 Thanks for the invitation. It's really a pleasure to be here. So in my lecture, I would like to give you an introduction to the field of tensor networks, which is something which has emerged in the last 10, 15 years, out of also the field of quantum information where people try to look at complex quantum systems from a point of view of quantum information, from a point of view of their entanglement and trying to understand what entanglement theory can teach us about the structure of these systems and how we can use it to understand these systems better. So I will try to proceed slowly and also give a bit of a background in motivation. This is a school, so please feel free to interrupt me if you have questions and ask your questions, and I will try to address them as far as time permits. All right. OK, so as an introduction, what are we interested in here? So this is motivated from starting systems in, well, condensed matter physics, high energy physics, maybe also quantum chemistry, and so on. And in all these fields, what we have is, well, matter around us is built of many particles, lots of particles indeed, we have many particle systems. And these particles, well, they're quantum. And generally, between these particles, they're interactions, right? So we won't be able, in a general setting, to describe this by simply studying the quantum mechanics of a single particle. We will have to take into account all these particles together the way they interact, their joint quantum states. So it maybe becomes already clear that if we really want to address this full problem, the quantum correlations between the particles are essential, the entanglement between these particles will be essential. So of course, if we would consider a problem like that where we don't have any extra information, this is really a daunting task, right? If we have many particles, the number of degrees of freedom grows exponentially with a number of particles, I will say a bit more on that later. So it seems there is no structure at all in that. But of course, that's not true in real matter. There is structure, right? Particles which are close, bi will interact strongly. Particles which are distant from each other will interact very weakly. So there is some notion of locality, some notion of spatial structure. And it's really this interplay which kind of ultimately allows us to say something about how these kind of systems behave, which ultimately also allows us to say that the fact that there is some structure in these systems limits the type of entanglement they display. It makes our entanglement still very special, despite being some complex quantum entanglement. And this allows us, from a point of view of entanglement, to address these systems, to study these systems. Now, to be honest, for most of the matter around us, actually we don't really have to think so much about the quantum interactions. So the entanglement is actually not that important. So most of matter is actually described by what is called mean field theory. And mean field theory basically tells you that you kind of neglect entanglement. You only think of entanglement as something which is a corrective level on top of something which is not entangled or not entangled in a non-trivial way. So basically it's saying we look at an entangled set of particles, try to understand the physics there, and starting from that point, we start to correct it. That's not what we're interested in this talk. That's kind of old-style quantum matter physics, if you wish. As I said, it's vastly successful. It describes most of the systems we have around us. And maybe one reason is also that many of the systems around us are at high temperatures, compared to, well, interactions, strengths, and so on. And at high temperature states become quite mixed. And in that case, entanglement becomes less and less important the more mixed the state is. So that's not what we're interested in. So in my lecture, I will try to look at things which are usually termed quantum matter, or sometimes termed quantum matter. So these are systems where the quantum correlations play an essential role, which we can't understand by neglecting entanglement, by neglecting quantum correlations, and which are kind of opposed to what one might call conventional matter. So as I already said, the regime we're most interested in is a regime of low temperature. And the reason is essentially that, as I said, the higher we heat our system, the more mixed the state of the system gets. And the more mixed the state is, the less entangled it will be. So if you have a state at infinite temperature, a maximally mixed state, the state has no entanglement whatsoever. Just everything is completely randomly correlated. So we kind of would expect, naturally, that the lower we choose our temperatures, the more quantum the system will behave. So if you want to see the most interesting physics, we should look at very low temperatures, or maybe ideally the ground state. That's a low temperature. So the ground state should be some of the most quantum. I'll make this more specific later on. What did I write here? OK, so now if you want to study these systems, consisting of many quantum particles, the general framework, well, it could encompass many different types of systems, system in condensed matter, in quantum chemistry, in high energy physics. In general, we would assume we have a continuous space. We have electrons and nuclei interacting there. Could be extremely complex. In my talk, I will try to address, or to mainly focus on a simpler kind of system, which is also closer to quantum information theory, but which still displays many of the features we're interested in, especially those which make the quantum matter special. And these are quantum spin systems on a lattice. So what do I mean by that? Well, what I mean by that is that we are given some spin, and I will not spin by a dot. And when I say spin, I really just mean that it's a system with a finite number of levels. I'm not really thinking about the fact that the spin might transform in a specific way under rotations. So it's really more saying that this is a system which can take a certain number of levels, but it doesn't have anything like a fermionic statistics or anything like that. So the simplest case would be C2, which I could think of as a spin one-half, but it's really just a two-level system. So we have this quantum spin, and then we make some lattice out of that. So we could, say, have a one-dimensional chain of these quantum spin systems. And each of these guys is described by Haver space CD. So the total chain is described by taking the CD and tensoring it at n times. So we have a chain of length n. So if one of these, if one of our systems is described by a basis i, 1, say, 0 to d minus 1, then the total system has a basis given by i1, i2, and so on, where each of these i's, again, can take values from 0 to d minus 1. So what you can already see here is that if I look at the total number of possibilities of states my system can take, then each of these can take d possible values. This can take d values. This can take d values, and so on. The total dimension of this space is d to the n. So this indeed tells us that it's exponentially difficult to describe the system with an exponential number of parameters to describe a general superposition because we need a coefficient for each of these basis states. So a general state would be of the form psi is equal to sum over ci1, i2 to in, i1 to in. And that's a general coefficient described in this state. And this is really a 2 to the n dimensional vector. So in order to describe a general state in such a system, we would need to describe it to specify an exponential number of parameters. And this already tells us why it's hard to describe how such a state behaves. Because if there should be a macroscopic system, n will be a large number, right? For some way, I mean, you know, we have something on the order of 10 to the 23 particles per kind of decent sized volume. Even if that's one dimensional, that's still 10 to the 7, 10 to the 8 if we have a long chain. That's a huge number, especially given that everything goes exponentially in this number. So if you want to describe anything which would call a long chain, this number would be prohibitively big. So we have no way of really addressing the full state. And that's kind of the problem we're facing, right, that we're having this exponentially big Hilbert space. OK, I'll get back to this exponentially big Hilbert space a bit later on. But let me first talk a bit more about the kind of systems we consider. So that's one kind of system, this one d chains. We could also consider two dimensional lattices, for instance. These are also particularly interesting. So again, each of these systems would be a d-dimensional space. And I should still kind of specify what I mean by saying the system is on a d-dimensional lattice. What does that actually mean? Or why is this lattice relevant? And this lattice is relevant because, of course, I'm, well, what's the point of just saying I have a number of spins? Well, something must govern the physics of these spins. So there will be some interactions in this system. And that's a relevant thing. So I will have, say, some two-body interactions between these guys, some Hamiltonian Hij, which could be, for instance, a two-body interaction. And this could act between adjacent spins but also between more distant spins. But the important thing is that I would like to have some locality structure in the interactions, which basically means that, well, it could mean different things, but essentially it should mean that Hij decays with a distance. So for instance, the most radical case would be that interactions only act between nearest neighbors. And it's really the structure of these interactions, which sets the lattice structure, which sets the dimensionality of my system. It really comes from the structure of the interactions in the end. So maybe I should stress, I mean, you might say that looking at quantum spin systems is something very contrived because we know that the world around us is, first of all, continuous. And second, it consists of fermions. So in that sense, you might indeed argue that this is a very special kind of system. But it also turns out that these systems can emerge very naturally as effective theories at certain energy scales or in certain regimes. Say, if you only look at the magnetic properties of a material, because in many, well, I mean, quantum spinor systems do form lattices. So it's natural to get lattices once you look at condensed matter systems at low temperature. And in these lattices, often electrons will form isolated magnetic degrees of freedom. So you can, for instance, have very tightly bound electrons, which sit in some kind of very localized shell, like d-wave electrons or so. And they will provide a magnetic moment, which is really sitting at that lattice side, which then via some mechanism, some indirect mechanism, usually will interact with close bi-magnetic moments. And you can also very naturally get, say, two-dimensional structures or one-dimensional structures if you have some layered material where you have, say, magnetically interacting layers decoupled by some non-magnetic layer. So these things can actually show up and also in different type of dimensions, even without engineering them specifically. But of course, one can also engineer them. One could take, for instance, optical lattices and in the optical lattice, trap atoms and then have a certain number of levels of these atoms, provide this D-level system, and have them interact. Oh, we can consider many body interactions. I should have stressed this better. I just started to draw this, and I thought, after I draw it, I should call it two-body. But it could also be many-body. Well, let's call them few-body. I don't like many-body maybe so much. Well, by few, I basically just mean that the body-ness should not depend on the size of the system. You don't want interactions between all constituents. Now, on the other hand, in physics, interactions are always two-body. I mean anything which we get as many bodies against some effective theory. Now, spin systems are also effective theory, so it's fine to have interactions between several constituents. But it would basically only go to the order of perturbation theory in which you get this effective theory. I mean to get an interaction between really many particles, that's something where you have to go to a very high order of perturbation theories. That's a very weak effect, typically. Oh, sorry. Well, indeed, a typical case is that T is equal to 2. That's why I tend to write 2. But yeah, that's correct. Thanks. So before getting into the entanglement structure of many-body systems, let me maybe explain a bit what kind of phenomena one can find in these kind of systems, which makes them special, which makes them interesting, which we don't find in conventional matter. OK, that was sub-point one quantum many-body systems. Point two would be to say something about quantum matter. Quantum matter is things which don't behave like conventional matter. So what I should really first say is how conventional matter behaves. So as an example, let me consider a one-dimensional spin chain with a two-level system. So I'll spin one-half, if you wish. And as a Hamiltonian, I will take the Ising-Ferro magnet. To have an Ising-Ferro magnet, so I have spins which interact via an Ising interaction, which makes that they would like to point in parallel in the z direction, so either both up or both down adjacent spins. And I have a magnetic field H, which I would like them to align along the x-axis. And now I can ask, well, and we look at the low temperature physics, so let's look at the ground state physics. And now we can identify two different regimes. We can identify one regime where H is very small. So say 0, to go to that kind of simplest point. In that case, well, there will only be the Ising interaction. So the spins will like to align in parallel. So what I could have is I could have a state where all spins point up, or I could have a state where all spins point down. Both are valid states. If I go to the limit where the field is very large, then this term is dominating, and it forces all spins to align the plus x direction. So all spins align like that. So now these two cases are very different. In one case, there's a unique ground state. In the other case, there are two different ground states. So this one is unique. Let me rearrange this so I can put things on top of each other. So I'll try to put the two cases next to each other. In the case where H is much larger than 1, we have a unique ground state. Here in this case, we have two possible ground states. But of course, in principle, any superposition of them will also be a ground state, at least if H is exactly equal to 0. But approximately again, this will be an almost degenerate space, so I will have something degenerate. So indeed, if n goes to infinity, it will still be exactly degenerate. So the physics is obviously different. There must be a point where the system changes from having one ground state to having several ground states. And how can we understand this? And well, this is what Landau realized, that this is related to symmetries. So in some sense, it's a very deep and certainly very influential discovery that symmetry is what kind of governs how systems behave. And so what's the point of the symmetry? Well, this Hamiltonian is invariant under flipping all the spins. So what we have is that H commutes with applying sigma x to all spins. I mean, it's easy to see the sigma x commutes with sigma x. And these guys anticommute, but they always come in pairs, so they commute again. Also intuitively, this thing would like to align spins in parallel, but it doesn't care if they all point up or all down. So if you flip all spins, it's still the same interaction. And now the point is the following. So this ground state respects the symmetry. This one is different. If I have either this state here or this state here, all spins pointing up or all spins pointing down, these states don't respect the symmetry. They break the symmetry. So that's the first insight. We can classify these different type of phases, these different type of behaviors by looking at the symmetry of our interactions does the ground state or a randomly picked ground state, if you wish, does it respect the symmetry of the interactions or does it break the symmetry of the interactions? If it breaks it, there must be a twin because then I can act with a symmetry on it. I get a new ground state, but it will be a different one. So it's intimately related to the fact that there's a degeneracy that I break the symmetry. So by looking at how our ground state of a system behaves relative to the symmetries of the system, I can try to understand, to classify what the physics of the system is, how the physics behaves. And in particular, a very important property is that this allows me to measure something which is called an order parameter. Let me very briefly say what that is. So what I will do is I want to measure the amount of symmetry breaking. So I will take an operator which doesn't commute with that one. So I will take something like sigma z at some position i or say the average over all sigma z. Let me call this o. Now you can see if I measure the order parameter in this state here, it will have expectation value zero because this is a plus state. But if I take a state which points like that, I apply sigma z, I get a state which points in the opposite direction. So it will have overlap zero. In this case, it's different. Sigma z exactly measures how much of the spin is pointing up or down. So in this case, the order parameter will be plus one or minus one depending on which state I have. So this way what we have is that once we understand the symmetry, this tells us something about the ground space structure, but it also tells us how we can identify, how we can detect in which of these phases we are. So there is a way to phases, kind of conventional phases, are characterized by how a system behaves relative to the symmetry, right? Of the Hamiltonian. And in particular, the question is, does a ground state break the symmetry? The second thing is that once we understand the symmetry, this also allows us to see how to identify in which phase we are and how to label the different symmetry broken states. So then we can find an order parameter which first of all allows to detect the state, sorry, detect the phase and also to distinguish if you wish label the symmetry broken states. So in some sense what it tells me is that if I know that the average behavior of a single spin, I have a good understanding of what's going on overall. That's why we don't really have to look at the entanglement because looking at a local property of a system at the behavior of a single spin averaged over all spins tells me the essential features of such a system. Okay, so what does quantum matter then? Well, quantum matter describes systems which cannot be described in this framework so we can't understand the physics in this language. And this leads to a couple of interesting effects. I will just describe the effects so we kind of know where this is going. We'll see some example later on in one of the next lectures. But quantum matter systems, well, they're also given by some Hamiltonian. We look at some ground state and these ground states can exhibit different properties. So for instance, for instance we will have degenerate ground states which are locally indistinguishable for which there exists no local order parameter. And by locally I really mean it doesn't matter if you look at two sites, at 10 sites, at 100 sites, they will always look the same. You really will have to look at the global behavior of the system to be able to make any difference between the different ground states. So in particular this means there's no local order parameter, right? If there would be any local difference, there would be a way to detect it, yes? Well, in that case it's indeed zero. So the point is rather that in the ground state, many fold their different state. Well, let's do it a bit careful. I mean, first of all it means that in the ground state many fold their states which is zero or non-zero. If you pick a random state it will always be non-zero. Now if you ask what is physically reasonable, you will always find the states where this is maximal. I haven't talked about that. But kind of the idea is if you have a state of that kind, this is what we usually call a cat state, right? It's like a big microscopic symbol position and it's very fragile to perturbations because you see if you put a very small magnetic field, say, on each spin which points up, these states will get a vastly lower energy than these states immediately. So you will see that this two fold the general manifold and the very small perturbations that will immediately go in one direction or the other. That's essentially the effect of symmetry breaking. But this is indeed exactly linked to the fact that there's a local order parameter which distinguishes them because if you would add a global perturbation which locally everywhere puts this field but not with this normalization, you would get plus infinity here and minus infinity here. So you see that under fluctuations coming from the environment these states are extremely susceptible to perturbations. So in nature you indeed expect to find these where this is maximal. But on a mathematical level it's actually a pretty subtle question, the symmetry breaking. And to the best of my understanding it's not fully settled how the different notions of symmetry breaking and so on are related. It's very small field in the lake. Exactly, yeah, indeed. Yeah, I didn't want to define it too thoroughly. I really more wanted to give the idea. But I agree, I mean, doing this properly is actually quite subtle. Okay, so these kind of systems will have different grounds that should look completely identical which especially means there's no local order parameter. It also means that the robust to noise, any kind of physical noise, physical noise will also locally change parameters in the system in a fluctuating way. But if all these states are completely indistinguishable locally, it also means that any noise which couples locally will also not make any difference. Right, it's this thing that if there's a way to read out information about the system I will also perturb it but if there's no way to perturb it in a specific way I also cannot read out information. So it means that these systems are also robust to local noise which is why they could be useful, for instance, to store quantum information because quantum information, well, should be protected from local noise. So one could, for instance, use it to build a quantum memory. There are a number of more interesting features. So for instance, this ground-stated generacy will depend on how I choose my boundary conditions for the Hamiltonian. So for instance, if I choose it with periodic boundary conditions or open boundary conditions or maybe if I deform my lattice and put it on a sphere rather than on a torus, normal periodic boundary conditions, I will get different ground-stated generacies. So for instance, on a torus, the system would look different than on a sphere. On a torus I might have a, well, finite number of ground states, say four ground states and here one ground state. Again, this is something which is completely inconsistent with this explanation because the way I can label the different ground states is exactly by the possible values this order parameter can take. So that's a local property, right? The possible values the order parameter can take doesn't care about the global geometry topology of my system. So again, this is completely inconsistent with this local description. Oh, yeah, they can shift. They can shift, indeed. There are a number of more interesting properties relating the structure of excitations in these systems we will get there. But this kind of hints that these kind of systems are really something which lies completely outside this conventional framework, which is basically a framework where we neglect the entanglement in the system to leading order and try to get a description of the physics just by looking at local properties, right, at the local order parameter. This is something which does not apply here. This is really outside of the Landau paradigm of local symmetry breaking, conventional matter, if you wish. And the point that quantum effects are really important. So this is why this is termed quantum matter then. Exactly. And kind of the idea is that, well, there's no local order parameter, but there is some kind of ordering in the sense that this degeneracy means that some structure is emerging, right? If I keep all symmetries, there's no extra structure coming out of the system. I have a single state. If I have several states, there must be some way in which the system orders. And the idea is that one could think of this as some kind of ordering in the entanglement. Usually these kind of systems are also termed topological phases because global properties are relevant. It's a bit of a fuzzy, well, I think it is the etymology of topology in that case is fuzzy. But it kind of makes sense in the sense that not only local properties, but the global topology of the system matters. And well, that's the kind of things we would like to look at in the following. See how we're doing. Okay. Why is there an echo here? Okay, questions as of now? Okay, so just to remind you what we're looking at, we're looking at ground states. So as I said, I already mentioned this earlier, that we're looking at the low temperature physics because we expect to see most of the quantum effects there. The second thing is if we look at the ground state of a system, the typically turns out the low lying excited states can be kind of, at least the essential features can be derived from the ground states. So understanding the ground state also allows us to understand states above the ground state. Most quantum and they contain features of the excited states. So what we're given is that we have some Hamiltonian on some kind of lattice system, which, well, in the simplest case, we might take a nearest neighbor or rapidly decaying. But maybe for simplicity, let's think about nearest neighbors. I mean, you know, if they're rapidly decaying and you can always block a few sites and they get even more local. So in essence, they're basically just between nearest neighbors. And what we need for this Hamiltonian is that it's, well, local, most importantly. And what we will typically also want is, it's not super important, but it's helpful that it's gapped. So what does it mean? Gapped means if we plot the eigenvalue, the spectrum of the Hamiltonian, we will get a number of levels. And what we mean by gapped basically is that between the lowest lying state and the next one, there is some gap delta. So this rules out states which have a degenerate ground state, but that's actually not so much what we care about. What we really want is that we want to know what happens when we make our system bigger and bigger. If we make our system bigger and bigger, we will get more and more eigenvalues of H because the dimension of the space grows exponentially. And this gap might change. So really, this depends on the system size if we take some uniform system. And what we really want is that the first excited state, regardless of how big the system is, stays at some distance from the ground state. So what we really want is that delta N is larger than some finite gamma for all N. So we don't want that as we make our system bigger, it gets easier and easier to take it away from the ground state because that means that a very big system, and we want to look at very big systems, will be unstable at any temperature or under any kind of perturbation. So we can't really expect them to show stable physics. So if you want stable physics, we want, but it takes some finite amount of work, of energy, of effort to get the system out of its state. And that's why we want a gap to the excited state. Now that's not so important, it's more technical condition, which we show up later, so I thought I stated. Well, it could be anything, right? If it's been one-half, I guess you can write on the polybasis indeed, but it could be anything, right? It's, we don't want to assume much, I mean what we want to assume that the way the systems interact is in some local way, right? So distant particles interact very weakly and maybe we can neglect it. But this could also be a higher-level system, right? This could be a 10-level system in principle, which could have a more complex interaction. I guess in practice, there will be some symmetries in the interactions from the way you build it or you might even want it to build it in a symmetric way, but we will not require that in the first place. I mean, for many results, it can also be algebraic if it's fast enough, which is, I think, typically something like dimension plus one or so. You could say Coulomb is a bit critical, but usually Coulomb shielded, so on any effective level, the Coulomb interaction will decay much faster. You will not have a one-over-R decay, right? But probably more like a dipole-dipole interaction type decay or so. Should be short-range in some sense, exactly. Okay, so maybe the last 10 minutes, let me actually make a nice point to conclude. Okay, so starting from these things and what I said, what we would like to know is what makes these kind of, okay, so we'll be able to, it's called the Areola. And so the problem we're facing is kind of the following. On the one hand, we want to describe quantum matter, so we know that the matter we describe is not described by a simple product state ansatz. By neglecting the entanglement. So we know we need to include the entanglement. Now the problem is if we include the entanglement, we have this, well, I just erased it, this exponentially big Hilbert space, right? So we have a Hilbert space which is CD tensor N. So it's D to the N dimensional. So we have this huge Hilbert space and in this huge Hilbert space, we need to describe a system and that's completely hopeless, right? We just can't specify so many parameters for sufficiently large N, right? If you have a computer, even if these two, you can maybe go to N equal 25 or 30 or something like that and that's when basically there's no way to go further. Now on the other hand, what we also know is that our system is described by this Hamiltonian here, right? So if we actually ask how many parameters describe the ground state, one answer is whether Hamiltonian describes the ground state and the Hamiltonian doesn't have many parameters, right? So how many parameters does it have? Well, worst case, if anything can interact with anything here, two-body interactions N squared, right? But typically if it's local, we will say it has order of N terms. But even if it's N squared, it's three-body N cubed, it's this case much, much, much better in N than something like two to the N. So what we kind of see is that we would expect that the Hilbert space should have a very small region which we could call the physical corner and that's the kind of states we would like to be able to describe. Now it's a bit misleading to draw this picture, right? Because it's not a convex set or anything, but the idea is that it's a very small subset of this very big Hilbert space which is what we're actually interested in if we want to describe ground states or even low energy states of a system just by parameter counting, we know that, right? The problem, of course, is we could just specify immediately the Hamiltonian, but the Hamiltonian, it's very hard to extract information out of it. So we want a more succinct way to actually describe the wave function of the system. Maybe that's also a bit of a quantum information approach to look more at wave functions and interactions. So if we want to describe this kind of physical corner, we need a more succinct, a more explicit way of figuring out what makes these states special, these ground states or low-lying states of a system with local interactions. And while it turns indeed out that they're very special regarding their entanglement properties, and this is exactly what the allele tells us, so what is the question we're asking? Well, I guess you already all heard some things about entanglement, right? So if you want to look at an entanglement, at the entanglement in a system, we have a system consisting of two parts, A and B, and they're in a joint state psi AB. If I want to compute the entanglement between A and B, I have to trace out the B system, and look only at the reduced state of the A system. And then I have to compute the von Neumann entropy. Maybe Barbara will say more about that. Sorry? But well, at least. Okay, so if you compute this entropy here, this will be exactly a measure of the entanglement between the part A and the part B. So we can try to do this for a quantum many-body system. Let's take some quantum many-body system and ask exactly that how much entanglement is there in such a system. So let's take one of these lattice systems and we cut some region out of it. And this region we call A, and the rest of the system we call B. So this region, say, has a size L by L. And now, well, the total system is in some pure state psi AB. Just the total ground state of our system, right? Which lives in this huge space, in principle. So now we can compute the reduced state of A, and we can ask what is S of rho A? So now one thing people have looked at in quantum information is if I take a typical random state, so a random state in this exponentially big space, what is S of rho A? So for typical, so to say random state, it turns out that, well, what I assume is that the total system is very big, right? So L is much smaller than the system size. I'm looking at a relatively small part of a big system. Then it will turn out that S of rho A is basically as big as a volume of the system. So it's some constant, which I guess is log D, times L squared, which is a volume, really, of the whole thing. Minus some sub-leading correction. I'm not sure how it scales, presumably log L or so, but it's something which goes to zero very quickly, basically. It's something which disappears. So basically what this tells us is that pretty much all degrees of freedom in that region A are entangled with all degrees of freedom outside. All degrees of freedom. So it's a huge mess, right? The entanglement is completely spread out. So that certainly seems like a very messy situation, but we are, of course, interested not in typical states, but in very special states, which live in this very small corner, which are ground states of some local interaction. And there are actually something remarkable that happens if we look at these systems, if we study such systems. What one finds is that this only goes like some constant alpha times the length of the boundary of the system, plus some sub-leading terms. So the leading term here only scales like the boundary of that region and not like the volume of the region. And the same we see in one D, or actually in any dimension. Well, like in one D, for instance, it would be a constant plus something which goes to zero as our system gets bigger. But in both cases, the point is that the entanglement really only scales like the boundary of the system, not like the volume. And this is what is called the area law. I suspect because it comes from the gravity community, where you're actually in three-dimensional space most of the time. That's my impression. I agree, it's confusing because indeed in the picture, I draw the area as the volume. So it's really the surface area of a bulk, but my impression is it really comes from black hole and holographic principle where you actually have three dimensions. Exactly, so area law, which states that the entanglement in the ground states of gap systems scales like the boundary rather than the volume. Now, this is something which has been proven for one-dimensional systems rigorously by Hastings. And there's some improved version by Vizic, Vazirani, Landau, I think it's also on some of these papers where people get better bounds in terms of the gap. How does a gap relate to this constant here? So in 1D, we have proofs of that. In 2D, I don't think we know of any proper counter-example to that. So in 2D, it also seems to hold. So it's generally believed, like in all systems, one looks at to be a general statement. And what does it tell us? That kind of tells us that the message is that the entanglement is only sitting at the boundary between regions. So in some sense, it tells us the entanglement is local. The structure of the entanglement in the system, despite the fact that there is a lot of entanglement, the entanglement is local. And well, starting from this point that the entanglement in quantum money body systems in their ground states should be local, should be distributed in a local fashion, I will start constructing some kind of answer, some kind of approach to describe these wave functions in the next lecture. Any questions? In 2D, I'm not sure we have any counter-example. I mean, in 2D, we know that critical systems have an additive log correction. Well, in 1D, we also know they have an additive log correction, if you wish. But, okay, so the point is that for a gapless system in 1D, the entropy scale is like log L for a region of length L, which is a violation of the constant area law. In 2D, what we have is that it scales like L plus log L, which is not a violation of an upper bound which scales like a constant times L. So it might be that in 2D, even for gapless systems, it holds in general. Now, I guess from a proof point of view, the question is what is the upper bound? From an intuition point of view, I think what people actually want to know is how does it scale? And I'm not sure that quantum metaphysis would call it an area law if it's L plus log L in 2D. They might actually say that there's a logarithmic correction. They would only call it area law if the subleading term is one over L or something which doesn't go with L. That's my feeling. But otherwise, yeah, in 2D, it might work as out-gap. Are there more questions? The log behavior comes from... In 1D, you can see it from renormalization if you kind of say at every length scale the system behaves. If you have a critical system, right? It's scale invariant in 1D. You say at every length scale, it behaves the same. And then you say how much entanglement does this give? You have a fixed amount of entanglement. The next level, you have half the entanglement and the next, you're going to have half the entanglement and then you see you get a logarithmic divergence. You get one plus one half plus one fourth as well. Well, the area law, the proof works for... It's about quantum Hamiltonians, right? It's about entanglement. But I mean, the proof works for any one-dimensional Hamiltonian like Hamiltonian acting on a 1D chain which has a lower bound on the spectral gap as the system gets bigger. I mean, it's all about scaling behavior, these things. Say again? For 2D system... Well, for 2D, we don't have a proof, but we still see it. So that's a point. This is a general law in the sense of that's what you see when you look at systems, but it's only proven in 1D. So in 1D, it's proven for gap systems and we know it doesn't work for gapless systems. I mean, just three fermions in 1D will have a law correction from the Fermi surface, basically. Okay, I should say that in 2D, actually fermions also have a law correction, a multiplicative law correction. So for fermionic systems, also in 2D, it doesn't hold. For spin systems, I'm not sure. Yeah, please. Sure, talk to me.