 Good morning, everyone. So let's start the morning session. In this session, we have two speakers. They will discuss the theory of quantum spin rigid. As mentioned, the questions divided in the talks encouraged. The first speaker is Professor Natalia Parkins, University of Minnesota. She will tell us, Proving Kitaev's Spin Rigid. Thank you. First of all, I would like to thank organizers for organizing this great meeting in a great location. It's always a pleasure to come here. And I'm very happy that my talk is in the morning. So listening about Kitaev's Spin Rigid and hearing about continuum in Kitaev's Spin Rigid will not take you away of watching the beautiful sunset in Trieste. So profit the evenings. Before that, I want to acknowledge my collaborators. So basically what I'm going to present comes from different work. And most of the results which I will be talking today was obtained in collaboration with Gabor Halish. But also, we have some contribution with Yerun Vandirbring, Brand Perot, Yonis Russo-Hazakis, Stefanos Cortes, and Ioannis Knoli. So different aspects of their talk were presented in the collaboration with these people. And I will start with discussing, I'm talking about quantum spin rigid. And during the years, this subject, first of all, was interesting to condense by the people already for many years. But the definition, what is the quantum spin liquid change with time? Quantum spin liquid is a state of interacting spins. It's usually the state in the insulator that breaks no rotational or translational symmetry and has only a short-range correlation. And basically, in this kind of older definition, you only say what quantum spin liquid do not have. It do not have any local order. And the idea of quantum spin liquids actually traced back to the pioneering work of Anderson, who proposed resonating valence bond state. And this is the prototype of the modern quantum spin liquid. And as I said, so unlike the states with long reach order, quantum spin liquids are not characterized by any local order parameter. And therefore, it's quite difficult to probe this quantum spin liquid. What are the experimental probes for this state? However, now we understand the development in the theoretical side. During the last, I would say, 20 or so years, people understood that quantum spin liquids are characterized by topological order and long-range entanglement. However, there are no probe known today which would be directly coupled to topological order or directly probe long-range entanglement. So we have to deal with the conventional probes, those probes, which were experimental probes, which were developed over years. And therefore, signatures of quantum order, which is mainly in the excitation spectrum, and excitation characterized with a fractional quantum number and anionic statistics. And what we want, we want to see the features of this fractionalized excitation in the dynamical probes. So the focus of my talk is on the ketide spin liquid. And there are two reasons for that. Ketide spin liquid is exactly solvable, and I will say a few words about the exact solution. But also, what is very important, that there are several candidate materials which can show at least dominant-ketide interaction. And the experimental relevance comes both in two-dimensional system and three-dimensional system. The expeditions start with sodium and lithium monodate. And Hiditakagi was one of the pioneers in this field. Then, from two-dimensional materials, the most discussed and probably one of the best candidate for ketide spin liquid are titanium chloride compound. However, both these two compounds are older below a certain temperature. And recently, it was a beautiful study of hydrogen-intercalated lithium monodate. Again, this study was done by a group of Hiditakagi. And this is the compound which shows no range, no long range, ordered down to very, very low temperature. There are also other materials which are a ketide system in three-dimensional, and I will talk about this later. OK, so as I said, let's leave with what we have. And therefore, quantum spin liquid, we have a hope that quantum spin liquid can be detected by looking to the signatures of fractionalization of excitations in dynamical properties. It can be done with inelastic neutron scattering, Raman scattering with visible light, resonant inelastic x-ray scattering, and also now there is some study about the ultrafast spectroscopy. Of course, few features can also be understood by looking at thermodynamics and thermal transport. And what is interesting that since now we have these candidate materials, there are a lot of experimental activity in all these fields, trying to look to ketide materials and see, do we indeed see some features of a ketide spin liquid? So since this fractionalized excitation, they carry a fractional quantum number, fractional to the local degrees of freedom, which correspond to the spin flip, which is a magnum and has s1 half, only multiple quasi-particles can couple to the external prop. Because the external prop, what it does, it flip the spins. And therefore, the response from quantum spin liquid is always in multi-particle continuum. And we know that this continuum can be very different. It can be constituted from different parts. It can have different shape. And to be sure that what we see as some broad features has some relation to the features of the ketide quantum to quantum spin liquid in general, we need to compare experimental results with theoretical calculation. And this theoretical calculation should be done in a reliable way, which is not always easy to do. So the simple example of what is a fractionalized excitation can be seen in a spin 1 half chains. And basically, what you have here, so if you have a 1D chain, imagine that the neutron is coming, flipping one spin. And then what you create, you create two unhappy bonds. And the energy depend on the number of these unhappy bonds. You can flip further on. And you see that these kinks or these spinons can freely propagate because you're not creating in one dimension any new unhappy bonds. So and then you can think about these spinons as a freely propagating particles. And what is also great that in 1D, we have some exact solution. We can use better ansatz, those who know how to do it. And you can compute what is the constitution of this continuum. And basically, in this compound, the excellent agreement was achieved between the experimental results, which were published in the paper by Belalake, and the better ansatz theoretical calculation. So they actually even compare what is really two spinon contribution, four spinon contribution. And basically, the agreement is really excellent. So they can do the full analysis of this. And this quantitative agreement for students, what would be the spectrum of the excitation if this is not fractionalized? What is the main difference in this picture from the point of view of theoretical prediction? If you don't have fractionalized spin, this is just a spin wave dispersion that we see there. So what is the remarkable new? This is the spinon dispersion. Exactly. But if the state is not fractionalized, what would be the spectrum? If the spectrum is not fractionalized, then you will have a spin wave excitation. And spin wave excitation will be in the probes, will be seen in this very sharp mode. So the difference between, thanks for the question, excellent question, so basically the difference between the response in quantum spin liquid and the response in magnetically ordered states that you don't see any sharp response. Instead, you see some continuum. And then you have to analyze the bounds of this continuum, but also what are the intensities in this continuum. And as I said, so in order to be sure that you understand where this continuum is coming from, you need to compare with some theoretical results. You need to have a model. You need to do the calculation. And then you have to check are you getting agreement with experiment or not. In 2D, it's much more difficult because for most quantum spin liquid which are available in 2D, there are no exact solution. And if you do the calculation, then the usual approach will be some part on mean field theory, except this Ketayev materials. So basically Ketayev materials are so promising to help us to understand what is the nature of quantum spin liquid in two dimension and three dimension because both in two dimension and three dimension, we have limits, the pure Ketayev model which has exact solution and therefore we can compute different types of dynamical response exactly. And therefore again, at least we have some limit how to compare experimental results with the theory. So what is Ketayev spin liquid? And here I will profit a lot from the talk by Eugene Matsuda who gave excellent introduction to Ketayev spin liquid. But what is important is that Ketayev Hamiltonian which was proposed in 2006 work in materials with free coordination. So it doesn't matter if it's two dimensional or three dimensional, what you need, you need to have three types of bonds which form 120 degrees in between and only three of them. In this case, you can write this Hamiltonian in the written such that in each type of bond, for example, in this vertical bond, this is nothing else as the honeycomb lattice, you have only one type of bond interacting. Let's say z component of spin is interacting. And on the other two, x and y component is interacting. And as it was shown that this model can have exact solution not only in 2D for the honeycomb lattice but it's different version in 3D. So this one is called stripy honeycomb and you can understand how to get these lattice from honeycomb lattice. You take every second layer of the honeycombs, you cut this z-bond, every second z-bond and you rotate this honeycomb stripe by 90 degrees and then you have new stripes of the honeycomb going in this direction. And then you can also get the hyper-honeycomb. These two three dimensional lattices are particularly important. I will talk also about other three dimensional lattices because there are experimental realization of the material which has exactly these structures. Where exact solution is coming from? Exact solution is coming because in this Hamiltonian there is a large number of conserved quantity and this conserved quantity are local plaket operators, this WP. So if it's just for honeycomb, let me explain everything for the honeycomb. Generalization for three dimensional models are very simple. So you have this honeycomb, what you construct here you take the, you go along this honeycomb, along this plaket and you multiply a particular component of the spin and actually this particular component corresponding to the type of the outgoing bond which is not belonging to this plaket. So importantly, all these WP for any plaket commute with the Hamiltonian. So it's the integral of motion and that's why it's, and that's why it's exactly solvable but it's also important that it's a good basis because this plaket operator from the different plakets they also commute. And basically this construct the eigen values for these operators are plus minus one so this is the G2 variables. And if WP is equal to one, we say that this plaket has zero flux and if it's minus one, we say that this plaket has flux pi. So the genius of Kittaya for showing that it's possible to map this spin operators to the Myorana fermions. So now each spin is represented with four Myorana fermions, three corresponding having some spin component and one Myorana fermions is just neutral Myorana fermions. And if you do this, you rewrite the Hamiltonian using this Myorana fermionization, then you see that the Hamiltonian can be written like this. What is important that there is another variable, so from two of these Myorana fermions, you can construct the bond variable. So for example, if this is the Z bond, you can take SZ to Myorana fermions with Z flavor. And what is important that these lean variables are also static variables and basically this plaket operator can be rewritten in terms of these variables. So it's also static and therefore you can put the expectation value here. So you can now look to the system as the different configuration of these U variables which define what is the flux configuration. And in each of these flux configuration, you have quadratic Hamiltonian for C Myorana fermions, which now on I will call Myorana fermions and this U I will call flux variables because they create fluxes. So basically how we can look to a k-type spin liquid is like this. So if we have either temperature or we have some perturbation away from the k-type model, what do we have? We have some amount of fluxes and actually if we have perturbation, then these fluxes are not anymore static, but if you excite them with temperature, there are still fluxes. So fluxes are this red where you have this WP equal to minus one. And on top of this, you have dispersive gapless or gapped Myorana fermions and gapless or gapped, it depends on these parameters. If you are around the isotropic point, then excitations are gapless and if you are away, then excitations are gapped. And in the case of isotropic point on the honeycomb lattice, the excitation like this. So this is the Myorana fermions and you see that there are two half dirac cones here and there and these are the excitation which we want to probe. So now this generalization to 3D materials and this was in detail discussed by Maria Hermans. There are many of them. Actually the number of three dimensional lattice is very, very large and this is the hyper honeycomb or very similar the stripy honeycomb lattice. This is the hyper hexagon and this is the hyper octagon. So what is different? In three dimension, we have a multi sub lattice system and we have more sub lattices. Nevertheless, we can still define what are the minimal plaquette, we can still define the plaquette operators and again in the pure Kittaric model, they are static. And for Myorana fermions, for the C Myorana fermions, we'll have some dispersive band structure and what is important that the nodal structure of these band structures, where are the zeros, depend on how the symmetry are acting on these structures. And what I should say because we are dealing not with the lectons but with the Myorana fermions, the symmetry like time reversal and inversion are acting projectively into the Myorana fermions and for these lattice, you have a closed line of the data cones. So here we have the discrete point of well, discrete well points and for the hyper octagon, we can have a Fermi surface. So basically we have very different type of the nodal structures. And another important, very important part in the development of this field come from the work by Jackeli and Haleulin, which actually told us in which materials we should look for this Kittaric system and they showed that what is important is to have strong spin orbit coupling and this pseudo spin one half degrees of freedom which can be realized in these ions. So this is this is a spin which in particular geometries like this, which was discussed in detail by Uji Matsuda, the effective Hamiltonian, the effective low energy Hamiltonian is the dominant Kittaric interaction plus some other terms. And these of course are the terms that perturbation they will kill Kittaric spin liquid and eventually will lead to long range order. So once again, few more words about the experimental realization. It's the sodium or the date, the NL temperature is 15 Kelvin, the ground state is zigzag. This is alpha aerudate, NL temperature is 14 Kelvin, the ground state is incommensurate spiral, rotating current, NL temperature seven Kelvin, zigzag long range order. Once again, this is very nice exception where no long range order was determined. So it's a potential spin liquid and probably in this compound disorder place a very important role. So you see there is a very complicated structure very complex long range magnetic order at low temperature. Still for some reason we are looking for Kittaric features in these materials. And these are three dimensional realization. As I said in beta aerudate with NL temperature 37 Kelvin, this hyper honeycomb lattice is realized and in the gamma aerudate, NL temperature is 37 Kelvin, this stripy honeycomb structure is, is realized. So you see that there is already a big experimental development in last nine years. So now basically the question is, is can we observe simple Kittaric physics in this very complex Kittaric materials with very complex low temperature magnetic orders? And in this case, the kind of goal of my talk is very opposite of what was discussed in the first days. Like you have iron selenium such a simple structure and you beautifully show very complex and non-trivial physics in this compound. So here we have opposite, opposite problem with complex structure. Still we want to look to something eventually very simple. So, and this is a view. So maybe some other people have a different view for the Kittaric materials, but here this is the temperature and here is the perturbation, this other interaction. So basically there, and that can be, this is shown for today, but the same idea works for 3D materials. So basically the idea is that this is the Kittaric T equals zero and no perturbation, this is the Kittaric point and that's exact solution. So what happens when we don't have perturbation? Then thermally at very low temperature because flux excitations are gap, the only excitation which you have are my run of fermions. Then flux that are excited and basically you have a region where you still have some disorder of flux configuration and you see that the energy scale is very large energy scale in this problem. So basically here and then my run of fermions are moving in the soup of some fluxes and then you go to conventional paramagnet. So what happened when you include perturbation? Basically this perturbation will set up these long range orders I was talking about and then of course, Kittaric T could have some stability but we see that it's quite fragile with respect to perturbation and then we have long range order. The excitation in the long range order will be conventional magnums or sharp quasi particles and what we will see, we will see some sort of near temperature which will grow with the strength of perturbation and basically we have two energy scales. One is near temperature and another this energy of the magnums and basically only above this temperatures in energy scale we should look for fractionalized features and in the Kittaric model, this my run of fermions, they are free particles. So they are infinite lifetime and if the perturbation are small we can still think that this fractionalized quasi particles are long lived. Okay, so... Oops, sorry. Oops, just a second. So here is crossover for 2D. So if you go here, it's a crossover. If you go here, it's most likely the first order but nobody showed this how to say... I think it's still an open question but the way how it was shown it's most likely the first order. Okay, so and what I want to say that that's what people, what experimentalists are doing they are looking for the features at temperatures above near temperature and at energies above the energy scale corresponding to the long range order and indeed this is the result from an elastic neutron scattering from Arnav Banerjee. This is the Raman scattering and there is also other probes. There is still no ricks and I will talk mostly about ricks. So and there is a very nice, very kind of pedagogical review in the paper by Johannes Knoll and Merzner which you can find here about the Kittai spin liquid. And what we also heard that thermal hole effect indicates that there are certain regime where we can see the features of the fractionalized excitation and that was told by Yuji Masuda on Tuesday in his very beautiful talk. So let me say what is our proposal. So we also proposed that one probe which was not yet used but should be used probably is ricks and ricks can probe both two dimensional and three dimensional Kittai spin liquid and what is important that contrary to inelastic neutron scattering ricks probes each type of the fractionalized excitation independently, myarana fermions and Z2 fluxes. And basically these are our main findings that there are four independent ricks channels and there is no interference between them. Three of them are non-spin conserving channel and basically they pick up a flux and they have little dispersion and they are very similar to inelastic neutron scattering and there is one which I will discuss the most is spin conserving channel which directly probes myarana fermions. And basically what we showed that spin conserving ricks channel is allowed to study the nodal structure in this Kittai spin liquid but also allows us to study the temperature evolution of momentum energy map of excitation in the Kittai spin liquid. So in brief ricks, yes, ricks is two photon process. So basically you have a photon coming, it's observed, the system is going to the intermediate state and then in other dipotransition the photon is released and the system is going to one of possible final states. So in short, once again, since the energy of the photon is very large, the electron which is pumped it leaves in the core level. So the electron from the core level is pumped into a valence band and the core level create the potential. And then there is a, the photon is going out and once again we can see what is the excitation of the system and what we observed that excitation of the system should have the q momentum K prime minus K and also the energy. So and if I skip this, so if we look to our sodium erudite, then how we can think about this. So in this case we have a direct ricks process and this is L free H. So and in L free H, eridium has not bad resolution. It's about a 3.8 milli electron volt. And in this case, the core electron is excited directly in the 5D level. And then the final state in the spin conserving channel is such that the spin in 5D channel remains the same. That's why it's called spin conserving. So we do not excite any fluxes but instead we can excite myorhuna fermions. So at zero temperature we can only excite two, four and so on myorhuna fermions. But if we are at finite temperatures then basically other processes are possible. In the rutenium chloride in L H the resolution is very bad and actually we have to look to the K H. In a K H it's amazing that Silicium analyzer give amazingly good resolution around 0.75 electron volt. But in this case we are dealing with indirect ricks process. So the electron is excited from core one S level to 5P. And basically it's created the core hole potential which modify the coupling around this particular site. And this excitation intermediate state is very short leaf. And then once again if the final state is like this we can see the myorhuna fermion excitation. So because I have really very few minutes let me skip the formalism it's very standard. You first in order to compute intensity you need to first realize in which polarization channel you stay then you compute the amplitude using the Kramers-Heisenberg formula. So you do this analysis of polarization and basically the spin conserving channel is, this one is A1G channel and these are the free non-spin conserving channels. And in the spin non-conserving channels again in the final state you have two flux excitation. So once again skipping this the only thing what I will say that in all these materials gamma disinverse lifetime of the core hole is very large much larger than the excitation and therefore we can do the perturbation in the calculation we can do the perturbation in terms of one over gamma. And let me tell you about results in spin conserving channels. So once again we are interested in elastic response and these come in the first order in one over gamma. Okay so results, results in 2D. That's the spectrum which we are obtained. So basically now we can address the whole spinon continuum. It's not only Q called zero continuum it has all this Q dependent this is energy, this is Q dependent and we can compute this exactly. And we see that gapless response for isotropic model appears exactly as it should. So there are few points, few gapless points. So this is the response and what is also important that intensity is suppressed around the gamma point. At final temperature what happening? So the evolution at final temperature is like this. So as I said that at low temperature flux are not excited and we are probing only myerone fermion in a zero flux sector. Then there is intermediate temperature where we are probing myerone fermions in the presence of disordered flux configuration and then at high temperature it's the paramagnet and that's how we should modify the formula for Riggs intensity. So these are the Riggs vertices which give us the contribution. So and this what I was showing in my first slide. So at T equal zero you only have a stock response. You can only create myerone fermions in the final state and you see all these sharp features which are coming from fractionalization of exact quitaeus pelliquid and then with temperature what you see that you start populating this anti-stocks channel and if you go to very high temperature then the stocks and anti-stock channels are nearly symmetric. Very quickly what is the result for this 3D quitaeus pelliquid? Once again what we want to see how this nodal structure line or points or Fermi surface appear in the responses. So in this hyper-honicome lattice we compute the response and we see that in most parts of this white dot corresponds to the place where we do have gap because it's not really well seen here but in most part of the brilliant zone we have a gapless responses. So this is the hyperhexagon lattice where we have this discrete number of a wave point and you see that in the response there is only a discrete number of gapless points and once again these features are all computed exactly and once again when you have a Fermi surface then almost all response is gapless. Unfortunately I have no time to go into more details here and the last comments, this is my last before last slide, so it's on generic quitaeus pelliquid. So basically the idea is that was the result, exact result computed for models which are actually not realized in the real material. So this is this other terms, this perturbation which are important. So the high energy response will be very robust. So all these features which we propose for high energy they will stay more or less the same for other terms perturbation. However the lower energy response of course will be sensitive to the perturbation but it will also depend what symmetry are broken by this perturbation and basically in this hyper honeycomb and stripy honeycomb lattice this nodal line remains in this plane as long as this perturbation do not break this two fold symmetry around the axis. Thank you very much for your attention. Thank you very much.