 We're going to talk about hydrodynamic description of transport in strongly correlated electron systems. So very much looking forward to your talk. Yeah, thank you so much. So I want to thank the organizers for giving me a chance to be here. It's really enjoyable scientifically and personally. So I prepared for you four stories. Let's see why it doesn't click for stories. So I'll talk about resistivity in strongly correlated two decks, thermal liquid anomaly in whole bar devices, same physics, but intervina devices with some aerodynamic paradoxes. And time permitting, I also would like to say a few words about most recent result on drug anomaly as observed in electronic double layers and graphene systems and the possible resolution of very puzzling observing features through near field heat transfer. And I also prepared a little homework for you to enjoy, which I think you may like. All right, so part one. So we usually ascribe RS as a quantifying parameter for the strengths of electronic interactions, which is just the ratio of coolant to kinetic energy into dimensions. Its scale is inversely proportional to the square root of particle density. So in dilute systems, RS can be large. We know from numerics that somewhere around RS 45 electrons are expected to crystallize and form big new crystal lattice. And yet it leaves us with a very broad range of RS parameters, larger than one and smaller than this critical value when system is in liquid state and yet very strongly correlated. So in experiments, this RS can be anywhere between 10 to 38. Some other prominent features that we will explore in our computations is that the systems have very small Fermi energies. Electron phonons scattering is typically really weak. And the systems are subjected to a smooth long range disorder potential. And I will explore this quite extensively in this talk. So sort of an experimental inspiration comes from this review article by Boris Pivak, Sergey Kravchenko, Steve Kilison, and Juan Gau, who observed and reviewed a large volume of experiments. And so I'll focus just on the resistances. They found that in very different systems, ranging from silicon germanium to silicon MOSFETs and gallium arsenide quantum wells, resistivity remarkably behaves very similarly. So what unites all of the systems is that they are all in very large RS values, again, somewhere from 10 to 35. So these three plots I took from the review, this one I just took from the Publis experiment and twisted bilayer graphene. Amusingly, it shows a very similar trend. So first part of the story is trying to perhaps understand it. One feature that I'd like to notice here is that resistance goes down with an increase of temperature. And if you think about this, this is sort of a little unusual. We typically think that if we increase the temperature, we increase the probability for electron to be scattered by something, by phonon, by impurity, or any other available excitation. So the fact that resistance drops requires an explanation. And historically speaking, we do have at least one candidate scenario how this could happen. And so I prepared one slide as a reminder and then we'll sort of elaborate on top of it. So back in the early days, Radji Gurji thought about the electronic hydrodynamic behavior. So he was interested to consider a regime when electron-electron means three passes, the shortest land scale and the problem. Shorter than any momentum or energy relax in mean three pass due to disorder, and phonons, and so on. So he, in this case, there are very frequent efficient electron-electron collisions. Electron fluid thermalizes at the short and fast short land scales times, fast time scales. And so he was interested in then exploring the transport of larger scale. So he started with kinetic equation and considered a herodynamic flow of this fluid. And from kinetic theory, he derived an effective equation of motion for herodynamic velocity. And perhaps not surprisingly, he got just the Newton's law that tells us that the force that exerted in the liquid that we're trying to push is balanced by the viscous stresses and also bulk friction, as he was also still considering bulk disorder. And so then complemented by the equation for the current. So this Newton's law gives us a new land scale and the problem. If we compare, so new here is the kinematic viscosity, which is shear viscosity divided by the mass density. And so if we compare these two terms and recall that Laplace and his units inverse length squared, so then the comparison of these two terms give us a land scale, which can be expressed as a geometrical mean between the electron-electron mean three path and electron relaxing mean three pass. So LEE sits in viscosity. So then if we would consider the flow in the restricted geometry, situation is different whether the, let's say in 2D case, whether the width of the channel is bigger than this Guruji lens or smaller than it. So if the width of the channel is bigger than the Guruji lens, viscous stresses are not important, we are dominated by the bulk friction. So then you from here is just proportional to relaxation time. And we get just to do the conductivity from here. But in the opposite case, we can ignore the bulk friction. Momentum relaxation occurs at the boundaries of the sample. And so we need to solve the Navier-Stokes equation, which would give us parabolic profile of the flow. We recalculate the current and get that effective conductivity scales quadratically with the channel and inversely proportional to the viscosity of the fluid. In other words, resistance would be proportional to viscosity. And in Fermi liquids, viscosity is inversely proportional to square of temperature. So once we increase temperature, viscosity drops, and that should lead to a drop in resistance. So that is the manifestation of the Guruji effect. All right. So I believe I'm not 100% confident, but it appears to me that the first experiment that contained words hydrodynamic and electron flow together was from Lorentz, Malin, Kapp and Nate. So it took a really long time since the proposals by Guruji to see some of these effects in materials. So first thing that I'd like to do is to sort of generalize this idea of the Guruji effect to the case of the system subjected to the long range disorder potential. And I will follow the footsteps of Anton, Steven, Boris. And I will guide you through a series of arguments that are very simple. Usually when we calculate transport, we have two options. We can calculate the linear response and establish proportional analytic efficiency between the currents and driving forces, or we can compute dissipative power and then equate it to the joule heat and extract resistance this way. In this talk, I will use both methods interchangeably. And sometimes it's easier to do it one way or the other. All right. So imagine, so for simplicity of it, I will consider a flowing 1D channel, again, with this long range disorder potential. Generalization to 2D requires a little more work, but conceptual is exactly the same. I will tell you what the changes would be. So suppose that we have electrons, a very short mean 3-pass, the swiggles of the disorder potential much bigger than the electron-electron means 3-pass. So let's see what happens. Assuming that we are in a dynamic regime, so we need to solve continuity equation, Navier-Stokes equation, an equation for the heat flux. So the continuity equation will tell us that the product of particle density and dynamic velocity is constant. Each of these quantities individually is coordinate dependent. We are dealing with an homogeneous system, but the continuity fixes their product. Now, the stress tensor in 1D contains only bulk component. There is no shear in 1D. So it means that the stress tensor is just a bulk viscosity divergence of the current. So in this case, it means that stress tensor is bulk viscosity current and derivative of the inverse density. And dissipative power, copy-pasted from Landau-Liefschitz volume 6, is just the product of the stress and the components of the velocity. It will give us yet another derivative from the velocity field. So we'll get current density squared, viscosity, and especially averaged derivative of the inverse density squared. So this would immediately, as you can see, give you something which is Guruji-like. So we'll have a result for resistance if I look at the left-hand side of this formula. So J squared would cancel, and the viscous part of resistance is like in the Guruji effect. So then I need to repeat this argument also for the heat fluxes. There is additional contribution to the dissipative power. And then a similar line of arguments give you another term in this resistant formula that would contain the entropy density per particle squared and also average toward the system and scales inversely proportional to the thermal temperature. So that is the formula that describes resistance. In two dimensions, you would have a factor of 2 in the denominator. And then you need to replace bulk viscosity by the sum of the shear and the bulk. So I'd like to make a couple of comments about this expression. So first of all, from the standpoint of view of electron interactions, this formula is non-perturbative. So all of the complications where interactions are set in this kinetic coefficients that we don't formally don't know. I mean, we don't have really reliable tools to calculate them. But that is the power of the aerodynamics that assuming that some quantities are given to us, we can express something else in terms of them. So still in order to extract some predictive statements, I'd like to take a pragmatic approach and sort of rely on experiments and people in the past ask questions whether Fermi-liquid predictions apply in the regime of large RS. And let's say from the work of Jim Eisenstein and others, they in particular from tunneling measured quantum lifetimes of particles in this strongly correlated fluids. And they say that at least in terms of the temperature dependencies, this lifetimes are consistent with the statements of the Fermi-liquid predictions. So let's take this and try to make estimates based on that formula that they just show it to you. So in Fermi-liquids, we know that thermal conductivity is the product of specific heat and mean three paths. So mean three paths by the usual phase space argument is one over T square. Specific heat is linear in temperature. So thermal conductivity should scale inversely proportional to T. And again, in 2D at the coupling, we can do better. There are some logs and et cetera, but I'm doing just an estimate without any extra logarithms. So the viscosity by similar arguments is inversely proportional to the square of temperature. So then we need also to do disorder average, because it's model specific. So I will assume that there is a doping layer somewhere in the distance d away from the plane of the 2-deck with uncorrelated coolant impurity. So that would be one of the model that I hope is experimentally relevant. So it will give us the power of the noise from this uncorrelated dopants. And that would enable us to do disorder averaging. So then, so if I go back, so as we just discussed, so this linearity kappa is one over T and entropy is linearity. So this term would be T to the four and viscosity is one over T square. But these terms would also have different density dependence upon average. So this is what you would get. Applicability of the aerodynamic picture in our case assumes that mean three passes shorter than d, meaning this distance to the gate, it plays the role of the correlation radius of the disorder potential, give us an upper bound threshold in temperature for the applicability of aerodynamic picture. It should be greater than the e-thermie divided by the square root of k-thermie d and experiments for typical densities, this parameter is let's say somewhere in range from being four to 16 or so on. Yes. I will come back to that. So infinite, so here I'm just considering the bulk transfer, so there are no, and I don't consider point like disorder. So it's all smooth flow on the profile of the large scale and the homogeneities. And so the only condition here I need is just to ensure that the mean three passes shorter than the length scale of the disorder potential. So roughly speaking, is that scale? Yeah, yeah. And I will elaborate on this a bit more later on. Yes. So basically what I'm trying to tell you here is that the competition of these terms gives this non-monotonic behavior. So first resistance goes down in this model case, not too dramatically, but it does go down. It's dominated by the viscous contribution, then it goes up, which is dominated by the thermal conductivity. So I hope to give sort of plausible argument that this non-monotonic trend is actually a property of the transport property of electronic flow. Yeah, again, we need to ensure that the electron, so they repeat the question for the audience. In Zoom, the question was, why do I need this condition on the mean three pass or on temperature? So this condition on temperature comes from the condition that electron, electron mean three pass should be shorter than the length scale of an often homogeneity. So I need to be in that regime of quick equilibration. Ah, sorry. That's right. That's what I'm saying that requires more work, but it goes, I will do 2D in few minutes, essentially. So we would split the flow in longitudinal transfer, so it's more work, but it comes, yes, remains with a modification of the factor of two and sheer viscosity that needs to be added. Yeah, of course. Yes. Yes. That's right. It's a minimum. Ah, yes, excellent. Thank you. Thank you. So in this plot, I normalize temperature to this threshold of applicability of hydropicture. So this is T1. So it starts from one. So this T1 is the e-firme or k-firme D. Now the minimum occurs at other scale, which is e-firme cubic root of k-firme D. So it is within where it's supposed to be furthermore. I can roughly see by how much resistant drops, assuming that I'm starting from somewhere from that peak and I compare to experiments and it's not that far for the density. It's assuming that the model applies, okay? So it's not infinitely far away, hopefully from reality. All right, so now graphene devices. So sort of, again, every story would start with experimental motivation and this would be like a logical continuation of the previous story. So these are experiments from the group of Philip Kim measuring thermal conductivity and the Lorentz ratio in the monolayer graphene close to the Dirac point. And so they observed, so the interest was to explore this Dirac fluid in graphene, its properties. And they discovered that from the measurements of the thermal conductivity that in the proper range of temperatures and this range of temperatures from this plot in the phase diagram you see starts from somewhere at 50 Kelvin to somewhere like 100 Kelvin, they saw an enormously high effective thermal conductivity which also translates that the Lorentz ratio normalized by the units of the Wiedemann-France is also giant, it's like a factor of 20. And so in the paper, in the abstract, they say that electron-hole plasma in graphene near charged neutrality point forms a strongly correlated coupled fluid and this effect is attributed to the decoupling of the charge and heat currents in the electron hydrodynamics. So that's something for me to comment and explain and see how this appears in the graphene devices and to tell you what this decoupling is and why it happens. So they had also a follow-up study on the Zeebeck effect. Again, please notice that resistance again has a similar trend that goes down and goes up. So hopefully it's sort of within the same picture. And they also comment that semi-classical mode formula doesn't work neither quantitatively nor qualitatively. And so our temp was made to explain these results also based on the picture of electronic hydrodynamics. So let's start with something very simple. Again, I will elaborate on this calculation more and more. So we need to generalize now things for graphene because electrons there are not Galilean invariant and so system and it brings a substantial modifications is that we need to describe the transport in terms of the pristine kinetic coefficients. And one of them are very important is so-called intrinsic conductivity. So York calculated it for us along with some other people early on it is of the order of conductance quantum times some logs if we do weak coupling. So the current electrical current in graphene is given by the convective part to the part that we already saw in the Guruji case but then it also contains the part what I will call in a relative mode, meaning that at the stationary fluid which is driving by electric field intrinsic conductivity and in principle thermoelectric coefficient and then the entropy current which is related by one power of temperatures to the heat flux similarly convective part thermal conductivity and thermoelectric effect. So now this decoupling that was mentioned in the abstract of Philip's work is seen at the level of currents. Let's say we go to charge neutrality and the charge neutrality convective part is out of the picture for the charge transfer that's all everything is quenched by the relative mode and intrinsic conductivity yet in the heat flux convective part hydrodynamic flow dominates because it's governed by the entropy density not the particle density and effective thermal conductivity would be giant. So let's see how this works out. Suppose that we drive the system in X direction there is a whole bar width we need to solve our Navier-Stokes equation here use just a vector units for n particle density entropy density this is my X density and capital X are conjugated forces electric field temperature gradients solving Navier-Stokes let's say simplicity for the no sleep boundary conditions again this parabolic profile we recalculate currents including this relative mode and this is what we would get. So resistance would have as a function so I would like to focus only at the regime near charge neutrality when the ratio of particle to entropy density is small in this regime viscosity is also known thanks to York he calculated again at weak coupling module logs and et cetera it's T square intrinsic thermoelectric coefficient is linear and temperature and linear and density. So resistance has a Lorentzian shape the widths of this Lorentzian is governed by the fluid velocity primarily in terms of temperature dependence as we discussed Sigma shows very little temperature dependence so Z back K efficient in our thermal conductivity it has this form if you tailor this formula to charge neutrality this term is gone so everything vanishes because it scales with density including gamma which is linear in density but this term survives it looks like a Guruji formula but in this case just for the thermal conductance rather than for electrical conductance it scales as a square of the widths of the channel and it's grossly proportional to the fluid velocity. So now if you work out the Lorentz ratio you would get roughly speaking the square of the Lorentzian again was the same width that I told you and for reasonable numbers and reasonable densities you can get an enhancement by the factor of 25 or so. All right, so now let's elaborate in this picture so graphene is of course not that perfect and I neglected everything in the first little calculation from STM it's known that it has this picture of electrical puddles the typical length scale of this electrical puddles is at the scale of let's say a hundred nanometers the scale of the potential fluctuations is about five milliliter volts. So we need to do better than that we need to develop Kiwis and Spivak like description for elect inhomogeneous system but now generalized for electrons in graphene that are non-Galene invariance. So this is what we did. So it's not pleasant calculation but the end result is sort of I hope intuitively clear this disorder does several things it renormalizes effective coefficient it leads to an effective friction coefficient which is governed by the local density and entropy density fluctuations and described by entire matrix of thermoelectric coefficients. So unlike in Guruji case the effective Guruji lens here would be very complex in a complex fashion depends both on temperature and density. And then there is another peculiar effect that is in this picture disorder actually enhances conductivity. So there are aerodynamic corrections to intrinsic conductivity which are governed by the fluid viscosity and let me try to give you an argument and hopefully in part it will answer the linear question as well. So when we consider inhomogeneous system which is charge neutral I'm talking about charge neutral globally in average locally we always have charges. So when I drive the system I have a local force density which is proportional to the local density variations. Then there are a few things happen so the vectors into D I can split in legitudinal transversal directions and they need to solve continuity equations into D and Navier-Stokes equations together. So this would give you both longitudinal components of the aerodynamic flow and transversal. Now the potential part of the flow which is longitudinal in the force balance conditions is balanced by the pressure gradients that are developed in the fluid but the transversal flow the vertical component of the flow is only balanced by the stress standard so viscous stresses. And so this leads to a big enhancement so the flow in sort of other direction and then from the current average if you take this expression for the aerodynamic velocity you get roughly an enhancement of the conductivity that scales with a square of the density fluctuations and inversely proportional to the viscosity of the fluid. So we could do crossover, Guruji crossover and the bulk regime so we can recalculate this coefficients. So I should say that I was not the first one to work on this problem to explain Phillips experiments. There are many papers before me and while we do agree on maybe big things there are some still differences in all of the theoretical approaches and maybe it will be nice to have a chance someday to get all together and discuss it with everybody. Yes. In the picture, how would you talk to people and actually look at that? This I don't know, I haven't looked at that but I would guess yes. Something to be looked at, I don't know. All right, so theory to experiment very quickly. So this is from Philip, this is from us and again there are like honest goodness estimates for densities and so on. So we've tried to match what we get to what has been measured. Now a couple of additional comments. Remember that Guruji formula in scaling with the square of the opening there was another experiment from a Manchester group creating point contacts in graphene devices, this whole bar devices. And they have measured what is called the viscous conductance by changing this opening of this viscous point contact and it seems to scale quadratically suggesting Poiseuille's profile of the flow. Again resistivity is a function of temperature describes as deepening and upturned temperature dependence and the extracted viscosity seems to fell really nicely to Fermi liquid. I should say that this experiment is done at high density. Okay, so this is not the near neutrality experiment. So my colleague at UW-Merison made a similar things but he had a sort of on-demand designed obstacles for electronic flow, this PN junctions also saw an enhancement of conductivity, effective conductivity in the whole bar device that scales with the opening of the channel. And just completely, I'd like to say there are many, many experiments these days from imaging and so on trying to see all of these features. This is sort of not an overview talk but I wanted to give a credit to everyone who are trying to do imaging. I think it's quite powerful to learn quite a lot about the systems. All right, so let me see on my time. Now I'd like to switch to the next part about the, so I'll discuss now the same effect. I will change only a geometry and quite remarkably there are some interesting puzzles appearing. So when I was in USPEN was a program on electronic aerodynamics and I've learned about this work from Grisha Falcovich and Andrey Shitov and Mikhail Shavit. So Grisha gave a colloquium about this paper which is called the freely flowing currents and electric field expansion in viscous electronics. So let me try to explain in my understanding what he did and what sort of paradoxical about the carbina device. So keep in mind the picture that we just had for whole bar geometry calculation, right? So what we did was said, all right, so first in order to calculate transport coefficient we need to solve now your stocks with some boundary conditions. Once we have the flow profile, we can recalculate currents and then we can reconstruct the kinetic coefficients. Now in the whole, in the carbina geometry when we inject, let's say current in the inner electrode and collected in the outer electrode, we have from the continuity equation of the current in the bulk of the flow continuity tells us that the ceramic velocity behaves as one over R, okay? It's fixed at just fixed at the step number one. And it looks that we don't even need Navier stocks in order to define the profile of the flow. Then you may ask yourself, all right, what then follows from the Navier stocks picture? Well, if you then look at the Navier stocks equation which has a Laplacian and a driving force, then let's take a ceramic velocity which is total current divided by two pi R E M, right? One over R and stick it in the radial component of Navier stocks in here. And so this must be equal to electrical field that we apply through a voltage between the inner and outer electrode. And so it's a simple calculation but you will be amused to discover that radial part of the Laplacian minus one over R squared applied to U over R which is one over R is zero, okay? So then it means that E is zero. And so this should tell you what he meant in the title freely flowing currents means that we have a current without electric field in the bulk of the device. Now it means that if field is constant which is gradient of the potential, potential must be constant. Then it leads to a problem that we applied voltage from inner to outer electrode. And where do I put this potential and how do I match this thing? And so there is no escape as to realize that there will be a discontinuous drop of this potential at the inner and outer electrode. So the potential in the pure hydrodynamic limit would be indeed constant with the jumps at the electrodes. And this is illustrated in this plot that he said. Now what is the paradoxical situation here? The paradoxical situation is that it seems that viscous force which is the left hand side of the Navier stocks equation does not affect the flow. Yet we have a dissipative coefficient. So energy is dissipated. You can check for yourself that the stress tensor is non-zero. It contains RRN55 components. And again, we can calculate the dissipative component of the flow, okay? So, but again, here you just need to remember that this Laplacian comes from the divergence of the stress tensor and sort of in the words of Falkovich that even though we have no net force, divergence of the stress tensor acting on any element of the fluid. There are non-zero forces acting on the fluid elements displacing somehow this fluid elements and the creating dissipation in the flow. I don't have a very intuitive picture but it does look like when you try to flow in a very viscous environment, you sort of need to change your shape to move around. And so this is how this flow occurs. And I will try to give yet another point of view exploring sort of a classical analog with the problem of a syringe. When we try to push the liquid through a channel. So when I'll present results for thermal resistance, I will mention this just in a second. All right, so I generalize this for the, again, entire density profile. It is more complicated than that. Grisha missed something. So he worked only in a high density limit. Now, if we go to charge neutrality you need to again include the facts of intrinsic conductivity and so on. So generalize the scheme. So what is changed is that it's not the electric field that is expelled, it's incorrect in general, it's there. It's fourth density that is expelled which is a linear combination of electric field and temperature gradient. It leads to a quite peculiar features. It means that if you drive the system only electrically you necessarily develop temperature gradients in the flow, it's unavoidable. So we recalculated all of the matrix of this kinetic coefficients. The contributions to the dissipative power comes both from the stress tensor relative mode. We have an expression for the resistance and thermal resistance. So in the resistance, this part which is high density when n is large recovers Falkovich result when n is small relative mode dominate you get something else. And we'll also get a thermal resistance which is also quite complicated. As you can see this resistances are expressed through everything basically. So that's the result. But I would like to tailor this analysis to charge neutrality. So in charge neutrality, Falkovich picture is gone. So resistance is just simply intrinsic conductivity. P here is the aspect ratio of the carbene disk. So R2 divided by R1. So this would be here only intrinsic conductivity because this var sigma would be just back to sigma itself. So let me try to derive a thermal resistance also charge neutrality. So this term was density would be gone. So we will only have a contribution from the viscosity. So the idea is the following. So from thermodynamics, we know that the jumps of temperature are related to density in local change in electrochemical potential. And then the change in temperature. And then the only thing that we need to remember from the previous slide is that force density expulsion condition. If we work at charge neutrality N is zero. So this part is gone. It means that the temperature gradient is zero, which means that the temperature in the fluid must be constant. Yet I apply temperature difference between inner and outer electrode. And the situation is similar. It means there has to be a discontinuous jump of temperature. It is different from, you may think it's analogous to let's say a pizza resistance where also a temperature drop secure. But this case is quite different. The amount of the jump in this problem is controlled by the dissipation in the bulk of the flow while the amount of the jump in the pizza problem is defined by the property of the interface itself. All right. So then we see that the extra excessive pressure is related to the temperature jump. So let me abbreviate TL would be temperature on the fluid and then force balance condition is matching this pressure excess by the radial component of the viscous stresses acting on the fluid. And I need to write this equation twice in the inner electrode in the outer electrode. So then it means that I can calculate by how much temperature drops if I know what my temperatures are in each of the electrodes. And if I take the difference of these two equations I find the global temperature difference applied. It scales with the entropy current and the ratio of the two give me thermal resistance which is this part of the equation. All right. So Weitzman group led by Shahalilani actually they've done this measurement with this carbon nanotube technique. Again, this device has this Guruji like effect in the resistance. And so they characterized the carbonadisk first at low temperatures at 640 and so on kelvins. These devices are not perfect. They're still they're not in a perfectly herodynamic limit. So you don't expect to see this perfect sharp drops and so on. But basically what they've done is that from the high temperature data they subtracted the measurement of the low temperatures to see how much of the hydrodynamic contribution remains. And they see that once you change temperature to a higher temperatures where a hydrodynamic window should occur, you see the flattering of the measured electrochemical potential in the bulk of the flow. And so this was sort of an indication of this force expulsion condition. And okay, that's experiment away from the test. Yeah. So that's part of that. No, as far as I know, no. So, and again, it's at neutrality most interesting thing will be thermal power. All right, okay. So this can be generalized to magnet resistance. Things get a bit more complicated, but it can be worked out. So we've done this with York and looked at magnet resistance in high density. So you get the viscous magnet resistance. And I also recently looked at the charge neutrality regime and sort of result is very similar, but the technical origin is sort of quite different. It contains the synthetic conductivity, but the rest is similar. So it scales inversely proportional to viscosity and scales quadratically with magneto-field. So it's positive quadratic magnet resistance. And I calculated not only the resistances, but thermal and the entire matrix of thermal electric coefficients. Okay, so that's that. So let me see. Do you have like 10 more minutes? Five plus. Five plus. At least maybe not everything, but I'll say a few words about the electronic double layers. Very big fields with lots of very interesting things. So back in the day, again, the Manchester group measured drug resistance. So when you have a bi-layer separated by an insulator, you drive the current in one system and measure non-locally induced voltage in the other system. So they measure drug resistance in double layer electron graphene monolayers. And so this is what they saw. So first they saw, so first of all, because of the independent gate control, you have sort of four options. You can consider drug between electron, electron, whole, whole, electron, whole, whole, electron, depending on what you do. They found, first of all, that it's non-symmetric. And secondly, they found, I think the most astonishing observation is that when you go to the double charge neutrality, meaning that each layer is at charge neutrality, they found a finite drug resistance. So then if you are in the electron, electron, whole, whole part, it starts to decrease, it changes sign and then decreases and dies out at high densities. And if you are electron, whole or whole electron goes up and so also dies at the higher densities. So then they pick the value of the drug resistance at charge neutrality has on itself quite complex temperature dependency in monotonic. It rises as a function of temperature and then dies out at higher temperatures. All right, so the pure momentum transfer. So usually we think about the drug is by transferring momentum from one layer to the other layer, but the charge neutrality, this mechanism doesn't work because all of the transport goes through relative mode and you could not drive the aerodynamic fluids impossible. So P, momentum transfer mechanism supposed to give zero drug resistance. So then Leonid Levitov came up with energy drug. So I'll try to give a quick sketch of like combined picture that P drug actually survives, but it requires a bit more complicated ingredients. So we again need to recall that the system is non-homogeneous. And by the way, I need to say all crucially important nuance. If you read the paper, then somewhere deep at the section which says anomalous drug neutrality, they say that we emphasize that the strongest peaks at the dual neutrality point were absorbed in the devices with the widest puddles regions, namely with the largest density variations. So we felt that this picture and the model that we have with this long range disorder potential could be quite useful model to try to address this experiment. And so essentially this is what I did just to look at the two copies of these graphene devices. And because the small spacing between them, I worked in the model, when electrons in each layer see the same disorder potential, they become disorder correlated. So this becomes very important in trying to incorporate both momentum and energy drug picture. Yes. Why? Well, because they also had a separate data on the magneto drug. I'm not discussing magneto or magneto whole. It's sort of separate, quite interesting, but I'm just zero field argument. All right, so let me roughly speaking, tell you about this mechanism of P-drag and E-drag. So the situation is the following. We learned with you in a single layer case that when I drive the system even electrically, I necessarily develop temperature gradients. This temperature gradients would be transported to the other layer as well. So it will be induced temperature gradients in the other layer as well. So that would give in the levite of terminology, a vertical heat flux from one layer to the other layer. And so this would be an E-drag mechanism and P-drag mechanism because of the disorder correlations is a usual coolant mediated electron-electron scattering interlayer. And so they can be incorporated together. So it gets a little messy, but maybe I'll just flash some results. So you could derive an expression for P-drag component momentum mechanism of drug resistance. It has a very similar form as the conductivity enhancement near charge neutrality that I've showed you earlier. It contains this correlator of densities fluctuations in each of the layers and it scales inversely proportional to fluid viscosity. It contains this drug coefficient that needs to be calculated. It's a little work to do this, but it's doable. It actually goes a little bit beyond of the pure aerodynamic picture. You need to promote aerodynamics to include Langevin fluxes and to work with that, but okay. And then as I said, there will be thermal imbalance between these two sheets and you can calculate an interlayer near field effect. So interlayer thermal conductivity also due to coolant interactions can be calculated and can be expressed again in terms of intrinsic conductivity in this fluid. If we are interested only in the regime of charge neutrality, similar expression exists for the energy component of drug resistance, the key part it has an opposite sign. And so then when you promote this to a finite density results, the competition of these two mechanisms explain you the sign change and non-monotonic dependence of the drug resistance. All right, so I'd like to thank my collaborators. So it's my student who just recently arrived but had a very quick impact working on this drug problem. My postdoc Sonchilin, my collaborators, Anton Maxim, my experimental colleague at UW Medicine, Victor Brarch in York here. And we came to a most fun part. So I have a homework for you to enjoy. To those of you who don't know this, Navier's Tox equation is a fun problem to solve. It gives you some rewards. If you go to the Clay Institute for Mathematics and the click on the link of millennium problems, you will find the problem of the Navier's Tox equation. And they formulate say that although these equations were written down in 19th century, our understanding of them remains minimal. And the challenge is to make a substantial progress towards a mathematical theory which would unlock the secrets of the Navier's Tox equation. So I hope that I unlocked some secrets for you. It doesn't mean that you owe me a million bucks, but so to your amusement, I suggest you to read the official problem description. You can submit your solutions to archive and then the price will wait for you. All right, with that, I'll stop. Thank you so much. Okay, thanks for this nice talk. I have a time for question. Going back to your Corbino geometry, I was curious. So you have the expulsion of force. What happens if you have a superconducting system and in a Corbino geometry? Temperature? I would be afraid to want to guess. I have no idea. No, we haven't even thought about it. Yes, another one. There are other mechanisms which can give you d-row, d-t-negative at low temperature. At low temperature. Like, yeah. Yeah, condo could be at low temperature, but again, you need magnetic disorder for that. What about Fridtel oscillations? Well, Fridtel oscillations would be quantum interference corrections if I'm sort of, but I don't think it gives. So yeah, you would get correction to conductivity, which is linear in t-tower log of t into d in this other case. But again, it doesn't give you this whole thing, right? And again, these are low temperature effects. And we're talking about here, what's in graphene case, hundreds of kelvin. Thanks. Sounds like you are working in the laminar regime, right? Correct. So how do you define the Reynolds number in this picture? Well, at similar estimates with viscosity in typical lens scale, I think this is pretty small number, given that viscosity of graphene is pretty high as well. So I think that any of the nonlinearities that I neglect in your stocks, I think is fully kosher, I think. Okay. Alex, are there any restrictions regarding the outer and inner radios for the carbene disc geometry? For instance, then you start to narrow the width of the carbene disc. You can cross to the diffusive region, for instance. Correct. So basically, again, because of imperfections of these devices, you have R2 minus R1. And again, you need to compare that to the Gucci lens. So then of course, if you go too far, then you could be, again, going to the sort of a bomb. Well, again, you still have these restrictions of the continuity. So some effects would still survive. Experimentally, I know the aspect ratio. So in Schachal's experiment, in Philippe's scheme experiment, in Corrigine's experiments, that I know aspect ratio is either three or five. And the sizes are about a couple of micrometers. Other further questions? I think. Yes, I think that it's actually conceptual. So are you saying that this Andreev's work and you're also seeing this lignite no disorder. So they had 1D, well, No, they had 2G. I just give an argument. No answer, but my question is about, actually you're understanding of their work versus Guruji. So in Guruji, I mean, there is no any lignite no disorder. There is just the different part of the speculative matters. That's right. So are you saying that basically this is the same thing and you'll get the same results or not? Because you somehow interchanged two things at some point. Yeah, so I was saying that the part of the Guruji, which is the viscous part, is quite analogous, but he missed all of this thermal conductivity component that is specific also to... Which basically that's what was producing the resistance in Andreev. So it looks like it's two different effects. Yeah, it's two different effects. Okay, so then in 2D, I just completely missed. So in 2D, I don't understand which one is kind of leading, right? Because in 2D, you have some profile. Yes. So a naive view is that there are channels and there is lots of Guruji effects. Yes. And another, maybe a bit more exotic view is that, okay, liquid has to go over the pumps. Correct. And it compresses and depresses and that actually creates what Andreev and Al had. That's right. Which I think is different from Guruji because it's just generation of heat out of the non-adiabaticity of compression and depression of liquid. That's right, yes. So your results are, is it... I mean, if I put it wrong, is it Guruji or it's Anton Andreev? No, that's Anton Andreev, yes. Okay, so why not Guruji? Why not Guruji? No, so let's say the example that I showed you for whole bar in the restrictive geometry, that would be a generalization, right? In the clean case. My point is that, okay, so somehow theory should know which effect is leading or somehow how to add them. And I didn't understand how you managed to consider the two effects together. No, not so that I... In fact, I mean, I just... The slide of Guruji was just a reminder about this DeuroDT and so on, but all of the calculations are extensions of Spivak Andreev to the case of graphene with long-range disorder. And this thermal effects are... Guruji said that you can do it for 2D and there is no problem, is it? Yes, it's mass, but it's... All right, okay. Final question? So is there some estimate for the variation in the density and variation of RS in this system? Is it in the sense in the low density regions is RS coming close to the Wigner crystallization? Oh, no, no, no, RS, for graphene devices, I don't know, RS is few. In the two tags that you were showing in the beginning? Yeah, two tags that are at the beginning, RS could be large. So is there any chance of pomeranchic-like effects in those low-density regions? Possibly. Okay. Good. Yeah, stick. Thank you. Let's thank the speaker again for the nice discussions. This also concludes the morning session, so thanks to all the speaker and see you this afternoon.