypothesis, forbid. OK. So that's the phenomenon in which physicists may try to understand. I mean, there's a very sort of minor contribution in me where we can actually say something. Now, what are interacting systems? So in physics, I mean, somehow the easiest interacting systems in which you can look at actually spin chains. And since we're dealing with quantum mechanics, then we want to look at quantum spin chains. Now, what is a quantum spin chain? So what you envision is a line of length 2L plus 1. And on each of the vertex segments of this line, you attach a Hilbert space. The easiest non-trivial Hilbert space you can attach is C2. And then the C2 somehow represents in the Hilbert space of a so-called quantum spin 1 half. OK. So quantum spin 1 half comes with a set of operators which represent measurable observables for this quantum spin 1 half particle. And they happen to be the generators of the rotation group, namely the Pauli matrices. So in the particular representation, where the third Pauli matrix is actually diagonal, these Pauli matrices are written on the slider bar. Now, so each Pauli matrix acts now on an individual copy of C2. We have 2L plus 1 copies of C2. So when you act, I mean, somehow on a single spin at side x, you can extend the action of these Pauli matrices canonically to the whole Hilbert space by just tensoring the action of 1's to these Pauli matrices. So this is a tensor product Hilbert space. We are defining operators on this tensor Hilbert space by telling you how they act on the individual components. Now, what is a Hamiltonian, namely a self-adjoint operator in which has some physical significance on such a tensor product Hilbert space? Well, since I think in the most of you when we are coming in from the probability crowd, you're certainly somehow seen in the Ising model. So the Ising model in me is in 1D and we would be a creature in which you can think of some spins on a line. But these spins, we have now values at each point of either being up or down. Now, here we are talking about quantum spins. And their values, and we are not just up or down. However, there's a basis in C2 as an orthonormal basis in C2, which would be 1, 0. Let me call this up and 0, 1, which is spin down. But that's not the only state in me this quantum particle can take in me, because we can then we put the quantum particle in a state in me, which is not one of those vectors in this orthonormal basis in me. But maybe in me, somehow, a linear combination of the two. OK, so that's the difference in between the Ising model in the end of the model which I'm talking about. But from the Ising model in me, you're maybe familiar in me that what you want to look at is coupled spins. And the coupling, typically, is translation invariant. And it's a nearest neighbor interaction. Now, in quantum mechanics, the spins are then replaced by spin operators. Therefore, the spin operators are just up to factors of 1 half, these Pauli matrices. So that represents spin in direction 1, spin in direction 2, spin in direction 3. And when you think about how an Ising model looks like, it would be a nearest neighbor spin-spin interaction. So the spin at side x interacts with the spin at side x plus 1. Well, since we don't only have sort of a z-component in any of those three components can interact, they can interact with a coupling J alpha. And translation invariants, we would require that all of these couplings are independent of x. So that is a conventional quantum spin Hamiltonian, which, so to say, is a generalization of your favorite Ising model. Now, as in the Ising model, we can take these couplings, ferromagnetic or anti-ferromagnetic corresponding in me to a plus positive coupling or negative coupling. Remember in the Ising model, when you couple in me, for example, with the three components, since I put an overall minus sign here, a positive coupling J3 would actually favor alignment of spins. And that would be the phenomenon of ferromagnetism. Whereas a negative coupling here, so J alpha being negative, would actually favor energetically in me that the spins and we are in opposite direction. Now, there's a zoo of quantum spin chains already for spin 1 half, which one can look at. The zoo and we basically correspond to choosing different versions of these coupling constants. They all come with names. So there's a zoology of names. There's the x-y chain, which is an abbreviation before taking the coupling in the three direction to be 0 and taking sort of generic non-zero couplings in the 1 and 2 direction. So what I will refer to in the following as a plane. There's the special case of the x-y chain, which is the x-x chain where we require the in-plane coupling to be equal, so without loss of generality, I can set them 1. And then there's the x-x-z chain, which is a generalization of the x-x chain, where we switch on the coupling in the three direction in comparison to the x-x chain. Now, what I will focus on for simplicity in this talk and it will be these two quantum spin systems. Now, no randomness has been added. So let me now add randomness. And this can be done in various ways. And we are a quantum spin glass. And we would be a creature. And we would take this J alpha random. But let me be more conventional. And let me now add to this quantum Hamiltonian to these quantum spins an external magnetic field, which points in the three direction and which is random. Think about if you know the Ising model and how you would do this. Well, you would couple, oops, I forgot the random variables, omega x here. Sorry. So there should be omega x up there. So multiply and be somehow the three component of the spin. And we buy omega x, sum over all lattice sites. And for simplicity, in the sense that I want to do the same game as we've done in the case of the Anders model, we want to keep the distribution of the random variables fixed without loss of generality. Again, somehow the single site distribution being absolutely continuous, compactly supported with a bounded density. We introduce a real parameter lambda, which is our disorder parameter, which we may switch off or just take positive. OK. Now, what is the question which physicists are after? So basically, I mean, this model of the XXZ chain in the way you take the J1 and J2 couplings equal to 1 in the J3 coupling, I mean, some other value. I mean, that's the XXZ chain, and we now with a random magnetic field. And that's in the physics literature really a laboratory for the phenomenon of many-body localization. I brought you some references. Now, what is many-body localization? The truth is, I mean, we're at a stage and we're where there's actually no hard mathematical definition of this phenomenon. So physicists are still searching in before a good definition. But let me bring to you, let me talk about at least two definitions of this phenomenon of many-body localization. OK. OK, so remember what we proved in the one particle system, and it was that the eigenfunction correlator decayed, and that in particular entailed, I mean, that the dynamics decayed exponentially. And this eigenfunction correlator, I mean, can be viewed and be as a sign of correlations of the one-body eigenstates. Now, so what physicists propose is, I mean, that in a many-body situation, I mean, what should decay our likewise correlations. Now, how do you measure, I mean, many-body correlations, where one of the simplest quantities in which you can look at, I mean, is you take local observables. Now, what are local observables? In the framework of quantum mechanics, a local observable, and it would be just a bounded operator on my Hilbert space, which models the individual quantum object, which in this case is a spin 1 half. So it's a bounded operator on C2. And then I can extend by tensoring once these local observables and we turn the full Hilbert space by positioning them at a side x and positioning b, for example, at a side y. OK. Then I can take, I mean, the quantum time evolution of these observables. Now, so far, we learned about quantum time evolution in the Schrodinger picture, where we time-evolved wave packets and there is a dual formulation of quantum time evolution, which is keeping the states fixed, but time-evolving the observables. And this time evolution, the quantum time evolution for observables, then we would be the so-called Heisenberg dynamics, where the full Hamiltonian gives rise to, at least in a finite dimension situation, where we are admitted to a unitary operator on our full Hilbert space. And you just unitary evolve your local observables. I mean, that's the Heisenberg time evolution. And then, so for example, this observable A, it could measure whether the spin points in the three direction. What observable would we take? Well, we would take exactly, oops, where is it? We would take, there's a minus sign, by the way, we would take exactly, I mean, this observable here, and extend it in the canonical way, I mean, to our Hilbert space. So you would ask, I mean, is the spin at x pointing in the three direction by considering this observable A x is equal to sigma 3? And what is its correlation? For example, if we have a spin in the three direction, it would be pointing in another direction. So the time independent correlation, then we would be then just taking the expected value of the product of these observables minus the product of the expectation values. And if you now consider time evolution, I mean, you would time evolve in one of those quantum observables. For example, in the eighth observable, then of course, what will happen is that if you plug in sigma 3 here at location x, then the time evolution will create a non-local observable. Because it's a horribly complicated spin chain, where if you apply it to a local observable, then even after an infinitesimally small time, and what this beast will represent, it will represent a whole series of spin observables around the location x. OK? Now, many body localization would be the phenomena that when you take these observables, spatially far apart, that even if you wait an arbitrarily long time, the quantum time evolution, and we somehow cannot really reach up there, and the correlations, we would still decay exponentially in time. OK? Now, when I write expectation values, I mean, what kind of expectation values could I consider? Well, I could consider expectation values in any quantum state. So the quantum expectation value would be a trace, in finite dimensions, similar with a density matrix. So a density matrix is a non-negative operator, which is normalized to be trace 1. OK? And this could, for example, be just a pure many body eigenstate. Just be some state normalized from my Hilbert space, and be of n 2L plus 1 spins. OK? It could also be a thermostat, and that's what you would look at in quantum statistical mechanics, where rho would be something like e to the minus inverse temperature h divided by, I have to normalize it, trace e to the minus beta h. OK, so that would be a thermostat. Now, what physicists claim, I mean, irrespective of whether you look at a thermostat, or a many body eigenstate, in these systems, at least at sufficiently large disorder, you will have the spatial decay of these correlations. Now, by the way, since I think you've probably never seen somehow these systems, let me ask you, in which situation would we immediately and we have even a total de-correlation of these quantities? Hm? Zero coupling? Zero coupling. Not zero Hamiltonian. I can still take, I mean, the magnetic field. But if this spins in my spin chain, so I have a spin, and we somehow minus l, and we have a spin, and we somehow at l and everywhere in between, if there's no coupling in between those spins, then the Hamiltonian itself, so if the j-alpha's are all equal to zero, then the Heisenberg dynamics of a local observable would be just equal to the effective dynamics on the individual component tensor 1. So what is h? Well, h in this case, or that is hx, would be just lambda, or minus lambda over 2 times sigma 3 omega of x. So in this case, I mean, you would have somehow a trivial dynamics in the sense, I mean, that local, the time evolution of local observable stay local. And also, I mean, if you take expectation values in, for example, I mean, the many body eigenstate or in the thermostat, these states, I mean, they would be product states, because they inherit, I mean, the product nature of our Hamiltonian in this case. Nothing is coupled. So if everything is a product, I mean, then, of course, I mean, the expectation value of this product is just equal to this guy. So unless x is equal to y, you just have plain zero here. OK. So in some sense, I mean, what's the proposal, I mean, which is behind, I mean, looking at these correlations is you're measuring to which degree the Hamiltonian and its eigenstate retain, I mean, the product nature, which would be present if all interactions are switched off. So localization is not any more localization in a position basis, but now it's the question of, I mean, whether the eigenstates of this many body Hamiltonian have actually are close to being a product state. That is, I mean, what this definition or this requirement asks you to prove. Now, here's even a more physical quantity, I mean, where we are now dealing with fancy things and we like entanglement. OK. So another quantity, which is quite popular in me and looking at this in this community, is the so-called bipartite entanglement entropy of many body eigenstates psi. So if you take just an arbitrary vector psi in this tensor product-herbert space, I mean, you can perform the following question. Calculate the reduced state of this one-dimensional projection. Now, what do I mean by reduced state? OK. So think about, I mean, this spin chain as being decomposed in a bipartite way of just carving out, we say, an interval of sites u and the complement of it. That's u complement. Now, how do you get the reduced state? What I mean by the symbol trace u complement of rho. So this is now an operator on the Hilbert space of the tensor product associated with this segment. OK. And as such, I can describe it by giving you the matrix elements in my favorite orthonormal basis of this Hilbert space. Now, my favorite orthonormal basis in this Hilbert space in me would be a tensor product of this orthonormal basis in C2. OK. So what would be in me somehow? So let me apply it in me to a state in me, which is a combination of, let me call them little sigmas and excuse for the second MBA, my physicist notation. So sigma would be a vector of sort of ups and downs, which is just a shorthand in me for a tensor product basis composed of these basic vectors here. Now, what is this reduced state? Well, by definition, it's the sum over where there's the tensor basis of the full Hilbert space. The full Hilbert space in me you can describe in me by tensoring omega's. So this would be sites in u and omega in me would be a basis in u complement. So we take the collection of u and u complement and you get a tensor product basis in the full Hilbert space. And what we do is we just take the state and calculate in me something in here which looks like in me somehow the trace. But the trace is only taken over spin configurations in u complement. So this is in me somehow what the symbol stands for. Now, these partial traces, something in which your linear algebra teacher should have taught you, but probably didn't. They have, I mean, this is really in me just again, in me pure linear algebra. They have a variety of properties. If you take a state in me, namely a row which is non-negative in trace one, then the partial trace in me would be a state on the reduced Hilbert space. So it will be still non-negative. It will still have trace one. That's in me somehow easy to see that it has trace one just take in me tau equal to sigma and sum over it. And then you get the trace of the full thing. And that was in me somehow set to one. So states in me are mapped to states and this operation of taking in the reduced state. And now in me you can look at something in me which as probabilists in me you're well familiar with, namely at entropies of these states. Now entropy in me somehow is usually you take the von Neumann entropy in me, but let me be a bit more fancy and introduce so-called reny entropies. Now what are reny entropies? Well, for reny entropies you take the state row u. This is now a non-negative matrix on this Hilbert space here. You take the alpha's power in me that's well-defined to the spectral representation. Alpha is being positive. You take the trace of this creature. You take the logarithm and you multiply by one over one minus alpha. That is called in me the reny entropy. Now it's a not entirely straightforward exercise in me to prove in me that for any alpha being bigger than one in me these quantities are actually non-negative. And so there are what at least physicists in me would perceive as the right sign of the entropy, in me namely the positive sign. Now you might in me somehow get nervous in me about alpha being equal to one. Now here's a little exercise to you I claim in me that when alpha goes to one this quantity actually has a limit. And indeed in me the limit is the well-known for Neumann entropy. So S1 of u of the state psi that is in fact equal to minus trace rho log rho. And it's the reduced state. OK. Now remember in me somehow what we're after in our first definition in me was a statement that these many body eigenstates of the Hamiltonian are actually close to being product states. Now what happens if I take a state rho or a state psi which is a product state and I calculate the reduced state. Here's a little exercise for you. So if pi p is equal to p of p u tensor p of c u complement then the reduced state on u would be just p of psi u. So if the state psi, so this corresponds to the state psi being equal to a tensor product of a state on u and u complement then the reduced state is nothing else in me but the rank 1 projection onto that component which lives on u. Now what is the entropy of your pure state? OK. Let's take this little exercise and you have a rank 1 projection. You take the alpha's power. What is this? That is equal to the same rank 1 projection by the spectral theorem. You take the trace, the trace of a rank 1 projection that's normalized to 1. The logarithm of 1 is 0, so the entropy is 0. So if the state would be a tensor product state then you would measure by considering the rainy entropy on any segment, if you would measure the rainy entropy of the reduced state you would always get 0. Now what do physicists propose? Well it cannot be quite, our eigenstates of the system cannot be tensor products because we couple the spins and so the eigenstates and we won't be uncoupled. However, it is reasonable to expect that they are close to being that. And one way to measure closeness, which is different in me then measuring these correlations, is measuring actually the entropy of the reduced state. Now if the entropy is positive and if it doesn't scale with the volume of the segment on which I consider the reduced state, but rather it only scales as the boundary of the volume of this segment, which in one dimension would be just two sides, then I'll call me somehow the system of satisfying an area law, an area law in one dimension and we would be just saying that the rainy entropy of the reduced state in fact stays bounded. So it's not anymore 0, but it stays bounded and it doesn't increase if we were the system size. That is another way of measuring correlations, in this case quantum correlations which are present in these many body eigenstates. This is an equivalent or close to saying that if we started the equilibrium, this entropy is the sum of the entropy. Yes, but being a product, if the Hamiltonian would factorize, then of course the entropy of the total system would be the sum of the entropy of the individual systems. However, here we don't have a complete factorization and we want to say, I should say somehow, what I'm measuring here, and that's important, I'm measuring in the entropy of a pure state, the reduced entropy of a pure state. So the full entropy of psi on the big system would be 0. So that's one of the wonders of quantum mechanics. You can have a state on a large system which has entropy 0, but then when I take the entropy of the reduced system, it is actually arbitrarily large. That is the essence of what is called the Alice Bob Bell inequality phenomena. That the entropy of a state on a segment can actually be much larger than the entropy of the state on the full system. That's one of the essence of the wonders of quantum mechanics. It's a way of measuring entanglement in this quantum system that's done by these rainy entropies. Now, what physicists and me have also done in me, they are of course a bit more ambitious than me than just considering maybe the static area long. They consider somehow time evolution. Now, time evolution of eigenstates, and that's boring because if you take an eigenstate, then the time evolution will be stationary. But you could perform maybe the following experiment, and that's done numerically in quite a few papers. One of them I featured here, which is this new graph. You could take a state which is not an eigenstate of the system in me, but rather take a product state. So some state on you, some other state can be on the complement. So the state in me, which is exactly as here on the blackboard, consider the time evolution of the full system on that state. Now, of course in me, since our time evolution is generated by a Hamiltonian which couples the interior in me with the exterior, on the time evolution will be the product nature of this state will be destroyed. And therefore in me, you expect that the entropy of the reduced state at time t is not equal to 0. And typically in me, this would grow in sort of typical system. This would grow in me actually fast and be with t in a polynomial fashion. But in a random system, in me, people measure numerically a very slow grow in me, which is proportional to the log of t. And that is in me what you can see in me somehow in the view graph. In the view graph, what you see is exactly in me the rainy entropy corresponding to different values of alpha and actually different values of the initial state. And you see in me somehow this logarithmic growth in me of the rainy entropy. Now, in the bottom here, what is depicted in me is exactly in me this quantity here. But for the xx chain in me, where we switch off the coupling in the three direction. And there you can see in me that it doesn't even grow logarithmically in me, but it saturates in me at a constant volume. So the task is in me, can you understand this? Now, unfortunately in me, this turns out in me to be a sort of a complicated question. And they are only very selected mathematically. Let me put it as such in me, some of these works in me are not completely mathematically rigorous, but some of them in me are completely mathematically rigorous. Now, what have mathematicians done in me so far in me? Well, as you will see in a second, these quantum spin systems, they can be actually mapped in me to particle systems. In me and people looked quite for quite a while actually at generalizing in me these proofs of localization. And we form one particle in me to just a fixed arbitrarily high number of particles. And for one particle in me, some of these are these old works. And starting in me somehow in the late 2000s, we have in me somehow came up with a scheme of doing an induction of particle number. Unfortunately, we cannot reach the limit of arbitrary number of particles in an infinite system because the localization length in this process diverged. Now, Imbry in me somehow gave a sketch of a proof of decay of correlation, something in me which I flashed in me on the previous transparency and be in certain one-dimensional spin chains in me this proof in me is not entirely rigorous. In me, it is a sketch of an argument in me which might or might not work. There have been other works in me in another formulation, but as you will see in me somehow in me this is close, Mastro Pietro in me looked at ground state localization and we saw what he took in me as Psy being the ground state in me of these systems. In me now, not spin systems, but fermions. In fact, a potential where, which is not random in me, but quasi-polyotic, it's the famous André Potential. And funnily, he could actually push through in me the decay of correlations in me at least for the ground state. There's a very nice work done here in Paris by Raphael Ducartes, which also contributed in me to localization proofs for weakly interacting from fermions, but not in me for the full system, but in so-called Hartree-Fox theory. And what I will talk about in me is now works in me related in me to this body of papers in me where we looked at integrable models which is the X-Y chain as I will explain in a second, and non-integrrable models in a particular phase of this non-integrrable model, which is the X-X Z-chain. Okay, now let me go now in me to this X-X Z-chain and describe to you why this quantum spin system is in fact equivalent to a particle system. Okay, now, instead of representing the spins on my line in my favorite basis, which is the eigenbasis of sigma three, the joint eigenbasis of all sigma threes, which I, as I said, in me somehow indicated in me by being spin up, I mean that's this eigenvector and spin down, I mean that's this eigenvector. So the tensor product eigenbasis in me can be represented in me by this string in me of spin up and spin down. I can now tell you another story in me for this string, namely let me mark in me the position of the down spins. Okay, I mark them in me in this very particular spin configuration in red. And I will now call the position of the down spins as position of particles. Okay, so in this picture in me I have a spin chain of length 13 and I have seven spin downs. I mean that would correspond to seven particles. Now clearly in me somehow there's a one to one correspondence of marking where the down spins are and marking in me the positions of the particles. The positions of the particles are encoded in a vector X where of course the first particle in me has to be before, I mean the second particle before the third particle and so on. So the configuration space of these N particles which emerge in me from this mapping is nothing as in me but the ordered N tuples on this segment of the real line. Okay, now as it turns out in me somehow there's a unitary equivalence of the full Hilbert space to the direct sum of Hilbert spaces of N particles. I have to allow in me for no particle being present in me because that would correspond in me to a spin configurations where all spins are up. That is N equal to zero here. And of course I have to allow in me for all particles being spin down in me that would correspond in me to a particle configuration in me where N is equal to two L plus one and everything in between. So there's this natural unitary equivalence in me of particle configurations and spin configurations encoded in me by this picture. And I can ask how does the XXZ chain now look in this particle representation? Okay, now there's a little calculation in me which comes along with this and surprisingly once you're unitary transform in me the XXZ chain you get in me on each of those sectors of a fixed particle number, a Hamiltonian which looks as the usual Hamiltonian. Namely, you have an adjacency matrix in me which just models hopping of the particles in me by one step. That is the term in which came from the in plane the one and two interaction. Now there's a term which measures interactions of these hard core particles. Now the interaction is, I should probably and we put up this picture in me again, the interaction is most easily described in this picture. Namely, the value of the interaction on this configuration, it's a multiplication operator is just equal to half of the number of boundary terms in me where a spin down in me meets a spin up. Multiplied in me by our coupling in the z direction. So this is this interaction potential. So it's half of the number of cluster boundaries in X where the cluster is just a naturally defined in me through neighboring spin downs. So neighboring positions of particles. In this case, in me there would be three clusters in this configuration. Well, and then our random magnetic field in the z direction, it turns out to transform to a random potential for those particles. Okay, so the random potential on a configuration X is just the sum of random variables in me where the particles sit. Now, as you somehow see in me some of this is not exactly just the unitary transform of the XXZ chain. But as usual in me when you work with finite volume systems, if you have to specify some boundary conditions and convenient boundary conditions would be the so-called droplet conditions where I'm adding a term in me to the Hamiltonian which prevents the spin downs to be actually glued to the boundary of my system. Okay, that's what this term here does. And its strength in me would be taken in proportional and be to the coupling in the three direction that is convenient in me so that the unitary transform of this Hamiltonian really works out as being a random Schrodinger operator on this particle configurations which now have an interaction which is proportional to the number of cluster boundaries. And this works up to a constant but remember, we're not interested in me in somehow energy shift. Oh, data. Data was somehow the very beginning here. Data is the coupling strength in the three direction and we have my XXZ chain, okay? Wonderful. Okay, good. Now, it is a fact worth mentioning that in this unitary equivalence these operators here, they're blocked diagonal in the decomposition of the full Hilbert space into particle numbers. That is because our original XXZ Hamiltonian in fact commutes and we were the total spin in the three direction and the total spin in the three direction is just proportional to the particle number. So there is no term in this Hamiltonian. I didn't need to include a term in the Hamiltonian in which involves particle creation and particle annihilation. So these operators act individually on the sums, block decompositions. Now, what's still sort of the basic message here is that this XXZ chain is really unitary equivalent to an interacting random particle system. But it's a quantum system maybe because I'm now considering a many particle Schrodinger operator with an honest interaction as long as delta is strictly positive or non-zero. Okay. Now, these Hamiltonians and we actually have a special structure and it's worth switching off this interaction before a while and considering the case of the XXZ chain. Now, as it turns out, the case of the XXZ chain is integrable and you switch off the inter cluster interaction by switching off delta. Now, but remember somehow in this system in this particle system there's still some sort of an interaction of particles present because you have a hardcore condition namely the configuration space of particles is such that there can never be more than one particle on a lattice side. So you switch off the inter cluster interaction by switching off delta, but you don't switch off the complete interaction. So the system may think that the system is not entirely trivial. However, as it turns out it is and the remaining hardcore interaction may be removed by what is known in this community as a Jordan-Wigner transformation. So this Jordan-Wigner transformation works as follows that you can associate in me to the spin operators on lattice sides other operators C in the following way in that you start and you from say the left end of the spin chain and attached to the left end can be the so-called spin annihilation operator which is this combination of sigma one and sigma two and as you work in your way up along the spin chain and you multiply products of sigma three and be to the spin annihilation operator. So that's just the definition. Now the magic is in me that these sets of operators which you define in such a way in me they satisfy this, the canonical anti-commutation relations for the algebra of creation and annihilation operators for fermions on the lattice. That is to say in me the C anti-commutes with a C. So the curly brackets I should probably say this in here because so this is the anti-commutator. So this is AB plus BA. The C's and the anti-commutator with the C's they adjoints anti-commute and C and C dagger they anti-commute to a delta, to a conical delta. So they anti-commute at different lattice sides but they anti-commute at different lattice sides but don't anti-commute on the same lattice side. Now if you write out the full spin Hamiltonian in terms of in me these fermionic operators in me then as it turns out in me it's actually a quadratic form plus a constant and the quadratic form is something which you recognize in me from the first part of the course in me namely the matrix in me which is associated in me to the vectors of the creation and annihilation operators is in fact your good old one-dimensional Anderson model on this segment, okay? So what do we know about the Anderson model? Well we know in me that the eigenfunction correlator indicates exponentially and that there's dynamical localization. I didn't prove it completely in one dimension but at least at high disorder and we know how to prove this now. Now the task is in me for example to understand in me what the correlations in this model do. So what you can look at in me is the first challenge in me namely the decay of correlations for that and you take a local observable A, put it on lattice side X, you take a local observable B, put it on lattice side Y, time involved one of them, take just in me a normalized eigenfunction of the X-X chain and compute a mere correlations. Now because everything is random, of course this correlation is now also random, so it's useful in me to consider the expected value of these correlations and what we could prove in a paper in me with Robert Sims in 2016 is that indeed in me some of this dynamical localization of the one-dimensional Anderson model implies in me that all these local correlation functions they do decay and they do decay in the very strong uniform way that even if you take a supremum over all normalized eigenfunction along the supremum over T, the expected value still decays exponentially in the separation of these local observables. Now the proof in me is a bit lengthy in me but to the very core what you have to do is you have to express these time-dependent correlation functions of local observables in terms of quantities of your one particle and the one-dimensional Anderson model. Now how does that work? This is not entirely trivial in me because when you take a local observable like and we want to those spin variables of course when you re-express this in terms of your fermionic observables since this Jordan-Wigner transformation is something which is utterly non-local. Remember we multiplied and we sum our bunch of operators in order to get these fermionic annihilation operators here. These quantities when expressed in terms of the fermionic creation and annihilation operators will be highly non-local and therefore it won't be just a fermion correlation function. It will be a bunch of fermions multiplied together and this you have to control. Now as it turns out that we somehow in the system since this is a quadratic system everything can be expressed in terms of fashions of basic two-point correlation functions in the Anderson model so the only deal in me which you have to go through is to understand and we why a pfafion decays once the entry decay. And that is of course in general not true but these pfafions in itself in me they have a particular structure and that's what helped me to push through this result. Okay, so there's also a result in me by Abdul Rahman, Nachtgerle, Simms and Stolz and we have the same year in me which answered in me the second question namely the question about the entanglement entropy. But let me not somehow put this up. Now let us go back in me to the XXC Jain and me because this is now in me somehow a model which is non-integrable so it cannot be mapped to free fermions in me namely a fermionic Hamiltonian which is quadratic in the annihilation and creation operators but rather in me you have an honest interaction in me in this particle system not just a hardcore interaction in one dimension. Now every configuration in me in this spin system in this particle system in me can be decomposed in clusters. So for example in me the configuration in me which I had on my previous slide looked about in me like that. In this case in me you have three clusters and if you have n particles in me you can obtain only all configurations in me by just taking the union in me of the configuration which are consisting of exactly K clusters. These are denoted by CK. Now remember in me the interaction for K clusters is actually constant in me in this set in me namely the interaction potential is just multiplication in me by K on a K cluster configuration. So what we wanna understand is the spectral structure in me and the properties of eigenfunctions of this many particle Hamiltonian which I put up in me again on this slide. This is an adjacency operator which describes in me one step hopping of the individual particles. This is my cluster interaction in me and this is the random potential which is common in me to all of these particles. So this operator in me can be now sort of acting in me just on the Hilbert space of a fixed particle number n. Now let's talk in me for the second and me about what happens if one switches of the interact the random potential. Now once I switch off the random potential and with a spectral structure of this Hamiltonian is actually quite well understood. And in particular in me something in me which you can easily see is that if I go down in me to the subspace of L2 which consists of K clusters or more then the energy of this Hamiltonian is bounded by the number of clusters which you have. And in case delta is actually greater than one and that is what is called the easing phase of this Hamiltonian. So delta being positive increases or makes the spins more likely to be aligned in the three direction. Okay and delta being strictly bigger than one and we makes the system more and more easing like and that's why this is called the easing phase. And as you see somehow in the easing phase where delta is greater than one if you decompose in me your configuration space into clusters and we then if you have a high number of clusters in me this is energetically disfavored linearly in K. Okay now I brought you in me some of our numerical simulations and we're back dating back in me to a paper by Nachtagel and Starr where I just shamelessly copied in the view graph from. This structure of the spectrum in case we switch off the randomness. Okay in fact and where do we know all this from at least in the infinite volume limit if there's another magic tool and which is good to learn about which is the so-called beta ansatz with which you can actually compute the full many particle spectrum of this Hamiltonian. Now in finite volume and with this beta ansatz and we as you probably know and it doesn't work quite as well and that's why one needs to resort and we term a bit of numerics and that was done in this paper and here's the view graph of how the spectrum looks in the easing phase. Now what should we take away from this view graph? Well if I fix the number of particles then there's a band for all n separated in me from the rest of the system which corresponds to just having one macroscopic cluster of size n. And because remember I mean we have a bound on energy which tells you in me that energetically all eigenstates in me which would contribute which come in me from two or more clustered configurations are energetically disfavored in me by this factor k two times delta minus one. And there you see in me somehow the clear energy gap in me which separates a configuration in me of just one cluster from the configuration from the configurations or from the eigenstates in me which intuitively correspond in me to more clustered configurations. Okay. Now let me stress in me that in me this band although in me in this view graph it looks like in me that there's just a single eigenstate and me that is in fact not true there's a full band of eigenstates in me so they do have if we resolve in me this band and be a higher multiplicity. Okay. They're actually formulas in me for how large in me this droplet band in me can be in terms of the interaction in the z direction. Okay but that's in me sort of irrelevant in me what is most relevant for us in me is that you have this interval of states in me which is clearly separated in me from the rest of the spectrum in me where the so-called droplet states of this Hamiltonian live. Now this is the states in me which I want to focus on. So intuitively you can think of these states as being composed of mainly as in the position representation in me of states in me which are which are living in me on just purely clustered configurations. Bear in mind in me that of course in me this is just a picture of me because this is a quantum Hamiltonian in me the presence of the hopping term requires in me that these product eigenstates are of course not exact eigenstates of the Hamiltonian. So this is just a picture of me what I'm presenting here. Now, so there's a bit of a cheat in me which I want to do and I admit it openly but that's the best we can do at the moment. So what I want to do now and what I want to focus is a system where I'm adding to this spectrum here a random term in me but with a particular kind of randomness in me namely that the random potential is in fact non-negative and compactly supported. Now what happens in me somehow in me to the spectrum intuitively if I add a non-negative random potential? Well, see everything is shifted up but the good thing is in me that the clustered configurations they still remain energetically separated from the configurations in me which have two clusters or more, okay? Because if I add a non-negative term in the Hamiltonian in me the energy estimate on the Hamiltonian projected in me to two or more clusters then we will just get worse. But nevertheless in me since I allow in me the random potential in me to take arbitrarily small values I still have states in me below this threshold. Because the random potential in me since the omega's and we can take arbitrarily small values the random potential can have arbitrarily large voids in me where I can accommodate a macroscopic droplet. So the bottom of the spectrum is certainly one of the things in me to bear in mind is that the bottom of the spectrum is certainly below above in me the one cluster threshold in me so above in me what we get for the non-random system. Where you add in me the value of the random potential at least in me to a minimum value the minimum of the overall clustered configurations. Now but that has an important consequence in me and that's really in me where our cheat is. That's why the system in me somehow in me becomes in me very much analyzable. Now here's an exercise for the probability audience. When you take in me the value of the random potential overall clustered configurations what kind of values in me can be min take? The omegas in me are supported in here let's suppose this is an equity distribution. Well as I said in me some of the minimum value in me will be zero right because all of these random barriers could potentially be very close in me to zero. However what is the typical value of the sum? These are IID random variables and we have hardcore particles so the XIs are all different. What is the typical value of the sum? Have we woken up? Wait it's order n right? Because it's a sum of n independent random variables in me they have a mean in me which is non-negative and the typical value in me will be n times in me that mean. Okay so the typical value in me of this guy will be of order n. So let me switch back in me to this view graph. So if I add in me some how a value of order n in me to states in me in the droplet band and we where we end up in me is somewhere up here in me on top of the mountain. All right so that's the typical value but there will be large deviations in this random potential. We have these large voids where the value of the minimum we are on clustered configurations and we will be order one or smaller. So I will have states in me filling up this gap here. However these states and we are large deviations of this random potential and since they are large deviations I can actually say something about the local density the behavior of the local density of states in an interval which is of size order one. So what is the behavior of the local density of states that's a quantity in me which for one particle we will have talked about yesterday. So that's the spectral projection now of this many particle Hamiltonian taken in the position representation of the particles in its expected value. And I claim in me that this is exponentially suppressed and before any interval of order one. Why is that true? I have a bit of a problem with the time but let me nevertheless in this amount give you an idea of the truth. How would you do this? Well see in me one cool method in me of estimating spectral projections is at the bottom of the spectrum is to actually employ semi-group methods. See in me by the spectral representation in me this is nothing else but the indicator function on I of the spectrum measure associated with XDE, right? But the indicator function that's bounded from above by an exponential well an exponential that we somehow modified and we buy e to the t times the supremum times this value here, right? So this can be bounded from above and we buy e to the t times the supremum times we now the integral with respect to the spectral measure of e to the minus TE but that is nothing else if you again use the spectral theorem the semi-group associated with this Hamiltonian. Now, but you now have a Schrodinger's semi-group what do we do for Schrodinger's semi-groups? Well we employ the Feynman-Katz formula. So what does the Feynman-Katz formula do? The Feynman-Katz formula and we now okay for the lattice but who cares represents in me somehow this matrix element of the Schrodinger's semi-group in terms of a Brownian motion actually a Brownian bridge and we which so these are Brownian path and we which started x and really returned to x of e to the random the potential the total potential now the total potential is a sum of only two terms so there's the random potential and there's the interaction potential okay so that's roughly in the form of me of this probabilistic representation of my semi-group and now one of the nice things which you can do is you can actually estimate I mean this is now a conventional integral and you can now estimate things since w is non-negative we can just throw it out right we want an upper bound and what do we want now let me not throw it out for the moment actually it's a bit of a silly idea not throw it out but what we want to do is we want to compute an expected value on the whole thing so take expectation values on the whole thing and we now take expectation value and be on just the random potential now this is still gonna be sort of hard to compute however I mean there's a neat trick in me in the Feynman-Kartz representation which is worth remembering namely that I mean you have an average up here which you can pull down and be using Jensen's inequality okay so by means of Jensen's inequality we can pull down the average the time average there's a t here and then we have this but now you just evaluate the random potential and be at one configuration so this is nothing else in me but the sum of the random variables associated and be to the lattice side and we where this Brownian motion passes at time s and this expectation then we can be computed it's just an expectation value of e to a sum of independent random quantities that can be done and of course in me somehow in me this is of the order of t times n okay and now you take in me somehow just some value of t and you see in me some other the whole thing is bounded and we buy a positive constant and be times n so that's the trick in me which you can employ in order to see in me that the local density of states of this many particle system is in fact exponentially suppressed for large number of particles that is nothing surprising as I said in me because when you think about and we what's the typical value of the random potential it's ordering now what quantity are we looking at surprise, surprise and we're looking at exactly the quantity in me which you wanna look at in me in the one particle setup and we namely look at the many particle eigenfunction correlator as you take configurations of particles you take a spectral interval and you just sum over the modularity of the spectral projections and just perfectly generalizes what we've seen in me now previous lectures and of course this quantity and we just buy means of a Cauchy Schwarz which you will also have on your exercise sheet is bounded by, whoops, that's actually not quite true in me it's bounded in expectation value it would be bounded by the x, x matrix element and otherwise you would have to put a square root at the end of the same quantity in me with y here and therefore we want to take an expected value of this quantity it's typically exponentially smaller in me by the estimate in me which we've just proven here with the help of the Feynman-Cartz formula. Now let me flash with some of what we can prove in me for this quantity we cannot only prove in me that it's exponentially bounded in me this we've seen but we can also prove in me that it decays in the distance of this particle configuration now the decay in this distance is encoded in me in terms of a function f which at first sight looks horrendous but it's actually a very easy function and so let me just talk let me just tell you somehow what this function f is in particular situations now if x and y are clustered configurations and we so beast some b of that form which is just one cluster considering four particles and you wanna know in me what's the decay of this particular cluster to that particular cluster in green then the decay rate in me which we predict is in fact given in me by the distance of these two clusters and the distance of the clusters is measured in terms of the distance of the locations of the first particle. So this is the decay which we get for clustered configurations with this horrendous formula. Now what do you get in me for general configurations? Here is a view graph of me of what this sort of signals namely if you're not on the manifolds of clustered configurations so think of me some other space of all configurations of particles and we as containing a manifold of clustered configurations and if you're not on the manifold of clustered configurations and what f does it actually gives you the distance in me which requires you to take the shortest path in me to the clustered manifold and then just take an underground line in me to the point in me which connects to the shortest path in me to the clustered configuration from y. So the distance of x to y in me would be symbolically in me something of that sort where the distance of the clustered manifold is in fact just the full L1 distance of particles. Remember in me the distance of particles in me one of what was contained in me on one of my previous slides in me that is in fact just naturally measured by the sum of distances of all the particles contained in the configuration. So of the clustered manifold and you get actually a much stronger decay in me than on the clustered manifold. Now let me in the remaining three minutes and we talk a bit about in me somehow what this entails and how this is done. Now this is a very detailed information on how the eigenstates of the many particle system at least in the regime of energies where we can prove such an estimate in me and that's the regime of energies which is basically in me below in me the band of two clustered configurations in the non-random system. Okay, so it's energies in me somehow below in me the threshold of four delta minus 12. It's a bit below that in me but not much below that. We can prove in me that the eigenfunctions have this particular structure in me that most of the eigenfunctions live on the clustered manifold and even on the clustered manifold the clustered configurations are stuck in terms of the natural distance of two clusters. And once they live on the unclustered manifold or on the unclustered part of the configuration space everything is exponentially suppressed towards the clustered configurations. So typical many body eigenstates in me they will have a large overlap in me with clusters and the clusters themselves in me will be localized in me that's the essence of this theorem. Now in typically in me in many particle systems everyone does not have in me a such a detailed information of how many particle eigenstates look like in me because you cannot in me just calculate things. So typically in me what people do and what they look at numerically are reduced quantities, right? It's like, you know, as a probabilist if you wanna keep track of probability distributions on a thousand parameters that's something in which you sometimes don't want to keep track of me if you're a statistician and we wanna keep track of him with just the parameters in me which you wanna focus on which might be three or so in me so you consider the reduced probability distribution and these core asking about these correlations in quantum system is a bit in me like asking about the reduced distribution. Now what we can do in me is take in me the reduced density matrix. So just focus in me on one particle in me configuration. So you take in me a position on the lattice psi a position on the lattice eta and you take your density operator and trace out over all particles aside from one. So you require in me that in the particle configuration X there's a particle like psi in the particle configuration Y there's a particle at eight that's called the reduced density matrix and using the previous expression on F one can actually prove that the reduced density matrix of any normalized eigenstate which contains n particles that decays exponentially in the distance of where you ask and we where these particles are and it even decays exponentially in the particle number. Okay, you now may wonder in me what is the normalized eigenstate corresponding to eigenvalue E and we will it is provable here in me that the spectrum is almost truly simple. So there is just one normalized eigenstate which you can associate and we turn. So eigen projection and we can associate and we to the state. Now this translates exactly into those spin correlation functions and in the paper and we which I mentioned at the beginning and we which also proves and we some of that and we they nicely illustrate and we how you can go and we from these quantities here and we turn the quantities and we which I flashed up on my slide and we namely that you ask about at least static spin-spin correlation functions in their exponential decay. Now, I think I have about two minutes and we for flashing the proof strategy. So see, I mean this looks like and we sort of infinitely remote and we from what I talked about in my previous two talks but it's actually not. What you do here and we is you actually combine the standard fractional moment analysis for localization for the natural object and we which screams at you in this problem. Namely, so let's talk about two particles. Let me give you a proof by picture. If this would be just two particles the configuration space and we would be X1, X2, and X2 and we would be bigger than X1 so we're living above and with the diagonal. Now the clustered configurations and the other ones that we would turn somehow close to the diagonal. It's the clustered manifold here. And what you essentially wanna prove is that the system of two particles which you can view only as just one particle with two coordinates now in that once you take in me somehow some configuration X that it decays to the other configuration Y. Now supposedly you have an excuse why if you go to from X to the clustered configurations things decay exponentially then you can think of this problem as somehow reducing to the case that X and Y are on the clustered manifold. But if X and Y are in the clustered manifold I mean it's really just one cluster that's like one particle moving, right? And so you try to prevent and be the movement of this monster particle which consists of n particles which moves like a train track. Now how did we learn to prove localization? Well once you're here and if I have an excuse Y decays here and decays here and we want some here then I can start a path expansion. Well this path expansion and we will somehow take me away and be from the clustered manifold. But if I have an excuse in me why somehow this doesn't matter that these are once I'm away and we from the clustered manifold I in any way decay the only thing in me which I have to prevent is in me that things move in such a way. But notice the following. I mean that one of those things in me which drove people mad in the analysis of many particle systems is that in fact the number of random variables in which you have in this system is very low in comparison in me to the number of lattice sites. In the one particle and the other model is if you consider a graph and you had that many random variables. Here if you have in me somehow two particles the random variable at that site will influence all configurations which are on that cross. So you get in me sort of awful correlations that we if you start any path expansion however if you start the path expansion in me in this direction you refresh your environment. So you can have me sort of start and we trying to integrate out of me in a similar way as what I showed you in the one particle situation and that's in fact in me what we do. So the proof is in me somehow does two step strategy to create an excuse in me why moving away in me from the cluster manifold and we doesn't matter. That is technically a bound in me which mathematical physicists we know at least in the one particle set up in me that's called a comm Thomas bound. A comm Thomas bound in general is a bound in me which tells you that well my greens function so in the one particle set up my greens function if I'm away from the spectrum it generically decays exponentially at least if the distance of Z to the spectrum of H remains bigger and bigger than delta. So that's a comm Thomas bound. These people let me know how to prove. Now here in me things are a bit more involved in me because in these comm Thomas bound if you look at them the way this is proven if you have a lattice in me which has a large number of neighbors and that's true if we are actually in an n particle configuration space the number of neighbors grows within then all of these bounds in me which you can obtain in me from the classical literature deteriorate you have to do something a bit different but decomposing in me the full configuration space in terms of these segments and we would contain at least K clusters and we can start an induction and prove nevertheless in me that the greens function even on the real axis once you restrict me to the Hilbert space of two or more clusters this is exponentially decaying in the L1 distance even on the real axis why even on the real axis because you know in me that this Hamiltonian and we don't have spectrum there so that's why I'm calling this a comm Thomas bound so this prevents in me somehow the expansions and we're moving too far away in me from the clustered manifold and on the clustered manifold I have an effective one particle problem and this I can deal in me with by the type of methods and we would try talked about in my previous two talks now do you go in me in this situation in me from decay of greens function to decay of the eigenfunction correlators and that's something in which Michael Eisenman and I and we learned to do in me somehow in our papers and we are in particle localization okay I think I should stop here and we are probably not talk about how to prove the other result in me namely that there's an area law for the system and in fact the time dependent entanglement entropy is uniformly bounded in a very strong way you can even calculate exponential moments in me and they are uniformly bounded and so this is a regime in me where everything in which the physicists so far looked at me in the XXZ chain is quite different in me to what we are looking at in me and that's because mainly we don't have any states available before the particle in me to live at okay maybe I should just flash in me the references and we to these papers and we that if you want to read more about this and there's a paper of me which just appeared this year in analogies Poincaré and we will indeed and with the proof of this droplet localization in the XXZ chain is written up thank you very much. Any questions? Is there any perspective in higher dimensional from those in chains? Okay so see I mean somehow basic okay basically I mean this this type of a system and we were for for the very reason I mean that the interaction in this particle system is attractive it forces things to cluster and it separates the states which are clustered and we from the non clustered configuration and therefore effectively reduces the problem and we to an effective one particle problem. You can think of as also as treating in higher dimensions then with a question of somehow what is a cluster and maybe you have to define clusters in a bit of a more relaxed way of saying that it's exactly and we sort of spins sitting next to each other but just being sort of close to each other and having a large overlap that could potentially be done. See I can also define and we sort of as an XXZ chain in the higher dimension and going in me to the regime and we where you have an attractive interaction and I believe in me this is potentially doable of treating in me this regime. However let me emphasize in me that this is not what you really wanna do as a physicist because this is a very special system we are in a regime and we where there are hardly any states the question effectively reduces in me to a one particle question for a compound particle which in quantum mechanics is sort of a bit hard to define but this is doable here. That's not what you really wanna ask about what you really wanna ask about is a regime and we where there are many states and nevertheless you have all this phenomena which I talked about namely decay of correlations decay of the entanglement entropies in a situation where the states are not rare. That is in some sense in me what physicists are predicting which would totally contradict any gotical hypothesis in statistical mechanics. So but that's extremely far away in me for that and presumably in the best shot which you have is looking at the paper by Imbrie and trying in me to make this rigorous. This is the scheme. It's a renormalization scheme which is much more based on the multi-scale analysis. Yes. All the results mentioned were for data? Yes and only for the droplet regime only at the very bottom of the so it's ground state localization in this very special system. For smaller than the smaller data? No, nothing works in me because see essentially in me as I said I set up a system in me which is ever so slightly cheating so that I can go back and be to a quasi one particle analysis. If I wanna do more, I mean if I wanna consider that is what frozen that I'll be for example where they do their numerics I would love to retreat and be the Heisenberg chain, right? I mean the Heisenberg chain and we would be data equal to one. That would be the Heisenberg point and we were the system and we sort of totally rotational invariant. Now let me warn you, it's not completely clear and I think every day are very good reasons to believe and we that many body localization is not true in higher dimensional interacting systems. It's probably a phenomenon which is bound to be to one dimensional systems. But that's not even clear and debated among the physicists. So I mean so this is really one of those situations where you're working on a front where you don't quite know what's true.