 So thank you for staying around till the last day, so we are going to change gears a little bit and talk again about in particular we are going to discuss today two modules that allow us to calculate conducting properties. So I will discuss about this and then the second part of this lecture will be given by a similar concept that will present the transport part. So since a lot of people have asked where is Dinghamton and just a quick view of the New York state, so this is New York, so we are somewhere here in the boundary of Pennsylvania and just kind of couple of interesting things that IBM started in Dinghamton and that's where also the place where the flight simulator was in fact. And the most recent thing is that in 2019, Professor Stanley Wittingham, that the Nobel Prize in Chemistry along with Jean Boudinat, then I'm going to show you a few more for here related to everyday life with being all of us affected. So just to start with, I will review slightly the, just briefly, the computational flow of electron form and matrix elements in it. And since not everybody here is doing superconductivity, I will just give a very brief overview of this year's theory and then a bit more details about Eliasberg's formalism of superconductivity, which is in Pennsylvania. In other words, in Pennsylvania, I will just show two short examples, one of which will be used in our first tutorial today. Yes, just to get started, you have already seen this Tuesday presentation given by Professor Stanley Wittingham. So what EPW does, it uses Bannier functions to convert for coarse grids that are calculated with quantum espresso, then we use Bannier 90 to the library. And then we go and we will see we will calculate things in the real space, and then from here we can convert back to then any basically mesh in very fast, very fine matches. So, as you have already seen, this is basically the quantity that we are calculating. This is the electron form matrix element. And this part here, that is parts, the periodic parts of the wave function, and this quantity in the middle, this is our deformation potential that we calculated with density function and perforation theory and quantum espresso, and it involves the measurement of every atom in the unit cell, according to a phonon frequency. So the good thing about the FTP is that you don't need to use supercells if you do it in your frozen phonon approach, but you can do all the calculation in the unit cell. This is because we have here this phonon modulation, which is shown in yellow. So once you have calculated the deformation potential, you need a phonon calculation, quantum espresso, you can use this information and we can, sorry, my mouse works very slowly. And then we can move on to the EPW code, and the first thing that we do is we estimate the electron phonon matrix elements in the real space. For this, we evaluate this object again, and as you have learned already on Tuesday, the good thing is that Vanier functions are localized. So whenever these three objects, if they happen to be far away with respect to each other, the electron phonon matrix element is going to be zero. So what it means, it means that you don't have to calculate hundreds of thousands of electron phonon matrix elements in order to estimate the electron phonon matrix elements in real space. This is going to be cheap. And once you have the quantities in the real space, we can go back to any fine measures in the block space and estimate the electron phonon matrix elements. So, you know, kind of, or what, how this will happen in the code, we basically have some keywords that you will see that what it does in practice, it leaves the Vanier, sorry, the wave function and some data from the EPW.x executable with quantum espresso, then we generate in our input file, you can do it directly, we don't do it in separate Vanier files, we have keywords, the same as in Vanier 90 to generate the inputs for Vanier 90 and perform vanierization. And the main kind of subroutine that is this electron phonon shuffle graph that calls first for a phonon shuffle and then read the VSEF, so this name you already suggest calculates the VSEF files created with quantum espresso. And in this subroutine, we first calculate the electron phonon matrix elements on the course grid, on the block course grid. Then we perform, so this is another main subroutine where we transform the electron phonon vertices and all the other quantities here I just wrote just electron phonon vertices like from the block to Vanier and then from Vanier back to block. So once you have this electron phonon vector elements in the confined greens in the block representation, we can calculate other properties like transport and superconductivity and the optical transitions and more recently all our own rules will be available in the code. So what you need to keep in mind that is when you are going to perform calculation with quantum espresso you are going to do a few expensive calculation, relatively expensive calculation when you do phonons, but then in principle you can go on very dense meshes to this Vanier Fourier interpolation technique. Now, just a few words about the superconductivity in general, and more particular about the VCS here, yes. So superconductors as everybody know, or have zero resistance below a critical temperature, and the first microscopic theory of superconductivity was proposed by Barney and Schieffer, and the assumption was that there must be some kind of attractive interaction between electrons that compensates the full impartial. And in conventional superconductors this attraction is thought about by forms in other words, by latest vibration yeah so this is just a feature I believe from Wikipedia where you can see this as an electron moves to a crystal creates the potential because the electron attracts the, the ions, and then another electron with opposite momentum and spin will be attracted in this region of the most positive potential and because of this, the two electrons become correlated and form what's So diagrammatically this is represented in the following form, you have an electron of momentum k, which absorbs a phonon, and then the same this one is, sorry, emits a phonon and the same is going to be absorbed by another electron, as I said momentum minus k, and the first electron is going to scatter in a new state k prime, and the first, so the first electron, the second would scatter in a new state of momentum minus k, minus k prime yeah so these two pairs we say that they let them scatter on each other by exchanging a virtual phonon. And for this transition to take place, we need to have the original pair and the final pair to be in the vicinity of the formula. So the first pair is to be occupied and the second pair needs to be emptied. So if, so an important, let's say, result of the DCS theory that because of this exchange of phonons, what we'll see is we will see an opening of a gap in the normal state of a matter. So this is a metal, this is how your friends actually look like, we have no gap here, everything is occupied, you know the Fermi level, everything is empty above, and now in a superconductor, the fermions do not behave, sorry the electrons do not behave the fermions anymore, since they form the superfairs that basically behave as bosons. So as a result of this behavior, we can have more than one electron in the same state, and this results in the opening of this gap and what we'll see is that because of the electrons that have initially occupied this state, now the cluster here, and then you have this sharp increase in the density of states. Yeah, so this is something that can be measured experimentally. And this gap that opens has a temperature dependence here, so this is the DCS expression for the superconducting gap, where here V is the sparing potential, and P in case just the quasi particle excitation energy that you can see is shifted by the gap, and the property is that the gap has a maximum value at zero Kelvin and vanishes at a critical temperature, so once you exceed to go above the critical temperature, the gap closes and you are again in a normal state of a matter. So the BCS is a phenomenological method or formalism, it's a descriptive theory, and in other words it's material independent, so it has this well known expression that the ratio of the gap, two times the gap to the critical temperature is 3.53. And another, let's say drawback of the BCS theory is assumes that the interaction between electrons and phonons is instantaneous, while in practice this is the target in time. So I'm going to move a bit gears and there will be a few slides with a lot of equations so you don't have to understand everything so don't get scared I just want to give you an idea about the, what's behind the equations that you are going to do that are implemented in the code and are solved in order to get the superconducting gap. So we are in this case what we are looking for is to find a two by two generalized matrix green function, and this greens function is basically used as you would see in a few slides to describe both the propagation of the classic article, and of the superconducting function. So that's a very similar to an expression that the channel showed on Tuesday, the only difference is now that they have these heads here, and this, in my case, points that these are matrices, yeah. And how T powers before this is the weeks and it's the time ordering operator, and sign is a two component field operator that contains the annihilation and the creation of an electron at a state of Nk. So if you just plug in this two component operator in this expression, you will get basically this two by two matrix. In this case, you can see that the diagonal parts are both in the same state Nk, sorry, the two components C and C plus for the diagonal parts in the same state, C and spin off. While for the non diagonal parts, you have the C and K, C and minus K and spin down, so the spin and the momentum are inverted. The diagonal part basically correspond to normal state green function and describe single particle excitations for electrons and poles, where the all diagonal elements are for the anomalous green functions, and they describe the two operators. So again, in the super if you are in the normal state, these two components of diagonal components become C. So we already seen in a few talks, I think, so please talk or also discuss this Fourier transport, it's more convenient instead of working in imaginary time, you can perform a Fourier transformation in which you can go in this matibata frequency or imaginary matibata frequency space. So this is just the transformation, how the matrix elements are written in the imaginary frequency axis. So, let's get a bit more detail so the basic in the bottom line is to find we need to find the greens function that describes the superconductor state. So the starting point, it's again the electrons, electrons self energy. So again, if I wouldn't have the heads here, this would be the same as, and this style matrices, which are the following matrices, this would be the same electron self energy that you've seen for the normal state. So the, this here I just wrote it in a condensed form is an isotropic electron-phonon coupling, but basically it contains the phonon propagator. The electron propagator multiplied with the electron-phonon multiplexer and square you will see the expression in a couple of slides. When you multiply this with the interacting greens function basically get this fun me without self energy that was also discussed on Tuesday. And the second term, this is the full on interaction. Yeah, so this would be the GW self energy. So you have the two components that you need to take care of the self energy is the electron-phonon interaction and the electron-electron interaction. And the, what we are looking to do right now is to solve this equation self consistently, but in order to do this, we need to find an expression for the whole G therefore the interacting green function. So in this case, we need to remember that the greens function obeys a Dyson equation in much better space. So in this case you can write the expression for or interacting with this function this we can plug in the genome is the non interacting and for Sigma, you are going to see a big the meaning again a bit later, we write the Sigma in terms of three scalar functions as a linear function of power matrices. Yeah, so Z here is the master normalization function. You have again already heard about the max normalization function a few talks this week. This guy is just an energy shift from the Fermi level in general it's very small. So when we are going to solve basically this formula is in the code you are basically calculating this quantities Z and delta. So now we plug in these two expressions in the Dyson equation. If you invert this G matrix, you will get a following expression for for the green function you again plug it in back to our original expression. And if you now equate the two terms from the, from the expression from the same energy, you will get what they are called an isopropic and after the question. Sorry. Okay, this is my part. So you can see that these equations need to be so it's kind of circumstances. Since we have Z entering both and that are on the left and on the right, the same year Kai and both left and right, as I told you this guy is simply an energy sheet you see this is how the energy levels with respect to the family level and this is just a sheet and the same for that. So we need to solve these three equations are consistent with the same time. And there is an additional equation for the electron number in order to find out to find the particular. And this is a new implementation that is not available in the source code that you are going to use today, but it will be available in the next release of quantum express which will be at the end of May. To start this calculations, you basically, you need this input flight. So one thing to notice here is that the column interaction only enters in one of this expression only enters in the expression for the superconducting yet because all the other effects that come from the electronic electronic interaction in the normal state should be already included in the, in the single particle. I don't know. One thing that I should also actually pointed out earlier but I forgot is that the phone on a little foreign interaction and the electronic interaction are on two different energy scale. So in other words, when we do this summation over the mass frequency, we need to have a cut off yeah we cannot have an infinite sum. And the cut off of the electron phone interaction is of the order of 5 to 10. In other words, you can take it the largest quantum frequency in your system, but the current interaction is of the order of 10 electron volts. Yes, we will go from a scale of many electron volts to a scale of 10 electron volts and I will come back to this one in the next slide. So now, as I said, this is a new intervention what was done before and what it was in general was standard is to assume that the density of states is constant, you know the family level. So this allows some simplification so you can do, you can solve, you can integrate this analytical and then you can show that this guy is basically zero. And then this is also serial because it all depends on time and what you are used to is only two equations that we call them now fairly sticky and you can see that there are these delta functions and in other words, these equations are very only within a family window around the energy level. And the key point and another thing that I have. The change here is that we don't have a full pool of interaction that will come in this in a moment, but we have a semi-empty part. Now, as I said before, this depends on the electron form on this element and this term here, this is just their form propagator. And that's the quantity that can be efficiently calculated. Now, regarding the current interaction that's something that will work in the future we are planning to implement this but currently you can. You can use this as a parameter as given in the code so for most superconductors it's considered that it falls somewhere between zero one and zero two. In principle, you can calculate it or using RPA or from GW codes like Berger, GW or Schrodinger, GW or L and then you can use this Anderson model potential to reduce basically to reduce the screen pool of interaction, this parameter which is called new start. Yes, so by making this strongly reduced pool of departure, what happens is that you are able to reduce that cutoff in the Matsubara frequency that I said that should be of the order of the 10 electron volts. You can reduce it at the same scale with the frequency of the electron form. I think I should think about it so just as a conclusion, you can this couple of equations need to be solved at every temperature. So you can see that you have here the T and they must be evaluated on dense in general dense electron and phonon key meshes to properly describe an isotropic effect. And then you need to have this sum over the Matsubara frequency. And then for this particular case if you only solve the two they are only meaningful for states or near or at or near the current surface. And you will see in the code this are the input slides to perform these calculations. So if you want other quantities like spectral representations then you need to go from the imaginary axis to the real axis so in order to do this you will need to perform an analytic continuation. In two ways, this is a very simple way to work by the approximants, and in most cases it works relatively well, or you can use an interactive procedure, but this is very heavy computation. Right now I will say that you need to use it on thousands of processors if you want, it's extremely heavy. So with this I said that I will give two examples and this is the example that we are going to use in the exercise today. This is the prototypical phonon mediated superconductor and the reason why I'm showing you this example because this is a two gap superconductor. So as you, your time, there are two sets of states of the family level for the show here in red, which are the sigma states, which are formed because of this forum layer the other as people hydrangeas level levels and then you have also high states shown here in green, and then you have these diracons one below and one above the family level. And the important characteristic of NGP2 is that it has this cold sigma, and you see that the levels are just above the family level. Yeah, if this state will be completely filled, then you are not going to destroy superconductor. So if you look at all the set of equation that I'm going to show, you will get two gaps. Yeah, so this is an old original course from 2013. And if you look at the distribution of the semi-serving, you will see that one gap, the lower gap comes from the high states, and this upper gap comes from the sigma. So if you go into the details of the study, but a number of recent papers with you to look at the hydrangeas, so this is an example of those papers from Ligia's Boerig group where she proposed that, so London Hydrogen was predicted to be a superconductor in 2018, and then it was indeed found experimentally, but I think the pressure there is way above 100 gigapascalets, 100 CPUs. So one idea to lower the pressure is to introduce another element that will basically give some internal pressure that instead is going to help reduce the external pressure that you require. So recently they proposed that basically using London Hydrogen, then they may reduce the pressure to 50 gigapascalets and still have superconductivity above 100 GPUs. So before I will let Samuel take the spot, I just want to mention a few things that if you want to estimate, you may not always need to do an isotropic superconductive calculation. At most times it may be not just to do isotropic, but you need to do at least a rough, let's say investigation of the anisotropic lambda to get an idea of how you may have anisotropic properties or not. But in order to do that, you need to use fine sampling of the electromagnetic cell. And I just want to mention that we are going to have, so I don't expect, if you haven't used TPW before, one hour to discuss transport and all, it's very short, but we are also going to have a TPW school in Austin. I mentioned here because all materials in general that you just found the school is going to be recorded and initially it will be placed on the website of the school, but later on it will be moved to the TPW website. In fact, if you go to the TPW website, maybe somewhere is going to be shown, there are already, there is documentation and lectures and presentations from the previous schools that we had. And finally, people that are interested in co-development, and I have a post-op position that I'm looking for. And this is basically on development TPW, and that's a collaboration with Ferigiano, and with professor with Berkeley, and then Texas Supercomputing Center. In this second, do you guys have any questions now, Antimo, or? You can, we can move to questions. Okay. Maybe we'll be kind of bigger first. You can find a green function which is this matrix of one particle green function. But if I think about who from there is about state of two electron, I would think that to describe it in this dynamic, I would need to use a two particle green function. And this way, you will be able to show that you can get around that by using the off diagonal green function. Is there any, what, is there a fault in how I understand this thing, or is there some approximation that you made along the way, and you haven't shown it yet? In this derivation, there is no other approximation. But there is another way to arrive to this equation to basically just dynamics. I think, if you look, Massibulus derivations, he has a couple of papers where he follows the other formula. So this is in the number of formulas. So I think mathematically it's easier to follow. Yeah, I tried to follow the mathematics. Yeah, it's just a physical physical picture. So if I have a full repair, which is the electron, I would expect that it's been explained by the green function that described the two particles. So instead of having full creation and creation operator, I would have to have four. Yeah, it's a new object. So you work with the pair here. Yeah, I would have one green function that described everything from it. So the property I would use, what is the, where do we have all of them? Yeah, this is why this is why you don't have two. Is there an approximation that you make, that you go from here to there? No. Yeah, yeah. Stop the recording and do it. So they know it. Stop. Yes.