 Hello. So welcome to the afternoon session on transports. And so in this session, we have three talks, two talks before the break. And our first speaker is Nigel Hasse from the High Magnetic Field Lab at Nijmegen. Please. Thank you, Mr. Chairman. Good afternoon, everyone. First of all, I'd like to start by thanking Dmitri Andre and Andrew for, first of all, for inviting me here to Trest again. It's always one of my favorite places to visit. But also for arranging such a great conference very much in the spirit of these workshops, and I'm sure where Piers here, you'll be very pleased with the way it's been going. And also, having just had the pleasure of co-chairing a conference with Piers a few weeks ago, I would like to extend my own birthday greetings to him. So we've heard a lot about this system, iron selenium, dope with sulfur. And the results I'm going to show today are new results where we've really asked the question, OK, there's clearly pneumatic fluctuations in the system. It appears that they go quantum critical at a certain fixed doping. And the question for us was whether this would manifest itself in the evolution of the transport properties at low temperatures across the sulfur doping range. So what we did was that we killed the super connectivity in a high magnetic field and studied the form of the resistivity with both temperature and sulfur doping. So really, this work is very much the work of my student, Salvatore Licchiadello. And he's been ably supported by members of my group, Jake, Jonathan, and Jan Ming. And the samples that we received very gratefully through our collaboration with Yuji, Taka, and Shigeru. OK, so I'll just give a very brief introduction for those perhaps less familiar about what we might expect in a material, a metallic system, that we can tune through a quantum critical point. Then just show briefly how that criticality, in this case anti-ferromagnetic quantum criticality, manifests itself in the transport properties of one of the ion-penic type systems. And then I'll introduce the ion selenium and how the resistivity has evolved there. And in time permitting, I will sort of have a preface to Shigeru's talk about how this is different from the behavior we see in the cuprates. So we're all familiar with this type of picture. And the original theories with quantum criticality, of course, were involving insulators, talking about tuning some parameter where it allows you to go between different ordered states. And at the intersection between those ordered states, you saw quantum critical behavior. But in a metallic system, you often tune from some ordered state through the quantum critical point that often is protected by some other, the emergence of some other state like superconductivity, then ultimately into the Fermi liquid in the disordered regime. But the two extremes away from the quantum critical point is normally you can describe the system in terms of product states. So you think of the magnetic product state here, and then perhaps your slater determinants on this side. But then as you approach this point, you can no longer describe the system as some mixture of these, or you can't describe the system as a mixture of these states. And what you end up with is a quantum entangled state that still defies a coherent theoretical description. But we often think of some associating the approach to this quantum critical point as the divergence of some length scale associated with the entanglement. And so for anything below, length scales below that this psi, you are essentially, the physics is dominated by the quantum critical fluctuations. And then, of course, you have an additional thermal length scale, which gives rise to the combination of these two, this diverging, and the other one gives you rise to this quantum critical fan. And perhaps then if you're measuring transport in this metallic system, where you have fluctuations associated with this ordered state, you may end up with some anomalous power law in the low-temperature resistivity, like 4 thirds or 5 thirds. Then as the system becomes ordered and those fluctuations gapped out, you then recover your Fermi liquid behavior. And on the other side, of course, we expect the canonical Fermi liquid t squared resistivity. But above that, in this quantum critical fan, you may then get other anomalous non-Fermi liquid power laws. Perhaps they are different exponents on either side. But right at the quantum critical point, you should see that anomalous exponent tracking all the way down to the lowest temperature. OK, so two aspects you see. So as you approach on the low-temperature side, you see essentially the quasi-particle is becoming increasingly dressed with this interaction, the quantum fluctuations. And that gives rise to a diversion to M star. And right at this along trajectory A, you expect distinctly non-Fermi liquid behavior. This has been beautifully demonstrated, of course, in heavy Fermion systems. The poster child here is the terbium rhodium 2 silicon 2, where the tuning parameter, in this case, is magnetic field. And you see the two domes of t squared resistivity and the quantum critical fan, where in this case, the resistivity is purely t linear over significant decades in temperature. And then the coefficients of this t squared term appears to diverge. So here I should say this is the anti-ferromagnetically ordered state. This is the disordered state. This is just purely metallic state. But on either side of the critical field, you see this a coefficient diverging. And this gives you an indication of the dressing of the quasi-particles in terms of their mass. OK. So the bearing 1-2-2 system really gave us the first clean system for high-temperature superconductors to demonstrate genuine quantum critical behavior, which is reviewed very nicely in TACA and UGIS and Tony Carrington's review article back in 2014. So here I show, this plot has been shown before, but this is the derivative of the resistivity at high temperatures above the superconducing state. And you see, again, the fan of, in this case, again, linear resistivity flanked by the t squared on both sides. And what was shown in, I think, 2010 was quantum oscillation studies found that as you approach the critical point where, in this case, the spin density wave order transition collapses to zero. It's around 30% doping. So here, I've got the scale. You saw a divergence of the effective mass determined from quantum oscillation studies. But then, very dramatically, what was measured inside the superconducing state in zero field was the penetration depth, the square of which gives you an indication of essentially one over the sous-fluid density times the mass. And because this is an isovalent substitution, one didn't expect, really, the sous-fluid density itself to be varying. So this very striking peaking of the penetration depth squared really suggested that it was indeed that the mass was being heavily renormalized on approach to this quantum critical point. The fact that it's exactly at the point where superconducing is maximized suggests, really, that quantum fluctuations are indeed, in this case, enhancing the superconductivity. OK, so what we were interested in, again, being in the high field kind of work, line of work, was to see how did this evolve if we killed and removed the superconductivity with the high magnetic field and just follow the resistivity down to the lowest temperatures we could. And this work was done in collaboration with James Annalytus. We found, indeed, taking the derivative of the resistivity in high fields that you basically got a linear line going to the origins for all the doping. So here, we're doping from 70% to 31% and just to orientate you. So that's really right out here, approaching very close to the quantum critical point. And you see, then, that this is symptomatic of a low-temperature T squared resistivity. And its slope, which is proportional to this A coefficient, was getting progressively steeper and, in fact, went up by an order of magnitude on approach to this putative quantum critical point. And, indeed, the evolution showed very good scaling with the effective mass squared as expected, say, from the famous Karawaki Woods ratio. So here, we saw very much the expected behavior. And what's nice about this system and quite surprising is that all of the effective masses that you determine, whether it's low-temperature high-magnetic field for quantum oscillations, low-temperature zero field for the penetration depth, or relatively high-temperature zero field for the jump in the specific heap at TC, they all gave very consistent ideas suggesting that this was really the effective mass fluctuations due to fluctuations was the same across the phase diagram. So what about iron selenium? So let's say we've heard a lot about this system. I don't need to really go into too much detail just to say we have, of course, this tetragonal to orthorhombic structural transition at around 90 Kelvin in the pure iron selenium system. And that, with some very detailed r-pares and quantum oscillation measurements, you find, first of all, if you just take a simple DFT calculation for the Fermi surface where you expect these three whole pockets to electron pockets, you lose some of those pockets suggesting those strong electron correlation effects. This is, like say, the high-T. This is, sorry, below the structural transition, above the structural transition, then below it, you get a distortion. So you get this actually elliptical-shaped Fermi surface indicating your electronic nomadic state. And the evidence for, say, criticality was coming from the observation of these elasto-resistivity measurements from TACA and UGIS groups showing that the nomadic susceptibility is most intense close to the point where the structural transition disappears and the effective Curie temperature goes through zero for this nomadic susceptibility. And while there are suggestions or experimental evidence for magnetic fluctuations inside this regime, there was certainly no evidence above TS for strong magnetic interactions. And so the idea is generally that this material, you can study the effects of nematicity in the absence of magnetic criticality. And what intrigue does before we started this experience was that the actual form of the resistivity is characterized above TC by predominantly T-linear resistivity over a wide range of doping. So not necessarily a crossover to T squared in the overdone regime like you saw in the penictydes, but more similar in essence to what was seen, say, in the cuprates above TC. And so we were intrigued to think, well, is this another indication of the cuprate-like strange metal, or does this show more conventional quantum critical behavior? And so what we wanted to do, then, is to study possible manifestations of this electronic nematic on the transport at low temperatures by killing the superconductivity with a high magnetic field. OK, now one of the nice things about Einstein's selenium system is that it's not as anisotropic as the cuprates or penictydes, and you've got relatively low TC. So normally what we have to do to suppress the superconductivity is to apply magnetic field perpendicular into the conduction planes, where HC2 is smallest. That induces transverse magneto-resistance, which somehow you have to subtract off to get the intrinsic electrical resistivity. But in this material, the anisotropy in HC2 is only a factor of 2, and that allows us to then suppress superconductivity with an in-plane magnetic field, where effects of magneto-resistance are strongly minimized. So here, I can just show you a whole set of sweeps, up to 35 Tesla, for a doped sample to illustrate this point that the magneto-resistance, right, there is some small positive magneto-resistance when the field is in the plane, but it shows essentially no temperature dependence, and in fact, really, you can see how flat it is. Yet we're still suppressing superconductivity down to about 1.5 Kelvin in this case. And so then you could actually, in these resistive magnets that we have at high field lab, you can then just turn on a 35 Tesla magnet and just cool down and follow the evolution of the transport across the entire temperature range of interest. OK, so that's what we did. We had a series of eight samples, and I just want to show you the difference for H-parallel to see. So here's data, early data, from Yuji and Takaka, that shows for pure iron selenium that if you put a field parallel to C, you induce enormous magneto-resistance, and then it becomes very difficult, in fact, to extract the intrinsic, effective normal state resistivity. And here's the iron selenium 35 Tesla data, where you see you've not really introduced any strong additional temperature dependence. But you can still get right down to the lowest temperatures in a 35 Tesla field. OK, so the bottom line is this is what we saw. So here is essentially the doping where the pustive nomadic quantum critical point exists. And what's nice here is that actually, once you get to 16% and above, there is no magneto-resistance above TC, either in, say, 25% here or 16%. And so really, you could have sort of good confidence that the resistivity measured at 35 Tesla is that of the intrinsic resistivity in the absence of superconductivity. And you see a distinct change here from T-linear, which we got measured down to about 1.5 Kelvin. Just below 1.5 Kelvin, we're hitting superconducting fluctuations. So HC2 is of the order of 36 Tesla at this doping range. So you start to see a small downturn here, which we've just removed from the data. But clearly, it's different from what's seen at 25%, where you have a quadratic temperature dependence of the resistivity. And I will show you now zooming in. And this region for pure iron selenium is also where you recover at T squared. Yes? The curves at x equal to 0 at 35 Tesla and 0 Tesla are offset, but then the other samples they don't see. No, no. So there is, as I said, there is a small positive magneto-resistance. Actually, for all x less than 16%, which vanishes at 16% and above. Yeah. OK, so here is the pure iron selenium. And I'm just plotting here the derivative. In fact, for the very lowest temperature points, we have had to take field sweeps and just show the 35 Tesla data. But what you clearly see here is that, again, as in the iron bearing 1, 2, 2, that the derivative appears to be going linearly down to 0, suggesting that the low temperature dependency is T squared. Even though, if I go back, there was a large region where it was effectively T linear above Tc. And so here plotting against T squared, you can see the range over which that is obeyed. And from the slope, we get a slope of around 260 nano ohm centimeters per Kelvin. And we can estimate, we have a formula for the coefficient, which we can just plotting in known Fermi surface parameters for the system. You can get a range between 19, 230, which is in good agreement with the measured value. OK, so here is the whole set of samples. So I think you can see that there is this T squared behavior, lowest temperatures. Here, again, there's a slight downturn for the resistivity at 13% crossing over to T linear. And then back to T squared. And here, you can really see how there's no magnetor resistance for the higher dope samples. But there's always some finite magnetor resistance for the small. And taking the derivatives, you can see there's a clear delineation in the transport behavior. So these are all, if you like, low dope systems show very similar behavior. And this dropping off of the derivative as you reach the T squared regime, at 16%, the derivative is now just flat up to about 15 Kelvin, so over a decade in temperature. And then crosses over again immediately to becoming T squared, which becomes flat again, suggesting you're returning to a T linear resistivity. So all of this shows a very systematic variation with doping. And if you take the slopes, which I plot here in these black triangles, the slopes grow gradually. So this is the strength of the T squared coefficient, but not dramatically as you approach 16%, but then drop very dramatically once the matic state is lost. And this was sort of reflecting the trend seen in, say, the strength of the superconducting gap. TC itself doesn't change very much across this regime, but the gap itself drops very sharply beyond 16%. And so this suggests there may be a correlation between the superconductivity and mass enhancement as manifest in the coefficient of T squared resistivity. But this is not the whole story because what has been shown in this isolinium, although it's isovalent substitution, quantum oscillation measurements by Amalia Caldea's group show that actually both the electron and the whole pockets grow in magnitude as you dope. And of course, the A coefficient is dependent on the carrier density. So if the carrier density is growing, then this will influence the A. So in fact, if m star stays the same, your A coefficient would just drop away. What it's actually doing is growing with doping. And by taking the frequencies observed by Amalia, we can then take out the growth in the carrier density and just focus on the part which we believe is due to the effective mass. So that's what we did here. So this is now a kind of a renormalized A coefficient where we've taken out this change in the carrier density. And you see that there's a much more of an increase as we approach 16%, but still the drop-off, perhaps it's less dramatic now because obviously there you have more carriers in the higher-dote samples. The range of the T-squared resistivity again shows this characteristic shape. And here I'm showing where your onset of T-linear resistivity comes in. And stress again that for the 16% T-linear resistivity extends over this whole regime. OK, so this was for us initially quite a surprise because if the system is only, if quasi-particles are coupling to, say, pneumatic fluctuations, quantum critical fluctuations, you typically think in terms of pomerantric instability that these will be Q equals 0 type fluctuations. And therefore they may not necessarily manifest themselves in the transport properties where you need large-angle scattering to decay or relax the momentum. And indeed there have been theories by several members of the audience about how you might get T-linear resistivity in a pneumatic system. But it's quite challenging to get there. In fact, it's very difficult in a clean system. You can get T-linear resistivity in a pneumatic quantum critical point in a disordered system where really the change in the resistivity is small compared to, say, the residual resistivity. But here we're seeing quite striking variation in the T-linear part of the resistivity. Of course, there is a residual resistivity here. But I think the whole evolution, even for very pure EINZ-Linium, you see the very much reduced residual resistivity here suggests that disorder is not the driving force for the physics that we're seeing. And so one idea that came to us in discussions with Chi Ma is, of course, that there are theories that suggest that the pneumatic state in EINZ-Linium is not just the simple Pomeranchal type, but there is also so-called anti-ferrocordiopolar interactions which are, let's say, coincident with these anti-ferromagnetic fluctuations in the system. And this is something which can coexist both in the Chalcogenides and in the Panic-Tide systems. But depending on the system, one may dominate over the other. And certainly here the idea or the argument of Chi Ma and its collaborator, you, is that if the strong coupling between these fluctuations and the itinerant quasi-particles this may suppress the magnetism but promote this type of pneumatic. And this gives you a finite queue scattering that potentially gives us the resistivity that we see. There are other proposals for possible finite queue. But we cannot forget the magnetism. I mean, there is magnetism there. We know that if you add pressure to the system, very quickly you generate a spin density wave and there have been mu-SR and NMR suggestions that will load T-S. There are significant magnetic fluctuations. So maybe that is what we are coupling to. And the evolution of the phase diagram here is very similar to that seen in the Panic-Tides. But there is one set of data that I think would argue against this. And this is the very beautiful series of measurements done by Euge's group, Antaka, who studied, with transport, the evolution of the different phases, both as a function of x in the ion selenium sulfur system and pressure. And although there is this low-lying spin density wave state near ion selenium, pure ion selenium, as you dope and you see how this pneumatic is being suppressed down to the quantum critical point, at the same point the spin density wave phase is pushed to higher and higher pressures. So really, while there's magnetism there, we don't see any evidence from this picture that it's the magnetic fluctuations that are going quantum critical at 16%. Really, you would argue that it is, indeed, the pneumatic fluctuations that are responsible for this behavior. OK. So in the sort of preparation for the next talk, we were also interested to notice that in all these systems, as I said at the beginning, there is a T-linear resistivity. And we know a lot about the Fermi surface, the Fermi surface parameters of this system. So we thought, well, we can perhaps estimate the strength of scattering in the T-linear regime and compare it to this very nice plot that was mentioned earlier on by Andy McKenzie's group that suggested for a lot of correlated systems, particularly those near quantum criticality, that you have, say, a range between 1 and 2, which is a multiple of KBT, this so-called Planckian limit that Sean alluded to earlier. And again, we can determine this alpha coefficient from both the strength of the resistivity coefficient and the same parameters that we used in extracting our A star coefficients. And here I show, basically, the value of alpha within the red zone, so within the T-linear regime. And in fact, it's very strongly pinned around 1, right up to 16%. And possibly, even these small departures from that, we may think about as due to the appearance of a third pocket, which changes. Here we've assumed the presence of two pockets all the way through, but there may well be the emergence of a third pocket at higher sulfur dopings. But in any case, within our experimental resolution, we can see that alpha is, again, between 1 and 2. So it does seem that within the quantum critical fan, there is a kind of universal value for this, as far as you can tell by this sort of Druder-like analysis. OK, so in the last five minutes or so, I just wanted to put this into the context of the system. It's still very dear to me that the whole dope cuprates, in particular this strange metal phase in the overdose regime. Now, when you look at the phase diagram, the cuprates, of course, the first thing that strikes you is there are a lot of potential on the critical points. And then you think, well, where do I start? But I think the one that universally people are most interested in is the so-called P-star critical point which marks the end of the pseudo-gap line, or where the pseudo-gap, this loss of states at the Fermi level, closes at around 19%, 20% in the whole doped cuprate phase diagram. And we know from many spectroscopic and transport properties that there is a very profound change in the fermiology of the system, you start out in this regime, at least, say, close to the edge of the superlative dome with a large hole-like Fermi surface, continuous Fermi surface that also displays quantum oscillations. But then as you go inside the pseudo-gap regime, this is truncated into these disconnected Fermi arcs, as they're called, where you now, and it's not clear even if these states seem to be lost down to the lowest temperature, and perhaps it's not clear even above T-star whether those states actually recover. So is this really a system which has a genuine transition in P, but not necessarily a transition in T, is a longstanding debate in the field? However, it's very clear that there are broken symmetries inside the pseudo-gap regime. In fact, there are many beautiful experiments suggesting that this T-star line is associated with some type of broken symmetry. But again, my question is, do any of these broken symmetries manifest themselves in an evolution of the transport properties in the same way that we understand, in, let's say, a more conventional, quantum critical point of view? And if they did, that might help us to identify which of these is really responsible for the pseudo-gap. But almost a decade ago, we did a similar measurement to the ones I've just shown you for panic-tides and chalcogenides in lanthanum strontium copper oxide. And really to our surprise, what we saw was that this is optimal-doped LSEO, and this is non-superconducting LSEO. And in the non-superconducting LSEO, you clearly see T-squared resistivity. At optimal doping, it's the T-linear. But in fact, the dominant form of the resistivity right across this regime is T-linear. So rather than there being a singular point at which you see T-linear resistivity, you get this sort of quantum critical phase extending in the zero-temperature limit. And that is very hard for us to understand within a simple quantum critical picture. Indeed, the strength of this T-linear term grows essentially from zero. Of course, we know it's T-squared out here, and reaches some kind of maximum very close to this 19% doping where the pseudo gap opens. There was also the T-squared term that we could delineate, say, from the derivative of the resistivity, which doesn't seem to do anything in contrast to what's seen in panictides and also now in the chalcogenize of this huge enhancement as you approach P star. So just sort of to end with a bit of speculation, again, we can estimate what this alpha is. And we find that actually when it reaches P star, the value is at or close to the value seen in other materials. But we'd argue that as you go below P star, of course, here we expect we still have a full Fermi surface. Then you start seeing Fermi arcs. And of course, that brings down the carrier density again. So you would expect, perhaps, that this would change dramatically here. But somehow it just seems to flatten out. And so our speculation is maybe, could the fact that you reach this limit at P star be actually the thing that's responsible for the opening of a pseudo gap? So not necessarily some kind of broken symmetry, typical quantum critical type scenario, but actually that you reach this bound. And this drives you into a completely different state where you lose effectively quasi particles at low temperatures. And Sean mentioned this earlier in his talk, that we know that there are two defining behaviors at high temperature in terms of whether material, whether that's resistivity saturates, approaches the so-called molybdenum for a limit, or whether it just crashes through it. And we see that you might think, well, is this just some kind of extension of some anomalous scattering? And I think Sean also mentioned this, that no, there is a clear evidence from an optical connectivity that as you cross this row max, you get a transfer of low energy spectral weight to high energies. There are many ways to explain this. One of the first really was within DMFT, where they showed how the spectral function for a correlated system near the Mach transition varies very dramatically with temperature. So here you have your quasi particle pole at low temperatures. But as you increase temperature, you see how that gets suppressed. And the spectral weight goes to the low and upper Hubbard bands. And this, therefore, there's no longer a scattering rate that you can associate with the drop in the conductivity, or the rise in the resistivity. And it may well be then that, as Sean did speculate, that there are two limits to one set by the absolute magnitude of the resistivity, in which you lose quasi particle states. But also possibly when you reach this planking limit, and maybe in all the quantum critical systems that we know about inside the quantum critical fan, this is the limit. This is the bound. But it might well be that cuprates are the only material to cross both of these bounds. And it's by crossing that bound that the pseudo gap itself emerges. OK, and with that, I'll just advertise a review that we've just published on some of these aspects and put up my conclusions. Thank you for your attention.