 after lectures for questions. Catch lectures after lecture and ask more questions and it's basically your week, so you regulate what to do. We'll have two questions and answer sessions later. Okay, thank you very much. Thank you, Andre and Dimitri. So welcome everybody. So the title of my lecture is Fluctuations and Quantum Criticality and Two-Dimensional Disordered Superconductors. Can everyone in the back read this handwriting? This is the rough length scale of the writing that you will be seeing throughout the talk. So if you cannot see, please move up. And also please feel free to interrupt me with questions at any time. So let me first start by asking why study the subject. You never, there were no talks on this subject last week, but to make some contact with what you saw last week, there was lots of talks and discussion on the idea of quantum criticality and in material systems. The problem that I will be discussing was one of the first settings in which quantum criticality was observed. And when I say quantum criticality, I mean not the kind of quantum criticality where you, the critical point is avoided by say superconducting domes or something like that. We actually have honest to goodness naked quantum critical points in this problem. So for that reason, it's worth understanding the setting, the context of this problem. It also, for me, I love this problem because it's a very rich intellectually problem. One of the signatures of the richness is you can approach it from many different points of view, maybe from statistical mechanics, maybe from diagrammatic or quantum field theory, et cetera, et cetera, and there's rich experimental connections. So before doing many fancy things, talking about dualities and all these very beautiful ideas that are used for the study of the system, I want to provide to you the basic background and context in which these systems have been observed. And the goal of my lectures, my two lectures will be to present this to you in as casual and accessible a manner as I see possible. Okay? And again, please feel free to ask me questions as we go along. So before we begin the study of fluctuations, we must start with mean field theory. And of course, the beautiful mean field theory for superconductors is the Ginsburg-Landau theory in which the free energy that depends on superconducting fluctuations takes the form, okay? So this is the Ginsburg-Landau free energy that was introduced by Ginsburg and Landau in where psi is the order parameter of the Cooper pair wave function. It's a charged system, so it's a complex order parameter and E star is twice the electronic charge and M star is twice the electron mass. There's a quadratic term, there's a potential which provides a cost for having large values of the superconducting order parameter. And then B is the internal magnetic field, which is a fluctuating variable, the curl of a vector potential, and H is the external applied field. Okay. And so the idea is that if you want to study the superconducting phase transition, you, so M star is twice the mass of the electron roughly, but you can always rescale things to change that. It doesn't really matter. E star, however, is twice the electron charge. Okay. Again, feel free to interrupt. So psi is the Cooper pair wave function. Electrons form pairs and condense. It gets a non-zero expectation value. I'm going to assume that you have seen this form of the free energy at some point in your lives. So let me continue with this. So this quantity R is a measure of how close we are to the phase transition. It is of the form A times T minus TC, where A is some constant, which is I will take to be positive. And U is going to be the only temperature dependence that the shear enters in the quadratic term. Okay. So one of the most important tools before doing any serious analysis is the notion of dimensional analysis, which I would like to use at this point in time to study the Ginsburg Landau free energy. So what do I mean by this? First I want to define some quantity which in brackets O to be the energy or mass dimension of any object O. Now usually as theorists, some of us set H bar E and C K Boltzmann to one. One is by definition dimensionless, so that won't have any units. On the other hand, the free energy is an energy. So it has positive dimension. Lengths have negative dimension. Momenta have positive dimension. Masses have positive dimension. Temperature has positive dimension. And so now all we have to do in order to decide what these couplings are doing, we just have to see to make sure that this free energy on the right hand side and the left hand side have the same units. And when we do this, let me say one more thing, the wave function of the Cooper pair will have dimensions of one over square root of length to the power D or one over square root of volume. And hence this has dimensions D over two. So with this information requiring the right hand side to have the same units as the left hand side, namely one, we find that in three dimensions, D is the number of spatial dimensions, D is three, because of course, experiments are done in three dimensions. We find that this quantity R has dimension one. And therefore since the temperature has dimension one, A has dimension zero, U has dimension one minus D. It's very easy to do this. I encourage you to do this on your own. And lastly, you can rescale the free energy as you wish, but it turns out one interesting invariant upon rescaling is this quantity A squared over U, which has dimension D minus one. And this is going to be slightly important in what I do later on. That's why I bring up this whole topic. Recall that the density of states, which I'm gonna call new, is the number of states per unit volume per unit energy. Volume is length to the D, and therefore the dimension of new is D minus one, which is the same as the units of A squared over U. So if you did a more microscopic and serious calculation that was first done by Gorkov, you would start with a system of electrons with attractive interactions, and you would produce this free energy by integrating over the fermion fields. And generating this theory. And because we're integrating over fermion fields, the couplings in the free energy here will be properties of the electronic system, in particular the density of states. So what can, since the A squared over U and the density of states have the same dimensions, at the casual level that we're produce, we're heading here, we will simply equate the two. And there are order one constants that come from a more microscopic calculation, but this is heuristic because they have the two dimension, the two have the same dimensions we're equating them. Of course, that's a very bad argument in general. The mass of the earth and the mass of the sun are both measured in kilograms, but we don't say that they're the same. This comes from a more serious calculation, microscopic calculation, and this invariant quantity comes from the density of states. Yes. Well, energy is K Boltzmann times the temperature. And so that's on a, in the classical world, that's the only choice we have, to set temperature to have a units of energy. Yeah, the fields themselves are not the same though, from high and low temperatures, so yeah. Okay, I will do that. Take a mic. Or take a mic, where's the mic? Well actually, you know what, I'll repeat the question. Dimitri, it's okay. Yeah. There was another question back here, yes. In quantum mechanics, we say that energy is HC over lambda. So the energy is always related to some Compton wavelength. And in units of H bar equals C equals one, energies will have dimensions of inverse lengths. Okay, so this will come back later when we talk about superconducting fluctuations. All right, so there's basically two important length scales in the Ginsburg-Landau theory. One is the coherence length, and the other is called the penetration depth. Basically since both psi and vector potential A are fluctuating variables, they will vary spatially. The characteristic length scale over which psi varies is called the coherence length, the characteristic length scale over which vector potential varies is called the penetration depth. And the expression you can find is C squared is H bar squared over M star, which is, let's just do it like this. Okay, now, and the penetration depth, lambda squared, is M star C squared. Okay, so these are the expressions for the penetration depth and the coherence length. If you haven't seen this before, or let's say you are seeing this for the first time, here's the hand-waving argument for obtaining these two lengths from the free energy. To obtain the coherence length, ignore the fluctuations of the vector potential and require the gradient part of the free energy, the variation of psi, to be comparable to the quadratic term here. So requiring the two quadratic terms to be equal is saying that at low energies, both are equally important in the cost-benefit analysis. So if you have the order parameter varying too much, then that costs a lot of gradient energy, kinetic energy, and there's a competition between kinetic and potential energy. So it's just like in freshman physics when you ask how far can you throw a ball, you have to equate the kinetic energy and the potential energy, and then we get this length scale. So in other words, if we say r psi squared is comparable to h bar squared over m star psi squared, and let's suppose that psi varies over the length scale xi, and this will become h bar squared over v squared, and psi squared. So requiring these two to be the same, one obtains this expression here. And similarly, for the penetration depth, there's a competition between the Maxwell free energy, v squared, and the coupling between the vector potential and the density of the Cooper pairs. And requiring the quadratic terms in a to be comparable, one gets this expression. The last thing that's very important is what we call the superfluid density, rho sub s. And that's basically a measure of how costly fluctuations in the phase degree of freedom are. Since this is a complex wave function, it can be written as an amplitude and a phase. The superfluid density is the characteristic scale for phase fluctuations. Let me explain. But actually, before I do that, I remember, I wanna tell you one thing here. One often also defines the zero temperature coherence length obtained by setting this, in this expression, t to zero, and so that's going to be h bar squared over m star a times tc. And we can express the coherence length at zero temperature in terms of the coherence length at finite temperature using these two expressions. So this expression here is t squared times tc minus t divided by tc. This right hand quantity will be important when I talk about fluctuations. This dimensionless measure, tc minus t divided by tc, is a measure of how close we are to the phase transition. I'm gonna call this epsilon. So c not squared is c squared times epsilon. Epsilon equals zero means that we're at the phase transition. Since this is a finite value, this is some number shown here, the only way this can be true when this is zero is if the coherence length at the transition diverges, which is what happens in mean field theory since the phase transition is continuous. Okay, so let me get back to the superfluid density. The superfluid density, if you take for psi, as I wrote before, an amplitude, which we can hold frozen times a phase, which is spatially varying, and plug this back into the Ginsburg-Landau free energy, you find for the kinetic term, psi squared over 2m grad theta minus e star over h bar c. Okay, so this is what you get from the quadratic term. So let me just say the Ginsburg-Landau free energy will be an integral over space, this plus a constant, which comes from the nonlinear terms, side of the fourth, and I've neglected the b dot h term for the moment because I want to extract what we call the superfluid density. So the coefficient of this grad theta minus a squared, e a squared is called the superfluid density. So rho s is h bar squared over m star, and I'll put a subscript zero on here to remind you that this is the low temperature or bare superfluid density, not necessarily the physical superfluid density that we measure in experiments. The nice thing about this expression, you can see the penetration depth is sensitive to psi squared. So if you want to know how the superfluid density is related to the penetration depth, we can look at these two expressions, and we find indeed that rho s is h bar squared, c squared over eight pi. So the superfluid density is inversely proportional to the square of the penetration depth in a bulk superconductor. The more distant the magnetic field penetrates the system, the smaller the superfluid density is. Can people see over here? Is there, oh, you cannot see this expression? Okay, so shall I move and rewrite it here? Please interrupt me if something is invisible or unclear, worse yet. Which parts can you not see? You cannot see the expression for the relation between rho s and lambda. So let me just say that here. So it's basically the key point is it's proportional to one over lambda squared. Okay, now we want to think about a thin film. We want to take this Ginsburg-Landau theory and ask similar question for a thin film and ask how the thin film differs from say the bulk expressions that I've written here. Well, so the way you do this is you imagine you start with a bulk three-dimensional superconductor. Let's say this thickness is, I'm gonna call delta and you shrink it down to a very small scale. So then basically d3r goes to something like delta d2r, the measure, okay? Now you can absorb in this free energy this factor of delta into the wave function. So this is a three-dimensional wave function. So from the three-dimensional wave function we can define a two-dimensional wave function which is equal to the square root of delta times the three-dimensional wave function. Okay, so that sort of makes sense because the three-dimensional wave function goes like one over the volume and the two-dimensional wave function should go like one over the area and that square root, so that makes sense dimensionally. But we cannot do this with the vector potential piece. So the coherence length, it turns out since we said the coherence length comes from comparing variations of the kinetic piece of psi with the quadratic piece here, that doesn't change in a thin film. The expression that I've written here is more or less the same in mean field theory. But the penetration depth will change because this extra factor of the thickness that enters from the measure. So if we follow the program I've outlined here to obtain the length scale by comparing the curl A squared term and this part of the covariant derivative and have them be comparable, that's the characteristic scale over which magnetic fluctuations occur. What we find is this, penetration depth in thin film. Ignore the spatial variation of psi. Then the first piece will just be E star squared over M star C squared psi 2D squared A squared. And we require that to be comparable to B squared times the thickness. Those order one factors don't matter. Sorry, I should have said delta. I changed notation in my notes. Yes, thank you. Thanks for keeping me honest. Yes, very good. So now this thing here will be some, since the derivative gradient is dimensions of inverse length, we can simply write delta over lambda squared A squared. So we can see here that requiring these two parts of the free energy to be comparable introduces this length scale, which is lambda squared over delta. By requiring these two terms to be comparable, we get lambda squared over delta is M star C squared over eight pi psi 2D. So if you compare this expression with this expression, they're the same except that for the thin film we have a lambda squared over the thickness. And so this is sometimes called the effective penetration depth. Notice it has units of length. This is length squared, this is length. This calculation comes from a more serious treatment of the problem than I have presented here. What you have to do is actually consider the fact that in a thin film, the wave functions terminate at the thickness. And suppose you had a vortex, suppose you had a spatially varying magnetic field, outside the film, the vector potential spreads out. And when you have to take into account the spatial variations and solve Maxwell's equation seriously, this was done by Pearl. And then you find that the variation of A in the film is over the length scale lambda effective. But in my hand-waving cartoon here, you can also obtain it from dimensional analysis by defining a change in the measure and the change in from three-dimension wave functions to two-dimensional wave functions. Now this is very important that the effective penetration depth goes like lambda squared over delta because what does that tell us? If you take a bulk material which has some penetration depth and try to shrink it, the penetration depth of the thin film gets bigger and bigger. And that in turn implies that the superfluid density is getting smaller and smaller. Are there questions? The limitation of this argument is that it uses mean field theory, nothing else. Oh, sorry, I should have said. The question was, is there a limitation to this argument? Sorry, yeah. One last thing before I move on beyond these preliminaries. Okay, is a final step where we relate the penetration, effective penetration depth to the superfluid density. Now, in the bulk case, I wrote here somewhere, I guess I've erased it, oh, here, that the superfluid density goes like one over lambda squared. So it might be tempting for you to think that in the thin film, the superfluid density goes like one over lambda effective squared, but that's not true. You can see that the superfluid density is h bar squared over two m star. So this is the 2D superfluid density times a two-dimensional wave function, psi squared. And so this thing is going to be proportional, this is going to be h bar squared, c squared over eight pi e star squared times lambda effective, lambda effective. So this is proportional to one over lambda effective to the power one, not power two. So that's something to keep in mind. Any questions? This is important because when you have these two length scales in Ginsburg-Landau theory, you can take the dimensionless ratio of them and define a ratio of the penetration depth to the coherence length. And if this is much bigger than one, we call this type two. And if it's much less than one, let's just say bigger than one and less than one, it's type one. So in a thin film, we have to look to compare lambda effective to xi. So this is lambda squared over delta times xi. And the definition of a thin film, which I should have said, is that the thickness is less than the Ginsburg-Landau coherence length, say at zero temperature. If it's bigger than the coherence length than the electrons in the material think they're in three dimensions for all practical purpose when it comes to thinking about superconductivity. But if it's thinner than that, the electrons now begin to see that they're in two dimensions. So because of this, this can be much bigger than lambda over xi squared. So if you had, say, only a weakly type two material, if you shrink it down into a thin film, it becomes even more strongly type two. Okay, now having said these things, okay, I will avoid this part of the board from now on. Any questions so far? I can erase the free energy. Let's look at the phase diagram as predicted by mean field theory. So you solve those equations of motion of Ginsburg-Landau theory. And what you find is that in the type one superconducting materials, there's a, if this is the applied magnetic field, this is temperature, there are two phases. There's the normal state, where the expectation value of xi is zero, we can even say it like this, and the superconducting state where not zero. Okay, so this is a second order phase transition. And in type two, there's an additional phase temperature magnetic field. This is sometimes called the Meisner phase because vector potential is not allowed inside the sample. It's expelled, the system expels flux. So here we have the normal state in a type two. There is the Meisner phase at low magnetic fields and low temperature. But now there's an additional phase, which is the vortex lattice, which was first obtained by Abracosov. And this is a system in which the vortices, so there's flux that penetrates the system at points and the vortex, if you look at it nearby, it corresponds to a place in which the amplitude of the Cooper pair is depleted over some length scale, which corresponds to the coherence length, it turns out. And if you look at the magnetic field at the vortex, so it decays exponentially over some length scale, which is the penetration depth. And the magnetic field can decay at length scales much longer than the size in which the condensate is depleted in the vortex. And so these vortices form a crystal. So Abracosov showed, actually he showed that it was a square lattice of vortices, but later on people corrected what he did and found that it's actually a triangular lattice for the simplest superconductors. Turns out the energy difference between a square vortex system and a triangular vortex system is so small that it's amazing that back then in the 50s when you didn't have powerful computers, you could still find this. So that's the vortex lattice. Any questions? So everything I've said so far is what happens in the absence of disorder. But you see that we're gonna be talking about disordered superconductors and I'll explain why disorder is important at this point in time. So but we, in order to proceed, we need to ask what disorder does. And the first question we can ask is how disorder affects the coherence length C and the penetration depth lambda. Again, in order to answer these questions in practice, one has to do a serious calculation, but for the purpose of these lectures, it will suffice to do a hand wavy type calculation that I will do now. So effective disorder, before I do that, let me say that disorder introduces an additional length scale, which is called the mean free path. And I'll denote that with length L. And basically what it is, is if you have a bunch of isolated impurities, charged particles will travel essentially following the equations of motion of a clean system until they go from one collision to another. And the characteristic length over which such collisions occur is the mean free path. And so we can define a dirty or clean superconductor as the limit where the mean free path is much greater than the coherence length. And we can define a dirty limit in the opposite sense where the mean free path is much less than the coherence length. Now the goal is to see how C and lambda are affected by this length scale L. So let's first consider the effects on C and C, the coherence length. So instead of following the Ginsburg-Landau way of obtaining coherence length, I wanna show you an alternate approach to thinking about the coherence length, which comes from a microscopic theory. So consider solving a Hamiltonian which has kinetic energy plus a bunch of static impurities. Let's for the moment take them to be non-magnetic impurities. So they only couple, they act like a random chemical potential. And in addition to that, we can have interactions which we can take to be attractive, say in mean field, BCS mean field theory. But let's suppose before we deal with the interactions, we diagonalize this part, the kinetic part and the impurities. And so if we didn't have impurities, we diagonalize the kinetic part of the Hamiltonian in a momentum space. But when we have impurities, momentum is not a good quantum number. They break translational symmetry and we have to work in a different basis. So let's suppose we have those basis states and I'm gonna call those single particle states size of N of R. Now one of the things we learned from early work by Anderson is that even though, so we think of superconductivity as an instability of a Fermi surface in momentum space where there's a pairing attraction between states at momentum P and momentum minus P, spin up and spin down, say, for a simple superconductor. We're not gonna talk about unconventional superconductors. I believe the next lecture will have some of that. So we're often used to thinking that this we require momentum space to talk about pairing between P and minus P. But what Anderson told us was that you don't need that. All you need is a set of states that diagonalizes the impurity Hamiltonian, the kinetic part and the impurity part and you have a bunch of wave functions and they're time-reversed partners and superconductivity occurs between a state, an electronic state and it's time-reversed partner. So if we denote the spin by sigma, then pairing will be involving phi up and phi down. And so the Cooper pair wave function in a self-consistent mean field calculation will depend on the coordinates of two electrons, R1 and R2. And this will be a sum over states times some coefficients. We don't really need to know what the coefficients are. So if we know the single particle spectrum of this impurity problem, we can in principle obtain the pair wave function. Now, the pair wave function will depend on the center of mass positions. You have two electrons here at coordinate R1 and R2. We can define a center of mass coordinate, which is R1 plus R2 and a relative coordinate, which is R1 minus R2. So this pair wave function can also be written as a center of mass piece and a relative coordinate. So this can be written as a sum over n. Let me not write it again. We can write this as capital R plus R and capital R minus R. Now, another definition, I bring this up here because we wanna know how the coherence length changes in the disorder system. Another definition of the coherence length is it's the characteristic size of the Cooper pair wave function. So we can say psi squared is actually in D dimensions, DDR. So if you look at, now, what we wanna do since we have disorder is we want to disorder average. That's what this overline will mean. We wanna disorder average this quantity. And when we do that, the center of mass coordinate sort of drops out because the system is imagined to be self-averaging. It doesn't depend where you are. It only depends on the relative coordinate. So if I plot the variation with respect to, let's say like this, it attenuates at the length scale C. And there's a very simple hand-waving argument for how this comes about. See, because the pair wave function can be expressed as a set of single particle states, now, single particle states undergo unitary time evolution. So since these are complex numbers, they have an amplitude and a phase. There's a time scale in which you start with a coherent superposition, but they defase, okay? And that defasing time, tau defasing, can be written using the uncertainty principle as h bar over some energy scale. Well, the only energy scale that these states know about since the superconductivity is a low energy phenomenon, it cannot depend on the Fermi energy, but instead it depends on the mean field gap function. This is the time scale over which they defase. So what that means is, in a weak coupling superconductor, electrons are not very faceful to their partners, you know, could prepare. They're moving around all over the place. They're moving around in the clean limit, ballistically. And in the time scale of this defacing time, we can ask how far do they move? How far do they move? If they're clean, the coherence length is just the velocity, the characteristic velocity of these electrons times the defasing time. So this is going to be h bar v Fermi over delta. Or if you wish, at zero temperature, C naught is h bar v Fermi. Zero temperature gap function can be expressed in terms of Tc, so we can write it like this. So this is another argument for what the coherence length is in the clean system. It's another expression, not from the Ginsburg-Landell. But now we wanna see how this expression changes when we have disorder. So in the case of disorder, we have collisions between these impurities. We don't have ballistic motion, but instead we have diffusive motion, okay? So instead of the electrons moving at a very rapid velocity v Fermi, they diffuse very slowly. And so in the same time scale of the defacing time, they don't get very far apart as they diffuse. As a consequence, the coherence length reduces. So let's estimate the coherence length from this simple physical argument. Any questions? By the way, I'm curious. Have you all, is this all very obvious? Have you seen all this before, or I see some yeses and some noes? So let me continue as planned. All right, so in the dirty case, the length scale, there's a diffusion constant, which is the mean free path times the Fermi velocity. And so in a time delta t, or delta, this tau, the dirty expression is the diffusion time, diffusion constant times tau defacing. So the diffusion constant, as you can see here, has dimensions of length squared over time. So this quantity here has dimensions length squared and its square root has dimensions of length. So if we plug this in and use the same expression for the defacing time, we get V Fermi mean free path H bar over delta. And so you can see that this is the clean coherence length times the mean free path to the square root. So C dirty is reduced in this way. So we can write this somewhat more cleanly, in a transparent fashion as C clean times L over C clean. So I told you that in the dirty limit, the mean free path is much less than the coherence length. So this number is going to be small. And so in other words, the coherence length of a dirty superconductor is reduced. But it reduces because again of this physical argument that in that defacing time, if you're moving only diffusively, you don't get very far in comparison to if you're moving with a velocity VF without colliding into anything. So if we don't have disorder, then the motion is dictated by the characteristic velocity, the Fermi velocity. But if we have motion, it turns out while the Fermi surface is not a meaningful quantity, what we still have charge conservation, okay? So if we have charge conservation, we can define a charge diffusion. So the motion of, if you look at say, the density-density correlation function, instead of the fermion-greens function, you'll find that it has a diffusive form rather than the standard form in the clean limit. Okay, so what that just says is that if you have a local inhomogeneous chemical potential, charge will pile up. But then at a later time, the system is undergoing a random walk. And if you ask how the charge density changes as a function of time, it's conserved, but it will change over position in a diffusive form. And this expression again comes from sort of, it comes from a serious calculation, but it also comes from dimensional analysis, yes. I've specified in order to have those wave functions that I used, I require, so I keep forgetting. The question was what was the nature of disorder? We assume that the impurities are non-magnetic such that they're still pairing between a state and its time-reversed partner. Yes, oh, if you have an anisotropic system with different coherence, so the question is how robust is this expression to anisotropy? Or what was the other thing? That was it. That's pretty much what you get from this hand-waving argument. But if you have an anisotropic system, you either do a more serious calculation or you require coherence length that enters this expression to be the geometric mean of the two-directional coherence length. But again, that comes from a more serious calculation. Okay, so that's the first expression. Now we want to know how disorder affects the penetration depth. And again, this is a serious calculation. If any of you have heard of this book by Abracossov Gorkov and Djeloshinsky, this is the calculation you'll find in the last chapter. In fact, the expression for this is in the last page of the book, right? So if you go through that calculation, you get the correct answer rigorously. But if you do it in a more hand-waving way, which as I will do now, you get a similar expression. Now the idea is instead of going to the Ginsburg-Landau theory, let's consider an alternate formulation again. And you look at the current response, you apply a magnetic field in the superconductor and ask how currents respond. That's the question of how you determine the penetration depth. So if you look, if you call the current J mu at some position r, it turns out in a superconductor, it's going to be proportional not just to the vector potential at this particular location r, but over some distance. So we can write this as K mu nu. So this is what, this is called the current response to a superconductor, sorry, to a magnetic field. And the fact that the current at position r responds to the vector potential at position r prime is one of the hallmarks of superconductivity. It's called non-local electrodynamics. But when you have disorder, it turns out the electrodynamic response becomes more and more local. And that's what I want to explain. And from that we will see why the penetration depth greatly increases in a dirty system and in turn why the superfluid density greatly decreases. So let's look at this. Yes, I'm coming to that. So the question was, is there some integration? I'm coming to that in a moment. Okay, in order to make progress, again I'm going to use only dimensional analysis. Since we know from Maxwell's equation, J is curl of H. And if you approximate H to be approximately B, so there's not much magnetization or diamagnetic response, this is approximately del squared times A. So what that means is that if you compare this expression and this expression, you see that this quantity K mu nu must have dimensions of inverse length squared. So I can write this as some one over lambda squared times a dimensionless quantity, which I'll call K mu nu tilde. And we will see that this length scale lambda that comes from rescaling this kernel, this current current response kernel, will be the penetration depth. All right, so as the question was brought up, insightful question, clearly this non-local electrodynamics can't continue arbitrarily. There must be some cutoff beyond which there's the vector potential doesn't affect the current. And that length scale must be anyone? In the clean limit, anybody know? The coherence length, exactly. So since K mu nu, so in the clean case, we don't know the detailed form of K mu nu, but it seems quite likely to suggest that it's going to be cut off like this divided by the coherence length. So it's just that beyond this length scale, you don't get any response. And so now what we're going to do is consider a following problem where we have a superconductor in a half space, okay? So let's say x greater than zero, you have superconductor, and you apply a small magnetic field at the origin. Now by symmetry, we can make progress by looking at this problem, it's a one-dimensional problem. The dependence, by symmetry, the solutions between the currents will respond in the system like this, the vector, sorry. If the magnetic field comes out of the board, the vector potential points like this and the currents point in the same direction. And the response only depends on one dimension, x. By symmetry, there's no y or z dependence. So if we solve this current response, we can say Jx at the origin is going to be one over lambda squared, and we want to average over this kernel. By the way, this type of thinking was first done by Pippard in a hand-wavy way. And I think what I'm doing here is essentially what he did, perhaps somewhat more simplified. So let's average this exponential non-local electrodynamics over some coherence lengths. So to do that, we just simply define this like this, e to the minus r minus r prime over Xe d r prime. But since we're in one dimension, this is just e to the minus x dx. So this integral is elementary, and we find that this is equal to, oh, there's a, sorry. So in general, we will need to know how A of X varies spatially in the superconductor along with the position. But we can make an even simpler approximation to say that the variation is over the length scale Xe. Suppose the variation in the vector potential is over a much larger length scale. If the penetration depth is much larger, then it's essentially over this length Xe, over the length Xe, A is essentially a constant. And so you can bring it outside the integral like this and simply take the value at zero and do this integral. And the Xe's cancel and we get one over lambda squared, A of zero. So what we found is nothing more than a tautology, which is we defined a length scale lambda to be the penetration depth and we found that the decay occurs over the penetration depth. But now the thing I will do is to ask, how does this thinking get altered in the presence of disorder? So in the dirty case, we expect the current, the non-local response to be cut off at a length scale of either Xe or the mean free path, whichever one is smaller. So this follows that K mu nu tilde goes like e to the minus the mean free path. So now we further cut short the response because again, if there's lots and lots of collisions a distance farther than the mean free path, we don't expect the currents to respond very well to the external magnetic fields to screen them. So we plug this back in. Essentially all that changes is you get another exponential function X over L and you do this integral. Now you get A of zero. So if we take the mean free path L to infinity, so this comes from just doing this simple elementary integral, okay? Now look at the limits of this thing as L goes to infinity we recover a clean penetration depth. But in the limit where L is much smaller than Xe, we get a different response. So from this we can call this to be a lambda dirty squared times A of zero. So gathering the terms we see that lambda dirty is equal to lambda clean times the square root of R, sorry, L plus Xe divided by L. So again as the mean free path diverges the dirty penetration depth becomes the clean penetration depth. And as the mean free path shrinks, the denominator diverges and we get a rapidly enhanced penetration depth. So in the dirty limit that is to say when Xe is much bigger we get lambda clean. So because now, so what have I shown you? I've shown you the physically intuitive statement that if you have disorder, first the coherence length decreases because they don't travel very far to de-phase. And the penetration depth increases because the motion of the electrons is much more sluggish and simply cannot screen out magnetic fields that ordinary superconductors do very well. So the length scale over which the magnetic fields vary is greatly enhanced. And that makes our assumption here self consistent. Remember we tried to solve the non-local electrodynamic problem by saying let's assume the vector potential is essentially constant over this length scale. Well we found indeed that's the case, the penetration depth is huge so it's self consistent. And so what that means is that in a type two superconductor disorder greatly enhances the penetration depth. Furthermore in the thin film the effective penetration depth goes like lambda squared over the thickness and that is even more further enhanced in the presence of disorder. As a consequence the superfluid density greatly decreases. And what that means is that because superfluid density greatly decreases in the presence of disorder we must go beyond Ginsburg-Landau theory and take into account fluctuations. And see how fluctuations alter our predictions for the mean field phase diagram. When does the lecture end, 1015 or 1030? 1015, okay. So I guess the last thing I'll be able to do in the first lecture is to ask the criterion for the breakdown of mean field theory due to fluctuations. And this was worked out by Levanyuk, how do I, am I pronouncing correctly? Levanyuk. But we call it the Ginsburg criterion since Ginsburg did it two years later. Ginsburg-Levanyuk, yeah. So the idea is simple. You consider homogeneous mean field solutions where the order parameter is constant, okay? So the kinetic term is ignored. And then you ask let's consider a fluctuation in which the order parameter is varying spatially. And it contributes a variational free energy, not variational, a free energy due to spatial variation which is of the form h bar squared over two m. Let's ignore the vector potential and magnetic effects for now. Okay, so now depending on how side spatially depends, very spatially, this will be the cost function for that, right? But again, in the simple minded spirit of these lectures, we can assume that the variation of psi will be on the length scale of the coherence length C. And hence from dimensional analysis, this integral will be over a coherence volume. And this will go like C to the D minus two, two because there's two powers of the gradient here which go denominator times psi squared expectation value. This is the change in the free energy due to a variation of the order parameter over the length scale of the coherence length. Now, such a variation will be forbidden if its free energy is greater than the temperature, right? If its free energy is much greater than the temperature, it's not going to be accessed in the Boltzmann weight. By contrast, if the free energy is much less than the temperature, these fluctuations will be occurring and they can completely change the conclusions of mean field theory. So the condition for mean field theory to hold is that the variation contribution to the free energy is greater than the temperature. Now, using the definitions for the zero temperature coherence length and the finite temperature coherence length that I wrote a few moments ago, we can write this expression here as 1 half h bar squared over 2m squared u is the side of the fourth term. How does this come about? This here from mean field theory comes from minimizing the potential, which is r times psi squared plus u times psi fourth. And we get the minimum condition psi is minus square root, sorry, minus r over u. And this, in turn, can be expressed as c to the d minus 2 over u. So that's how I got this expression. And this becomes d minus 4. We can further make more progress. And recall that I said that this Ginsburg-Landau a squared over u has dimensions of density of states. That's how we're going to use that expression. And rewrite this as we can express the coherence length in terms of the zero temperature coherence length. Then we get tc minus t over tc, density of states, squared. If this expression is yes, OK, I will get to the punch line and then get to the questions in a few minutes. So you may think like I pulled this out of a hat, but it comes from just the mean field theory expression for coherence length in terms of the zero temperature coherence length, which introduces this power epsilon, which I defined earlier, a dimensionless measure of how close we are to the transition. Oh, I'm sorry. Yes. Thank you very much. So this condition applied near the phase transition will be that this should be bigger than tc. So when you do this simple manipulation at the end of the day, the condition for mean field theory to be valid is that epsilon to the 4 minus d over 2 times the density of states times tc squared p is OK. Now Levanyuk and Ginsburg introduced a measure of how much fluctuations are important as how big should this be, this epsilon, how big should this be such that this condition is violated? And that's the beginning of the breakdown of mean field theory. And what you find after doing a simple manipulation, epsilon goes like 1 over density of states tc, c0 to the d over 2 over 4 minus d. Now we can make further progress by recalling that c is h bar v Fermi over tc. And the density of states in any dimension goes like k Fermi to the d minus 1 divided by v Fermi. Plugging this in, you get that epsilon is equal to tc over ef to the power 2 times d minus 1 over 4 minus d. So let's look at this. This is, by the way, for the clean case. So in three dimensions, epsilon, the Ginsburg parameter, sometimes this is denoted as the symbol gi. This parameter goes like tc over ef to the power 4. And that's a ridiculously small window of the critical point where this will break down. In two dimensions, we get a little more window fluctuations. Epsilon goes like tc over ef to the power 1. And in one dimension, it's an order 1 thing. So that goes back to our understanding that in one dimension fluctuations are always important. In fact, you don't have longer in superconducting order. So that's for the clean case. The last point I want to make is, if you go back and fix this expression, taking into account how the coherence length is varied in a disordered system, you will find that, and again applied, let's just do it for two dimensions, since we're running out of time. I want to get to the conclusion. For the dirty case, epsilon for the dirty two-dimensional film goes like epsilon of the clean system times the clean coherence length over the mean-free path to the power d over 4 minus d. And when you apply it to d equals 2, this becomes 1 over kf times the mean-free path. Now, in the clean limit, the mean-free path is much bigger than the inverse k-pheromy. And so kfl will be much, much bigger than 1. And then we recover the fact that in the clean case, the window of fluctuations, the Ginsburg-Levanuoc parameter is quite small. But when kfl becomes of order 1, fluctuations become of order 1, and we must look at the effects of alteration of Ginsburg-Landau theory. So one of the reasons the disordered superconducting problem is interesting is that it enhances the effects of fluctuations. It can even destroy the superconductivity. And if that destruction occurs such that Tc goes to 0, we get a quantum critical point. And we call that critical point a superconductor to non-superconductor transition, usually in two dimensions that's a superconductor to insulator transition. OK, I can stop here and take questions if there are any. Yes. Let me start here if you don't mind. Yeah.