 He has worked on also on spin drag and other related effects. More recently, well, more recently, since probably 10 years, 15 years, he has been working on graphene and other 2D materials, and so with no further delay, I would ask you to start your talk. Okay, okay, very good. Thank you for the introduction. I'm really happy to be back in 3S, a beautiful 3S after last time was in 2013. Okay, so it's almost 10 years. So I would like to give you a bird-eye view of some recent work on hydrodynamics of quantum liquids, which I'll try to show you. It covers many, many different physical systems going from the more familiar electronic systems all the way to quark, blue and plasma, passing through things like helium-3 and cold atoms and so on. But first, let me show you, I want to point out to you my new affiliation here. Let's see. Okay, this is how it works. So I am right now in Singapore, at the National University of Singapore, and specifically in this institute, which is called the Institute for Intelligent Functional Materials, and this is the most iconic picture of Singapore, the three Marina Bay fans towers with a swimming pool on the top, and you can see also this lotus-shaped building, which is really the Museum of Art and Science. Now, on a more serious note, this is the building where the work happens or is supposed to happen. So if you are interested in visiting, I think it's a nice experience and a nice place to visit. Okay, so let's start at the beginning here. Now, I'm sure everyone realized that hydrodynamics, as the science of liquid flow, is one of the most ancient branches of science. So it certainly was, to some extent, cultivated by the Greek, for its many and obvious technological applications in the transport of water. And here, for example, is the beautiful Archimedes crew 300 BC, which with little exaggeration, one could say, has anticipated by about 2,000 years the discovery of a diabetic pumping of charge in mesoscopic structures. So it's an absolutely brilliant idea to adiabatically transport water from a place to another. Now, much later, in the 1400s, Leonardo seems to have been fascinated with liquid flow, in particular, turbulence. You see, there are lots of drawings like these, which show vortices in all shapes and dimensions. Okay, then we have, of course, contemporary applications. This comes close to the place where I've lived for many years, the Missouri River. It's a dam on the Missouri River, actually, up in Nebraska, not very far from Columbia. And here, much less known, perhaps, is this so-called Marangoni Gibbs effect, which is responsible for the formation of the so-called wine tears. Okay, so if a wine is good, it's supposed to show these arcs, you know, which can be explained by differences in surface tension, an indicator of the quality of wine. So I hope that tonight we will have an opportunity to demonstrate the Marangoni effect. Okay, Marangoni Gibbs, actually. Okay, anyway, so now, but in the modern sense of hydrodynamics, the story started with the atomic theory, okay? So with the atomic theory and Boltzmann adopting it and trying to give a statistical description of many, many particle systems. And I would say that, in a way, the classical hydrodynamics is the mother of all effective theories. It has an incredibly high degree of universality, meaning that no matter what the elementary constituents are, as soon as you look at this system of atoms or molecules on a sufficiently long length and time scale, in lengths which are much larger than the, you know, the mean-free path traveled by molecules in between collisions and also times longer compared to the typical mean-free time, then, you know, you get a behavior, a fluid dynamical behavior which is completely independent of all details, okay? And this, in some sense, is a metaphor of physics itself which always tries to abstract from details and give you universal descriptions of things. So, in a sense, you can think of, it is absolutely correct to think of hydrodynamics as a kind of generalization of thermodynamics in which you allow the thermodynamic variables. For example, you could choose the density, the temperature, and the velocity of the flow. These are appropriate thermodynamic variables and you just simply allow them to change from point to point to become slowly varying functions of position and time. They are still locally connected to each other by the local equation of states. This is a very important point of hydrodynamics that you assume the system remains locally in thermal equilibrium because of these very frequent collisions between the atoms and so on. They must be very, very frequent. But otherwise, you know, so you basically, you let the thermodynamic variables fluctuate and each of these variables here is associated with a conserved quantity. So the variation of this conserved quantity since they are quasi-conserved, you know, they would be conserved if the system were exactly uniform. Since they are quasi-conserved, they evolve slowly in time and that consistently justifies the idea of looking at the system on long time scales and long length scales. So let's then, okay, I will not show you much math in this colloquium because that's not the purpose of the colloquium, but I just want to show you that these basic ideas that I try to explain in words are encoded in a few mathematical equations which essentially express the conservation laws. So the first one is the local conservation law of the particle number, it's known as the continuity equation. Okay, so you see the derivative of the density with respect to time is the negative of the divergence of the current density. And then there is a second equation which is known as the Navier-Stokes equation which essentially is a modified conservation and modified for the momentum density conservation. So it says that the density of momentum here is equal to the divergence of an object known as the stress tensor which is in fact the current of momentum density. And plus any external force that might be present and would also change the momentum in the volume. Okay, so it could be for example an electric field if you are doing hydrodynamics with electrons. And then of course there is a last equation that in a certain sense is the generalization of the second law of thermodynamics. Okay, so it says that the derivative of the entropy density is s and s is the entropy density. So the entropy density can change either because it is transported out of the volume. So here you have the usual divergence of the entropy current or because of various dissipative processes which are all lumped mercifully into that queue and which include for example internal friction, viscosity, joule heating, heat transfer. There are a number of complicated things. And so the entropy of course has to increase according to the second law of thermodynamics. And this stress tensor which controls basically the force that different parts of the fluid exert on each other has an ideal part which is, you see, is this first two terms. Okay, so this is basically the naive current. I think, yeah, there should be an m probably here. And the pressure here, this is what defines the so-called perfect fluid. But then there is terms which are proportional to the derivatives of the velocity fields and those involve this constants of viscosity, eta shear viscosity and zeta so-called bulk viscosity. Usually the shear viscosity is essential and most important for what we have to say. So let me say a little bit about the viscosity because the viscosity will play a central role in this talk, okay? So I will talk about viscosity in electrons and viscosity in work gluon plasma and viscosity in electrons in graphene and so on and so forth, okay? So let's, first of all, let's try to get the feeling of what it means, really, this viscosity, okay? So, you know, if you have liquid flowing in a pipe, like I'm showing you here, right? Typically the velocity is, tends to be low near the walls of the pipe because the fluid tends to stick to the walls, right? And it is maximum in the middle, okay? In the central lane, so to say, of this thing, okay? So what happens is, yeah, this is... So what happens is that because the momentum density is the largest at the center, you know? So this momentum density tends to diffuse towards the walls, okay? And the rate at which this diffuses, so it generates a... as I said, a momentum current. So it's the momentum in the x direction that flows in the y direction, okay? And this current of momentum density is described as Txy. And as you can imagine, it will be proportional to the gradient of the velocity, right? So if the velocity is changing, decreasing sharply, there will be a larger current, you know, of momentum. So you see that you have a flow of momentum from the center to the walls, and eventually this momentum is dissipated at the walls, okay? So you have a dynamical equilibrium between the pressure difference that drives the flow and this viscous process that, you know, takes the transfer, the momentum transversely relative to the flow. And this is the meaning of that data. In fact, let me tell you that a very nice way to think about the viscosity is to consider this particular combination, the viscosity divided by the mass density. This is called kinematic viscosity, okay? And it has a very, very simple physical meaning. It is the diffusion constant for the momentum density. Just, you know, I think most of you are probably familiar with the diffusion constant for the density, right? Which is typically v squared times tau, where tau is a momentum relaxation time. Now, you know, you multiply that by density times m and you get the diffusion constant. So you get the viscosity. But this nu has exactly the dimensions of a diffusion constant. That is meter squared per second, okay? And so you see water has very little viscosity, kinematic viscosity of 10 minus 6 meter squared per second. But then, you know, honey is about a thousand times more, you know, a thousand times higher kinematic viscosity. And then, if you keep going, we will see later that graphene, electrons in the graphene fermi liquid are about a hundred times more viscous than honey, okay? Believe it or not. Okay, so there is a field theoretical expression for the viscosity, the shear viscosity, which we will become important later when we start talking about black holes, et cetera. Which is this, okay? It's basically a response function or a correlator, you know, of the stress tensor with itself. Okay, so there's this stress tensor field, T x, y. Its correlator Fourier transform basically gives you in the appropriate limit the shear viscosity. Okay, very good. So just for orientation, what happens if I take an ideal gas? Okay, that's the simplest statistical system that we study in school, okay? Well, you know what the thermal velocity is, right? It's kBT over M, the square of the thermal velocity. And also, you know that the scattering time is related to the cross-section of the collisions between atoms. So you put all these things together and you arrive at this formula, which I think Maxwell derived first for the viscosity of an ideal gas, okay? It is expressed in terms of the temperature and in terms of the cross-section of the collision. And there are several very interesting and puzzling features about this simple formula. Perhaps the most striking is that it does not depend on the density of the gas, okay? So the story is that when Maxwell saw that, he just didn't believe, you know, he must be wrong. So he went to the lab because he was a serious scientist. So he went to the lab and tested it and indeed he found that it was independent of density, okay? That is one thing and it comes basically from the cancellation of the N in the scattering time with this N here that appears in our definition of the density cancels out of the formula. An even more puzzling thing, which takes a long time to wrap your mind around this, okay, is that somehow the viscosity increases as the collision rate between the particles decreases. You see, if I make my cross-section smaller, the A smaller, you see, the viscosity goes up. So it seems that the more interacting, the more strongly interacting your system is, the lower the viscosity, okay? So it seems to make no sense at all, but we will see that this, in fact, one of the most striking things that have emerged are these extremely highly correlated electronic system as well as the quark-gluon plasma, in fact, which have an almost negligible viscosity, okay? So almost near to, you know, the conjectured lower bound for the viscosity and that is really, really very strange in a way defeats intuition, but it is obvious if you look at the math, right? Because increasing the strength of the interactions, meaning that the scattering time here, the time that the particles travel alone and this goes down, right? So you see immediately that, you know, the eta has to be going down, okay? So this effect is highly non-perturbative in the interaction, okay? So, and I leave it at that, but we will see many examples of this point. Okay, so, okay, so all this was classical, okay? And even at the classical level, there are things to understand clearly, but what about quantum liquids? Okay, so for both the liquids which go superfluid and so on, then it's quite clear that hydrodynamic description, you know, is sensible, but for Fermi liquids, it is much more tricky, I would say. Let me explain why it is so difficult. It has been really very difficult until recently to really, you know, achieve a hydrodynamic regime for Fermi liquids. Okay, so the big problem is that under most general conditions, although there are exceptions to this, okay? So the Fermi liquids at low temperature are described by a paradigm known as the Landau theory of Fermi liquids, okay? And the key idea of this is that because you are at low temperature and there is this big sphere of occupied states in momentum space, the particles, the fermions have very little room left to scatter against each other because almost all the final states are occupied. There are severe restrictions, okay? There are severe restrictions on basically the scattering processes and the result of this is that these particles at low temperature, no matter how strong the interactions, the physical interactions are, they effectively, in an effective sense, they become non-interacting, okay? They behave like non-interacting. Now, this is another amazing thing, okay? The interaction can be incredibly strong as it is, for example, in Helium III, you know, very strong interaction. There are even more dramatic examples, okay? But so the theory of Fermi liquids is non-perturbative, okay? No matter how strong the interactions are, when you approach the low-temperature regime, there is this Pauli blocking effect that makes the particles effectively non-interacting. So tau goes to infinity. Now, that spells disaster for hydrodynamics, okay? Because hydrodynamics requires frequent electron-electron or particle-particle collisions and if that doesn't happen, you are not able to establish a hydrodynamic regime. And this is very nicely reflected in the fact that you see the formula for the viscosity which you can derive for the Fermi liquid. Look what it does, okay? So first of all, it depends on density, like the ideal gas, but the most important thing is that it diverges at low temperature, one over T squared. Okay, so you know, this is again an expression of that apparent paradox that you make the system non-interacting and the viscosity goes up. You make the system strongly interacting, the viscosity goes down, okay? Go figure. Anyway, so this in practice means that most of the time you are not in the hydrodynamic regime with Fermi liquids, okay? Now, I had in mind the canonical Fermi liquid is helium-3, okay? But what about electrons? Electrons are also under broad conditions described by Fermi liquid theory. For electrons, there are, let me put them all together here, there are essentially three modes of electronic conduction, okay? And the most common one by far, the one that operates in the electronic devices that we are holding our hands and so on, is this one in the middle, the diffusive regime, okay? So in the diffusive regime, there is typically some external force like exerted by an electric field. I'm sorry, I'm not used to. And basically, the electrons move, drift collectively under the action of this electric field, but each electron you see scatters against impurities or against lattice vibrations, okay? And so this limits the amount of current that flows. So they perform a kind of random Brownian walk whose net result is that there is a net drift in the direction of the electric field or opposite to it. So this flow is essentially uniform across the pipe, okay? And so basically, the electrons don't see the walls, okay? The boundary conditions are irrelevant in this regime. They see mostly the impurities and the phonons, okay? So this is, of course, not a hydrodynamic regime at all. Now, there are two additional regimes of transport. One has become very, has come into the center stage recently. That's the purely ballistic mechanism of conduction, okay? So that happens when you have, you know, devices at the scale of nanometers in which basically electrons can flow or rather can fly without any encountering any obstacle, you know, from a source to a drain, okay? And that happens because they're small, these devices. So nothing has the opportunity to get in the way of the propagation of these electrons. So typically, the conductance of these devices, you know, is a small multiple of the fundamental quantum of conductance that is square over H. In fact, you might even ask yourself, why is this conductance finite? Shouldn't it be infinite considering that, you know, there is nothing at all to hinder the flight of the electron? And the answer is that there are the contacts, okay? So those contacts between the source and the channel and the drain and the channel, those are very tricky places where a lot of irreversible process, physics happens. And essentially, they are the ones that are responsible for this residual finite conductance, okay? But now what we are interested in here is a hydrodynamic regime in which, you see, the electrons scattered, they see mostly each other and very little anything else, okay? They don't see impurity, they don't see walls, they just see each other, they huddle together, they are in a kind of local thermodynamic equilibrium and you see they flow according to this, in a pipe, for example, you have this typical Poise flow which leads to this kind of parabolic, you know, profile, okay? So you see the flow is highly non-uniform in the transverse direction, okay? And so this is really what we would like to realize, but it's not easy, it's not easy because you have to satisfy that condition that the electron scattering time has to be much larger, I should say the electron scattering rate, okay, should be much larger than all the other scattering rate, including impurities and phonons. And this means not only that the materials must be very clean, obviously, but also that the temperature can be neither too low nor too high because if it is too low, you fall back into the Fermi liquid where they think, you know, the lifetime diverges, there are no collisions and if it is too high, then you are killed by lattice vibrations, phonons and so on. So there is a so-called hydrodynamic window, okay? And this has been now in many systems, this is an example, the red region is for electrons in graphene, shows a hydrodynamic window which you see covers a very interesting range of temperatures, you know, the red region extends from 150 to room temperature, 300 Kelvin, that's a very interesting regime for electronic devices, right, when you start approaching the room temperature and so on. So it's not unconceivable, but can it really be done and how would it show up anyway? Okay, so suppose, so I give you already a partial answer that, you know, you have that kind of parabolic shape, the flow and so on, but perhaps there are things that are more compelling there. So one effect that was known since 1963 was the so-called Gursi effect, predicted by this Russian scientist, Gursi, back in, as I said, in 1963. And basically he predicted that the resistivity of a metal should drop significantly when you increase the temperature as soon as you enter this hydrodynamic regime, okay? Now this is in itself pretty strange because now barring things like the condo effect and so on, normally the resistance tends to increase as a function of temperature, right? If nothing else, because of lattice vibrations, for example, and so on. But here you see, so it is a theoretical prediction, okay? It goes down, okay? And this region is essentially the hydrodynamic region. Then it will eventually start increasing again. So it looks like this strong electron-electron collisions reduce the resistance, okay? Which is again, you know, counter-intuitive because you would imagine that if you have a lot of electron-electron collisions that also would get in the way. But it doesn't, okay? And we'll explain it in a moment how this little miracle is possible. So there was this prediction back in 1963 but nobody knew how to verify it for a long time. And there was a valiant attempt in 1995, okay? By Molenkamp, who's now very well-known for the discovery of the topological insulators, et cetera. And he did this experiment using gallium arsenide wires. So he drove a current in those wires. And the current, you see, the current is what is plotted here on the horizontal axis. So the current flowing through the wires heats the electrons by joule heating. So basically this current here is a proxy for the electronic temperature, okay? On the other hand, the lattice temperature remains, or so they claim, low, okay? So you see that lattice temperatures stay pretty low in this series of, and this is important because it means that you are not running into problems with collisions with phonons, for example. So you get rid of collisions with phonons by staying at low lattice temperature, but you can increase the electronic temperature by driving a current. And lo and behold, you see that the resistance, the differential resistance, the VDI, indeed shows the Gucci behavior, predicted Gucci behavior. So they claimed observation of the Gucci effect. But of course, as soon as they announced it, Gucci himself disagreed that this was an observation of his effect. It's a very remarkable position to take, I would say. And because, you know, he argued that the conditions for having entered the hydrodynamic regime were not quite satisfied, not to his taste. All right, so that remained a little bit in a limbo for a while, and it's only more recently as I've shown you that the thing has been convincingly demonstrated. I want to mention just on the fly that in that period also some ideas were put forward of using hydrodynamic effects for interesting technology, or gedanken technology. So this, for example, is the so-called MOSFET, the typical field effect transistor, which operates with a two-dimensional electron gas here at the boundary. And basically, Diakonov and Schur in this beautiful paper, they argued that if the density of carrier is sufficiently high and you enter this hydrodynamic regime, then you can have an instability in which the electrons will start spontaneously to oscillate and you will get a generator of electromagnetic waves in the terahertz frequency range, which is a very important technologically, okay? Anyway, so some work in this direction has been done based on graphene devices more recently. But this brings me really to graphene, okay? So graphene has, in a sense, revolutionized this field because it's the first material in which many of these hydrodynamic effects can be seen, okay? Well, you all know, although, you know, the electronic structure was known since 1947, and in a sense it occurred everywhere, every time you write with a pencil or a piece of paper, there is some graphene forming there. But it was isolated only in 2004 in this landmark experiment with this Koch tape, metal, actually, by Kostya Novoselov. And by the way, Kostya Novoselov is now the director of the Center for Intelligent Functional Materials, okay? So he's my boss right now, okay? So I definitely have to show this, okay? So the characteristic of graphene is that it has these two bands, you know, valence band and conduction band that touch at just one point. So in the pristine state, okay, before you start playing games and so on, all these states below this point would be occupied and all these states above would be empty, okay? So what I'm showing you is already something in which I played some game, you know, either by doping or by applying some gates and so on. All right? So keep that in mind because this is important because this system is very interesting because it exhibits two very different regimes, okay? Depending on basically where your Fermi level is, okay? So if you are here on the right, okay, just as in the previous slide, so, you know, you have, you know, substantial density of electrons, let's say, in the conduction band, okay? Then this is essentially a normal Fermi liquid, okay? And by which I mean that the inverse of the quasi particle lifetime scales as the square of the temperature, okay? So it tends to zero, you see? It tends to zero when the temperature goes through this typical Fermi liquid behavior. Also, you can calculate the viscosity here and you'll find that it shows the typical Fermi liquid divergence of the viscosity, which in a certain sense tells you that, you know, hydrodynamics, you know, it's difficult to, you know, so if the viscosity is too big at some point, this eta loses its hydrodynamics significance. But then there is another phase which we call Dirac fluid, okay? Which is quite different. So this phase happens when the system is kept at charged neutrality. So the chemical potential is here at zero and these orange electrons and holes that you see, they're not externally induced or doped, they appear naturally because of thermal fluctuations, okay, across the, okay? So this is a different phase and you can see how different it is if you look at the behavior of the quasi particle lifetime, for example, one over tau, it scales as t, not as t squared. So the width of these particles is comparable to their energy. So these are no longer Landau quasi particles. So people describe this as a Planckian regime, okay? So it's not the Fermi liquid. It's also characteristic of a class of systems that are called strange metals, okay, in condensed matter. Okay, also the viscosity, look what happens at the viscosity because, you know, interactions are so strong here, it goes to zero as t squared basically. So it's very low viscosity. Again, that paradox, right, that the strongly correlated system has low viscosity, t squared, okay? So you see, this gives you a lot of room to play and in fact, and here is, we can skip on this, the lots of experiments have been made on hydrodynamics in graphene, particularly after the discovery that you can encapsulate a sheet of graphene within HBN and that essentially suppresses the form, it makes them very, very stiff, very energetic and so you don't have to worry so much about phonons anymore, okay? And I see that, okay, so basically the conditions for realizing the hydrodynamic regime are much easier to achieve for a variety of reasons in graphene than, let's say, in the traditional 2D semiconductors like gallium arsenide, for example. Okay, so let's go far. So now I want to show you, so finally, you know, with the help of encapsulated graphene, okay, it was possible to observe the Gurzi effect, okay? So here what they're measuring is the, you know, is the resistance through these point contacts, okay? So here are basically the graphene, the encapsulated graphene and the current is driven through these point contacts and they are able to measure the resistance of these contacts as a functional temperature and you can see very clearly now that the, you know, that the temperature is causing the resistance to decrease, okay? Rather than increase, strong electron-electron interactions are considered responsible for this behavior but there is something even more exciting technologically speaking also conceptually about this and that is that if you look carefully at this resistance you see that it drops below the value expected for purely ballistic conduction through the point contact. Okay, now that's really surprising because you could imagine that nothing can beat a purely ballistic flight from one point to another, right? The straight line is the shortest distance between two points, right? So you would say, okay, so ballistic is the absolute limit, but you can see that here you are getting because of strong electron-electron interactions somehow those interactions are opening up in some way another channel of conduction which facilitates the transport of electron and leads to this drop of resistance. And in fact here, by the way, this is a very nice review. There is a very nice review article here by Polini and Andre Geim in Physics Day which I recommend to you, you know, for an overview of this field. But let me give you a qualitative feeling for what's going on. You know, how is it possible that the electron-electron interactions de-conductance through a point contact, okay? So what's happening is that, you see, if you had no interactions, the flow would be, you know, ballistic, so it means essentially a straight line. So you see, only the electrons coming, you know, within the opening of the point contact would go through and the others, these two guys and these two guys, for example, they would just hit a barrier and be reflected back, okay? Okay, sorry guys, okay. But now that you have electron-electron interactions, so you see that there is this possibility of transporting momentum in a direction perpendicular to the flow. So the flow is no longer uniform, okay? So that's the key, okay? So it's possible for some electrons that normally would have not contributed to the conduction to be sort of, to start contributing to the amount of, you know, momentum that flows through the opening, okay? And this is a very, very qualitative thing, but it has been made very precise mathematically by people like Levitov and Falcovich. And basically they have presented a very nice, what they call an anti-Mapiason rule, you know, because normally when you have two scattering mechanisms, let's say the walls and the interactions, the resistances add in series, but in this case the conductances add in series, okay? It's the opposite of the conventional Matiason's rule. Okay, so this is really a very nice and striking result in electronic hydrodynamics. How much time do I have here? Okay, so I think I better move because, okay, so there has been here another rather striking, you know, smoking gun you could call it of hydrodynamics is this so-called negative non-local resistance. So schematically what you do, you have a device like this and you drive a current in one part of the device and you measure the potential drop in a different part of the device, okay? So these experiments can be done, okay? And now if you did a normal calculation without viscosity and so on, you would discover that because the flow looks like this. So the red regions are high electric potential, the white regions are low electric potential. So you find the positive potential difference here in response to the current that you have injected. But if you solve the equations of hydrodynamics with viscosity and everything, and this has been done, then you get these whirlpools or vortices, right? So the flow you see is high in non-homogeneous and the flow, and then all of a sudden you realize that the potential difference between these two contacts is no longer positive but is negative. So there is a switching in design of the potential difference, okay? And that is, again, of course it's not as simple as I'm making it because there are other mechanisms that could cause this even without relying on hydrodynamics, but definitely hydrodynamics is a possible mechanism for this. And so there is an article by Marco Polini and collaborators and so on which is entitled Whirlpools or No Whirlpools, okay? Which addresses a little bit these questions. I refer you to that for later. Okay, so let's skip this one. I just wanted to mention very briefly because this work on which I have been personally involved very closely, that there are some really striking manifestations of the hydrodynamic region in thermoelectric transport, okay? So in particular there is this thing called the Viedemann-France Law which says that the ratio of the thermal conductivity, k, to the electric conductivity, sigma, okay, scaled by T, this is called the Lorenz ratio, is in most cases, you know, very close to a universal value which I call L0 which is given by this number 2.44 times 10 times 8, okay, in SI units, okay? What is the significance of this proportionality between thermal and electric conductivity? Well, the idea is that the same carriers are responsible for both, in this case, electrons and the scattering mechanism, the relevant scattering mechanism affects the two conduction processes, thermal and electrical, in the same way, okay? And that's why essentially all the details of the scattering mechanism disappear and you get almost universal number, okay? It's not really universal, I must say. There are lots of exceptions, but they're usually of order 1, okay? So if you put, you know, you can tweak the result by factor 2 easily, but not by much more than that, okay? But now, and let me remind you, and this is very important, that when you measure the thermal conductivity, it's very important that you don't allow any electrical current to flow through this. So you apply a gradient of temperature, you make sure that no electric current flows and then you measure the heat current, okay? So the electric current is zero in the measurement of thermal conductivity. Now, why is this important? Because now here you can see what happens in graphene. First, let's consider the thermal liquid, the normal thermal liquid regime. So you see you have a Fermi surface up here, okay, in the conduction band. Now, in electrical conduction, what happens is that you have these electrons and they all flow together in the same direction. So assuming the system is very clean, so there are no impurities and no phonons, so you see that in this situation, electron-electron interactions don't do very much because they conserve momentum, okay? So there is no way electron-electron interaction can change the momentum. So you get a very large conductivity, electrical conductivity. On the other hand, if you look at the thermal conductivity, you see that is going to be much lower because you see you cannot, so you might think you can do the same thing to transport heat, but you cannot because the current, the electric current has to be zero, okay? So you are forced to do something more complicated like electrons above the Fermi level go to the right and electrons below the Fermi level go to the left. The result is that some internal friction between electrons is turned on and that creates a finite thermal resistivity, okay? So under these conditions, the electric conductivity is larger than the thermal and you get L less than L zero. So it can be significantly less than L zero because you see the electron-electron interaction treat electrical conduction different from thermal conduction. That's the key idea. On the other hand, this Fermi liquid, but if you go to the other regime of Dirac fluid, then the situation is exactly the opposite. It's reversed, okay? Because now you have the carriers are electrons in the conduction bands and holes in the valence band and now you see that they can, in thermal transport, they will travel in the same direction. So you see that this car has no current whatsoever, no electric current because they have opposite charge but it carries an energy current. So you have high thermal conductivity at the same time for the electric current, electron and hole will travel in opposite direction and that will turn on the frictional effect. So in this case the electric current is low, the electric conductivity is low and the thermal conductivity is high. So this results in L greater than L zero. So it's very clear and this has been experimentally verified. So you see you get ratios of L over L zero which can be as large as 20. There is no way this can be explained by little adjustments in the band structure and this and that. So it's a very striking experiment but this happens only in the Dirac liquid regime which is very close to charge neutrality point. If you move away from the charge neutrality point you see it drops below one consistent with what we found for the Fermi liquid. Okay, so I think I will now skip all this because I promised to say something about high energy physics. Now I definitely leave my comfort zone here. Okay, so I am a condensed matter physicist and I don't know a whole lot about this but this developments definitely catch the attention of condensed matter physicists because the main development is this so-called principle of holographic duality which basically states that there is a correspondence between a strongly coupled quantum field theory in the usual four dimensional space time. It must be a field theory at finite temperature. The typical example is this Young Mills theory which QCD is based. And the idea of the holographic duality is that this strongly coupling couple theory that lives on a four dimensional space time is connected through a renormalization group transformation a renormalization group flow to a gravitational a weakly coupled classical gravitational problem in general relativity. But you see the important point is that so the initial field theory lives in four dimensions but the scaling, the flow variable of the renormalization group introduces an extra dimension so that the space time that you generate under this renormalization group has one more dimension of five in this case. And so I'm sure that all of you, not all of you but many of you and even myself, you know, I knew that the black holes have a very nice thermodynamic interpretations with the entropy corresponding to the area of the horizon and so on. Now in a certain sense this duality extends that connection between black holes and thermodynamics it extends it to hydrodynamics. Remember that hydrodynamics is a kind of generalized thermodynamics so that's essentially what's going on here and let me be a little more precise. So the canonical example as I said is this Young Mills theory finite temperature but also it's very important so this is the structure, the action for these theories so these are generalizations of the electromagnetic field tensor but with non-Abelian fields but the important point is that in order to make this correspondence work you have to go to a limit. It is the limit of the n corresponding colors of the theory so n equal 3 in quantum chromodynamics but you formally must assume that n is very large and also this coupling constant G times n must be also very large. If you do that you discover this strongly coupled you see G squared n tends to infinity this strongly coupled field theory is equivalent to this relativistic space time in five dimensions so if you count you see that the four standard dimensions time x, y and z and then there is an extra dimension that corresponds to the flow the renormalization group flow that you have introduced and most important this space time which is called an anti-deceptive space time because it's a solution of Einstein's equation with a negative or attractive cosmological constant it has a black hole in it you see there is a singularity at r equal r0 at r equal r0 this thing this has a black hole in it and the horizon this black hole is crucial because it's the thing that encodes the temperature the finite temperature of the field of the initial field in fact this is the so called Hawking temperature the temperature that is responsible for the evaporation of the black hole and yeah it's given by this expression so here you have this correspondence why is this interesting? I think I should probably move a little fast because now you can calculate the viscosity of this field theory okay so what is the viscosity of a field theory first of all well it's just exactly formally the same thing that we introduced earlier for condensed matter so it's the correlator of the stress tensor which is also perfectly defined in field theory so it's exactly the same formula that we had before okay so this can be calculated by observing that this viscosity under the duality transformation corresponds to the cross section for gravitons at the event horizon of this five dimensional space time sounds really amazing to me so it's not so surprising if you accept this proposition which of course is highly nontrivial I don't know how to prove it to you but if you accept this proposition I think it is quite intuitive that this cross section which is denoted here by sigma abs of zero would be simply the area of the horizon that I find very intuitive right because that's the idea of a black hole that absorbs the thing that hit upon it so the cross section is a the area of the horizon but the A is also the entropy the Hawking entropy you see here it is the Hawking entropy the well-known result Beckenstein Hawking is proportional to the area of the horizon so if you take now the ratio of viscosity to entropy I should call it entropy density to be more accurate you get that the area of the horizon cancels out and a beautiful universal number emerges which is that 6.08 times 10 minus it's h bar over Kb divided by 1 over 4 pi and it is a very very small number just look at the 10 minus 13 it's a very low viscosity in fact it's so small that these guys Sonne, Copwoon and others who really did this beautiful piece of work they suggested that this must be an absolute lowest bound on the ratio of eta over s for any system and here they show a number of systems helium, nitrogen, water which is all well above that viscosity bound in a certain sense I don't think I have very much time to go into this but this idea of the minimum value of eta over s is very similar to an old idea of condensed matter physics that there should be a minimum metallic conductivity based on arguments of uncertainty principle and so on and that's how they justify it in their paper why they say this should be a lower bound I must warn you though that the concept of minimum metallic conductivity in condensed matter physics should not hold after the discovery of the scaling theory of localization so take this with a grain of salt because there might be violations so anyway I've almost finished now since I promised to say something about quark I want to say that so there is a lab in Brookhaven it's called the relativistic heavy ion collider in which they make heavy ions like gold atoms collided incredibly high energy electron balls and so on and they generate some fireballs which have been compared they're called sometimes small banks they're similar to the situation in the big bang just before the formation of the hadrons of course there are some little differences in detail like at the time of the big bang the fireball was the whole universe not just the lab in Brookhaven but apart from these minor differences you can really take that as a good modeling of the situation and then they have ways to measure the viscosity and the entropy of this by looking at the propagation of the fireball I'm not really showing you this now but they find that the ratio at over S is not far at all from the theoretical minimum it's just a little bit above it which is good but not far at all now surprisingly that is also good it's also true for electrons in graphene in the quantum Dirac fluid regime so this is the right so it's really this green line which is almost invisible which is the conjectured lower bound and you see as you go into well into the hydrodynamic regime here you see graphene comes really very close to that so that's why graphene has been called an almost perfect fluid in fact one of the authors of this book was in Trieste until not too long ago from Marcus Müller anyway also another situation in which you find that you can extremely close to the limit is in these experiments on fermionic cold atoms these are lithium-6 atom fermionic very low temperature so again you produce these clouds and what makes them interesting is that you play a trick called the Feshbach resonance to make them interact very strongly in fact so strongly that they are close with so-called unitary limit so what is the unitary limit is when the phase shift of scattering theory in S wave let's say reaches the value pi over 2 that's the maximum scattering cross section allowed by quantum mechanics it's called the unitary limit so when you go to that particular limit you can show that eta over s becomes incredibly close as this figure shows to the bound so this seems to be something interesting here but I don't know there are probably some exceptions known to this but I am under the impression that they are rather artificial they do not emerge under ordinary conditions ok so I can finish now so one should ask at the end of all this why is hydrodynamics interesting so I would suggest that first of all I hope I've shown you that it's really a universal beautiful universal theory it's strongly interacting degrees of freedom and it creates you see an incredible bridge between everything condensed matter physics quantum field theory strain theory generativity everything we know practically on a more down to earth point of view well it offers a new paradigm of electronic transport which may have some interesting applications to technology think for example of that super ballistic conduction where you can beat the ballistic limit by exploiting interaction and here I would like to add as a final point something I've been thinking about for a long time but it's very speculative that possibly some kind of hydrodynamics might also be the ultimate resolution of the problems the foundation of quantum theory because you probably know that the Schrodinger equation is exactly equivalent without approximation to a set of hydrodynamic equations like the continuity equation and there is a kind of Euler equation with a Bohmian stress tensor the big problems of course is that you don't know how to write that stress tensor as a functional of the current density so you don't know how to close the theory but you know that that functional exists that's what current density functional theory does for you it proves that that function exists so my feeling and already Bohm tried long ago to explain quantum mechanics in classical terms using the so-called pilot-wave approach which is form of hydrodynamics very non-local so it doesn't conflict with Bell's theorem and so on so it's quite possible that ultimately the measurement problem and so on may be approached by including the viscosity in those equations so that's my speculation so the viscosity might be the key to decoherence in quantum systems okay so on this note thank you very much for your attention yeah so in margin fields is experimental theoretical understanding of spin liquids of spin liquids and although experimental verification is a bit far from being agreed by all group do you know anything about hydrodynamics of spin liquids and can hydrodynamics lead to some striking predictions which will lead to verification of the state of matter thank you very much yeah yeah you are absolutely right there is in fact a lot of activity going on in spin hydrodynamics okay I think here I think Duin in Utrecht is one of those who work on this thing also Maikawa has predicted some really interesting effects related to spin orbit interaction in the spin liquid transfer so I am sure that lots of exciting things will come out of this now here I stayed completely away from the spin liquid because at this moment hydrodynamics has been spinless but that's definitely a very interesting question to go and you are right there might be some interesting transport signatures of the spin liquid which would be very exciting to discover that's all in the future for me at least I have a question and curiosity the question is what are the prospects or situations about turbulent flow yeah, turbulent flow that idea has been has been considered in fact there are several proposals to no, no, no there are so the way turbulence would be measured is through the magnetic field that this turbulent current so you need very very sensitive magnetometry and apparently now the NV centers are providing that kind of so there is a lot really of several people are trying so the main question is what are really the issues so what is that one would like to learn from this hydrodynamic turbulence so I would maybe the effect of quantum mechanics so turbulence as you know is not completely classical level and now the question is how would the quantum, how would each bar show up you know, maybe it would give us I say a minimum microscopic size of the vortices that could be an interesting way so that you cannot the chain of transfer of energy from large to small would terminate at some point see the idea has been certainly around for a while and then I have a curiosity so in this experiment how do they check that they are in the hydrodynamic regime yeah I don't think they check it so they assume it they calculate the evolution of the fireball using hydrodynamic equations it works and then they analyze the data and they extract the data over s the whole thing seems to be quite consistent viscous the last picture you showed reminds people like myself as the flocks of birds looks like a very highly viscous system what consequences would that have on you know I'm a former hunter so I'm interested to know what happens in a case like that I would say that this is an answer that should be asked of Giorgio Parisi he probably has a question that he could address I think there are lots of differences between flocks of birds and and generally hydrodynamic because you know basically the way that those living beings communicate to each other must be profoundly different right from the way you know particles interact mindlessly but it is known to be very intense so in that sense there is an analogy the interaction is very strong they want to stay together I don't think they have a Hamiltonian theory for that so maybe one can make you understand there are analogies so people for example if you if you if you start with a chamber and you put all the gas the molecules in a corner and you let them go then they spread out and they feel uniform now if you do the same with people they will also spread out for very different reasons right so the you know the molecules spread out for purely statistical reasons even if there is no interaction between them simply because there are many more ways to be spread out and people would spread out because they don't like they want to do social distancing they don't like to be too close so I mean the psychology is the consciousness so out of the Joker this would be relevant to active matter I imagine right and journalist particles that play like flocks of birds would have motions and things that this kind of thing would enter very much look I think this is a fascinating field yeah I agree so yes one question thank you for the presentation in simple words I would like to know what is a quantum liquid quantum liquid oh a quantum liquid I get it well so a quantum liquid is like a classical liquid is a collection of particles but with this difference that typically the temperature is very low and the wavelength so the particle in quantum mechanics you can associate a wavelength right so the particle which h bar over square root of m k t so the debris wavelength in a quantum liquid is larger than the distance between the atoms so it covers many many within a debris wavelength you find lots and lots of atoms so that means that quantum mechanical effects begin to play role like for example if the particles are identical they can exchange that's the reason why you have this exclusion principle in the case of fermions which is crucial to fermi liquid and yeah so generally I would say that that's the criteria so typically happens at very low temperatures so the most materials become solids before that happens so there are not a whole lot of quantum liquids in there but there are many notable examples helium tree for sure electrons in metals are also very important and then sometimes even when the temperature is very high so electrons in white dwarfs you know electrons in white so the temperature in white dwarf you can imagine is pretty damn high but still those are quantum liquids they generate because if you calculate so the density is so high that the distances between the particles are smaller than in the debris wavelength okay so I think there are refreshments outside except for the diploma except for me and the speaker yeah I understand I'll join you later okay alright I think I should come here something I need do you think you need a computer for the photo session okay perfect