 I would like, first of all, I thank the organizers of this wonderful meeting and so many diverse, interesting topics were discussed and basically I'm very happy to hear that there is discussion about everything, time and so on, so people really take attention. And thanks for letting me talk about the Kitaev interaction here. So, dear Boris, first of all, congratulations, happy birthday and Tanti Aguri how they say in this part of the world, which I know you like a lot and I was really wondering why you like it so much and now I understood. So, basically 60 on that part of the world where you spent a winter is actually 16 incenses in this part of the world, so happy sweet 16. Okay, and I'll go to the topic of my talk. So, let me start with acknowledging my collaborators. The most of the work which I'm going to present today is done with my postdoc, Ernest Russo Hadzakis and with the Craig Price, who is now in Penn State and some earlier work which motivated actually this recent study was done with Yuri Suzuk who is also in University of Minnesota and with Peter Wolflet. And I will also acknowledge Stefan Rochelle, Johannes Reuter and Roni Tomalia because they were interested in this model and they did FRG study of this model. Okay, so here's the outline of my talk. As you can see, I'm talking about something quite different from the rest of the conference. So, we started with introduction and I will try to explain why anisotropic Kitaev interaction are in principle possible in system with strong spin orbit coupling and a certain type of geometry. For example, you need a honeycomb lattice or hexagonal geometry. And I would also argue that in some particular system not only nearest neighbor Kitaev interactions are possible but also second neighbor interactions are possible and actually very important and at least they help to resolve several puzzles related to one of the most studied irradiate which is sodium irradiate. And basically the main topic of my talk is I want to present a new frustrated model which we call K1, K2, K, stay for Kitaev interaction on nearest neighbor and K2 for Kitaev interaction on the second neighbor on the honeycomb lattice. And I just want to show you that this model is very interesting, non-trivial and you will see that this non-triviality is actually reflected in a very different classical and quantum phase diagram for this model. And basically I will conclude at the end. Okay, so I will start from the beginning. I will start from why we are interesting and worry about these materials and basically in the strongly correlated electron community people were working a lot for 3D system where the main interactions are common interaction and basically something else like spin orbit coupling relativistic interactions are sometimes really important but in any case they are significantly smaller. If we are going down in the periodic table so correlations decreases, they become weaker and weaker while the atoms become heavier and heavier and basically when they reach age of 77 this is another happy age for elements then the spin orbit coupling constant can be more or less of the same order of magnitude as our quantum interaction. And basically irradiates and I will show in a moment why irradiates who are attracting a lot of attention in their community they are basically it can be explained using their phase diagram which was suggested by Leon Ballets and Mitropasin these irradiates they sit in between. So basically they have significant row of correlation U is not small but spin orbit coupling is already large. And therefore in this phase diagram where they plot so here is the ratio of coulomb repulsion with respect to the bandwidth and you know that if you go to 5D elements they become bigger and bigger. So and then here is the spin orbit coupling also normalized by bandwidth. So they sit in the middle and somehow in different irradiates you can see the physics which is belonging anywhere around them. So basically some of irradiates was suggested to be showing like topology of insulator properties or it's also it's more insulator and it's actually a spin orbit coupling assistant more insulator where some of the traditional world physics can be studied in the presence of a strong spin orbit coupling and what it gives. So basically this is just to remind what are the degrees of freedom for irradiates. So it has five T2G electrons and if spin orbit coupling would be small we would be dealing with spin one half moments while because spin orbit coupling is large these levels are splitted in the low levels quadruplet of J-effective free half and then one electron or one hole is occupying this J-effective one half state and basically we again have kind of quantum physics in these materials. And then it was a really very great work by Georgia Kelly and Gignard Halleoulin who was actually showing that this is a spin this J-effective one half they are a very complex object because they are it's a mixture of the spin degrees of freedom with one orbital index and in others so spin up with orbital angular momentum equal to zero and spin down with orbital angular momentum equal to one. So basically we are looking how this interesting and strange objects are interacting in different lattices and what is important since in the wave function describing this is a spins now both spin and orbital angular momentum are present then these magnetic moments now know about the lattices and we can get different anisotropic interactions on different lattices. And already in this work basically what they show that if we have H sharing octahedral so here in the middle are iridium atoms and they are surrounded by oxygen octahedral so in these geometry interactions remains roughly speaking isotropic Heisenberg like with small dipole-dipole interaction. And if you have some corner sharing geometry so you have this 90 degrees super exchange then more interesting anisotropic interaction can appear and for example one of them is the Kitaev interaction so just to give you some materials which were studied so for this type geometry the most studied compound was strontium 2-iridium mophore and the recent data that if it's doped it shows some features resembling superconductivity though superconductivity will not yet confirm but this compound is in a sense very similar to the parent compound for cooperate. And for 90 degrees geometry the compounds we have studied quite a lot of sodium iridate and lithium iridate in all of them there is this geometry and for lithium iridate nowadays there's two dimensional and also three dimensional versions. Just to once again remind you why Kitaev physics is attracting a lot of attention. Basically Alexei Kitaev he proposed this model in 2006 and originally this model looks kind of artificially because in this model each component of spin interact along certain type of the bonds. For example along all red bonds on the x component of spin will be interactive. This model is exactly solvable and we know that there are not so many exactly solvable two dimensional model moreover it's also exactly solvable in three dimension if it's realized on this hyperhanicum lattices then the ground state is spin liquid and there are fractionalized excitations which might be a problem in some experiments. So it's a very interesting playground because due to exact solution we can really compute a certain quantity and compare them all with the experiment. Just to remind what I mean by this spin liquid so basically once again here my Rana fermions are coming into play in a slightly different context when it was discussed in this conference but we can substitute, we can express the spin degrees of freedom in terms of my Rana fermions so on this honey cum lattice that's how they're introduced where A would be the real electron operators and basically the B my Rana fermion can be formed into the bond variables. So what do we have? We are basically, when we rewrite this Hamiltonian in terms of the spin Hamiltonian, Kittai spin Hamiltonian in terms of the my Rana fermions we can combine here as a bond variables multiplied by C operables. These U variables are actually static quantities, these are bond variables and it comes from the fact that there is one true physical quantity which is a placate operator which is nothing else as a product of spin operator around one of the hexagonal placate of the honey cum lattice and this quantity is conserved, it's commuted with the Hamiltonian and since this quantity can be rewritten in terms of these U operators, this quantity are also static. Basically what one can do, one can just put the expectation value here and then the remaining Hamiltonian for the C my Rana fermion is nothing else as they are hopping Hamiltonian with a certain hopping parameter which depends on the flux state. So once again, it can be, look what is the ground state? So since all the flux sectors are orthogonal to each other then basically in each of the flux sector we can compute this spectrum, the ground state is flux-free and basically you can see that the flux can be either zero or pi because this product of this placate operator is either one or minus, it's one or minus one. Okay, so basically there is a certain spectrum and it's all exact results and that's why people were quite attracted to this model. So this, the ground state is spin liquid with the well-defined excitations. Why people look to their materials is because in this paper which I already discussed, they showed that the spin liquid ground state which is exact in the quitaif limit. So here there is a certain parameterization you can parameterize two parameters. So it was suggested in this paper that in real materials in the irradiates it's not only the quitaif interaction which comes from this 90 degree super exchange but also Heisenberg interaction which comes from direct overlap of iridium ions and it's Heisenberg like. So they showed that, so here one corresponds to only quitaif interaction and if you go in this direction you can see that spin liquid survives with respect to small perturbative Heisenberg interaction and there was experimental interest to this system because people were thinking if it would be stable just in one point it would be no experimental interest but since it has the region of stability then that caused this interest. Another phases which were obtained in this phase diagram so it's basically Niel-Antifera magnet which we know should be the ground state when we have only Heisenberg interaction on the honeycomb lattice and in between in the intermediate values of quitaif interaction there is a stripy antifera magnet. So when people study as I said sodium aridate was one of the mostly studied compounds. So first of all what they see that up to now none of the compounds show the spin liquid ground state probably spin liquid somehow show itself in the excitation spectrum but in sodium aridate in addition to that what was shown that the ground state is not stripy phase which is next to the spin liquid phase in the simplest quitaif Heisenberg model but it's another antifera magnetic zigzag phase which is shown like this. So basically you have two types of the ferromagnetic bonds and one type of antifera magnetic bonds. So and it was some neutron scattering data and Riggs data which showed that in fact the excitation spectrum are compatible only with this state. The question is what is the model which would produce this phase? Obviously the minimal quitaif Heisenberg model does not do the job and that was suggestions how to extend this model to include further neighbor interactions but in order to resolve all the puzzles related to this compound this further high like second neighbor and third neighbor Heisenberg interaction should be rather large. So basically what we did we revised this super exchange model based on sometime binding model by Katerina Faisova et al. That's the Rosa Valenti group and we just saw that the second neighbor hopings are quite large and basically what we did we looked all possible super exchange passes and we saw that if we compute the second neighbor coupling it has not only Heisenberg component but it has exactly the same symmetric quitaif component. Basically you can see here there are four passes and they add to each other. You can go from this iridium to oxygen to sodium to oxygen and back to iridium and it couples the same orbital so it's produced the same kind of quitaif physics and this interaction for this compound is actually not small. The reason for this is very natural is just the sodium iron which sits inside the structure is very big diffusive iron so basically the hopping processes are quite large. So what we did, then we just add this quitaif interaction to the model. Actually what we also found that for sodium iridate quitaif interaction on the second neighbor is anti-ferromagnetic while the quitaif interaction on the nearest neighbor is anti-ferromagnetic and with these realistic numbers which we obtained from the super exchange calculation basically we got this exact state. We got anti-ferromagnetic quitaif temperature and that was a puzzle for experimentalists for quite a long time how with the dominating quitaif interaction which is ferromagnetic how to obtain large anti-ferromagnetic Q-revised temperature. And what is also important that in our calculation in this model, this K2 interaction let me explain this figure. So here the circles are for iridium ions and basically if we take two iridium ions which are second neighbors and which are coupled by green and blue bonds means that nearest neighbor interaction would be either XX and YY for example then the second neighbor coupling is always red so the K2 anisotropic interaction does not load the symmetry of the model. So it was very recent experiment about the ordering of the spins in this compound and what they show that the spins actually play in the phase diagonals which is also in contradiction with any of the nearest neighbor models like the quitaif Heisenberg model which would choose cubic axis as the direction of the spin variables. So basically one can understand how it works. So if you look to one of the zigzag then basically you have in the already in mean field you choose the easy plane because since the strongest interaction is nearest neighbor ferromagnetic quitaif interaction then you want to satisfy this bond. And if you do a fluctuation analysis then you see that the cubic axis is selected but now we did this inclusion of fluctuations and basically due to this quitaif interaction where so the four second neighbor bonds would be satisfied for the second neighbor quitaif interaction you can show that actually this minimum of fluctuation of energy are in this diagonal direction so K2 is important. Okay, so let me just say that quitaif interactions both nearest neighbor and second neighbor interactions are important at least for sodium two iridium of free but actually for other compounds as well. So this is basically the summary of what I said that the dominant interaction in all of these compounds nearest neighbor quitaif interaction and the quitaif interaction on the second neighbor is actually the second largest energy scale at least for sodium iridate and the interplay of K1 and K2 already stabilize the zigzag phase and many puzzles are resolved. So okay, so this is just this justification why K2 is important. So this is the model we decided okay let's read off all secondary interaction just look to the essence. So this is the model, this new model on the honeycomb lattice where we only have an isotropic interaction. So quitaif K1 interaction would be quitaif interaction on the nearest neighbors and K2 would be the quitaif interaction on the second neighbors and once again which component of spin is interacting it's corresponding to all these plots. Let me tell results first. I don't know how well I'm doing with time so the results are different. So this is the results of our work. So first of all what we found that this model holds many unconventional properties. For example the formal and so classical and quantum spins behave completely different and therefore the role of formal and quantum fluctuation for order by disorder in the selecting of the states is very different. Second is that magnetic phases the true long range magnetic order is stabilized only for quantum spins. It has very classical characteristics actually ising like order but it cannot be stabilized for classical spins and that's related to the different symmetry of the classical and quantum K1, K2 symmetry of the model. So the classical spins can have only nematic order at a finite temperature and what is also important the numerical calculation show that KetaF spin liquid so now we are looking for the competition of KetaF spin liquid with the K2 perturbation. So it's very fragile with respect to this perturbation much more fragile than against the Heisenberg terms and that basically show that sodium is deep into a magnetic phase. And it's actually tells experimentalist to which compounds to look if they want to find this KetaF spin liquid phase. So basically the idea is kill K2 interaction by making the central ion small. Okay, so the classical analysis let's look not for quantum speed but for classical spins then we can use the Lachlan-Jersey theorem and basically look what is the degeneracy of the classical ground state. And one can show that the degeneracy is actually extensive. So because you see there since the couplings are anisotropic so when we do using the Lachlan-Jersey theorem we go to the Fourier space and basically we are looking to the eigenvalues maximum eigenvalues for this matrix lambda ij it has also the spin index. So basically here are the eigenvalues and for each direction of spin x, y and z they are minimized or maximized along so the minimum of eigenvalues is along a certain lines in the K space. So we have line degeneracy and at t equals zero basically all these lines are degenerate. So the degeneracy of the ground state is free free coming is from the direction of spin and two to the power of L and I will explain where it comes from is basically the number of letters which scales with the system size. So this degeneracy is not accidental but it's actually related to the sliding gauge like a symmetry of the classical K1, K2 Hamiltonian. Let me explain what I mean by this sliding symmetries. So here's the picture. So let me assume one of the states. One of the states, for example, I assume the states are looked to the eigenvalues of lambda z. So I put all the spins along z direction and just for simplicity, I choose this combination of both K1 and K2 interactions are ferromagnetic. So one of the states are here is this nearest neighbor bonds. So these are the nearest neighbor z bonds. These are nearest neighbor x and y bonds. And then you can form the second neighbor bonds and see what is what. So basically its interaction is ferromagnetic. So I will have along the z bonds I will have the order. I would put all the bonds along blue circles, along plus z direction and along these yellow circles minus z direction. So classically this state and the state where I took one of the letter and flip all the signs classically they are degenerate. So basically I can flip one letter or I can flip all of them and then it would be this situation. All these states are degenerate and basically they correspond to these blue lines in this brilliant zone. So all these states are degenerate. And that's what I call by sliding symmetry. And for classical spins it's clear why we can do it because all of these letters for the nearest neighbor you can see that they are coupled by x and y bonds but I put all the spins along z direction. So classically there are no processes. So therefore at t equals zero all these states are degenerate. I can also put the spins with x or y direction. If I go, so this is just another example but the idea is exactly the same. I just look to the different part of the phase diagram looking for different signs. Now I have anti-ferromagnetic couplings. So I will have these anti-ferromagnetic letters. If I go to the finite temperature what I see if I go to the very high temperature then and this is our result of the classical Monte Carlo then I will see and okay and then what else we did. So we parametrized this K1 and K2 interaction with angle phi and basically these line corresponds to K1 interaction only. So K2 interaction here is zero because K1 is cosine and that's vertical with corresponds to K2 interaction. So here I do have spin liquid and basically in the spin liquid if I go to the high temperature then all the points in the brilliant zone would correspond to my classical degeneracy for this state. And then I go away from this so when K2 is non-zero I see these lines and these lines corresponds to this minimum of edge. And then what happens is if I go to low temperature then I see very sharp pneumatic like order and then only certain types of the lines are chosen. So basically thermal fluctuation choose only one combination, one direction of spin, x, y, or z direction of spin. And once again it's first order phase transition quite sharp you can see it in the specific heat. Okay so this is the classical picture at find at zero temperature we have classical spin. Okay so basically I'm going to the quantum case and what I want to say that quantum case is very different and the difference comes from the fact that the model itself does not have this sliding symmetry. What I mean by this? So this is the classical picture which I showed you where I flip compared to this case only one letter. In order to do it I should combine the pi rotation in space and time reversal operation. But I should apply time reversal operation locally and for quantum system there are no local time reversal operations. So what does it mean? That means that these letters must couple due to some quantum mechanical processes and this quantum mechanical process we can actually look in perturbation theory and we will see that it gives possibility for long range order both at t equals zero and at our finite temperature. Okay so there are three different types of processes here and the idea is the following. So we, oops. So we assume that X and Y coupling are smaller and that's our perturbation theory. So X and Y coupling would be a certain number R which is smaller than one from the Z coupling and these couplings are three types so it can involve only a nearest neighbor like this. They can involve nearest and second neighbors and only a second neighbors in direction. Okay three types of processes give us three different types of couplings but what it does actually, let me just quickly say that it couples these letters. So details apart what left is only some discreet degeneracy, okay? So what I do here, I compute these couplings from perturbative theory. So R is this perturbative parameter. So then this term, the first term basically which I call JW, this is like plaquette's term and this term supports spin liquid and you can see that this is the parameterization in the parts where we expect to see ketive spin liquid near zero and one and two so these terms are dominated. And in between in these magnetic ordered phases there is two other processes which coupled like this so these terms involve only a second neighbor ketive interaction and like this which involves both nearest neighbor and second neighbor coupling. So these terms can be computed perturbatively and we see that in the biggest part of the phase diagram J1 interaction is dominating. So it leads to a certain type of long range order. So we obtain it first perturbatively and then Ionis did exact diagonalization so this quantum phase diagram is the result of exact diagonalization and where you can see that quantum spin liquid is extremely fragile so it's break at when the K2 interaction is actually 0.025 pi so it's very close it's very, so this phase is narrow and then what we get already for these phases just without any high break interaction we can get zigzag interaction in the part of the phase diagram where K1 and K2 have different size. There are many interesting properties of this phase diagram but in any case, long range order is possible for quantum spins. Last thing which I want to show is the result from our exact diagonalization so here a few things are computed. So here what is computed is the expectation value of plakete operator and you see that when we are at the point where we have only nearest neighbor ketive interaction so in the quantum ketive spin liquid is equal to one and then it's very fastly dropped to a very small value basically zero value in the magnetically ordered phase. So this actually shows that the phase transition is of the first order. What else we see and here is both the result from the spin wave calculation and from exact diagonalization that the spin lengths remain very close to one half basically fluctuations are quantum fluctuations are basically negligible so the order is very classical. Nevertheless it can only be stabilized by quantum processes and here is the plot of the spin-spin correlation function in the real space and what you can see that if you are inside quantum spin liquid so that's the left column then the interactions the correlations are extremely short range as it should be in the ketive spin liquid. This is a known fact and basically here the size of the circle corresponds to the value of the correlation function. Once you go into magnetically ordered phase then you truly see a long range order. So basically that's what I want to tell you so thank you for your attention.