Investigation of the chiral antiferromagnetic Heisenberg model using PEPS
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Transcript: English(auto-generated)
00:16
First I would like to thank the organizers and especially Slava for allowing me to talk, to sneak into the program.
00:26
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.
00:43
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.
01:00
So it's it's a Heisenberg antiferromagnetic model, but which has a very particular feature. It has chirality, that it breaks time-reversal symmetry and it is supposed to host new type of state which are topological ordered states.
01:20
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 from the
01:43
for testing the ability of IPEPs method. Then what I also present is tell you a little bit about the type of PEPs and ZATs 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.
02:06
So there will be less variational parameters in the in the ANZATs 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 with using the
02:25
particular symmetry of the the PEPs ANZATs I'm using. And then I will go to the to the result considering this particular model. So the actual motivation comes from the
02:41
physics of the fractional quantum Hall effect, I guess. Which give actually beautiful example. So that deals with the continuum 2D plane.
03:00
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 Laughlin wave function with a well-known Laughlin wave function
03:23
which has a very simple form. So this is Psi Laughlin. So it's just a simple function of this sort. So ZR is just the
03:41
complex coordinate of the particles. And then there is some exponential factor to confine the particle on a disk. And M is actually connected to the filling fraction. So the filling fraction which is just the number of particles
04:03
divided by the number of fluxes in the system is basically 1 over M. So for the most well-known fractional polynomial state the filling fraction is one-third, so M is just 3. And so obviously you get a
04:22
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 one-half.
04:42
And so this is a beautiful, very simple wave function that captures very well the physics of the fractional polynomial state. In the late 80s, Kallmayer and Laughlin actually wrote the
05:01
version on the extension on the lattice. So this is the Kallmayer... So I'm not sure I get the spelling right. Is that correct? And the idea is basically to take this bosonic M equal 2 nu equal one-half wave function and just to put on the lattice.
05:24
So now the coordinates here z, zi are now leaving on the lattice, for example square lattice. And to map the wave function, the Laughlin wave function, on a spin one-half
05:44
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
06:02
to a spin down. And if you do this simple prescription, basically the wave function you get is just a global singlet wave function, which doesn't show any symmetry breaking, no longer in order, but it has topological order.
06:26
And I will be more precise by what I mean by topological order. And so this is the simplest realization and most beautiful representation of what is called a chiral spin liquid.
06:40
So this chiral spin liquid somehow can be viewed at the analogs of the fractional quantum Hall 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
07:04
chiral 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, and let's see,
07:20
construct some, let's parent Hamiltonian for this chiral spin liquid, and then try to investigate them with some techniques, which here is going to be the IPES. To be clear, this is for odd filling fraction, right? So this is for M equal 2. This is a bosonic state. This is a spin 1 half state. So it's, this is for this case, M equal 2 case.
07:50
So that corresponds to half filling, actually, there is a commensurate relation. And so the number of spin up and the number of empty side is actually equal. So this is why you get a global singlet, which is invariant under spin rotation.
08:08
So this is what I just said. So this Kamil-Lauflin state is, sorry, mix up, is really a paramagnetic example of a chiral spin liquid.
08:23
And the goals are the following. So what we like is to search for some simple projected entanglement per state, PEPs, ANZATs, that will describe such states. And we want to apply these and optimize those wave function
08:41
to study simple parent Hamiltonian that will, you know, which have the potential to have such phases. And we might also would like to have such Hamiltonian to be local. And there might be a problem here that really to construct such model.
09:04
So if we look for topological spin liquids, really where we have to go is to go beyond the Ginzburg-Landau paradigm. So the order parameter paradigm, because those state are characterized by the absence of any spontaneous broken symmetry.
09:25
They don't have local order parameters, but they have what was actually induced by when topological order. So by topological order, I mean, I can maybe take the definition of when is a state whose ground state
09:42
degeneracy will depend on the topology of space. So if I put, for example, the Cameo-Laufin state on a sphere, on the torus or on another manifold with higher genus, I will get different number of degeneracy of the ground state.
10:04
So this will have really a ground state topological degeneracy here. And those state will be characterized in principle by a gap. So they will be incompressible. And then above the gap, the excitation will be
10:24
fractionalized and unique excitation. So this is really where we target. Yes. So this Cameo-Laufin, that's a wave function. Yes. For a spin system. Yes. And it does not come with a Hamiltonian, does it?
10:42
So I will go to exactly to this question later on. So there is a Hamiltonian that has been constructed by your friends that you know, Ignacio and company. And I will describe that. But no, I'm saying the original proposal by
11:02
Cameo-Laufin did not come with a Hamiltonian. No, but they were actually suggesting that it will be the ground state of some frustrated spin model. I think, yeah, in the original paper, maybe the triangle lattice or something like that. But they were suggesting that it could be
11:21
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 paper. In their suggested Hamiltonian that would sterilize this wave function, did they suggest that this would be a gap system? Do you know? Yes, yes, yes, yes. Yeah, because I think it's known that this wave function
11:42
should be exactly like that. It should have a gap. Yeah, like the fraction of quantum Hall state on the plane as 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 ground state of the Hamiltonian,
12:01
it means the Hamiltonian 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. OK, let's say this way.
12:20
But is 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, yes. OK, so this is one feature you would like to observe.
12:42
And the other feature, which is characteristic of this topological state, is the fact that they have edge states. So, for example, if you put your topological Carroll state on a sphere like that, and you just imagine you cut the sphere into two hemispheres,
13:03
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
13:21
that the bulk has long-range entanglement. And they are also described by simple CFTs. So, for example, in the case of the Cami and Loftin state, what you would observe is the fact that these edge states are described by simple SU2 level-01 CFTs,
13:42
so just central charge equal to one. OK, so now, 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 ground state exactly as,
14:03
this wave function as the exact ground state. So, there have been an attempt to do that in this paper with Inassos Hirak and Hermann Schirra and Ann Nielsen. And what they did here, they used a clever rewriting of the Cami and Loftin state
14:24
in terms of a conformal field theory collator that involved the primary fields of the theory. And from this formulation, they were able... So, don't ask me the details, because I can't really answer how they do this,
14:42
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
15:03
with the distance between the sides. So, this is very complicated, but this is exact, I believe. Now, what they do in this paper, they say that they can truncate. So, they truncate and they just keep the short distance term.
15:20
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 spin-one Hamiltonian, spin-one-half Hamiltonian. And then it will have some frustrating term. So, the J2 is an antiferromagnetic coupling,
15:40
but now between next nearest-neighbor side, so along the diagonal. So, I 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
16:00
and J2 is also positive. So, these two terms are actually frustrating. So, this J2 actually will eventually kill the near state, if for a large enough amplitude. And now they have a term which is very important,
16:21
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 operators do, Pijkl, just make a cyclic permutation of the spin on the four sides.
16:42
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 Karel term as also,
17:02
if you have a plaquette, site 1, 2, 3, 4, you can rewrite it as the sum of scalar chirality. So, you can write it as S1 dot S2 cross S3,
17:20
plus the other triangle like that, S1 dot S2, so this one, no, this one, S1 dot S2 cross S4, et cetera, plus the two other ones. Okay, so this is just the,
17:41
so when you write that, you have to pay attention to which order, to which direction, you write the triple product between the speed. So, this is a simpler Karel 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,
18:02
and the calculation they do, although quite interesting, is maybe not completely reliable, in the sense that what they do is they do exact linearization, so sorry Andreas, but also pretty small size, much smaller than yours, and what they try is to maximize the overlap
18:22
with the Camiel Orphan state. Okay, so they get the ground state, they look at the overlap, and then they tune the parameters and 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,
18:41
so this is two parameters, J2 over J1, and this coupling lambda C divided by J1, and there is a region here where the the Karel-Spin liquid phase will be stable. Okay, but this is based on this
19:01
small cluster calculation. So, now what we want to do is try to attack this problem, so consider this simplified truncated model, and now see what we get with using IPEPs method. Yes? So, in the original untrunked Hamiltonian,
19:24
J1, J2, and lambda 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 not only truncate, but they also allow for changing the parameters they keep. Yes, yes.
19:40
Otherwise it would be too brutal. I don't know where you will be, but there will 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,
20:01
and this is supposed to have, you know, to get the best overlap with the can be a laughing state on small cluster. So, this is the one that will, the parameter I will consider these two points. So, this one has J1, J2, and lambda C. Okay, now the issue is,
20:21
actually if I want to use PEPs to address this question, is really whether there is... Can we actually do that? Can we actually describe a Carol spin liquid with PEPs? And there have been a number of arguments in the literature.
20:40
The first thing comes from a no-go theorem by Dubois and Reed, but that concern actually tensor network that describes free fermions. So, this is tensor network that are grand state of free fermion Hamiltonian. And what they say is that,
21:00
I think this is exact, they say that this Carol tensor network for free fermion have no gap local parent Hamiltonian, which means that, if you take such of these tensor network, basically if you try to build a parent Hamiltonian and the parent Hamiltonian is not unique,
21:21
then you can build an arbitrary number, they will have two properties that could be, if you insist that there would be local, then they will be gapless. Now, if you insist that they are gapped, then the hopping amplitude will decay according to a parallel.
21:40
So, you cannot have both. You cannot have a parent Hamiltonian that will be together local and have a gap spectrum. Now, whether this applies to the grand state of interacting spins model is not clear.
22:02
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 Gauss-Willel projection on the site.
22:21
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. Are you going to define Carol PEPS?
22:40
What is the definition? Yes, I will define it. Yes, sorry. So, what's the meaning of not fully in red? Maybe I should have removed it, yes. Yeah, not fully in the sense that I can define some, but still I have to... But it's another example to what?
23:00
I mean, there is a no-go theorem? Yeah, he will agree with the no-go theorem. Yeah, you're right. So, next time I will remove that. Okay, good. Thank you. No-go theorem of Jerome and Nick. Yes. I mean, looking for the chiral
23:23
free fermionic seems 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 this Gaussian PEPS. I guess this is the reason why they managed to get exact theorem in this case
23:41
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 that... I don't think they have a... Free fermionic theories on a lattice. I mean, I could imagine adding complex Hall things
24:01
that would break time reversal. Yes, yes. So they're saying that that won't ever... Yeah, yeah, yeah, exactly. Okay. Okay, so now what do I mean by chiral PEPS? Just to be, let me be more precise. So what I want is I want a state
24:20
that actually breaks time reversal symmetry in the following way. So assume I take my square lattice and now I look at the point group. So I just specify a 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,
24:41
the reflection with respect to the diagonal direction. Then what I want is that for all these symmetry of the discrete C4V group, I want that when I apply this symmetry on the state, I get the time reverse partner.
25:02
So this is my definition of a chiral 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.
25:20
So I will just get the complex conjugate. It's like changing the magnetic field, if you want. So this is my definition. And so this is the condition I want to realize. And now I have a simple prescription which might not be necessary,
25:41
but at least it works. So it might not be the most general way of constructing chiral 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, so a real part and an imaginary part here.
26:01
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 irrep of C4v. So it's completely symmetric. It's like an S wave, if you want, state. And the other part here will transform
26:21
according to the A2 symmetry. So it means A2 symmetry is like a, 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,
26:44
which are eight zeros. So it basically belongs to the A2 irrep 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
27:02
locally at the level of the PEPS. This is why I find it beautiful, because you need only to enforce it at the level of the PEPS, of 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 PEPS tensor.
27:21
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're looking for a SU2 symmetric singlet state,
27:41
and also the lattice symmetry, because I want to separate between A1 and A2 tensors. And so what we do is possible to classify all the PEPS according to these two symmetries. So basically what we do is, so this is your tensor here, this is a physical space S, and this is the virtual spaces here.
28:02
So the virtual spaces, we can basically enumerate all possible virtual space in terms of direct sum of eRep of SU2. So this would be pin one half plus zero, and then this one pin one half plus zero plus zero, one half plus one half plus zero, and that would correspond to different dimension,
28:23
bond dimension, D3, four, five, 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, the physical spin is one half,
28:41
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 one half. And we can just count them. So it can be done efficiently with Mathematica, for example. And then you count the number of tensors you can generate.
29:04
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 four times small d,
29:26
look at the physical dimension, which is two here. So the number of tensor I can realize is much more than that. So this is for the A1 symmetry, I have actually two tensors, eight, 10, 21,
29:43
and for A2, one, four, eight, 12. So 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.
30:03
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 eight parameters here to optimize.
30:22
So it's a very small number of parameters. And so that means that now I can do a different type of optimization than the one described by Philippe, which is the imagery time evolution. I can actually do brute force minimization.
30:41
Just minimizing over a small number of parameters. That's possible. So what I'm going to do... So you start with the Hamiltonian, which is the truncated Hamiltonian. And that Hamiltonian has some symmetries, space symmetries.
31:01
These ones. 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. But you're also fixing the one we mentioned, the SU2 representations on the one inside. And this is your restriction.
31:22
Yeah, but if I go further and further in D, if I crank up D, then I will have to look at separately all the possibilities for the virtual space. So let's say, you see here, for example, for five, I can take this virtual space or I can take this virtual space.
31:40
They will correspond to two different solutions. They don't mix. I 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, right? What is the ansatz? Oh, the ansatz is saying that by doing this prescription here,
32:02
first taking the same tensor on every side, and by writing the tensors in this form, I have the most general tensor that gives rise to a translation invariant state, which has the chirality, the chiral property I want.
32:22
There might be, you know... But this is not ansatz, right? I mean, you assume translation invariant, that's making an ansatz? Yeah. No, but you know, it's not clear that the family of all these states span the whole space of tensors which are translation invariant.
32:42
You might have tensors which are defined with two sides, for example, with two different tensors. And which would be still translation invariant, but that you would not be able to rewrite with a single tensor on the side. I mean, that's a complicated problem.
33:00
I mean, you agree with that, you know? But it's reasonable that it will capture most of this... But the main limitation of these answers remains defined upon them, is the... Yeah, yeah, yeah. I agree, I agree. Yes, I agree. Okay, so now how are we going to do?
33:20
So we have these peps that depend on a small number of parameters, and the method 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.
33:41
So basically, the idea is, you know, I have a plaquette here of four sides, where my Newtonian acts. So I really have to consider four sides. And then I have the tensor on the top, the bra and the ket. And I want to investigate the expectation value of this operator,
34:02
of 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.
34:26
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
34:43
based on this corner transfer matrix. So this corner transfer matrix is this matrix here on the corner. And then also, the environment involves these T tensors here that build the edge of this box here. And so once I have the energy for a given set of tensors,
35:03
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 multi-dimensional gradient. And I do the full optimization with conjugate gradient method. And this is feasible because just I have reduced enormously
35:21
the complexity of the problem to a small number of parameters. So it's by using all the symmetries I can. Now, compared to what Philippe described, there are a simple simplification that can be used.
35:40
The fact that first the corner transfer 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 transfer matrix would be the same. I have only one corner transfer matrix. And all these T also would be the same. So I need only to do a renormalization of just one corner.
36:04
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 transfer matrix now is a mission. It's a mission, so instead of doing SVD,
36:21
I can do more stable exciterization to get the, to truncate. And actually that was suggested by Philippe to me at some point. And it actually works well. And also the last thing which is being done in this renormalization process
36:45
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
37:02
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 exciterization,
37:24
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 CTMRG real space renormalization.
37:42
So maybe I should, unless there is some question about the method, I will go to some result. Okay, so what I now I do is I go to the previous truncated Hamiltonian I described.
38:02
And here I will focus on this point in the parameter space. So for this particular value of the coupling, which in the original paper was claimed to be a point where the overlap with the Cami-Laufen state is maximum. So it's potentially the point, the parameters which are best to obtain such a state.
38:27
And so what I draw 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.
38:42
So this is a 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.
39:00
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 choices of the virtual space. So I think this one is for one half plus zero.
39:22
So it's very small bond dimensions, d equals three. And this one is d equals four. This is one half plus zero plus zero. And this one is one half plus one half plus zero, which gives this point here. But I've been unable to get further points in this because of instability problem.
39:41
But what you see already for this small bond dimension that if I just increase chi, then I have to extrapolate linearly with one over 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
40:02
Kammerer-Laughlin energy in the thermodynamic limit. So it's reasonable to compare. I mean, both calculations are in the thermodynamic limit. And this Kammerer-Laughlin energy you can get very accurately in Monte Carlo. And what you see is that the energy even for the small d parameters,
40:21
I get energy which is below the Kammerer-Laughlin energy, which is the targeted state. I mean, this is why this term was constructed. So I get something better. So which means that suggests that the peps is a reliable description of the ground state of this state.
40:46
So now what I want to see is whether this state has one important feature of the Carroll spin liquid, which is the existence of Carroll edge modes. So this is the first thing to check. And Monte Carlo is done on the same Hamiltonian.
41:03
Exactly. No, no, there's no Hamiltonian. It's for the Kammerer-Laughlin. It's just Monte Carlo for the Kammerer-Laughlin state. There's no Hamiltonian, just the energy. I know, but this is energy in this Hamiltonian for the Kammerer-Laughlin state. And then the exact diagonalization.
41:20
Okay, so this I didn't talk about that because I thought it would be. The externalization is much lower, but for the reason that usually you have in small clusters, you have very strong quantum fluctuation 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 when you increase the number of side,
41:45
it would just go up dramatically. So this is difficult to compare to this value. If you don't have finite size scaling, I don't see when it means anything. It's a gapped state, so the finite size correction. Well, how do you know it's a gapped state?
42:02
That's the point. It's a Hamiltonian that was constructed in order to indeed get a gapped phase, but at the end of the day you don't know what you get after the truncation.
42:20
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 certain sides. And I don't think it's very good because it's five times six, so it has an odd number of sides in one direction, which I think is not too good when you have...
42:42
It frustrates somehow the entry from magnetic order. So it's not a very good cluster shape. Okay, so let me come to the issue of the edge states.
43:04
So now there is a conjecture which has been put forward by Lee and Aldein in a famous paper. It's a recent paper, that actually the entanglement spectrum
43:28
will capture all the physics of the edge modes. For example, if you take a cylinder like that, and you calculate, you make a partition into A and B,
43:41
so it's a mathematical partition, and then you calculate the reduced density matrix by tracing over the degrees of freedom of the half-cylinder of this projector, psi, psi. And then you write rho A as the exponential of minus some Hamiltonian
44:09
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 Hall state
44:23
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 the...
44:44
Basically, we should get something which is in one-to-one correspondence with the actual edge states of the system. Just a general observation or it's... I think it's a conjecture. I don't think it has been... Or this is true for these specific systems. Or they have shown that for the, I think, for the new equal one-third Loflin state,
45:02
and then it has been used in many other cases for non-Iberian fractional quantum Hall state. I think these examples always have in common that these are chiral phases with a gap. Yes, exactly. And maybe that's relevant here, if you have a gap. You will see, because...
45:23
Yeah, exactly. So now there is a puzzle still I'm coming to. Will you explain how these correspondence work, this one-to-one correspondence? Okay, so this one-to-one correspondence is... Well, maybe I can show you here. Okay, so what I draw here now with this entanglement spectrum
45:41
as a function of the momentum along the circumference of my cylinder. And for PEP, there is a very simple way of computing these guys. 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
46:02
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 particular.
46:23
What you find is really linear dispersion. So you really find Carroll modes. You don't find the CFT spectrum that Guifrey was showing, where you really have two branches and everything is filled inside. Here you just have one branch. So it's really Carroll.
46:40
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 will expect for the CFT that characterizing the edge states.
47:03
So maybe the energy levels will not be exactly at the right position, but the counting will 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, two one plus zero means I have two triplets here,
47:21
which are these symbols, plus one singlet. And then if I go up, I go here, then I have all these states. I have a quintuplet plus two triplets plus two singlets and so on. And now 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
47:40
and you look at what you should get. So this is for the SU2 level one CFT and you look at the table in the book and what they provide is the SU2 decomposition. So all the quantum numbers for the tower of state that you get. And so here they actually use different notations. So two means actually spin one
48:01
and one means spins one half. I don't know whether 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.
48:23
One triplet, one triplet, singlet and so on. And of course here there are some dispersion because the size, when you go further in momentum, 10 minutes, okay, thank you. Sorry, what is method even in odd sector in your case? Yeah, so this is something I forgot to say
48:42
but there is a gauge symmetry of the tensors that allow me to construct two different sectors. If I can play with the boundary, so the way to compute this
49:02
is just to iterate the transfer matrix basically from infinity. I can fix these topological sectors using the z2 symmetry in the initial state and then it would be preserved and so I can just diagonalize this,
49:21
reduce the matrix in these two different sectors. Why is the momentum more or less pi and not two pi? You notice that, okay. Is it because you can put two? Okay, I tell you, okay, these are technicalities.
49:43
Because in fact, in order to have only one tensor per side, what I do is I do a spin rotation on the b-side. Otherwise I would have to use two different tensors. And so that somehow breaks the translation symmetry.
50:02
So there is some translation symmetry breaking here and it means the spin multiplets will appear at momentum k and k plus pi. So I could use momentum modulus two pi but then I will have the multiplet coming at two different momentum. So what I do is I just write everything
50:22
in terms of modulus pi and so just to bring back all the different terms of the multiples together. But it's not the property of the boundary theory. No, no. Yes, so you see here
50:40
I have pretty good evidence that actually the edge, these states have well-defined cowl edge modes which are described by this SU2 level one theory. Now the surprise comes from the
51:02
investigation of the correlation function. And as you say, I would expect that my state would be gapped. I have nice edge states and 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
51:23
and neither of them actually turn out to be short range. So the first correlation function you can compute is basically the Daimler-Daimler correlation. So what you do, now that you have your environment tensors, what you can do, you can construct a strip here.
51:44
And here look at the correlation between s.s on these bonds with s.s on these bonds. So it's like a four-point correlation function as a function of the distance between the two objects.
52:02
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.
52:23
And so in principle, you can calculate correlation to arbitrary, basically arbitrary distance. And so this is what I show in a semi-log plot. So this is the log of the correlation as a function of distance. And what you see is something that you would expect that for large distance, for finite chi of the environment,
52:43
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
53:02
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,
53:21
you know, going up to 20 times d squared. So it's not small. And it seems to linearly, to vary linearly, which means that the correlation length diverge and that's consistent with the parallel decay of the Daimler-Daimler correlation. And actually, if you look at short distance,
53:41
below this exponential behavior, you can actually pretty much feed the data with a power law. Now you can also look at, five minutes, yes, I would be almost down. So this is the last plot I want to show.
54:01
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. So you construct this strip here and you put the environment on the side. And then here now you have
54:21
the spin operator and spin operator, and you look at the correlation between these two sides as a function of d. And again, you can plot the data on a semi-log plot, the log of the spin-spin correlation versus the distance d here between the sides. And the first thing you find
54:41
is that at a short distance you find a very short correlation length, a very sharp decay with a very short correlation length. And actually, this correlation length is compatible with the Kamielov instant. It's basically what they get in Monte Carlo. So it means that a short distance, the wave function,
55:01
the ground state we get, I mean the ansatz we get, is as the same property as the same spin-spin correlation at short distance, maybe up to distance, I don't know, 10 or something like that. It's very similar. So at short distance, I think we cannot distinguish our ansatz from the Kamielov state,
55:21
but now at 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 chi. So it's again chi up to 16 times d squared
55:42
and we don't see any sign of saturation. But now the difference with the Daimler-Daimler 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,
56:02
which is basically what I can get from this plot, then the weight here associated to 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,
56:20
all distances will be included. Sine max will go to basically infinity with chi going to infinity, but the weight will be becoming very small. So I think the decay, I don't know the analytic form, but I think the decay will be actually, it will be slower than, of course, a pure exponential,
56:41
but it will be faster than algebraic decay. This is something in between, maybe a stretch exponential or something complicated like that, but it's clearly not simple to... Okay, so now this is the big issue that is left,
57:01
is whether these features, first, are they really generic features of the ground state of this model or are they artifacts of the PEPS representation? And also, if they are really generic features of this model, are they really generic features of all karaspi liquids of this type?
57:22
So this is basically the conclusion of the study. And so I would like to summarize and then propose some outlook. So to summarize, I would say that this method offers a really new conceptual understanding
57:40
and a quantitative description of many quantum antifera magnet with the exotic ground states. So not only the one that were described by Philippe, which show spontaneous symmetry breaking like stripe order or this sort of order, but also topological state, which are maybe
58:00
a more involved type of ordering. Also what is important is that the virtual degrees of freedom in fact play really a physical role at the boundary, because the virtual degrees of freedom really are the ones that actually build the edge states.
58:21
So 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
58:41
is extend this to many other cases. So one obvious extension is to look at Hamiltonian with larger spin, like for example spin one, where we might realize some karaspin liquid
59:00
which would be some non-Habelian karaspin liquid with more complicated edge states, maybe SU2 level 2 type edge states. And we know that in the field of fractional quantum Hall effect there are such more involved topological states.
59:20
So the aim is to try 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 karaspin liquid so far.
59:43
No, there is? Okay, so I have to tell you. And maybe it could be applied to this crystallized field theory, but this is really not my field, so I will not...
01:00:00
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 Romain Horace, from whom I learned a lot of these techniques in the last, say, seven or eight years.
01:00:24
Also, I'd like to thank my collaborator, Matthew Mambrini Toulouse for his major role in the classification of SU2-PEPs, and also Jan Affleck, who collaborated at some point on the Kerospin liquid issue.
01:00:42
OK, and I'd like to thank you for your attention. Questions? Yes. So if you take the point of view that this, for some reason, is gapless, then do you think there's a chance to have emergent Lorentz invariance?
01:01:06
Yeah, I thought there would be some emergent gauge symmetry that would, you know, like U1 gauge symmetry at some point. Because we know that there are simple PEPs which have U1 gauge symmetry, and we know why they are gapless.
01:01:23
So maybe there is some emergent gauge symmetry that's appearing. Well, I was wondering what is the dynamical critical exponent of this. Could it be one? Yeah, I don't know. How do you compute that?
01:01:41
You would have to look at the spectrum of the excited states from the very hard and final point. Yeah, exactly. I mean, constructing excited state for this Carol Hamiltonian, I'm not sure it's... Well, it could be attempted, but I don't have any idea of... I mean, maybe with the method you suggested, you know, with some B tensors, but that's super hard.
01:02:03
I didn't understand what the conclusion is for the ground state. What's the conclusion 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 than the cameo laughing state. So we point to the fact that it is a good representation of the ground state.
01:02:25
And which has both features. The first feature, it has well-defined Carol edge modes, which are described by this SU2 level 1 CFT. I think this is very, very good evidence. I'm a little confused about that. You've extracted those edge modes from this entanglement formula.
01:02:43
Yes, yes. Which is still valid, even though... That's why I'm confused. What is it that goes into this? OK, the conjecture. Oh, you want to mean... Do you mean whether I can use a conjecture in this case? Yes. The answer is, I don't know.
01:03:00
I have to ask Duncan whether he thinks it's still valid, but I don't know. It's a conjecture. 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 whether... There are edge modes for certain, you would want to know the... Yes, but what does he mean edge mode? This is dynamics, right?
01:03:21
And we haven't established whether this is a God Hamiltonian or a Godless Hamiltonian, that may have an important impact on the dynamics. But what's clear is that when you compute this entanglement spectrum, it has a structure that is compatible with a kind of siesta. Yeah, OK, so I should rephrase, maybe don't say edge modes, but entanglement spectrum which are well described
01:03:41
by this SU2 level 1 siesta. And the question is what to make with that. Yes. And 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 trippet centers. Which would indicate that this is a Godless Hamiltonian?
01:04:01
Exactly, exactly. Exactly, because Hamiltonian is local, no? 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 Godless. Yes, sorry, Andreas. Sorry, I missed it, it's my fault, but you're the state to construct, do you do that for fixed t, like t equals three,
01:04:20
or what is the bond dimension of your state, so 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. You see, so for example, the blue curve corresponds to this d equals four case,
01:04:41
one half plus zero plus zero, for the largest chi I could handle, which is 256. And the green one corresponds to this very simple one half plus zero ansatz, and there are two sets of curves that correspond to chi 36 and 144.
01:05:04
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 they correspond to two different points in this plot.
01:05:22
I'm just wondering a bit, I mean, I'm personally thinking that this Hamiltonian, if it's truncated, it's very likely that it's a gapped Hamiltonian, and it's in the same, the ground state is a chirospin liquid, but a genuine one without the gap. Then if you take these pegs, theorems, no-go theorems for grounding, you can still ask, okay, it might be that they're not at finite bond dimension,
01:05:42
they're not able to exactly reproduce a chirospin liquid, but how well do they approximate that? And since you said yourself that initially at short distances, it seems like reproducing... Yeah, yeah, this I believe. This I believe at short distance, I am okay, basically. I get the right shot.
01:06:00
It's hard, it's not in the pitch, 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 d, the more... You're right, you're right. There are two options.
01:06:20
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 short-range state, but if I crank up d, I will approximate better and better and I get shorter and shorter correlation.
01:06:43
Because I mean, there's an argument, which I myself I did not fully understand, but the things you share on Dubai and the Greek, they have some understanding that these boundary theories, they're actually like living on the edge, and then I think there is some connection with the fermion doubling problem or no-go theorem, that you cannot really write down some local Hamiltonian,
01:07:00
which is completely chiral at the edge, so I'm wondering whether this is something... I wonder whether this thing is not the fact that there might be another mode. If you say actually I don't have one chiral mode, but I have two, but the other one with a very steep velocity, then I think I'll be alright with this,
01:07:21
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,
01:07:41
I mean, conceptually there could be another mode with a very steep velocity. Coming out of your path. Yeah, yeah, coming out of the path. How may your lovely movement happen? Yeah, yeah, yeah, yes, yes.
01:08:01
If I can comment, so the first comment is that if it's godless, for this Simon's collaboration, it's more interesting. And the second, it's amazing that it's already a second example of a very simple Hamiltonian, for which it seems there is a controversy. It's not clear if it's godless or if it's godless.
01:08:21
Right. 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 gapped than it is in the real life,
01:08:41
not less gapped. Yeah, right. 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 call term. And there is a controversy whether at J2 equal 0.5 of J1,
01:09:03
whether you get a symmetry breaking state, whether you get a dimer state, a very translation symmetry, or whether you get also a gapless... I actually don't understand this comment about chirality. So we have these plots which show that chiral states, they exist chiral states,
01:09:21
which are gapped. So chiral state at finite bond dimension is gapped. I don't know what that means. Well, we saw the plots... I got the property of a Hamiltonian. We saw the plots which are exponential, we saw the plots for correlation functions at finite bond dimension, which are... No, no, no, you mean... No, no, which plot?
01:09:43
You mean my plot? This one? No, no, no, but this doesn't mean exponential. You mean that if you approximate the exact contraction by an environment
01:10:01
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 diverge.
01:10:21
So even for a finite D, if you would be able to crank up chi to infinity, then probably you will see no exponential... You will see purely algebraic decay. So it's a finite chi, it's an artifact of the fact that you truncate the contraction when you do the...
01:10:41
I think the following is even true. Is it correct that the challenge now is to find a chiral PEPS with a finite correlation line? Yeah, whether it comes or does not come from... No, just you enforce chirality and now look if, genetically, this seems to have power over here of correlation. So there seems to be a theorem out there that any chiral PEPS,
01:11:01
there might be a theorem that any chiral PEPS has infinite girlish alert. Yeah, but I still think there's an idea that here that as you crank up the physical bond dimension D, it could be that on longer and longer scales the system looks exponential and there's an algebraic tail in the end, which is due to a no-go theorem. And so you try to act
01:11:21
against the no-go theorem, but as you crank up bond dimension, you do a better job in being gapped or looking exponential over some scale and then the algebraic tail takes over, which is a bit weird, but that might be what's happening. It would mean that you're trying to reproduce an exponential decay by summing power over the case, right?
01:11:40
So, yeah, you're trying to to simulate correlations, but you are using answers that at finding bond dimension will necessarily have power over the case. So at short distances, you manage to superpose many power over the case so that it looks like an exponential over there to be the true power over the case emerges.
01:12:09
I just have a quick question. You mentioned that you would like to realize SU2 level k. Yes, yes. So now we have a model. Can I just take the Nielsen Sierra...
01:12:21
Exactly. This is what I... Replace what spin has. No, no, no, no, no. So it's a more complicated spin one. So it includes, well, maybe I can... 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 the parent time and time, which again for spin one is long range and they truncate again and they play the same game.
01:12:42
And eventually, they came up with a simple Hamiltonian, which has bilinear interaction. Nearest neighbor. So maybe I can write it down. Do we have one minute to write it down? So we are actually
01:13:01
working on this one. So it's a bit more complicated. So it's spin one and it has a J1 term. So it's nearest neighbor. And the difference now is that there is a bi quadratic. So there is a between nearest neighbor side.
01:13:22
So it's SI dot SJ square. And then it has again next nearest neighbor bilinear term SJ. So this is on the diagonal.
01:13:41
And there is a bi quadratic term on the diagonal. So the same SI SJ square. So this one was not allowed for spin one half. So now it comes in. And there is the counter.
01:14:00
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 large parameter space that would be terrible. But they have a proposition for what are the optimal value of these.
01:14:21
And they claim basically the ground state is like the married non-abelian fractional polynomial state on the cross spin one. So the game is to try to attack this again with IPEPs.
01:14:43
Thanks a lot.