 OK, pa z taj dve lekti, ki najbolj zelo vsega, sem začal tukaj, na tom, da je vsega vsega vsega inzavljena superkondakta, in sem zelo vzela vsega vsega vsega, In v zelo, nekaj sem gleda, da smo predzivali se všeč, da so vam se načali, da se je vsešte, in zelo nekaj je izgleda, da je tjela vso, da je stavil, da je več, da je zelo, da je tjela, da je zelo, da je vso, da je vsah zelo, da je vso, In oče sem tukaj pravo izgleda, kako se veče izgleda in izgleda. Me raci sem tukaj prav, da imaš več kot, da so čekali, vsega, da mu povalo o zrigeni in jevalo, ki se je skupaj, vsega, kako ne nam zvom angeli, zrejte v dobeli tanks, v jeljih taj nešta, tudi je nešta lahko pomembnili, In menej različi, kako jih se zvonila z nekaj instrumentov, nekaj različi, nekaj različi, ki so prišli, da so prišli, da so prišli, da so prišli, da so prišli, da so prišli, da so prišli. So I will actually try to emphasize this as much as well as the physics. And in particular you've been hearing theory talks, theories like to see, oh we can sort things using symmetry, Oč bomo tudi tudi načo fez na postavlja in zahvali vzvojne odgledenje. Je to seboj. Na primer vso schod bo na postavljanje. Oporavim do vse bo, ki je napadaj s veče in vse zmah ne bo. in počkaj sem zrešal, da je vso in m tudi, nekaj je, zelo da je, da je začel, tudi da tega, da počkaj je sredan, nekaj je nekaj nekaj lepčer. Zelo taj, zelo nekaj lepčer. So, timel vev sem simetri, T dedicated method is to the way the physical laws behave under the time reversal transformation, t goes to minus t. Time reversal operator applied to the velocity gets you to the minus velocity, So if you apply it going right, you will get now a runner going to the left. This allows you also to do the same for current. Current goes to the right, you apply the time reversal operator, you get now current goes to the left. Now a pause and something that has been an issue between the high energy theory in konvensem način, posledo je, čeče ga vsimetri, tako zrejozno, standat model, koraz sem, gaj taj vsego zrejva subjetel Seymetra. In jaz je taj vsega interaktija, kaj nis enough se metra vsega vsega. in zelo je to zrteveno z koning in fitch. Zelo, zelo se rečimo, da ne bi vseh teori vseh, in ki se vseh ležite vseh teori, bo tudi je neseljne opcije, nekaj, kaj bi to tudi, tudi, kaj bi teori vseh teori, ne zelo vseh taj vseh vseh. Na to, taj vseh vseh vseh vseh vseh izgleda izgledan fenomenon. Why do I mention that? Well, first of all, there have been debate for many years between high energy physicists and condensed matter physicists on the fundamental versus emergent. Well, there is still a possibility that time over symmetry breaking is an emergent phenomenon in the standard model. And second, it's because in condensed matter physics time over symmetry breaking is always an emergent phenomenon. That is because if you start from the basic Hamiltonian of a solid, it is basically realized on electromagnetic interactions between the ions and electrons. And obviously, such a Hamiltonian does not break time over symmetry. Time over symmetry then appears as an emergent phenomenon through different types of physics. For example, if magnetism appears below some energy scale, below some temperature, then it is an emergent phenomenon. So, whenever we talk about time over symmetry breaking in condensed matter physics, we are talking about some emergent phenomenon, some low energy physics below the physics of the fundamental Hamiltonian of the system. So, this has to be kept in mind. So, going back to basics, if you start with the Schrodinger equation and you apply the time reversal operator, then the condition for time reversal symmetry to be satisfied is that you come back to the original Hamiltonian. And without any spin considerations, as you all know, time reversal operator is simply the complex conjugation. You can see it immediately from the Schrodinger equation. When you take the complex conjugation, then you will get minus here, and it will come back. If h under the operation comes back to h, it comes back to the same Hamiltonian to the same Schrodinger equation. On the other hand, if you do have a spin consideration, if there are magnetic moments or spins in the system, then time reversal operator becomes an anti-unitary operator, which is a combination of the complex conjugation and the spin rotation operator. So, this needs to be kept in mind. So, the simplest, which is going to be very useful for the talk later, is what happens to a plane wave of light that you are shining on your sample. Suppose the light goes in the z-direction, so the electric field is proportional to the e to the minus ikz. And if you apply the time reversal operator, since there is no spin consideration with this beam of light, you come back. So, applying the time reversal operator on beam going in the plus z give you now a beam of light coming back in the minus z-direction. This is very simple, but as you will see, it is very deep and important for understanding experiments. OK, what happened with spin rotation? I think the easiest way to see it is in a classical picture, because in a classical picture I know how to create a magnetic moment. I'll take a loop of current. The magnetic moment is simply the current times the area enclosed by the loop, OK, and it's in the direction which is perpendicular to the loop. But current, as you remember, if I apply the time reversal operator, gets me the reverse direction of the current and the reverse direction of the current, therefore, is going to reverse the magnetic moment. So, applying the time reversal operator on a moment or on a spin gets me the spin in the opposite direction, which is exactly applying this spin rotation on that magnetic moment. Well, obviously, if you have a ferromagnet and you apply the time reversal operator, then you get a ferromagnet with the moment pointing in the opposite direction, OK? OK, so that's for time reversal operator now for unconventional superconductors. Well, as you will know, we've seen it. Superconductivity is the phenomenon that below a certain temperature the resistance goes to zero, identically to zero. And important also for this talk is the Meissner effect that below the critical temperature, below the superconducting transition, then magnetic field lines are going to be expelled from the interior of the superconductor that is called the Meissner effect. Superconductivity, it's a thermodynamic phase, it involves a thermodynamic transition and an order parameter that can be described by the pair wave function that has an amplitude and has a phase. And in general, you go through the superconducting transition, you break could be other symmetries as well, but at the minimum you break the U1 gauge symmetry and you condense electrons into Cooper pairs. So if I start now with the pair wave function which is made with operators that describe both the momentum and the spin, I can then rewrite it in terms of the orbital part, of course there is the center of mass which I'm going to ignore, and then there is the orbital part in the spin part to remind you that the pair wave function symmetry is the same as the gap symmetry, sometimes people use pair wave functions, sometimes people use the gap to describe the symmetry. And obviously since I need for any pair of electrons to take care that the wave function will be totally anti-symmetric under particle exchange, then if k goes to minus k, then s will go to s prime and there are two possibilities to do that. I can have an even parity for the orbital part and this can be described using angular momentum indices. So if the angular momentum index L is 0, 2, 4, that is it's even, then I get a spin singlet, a spin singlet superconductor, or Cooper pair, and if it's odd parity that is L equal 1, sometimes people call it p-wave, equal 3 that's f-wave, I mean here 2 is the d-wave, then we get a spin triplet superconductor. Now even versus odd parity sometimes also bring about different type of order parameter that people use, so I'll just flash it briefly, I will come back to it if I'll have time when I discuss specific models, specific materials, but even parity spin singlet, simply the gap function is a matrix of diagonal terms that have the delta like that with delta of k like that, where if you have an odd parity spin triplet, then the order parameter can be described with a vector and this is called the d-vector, I will not use it, at least not in the first lecture again, so don't confuse d-vector with d-wave, d-wave is the L equal 2 symmetry of the angular momentum, it just happened that people have been using d to describe this director of the vector that is associated with the order parameter of such superconductor, I will come back to it when we talk specific examples. So some hallmarks and then classification of these superconductors, I said unconventional superconductors, but obviously I need to distinguish conventional from unconventional. So in most classification conventional superconductors are associated with the BCS-type superconductors for which the angular momentum is zero in the center of mass of the cooper pair and it's a spin singlet superconductor. Such superconductors when it's a perfect superconductor, of course these are pairing of opposite momentum states, but in general it was shown by Anderson that such a superconductor is basically pairing time-reversed states and a hallmark of that is that if you average the wave function over the Fermi surface you get something which is finite. You have a gap everywhere on the Fermi surface which, therefore, scattering of electrons in non-magnetic impurities is not doing much to the superconductor, this is called the Anderson theorem. In fact, Tc remains the same. Tc in fact remains the same and it is still the way to very, very close to Tc. I don't have time to talk about it today, but in the presence of non-magnetic impurities then there is a criterion that tells you when Tc starts to change, not the critical behavior which is a mean field like as you all remember, but just Tc. If somebody will ask, I can talk about that, but otherwise it's still very, very close to Tc for almost all standard superconductors that Anderson theorem is fulfilled. Opposite to that, unconventional superconductors will be those that do have internal structure in the center of mass of the cooper pair. L, the angular momentum, therefore, is node zero and then many times there are nodes. I mean, this is a cartoon for the high Tc superconductors, the d-wave superconductors. This, by the way, does not break time over symmetry. It is simply a simple d-wave with spin singlet, but if you average now the pair wave function on the Fermi surface, you get zero. So many times this is used as the criterion between conventional and unconventional, although there are other criteria that people use, simply average the pair wave function over the Fermi surface whether you get zero or not. Now, in general, when you talk about superconductors, unconventional superconductors, those that have internal structure in the center of mass of the cooper pairs, then the key symmetries are time reversal and inversion symmetries. And these can be fulfilled or broken and a variety of novel phenomena can appear in the superconducting state as you analyze these symmetries. So today I will concentrate on time reversal symmetry. Inversion symmetry is also very important, but it's not going to be the main topic of today. I think that in talks that involve stripes, et cetera, and you may have heard some manifestations of this. OK, so I mentioned that this type of wave function does not break time reversal symmetry. It's a simple d wave, like for the cooperates, and therefore, as it is known, it's the dx square minus y square. However, if, for example, for some reason, in the center of mass, you will acquire an imaginary part, OK, like dxy, which have been proposed for quite a few systems, including the cooperates in the early days, then, obviously, plus idxy and minus idxy are degenerate. As they are degenerate, you have kind of angular momentum that on the average points one direction versus the opposite direction. This reminds you already of like an icing system, and therefore time reversal symmetry is broken. So whenever there is an imaginary part, then this is a signature for time reversal symmetry breaking. I'll come back to it in a minute. So if you now, and again a reminder, because I'm going to use it later, if you write the gap function in the normal way using the Fermi functions with this, the energy of the state that's the gap, then in general you can write, this is for the cases of internal structure in the pair wave function or in the gap function, then there is going to be some amplitude, which will represent really the size of the gap, which is going to be associated also with Tc of that superconductor. And then you can have whatever there is in the internal structure, and if an imaginary part appears, then it's going to be a time reversal symmetry superconductor. Yeah, then time reversal symmetry is broken. Right, yeah, if it's only, and actually I will even show an example of that, probably in the second lecture, UPT3 has such a phase. Yes, obviously, but if it's just the imaginary part, I can always rotate it. Anyway, so again to emphasize the fact that high Tc dx squared minus y squared, the gap function can be written as cosine kx minus cosine ky, and this does not break time reversal symmetry. Usually when time reversal symmetry is broken, then there is a tendency to gap the nodes. Okay, now a little bit of history, early searches for time reversal symmetry breaking. So in fact time reversal symmetry breaking in superconductors, at least to my knowledge, started with a proposal that high Tc superconductors exhibit any unsuperconductivity. This goes back to 1987, and in fact I think that one of the very early places where possible any unsuperconductivity was announced was here, there was a conference here in Trieste in 1987. I was here, which means I'm old. And this came quite early in early papers of Vadim Kalmayer and Bob Laughlin, and then became a very interesting theory to discuss. So the basic, and again I don't want to go too much into it, I mean it goes way beyond the just searching for time reversal symmetry breaking, but the idea was to write the many particle wave function of the system, and then in exchanging two particles, of course you acquire a phase, which we know how to take care of in the case of either bosons or fermions. So if you write the phase that is acquired as pi 1 minus 1 over nu, then nu equal 1 is going to give me zero here and therefore it has to do with bosons, but if nu is equal to infinity, then I get pi and e to the i pi is what you get if you exchange two fermions, but then if you write it this way you can have any other value for nu and for any other people called it anion. So if you didn't know the origin of anions, it's the any that's what started the name. So such a superconductor, which again I'm not going to discuss the nature of the superconductivity in this state, but I would say that the ground state of the superconductor was shown to break time versus symmetry and in fact break time versus symmetry to order 1 and I'll say in a minute what it means. It means that the effect is very, very large. So very early on experiments were proposed and you can see from the cast of characters that proposed these experiments in theory really captured the imagination of almost every theoretical physicist. First experiment to be proposed was the muSR experiment in which you shoot a muon into that superconductor and then you destroy locally superconductivity or alternatively there are impurities that destroy superconductivity that comes reside in that site. So if superconductivity is destroyed, it is shown that there is going to be a local magnetic field, the muon will process in this local magnetic field, you'll get oscillations of this procession and then this is detected via the positon that is ejected and you do the proper statistics and then determine what is the local magnetic field. And this is, it's not a very complicated calculation to determine what should be that local magnetic field and it was supposed to be very large. I think in the early calculations it was supposed to be something of order of 100 Gauss or something like that, easily detectable by this method. Okay? So that's basically the... Now these experiments were done and in some materials they were shown to exist. Stonsum ruthenate, I'll talk about Stonsum ruthenate later and then indeed there is a difference between above and below, but in the cuprates there wasn't. Then another experiment that was proposed was a spontaneous whole effect. Just like anomalous whole effect in a magnetic material that is whole effect without the application of magnetic field, remember whole effect is a hallmark of time reversal symmetry breaking. So therefore an anomalous whole effect or spontaneous whole effect means that time reversal symmetry is broken just like in a magnetic material. Such experiments failed in a big way because of misalignment, remember, in the normal way you measure whole effect is now with magnetic field you apply the magnetic field in the z-direction and you have a current going say in the x-direction and you measure voltage between two leads in the y-direction. Now you have to prepare those leads and you have to prepare them exactly to be across from each other. You can never do that in real life. There is always misalignment and if you are trying to find a tiny effect as a large effect was not found experimentally, there is a problem. So experiment, this experiment failed because of misalignment and edge problems. Also if you take a small crystal or film, the edges are really rough and this experiment becomes a mess. Well actually, and I thought since experiments are as important in sorting theories, we decided to come back to the issue of whole effect. So let me show you one way how you can measure the whole effect with absolutely eliminating any such misalignment and edge effect. So I'm pausing that. It is still time over symmetry breaking and it can be applied to superconductors. So remember the Corbino disc. So here is a Corbino disc. The gray is the material I want to measure and what I'm going to do is, I'm going to put a very high conductivity material on the outside and the inside and keep it at a constant potential. So I'm going to measure the outside at ground and the inside at some voltage V. So if you remember your E&M, the electric field is radial and the radial electric field is the voltage divided by, I mean, this R is wherever at any radius R, this is the electric field and the electric field in the phi direction is zero, right? That's symmetry. There is going to be a current, though, and it's only a radial current. I mean, there is a zimutal, it's not, I mean, if you connect it through instrument, there is going to be current flowing, but let's say that right now I'm just putting it at a constant voltage without drawing current, then the only current flowing is a current going around the disk. You can easily calculate that this current density, so at any radius R, is simply sigma xy, that's the off diagonal term of the conductivity, therefore the whole conductivity times the radial electric field. The radial electric field is what you put because you put a voltage in the magnetic moment. What this current does is it gives you a magnetic moment, remember from before. The magnetic moment can easily be calculated when you integrate from the inner to the outer radius and therefore you get, if you want, you can relate this magnetic moment to an average current and just like in the simple way I showed you earlier, so you have a moment that is proportional to the average current and to the average area enclosed by this loop. Now, if you remember your multiple expansion in E&M, I don't remember which chapter in Jackson, then for such an arrangement, the first moment that appears is the dipole moment, therefore you are only approximately just this moment. Well, this moment does not have any contribution from longitudinal direction whatsoever and therefore you measure sigma xy period, nothing else. So, here is a configuration you simply, how do you measure it? Well, the easiest way is you take your disk, you put it on a cantilever and you look at the torque that is exerted on the cantilever by a magnetic field and if you have a spontaneous field then you put a pure parallel field in order to produce the torque and you are not sensitive, you are not contributing to the perpendicular one. So, that's a math. I have it because I'll put the slides on, you can see and check me and you can calculate what is going to be the moment and this if you look at the sensitivity of these apparatus then you can get sensitivity of sigma xy as low, sigma xy, remember it's not rho xy, in order to calculate rho xy you need to sigma xy divided by sigma xx squared believe me nobody ever measured such small sigma xy we made these cantilever that's the first generation that's the important generation because that's the one that produces the data there is a symmetry here these are the leads that go to the side these are the leads that go to the center this is showing how we excite the cantilever remember your radiation pressure is going to push the cantilever proportional to the power of the light we can do that it produces a very nice resonance that's the first measurement with current in one direction versus the other and we could calculate what is that whole current okay, that's not the subject of this talk I just wanted to demonstrate yet again that when there is a problem you need to think about what from the experimental point of view what is the issue and how can I use symmetry in particular symmetry in order to mitigate that and design a new type of experiment that will allow me to measure something with better in a better way and better sensitivity that was one example but not the main one I'll come to the main one back to our topic I talked about at the time spontaneous whole effect but then there was another proposal by Shahgeng Ren and Tony Z that said basically we are talking about an effect that should be very similar to magnetic to magnetic material and therefore magnetic materials have so called magneto-optic effects such as care, far day, dichroism et cetera so let's calculate whether these anion superconductors also produce much magneto-optic like effects and they did and they came up with an idea and this is 89 that indeed this effect should be there and in fact it should be there if any on superconductivity exist this effect should be there of order one means that as strong as for normal ferromagnets what it means because I am now introducing magneto-optics let me say a few things about it magneto-optics it investigates the response of a sample that breaks time reversal symmetry to polarize light and it's important like whole effect it measures sigma xy but now because I am using light at the finite frequency it measures sigma xy at the finite frequency omega so most optical measurements that you are familiar with look at sigma xy but the longitudinal part if you want that's the transverse part in general it's more difficult to measure it can detect magnetic effects obviously and it appears to be for example if I am looking at recent publications recent current physics it appears as a diagnostic for topological magneto-electric effect the in topological insulators et cetera and as I'll show soon it detects time reversal symmetry effects in unconventional superconductor and many more so to remind you how does it work well let's start with a classical picture you write the equation of motion the Lorentz force for an electron that is moving with a velocity v in the presence of magnetic field B and to go in a circular motion and if you look at the direction of the force you see that depending on whether it moves clockwise or counterclockwise you get the different force so if you start with a circularly polarized light that is an electric field that has circular this circular electric field then depending whether it is counterclockwise or clockwise you are going to get a different direction of the force which means that you get a different direction a different difference in the conductivity for right and left represent the circular polarization whether it's right or left and in turn if the conductivity are different obviously the indices of refraction are different if you look at the quantum picture then remember that a circularly polarized light the circular polarization are equivalent to the photon spin which is plus minus one so if you now look at some transitions you start from say L equal 0 to L equal 1 then you split the Lz and then you split the m states then as they align like that you see that there are different allowed transitions for left circularly polarized photons in a way the same as if the photon has a plus or minus spin so if you are at jz of plus one half you are here you can only go to three halves because you can only add one so we subtract one and if you add one you go to the one half here and these are different transitions for again this means that you are going to get differences in the optical conductivities and you are going to get differences in indices of refraction ok coming back to the equation of motion now in the presence of an electric and magnetic field you can then go through these equations again I'm sure you did it many times I'm not going to go too much on that I just wanted it to be on the slide so you'll have it when I publish the slide so you can calculate the polarizability based on that electric field you put it back you get the displacement vector d which can be written in terms of a dielectric function epsilon omega that's the electric function is a function of frequency and d is of course e plus 4 pi p the polarization now the dielectric constant can be written in terms of the frequency as 1 minus you can see here and you've done this many times including calculating the plasma frequency again skipping that but the important thing is that if we add the magnetic field usually in your E&M classes you write that but then you write epsilon of omega you forget about that part the magnetic field you just go through that you write the dielectric function as a function of frequency and that's it but if we now add the magnetic field in this way then simply through the equation of motion then it adds a term we can rewrite it in terms of the electric field with a new vector g g is called the gerotropic vector and you see basically what I do here I exchange e and h in the cross term then I can write everything in terms of e and then I give f times h is g writing it this way so you see that in addition to the dielectric function I have this new term this gerotropic term now there is a comment here which I don't have time to discuss but I think it's a very deep comment that I suggest all of you to go and look at it it's mentioned only in passing in Lauda lift sheets electro dynamic of continuous media in the second edition that's without pitaevski and there is a little bit of more elaboration in the paper of Pershan from 67 you notice that you here I only involve the dielectric constant as a function of frequency I don't involve the permeability as a function of frequency I mean there is also a magnetic field term I apply light, I have an electric field and magnetic field and I completely ignore that well there is a very good reason for that everything is shoved into the dielectric function and I strongly suggest to go and look at this paper in any event I go back to the equations and now I apply an electric field that a plane wave type of an electric field with k the k vector describe the direction and the frequency and I use the Maxwell equation and I get that and now in the presence of this plane wave and using these equations I can get an expression for the displacement vector and n square is going to be the eigenvalues of such vector which I can calculate because I can rewrite this vector equation as a matrix equation and then the components are going to be given in terms of e x, e y, e z with n square being the eigenvalues I can calculate them again I am not going to go through the steps but if you calculate them you find that when diagonalizing you get two values which are I will call them n plus minus obviously these are going to be for right and left which are epsilon plus minus g where g is this gerotropic term I talked about I showed you earlier so you see in the if g does not exist then n square plus minus and you remember that the index of infraction is just the square root of the dielectric function so it's the same it's the same here now if you do some further manipulations for a plane wave you can get that for the x and y components in the displacement vector you can look at the angle of rotation which is just the ratio of them you get this tange of this angle and in the presence of g that angle the tangent of that angle is the tangent of the so-called Faraday angle what is the Faraday angle therefore it's simply if you take a linearly polarized light and if you go through a material you get rotation of the polarization and the amount of rotation is this angle that I calculated here for a plane wave that is linearly polarized so from here I can calculate where I go z is now the thickness of the material because I'm going through the material and therefore that's the thickness of the material I acquire this component and that's simply the real part of this difference you can check that what I wrote here is exactly that so for small angles I can expand that and then what you get is that the Faraday angle is given in terms of the gerotropic piece was simply f times h times omega divided by square root of epsilon which is the average index of refraction and then the thickness of the material ok there are further manipulations in order to connect it to the optical conductivity but in general the electric function if you remember is one or epsilon infinity if I have other contributions that is a constant plus 4 pi i over omega times sigma that's in general the relation between the dielectric function and the conductivity so it continues to be the same relation now between the dielectric function for right and left circularly polarized light and the optical conductivity for right and left circularly polarized light and now if I calculate this and I take the very small gerotropic for example very small magnetic field in the presence of magnetic field then I get this relation that's some index of refraction which is I can write in terms of real and imaginary part and this is probably what you remember from your optics classes that the index of refraction has a real and imaginary part but if you trace back when you have these notes you can see where it comes from alright so if I now write the tensor for the conductivity and let's say that everything I do say apply magnetic field et cetera in the direction then the action is in the xy and I'm going to measure the xy of diagonal terms of the conductivity these are the terms that tell me whether time reversal symmetry is broken or not you can write it in terms of right and left circularly polarized light which in turn can be written in terms of sigma xx that longitudinal conductivity sigma xy and then if you further continue you will get the relation to the previous to the n plus ik as I showed before now there is another phenomenon obviously if I have linearly polarized light that I go through the material I can also have a linearly polarized light that is reflected for my material but I assume first that the thickness of the material is larger than the optical penetration depth of the material I'm not sure by the way ask me questions or stop me throw tomatoes which as you know in Italy are very good so I will not I will not mind but I feel like so if there are any questions please because I think I know this so if anyway so I can do the same with reflection and obviously for thickness which is larger than the optical penetration depth then especially in normal incidence I'm talking now about a different type of phenomenon which is the care effect now for almost all the measurements that are relevant for these superconductors I'm going to talk about it's the care effect which is important and the reason is that we are talking about bulk materials people are making very small crystals still much larger than the optical penetration depth I'm not sure if you guys are familiar with numbers but for example what's the optical penetration depth in the visible of aluminum anybody knows of gold it's larger than nanometers but it's in the right direction what's the optical penetration depth of the cuprates say people are talking about similar oxide superconductors how is the optical penetration depth related to the conductivity if the conductivity is yes that's in general this is correct so the optical penetration depth goes like one over square root of the conductivity similarly with the frequency but let's say we are talking about visible light for good metals it's going to be of the order of say 10 nanometers or so ITC and other oxides it's 10 times larger so it goes pretty deep into the material so if I want to measure care effect I need to be at least a few hundred nanometers if not more so typically thin films that people have been preparing of high quality they will be thinner than the optical penetration depths but that's something to keep in mind but I will talk about those so these are of order a few hundred microns at least and therefore we are safe when we talk about the care effect so if the further effect was simply the difference between the indices of refraction for right and left circularly polarized light the care effect is the imaginary part of the combination of these two and I'm not sure if I'll have enough time to go through it that the care effect is if in the approximation of g, namely very small gerotropic compared to the average dielectric function when it's small then it's basically just measuring the imaginary part of the diagonal term of the conductivity of omega so that's actually very important by the way, if you didn't notice then in the case of the Faraday when g is much smaller than epsilon you mostly measure the real part I think it was here somewhere yes, it's the real part so Faraday and Kerr angle can be then related via karmaskonic relations in that limit so what do we conclude we conclude that when time over symmetry is broken then we have indices of refraction for right and left circularly polarized light that are different and we can measure one of these two effects the Faraday effect where I go through the material and look at the rotation of the polarization and the care effect in which I go in reflection ok so I want also to I don't remember when I started ok, excellent ok, so I want to now talk a little bit more on the care effect care effect has been now very popular topological materials people measure that so I want to talk about the conditions for care effect and care effect as measuring time over symmetry breaking so in the presence of magnetic field we have a finite gyro tropic term and and therefore we have differences in the indices of refraction, we went through the steps or you will go through them later when you look at the notes so ok, we understand that but there is a question is the general requirement if the gyro tropic term is non-zero also that the care effect is non-zero because I showed you a gyro tropic term that originated from the magnetic field there are other reasons for other ways to generate gyro tropic terms that is for example optical activity, chirality quotes if you didn't know there is a direction which has chirality in fact the rotation of the polarization of light passing through quotes was discovered what 200 years ago by and so people knew about chirality already back then and Louis Pasteur actually did experiments in which he showed that if you use circularly polarized light then he can distinguish chirality of the same molecule but with different chiralities you know, sugar molecule we are sensitive that is this kind of sensitive to one chirality and people have been talking about producing sugar with the opposite chirality that we will not digest there are other molecules that appear in nature with both chiralities and Louis Pasteur was the first in fact sodium ammonium tartrate was investigated by him and he showed that there is opposite chirality and that the index of refraction for them are not the same it's different so obviously if I now go back I will say well there is a finite gerotropic term will I be able to measure a care effect? that's a question right? so as we said the question is is this enough for a care effect we showed that the time of the proximity breaking produces n plus different than n minus this in turn gives you care effect now we go back if these two are not equal does it mean that we are going to have a finite care effect? well in fact this is a non-trivial issue that was even mentioned again in Landau and for the younger people unfortunately you are probably using the third edition and in the third edition this is not treated properly in fact there is a mistake in the second edition which is my favorite edition that's the Just Landau Leafschitz it already mentions that there is an issue with what constitutive relation you use in order to calculate the care effect of a material that has opposite chirality and therefore different difference in the indices of refraction of the dielectric function in Landau Leafschitz the current version of electrodynamics of continuous media one starts with this constitutive relation actually relates the electric field that you apply to the material with the electric displacement that you are going to measure as a consequence of whatever is happening in the material and then there is this normal term that's the dielectric function that is the proportionality between the displacement and the electric field and then there is the second term which can be written in this way remember the dE dx is equivalent in Fourier space to k so this is going to be something like k with the electric field ok so just to show you how deep is this issue on whether one should or not use this constitutive relation in the early days of high tc as this is part of what we are talking about today Gorkov wrote a paper in which after some results on care effect he used this equation to say that the finite effect that people measured is probably coming from chirality and not from time reversal symmetry breaking so that's an issue so let's see the difference between these two effects ok now as you know I can have a situation so we are talking about circularly polarized light that goes through a chiral material and you take a screw a screw is a representation of a chiral material but as you know if you take a bolt which is say right handed bolt and you take the nut that goes through then it doesn't matter from which side of the bolt you are screwing the nut it will work either way in other words if the bolt is right hand chiral then it doesn't matter from which side you are looking at it it stays right hand chiral right can you imagine that ok but let's look at the magnetic material the magnetization goes in one direction ok that's like a magnetic field goes in one direction an electron goes in one way or another is going to be very different in other words if I'm looking at the now magnetic material from one side and I can do it for example by allowing electrons to circle they are going to circle in one direction I will call it say right handed but if I'm looking at a magnetic material from the other side and I shoot an electron now to go in it's going to circle in the opposite direction so this is like saying oh I can see a right hand screw this one says I can see a left hand screw that's different than this both are chiral this one does break time reversal symmetry this one does not break time reversal symmetry ok I can now reflect light on either of these materials ok I apply an electric field and I as a result I get the electric displacement ok now a very fundamental issue in all these experiments is to remember that you need to respect on zager reciprocity so let me remind you on zager reciprocity when I do that on zager reciprocity basically says that if I say usually it's used by mostly used in this form by electrical engineers who are looking at an antenna you shine light from one end and you look at the antenna that picks it up and then you shine from the antenna side and you pick it up at the other and basically on zager reciprocity says you need to get the same thing so let's write it in terms of our our effect and response the effect is the electric field the response is the electric displacement that I measure that's after the interaction with the material ok so if I start with point one I send the electric field I get at point two the electric displacement after the interaction here now I send from here and then I get at point one the electric displacement ok so here I send and get here and then the opposite here and reciprocity says that the integral over all space of d1 dot e2 is the same as the integral over all space of d2 dot e1 this is on zager reciprocity ok so I need to apply that I always need to apply that in order to see where the reciprocity when reciprocity needs to be preserved yes I don't need to, I mean it's a thought experiment so I put the source here and the detector here and then I exchange them and I integrate over all space obviously over all time there is no time issue, I mean actually this is a good point because if you measure at finite times and we can then negotiate what does it mean to be short time or long time then reciprocity does not need to be preserved in spring glasses for example you can if you measure over short short enough time you will not have reciprocity simply because the system is not equilibrated well that just just following on zager's derivation so the point is that if this is if this is the condition I think there is no argument that this is the condition for reciprocity and reciprocity means that time over symmetry is preserved so I'm using now the Landau and Liftschitz constitutive relation I apply it here and here that's okay and I calculate and what do I get well I get an extra term okay I get that instead of getting this equal that I get an extra term which is here and as you can see this term which is ddxk of that it can be written as a divergence of this piece again you will look at the notes later and the divergence calculator of all space is actually a surface integral so it turns the difference why they are not equal they should have been equal by the way I could also manipulate the other side and I would get the exact same thing it would be divergence of e2 in course with this tenso gamma tenso times e1 it's the same but the point is that I'm getting a surface integral and this surface integral basically says that what I forgot with this constitutive relation is the fact that I have an interface say between vacuum and the material and you cannot not include the interface in the calculation okay so this is actually very deep always always being shoved under the rug but you need to understand that well if you take that term then you find that the correct constitutive relation is the one that we showed before the one that appears in Landau-Liftschitz plus a different term which takes care of the surface in that if I have a step function and the step function let's say zero here and one here and then I want to go and reflect exactly from the wall that separates the zero and the one I need to take care that the average between the zero and the one is one half in the middle and this is exactly kind of heuristic argument but that's exactly why this one half appears here so if you go through this calculation you can now correct the section in Landau-Liftschitz and find what is the correct constitutive relation in the presence of time reversal symmetry okay now I'll go fast because I want to be able to make some progress I can now use scattering theory to calculate the care effect with the correct boundary condition the correct constitutive relation so I can define then the state for light with a k vector with a circularly say right circularly polarized light and then coming backwards with a right circularly polarized light I can use two types of conventions one in which the circular polarization is in the direction I always measure it in the direction of propagation and one which is fixed in space and then I propagate in that here I'm using the one that goes with the with the light so I'm scattering light from k right to minus k right and then k left minus k left and then I can use all that machinery to calculate the transition amplitude using Green's function you can do that or just go over the over the the nodes to then define the reflection coefficient of right circularly goes back into right circularly and left circularly back to left circularly so these are these two you can then show that the care angle is just the difference of the arguments of these two and it is directly related to the reciprocity again if I'll spend time on that it's going to be tedious and you'll probably lose me you will be able to go over these calculations and see what it means so that is that care effect and it turns that if indeed reciprocity is obeyed then the interaction term has this form the transition amplitude has this form and the transition amplitude for k right goes into minus k right is equal to k left goes into minus k left that is these two are the same and therefore the care angle is zero so if time reversal symmetry is preserved or if reciprocity if I want to be more general is preserved then there is no care angle so we are safe we started with a care angle as a consequence of time reversal symmetry breaking and then we discovered that there are materials that have different dielectric constant so different indices of refraction for right and left circularly polarized light and then we were puzzled whether they will or not produce care effect well they will not because when time reversal symmetry is preserved care effect is identically zero however if time reversal symmetry is not preserved I can show that that transition amplitude and you will just go care effect at home when you look at it you will find that these transition amplitudes are different and these two amplitudes of reflection are different and therefore the care effect is finite so care effect therefore as a consequence only measures breakdown of reciprocity time reversal symmetry breaking is breakdown of reciprocity now ok that that's just the summary of that because you can show that this expression goes into this expression which I showed you first which goes into this expression when the gyrotropy is small that's just the summary but it's always in the case of time reversal symmetry breaking ok what about size of effect I should stop just finish here so what about the size of the effect the care effect well here are two examples strontium ruthenium O3 is an itinerant ferromagnet oxide similar to many of the oxides that were discussed in fact it is similar to some of the cooperates it is a ferromagnet below about 150 Kelvin and as you can see time you get to low temperatures the magnetization saturates and the care angle is about 10 milli radians 10 milli radians that 0.01 radian this is enormous iron for example in photon energy of order of 1 EV that's visible light 1.5 visible light also about 10 milli radians doesn't matter plus or minus we can talk about that but it's a large effect 0.01 radian in some ferromagnet it can get all the way to 0.1 and maybe even more so let me just mention that at the time when this was proposed for any on superconductivity immediately after people measured effect of order 200 milli radians at Bay Labs in other places in the world so this is enormous this is like a very very very strong ferromagnet with very large spin orbit interaction which is usually the way you couple to the magnetic component which the world celebrated any on superconductivity the problem is that a good experiment should reject all reciprocal effects which these people failed to do and we showed just before that care effect is finite only when reciprocity is broken ok so you need to check whether you don't have any contamination by reciprocal effects and this is not very easy you need to measure the absolute value of the care effect you cannot use magnetic field to modulate the care effect in a superconductor because you apply magnetic field you broke time reversal symmetry and therefore a cross polarization method which is usually what people have been using will not work so I will stop by just telling you the way that we measured and I will start the second lecture with that is we invented a new apparatus this apparatus is based on a rotation a fiber optic gyroscope this is what is used there are fragments of such gyroscopes everywhere in Afghanistan and Iraq because you can make them very compact by nose of missiles and explode them these fiber optic gyroscopes so in a gyroscope measures rotation so I think this is the only experiment ever made in which an apparatus was calibrated using Earth rotation this is the calibration of the apparatus that I am going to discuss in the next lecture it was calibrated using Earth rotation this is a 100 micro radian precisely calibrated and then when the high temperature superconductors that were supposed to produce many milli radians here I calibrated it at 100 micro radian to within 1 micro radian if you want this is zero showed nothing so despite the fact that the effect was found experimentally we showed with this apparatus that it does not so the question is why and I'll start the second lecture talking about that and then move on to where we do now using this apparatus we do find effects in other unconventional superconductors oh questions questions