 I just announced to announce Natalia from University of Minnesota, who will speak about disorder and perhaps been like this. Thanks Laura. Good afternoon to everybody and let me continue saying that it's great pleasure to be here and thanks for organizers to give me the chance to present our work. Maybe as you can guess from the title of my talk, I'm going to talk about the interplay of some randomness which can, which exist in all the materials and some quantum fluctuation which bring the system and probably some quantum disorder state and the playground which I'm going to discuss is there. So today I have spin liquid which was very nicely introduced on the tutorial yesterday by hanging. So with that, let me start. So quantum spin liquid, the definition, the most robust definition of the state is a negative one. This is the state which doesn't break any symmetry and does not order. So basically in the quantum spin liquid we do have intrinsic disorder. And this is one of the, probably the prototype of the quantum spin liquid is there are under some violence bond state the so resonant violence bond so state which was proposed in 1973 by Anderson. So we see that the state is the super position of covering in this case of the triangle and that's why I'm not saying what is the model behind the state. So it's a triangle that is covered by various position of this spin singlet, and the state is the super position of this singlet, perhaps only nearest neighbor one, or maybe it has all possible lens the singlets of all possible lens. So, these are what we mean by intrinsic disorder, which might happen in a quantum spin liquid. So once we go to the lab or somebody is going to the lab, Gersh goes to the lab. So, then, in any real material, there is some randomness and this randomness in the form of extrinsic disorder can come in all possible ways. There can be some impurities it can be some defect in the lattices it can be some bond disorder. So, whatever it is so and sometimes we see that the system in which there is a lot of disorder does not order, but then the question is, is it spin liquid or not. What happens if we take the really beautiful pure material which is in the quantum spin liquid state. Imagine that we have one, and then we put by hand some randomness in this material. So what it will do we would kill that quantum spin liquid state with support the quantum spin liquid, what will happen so and basically that is what we are going to discuss today. Okay, so my playground is a catastrophe liquid and I will tell you later how we can apply this playground to some real materials, and very very briefly just to set the notation. So with this Hamiltonian we have a honeycomb two dimensional lattice and on each bond only one component of spin is interacting while people was so interesting at this model because it's exactly solvable. This seminal paper like say guitar showed us that we can find this beautiful exact solution by representing each spin degrees of freedom with the help of my run the parents, and it's also clear even from the classical theory that it's very difficult to satisfy all the bonds that it will be this money for the degenerate state from which we can construct this superposition which will be at the end, the quantum spin liquid, which is not a product state, but should be as a superposition so this is my kind of very brief introduction to quantum spin liquid but now I want to discuss some characteristics of this guitar spin liquid and that the easiest can be done through the excitations. So, and this is from the paper I have referenced some of it's not seen now so once again, if we try to understand this quantum spin liquid state in terms of the spin is very complicated. So using this my run a representation from a key type, we now can see what are these fluxes and fluxes are the spin, spin flip fractionalized into a fluxes which are static and gap excitation and my run a permanence, which are the dispersive mode. So, in this mode we do have a fractionalized excitation and that is the means how we can actually study this, the property of this spin liquid state. And what is also important and related will be important that if we flip one spin, then we will create two fluxes on the nearby surface. And another thing what I will need for my later discussion is how I characterize this excitation so basically, once again since I'm going to discuss this order. Let me first say when I'm going to high temperature and I will show another plot for this, then, despite the fact that this gap and static fluxes, then, above certain temperature which will call flux gap. I will go to the random flux sector, but at the zero temperature. Leap theorem tell us that the system in the zero flux sector, meaning that all these plaquette operator, which are conserved quantity and which are the reason why the model is exactly solvable has a value one. So, in this sector we can discuss the thermionic excitation, what is plot here is the density of states for the my run a permanence in this zero flux sector so if I'm talking about thermionic excitation, I'm talking about excitation inside the same flux but I can also consider going from one flux sector to another flux sector, and then I can define what is the flux gap. So, if I put the energy scale on the vertical axis here. So, and then that would be the energies, the fermionic energies in different flux sectors. I can define what is the fermionic gap like from the ground state to another or what is the energy of the fermionic excitation. I have time to go so if I'm inside. This, this is basically the key trial triangle so if all my couplings on x y and z bonds are equal to each other or almost equal to each other so if I'm in this part of the parameters space then I have a gapless fermionic liquid. Then I can also define. So I go from one sector to another, and my flux gap can be either very small or very large so I will use this terminology, thermionic excitation and flux gap later on so I just wanted to introduce this notation. And as I said, our fractionalization differ at zero temperature, and at finite temperature so if we are very low temperature because flux excitation that gap. So there is a low temperature, the excitations are mostly this my run a thermons and the situation is very close to the graphene physics, and then at the temperature around 10% of the guitar interaction. I go over to the, to the face, in which I have a random fluxes. And then I still have some in this in this random flux of sector. I have this my run a thermons and then of course, there is a once all the fractionalize excitations are excited. At the energy scale which is my th, which is of the order of key type interaction, then finally I'm going to conventional paramagnetic state and then there is no sense anymore at this temperature there is no sense anymore to talk about the fractionalize excitations. And that was all the theory at the beginning, and then, once again, I'm not going to discuss it in detail but studying from 2009 from the paper of jacquely and halloween, where they told us a recipe how, how to make these materials. This field become very active it has both experimental pattern theory, and people. It's become really the field, which maybe one can call key type of material. And once again what is the microscopic mechanism, you need a spin orbit coupling in order to have key type interaction it. There are also other non key type interaction and once again, in the tutorial yesterday hanky discuss this in detail. So what I want to sell with this that we do have materials so after this are 13 years. Basically, the list of material is very long and crystal growers are telling us, often, how we have a new materials which might be a candidate material. At the beginning, so the study started with the sodium and lithium irradiate materials, and then of course this alpharotenium chloride, which is the most studied key type materials. So, all these materials order at low enough temperature. We know that because of this non key type residue and actually which are probably smaller than key type interaction. And that's why we are calling the key type materials are materials in which key type interactions are the dominant direction. So from this plot from the review of he did a copy and others that in all these materials except one, and this is the last one I'm going to talk in more details about thermodynamic quantities show some transition present and basically the low temperature face is, is magnetically ordered. So people start trying to see how to kill this residual interaction and how probably find the two key types in liquid with no longer in shoulder. And then there's another list of materials. And this is once again the second generation of key type material this idea to go first and the second generation is taking from this paper of Simon traps. So these second generation of key type materials are more disordered. So it's clearly there is a disorder there. And at least, we can say for sure about this hydrogen and the collated lithium ready that this material does not order down to very low temperature so it does have disorder because this position of this hydrogen atom, they are in between the layers so it's not in this honey between, but their positions are not at all well defined so there is certain disorder, and people from the beginning understood that this the absence of the long range order is related to this quench disorder. And then the question is it just this ordered magnet or it is in liquid. And once again so this is the material of interest. So, in short, this material does not order at least down to 0.05 Kelvin, but it also have some divergent specific heat at the low temperature which actually tell us that there is some pile up of low energy states. And once the magnetic field is applied to this material. In this case it's applied along one one direction. So, this upturn of this, the suppression of this upturn in the lower temperature. The upturn is suppressed very very quickly with the very small magnetic field. And in addition, there is probably some scaling behavior so that was very interesting data and many people look to the disorder in this particular material trying to answer to the question. So, what is this a Kitayev still liquid. But I think in all these lists and maybe I need some work, and works I'm sorry, in that case. So the people asked the question, starting from the earlier work of John Jolket group. So, what disorder does for the Kitayev skin liquid behavior. So, and that we also look to that so basically let me tell about our contribution what I'm going to talk today, the driving force behind this especially numerical calculation is of my student then Hong Kong was in the University of Minnesota and absolutely bright day. So, let me first say how we describe disorder in the Kitayev skin liquid. So this is the pure model so basically, you have here different colors which correspond to x, y and z type of the bond. So, as I said, if you go above certain temperature, then you have a soup of different flux configuration in the pure Kitayev model they're still static. And basically, one of these configuration is shown here. So, the thermal disorder is a particular characteristic of the Kitayev skin liquid. And in that we can engineer different types of disorder for example we can look to the bone disorder. So some bones here, shown in the zero flux can be weaker or stronger than the remaining both. But we can also introduce a certain number of the vacancies or quasi vacancies. In that case, there will be a certain density of the sites, which would be either completely decoupled from the rest, or will be very, very weakly coupled and in that case we talk about not vacancies but quasi vacancies. And at the end of my talk, if I have time I will also talk about the strong disorder. And what kind of effects we are going to look surprisingly or maybe not surprisingly in this Kitayev skin liquid, the concept of fractionalization actually bring together different ideas of Anderson. We can look for Anderson localization in the Majorana fermions. And basically that comes together with the idea of the state which cannot be written as a product state. But we can also look to the leafships they also this all this kind of localization phenomena can be studied in the Kitayev skin liquid. And also, if we're looking to the strong disorder, probably we can look to some critical phases and some universal behavior. And of course, these order can induce a particular flux structure. Okay, so let me start with the weak disorder and compare two different type of disorder. So when we have a bone disorder and decide disorder, and you will see that in this case, we have completely different behavior. So when we are doing Kitayev, his original paper I just rewrite the Hamiltonian in terms of the Majorana fermions, and that is a straightforward way of doing. And this is just the Kitayev model. And the second term is the way how we can perturbatively describe the effect of magnetic field which break time reversal, and they will have for the students here just have next slide to explain how to obtain this term. Okay, the disorder is introduced here in the model. So now we have this j i j which depends on the bond. And with that, we can mimic any type of disorder which we want we can either have a binary bond disorder, or we can also do a uniform bond disorder when the bond is kind of distributed. And we can also divide all the sites into sites which are not connected to the vacancy and those which are connected to the vacancy and basically put his here this interaction j prime, which for vacancies simply zero, and for vacancy vacancies significantly smaller than j. So this is just one slide how this free spin interaction turns is appearing in the Kitayev model so you start with magnetic field which is, in this case in one one one direction. And then you treat that this Hx is small compared to the Kitayev interaction but what is more important that it's small compared to the flux, get the energy which goes to create fluxes in the lattice. And then what you do you just apply this term one after another, each spin clip introduced two fluxes. So if I apply Sigma X here then Sigma Y here and then Sigma Z here, then I'm going from the zero flux back to the zero flux. So, and this is basically the leading term in this perturbation what I have here in terms of the minor unfairness. Now I have a second neighbor hoping and these models now with the presence of couple look very much like how they model. What it does for so it doesn't change the flux sector if we start from the zero flux sector we remain in the zero flux sector. If we start for a particular vise and crystal so a particular flux configuration, we still remain in this flux configuration, but what it does for minor unfairness in the zero flux sector for example, just open the gap. So we will use it later so I just want to show how it. So the problem is, now, once we have this disorder, we cannot momentum is not any more good quantum number because translational symmetry is broken, and therefore we should do all the diagonalization finding the eigen energies at a finite system. Luckily, it's just a hopin Hamiltonian it's not so difficult and we can go to rather big system and basically, we can diagonalize this Hamiltonian in the real space using single value decomposition. Now, we can rewrite the diagonal Hamiltonian in terms of the complex fermion, but here and is not the momentum, but it's simply all this eigen energy in a given flux sector. So, now, how we started this order. Okay, so basically, let me start with a bond disorder. So, and in this case, as I said, I can look to a certain number for bonds which here is written by raw. It's like a binary disorder. And on a certain density of bonds, I can either put J prime which is smaller than J. And in this case, I dope my system with the weak bonds, or with the stronger bonds like J code to where all the rest bonds is equal to one. So what is important here is the density of states, and on the background, you always with the dashed line, you see the density of states for the pure kittles model. And you can see that, in principle, at least by looking to the density of states. Okay, so what is depopulated the states at the bump hope singularity is depopulated and somewhat distributed in the energy. So if I have the strong bonds, then I have more state at the higher energy, but overall, one can say that if this order is pretty weak, then this modification of the state is not that strong. Okay, so this is just the density of states and the blue color here means that I compute everything in the flux, zero flux sector. So, this is reasonable at temperature below the flux gap. I can also compare for the bond disorder this for the random flux so when I go to the temperature above this energy scale, and I consider, once again, that the probability for each packet to have either WP equal to one or minus one or equal to all these random flux. And then I can see that all these features in this random flux goes away. Okay, so the reason why I do not compare it with any other kind of flux configuration only zero flux or random flux. Once you have a bond disorder locally, your flux gap can be very small. And basically, the transition from zero flux to random flux is happening pretty fast. It can only tell me about the state it doesn't tell me about the way functions are there localized or not localized. So, and for that we can, we can introduce another quantity, which can illustrate the localization and that's usually done with the help of inverse participation ratio this PN, PN is computed for each eigen mode and, and here I is the summation oversight. So this inverse participation ratio, if we have a delocalized mode, then we will have a contribution, a small contribution here from many, many different sites. And then you'll see that this PN will scale as a one over and because and many sites are contributing where and scales with the system size. And that would be very small, however, if we do have a localized mode, then this PN would be larger so if we have just one stated would be just PN would be equal to one, or in any case that would be significantly larger value kind of order of one. And here this IPR inverse participation ratio everywhere is shown by red, and you can see that if we have a bone disorder, then, even if we have some low energy state low energy states remains delocalized in all possible cases. We do see that this, especially when we doped with the stronger, stronger bonds, we see that IPR is significant on the tails, and that is in fact what is called the listed sales, when the states at the age of the band becomes localized. So here's just a comparison of IPR for weak bond and strong bond cases. So here we have 10% of the weak bonds and, okay, so there is no listed sale there is no localization here. And that is, if we take, for example, one of the high energy modes from that area. So, since the majority of bonds which contributes to the density of states here they are coming from the strong bonds, which are the majority, then these bonds can be easily delocalized. However, if I doped with the strong bond so here it's J prime equal to two. If I take the state from close to the age, then this state has their mostly dominated by this weak bonds with the strengths are equal to two, but they are surrounded by the weaker bond, and that leads to the localization to localization. So it's kind of very intuitive way to understand why these states are more localized. And basically, this is nothing else as a leaf shit sales. Similarly effect was also the coalition sales but if you go back to original Anderson model about the localization, then he also mentioned this. Okay, so this is about the bond disorder. In any case what you saw that you don't have a pile up of the low energy states in this case. Maybe you see some only if you consider the random flux configuration. But now I want to consider the side disorder. And basically I want to consider my back again he's the Hamiltonian where I divided into parts. This is for the bulk and this is basically for the states around the vacancies. And what do I see so around each of these vacancies. Once again this J prime is zero for the true vacancy and very small for the quasi vacancy. So first I have like either missing side or inside or very weakly connected side. And I have some mode which is localized on the opposite sub lattice so if this is a site that on the B sub lattice around this site. And basically which was shown first by the group of john Joker already in 2010. If you have a vacancy, then energetically it's prefer to attach the flux to it so that is called flux binding the energy load if you attach the flux to the vacancy. Okay, it's always what you do you always introduce the flux in the pairs. And in the ground state you will not be in the zero flux but you will be in the bound flux state. So now just to have this kind of color coding so blue is always for flux free orange is always for random flux and green I show for the case of their side disorder in the form of the vacancy. So here, for example we have the density of vacancy 2% here the 5% that is the case of the true vacancy. And here is the case of the quasi vacancy when J prime is equal just 1% of J, and I have 5% of the state. So immediately see that in this case, you have a pile up of the low energy state. Moreover, these states are localized so, or at least they are quasi localized and we need to understand this, what is this localization. So the spectrum weight is transferred from around the one whole singularity region, it's transfer both to the low energy region but also a little bit. So it's transferred to the low energy, but you also have localization so red is always the IPR. So you see that both the low energy states, and the state at high energy are quite low, quite localized. So, again, so this is delicious. And, once again when we talk about this quasi localization you see it's not one, it's large but not one. So, in fact, these low energy modes are only quasi localized, and they decay with a distance as one of our to the power alpha so it decays the way function decays when you go away from the vacancy. So, in fact, I can consider this vacancy also in the random, in the random flux. So basically I do not attach the flux to vacancy but I consider the random but that's why it's in orange. And then you see that in this case, the vacancy still give you some pile up of the low energy states, but they are not localized. Basically, we see that one ingredient which we need to explain this experiment and I will come to the experiment in the moment that the vacancy introduce a certain number of low energy states and these low energy states will be seen in the thermodynamics. So this is the conclusion of part one. And basically what I wanted to show that there are different type of disorder, side disorder, bone disorder and also thermal disorder. And once again in the Kitai quantum spin liquid, as I said before the ideas of quantum spin liquid and the one and one body Anderson localization kind of intertwined through the concept of spin fractionalization. And now basically what I am going to discuss, can we use this vacancies in order to explain experiment by Takagi when lithium adidate was intercalated with hydrogen. Okay, so here's just the collection of data. I don't know so maybe everybody saw this before, so many different type of experiment were performed and they're all reported in this, in this paper what is important that it's indeed an insulator. And you see this NMR there are no splitting or broadening down to the very low temperature, but also here you see in the NMR you also see that there is a low energy excited. And then, once again this particular Paulo divergence of the C over T in the specific heat, and some scaling behavior in the presence of the magnetic field. You can see how we can explain this with our theory. So, once again, let me consider now this 2% of the vacancy, and I can see that this bound flux bound to the backers. So then this is the density of states. So it's not really universal because the way how the slope is starting depends on the on the density, but I can clearly see that I have the corresponding upturn in the C over T. And what is important so it just basically comparing in this insert, what is compared is how the specific heat behave in their, in their pure model, and then how it behave in their bound flux. There's some questions so maybe you can read it to me but let me finish just this slide. So you see that this upturn you get both in the bound flux and in the zero flux. And once again it's how this upturn depend on the system size. And basically you can see that when you go to bigger and bigger system size, then the slope more or less saturate to, you can saturate to a particular value of like 0.5 which is very close to the experimental result. So what is the question. Yeah, the question is what is the physical significance of the random flux sectors at zero temperature. The ground state isn't zero. In zero temperature, there is none. So, in principle, you can think about the disorder but what is happening that in the cataract model we have all the flux sectors. So zero flux at the ground state and these random fluxes these are all different configuration these are excited states, but once you have a temperature, you can, your system is actually the soup of all these different configuration. In any case, we are interested in the term of dynamics so we need to consider what is the exciting stage of the model. Okay, so this slide is just to show that the vacancies are kind of good because once again flux sector is one thing, and also the density of boxes is a kind of the external thing which we control. But you can see that in all these three cases green that is the vacancy with the bound flux, zero flux and the random flux. You can see that you have upturned. So the question which you need to know to answer is how to suppress this upturned so what do you need what are the ingredients, you need to suppress this pile up of the low energy states in there. And here it becomes a bit tricky. So, one can do a very careful numerical study. And basically you can show that when you apply magnetic field and we always remained in the exactly solvable limit so we never say magnetic field we mimic it with effect of copper. And that is this free spin interaction which I show you before, because in that case the model remain exactly solvable. But you can see that here is the transition so when copper is growing there is a critical value of copper, where above which it's not anymore energetically preferable to attach the flux to the vacancy. So the system goes from the bound flux state to the zero flux state. So we just look in this particular kind of either in the bound flux state or in the zero flux state. We're now switching up copper. And what we see what copper does copper opens the gap in the bulk of the my run effluent. What we see here when we go down so these green once again it's a bound flux blue is the zero flux. What we see that the structure of the states are different in these two cases. And there's difference in the, in this in gap state in gap spectrum is actually responsible for for this behavior. And I would want to describe it in the middle. I don't know if there are more details, but you see that when we have a bound flux, we know when the time reversal symmetry broken, then for each flux in the time reversible in time reversal broken face, it will be a zero and this. Yes. And the zero mode is basically will be very important. Yes. You're back on the particle both. Yeah, it's remain my run effemence. I will always have particle false huge disorder with even disorder. And in my case, my physical states are actually I mean it's just shown here also so in the guitar model when you're dealing with my run effemence, you redefine your vacuum so in the vacuum, you feel all the states with the negative energy, and the excitation only with a positive energy because in my run fermions you have extended human space. So, but the particle whole symmetry remains all the time because my run effemence creation and regulation operator is the same so. I will just discuss, I will discuss the zero energy in the next couple of slides. Okay, so what I wanted to say that we do see this or if you follow my stars. So you do see that when copper is growing I should go from one sector to another, but you also see here that the. Basically, the idea is that once I have big enough copper. So, and my gap is big enough. What is responsible for the behavior in specific kid is what is the spectrum of this in gap state. And if I have a bound flux, because once I have a flux I have a zero mode, which will be broadening and I can show it in the like toy model. These states here would still give you some upturn, even when copper is growing, however, because when the copper is growing you have to go from this picture to that picture. And then you see that basically the absence of the zero mode in the zero flux sector. Actually, you start having some less and less state inside the gap. And the difference is once again, what are the low energy modes which we can see this so for each vacancy in the zero flux state. We have something what we'll call be this mode is like either the absence of this is the vacancy mode, which would be localized. And then we will have this mode, the peripheral mode, that's the same stories and graphene so, and these mode is only quasi localized. So once we have a flux attached to this, and this is our flux mode, then when copper is broken then we'll have additional low energy mode. So basically this is kind of this toy mode for explaining this in gap modes. So, again, so this is basically what is written here already said that I have these two different vacancy I have interaction between these low energy modes inside the vacancy. So I can, sorry, it's not really seen. But in case so this, I don't know how to remove this. Okay, maybe it's because here I remove everything so maybe I should also remove. Yeah, I want to remove because otherwise. Okay, in any case so what is written here, maybe the simplest isn't optional. Sorry. Just bring it up. And I will show play. Okay, so what I want to say that we can write the minimum or if the gap is large we just write everything at the energy scale below the gap. We can write what is the interaction between these modes inside the vacancy, and that is just proportional to this day prime, and that is the quantity which is basically it's zero if it's, if it's a true vacancy and it's very small, but I also have this coupling with this vacancy because all the vacancies are created in pairs, and this coupling is weak. It's weaker and weaker if the vacancies are further apart so it's proportional to one of them. Okay, so, and then, basically when copper is getting larger, what is happening this my zero mode is depopulated, and then I have only this interaction inside which is basically I have a gap so it's very difficult to communicate in between these two phases. So, and there is nothing to protect this zero mode and they become gapped because of hybridization. The situation is very different when I have a bound fuck, because now I have three degrees of freedom and I need to write this interaction inside the gap for these three different modes. So this zero mode, it's hybridized of course a little bit with another low energy mode. So it's broad and somewhat. So I do have a macroscopic number of these, I have some number of states around zero, and this state still participate in the thermodynamics so they should be seen in the thermodynamics. So in short, the answer is that the fact that the specific it's you were T is suppressed when we increase copper is because of two things. And because we have a transition from the flux three sector. So sorry from the bound flux sector to, okay, to the, from the bound flux sector to the zero flux sector. And then, once we go into this transition, then the structure in gap states changes. So, and once again, we look back to experiment, and we just compute this curve for different copper, and you can see that. Okay, so you suppress it, and then you have this behavior so which looks very similar to what was observed experiment. Not really exactly because couple of course is not the magnetic field, but assuming that copper is proportional to h cube, we can still try explain this scaling behavior. So I understand that I have only five. So, basically, I know that he did a colleague is not completely buying our theory and I don't really understand the reasoning. But I do think that this kind of vacancy of some sort of these local defects is actually if some reasonably some reasonable understanding of the experiment. So, we are kind of happy with that. Okay, so, let me, I have really few minutes but let me tell you where we were going recently from that so what I was telling before it's all about the week disorder. And then, by reading this paper from it a mark in Chi. So, at the same time they were also thinking about the same experiment. And they were thinking that, in fact, this kind of scaling behavior in the specific heat, which was seen in with the magnetic field can be understood with the help of the random singlet face. So we wanted to understand this. So first let me say that random singlet face. That is the face which naturally appears in the one D model in the disordered one D model and they were studied to death with the help of a real space, strong disorder and realization group. Basically, the idea is when you do dissemination when you get read at each step of iteration you get rid of high energy degrees of freedom at at the end you go to the low energy phase, where each side participate in the formation of the single. So, but, and basically in one day, that can be the exact solution. So, but a random singlet is not a quantum spin liquid and if something like this is happening here, then perhaps we should not talk about the guitar spin liquid. But we should talk about some strong disorder which leads to a formation of something which will understand in one day, and in 2D that's very difficult to study. So, with them, what we did we tried to look to the minimum model in which we can still study the physics of of the guitar model into the what do we need, we need to have one of the important thing for exact solution. So, we need to have this placate operate. And then we wanted to have something quasi one day that we can apply this strong to the disorder and realization group. And for that we just took a stripe of this honey combs. And basically, in order to, to employ the periodic boundary conditions in this direction so you need to add this additional coupling here. And then this stripe of the honey combs is nothing else as this letter, and the guitar model on the spin letter is exactly solvable you still have a placate operator, and placates are still exactly so. Basically, what we first look, and that is for the pure model, we look to the flux distribution flux gap distribution in this model. This is a honey combs model, and you can just put it jacks over jz here j y over jz. So, this is some sort of Kitayev triangle but computed for the flux degrees of freedom. And we computed also for the letter, and there is a certain similarity between the two so without okay so it's difficult to study this problem is difficult to study, but let us study the letter. There are two limits which we only managed to study. It's this xx limit. So here everything is in terms of jz. So, in the x limit j x and j y equal to each other. And in the icing limit we considered j y very small so these are two different limits. And really two minutes so what I need to say quickly is that the people before us in this in this paper they make wonderful transformation so there is a duality transformation which map this. So it's better to the one dimensional chain, but in this mapping so it's exact mapping what you have is now your placate operator this WL is inside the model. And this is already one D model and this is amenable to this strong disorder and realization group analysis. And skipping all the details. What I want to show is now you can in this model in this one D model you can study in these two limits at least you can study the scaling of the spin and floods gap. Again, I understand that I don't have time let me skip details and I have them in the conclusion. So, basically what is important there is another model one D model where you can study when you can apply this very well developed as the RG approach. And basically what you can see there that you have a spin gaps. So we look to the extreme statistic to the minimal spin gap and minimal flood gap. So, you can see that the spin gaps and floods gap. Remember my definition at the beginning of the talk. They show different type of the criticals they show some universal behavior in some cases, but this universal behavior depends are you looking for the spin gaps or you're looking for the flux gap. We cannot immediately extend our one day results to the to the result, but I think we can learn something. It's something non universal is happening at really low temperatures in this 2D, perhaps we can understand it with the help of this one day analysis. Okay, thank you very much for your attention. Thank you so much for your questions. So, the experiments then I reproduce by back and see disorder, no bone disorder. I mean, yes. I mean, in order to have this for this experiment you're talking about this, so you can still are somewhat explained with the help of the bond disorder but for these units to assume the random flux configuration. But if you have a random flux configuration that means that your flux gap is very small. It's not clear what protect quantum spin liquid, and is it really the spin liquid behavior or not in vacancies, the gap for the fluxes for most of the pocket is still the same flux gap so spin liquid behavior is preserved. Let me read. Okay, I just saw the, it's already disappeared. Okay. Please. No, we'll do it later so. Thank you, Natasha. I came in late so I may have missed this, but do you ever explore the possibility of non ergodic phases like glassy phases where the dynamics are frozen, because at least the phases that I heard about tended to be ergodic phases. I did not discuss dynamics at all so we only consider thermodynamics from the statistical point of view trying to explain this specific heat and so on so to some extent no. Okay, but in principle, you could have them in principle can have and you could have some very interesting glassy behavior. Okay, thank you. Okay, let me read the question. The question was, why in one day the random singlet phase is not a quantum spill liquid, because it's a product state. So basically the definition for quantum spill liquid is that it's a not product state. And here, even if you have all these different singlets, then, in any case you're writing the product state which is not a constant. Are there any other questions. No, that was the question. Okay. Okay, then let's thank Natasha again.