 All right, first of all, let me thank the organizers for inviting me here. I'll tell you how the work we have been doing with these two gentlemen. One of them is in the audience, which probably do not need much of an introduction here. And it's related to quantum criticality in topological insulators with disorder. So I thought it is appropriate for this occasion to start with this. So what we have been told back then is that to think about localization transition, we need one single parameter, which is the conductance, longitudinal conductance. And then everything is given by that. And according to this famous picture, there is a localization transition in dimensionality larger than 2. We can do something with that analytically in 2 plus epsilon dimension. That's this critical point in 2 plus epsilon dimension. Not much can be done in three dimension. There are marvelous, exact results for one dimension, starting from Ithieta and Larkin and then many others. But there is no transition in one dimension. So we do have exact theory. But there is no much of interesting things to look at. Now, of course, the situation very quickly changed after that due to advent of a quantum Hall effect. Today we understand that it's because there is a different symmetry in the system. It's a class A Hamiltonian. And what it has, it has a Landau levels. And in clean system, there is a marvelous picture of Landau quantization. And according to that, sigma x, y should be quantized. And sigma x, x should be zero unless Fermi energy is right at the Landau level. Now, of course, disorder changes these things quite a bit. And Landau levels become broadened. The density of states is smeared. It's still a symmetry probably, but very weakly. Nevertheless, the quantum Hall, of course, survives. And the reason for its survival, as was understood again soon after, is the fact that disorder actually stabilizes, as we would say today, topological index. And here, the topological index is nothing else but quantized sigma x, y. So the way it was internalized in the early 80s, first by Khmelnitsky, and then substantialized a little bit by Pruskin, is that we actually have to think about two parameters, not one parameter. One of them is our old friend, Legitudinal Conductance. But there is the second one, sigma x, y, which in today's language, we understand it as an average topological index. Now, sample specifically, it is still quantized. But once we think some ensemble average of different disorder realization, we lose this quantization. And it may be anything. But once we go to larger and larger system sizes, or equivalently smaller and smaller temperature, the topological index, the integerness of the topological index, it restore itself. So it returns to self-averaging situation. And the way to understand it is to follow this flow diagram, which is a flow with increasing of a system size. According to this, sigma xx goes down, and that's localization. And simultaneously, sigma xy approaches the integer values of 1, 2, and so on and so forth, which is nothing else but feeling factor. Unless you happen to be right in the middle with a half integer feeling factor when the system is frustrated, if thermal energy happens to be right at the middle of the Landau level, and there is a delocalized state conjectured there, so conductance supposedly flow to a finite value. Now, not much have been done with that since this groundbreaking works. The only thing probably which happened is that from now on people decided to rotate this phase diagram by 90 degrees. And ever since, after Chmielnicki, it was plotted 90 degree rotated. So I will show quite a few diagrams like this. But they all will be all 90 degree rotated, so don't be confused. All right, so meanwhile, what was understood that there are many more symmetry classes. Actually, there are 10 of them. And depending on dimensionality and which is this horizontal line, count dimensionalities, the vertical line labels symmetry classes. And what stays in this periodic table is a homotopy group, which may be either Z or Z2 in some of the symmetry classes. Again, depending on dimensionality. And you see it's arranged in this beautiful periodic table. So few famous examples is integer quantum hole effect in D equal 2, spin quantum hole effect in D equal 2, and then three-dimensional topological insulator in D equal 3. Now, the topic of my talk today is actually one dimension or rather quasi one dimension. All what I will be talking about is quasi one dimensional, not strictly one dimensional. Actually, number of channels will go to infinity. And according to this periodic table, there are five symmetry classes in one dimension, which may have non-trivial topology. And three of these classes has topological index, which belong to Z group, pretty much like integer quantum hole effect. And two of them has Z2 topological index. Most famous of them is Qtive chain. Actually, either that or that, depending on details. All right. So what I will be talking about is this quasi one dimensional situation with symmetries appropriate to non-trivial topologies. Now, we are definitely not the first one who tackled this question. People started to realize that something funny happens in these symmetry classes back in actually late 90s. So that was a bunch of works mostly with DMPK technology done by Brower and Moudre and Altland and for Rosaki and many others. Reed, Groosberg and Smita should be commented by doing very insightful work there. Also in two dimension, in addition to integer quantum hole effect, people realized that class C can be solved exactly under some work in class D. So what was common to all this work is that people stumbled upon the fact that in this low dimensional system, one may have delocalization. For two dimension, it was sort of after quantum hole effect not that surprising. For one dimension, delocalized situation for disordered one dimensional system with delocalized situation was kind of odd. Now, what we understand today and what will be sort of unified melody of my talk is that we understand today that what happened here is that they accidentally happen to be in a quantum critical point of topological quantum phase transition. So all these examples, they have quantum phase transition between different topological classes. And if you happen to be by accident right at this critical point, you see delocalization. Pretty much like in integer quantum hole effect, if you happen to be at exactly half integer feeling, you see delocalized situation. There is a delocalized state. All right, now, so how one goes about it. So we're dealing with topological insulator. So first of all, we were told that it is characterized in quasi one dimension by presence of H states. It may be zero or one, if it's Z2 class, or it may be zero, one, two, three, five if it's Z topological class. And we were told that because of topological definition of this H states, they should supposed to be robust to small perturbation. So if we add small perturbation, they are protected. Now protected by what? Protected by the presence of the gap. We are talking after all about insulators, it has gap and this gap protect the H states. So if you add the small disorder, sort of the natural tendency is to say that nothing happens. So you can't violate topological protection. Now if you think a little bit, look at a little bit closer on that, that's not that simple. Because even the small amount of disorder immediately destroys the gap. First it does it because of the Liffschitz tails, but if you increase a little bit disorder when already in kind of mean field, naive level, gaps are closed and you have a nominal metal. So since gaps closed, there is nobody actually to protect the holiness of topological index. So one should to sort of discuss what happens within H states in the presence of many other states which live in a gap due to disorder. Okay, so now H states by no means not the only one who lives, whose energy is inside the gap. Now another sort of difficulty or unusualness is that 99% of texts will explain topological index in a case space. You're supposed to look in a brilliant zone, you calculate certain topological invariant like very face or chair number that you integrate over the brilliant zone and here you go, it's either integer or non-zero integer. Now what do you do about it with disorder? With disorder you immediately lose your case space. K is not a good quantum number. So even the definition of a topological versus non-topological supposed to be somehow refilled. So if you'll make a small table and compare clean and disorder situation, so in clean you have a case space in disorder, you don't. In clean you may be lucky and have been get which protects you in disorder, you immediately lose it. Nevertheless, the topological index can be defined even in disorder situation but with some care. Now the picture which emerges out of this, let me jump straight to the answer, so to say, is the following. So suppose you have a certain, in a clean system, suppose you have a certain control parameter in, let's say, in the Chinese chain, it's typically assumed to be chemical potential or an experiment, it's usually a magnetic field which drives you between topological and non-topological phases. So you have a certain control parameter which will switch you between n equals zero, topologically trivial and then n equals one, maybe n equals two if it's that symmetry class and so on, so forth. So now the transitions between these points, between different topological classes happens when the gap, the band gap, the clean band gap closes. The only way to go from one topological class to another is by closing bulk gap. So in these points of the transition where in clean system the system goes to, from one topological state to another, the band gap closes. Now if you add the disorder as an additional direction in my phase diagram, so you would think that these points should be not an isolated points but as basically termination lines, termination points of a certain lines in this phase diagram. So the picture which emerges out of this is that in a phase plane of disorder versus some control parameter there are lines which distinguish between topological situations. Again, in a clean case that's the lines, that's the points where the gap closes. In a disorder case there is no strictly speaking gap anywhere but nevertheless there is a line which distinguish between different topological classes. So what happens when you cross this line that's a genuine phase transition. It characterized by divergent correlation length and in this case it's localization length. So anytime you cross one of those lines, localization length diverges and there is a delocalized state at zero energy. So I didn't mention it but in all these symmetry classes zero energy plays a special role. There is a symmetry of a spectrum with respect to up and positive and negative energy. So the delocalization I'm talking about happens always strictly at zero energy and it happens at one point in a phase diagram you have to tune disorder and your control parameter to a certain special point where there is a delocalized state at zero energy. So once you have this diagram the natural question is what's the critical theory which describes it and whether one can make it more quantitative and not just fairytale like that. And here the miracle of quasi one dimensional system helps we can solve things exactly in one dimension. So the way to solve it exactly is to use the sigma model machinery I'll jump to this in a moment but first let me just tell you what is it all about. So we'll be driving to universal theory describing this quantum criticality. It will be described in terms of two parameters pretty much like integer quantum Hall effect. One is longitudinal conductance as always. The second one is a configuration disorder average of topological index which again initially may be not necessarily integer because of configuration averaging but then it will flow to an integer. So there is this two parameters which are functions of all microscopic control parameters of my theory, the chemical potential, magnetic fields, strength of disorder, all that. So they all define these two parameters. And then these two parameters plays a role of effective charges in my sigma model in my field theory and I can look how these two parameters, how these two observables flow as a function of the system size. So as a function of the system size they will flow to one of the fixed points and these fixed points are characterized by vanishing longitudinal conductivity because it's localization after all and integer value of the topological index. So we will find this Prouiskin-like, Chmielnitsky-Prouiskin-like phase diagram but we'll be able to do it sort of exactly because now it's not in two dimension but in quasi one dimension, okay? So that's pretty much what I'm driving at. Now the simplest model to consider in that classes I'll talk a little bit more about the two classes is so-called Suss-Riefer-Hieger model which is kind of a polyacetylene dimerized situation. You have two sub-latices A and B and you have two different hopings. You can generalize it immediately to quasi one dimension you still have two subspaces. So the symmetry which protects your topology is this sub-latice symmetry. Hamiltonian supposed to have only entries which goes from sub-latice A to sub-latice B but not zeros on the diagonals. Then in clean system the topological index is defined as whether this thing has a function of a momentum you go around the brilliance on, it goes around closed loop and this closed loop may encompass or not encompass the origin in a complex Q space. Now, so that's scientific way of writing I think it integral over the brilliance on of a phase essentially and if this phase goes around the origin that's an integer if not that's zero. Now, in disorder case, you still can define it. The trick to do is to put your disorder system in a very big loop of size capital L and subject it to a fictitious gauge flux. This gauge flux acts in a different ways in sub-latices A and sub-latices B. So once you do that, you essentially take a unit cell of your system to be the entire system of the size L and the fact that it is in an on a ring kind of give you momentum space but the very, very tiny brilliance on it's now size of one over eight but to define topology that's okay. You can have as small brilliance on as you wish and then calculate your topological index. So the way it's done formally you don't need to go into this detail but the thing is that the formalism actually landed itself very, very nicely into the supersymmetric kind of type of thinking. You need to calculate ratio of the determinants of the two green functions with different gauge fluxes fictitious gauge fluxes inserted and then your topological index and your longitudinal conductance may be easily calculated once you know this ratio of the two determinants. So the trick is actually 20 years old so it was very well explored in this community. I probably don't have to go deeper into it. The nice thing about it that since we have a ratio of the two determinants we can encode one of them through the bisonic degrees of freedom in our through the fermionic degrees of freedom and all together it's maybe represented as a certain supersymmetric field theory. Now you go ahead, you crank the machinery. You end up with the sigma model which has as was promised two parameters. One, so one term is a gradient term, quite familiar so the coefficient in front of it is a longitudinal conductance which is the same as bare localization length in quasi one dimension, that's the same thing. In addition you may have a topological term the symmetries of this models allow to have non-trivial term like this and the coefficient in front of this term is going to be our topological parameter. And then you parameterize somehow this supersymmetric manifold. It has non-compact and compact coordinates and your fictitious flux enters as a gauge field. So you which you can gauge out to have a twisted boundary condition. So you consider this supersymmetric field theory with twisted boundary condition and the boundary twist is your fictitious flux. Then you take derivative with respect to these fluxes and you have your observables. Now that can be understood as, let me see how much time I have. Yep, I probably should accelerate. So since it's a quasi one dimensional field theory it can be solved by through a transfer matrix technique by mapping on a certain quantum mechanics and this quantum mechanics live in this on this supersymmetric manifold parameterized again by certain coordinates. So this quantum mechanics actually is quite similar to the usual a Rohn of Bohm type quantum mechanics where topological index plays the role of the vector potential. So as a function of a topological index the spectrum of my transfer matrix changes and there is a special points at half integer value of the topological index where I did generously in the spectrum. So essentially what we have is slightly more complicated version of the same story. It has few coordinates as opposed to just one but by and large it's this story. So you end up with certain sharing your equation in a curved space where are Jacobians and whatnot. Now luckily to us in all symmetry classes this quantum mechanics can be solved exactly in terms of certain sometimes simple function and sometimes in terms of hyper geometric functions but it can be solved in all symmetry classes. And as a result we have an exact solutions for conductance anthropological index as function of the system size. So here is a sort of Pruskin phase diagram. Points are for different system sizes versus a equal psi, two psi, four psi, eight psi and so on and so forth. And you see that depending where you start you go either to integer topological index or zero topological index or you may go to next integer, it's out of the screen unless you happen to be at a quantum critical point and then your system keep going along the critical line and topological index remains half integer no matter how big is the system size. Moreover, for all those situations longitudinal conductance become exponentially small. Here it goes to zero but very very slowly in a power low base so there is no localization along the quantum critical line. So that essentially what I'm trying to say is that there is this phase diagram in a space of bare parameters you can be either in one of the phases topological underson insulator or trivial underson insulator. You may also be in a transition line and then depending where you start you float to one of those points. So I'll skip numerical calculations, we can do it. Now let me say few words about the two classes. Yes, okay. So in the two classes the story is quite a bit different in the sense that you cannot introduce topological index. You cannot, I take it back, there is no topological term in your field theory. Instead the group manifold is doubly connected. There are two parts of the group manifold one with determinant plus one another with determinant minus one and what your field theory does it discontinuously jumps between the two group manifolds. You still can run this transfer matrix calculations. The funny thing about it that is your effective quantum mechanics and now spinor quantum mechanics because you have an amplitude to be in one part of the manifold and there is an amplitude to be in another part of the manifold and this spinor quantum mechanics happens to be by the reasons which we do not understand nicely genuinely supersymmetric not in a sense of the effect of supersymmetry but it's just textbook supersymmetric quantum mechanics. And because of that again you can solve it exactly with all the details and again I'll skip technicalities but so this phase diagram legitimate conductance versus topological index that's now not an artistic view. This is an exact result as function of a system size. You flow either to trivial phase or to a topological phase and you know really exactly how the system does it. Now in remaining three minutes I want to say something else. So imagine you're sitting in this quantum critical point where the system is nominally delocalized and you may ask now dynamical question so okay it is delocalized but how the propagation in this delocalized phase look like? And let me give you the answer. The answer is that what happens is a Sinai diffusion and the Sinai diffusion is an observation which was made quite a long time ago that whatever is associated with Sinai diffuses painfully slowly. So displacement scales not like square root of time as a custom in a normal diffusion but it scales like logarithm square of time. Now the specific model for that is known again for long time due to Yako Sinai Sinai and it's a classical model where you have a long-term in process and the force is random. Not the potential is random but the force is random. So if force is random then potential is being an integral over force over plus minus whatever random process. So potential exhibits random walk and that means that if you want to move distance x you typically encounter a barrier which is square root of x and that means that typical expectation time to move distance x scales like e to the square root of x the height of the barrier divided by the temperature and that immediately bring you back to this formula and the coefficient in front of this logarithm square is nothing else by temperature square normalized by the amplitude of these fluctuations. So in classical physics it's a very well-known process and mathematicians proved that that's indeed the correct asymptotics for this process. Now the funny thing which we found is that at the quantum critical point the same exact thing happens in a quantum system. We don't have a good interpretation for that. We don't have a hand-waving model like the classical one which I just showed you but that's an unmistakable fact once you do this machinery you immediately find that the nature of the diffusion is a sign of the diffusion. So that's supposed to explain it in a little more technical details but probably I'm running out of time so I'll skip technicalities, I'll skip mod-virezine-ski physics and I go to my summary. So what I try to tell you is that there is a real space approach to non-translationally invariant disordered topological insulators. It exhibits quantum criticalities at the topological phase transitions. It may be described by two-parameter field theory which exhibit universal scaling and the topological index is stabilized by localization actually it's the localization pretty much like in quantum Hall effect that is localization which makes topology stable. And the second part which I didn't have much time to discuss is that the dynamical rules which govern propagation at the quantum criticalities is Sinai diffusion and there is a generalized Mott's law which I didn't have time to discuss but maybe for private discussions. All right, thank you very much. Thank you very much. Thank you.