 Ddweud. Ddweud. Ddiolch i mi. Ond yno, mae'r ffordd yn fwy o'r cyflwyno a'r sesfynol gyda'r 60 ysgolion yng Nghymru, a'r chweithio'r ysgolion er mwyn o'r Ffyrddol o'r ffyrddol i'r ffyrddol gwahanol i'r gwleidio. Rwy'n meddwl i ddweud, oherwydd i ffyrddol o'r ddweud o'r ddigon a'r ffordd ym mwyn o'r rwy'n meddwl i'r ffordd o'r ddweud o'r ffordd o'r ddweud o'r ddweud o'r yw'r rhan i'r cyfnod erbyn wych yn cael ei maen nhw yn ei gyfnodd. A os oeddwn i'r cael eu ardangos o gyda ni'n ffrifoedd yma, oedd yn eich huw hyn, Rydych chi'n cyfun o'r gwybod, o Ahmadog o'r meddorol, oherwydd o'r hyn o'r Chondduryd mewn ffii, o'r cyfnod i ni'n gwneud, oherwydd o'r brifannig oedd drylfaethon ar hyn. Mae'n byw'r cy maggol rannu gwiriant a'r pryd, oherwydd o'r pryd deoedd yma oherwydd o'r Friedeg. felly fawr oedd am 5 oed. Felly, rydyn ni wedi'i hynny ymddangos i'r ddweud, y gynhyrch yn ymddangos i'r ddweud, ond ychydig yn ymddangos i'r ddweud, rydyn ni'n credu bod yn siaradol bod yn cymryd ar y cyfathol, ac ond yw'r amdano i'r amser i'r rhain. Felly, e'n gweithio'n achos i'r awr sefydig ar gyfer y ddaeth, ymdyn ni'n gweld yn ysgrifetio, ac mae'r cyfathor i'r ddweud, Na'r wahanol sy'n cyflaenwyr Iesaf wedi chi'n ei gwrtho'r rhan, yn gwertho'r drwy am y briech, oedd y gallwch ei ffrinog yn ddifhau, sydd ymateb i chi i'w iawn, os ydydd eich o'r wahanol o'r wahanol auh, mae'r ffordd o'r difnodau os ydydd yng nghymru yn y blaenwyr, oherwydd mae'n gweithio'n effaith, ond mae'n bryd o'r marg, nifer o'r rhan o'r brныdd. Rydw i chi eich eimlaenwyr i chi fydd y gallwch chi'n mynd i chi Ac mae'n gofio i'r cyflawni am y cyflawnu'r ystod i'r cyflawni'r Cymru, mae'n cyflawni'n Cymru, Ddodd wedi'u gwybod, ddodd yn fwybodaeth bod y cyflawni, Dimitri ac Andrei, i gydig i mi'n bwg i'n cyfrannu i'r tristau, mae'n wneud yn y gwaith i'r ffordd o'r cyfrannu ar gael yr hyn o'r cyfrannu o'r cyfrannu mewn wneud. Mae'n gwybod i'n nhw'n ddweud o Y Llyfrgell Ffrandslaw yn WP2. Mae cyfnodd ymlaen iawn yw Alexandre Jaby, mewn gwahanol yn ei ddechrau eich drwsgwrs. Felly, gweld i'i ddweud am gyfnodd, ac mae'n ddweud o'r wych yn Cymru, o'r cymdeilio o'r Chlodiol Felcer yn CITP Santa Barbara, a'n ddweud i'n fath o'r ddweud i'r newid arnynneu'r gyfer y lleon, Dimitri, ac yr Alaskar Subedi. I'm going to refer to other stuff during my talk which has been done all around the world in Tokyo, Yomachida, measure thermal conductivity of black phosphorus, I will show a slide about this, and also some work in Rio de Janeiro done by Valentina Martelli on the strontium titer. So thermal conductivity is very, many of you are not familiar with it, but conceptually it is as simple as electrical conductivity. Fourier's law is the equivalent of Ohm's law, it's something you expect in any diffusive medium where you can put, you know, under scrutiny what happens if you put a force, it could be electric field or a temperature gradient, if you have carriers of charge and entropy you have basically a linear response and you have, you can measure it, and this is basically the root picture which is with us since the beginning of the 20th century. Now ironically the law of Whitman and France was discovered 50 years before and it is very simple, if you measure elements at room temperature and you look at the ratio of these two conductivities, this is what they discovered in 1850 something, actually you come close to a number which is not exactly the same, you know, it's important to see that you have at room temperature it is an approximate law, you are not very far from one number which is of the order of 2.45, it took 50 years and the quantum revolution for people to understand what are these numbers, this is basically the Sommerfeld number with this pi 2 over 3 and the other guys are basically the quantum, the quantum of entropy, the Boltzmann constant and the quantum of electric charge, the charge of electron. Now the Whitman-Franc's law, one of the oldest law of physics has been one of the most attacked law during the last 20 years for different reasons in different contexts, but all of them, maybe not all of them, but most of them has turned to be wrong, you know, to be honest at this moment we don't have any solid evidence that in any system the Whitman-Franc's law is broken at zero temperature for different reasons and now I'm going to talk about another aspect but before going there we know that Whitman-Franc's law should be valid or expected to be valid at zero temperature but we know also that there is a finite temperature departure at intermediate temperatures as soon as you put some inelastic scattering then the fate of heat transport and the fate of charge transport don't coincide and well one simple way of thinking about it is that when you are measuring charge transport you are looking at the momentum flow and there is an angular filter one minus cosine theta which means that you are if you have a lot of small angle scattering then that doesn't matter that much for charge transport but that matters for heat transport which is basically about the energy carried by particles and you see in the old school like in in Zeiman this is called horizontal versus vertical scattering in the sense that in one case you change drastically the energy of the system and the other case you don't and what is expected is a downward deviation okay and actually we have a good theory about this if you look at electron phonon scattering as you know at low temperature this block grunaisen law gives you a t5 resistivity part of it is actually part of this large power law comes from the fact that as you cool down the system most phonon scattering become small angle so you have something which is not the familiar t-cube law that you expect according to the doby picture and basically the phasor space for scattering for heat and charge are different there is also a difference for electron electron scattering both are suggested to give t2 are you know expected to be t2 because because of the poly exclusion principle you expect the phasor space for scattering electrons to be always proportional to square but for the same reason that you some of these electron electron scatterings are vertical processes some of them horizontal the two numbers the two prefactors are not going to be the same and this has been checked for example in this particular case of cerium rhodium indium 5 by louis typhers group you see this is a heavy fermion system which becomes anti-ferromagnetically ordered and you can measure the t2 both in the heat channel and the in the charge channel and the two slopes are not the same okay up to this is just firmly liquid transport one thing so there is a new angle of attack to ritman franc's law and this comes from hydrodynamics so you have probably heard about this during the last couple of years and I think the next speaker shun hartnol is going to talk about hydrodynamics more but let me give you this very very simple picture so this is a very now classical paper by gurzi four decades ago and you know it's a very interesting paper and actually it is it reads very well I would take this out as the main message of the paper that when we think about transport in metals we are most of the time almost always interested in what relaxes momentum during a collision the whole Boltzmann picture is based about this quantifying what happens to current flow to have to flow of charge to flow of heat when electrons are scattered and during this scattering they lose some of their momentum or or the energy flow is decay but this is not what happens when water flows and we talk about Fermi liquids but we don't at all look at it the way people in hydrodynamics think about the flow of fluids because in in the case of a liquid what predominates or momentum conserving collisions and actually this can happen so he mentioned two particular cases the case of electrons in metals and the case of phonons in insulators if by some particular process for example by making the system very pure you just abolish any kind of momentum relaxing collisions then the whole fluid of electrons or or phonons can drift under the influence of an electric field or a thermal gradient and in that case what you expect to see is that the viscosity is going to drive the whole flow picture which is very very different from what we see in the case of metals so very recently Andy Mackenzie's group in Dresden have seen the you know evidence for sorry for this kind of behavior in this very pure oxide palladium cobalt oxygen too but and this is one message of my talk and then we come back to this this is a very subtle effect they change the sample size so this has been done thanks thanks to Philip Moll's you know macrostructuring they change the sample size by factor of 75 and instead of seeing just a variation which corresponds to the standard Boltzmann theory they see a difference by a factor of 30 percent this is small this is not a huge effect now very so this is the work done also in Dresden by claudioffencer and her collaborators and you see again they measure the Widman France law this time so they looked at another system WP2 is a very pure system the residual resistivity is very low normally we discuss in micronsentimeter this system has a residual resistivity of the order of four nano ohm centimeter the triple R is in five digits so the mean free pass is extremely large extremely long and then if you measure thermal conductivity and electrical conductivity then they found that actually they are violating the Widman France law and of course when she gave this talk I told her we want to measure that and actually even if this paper is not still published it has created many many theories about how can hydrodynamics break the Widman France law so as experimentalist we have our own setup you know the measurements I showed were two point two electrode measurement and you can have thermal resistance of your contacts they they can do it because they are working on very small samples but if you get a regular crystal of WP2 and we use our four electrode setup one heater two thermometers so it's not a it's a tricky measurements because thermal conductivity becomes very large because electric resistivity as you see changes by many orders of magnitude so it can be measured and here is the result well that was the result by two contact measurement and the Widman France law again is saved we find again like any other system that if you go to low enough temperature and you have error bars within 10% you recover the Widman France law but the main message of that paper was not wrong in the sense that the deviation from what you see well is not maybe as large as what they are saying is much larger than anything known so cerium rhodium indium 5 is a strongly correlated electron system where this deviation downward is driven by electron electron scattering this is just a silver wire we measured just to test our setup and we find basically what other people have found there is these two guys both show a deviation less than a half 40% 50% maybe here here the the ratio drops to one fourth of what you expect at the temperature of over 10k now what is nice about this system is that we can measure both t cube sorry both t square resistivity and t5 resistivity it it has a carrier density in the intermediate range where the t2 term is large but it doesn't completely abolish the t5 behavior so if you do it you can basically plot your resistivity as function of t2 you see that at some temperature it deviates upward and then if you subtract this part you see that it nicely follows a t5 behavior and if you measure your thermal resistivity basically you have also a t2 behavior then the upward deviation but this upward deviation is fitted by a t cube behavior so we are back in familiar territory we can quantify these two terms and the first question is what is strange in this system is it electron electron or is it electron phonon and the response is it is actually electron electron there is a five time mismatch between these two prefactors and so the whole thing okay and then we can compare the t5 and the t cube of this system with let's say silver and we see that actually there is a normalization factor which is by the way interesting because in spite of having a dubai temperature larger than uh than noble metals it's a t the electron phonon coupling seems to be larger but it is scaling in the same way for thermal and electrical so there is nothing particular about heat transport being affected by anomalous electron phonon coupling but something is fishy about electron electron scattering well this is a semi metal this is a vile semi metal if you wish it has vile points but far from your fermi energy i don't think it matters at all for the discussion i am going to have here and the first thing we can say that actually what's happening here is that we have a lot of small angle electron electron scattering and you see this is one of the processes we can imagine you can imagine an um clap uh small angle scattering there are two electrons one of them is going to suffer an um clap scattering so it is going to move from one part of the brilliant zone to another one but the other one is a small q scattering and you see the whole the only thing is based on this that if the collision rate has such a distribution very much skewed towards small q then basically you can explain our results now actually this kind of experiment has not been done on many systems we found in all literature actually there is a old work on tungsten which beats our record it is even lower than our case both these systems tungsten pure tungsten and tungsten diphosphide are very pure look at their residual resistivities and uh the other systems which are not as pure as that don't show such a strong deviation so the whole question is is this something to do with hydrodynamics i come back to this or is it basically again uh something that you can explain in in transport picture so uh dimitri uh has a paper which is not yet published which shows that actually what was suggested in this old paper that there should be a universal boundary about what b2 divided by a2 can get is not correct actually you can get this number as as small as you want if you have a very long screening legs so maybe the whole thing is that the screening legs in different systems are different okay now i tried to be more sympathetic to this point of view does hydrodynamics play any role in this and the main message is that we cannot really talk and you should bear with me because it's it's not so anyhow don't forget the first message if there is anything about hydrodynamics it's a subtle discrepancy with the standard picture and for the moment in none of the systems are talked about anybody has calculated either a2 or b2 b2 from first principles normally it should be accessible through theory because we know the Fermi surface in great detail dft works and quantum oscillations are there so normally this should not be something beyond standard theory it has not been done and it may be that there is something else for this i want you to look or recall or maybe you know you don't know because we are working on metals we are not familiar with quantum fluids like helium tree in helium tree thermal conductivity at first approximation is one over t viscosity it's a fluid is actually behaving like t to the power of minus two and actually if you translate this kind of data into our language language of people working on electrons it means that actually the same data can be plotted in this way the wt the thermal resistivity this is basically the inverse of cup over t is t square and this t square goes up to some temperature and then you have a deviation and actually you can put helium tree on the kadowaki woods plot nobody has done it before but it does this is basically electron electron scattering now the red points here are our b term which is always larger in all systems than the a term but you see that basically this is also electron electron scattering but this is sorry fermion fermion scattering but these are momentum conserving events these are not momentum relaxing effects so before coming back now to electrons let me talk a little about for hydrodynamics of phonons because in this case actually i think both the experimental data and you know is more a is let's say wealthier i have has a longer history and the theory is maybe simpler in the case of phonons we know that actually you have no electrons to be bothered with if you have an insulator basically the thermal conductivity of an insulator has a peak and you go through different regimes and it is possible that you have this regime first you know highlighted by gurzi which is the poiseu regime now this poiseu regime has been seen in several insulators first in helium three and helium four bismuth hydrogen and just during the last year in two different systems where actually the normal scattering between phonons is large enough to see this so the reason actually doing phonon hydrodynamics experiments is easier than electron hydrodynamics experiments is that in the case of electrons it is rare or unusual to be ballistic wp2 or palladium cobalt oxygen are rare examples in the case of phonons this is the default case for any good crystalline insulator so the whole question is does the poiseu regime popping up between the ballistic and the diffusive regime so here is for example the case of strontium titanite in the case of strontium titanite between the ballistic regime at low temperature and all these different regimes we have a neural region where the thermal conductives behaving faster than t-cube and this is you know the symptom you you can only explain at the moment with the fact that the whole phonon liquid is drifting under the influence of a temperature gradient momentum conserving collisions outwey momentum relaxing events same is true for black phosphorus again we saw this t-cube behavior in both cases there are reasons to believe normal scattering among phonons is very large now let's come back to before coming back to electrons let me put it in this way what we know about the phonons is this if we want to be in this gwrzi regime we should find a region between the ballistic region where what matters most is collisions with boundary and the diffusive regime where what matters most are momentum sorry is yeah when we go from momentum conserving smaller than momentum relaxing than than top boundary a region where you have this intermediate hierarchy where momentum relaxing can be neglected but boundary collision is not also as frequent as momentum conserving events no this gives rise to something which is of the order of 30 percent rise in thermal conductivity between different systems even in sodium in regular crystals you don't see a very drastic effect it is a very subtle deviation which says that you have an extra contribution to thermal conductivity which cannot be explained in collision-based picture just in the same region people have done second sound so this is maybe even more compelling than a steady state second sound means that you send a heat pulse and now the temperature propagates as a wave not diffusively as you can see you can see the ballistic peaks here and then in this narrow region there is this second sound peak which has another velocity its velocity is different from ordinary sound and it vanishes both in the diffusive and in the ballistic region now the question is can we these things in metals so if now i use the data we have we can build up a region where the gurzi hierarchy if you wish can be accomplished but first of all this is a very very narrow region okay incidentally it is very close to where we are seeing this minimum in the lorence number but as soon as you change a little the parameters because what i showed you was based on this idea that our four nano ohm centimeter residual resistivity is uniquely ballistic which makes sense because the mean free pass actually we get from this number is 140 micrometer and the diameter of the sample is 100 micrometer but who knows actually we can change this repetition between residual resistivity due to ballistic transport and residual resistivity due to impurity transport by factor of two and as you do this actually the whole window of hydrodynamics closes up so at the moment if you have a conservative you know approach one can say that the deviation from widman frost law can be explained entirely in the boltsman picture on the other hand since we don't have any kind of quantitative description of our data in the boltsman picture we cannot rule out that the presence of this particular hydrodynamic regime is something which pulls down partially your lorence number compared to the somerville i want to finish my talk with saying that you know i'm talking about very old subjects i'm one of the reasons that i'm not sure that we really understand what's going on even in the simplest case so i think most of you know a family which should have a T square resistivity it has been known since 1937 and it's very simple because uh you know it's uh it's basically comes from argument of poly exclusion the phases space of scattering between electrons should uh scale with T square now the problem with this is that if two two electrons bump to each other and momentum is conserved it is not clear why there should be any effect on transport and we tried to highlight this by looking at the case of strontium titanate where resistivity is T square it goes you know down to very low carrier densities as you can see you can uh you can change the carrier density by four orders of magnitude and basically you see a smooth behavior of this T square behavior and you have a region where as i'm going to tell you in one minute none of the mechanisms we know about momentum relaxation works now this is not particular to strontium titanate many of these systems you know them because they have been considered to be topologically interesting you can look at the resistivity data it is T square some of them are compensated some of them are uncompensated some of them a low enclap scattering some of them don't what is funny is that if you just know the fermi energy of this system you can guess within the order of magnitude your T square resistivity pre-factor and the fact is that we know the bubble mechanism which needs two reservoirs we need the enclap mechanism in the case of strontium titanate at low doping none of them works somehow when you put electrons in a lattice the lattice taxes any exchange of momentum between electrons and the reason i'm putting this uh you know on the stage is that since the order of magnitude i show this kadowaki woods region seems to be within one order of magnitude the same it is very unlikely for each of them you need one mechanism the other mechanism and this is something that i think the chairman of this session has many things to tell us he has a very nice review paper about this but that is my last you know message the summary of talk is that okay the widman front slot zero temperature again is preserved so another attack to the widman front slot leta zero temperature has been you know that doesn't work it seems to be very robust uh we see a strong departure at finite temperature and we can identify it is caused by electron electron scattering and not by electron phonon scattering qualitatively this behavior can be explained within the boltsman picture just by invoking a lot of small angle low q scattering between electrons but we don't have any uh quantitative handle on that what comes from maybe this is the most interesting part of this if you put t square thermal resistivity which is there in helium tree and it should be there in all of these Fermi liquids you have another handle of the system because you can separate momentum relaxing and momentum conserving collision between electrons and this is at the heart of hydrodynamics and finally to be optimistic in this system like other systems there is a narrow temperature window where you can actually be in the hydrodynamic regime but marginally thank you for your attention