 The interactions are attractive in order to favor superconductivity. You decouple the interactions, there are four Fermi interactions, using the Hubbard-Stratanovich trick, and then you proceed to integrate out the Fermions, and that produces the Ginsburg-Landau theory. Now what one does is if you look at the coefficient of the quadratic term and set it equal to zero, remember I said R was A times Tc minus T. And so when this term vanishes, we're at the critical point. So if you do this formal integration and ask what is the condition for the quadratic term to vanish, R equals zero, in terms of the microscopic model, you find that let's suppose the interaction is characterized by an attractive coupling g. You find that there's an equation of the form, one over g is a sum over Matsubara frequencies, the omega n are Matsubara frequencies, and you sum this up and you get the expression for the critical point. Okay, so this should be thought of as like a stoner criterion for superconductivity. The left hand side is an inverse interaction and the right hand side is the pair field susceptibility. Because the susceptibility in BCS theory diverges logarithmically, even for infinitesimal interactions, you get an instability. And so this is the condition for the vanishing of the quadratic term of Ginsburg-Landau theory. Now this was written in the case of a clean system, but even in the case of disordered systems for non-magnetic impurities, so long as an electronic state near the Fermi energy is being paired with its time-reversed partner, we still have a complete set of states for which we can express this integral. And so long as the density of states at the new basis is not altered significantly by disorder, this expression carries through even for the dirty case. And therefore, this is to leading order saying that TC does not change with disorder. Is there a question? Yeah, yeah, I mentioned that, yeah. Extended S wave and trivial S wave are same thing, so yes. So if you have an onsite or simple S wave and an extended S wave, maybe the extended S wave part will be affected, but the onsite part will not. By symmetry it's the same. So then the question is how does one go away from this? So we have disorder and temperature, Anderson's theorem would say that TC doesn't change, right? But I told you in the previous lecture that fluctuations and mean field expectations will be significant in the presence of disorder and especially in two dimensions. So we must find ways of deviating from the simple way. Now I should also say that Anderson's theorem is an assumption, underlying assumption is that the disorder is weak. The mean free path is long compared to the inverse K Fermi. So in that case what Anderson had in mind was a typical system like this. If you look at energy and density of states, if you are in three dimensions, you find typically at the edges of the band, states are localized, whereas in the middle of the band, states are extended. By this I mean that if you have a disorder potential and you, this is basically just particle in a box physics, if you put a state here, tunneling will be forbidden if it's strong disorder and you expect psi of X to be exponentially decaying away from some center of position. So X over lambda sub L. Lambda is not the penetration depth that has a subscript L. And so this is called the localization length. Now Phil Anderson's theorem assumed that we were sitting somewhere here in the density of states. And then if you switch on, so we start here, so long as states remain extended, particles are not ballistically moving, they're diffusing, but they can still pair and form superconducting state. But later on people suggested to push the Anderson theorem beyond its original intention of validity and across this boundary between extended and localized states, which is called the mobility edge. And so as you go into the localized regime, states, the single particle states are localized, but can you still get superconductivity? And the answer is you can in three dimensions, at least in this case, because they are both extended and localized states. But if you're in the localized regime, you can still get superconductivity. Can we estimate the condition for which superconductivity persists even in the localized regime? And again, we can estimate this using dimensional analysis. So here's what we do. Suppose all states are localized. So assume that all the states are localized and you break up your system into little boxes, each which has a characteristic scale corresponding to the localization length, roughly. Since they're localized, we can say that there's a localization length, which is finite. Now, if you're looking at the system at length scales smaller than lambda sub L, the electrons are still moving. They exponentially decay far away for length scales much bigger than lambda sub L, but for length scales less than lambda sub L, the electrons don't quote unquote know that they are localized. So in principle, they're moving around and they can still pair up and form a superconductor. But for that, the gap must be sufficiently strong such that the density of states times, since the density of states is an inverse energy per unit, it has dimensions one over energy times volume. The condition for superconductivity to survive even in the localized regime is the density of states times the gap times a localization volume bigger than one. Now, intuitively, what this means is that if you have some density of states, and here's e-ferme, and this is a blown up picture of localized regime. So if you wish, take it to be a blown up region of one of these sides. And the point is that if you have many, many states within a window of delta of e-ferme, there's sufficient number of electrons to gain condensation energy from opening up a gap. If this condition is not satisfied, you lose the superconductivity, right? So what this suggests is that you can have a situation in which in a more realistic system, the system breaks up into sort of grain-like sizes, each having a characteristic length of the localization length, in which you have a well-defined phase, okay? So remember, the superconducting order parameter has an amplitude and a phase. There's a well-defined phase characterizing each of these grains. And if you look at length scale small compared to the land-versa-bell, if this condition is satisfied, for this length and for a fixed delta, at those short lengths, it looks like we're in a superconducting phase. But then if we look at big distances, there's not in general superconductivity. The only way that the system will be macroscopically a superconductor is if the Josephson coupling between grains favors the phases to align, okay? So by Josephson coupling, I mean that the coupling between these grains will be of the form some jij times cosine of theta i minus theta j. Okay? So this is sometimes called the granular limit. Now, it turns out that nature naturally makes films like this. If you don't work too hard, as you make your film by your favorite growth technique, if you're not careful, the system will sort of face separate into this grainy-like structure, each of the width of maybe 10s or 100 angstroms or so, and you form a granular-like superconductor. And what you find, let me just show you since this is real physics, let me just sketch some experimental data, what you find is that if you look at temperature and resistance, resistance in a square system, actually let me back up before I do this. The point I wanted to say is that this is three dimensions. There are both extended and localized states, but I told you that if this condition is satisfied, superconductivity survives in the localized regime. Now it turns out in two dimensions, for weekly interacting electrons, if you look at a density of states like this, all states are localized. If Anderson, the original theorem, only applied to the weak disorder case where states were extended, yeah, J is the exchange coupling, Josephson coupling. Yeah. It's an oscillator function. No, no, no. The ground state of this corresponds to all the phases aligned. The cosine of the relative angle is zero. So because in three dimensions, studies showed that superconductivity survived into the localized regime, it's encouraging that in two dimensions also, where all states are localized, superconductivity can still survive. Now in thin films, this condition is satisfied if you don't work too hard and grow such that the disorder is very inhomogeneous. These are called granular superconductors. And in two dimensions, they exhibit superconductivity. And what you see is typically, so if I plot the resistance in a square sample, that means essentially for us, resistivity. So this is nonzero resistance and zero resistance. This is the superconducting transition. The normal state had different values of resistance, which means that they had different values of disorder. But for a range of disorder, the transition temperature is insensitive to the disorder. So that's consistent with the Anderson theorem, applied well beyond its original intention of validity. However, if you now increase the disorder even further, you find divergence of the resistance and eventually superconductivity will be destroyed. For more resources, references, this was done first by Alan Goldman. I can provide more references if you'd like. Now, are there questions? Yes. This is a, the question is, is this two dimensions? Yes, this is for a thin film where all states are supposedly localized. Okay, so what you have here is the normal state is an insulator morally. I mean, at finite temperature, it's not a sharply defined thing, but electrons are well localized, but they form a superconductor nonetheless. But if you make the disorder strong enough, they cease to be superconducting. And furthermore, in the granular limit, people did tunneling measurements, and in tunneling, you are sensitive to the local density of states. In a superconductor, you open a gap at the Fermi energy and you have a peak at a scale of roughly delta. And the extent to which delta changes with TC tells us the extent to which amplitude fluctuations are important. And in this system, the gap remains rather smooth across the superconductor to non-superconductor transition in a granular limit. By contrast, if you worked very hard and made the films much more homogeneous, by this, I mean, you work very hard. The experimentalist works very hard to grow the films in some way, such that they are not granular, they're more homogeneous. In this case, what happens is that, first of all, TC is sensitive, normal state resistance. So you typically will find something like this. So this is the superconducting transition. This is resistance. As the normal state resistance increases, the superconducting transition temperature decreases. Okay. Furthermore, if you look at TC versus the normal state resistance in a square sample, the data fits well to a theory which involves loss of superconductivity primarily due to the fluctuations in the amplitude. How does that come about? It comes about because in a real system, superconductivity has both attractive interactions that give rise to superconductivity but also repulsive interactions that disfavor superconductivity. And the repulsive interactions, it turns out, if you work hard, it was shown by Finkelstein and others, I think, Maikawa. If you compute by using the diagrammatic technique applicable at weak disorder, you find that the Coulomb interaction and the density of states are both suppressed. The density of states is suppressed by disorder. That's the work of Altschuler and Aronov. But also the transition temperature is suppressed because the Coulomb interactions are now stronger, effectively stronger. Why are they stronger? It's because instead of ballistic moving electrons, we have a diffusive motion of electrons and the diffusive behavior screens the Coulomb interactions somewhat more weakly. So this theory fits rather well to the homogeneous limit and in such systems, if you do tunneling, what you find is that as you approach the destruction of superconductivity, as TC decreases, the gap scale also decreases. And what that means is that the amplitude fluctuations are very important in such systems. So we are quite lucky that we can think of two types of systems, one where the amplitude is very important and one where the phase degrees of freedom are very important. Now these are sort of two extremes, maybe in the real world, both are important. Now along with the suppression of the gap, there's also the suppression of the superfluid density that I already talked about in the presence of disorder because the penetration depth increases. So all of these effects are important. Yeah. In homogeneous films, no, typically no. Oh, in granular, yes. In the granular case, yes. In three-dimensional homogeneous systems, you actually have superconductor to metal transition and at the metal transition, the gap vanishes. But in 2D thin films of homogeneous case, you have a superconductor to non-superconductor transition and the gap doesn't survive. The gap vanishes at the TC, at the critical point. So given this choice and given the time that I have left, I'm going to consider one class, namely a class of theories where the fluctuations in the phase are more important rather than the amplitude. Okay, are there questions? So let me, again, mention the roadmap. We started with Ginsburg-Landau mean field expectations. We found that the coherence length and penetration depth were strongly affected by disorder and we have to say that phase fluctuations are very important. There are amplitude fluctuations, there are phase fluctuations. Some systems, one is more important than the other and now I'm going to focus on a class of systems, the granular systems in which the phase fluctuations are more important and that enables me to get much further in the time that I have left. So I will start talking about destruction of superconductivity due to phase fluctuations. Any questions? Interesting. Okay, so now since we are theorists, we like to study the simplest situations and hope that they make contact with experiments. We can consider a system in which the amplitude is frozen, held frozen, and only the phase degree of freedom fluctuates. And that system is the 2DXY model. So how many of you have studied the 2DXY model and the costal and stylist transition? May I see a show of hands? So let me back up for a second and talk about superfluids versus superconductors and get in 2D. So if we have a superfluid like Helium-4, they don't couple to the vector potential and the Ginsburg-Landau theory of such a system in two dimensions would be as before, there's no covariant derivative. So again, psi is a complex order parameter. And now if psi of X, psi of R is an amplitude, which I'm gonna call psi naught times e to the i theta of R, then this free energy rewritten in terms of these variables will be the stiffness, which we talked about last time of the superfluid density, d2R grad theta squared plus constants, which don't matter. And this problem is well studied. It's something that you should, if you haven't seen before, you should study at one point in your life in your statistical mechanics course or something like that. And let me just summarize what is known. So the phases of the system can be understood by looking at the correlation function, psi of R. Now at high temperatures, this is the order parameter psi. At high temperatures, we expect there not to be any, they expect to be fluctuations dominate, the superfluid state is destroyed and we expect exponential behavior. Now usually with long range order, what happens is that at low temperatures, this correlation function goes to a constant, okay, non-zero constant. And this is what we mean by long range order. But in two dimensions, we have something called quasi long range order, where at low temperatures instead of a constant, G of R is a power law, some power, and this power eta, it turns out, depends on temperature. Please feel free to interrupt me as I go on. So in order to determine this exponent, we can use this free energy and simply compute this correlator, right? Because in this approximation of ignoring phase fluctuations, this is just psi not squared times expectation value of e to the i theta of R minus theta of zero. Now because this is a Gaussian theory, a Gaussian theory has a nice property that the mean and the variance dictate all higher moments, particularly the exponential of any moment are dictated by the mean and variance. And so you can write down this thing, expand the exponential, and write down all the moments in terms of the mean and variance. This is called Wicks theorem by the way. And after doing that, you do this calculation, you find G of R is psi not squared e to the one half. So you see what we have done. We have written the average of the exponential in terms of the exponential of some other kind of average. And now this quantity here, theta of R minus theta of zero squared, can be simply obtained by Fourier transformation of the original problem. So in Fourier space, this is row S over two e to K, K squared theta squared. And so we essentially want the propagator of theta zero. And since we're in two dimensions, let me just write it out. We can write this in terms of the propagator. This is two times the temperature divided by superfluid density. There should be a two pi squared here. It doesn't matter. So again, from our favorite method, dimensional analysis, you see that in two dimensions, the numerator and denominator both go like Q squared. And that usually means that logarithmic divergence is a car. And so this quantity here, when you do the calculation more seriously, is a T over pi row sub S log of R over A zero, where A zero is microscopic length, say a lattice scale or something like that. Now, because this object is logarithmic, when you exponentiate the logarithm, you get a power. And so from this, we find eta is T over two pi rows of S. Now, if you haven't seen this before, I'm just sketching what you would do to obtain these results, not obtaining them in any serious way. But let me just point out that the exponent is smoothly varying at temperature-dependent exponents. But notice that as the temperature goes to zero, the exponent vanishes. And you do indeed get long-range order at zero temperature. But any finite temperature, you've lost long-range order. Instead, you get this quasi-long-range order. So notice that we've already obtained our first deviation from mean field theory. So if you take this phase perspective, life becomes fairly easy. Ginsburg-Landau mean field theory told us that they would be a superconducting state and a normal state. A superconducting state will always have long-range order in mean field theory. But a phase fluctuating model, which treats fluctuations exactly, shows us that instead of long-range order, we have algebraic or quasi-long-range order, yeah. When is it, the question is when is it safe to compute a finite temperature correlator and to take the zero-temperature limit? I think that depends on situation. In this case, it's perfectly no problem. We know from other means that in zero temperature, the 2D-XY model does order, so. I told you that at high temperatures, you have exponential decay of correlations. And at low temperatures, you have this power law decay of correlations. So at some temperature, you must go from exponential to power law. And this is the costalist style of transition. And the way it works is that one has to go beyond the simple Gaussian-like model. And remember that in two dimensions, you not only have smooth fluctuations of the angle, the phase, but you also have topological defects, which are called vortices. A simple vortex I already mentioned last time where the amplitude is actually depleted over some length scale, which is C, the coherence length. But the phase, theta X, comma Y, around the vortex, winds by two pi. Less or minus tangent. So if you look at this from above, the phase sort of circles around, gradient of the phase circles around. So a vortex is defined by this. If you take a line integral, you must get two pi times an integer. And if this integer is non-zero, we have a vortex. Now, if you work out the energy of having this phase distribution into two-dimensional system, you find that it diverges logarithmically. So the energy of a vortex is pi rho S. If you plug this expression in into this and compute the energy, you find a logarithmic divergence. But we also know that the entropy diverges logarithmically because the number of positions the vortex can be is the number of, say, positions divided by a microscopic. So imagine you have a lattice, break it up into lattice sites, the number of distinct lattice sites you could put the vortex anywhere. And so the log of the number of sites, places you could put the vortex is the entropy. So this is, actually it's the area, right? So because you could put it in, it's a two-dimensional system. So you have a N squared options. And so that's why you get the entropy is log of R over A naught squared, which is the same as two log. So if you look at the free energy, it's going to be E minus T S. And this is pi rho S minus two T times log R over A naught. And so now this is the crude estimate for the phase transition above which the entropy always wins. The vortices are entropically favored. And that system has short-range correlations. And a low-temperature phase where the vortices are entropically unfavored, but the energy dominates, and instead you have quasi-long-range order. And so this crude estimate from the free energy, this argument goes to, I think goes back to Berezinski. And this is called, what we could say BKT, the Berezinski Cauchy-Listalis transition, where T, this is equal to pi over two times the superfluid density. Now let me just say one last thing. If you have exponential correlations in the disordered phase and power-law correlations in the ordered phase, at the critical point, the correlation length diverges exponentially. Turns out, E to the minus some constant over, is it absolute value? I believe so. Yeah, T minus TC square root. So this is an unusual form for a phase transition, usually the correlation length or coherence length diverges as a power-law at a phase transition, but here we have the unusual situation in which the coherence length or correlation length diverges exponentially. Okay, I bring all this up, not only because it's simple and elegant, but also we wanna use it. Okay, good. But this was all superfluids. In a real superconductor, we have not just, we also have the vector potential, so we have to go back and fix that. So how is the real superconductor different than the 2D superfluid? So again, working in a phase-only representation, you find that the free energy, the Ginsburg-Landau free energy, is, right? So we had, instead of just the gradient term, we have a covariant derivative now, and then we also have the vector potential as a fluctuating variable. This is the difference between a superfluid and a superconductor. So now we can ask what happens when we have a vortex? Because in the presence of a vortex, the phase field looks like this. If you look at the gradient of the phase, grad theta, at large distances it goes like one over r. So now we have an option that the vector potential can screen the gradient of theta by also varying as one over r. If that happens, this term vanishes, it costs no energy, but the price we pay is that there's magnetic energy that we have to include in the system. And so I already mentioned in the first lecture that there's a length scale which tells you the scale at which the vector potential fluctuates, that's the penetration depth, and in two dimensions that's the effective penetration depth. So what you find is that for lengths much less than this effective penetration depth, the screening does not work very well, and the system behaves essentially like a superfluid. So the vortex energy goes like a log of r over a naught. But for lengths bigger than lambda effective, sorry, this should be r, lower case. So for lengths much bigger than this effective penetration depth, the vector potential is very good at screening the gradient of the phases. And so essentially then, a vortex doesn't cost any further energy. So actually it does cost some energy, it goes like a power law, one over r, but that's not so interesting in the large distance limit. So what ends up happening is that now the key difference between a superfluid which has logarithmically diverging energy costs to create a vortex and a superconductor is that in the superconductor, the vortex energy is finite. As a consequence, however, the entropy is always gonna be this. As a consequence, the free energy is not this form, but rather the free energy for the superconductor goes like we can replace, as a crude estimate, we can replace r over the logarithmic range of the energies by the lambda effective, okay? Over a zero, which is a constant, minus two t log of r. So if you took a strict point of principle perspective, then this would always dominate at any finite temperature and it would imply that there's at any finite temperature vortices are always present in a superconductor. Yes? Oh, thank you, yes. Although this is a rough estimate, this is not the exact calculation, okay? But the statement that is exact is that the vortices are finite energy excitation. So therefore, in the infinite volume limit, there is no costal etymol transition in a superconductor. There's merely a crossover from exponential attenuation of correlations to what looks like quasi long range order and this true phase transition is only at zero temperature, okay? We have not put in disorder, but we now have to put in a simple fact that is important to keep in mind. So you can always make statements of principle, but they may not be useful in the real world. So it turns out in the real case lambda effective is of the order of centimeters, or one centimeter. So this expression, this logarithmic cost of having vortices interact with one another is true up to a centimeter scale. So for all practical purposes, since samples are not as big as the Milky Way Galaxy, this is essentially a macroscopic, this is the sample size. And so the finite size effects that cut off this behavior are utterly negligible. So what one finds in experiment is that there is a costal etymol like transition, but you must go very, very narrow in temperatures to see the breakdown of the costal etymol transition due to the effects that I've described here. So as a consequence, I'm going to take on the practical point of view that there is still a costal etymol. So no real effect on TKT in superconducting films. So what we find is, so we're gonna add disorder in a minute, and this is temperature, we can also add magnetic field, but what we find here is a costal etymol transition, at least in the Kali limit, this is all I've told you. Now before I go this way, let me go out to this way and ask what happens when we switch on a magnetic field. Then it turns out we don't have a costal etymol transition anymore. We have a different physics. Okay, are there questions? If I'm confusing any of you, please let me know. Recall in mean field theory, we said that there's a lattice in a type two system. The phase transition at any finite magnetic field is always into a lattice from a normal state, a bricose of lattice, so long as there's no disorder. So now we wanna ask what does fluctuations do to the lattice? And so lattice, so let's consider the vortex lattice transition, melting transition rather. So for that I need to tell you what we mean by a lattice, right? The order parameter for the lattice is the density operator at some wave vector G, which is e to the i g dot r. And G is the reciprocal lattice vector of what would be the crystal. So clearly the crystal breaks translational symmetries, but it also breaks the rotational symmetries, right? So for instance, I'll do it for the square lattice since I can't draw hexagonal lattices, which is what the abricose of state is. But if we look at the square lattice in one region and rotate and consider a domain where the lattice forms at some angle theta. Now what happens to this is that the translational correlations are broken by this rotated crystal, but so are rotational correlations. But by contrast, if you have a crystal in one region which is oriented this way and in another region it's oriented in the same way, but is shifted like so, then the translational symmetry is distorted by this, but the orientational symmetry isn't. So what I wanna say is that if you break the orientational symmetry, you affect translational symmetry. But if you simply slide along in the translational axis, you don't necessarily affect the orientations. So there's possibility in principle of breaking translations. And when you break translations you also break orientational symmetry. But there's a possibility that you can restore the translational symmetry but not restore the orientational symmetry. And that's called a hexatic phase. But I probably don't have time to go into that unless you guys want me to. So let me continue with just the vortex crystal. It turns out that the vortex crystal also breaks a continuous symmetry, right? Because the crystal could have formed the say one position could have nucleated here or any continuous sets of positions. And hence the crystal also breaks continuous symmetry. And as a consequence, it turns out to behave very much like the Kostelitz-Thoula story, okay? So we can look at again the correlation function which for the crystal looks like this. And we wanna ask how does this behave at different temperatures? And that will tell us what happens to the crystalline transition in a fluctuation theory. But before we do that, let me just tell you briefly what the mean field theory would look like for the crystalline transition. How much time do I have? Okay, so I have half an hour. Okay. Okay, slightly less, yes. All right, so in that case I'll be more concise. And I will tell you that the mean field theory, there's a term which is cubic for a hexagonal crystal. There's a term in the free energy which looks like this, cubic in the order parameter for the crystal. Okay, so as a consequence, whenever you have cubic terms in Ginsburg-Landau theory, third order terms always imply a first order transition. So the transition from the crystal to the liquid of vortices is first order. In mean field theory. Now when you have a mean field transition that's first order, fluctuations will not really change this conclusion very drastically. So even in a fluctuational approach, the transition will be first order. Since I'm running out of time, I won't talk about the hexatic phase unless someone wants to know about this. It's a fascinating story, but I'm afraid I don't have much time to go into this. So let's fill in this phase diagram. So as we look at zero disorder and we look at this transition, we have a vortex melting transition which is first order. And below here we have the crystal. It turns out, this was work done by Daniel Fisher, that the line of transitions, TCH is not equal to TCH equal to zero, which is TKT in the clean limit. So there's actually a separation between the line of melting transitions as the field goes to zero and the zero field costal ethylis transition which is actually above it. So now let me ask what disorder will do to this phase boundary? Yes, in between? So in between, you have power law correlation for the superconducting state and there will be no crystal, I mean vortices will come in pairs. So you have power law correlations and as you pass this temperature, you're asking what happens at zero field? Is that what you're asking? Yeah, below TKT, what happens is you're in a superconducting state but there's a critical enhancement as you cross so there is some critical slowing down that comes from this melting line here, but otherwise nothing happens. The phase transition is normal to quasi-long range order at TKT at zero field but that's a point of set of measure zero. No, no, no, this second transition never culminates here. It's always at non-zero field. It's a melting transition. All right, so now we wanna see what happens in the presence of disorder. We can ask what disorder does by coupling it to our Ginsburg-Landau free energy and there's two types of disorder we can imagine. So there's a term which adds to the free energy, something like a linear coupling to the order parameter and another term which is a quadratic coupling. This is sometimes called random fields because it couples like a magnetic field. If psi were ferromagnetic order parameter, this is like the field that couples conjugate to it and this is sometimes called random TC, sometimes random mass. So which one is applicable to which problem depends on symmetries and so on but in general the rule is if there's no rule, if there's nothing forbidding, a coupling to disorder at the lowest power, it will happen. But with superconductivity we have protection because the order parameter is a complex scalar and under gauge transformations we can always, we have to be free to make gauge transformations. So gauge invariance requires that a random field cannot couple to the superconducting order parameter. But for the vortex crystal, so this would be the case for the superconductor but for the vortex crystal, there's no such symmetry principle that forbids a random field. So the appropriate disorder to consider for the crystal would be a random field. And now the question we can ask is, does the story that I told you in the absence of disorder continue to hold in the presence of disorder depending on whether we have a random field or a random TC. And there's simple physical arguments that tell you for both cases what happens and let me just talk about that. So let's talk about the random TC first. So when you're confronted with a problem, a disorder problem of the form random TC, what you need to do is use something called the Harris criterion. And this idea is actually very simple and very beautiful. The idea is take your system and you break it up just like I did before but in a different context. Now break up the system into regions of size coherence length and focus on, since the TC is random, you have a TC at each of these boxes that are different from one another, okay? And so there will be some characteristic variation of TC which is how much TC changes from one to the other. And the idea is, okay, so as you approach from above the transition, we want delta TC to be less than T minus TC for the system to be stable to disorder. What that means is that in some regions, there might be some variations in TC but as we approach the phase transition, we want those variations to be less than the deviation from the clean critical point itself. If that's the case, in the presence of disorder, there won't be any change to the phase transition in the properties. So let us see what happens. Let's slightly manipulate these things. So to obtain how delta TC depends on the coherence length, what you wanna do is add up a bunch of random numbers in all these boxes. And the error in adding up such random numbers, it's like doing a random walk. And so delta TC should be proportional to one over, there's a one over N, one over square root of N effect. If you add N random numbers, the change from the average is one over square root of N. So similarly here, this goes like minus C to the D over two. So this is the number of, so within each box we're adding up TC locally and asking what the characteristic variation is and there's C to the D over two such values. Sorry, C to the D such values. And so the error in that is one over square root of C to the D. That's what this is. On the other hand, T minus TC can be obtained because we know that the correlation length itself goes like T minus TC to the power minus new. So therefore, T minus TC is C to the minus one over new. So the Harris criterion, the criterion for when the disordered system is stable, sorry, in the presence of disorder does not affect the fate of the system in the clean limit is that C to the minus D over two is less than C to the minus one over new. Or rearranged slightly, new times D is greater than two. So this is a very simple condition. You obtain these exponents in the clean limit and ask, is it bigger than two? And if it is bigger than two, the system, the original phase transition still survives. Okay, so, but in the costal et stylist transition, this is extremely well-satisfied. Why is that? Because I told you that correlation length diverges exponentially. So new is essentially infinity for us. So this is always going to be satisfied. And hence, the superconducting transition in the disordered case continues in the, as it did in the clean state. So transition in zero field stable to random TC. So what this tells us is that as long as we're in zero magnetic field, the transition in the plane of the board continues down, okay? Furthermore, I told you that the superfluid density will decrease, phase fluctuations become important, and eventually the transition temperature will go to zero. So this is the first example of a quantum critical point. This is called the superconductor insulator transition. So this line here remains costal et stylist, okay? Modulo, the caveats that I mentioned earlier that the vortices have finite energy excitations. Yes, yes, oh, oh, sorry, yeah, you're absolutely right. Yes, thank you, yeah, very good. Okay, but now if we wanna know what happens to this thing at finite field, we must consider the problem of the crystal in a random field and ask what happens in that case. Of course, we haven't gotten to that yet, but it turns out it's not in the KT universality class. It's in a distinct universality class. This is quantum critical point. It's at zero temperature. So let me actually break the line here. It is continuous, but not because it's related to TKT. Given how we're doing on time, I may not get to the actual quantum critical point. But anyway, I'm still giving you the overall global phase picture, phase diagram of what you expect in a phase-only model. All right, so what happens with a random field? Well, we use a different criterion, which was first worked out by Larkin and of Djinnikov. I would probably spell this name wrong. Correct me if I made a mistake, but equivalently by Emory and Ma. All right, so now what we have is a slightly different problem. Because there's a random field like this, so now imagine we're talking about the vortex crystal. We will have disorder that couples linearly to the order parameter. And now what we wanna do is because of the linear, the magnetic field will produce domains where the field takes on a certain value. And within a certain domain, let's call it the domain of size L. Within a domain, if you think in terms of a magnet analogy, if psi were a magnetization that couples to a field, the magnetization would align with the field in this region and they would be gain in energy. However, they would be loss in energy because the order parameter would be varying as a function of position. So we have to compare the gain in energy of aligning with the random field with the loss in energy for varying in space. And so the energy gain is very much like what we found before, is just sum over the domain of the value of the random field at each position. But now again, we're adding a whole bunch of random numbers in a volume of L to the D and that goes like one over L to the square root of D, right? So this goes, sorry, one over L to the D over two. So this goes like L to the D over two. To the D over two. On the other hand, the energy cost is coming from the gradient part of the free energy and this goes as before, this goes like the length to the D minus two. And so the energy gain and the energy lost are equal at a certain dimension of space when D over two is equal to D minus two, okay? So when L to the D minus two is greater than L to the D over two, what happens is that the random field doesn't affect the phase transition because the energy cost is too great, is greater than the energy gain. And this condition is just D bigger than four. But we're in two dimensions, right? This is less than four. As a consequence, the energy gain will always win over the energy cost and we will always lose the phase transition of vortex melting. In fact, we lose more than that. We lose the whole phase itself. So disorder has a catastrophic effect on not just the phase transition, but the phase itself. Yeah, was there a question? So we lose the translational symmetry. What do we get instead? So let me just say again, D equals two, abracausa of vortex, abracausa of lattice, unstable to disorder from the Imrima argument. So you see that disorder makes a huge difference. An entire phase of matter has been wiped out. So what do we get instead? Instead of, as viewed above, a hexagonal crystal, so on, we get a system where the vortices are in completely random positions. And so long as there's disorder, they're trapped in minima, they're pinned on random positions, locations. So this phase has exponentially decaying correlations of the crystalline order parameter. And this is known as a vortex glass phase. What that means is that the positions are completely random. It's analogous to a spin glass. So that's what happens with disorder at finite magnetic field. But now let's add one last thing, the effects of temperature. Let me do it here. Let me back up and say, the effect of disorder wiped out the entire vortex crystal phase, in particular, even at zero temperature where you would have long range crystal in order, it's been wiped out by disorder. And if you march along this axis, you have the vortex glass phase. So in this region, you have a vortex glass in the plane H disorder plane at zero temperature. So now I wanna ask, what happens when you raise the temperature slowly? Is this a superconductor or is it not? Now what you need to know is that if vortices are pinned, the DC resistance is zero. And if not, the DC resistance is not zero. Why is this? Because we have the Josephson effect, Josephson effect, which are two equations. The first is that the super current goes like the gradient of the phase. And the second, that the voltage is related to the time derivative of the phase. One is, they're both the equations, right? So you can see, for instance, that if you have a superconductor, where if you imagine setting boundary conditions such that theta is zero here and theta is non-zero across here. So you impose a variation in the phase. What ends up happening is that in order for this constraint to be preserved, vortices move parallel along this way. Because as they move, the phase is changing from one position to the next, but it's not changing from here to here. If the vortices move upwards, the phase would change because they have a two-pi phase winding. Every time they exit, the phase would jump by two-pi. So they must move this way. But if they move, they also introduce a voltage this way along the direction of the phase winding. But since I told you that a gradient in the phase is a current, the way you do this in practice is you drive a current across a superconductor and the vortices move to the right and they generate voltage fluctuations in the same direction of the current. And now since the voltage of the current are non-zero, there's dissipation. This is the I squared R, R dissipation, joule heating. So as a consequence, whenever you have vortices moving, the DC resistance is non-zero. So what we must ask is if we're sitting in the vortex glass phase, where the vortices are pinned, at zero temperature they're frozen in a glassy state, what happens when you have a finite temperature? So the energy, so let v equal the pinning, a characteristic pinning energy. By this I mean that if you have some potential, so you wanna view the vortex as a little charged particle. This is somewhat a different perspective than thinking of the superconductor in terms of the phase variables, but think of the vortex as a particle in the thin film that's trapped in the disorder landscape and it has to overcome some barrier of the order of V to exit this pinning potential. So the Boltzmann weight, at any finite temperature, it can overcome this barrier through thermally assisted tunneling. And so then the vortices will start moving and the vortex glass, because this is a finite energy, the vortex glass is no longer a superconductor. So the conclusion is that vortex glass, superconductor only T equals zero. So what does that mean? If we were in zero disorder, we would melt, we would freeze the abracosa of vortex and so long as we have exponentially small disorder such that we localize the crystal, that's a superconductor. But now, as we switch on the disorder, if we're at any finite temperature, the vortex glass phase has mobile vortices and superconductivity is lost completely. So what we see here is that disorder, as I advertised in the beginning of the talk, has spectacular effects on the phase diagram of the system. I have two minutes, so let me just sketch what happens at zero temperature as a function of disorder and magnetic field. I keep losing my eraser. So at low magnetic fields and low disorder, we will have a vortex glass phase and this is at zero temperature. But as we increase the disorder or increase the magnetic field, there will be a phase boundary where superconductivity, so this is actually a superconductor because we're at zero temperature. And then at high enough magnetic field or high enough disorder in two dimensions, we will have an insulator. And so this line of transitions is called the superconductor to insulator transition in a function of magnetic field. Now it exhibits very beautiful and spectacular properties and the phase diagram in actuality is far richer than I've drawn here, but I hope to give you a very simple-minded overview of what this problem is about. I think I've run out of time, have I?