 First, I would like to thank the organizers and especially Slava for allowing me to talk and to sneak in into the program. So I wasn't actually prepared to give a blackboard talk because I didn't get any instructions, so I hope it is okay to give mostly talk on the slides. So it's some kind of follow-up of what Philippe told us. And what I would like to describe here is investigation of a different kind of model that I don't think was included in the list of all the models that have been studied with IPEPs in Philippe's list. So it's a Heisenberg antiferoractic model but which has a very particular feature. It has a carality that it breaks time reversal symmetry and it is supposed to host new type of state which are topological ordered states. And for this reason, I think it's a particularly challenging problem for the IPEPs method to see what's coming out in the investigation of such model. So what I will do is first give you some motivations. Why studying this model also from the physical point of view, not only for testing the ability of IPEPs method. Then what I also present is tell you a little bit about the type of PEPs ansatz I will be using. So there are actually important modifications compared to what Philippe described. There will be more constrained PEPs which are designed to address this type of problem. So there will be less variational parameters in the ansatz to optimize over them. And then I will tell you about the modification of the IPEPs method I'm actually considering which is somehow simpler which using the particular symmetry of the PEPs ansatz I'm using. And then I will go to the result considering this particular model. So the actual motivation come from the physics of the fractional quantum Hall effect I guess which give actually beautiful example. So that deals with the continuum 2D plane. So there's no lattice in this problem. It's just electrons and magnetic field and they are interacting with a Coulomb potential basically. And this system hosts topological states which are described by the Loughlin wave function with a well known Loughlin wave function which has a very simple form. So this is psi Loughlin. So it's just a simple function of this sort. So Zi are just the complex coordinate of the particles. And then there is some exponential factor to confine the particle on the disk. And M is actually connected to the filling fraction. So the filling fraction which is just a number of particles divided by the number of fluxes in the system is basically 1 over M. So for the most well known fractional quantum Hall state, the filling fraction is 1 third. So M is just 3. And so obviously you get a fermionic wave function. If you take even values for M, basically you describe bosonic wave function. For example, the simplest bosonic state would correspond to M equal 2. That is filling fraction 1 half. And so this is a beautiful very simple wave function that captures very well the physics of the fractional quantum Hall state. And in the late 80s, Carl Mayer and Loughlin actually wrote the version on the extension on the lattice. Okay, so this is the Carl Mayer. So I'm not sure I get the spelling right. Is that correct? Carl Mayer. Loughlin. And the idea is basically to take this bosonic M equal 2, nu equal 1 half wave function and just to put on the lattice. So now the coordinates here Zi are now living on the lattice. For example, square lattice. And to map the wave function, the Loughlin wave function on the spin 1 half state, one can use a simple prescription that basically at the places where there is a particle, you say the particle corresponds to a spin up. And if there is no particle, for example, these places that correspond to a spin down. Okay, and if you do this simple prescription, basically the wave function you get is just a singlet, global singlet wave function, which doesn't show any symmetry breaking no longer in order, but it has topological order. And I will be more precise by what I mean by topological order. And so this is a simplest realization and most beautiful representation of what is called a carousel spin liquid. So these carousel spin liquid somehow can be viewed at the analogs of the fractional quantum state for spin system on the lattice, basically. And so this is what we are targeting. We want to investigate models that will have the potential to host such exotic carousel spin liquid states. So that's basically the motivation. And so we want to first try to consider, I mean try to investigate, consider some model. I mean let's see, construct some, let's parent Hamiltonian for this carousel spin liquid and then try to investigate them with some techniques, which here is going to be the IPAS. This is for odd filling fraction, right? So this is for M equal 2. This is a bosonic state. This is a spin one half state. So this is for this case, M equal 2. So that corresponds to half filling actually. There is a commensurability relation. So the number of spin up and the number of empty side is actually equal. So this is why we get a global singlet which is invariant under spin rotation. So this is what I just said. So this Camille Loughlin state is, sorry, mix up, is really a paradigmatic example of a carousel spin liquid. And the goals are the following. So what we like is to search for some simple projected integral per state peps ansatz that will describe such states. And we want to apply these and optimize those wave functions to study simple parent Hamiltonian that will, you know, which have the potential to all such phases. And we might also would like to have such Hamiltonian to be local. And there might be a problem here that's really to construct such model. So if we look for topological spin liquids, really what we have to go is to go beyond the Ginzbo-Lando paradigm. So the order parameter paradigm because those states are characterized by the absence of any spontaneous broken symmetry. They don't have local order parameters. But they have what was actually introduced by WEN, topological order. So by topological order, I mean I can maybe take the definition of WEN is a state whose grand state degeneracy will depend on the topology of space. So if I put, for example, the Camille Laughlin state on a sphere, on a on a on a Taurus or on another manifold with more, with higher genus, I will have, I will get different number of degeneracy of the grand state. So this is will have really a grand state topological degeneracy here. And those states will be characterized in principle by a gap. So they will be incompressible. And then above the gap, the excitation will be fractionalized and unique excitation. So this is really what we target. Yes. So this Camille Laughlin, that's a wave function. Yes. For a spin system. Yes. And it does not come with a Hamiltonian, does it? So I will go to exactly to this question later on. So there is an Hamiltonian that has been constructed by your friends that you know, Ignacio and company. And I will describe. But no, I'm saying the original proposal by Camille Laughlin did not come with a Hamiltonian. No, but they were, they were actually suggesting that it will be the grand state of some frustrated spin model. Oh, they suggested that. I think, yeah, in the original paper, maybe be the triangle lattice or something like that. But they were suggesting that it could be, and they were suggesting actually that it is equivalent to the RVB wave function of Anderson. So that's actually the main point of that painting. So they have suggested Hamiltonian that would sterilize this wave function. Yeah. Did they suggest that this would be a gap system? Do you know? Yes. Yeah. Yes. Yes. Yeah, because I think the wave function, it's known that this wave function should be exactly like that. Should have a gap. Okay. Yeah. Like the fractional quantum all state on the plane is incompressible. Yeah, but that's a statement about the Hamiltonian. No, no. But if you want to say that this wave function will capture the grand state of the Hamiltonian, it means the Hamiltonian will be, will be gapped. I think if it's local, if the Hamiltonian is local and the wave function has finite correlation lengths, then it has to have a gap. Okay. Let's say this way. But does it, does that, does that Hamiltonian going to have anionic excitations that are the same as the anionic excitations of the original fraction? That's the wheel. I mean, that's what you want. Okay. Yes. Okay, so this is one feature you, you would like to observe. And the other feature which is characteristic of these topological state is the fact that they have edge states. So for example, if you put your topological Carol state on a, on a sphere like that, and you just imagine you cut the sphere into two hemispheres, then you will observe two counter propagating edge modes on the two hemispheres. Or if you do the same on a torus, then on each part you cut here, you will also observe counter propagating edge states. And these edge states are actually protected by the fact that they, the bulk has long range entanglement. And they're also described by simple CFTs. So for example, in the case of the Camille Loft instead, what you would observe is the, the fact that these edge states are described by simple SU2 level 1 CFT. So just central charge equal to 1. Okay. So now, so let's come back to the question of the parent Hamiltonian. So in principle, I have the wave function. And can I construct a parent Hamiltonian that would have this grand state exactly as, you know, this wave function as the exact grand state. So there have been an attempt to do that in this paper with Inaso Sirac and Hermann Sirac and Anne Nielsen. And what they did here, they used actually a clever rewriting of the Camille Loft in state in terms of conformal field theory co-relator that involved the primary fields of the theory. And from this formulation, they were able, so don't ask me the details because I don't, I can't really answer how they do this, but basically with this technique they are able to derive an exact parent Hamiltonian. Now the problem is that this parent Hamiltonian is very complicated and is long range. So it has a sum of many three body terms at all possible distances. And the weight of this term they decay algebraically with the distance between the sides. Okay, so this is very complicated, but this is exact, I believe. Now what they do in this paper, they say that they can truncate, okay, so they truncate and they just keep the short distance term. Okay, so if you look at the short distance term, which they claim is the most relevant, there will be just a simple nearest neighbor Heisenberg coupling. So these are just spin one, this is a spin one Hamiltonian. Spin one half Hamiltonian. And then it will have some frustrating term. So the J2 is an antifromagnetic coupling, but now between next nearest neighbor site, so along the diagonal. So you have the square lattice. So J1 is a term like that. And the J2 is a term like that. And now what is important is that J1 is positive and J2 is also positive. So these two terms are actually frustrating. So this J2 actually will eventually kill the nearest site for a large enough amplitude. And now they have a term which is very important, which is the term that actually breaks time reversal symmetry. And which is this term. So this term is defined on a plaquette. And it's basically what these operators do. So this plaquette is RJKL. And what the operator do, PIJKL, is just make a cyclic permutation of the spin on the four sides. So it makes a cyclic permutation. And then this guy is the inverse permutation. And then this term is intrinsically complex. So it does break time reversal symmetry. And you can even write this term. Maybe it would be clearer. You can write this carol term as also if you have a plaquette, site 1, 2, 3, 4. You can rewrite it as the sum of scalar carolities. So you can write it as S1 dot S2 cross S3 plus the other triangle like that. S1 dot S2, no. So this one, now this one, S1 dot S2 cross S4, et cetera, plus the two other one. So this is just the... So when you write that, you have to pay attention to which direction you write the triple product between the spin. So this is the simplest carol term you can think of on the square lattice. And so now what they do is they try to map the phase diagram for this Hamiltonian. And the calculation they do, although quite interesting, is maybe not completely reliable. In the sense that what they do is they do exactingization. So sorry, Andreas, but also pretty small size, much smaller than yours. And what they try is to maximize the overlap with the Camille Lofen state. So they get the ground state, they look at the overlap, and then they tune the parameters to get the maximum overlap. And they say that there is a region here where the overlap is very close to one. So they would say that in the thermodynamic limit in this region of parameter space, so this is two parameters, J2 over J1, and this coupling lander C divided by J1, there is a region here where the carol spin liquid phase will be stable. Okay, but this is based on this small cluster calculation. So now what we want to do is try to attack these problems. So consider this simplified truncated model. And now we'll see what we get with using IPEPS, IPEPS method. Yes? So in the original untruncated Hamiltonian, J1, J2, and lander C were fixed? Yes, yes. And so what they do is they truncate, and then they say let's compensate for that with... Exactly. They're not only truncate, but they also allow for changing the parameters they keep. Otherwise it would be too brutal, I don't know where it will be, but there would be one point here which correspond to the truncated model. But it might be out of this blue region, I don't remember. So the two points here is the one they study more closely, and which I will also look at where they have a lot of data. And this is supposed to have, you know, to get the best overlap with the Camille Loflin state on small cluster. This is the one, the parameter I will consider these two points. So this one has J1, J2, and lander C. Okay, now the issue is actually if I want to use PEPPS to address this question, is really whether there is... Can we actually do that? Can we actually describe a carol spin liquid with PEPPS? And there have been a number of arguments in the literature. The first thing comes from a Nogo theorem by Dubai and Reid that concerns actually Tansen network that describes free fermions. So this is Tansen network that is a grand state of free fermion Hamiltonian. And what they say is that, I think this is exact, they say that this carol Tansen network of free fermion have no gap local parent Hamiltonian, which means that if you take such of these Tansen network, basically if you try to build a parent Hamiltonian, and the parameter is not unique, you can build arbitrary number, they will have two properties that could be, if you insist that there would be local, then there would be gap less. Now if you insist that they are gapped, then the hopping amplitude will decay according to a parallel. So you cannot have both. You cannot have a parent Hamiltonian that would be together local and have a gap spectrum. Now whether this applies to the grand state of interacting spins, the experience model is not clear. And there have been one example in the literature of a particular peps of this kind, which they construct from two layers of free fermion peps by using some Gershwin projection on the site. So it's really an interacting version, but it's true that in this case they get diverging correlation legs. So for the moment there's no example of a carol peps that will have finite correlation legs. I'm sorry, but are you going to define carol peps? What is the definition? Yes, I will define it. So what's the meaning of not fully in that? Maybe I should have removed it. Not fully in the sense that I can define some, but still I have to... But it's another example to what? There is a no-go theorem? Yes, it will agree with the no-go theorem. You're right. So next time I will remove that. Okay, good. No-go theorem of Jerome and Nick. Yes. I mean, looking for a carol free fermion, that's kind of the wrong place to look. Free fermions are going to be trivial, aren't they? Well, this is a class of peps which are called these Gaussian peps. I guess this is the reason why they managed to get exact theorem in this case, because it's very simple. I mean, I'm sure they're trying to find a more general theorem. They have been trying for several years, but I haven't seen anything like that. I don't think they... You can free fermions on a lattice. I mean, I could imagine that it would be almost Hawking if that would break down the reversal. Yes. So they're saying that won't ever... Yeah, thank you. Okay, so now what do I mean by carol peps? Just to be more precise. So what I want is I want a state that actually breaks time reversal symmetry in the following way. First of all, I take my square lattice, and now I look at the point group. So I just specify given site here, and the point group is characterized, for example, by some reflection symmetry. So rx along the x-axis, ry and the diagonal, the reflection with respect to the diagonal direction. Then what I want is that for all these symmetry of the c4v group, I want that when I apply this symmetry on the state, I get the time reverse partner. So this is my definition of a carol peps. So if I do any reflection with respect to any axis, then I will get the complex conjugate. So this is my definition. So basically, if I do a reflection, I will just change the circulation of the edge modes, for example. So I will just get the complex conjugate. It's like changing the magnetic field, if you want. So this is my definition. So this is the condition I want to realize. And now I have a simple prescription, which might not be necessary, but at least it works. So it might not be the most general way of constructing carol peps, but at least it's a prescription that gives the result I would like. So the idea is basically to realize that I can construct a wave function, which has a sum of two terms, a real part and an imaginary part here. And these two terms will transform differently with respect to the point group symmetry of my lattice. So for example, this one will transform according to the A1 EREP of c4v. So it's completely symmetric. It's like an S wave, if you want, state. And the other part here will transform according to the A2 symmetry. So it means A2 symmetry is like, I think, a G orbital in atomic physics. That is, if I do a reflection with respect to any of these axes here, I get a minus sign. So it's like an orbital which has many zeros, which are eight zeros. So it basically belongs to the A2 EREP of the c4v group. And now the nice thing about peps is that this is a symmetry of the global wave function. But now I can enforce this symmetry locally at the level of the peps. This is where I find it beautiful, because you need only to enforce it at the level of the peps, the unique peps tensor. So I will construct a peps tensor with the same tensor on every side. And I will enforce this symmetry just at the level of each pep tensor. So how do we do that? To do that, we need to make a classification based on the symmetries of the problem. So the two symmetries we want to include are the SU2 symmetry, because we are looking for a SU2 symmetric singlet state. And also the lattice symmetry, because I want to separate between A1 and A2 tensors. And so what we do is possibly to classify all the peps according to these two symmetries. And basically what we do is, so this is your tensor here. This is a physical space, S, and this is the virtual spaces here. So the virtual spaces, we can basically enumerate all possible virtual space in terms of direct sum of EREP of SU2. So these would be pin 1 half plus 0, and then this one pin 1 half plus 0 plus 0, 1 half plus 1 half plus 0. And that would correspond to different dimension, bond dimension, D3, 4, 5 and so on. So we can make a full list of all possibilities on the virtual level. And for each of them, for let's say if we say now the virtual spin is 1 half, then we can actually look at how many different ways can we actually project this virtual space that belong to these different spaces to the spin 1 half. And we can just comb them. So it can be done efficiently with Mathematica, for example. And then you comb the number of tensors you can generate. And actually surprisingly the number of tensors is not very large. It's much smaller than the total number of tensor elements. The total number of tensor elements would be just D to the 4 times small d, the physical dimension, which is 2 here. So the number of tensors I can realize is much smaller than that. So this is for the A1 symmetry, I have actually 2 tensors, 8, 10, 21, and for A2, 1, 4, 8, 12. So the total number of parameters I will have is this. Because how does it work? Now what I do is I implement the constraint I want, the symmetry I want at the level of the tensor. So I write the tensor now is a real part plus some imagery part. And this real part now is a linear combination of all the tensors of this A1 symmetry. For example, for this case I will have 10 tensors here. So I will have 10 parameters in front here to optimize. And for A2 I will have 8 parameters here to optimize. So it's a very small number of parameters. And so that means that now I can do a different type of optimization that the one described by Philippe, which is the time evolution, imagery time evolution. I can actually do brute force minimization, just minimizing of a small number of parameters. That's possible. So what I'm going to do... So you start with the Hamiltonian, which is that truncated Hamiltonian. And that Hamiltonian can have some symmetries. Space symmetries. These ones. So you are not making any assumption here. You are actually... Yeah, I'm using the symmetry of the Hamiltonian. But I assume the ground state doesn't break those symmetries spontaneously. Yeah. But you are also fixing the one I mentioned, the SU2 representations on the one in C. And this is your restriction. Yeah, but if I go further and further in D, if I crack up D, then I will have to look at separately all the possibilities for the virtual space. So you see here, for example, for five I can take this virtual space or I can take this virtual space. They will correspond to two different solutions, they don't mix. They don't want to mix them. And then I will look at the energy, which is the lowest. I was trying to understand in which sense you are making an ansatz. What is the ansatz? The ansatz is saying that by doing this prescription here, first taking the same tensor on every site and by writing the tensor as in this form, I have the most general tensor that gives rise to a translational invariant state which has the carality, the carol property I want. But there might be, you know... But this is not ansatz, right? I mean, you assume... No. You assume translational invariant that's making ansatz. Yeah. No, but you know, it's not clear that the family of all these states would span the whole space of tensors which are translational invariant. You might have tensors which are defined with two sites, for example, with two different tensors, and which would be still translational invariant, but that you would not be able to rewrite with a single tensor on the site. I mean, that's a complicated problem. I mean, you agree that, no? I mean, it's reasonable that it will capture most of this... Yeah, yeah, I agree. I agree. Yes. I agree. Okay, so now how are we going to do... So we have these peps that depend on a small number of parameters, and the method I will go... I will use is basically very much the same as described by Philippe. So this is the infinite projected integral per state method. So I will use the environment. So basically, the idea is, you know, I have a plaquette here of four sites, where my Hamiltonian acts. So I really have to consider four sites. And then I have the tensor on the top, the psi, the bra and the ket. And I want to investigate the expectation value of this operator, the Hamiltonian operator, which I have in yellow. Now, of course, what I have is all the tensors up to infinity around this plaquette. So what I want to do is I want to use an approximate contraction of all the tensors from infinity up to some environment here. So I want this environment here, tensors, to capture all the contraction of the tensors from infinity. And as Philippe explained nicely, this is done with real-space renormalization scheme based on this corner transform matrix. So this corner transform matrix is this matrix here on the corner, and then also the environment involves this t tensor here that build the edge of this box here. And so once I have the energy for a given set of tensors, what I do is just brute force optimization. So I can compute the gradient by varying a little bit each of these parameters. So I compute the multidimensional gradient and I do the full optimization with conjugate gradient method. And this is feasible because just I have reduced enormously the complexity of the problem to a small number of parameters. So it's by using all the symmetries I can. Now there are, compared to what Philippe described, there are a simple simplification that can be used. The fact that first the corner transform matrix here that I have on every corner would be exactly the same because those tensors by construction are taken exactly the same. So this corner transform matrix would be the same. I have only one corner transform matrix and all these t also would be the same. So I need only to do a renormalization of just one corner and then I can just copy and paste the other matrices on the environment. The other thing which makes some simplification is that this corner transform matrix now is a mission. So instead of doing SVD, I can do more stable excitedization to get the, to truncate. And actually that was suggested by Philippe to me at some point. Long time ago. And it actually works well. And also the last thing which is being done in this renormalization process is that the SU2 symmetry is preserved at each step because at the truncation step, one avoid to cut within the SU2 multiplet. So you have really to look at where are the SU2 multiplet and just keep the full SU2 multiplet when you make a truncation. But apart from these simplification and I think make the method more efficient, the algorithm is very similar to what Philippe explained that basically what you do is you have the corner here, you just add one side and then by doing this excitedization you do a truncation and you introduce some isometry here and then this isometry you absorb in the t-tensor to get the new t-tensor. So this is a standard CT-MRG real space renormalization. Okay. So maybe I should, unless there is some question about the method, I will go to some result. Okay. Okay. So what I now I do is I go to the previous truncated Hamiltonian I described and here I will focus on this point in the parameter space. And also for this particular value of the coupling which in the original paper was claimed to be a point where the overlap with the Camille Lofen state is maximum. So it's potentially the point, the parameters which are best to obtain such a state. And so what I do here is the energy as a function of what? As a function of d squared divided by chi. So chi is the environment dimension. So I think I use the same notation as Philippe. So this is the dimension that enters here in blue of the environment here. And this dimension you should scale as Philippe said you should scale it like d squared. So the relevant parameter is really d squared divided by chi. And what you want of course is go to the limit where chi is going to infinity. So you want to go here. Do you want to extrapolate here on this axis? And so here I show two examples for two different choice of the virtual space. So I think this one is for 1 half plus zero. So it's very small bond dimensions d equals 3. And this one is d equals 4. This is 1 half plus zero plus zero. And this one is 1 half plus 1 half plus zero. Which gives this point here. But I've been unable to get further points in this because of instability problem. But what you see already for this small bond dimension that if I just increase chi, then I have to extrapolate like linearly with one of a chi. So this is very important to do the extrapolation. And I can get quite accurate estimation of the partial energies for these two cases. And what is interesting now if I compare to the Kalmeyer-Loflin energy in the thermodynamic limit. So it's reasonable to compare. I mean both calculations are in the thermodynamic limit. And this Kalmeyer-Loflin energy you can get very accurately in Monte Carlo. And what you see is that the energy, even for this small d parameters, I get energy which has below the Kalmeyer-Loflin energy which is the targeted state. I mean this is why this sometimes we're constructed. So I get something better. So which means that that suggests that the peps is a reliable description of the ground state of this state. So now what I want to see is whether this state has one important feature of the carol spin liquid which is the existence of carol H-modes. So this is the first thing to check. Monte Carlo is done on the same Hamiltonian. Exactly. No, there's no Hamiltonian. It's for the Kalmeyer-Loflin. It's just Monte Carlo for the Kalmeyer-Loflin state. There's no Hamiltonian. Just the energy. No, but this is energy in this Hamiltonian for the Kalmeyer-Loflin state. And then the exact diagonalization. Okay, so this I didn't talk about that because I thought it would be. The exact diagonalization is much lower but for the reason that usually you have in small clusters you have very strong quantum fluctuations that lower the energy very much. So if you would be able to do finite size scaling what you would see that actually the energy for 30 sites, when you increase the number of sites it would just go up dramatically. It's difficult to compare to this value. If you don't have finite size scaling I don't see when it means anything. Is it a gap state so the finite size corrections? Well, how do you know it's a gap state? That's the point. It's an Hamiltonian that was constructed in order to indeed get a gap phase but at the end of the day you don't know what you get at that location. So I think that's an interesting problem for you to try to crank up the... maybe we can get reasonable energy. I think this is the limit they've gone maybe 30 sites. And I don't think it's very good because it's 5 times 6 so it has an odd number of sites in one direction which I think is not too good when you have... it frustrates somehow the antifermagnetic order. So it's not a very good cluster shape. Okay, so let me come to the issue of the edge states. So now there is a conjecture which has been put forward by Li and Aldein in a famous paper. It's a recent paper that actually the entanglement spectrum will capture all the physics of the edge modes. So for example, if you take a cylinder like that and you calculate... you make a partition into A and B so it's a mathematical partition and then you calculate the reduced density matrix by tracing over the B... the degrees of freedom of the half-cylinder of this projector, psi-psi. And then you write row A as... you can write row A as the exponential of minus some Hamiltonian with some normalization. Then the spectrum of this guy is in one-to-one correspondence with the actual edge states. And it has been checked for fractional quantum whole state and it works extremely well. So then the idea is to compute the entanglement spectrum. So the entanglement spectrum is the spectrum of this guy, of this Hamiltonian. So it's the spectrum of minus the log of the reduced density matrix. And from this we should have... basically we should get something that is in one-to-one correspondence with the actual edge states of the system. Is this just a general observation? I think it's a conjecture. I don't think it has been... Or is this a true for these specific systems? They have shown that for the... I think for the... for the... and then it has been used in many other cases for non-nibalian fractional quantum whole state. I think this example is always having come under these archival phases with a gap. And maybe that's why I'm here. If you don't know, if you have a gap. You will see, because... Yeah, exactly. So now there is a puzzle still I'm coming to. You explained how this correspondence works, this one-to-one correspondence. Okay, so this one-to-one correspondence is... well, maybe I can show you here. So what I draw here now with this entanglement spectrum has a function of the momentum along the circumference of my cylinder. And for PEP there is a very simple way of computing this guy. So I will not go into the detail, but there is some bulk-edge correspondence that we have established that allow to a very efficient calculation of the entanglement spectrum for an infinite cylinder. So what you keep finite is the circumference, but you can let the cylinder become infinite. And then you can get the spectrum of this. And what you get as a function of the momentum along the circumference is something very very particular. What you find is really linear dispersion. So you really find carol modes. You don't find the CFT spectrum that Giffrey was showing, where you have really half two branches and everything inside. Here you just have one branch. So it's really carol. You just have one branch going in one direction with one velocity. And the correspondence is the following is that now if you do the precise counting here what you get is basically the counting that you would expect for the CFT that characterizing the H states. So maybe the energy levels would not be exactly at the right position but the counting would be correct. And by counting I mean now if you look at the quantum numbers of all these states for a given K, for example here 2 1 plus 0 means I have 2 triplet here which are these symbols plus 1 singlet. And then if I go up you know I go here then I have all these states. I have a quintuplet plus 2 triplet plus 2 singlets and so on. And if I look at what I should get from the CFT so if you're like me you don't know much about CFT so you take the yellow book and you look at what you should get. So this is for the SU2 level 1 CFT and you look at the table in the book and what I provide is the SU2 decomposition. So all the quantum numbers for the tower of state that you get. And so here actually they use different notations so 2 means actually spin 1 and 1 means spin 1 off. I don't know when they do that. So you have all these countings for the different levels of the tower of state and you can check that actually the counting is exactly what you get here. So this is exactly the right counting here 1 triplet, 1 triplet singlet and so on. And of course here there are some dispersion because the size when you go further in momentum is 10 minutes. Okay, thank you. Sorry, what is meant by the even non-sector in your case? Yeah, so this is something I forgot to say but there is a gauge symmetry of the tensors that allow me to construct two different sectors. So if I can play with symmetry, so the way to compute this is just to iterate the transfer matrix basically from infinity. And so I can fix these topological sectors using this z2 symmetry in the initial state and then it would be preserved and so I can just diagonalize this reduced density matrix in these two different sectors. Why is the momentum more high and not too high? Ah, you notice that, okay. Is it because you can put... Yes, okay I tell you, okay, there is technicalities because in fact in order to have only one tensor per site what I do is I do a spin rotation on the B site otherwise I would have to use two different tensors and so that breaks that somehow breaks the translation symmetry so there is some translation symmetry breaking here and it means the spin multiplates will appear at momentum k and k plus pi. So I could use momentum modulus 2 pi but then I will have the multiplate coming at two different momentum. So what I do is I just write everything in terms of modulus pi and so just to bring back all the different terms of the multiplates together. But it's not the property of the boundary theory. No, no. Yes, so you see here I have pretty good evidence that actually the edge these states have well defined kaol edge modes which are described by this SQ2 level 1 theory. Now the surprise comes from the investigation of the correlation function. So as you say I would expect that my state would be gapped. I have nice edge states so I would expect the bulk is gapped and actually this is not the case. So for example you can calculate two types of correlation function and neither of them actually turn out to be short range. So the first correlation function you can compute is basically the dimer-dimmer correlation. So what you do, now that you have your environment tensors what you can do you can construct a strip here. And here look at the correlation between s.s on these bones with s.s on these bones. So it's like a four-point correlation function as a function of the distance between the two objects. And basically this environment here takes into account the rest of the system. You have to think that basically you have contracted all tensors from infinity to there and the effects of the environment is taken care by these red tensors here. And so in principle you can calculate correlation to basically arbitrary distance. And so this is what I show in a semi-lock plot. So this is the log of the correlation of the function of distance. And what you see is something that you would expect that for large distance because for finite chi of the environment it will all be a straight line which means that there is an exponential decay at very long distance. But now what appears is that if you now compute these correlation lengths from the slope here and you draw the correlation lengths as a function of chi of the environment dimension you see that actually it seems to scale linearly with chi. So that suggests that actually it doesn't saturate. For the value of chi I've considered it will never saturate and these are pretty large value of chi, going up to 20 times d squared. So it's not small. And it seems to vary linearly which means that the correlation length diverges and that's consistent with a parallel decay of the dimer-dimmer correlation. And actually if you look at short distance below this exponential behavior you can actually pretty much fit the data with a parallel. Now you can also look at 5 minutes. Yes, I would be almost down. So this is the last plot I want to show. So now you can also compute the spin-spin correlation. So the first correlation was basically the correlation in the singlet sector and this is now the correlation in the triplet sector if you want. And you do the same, you play the same game to construct this trip here and you put the environment on the side and then here now you have a spin operator and a spin operator and you look at the correlation between these two sides as a function of d and again you can do the plot the data on a semi-log plot the log of the spin-spin correlation versus the distance d here between the sides and what you find is the first thing you find is that a short distance you find a very short correlation length a decay, a very sharp decay with a very short correlation length and actually this correlation length is compatible with a Kamiya Loft instant. It's basically what they get in Monte Carlo so it means that a short distance the wave function, the grand state we get the ansatz we get is as a same property as the same spin-spin correlation a short distance, maybe up to distance, I don't know, 10 or something like that it's very similar so a short distance I think we cannot distinguish our ansatz from the Kamiya Loft in state but now a long distance it's more tricky because now we get again this exponential decay but now again the length scale we extract doesn't seem to saturate with Kai again Kai up to 16 times this square and we don't see any sign of saturation but now the difference with the dimer-dimer correlation is that if I look at the weight here corresponding to this exponential decay it's very very small so actually if I write my correlation function in terms of the sum of exponential which is basically what I can get from this spot then the weight here will be infinity to the the largest correlation length is becoming very small so in this decomposition I will have a very exponentially small weight so all distances will be if I do this expansion all distances will be will be included, side-max will go to basically infinity with Kai going to infinity but the weight will be becoming very small so actually it will be slower than of course a pure exponential but it will be faster than algebraic algebraic decay because this is something in between maybe a stretch exponential or something complicated like that but it's clearly not simple to to so now this is the big issue that is left is whether these features are the really generic features of the ground set of this model and or are there artifacts of the PEPS representation and also if there are really generic features of this model are the really generic features of all carousel liquids of this type so this is basically the conclusion of this study and so I would like to summarize our outlook so to summarize I would say that this method offers a really new conceptual understanding and a quantitative description of many quantum antiferromagnets with exotic ground states so not only the one that were described by Philippe, we show spontaneous symmetry breaking like stripe order or this sort of order but also topological state maybe a more involved type of ordering that also what is important is that the virtual degrees of freedom in fact play really a physical role at the boundary because the physical the virtual degrees of freedom really are the one that actually builds the edge the edge states the edge Hamiltonian is really an Hamiltonian that acts on the virtual degrees of freedom at the edge of your system so they acquire some physical meaning in a way now what can be done if I have 30 more seconds is extend this to many other cases so one obvious extension is to look at Hamiltonian with larger spin like for example spin 1 where we might realize some carousel liquid which would be non-habilian carousel liquid with more complicated edge states maybe SU2 level 2 type edge states and we know that in the field of fractional quantum there are such more involved topological states so the aim is to try to to construct this in the field of quantum magnetism of course we can use other lattices we can look at other symmetries like Philippe mentioned and I don't think so far there's been any discovery of SU3 or SU4 carousel liquid so far no there is and maybe applied to discretize field theory but this is really not my field so I will not go elaborate more so and then finally I would like to thank my collaborators over the past years actually especially Ignacio Sirac, Norbert Schur and Roman Horus from whom I learned a lot of these techniques in the last say 7 or 8 years also I would like to thank my collaborator Mathieu Mambrini to lose for his major role in the classification of SU2 peps and also Yana Fleck who collaborated at some point on the carousel liquid issue ok and I would like to thank you for your attention questions so if we take the point of view that this for some reason is gapless then do you think there's a chance to have emergent Lorentz environs I thought there would be some emergent gauge symmetry that you want gauge symmetry at some point because we know that there are simple peps which have gauge symmetry and we know why there is no gapless so maybe there is some emergent gauge symmetry that appearing I was wondering what is the dynamical critical exponent could it be one yeah I don't know how do you compute that I mean you would have to look at the spectrum of the excited state from the very heart yeah exactly I mean constructing excited state for this carousel well it could be attempted but I don't have any any idea I mean maybe with the method you suggested you know with some beat answers and you but that's I didn't understand what the conclusion is for the ground state the conclusion is that I have you know I have a state I have constructed a state for this Hamiltonian which has a better energy that the Camel often state so with point to the fact that it is a good representation of the ground state okay and which has both features first feature it has well defined carol edge modes which are described by this S2 level 1 CFT I think this is very very good evidence and and this you extracted those edge modes from this entanglement formula yes yes which is still valid even though that's why I'm confused what is what is it that goes into this the conjecture oh you want to me do you mean whether I can use a conjecture in this case the answer is I don't know I don't I have to ask Duncan whether he thinks it's still valid but I don't know but it's a conjecture yeah so it has been yes no but I mean no no you might have a feeling you know you might have a feeling there are edge modes for certain you would want to know that you mean edge mode this is dynamics and we haven't established whether this is a gap Hamiltonian or a gapless Hamiltonian that may have an important impact on the dynamics right but what's clear is that when you compute this entanglement spectrum it has a structure that is compatible with the carol CFT yeah okay so I should rephrase maybe don't say edge modes but entanglement spectrum which are well described by this S2 level 1 CFT and the question is what to make with that yes at the same time so this is one feature and the second feature is that it seems I seem to have good evidence that the correlation functions are long range both in the spin and in the triplet sense which would indicate that this is a gapless Hamiltonian exactly exactly because Hamilton is local so I think you can immediately would say if this is really a good representation of the ground state then the Hamiltonian should be should be a gapless yes sorry I missed it it's my fault but your state to construct do you do that for fixed D like D equals 3 or what is the bond dimension of the states or do you have a family of yes yes exactly I have several families so here I only showed so each of these curves correspond to a different family which is optimized so for example the blue curve correspond to this D equals 4 case 1 half plus 0 plus 0 for the largest I could handle which is 256 and the green one correspond to this very simple 1 half plus 0 and there are two sets of curves that correspond to chi 36 and 144 so this one is for 36 and this one is for 144 and you see the decay is here the decay is less severe so the correlation length is bigger so there correspond to two different points in this plot I'm just wondering a bit I mean I I'm personally thinking that this Hamiltonian if it's truncated it's very likely that it's a gap Hamiltonian and it's in the same the ground state is a chiral spin liquid but the genuine one without the gap and then if you take these peps theorems no-go theorems you can still ask okay it might be that they're not at finite bond dimension they're not able to exactly reproduce a chiral spin liquid but how well do they approximate that and since you said yourself not initially at short distances it seems like reproducing this I believe this I believe at short distance I am okay basically I get the right shot it's not in the fix but it could be that as you crank up T and you do a heavier job that actually these tails are somehow the compromise between the fact that at finite bond dimension you're not able to get everything correct so this is the approximation thing but actually the further you go in T the more you're right there are two options I mean one option is I get the correct physics and this Hamiltonian does have this type of ground state and the other option is there is some kind of no-go theorem that prevents me to really for finite D get a fully transparent state but if I crank up D I will go I will approximate better and better and I get shorter and shorter correlation because I mean there's an argument which I myself did not fully understand but I think Sharon Dubai and the they have some understanding that these boundary theories they're actually like living on the edge no and then I think there is some connection with the fermion doubling problem or no-go theorem how you but you cannot really write down some local Hamiltonian which is completely chiral but I wonder whether this this thing is not the fact that there might be another mode if you say actually I don't have one carol mode but I have two but the other one with a very steep velocity then I think I can I'll be right but you would not be able to see it because it's so steep that the first excitation at 2 pi over L would be beyond the roof so I'm not claiming there is only one mode but at low energy you don't see the other mode but there could be another I mean conceptually there could be another mode that with a very steep velocity yeah coming out of the path how may your lovely bullet happen yeah yeah yes if I can comment so first comment is that it's godless for this Simon's collaboration it's more interesting and the second it's amazing that it's a really second example of a very simple Hamiltonian for which it seems there is a controversy it's not clear if it's god to if it's godless quite amazing it would seem like if you take a random Hamiltonian if you do something random and maybe not particularly precise like peps then it would be much more likely it seemed like it would be much more likely to fall into something more gap than it is in the real life not less gap not once you factor in chirality that is being enforced exactly so chirality seems to be no but I know another example which is the J1, J2 the same model but without the counter and there is a controversy whether at J2 equal to 0.5 of J1 whether you get a symmetry breaking state, whether you get a dimer state a break translation symmetry or whether you get also a gap place I actually don't understand this comment about chirality so we have these plots which show that chiral states there exist chiral states which are gap so chiral states that find that one dimension is gap I don't know what that means I got to the property of a Hamiltonian we saw the plots which are exponential we saw the plots for correlation functions and find that both dimensions which are no no you mean those states were carol no no which plot you mean my plot or this one no no but this doesn't mean exponential you mean that if you approximate the exact contraction by an environment with a finite bond dimension then it is exponential but if you just imagine you crank up the dimension of the environment here then what you will see is that this decay is less and less severe and then eventually this length scale here will just go diverge so even for a fun idea if you would be able to crank up chi to infinity then probably you will see no exponential you will see purely algebraic decay yes so it's a finite chi it's an artifact of the fact that you truncate the contraction when you do the I think the following is even true is it correct that the challenge now is to find a chiral peps with a finite correlation one yeah whether it goes or does not come from just you enforce chirality and now look if genetically this seems to have power or decay of correlation so there seems to be a theorem out there that any chiral peps there might be a theorem that any chiral peps has I still think there is an idea that as you crank up the physical bond dimension it could be that on longer and longer scales the system looks exponential and there is an algebraic tail in the end which is due to a no-go theorem and so you try to act against a no-go theorem but as you crank up bond dimension you do a better job of being gapped or looking exponential over some scale and then the algebraic tail which is a bit weird but if you mean that you are trying to reproduce an exponential decay by summing power of decay right that's so yeah you're trying to simulate challenge correlations but you are using answers that a finite bond dimension would necessarily have power of decay so at short distances you manage to superpose many power of decay so that it looks like an exponential but it will be the two power of decay emerges I just have a quick question you mentioned that you would like to realize SU2 level K so now we have a model can I take the Nielsen Sierra exactly this is what I no no no no so it's a more complicated spin one so it includes so they have an equivalent paper for spin one when they do the same thing they just start from the CFT correlator and they deduce apparent Hamiltonian which again for spin one is long range and they truncate again and they play the same game and eventually they came up with a simple Hamiltonian which has bi-linear interaction, Nielsen neighbor so maybe I can write it down do we have one minute to write it down so we are actually working on this one so it's a bit more complicated so it's a spin one and it has a J1 term so it's Nielsen neighbor and the difference now is that there is a bi-quadratic so there is between Nielsen neighbor side so it's si.sj square and then it has again next Nielsen neighbor bi-linear term so this is on the diagonal and there is bi-quadratic term on the diagonal so the same si.sj square so this one was not allowed for spin one F so now it comes in and there is the the the character and they have a proposition for the values of all these coupling so I think it's nice to have this proposition because now the parameter space is so large that if you would just do searching in such a large parameter space that would be terrible but they have a proposition for what are the optimal values of these and they claim basically the ground state is like the Moorid non-nibalian fractional polynomial state for spin one so the game is to try to to attack this again with IPAPS Thank you