 Since I'm still wearing my hat of an organizer, I wanted to use this opportunity to thank you all for coming here, although for some of you it wasn't easy, and also for being so active in making the school a vigorous event. Thank you. And on a more mundane note, if any of you need some kind of letter from ICTP for your home institution, which you need to show that you've been here, let me know and I will issue such a letter. All right, so Professor Kamenev raised the level of the discussion during his last lecture very high. I'm going to bring it down. I'm going to talk about very simple things. Professor Chibukov promised that I'm going to talk about Adrude model. I will. So if you have a feeling that you are back to your solid state one class, that's okay. My purpose would be to go through very basic stuff, emphasize certain pitfalls which one may run into, even on this simple path, and then take it a little bit further and talk about the behavior of certain non-fermal liquid metals. I'm not going to really split the lecture into Fermiliquids and non-fermaliquids. I will bring in the material as we go. The first lecture is going to be PowerPoint mostly. The second lecture is going to be mostly blackboard. Okay, so this is the idea of my Agedankin experiment. This is a Fermiliquid in here and one way to measure it just to stick in our context and to read what Omitra says. But I will be also talking about another way of measuring conductivity, and this is the optical one. When you shine light on a sample and you measure a reflection are coefficient and then wire some transformations, you get both the real and imaginary part of the conductivity. This is a list of people with whom I collaborated for many years on the subject of transport and they're not responsible for anything I will tell you today, but they did help me to form my current somewhat poor understanding and there is a subgroup of people which work with me or used to work with me at the University of Florida. And then I think the rest of them, you have a chance to see either during the workshop or during this week. Well, you certainly recognize this person here. Vladimir Yudson is on and off. I think I saw him. Okay, so he's here today. Cameron Benia was giving a talk on transport in Tungsten, Force 4-2 and actually I will probably touch on this. Gil Lanzerich was giving a talk on, I don't remember what, but we did talk about transport and also to other people, Catherine Pippa, she was giving a talk on quantum criticality and Indranil Pol was talking about electron spin phonon or coupling in iron-based superconductors. And there is one person who is not with us anymore, but he's well known to the senior members of this audience. This is my PhD advisor, Yasha Lemonson and all I learned from a transport. I learned from him and many other things to which I'm eternally grateful. And his book on transport is a classic, so if you really want to learn something about transport, go and find this book. I've listed here a kind of random collection of references, which I think form the basis for understanding transport. I will put these slides online so you can take these references and in brackets I specified what particular reference can be useful for, because we'll be touching upon different aspects. And the last two, not that they are informative, but that's something which I wrote in a collaboration with some people here, and at least I'm supposed to know what these papers say. All right, so what is the challenge we are facing in condensed metaphysics? There are many, but one of them is that there is a large number of conducting compounds, of literally speaking metals, that do not conform to the predictions of the Fermi liquid theory. Whereas if we look at scaling, for example, of the specific heat, or the spin susceptibility of resistivity, the scale, the temperature, the frequency in some way, which is not in agreement with predictions of the Fermi liquid theory. And among these variations, the transport data perhaps is the most extensive and abandoned, because you can do many things, and you can do it without spending millions of dollars on, for example, synchrotone or arpys. But they are notoriously difficult to interpret. And one of the reasons is that there are additional conservation law, which is momentum and velocity, which you need to figure out before you can claim that your particular non-formal liquid model produces resistivity, which you see in other experiments. And as a result, the outcome of your theory is usually not an intrinsic property of a Fermi liquid, but it shows you how your Fermi liquid or non-Fermi liquid couples to reservoir, which sinks energy and momentum. And so sometimes we cannot get the information about the intrinsic interaction in the system because we have to worry about things such as disorder, lattice, phonons, et cetera. Okay, so this is an outline. My lectures roughly split into three parts, and I shamelessly borrowed a terminology from Sean Hartnell, who was speaking on coherent transport last week during the workshop. The first part is going to be a coherent transport or coherent quasi-particles. That is basically how transport of Fermi liquid looks like. Then I will make one step forward. I'll be talking about coherent transport of incoherent quasi-particles. So there are situations when you cannot define quasi-particles as well-defined objects, but you can still talk about coherent transport, which is characterized by some time. And a particular example I'll be talking about will be a metal which is tuned to a proximity of the ferromagnetic quantum critical point. I hope you still have some pre-collection of Professor Rubik's lecture in mind from yesterday, so he was mentioning these two paradigmatic scenario of quantum critical point, ferromagnetism and anti-ferromagnetism. In this part I'll be focusing on ferromagnetism. And then basically I will switch to a mode when I will be mapping out problems and puzzles rather than telling you why, for example, are the high-temperature superconductors have linear resistivity in normal state. I don't know, and I think no one knows, but the best I could do is to formulate equations. All right, so just a reminder of the Fermi liquid theory. Since we'll be talking primarily about the scattering rate, let me just jump to the argument, which is usually ascribed to Landau, although I think in his first paper on Fermi liquid theory he says, as it is well known, and then he quotes this formula. So this formula tells you that if you try to organize a collision of two fermions, whose energies are distributed according to the Fermi Dirac distribution, there is a what is called Pauli blocking, that is you can only use other states in the vicinity of the Fermi energy and in the interval of its temperature. And because you have two of those, it's like multiplying probabilities and so the widths of each energy interval being squared gives you a T squared dependence of one or two. I would also like to emphasize that actually this is not really an outcome of the Fermi liquid theory. This is an input. What you say is that by this argument the quasi-particles in our Fermi liquid live long times because as you go to low and low temperatures the lifetime goes infinite and based on this argument I say, I'm going to treat them as free particles. So the Fermi liquid theory is really a theory of free quasi-particles. Okay, that's my range actually. Okay, so what allows you to do, it allows you to replace an ensemble of strongly triton particles such as helium free atoms or electrons in the solid by an ensemble of essentially free particles but with different parameters. So these particles will tend to be heavier as a rule. They may have a different g-factor and the Fermi liquid theory allows you to formulate every property of a Fermi liquid as of a Fermi gas of these particles. Sometimes it's kind of better to think in real space rather than in the energy momentum space. So I just wanted to highlight one fantastic feature of a Fermi liquid which you get by looking, for example, at helium-3. So helium-3 is a canonical example of a Fermi liquid and it is a liquid. It's not a gas. Okay, it has a surface tension, you can pour it out which means that the interatomic distances are on the same order as the size of an atom. It's dense in this sense. Now suppose that we take helium-3 and we heat it up to temperatures which are larger than the Fermi energy but smaller than the boiling point so it is liquid and then actually we would find that viscosity of this liquid is not that dissimilar to the viscosity of any classical liquid such as water and it changes with temperature exponentially which tells you that if temperature goes down you have tendency to freezing. Now it's not really practical here to talk about mean-free paths but if you do want to talk about mean-free paths it's going to be on the order of the interatomic distance because the atoms are packed. One atom moves by one or two interatomic distances and a run into another atom. So now suppose that we cool the same system, the same liquid to temperatures which are smaller than the Fermi energy and because of the argument which I gave you on the previous slide the mean-free time becomes long but in real space it means that the mean-free path becomes long and becomes much longer than the interatomic distance. So think about this way. You have dense liquid, a particle starts motion but it goes through over distances which is much longer than the interatomic one and so that's the property of Pauli principle. It tells particles that you cannot really interact with your neighbor although you see your neighbor you have to make sure that the energy of this neighbor is in the right interval and if your energy is not in the right interval I will bypass you and I will go to the next neighbor I will bypass him and I will go to the next neighbor and then finally I will find the neighbor in the right energy interval and viscosity of a Fermi liquid will be like viscosity of air in this room it's going to be proportional to the mean-free path oops, sorry and the mean-free path is one of the squared that's what some of the experiments done on actually conducting systems show. So of course the original theory of Fermi liquid was developed for a Galilean invariant system with adiatropic interactions in neutral particles while in condensed matter we deal with electrons in solids and they're also charged. So we have somehow to make a mental jump from a system with an adiatropic Fermi surface which harbors neutral fermions to a system where the Fermi surface can look like that or even like that, this is copper, this is an aerobium and at the same time we need to take into account long-range air cooling forces and this procedure has not been ever done rigorously. What we have is something which is known under the name of an adiatropic Fermi liquid that is, it is a belief that if we morph our Fermi surface in this particular way and we keep the cool of interaction the physics will not change much and by and large this is true but here are the pitfalls which I mentioned before and I will walk you through these pitfalls. All right, so in the original Fermi liquid theory as formulated by Ablandow the quasi-particles are just free. This is Fermi gas theory and the predictions are made only for the 3D dynamic properties. Specific heat, which is predicted to be abnormalized by an interaction in a proportion to other ratio of the masses. The spin susceptibility for example is abnormalized because you have abnormalization of the g-factor and of mass and these abnormalizations are protected and follow from certain symmetries which we have. This would be Galilean, SU2 and U1 and U1 symmetries and all of them we have in a system like Archelium III. When we go to metals we first lose Galilean because we have lattice and quite often we also use SU2 because of a spin orbit interaction. So what we really have is U1 and certainly point groups. Then formally we can make predictions which look similar to what we have before but now the prediction would be that the mass is going to be abnormalized with respect to the band mass and the band mass is not the mass of a free electron. So we need to know this band mass, we need to rely on DFT to tell us what would be this mass without other interactions and so in this way the prediction becomes less quantitative compared to Helium III. This is hard to quantify and this is why people turn to transport because in order to have transport you need to bring in some residual interaction between our quasi-particles. You need to take into account the scattering and what is usually considered to be a hallmark of the formulaic behavior is that the t-square dependence of the scattering rate is being translated automatically in the t-square dependence of other resistivity and this is one of the pitfalls I'm going to walk over. All right, so this is just to remind you how the resistivity of a conventional or firmly-lipped metal look like. This is very old plot. This is aluminum. This is pure aluminum. Aluminum of resistivity as a function of temperature and then you add 0.5%, 1%, 3% of manganese. Aluminum by itself is a superconductor. You see a supernatant transition but then you kill a superconductivity of magnetic impurities and you study the normal state of resistivity. So there are two regions at low temperatures and at high temperatures which we understand well. At low temperatures, resistivity is determined by disorder, chemical substitutional, et cetera. At high temperatures, resistivity in conventional metals, again, is determined by a scattering at phonons and phonons at these temperatures, much larger than higher than other-day-by-temperature, behave as a classical bosons. The number of bosons in a mode with frequency omega-d is proportional to temperature. Well, resistivity is proportional to the number of these phonons. The resistivity is linear in temperature. That's a very simple argument. And then in between this constant region and the region of linear R-behavior, we have a region where resistivity is obviously nonlinear. Theory tells us that it should be a combination of two terms, T squared and T to 5. T squared is coming from electron-electron. T5 is coming from electron-phonon. To be telling the truth, it's not very easy to disentangle the temperature dependence here and really show that it follows the sum of these power laws because the interval, as you see, is fairly narrow. But that's lower. It is supposed to be this way. Okay, so this is the same aluminum, but on a smaller-temperature scale. And you see that other resistivity is linear as a function of T squared. So we do map out the first term. The range of variational is not too big because this is T squared. More of this. This is a heavy-firm compound, which is a heavy-firm liquid. I don't remember the R-mess in this particular compound, but they can run up to a thousand of R-mess, which is a mass of a proton. And again, at very low temperatures, below 100 of millikelvin, we do have the conformation of T squared long. So everything is fine. Everything is harmonious. As in this Abbotticelli painting of Madonna, it's beautiful. It's proportional, harmonious. But then we go to something which is known as the name of bad, strange, strongly correlated, non-firm liquid metals. And there are certain characteristic features of this strangeness. This is the system I will spend some time later on. This is a palladium, which is doped with nickel. Actually, sorry. No, sorry. It is palladium, which is doped with nickel. So palladium is intertic metal. It's on the verge of ferromagnetism. But it doesn't quite make it there. It remains a paramagnet. But if you add a little bit of nickel atoms, which are magnetic moment, it does go our ferromagnetic. And it does so at a particular concentration of nickel atoms, 2.6% in here. And you can measure the resistivity for different doppings. And the claim is that if you do it precisely at the doping where you are at the quantum critical point between a paramagnet and ferromagnet, the resistivity scales as t to the power of 5 thirds, not 2. This is a probably better known example. This is just one of the many plots of resistivity in the particular high temperature superconductor. And this is the famous linear scaling of resistivity, to which I will also come back later. This is a heavy fermion compound, which is famous for being able to produce a linear resistivity down to milli Kelvin range. From time to time, people say, well, what's so strange about the linearity of resistivity, because I just told you that aluminum has linearity of resistivity. But aluminum has it above its high temperature. Well, maybe there is some soft mode. Maybe at a temperature of 100 Kelvin, maybe it's 10 Kelvin. When it goes down to milli Kelvin, there is no way you can explain the linearity of scaling by electron-phonon interaction. So this is kind of a poster child of forget about phonons. This is a compound, which is very interesting because it really doesn't do anything, except for very low temperature superconductivity at about 1 Kelvin. Aaron Kapitulnik in his talk was telling you about the time reversal symmetry signatures of this compound. But above 1 Kelvin, this is a vegetable. It doesn't have any phases, but it is a non-firm liquid vegetable in the sense that it does show you a t-squared behavior of the CVT at sufficiently low temperatures, which then turns into a linear and they went to 1200 Kelvin. And as you see, it goes without any signs of our situation. And in the process of doing so, it steps over what is called motive irregular arc criterion, which can be defined in somewhat different way, but it tells you that the mean free pass becomes shorter than either the wavelength or the interatomic spacing. So you can cook up three versions of this criterion and you will step over all of them in other material and not only in this one. And now this is like another Madonna. This is Madonna cololunga, the long neck. And it's also beautiful in its own sense, but look at the proportions. It's all distorted. So it's called long neck because the neck is very long. And there is, of course, a baby Jesus in here, but also look at this child. They don't make our babies like this. And then there's a pillar in here. This pillar is roughly the same size as Madonna, et cetera, et cetera. So this is actually Parmigiana, those of you who went to Florence last week and had the chance to see this painting. In the Fizzi, this is at the beginning of Manerism. And the strange metals, they're beautiful as are the Madonna, but they're also strange. This plot was shown, I think, several times during the workshop. This is to emphasize that the linear scaling of the resistivity is ubiquitous. Not only you see it in conventional metals, aluminum, copper, silver, where we think we know that it's coming from written phonon scattering, but also in a bunch of other materials, heavy fermions, organics, high temperature superconductors, iron-based superconductors. But the claim, which comes from Andy Mackenzie group, is now more quantitative. Not only it is linear in temperature, but if you convert your resistivity into one or two by using some version of the formula, then the pre-factor in one or two, which is this dimensionless number alpha, is actually not just order of one, which can mean from 0.5 to 5, let's say, it is actually near one. Within interval between one and two. And that's all we can say about it. Totally mysterious. I think Sean Hartnell has some ideas about it, which go into the name of bounds. We'll probably spend some time on this later on. But we are still in the part of the talk when I'm talking about coherent transport of our coherent composite particles and how do we want to describe transport? Well, it's all very complicated. You need to know Boltzmann equation, you need to know Kubu formula. How about the good old Drude formula, which we all know. Okay, this is the same picture which Andrew showed you yesterday. This is Paul Drude, and this is his model, which, as you certainly know, is nothing more than the second law, complemented by the frictional term, which is responsible for scattering. So you pull the electrons with the electric field, that's as your MA is equal to the force, but the force has a pulling part and the frictional part. What is tau? I don't know. Actually, Drude thought that tau is due to scattering of electrons from ions in the lattice because the Bloch theorem was not proven yet. So now we know that this is at U2 or due to scattering from deviations of ions from the equilibrium are positions. Now, if you want to solve the DC case, we just say that the time derivative is equal to zero. Here is your velocity proportional to the electric field to plug it into the current which is charged times number density times at the velocity, out comes your conductivity. All right. If you think that this is simple, I'm going to fool you specifically deliberately. I'm going to give you totally false argument and I will ask you to figure out what goes wrong. So there is something which is known in the name of Bloch-Drude model. And it's in some textbooks. Drude thought about references free with the dispersion of p squared over 2m. But we now know that we have Bloch wave functions, we have bands, right? And we still know that in the absence of scattering we have a semi-classical equation of motion which tells you that still dk dt with an h bar is equal to a applied force. That works. That's how we study our dynamics of electrons in solids. But let me put in the same frictional term as Drude did, which is proportional to the same h bar k at the momentum over some tau. I will also solve it in dc, right? I will find the component of the wave number along the direction of the field. And then instead of doing what Drude did, I will say that the velocity, the zr component of it is not kz over m, but because I have a lattice of this kind, for example, this is the derivative of the dispersion with respect to the wave number. I will calculate the velocity as a function of kz. I'll plug in the value of kz and then I will find my occurrence as charge, number density, and velocity. Done. I have related the current, the magnetic field. Who needs Boltzmann? Or who needs Kuba? Well, let's apply it to a simple case of a one-dimensional 10-manif model. Forget about localization and all this stuff. So we just want to study the motion of electron in a 10-manif model with some friction. So this is my dispersion. This is my r velocity. I solved for k on the previous slide. I will plug it in. So now our velocity is a periodic function of the field, but I want to do what is called linking up-response. I want to do the weakest electric field as possible. I'll replace the sign by its argument, and here you go, you have the velocity which is a proportionate magnetic field. Now I will do the same as I did. I will multiply this velocity by charge, number density, and I will read off my conductivity which will be proportional to tau as in the formula. It will be proportional to number density and lattice constant. Done. What's wrong with this? At small number densities, the conductivity is small. At large number densities, near the bottom of the band, the conductivity is large. I almost filled my band. It's an insulator. Not in the model. Well, firm statistics is where I'm putting my affirming energy. And we certainly feel that this is wrong, right? And we also feel that the correct appraisal for the conductivity should be that when we are in almost empty band, it's a small conductivity. In almost full, it's also small conductivity and the maximum in this model is at half filling. Okay? And where is the mistake? It's in here. On a lattice, I don't know what number density is. Full number density, in this case, it's equal to one over lattice over lattice air spacing. It doesn't mean that the whole number density participates in transport. And the right result can only be obtained if you solve, even in this simple case, a Boltzmann equation, then you will see the maximum of the conductivity at half filling. All right? So there is no way to get around for getting into a Boltzmann equation or using the Kubo formula. Okay, so there's one more thing which is normally associated with the Drude model, but it comes from top of it, and this is called the Matissein rule. Namely, if I would upright the result for the conductivity in terms of resistivity, resistivity is going to be proportional to one over tau to the scattering rate. And if I have several scattering mechanisms, impurities, phonons, electrolyctin, then we understand that the rates should add up because the more of them we have, the smaller should be the conductivity, the larger is resistivity, and we tend to think that it can be done in an independent way. So this is called the Matissein rule where pretty much are confirmed, but when you start to think about electrolyctin, an interaction that fails you. And according to this rule, you should just add up all sources of scattering, while if you go to low temperatures so that you can forget about phonons, the only two sources of scattering which you have is impurities and electrolyctin interaction. So let them up, and we saw the Landau formula for the scattering rate which is proportional to t squared, so let's lump it into the resistivity. Simple. But it's based on something which is called SCA. Remember SCAs? Like RPA? SCA? It's a spherical cow approximation. What we did, we took our Fermi surface, of a metal, and we morphed it into a back into a sphere because we wanted to use second law. And if we wanted to use second law, we have to recall that the second law is based on Galilean invariance. And because of that, the internal forces do not affect the emotion of the center of mass. And if we forget about it, then we will recall the story of Berenin housing. So for those who read this tale, you know what it's about. For those who don't, this was a fictitious hero, a hunter, and a hero in many other ways. But the tales which he used to tell about himself were somewhat more than what Archie actually did. And one of the tales was that when he was hunting, he got stuck in a swamp on his horse, and being so strong, he pulled himself and his horse out of the swamp by his hair. Okay? Internal force cannot do that. Well, this is just the algebraic formulation of this principle. If we have an equation of motion in our solid with a with a spherical Fermi surface, we have external force, we have forces of electron interactions, but they all add up to zero when with some other equations of motion in the particles, such that the center of mass doesn't know anything about other forces, and in the presence of steady-state electric field, it's going to be accelerated in such a way that the momentum is going to be a proportional to time. So now we need to think how we bring in lattice, which breaks Galilean invariance, and there are three ways in which lattice can provide, can abrender the conductivity finite. One of them is omklop. The other one is Beiber, or interband scattering, and the third one, which is not widely known, is that you can actually get away with just having entropic Fermi surface, but you need to add disorder. But if you ask a more detailed question, if you have only these are processes and nothing more, no phonons, no disorder, can you get finite conductivity due to either of these three? And the answer is yes for omklop. You don't need anything else. For interband scattering, it's somewhat more subtle, and yes, no. And it depends on whether you have a generic multiband metal, or you have what used to be a special case, but actually this is now a very common case of a compensated semimetal with equal number of electrons and holes. In the first case, the answer is no. If you have an agent, phonons or disorder, in the second case, the answer is yes. If you are compensated, bismuth, graphite, iron-based superconductor in its permanent state, then you don't need to bring anything else. And if you just play the game with entropic Fermi surface, no, you need to bring something else. Okay, so first, omklops. Again, I'm bringing you to solid state one. Omklop is scattering of either two phonons. That's actually how it was formulated by Pyrals in 1921, who was thinking about a similar paradox in thermal conductivity of insulators. And then later on, it was brought into a context of charge, a transport by Abel-Andal and R.R. Chuk. The word is German and means jumping over. So you're jumping over a fence in this case, and the fence is at the boundary of the Brillouin zone. So because you have lattice in the system, your wave function is not a flattened wave function. It has certain symmetry. It is a periodic in real space. And therefore, your momentum is not really a well-defined quantity. You can only speak about quasi-momentum, which is defined modular and integer number times the reciprocal lattice vector in this one-dimensional case. So then in general, the momentum are conservation law in a sort that looks like this. You have two particles which come in with momentum K and P. They go out with K prime and T prime, but you can add an arbitrary number of reciprocal lattice vectors. If n is equal to 0, this is your normal process, which can also happen, but by itself it does not contribute to resistivity. It is not equal to 0. This is your own club, and by itself it does contribute to resistivity. Okay, so what enters A is a combination, as you said, of two probabilities, normal and own club. Because it works in such a way. If I remove all own club and keep only normal, A will be 0. If I keep own club, these are coefficient is going to be affected by normal scattering, as well, because there is an interference of two processes. I can repeat. Yes, I should say that if you have pure metal, for example, alkaline metal, so the temperature will not be too square, it will go by exponential. Coming to this. This is my next slide. No, actually, sorry. I'm coming to this in two slides. Because the rest of the discussion is based on a Boltzmann equation, I need to introduce the Boltzmann equation, and we're going to stay with it for a while. In the Boltzmann equation, we describe a system of, let's say, a many electrons by the distribution function, in some sense a probability of finding an electron with momentum k at point r in space and at time t. And because we have something which depends on the momentum and the distance, you can tell that the nature of this approximation is semi-classic. All the observables which we want to know, such as number density, charge current, thermal current, are the averages of the distribution function either given by one for number density, the velocity times charge for the electric current, or the energy times the velocity for the thermal current. The left-hand side of the Boltzmann equation is nothing more than the Liouville operator which describes the evolution of the distribution function in space and time in the presence of the classical force. I'm giving you a very simple version which is called a gas Boltzmann equation. We'll talk later about the modifications in the case of a Fermi liquid. And if particles would not interact with each other, the right-hand side of the Boltzmann equation would be equal to zero. So the right-hand side captures the information about the scattering. It's known as a collision integral and it describes scattering events, or various types. For example, if we want to consider a scattering disorder that is elastic, then the collision integral is proportional to the deviation of the actual distribution function from its average value. Typically what you see in the textbook is that this is the deviation of the distribution function from its equilibrium value. This is right because electron impurities scattering alone cannot form equilibrium. It is elastic. But for all practical purposes you can replace this average F by an equilibrium F. Or what we'll be more dealing with is there is a collision integral due to electron interaction which is nothing more than an algebraic representation of this picture. When you have two electrons, momentum K and P, they go in, they go out with K prime and P prime. There is some probability that they really need to be articulated separately. But the statistics of this process goes in as the product of these Fermi functions. There are two Fermi functions which tell you that in this particular process the states K prime and P prime must be occupied, but the final states must be empty. And there are also two delta functions which reflect momentum conservation law. So in here I also added the reciprocality vector to take account omklabs. And finally there is energy conservation. So we'll be interested in the electric fields. And actually I don't think that in this talk I will touch on electric field. So in this case we would say that the perturbation on the equilibrium function produced by the electric field is small. And I can parameterize this perturbation by this unknown function G and just for our convenience I will pull out a factor of temperature and the derivative of the Fermi function. Why? Because I can. Because my function G is arbitrary but it pays in later analysis to make formula simpler to pull out these factors. Now the product of the electric or the classical force times the momentum gradient of F is now the electric field times the same derivative of the electric field is weak I can forget about the deviations of this function from the equilibrium one so I will put an index 0 in here. And then I will recall that this function depends on momentum only through the dispersion. I will differentiate with respect to dispersion and the derivative of the dispersion will give me the actual group velocity of reference on my given lattice. And in this form the left hand side becomes somewhat more palatable and if I do a linearization of the non-linear collision integral for electron-electron collisions it's not linear because I have autocollide 4 particles but if I do a linearization that is I will expand this in the deviations from the equilibrium I will form this nice combination of the non-equilibrium parts of the dispersion function or each with the momentum of the corresponding incoming fermion. And then I will have a bunch of fermions functions and then I will have the same our probability. What's really important for the rest of this argument is to have this nice our combination of the equilibrium parts of f. Alright, so now why do we need umclubs in the first place if we don't have anything else? Suppose that I disallow umclubs for one reason or another then the momentum our conservation law becomes a conservation law of actual momentum there is no shift by an arbitrary number of the reciprocal lattice vectors and suppose so I'm also looking at static case, DC conductivity and at uniform electric field so I delete the time derivative and I delete the spatial derivative my left hand side is a known quantity proportional to the field and here I have something which I need to solve for and suppose that I solve it. Suppose that somehow lo and behold I found a our solution of the integral equation let's let me call G with index one but then I can add any function which is a scalar product of momentum and any constant because this guy by itself will nullify the combination of G's in here if I add something which is proportional to K and something which is proportional to P these are the same constant minus K prime minus P prime take out this constant vector and you'll have K plus P minus K prime but I have the conservation law so in mathematical terms it means that I don't have a unique solution in physical terms I can say that I can choose this vector C to be infinitely large and therefore because the deviation from the equilibrium controls the magnitude of the current I can produce an arbitrary large current with an arbitrary weak electric field that means that my conductivity can be arbitrarily large and that is infinite so that's why if we have only electron-electron interactions we do need umklabs and notice that the argument here is not for a Galilean invariant system I did not say that my thermosurface is a sphere my thermosurface is parameterized by the velocity which is some group velocity of a block electron of course I need to work hard to find the scattering probability of the block electrons but the statement is true for any thermosurface okay so we need umklabs but they don't come for free we need to specify two conditions one of them is in textbook the other one I think is not so first of all of course we need to have a rather large our thermosurface we need to organize a scattering event in which the difference between the initial the initial moment and the final moment is equal to the reciprocal of lattice vector that is the size of the brilliant zone well let's say B is a Eura and K P and K prime and P prime in magnitude are one cent I do it I need to have enough change to change my dollars or my Euras in coin and in particular two dimensional case I need to make sure that the maximum of this difference is larger than the reciprocal of lattice vector which means that the thermosurface has to be large enough KF in some sense has to be larger and then a quarter of the brilliant zone only then I can have unclobs scattering if I don't have this the probability of unclobs scattering will be exponentially suppressed and resistivity would not fall which is squared it will be exponential in temperature and this is seen or was seen in ultra pure alkaline metals I believe but there is and the second one is actually that we need to have a sufficiently short range interaction that is we need to think about screen coulomb and the screening clings of the coulomb interaction there to be not to be much different from other Fermi wavelengths both of these are conditions are satisfied in conventional metals you have large Fermi surfaces you have large number of electrons at the screening clings are short on the scale of the Fermi wavelengths which are the same as the inter-time distance so there is no problem to get umklap for example in aluminum on the other hand you have semiconductors if the electrons there are degenerate then you can talk about our Fermi liquid here but the Fermi surfaces tend to be very small because we get these electrons by doping so in semiconductors for example in two-dimensional electron there is no way to get umklap for the first reason but if we also have long range interaction in the system then umklap scattering is suppressed I will tell you why in the second but what are the examples of systems with long range interaction there are some systems which were also mentioned during the workshop for example Astrone centertainium oxide STO which has a huge electric constant of 25,000 because it's almost ferroelectric now you can imagine that the charge which you put into this insulator its magnitude is going to be divided by 25,000 so at the coulomb interaction is very weak that also means that the screening of this interaction by other electrons is also very poor and the screening length is going to be long but more in line with strange metals I will argue that the effective interaction which electrons which electrons experience near a particular type of a quantum critical point in general pneumatic that is a quantum critical point into a state where the actual order parameter is also very long range why well because you interact via an exchange of fluctuations of the incipient order parameter and this is second order continuous quantum phase transition and is any quantum phase transition as you approach it the accolation length becomes longer and longer and that sets the scale of your interaction and this is going to be relevant for the example that we are talking later so why is it so why long range interaction is bad suppose that I want to organize an un club scattering event on this on this rather generic our furnace surface it's a time banning model above our feeling I need to specify the conservation law that k plus p plus b but I have an extra condition that scattering can only occur by small change in other momentum ok so let's choose for example the the change between the initial momentum k and final momentum k prime to be small let's make it just equal let's say k is equal to k prime I still need to satisfy the conservation law such that the difference between the other two momentum p and p prime has to be large it has to be equal to the reciprocal of the lattice vector so if I make this electron to scatter by small angle to compensate the other electron has to scatter by very large angle but if you look at the furnace surface you realize that you cannot do it between two generic points on the furnace surface you can do it only on a subset or special points where the furnace surface intersects at the brilliance on so you have umklap hot spots and then at the scattering crate in this case is going to be suppressed in proportion to the volume of the area of other spots and what you see is not that t squared but t squared are multiplied by something small and the small near a criticality will be also temperature or dependence so you wouldn't get the interesting or a temperature dependence of the restivity you'll get something sub-leading if we suppress umklaps let's say by this argument then we need two things we need either a compensated ceram metal or disorder to render the conductivity finite why well because I told you that if we only have electron-electron interactions on the lattice but without umklaps the conductivity is still infinite ok so that was umklap let's move on to the second our possibility this is Weber scattering it goes back to the name of Weber 1937 and I'm going to do it at a very simple level so I told you that it's not a good idea to use a spherical cow approximation for a metal but one can use approximation of two spherical cows so if I have a two-band metal multiple for mimins too ok that would work I will morph the actual thermosurfaces back into spheres you'll tell me that well I distorted my Galilean invariance and I'm subject to all the penalties which I'm supposed to incur but the very fact that I have two bands well it's Galilean invariance empty space has one band with dispersion p2 over to m period bands it means that somewhere I brought in lattice I produce a different type of electrons number one and number two ok so then under Galilean invariance is broken but because I'm going to use a parabolic model for each of the bands I will get away with the Drude model in which I will have the usual ma term the pulling force and I will allow myself to treat the charges e1 as a two as parameters plus minus e then I will have interband of relaxation think about disorder for example but in addition to the single band case I will have a frictional force which is of the stocks type so the change in the momentum is proportional to the different of the velocities between the two bands if the two flow at the same speed there is no force but once they start to move at different speed then there is a frictional force and it's also proportional to the number density of other band and there is something logical are coefficient which from a detailed Boltzmann equation we know is in a fermi-licate case proportional to t squared so there are two cases one is when the signs of other charges here are the same positive or negative doesn't matter it can be two latent bands or it can be two whole bands and in this case I still need interband of relaxation I still need disorder for example and the second one is when other signs of other charges are opposite that's electrons and holes and one is equal to N2 which is a compensated semi-metal does not require interband of relaxation and it used to be in a very special case people would think about Bismuth people would think about Grafite and other group 5 semi-metal antimony but actually if you now scan over the systems of current interest except for iron superconductors and probably some here from metals everything else is a compensated semi-metal as long as you have a metal with an even number of electrons per unit cell even number this is a compensated semi-metal so this family is actually white and particular examples iron based superconductors in the non-doped state a variety of wild semi-metal it's called wild type 2 where the Fermi energy is not at a direct point but it is actually intersecting other bands so there are many conductors in other's class so it actually pays to think of this case in more detail so first suppose that I have a 2-band situation like this it is a 2-band case it is not compensated and how can I understand the solution of this Boltzmann equation well this friction which I have within the 2 bands is determined by temperature the higher the temperature the stronger is electron-electron interaction by the kinetic argument I don't care whether it's T squared or something else but the rate with gamma which I had in adéquation of motion is somehow proportional to temperature which means that if I go to 0 temperature this rate drops out and what I have is 2 bands and this equivalent scheme of 2 resistors each of them conducts in its own way it is usually discussed in other context of transition metals where one of the bands is light it is an S-band and the other band is heavy it is a D-band so let's talk about S-band and D-band ok so at 0 temperature I add up conductivities not resistivities and if they are conductivity of one of the band which is lighter is much larger than the other the conductivity at low temperatures is controlled by the most conducting band ok so this is my resistivity as a function of temperature I am somewhere here and in this case my resistivity is small in some sense now let's go to very high temperature by going to infinity I don't mean that I am going above the Fermi energy I will define in a little bit what does it mean to be at high temperature but if I at high temperature and create it is very high so this frictional force is the strongest force in other system and it locks two bands together into one system so now if I take a momentum from the field I cannot bypass one of the bands it will go first into one band and then while the frictional force it will be transferred to the other band so the equivalent scheme in this case are two resistors in series in which case I will add resistivities ok and this resistivity is going to be determined by the lists are conducting band let's say the other band each of these conductivities and resistivities are determined by interband of relaxation so the frictional force drops out at zero temperature because it is too weak it also drops out at high temperature because it is too strong this saturation saturation values at low and high temperatures are determined by interband interaction and if this is coming from impurities then you have two limits to saturation values of resistivity electron-electron interaction takes you from one limit to another limit now if these two limits are really separate which can happen if the two bands are really different in masses one is very light the other one is very heavy so the ratio of resistivity here to resistivity here is large then you can talk about an intermediate asymptotic range where resistivity follows some power so that's one way but if we actually solve the system without specifying whether we are dealing with two electron bands or two hole bands we'll find that this upper limit on the resistivity at high temperature contains something which has the units of the plasma of the frequency so it's some sense it is an interband plasma frequency and it contains the number density of one band times the charge of this band plus number density of the other band plus the charge of this band so now if I go to the case of a compensated semi-metal where the sum is equal to zero the upper limit here goes all the way to infinity so I don't have a bound but it's probably easier to look at this so now I'm going to the case of a compensated semi-metal it's a DC case so my time derivative is equal to zero I have a pulling force and the pulling force comes in with opposite signs because I have electrons and I have holes I still have frictional force but now the number density of the other band is common because I have fixed and common number density add these two equations oh sorry, subtract these two equations and you will find that the difference of the velocities is proportional to the electric field I did not solve the problem I did not find the velocity of band one and velocity of band two separately but in order to find the conductivity I don't need to because the current is proportional to the difference of the velocities and I found it here that it is proportional to electric field I can read off the conductivity from here which is inversely proportional to the frictional arc coefficient which in Fermi-Liquid is going to scale as T squared so in T squared the conductivity is one with T squared and I didn't need any help from disorder phonons, etc etc all I have to do is to break Galilean invariance by saying that I have two bands in this space but also saying that I have perfect compensation and the third way to get around of how to bring in electron-electron interaction into a transport is to do something like this so now I'm looking at an actual Fermi surface which is complicated, futuristic surface in three-dimensional space so it is very endotropic and I do bring in a disorder to render my connectivity finite but I'm also going to bring in electron-electron interaction without allowing for un-clubs that's the model electron-electron on a given Fermi surface plus disorder I will claim that it would be enough to produce a T squared in the Fermi-Liquid term in resistivity however it will be bounded from or from above now because I don't have un-club this electron-electron are part of the collision integral can serve the momentum which means that if I multiply this integral by the momentum and integrate over all moment I will get zero, why? because imagine that I would restore the time derivative here under this procedure the left hand side will give me the rate of the variation of momentum in time and it better be equal to zero if momentum is conserved now suppose that for a second I have a Galilean invariant system with disorder in this case the velocity is just KOM this is again a definition of a Galilean invariant system so if my collision integral vanishes upon being multiplied by the momentum and the velocity only differ by a constant factor of mass the same will be true if I multiply the collision integral by the velocity velocity is current so now if I take my Boltzmann equation I will multiply the left hand side and the right hand side by the velocity also times charge the electron-electron part of the collision integral will drop out because of this but I still have an electron impurity part that will drop out if I multiply this by the velocity and integrate this is the current now I can read it backwards current is proportional to electric field so I solved my problem I found the conductivity and this conductivity is determined entirely by electron impurity scattering rate there was no contribution of the collision integral did I solve the problem what the actual distribution of electron is no I did not did I solve the problem what the conductivity is yes I did so in Galilean invariant disordered system electron-electron interactions do not affect resistivity now in this building I cannot get away with the statement without some penalty there are other ways so this statement is true within the formalism of Boltzmann equation which is semi-classical there is truth beyond semi-classics and truth is that the truth is that there are effects which do affect the resistivity due to electron-electron interaction but they go beyond the level of other Boltzmann equation there are two kinds of them one is hydrodynamic type which I hope to tell you a little bit later but the other one are so-called quantum interference effects and they come into our categories weak localization which is a precursor of Anderson localization and the interaction which is known under the name of Alchulier Aronov and Zala what happens here for example in localization our mechanism when we talk about Boltzmann equation there is an implicit assumption that the phase of an electron is being randomized after every collision there is no memory but electrons being a quantum mechanical particles maintain a deformation about their phase and if we organize a special type of scattering of a trajectory where we go through the same zigzag path twice first clockwise clockwise there will be interference effect similar to interference of waves and that will lead to a suppression of the conductivity it is somewhat more difficult to understand what's going on in the interaction or correction but the ballistic version of it is simpler to Zala, Alaneur and Amnarozny you can think about it as an interference in scattering from a core of an impurity which you put into a metal and Friedel oscillation produced by the same impurity the total potential acting on the incoming electron is the sum of the two and the potential of Friedel oscillation being a periodic in space has certain interesting property that is you get an enhancement or suppression actually depending on the effective sign of the interaction of back scattering amplitude all of these lead to corrections and they need to be appropriately taken into account but the outcome is this there is a certain a scale which is the impurity scattering rate right and my temperature can be larger than the scale or smaller than the scale this is called diffusive abrasion and this is called ballistic abrasion in this sense so if we compare the thermo liquid corrections which is coming from the deviations of the thermosurface from Sphere with the interference effects we will lose in the diffusive regime that is at low enough temperatures but in the ballistic regime we will win so it makes sense to talk about our corrections to resistivity of order t squared which you get within the Boltzmann equation and forget about the limitations of the Boltzmann equation itself given that your temperature is high enough probably better for the thermosurface all right yeah, why don't we stop here and talk the equations