 All right, good morning, everybody. Welcome to the second last talk. Before we start, maybe we can agree on a change of rules. Can you ask questions during the talk and not after the talk? It may make more sense, perhaps. Yeah, fine. So I noticed most people here start or end their talks recounting cheerful anecdotes relating to boroughs. That puts me in a somewhat tight spot because the only halfway entertaining anecdote I could possibly tell had a deep and, in fact, somewhat traumatic influence on my life. But he is not yet here, so I can tell you. It's about how I ended up in Borosin's car on two separate occasions, you remember. And I don't know how many of you share with me that experience. But with me, it had been enough to lose all faith in that means of transportation. And ever since, I have to rely on two-wheeled vehicles as an alternative. And you can imagine what that means in the German winter. So that's a somewhat sad story. There is one more personal account I would like to share with you before I start going. It goes back 1,000 years when I was a young, just beginning research student at our institute doing my first own research. And from one day to the next, the Russians came to our institute that before had been rather quiet. So not just Boris, but also Igor Lerner and Volodya Kraftsov, Volodya Jutson, several others more. And I vividly remember how that introduced me to a style of doing physics, which I hadn't known so far. I mean, a style distinguished for vibrancy, for energy, spirit, degree of healthy roughness at times. I mean, all things which were very influential. And I also understood that I could never possibly be up to it. I mean, not being breastfed in the Soviet Union don't have the proper education. But still, it was enough to have a strong influence. And I'm profoundly grateful for that. And I hope some of that reflects in the subject of the talk I want to give. It's about vial metals, topological metals, in connection with disorder. A subject many of you here in the room have, in fact, defined and shaped. And I understand you had an entire session on vial metals. So the way I want to organize this, I start with a concise, yet hopefully self-contained introduction to the clean vial metal, essentially retelling things you maybe already know. And then we turn to the disorder case. And I motivate the story I want to tell you. So the vial metal, I mean, we are dealing with three-dimensional fermions. And in the Brion zone, we have a situation where two Dirac cones sit, two linearly dispersing Dirac cones. These cones can be manipulated. They can be split relative to each other in momentum space on the count, I mean, on the expense of breaking certain symmetries by some momentum vectors. They can also be shifted in energy at the expense of breaking other symmetries, such that we create topological Fermi surfaces, as shown here. But they cannot be individually destroyed because they carry topological charge. So let's wrap up what we can say in general terms about this type of systems. Already mentioned, the nodes are protected against being kept out. They can be moved. And in this way, we can create Fermi surfaces. These systems can be physically conceptualized in different ways. And one way I want to bring up briefly, we can think of them as theories in terms of stacked topological insulators. I mean, if you don't know this picture, you will not be able to understand it on the spot. It's just meant to alert you to one thing, this two-dimensional layered way of thinking about the system. In terms of topological insulators, tells us that they share properties with layered quantum-hole systems. So each topological insulator is similar to a two-dimensional quantum-hole insulator. And if you stack them up, you generate a layered structure which, in fact, has direct nodes in the dispersion of this environment. They are also real. I mean, of course, you have to couple these guys to achieve this. They are real. I mean, in reality, they have been realized in different ways. And there's more materials coming. The quantum-hole analogy also tells us that something is going on at the surface. I mean, quantum-hole physics, right? Surface physics going on. The key word in this connection is thermi-archs. And the most important physical mechanism at work in the system is the non-conservation of charged individual nodes, which is known as the axial anomaly. Now, being condensed metaphysicists, we can think about it as follows. If we perturb the system by application of an electric and or magnetic field, we induce spectral flow from one node to the other via a high-lying portions of the band, which are not shown in this picture. There is an equivalent way, another way of thinking about it in a more particle physics-like way, where we understand the same phenomenon as the non-conservation of charged mean axial anomaly known from relativistic fermions, Dirac fermions. Now, this phenomenon, in turn, that generates phenomenological effects, consequences. And in the first place, several types of transport coefficients, transverse transport coefficients, and the two famous ones, or most famous ones, most distinguished ones, go by the name chiral magnetic effect and anomalous-hole effect. And I will explain or capitulate what that is. So the short, fast track to understand these effects, or at least hint at their existence, is to write down an equation which expresses the non-conservation of currents and densities at individual nodes. So we have an axial current density, the difference in charge and current between the two nodes. And this guy is non-conserved, and the non-conservation is expressed by a certain topological creature involving electromagnetic fields. And if you think a little bit about this relation and ask what it means, then you are led to conclude that it leads to transport effects. And in particular, we have the anomalous-hole effect, which means the following. If the nodes are split in momentum space by a certain fixed offset vector, this B-splitting vector, which I mentioned in the beginning, and you apply an electric field, you generate a current transverse to the field. So it's like a whole effect. And this guy here, the splitting acts like an intrinsic magnetic field. That's one effect. And the other one, which is arguably a little bit more interesting, goes by the name carol magnetic effect. And that tells us the following. If the nodes are split in chemical potential, they bias the chemical potentials, and you apply a magnetic field, you will observe a current flow parallel to the magnetic field. And these two are more or less immediate consequences of this relation there of topological origin. OK, so that wraps up in a nutshell what I wanted to say about the clean vibe. And what I want to do now is I ask the question, what happens if we take the idealist clean system and subject it to disorder, which will always be there? And of course, I owe you a motivation why one should care, I mean, why this is interesting. The way I want to organize the discussion is I first discuss entirely on qualitative terms what disorder can be expected to do. Then in a very, very brief intermediate term, I construct an effective theory which uses field theory. I will argue why we need field theory here. And then I want to discuss with you the ramifications of disorder in the two transport coefficients which I just mentioned, and I promise there will be one surprise. At least for me, it came as a qualitative surprise. And that is work done jointly with Dima Bagdetz. So to motivate why we might want to look at disorder, let me briefly review for you two strands of previous work, which appeared recently. And they are hooked to two questions you can ask. The first question, and it was asked by Spivak and Zon and independently by Borkov, is a follow. Suppose we consider the metal tuned to a configuration where we have two nicely developed topological family surfaces. And now we put some disorder, and in particular disorder, that will scatter from one cone to the other. So that will actually make the two cones talk to each other by scattering. How much of these beautiful topological textures will survive? I mean, will it all get washed out, or what is going to happen? And the way they approached this question was because they have a small parameter here, large family surface, small disorder. They used kinetic equations and diagrams respectively and independently came to the following answer. There is a kinetic equation which tells us how density relaxes in the system. I mean, when we talk about density, we have to discriminate the actual total charge density and the axial charge density, which is the difference in charges sitting at the two nodes. And both of them satisfy a diffusive kinetic equation. So they diffuse. No surprise. The axial charge density relaxes due to inter-node scattering. Also no surprise. It gets equilibrated. The interesting thing is that these equations are coupled. And they are coupled by a term of topological origin, which is proportional to an external magnetic field. And that one here is a remnant. This is a remnant of topology in the system. And if you now work a little bit on these equations and solve them under mildly idealizing assumptions, you find that there is the chiral magnetic effect manifests itself in a contribution to the magneto-conductance. And that contribution is quadratic in B and proportional to the inter-node scattering time. So if you have strong inter-node scattering, you kill it. But for any finite amount of inter-node scattering, it is there. And this has been seen experimentally in the recent experiment by Ong. And they say that the ratio of the elastic scattering time to the inter-node scattering time, because the disorder is somehow weak, soft, excuse me, is all that tend to the two. So we have a small parameter. Yeah? Yeah, very good. That's my point. I come to that. That's the question you want to ask at the end. Yes, so it cannot be correct. Very good. Thank you. So it doesn't have a sensible limit. So we have to understand what happens for infinite tau in the clean limit. Yeah? Yeah? Oh, that's it. They're really linked topologically. And I'll explain that later on. Yeah? Yes? It's not my formula. It's their formula. And I'm telling you there is a better formula. So this formula, that's a harsh formulation. Yeah, but inspiring. OK. That was the first line of work. Then there is another one, which is actually equally controversial. And that now asks you a different question. It started many years ago, and the latest reincarnation is recent work by Cisronov and Gurari. And they ask, how will disorder affect these delicate nodes sitting down here? Yeah? And they do that as follows. Well, it can be done as follows. Suppose you have some Gaussian distributed disorder in the system, you will generate an acuity scattering matrix on average. Now, if you do dimensional analysis, you find on dimensional ground that this is strongly irrelevant in three dimensions. So it scales to zero. It suggests that disorder doesn't affect the Dirac node very strongly. Then you think a little more, and realize that dimensional analysis may not suffice here because we are being Dirac fermions in high dimensions, and we should expect strong ultraviolet fluctuations. So you work a little harder and go beyond this approximation, and in fact beyond the non-crossing approximation, and find ultraviolet divergent contributions to the scattering matrix, and what it amounts to is an additional term in the flow equation of next higher order, similar to Koldeau perhaps, and what you are led to conclude is that there is a quantum critical point. So if the disorder is stronger than a certain critical amount which is of the order of the bandwidth, there is no other parameter at this point, you generate a runaway flow to strong disorder, and for weaker disorder you flow back to the fixed point, the quantum critical point. The question then is what happens at, I mean that's the question I want to address in this talk, is what happens here in the strong disorder regime. I mean what happens if naively the disorder scales up to what kind of phase you will flow, and Gourary and Sucronov came up with all kinds of predictions, and all of them are controversial, so it's a bit grounded. So the question I want to address now in the rest of the talk is what happens if we are, yeah? That's a bare disorder strength. You say you have some disorder characterized by some variance. Small gamma? Yes, and I will. This is just the bare, I mean, so it tells you that much like in graphene at the diracon the disorder becomes stronger, then you reach length scales where the scattering, you have multiple scattering and you go diffusive, and that's what I want to turn to that. So what I want to do now is I want to discuss the physics at large distance scales, in the supercritically disorder system, large distance scales means multiple scattering, and what I suspect is, and I will show how it comes, is that this is a 3D Anderson metal. But that metal must contain some topological signatures built in. I mean, these topological features survive as we learn from the first line of works, and the question is how and how the system does all that. And to answer all these questions I use a bit of field theoretical machinery. The construction is actually quite technically, but I don't want to torture you with that, I just want to highlight a few conceptual steps in the construction of an effective theory which are conceptually important. And then we quickly jump to reading out the predictions. So we want to do a theory of disorder in non-interacting fermions system, complete green functions to extract transport coefficients, and the way to do it is to start from a replica functional or supersymmetry or Keldisch, doesn't matter here. So we have a replica index and here is a Hamiltonian, and then we need to discriminate between advanced and retarded green functions. So there is a bit of symmetry breaking on that level. One important thing, and that will be very important in the following, is that this action here, it sounds technical but there is physics sitting there, has a huge inbuilt symmetry. You can rotate these sides by rotations in replica space and in advanced retarded space by unitary matrix, and if this is constant the action stays invariant. So there is a huge symmetry. Okay, now we play the standard games going back, I mean, in endless times, we average over disorder, and then we get an effective fight to the four fermion vertex, we decouple it to Habat-Srotonovich and we run a mean field program. That's what everybody does in this business. And it was done for us in this context by Fredkin in 1986. And what Fredkin found is that there does exist a finite mean disorder-induced mean field scattering amplitude. This has the significance of the mean field, of the mean scattering field, for example, if and only if your disorder is supercritically strong otherwise you don't get it. So if the disorder is too weak, you don't find a mean field and there is no field theory to talk about. But if you are supercritically strong, you induce this term, it gives us a mean scattering rate physically, and it breaks the symmetry which I just mentioned. It breaks it. By that I mean the following, there is not just one solution to these mean field equations, but many are distinguished by the symmetry which I just mentioned. And the situation is if you have never seen that before, really analogous to what happens in a magnet. So this infinitesimal i delta, I started out as like an infinitesimal magnetic field in a magnet. There is a phase transition. We break the symmetry. It's like a mean magnetization, but then we can rotate it homogeneously by changes in the magnetization axis and they generate goldstone modes. Physically, these will be the diffusion modes in the problem. Now, we want to understand the physics of these generated by these soft rotations here. These are the lowest lying energy, degrees of freedom in the problem. And much like in a magnet, we want to expand somehow conceptually in soft generators of magnon-like diffusion excitations. And we want to know what is the effective theory of these skies, and they will be the main players in the game. And one important thing is that we can conceptualize these in two different ways. We can think of them as goldstone modes. That's the way I introduced them. But equivalently, they have a lot in common with gauge fields. Like a non-Abelian gauge field. And that's important because if we deal with a gauge field in three-dimensional Dirac fermion materials, we expect massive manifestations of the anomaly. And that was all worked out by the particle physicists in the 80s for us. And the keyword is parity anomaly. So we expect a manifestation of the anomaly due to worked out by Redlich. And I'll tell you how it comes and what the phenomenology is. Regardless of the interpretation, what we want to do now is we want to integrate our fermion and then expand the action in somehow in gradients in ascending order in these skies. And I will tell you now how this piece here looks and what kind of physics follows from them. So the way I want to organize that part of the discussion is I directly link it to the physically observable effect. So the first part of the discussion now connects to the anomalous whole effect to recapitulate the effect that the system has somehow like an inbuilt magnetic field, the splitting of momentum modes and has transverse response. And the question is how do we get that from this type of description. So it turns out if you work very seriously hard that the first two terms in the expansion in these A's look like that. I mean they will not mean anything to you, don't worry, just notice they are of first and second order in A perspective. These are the first two terms. And they look utterly alien in this way. But we can introduce variables which are frequently used in this context, the famous Q matrices of Hefetov and Wigner. And if we express this action in that language we obtain this here and that's known to the diffusive nonlinear sigma model which in this business contains two terms. One is an ordinary diffusion term, that's what it physically describes and there is a pre-factor which is a longitudinal bear conductivity. And then there is a term which looks like a Poiskin term, like a quantum whole term. This is a term we get in the quantum whole effect only that we are in three dimensions. So it looks like a layered stacked up quantum whole action. And that connects now to this interpretation I mentioned in the beginning that this system can be understood in terms of layered quantum whole species. And the corresponding physics was discussed by the next speaker, so I don't have to discuss it long ago. So there is a system known as the layered quantum whole effect and the action above which we derived is the effective action of that one. What does it mean physically? It means the following. We have this action with some bear coupling constants which respectively define the bear conductivity which depends a little bit on where we are or whether we are at the nodes or far away. And then there is a bear whole conductivity which is essentially set by the splitting vector. And now you want to know what happens, I mean how do fluctuations, quantum fluctuations, diffusion mode fluctuations affect this and that too has been worked out long ago in this paper here by two loop renormalization group methods and the physical conclusion is that the situation is boring. So what happens is that the longitudinal conductance in three dimensions scales omic, so we have metallic behavior, we have a good metal, an omic metal and we have a stable whole conductivity, the whole conductivity is also omic and it's not affected in any way by disorder. It's just set by the splitting vector in momentum space, it doesn't renormalize unlike in the quantum Hall effect, the whole conductivity does not renormalize. So that's the first conclusion. To recapitulate the bulk, I mean to zero order approximation the physics of this system is a good metal, it supports the whole conductivity or whole conductance which is not affected by disorder significantly and that's pretty much it. Now that here I mean this whole conductivity describes the anomalous Hall effect. Now what about the chiral magnetic effect? And that one is more interesting. Question? Yes. No, that is because I have to go back. Oh, I have to go back quite a bit. Yes. No, you don't see it with your naked eye, I agree. But believe me, if you plug this here, if you plug this in, this representation you get it. I can do the calculation in five minutes. Yeah. You mean why there is no sigma yy? Oh, but the magnetic, you should think of this here. Yes. I mean there is more coefficients but there is a lot of degeneracy in the conductivity tensor. So in fact you have a three-dimensional metal. I mean one direction is singled out by the magnetic field and then I have a sort of boring isotropic longitudinal conductivity, this is sigma xx and if I now apply an electric field perpendicular, this guy then I will get something transverse and that's the sigma xy. Sigma yy is the same as sigma xx. Yeah. Yeah. But there comes something very good. I mean if we go to the chiral magnetic effect that stops. There is another indeed, very good. I mean how much time do I have? People are asking lots of questions. Oh yeah, very good. So now I want to turn to the chiral magnetic effect and just to recap that one, that's an independent transverse effect and it means that what it tells us if we bias the nodes, give them a relative chemical potential, apply magnetic field, then we will obtain a current in the direction of the magnetic field. So this will be sigma zz. Where does this information sit on this effect and the answer is we have to go one order higher in A. And why is that no surprise that we have to go one order? This is now, remember gauge field in three dimensions, that's what we are dealing with and what we should expect is a chance-heimens action. So chance-heimens is a natural action to expect for a gauge field in three dimensions and indeed there is one in this system. So there is a non-Abelian chance-heimens action which appears at the third order in the expansion. Oh, that was no good. Here it is. It looks a bit strange, there are some projectors on the target advance space, forget about them, but it has a typical form A, DA and A cubed. And now what is the physics of this one? To build some faith I mean into the existence of this, some theory nonsense, we can run a little sanity check. We do the following. We assume that localization effects don't play a role and that justifies to take these t's, these rotation, these guys and to expand them to leading order and magnons if you want. I mean quadratic order in rotation generators and if we plug that in we get something which is for this A which is still a non-linear object which is quadratic in these nodes. They have some internal structure don't worry about that. They also have a structure in nodal space. We have two nodes and we plug that into our action in the presence of a magnetic field, an external magnetic field and we work a little and we get this kind of effective diffusion propagator. So what you see here is an effective diffusion of the system. And again it has a structure in node space which they get coupled by inter-nodes scattering and there is a magnetic field which effectively there is a diffusion term and a frequency term. I'm showing this because if you now on that basis work out the what densities how they diffuse you get these work of zon-spivak equations. So it is absolutely essential I mean the Chan-Simons term which effectively sits here. You can't see it so quickly why but believe me it's there. It generates these coupling terms. So that that term is definitely an essential player in the game. But now it was already pointed out by the audience that the physics that derives from there, the quadratic in B magnetic conductivity can't be quite right. So the question is how do we fix that. And to this end let me now ask a physical question. Suppose we do a theories cartoon of a quantum transport experiment I mean we take a quasi one-dimensional vile vile wire the splitting is in this direction so this is along the axis we apply magnetic field along the axis and we apply battery and we ask what will we measure. And work of zon will tell us there is a correction to the omic conductance which is B squared. But then it was pointed out that this I mean suppose we tens out tau a to infinity clean limit it doesn't make sense right. You have a diverging contribution can't be right. Then comes another group of people Parameshwaran and Vishwanath they tell us something else they consider the clean limit. So forget about this order for a while. And they argue as follows. If you apply magnetic field what happens is I think Adi mentioned it in his talk you get a quantum hall effect at the Dirac nodes. And in particular there is a zero energy quantum hall layer bent and it has it is chiral and so that quantum state contributes to transport. And it gives us a contribution to current flow which is proportional to an effective number of channels and that effective number of channels is proportional to the degeneracy of the quantum hall states. So you would say there is a ballistic conductance in the absence of this order and the ballistic conductance is linear and B. And now is the question how do these guys match in what sense. I mean some of they should, right? I mean if you put this order here, I mean some of they should meet and I want to tell you how they match and that what the system has for us in store is really a surprise. At least for me it was surprising the way how it does it. So let me try to explain that. So the way to describe this in comprehensive terms is I mean what we do is we take our frenzy A's we couple some external magnetic field and maybe some source fields required to compute something and plug this animal here into this complicated Schoen-Simon's action and then we project onto the quadri-one-dimensional limit. That's what we do and we obtain this action which at first sight looks complicated but just by looking at it you can understand that there is interesting physics. So what this action has, this is now the action of the quadri-one-dimensional wire. It has an ordinary diffusion term here which we already had I mean this is the diffusion term and then it has a term linear derivatives and this pre-factor here is independent of this order. So what is surprising about that is that just on power counting we know that a term that is linear derivative will always win at large distance scales. So we are led to the conclusion that the system diffuses, will diffuse at short scales and then become ballistic at large scales. Normally it's the other way around in a metal, right? If you have an ordinary metal you are ballistic at short scales then you scatter and go diffusive. Here it's the other way around. So you first diffuse and then you cross over into a drift dominated transport which is this order independent. For me it took a time to accept that. That sounds like Halloween physics but it's all cut off eventually. If you have inter-node scattering the two fields at the nodes they lock to each other by this term and that kills this kind of guy. So we are led to the conclusion that there are at least three independent length scales in the problem and you can adjust them at arbitrary, at will. I mean you can tune them to whatever you want. There is one length scale I call it LB which determines the crossover from diffusion at short scales to ballistic drift at longer scales. And this ballistic drift is pushed by the topology. Then eventually at some point your nodes hybridize and you cross over again into some diffusive regimes and eventually at some point you have Anderson localization in quasi one dimension. So that's what kind of the rough picture is. Now you can tune these parameters to whatever you want and that's complicated. I mean three independent scales and it's too much to keep track of. Let me just show you on a few examples what that means now for transport coefficients. So what we do is we sit down and work out the conductance you have to be really careful here with boundary conditions it's a bit more tricky than usual and what you then find is an expression for the conductance in terms of two independent parameters. One is system size in terms of this nodal hybridization length elastic free pass due to nodal coupling and then there is this drift diffusion crossover scale again in the units of the same length scale and you get this little monster here. And that one here now has Burkhoff and Vishwanath as limits. So that's the same equation. So for example if you consider this limit length shorter than this crossover length I mean where you still diffuse and all that shorter than the nodal scattering then you get the Burkhoff and Zohn approximation. But if you plot now just this conductance result as a function of an effective magnetic field so you crank up the magnetic field then you find that at short I mean for small fields where you effectively in the diffusion regime you get the Burkhoff and Zohn quadratic increase and then for larger fields you go ballistic and you cross over into this Vishwanath regime which is linear in the field. A kind of more entertaining way of plotting this is this here. This is now the conductance as a function of length in double logarithmic units and what we have is that for short length lengths shorter than the drift diffusion crossover you have a 1 over L behavior I mean it's omic and then we cross over between here now here now drift takes over makes the system longer it effectively becomes clean there actually we check the noise here this is a noiseless I mean the system doesn't support noise here you have ordinary omic noise but this is really ballistic and eventually I mean length independent eventually you couple the nodes and then you get again cross over into omic behavior so this kind of behavior comes out of that theory. Yeah and at least I find that somewhat unusual. Okay so that's pretty much what I wanted to tell you to wrap up I mean to a zero's approximation the disordered wire we believe is a 3D Anderson plus a bit of topology mixed in it supports a rather stable oh you can read here it supports a rather stable anomalous hold effect and layer quantum hold physics but then there is also the chiral magnetic effect which sits in the Chan Simons term and the most interesting to me at least conclusion of this is that the Chan Simons leads to this diffusion drift for this ballistic at large scales small scale so that I have not seen before. Okay thank you for your attention.