 Okay, great. Don't use this. We have to often. I see, but this one. Yeah. Oh, I see. Right. Okay. Okay. Okay, it works now. Thank you to talk with English use the mic. Okay. Oh, this one. Oh, yeah. So I was on this one. Okay. But not your. Right. I see. Okay. Okay. It seems to work now. Let me again. So there are more people in the audience. Let me introduce the speaker once again. It's a pleasure to introduce Professor Dmitry Maslov from University of Florida. Who will be talking about transport properties of correlated electron systems. You can take from here. Thank you. I should probably think the organizers of this conference for placing me talk here. Being one of them. I'm sure that this is going to be a wonderful conference and you'll have the time. Centrically and. And otherwise. I know that you had a long day. You have. Tutorials about. This is already complicated. And I'm going to. Talk about. A few simple things. Including second law. The formula. And all the stuff. So you can relax and view it as a transition to a. Seasoning session. In the afternoon. And I'm not going to go with this, but this is my. Favorite list of sources. On transport. And because the talk. Will be somewhere. Will be posted somewhere. You can. And this is where I usually go to. If I need to. Okay, so. I'll start with. A general statement that one of the most serious challenges. Intendance matter is that a large number of. Conducting compounds. Certain metals. Do not confirm to the predictions of the term. And. The transport data. The most extensive. And abandon. Because we have. Various. Means here. The series is. To. But it's. Difficult to interpret. And the reason are that there are. Additional conservation laws. Mainly. And current. And the result. Which we see. If they're not in this property. Of a family. But it depends on coupling. External degrees of freedom. But. But it depends on coupling. External degrees of freedom. Such as a beauty. So they'll find the top will be that I will. First briefly. Remind you of the family. And how it breaks down. And then. We'll talk how to discuss. How to. Relate this property. This. This feature of the transport properties. We'll talk about. Skittering. Then. I was catering. And then about. The. The. Okay. So. Starting. Starting from the. The. The idea. Of. I. So. Very much familiar. To everyone. Reminding. That. It's based on the notion. That when. To. In fact. And. They. They have the general. It means that they have to. Compete. For. The. Improvisal. Of the energy space. In the energy space. Which are. Of temperature. And as a result. The. The. And as a result, the scattering rate, which is proportional to the scattering probability, is proportional to the nominal cross-section of these two particles, as if they were interacting in vacuum, to a statistical factor, which comes from Pauli's principle. And the final temperature, just multiplying these two factors of temperature to get the factor of T squared, and if we do the particle, which is an energy, omega, above the, or below the fermion energy, then it is extended into a symmetric form, omega squared plus cos squared T squared. And so what it tells us that if you go with very low energies, then quasi-particles are essentially free, and a bunch of strongly interacting particles, such as feelings with atoms or electrons in a network, can be deployed by quasi-particles with different masses, different de-factors, which interact only with lower fermions. And this is beautifully demonstrated by the notion of viscosity, just to remind you what viscosity is, basically the pneumatic viscosity in a gas is the same as the diffusion coefficient in classical gas, such as air. It's given by the product of the RMS thermal or linear velocity and linear pre-pass. The linear pre-pass is fixed by the distance between the molecules and the cross-section, while the velocity is proportional to the scale of T. And therefore the velocity of gas is supposed to scale a scale of T. And this is indeed confirmed, let's say, for air fuel. But if you think that classical liquid is anything like classical gas, just this high density will be totally wrong, because the viscosity of classical liquids follows totally different law. It's actually an Arrhenius law, and here is the viscosity of water, and this is because the molecules of classical liquid cannot travel long distances, there is no motion in the glass. And people want to talk about pneumatic pre-pass here. Sorry, Dmitry. Dmitry, can I ask you to speak louder? Sure, I can. Sound on my screen. Now, in a quantum thermo-liquid, the viscosity is like in a classical gas, except for now it is a quantum gas. It's still the diffusion coefficient of fermions. Now the velocity is the thermal velocity, so it's temperature independent. And the Dmitry pass is proportional to the Dmitry time, and this is the inverse T squared factor of temperature. So something really neat happens here. If you think, for example, of helium-3, that's an liquid. It means that the distance between the helium-3 atoms is of the same order as the size of the atoms itself. Here due to the power principle, the Dmitry pass is much longer than the separation between the atoms. So a particle starting here will fly over long distances before a collision will happen. This is the viscosity of helium-3 measured in the 1950s, which follows one of its squared law. And this is more recent measurements of another thermo-liquid in graphene. And this is from under the gain group, where it also follows one of its squared law. Okay, so the thermo-liquid scattering rate, which I mentioned before, can be obtained in the prediction theory, but it can also be obtained for any order in the prediction. And this was done by Lackinger and Beliage in the 1960s. The question is, what do we mean when we say that it's valid at low frequencies and low temperatures? So, formally speaking, they go to zero, but we know that physics doesn't know zeros. So we need to ask how small omega mt should be. And, naively, they should be smaller than the thermo-liquid. It's a necessary condition, but not always sufficient. And to see what the actual condition is, let's do a quick calculation in the thermo-liquid rule, where I will consider scattering of two fermions, depending on the k and p, and to final states of this different momentum. And I will choose an interaction to be of this type, and it may remind you of a spin-pulling potential in three dimensions, but I will use it in dimensions two and three, and I will tell you why I did it later. So the thermo-liquid rule contains integrals over energy transfer between the fermions, and it's limited by temperature by power statistics. Over the energy of the second thermo-liquid, also limited by power statistics, there is an angle integral, which is related to the momentum transfer of the square or the scattering potential. And then there are two delta functions, which represent energy r-conservations. So because I have two integrals here, which are bounded by temperature, are already connected to the peak of the square. And therefore, if I want to keep this up, result to be proportional to the square, the frequency should disappear from this formula, because otherwise it will change it. How can we make sure that it can disappear? Well, suppose that the typical momentum transfer is much smaller than the thermo-liquid momentum, then I can expand the difference of dispersions to first order in Q. And then I can neglect the energy transfer if it is much smaller than thermo-liquid times typical momentum transfer, and because the frequency is over the temperature, this is the same to say that the temperature is much smaller than the F times typical momentum transfer. After that, each of the integrals are related to the function of the angle, leading factor one with Q from here. And then if we do a power counting with our interaction, we'll end up with scaling here. And this parameter psi minus one, which I chose to characterize the mass of the non-pasonic field, will enter with its only six minus d. And at the same time, we'll see that typical momentum transfers are of the order of psi to minus one. Okay, so what we'll have here is a nice formula. We have our Q squared factor, but we have this pre-factor psi to six minus d. And if this parameter psi minus one is of the order of QF, then the condition to have this formula work is that temperature is just much smaller than the thermo-liquid. That's what we're assuming from the very beginning. But what happens if this parameter psi becomes very large, goes to infinity, surely this formula cannot survive. And this is one of the ways to absorb a breakdown of thermo-liquid. In summary, I'll mention that it can also happen. The deviation from the Q squared independence happens even in a casebook case with a spin pool in gas. In this case, the parameters of the interaction, the pre-factor lambda is proportional to the square of the charge. And this one is just in yours, our screen radius, which we're coupling is much smaller than our QF. And so the condition for the thermo-liquid to work would be that temperature is much smaller than the plasma frequency, which by itself is much smaller than the thermo-energy. And if temperature is larger than the plasma frequency, yes, then the thermo-energy. Actually, we wouldn't have a thermo-liquid. The scattering grade will be proportionate to temperature. So this is the thin pool in gas, but we're really interested in a situation when we can arrange for long range and probably diverging range in interaction. And this happens near a second order of one-to-face transition into phase with spatially uniform order parameter, which can be, for example, a thermo-magnet or a pneumatic. And we've already talked about thermo-magnets. What happens is that on the right of the screen, we're far away from the quantum critical point, the acceleration lens is short on the order of the interacting distance. As we start to tune our system for this red dot, which is the quantum critical point, the acceleration lens increases, and finally formally diverges at the quantum critical point. And this scenario, which I just described here, is known as And this is the interactive interaction, which I would want my electrons to interact with, and when the electrons are coupled with bosons. In the kind of RPA scheme, you can see that the effective susceptibility will be dressed by the electron polarization bubbles. And that means that the susceptibility after this process will be the bare bosonic susceptibility minus the polarization bubble of thermo-magnets. And one property of this bubble is that if you go to frequencies which are much smaller than typical Q, this is a damping term, a low damping term, which is a medical overview. And that's criticality when psi is equal to infinity, it drops out, and then we see that scaling tells us that Q is proportional to 1.3, that's why it's called Z equal to 3. So the dynamical exponent is equal to 3. And then crossover between thermo-magnets and thermo-magnets regions, and non-termo-magnets regions occurs at energies which by itself is proportional to psi minus 3. Okay, or it means that I can replace psi minus 1 to a negative by 1.3 if I want to go to a particular. So if I go back to my thermo-magnets formula, I cut a factor of this square. And then the divergent correlation events will be inversely proportional to temperature to the power of 1.3. And that will give me a non-termo-magnets exponent of the specter ink rate, which is d over 3, which means that it is a linear in three dimensions, and it's 2.3 in two dimensions. So this is just a quantum lifetime, quantum specter ink rate, which is the width of the thickness in the special function, which can be measured in four dimensions. So at any fine correlation events, if I see, let's say here, that's a distance from at the quantum critical point, because we go down in temperature, then I would first measure the specter ink rate, and then I would measure the non-termo-magnets form, t to this point d over 3, and then it would cross into a thermo-magnets regime for temperatures below this line. And if I'm just at the quantum critical point, well, then the thermo-magnets is gone, I would never see this square. This is all wonderful, how to relate these results to the transfer. So the start is level minus one. So life is complicated. Solids are complicated. They have complicated thermal surfaces. What happens if we replace our metal by a single parabolic band at the gamma point, and these are the specter, which is proportional to p squared. We better don't do it, because by doing so, we said that our system is Galilean invariant, and as it happens in Galilean invariant systems, internal forces cannot affect the motion of the center of mass. So we can take, for example, two drops of liquid, water and honey, and they have totally different dispositive, and the dispositive of water is much smaller than that of honey, but if we let them drop, they will be moving to the same direction, because the gravity force on G will be acting from the center of mass, regardless of how strongly the molecules will interact with each other. We can do another high school experiment, we can place same drops onto an incline with some rough surface. Well, in this case, the drop of honey will not probably move at all, but the water drop can move, given that the incline is large enough. But in this case, its acceleration will be determined not only by the cooling force, which is gravity, but also by friction in between the surface of the incline and the drop. Okay, so if we try to translate it into this piece of solar, so we have a bunch of electrons here, which we replaced by Galilean invariance system, but why electric field, what was going to happen, that the momentum of the center of mass will be directly opposite to the field because these electrons are nearly charged, but the internal forces will not have any effect on the motion of the center of mass, and the momentum of the center of mass in magnitude will just increase linearly with respect to this time, which means that we wouldn't be able to establish any kind of statistic. So if we try to describe this situation at the level of linear response, we'll have to conclude that our conductivity, the real part of it is actually infinite at zero frequency and zero otherwise. And this would work for any frequency, any temperature, which can change the interactions and the whole response. Okay, now we can introduce lattice. This is the case of friction between the incline and the water drop. Conductivity on lattice will be certainly finite, but it may be controlled by friction that is forces which are external to the system of electrons, which can be phonons, utilities, etc. And what we want out of this measurement is to extract the information about electron-electron interaction. And so we need to understand under what conditions electron-electron interactions can control the conductivity by themselves without the help of external agents. Okay, so this was level number minus one. If we go, suppose that we somehow found this abrasion, and the level zero will recall through the formula, which contains for the resistivity, we can have an upside down form for resistivity rather than conductivity, and resistivity is proportional to some scattering rate. And if I replace this scattering rate by the quantum scattering rate, which we estimate using the third golden rule, the resistivity of the liquid scales is squared, and if a non-firm liquid appears, then it has this non-firm liquid spawn due over three. Okay, that's probably also the graph of approximation because through the formula, it's supposed to work even for entire classical systems, and therefore it cannot contain quantum time. So we can look at a more complicated example. For example, we can take Boltzmann equation for electrons which are moving in the presence of disorder, such that our relation function with disorder has finite range. That's about the only Boltzmann equation we can solve. And what we'll learn from here is that the proper time which enters the conductivity is not a quantum time, but what is called transport time, which contains the integral over the scattering angle, over the scattering probability multiplied by the transport factor, which eliminates collisions with small angle. If the scattering is isotropic, then the average of cosine here vanishes, so quantum time and transport time are the same. If the scattering probability is picked at a zero angle, then we expand our cosine, we get the factor of theta squared, which is related to momentum transfer. And then we conclude that 1 over tau transport is suppressed compared to 1 over tau quantum, so this is supposed to be tau quantum, in proportion to smallness of momentum transfer. It also works for phonons. For example, for electron phonon interaction, temperature is much smaller than the Dubai temperature. Typical momentum transfers are small, they are of the order of the momentum of the formal phonon, temperature was to the sound. And one of the quantum time scales is TQ, one of the tau transport scales is TQ5 and the difference is precisely the square of the typical momentum. Okay, if we do similar procedure for our nearly apocalyptic thermal liquid, we know that typical momentum transfers are of the order of the inverse convolution events, so we need to multiply our quantum time by a factor of Q squared. And in the thermal liquid, we still have a T squared scaling, but now the dependence on site is different, and we can do our smooth crossover to non-fermal liquid behavior, the exponent for the transport time changes to T to D plus 2 over T. Okay, so then this would be our first order deductions, we would expect that the resistivity of a thermal liquid, sorry, of a thermal liquid, not non-fermal liquid scales as T squared. If we have a non-fermal liquid of this type into dimensions, then it's going to describe thirds, and if we have a non-fermal liquid in two dimensions, then it scales as four thirds. So I can do a similar analysis for the optical conductivity, and then again I will use the Drude formula for the conductivity, for the real part of the conductivity at our final frequency. And then I will make a hundred argument that if I know the transfer time at final temperature, then the transport time at final frequency will be obtained by replacing temperature by frequency. And then I will also take the conductivity into the high frequency regime when frequency is much larger than this factor in three. In this case, the formula will give me something which is proportional to omega squared and one or two times. And then again, by the same logic, I will have a constant optical conductivity in a thermal liquid, this is known as thermal liquid food. In three dimensions, the conductivity will scale as omega to minus one third, and in three dimensions it will scale as omega to minus two thirds. And the rest of the talk will be about the conditions under which this can be done, because it wouldn't be very important. Alright, so we want the conductivity which is controlled thoroughly by electron-electron interactions, and doesn't mean that it is dominated by electron-electron interactions because some other sources can also contribute, but we want the situation when at least electron-electron interactions are allowed to control the conductivity. And the answer in both samples is that, well, we need two cases. If I'm talking about DC resistivity, this is possible only if we have either unclubs, which I'll talk about later, or compensated bands. For the optical conductivity, the conditions are somewhat less restrictive. It will be controlled by electron-electron interactions and compensated bands, but also for non-probability bands. And by non-probability, I mean non-electrolable, because you can morph them back into spheres. And if I talk about smaller samples, let's say, narrow wires with rough walls, then there are other situations. One can have a situation when you have a Navier-Stokes flow with the selective liquid. I was introduced by Guruji in 1968, and I believe that Alex Levchenko will tell you more about this on Friday. Or you can have a hydrodynamic flow in systems with very long-range disorder, as shown by Andrei Pilosov and Spivak in 2011. I will focus on both samples from this point on. And so, as the first condition I mentioned, unclubs, unclubs is an ocean, which was introduced by Google Files. I then finalized the form of the series of oscillators, and then later on at Landau in Tenderin 2, introduced into the electron transport. And the word literally means jumping over in German. So we have electrons only like this. The momentum is not a good quantum number. What we have is a quasi-momentum, and the momentum conservation can be a date up to a number of reciprocal object selectors, which is d. So if we don't have the reciprocal object selector on the right-hand side, this is known in the normal process, which can serve the momentum as if light is absent. And if n is not equal to zero, then it is an unclub. And normal processes normally don't contribute to the stability, unclubs processes give you uniformity of the key square scale. So how we see that we need unclubs, we need to introduce the Boltzmann equation, because that's one way to see the other ways. The Boltzmann equation is basically a legal equation for the distribution function under the action of a classical force. And the effects of scattering of electrons from each other, from phonons, etc., parameterized by what is known as collision integral. And for the effects of electron-electron interaction, collision integral has this Fermi-Guggen rule form. You have scattering probability from initial state to final state. This bunch of Fermi functions simply impose the power principle. And this is momentum conservation constraint, and this is the energy conservation constraint. So if we have time independent unit for the deflected field, then we expect that the deviations from equilibrium are very small. In this case, first of all, we don't have the time derivative, we don't have the spatial gradient. And the force term is produced to be the product between the directed field and the velocity. And we can now linearize our nonlinear Boltzmann equation by expanding the distribution function around the equilibrium function. And we can just choose parameterization that the new molecular function is called G. So if we do so, the collision integral will take a much simpler linear form. And the combination of the monoclear ground parts of F will enter in this linear form, which will be the sum of 2Js in the initial state minus the sum of 2G in the final state. Okay, so why do we need, in general, unclubs in order to maintain finite conductivity in the case? So suppose that I have lightness, I have arbitrary distortion. My velocity is some gradient of my band, and it's not equal to momentum divided by mass, so I don't control any constraints on the lattice, but I disallow unclubs for a second. And in a single band system, electron-electron interactions alone cannot control the disadvantage of finding. So why is it so? So what we need to solve is this integral equation on function G, once we know G, we know everything. And suppose we found a solution that's called this G1. The point is that because momentum is conserved, I can add a scalar product between momentum K and some arbitrary vector C to this solution. And this will be, again, a solution because the added term will vanish the collision integral. So this vector C is completely arbitrary, its magnitude can be infinitely large, and therefore the conductivity or current can be infinitely large, and the conductivity can be arbitrarily large, even if I apply it because it's small. That means that the conductivity still remains its delta function term, as it was for the case of the DeLorean invariant system, and the only difference is that it's now, and it's a drop it, so it has some drool, drool the weight, which knows about the symmetry of lattice. Another way to see it is that in assisting with conserved momentum, the integral of the collision integral with momentum vanishes, and they can just take this equation, multiply by momentum and integrate, and it will give me momentum conservation law, by which the current derivative of the momentum will be proportional to the total force at the point of my electrons, and that means that the momentum of the system will move under the action of all this force, and there will be nothing to stop. So what I gave you is one way to see that the necessities are for one plus, there is another way that follows straight from the tubal formula, which was done by Maybach and Tokuyama. And for people who are more familiar with the memory matrix formalism, this is a statement about the existence of zero mode of the memory matrix. So we need them plus and then plus can occur on the two conditions. One is that the Fermi surface is large enough. And this is kind of obvious because of what we need to satisfy is that the mismatch between the final and the initial momentum has to be on the order of the reciprocal, has to be equal to the reciprocal of a lattice vector, which is on the order of the inverse of lattice. So if I have a tiny Fermi surface, let's say at the demo point of the building zone with size B, then all momentum on the Fermi surface are much smaller than the size of the building zone and I cannot satisfy it. I need to go for states which are away from the Fermi surface, but there are only few of them. And as a result resistivity is fine, but it is suppressed explanation. So here's an example of a process on a honeycomb lattice, which is what we need to hear more about the second week of this workshop. So imagine that I have a time model where I have a honeycomb lattice such as for caffeine, it is doped and it's doped rather heavily, but it's still a neglect the effect of freedom of working so I will model my Fermi surface by the circle. One is a key point, another one is a key point. So how we can, how we can make an input in this case, we can take two electrons that say from this side of one Fermi surface and transfer this pair to the other Fermi surface. And if we count the momentum, then the minimum condition for the inputs is that if I start from two states here. Then the total momentum is equal to minus two pf along this axis. And I need to bring them all the way here. That means that the momentum changes by the distance between p and p time point, which on a honeycomb lattice is one third of the size. And then it has a two pf because I placed them here on the top of the Fermi surface. So the change in the momentum is 4 pf plus two thirds of this apoplectic vector, and that has to be equal or larger than this vector on itself. That means that the fermenting has to be larger than the over 12. And that's very hard to realize the real one over here because it applies very strong. But it obtains in PBG because the superlacids are constantly much larger than the atomic lattice constant, some of the scale of 1500 electrons, and this condition can be really hard to realize. Okay, so that was one condition, which is what's kind of obvious that we need to have a large enough Fermi surface. Another one which is not the produce is that the momentum transfers a large amount. Why? Well, suppose that we have a case of forward scattering when typical momentum transfers are much smaller than kf, which by itself on the size of PB. And we have a comfortably large Fermi surface, let's say the copyright type, which I stitched here. But to make sure that the mismatch between the initial and final momentum, which I now wrote in a different way, but it's the same kf plus kf minus kmp equal to b. But the momentum transfers in each collision act, let's say between kf and kf, is equal to q and qs. So if I neglect q in here, that means that the first conventions, and that means that the difference between momentum to time and p has to be close to the reciprocal lattice vector. This can be satisfied, but only at special points, which are unplugged special points shown by these dotted lines. And therefore, one would expect a suppression to unplug the spectrum. In conventional metals, the Fermi surface is large enough, and momentum transfers are also large enough, both out of the order of kf. Therefore, on clubs are large, and when we sit to square in a conventional metal, such as an aluminum, and the less conventional, but still conventional thermion metal, which is Syrian aluminum three. When we sit to square in the spaces, we don't think twice, we say on clubs, and we are very happy to see. However, our metal near PCP provides us with a problem, as I just told you, we have suppression of on club scattering, and therefore, one would expect that the actual scattering rate, which is the on club rate, which enters the resistivity is smaller than an estimated transport rate, which scales as 2 over 3. How small is small? Well, that actually, it is convenient to consider a special case of the surface surface, calling Patrick Lee to introduce it to the program. So I have this mismatch. The upper momentum space, which the electrons have to color in order to produce an input process. The Delta B is of the order of B, which is the case of a generic thermo surface, then we can certainly say that the inputs are separate. We know that the distance between the thermo surfaces, the distance between the hotspots is small, smaller than B. Then we have another energy scale, which at quantum physicality by itself scales as the input mismatch to the power of three, and power of three is again the dynamic of the quantum phase transition. So at higher temperatures, if it was smaller than the thermo energy, but much larger than this intermediate scale, the on club scattering rate is actually proportional to temperature. So if we go back to temperatures, which are much smaller than this input energy scale, well, then the input rate scales as the square, which means that they are back to a terminal. So this is one of the ways to produce a linear entity conductivity, a linear entity resistivity. It requires two conditions, one to have a zero to three quantum critical point combined with a special kind of a thermo surface, where the distance between the input points is small. All right, so that was about complex, but there are two warning signs here is that the T square resistivity is also observed in low density semi-metals such as bilma graphite antimony. And that relates the condition that the thermo momentum has to be comparable to that is constant, it's actually much smaller than the size of the bilma. And one critical scaling, which we want to see in two dimensions, T to five thirds, is observed in quantum critical thermo magnets such as palladium drop is nickel, or zinc as in a zirconium two, which relates the condition that momentum transfer has to be large, if they are quantum critical thermo magnets, we expand their community transfers are smaller than that. So this is a compilation of the data on the five thirds scaling in quantum critical thermo magnets. This is a system of palladium drop is nickel, palladium by itself is almost a thermo magnet, and it becomes thermo magnetic and a little bit of nickel or 0.426. And this critical concentration is shown here. And the exponent is very close to five thirds, which is in here because it is a temperature of five thirds. This is another system. This is zirconium two, again, if you plot resistivity, so you could use some other reference resistivity at scales as T to five thirds. So why does it happen? There is one more condition, which I mentioned that we either need unclubs or we need specific metals. And there is another specific mechanism, which is specific to metals which are compensated, that means that we have people number of atoms and poles. And it's known as Baber scattering, Baber who came up with this idea in 1937, and understand this idea. So I can go to a very simple level of two parabolic bands. And because we know that parabolic bands by themselves don't allow for current polarization, I will need to focus on the interband electric electron scattering. And the bands can be of the same sign means two different bands or they can be opposite sign in one electron and one for them. And because I broke the Galilean invariance by saying that I had two bands, Galilean invariance is simply one band. I can allow myself to go back to the simple level of fluctuations of motion, where the change in momentum in the third band is proportional to the driving force which is a rapid field, and charge E1 is charging the first band. And I will describe the interaction between the bands by this term which has a friction nature, it's proportional to the difference of the velocities in the bands and to the number of the velocities in the other band. And now my this level is something logical coefficient, which quantizes the strands of this friction. And the same equation for the second band except for the change one minus the plus so if we add up these three equations, the friction course will, will, will cancel as it should because this is the term of this. I will multiply the cost equation by the charge of the second band, the, the second equation for the charge of the third band, separate them off. And then we'll have a condition that the product of the total charge density times the difference of velocities must be equal to zero. And because we want to have some difference in the velocities. This is only possible the total charge density is equal to zero, which means our content section. So, because we're talking about electrons, you want to be to must be to the charge. So that means that one of them, one of them, not, not the electron like and the other one is holy. And then means that the number that's the same. The current is the sum of the current image of the bands. And under this condition of compensation, it just proportional to the difference of the velocities. So we can find the difference of the velocities from the equations, it's proportional to the field. Therefore, the current is also proportional to the field divided by by deflection. Now, if they are in the liquid, we would expect that the friction are efficient is proportional to the scattering trade, and then the conductivity is proportional to the square. And therefore resistivity for portion to be square. And if we are in non-formal liquid, well, we can expect this to square to change according to the type of the quantum political point and the connection. So it is in this case, when the metal is compensated. We don't need any type of non-clubs. We don't need the purities, we don't need phonons. In this case, resistivity scales as it's supposed to form in our very naive history. And actually, this case sounds like a special case, but you can apply it actually involves the wild class of our materials, because all metals with even number of different cells are compensated. They have equal number of atoms and are holds. And there are also semi-metals, these moves into one in graphite, while two semi-metals such as when I did course for us to parents states of iron based superconductors also compensated and the remain compensated on Isabelian booking. Big for madness, actually, which I mentioned before, which exhibit the quantum physical phenomenon is this to also happen to be compensated. So, maybe the evolution of the paradox. So, what we see before is that when we see a T squared in the load density system and two to five thirds in the system, this is the death of compensation. Okay, so to summarize this part. If you talk about this resistivity, it can be controlled by the interactions only in the presence of full blocks or compensation. And we understand all cases of this resistivity, because nothing is perfect, of course. And the theory of some are cracks in the life of the previous will be simple and present. It's not for experimentalists, which keep challenging our beloved notions. In this particular gentleman's gentleman very good at challenging our notions, they come up with examples when the T squared is observed when it is not supposed to be observed and the game goes on as in tennis match. So there are several balls thrown back and forth. The first one is the docked strontium titanium oxide, or also known as an STO, it has a tiny surface, there is no possibility to open clubs, the structure is well known, there is no compensation, yet it shows to square it over the white temperature range. This is where that it goes to the energy and therefore can really be the future of a family liquid. So theorists are the first shock. So little bit and recall that this is actually one of our electric it's very close to it but it doesn't quite make it, which means that it has a sort of two mode. A single two-of-one spectrum is forbidden because it's more distanced first, but one inspector at two two-of-ones. And the temperature is larger than the two-of-ones and the number of two-of-ones is proportional to temperature. And because I have, I need to have two phones rather than one, the resistivity is proportional to this number on the square, which is T squared. Over some temperature range. Theory kind of works in the sense that one has to ignore the fact that the frequency of a soft mode is small on the order of 10 Kelvin or 10 perches, but it's not zero, it is finite. And we should expect to see an exponential freeze out of resistivity in this range. But if we go above the two-of-ones frequency, then indeed the theory based on two two-of-ones scattering works pretty well. And so for a while theorists feel good about it. And I will just mention that superconductivity between two two-of-ones mechanisms will be discussed by Krivin Chandra and Dmitri Kicelyov on Friday, the last day of the workshop. Well, experimentalists don't keep us crazy. They came up with another system, which is some gizmo oxides, and it's just a garden variety, the semiconductor doesn't have soft formal modes. And yet it shows a really convincing T squared scaling, and our friends finished their paper by saying that our results imply the absence of a satisfactory understanding of the T squared resistivity in terms of liquids. And this is how we theorists feel about this. The ball is on our side of the court, and for the time being there is no explanation of this experiment. Okay. How much time do I have? You have around half an hour, Dmitri. Since I talked about DC transport, maybe, and then I'm going to switch to optical conductivity, maybe it would make sense to pose for a while and ask if there are any questions. I don't know if I would hear the audience, but maybe you can give me some of the questions to me. Any questions from the audience? Oh, can I ask a question? If it's severe online. Hi. The very nice presentation, Dmitri. The issue of compensation, if I take the memory function point of view, it would seem, you know, you need the momentum and the current to be orthogonal. And that would require some symmetry between the two bands, not just equality of densities. Is that correct? No, not really. No, I don't know how to formulate this particular one in terms of methods formalism, but from the Boltzmann equation, there is no symmetry other than just compensation. Okay. I mean, but you're also assuming parabolic bands or that's not important either. This is just a toy model. Yeah, but it's not, you don't think it's important. No, no. Okay. All right, thank you. Yes, let's go ahead. I have a question about the experimental data. I noticed that the power, the T square power, the data was over about one decade and a little bit. Is that convincing enough? Okay, are you talking about the data issues now? No, the trace on the T square. Yeah, yeah. Well, it's T square. It's one decade in T square. Right, so if you convert it into T, it's somewhat better than it looks, but it has to be compared with other cases of T squared when, which we see in conventional metals, such as the one which I showed before, the range for T squared is always narrow in real systems, because at higher temperatures, you have a contribution from phonons, let's say to five and then at low temperature that it's situated such a way. So in many cases, you don't see T squared at all. That is in copper, for example, or in our goal. And when you see T squared, you may have a fraction of a decade, half of a decade. So compared to conventional metals, this T squared is better to square than you see there. Yet, in conventional metals, it has to be there and doesn't have a place in this field. Mitya, what is T squared in this material? What is, sorry, what is, sorry, what is for my energy in this material? Okay, so as opposed to STO, the temperature goes up to the fermion, because the fermion energy is about room temperature 300 Kelvin, and that's roughly the scale of the fermion. So it doesn't have this feature which STO has that STO to square goes to the fermion energy without noticing it. So here we only know that it's the square below the top. So it may be that this material is more conventional explanation works. Except for we don't have one, because it's a tiny thermosurface at the center of the brilliance zone, and there is no temperature. Okay, yeah. No questions. No questions. You can continue, Dmitry. Okay, so now I'm going to talk about optical conductivity. So we're focusing on high frequency regime when frequency is much larger than the inverse transport time, it by itself is a function of frequency and temperature. And in this region, we expect the conductivity to scale as one or made a square, we suggest them to do the formula proportional to the scattering rate, which we need to figure out. I will start with the formula, which relates the real part of the conductivity to the major part of the current current of relation function, but instead of doing diagrams, as we normally do, I will use a trick, which was probably by people who work in matrix formalism in memory matrix formalism, and in particular by Akin Roche, where you start with this operation function which is integral over time, and you integrate over time twice. That gives you a factor of one omega square, and you replace the derivatives of the current under the operation function by the commutator of the current with the r-commonitonium, which is the exact operation with motion. And then, identically, the real part of the conductivity can be written as the imaginary part of the different arc allele later between the current and the r-commonitonium and the factor changes to one over one. Why it is convenient? Well, it's because if some part of the current commutes with some part of the Hamiltonian, we don't need to worry about this process if drops out automatically, which somewhat simplifies our life here. So the model which I am going to consider will be pretty generic. I will have the Hamiltonian, which contains some left and then is arbitrary, so for our dispersion. And then I will write the interaction as some general function of k, e and u, as usual, for the normal form, and I will not assume that the interaction is still on the function of Hamilton transport. If I do, that means that I am back to the density interaction, but I can also allow this interaction to be proportional to some form factors, which depend on p and p. For example, if the direction is p in the d-wave channel, then this form factor is the familiar for sine px minus cosine p. I single out a dimension of constant g out of the r-commonitonium, just to make the power constant center. So for this Hamiltonian, I can write other current by using the continuity equation, and it's the sum of two parts. One is just the free-electron w, free-electron operator, which contains the velocity. And for this kind of interaction, which is not density, density is another part, which contains the gradients of the interaction, which are non-zero if I have this point. Now, if I want to obtain my real part of the activity of the second order in the interaction, I need the correlation function. I need the r-commonitator between the current and the n-other Hamiltonian to linear order in G, because the conductivity by itself is a product of the piece. So if my current is a 3-1 plus the interacting part and the Hamiltonian is the 3-1 plus the interacting part. Now, to linear order in G, I have two r-commonitators of the free-current here with the interaction part of the Hamiltonian and of the interaction part of the r-commonitonium of the current with the r-commonitator. Let me focus on the first one. I'm going to start with the r-commonitator, which is very tending. So, and what is tending is this combination of the velocities, which is total velocity in the final state minus total velocity in the initial state. As you see here, the interaction is outside of this bracket, so it can be anything here. Now, if the system is brilliant invariant, just for a second, that means that the velocity adjusted to the momentum over the mass, and in this case, this velocity imbalance vanishes identically. Okay, that's what we already know, the real part of the conductivity in this case will be just another function of the frequency. The interaction is not parabolic, but isotropic. Suppose Dirac as linear, the imbalance of the velocity is not zero, but it's small. It's proportional to a factor of either frequency or temperature, whichever is larger. Why is it so? Well, if the dispersion is isotropic, I can write the velocity as a vector of a momentum multiplied by the derivative of the dispersion divided by the magnitude of the momentum. But if I project all the momentum of the initial and final states onto the Fermi surface, make them equal to tf, then the common factor which depends on the magnitude is taken out. And what I have now is a vanishing combination of momentum and dispersion. So my system is non parabolic, but I project it onto the Fermi surface, I still have zero for the conductivity. One needs to expand near the Fermi surface. And understandably, I can't understand the difference. So, if we expand near the Fermi surface, well, we need to go in energy away from the Fermi energy, and that means that the imbalance of the velocity just will be proportional to let's say a factor of frequency. So if the conductivity is a product of two acrylaters, then it gives an extra factor of omega square in the conductivity. And this is the result of the conductivity, which has an additional omega 4 in here, or t4 depending on what is larger. And then we can go into the thermo liquid operation where we have omega squared or t squared. So if I can interrupt for a minute. You had a two slides ago you had a K one and a K two what happened to the K two. Ah, okay. So this is K two, but K two contains just the just just other difference of the final energies and initially, because you can use the Hamiltonian which contains the distortion with the current or other current, but this interaction happens in the presence of a photon. So the difference in the energy is just people to the energy of the form. And so in this case, the tool is proportional to the factor of frequency and the conductivity has an extra factor of frequency square. Okay, thanks. Um, this factor lambda is a measure of non probabilistic. It contains some combination of other derivatives or other discussions such that if we go back to the parabolic case zero for the Iraq case to get one and whatever the distortion is as long as as it deviates from the non parabolic, it's a number. And then if we use our quantum physical interaction here. I would count three factors of the appellation lens in three dimensions in four factors of the appellation lens in two dimensions, which is simply the scaling dimension. The two dimensional case also has a log, which is kind of interesting because it's a transport, and we normally say that transport doesn't look like logs. I can explain it. Okay, so in terms of transport spectrum trade, and forgetting about law, then the transport rate scales as I make a four, 44, both in two and three dimensions. This suppression is common for any, or any dimensionality. It's only a factor of the log. So if I would not bravely continue this result to the quantum physical point, then I have to replace omega four by omega two minus four thirds and then found two powers. So this is the conductivity, which scales as a negative field times log omega three dimensions. So let me three, can you talk louder please. Okay, from this point. From now on. From now, I don't need to repeat anything. I think we're good. Okay, thank you. So that's really omega two thirds not minus two thirds. Okay. Okay. Yeah, it's really omega two thirds and the naive scattering rate would be minus two thirds. Okay, and then there's a special case of a convex fermi surface in two dimensions. And this subject actually has very long history, as many things in transport. It started with Guruji back in 1980, to notice that the taxation of odd harmonics for the distribution function, including the first harmony, which gives us the current is slow on the dimensional candidates for Mr. And then there was lots of people who work on this. And even recently, we have two papers from central group and super group. We just got this subject so I wasn't going to talk about it originally, but because the subject still seems to be open interest, I do. And if you are like me, always confusing convex and cave. So just to just avoid my mistake. This is a convex premise of this is a complete premise of this project. And the difference is that the second one has action points. And the first one. Okay, so we want to have fine conductivity. And ideally, we would want to have them from elections right on the pharmaceuticals because if you go away from the pharmaceuticals to pay energy. Which means that we have to specify three conditions. The imbalance of velocities must not be visible. Otherwise, we will know current at the last section. The initial state and the final state of two elections must be on a different. So that it is a purely a genetic problem. I can put it up a label on P into minus people with your bar. And then because I have time it was a symmetry. The equation for the bar is the same as the equation for the tools that it is the same. So genetically when we solve, let's say first equation for K, but we do we take a firm contour, and we shifted momentum space momentum transfer you. And then once we change our view we shift again and we find the solutions that is the positions of the initial space on the thermosurface, which allow for energy momentum conservation. Well, the property of I can give her my surface. Did I say can take an X. Okay, so it can be that it has only two self intersection point. If you move within a place, which means that the other equation for a has two roots. Let's call them one of them is going to be K naught, but then the other one by seemingly is minus cannot last year. But the question for the other formula, which we now label by bar is the same. Which means it has also two solutions to not bar minus the number plus few or if I go back to the original momentum, it's minus the number. But it means that the two sets must answer. Because they came from this image. And there are only two choices. The first member of the set has to be equal to the first member of this. But that means that we're an equipped channel or head on our team. When the momentum of the initial state opposite to each other. That means that the total our velocity is equal to zero, and the final our velocity is equal to zero. There is no current in the initial state, there is no current in the final state, not nothing to be relaxed. So the factor of the TV is equal to zero. Or we can swap our momentum. The first member of this set is equal to the second member of this set and vice versa. And then we arrive my velocity imbalance in a slightly different form just of the range of terms. Here is the difference between the initial state and the final state of the second electron, but because we had swapped the same. So this is a zero. And the same for this term. So it means that if I have this final state, this initial state, it's final state and science is the initial state. That means that the reading term in rotation or eight or in optical conductivity or in the second activity in the presence of these languages. If we are in the family with it. Then, we would expect to have a mega square T square term, while it doesn't happen on it on the surface goes away by this hidden integrity of the service. And the next sort of term is a mega four T four, which is the same. For a non parabolic, but as a prep exception, which I talked about. Subleading term for the non for military was obtained recently in super group who said that it is as in a family with it's proportional to a meter squared. Will not comment on this. This is to be discussed. So what happens if they can if the thermosurface is complete. Well, if we slide a country for my mentor my mentor in the plane, we are guaranteed to have more than two self intersection points. So let's say there are n of this point that means that there are n larger than two solutions to the energy consideration. Two out of the same will be people who parent swap channels, they do not complete. And there is no current other section, the remaining channels will contribute and then the family liquid, I will recover the, the usual mega square T square. And if we are if we continue with this to a political point, once we have a family liquid and maybe square T square and we know the time to power, then they should give us a negative minus two third for the conductivity, which was which was put back in the 90s was calculated by some partial set of diagrams by Andre and himself and it was done in our case on the assumption that the thermosurface does not allow for a consideration. The same result would be for optical conductivity will be related or will be obtained for a compensated case. In this case, even even convexity, the thermosurface can be a circle, but it's enough to have two circles. Two circles simultaneously, we get more than two solutions. And in this case, the conductivity scales as predicted by naive estimate of the student. So, I wanted to stop here by thinking people who work with me with a number of projects, which I mentioned here, and this is Andre Chebrykov, Vladimir Lutson, and with Sandra Pichipi and also for the members of my group. This is Paul, I wish I could come on some simply in Prachi Sharma. And with this, I will put my summary in. Okay, so what was thanks for speaking. Questions, please. So, there is a question in the chat. Yeah, I see. I probably need to stop sharing. Okay. Right. Question is, can we generalize this effect? This probably means consolation on the convex thermosurface. Right. Okay. Between two concave bands, for example, does it affect the conductivity as well. Well, so if you have concave thermosurface, then consolation doesn't happen. And as I just said at the end, probably too fast. If I have two convex thermosurface, and this is enough to avoid a constellation. But to compensate that contract. No, actually, for the optical conductivity, you don't need compensation. It's enough to have two bands. You wouldn't get finite DC resistivity without our compensation. But optical conductivity would be fine. Other questions? No questions. So let's thank the speaker again. Thank you. This concludes our session today. Thank you everyone for participation. Have a good series session. We'll see you in a couple of minutes.