 This is the team for this project and title has been slightly changed from what you saw originally in the program, simply because Ichikawa also has his own point of view of project, unfortunately Alexei could not come, so I will tell you about our recent results. And then his female is PhD student from our group in Munich. As Volodya Yudson mentioned yesterday, indeed the main idea of this project appeared after careful reading this paper and numerous discussions with authors including of course Boris Alishulio. So I will very briefly summarize what was the main point of Volodya's yesterday talk. I will address one-dimensional systems. And we all know that transport in one-dimensional systems with or without interactions is carried by collective charge density waves, which propagate ballistically in ideal systems, which are clean, but if you add disorder, it can easily pin charge density wave and it can drive system to enslave. And the question is what are exceptions from this general picture? The exception, which Volodya explained yesterday, is related to H transport in two-dimensional time reversal symmetric topological insulators, where there are so-called helical H modes where chirality is locked with spin. So right move has spin up, left move has spin down. And the four, if you put potential scatterer, it is unable to mix these two modes, simply because of spin conservation. So there is no back-scattering from right-moving mode to the left one. And we hope that conductance is immune to localization effects and can remain ideal at least up to some extent. And the question we addressed is, can we expect other setups where a symmetry-protected transport with suppressed back-scattering can appear in purely one-dimensional systems without presence of topological non-trivial bulk? This question I'll address. I will take the model of Conda-Chain, explain physics of Conda-Chain and different regimes in this model. I will explain you how we derive low energy effective theory for it with quantum phase transition, where discrete symmetry spontaneously broken. And at the end, I will address the properties of these two phases separated by quantum phase transition. Let me start with the model. The Conda-Chain is very well known. We have band electrons, which interact with local magnetic moments. So I have exchange interactions. There are many reviews about this model. I will study this model and the following assumptions. Capon is anisotropic. It is of XXZ type. And it is weak. Spin multiplied by any capon constant must be much smaller than the bandwidths of conduction electrons. Must be much smaller than the ultraviolet cut-off of the theory. I will further assume that band is far from half feeling. The array is dense and I will talk mainly about zero temperature limit. Okay. My goal is to develop effective flow energy theory. And to start with, I might go to continuum limit. So first of all, I change from thermionic operators to smooth chiral modes, right-moving particle with a spin either up or down and left-moving particle with any spin. This is absolutely standard. We end up with two direct particles, right-moving particle with plus thermo-velocity, left-moving particle with minus thermo-velocity. And this is, we'll know the grandian of band electrons where the first matrix acts in spin space and the second in chiral one. Okay, now let me analyze most relevant terms in spin-local moment interactions. There are two types of such terms. The first is related with forward scattering accompanied by spin flip. It is nothing else but origin of quantum physics, as you know. The second type of most relevant terms is related to back-scattering. Here I collected all terms. And back-scattering can also be accompanied by spin flip, or it can occur with forward spin flip. And you see that I have put two different captain constants for these two different back-scattering terms. Now I have to assume something about spin configuration and let me at first assume that spin variables are slow as it was done many years ago in well-known paper. What I have after this? All back-scattering terms in the Hamiltonian are multiplied by oscillating exponential. Under my assumption that spins are slow, this exponential is very fast. So it effectively suppresses this term and we are left with forward scattering spin flip, nothing else but quantum physics. But in principle, there is no guarantee that other spin configurations are absent and I may consider the case where spins have 2K Fermi components. After proper choice of spin configuration, I simply put these fast oscillations into spins. Effectively, they reappear in quantum term such that quantum term becomes suppressed and back-scattering term dominates. Back-scattering can open the gap. So this is the regime of gap physics which I will address. The gap physics may dominate, for example, when spin is large and there is no conduit at all, or I can switch on additional electron-electron interactions, repulsions, and if interactions are almost as you to invariant, then the first loop of the RG will tell you that the back-scattering dominates. So there are such cases where this is the main term and I'm going to address these physics right now. So, I expect opening the gap in the Fermi spectrum and after this I have scale separation in the system. There are three scales in space. The smallest scale is related with 2K Fermi oscillations which I'm going to eliminate at first. Then after gap is opened, a coherence lens appears and there are two other regions. Gap fermions live at smaller scales and this is the region of low-energy physics which is my goal. I expect that maybe remaining massless fermions and slow spin components live here and as you will see these two regions may be coupled by massive fluctuations of spin degree of freedom. Now, necessarily, I have already a single out smooth electron degrees of freedom. Now I have to separate scales in spins. I may start with usual parameterization of spin by S2 vector on three-dimensional sphere and let this be one of vectors of orthonormal basis. Then I will do the following trick. I rotate spin, introducing two components, longitudinal and transverse components of new spin and they are characterized by two new angles which I call alpha parallel and alpha perpendicular. Next, I say that let me duplicate variables namely in addition to these two initial angles I allow two new alpha variables to be dynamical degrees of freedom. So I will treat them as independent variables. And I assume that I will be able to single out the fastest oscillations in spin degree of freedom by the composition saying that alpha perpendicular consist of two k-therma oscillations plus some smooth component. What happens now I have four spin degrees of freedom or standard of initial two and of course this is possible if and only if I end up with separation of variables. So at the end, you will see that alpha and psi will be slower spin variables and alpha parallel and theta will be massive spin variables. So nothing bad with this duplication of variables is just the way how to separate degrees of freedom. Okay, now I write down full back scattering term my basic equation with which I will play further and further. I remind you that there are two different types of back scattering without spin flip and with spin flip with two different coupling constants and after I did shift of a perpendicular alpha variable faster oscillations simply disappear. I have included them in alpha. So what remains here already contains no two k-therma oscillations. Now what I expect back scattering must open the gap and the spectrum of fermions after opening the gap I effectively decrease energy of fermions and the ground state energy is minimized when the gap is maximized. So I must find spin configuration which allows me to maximize different back scattering terms. And you see that this effective spin requires that alpha parallel must be close to zero and depending on which term dominates I can have three options for theta variable. It must be close either to zero or to pi over two and which minima is realized depends on ratio of two coupling constants. Let me start with a so-called case of easy access anisotropy when that component of coupling is much, much larger than perpendicular one. So the term with our spin flip dominates and as far as it links all fermions it hybridizes all fermions from left to right and vice versa it can open gaps in spectrum of all Dirac fermions. I expect that this term dominates so alpha is zero, it will be always zero and this term requires theta to be close to pi over two. Next I have to do some technical steps which I will not explain, I will just leave them. I do gauge transformation to gauge out remaining phases from back scattering. I calculate fermionic gap. I integrate out massive fermions which appear to be massive due to back scattering. And finally I integrate out remaining massive fluctuations of spin phases to take into account the normalizations of parameters of my new collective degrees of freedom. So I assume that gauge transformation is done. This is a fermionic gap which comes simply from four by four matrix. Gaps determine fermionic contribution to ground state energy. It is a minus sign so the larger is the gap. Ground state, ground state. The larger the gap is lower ground state energy is and from this cartoon you see that minimum of this contribution is indeed achieved at theta being close to pi over two. Further I can tell or expand ground state energy around this minima and you see that fluctuations of theta and alpha parallel are suppressed. So indeed those two are massive spin variables as I expected in the very beginning. My approach has been preliminary justified. Now I integrate all massive modes, massive fermions, massive spin fluctuations and I end up with effective low energy theory for the case of easy access anisotropy. There are two contributions here. Both are of type of flotting liquid, L stands for flotting liquid and you see that there is pronounced spin charge separation. I have spin density wave which propagates with bare velocity, fermi velocity and I have charge density wave which propagates with velocity being much, much smaller than bare velocity. Correspondently compressibility of charge density wave is also small. The physics of this renormalization of parameters for charging density wave is very simple. Volodya explained it yesterday, I can briefly repeat. Spin phase can rotate freely. I don't know how to draw it better but the main thing is that spinons feel presence of localized electrons and because of this coupling, charge density waves become slow. This is the same picture as Volodya explained yesterday. As concerns configuration of spin waves, we have spin oscillations both in X and Y directions but if I calculate cross correlation functions, I see that X and Y components are not correlated to each other. Now let me add a potential disorder and study stability of charge density wave with respect to this potential disorder which has no spin structure and therefore it links only particles with the same spin. Fermilons which stand here are all gapped fermions. I would like to study property of charge density wave destroyed by alpha so I integrate out massive fermions and after this I end up with sign Gordon model for alpha. Now I recall that Latvinger parameter for alpha degree of freedom is small. Therefore certainly this term is relevant and disorder can pin charge density wave very, very similar to how it happens in usual one-dimensional bias. We come across usual Anderson insulator. The charge density wave is unstable with respect to additional added potential disorder. Now I would like to study the opposite regime where J perpendicular dominates. I remind you equation for the fermionic gap in easy access case and I ask myself a question what happens with this expression at the SU2 symmetric point where two coupling constants become semi equal to each other. If two constants are equal then cosine quadrat plus sine quadrat is one. So for M minus term square root exactly cancels J perpendicular and I see that some gaps go to zero as the SU2 symmetric point. Now I can assume that probably if I go further to the case where J perpendicular dominates only that gap survives and if M minus shrinks to zero it will remain zero in the second. Easy access is a plane. Easy access is a plane. Easy access is the case where JZ dominates and vice versa. So my guess is that this point will be point of quantum phase transition which will separate two different phases. To simplify my life I say that let me consider the case where JZ goes to zero. This is the same equation for back scattering term but now I cross out back scattering without spin flip and I consider back scattering with only spin flip. Sigma minus and sigma plus tell me that here there are two contributions which can be rewritten as helical contributions to back scattering. Again I recall you reminded of helical modes are modes where direction of propagation is locked with spin. This is taken from cosine and if this term dominates in this limit then as usual alpha parallel is zero and I must maximize cosine to open gap as large as possible so I expect theta to be zero. There is second contribution which comes from the sine and the only difference is that as far as I have assigned here then theta must be close to pi not to zero. So I calculate ground state energy, a freemonic contribution to ground state energy in the easy plane configuration and I see that there are two degenerate minima with theta being close to zero and theta being close to pi and depending on which configuration system chooses I can open gap in one or other helical sector, the second helical sector remaining gapless. This is the same situation which I redraw in chiral representation so you see that for one spin right moment particle is gapless, left moment particle is gap and vice versa for second configuration. We can say that there is partial gap opening in the system. Instead of gap for gaps in all branches of chiral freemons now I have only two gaps. Again, I must verify scale separation. I choose some minima. I calculate fluctuations of spin variables around this minima. I see that fluctuations are gap and suppressed so everything is self consistent. I have two massive spin variables as expected. I can go ahead, I can integrate out all massive degrees of freedom but now situation is drastically different from what you saw before. I have two different charge carriers, three helical fermions which are gapless and collective wave which is now coupled to everything to charge and to spin so I can call it charge spin density wave. Three helical fermions of course propagate with bear velocity fermi velocity and collective wave is again slow because of same reasons I explained yesterday kind of polaronic effects. The principle difference between these two phases is also reflected in spin correlations because if now I calculate correlation between Sx and Sy spin components I see that there is local helical order and this local helical order reflects broken helical symmetry. Similar effects were also addressed in talk by Daniel Loss one week ago. Now I'm interested in the question whether after breaking helical symmetry the system can acquire some robustness again localization effect whether we may expect back scattering to be suppressed. So I add again weak scholar disorder which has no spin structure and I see that disorder is able formally link only gap and gapless fermions but matrix element of single particle back scattering would be zero simply because there is no density of state inside of the gap of gap particle. So first conclusion is that single particle back scattering is impossible if I assume spin conservation in the system. Then I do the same as for the case of the axis I integrate out gap fermions. After this I obtain effective back scattering which is multi-particle back scattering you see that I have more fermions those are massive fermions, external mass less fermions and you see that there is spin flip which is hidden inside of a propagator of massive fermions because the gap is opened by scattering with spin flip. I have no full theory for effect of this big scattering in my hands but for simplest estimates I can do the following trick. Let me put expectation value for this exponential and ask whether such multi-particle effects can be strong. We think no because this exponential is always either vanishing or small and before we see that not only single particle back scattering is absent but multi-particle back scattering is also suppressed and this is very similar to suppression of back scattering in time reversal invariant topological insulators so I emphasize that I obtain it due to interaction effects instead of presence of topologically non-trivial bulk. Chairman advices that my time is almost over so these are my conclusions. I have explained you that quantum chain may show quantum phase transition depending on anisotropy in the system. In the case of either plane anisotropy helical symmetry is spontaneously broken and this may lead to noticeable suppression of back scattering which is very, very similar to suppression in edge transport of topological insulators. These are questions which we plan to address further. We still have to explore better robustness of this absence of back scattering deep in the infrared limit and we still do not have full theory of phase transition in the system. Okay, and I see Boris came. So the main conclusion that on behalf of all speakers organize us happy birthday Boris.