 All right, welcome back everyone to the last lecture on tensor networks So in the beginning that we just super briefly review what we did the last few lectures just to remind you we started by talking about the Structure the nature of entanglement in many body systems and We learned about the area law for entanglement Which basically told us that in many body systems entanglement is distributed Distributed locally and starting from that point what we did is was to Come up with this ansatz of matrix product states. So MPS ansatz Which could be understood in different ways it could be understood as placing entangled pairs in our system And then applying some map such as a projection To single out some physical degrees of freedom starting from these auxiliary entanglement degrees of freedom it could also be understood as a way of expanding the Coefficient in the computational basis as a product of matrices or it could also be depicted graphically By using a graphical notation for tensors as a box with three legs so we could basically write our Expansion coefficient up here as some kind of network of tensors for instance with periodic boundary conditions and Based on that we saw that we can do efficient calculations with this kind of ansatz so So in particular what we saw is that if we wanted to compute things like normalization expectation values and so further We had to take two layers of the system and contract this network Putting some operator somewhere and maybe some other operator somewhere else when we went to the correlation function And then what we could do is we could cut this into slices Which we termed e and just by multiplying these e operators Which would be something which wouldn't scale unfavorably with the system size we could compute expectation values And well one thing which I would like to point out we have this picture here I'm well I don't think I will get there towards the end if yes I will explain in more detail but one thing you can see is that if your system is translation variant All these tensors are equal and you want to compute a correlation function between some operator X here and some operator Y here And you want to know how this scales with the separation of these operators L What you see that between these two guys says e appearing L or maybe L minus 1 depends on you define L times and Intuitively you would think that e kind of loses discards a fixed amount of information About one operator when it kind of passes it on to the other one So what you see is if you lose a fixed amount of information each time you multiply these operators The information about X will decay exponentially in that direction So what it means is that correlation function in these systems decay exponentially? One can phrase it more clearly by doing an eigenvalue decomposition of e. I don't want to do that because it's a little bit lengthy It's not very complicated, but a bit lengthy But a main message one can kind of already infer from that is if I lose a fixed amount of information in each of these ease Then what what this implies is that correlations decay exponentially so indeed if you know about many body physics You might know that well their their systems which are critical which usually have algebraic correlations They're scaling variant and these are the systems where the energy gap closes So the fact that the correlations decay exponentially certainly Consistent with the fact that these states describe ground states of gap Hamiltonians well because for gap Hamiltonians as actually a theorem saying that all correlations decay exponentially and Conversely for critical systems. We would not expect an exponential decay. Okay, so What I explained the last lecture was that matrix product states are a very useful answer for numerical simulations and indeed they're extremely powerful What I would like to look at today or to discuss today is a different application of matrix product states Namely using them as a way of coming up with solvable models with models where we can kind of construct a Hamiltonian and a wave function Which is its ground state which has exact matrix product form and starting from there We can try to understand the physics of these systems and maybe classify the behavior the way in which this quantum system behaves from this Ansatz that would be a sectional study of quantum phases with matrix product states and To start with let me get back to something. I already discussed like Yesterday morning. I think it was right name the AKLT model, but now in a bit more detail named after Affleck Kennedy Leib and Tazaki Okay, so so basically I will start by restating what I explained the other day So what we have is we take maximally in that we take entangled states. Oh our entangled states are slightly different They're singlets and and these systems here are now supposed to be spin one-half systems So there are two level systems But really with a notion of being a spin one-half system was having a well spin up or a spin down state and This state omega hat should be a singlet state Now you might wonder if that's a fundamental difference to this construction here And the answer is it's not because if I have a singlet state I can always understand the singlet state as A normal state which is 0 0 plus 1 1 and Then acting with some operation right if I have 0 0 plus 1 1 and then I act with something like sigma y or I sigma y tensor identity On it then I will get exactly a singlet state, right? Because it flips a spin and it adds a minus so I get 1 0 minus 0 1 So it means that up to the action of a sigma y. This is completely equivalent to the original construction I proposed But then if I have to act with a sigma y on this side here I Can equally well absorb the sigma y in the P, right? So all I have to do is to say okay. I could as well start so if I have this guy plus some map P I could as well take this guy and add some map P times Identity tensor I sigma y Because this will exactly put this I sigma y here and convert it into a singlet So it's really the same thing, right? I can just move this sigma y from one part of the construction to the other So we have this singlet here And well, it's a singlet. What does it mean? It means that whenever I act with any u in SU2 I have that u tensor u Applied to omega hat is omega hat So it's SU2 invariant which is not surprising because that's what the thing that is defined to be So that's part one of the construction part two is to say what is this map which we apply? So what is a map which we apply? Well, we have these two spin one-halves Now we want to apply a map and the point is if we have two spin one-halves, we know that we can Decompose them as a spin one component and a spin zero component So this is a singlet state and these are the three triplet states triplet Bell states for instance, which span the space is So now what we can do is we can choose a map which only keeps a spin one component So P would be the projection onto the spin being one Which is nothing but saying the sz equal plus one state corresponds to up up The sz equals zero state In the new basis corresponds to a singlet As sorry to a triplet and the minus one to both spins pointing down Which is exactly the basis of the spin one state, right? It's all but the singlet state So now if we do this construction what we can see so if we have a rotation in SU2 We can of course find some representation of this rotation on spin one, right? It's just a spin rotation and what we see in that case I mean you can explicitly go through it, but basically it's a way of defining what it means to Act with a rotation on a spin one if you know what it means to act on the spin one half namely So if I have some are you which is a spin one representation of that rotation you I Will see that rotating the spin one system is Nothing but doing the same rotation on the spin one half system. So what I'm saying is whether I rotate here or Whether I rotate here Doesn't make any difference. It's exactly the same just in one case I have to act with a spin one rotation while here I have to act with a spin one half rotation on both sides So this implies that the overall wave function is invariant under rotation, right? So this implies that the overall wave function so if I rotate all the spins and Psi is the wave function constructed from this P Well, then what we have is that we have r u tensor n and Well psi is p tensor n acting on omega hat tensor n Now what we know that each of these r u's Well satisfies this equation here So what we have is we have r u times p tensor n acting on omega hat tensor n and This again using the condition is nothing but p times u tensor u times n So overall we have p times n u tensor 2n Acting on omega hat tensor n But now using the other condition namely that the state is a singlet We find that the state is invariant, right? So what we find is that the omega hat is left invariant So this is just p times n acting on omega hat times n and this is just exactly our original state psi Ah, no, it doesn't really commute, right? I mean the equation is that right if it's on the left It's a spin one rotation and If I try to move the spin one rotation through the p so the p maps to spin one-halves on to one spin one So if I do a rotation on the two spin one-halves in the same way that's equivalent of doing the same rotation on the spin one But I mean same rotation means, you know representation of the element of the rotation group Say again. Yes Yes Oh what the tensor product, right? I mean what's what's written here is Right, what's written here is or here is something like p times u tensor u tensor product with p And so further, I mean that's supposed to be a tensor product here, right? Otherwise indeed this wouldn't make much sense Okay, so what do we learn from that we learn from that that this way function psi is invariant under spin rotations, right? So if you construct our wave function exactly that way with these entangled states and The circles we will obtain a rotation invariant wave function That's of course nice because in physics we like systems with high degrees of symmetry, but of course we would like maybe to have a bit more We might want to be interested in seeing if this wave function maybe also appears in a natural way as a ground state of some Interaction which also has the same symmetry So we would like to construct a Hamiltonian for this wave function Okay, so if you want to construct the Hamiltonian, so how will we proceed? So let's start by looking at just a few sites. I mean the Hamiltonian acts locally, right? So we we should start by understanding what local properties a system has which we might be able to single out using some Hamiltonian So we start by considering two sites. What do we have? We have a spin one-half particle here We have this entangled state of two spin one-half particles Which is overall a singlet, right? So the total state has actually spin zero Consist of two spin one-half, but if the singlet it's been zero and we have another spin one-half And now we apply this map, right? So we take this guy here and we take this guy here and We apply the projection onto their joint spin one space so we obtain a spin one particle and we obtain another spin one particle So now what can we say about this system? Okay, so so let's look at the at these two sites, right? Let's I mean this chain goes on here But we we don't want to think about it right something is happening there Which means that these states will have some value which they get from well the rest But let's not think too much about it. Let's say we just don't have any prior information on it So what can we say? Well if we look at these two states, right? That's two spin one particles So they live in a total space built by two spin one particles, so it's a spin one times a spin one So over all two spin ones decompose as either a spin zero component or a spin one component or a spin two component That's something which is well well known from spin theory So these are the possible values this can take Okay, so now let's look up here. So what can we say up here? Well up here we have four spin one halves on the one hand which actually decomposes in a very similar way But if we look more closely, we know more right what we know is that? Here the thing in the middle is actually a spin zero state So what we do have in fact up here is the left thing has a spin one half Then the two middle ones jointly have spin zero and the right one again has been one half So what this is in total is just the same as been one half times been one half Which is been zero plus been one so we start from something which has been zero or spin one We apply this projection. It's a projection. It doesn't change the spin. It removes certain spin sectors But it doesn't change it right so it must be the same afterwards. We start it in zero or one We end up in this space which can support zero or one or two But well we start it in zero or one so we cannot get this to right So this is forbidden So what we see is that over all in the system the spin two state on two adjacent sites cannot appear So now we can use this as a way of constructing a Hamiltonian We can we can we can do the following we can define a two-body Hamiltonian on two adjacent sites I and I plus one Which is a projector onto the joint spin two state right this joint space has been zero spin one or spin two It's a projector onto the spin two states So what does it mean it gives energy one to the spin two state and energy zero to the other states So if we now take this guy and we apply it to our wave function Psy this AKLT wave function. What does it mean? Well, we have seen that for the AKLT wave function We cannot have spin two So well we are in the kernel of the projector so we get zero right At the same time, of course, we know that this thing because it's a projector It's positive semi-definite right because well it has eigenvalue zero or one So now we can make a full Hamiltonian out of that by putting this term everywhere on some chain with periodic boundaries And well what we will find is of course exactly the same right acting with a full Hamiltonian on Psy We still get zero because each term individually gives zero and Of course if we sum up a set of positive semi-definite operators We again get a positive semi-definite operator. So the total Hamiltonian has only well zero or positive eigenvalues So what does it tell us well it tells us It tells us that Psy is a ground state of that Hamiltonian right because we know that this Hamilton has eigenvalues zero and larger And this is an eigenvector with eigenvalue zero, which is the smallest eigenvalue. So Psy is a ground state Okay, so now what it actually turns out is that I will not discuss this here, but one can prove the following thing namely First of all One can show that this Psy in this case is the only ground state of that Hamiltonian So it's a unique ground state and Second of all There's a gap above this ground state So what we have this way is that we really obtain a Hamiltonian Where we well we have a Hamiltonian, but we also understand the ground state wave function It has an exact matrix product form. So we really basically found some way of constructing a solvable model so we have we obtain a Some kind of solvable model namely a gap Hamiltonian Plus a ground state wave function. So one thing I should maybe add just very briefly is that I Constructed a very specific example So you might think it's indeed specific to this example that I can construct a Hamiltonian given a matrix product state But this is not the case what I really did in some sense is the following One thing I asked is how big is this space here? This space here is well capital D. Sorry small D physical dimension to the two-dimensional In this case the physical dimension is three for a spin one system So this is three three to the two-dimensional Then I asked how many states can I find in this space in this construction? Well, I said this state is completely fixed. So one degree of freedom. I have is here choosing two possible values and One degree of freedom. I have fear is choosing two possible values right and Then basically what I said is well, how many values can I obtain? Well, I can obtain D square equal to square equal for states in what well in a D square equal Three square equal nine-dimensional space and from that I said well I can only get four states in a nine-dimensional space five are missing I can define a Hamiltonian which gives a higher energy to those which are missing right? So I can define a Hamiltonian Which is a projector onto the missing ones now now it already sounds much more general all I need is that kind of This space is bigger than that space Now you might say okay, but this depends on the choice of these parameters. This must be smaller than that one But even that need not be the case because there's no no specific reason to consider two sites I could do the same for three sites or for some number L of sites, right? So if I do this for L sites, I get D to the power L here So in the general case what I would have I would have capital D squared states and It's always square because that's the size of the boundary the boundary is a fixed size independent of the length of the block So I would have capital D squared in a small D to the l-dimensional space And you can easily see that at some point of course this space will be bigger than that one if I increase L And since this goes exponentially it will happen rather quickly and then that number of sites You will always be able to construct such as such a Hamiltonian And it actually turns out that this will always under generic conditions have a unique ground state and be kept So in fact, this is a general Well, it depends what you start from right if all initially ingredients are rotation invariant say you said rotation invariant If all initially ingredients are rotation invariant what you get finally will also be rotation invariant Because you see what you're looking at is the the orthogonal complement of all the states you can generate with general boundary conditions So you could as well choose rotation invariant boundary conditions Well, not as vectors that they might transform like non-trivial irreps, but the projector will be rotation invariant So in the end the projector onto all states you get will be rotation invariant if if the object to use So this state here and these states here have any kind of symmetry in fact Then the same symmetry will also be seen in the space of states here Not an individual state at the space and this is just one minus a projector onto those states So it's all it also has that symmetry So indeed you can generally construct any kind of symmetric Hamiltonians by starting from a correspondingly symmetric wave function And it's indeed an important point that you can encode all these symmetries locally, right? It's really like what happened here. You want that that P has a nice behavior relative to the symmetry and I guess I erased it the singlet so the the bond you use also has a nice behavior Then the overall thing will have a nice behavior and the remarkable thing is that there's a theorem Which says it's the only way of getting symmetries if you want to get a symmetry you can always get it by encoding it locally Of course, you I mean obviously you can cheat around and make it look like it's not local But in fact there's always a way of getting it locally So you don't get any extra cases by considering weird ways maybe of rotating the symmetry It's always encoded in the P and in the omega individually Okay, so before discussing some of the features you get from that let me Briefly say where this comes from and why people who are very interested in this construction So so one thing if you just work out the semitonian for the AKLT case It's actually a nice exercise to try to think how can I re-express this projector in terms of spin operators? I mean you can do it brute force, but there's some kind of more elegant argument Based on well spin addition and things like that But in the end what you get is that this projector to the spin to subspace is of the following form It has a term which is a Heisenberg interaction It has a well you might argue comparatively small correction to the Heisenberg term and well a constant which doesn't really matter So this is kind of close to Heisenberg, right? And this was really the original motivation of Affleck, Kennedy, Lieben, Tassaki to construct this wave function Namely what is known as a Haldane conjecture So this would really give a rigorous proof or an example of a variant of the Haldane conjecture which is among the things I think for which Haldane actually got awarded the Nobel Prize and The thing is the following so what Haldane considered was let's look at a pure Heisenberg antiferromagnets and Well now there are two cases so the spin could be half integer or The spin could be integer So spin one half spin one spin three half and so on and for half integer speed There had been a result by Liebschultz and Mattis Where they show that these systems must be gapless in some sense either they must be critical so a whole bunch of low-lying excitations or They must be symmetry breaking so having at least a second degenerate state And what Haldane showed by mapping this to something which is known as a nonlinear sigma model He showed that in the integer case It's very different that integer and half integer behave very different and based on that he stated that in the integer case The Hamiltonian has a unique ground state with a gap above It was really mostly based on the rotation symmetry. So Affleck, Leap, Kennedy and Tazaki Took this model which had the same kind of symmetry properties and for that one They could really rigorously prove that this system has a unique ground state and a gap above Okay, so in the last 10 or 15 minutes I would like to briefly talk about some consequences of some kind of Special features one can see on this kind of AKT construction relating to kind of non-trivial Phases non-trivial behavior of systems and the symmetries in one dimension. So this would be about Edge modes and fractionalization so For one thing let's so let's again take this AKT chain and Let's ask if we look at a specific position and we cut the system in two halves at that position What can we say about the way in which the left and the right half is entangled? So the entanglement here Comes from a spin one half right it comes from this singlet which sits between these two spin one half degrees of freedom so What this kind of suggests is indeed that the way if I would act with a symmetry on my system with some rotation It's a spin one chain and I would ask what happens to the entanglement I'm being a bit way vague I will try to make it more concrete in a moment if if I act with the intact if I act with a rotation on my system I could ask what happens to the entanglement does it also transform in some way and Well, what you certainly sees if we act with some rotation here everywhere This will amount to some different rotation acting on the entangled states right, but this other So here I would act with a spin equal one Whereas the rotation would act like a spin one half on the entanglement and That's kind of curious in the sense that what we actually have is we have a physical spin one half But the entanglement of in the system behaves like a spin one half, right? So what we have is that we have a spin one chain But the entanglement Transforms as a spin one half So this the spin one particles in our which which build up our system Break up in some sense and to spin one half particles which are fractional particles in some sense Right spin one is the elementary building block of that thing I can't really create spin one half objects in the spin one chain, right? I could you know flip a spin I can create a spin one excitation I can't create spin one half So there's some kind of fractionalization of the physical spin in the entanglement If you want to see this a bit more concretely you can use this entangled bond and make a Schmitt decomposition of your chain And then ask how do the Schmitt vectors transform and you will see they do exactly transform like a spin one half So another thing related to well, not entanglement, but the edge Is what I would like to explain next that's another instance where we see fractionalization So what we could consider is we could start from our original Hamiltonian We construct it on a periodic boundary system, but then remove the term which causes a boundary. So open up the boundaries So for periodic boundaries, we have a unique ground state So now we could take our Hamiltonian and put in our system with open boundary conditions and ask what happens and well What we have in this chain if we put in our system with open boundary conditions is that we have Hamiltonian terms acting everywhere So we have a Hamiltonian term acting here and the Hamiltonian term acting here And the Hamiltonian term acting small h acting here, but we don't have Hamiltonian terms acting on the edge Now what kind of each Hamiltonian term make sure is That our state looks like it's constructed from putting an entangled bond here and applying the correct map here and here But it doesn't Hamiltonian doesn't care about these two sides at the edge It was exactly constructed in such a way that whatever I put here and here This two-body Hamiltonian would have as a ground state any state I can get by putting anything here and here I want But leaving the rest the way it should be So what it means is that there is no Hamiltonian term here Which would impose any boundary condition at the on the term at the corner, right? So This term could be anything. So here That's Here I could put any state phi left and I would always have something which is a ground state of this Hamiltonian, right? Because that's how it's constructed Same here. I could put any state phi right and I would always have a ground state of all the Hamiltonian terms again Because even the last term is constructed in such a way that whatever I put here I will have a ground state So what I obtain this way is I obtain a set of states all of which are ground states of my My chain so all states with any Psi left and psi right our ground states and Using the same techniques one can indeed show that these are the only ground states we get So what does it mean? Well? It means that kind of well if I remove a term coupling a ring I get new excitations right because I want constrained one Hamiltonian term is missing I get new states which can also be ground states Now the ground states the new ground states depend on parameters which sit in the left and on the right edge But each of these guys again corresponds to a spin one-half right so again phi left and phi right Transform like spin one-halves and the other thing is that indeed when I change phi left I will only see this effect close to the boundary if I do any measurement on my system Again because all the all effects decay exponentially So what we have is what people would call edge modes so excitations with which lift close to the edge and The edge modes are again spin one-half edge modes in this case So again what we have is that the physical spin one system if we introduce an edge to the system Fractionalizes into a spin one-half on the left end and the spin one-half on the right end And that's something which is kind of very peculiar. It's it's it's really something which Which doesn't show up in conventional matter if you would make kind of a product and that's not have entanglement You couldn't have excitations who's charged in the sense of how do they transform under a symmetry under say spin rotation? It's a fraction of the Transformation property of the original system right if you have an integer spin system You can only change the spin by integers the new thing will again transform like an integer It will never transform like a spin one-half So that's something which is really only possible because of this non-trivial entanglement structure in our system And that's something which indeed gives rise to a classification of phases of Inequivalent phases in the presence of a symmetry Which is something again, which is not present if you do mean field theory if you do learn our theory So let me just say a few words about that and it's really relate to the following fact There's something which we termed the fundamental theorem of MPS and Well, what it states is it tells us if we have two Inequivalent ways of describing the same matrix product states what is a relation between them? But an important consequence of that theorem is the following If I have a matrix product state Psi with a symmetry so there's some ug some representation of a group Which leaves this invariant By acting everywhere Then what we have is the following we have that In a graphical language if we take our a and now we're talking about translation invariant systems all the time We want translation invariant Hamiltonians and so on So there's only one tensor a Then it tells me that if I multiply this with the action of their physical transformation This just amounts to multiplying A with a different action VG and VG dagger, which is a unitary on the auxiliaries degrees of freedom So you can actually convince yourself immediately that if you have a chain of these a's with periodic boundaries if you act with you Everywhere you will get a symmetric wave function because the V and the VG on adjacent a's will all cancel out nothing will change But the really important thing is that So you might say whether VG transforms exactly the same way as a UG so it's a representation of the same group, right? So what we have is that we have that UG? UH is UGH because it forms a representation of a group like the rotation group But now what does this mean? I mean this a establishes an isomorphism between this here and these two guys here So it really tells us that VG tensor VG dagger Times VH tensor VH dagger is something like VGH tensor VGH dagger So this means that indeed the first tensor component must again obey the same kind of condition But the point is a phase is not well defined because that's a dagger It could compensate for any phase which I have on the other side So really I only get this equivalence up to a phase And this is what is known as a projective representation So a particular example would indeed be the case where UG is a spin rotation of an integer spin So specifically there should be the rotation group in real space SO3 And then I can build integer spin representations which are normal representations or I can build half integer spin representations So if the VG would be a half integer spin Then this would have a non-trivial phase. I mean If I cheat a bit it's due to the fact that if you rotate by 2 pi you get minus 1 That's not really the point because this can be compensated by a gauge But if you combine several rotations Then it turns out that the spin one half has a phase which you cannot remove regardless of how you choose your phase conventions So it turns out that they're in equivalent classes of these projective representations And well one one example of in equivalent classes are integer and half integer spin rotations And it turns out because they're in equivalent. There's no way of combining them right in equivalent date this here Well, it's a normal representation wouldn't have a face here, right? I think it's called projective because in principle if you were a mathematician you wouldn't write it like that But you would write in a projective space or in a space modular C. Exactly. Let me just state the Okay, so what we have is that we can have in equivalent projective representations and it turns out they cannot be combined and This implies that there's no way to kind of smoothly deform a state where this transforms in one way To one where it transforms in the equivalent ways So different MPS cannot be connected with in equivalent So they cannot be connected smoothly Meaning within the set of matrix product states meaning within the set of gap phases So it means that these indeed classify in equivalent phases under symmetries And this is what is known as symmetry protected phases or short SPT Where the T stands for symmetry protected topological or some people claim trivial because I'm not topological So I just prefer to just call them symmetry protected phases And that's really a quantum phenomenon these states why they don't have long-range entanglement We in one dimension have an entanglement which transforms in non-trivial way under the symmetry So as long as we keep the symmetry, they cannot be transformed into systems Into other systems where the entanglement transforms differently say like an integer spin And that's why the whole day in phase and the AKLT model are in one of the symmetry protected phases All right. Thanks for your attention