 Well, thanks, many thanks to the organizers for this chance to talk and it's always a pleasure to of course be in in Trieste So I want to talk about diffusion So but before I start I want to give a few this is going to be mostly a sort of theoretical talk concerned with some theoretical problems, but I want to spend a moment grounding it in in Experimental facts first so so some preliminary big picture questions that this talk is related to the following three They're gonna be three. So one is so bad metals are metals that have That have conductivities well above the multi of a regular limit, maybe you know 200 or 300 micro I'm centimeter and so here a couple of cooperates here as vo2. These are going up to very high very high temperatures in the resistivity Just just keep keeps going up. It's it's been argued often that transport with such a high resistivity Is not compatible with a simple quasi particle picture And if that's the case one would like to know what is the physical mechanism? What replaces the Druto formula for these for bad metals so transport above the module for regular limit, okay? another question You might like to ask is why is T linear resistivity so so widespread so the I won't go into these plots in details, but we have cooperates nictides Organics heavy fermions strontium ruthonates and these these bits in the middle of the phase diagram By definition here in the middle are all showing T linear resistivity and furthermore many properties as maybe we'll hear more later today Many of the properties of these T linear resistivity such as this the scattering rate that you might extract rather similar across these quite Different compounds where you might expect the microscopic scattering mechanisms to be Different, okay, that's another another question and finally as I just mentioned in fact Not only does it as a T linear? So the resistivity does not have units of time okay or inverse time And so if you want to associate a rate to to to a resistivity you have to do some more work The best thing maybe is to measure an optical connectivity and look at the width of the Judah peak for example And so it's a fact that many of these T linear resistivities associated with what's Sometimes called a Pantheon scattering rate set by KBT over H bar with just all the one number like one or two in front of it So there's sort of a universal scattering rate that seems to appear in many different compounds That's what this plot is showing in a rather non-intuitive way And these are old plots about this is a self-energy showing scaling Omega over T and this is an optical connectivity also showing an omega over T scaling But there are many many experimental instances that the T linear resistivity in many of these materials is associated with the H bar over KBT timescale, okay And you could ask where is this universe why why is this timescale appearing in so many different places and furthermore Even if you know you're gonna grant yourself maybe quantum criticality or something to explain this scale It's not at all obvious why this timescale should determine the resistivity of course in the Judah formula The resistivity is set by a timescale, but the Judah formula may not be valid for these systems Okay, because they may not be quasi particle systems So right so even if you grant yourself this timescale, why why does it determine the resistivity? So those are three big questions that we're not gonna answer, but they're gonna motivate what we're about to do So the objective for today is to Establish results and transport theoretical results that relate to these questions, and I want to set some rules Okay, for myself, which I have not generally followed in the past. I'd like to really prove something Okay, and by which I mean I don't want to make any uncontrolled approximations And I also don't want to use Boltzmann theory slash quasi particles, which I have nothing Against Boltzmann theory, but it is conceivable that it doesn't apply to some of these materials So I'd like to set myself the goal of showing what can you prove without assuming quasi particles and more generality Okay, and secondly, I'd like to be as realistic as possible, which in this case means I don't want to do Large n things like that. I don't have infinite coordination number as in as in DMFT and and so because both of these things Fundamentally change the nature of scattering for example large n tends to add a bath You can scatter things into and that really changes that the nature of the calculation and the results. Okay Right, so I don't use Boltzmann theory. I don't want to do large n. So so what what what what what can we do? Okay So I'm gonna do two things. So the first thing is gonna be a I'm gonna argue for a bound on on diffusion In fact, I've talked about several bounds over the years But I really want to try to prove something. Okay, so there's not going to be a conjectured bound It's gonna be a proven bound and we had a physical arguments actually over a year ago and more recently We've actually managed to prove this bound on diffusivity within a more restricted context that I'll come to and so I'm very fond of bounds because If I if you want to throw away a lot of the machinery that we're used to you Don't really have much left and it may be very difficult It may be too much to ask to be able to calculate the resistivity But putting bounds on things is a little bit easier and you could ask well How does quantum statistical mechanics constrained transport? Okay? What what can you just show and of course the fact so many different materials behave in a similar way? One way that can happen is that they all set pushing up against some kind of bound Okay, so I'm from the bounds and we'll try to prove something and Secondly, if I have time I'd also like to discuss Diffusion in a particular sort of quasi solvable model that we considered recently All right, so that'd be these are the two parts of the talk Let's talk about bounds. Okay, so this is this is again. There are many different bounds that I've talked about at various points This is probably not the same bound that may come up later in other people's talks Okay, and and so this is a bound related to causality and again So first I'm going to give a heuristic argument that we came up with about a year ago With Tom Hartman is a Cornell and my ex-student a ragu who's now at Princeton Okay, so before I do that, let me say something about hydrodynamics And so I'm going to mean hydrodynamics here in a in a weaker sense than what Cameron was just telling us about a few minutes ago So a strong sense of hydrodynamics is that it's the theory of water and that means the Navier-Stokes equation And and so on. Okay, a weaker sense of hydrodynamics means it's the collective dynamics that describes the approach to equilibrium Okay, after all the microscopic stuff is decayed what you're left with is hydrodynamics And so and in the simplest case that could just be diffusion for example So I'm going to call diffusion a hydrodynamic equation. Okay, some people Just to fix what I mean and include diffusion. So let me say this so conserved densities are special because The total charge is conserved and that constrains so the total charge is conserved Then that constrains a very long wavelength fluctuation of the charge Okay, because in the limit where the wavelength goes to infinity That's that's the whole charge and that cannot change as a function of time And so by continuity very long wavelength fluctuations of charge have to change slowly with time Okay, that's that's the essence of hydrodynamics And so the correlation functions of conserved densities about thermo equilibrium are very strongly constrained and a very nice classic paper on Is this one by Kadunov and Martin? So hydrodynamics holds at times later than Something called the thermalization time. Okay, and so the idea is if you take your your sample and you hit it You do something to it. Okay, it'll be out of equilibrium a lot of crazy stuff will happen And then what will happen is all locally reach thermal equilibrium So you'll locally establish a temperature at every point, but the temperatures won't be exactly the same And then a much later times after this thermalization time heat will diffuse and it will establish global thermal equilibrium Okay, so the hydrodynamics is this late-time diffusive process that happens after you reach local thermal equilibrium You have to have a notion of local temperature before temperature can diffuse, right? It doesn't make sense otherwise Okay, so there's a separation of time scales So hydrodynamics by definition is what happens after you reach local thermal equilibrium at some time scale All right, so at long times the non-conserved quantities have all decayed and all that's left of these conserved densities However, the dynamics of these non-conserved quantities the fast dynamics it enters into hydrodynamics Should have done this before The dynamic the microscopic dynamics enters enters hydrodynamic through the so-called transport coefficient So for example in the simplest case that we're going to be talking about where let's say there's only one conserved quality energy or charge It will diffuse and all the all of the short distance physics goes into determining the diffusivity Which is the diffusion constant so the form of this equation is determined by the late-time The structure of conserved quantities, but the value of this coefficient D is determined by microscopic dynamics. Okay Very good. So that's hydrodynamics So alright, so now I want the objective is to bound the diffusivity. Okay, because the diffusivity controls the conductivity So so we're going to use causality now causality prior to 1970 the only people who cared about causality were Relativists, okay people studying special relativity because as you all know in special relativity, there's a light cone There's a strong sense of causality that you can't propagate signals outside the light so it turns out that Non-relativistic lattice systems also have a light cone and and the velocity is called the Lieb Robinson velocity And it goes like so that the statement is this one that if I take an operator At time x and time t its commutator With an operator at time zero and space zero is exponentially small if you're outside of a light cone, okay and So this this right so this is a lit this is a light cone and this V is the is the velocity of a light cone and This velocity is called the Lieb Robinson velocity I'll give you the formula in a second, but let's understand where it comes from because it's actually pretty simple Okay, so these these people are Mathematical physicists and it all looks very complicated with these operator norms and so on but the physics the physical intuition and in fact The proof is is quite straightforward. All right, and so it goes like this Here you have a lattice Spacing a so this is the lattice spacing and imagine at time zero. I start with an operator here Okay, so maybe this is the spin z component s of z right some some operator at this site Okay, and now I evolve it in time with a local Hamiltonian, right so perhaps the Hamiltonian is something like s i z s i Plus one z okay, and let's say I have the operator s x at site 20 whatever okay, so When I commute so how do operators time evolve in the Heisenberg picture what you have to commute them with the Hamiltonian So if I commute to this operator at site 20 with this Hamiltonian, I'm going to get an operator at sites 19 20 and 21 right so the operator as you as you commute with a local Hamiltonian the operator grows Okay, does that that makes sense? So if I take a local Hamiltonian The the operator equation of motion as time passes times going this way every time you at each dt Small interval of time you commute the the operator the Hamiltonian at the Hamiltonian's local The operator will start growing and this and essentially this is the light cone Okay, as time passes the operator only grows the number of times you act the Hamiltonian and this defines a velocity And what is the velocity? Well, it's set by dimension analysis It's the coupling J of the Hamiltonian times the spacing a divided by h bar. Okay, so this is a Lieb-Robinson velocity Very good Okay, so so and and so you cannot it's not it's not as quite as strong a light cone as in as in special relativity But it's a light cone nonetheless So if you do something here up to an exponentially small tail you can only influence stuff Inside a light cone. All right. Now. What's one way that you might try to yes Velocity is not universal it depends on the operator Microscopic velocity and the energy scales are involved in the problem It's a microscopic energy scale If you have multiple energy scales in the problem, how do you fix this? That's it. Sorry. Thank you. So Very good. It's bounded by You take you take all possible couplings between neighboring sites and it's the biggest You take like the local Hamiltonian that couples one site to the next one and it's the biggest eigenvalue of that Hamiltonian So the biggest energy scale wins. Yeah, that's because that's the fact that's what lets you go the fastest Yeah, thank you, but absolutely. This is it's a it's a microscopic velocity It's not unreasonable to hope that this would not be too different from the Fermi velocity in many systems, but but you're totally right So now okay, so we're not allowed to go outside. It's like we're not allowed to send signals outside It's like on certainly not big signals. And so what's one way of sending a signal? Well, I Just wondered, you know, you wrote down with it. That's an exponential decay. Why is it not t-squared? I mean you would have thought based on knowledge of diffusion that No, there's no there's no diffusion yet. Well, I know there's no diffusion. No, no diffusion is coming No, so this is a this is a this is a linear So I think what you're asking is why yeah, no, this is what they proved This is it's this it's this fact that so it's essentially this picture that that the Hamiltonian acts sort of linearly in time And so the operator grows only in time. I mean that's it is a proof and those are words But but it is but indeed indeed. This is the whole point So one way I could send a signal to you is by standing here and making some ripples and the riffles would diffuse out to you Okay, so how fast does the fusion go? It goes like the square root of t right and in diffusion the distance you travel is Goes like the square root of time. Okay, so now we see that we're gonna have a problem It's one of the lattice spacing Essentially, I can't hear yeah the coulomb interactions will spoil this. Yes. Yes. Absolutely. Yes I mean they're screened in practice, but indeed so Very good for power law tales think essentially what you want to do is think of the power law as Imagine it as sort of a sum of short range interactions of different lengths and with different strengths and then you sort of have to Replace this by a sum of Like, you know as you go further and further away and sometimes I converges sometimes it doesn't and and so it's tricky Yeah, indeed long range stuff is tricky Okay, so I try to send something oh before I get to diffusion So most of the time in most discussion of the Leibovism bound it's treated in spin systems For example, it's treated synonymously with a spin wave velocity And so it's obvious that it fix up if you're if your spins are ordered and you have a spin wave that propagates Ballistically then the spin wave velocity had better be less than a Leibovism velocity Okay, that's that's clear, but now suppose you have a diffusive case So diffusion propagate sends things out I can send you a signal in a square root of time like like this And so let's put this on the on the space time diagram So here's his time and space. This is our like on and this is the square root of T All right So late times you're fine diffusion is subliminal if you like, okay, but the early times diffusion is too fast It takes you outside the like on all right So is this bad does this mean diffusion doesn't exist? No, we should all be happy because I told you that hydrogen diffusion is hydrodynamics And that only happens at late times after you establish local thermal equilibrium. So for consistency with causality T thermal had better be at least here if thermalization if thermal equilibrium local thermal equilibrium is established too quickly Then diffusion is too fast. If you have to wait up to here before diffusion starts, then it's always subliminal Okay, so you do a simple thing you take this picture You you put a line across here for T thermal and you require that it be above this point That that that makes sense Right, so so let me do it again We do this we have two lines We intersect them and this point T intersection should be less than T thermal T thermal equilibrium Okay, simple. I like simple like the more the older I get the more like simple simple I'm sorry in order to get a diffusion you need to couple your system to something to the zero or a disorder, etc No, no, so this no no no no no and so this is very good. Firstly this diffusion This is not a diffusive so like with disorder like that the electron propagator actually becomes diffusive Which is not what I'm talking about. This is hydrodynamic diffusion that The only things that diffuse are conserved quantities at a late time and interactions are crucial So so talk to me. It's sound Oh, no, no, no, but that's because there's energy and number density, so there's not diffusion, right, right, so Fantastic, I'm what I'm talking about now is the case with is only one conserved quantity when you have more than one You get sound waves and then and then the right thing to think about is is this not in terms of diffusion So I'm talking about for example camera and systems insulator Maybe spins in a disordered a disordered phase Thermal transport and insulator charge transport if you can neglect them electric effects So diffusion happens when there's only one conserved quantity scalar conserved quantity at late times Very good. Okay, so so this is a naive argument Which is does have loopholes and so that's why I'm going to give you a rigorous argument in a minute Okay, but this is the physical the physical picture And so this this naive argument will tell you that the diffusivity is bounded above By this leave Robinson velocity squared and this termization time all right so so Okay, and so this tells you that the diffusivity cannot be too fast given a termization time And also if you know the diffusivity you can't thermalize too quickly So if we go ahead and apply this to electronic transport Neglecting thermo electric effects. So just think about charge it bounds the resistivity. So the resistivity is Given by the Einstein relation. It's one over the susceptibility kind times the diffusivity And so dropping all the other stuff this bounds it by one of a tau thermal So I like to think of this as a quote generalized Judo formula in the sense of the Judo formula tells you that the resistivity is equal to one of a tau You know transport time with some prefactors of course and What I'm saying is that without assuming the existence of course hardly assuming anything There is still exists a relationship between a time scale and the resistivity which the resistivity Is bounded below by this thermalization time If I have a weakly coupled system the thermalization time is in fact the transport lifetime Okay, and this and this inequality actually becomes an equality. Okay, but more generally there's still a relationship between Transport and and and a time scale that holds with or without quasi particles. Okay, so Good. All right. So another way of saying that the fact that there is a timescale that controls the resistivity Does not imply that there are quasi particles because it's a more general thing And also you see for example in this formula if this time still starts getting very short So the thermalization starts getting faster and faster for example at high temperatures The resistivity is necessarily pushed up and you necessarily get a bad metal So for people who there are loopholes in what I just said and so We finally mainly actually my student DJ actually managed to to prove a version of this and I very quickly Want to give the outline of how how it works because I think it's interesting Okay, so I would call a heuristic argument what I just gave you and and here's a sort of proof Of course to prove something we had to be a little bit more restrictive So let me very quickly tell you how this worked So the right so thermal equilibrium is a very complicated complicated place a Simplification of it is in so-called Limbladian Dynamics where you basically imagine integrating out the thermalizing bath and just focus on some subset of so you imagine you have some Electrons the electrons a couple to some phonons and you just trace out the phonons and and the role of the phonons They give you decoherence of the electrons. Okay, and you just look at the electron system The shorting your equation is replaced by the limblad equation And so and so the and the Heisenberg equation is replaced like this or some operator Dot is what it would be. Okay, this would be the Heisenberg the normal thing But now you have these L's which are these decoherence operators. Okay, and so what this equation is is it's the most general equation So in particular You could evolve the density matrix like this It's the most general equation that is first order in time Linear and preserves the positive the complete positivity of the density matrix because so in quantum mechanics Right the Schrodinger equation is unitary because you want to preserve you want to preserve the normal the states Okay, but actually in an open quantum system. There's a generalization of that Which is you want to preserve the trace and the positivity of the density matrix It's a slightly weaker assumption and this is the version of the Schrodinger equation that applies in that context an open quantum system it's simpler than a thermal state because Typically if you integrate out the phonons the phonons are going to have a lot of time scales And it's going to make the effective electron dynamics non-local in time and the limblad equation throws that away with the so-called Markovian property, which is that it's first order in time now another nice thing about this limblad dynamics is that in in in a unitary evolution in a thermal state of course norms are preserved, okay, and oh So diffusion with something you'd see at the level of expectation values like the expectation value of the charge would diffuse But actually in these Lumbadian dynamics the whole operator just decays so diffusion They can arise as an operator equation in Lumbadian dynamics, so it's simpler Yeah, I don't know how much detail want to go into but the idea that basically there's there's going to be an operator C of K, which is that the Kate there fluctuations the charge on wavelength K that obeys an operator equation, which is going to be diffusion and by Doing sort of perturbation theory in small k. We got a formula for D Yeah, I think I'm just gonna flash this. Okay, so for people So we got a formula for D. We calculated in the xxz model Some simple model and it agrees very nicely with previous calculations. So it's correct See how much let me do I want to Yeah, I think I'm gonna maybe yeah, I want to get to the next point So I'm gonna go through this very quickly. Yes Not with the phasing. Thank you. Sorry. So this is the xxz model with on-site the phasing which which which is diffusive Yes, he was the The question slash complaint was that the xxz model is integrable But this is a X model with this on-site the phasing So you imagine you've coupled it to some phonons and then it has diffusion and actually this has been studied by by by by prosen and collaborators in Slovenia and We've reproduced their results. So, okay, it turns out as a formula for diffusivity We do various mathematical steps and you usually Robinson bound And and we prove the balance diffusivity, I'll just I'll leave it at that Interesting thing happened. So this is the lesson is that it's good to do rigorous arguments because in fact the bound changed a little bit So there was the diffusivity is has it has to is bounded by this timescale and these velocities But there's actually a sort of a constant shift by this quantity zeta, which is the range of the interactions Okay, so the bounds that we had before the sort of heuristic bounds the diffusivity was less than Some velocity squared times the thermalization time But in fact, there's an additive contribution That goes like a velocity times The range of the interaction. This is a microscopic range. So this is probably like a Okay, and that's quite interesting because what that means what this bound allows is that if you imagine increasing temperature This thing presumably Gets shorter with the lengths to get things get faster at high temperature But there's there'll be this microscopic length scale and when you go above that It'll the bound becomes this and so it allows this region. So this is resistivity, which is one over diffusivity. I'm sorry And so what what this term actually does is it actually allows? Resistivity saturation I'm sorry. I went through that quickly the main takeaway message was that you can put some meat behind this heuristic picture And if people are interested in the body and dynamics, I'm happy to say more about it So I want to spend the last five or ten minutes talking about now something a bit different, which is a microscopic model of bad metals with diffusion and this was with my student Connie Musetov and Ilya Steres is a student of Steve Kovacins All right, and so again the idea is to study diffusion in a bad metal Where we can actually calculate things properly And so let me make the so we start with the following observation So a good candidate for bad metal would be a strongly correlated Let's say 2d Hubbard model where you is much bigger than T And the first thing that would come to your mind and in fact did come to people's mind in the 70s Is that maybe you could do perturbation theory in little T However, that's a very bad idea because in the Hubbard model if you set little T to zero the interactions are purely on site Okay, and so each electron just doesn't even know about all the other electrons and so the spectrum is massively degenerate Okay, it has an extensive degeneracy. Is that that makes sense if I turn off the hopping term But there's only an onsite interaction. I mean, yeah This is there's an extensive degeneracy and perturbation theory about that state is essentially impossible Okay, everything will be infinite All right now, however, let's There's a simple thing you can do which is I guess to undo probably what what mods so I Hubbard Wanted to do originally so of course this onsite interaction is an idealization It's not what actually happens. What actually happens is maybe that there's an exponential interaction between sites Okay, so the first point that did surprise me a few months ago, but maybe it's not surprising in retrospect That an exponentially decaying interaction with a very short range Okay, a range. Let's say if the lattice spacing is one the range only has to be like two lattice sites is enough to Completely lift the extensive degeneracy and give you a continuous spectrum Okay, so I could so as a physicist you would say what possible difference could there be between an exponential interaction with Range two, okay, and the onsite and maybe a next to nearest term. Okay. Well mathematically, there's a huge difference Okay, but if you keep to purely a finite range interaction the t equals zero Model will be always be extensively degenerate and exponentially short range interaction Is not degenerate. Okay, and you can just prove that. Yes Must be degeneracy associated with spin if you're yeah, of course absolutely. Yes. Yes, but it's not it's not extensive in space I think It's totally right that the sorry Yeah, this doesn't mess things up for what I'm doing. Yes. Yeah, you're totally right of course if this but I mean you could trivially lift that right by Coupling spins Right, but yeah, of course. Yeah, this doesn't lift any spin spins are gonna be quite inert and everything I'm about to say Yeah, so yeah, yes the question was are we gonna take you and V or just keep V so just To be as convincing as possible what we did is took both and for the more we took V to be one-tenth of you so there's a small well to be small perturbation of the hub and model but but Obviously the V this includes well actually we separated out the you but I mean It's not very important. Yeah. Yeah, so but indeed we're gonna we're gonna take V to be zero point one times you just Make it a small effect. Okay, but so just again You know the habit model is supposed to make life easier Okay, but I'm putting an onsite interaction But if you want to do small t perturbation theory the habit model makes life harder than than it would have been with the Exponentially short-ranged interaction, okay So that's just a technical point So and so you can prove that this thing has a continuous spectrum. It's not difficult when when else to okay So let's take L to be two and so now we can just take this model and start doing small t perturbation theory on it And the nice thing is so the t equals zero model It's a bit complicated, but you can find you can solve it by doing classical Monte Carlo because all of these terms Commutes, okay, and classical Monte Carlo is essentially trivial. Okay. I mean, you know you can Put on a computer you put on your laptop and you do it. Okay. It's not corner Monte Carlo. It's it's it's doable All right, and in particular you can start counting the connectivity. So so again, you could do classical Monte Carlo In small t perturbation theory, so this you can just derive this formula for the connectivity So what is it? It has a t squared because the current operator Has a t in front. Okay, and then but when you calculate the expectation value Then you just put t equals zero because you're doing perturbation theory And so you have a sum over configurations of occupation numbers with a Boltzmann weight And then the connectivity is given by some spectral weights, which is basically how many ways Can you hop from one site to the next site such that the total energy changes omega? Okay, and you sum of all of those and that's a connectivity, right? So you do classical Monte Carlo you generate configurations like this So this is like zero one or two right the the the guys on the site This is now the energy right because the energy has this on-site term, but also this long-range term All right, so this so you sort of see that to Hubbard back There's an upper band a lower Hubbard band, but now they're smoothed out right that they're they're They're smoothed out by by this long-range interaction, and then you do the sum and this is your druter peak except it is calcium As I explained in the second and this is the optical connectivity over a wider range So there's some Hubbard bands which have been smoothed and there's this interaction Of course in the full Hubbard model the Hubbard bands are smooth But you have to work non-perturbatively in little t to see that here We can work perturbatively in little t and the v interaction smooths out everything for you okay, that that's the name of the game and Of course, we have to be at high temperature. There's a cost to doing this which is that Everything has to be bigger than little t so that includes the temperature Okay, so the temperature doesn't have to be big compared to uov, but it does have to be big compared to little t So this is not in your degenerate, you know, it's not quite where you want to be but this is what you can do easily So you just calculate it and you do run a computer and you you you can calculate Connectivities resistivities Alright, so there's there's a nice t-linear resistivity ish In calculate time scales and so on so I let me very quickly say a few the whole point of having solvable models Or quasi, you know, I think classical Monte Carlo is essentially solvable as far as I'm Concerned so it's to get some intuition for what's happening. Okay, so so this is a very bad metal Okay, the resistivities this is a u over t here and so t is much less than you so this is a super high resistivity What happens essentially is that you get this is like the on-site energy. Okay, so there's basically an emergent There's an emergent disordered landscape due to the interactions. Okay, so There's no there was no disorder in this problem But this is a typical configuration and as you can see it's quite disordered and so essentially what's happening is electrons are hopping around In in this very disordered configuration and because of this hierarchy of scales The landscape doesn't change The current decays more quickly than the landscape changes So it's a lot it becomes a lot like a disordered problem Which you could then worry about localization, but I don't have time to talk about it So you just go ahead and calculate right so so this till in your city in this range Actually, it doesn't come from that. This is the lifetime the The width of the Judah peak. Okay defines a lot of thermalization time and actually it doesn't it changes by a factor of two But it doesn't change very much what what's driving the till in resistivity is actually the the Judah weight is spectral weight transfer And that's reasonable because we're at very high temp. We're high temperatures But we're not at super high temperatures. Okay, there's like this again There's sort of trivial regime when T is much bigger than you then that's just everything's completely classical So we're not quite that high. Okay T is small compared to you, but it's it's not degenerate I think I'll be one more minute. It's three one this this This linear with this festivities is not exactly linear or approximately near is quite smooth But actually if you look at the diffusivity it actually shows a crossover at at you from being constant In the high-temperature regime to having some temperature dependence here And this doesn't come from the lifetime actually comes from a temperature dependent velocity Which is related to the spectral weight changing with temperature, right? Is it the spectral weights like the kinetic energy which defines a velocity which is temperature dependent? So this is again where temperatures much above T So this is not what is happening, you know in most of the metals that you care about but it's a model for bad metal transport and curiously It's too high temperature for metals, but it's not too high a temperature for cold atoms Which is hopefully you all know and not that colds. Okay in absolute absolute units and so More or less in parallel with our paper This is this is supposed to be a cold atom realization of the Hubbard model with you basically eight T And so this is almost exactly the same temperature range that we're looking at. Okay, of course our T is much smaller and So look the defeat unfortunately didn't plot one of the diffusivity, but this if you if you take one over this Okay, it really looks like it looks like that And this is the susceptibility if you plot one over that it really looks like it looks like this Okay, and so here you see that the diffusivity has a temperature dependence But the resistivity is actually fairly fairly doesn't have this feature. Okay, so there's an interesting thing which seems to be happening which is that in This case that T linear resistivity sort of masks and not a non-trivial change in behavior that happens at this scale you Actually similar things from quantum Monte Carlo that I will not talk about. Oh and just to connect also Vitamin franz is super violated In this regime. These are high temperatures. So you don't expect vitamin franz to be true. Okay, however L zero the Sommerfeld value is still a rather useful ballpark and you can get L Much much less than L zero in this in this high-temperature regime That that's this plot and actually possibly connected with Something observant via to all right. I'll I'll just stop there