 Hello. Yeah, so good morning, everyone. So welcome to the morning session of our workshop program. And in this morning, we have three talks. And the first talk is given by Eris Berg. And he's telling us about the basement of France law in strange metal. So let's welcome. Okay, right. Thanks very much. Thanks to the organizers for the opportunity to speak here. So I'm going to talk about the minimum France law and it's an application to strange metals. And these are the collaborators on the works that I'm going to mention and I'll highlight them as I go along some of them are here. So, yeah, so this is my outline so I'm going to start from basically the most conventional strange metal confined. That's a that's a metal tune to the vicinity of a van Hof singularity just as a test case. And it turns out that just just from that you can actually learn something interesting about the minimum France law. It actually is predicted to break down in an interesting way close to a van Hof singularity. And then I'll move on to a to more exotic cases, a strange metals like a marginal family liquids, and I'll discuss what the tickets recording. And I'll discuss what the minimum France law can actually tell us about the mechanism of of a strange metals a little bit more, more broadly. All right, so I'll start with this example of tuning a metal to the vicinity of a van Hof singularity that's a relatively conventional case but beautifully it was actually realized recently in a search to one for very, very clean metal so it's a very good test case. And what's been done is actually applying uniaxial strain to the system so that that can tune a one of the three bands in the system to the van Hof singularity this is a very quasi 2d metal so we can really think about the 2d band structure. But this, this band is a function of strain actually reaches touches the, the zone edge. Okay, and this is a plot of the logarithmic derivative of the resistivity as a function of temperature, and as a function of strain. Okay, so here's zero strain. And this is where the van Hof singularity is that's actually been confirmed by our first measurements as well. And this is temperature. Okay, so you see this beautiful fan shaped a behavior. So it really looks like a quantum critical point today is a quantum critical point but presumably a very trivial one it just it just the van Hof singularity. But you see here very clearly. Okay, so if you look at the exponent of the resistivity as a function of temperature. It's, it's close to T squared on either side of the van Hof singularity, but close to the van Hof singularity and in this quantum critical region. It actually deviates and it looks like somewhere close to 1.5. Okay, and this is the actual resistivity as a function of temperature. Okay, the black one is is just shifted. Okay, this one is the actual data. Right at the van Hof strain. So this is as a function of temperature at the van Hof strain. And in this paper, they actually try to fit it either to T to the power one half, or to T squared log T. And over this range of temperature. Okay from from DC which is, which is enhancing this close to the van Hof singularity three Kelvin up to 40 Kelvin. It actually fits more or less equally well both both forms. And there's a more recent experiment actually by the same group by by a Clifford Hickson company, where they suppress superconductivity by a field, and they actually find that T squared lock T actually fits the data visibly more than than a T to the power of 1.5. Okay, so where is this actually coming from. And if you think about a little bit this is actually somewhat surprising so think about it in the following way. Okay so the only change from zero strain to the van Hof strain is that there's this small area of the Fermi surface and one of the bands that's going through the van Hof singularity so in that part of the van Hof singularity the Fermi velocity is going to zero, but other parts of the Fermi surface are actually not affected but much, at least in terms of the band structure. So it's somewhat surprising that the transport, which presumably would be short circuited by the parts of the Fermi surface that undergo the small scattering rate is actually affected at all by this by this minute change in the Okay so the question is why isn't there no short circuiting of the van Hof singularity by all the parts of the Fermi surface that are that are far away. Yeah. So, so you know I mean it's a, a, a, I think that it is elastic but for this purpose yeah I mean there's no plastic effects the residual resistivity is actually not affected much. Okay. All right. So, yeah, so the question is, you can imagine it in this way, the parts of the Fermi surface that are undergoing the van Hof singularity are somehow becoming becoming hot. Okay there's a very high density of states there, but why. Okay, so maybe there's an enhanced scattering there also, but why aren't they short circuited by all the all the rest of the Fermi surface that actually, which is actually not undergoing any dramatic change. Okay and to answer that you have to think about different scattering processes. So, here's the system where one of the bands is tuned into the one of singularity. And we can, it's useful actually to classify the different scattering processes into these different groups. Okay, so we'll distinguish the cold electrons which are which live far away from the van Hof singularity and the hot, the two electrons which are living in the vicinity of the window of singularity. This would be a cold cold cold cold, a scattering process process of two, a, of two electrons. Okay this would be classified as a cold cold to cold hot scattering process okay so we start from two electrons, which are far away from the van Hof singularity. One of them is scattered somewhere over here, and the other one is scattered into the van Hof singularity Okay, so that's another type of scattering process. And thirdly we can have a cold hot to cold hot so we have we start from an electron far away and an electron at the van Hof singularity, and they scatter by small momentum, such that the hot one remains hot and the cold one remains cold. Okay now we can ask about the contribution of these different scattering processes to transport. Okay so the cold cold to cold cold. Doesn't know about the van Hof singularity so we expect this to have the usual Fermi liquid type scattering, scattering rate of T squared. Okay this cold cold to cold hot process picks up the logarithmic singularity of the density of states from the vicinity of the van Hof singularity. And two other important points to note this process involves a large momentum transfer. Okay, actually might as well be a process so it does show up in the electrical resistivity. However, it turns out that in the presence of multiple bands this process can actually occur anywhere on the premise surface it's not limited to any particular portion of the premise surface so it's not short, short circuited the scattering rate from this process if it's large would actually appear anywhere on the firm surface so it will appear in the resistivity. And finally there's this cold hot to cold hot process, which can occur anywhere on the Fermi surface but always involves a very small momentum transfer because the hot electron has to remain hot. Okay so this would show up a very strongly in the single particle scattering rate but would not contribute much to the electrical transport because it involves a very small momentum transfer. Okay so from the cold cold to cold hot is scattering rate you actually get this T squared log T behavior. So in this paper with Sean and Connie from a couple years ago, we actually applied the same mechanism to a slightly more exotic in a behavior okay in the three to seven the bilayer structure mutate material. Apparently there's a stronger van Hof singularity than this this is just the ordinary logarithmic van Hof singularity. So we invoked that to get to the behavior which is observed there which is actually leader in T resistivity in a specific heat diverges like T log T, okay and possibly that can explain that material. Okay but I think that here things are much clearer. And we believe that this is actually the right mechanism. Yeah. Yeah, so, you know, one, one thing you can do is scatter right from here to here. Okay so that's a very particular momentum. So that that won't won't allow you to scatter cold electrons from anywhere to anywhere on the Fermi surface. Okay, so that that again was not expected to show up in travel. Okay, any other. What's that. Yeah, so. Yeah, so, so in the quote marginal Fermi liquid applies to the three, three to seven not to this material. Okay so there it looks like. Okay, the, the, the resistivity there in a special value of the magnetic field scales like T, and the specific heat is T log T. So that's, that's a behavior which we associate with a marginal Fermi liquid, where the self energy goes like omega log omega. But but this is not here. I'll actually discuss that later, but in this situation what you get is not the marginal Fermi liquid. Okay, it's from the usual from adequate behavior but it's not the marginal. Yeah, any other questions, comments. Okay. So, yeah, so so now the question is, well, there is this, if these cold hot to cold hot processes that they're actually much more singular than the cold cold to cold hot processes and the question is where do they show up so this is small momentum transfer to show up in the electrical resistivity, but these are inelastic scattering processes you might expect them to show up in the electric in the thermal connectivity. Okay, and that's that's what we actually decided to look at. So, just as a brief reminder, the, a widemann France law is is is a relation between the electrical and the thermal connectivity that was actually observed empirically very long ago. That appears in metals. And it's basically the following statement. Okay, so you define the Lawrence ratio, which is the ratio of the thermal connectivity over temperature times the electrical connectivity. And in a very, that in a very broad in variety of metals at at low temperature, this ratio actually approaches a universal value. Okay, so it's, it's pi square over three times a k Boltzmann over e to the power to. Okay, and here are some examples. Okay, so the open squares here are nickel. This is a, a quasi one dimensional metal. The ratio divided by L not as a function of temperature and you see this nice approach to one. Okay, this is a, this is silver. Okay, so so again at the lowest temperatures approaches one. And here's even a heavy Fermion compound. And this is very typical behavior of it starts from one at the lowest temperatures. It deviates down and it actually recovers back to one at higher temperatures, but here I mostly focus on the, on the, on the low temperature bit. Okay, and what's kind of striking is that in these different materials is this ratio really approaches one within a few percent so it's really a quantitative, a quantitative prediction. So what's the origin of this of the of the Whitman-Franc's law. It's basically elastic scattering. Okay, so it comes very naturally out of the of the Boltzmann equation when the scattering is elastic, and the argument is very very simple. Okay, so the idea is that at low temperature both the thermal connectivity and the electrical connectivity are both basically carried by electronic quasi particles. So to get a resistivity of any kind you need to relax either the electrical or the thermal current. Okay, so here's the expression for the electrical current in terms of the quasi particles. And here's the thermal current. Now, if the scattering is elastic, it cannot modify the energy of the quasi particle it can only modify the velocity. So if you see that to relax the electrical or the thermal current you have to relax V of k. Okay, you cannot relax the energy. And therefore it's exactly the same scattering processes that contribute in the same way to both electrical and in thermal resistivity. So that's the origin of this law. Okay, so the explanation of this behavior. Okay, of the of the Lawrence ratio is that at the lowest temperatures, the scattering is elastic, okay it's dominated by purity scattering, then at higher temperatures phonons kick in, and they cause inelastic scattering this is why deviates down. As you as you raise the temperature, as soon as the temperature exceeds some typical if the frequency of the phonons, the scattering from phonons becomes quasi elastic again. Okay, so the phonons can only absorb a energy of order their frequency. So when the energy of the of the electrons becomes higher when the temperature is higher than that, then the scattering from phonons is essentially elastic. And this is why the Lawrence, the Widman-Franc's law actually recovers at high temperatures again. Okay, this is a very, very difficult behavior in many metals. Okay, so coming back. Yes. Yeah. No, so yeah, so what what happens here is that this is a this is a heavy fermion compound. Okay, and at high temperature you have quasi elastic scattering from from the local magnetic moments, and this is the reason for this recovery. Okay, so it's actually quite different physics, but yeah, it's not phonons here. All right. So, coming back to the example of a metal tuned close to a venous singularity. Okay, so what's expected to happen there. So we saw that there are these ch to ch scattering processes that are that involve a very small momentum transfer and don't show up in the electrical resistivity. But these are in elastic scattering. Okay, so they can relax the quasi particle energy. And therefore we expect them to show up actually very strongly in the thermal resistivity. And this might lead to a very strong violation of the Widman-Franc's law. So we calculate the, the cold electron cross section to scatter off hot electrons. Okay, which is described diagram. And that it turns out to scale like the like t to the power three halves. Okay, so very, very different from the usual Fermi liquid behavior, which is, which is the squared. If you, if you calculate the thermal connectivity, kappa over t, that's that actually scales like one over t to the power three halves. Okay, so it is actually dominated by these processes. And you compare that to the electrical connectivity that goes like one over t squared log t. And the ratio, which is the Lawrence ratio actually scales like square root of t times log t. Okay, so as a function of temperature, this is what happens the red curve here is when our system is exactly at the one of singularity. Okay, so you, there's this, this very, very dramatic drop of the Lawrence ratio, basically to zero. And then if we tune away from the one of singularity, this is what we expect. Okay, so, by the way, this is a perfectly clean. So the calculation is for a perfectly clean system will discuss what happens in the presence of impurities shortly. But in the clean limit, this is what we expect to find this with this very, very dramatic violation of the vitamin front slot. Okay, and a similar effects were actually predicted in a variety of situations where there's a source of very strong inelastic scattering that involves a small momentum transfer. And some other examples are just an ordinary to the firm liquid is also expected to be a violation of the demand France law but only a logarithmic. Okay. And, yeah, and some other examples are a hydrodynamic metals, or metals close to specific quantum critical points, these all are expected to have similar violations of the vitamin front slot. Yeah. Yeah, yeah. So it comes from this. Okay, so, so, if you just calculate the spectrum of chart fluctuations close to a but no singularity. Okay, so, so that turns out to have a stronger singularity than log. Okay, so, so, so, so, so, yeah, so, so, so, this is coming from Q, a Q smaller than, sorry, Q bigger than omega. And over a small range close to the vinyl singularity. This is where this, this, this singular behaviors is coming from. Yeah. Given the high density of states of singularity why don't we also consider the hot hot and cold cold scattering process. Right so so in the question is why, why not consider a hot hot. Why not consider a hot hot to hot hot scattering process. That's because most of the electrical and the thermal current is carried by the cold electrons. Okay, so this would occur this is what what you might measure if you measure the the single particle lifetime of the hot electrons, but we're interested here in the transport properties. Okay, so it's really mostly the cold electrons. Okay, so. Okay, so, so we see that the Lawrence ratio actually carries some interesting a information about the scattering processes in the system the inelastic scattering processes in particular. And now we'd like to apply that to understand the mechanisms of strange metals more more broadly. Okay, and here's just one example of that. So, this is a recent example from Andy Mackenzie's group. So this is a palladium chromate material, and that over some range of temperatures shows a very nice linear and T resistivity, and from comparing that to a sister compound to a palladium cobaltate. They argued that this linear and T may not be due to phonons that's usual mechanism to get linear and T resistivity at high temperature. And the reason is that a palladium chromate cobaltate actually has a identical structure. Very similar, very, very similar a phonon spectrum. And the only difference between these two materials is that the chromate actually has low localized magnetic moments on the chromium ions and the cobaltate doesn't. Okay, and you see the big difference in the resistivity and also the resistivity and the cobaltate is not quite linear. It turns out that if you look at the Lorentz ratio in the chromate material in the region where the resistivity is linear that actually approaches one within a few percent. Okay, so this is a very strong hint that the source of this linear and T resistivity is coming from elastic scattering processes from some slow degree of freedom presumably the phonons, never the list. Okay, so why, why the big difference between the cobaltate to the chromate materials. Also, the slope of the scattering rates as a function of temperature here turns out to be close to the so called Planckian limit. So it's h bar over KBT. And the question is why is that a okay so these puzzles, I think are still open. So this is just an example of how, how you can learn about the mechanism of the strange battle. Okay, from from the Lawrence ratio. If it's one, it's a strong hint that it's all elastic scattering. Okay, but now we get to the main puzzle. Okay, so we'll, yeah. Yes. Yeah, so in in right so the Lawrence ratio is is Kappa over T sigma. So if Kappa for some reason is much bigger. It can go above one for instance in an insulator it will go infinitely above one because sigma goes to zero when Kappa is carried by phonons. Okay. Yeah, so in in metals usually what happens is that even at quite elevated temperatures the electrons are dominating both the both Kappa and sigma. And that's because the Fermi velocity is so much bigger than the phonon velocity, the speed of sound. Okay, and then typically what you what you find is that it's either below one or one. What. Yeah. Yeah, yes, yeah, right, right. So yeah, so there are various reasons for that. The electrons are not so fast. And they're probably strongly coupled with the phonons, but yeah. Okay, but now let's let's get to the corporates. Okay, so now we get to the maybe the strangest of the strange of the metals. So here's a particular corporate. This is a new demium doped LSEO. And a close to the so called critical doping. The resistivity is actually linear on one hand down to extremely low temperature and on the other hand up to quite high temperature. This is a clearly a violation of what we expect from a Fermi liquid. So we're over at this value of the doping the specific heat actually has a logarithmic is a singularity it goes like T log T. And the question is what can we learn from the weight of our front ratio on this on this behavior. It turns out that at at the lowest temperatures measured. Surprisingly the weight of our front slot is actually obeyed within a few percent in this system. And most strikingly, even in this magic doping this critical doping when the resistance is linear down to the lowest temperature, the real one France is actually obeyed. Okay, it's also obeyed for the transfers, transport a coefficient sigma xy and cap xy. Here's a different corporate is this is a thallium corporate also that shows linear resistivity down to the lowest temperatures over a range of doping actually, where super connectivity is suppressed by high field. So the lowest ratio is a obeyed up to up to one percent. Okay, so yeah. What's the temperature here. Here. Yeah, so so so this is trying to extrapolate to equals zero. Okay, so they measure they measure the lowest ratio. So, so super connectivity is suppressed by high field. Okay, and, and, and this is going. This is trying to go to the lowest temperature, because because really the, the way them on front slot should only be obeyed, even in a normal metal only at zero temperature. Right. Yes. Right. Yeah, so that's the question. Okay, so presumably it is an impurity dominated regime. Okay, but the question is if you have a some kind of nonfiber liquid, which this presumably is. Okay, so should the we run front slot be obeyed even at zero temperature. Okay, the argument for a normal model was all relying on the existence of essentially free quasi particles. Okay, so what what does this tell us does this tell us that there are quasi particles or just that it's obeyed because of disorder or what is it said. Yeah, so as far as I remember it's always obeyed within our bars. Yeah, yeah, they never saw a clear large reproducible violation anywhere basically at low temperature. Okay, so what what does that tell us. Okay, so that's that's going to be the question for the reminder of the stock. So maybe it tells us that actually underlying this, there is an ordinary Fermi liquid, and just the scatterers are quasi elastic but for some reason the scattering rate is linear in temperature. Okay, there's some low energy degree of freedom that we didn't know about that scales linearly with temperature. Okay, and in particular what we'd like to know is, for instance, what's the lowest ratio of the marginal firm liquid. Okay, so suppose we have a system where perma liquid is just marginally violated. And that's a that's a phenomenological model that was proposed a while back to explain the behavior of the of the corporates. Okay, so this would be a system where the single particle self energy scales in this way. Okay, the real part has a logarithmic correction to the usual Fermi liquid behavior. Lambda here is some coupling constant, and the imaginary parts a scales like the maximum of T and omega. Okay, and this implies in particular that the single particle scattering rate at zero frequency scales like temperature. In some situations, if this scattering of single particles is a large angle scattering, that also implies that the resistivity actually scales linearly with temperature. Okay, and the specific heat would get this logarithmic behavior both of these behaviors actually seen at this magic doping in the in the corporates. Okay, now in a couple of comments here. Okay, so a in the marginal firm liquid is really a phenomenological theory. Okay, to, to make predictions we really need some model that we can solve that gives this behavior and allows us to calculate any other quantity, for instance, we can write models that give this single particle scattering scattering rate or self energy, but the resistivity would not scale linearly in temperature because the scattering all comes from small angle scattering. Okay, so this is actually what happens in the model that Avi mentioned yesterday. If you have a 3D Fermi surface coupled to critical boson, this is this is precisely what happens. Second, it's kind of interesting to note that in the marginal firm liquid has a nice property if you look at the single particle scattering rate that's the imaginary part of the self energy divided by the derivative of real part z factor. Okay, and that's a from this the coupling constant lambda, if lambda is big, it actually drops out. And this is T over log T. So it's, if you like this plankton behavior, it's bounded by a KBT over H bar. It never exceeds that value. So, in that's worth, worth noting a property. Okay, so, right then the question is what would be the Lawrence ratio of this type of system. And for that will actually need some concrete model that we can solve and gives us this behavior. So, to construct such a model, I'd like to make a small detour. Okay, and just describe the strategy. Okay, so in general, we're interested in a some kind of lattice model of electronic degrees of freedom. So suppose we have an n band hybrid model, and on every site we have some number of orbitals, and the type of Hamiltonian we'd like to solve is a each each orbital has some hopping into some interactions between the orbitals on different sites and so on. And, of course, we're interested in, in these strange metal phases that are difficult to get perturbatively we know that if we just do perturbation theory and the coupling will typically get a firm liquid phase. Okay, and in addition to that we'd like to include disorder lattice defects phonons and so on. Okay, so these are of course, of course very very hard to solve. So that's the interactions. Okay, the strategy I'll take here is to imagine that the number of orbitals on every site is actually very large. Okay, so that provides us with a small parameter one over N, which is not the interaction strength so that can potentially give us some access to these strange metal phases. Okay. So that comes of course with the price. Okay, we have to introduce a symmetry, at least statistical symmetry between these different orbitals in order to be able to solve it. Okay, so there's the worry that this large number of degrees of freedom on every site provides a bath for dissipation. That's not something that we that exists in the real system. Okay, but this can give us access to these non-quasi particle regimes that are hard to access otherwise. Okay, so this could give us at least some window into these behaviors. Okay, let me just give you one example that we worked on and kind of relates to something that Nikolai talked about yesterday. Okay, so we wanted to study a scattering of electrons due to phonons close to the material for Reagan limit where the mean free path of electrons becomes a photo one. Okay, and that can be done using this approach. Okay, so we introduce a model which has N electronic orbitals or bands on every site, and has n squared flavors or modes of phonons on every site. Okay, so this is our Hamiltonian this is the electronic part. This is the phonon part and this is the interaction. And as Nikolai described, we considered two types of interactions one of them is on site. Okay, and one of them the phonons are really coupled to the electron density on the same site. And the other one is the SSH type where the phonons are coupled to the bond density. Okay, we can solve this model when the electron phonon coupling is of order one or even exceeds one as long as n is big enough and we're interested in this context in the in the in the regime where the temperature is much bigger than the by frequency, but much smaller than the permi energy, permi energy. Okay, and it turns out that these two models actually give a very different behavior in this case. Okay, and the hosting type model, the, the resistivity grows linearly with temperature. And then when you reach the material for Reagan limit a mean free path of the electrons becomes order one. There's a kink in this slope, but it just continues linearly with a different slope. It's not a crossover, but there isn't the so called resistivity saturation which occurs in many, many real metals when, when a KFL approaches one. Okay, on the other hand in the SSH type, a variant of the model. It starts linearly as well. That's the usual block a runeism behavior, but then when KFL becomes of order one time some function of the density. And this is the resistivity saturation to a value of the order of H over E squared. Okay, this is a 2d model. Okay, so this is the so called resistivity saturation is a possible explanation of that behavior which is actually seen in many metals. And in this particular case we can test a how bad is the approximation of setting end to infinity. Okay, so in this temperature regime where the temperature is much bigger than the phone and frequency we can treat the phone as relatively static. Okay, and we can solve the problem in numerically and calculate the resistivity using Monte Carlo. Okay, we can calculate the resistivity for finite end so here's n equals the same model n equals two, four, six, eight, and here's infinity. So at least qualitatively, the end the large end limit actually captures the behavior of this model, even for in a small as four or four. There are other other predictions I won't get into that. Okay, but now what we want is basically a controlled theory of a non firm liquid at low temperature. Okay, where we can ask about the Lawrence ratio. Okay, there is a model that does that's just that that's the such the vehicle type model. So that's basically a model of quantum dot a zero dimensional system that has an electronic orbitals, and they're interacting all to all. Okay, so there's only interaction in this model, no hopping. Every four of these orbitals are interacting via this, this type of term, and the interaction matrix element is random, we take it to all these interaction constants to be independent random variables, such that their average is zero and their variance is some value use Okay, and it turns out that this problem can actually be solved in the larger limit. It gives a very interesting non firm adequate behavior, pay the single particle self self energy scales like like one over square root of Omega. So this model is argued to be self averaging meaning that in what we can compute is only the, the configuration average self energy over all realizations of this interaction matrix, but the claim is that even a typical realization would actually show the same behavior and that's been checked numerically. And that's the maximum chaotic but I, which I won't talk about. Okay, so the idea is to try to use this model as sort of a window into non quasi particle transport. Yes. A green function. Sorry. Yeah. Okay. To study transport we actually need to generalize this model into a higher higher spatial dimensions. Okay, so here's a lattice lattice generalization of this model again on every every unit cell we have now these n orbitals. Okay, and there's hopping between the sites and there's a random all to all interaction on site. There's a single particle dispersion and there's a local interaction in real space that has this s a s y k form. Okay, so all the orbitals within a site are interacting in this way. And we also want impurities. Okay, the solar scattering. So for every realization of this random interaction. If we didn't have this last term of the Hamiltonian the problem would be exactly translational invariant. Okay, so a this w equals zero that's that's like a clean metal in every realization okay it's random we only know how to compute things averaging over these years, but for every realization there's actually exact translational symmetry. So we also want to introduce controlled disorder so this is, this is a single particle potential on site which is actually random inside dependent so this breaks translational symmetry. Okay, so this kind of model one can actually solve in the larger end of it shows a crossover from a firm liquid at low temperature to a non firm liquid at a high temperature. It's not quite what we want. Okay, so this is the translational symmetry. It's not what we actually want is something that shows a non firm liquid behavior even at even at very low temperature and for that you need to generalize this model a little bit. So there's a variant of this model which is analogous to condo lattice problem. Okay, there are two types of electrons the sea electrons, whose Hamiltonian is shown here, and they're much heavier so called f electrons that are dispression less. They also have their own local view, but they don't hop at all. And there's this interaction between them which is like a density density on site interaction between a between the sea and the f electrons that also has this is like a platform. Okay, and yeah. Okay, yeah, I mean I'm almost done. So it turns out that this model this condo lattice type model actually shows a marginal firm liquid behavior down to the lowest temperatures. Okay, you can you can solve this model exactly in the larger limit that's equivalent to solving basically these self consistent equations for the self energy and then you can compute connectivity. This model actually realizes exactly the type of marginal firm liquid proposed by by Varma et al. Okay, so it has a electrical resistivity that scales linearly with the temperature down to the lowest temperatures, an optical connectivity that go like along squared, specifically to just T log T and so on. Okay, it even has interesting one hope singularity. Interesting, the husband up in oscillations that a display the Fermi surface of the sea electrons but don't have the difference cost of which behavior. Okay, so what about the Lawrence ratio in this model. Okay, so we can compute the electrical resistivity we can compute the thermal kind of a resistivity. So we start from the clean limit that is without the single particle disorder term. So that gives us that the electrical resistivity is just linear and T. Okay, and it turns out that in the clean case the Lawrence ratio as far as temperature independent, and it has a value which actually deviates from the firm from the conventional value. So it's a some number this number was computed in this paper, but this number is actually not universal it depends on on details. Okay, but now we want to introduce impurities. So in the disorder case. Okay, so a somewhat to our surprise it turns out that at zero temperature the Lawrence ratio actually approaches one, even though this isn't. A marginal firm liquid. In fact, there's a slight generalization of this model which is really a non firm liquid and nevertheless the Lawrence ratio approaches one in the limit of T going to zero. But what my students, if it downloaded is that if you look at the first correction as a function of temperature to the Lawrence ratio, it's actually linear in temperature. So this might actually serve as a sharp, a sharp signature for marginal firm liquid. If you do the same calculation and an ordinary firm liquid with impurities and with electron electron scattering. The first correction would be T squared. The electrons would give you even even higher power at low temperature. Yeah. Okay, so so this is our main point. Okay, this is what the Lawrence ratio of off of a marginal firm liquid, at least within this model should show okay the Lawrence ratio to the approach one at T goes to zero. So it's a non firm liquid. But if you deviate from zero temperature the first correction is actually linear, and the scale of the variation of the Lawrence ratio is off the scale of the elastic scattering rate. Okay, and when when the temperature is much bigger than the elastic scattering rate that reaches some non non universal constant value. This is what happens in this month. Yeah, yeah. Yeah, yes. Yeah. Yeah, this this model actually satisfied Madison rule. Yeah, yeah. Square root of temperature. A good question. I need to think about that I don't think so but Yeah, so that's a good question. Yeah, let's let's chat about that later. Yeah, yeah so I'm basically done. Okay so so this is this is what looks like in silver. Okay so much more and more flat so this is really the, the prediction here. Okay, that's that's my, that's my summary slide. Okay so the winner my front line is violation can tell us something interesting about the mechanism of ordinary and straight metals. And this is what it done. It's doing it a clean metal near a no singularity. And this is a disordered marginal family equity. Okay. Thanks very much. Yes. Okay, thank you. Can you explain a little bit. How the disorder happened is between the site is among sites. Yeah, so, so the model we studied is a model where the disorders, the disorder is on site. It's just, if you like it's a it's a random hybridization between these different orbitals. I don't expect it to be different if we added a little bit of randomness and the hopping between the sites. Okay, but that's not what we did. I'm wondering in detail why there's this minus alpha T correction in the disorder. Yeah, yeah, yes, so, so right so I can give you an intuitive my intuitive understanding of that. That is based on sort of a Fermi liquid picture so I don't know what extent that's really correct but the idea is very very simple so you say in the scattering in this model of electrons on near the Fermi surface essentially momentum independent. And that's coming in both to the thermal and to the electrical resistivity, but it's always in elastic. So these scattering processes are weighted differently for the electrical and the thermal resistivity, unlike the elastic case where they're weighted equally. Okay, so if you go to zero temperature it's all elastic, and then the Lawrence ratio is obeyed. If you look at the first temperature correction, you get a different correction to linear but different for the sigma and the kappa. And that's why if you look at the ratio you get the linear. Yeah, about the model, you had the interaction on sites to be same for each of the sites. Yeah, so yeah, but wouldn't that lead like after this order averaging to some non local interaction between different sites. Yeah, it does. Yeah. So, right, if you disorder average it looks like a non local interaction. Yeah, but you can still solve it turns out. No, there's an online question. Yes. Right. Yeah, if all the scattering in the system is elastic. Is the way we learn front slow violated. I have not found an example of that. Yeah, I mean, I think that's, that's an interesting question. If there is, if we can construct a model with that is the case. I don't know if an example. What is a, what is the experimental property of a firm liquid and non firm liquid. Okay, so, right, I mean a firm liquid we know a lot of properties right so they're well defined quasi particles and that has all sorts of experimental consequences. So scattering the resistivity is D squared in a specific heat is constant and so forth a non firm liquid is anything that's not that okay a metal that's not that it's not, it's not just one thing. And I presume that the Fermi surface of your marginal firm liquid is small, not, not necessarily, I mean it's, it could be large. It's not essential I think to anything that I discussed. Okay, so, so I understand that you have okay one minus alpha t and then presumably, you know, minus beta t squared minus gamma t cubed, etc. Yeah, but then the question is about the pre factors, because I'm guessing that alpha goes as one over one over w or one over w squared. The pre factor itself of the ratio between one over one over w squared and your interactions right. So the question really would be, what would with the next order is it, which come from phonons actually completely dominate over this. Yeah, so, so in a clean metal, you want to look at the relatively clean marginal firm liquid and then the slope is very large. But it would also have weak electron phonon or then it's weak electron phonon would also help, which may or may not be the case. Okay, so I think you need to move on. Thanks. So the next speaker.