 Welcome everyone. So today we have the pleasure to have with us Andrei Bernovik from Princeton. So Andrei is a world leader in everything which concerns the political properties of electrons and solids, be that insulators, superconductors, whale semimedals, and his work also is characterized by close attention to experiments. In fact, some many of his theoretical predictions have been confirmed experimentally. And recently he's been doing some intriguing work on some special quantum SCAR states in various condensed matter systems, which also have been searched for experimentally. And that's what you're going to be talking about today. So please, Andrei. Many thanks. So yeah, thanks for having me. And this is the first seminar online in Paris, I wish I was a different person, but such is life. And what I'll be talking about today is work that was done in collaboration with my long-term collaborator, Nicolas Renaud, and a student who's now postdoc Sanjay Mulgalia, and more recently Frank Schindler. I'll be presenting some results, some new results from work that Nicolas and Frank Schindler and I have done together. And some results from the collaboration with the second row of people, in particular, Stefan Rachelle, Rahul Makishar and Paul Fendley. These are the references. And let's get into it. So basically, to motivate the talk experimentally, if needed, it turns out that, you know, while we didn't really know how to study non-equilibrium dynamics in, let's say, 20 or 30 years ago to the extent of controllability that we do today, cold atoms basically have solved this problem in the sense that they have enabled the study of non-equilibrium dynamics of interacting quantum systems. And of course, this brings about realization of phenomena or states of matter that can only be realized out of equilibrium. For example, one of these is time crystals. And hence, there's been a lot of effort on strong theoretical foundations of interacting systems. And this work is, you know, sort of a very small piece of that in the sense that it tries to review whether the eigenstate thermalization hypothesis in its strong sense is violated or not. And this is the outline of the talk. I'll basically be presenting the first exact model where the strongly T H violation was observed and then connected to what's later known as scars. And then I represented a lot of other systems where scars exist. So it turns out that this phenomenon is quite, it's not, of course, it's not entirely generic in the sense that it won't appear in any disordered non-translation environment. But it's quite generic for some set of Hamiltonians. And I'll make some connections. We still don't know the entire theory, but I'll make some connections to what we know. So I'll give a quick review of ergodicity. And I'm not an expert in this, but the one thing that I'll be focusing on is entanglement entropy. And basically, the property of ergodic systems that I'll be using is entanglement. Entanglement of states in the middle of the spectrum. And these are some properties here of ergodic systems. But really, what I'll be using is that the fact that the eigenstate termination hypothesis says that in the middle of the spectrum, so in this region of energy where there's a finite density of states of levels, where it's basically exponential density of states of levels, then the eigenstates are thermal and their entanglement entropy obeys a volume law. So you take one of these eigenstates, a generic eigenstate in the middle of the spectrum, and the strong ETH hypothesis says that any state you take here, you cut half of the system, you compute the reduced density matrix. Once you compute the reduced density matrix, you can compute the entanglement entropy. And this obeys a volume law, which basically means that any matrix per state representation of this state would have a bond dimension that is basically exponential in the volume of the system that's traced out. And this is very, very different from the ground states of or highest excited states, because the highest excited state is the ground state of minus a Hamiltonian. And the ground state of the system and the low lying states, and by this I refer to, I'm not sure if you can see my cursor, my mouse, but for this I refer to the ground state and the ground state like states, which means the low energy excitations before the middle of the spectrum or before the density of the level spacing between the states exponentially small. So before that, before you reach that, just the ground state and take like excitation or the upper, the highest state and excitations are around below the highest state. These ones obey an area law. And this is, of course, people misogynists know this is the area law entanglement. Now the strong ETH says that whatever ion state I pick here in the middle of the spectrum, they will satisfy ETH and they'll have the one thing that will be focusing on the one property that will show it's violated in these new states is that their entanglement entropy would be proportional to the volume if they respected ETH. Now there's many Hamiltonians that were thought to obey ETH and only a few examples that were known not to obey ETH and one of these examples are, of course, integrable systems. And another example is many body localized systems. Many body localized systems can be sort of thought as, this was the work of Babanin as integrable systems with stable integrability. So in many body localized systems, of course, you need disorder. But once you have disorder, you can write quasi local or local constants of motion. And then if you perturb the Hamiltonian a little bit, they still remain integrable. So you can still write integrals of motion. So that's why we call stable integrability. And these ETH violations, ETH violation was known and thought was to occur only in these two types of systems. So now the question we wanted to pose is can ETH be violated in the absence of an extensive number of conserved quantities in the absence of an integrable systems and in the absence of disorder. So no many body localization also. So in a purely translational invariant system interacting, of course, without disorder and without a number of, without integrability. And we solved this question analytically. And this was later known by the independent work, kind of these two works basically didn't know about each other. This was later given the name of many body scars by Serbin Papin and Papich and Abanin and Turner at all. And these, in this paper, and they were coming at this question from trying to model an experiment by Misha looking. And these quantum antibody scars basically, the name was given because of similarity with classical systems trajectories of the billiard ball that are non-chaotic. So the questions, is there a paradigm of ETH breaking beyond integrability and many body localized states? And what are the issues? Yes, so when it's beyond integrability, how would you test non-integrrability? Yeah, so the one way we test it is, so of course, as you know, as you're thinking, it might be, maybe it's not visible, it's not clear, integrability, maybe there's something hidden. But the one thing that's done in these is large scale level statistics. And there, now I know there's some issues with, there's some very special cases in the level statistics where you have Wigner-Dyson, but it might still be, I think, Shaman was pointing them out to still be integral, but these are not those special cases. So they're very simple models that have perfect, even on small size numerics, not small size, but, you know, 10 to 5. So by testing the distribution of eigenvalue, the statistics. Exactly, exactly. And then you can do more. Yeah, I'll show you the plots in two slides. Thank you. But indeed, you know, that's tested numerically because, as you pointed out, it's, you know, there might be, you might be missing analytically the integrability, but all these have perfect Wigner-Dyson distribution. So I'll show you the plots. Thank you very much. So then there's, so what can you do to find a paradigm, to find the ETH breaking? Well, the certain issues, strong issues arise in the fact that we don't know how to solve for eigenstates in the middle of the spectrum. Obviously, these are very complicated in many body systems. The Hilbert space curve is exponentially, and you don't have control over the eigenstates of spectrum, even for models where you do have control over the, over the ground state, for example. So hence, what most people in the field, or almost all the papers in the field of many body localization do, is basically just very smart conjecture supported by numerical calculations. And these numerical calculations are as big as, as one can, one can, one can do. It really just depends on your numerical skills. Roughly, there are 20 sites of the order of that order roughly. So, and they're mostly one D because in one in two D, you immediately get swamped by the Hilbert space dimension. So then what I want to claim is that we found a simple set of states that span from the ground state all the way to the highest excited states. This, the kind of peers through the spectrum of a non-integral systems, a non-integral system, and we're able to give their expression analytically, and then we're able to calculate the entanglement entropy analytically, and to show that they don't, that they, you know, it doesn't, that the entanglement entropy does not follow a volume log, even though these states kind of interpolate between the ground state and the highest excited state, not interpolate, but they appear at energies that form a tower. And since then there's been many examples of systems, of systems where this happened, and these are the so-called exact scar states. They should be kind of sort of distinguished from the, from the approximate scar states, which are the ones in the experiment, which, you know, they have a volume component of the entanglement entropy, but it's very small. They're very small, and these ones have no volume component of the entanglement entropy. Okay, so what's the model that we used? Well, the model that you use is the AKLT spin chain, and this is a famous model, where you have a sum of nearest neighbor projections, which, for example, cannot have, two nearest neighbors cannot be in a spin two state. This is spin one model, and two nearest neighbors cannot be in a spin two state, and the Hamiltonian can be written more, more familiarly as this, basically, as this bi-quadratic Hamiltonian, and this is, of course, the model where you, so the camera being spin two state in the ground state. This can be, it's a projector Hamiltonian, and this has been used to prove the holding gaps by, you know, these, of course, the AKLT paper. It's also, this ground state serves the prototype for symmetry, technological states, matrix product state wave function. It's got a matrix product state expression, the ground state with a bond dimension two, etc. So now, how do you find the ground state in the highest excited state? Well, the ground state is actually very simple, in the sense that you form, you take every spin one on one side, and you basically say that, oh, I'm going to form the spin one out of two, spin one half degrees of freedom is symmetrized, and then I'm going to dimerize the spin one half degrees of freedom in between sites, so that they form a singlet, and this is the, you know, the pictorial representation of the singlet, of the singlet is denoted by a line here, and that means that in the ground state, for example, since these two spin one halves, the right one on this side and the left one on this side have formed a spin singlet, I remain with two spin one halves, they can only be in spin zero or spin one state, so they cannot be in spin two state, and hence the projector term on sites N and N plus one, out of which the Aklity-Cannuptonium is formed, which basically projects onto the spin two, gives you zero on this state, okay, because this state cannot be on spin, this is the state that two spins one on one on two ADS in size, they cannot be in spin in a spin two state, and of course the highest state is the ferromagnetic state, and this is, you know, this, this is the expression because it's, it projects onto spin two, and the ground state of this Hamiltonian has energy zero with this, with this, of course you can shift it, but the one that I wrote before, which is the sum of these projectors says energy zero, and the ferromagnetic state is the highest excited state, so you can write the ground state exactly, and despite this fact the model is non-integral, so you can see very clear GOE level repulsion being done, so which is the rate to the, to the question before, that's, you know, you can solve, even this is the thing for 10 sides or something like this, and it satisfies very neatly the Dyson level repulsion. Now the other projectors Hamiltonians that we know, there's many of them, these are, these are sort of called frustration-free Hamiltonians, where you just have a Hamiltonian being a sum of projectors, each of which annihilates the ground state, these projectors do not commute, so it's not kind of a stabilizer code, like the other, like, like the exactly soluble Hamiltonians, so they don't commute, so it's not exactly soluble, and there's many examples of these in any dimension really, for example, the Halding-Südder potentials are one example for the fractional quantum whole state, the Schastri-Sullin model, the Madrugnagosh model, and all of them that I know of are non-integrable, these, and then how do, so then what we said, well, was let's look at the spectrum of this model in finite size, and you look at the spectrum of this model in finite size, and this is for helical 16 particles, okay, and this is for helical 16 sites, so it's, you know, quite a large system, you've got, you know, 3 to the 16 by 3 to the 16 matrix basically cut into symmetry sectors, and what you see is that, you know, you've got a bunch of energies, which is, of course, a huge amount of energies, but then what you observe is you observe some remarkably, some remarkable energies which are integers, and this is kind of like the way we, Nikolai and Sanjim, myself, initially thought about this, you know, the Hamiltonian basically has an integer, it's an integer matrix, so really if you're going to get out of a 3 by 3 to the 16 times 3 to the 16 integer matrix, you're going to get some integer energies, and those energies, the specific polynomial is certainly not going to give you, you know, integer numbers as roots generically, and if you get integer energies basically out of a huge integer matrix, those basically define some very special Krilov subspaces, and out of these integer energies, I want to point out that or they're not all integer, but they're some rational, okay, and you can get this out of high-scale numerics by doing some, by basically, you know, converting numbers to ratios, and then, and then, and then, you know, you can see within machine precision that they're these numbers, and what I'll be focusing on in this talk are these integer energies denoted here in red, which as you can see, formal ladder, so you can see zero as the ground state, 16 is the highest excited state, but then you've got two, four, six, eight, ten, twelve, fourteen. So after numerical observations, has anyone proved algebraically or analytically that they are integers, true integers? I mean, yeah, yeah, I'll give you the expression in the next slide. Okay, thanks. Yeah, yeah, yeah, very good, yes, so this is how you basically, this is how you hunt for them, right, because given a model, you're not gonna, just by looking at it, it's not entirely, I mean, in retrospect, it's easy to see that these are the excited states, and you'll see them immediately, but it's not, it's not, given a generic model, it's not easy to find them, and this is how we hunt for them, and you know, this procedure has been kind of now used by some people, but it hasn't been used to the extent that I had hoped for, so even now we find using the same procedure, other models that have exact eigenstates, and I'll show you in a little bit, but yeah, this is how you hunt for them, so now let me give you their expression, so their expression is the following, let's define the basis state mn, which is a S, a spin on site n, raising operator, applied twice, so it's a spin to magnon on site n, and this what it does is it breaks the bonds on site n minus 1 and n in the ground state, acting on the ground state, based on the bonds between site n minus 1 and n, in between site n and n plus 1, and it has to create the ground state of spin 0, it has to create the spin 2, so it just polarizes those spins and you can see it here, so now you can immediately see that the beauty of these projector Hamiltonians, of these frustration free Hamiltonians, that you know, the only thing that will matter in the scattering problem is what happens around this site n, where I hit, where I flip the spin twice, and this is because you know, the projector Hamiltonian will kill everything that's, that's to the left of site n minus 1, is to the right of site n plus 1, so that one you don't care about, and you find out that the Hamiltonian scatters this mn configuration to bond inversion symmetric configuration, so it scatters them to, for example, this configuration plus the left configuration plus the right configuration, okay, and now you can immediately see that, so you have the scattering, a scattering equation which is, which is h times mn is equal to some eigenvalue times mn plus this n, n states are the left one and the right one, okay, but you can see that the left one here, I'm not sure if you see my cursor, but the left one in the second row is, is a translation or the right one is the translation of the left one, okay, so now if you make a momentum pi excitation of this, then this scattering cancels and you're left with an eigenstate, okay, and this is an eigenstate that energy 2, because of this 2, spin 2, and it happens for all even l, by the way these states happen for even, even l because of this, this, this issue, so, so okay, so good momentum pi you have to hit, you have to have even l, so this is the first eigenstate of energy 2 is minus 1 to the ns n plus squared on the ground state, which we'll call this an operator p, so you act with p on the ground state and the energy is, is, is 2, now, you know, I've, we've thought and a lot of people thought so for a long time that, that only momentum pi scars, it can exist or, you know, momentum zero or pi, but recently there's been a paper by Nusha Chandran and collaborators at Boston University basically found scars, they were able to come, kind of reverse engineer scars at every momentum, so okay, so now, so now you take this operator p, which is again, is this momentum pi spin 2 magnon, you take this operator p and you act it n times on the ground state, okay, and this is the tower of states that we had found numerically and you can kind of see that, you know, if you acted once it gives you one spin 2 magnon and momentum pi, if you acted twice it gives you two spin magnons which scatter a little bit, but only very close to each other at total momentum zero, so they're, each of them is a momentum pi, etc, and if you acted l over two times, you get, you get to the ferromagnetic state, so these are the exact expressions, so now having the exact expressions, we ask what are the implications for the eigenstate thermalization hypothesis, well the implications are immediate because you can, having these exact expressions and having this operator and having this very simple form of these states, you can immediately estimate their entanglement entropy, so what you can do is you can write this s2n as a matrix product state with a small bond dimension, and the bond dimension, so this n here is the number of times you hit you apply this operator p, okay, and basically you can write this operator p as a matrix product operator, etc, but this, but the fact that it's a local operator that you apply, you know, n times, so some local operators that you apply n times, basically gives you a matrix product state with a small bond dimension, you can, you can find exactly this bond, an upper bound on the bond dimension, and this bond dimension is two times n plus one, okay, and now you can basically see that the entanglement entropy, because by bond dimension you can, you can bound the entanglement entropy is, has to be less than log of this, which is two times n plus one, log of two times, two times n plus one, so it's got log bounded, log of n, where n now is the number of times you hit, you know, the, the, the, the ground state with this projector, if I'm in the thermodynamic limit and one states in the middle of the spectrum, n over l is finite, is constant, is some fraction, n over l, so that means that s is bounded by log l, so these are log l states as opposed to s being volume law as predicted by the age, so these states analytically violate the eigenstate organization hypothesis, you can see them in a spectrum where you, where you compute the entanglement entropy of all the states in every, of all the states of, for example, this is 16 site AKLT model, so the entanglement spectrum of all the states has this kind of nice semi-circle law, but then you have these states, these cars, these x's, which have much lower entanglement entropy, they kind of, they kind of, you know, they, they, they, they, they're, they're much, much lower than, than the ones that satisfy DTH, so all of these states that sit on this, on this semi-circle is, I mean, semi-circle is parabola, some, some curve is not, is, are, are not, are volume law and then you get these cars, which are area law, which are log l entanglement entropy. Okay, so if there's questions now, I'll take some questions, if not I'll go to the connection between these and the matrix rock states and try to get to the spectrum-generating algebras that describe these states. Okay, I can ask a question, I mean, if nobody else has to, but it's just too similar to the coordinate beta ansatz to me, so how do you explain this or, or maybe it's the recognition, maybe like the sub part, subsystem is actually integrable or like? Actually, related question, maybe, maybe could, so yeah, on the landscape of spin one, spin chains, there is an integrable system, right? There is a way to write the Hamiltonian using gamma functions probably, which will be integrable, which will be solvable by beta ansatz. So how far is the AKLT model from that integrable point? Yeah, so I'll, so I'll answer both of these questions in the matrix cross-state because I'll construct the pathway to integrability. So, so the very good, I mean, it's quite the model, the AKLT model by itself is quite far from integrable, you can see it from the vignetization level statistics, but there is a path, and this is another standing question, whether, whether these cars are remnants of what was, you know, like a one of the beta ansatz states which remains. So the answer is probably yes, you know, but they're remnants very far away from integrable. So, so, so, so I'll answer both of these questions right now. You can, you're just preempted basically my next five slides. Thank you. So, so, so yeah, so if I can take 10 minutes to basically, or five minutes to, to point out this matrix cross-state connection, then construct the pathway to integrability. So basically, the ground state AKLT is a matrix, can be written as a matrix cross-state. And this matrix, these A matrices are two by two matrices. And then you can create quasi-particles and mental k, kind of excited states are on top of this, by this operator, you act with some operator on side j, and, and this is the simplest quasi-particle that you can create. And, and you put a momentum k. And this basically acting with this operator basically modifies the MPS of this A matrix in the AKLT case, its translation invariant, so all the A's are the same. It modifies the A matrix by basically putting a matrix block operator on them. And, and you create this new B matrix, which is this operator O on the MPS matrix. Okay. So now the question is, how do you construct a, how do you, you know, backtrack to find B from a given A? Okay. And the reason is, if I, if, if we manage to find this, we can have a pathway so you can tune the A matrices, the MPS matrices, to find a pathway between the AKLT and the integrable model. And then if you find a prescription to find a tower of quasi-particles, kind of scars in this matrix cross-state prescription, you can kind of follow it all the way to integrability. Okay. And I won't bore you with the details, but this is very similar to how you construct exact Hamiltonians from MPS ground state. So it turns out that if you have a ground state or eigenstate, which has an MPS description with a bond dimension that's finite or that doesn't grow, that doesn't grow as the volume. So let's take it to a bond dimension that's finite. Then you can find a frustration-free Hamiltonian that, that, that has this as a ground state. And the prescription is simple, is simpler. This is a simpler way of doing it. Then there was first done by Verstrasse and Sirach and other people. I think there were two or three groups that did it at the same time. But basically, what you do is you want to find an operator H that annihilates some parts of this matrix contraction. And then given that you have, that you have a finite bond dimension, all you need to do is count how many, how many times the Hilbert space onsite, how fast it grows. And once it grows, once it's become larger than the span of the bond dimension matrices, then you can write the Hamiltonian. So for example, for the AKLT, these matrices are two by two. And then the image of the, so since they're two by two, I can't write a one-body Hamiltonian because a one-body Hamiltonian, the physical dimension would be three. So three is less than two. But if I have two sides, now the physical dimension is nine. Whereas, whereas, whereas actually the bond dimension is, I have two here and two here, so it's four. So I can project into the complement of, of, of, of the image of this image of the two-side MPS. And this projection Hamiltonian will basically annihilate the, will annihilate the ground state if this projector here is Ej matrix is positive semi-definite. So very easy construction that you can build it actually, and you can generalize it to excited states. So this is some sort of, you know, MPS generalization to excited states. This B operator is the operator of the, of the excited states. So it's the, this operator on the matrix A. Okay. And you can find out the conditions for which, for which this excited state is now the ground state of the Hamiltonian that, that annihilated the way function with matrix products, state A. Okay. So, Bogdan, can I ask a question? So are you saying that for these parent Hamiltonians, you can construct course particles, eigenstates analytically? So, so I, so there's, yeah, so there's several, so I'm saying there's, there's, you can find the conditions, and then you have to solve the conditions. So, so given, so given a matrix product state A, if the matrix product state A has a finite bond dimension, then you, then you can find the projector Hamiltonian. You just, you basically just count how many times, how many sites you have to, to, to get to the space of those sites is larger than the bond dimension squared, for example. Yeah. Here, and then you project into that, and then you found, you found, you found Hamiltonian that annihilates that state, so you can find the Hamiltonian that, this is, this is, this is the frustration. That's, that's simple, but now you're in the second slide, you were saying how you can find quite, quite a particle situation. Yes, and now there's a more complicated condition, and the condition is the following. The condition, the condition is, is, is, is this one. You have this, this, this one here, so now you have to solve it, and you have to find out if this, if this exists or not. So, so it's not, so it's not immediate that it will exist for all the frustration free Hamiltonians even. But it doesn't exist, where's the mistake? So we know that the quasi-paric, for example, let's take ATLT, as far as I, I thought that for ATLT, quasi-paric excitations are not exactly solvable. So something is well, but the Hamiltonian is of this parent Hamiltonian form. Ah, right, right, right. This, this is not, this is, okay. So first of all, this quasi-particle, so this, I'm claiming that here you can find this, you can find a solution that's solved for this tower of states, not for every quasi-particle, right? So this is, so for example, you can immediately see that this is a very simple solution, just takes an operator, this, the, the, the ansatz for this solution is an operator, local operator on site J, times a momentum, right? This is not the most generic form of all, right? I could have operated that act on two sites, right? So this is how, so we try to kind of backtrack from, so we know the form of these tower of states in the ATLT, and we try to backtrack from there. Okay, so you're saying these ansatz would not be sufficient state to describe the, the first quasi-particle, but it might be sufficient for some states in the middle of the spectrum. That's right. Well, it's certainly sufficient for this tower of states that I, that integer energies that I, that I, that I showed. So this is, so yeah, so it will certainly want to describe all the, all the, well, it will certainly want to describe both of the states, right? It will just describe these, these, these ones that are local operator, times a momentum, and then, and then you want to apply this operator in times, okay? So, so for example, right, so, so, so you find these conditions, and then you know the matrix A, and you can kind of do a numerical search in this matrix A is two by two, you can kind of do a numerical search in the space of operator satisfying this condition for both the momentum and the operator, and the AKLT reproduces, you find the AKLT tower of states. But what's more interesting here, and this coming to your question, is that, so for example, this, the way it works for the AKLT is you have this AKLT matrices, this is A for spin, 1 for spin, 0 for spin minus, then you find the subspaces that I was talking about that are spanned by, by the, by the bond MPS, and you find three subspaces. This is the subspace of, of, of, if you project into the complement of this of A, you find the ground state, but if you project in the complement of, so this is the B is the quasi-particle space, because we chose the operator to be S, S, S plus squared, so it's spanned by S plus squared on, on, on the ground state, and this is another complement space, which is the complement space of A minus B, and then you find the most general nearest neighbor Hamiltonian that has this, so the tower of states that was, it's the exact eigenstates that I presented is a tower of states not only for the AKLT, but for this more general Hamiltonian, okay? So for example, this more general Hamiltonian would be AKLT if this, I guess, if these ZJs and if these, if Epson was two, and if these ZJs were one, for example. So this is kind of a more generic Hamiltonian that has the exact same tower of stars. So now going to answer the question that was posed before, you can start, you can change these matrices, right? You don't need to start from the AKLT, from the AKLT matrices which are these ones. You can change these MPS matrices and you can change them to be, for example, these constants, C plus, C0, C minus, okay? And you can tune these C's around, okay? And you can ask, when is this operator P, which is the momentum pi spin 2 magnon, when are P to the n times the ground state, exact states of a Hamiltonian that also has this as the ground state, okay? And you can find from this, by tuning the C plus and C minus, you can find the continuous family of spin 1 Hamiltonians, which have these states times the ground state. Now notice that as I change C plus, C0, and C minus, the ground state changes, but the scars are just P to the n powers times the ground state, whatever the ground state changes, the scars will change, but there's still P to the n times the ground state. And then there's a continuous family of spin 1 scarred Hamiltonians connecting the AKLT point to the integrable pure by quadratic point, okay? And then this raising operator becomes a symmetry at the integrable point. It's not a symmetry, it doesn't commute to the Hamiltonian away from the integrable point. It's, even though, you know, applying it on the ground state gives you exact excitations. There doesn't mean commute to the Hamiltonian. I'll tell you in like the next five minutes, the last time I talk what it actually means, it's a restricted spectrum generating algebra, but at the integrable point, this operator does transform into the one of one of the symmetries of the pure by quadratic chain. So I think that this answers the two questions that were posed before. So, you know, even though the ground state changes over this interpolation, to this, you know, you find interpolation with an angle, the ground state changes over this interpolation, the P to the n times the ground state are still exact eigenstates. And they're eigenstates that interpolate between the ground state and the highest excited states. And as I go to the integrable point, this operator P that was used to create the states becomes a symmetry. Okay, and there's more involved scars you can do scars in the perturbed POTS model that involve more more you can sort of generalize this is Slava question you can generalize this this construction to more complicated operators, you can't get all the excited states, but in the case that when there are scars, it works for more complicated operators. For example, for the perturbed POTS model, you find this operator that creates scar states. Okay, so now I want to basically for the last part of for the last five minutes or three minutes of the talk, I want to basically sort of try to give a more algebraic description of these of these of these scars. And we do this by looking at the Haber model in any dimension really. So the Haber model or translation symmetry, this is my graph, for example, but let's for right now consider translational symmetry. So the Haber model is nearest neighbor hopping plus Hubbard and it was known very since, you know, early work by Sien Yang that it this model besides having, you know, st2 symmetry spin symmetry also has an eta bearing symmetry, which is a pseudo spin symmetry. And it's a it's it's an operator that doesn't commute with the Hamiltonian, but it commutes up to itself. Okay. And this is kind of the spectrum spectrum generating algebra. It's a very powerful right. I'm sure this audience knows very well, very powerful construction in the sense that once you found the state of the Hamiltonian, you can hit it with eta dagger and and find a number of times and find eigenstates at equally spaced energies. Now, this of course reminds you of scars, except that in this case, eta is an exact symmetry. Okay. And it's an it actually gives you another recipe to the tree. And, you know, of course, as I said, at some spacing starting from any any state. Now we know that the AKLT model or the models that I presented before do not have this symmetry because the spectrum does not come, you can find only one tower of states, even, you know, if it had the symmetry, the spectrum for any state that you that you start with for me for any eigenstate, you could find this tower of states. You'd find many towers, but you know, you find only one. Okay. So, so the, of course, the difference is that in the Haber model, eta is a symmetry, whereas in the AKLT, this p operator that I used to that we used to define the states is not a symmetry. So what is it? Okay. Well, it turns out that it's not a spectrum generating algebra, but it's a restricted what we call restricted spectrum generating algebra, which means the following thing. It basically means that you have spectrum generating algebra property, but only when acting on one on one state. So for example, you focus on some tower of eigenstates and you ask, which is eta or p in the AKLT state to some power n acting on some on some state be the ground state or another state, and you can prove that the tower of eigenstates exists when the following things happen. Psi zero is an eigenstate of the Hamiltonian. The commutator of h and eta gives you epsilon, some energy times eta, but only when acting on this psi zero. And this is the case of the AKLT state. And, and the second commutator of eta vanishes. So you can see this is kind of like, it's kind of like, right, kind of like a symmetry, but the symmetry that's cut when in, in, in the expansion of commutator. Right. So if I tried to transform a Hamiltonian by a symmetry, I would have, you know, a lot of commutators, but at some point this commutator cuts. Okay. And this we call restricted spectrum generating algebra of order one. And I'll show you why. And you can actually modify the Haber model by adding extra terms to the Haber model so that this eta pairing is not a symmetry anymore, but you can still find the tower of the exact states. And for example, one of the interactions that you can put on is nearest neighbor spin-spin interaction, which would kill the eta pairing as a symmetry, but would remain what would still have scar states because all these conditions would be satisfied. Okay. And you can generalize this, of course. Schematically, what does the commutator of h and eta dagger looks like? It's, so you're saying it should be something which commutes with eta dagger. So it's, it looks like what like schematically? It's zero. It's not just epsilon eta zero. So it looks like what, a bunch of other eta zero, eta dagger. That's a good question. So, so I didn't think about exactly how h eta dagger is not eta dagger, because if it was, you'd have a spectrum generator. Yeah, it's not eta dagger plus some extra terms. Exactly. It's eta dagger plus something that commutes with, plus something that commutes with eta dagger. Right. So, yeah. So it's a good question. I don't, I don't know a priority answer to that. But there are other many things which commute with eta dagger, which could appear in that side. For example, for this simple model that you present here by adding this spin-spin interaction, probably you know what in this case the commutator is. Yeah. Now I know this, it would be easy to write it down. I don't know in my heart what the commutator is, but it's easy to write it down. Yeah, let me, yeah, I'll get back to you on that. That's a very good question. Yeah, it's a very good question. But okay, anyway, it's clear to me that it's a more general condition. Yeah. Yeah. And I mean, you can generalize this condition to the following condition, which is, which is, you know, you act, you act with, you have a, you have a psi zero state, which is, which is an eigenstate of Hamiltonian. You have a commutator with eta, which is, which gives you a spectrum generating algebra, but only when acting on psi zero, then the commutator n times, small n times of eta with h is zero only when you acted on the, on the, on the state psi zero, but then from n plus one, it's, it's, it's, it's non-zero. It's, it's for any state that you act on. So I don't put a state here. And for example, the AKLT, the AKLT state has, so this is, this is called the restricted spectrum generator of order m. And the AKLT state is when m is equal to two in this case. So only when I, when I do the commutator three times, does it kill the, the, it's, it's commuting with the Hamiltonian in the AKLT case. Okay. So there's many scars in other things such as non-linear messenger liquid, but I will basically stop here because I've taken, I think enough of your time with this, but the main message of this is that the scars are, I think generic, at least in these simple Hamiltonians that we look at, certainly they exist in the Haber model, they exist, you know, in, in, in, in many other models, they exist in this projector, frustration-free models, and the relationship with integrability is not, not, not yet known. So that's, that's what I wanted to say. Thank you. Thanks, Andrei. Thank you. Are there any questions, including from guests from outside the YGS? I actually have a question. Andrei said this, you know, of course I'm not an expert, but I know I heard about this parent Hamiltonians and like AKLT and so on. And I know that people usually say that for many aspects, for understanding the properties of the ground state and maybe the first excited states, this parent Hamiltonians are as good as more realistic Hamiltonians, which don't have this property of being made of projectors all annihilating the ground state. But usually they say that, okay, well, for, for the excited states, it's clear not the case. These guys are very different. In the sense, your work is showing, is giving some very dramatic example of how these Hamiltonians made of pieces annihilating the ground states are different for the excited spectrum. But in your, in your abstract, you were mentioning that these scar states were actually observed in experiments. So how is this possible that in experiments that they were able to reproduce this very special Hamiltonians, how did this work out for them? Yes, they reproduce, they didn't reproduce. They're not exact scars in the, they're, they're these things called the PXP models. They're not, they're not actually exact scars. They're basically, they're basically Hamiltonians that where you, where you have the kind of, I don't know if you know this friendly types Hamiltonians or whatever, say a Sigma X and some Sigma Z operators on, on, on the ADS and sites. So basically this is kind of like, this is projector one minus Sigma Z on sites, one Sigma X on site two and one minus Sigma Z on sites, on site three. So these things, they can, they can, they can kind of get them in these Rydberg atom experiments where, okay, so, so basically where you, where you kind of tune to some resonance between, between two different levels and kind of forbid, forbid some, some spin flips, depending on whether the spins here or here are up or down. So, so these are, so these are basically these Rydberg atom experiments. I don't know exactly how they, you know, the details, but they're basically, they basically realize this PXP model. Now in this PXP model, what you can show, so this, this is, you know, this is the model where this, this P is the, you know, one minus Sigma Z, for example, and, and they don't commute between themselves, some of operators don't commute. Now this model, you don't know the exact ground state, but what you do know, and, and the scars are not exact, but what you, what you do know is that you can find some, you know, almost exact expression of the scars, they don't, they still scatter, scatter very weakly, and this is what Serbin, Pappic and Habanind did. And these ones have a volume law entanglement part, but it's very small. So normally the ETH would predict there, there's, you know, basically some, some coefficient to the volume law of the entanglement entropy. And these ones, because they're not exact scars, they have that, but it's much smaller than the ETH would, would, would, would predict. And, you know, if you really, if you want to, if you want to think of these models, they're actually, they're actually just some version of pair hopping models, actually. So this type of model where I have, where I have, so this is, this is a simple model that, for example, you also don't know the ground state where you take two particles on site one and two and then hop them on sites zero and three. Okay. So this is the process, the pair hopping. And these appear in like, you know, you can realize them, for example, in strong, in strong electric fields, like this is with interactions in the strong electric field, you realize exactly this Hamiltonian plus some, some, plus some electrostatic terms, or you can realize in quantum whole on the thin torus plus some electrostatic terms. Okay. And this model in, in some, so this model is actually very special because, because it's Hilbert space breaks up into Krulloff subspaces that are, that are not generic in the sense that, you know, Krulloff subspace, if I start with a state with a random state and hit the Hamiltonian many times, hit the state of Hamilton, I mean, does I'll go into and cover the whole Hilbert space. For this Krulloff subspace, I started with a special with the product state, I hit, I hit it with the Hamiltonian and the Hilbert space does not cover the whole, the Hilbert space, such form by hitting the Hamiltonian does not spend the full Hilbert space. So if you do this, you'll find out that these Hamiltonians have, you know, what's called, have, now there's a fancy name for this in this, in quantum evidence, it's called Hilbert space fragmentation, but it's really just Krulloff subspaces that are not, that don't spend the full, the full Hilbert space. So what you find out is that this, this pair hopping Hamiltonian some integrable, has some integrable subspaces and some finite dimensional subspaces and has some non integrable spaces. And this PXP Hamiltonian, this Hamiltonian for the Rydbeck atoms belongs to one of these non integrable subspaces. And here you don't, you know, there's nothing exact here in these things, there's only approximate, you can only approximately argue that there's cars and you can only find them numerically. So it's not, so it's not a, you know, but, but so, no, so they're not, so my point is cars are not, and this was, you know, a bun in, in Papich and Serb and Turner that showed this are not particularized to like these exact models, they exist even in models where you can't find the exact ground state, when you can't find almost anything exactly, but you still see some, some, some scar states, and you can see them numerically, like what they did is, you know, you can see them numerically here. So you can see, you know, this, these are the entanglement entropy with the one, two, three, four, five, six, seven are the scar states, very low entanglement entropy. Now, they're not exact, so they have a small volume component, but it's much smaller than it should be. It's much smaller than it should be based on, based on the TH. So my point is that, you know, an experiment can create something that's close to, you know, you don't need these fine tuned models to get scars, that's my point, or to get, to get the same phenomenon, of course, you know, as I said, they're not exact scars, but, but, you know, an experiment over some time scale doesn't matter. So they, they see, what they see is they prepare a state, a special, a special state, and then they evolve it, and they see revivals of this state, which basically tells you that this state has a large component on some thing that's almost an eigenstate of the system. Okay. Thanks. Thanks. Any other questions? People are not leaving, so presumably somebody has. Yeah, I have a question. So that's one way, whenever you hear the ETH hypothesis, you normally you hear that you take a generic state from the middle of the spectrum, and you have certain properties. And then to me, not, not, not being an expert, that the word generic encodes maybe some complicated tests that you should take. So in one sense, do these scar states fail some of these tests? Or is it just that you can't compute their energy? Is this the main reason why then everything fails? Yeah, I mean, so basically, so the main thing that you can do is you can, you know, prove that their entanglement entropy is not volume law. So that's one thing, but there's many other things that you can prove. So you can, for example, prove that the matrix elements of these states to ADS and states, okay, are of local operators are zero, you know. So these states are basically don't basically kind of don't mix with states, even though the other states are, right, exponentially close, they sit in a background of those exponentially close to them, right. So this, they sit there, they're like, right, there's states here, just there's an explanation, the system has a finite bandwidth, there's an exponential number of states here. So there's going to be some states very, very close to this to these scar states, but they still won't mix, even though they're exponentially close, they still won't mix with these states by local operators. So that's kind of one way that, you know, they don't satisfy ETH, but the main thing that you can compute here, so that's a hard thing to compute the matrix elements, because you don't know the expression of these other states, but the one thing you can compute is their entanglement properties, which are completely different. So basically, the volume law would tell you that if I cut a state in half in the middle of the spectrum, I have connections between every volume on this side, every part on this side, and across the cut, right, whereas these states have log L, right. So they look kind of like gapless ground states, right, this log L. Maybe I can rephrase a bit. So even before you found these states, let's say you take some spin chain from Newtonian, you diagonalize it numerically, maybe for finite size, you compute the entanglement entropy, and you will certainly find some non-generic states that will have low entanglement entropy. Not in the middle of the spectrum, you shouldn't. Well, you could have done this before finding the scar states in this model, for example. Right, right. And you can imagine in some other model, you could find the same. If you find the same, you probably found scars. So you're saying that it is the only way to break KTH for now with scars, so you should be looking at scars and maybe trying to prove this other property you just mentioned. Yeah, the only way to break KTH if you don't have many body localization or integrability is this way. That's the statement. So yeah, so you're certainly right. If I can, you can compute, and this is what, you know, for non-exact scars, this is what people do, right. So like, so this is a plot from the original paper of the PXP model, where is the non-exact scars and, you know, they compute the entanglement entropy and they saw this low line, right. Most states are here. There's a huge amount of states here at large entanglement entropy volume, but there's tons of them. There's this low set of scar states. So that's how they basically found. So if you find some of some low ones, like that look like this, those are scars. And they might not be exact scars, like in this PXP model, this is the puppet serving up on plot. In this PXP model, they're not exact, but that's, you know, that's how they found them. So yeah, so that's exactly what you pointed out. You can just compute and if you find that low, then you find scars. Thanks. We have some younger people, students and so maybe they have some questions. So they shouldn't be afraid to speak up. Let me ask something because probably I didn't fully get the talks. So what are the scars in the continuum limit? What are what? What are the scars in the continuum limit? Yeah, so they're basically spin to momentum pi magnons. That's the first state. I mean, right, so let's not talk about lattice, but there, you know, you can still build a spin. Okay, so it's a spin to momentum pi magnon. Okay, this state in particular is a low lying state. Okay, because it only has energy to not energy. Right. So the bandwidth of the model is order L, okay, the bandwidth of the model. This L grows in the continuum limit. It's L is the number of sites. Okay. And this is a low state, but then all the other states, you just take this spin to magnon and momentum pi and you hit the ground state with it. Right. So it's some it's some combination of spin to magnons with some scattering between them. Okay, so for example, this one schematically looks like this. You create two spin magnons when they're far away from each other. They don't interact. They interact when they're close to close to each other. Okay, so this so each of these p operators contains a spin pi magnon in real space, you kind of act one and then you act the other one. There's going to be, you know, they're going to act differently on the in the real space when they're far away. They're not interact when they're close together interact now, as you get to P to the N where N is a fraction of L, those are in the middle of the spectrum when it is a fraction of L, then you get a lot of them. Right. So their interaction is not particularly easy to to to express like this in in in real space when there's a thermodynamic limit, thermodynamic number of them, but you can express in matrix product state form. But but yeah, so this is, you know, this is their their their expression in the momentum space, you just in the in the sorry in the in the in the real space in the continuum just spin to spin to meant to pi magnons on top of each other. And I propose that that we thank Andre and and then people who have more questions like myself can stay for a little longer so that we can say it says some people who want to leave they should they shouldn't be you. Okay, so thanks Andre again for thank you. Great job. Thank you.