 and talking to you guys. So what I want to talk today is actually, my introduction is going to be very similar to Natasha's. So I'm also talking about spin liquids, but on a totally different type of lattice. So I'm going to be more interested in a simple Heisenberg, finding spin liquids on simple Heisenberg models. So essentially we've introduced this new lattice called the stuffed honeycomb lattice. This is what it looks like. It looks somewhat complicated here. We'll go through and build it up. It's not that complicated. And we've gone through and solved this both classically and quantum mechanically. And this, I think, gives us a lot of nice insight into spin liquids on more familiar lattices, like the triangular lattice. So at first I want to acknowledge the people who've done a lot of the work. So my student, Jewishman Sahou, has done a lot of the analytical work. And Brian Clark and his student, Dmitry Koshkov, Banner Champagne, have done a lot of the heavy-duty numerics to understand especially the quantum phase diagram. And the classical model has already been put on the archive earlier this year. And since this is a conference in celebration of peers in part, I just want to also acknowledge peers because actually my first project with peers was working on frustrated magnetism. And actually saw Christian talking about some results extending from that. All right. You know, he did not fall down on this one. But you could really see on this hike, this was with Colin Brougham, so doubly on frustrated magnetism, that his enthusiasm and optimism was leading him to think that we had passed the trail junction because we should have been there already. Yes. I think that didn't come till later in the hike when we were still hiking until 7 o'clock at night. OK. This is from Boulder back in 2008. So I just want to start off with sort of the broad, motivated question of what are spin liquids? More for the students, which has already been partially addressed. So I'll go fairly quickly through that. And why should we care? Why should we look for new spin liquids? So the simplest definitions are possible quantum ground states and insulating spin systems. And more specifically, these are strongly correlated phases with topological order and fractional excitations. And these are really some of the simplest examples of these. We have totally gotten rid of the charged degrees of freedom and you're left with only spins. And so if you want to understand all of these more complicated problems and correlated electrons, it might be sort of easier to start with these spin liquid systems. OK. So to give the earlier definition for spin liquids, which came from Anderson is sort of a very simple picture where you can think of a spin liquid just like you think like a normal liquid. So it's a case where you have essentially a magnetic liquid. So the analog of a spin solid would be an ordered magnet. The analog of a spin gas would be a totally disordered paramagnet. And a spin liquid is something in between, where the spins know about neighboring spins, but not about the spins at infinite distances. So they're correlated, but they haven't formed any long range order. And so you sort of always get this kind of region of correlation at finite temperatures if you have some kind of short range magnetic order. But what we're really interested in is when you can actually kill this magnet and find your quantum spin liquid as the ground state. And a lot of the description we have of these is in terms of these valence bonds, which are spin singlets on neighboring sites or further sites. And you can think of a spin liquid as made up of these resonating valence bonds. And so this is just an example of what a spin liquid is not. This is something I can write down a ground state, which is an ordered state of these valence bonds. So this breaks both rotational and translational symmetries and is not a spin liquid. It doesn't break spin rotational symmetry, so it's not a magnet. But spin liquids are even more complicated than this. So as you see, spin liquids are made up of essentially resonating valence bonds. So I will have this state, but I will also have this state, and this state, and an infinite number of other states. And these are all existing in a quantum superposition, and that gives me my spin liquid. And one of the reasons we really like this picture is because it's easy to see that you have these spin one half excitations. So I can take this valence bond and I can break it, and then I can move these two excitations apart. And we see that these are both spin one half. I can move them freely just by flipping valence bonds. And so they're fractional particles. They carry the spin of an electron, but not its charge. Okay, so, and just sort of to connect to the early motivation for these was the connection to high temperatures cuprates where then you add in charge and you have fractionalization, not just into spin, but also into charge only degrees of freedom. Okay, so our first definition then is sort of this absence of symmetry breaking. This is kind of a negative definition. It's not particularly, it is somewhat useful experimentally in that you look and you don't see anything and then you think, well, maybe this is a spin liquid. But you would really like to see these positive definitions, which Natasha talked about a lot, this idea of fractionalization. So you have this spin on, all right, okay. So essentially your low energy excitations are these spin ons. You can think of them as fractionalizing an electron or fractionalizing a magnon into two spin one half particles. And since neutrons can only flip spins and it gives spin wave, a spin one excitations, you get these spin on continuance. Which are one example of a positive signature and of course we struggle in models like Heisenberg models where we don't have any nice exact solutions like you do in the Cataya to connect these in a very clear and obvious way to experimental signatures. The second definition of spin, sorry, this is, the more modern definition of spin liquids is in terms of both this fractionalization, but also this idea of topological order and long range entanglement. And this actually gives us the possibility for a positive definition, which is not particularly experimentally relevant yet. We'd love to make it so, but it's very useful when you want to actually look at numerics. So you can actually now look at your numerical solutions and say this is a spin liquid. So the idea is that you have some kind of topological order parameter like you do in quantum Hall states that differentiates different types of spin liquids. And you have this idea of long range entanglement. So a singlet in general is short range entanglement if they're nearest neighbors. And the sort of, I'm gonna go through this very quickly. So you can calculate essentially the von Neumann entanglement entropy. So you split your system into two parts, part A, part B, you have some boundary between them. You take your reduced density matrix where you trace out all the states in B and find your reduced ventrary matrix. Now just for A. And then you can calculate this von Neumann entanglement entropy by taking the trace of row A, log row A. And what you find generically is your first term is what's called an area law term. This is proportional to the length of the boundary. And this generally depends on the microscopic details. And then the next term here is a schema which does not depend on the length of the boundary and any other further terms are gonna be in one over L. And so we drop those entirely. And this is known as a topological entanglement entropy and it is universal. And I'm still thinking here about gapped phases. Gapped phases are a little bit more complicated. And if you have a trivial phase like a valence bond solid or a magnet, you're going to get zero topological entanglement entropy. If you have a topological phase, you're going to get a non-zero topological entanglement entropy. So for example, the two spin liquids we seem to find the most that are gapped are Z2 and chiral spin liquids. And these both have this topological entanglement entropy of log two. And you can define many other entanglement entropies and other quantities. I just give this sort of as a simpler example, but with these you can actually differentiate with higher, with more complicated terms, you can differentiate between say the Z2 and chiral spin liquids and you can start to see, sort of get positive identification from your numerical simulations at least. Okay, so to sort of summarize, we have these quantum ground states without long range magnetic order. You have no symmetry breaking in your original definition, in your topologically non-trivial modern definition, you must have long range entanglement. And this actually allows us to say, well, you can actually break some symmetries and still be a spin liquid. So there's chiral spin liquids, break time reversal symmetry, there's proposals of pneumatic spin liquids, just break rotational symmetry. As long as you still have this topological order, you're still a spin liquid. Okay, and your spin ons can have these non-trivial statistics and can be either gapped or gapless. And you have a whole big range of possible spin liquids. So Katayev is one example and there's essentially thousands of different kinds of spin liquids. And this then sort of brings us to the question of, okay, well, this is all very nice and completely theoretical. Most of these spin liquids do not have concrete realizations, at least not yet. That was certainly thought the Katayev model would not have any concrete realizations for a long time. So, okay, so what we wanna do, our goal is to sort of get to the point of first a concrete model realization and then second a concrete experimental realization. So where you want to go to find a spin liquid in the wild is, first of all, your key ingredient is this idea of magnetic frustration. So here I have a picture of a triangle and I'm imagining ising spins here and they have an anti-pheromagnetic interaction. And so if I put down two spins, I put down one spin up, the next one I put down anti-pheromagnetic to make this bond happy, then I don't know what to do with this third spin. So this is sort of the fundamental idea of frustration and triangles are sort of the fundamental unit of this frustration. In general, we want low dimensionality. One dimensional magnets are always, there's no language magnetic order there and 2D is much more interesting and you always want quantum fluctuations. So Heisenberg spin one half is sort of your best place to look. So, and then where you find them experimentally in the wild, the best example I think is the Kagame lattice. So it's named after this type of Japanese basket work and this is corner sharing triangles in two dimensions and this is sort of the simplest, the most frustrated possible two dimensional lattice because you have triangles that are as weakly connected as they can be. So from numerical simulations, GMRG results and exact diagonalization and many, many others, we know that we have a spin liquid ground state with a very small gap. Probably this has actually come back into being debated and it's a fairly stable robust spin liquid and experimentally there's a number of materials that realize this lattice but the best candidate for spin liquids is this material Herbert Smithite. Which is actually a naturally occurring mineral. It exists on the moon or where the temperatures are low enough so that you can actually have a spin liquids on the moon. But if you actually want to do measurements, you have to grow these tiny crystals that Youngly has done and you can actually see the spin on continuum in the neutron scattering and there's potentially a small gap seen in NMR. This has also again become more controversial since I first wrote this slide. Okay, so this is sort of the nice, a simple example on one of these spin one-half two-dimensional Heisenberg models. And there are many other possibilities. Sort of the next set of fairly robust possibilities are these organic triangular lattice materials which may be gapless spin liquids but are pretty clearly not a simple Heisenberg model. They're close to this Mott transition. And so I'm gonna sort of stick with the simpler Heisenberg models. And if we want to look now at models and not necessarily materials, we can look at a variety of sort of simple two-dimensional models. So if we look at the J1, J2 triangular lattice, we have 120 degree order for zero J2 and as I turn on a second neighbor coupling, I go into this spin liquid region between about 0.06 to 0.16 and then beyond that, I go into a collinear order. I also have the J1, J2 square lattice. Again, this J2 introduces these triangles. Still unknown whether or not this has a spin liquid for regions around J2 over J1 around one-half. And in the honeycomb lattice, you can again introduce J2. Again, it introduces triangles and frustration. And there you have a tiny region which may or may not be a spin liquid, probably not. Basically two out of three DMRG results say it's not a spin liquid and one says it is. So I think it's still up in the air. And you also get some valence bond solids here. But there's no experimental candidates for any of these spin liquids in part because it's sort of hard to get into this narrow amount of phase space where you have a spin liquid. So this is sort of the general idea of this talk is that if a spin liquid is in a fairly narrow region, can we do something to enlarge this region? Yes. I think it's fairly firm at this point. There is like seven or eight studies. So several DMRG studies, exact diagonalization studies, all show a spin liquid. The debate is more about what kind of spin liquid it is. Okay, so this question has then led us to introduce this model, which we call the stuffed honeycomb lattice. There's sort of a threefold motivation for this which has evolved over time. The initial motivation, which I'll talk about briefly is a material in the Themes Agatou, Militium III, Oxygen VIII. You can also engineer this in different ways. But there's also just sort of the pure theoretical interest in that it's a way to interpolate between the honeycomb and the triangular and also out to the dice lattices while keeping this hexagonal symmetry. So it's a way of taking your triangular lattice and making it anisotropic in a different way than is seen in the organic triangular lattice materials. Okay, so this is the model. I have a honeycomb lattice where I put a J1. And then I put an extra spin in the center of each honeycomb. So in the dual lattice to the honeycomb lattice. And then I couple these extra spins to my honeycomb sites. So if these bonds are all equal then I'd get the triangular lattice back again. And then I had J2s on both the honeycomb sites. That's where they are. And then also on the triangular sites. And for simplicity, I'm keeping these all the same. In some materials, in this material they probably wouldn't all be the same. If you're engineering it in other ways this is sort of a natural assumption. Anyway, it makes my face face two dimensional and makes it much nicer to present. Okay, so, and there's three sublattices. This is a non-Bravais lattice which makes our classical analysis a little bit more difficult. Okay, so our first motivation is this interpolation between the honeycomb and the triangular lattices. So we see there's a clear spin liquid on the triangular lattice. There might be a spin liquid on the honeycomb lattice and maybe the original idea is maybe you go a little bit away from this by turning on these J primes and you enlarge a spin liquid region. Okay, and this is, I should also also want to note especially since we're talking about Pierce this week that this is related to the windmill lattice that he has looked with along with Peter Orrith and Pregme Chandra and Yorick Chimalyan and which has all sorts of interesting classical order by disorder behavior. And what we find is that this is actually most interesting when we go to the triangular lattice limit and move off of it and we find essentially that what was a weak first order transition in the, on the triangular lattice becomes actually now a multi-critical point. I'll talk more about this later. So I just want to talk very briefly about this really interesting and totally inconclusive, not understood at this point material. So this is lithium zinc-2, molybdenum-3, oxygen-8. And the basic idea here is that you have clusters, especially molybdenum-3, oxygen-13 clusters are then forming molecular orbitals where they have a single spin on half and these form the triangular lattice. And what they see in this material, let me bring up, oops. Okay, this is the susceptibility, the inverse susceptibility versus temperature in this material. And you see two different regions of curie behavior. And if you look at the first region above 100 Kelvin, you see an effective moment of 1.4 magnetons, a fairly large curie vice temperature and you can essentially understand this as you see all of the spins are all in a paramagnetic state. But below 100 Kelvin, something very interesting happens, which is two thirds of the spins completely vanish. They're not seen in neutron scattering. If you look at there, so if you look essentially at your curie constant out front has become one third and your curie vice temperature becomes significantly smaller. And what you're left with is essentially just these central spins. And you can look, you might worry this is a single ion phenomenon but they've looked at the ESR and they see that the G factor is basically not changing. So the natural explanation is that two thirds of the spins have disappeared. And then that leads to the question of why on earth would they do that? And so this is sort of, the idea is that if you have this triangular lattice where you have these clusters, here's a cluster. What happens at 100 Kelvin, it turns out is that the lithium above 100 Kelvin is very light atom. So it's actually mobile those lithium zinc layers in between the molybdenum oxygen layers are disordered but they order at 100 Kelvin. So this then changes is gonna change the structure of your lattice. This is one possibility. There are other possibilities that Gang Chen has also worked on the breathing Kagame lattice. But here is sort of one possibility is that it can decouple or it can distort into this stuffed honeycomb lattice. So it can do this if you have basically your sites ordered in such a way that they couple to some breathing mode of this octahedron and then you can shrink or enlarge the oxygen sites and affect your super exchange. So this is one reason you might be interested in what this model does. I don't think it provides a natural explanation for this but I sort of, this is something that can happen is more what you should take away from this. What I think might be also interesting is to try to engineer these things. So you can engineer honeycomb lattices by just having bilayers of triangular lattices. You could do the same thing for this if you make just a tri-layer of triangular lattices. So if you have, basically if you look at your A sub lattice, all the red spins form a triangular lattice, your B's also form a triangular lattice and as your C's. And so you just stack them in this ABC fashion. And here you get your J1, J prime, J2 model where your J2 really is the same between the C's, the A's and the B's because that's your sort of intra-layer coupling there. So I don't know any materials realizations of this but it would be really interesting if someone could do this. Okay. So here's our classical solution. As I said, this is a non-bravellatus. So we have a very complicated phase diagram. I'm not gonna talk about all of the phases. If you want, you can ask me afterwards. There's some strange ones, but let me just take you through the important ones. So first of all, the axes here, I have my J prime over J1. So over here is the honeycomb limit. This, when they're equal to one is the triangular limit. If I go out to infinity, that gives me the dice lattice. And then the Y axis here is this J2 over J1. So the first phase down here is what we call an interpolating phase. It interpolates between a nail phase down by the honeycomb limit out to 120 degree order exactly at this triangular limit and then out to this very magnetic phase beyond J prime equals to 2J1. Okay. The next one is we have a nail phase here where the C spins are now forming 120 degree order. We have a collinear phase, which is the same one you find on the triangular lattice. And now we have two new non-collinear phases. And this is the most interesting region here because if you look at this now, instead of just having a weekly first order transition between your triangular lattice, 120 degree order and collinear order, you now have these two new phases that come right into this point here. And these two transitions are still first order, but these two transitions are now second order. So you actually have a lot more fluctuations than you would have naively thought if you're just looking at the triangular lattice. So this might be a very good place to look for a spin liquid. And in fact, we know that there is a spin liquid there. So what we looked at next is what happens when you go to the quantum phase transition. And so here is our quantum phase diagram. This is for spin one half. This is obtained with exact diagonalization, mainly done by Brian's group. And so we did this on 18, 24 and 36 site lattices. And you can find, understand what the phases are by looking at your spin-spin correlations and also looking at dimer-dimer correlations and the like. And you can identify phase boundaries by looking at overlaps of your wave functions or level crossings. And so what we see, we understand all of the classical phases. There's also some valence bond, solid region up here. And then we have this large spin liquid region. And that brings us to the question then of what kind of spin liquid is this? What can we understand about this? And what can we gain by going away from just the pure triangular lattice to the stuffed honeycomb lattice? So we see that we go from, instead of having this one line, we actually stand out to a fairly big bubble, which principle might make this easier to realize experimentally. So what do we know about the triangular lattice spin liquid? So people have done a lot of work on this, mainly DMRG and exact diagonalization. And so the current proposals are, it's either a gapless U1 Dirac spin liquid or a de-confined critical point. We can't resolve between these two possibilities at this point. We also know that it's very close to, okay, thank you, very close to chiral and nematic spin liquids. But is not itself chiral or nematic. And from our exact diagonalization, we can see it's not chiral. You need a finite chiral field to go into a chiral spin liquid, which is then very obviously a chiral spin liquid from these topological signatures. There's no obvious gap, which in exact diagonalization, you always have a gap, but then you wonder what happens as it goes to the thermodynamic limit. It's not clear that it vanishes. It doesn't look like it's nematic. And so this expansion to the stuffed honeycomb lattice has given us more face space to play with to sort of see how it might, what we can get out of going off of this triangular axis. So I'll go through this next part fairly quickly because I'm almost out of time and the conclusions are the interesting part anyway. So essentially we can do a projective symmetry group analysis where we look at fermionic mean field theories and look for the sort of finite set of topologically distinct spin liquids. So I'm gonna skip the details of how we do this. But okay, what it tells us is some set of spin liquids that are possible. So we find some 15 distinct Z2 spin liquids. We didn't look for chiral spin liquids, just symmetric spin liquids. And then what you can do, this tells you what's possible. If you do another numerical technique like variational Monte Carlo, this can give you some sense of what spin liquids are competitive now that you're looking on an actual lattice with particular values of J2 and J prime. So what the variational Monte Carlo tells us is if you look on the triangular lattice, you have this U under X spin liquid. This is the single spin on dispersion. This is something described by only nearest neighbor hopping without any on-site potentials. And what you find is you have these direct cones. They have a six-fold symmetry. They're at, so I called them at the gamma point. They're obviously not shown at the gamma point here. And you expect them to be unstable to a gap Z2 spin liquid just sort of by their nature. They can gap out. They're allowed to by symmetry. So it's sort of naively what you expect. And so it's a bit of an open question of why they don't do this if this is in fact the ground state. So what we see when we go and break the triangular lattice symmetry down to the stuffed honeycomb symmetry is that we get two possible spin liquids. So the simplest possible thing we can do is we can gap this out. So you get second neighbor hopping. This is now allowed by symmetry. And you still, well, you have what were direct cones but are now gapped out. But again, they have the six-fold symmetry. This is sort of a contour plot that's meant to show you that. And this gives you now a gap to Z2 spin liquid. So this is kind of the boring story. There's a more interesting potential story which I can't tell the difference between which of these two is happening. But I will tell you now the interesting story which is the second possible projective type of spin liquid. Again, you can have second neighbor hopping and on-site potentials. But your direct cones here are stable. But this is no longer a U1 direct spin liquid but a Z2 direct spin liquid. So I didn't really explain what that was because I skipped that. It essentially means that your gauge degrees of freedom or your gauge bosons are gapped. And so in the Z2 case, I'm not in the U1 case. So the Z2 direct spin liquid is much more stable. We've actually broken the, okay. So the direct cones here are located at different places in momentum space. And so there they have a three-fold symmetry. So here you can see them as well. And so this would provide a way to sort of understand the results from exact diagonalization that you have a very small, either a very small gap. This is sort of an extreme picture in the variational Monte Carlo results. This gap is really tiny. And so you have either a very small gap or even fact distorted your spin liquid into this stuffed honeycomb spin liquid which is now has a robust direct cone. And all three of these are competitive right on the triangular axis. Once you go off of the axis, these two have significantly lower energy than the U1 direct spin liquid. Okay, so we're still working to resolve the difference between these two. You have to worry about the finite size scaling. So it's a little bit more complicated. So I'm just gonna put up my conclusions and I'm happy to take your questions at this point. Thank you.