 Okay so can you hear me? Okay so I think we are gonna start with the second part so this as I said I will finish the slides from the first presentation to show a few examples from calculations and how this compares with with experiments. So since you've seen this example in almost all the tutorials that we done in EPW and in fact the first tutorial on first exercise in the tutorial on superconductivity it's also going to be on lead. I will just give a quick overview of superconductivity in lead. So this it's a so it can be lead can be treated with the isotropic mid-dialyliashber formalism. Of course you can treat it with an isotropic as well but I'm just gonna in this study I just did the isotropic mid-dialyliashber formings and this is formally if you were to solve the two if we go back here if you were to solve this two okay sorry this two equation in the superconducting state a different temperatures once you have lambda so you will solve this at given t you can you will obtain two solutions a solution for z and the solution for delta and in this plot I'm just showing sorry here I'm just showing the solution for d as a function of the imaginary macho-barra frequencies this in fact are discrete points but I am just showing them as a line but effectively there are if you are if I were to show the data that this should be discrete points along this curve and only the whole if the only information that you get in the on the imaginary frequency or useful information at this point is basically this superconducting gap edge and this is basically your gap that if you were to take to make this plot a different temperature so this is a plot at a specific temperature if you were to redo and solve the equation at different temperature you are gonna get different difference of this course and from all those curves you basically take this value of the leading superconducting gauge at i omega equal 0 and later in the next talk I will explain why this this works if you plot this points at different temperatures that you solve the Mikdal Eliashberg equations you will get this curve which is basically your superconducting gap yeah so this is in the limit of course you were in practice we cannot do t equal to 0 yeah so we go as close to 0 as possible but you can see that you always get this kind of flat behavior so we can extrapolate to to the zero temperature and then the point where the critical temperature becomes zero the gap goes to vanishes yeah so tc the critical temperature as I said is defined as the temperature at which this delta naught is going to be equal to to 0 so I you can also see a comparison between the Bington Eliashberg formulas and SCF DFT so in this case I took the data from so this is what will happen in let's say in EPW if you were to use a different parameter for the Coulomb interaction you can see that this blue gap a blue curve is the gap with mu star 0.1 if you change the mu star to 0.09 you get a different curve and so on and and this are the experimental value so of course what a lot of people do in practice but that's not necessarily a good idea they change this new star till they get a value that matches the the critical temperature if if you were to calculate a bin issue then you will have a value for mu star that you can it's from first principle and that most of the time and I would say from my experience it will never match your tc experimental tc so if somebody tells you that they've done first principle calculation and they predicted an exact value of tc that match exactly the experiment that's just fortuitous so it you will never get the the exact value at this point so and this plot as I said it shows data for SCF DFT so if you are interested this is from this studied by by Flores but you can see that there is very good agreement so this indeed mu star it's of this order in in in glad yeah so always people say it's usually you take a value between point one point two so in this case indeed a value of around point one works very well for for lead and in case I think in the study on SCF DFT they even did an isotropic calculation so they fully in fact it's not completely full but I will say yes the two black curves it's an fully k resolved so if in other words it shows that there is some an isotropy in the superconducting gap so in principle if you would have solved this instead of the I was they've solved instead of the isotropic equation that an isotropic once I would have seen some smearing of this thing so it all have not been just a simple car one dot but just some gap and you will see in a next example what that would look like so another example that I want to to show briefly is because this is the large the funnel mediated superconductor with the largest critical temperature I will briefly discuss magnesium diboride and this is a classical example of a two-gap two-gap superconductivity yes so magnesium diboride has this layer structure and there are two sets of bands at the at the Fermi level that come from the so from the Sigma so in plain modes sorry in plain bones and then you have PZ from the Pi orbiters yes so this two set of bands appear at the Fermi level as a result the Fermi surface have multiple pockets yeah so the cylindrical Fermi surfaces I think come from from the from the Sigma bands and these three-dimensional surfaces come from the Pi bands so if you if you look at the anisotropic lambda average of over the Q you can see that the structure of lambda has this two basically two peaks one corresponding to a set of Fermi surfaces and a set the second to the second set of Fermi surfaces so this already suggests that there is probably some anisotropy in the gap so indeed if you solve the anisotropic gap equation you this is the superconducting gap on the Fermi surface this is from somewhere else study then you will automatically get these two gaps one on the Sigma and one on the Pi yeah so and that's what I said of course this is this meaning yeah you see some you get some broadening it's not just just a point if for example you you were to solve this using the isotropic gap equation you will just have gotten one band somewhere here in between yeah it's almost like an averaging of these two gaps so that's why I'm saying that for example Macmillan expression will have never can never give a reasonable description of a multi-gap superconductor especially if there are the gaps are very far apart yeah so you can see here that I said we are also overestimating the TC it's basically here it's 50 Kelvin in experiments is 40 so if you were to increase in principle this column B interaction you can lower the critical temperature but again this is just an empirical parameter in our case and that's also a comparison of superconducting DFT so you remember that I said that superconducting DFT both the electron phonon and the electron electron interaction are on a treated on the equal footing so in their first study when they they looked at the super conductivity in image V2 it seemed that they got a very good agreement with the experiment but then later on the same group published or I think it's in the same paper they went into do more details and looked at the behavior of the superconducting gap and prediction of TC by varying different parameters so if we just take a quick look at this table basically what they did they here it's they calculated the Coulomb electron electron interaction in different approximations and then for the electron phonon interaction basically the Liashberg spectral function again they just assume so they removed the anisotropy in K so basically they average over the K but they still kept a two-band model yeah this in order to since they knew that they need to have two superconducting gaps so this just shows how the TC varies by varying the Coulomb interaction yeah so they went from a value of 30 to a value of 50 so this will be an average RPA which is kind of close to the value that we predict using just an average Coulomb interaction and this is also the behavior of the delta function now what's interesting when they did a fully anisotropic calculation when basically they included anisotropy in electron phonon coupling the prediction is way below the critical temperature is 22 Kelvin so it's not really clear there may be there are effects that are missing unharmonic effects adiabatic effects but that's the best that that we can do at this point yeah so that was just kind of to understand what are the limitations of what we predict and that's why when I said that we cannot claim and we shouldn't claim that we are going to predict exactly the the critical temperature and as a final two examples this is superconductivity in in graph intercalated bilayer graphene so graphene is not a superconductor but people have been looking for many years to make graphene a superconducting material and just a few years back it's been showing that this are this is a resistivity plot sorry in intercalated bilayer graphene and it shows an onset of superconductivity around 4 Kelvin and about the same time there was another study where they did magneto resistance measurements in calcium graphi laminates so this is not really graphite but it's also not bilayer graphene but the interlayer distance between the the carbon layer is close to the bilayer system and in this case they also seen an onset of superconductivity at around 6.4 Kelvin so we in fact why when these papers have were published we are in fact almost ready we are looking at the same system using our model and in this case in fact we calculated the abinish of coulomb interactor abinish sorry the coulomb interactor abinish or using GW approach and what we predict was that we may have there were multiple gaps and we predict that you can see here that there are different fermic pockets so on the there are two electron pockets centered around gamma and this will have these two separated gaps and then there is another fermic pocket that is on the sorry whole pockets around the K point so this is unlike for example calcium C6 where only one superconducting gap is found and indeed when we applied our I'm not showing here the result but when we applied our method on calcium C6 we are seeing one gap so our base experiment may be able to to confirm if indeed there are two fermi surfaces on the electron electron pockets and as a so here I said that we have calculated mu star so we have used we have done it using this random fame approximation so this is a plot that shows the screen coulomb interaction average over the fermi surface and this has the same structure basically at lambda we it's just not separated but if you were to put these three fermi surfaces together on a single plot you will see basically the same the same structure for lambda and mu and so in this case we found that's a new value of 0.25 and then we using Anderson Morrell model for an electron for a plasma energy of 2.5 eV and then we use the largest phonon frequency in our system we estimated the mu star and they found the value of 0.155 so this just shows the behavior then we took different values for the phonon frequency and we saw just to understand how new value of mu star changes I think on the right this black curve this is how TC the prediction of TC will change as a function of this phonon frequency in the expression for the mu star and this will be the new star as a value of again omega pH just to get the feeling of how sensitive our results were on the with the new star parameter and and finally the last example that I'm going to show this is also graphene but this in this time is a lithium decorated monolayer graphene so in this case there are low temperature are felt experiments that show that a gap opens at around 5.9 Kelvin and the value of the superconducting gap it's about 0 points so about 1 million electron volts and the measurements were done for this point on the Fermi surface when experiments were repeated for another point on the Fermi surface there was no superconducting gap so this suggested that that there is some anisotropy in the system yeah so again we did our calculation with the Mikdal Eliashberg and isotropic Mikdal Eliashberg formulas in EPW and that's the superconducting gap on the Fermi surface and then we look at the solution we indeed find that there is one gap but there is anisotropy so this it's in agreement with the experimental findings so I think with this I'm this should have been the end of my first talk and just kind of to like take home messages that we should have that we can obtain measurable superconducting properties with anisotropic resolution using Mikdal Eliashberg theory however you should be careful when you claim what you are doing yeah that you still cannot say that your in most cases your TC would be not spot on with the experiment and you kind of need to understand that the limitations of the method now when it comes to converging or practical issues you will need to do calculate this electron phonometric elements on very dense matches yeah and we'll see more in the in the next talk that I will give and second the Mikdal Eliashberg theory and this superconducting DFT describe the same physics yeah so and in principle if one introduces the first principal approach to calculate new star they they should be in very good agreement questions about these parts before I move to the second talk okay so now it's with more detail what exactly or how exactly these equations are implemented inside EPW and I will go over the structure of the code and then also mention a few more technicalities when it when it comes to what parameters we really need to to converge so as I will describe the implementation of the anisotropic case this is more difficult but the same principle is behind the isotropic case so so the key quantities that we will need to calculate is this lambda and this we already shown is done once we know the electron phonon matrix elements so in order to get the electron phonon matrix elements on the Fermi surface the flag that you need to set up in the input variable this is ETH right through and the reason why we write this on file is because we want to solve these equations at different temperature yeah and the electron phonon matrix elements are temperature independent so usually you will calculate them once and then you can reuse them when you rerun your the code for different temperatures since this is quite expensive and it takes a lot of time it's better to do it once and then you can you can repeat your calculations then you can so when you will do this you will create a lot of files and these files are called EPH math and X and this X stands for the number of files and this is CPU dependent yeah so depending on how many CPUs you are gonna run your original calculation this is how many files you are gonna have with this electron phonon matrix elements and the files originally they are not the same size however when you reread them the code redistributes them equal on the number of CPU that you are gonna use when you solve the the equations and more specifically in the code this writing is done in the so the central file that Samuel has already gone through is this EPH F90 and inside we have this main subroutine EPH L phonon shuffle wrap and more specifically the writing is done in this EPH van shuffle yeah and I put here some comments some not all the comments are for example if you open the code so and not all the lines I may have excluded some of the lines so if you are gonna open the code there may not be one-to-one correspondence between the slides and and the code because I may have some I have some extra comments here but basically this is what the code is doing at that point it's it finds the irreducible k points on the fine grid within a Fermi window that you specify and it's gonna write a series of files so besides this EPH math you are gonna write three more files and I'm gonna describe them in more detail later and this are the file with the frequencies on the fine Q mesh this is a file with the eigenvalues on the again the file of fine K mesh and this is a mapping between the K and Q flight Q plus Q points so once so more specifically these are the input variables that you are gonna have in the code so as I said EPH write just it's gonna instruct the code to write these files and this is you are providing information on which meshes you are really gonna write the file so this is the fine mesh for electrons this is fine mesh for phonons and particularly for the superconducting implementation of the super big Dalai Liyard equation you can specify this flag MP mesh K2 so in other words instead of using the full K mesh you are gonna work with the irreducible k points and that's tremendously decreases the computational resources now in order to be able to do this and in fact because of the structure of the superconducting equations since we need to solve them solve themselves consistently till we get convergence we can see here if we go if I go back that we have K and K plus Q yeah and this in order to have this mapping you will need to specify the K and Q meshes need to be commensurate and that's what allows us to to use the MP Mesh K yeah you cannot solve the anisotropic equation using random meshes in a EPW yeah and because of this self-consistent problem that we will need to solve to resolve and this is again a description of the files created by EPW already mentioned them in the in the previous slide so once you have the electron phonon matrix elements you can on the Fermi surface you can have done it in a separate run or you can have done it in the same run where you want to solve the Eliashberg equation this is the flag that will tell the code basically in fact it doesn't tell yet the code to solve the Eliashberg equation it just tells the code to calculate this anisotropic lambda yeah so once you have this G the electron phonon matrix elements you can calculate lambda and now you can finally solve the the anisotropic equations and for that you will need to specify a few more flags yeah so this Eliashberg flag holds this subroutine and inside this subroutine okay this is only okay yeah sorry I remember in the subroutine if if you don't specify that you want to solve the anisotropic equations the only thing that the code is going to do is evaluate your spectral function and then it's going to estimate the TC based on McMillan formula so in principle you can do this also through the phonon self-energy so this is equivalent yeah but the difference in comparison to the phonon self-energy calculation is that right now you also calculate this anisotropic lambda so in order just to understand if you have let's say an isotropy in your system maybe you don't want maybe to go directly and solve the anisotropic Eliashberg equations since that's very expensive maybe you just want to look at lambda on the Fermi surface and that already is gonna give you some hint if you should expect an isotropy or not so that's the reason why this flag was introduced to be able to work around without really solving the equations yeah now just by running that flag Eliashberg true you are going to generate this set of files yeah so you are going to get an isotropic Eliashberg spectral function you are going to get also a second file which is called and this will have different smearings and the default I think are 10 smearings values yeah you can change that but that's the default then you can or you are also going to get another file which is 8 to f ISO this is basically the also the Eliashberg spectral function as a function of frequency and the second column is just the second column from basically this smearings from this file while the rest of the columns you are going to have three and columns where n is the number of atoms is just the mode resolve Eliashberg spectral function now you need to keep in mind this mode resolve doesn't have anything to any information about the specific modes to which atomic species they represent for that you will you should do you'll need a projection yeah so it's just or in the order that the modes modes appear so it's not necessarily it particularly useful but sometimes you you you may want to know just in some region or another file that you are gonna get this is this file and this is basically this lambda nk on the Fermi surface so that's the file that you should plot if you want to look and understand if there is any an isotropic yeah then that's the same thing but in a different format if you want to to plot it like I think this may work with the is this the work with Vesta in the format that works with Vesta probably not but that's basically the same the same information yes it yes I think this is the file that you can plot directly with Vesta since it has the k-point in Cartesian coordinate and then the lambda and lambda nk and the same as for the Eliashberg spectral function you can get the phonon density of states and the phonon projected phonon density of states yeah the same with the know that the mode resolve do not have does not have information about the specific atomic species questions okay then if you have so this is the default now if you have i verbose it is set to two in the code you are gonna get another set of three files no this cannot be plotted directly with QB this is sorry this is the file but it contains the same information so this file if you print this can be used to be visualized directly with Vesta yeah so we can read this in Vesta directly and this is the full an isotropic Eliashberg spectral function yeah if you have both indexes nk and mk plus Q and this I think this I'm repeating this is just lambda mk okay this is also basically the same information as this but just in a different in a different format and in the tutorial it explains more which files are plotted so okay so coming back right now if you go with all these files that's basically if you plot what you will get if depending which one you are plotting yes so this is the a to f and then if you just do an integral you will get the lambda this is this lambda k pairs if you were to plot so this is the example for mgb2 you will see again that you have two structures and this is if you were to plot the full anisotropy what you will get so these are the basically the values for for lambda so so far we haven't solved yet the the equation so if you want to solve the equations you need to put two extra flags in the input files and if you want to solve the equation in the anisotropy case is this l and iso true and since we are solving initially on the imaginary matzubarra frequently imaginary axis on the matzubarra frequencies you need to specify this elimag true and this in the code if you want to look at the structure this again is through this subroutine called eliasberg and it's very similar this loop this if is very similar to the sec to the previous part the only difference is that right now the code we also call this and eliasberg and iso imaginary axis and that's the subroutine where basically the anisotropic mcdal eliasberg equations are solved yeah so this is in this this file and the steps are all following so first you see that there is a loop over temperature so this will be the temperature that you are going to see is set up in your input file and I think on the next slides I will show that first you generate this call generates a grid of the on the frequency points that you are going to calculate your eliasberg equation for a specific temperature so you can see that this is temperature dependent and then that's the code the subroutine where the that will solve the mcdal eliasberg equations yeah and this is just some internal thing where you just call here you are going to call the lambda yeah this anisotropic lambda so so this is done self consistently you can see that there is a do loop either until convergence is reached or the number of iteration have been reached so if you and then the code is going to stop and it will calculate the free energy I will give a bit more details about this in the next step in principle you can also have a restart option so let's say you have already calculated t equal 10 yeah and you have stopped your calculation and now you want to restart at the temperature of 15 in principle you can reread your gap function and delta and z on 10 equal 10 and use that as a feed in your as a first guess for when you solve the equations and that's flag can be with this stop restart option EMAG read equal to so first what the code is gonna do the temperature that's a 10 10 Kelvin it's just gonna read these files and then it's gonna proceed to the next temperature and solve the the equation and this are so this flag as I said this are this will just tell the code basically to calculate lambda this will tell to solve the then isotropic equations on the imaginary axis and these are the temperatures at which you want to calculate yeah so one way to specify the temperature then you will give a mean and max temperature and the step a number of steps and then that will be equally spaced step from mean to max and this is the threshold for the convergence in the self-consistency for the solution on the imaginary axis or they said you can he also provide the number of iterations so depending which one is reached first the code is going to stop and this is the cut of frequency for the Matsubara frequency yeah so you they said you cannot have that some I mean you want to have as many terms as possible but you need to truncate it at some point and new C is just the Coulomb parameter that's just to show that you can provide the temperature so this as I said it will give you know uniformly or the temperature are going to be uniformly spread if you don't want uniform spread temperature and as you remember in my my plot I had something like this so yeah here you can have like larger distance between your temperature point but here you want a closer sampling yeah so that's another way to to input the temperature and then you can specifically give your temperature value now again when you are gonna run this in isotropic calculation the code is gonna generate many many files and some of them are quite big so you need to pay attention especially yeah to the disk space when you are gonna generate these files so xx this time indicates the temperature so for each temperature that you are gonna evaluate the the equations you are gonna have this this type of file so the first file and this is one of the most useful file is this imag underscore an iso underscore xx and this file has five columns the first is the the frequency the second is the energy the eigenvalues with respect to the Fermi energy then this is the Z this is delta and this is the Z in the normal state so if you will see why this is needed later on and then if you want to make those plots that I show with the delta oh no sorry this plot since you want let's say to plot something like this as I said that's every temperature you just take the limit when i omega n or j goes to zero this is basically what this file contains yeah so is the limit of delta and k for the zero Matsubara frequency and that again this you will have one file per temperature and finally this is just the same information like this but only on the providing also the Cartesian coordinates if you want to plot on the Fermi search this file and this file are equivalent the only difference is that this can be plotted directly with western and this will only be generated if you specify i verbosity 2 in your input file yeah this will be by default this won't be by default and here you also have an extra term which is for the bands yeah depending on how many bands like for example for magnesium diboride there are a few bands that are on the Fermi surface this will be the n index yeah this why why so that that's kind of just look a bit what what we are getting just to get an idea this will be if you plot this EMAG and ISO file and this is when I plotted the Z column as a function of this should have been i omega the imaginary frequency and this is if when I plotted the delta column as a function of i omega so and this is the magnesium diboride case so this is why you see this an isotropy and two two gaps and that's what you will get for the convergence yeah so if you just take now this value in this limit that are written in this input sorry in this input file you can extract this plot yeah so this is for specific temperature and if you were to plot the cube file you will get this plot and somewhere has a YouTube tutorial explaining how to get this this nice nice plot so if you have questions about this Samuel is the best person to to ask finally what do you need to converge you can they said you only need to converge a few quite a number of parameters so first of all are the k and q meshes this vs cut and then there is this additional the Fermi window yeah that you can specify in the input file and just to get an idea how important all these parameters are this is just an example of the superconducting gap dependence this is just the highest gap in mgb2 by varying the k and q meshes yeah so we can see that it's not just let's say that you need to converge the k mesh but you also need to converge the q mesh and that's the same kind of the same plot for the same gap as a function of this cutoff frequency yeah so you can also see 5 10 15 and so on so principle you should go to to a cutoff as large as possible till this doesn't change in practice it's the calculation are getting too demanding so I I have mentioned this many times but anyway this what I say that indeed describing an isotropic quantities it will require dense meshes and quite a lot of computational resources and on the other hand if you just look at isotropic cases you still need to converge but you will see that you can go around with maybe not such dense meshes like for example this is again from the same work well for this the same meshes that here that shows that delta vary a lot this case it shows that lambda was pretty much converged yeah but this is just an isotropic quantity but yeah now you can also extract so for now we just I just discussed Tc and delta yeah but there are the once you solve and you have a solution for for delta in the Z you can extract some other properties and compare them with experiments so one of them you can get the specific heat superconducting for specific heat and for that you will need to estimate the free energy and that's the expression for the free energy and then the superconducting specific heat is just a second derivative with respect to temperature and this is a plot that somewhere has has calculated and I think the blue curve is the superconducting gap evaluated using the solution from the Eliasberg calculation and this is comparison with to give BCS model and this one gives BCS model yeah so if you want to learn more detail about this plot you can find it in in in this reference okay I think this is the I don't know why I put this it's just a repeat of this is just to show where in the code yes where in the code this free energy is calculated and it seems as it's automatically done once the converters is reached there was a question so I guess I should stop for just a second here to see if there are questions since I will move on from the and I thought imaginary axis to the real axis and that was a question that somebody asked in the first part of the talk so so far what we've just seen indeed is that the MiG-Dalevian equation on the imaginary axis can be solved and only from the numeric computational point of view we are doing this because numerically it's very easy or efficient to do just sums over finite number of Matsubara frequency but in the end they are gonna only provide information about critical temperature superconducting gap and the free energy that will give you the specific heat yeah if you want to extract additional information about the spectral function like for example the quasi-particle density of states or the single particle excitation spectrum then you need to solve the equations on the real axis and formally you can do direct evaluation on the real energy axis but this will be very demanding computation I don't think this has been done in the anisotropic case directly and the reason being because if you work out the math you'll see that you'll have if you will need to evaluate many principle value integrals and in fact as I said earlier this is implemented in Iliashberg in the code but just for the isotropic case but it's not it has not been tested in a while so I it should work but anyway it needs maybe more testing as an alternative and that's what it's I usually use is to solve the solutions as we done so far on the on the real axis by doing an analytic continuation on the imaginary frequency axis and this can be analytic continuation can be done in two ways you can use by the approximate and this is very cheap and that's the recommended method if you use when you solve the anisotropic equation or you can in principle you do it by an iterative precision but the procedure but again this becomes very heavily computationally so as just to give you as an example this is seconds this could be hours to days depending on how many k points you you have so to do this analytic continuation that's the part it's also done in this subroutine and this is the flag El Pade will tell the code to do a Pade analytic continuation analytic continuation is in Pade or if you do an analytic continuation using this iterative method and the reason why this is cheap because this is basically just one shot you just do it at once this analytic continuation the same in the same way as you solve it on the imaginary axis it involves a self-consistent solution yeah and the equations I am not gonna show them here but if you go in in the mgb2 paper that I am showing I showed this reference in my first talk you are gonna find the equations in that in that article and as you can see here once you solve this equation in the on the real axis the core the code with once it reaches convergence it will also calculate the quasi particle density of states yeah so you will have an extra information and the only the additional flags that you are gonna need to introduce this is El Pade this will tell you tell the code to solve the Pade or to the Pade method El continuation will tell the code to solve this analytic continuation method and only this has a threshold this instance one short doesn't have a convergence parameter so you don't need to specify anything else and again similarly I'm not gonna describe this but it will be a set of files for each for each run and again they will have the information about the z and delta but now on the on the real axis so how would this look in practice let's say again isotropic case of lead this was the solution that we saw at some specific temperature for delta as a function of imaginary frequency if formally what you do you just go from a complex number to to from the complex plane to the real plane but you still need to add the small function and what you are gonna get while here you had just one thing that it was just the you will get the real and imaginary part yeah so this is using this Pade approximate method so this is the real delta function and this is the imaginary delta function so and that's a comparison so in the isotropic case between the Pade approximants and this analytic continuation the iterative method so we can see that it's very good agreement there is a bit more structure in the analytic continuation but overall taking into account the huge difference in the computational cost Pade approximants method that's it's very quite robust and another key thing that you should notice while there was no structure in the solution on the imaginary axis now in the real axis solution we see that we have structure and this happens at the scale of the phonon energy yeah so in lead the the maximum phonon it I think it's about 10 milli electron volts so this is while the place where you are gonna see see the structure yeah so this method carries additional information and that's just a plot in the case of mgb2 where we had two gaps so this is delta again on the imaginary axis this is how it will look on the real axis this is again is Pade approximants I never used analytic continuation in this case since I yeah it's always been just too expensive but again you will see structure at the representative phonon frequency in the mgb2 and in mgb2 the maximum I think the largest phonon frequency goes to 200 mv so you will see it's a difference scale compared to lead now in the real axis you can also look at the I'm just gonna quickly discuss this you can also look at the single particle greens function on the real axis and that's I in my previous talk I had this expression for g as a function of i omega so in principle if you just change i omega to omega you can rewrite this this equation and again tau not tau 1 tau 3 these are just the Pauli matrices and the poles of this die of the diagonal diagonal components of of g give the elemental excitations of the superconductor so the you can take the 1 1 component you remember this is a 2 by 2 matrix I showed in the previous talk so the diagonal elements will be the excitation while the non-diagonal are related to the cooper pairs so if you look at the poles of this function you you can find this this expression and then you can see that this is enk appears here and also appears here and here so this is delta is a function of enk yeah and this can be directly related to what we've seen before for the equivalent to the normal state this real part of enk is the quasi particle energy renormalization by but for the superconducting pairing and the imaginary part is the quasi particle inverse lifetime so this is equivalent to what Feliciano has been showing in for the normal state but only that is for the superconducting pairing and the Fermi level basically when enk is equal to EF the quasi particle shift you can see that is just real part of since this is zero this is going to be just the real part of delta nk of enk so in other words this will identify to the your leading edge of the superconducting gap so it was that value when I show this will happen only at very small value so in that plot of delta as a function of omega this is basically your limit at omega equals zero yeah or very close to omega equals zero yeah and this is the binding effect 2d this is incorrect 2d of 2 delta e is the binding energy for electrons in a cooper pair so that's as much energy this is the energy that you will need to provide in order to break a cooper pair in principle you are not going to break just one pair you will need to break all the pairs in the system so but once you have this diagonal element you can also so besides that you get this information about the pulse and quasi particle energy and normalization and the shift you can also get the superconducting quasi particle density of states and this is just the imaginary part of this element that you will then need to integrate it over enk and multiply by minus 1 over pi so in the BCS limit you can take Znk equal to 1 and if you perform an integral you are going to get this expression so this again you can see in fact this is integrate average already in the code but in principle you can also look at an isotropic and K and you will get a bit of an isotropic information but what you usually and this is what the code calculates directly it doesn't print this out but it's already it's calculating the code is just not printed on file you get an average over the Fermi surface so you get this this thing and that's the plot for in case of mgb2 again this is the superconducting quasi particle density of states calculated a different temperature and you can see that you can this is the dose yeah so okay if we look at 15 Kelvin you can see the two gaps and if you go and look compare with the delta over T temperature you'll see that these are the values at the average values of the pi gap and sigma gap and you see that as you increase the temperature you close this this this gap yeah you will get tower and towers the normal the metallic state another thing that you can use it's basically you can get the spectral function and this has been done this is not our study but has been done in this paper but by sauna but if you were to calculate this spectral function this is in the case of the normal state you cannot really see very well in this spot but there is a line here yeah that is continuous and this is the Fermi Fermi energy and this is the same plot so this was calculated for the normal state this is calculated for the superconducting state and here you should see there is a gap opening so it looks like this and like this yeah so this is your superconducting gap but just that the resolution in this picture is not so so great so I think with this I am almost done so just a few more things just ideas about different parameters that you need to pay attention to kind of more like a summary is this the cutoff frequency as I said this will be the cutoff frequency in the matzubara for the matzubara frequencies and you generally you set it somewhere between these values there is another option if you for example since the matzubara frequencies if you remember there was an expression is I don't I think it was function of j I don't remember 2 pi something but it was a function of J and temperature basically as you increase the temperature and if you have a fixed cutoff you are gonna have fewer and fewer discrete points so there is an alternative option if you want for every single temperature to have the same number of matzubara points you can set this this parameter in the input file NSWI and then VS cut is gonna be ignored so this will mean that basically you are gonna have a different cutoff a different temperature but you will have the same number of discrete points if you use LN ISO or L ISO you will need to always have specified this parameter usually the code I mean if this is not done the code is just gonna stop and it gives you a message with a with a warning. LPAD requires LIMAG L a analytic continuation requires both flags and the reason why it requires LPAD because it uses the solution from LPAD as the first guess when it solves this self-consistent equations. Just as this is more like an advice sometimes you may not know at which temperature you want to calculate your Eliashberg equation so you can first just evaluate you see using Elandine formula and this can be just as a guide for defining the temperature that you are interested in. This flag will be very important this will require to write the the matrix elements on the fine meshes and if you use Eliashberg and these files are do not exist again the program is gonna stop with a with a message that you need these files. Another thing if you change the fine and QK and Q meshes obviously you will need to regenerate these files so you will need to rerun another calculation where you where you generate the files. If you want to have this kind of email which is read which is a restart option in fact this does I just mentioned just one thing when you can use this email grid but in principle it can do a few things. So what the code will do read the input files on the imaginary axis at the initial temperature that you provide in the code and it can be used to do the following. You can you can solve the anisotropic equation at some temperature larger than the T where you are reading and you just are gonna use this file as your starting guess but for example let's say that you have solved the Eliashberg equation just on the imaginary axis and now you want to do this padé approximants you can just give the temperature where you want to calculate and then it will solve the padé solution sorry it will find the solution on the real axis or maybe you haven't written this Fermi surfaces and you want to redo something with IVAR positive 2 for a specific temperature. So you will find all the references in fact the other talk also have references if you want more information about all this study like this paper it's a very nice paper if you want to learn more about anisotropic Mikdal Eliashberg equation this is also a very nice study that describes the formalism and in this study we are describing the implementation in the EPW code. If you are interested about this padé approximants this is the reference you want to look at and this is a reference for the analytic continuation and with this I think you can have questions.