 Oh, I'm muted. OK. OK. Great, great. So, yeah. So I'm actually not going to be talking about spin liquids. I'll be talking about superconductivity. And I'm going to ask the question, can neural networks capture superconducting order without really any information that the state is going to be superconducting? So we're going to try to start from sort of a bare bones neural network and see if we can stabilize superconductivity of various types. I'm going to be doing show and calculations from variation of Monte Carlo. There's been several really nice introductions to it already, so I'm going to skip that part. But I do want to talk a little bit about my wave function or our wave function onsets, which are these hidden for me on the terminal states. And actually I'm going to add some lattice symmetries to these states. And I'm going to show some benchmarks on the half-filled Huber model. So this is a model that's pretty well understood, actually. And there's some nearly exact results on this predictor system. So it's a great way to benchmark your calculations. And then I'm going to discuss the attractive Huber model, where we're expected to find S wave superconductivity. And in fact, we do stabilize it. And then I will eventually move to the repulsive Huber model, which describes Cooperate superconductors. And show that we do actually see the S wave superconductivity. So let me start with this hidden for me on approach, which is what I'll use. This is an extension of a Slater-Jastrow method. So the idea of Slater-Jastrow method is you have some mean field orbitals for your electrons. So you have one orbital for every site. In this case, we have spin up and spin down. And so you take some determinant over those orbitals. But in this configuration, we actually also have these auxiliary orbitals. And these auxiliary orbitals are output by a neural network. And so what you end up having is your wave function is the determinant of some matrix. The top half of the matrix are these mean field orbitals. And then the bottom half are actually output by a neural network. And so again, as I said before, it's an extension of this Slater-Jastrow approach. And actually, if you expand up the determinant this way as a product of determinants, you can see that actually the first term is just the Slater part of the onsets. And the second term is this Jastrow factor. But you can see this Jastrow factor is actually informed by the mean field configurations. So you have this interaction between your neural network orbitals and your mean field orbitals, which gives you this really, really rich expressibility, expressive power. And so what's great about this method is actually that this mean field part of the wave function can be computed very efficiently for the ratio of this part can be efficiently computed for single fermion hoppings. This is known as a low rank update to the determinant. And so this is very efficient and actually makes it probably more efficient than backflow in some ways, which is a sort of competing method. I will say that I do not, I don't do these low rank updates for the simulations I'm showing. So I won't scale up the huge cluster sizes, but just know that these can be approved a bit. And I can do bigger clusters in the future, I think. So yeah, so why add symmetries to this hidden fermion state? The main idea of adding symmetries in a sort of variational problem is actually you want to take the Hilbert space and you want to divide it up into pieces. So symmetries, what they give you is they give you good quantum numbers. So here we have a Hamiltonian defined in a lattice A. We have rotational symmetry. We have translational symmetry. We have reflection symmetry. And these quantum numbers divide your Hilbert space up into pieces and it makes the sort of search space for the ground state a lot easier. And the other great thing is you can also find low lying excited states that as long as they have different quantum numbers, you can find these excited states. So I'll be showing that this is a bit of a blessing and a curse because you have to try different quantum numbers to find the ground state. You often don't know what the quantum numbers are and you don't know what the quantum numbers are. But ultimately, it improves the variational accuracy by enough that I think the trade-off is worth it. So the strategy here is to start with all the symmetries, start as symmetric as possible, and then try breaking some symmetries and see what happens with the wave function. So this is kind of how we add the symmetries to the hidden fermion approach. And this is rather technical and I think I'd rather discuss this mostly offline if anyone's interested. But our neural network orbitals are now this thing called a group convolutional neural network. And we also have this kind of string we have to put, this sign with this pi thing is a fermion sign we have to put into our wave function. And this is because if we define a Hamiltonian in Fox base, basically the ordering of the fermions will break lattice symmetries. And so we kind of have to stick this sign by hand back into the wave function if we have the Fox state. But this is generally how we do it. And I'm happy to discuss more. But I really want to talk about the results because I'm excited about them. And so first I'm going to talk about the square lattice Hubbard model at half filling. So this is the model right here. And this is a paradigmatic model in quantum antibody physics. It's kind of the simplest model you can make of sort of a mod insulator or interactions among electrons. And so basically there's this potential energy term which says that two electrons don't like to share the same site. They interact repulsively. And then there's this kinetic energy term which is the term on the right with the T. And that's just the desired electrons to move around. And so when we do this model, when we study the square lattice Hubbard model at half filling, we can compare with this method known as auxiliary field quantum Monte Carlo. This method doesn't have a sign problem. And so the results are somewhat exact. They're sort of exact to statistical accuracy. And what's kind of great about this also is that there's this particle hole transformation that you can make on the fermion operators. And this actually maps you between the attractive and repulsive Hubbard models. And so we can kind of, even though these two models are the same in some sense at half filling, the ground state looks very, very different in the space of fermions. And so actually if you have a repulsive Hubbard model, the electrons are localized. And you have this anti-ferromagnetic super exchange between the electrons. So they want to be anti-ferromagnetically ordered. So you have this anti-ferromagnetic mod insulator. But when you turn U from positive to negative, you actually get this charge density wave where the electrons are sort of paired at every other site. So even though the model is the same, the physics is very different. And so what we do, what we did actually was we tried both this attractive and this repulsive Hubbard model. And with our neural network approach and variation of Monte Carlo, and we compared our results with AFQMC. And we found that indeed our results are really, really accurate. I mean, on 16 sites, they're nearly exact. And then on 36 sites, they're pretty similar to AFQMC. And so of course, a 36 site problem Hubbard model is the Hilbert space is still way bigger than anything you could do with ED. And we're demonstrating nearly exact results here. The other thing we find though is that it's actually easier to represent this spin density wave, this mod insulator than it is to represent this charge density wave. And so, so actually, yeah, yeah. This is the ground state energy, yeah, no. Yeah, so these are the ground state energies. And on a torus with periodic boundary conditions in both directions. And so everything I show is gonna be on toruses, tori with periodic boundary conditions. So yeah, so anyway, so I just, I also wanted to check to see if we were kind of physically getting the same thing for the attractive and the repulsive Hubbard model. So these are some correlation functions here. So on the left-hand side, we're showing some correlation functions of the attractive Hubbard model. And the right-hand side, we're showing correlation functions of the repulsive Hubbard model. And these correlation functions also transform under this particle hole transformation. And I mean, the main thing to take away from this plot is actually that they look exactly the same. So on the top, we're kind of seeing these charged correlations and we're seeing this sort of, this charged density wave for the U equals minus six. And then we're seeing the spin density wave for U equals plus six. And then similarly for the superconduct, S wave superconducting correlations on the left and what I call spin flip correlations on the right, which is just how they transform under these symmetries. So this is a really good check that actually we're getting the right results in the half field Hubbard model. So I think with this, armed with this knowledge, it's worth it to look at the hold-up models. So first we're gonna look at the hold-up attractive Hubbard model. So actually, and you can see kind of in the previous slide, there's actually this degeneracy between this charge order and this superconducting order. And so actually doping the Hubbard model with holes, this attractive Hubbard model with holes will split the degeneracy between the charge density wave and the superconducting order and it'll favor superconductivity. So we expect to see superconductivity in this model. One kind of nice way to think about it is basically when electrons share a site here, when you create a pair of electrons in a site, they have to form a singlet state. So they're kind of forced to form a singlet state by sharing a site. So when one of them hops away, they kind of stay in this singlet state. And so what you can look for is this idea of off-diagonal long-range order, which is if you kind of create a coop repair at a site and then destroy it infinitely far away, you expect this expectation value to sort of stay finite as you move these coop repairs as far as possible away. And this is associated with sort of breaking a particle number symmetry locally, which is really the hallmark signature of a superconducting state. So we look at these plots of the whole Dove-Tubborn model. We can see that they're blue. So what that means is that the superconducting order parameter you can see is pretty large, first of all, and that it's also persisting all the way across the cluster. And I should say actually that the reason the center is white is just because I set that value to zero so that you can see the scale of the correlation functions. But yeah, so you can see that basically, there's not really, these plots aren't that exciting. They're just kind of, they're blue plots, but it shows that we have this S wave superconducting long-range superconducting order, and it's already kind of stabilized on a 36 site cluster. So we've stabilized S wave order, and now let's talk about D wave order, which is really the more sort of exciting and interesting problem. And so the repulsive Hubborn model is well known to describe the cuprates. The left plot here is sort of a cartoon of the phase diagram of the cuprates. We have these two domes. We have this one, the small dome on the left, which is electron doping, and this sort of larger dome on the right, which is hole doping. And actually the right figure is from a recent paper that kind of combines AFQMC and DMRG, and this paper, they get a lot of things correct. So the two main things they get correct are sort of where the domes are located, the sort of fillings that allow for superconductivity, and then also that the hole doping dome is larger than the particle doping dome. So this is really, really, really exciting, this paper, and it basically shows that the Hubborn model was really, really describing the physics of the cuprates well. Now this paper uses these two methods that are much more established than deep neural networks. They use AFQMC and DMRG, and this paper is often termed the handshake between the two methods, and the idea is that DMRG is, AFQMC is cheaper calculation, but it's uncontrolled and it separates from this Fermion sign problem, whereas DMRG is a more expensive calculation, but it's accurate in some sense, like if you extrapolate the infinite bond dimension. So the idea is to use DMRG to sort of verify that AFQMC is working on smaller clusters, that the correlation functions look the same, that they have the same physics, and then maybe use AFQMC to get the bigger clusters. And so that was kind of the strategy employed by the paper I showed before. The question is like what can neural networks bring to this story? And really the argument for neural networks has been that they can kind of cover these blind spots of DMRG. So DMRG is kind of a quasi 1D algorithm, and it also, the matrix product states always have a finite correlation length, and so if the entanglement scaling is either super area law, like in some gapless spin liquids, or actually which is less discussed is just if the physics is really two dimensional, in the sense that there's kind of long correlation lengths in both dimensions, both of these cases are cases where DMRG might fail, and we might even need neural networks to check DMRG. So we're kind of checking DMRG, which is checking AFQMC in some sense, I think. And the good news is that the NQS results really, really strongly agree with DMRG actually. And I basically have kind of looked at different things, and I haven't really been able to find a stronger discrepancy between NQS and DMRG. And so yeah, first let me just show that we stabilized this D wave superconducting order in the repulsive hover model. So we've added, there's a, the cuprates tend to have this T prime term, which is a next neighbor hopping as well. And so we're kind of looking at typical parameters for the cuprates, and we're doping with about 20% holes. And now I'm defining my pair creation operator differently. So now the pair creation operator acts in a bond, right? So this is kind of two electrons being entangled in a singlet state that live at nearest neighbor sites. And the kind of mechanism which they become entangled by is this, again, this anti-thera magnetic super exchange interaction. And so defining this kind of D wave average pairing function is just averaging over this pairing function for the, you know, adding the X component and subtracting the Y component, because this is sort of the D wave symmetry that's seen in the cuprates. And what we see basically is, again, this plot is blue. And so, although there's still obviously some, some kind of structure in the plot, so it's clear that we haven't quite gone to big enough clusters, but we've gotten to 64 sites. And that's about the limits without doing low-rank updates. So we do see this D wave superconducting order, which is really exciting to see this with neural quantum states and kind of confirms what DMRG and AFQ and CC and also, you know, what's seen in real life. So this is all great news. Yeah, so actually, so what I showed you there was one-fifth whole doping. So I just want to confirm some of the other findings with DMRG, so the other thing you kind of see with the superconductivity is actually these spin stripes. And these spin stripes are kind of thought to have some kind of relation. Again, with this anti-ferromagnetic super exchange that that's really causing the Cooper pairs to form. So to find these spin stripes, actually we have to break a symmetry. So if you look at the left, that's what the actual ground state spin-spin correlation functions look like. And so they're, of course, they have to be symmetric because the ground state's symmetric, but you can kind of break the symmetry at a very, very small energy cost. Which is shown on the right here. And when you break the symmetry, you in fact do see these superconducting stripes and the spins correlation functions. You can see kind of as you go vertically on that right plot that you see this kind of this fluctuation of up and down spins. And actually, so yeah, so my thought about why DMRG is actually working really well on this problem is actually that these stripes, these stripes can kind of form the stripe pattern is a very short wavelength pattern. So even if there's some long wavelength physics that's stabilizing the superconductivity, the short wavelength physics, DMRG can kind of capture, even if the clusters are not completely symmetric. So yeah, so this is what we see with the spin order. And then again, the charge order, we don't see much. Which is what's expected with DMRG. Now, here I'm plotting again the number of, the number of electrons at a site times the number of electrons at different site displaced by some translation, minus the expectation value squared. And so we don't really see much structure in sort of the way the electrons are ordering. So again, there's not really any charge ordering. We see this superconducting order seems to be the kind of strongest thing we see. And yeah, so there's also this one-eighth hold up case. And so actually the green here, I realize I never explained this, but the green here is the charge, the charge sort of the charge. So you can see the charge is pretty evenly distributed for the, or actually it's the whole density of the green. The green's the whole density. So you can see the whole density is pretty evenly distributed for the one-fifth state which I just showed. And then you have this one-eighth state where there's also some superconductivity, but you can look at the charge density and it's actually not, it's not uniform. It looks like there's kind of these ebbs and flows of the charge density. And so first we can look at the superconductivity. And what we find is actually a little strange. The superconductivity kind of only seems to form along one axis here with this, along the short axis. This is a 32 site cluster, so it's actually already, I'm actually, the rotation symmetry is broken here already. And the, but again you see the superconductivity kind of is in line with this, these kind of spin fluctuations and the superconductivity that kind of match up together. And what's going on in the x-direction. So actually if we look at the x-direction, we can look at this sort of charge correlation functions which don't really show much actually. And then what you can do is actually you can just break one of these symmetries. And so once you break a symmetry, the charge is no longer uniformly distributed. If you have the symmetry it has to be sort of uniform on average. But when you break the symmetry you can break that. So when we break the symmetry we actually see these stripes here on the right. And so this again is the charge density wave that DMRG sees. Okay, so yeah that's basically my talk. The, yeah I mean the main thing is that we were able to stabilize these both S wave and D wave superconductivity using neural quantum states. And our findings really back DMRG. And the kind of potential issues of the DMRG seem to be avoided because these spin stripes are kind of formed at the very short period. And so you don't really need a huge two-dimensional cluster to really see the superconductivity in action. Thank you. Thank you, we have time for questions. All right, thanks a nice talk. From the methodological point of view I just have some questions with this, the hidden ansatz and the symmetries with respect to, I mean you made the statement in the beginning that it's more expressive than Slater, just to, I mean of course you can go beyond that with the backflow transformation. Now with backflow transformations the problem is that you can't really describe anything that is very, I mean that is far away from filled shells. So as long as soon as you have some open shells it basically doesn't really, it has a hard time representing those things. So I'm just wondering, you seem to be able to do any kind of doping unless you chose those in a specific way. I don't know how you came up with these numbers but I'm just wondering, you seem to be having no problems with that. So do you think that would be due to the symmetries and the fact that you take into account all the rotational symmetries in your transformations as well or do you think it's rather because of the ansatz that you're choosing of the hidden fermions? Yeah, so I know hidden fermions improved upon backflow by a good bit when it came out but also the symmetries also improved upon hidden fermions. And I mean the symmetries really, really help on small clusters so you know because the Hilbert space, you're cutting up the Hilbert space into pieces which is really, really great when your Hilbert space isn't that that large but then as your Hilbert space gets larger and larger these symmetries kind of become less and less effective in some sense. So I think symmetries are really, really doing a lot of work especially on the small clusters with getting good, good variational energies, yeah. But you didn't look at, for example, the performance as a function of the doping specifically whether this had an effect or? No, I didn't compare symmetries and no symmetries as a function of doping. No, I mean I did compare it a few points like and no symmetries seems to work better. I mean symmetries seem to work better, yeah. Just a curiosity, when you compared with the affilling case in the quantum Monte Carlo for the six by six in order of magnitude of the parameters you used just to have an idea. How many parameters did you have to optimize to get the energy you showed in the six by six? Oh, wow. I don't want the precise number 10,000, 100,000, a million just to know. Yeah, let me just try to get a ballpark estimate and I think 100,000 of that order, yeah. Hi, thanks for the nice talk. You said at the end that here you don't get like so much different physics and energy because the superconductivity shows up already at short scale. So can you imagine any other systems where you might get qualitatively different results with that's not the case? Yeah, so I mean, I think one sort of example would be maybe like a gapless spin liquids, you know, they have long or they really have infinite correlation length in both directions. So that's a place where DMRG should absolutely completely fail and actually DMRG can kind of work if you're like really, really clever and like, you know and like you know the answer already and but but I think it's a place where the MRG fails in terms of superconductivity. Yeah, that's something I'm interested in thinking about that people have answers and the audience, that'd be amazing. I think. Thank you, so let's thank the speaker again.