 OK. So I had two things I was trying to decide about, which one to talk about. And I had thought my collaborator had talked about one at this conference, but it turns out that it was at a previous conference. So I'm going to talk about this topic. And then if I have time at the end, I'm going to talk briefly about another topic. So let me start with this. This is work I've done with Boris Spivek. And it's also based on earlier work in these papers. And the question we asked ourselves was what is the nature of the superconducting state when one makes a wire in which the wire is not a single crystal of a high temperature superconductor but made out of a composite of many grains? And this was actually a problem that was suggested to us by Aaron Kepitonik. So what we're going to be doing is looking at physics, at scales much larger than either the separation between grains or the size of grains. And we're always going to assume that the grains are, in turn, much larger than the superconducting coherence length. So all the microscopic physics of superconductivity is irrelevant. All that happens is we have a superconducting state on each grain. And then we ask, how does the superconducting state propagate from grain to grain? I apologize. Everything in this talk will deal with systems in thermal equilibrium and nothing small. Everything will be in the thermodynamic limit. And in fact, essentially everything I'll talk about will be classical. On the other hand, I'll discuss collective phenomena, disorder, and frustration. And so when physicists, especially theorists, talk about the practical importance of their ideas, chances are they're either lying or totally misled or both. Nonetheless, I have a fantasy that this may be useful in designing new high-temperature superconducting wires. If you look at papers on high-temperature superconducting wires, the situation I'm describing is mentioned in words like trillions of dollars are in the same paper. So OK. With that introduction, the thing that makes this problem a little bit non-trivial is that for the cuprates, the superconducting state is a de-wave state. And this is one of the things that we know for absolute certainty on the basis of phase sensitive measurements. So there are two types of phase sensitive measurements that are done. One is that one takes a piece of high-temperature superconductor and then makes a squid loop by completing a circuit with a conventional superconductor. And if this squid loop is a corner squid like this, then the ground state spontaneously traps a half flux quantum if the squid loop is connected either opposite sides of the sample or the same side of the sample, then the squid loop encloses no flux. So this is the phase sensitive measurement that macroscopically measures the de-wave character of the superconducting state. The problem also arises if one tries to connect several grains of high-temperature superconductor. There were famous tricrystal experiments done by Kirtle and collaborators where you took three pieces of high-temperature superconductor, rotated their crystallographic orientations such that the Josephson couplings around this circuit can't all be satisfied. And again, what one finds is that a half flux quantum is trapped in the middle of this tricrystal under appropriate circumstances. The important point here is that the Josephson coupling between two neighboring grains is a function of the relative angle of their crystallographic axes. And so the sign of this depends on the relative orientation of these two grains. So this means that when we try to put together a collection of such grains, there's frustration in the problem. Now, the superconducting wires, at least the ones made with the Kuprate superconductor bisco, are not made by just jamming together grains of high-temperature superconductors, but by embedding them in a metal, I think silver in particular, although it's a little bit hard to find out exactly what the wires are made out of, because that's proprietary. And so unlike most things in science, people aren't eager to tell you about it. But if we think about embedding a grain of a D-wave superconductor in a metal, then, of course, superconducting pairing is induced in the metal around the grain by the proximity effect. And close to the surface, you don't know that the system is a D-wave superconductor. You either see near this surface, it looks like a piece of S-wave superconductor with a phase pi. Or near this surface, it seems like an S-wave superconductor with a phase zero. And so pairing correlations are induced in the surrounding metal with some pattern that I've sort of sketched here, positive in some regions, negative in others. This dashed line is supposed to show the diffusion of electrons after having been Andreyev scattered off of the surface, which are carrying their phase information. And so what one could see is if one took two such grains and put them close to each other, then depending upon whether you're overlapping patches that are of the same sign or opposite sign, you would get different signs of the Josephson coupling. That is to say, again, for two grains that are close to each other, just as when we were looking at the junction through the insulator, the sign of the Josephson coupling would be determined by the relative orientations of the grains. But if the grains are far apart from each other, you can see that this grain, far away since there's more surface oriented perpendicular to the plus lobe than to the minus lobe, far away from this, this grain will look like a piece of S-wave superconductor with sign plus. And so the sign of the Josephson coupling between two grains will be determined by the product of two factors that have to do with the shape of each grain, whether the two grains have more plus surface or more minus surface. OK, so this leads us to a simple model of how phase coherence propagates in a wire made out of many grains. We have imagined each grain is big enough that quantum effects can be ignored so they can just be described by the phase of the superconducting order on each grain. There is some Josephson coupling between each grain. And the Josephson coupling is of this form. There's three different terms in this random ensemble. Of course, the magnitude depends on the distance between grains i and j. But the sign, there are three sorts of terms. There is a usual unfriestrated term, which has some random ensemble. There is a term that has a sign that depends on some mutual information about each pair of grains and is somehow randomly distributed for each pair of grains. And then there is a term whose sign is determined by the product of two factors, one associated with each grain. So if we only had this term, this would be an example of a model called the Mattis model. It would be an xy Mattis model. If we only had this term, this would be a model of a usual xy spin glass. And of course, if we only had this term, this would be some sort of random magnitude xy ferromagnet. So now let me just take a minute to define the different phases that I'll discuss. The superconducting phases all have some expectation value of the pair field operator. Now, since we have a random system, we may ask about the ensemble average of this quantity. And that's what I'm going to call positive phi. So that's the ensemble average of the anomalous expectation value of the pair field operator. There is also something that will be useful for constructing an Edwards-Anderson order parameter, which is the anomalous expectation value of the pair field operator squared, and then configuration averaged. And I've kept this as a two-point function because I'm going to also want to distinguish between S-wave and D-wave superconductivity. On the other hand, this quantity we can evaluate at zero separation, even if the microscopic physics is D-wave. We can discuss that if anybody's interested. Then we also, since the states involved might break time reversal symmetry, one might ask, what is the expectation value of the current operator in the state? The little trapped flux quantum means we have states with non-zero current in thermal equilibrium. And again, we may want to make a Edwards-Anderson-type combination of these things where we take the product of these expectation values and then configuration average it. So an ordinary superconducting phase, just the effect of weak disorder on a regular superconducting phase is one in which the configuration average superconducting order parameter is non-zero. And moreover, and all the currents are zero, this is just the disordered version of the ordinary phase. And as long as the disorder ensemble has a fourfold rotational symmetry, we can still distinguish D-wave and S-wave states by the symmetry of this configuration average quantity. Then there's an XY spin glass phase in which this configuration averaged quantity is zero, but the Edwards-Anderson version of this is non-zero. And in general, then currents, the configuration averaged square of the currents will be non-zero. This state breaks time reversal symmetry. I've been trying to understand what the declarative statements are that are widely accepted in the spin glass literature. But almost every statement seems to be controversial from the best Monte Carlo calculations, what seems to be more or less agreed upon is that this phase exists for D greater than some critical dimension and that that critical dimension is probably between two and three. But even if it's less than three, it's clear that the critical current is much less than any sort of mean field, I mean, critical temperature is much less than any mean field critical temperature. And it's unclear, at least to me, whether there's a critical current in this state. So even if this state exists, it's a pretty terrible state to use for a wire. So it's something to be avoided. There's also, one could have a state where you have coexisting uniform superconducting and spin glass order that would be characterized by having a non-zero value of the configuration average superconducting order parameter but also by having non-zero equilibrium currents in the state. And again, this state exists above some lower critical dimension which is probably between two and three. And can again be classified as D wave or S wave based on the symmetries of this quantity. And finally, there's been a lot of discussion of the possibility of just a chiral glass phase in which both the Edwards Anderson order parameter is zero and of course the average of the superconducting order parameter is zero but there are still breaking of time reversal symmetry and non-zero currents. And this is hotly debated and I don't think there's a resolution of this as far as I know. Okay, so those are the phases in mind. Now let's look at how these play out in the context of this simple model that we've defined. So I'll take several passes through this phase diagram and sketch them for you. So first, so I'm going to define a everything in dimensionless value. So I'll hold fixed the magnitude of the Josephson coupling and then vary the various amounts of these different types of couplings and sketch for you what the phase diagram looks like. So if we, oh yeah, if there were no frustration in the problem say if we only had the ferromagnetic couplings then the mean field TC would just be J bar over two and since we're in three dimensions and if there's no frustration this mean field estimate is, you know gives us an idea of the magnitude of TC. So it's not so bad. Now there's a symmetry of this model which is the symmetry that was noted by Dan Mattis long ago which is if I define in terms of new variables by shifting every grain that has a negative eta shifting the phase by pi and leaving the phase on the other grains unaffected then what I do is I interchange the role of this Mattis model interaction and the ferromagnetic interaction while I leave the usual spin glass interaction unchanged. So it changes in the ensemble the role of J1 and J3, yeah. What was the mechanism for generating a uniform thermo magnetic couplings? So I'll come back to that in a minute. I mean if we took, if we had grains that were oblong and were oriented say a pneumatic phase of grains then we would generate primarily couplings of one sign. Yeah it's through the metal but if our grains were all oriented in the same direction then we would have some net ferromagnetic component. Okay and so this reflects the fact that while the Mattis model has disorder it doesn't have frustration. So it's really a ferromagnetic model in disguise. The reason it's a ferromagnetic model in disguise is that it's really a ferromagnetic model and the way to see that is suppose we had grains that were oriented like this and so even though inside each grain we have pure D wave superconductor looked at from the outside since the shape of the grain itself breaks the four fold rotation symmetry what it's as if we had as a grain which had some mixture of S and D wave superconductivity either S plus D or S minus D and what this Mattis model or ordering corresponds to is even though the S wave component might be the subdominant component it's ferromagnetic ordering of the S wave component. So the hidden ferromagnetism in the Mattis model is actually that this thing despite the fact that all of the microscopic physics is that it's a D wave superconductor is macroscopically an S wave superconductor. Okay good so let's just see what the model is if we ignore the ferromagnetic interactions. So here's a sketch of the phase diagram the parts of this phase diagram that I'm prepared to defend are the parts on all the edges of the phase diagram in these regions one can do a respectable asymptotic analysis the dashed lines are just a way of completing a phase diagram with the minimal number of additional assumptions. I think there's no reason to even believe that the middle of this phase diagram is universal so don't take that too seriously but the point is that if we have dominantly Mattis model interactions then we're going to have a high transition temperature because this is basically a ferromagnet with globally S wave symmetry whereas if we have dominantly spin glass interactions we're going to have this terrible spin glass phase and with a very low transition temperature. Okay I won't go through the arguments that lead to this phase diagram they're fairly straightforward. If we included quantum fluctuations we would lose this over here. Here's the answer to Vadim's question if we oriented the samples like this we could generate some dominantly ferromagnetic interactions in the D wave channel and so if we take a pass through the phase diagram in which we suppress the spin glass interactions well the phase diagram has to be reflection symmetric about the middle because of this transformation that exchanges J1 and J2 and again we'll have globally ordered phases on the two edges of the phase diagram. Okay so here's the point even with local D wave superconductivity in the grain we can get globally S wave superconductors if the distance between grains is larger than the size of the grains. If we either have to orient the grains well or work in this regime if one's going to generate a wire that's going to be useful for anything. So what one would like to do is to maximize J but at the same time minimize frustration which one can do well to have J be reasonable one certainly needs the spacing between grains to be small compared to the thowless length in the problem so that there's coherent coupling between them and then one can estimate the size of the Josephson coupling by something that looks sort of like an Ambeko-Karbertoff relation which says that the Josephson coupling goes like the conductivity of the metal times the square of the radius of the grain divided by the cube of the distance between them so what we want to do is we would like to make this quantity as big as possible while keeping L bigger than R and so obviously that can be done by making both R and L smaller that is to say making an array of smaller grains at higher density. Good, so how much time do I have left? Five minutes, okay. What? 10 minutes, okay. So let me switch to another topic and I will go through this quickly but this is a experimental talk, it's numerical experiments. I've been spending a lot of time of late being frustrated not understanding anything at all about the theory of metallic quantum critical phenomena and about a year ago we realized that we could do sign free Monte Carlo calculations of pneumatic metallic quantum critical phenomena so I'm going to report to you the results of our numerical experiments, the results are very crisp and very clear and we don't understand them at all. So first we can do sign free Monte Carlo calculations on this problem, we've gone up to system size 24 by 24. When we couple the pneumatic modes to the electrons through some Yukawa coupling of strength alpha, if alpha is big enough the transition becomes first order but we've found that if alpha is somehow intermediate the transition remains continuous which means there is a quantum critical point to study. That's good news. The model we've looked at, we define a pseudo spin that lives on the bonds of a lattice and when that pseudo spin is up we strengthen the electron hopping on this bond and when the bond is down we weaken the hopping on this bond and then the pneumatic state is a anti-ferromagnetic ordering of these pseudo spins as shown in this picture here we've strengthened all the bonds in this direction and weakened the bonds in that direction. And then we simulate this and well okay, here's the phase diagram. So here's temperature, here's the transverse field that controls the quantum fluctuations of our pneumatic mode. In the absence of coupling to the fermions here's the phase diagram, the pneumatic phase boundary is here. There's a crossover to a out of the quantum critical region that comes up like this. When we turn on a Yukawa coupling to the fermions the phase diagram changes. The pneumatic phase is stabilized but most importantly you can see that the critical exponents characterizing this phase boundary have changed. Yeah, I did, I sketched the Hamiltonian. This is the Hamiltonian. So there's electrons that hop on a lattice. If this pseudo spin is up we enhance by a fraction alpha, the coupling, the hopping on that bond. There's an anti-ferromagnetic interaction between neighboring pseudo spins which favors this ordered phase and there's a transverse field that gives them dynamics. Okay? So at alpha three comma zero is just the 2D transverse field? It's just the 2D transverse field ising model and so if you look at this I'll show in a minute you see exactly the critical exponents of the 3D classical ising model. So we can characterize the phase boundary by as it comes down by a critical exponent b minus and this exponent is one within our numerical error in contrast for the decoupled problem this value is as John guessed those of a three-dimensional classical ising model. We can define this crossover line we defined it as following follows you find the value of the pneumatic susceptibility at criticality then you move over until you find that the susceptibility is dropped by a factor of two which is an arbitrary choice and then you mark this crossover here and again you can ask with what exponent does this crossover line come in and well of course you would expect it to be the same but we measure it separately and indeed this line comes in with slope one whereas the similarly defined crossover line comes in with slope with exponent two thirds in the decoupled case and so the first thing you learn which is actually sort of good news if you're interested in quantum critical phenomena is that the quantum critical fan is substantially wider in the metal than it is in the insulator. Then we calculate the pneumatic susceptibility this is in general a func this is equal to the imaginary time pneumatic propagator at q equals zero and Matsubara frequency equals to zero and again there are various critical exponents that we can define if we sit at criticality and lower the temperature we can ask with what power does the susceptibility diverge and that exponent also turns out to be one and we can go to zero temperature on the disordered side and ask as we reduce the transverse field and approach the quantum critical point by tuning the quantum parameter with what power law does the susceptibility diverge and that also comes out to be one these are very non-trivial critical exponents you shouldn't be deceived by the fact that they look so simple and in fact these are critical exponents that have been seen in experiments on pneumatic susceptibilities in the iron-based superconductors that, well that I shouldn't have said that that's left over from an earlier when I gave this talk at the mechanisms of superconductivity meeting that people cared about that there so more generally we have full information about the imaginary time propagator of the pneumatic mode as a function of Q and Matsubara frequency or as a function of position and imaginary time and what I will try to show you is that the full imaginary time propagator is very well approximated by this simple scaling function which has all sorts of critical exponents in it which happen to be integers and correspondingly in real space and time the zero time propagator as a function of position falls like one over R squared so let me just show you oh and this corresponds to the normal in terms of usually defined critical exponents new equals a half gamma equals one the anomalous exponent zero but most surprisingly and unanticipatedly the dynamical exponent z equals two and also we have an emergent rotational symmetry that is to say that for instance this fall off is isotropic at long distances so what I'm going to do is plot our measured values of D versus this function with these constants taking on these particular values and so here's all our data that's through almost three million data points at different temperatures, different system sizes different values of the critical field and all values of Q and omega and okay other than the fact that it's monotonic that doesn't tell you much but if we zoom in on the asymptotic region at lower energies and longer wavelength it begins to look like something this is the lowest half a million points and then if we zoom in again we begin to see something in asymptopia that looks a little bit like a scaling collapse this is over a decade of dynamical range and almost 11,000 data points and finally if we zoom in on our version of asymptopia which is as small as far as we can get with our system size we really see quite a spectacular data collapse of all this data this is the lowest 7,000 data points here's a picture of the fall-off in real space so this is different system sizes so red is 24 by 24 the purple stars are eight by eight there is here substantial size dependence of the results but as we go to the biggest size systems you can see that this fall-off is described very beautifully by a simple one over r squared the data points are from all different directions so you can see the isotropy of the result as well so I guess I'm almost out of time so I'm going to skip fermionic properties don't understand them as well anyway and I will conclude so at the pneumatic quantum critical point we see critical phenomena we see beautiful scaling collapse but with critical exponents so once we say z equals two you might be tempted to say this is in two plus two dimensions and so seeing sort of mean field exponents for nu and gamma doesn't seem so surprising but z equals two is certainly not what you get out of Hertz-Millis theory or any of its more sophisticated descendants and with that I'll stop thank you thank you very much