 Okay. Yeah, thank you. I'm Nevin Gornic and today I'll be talking to you about the veneration of charge and current vertices and the GWBC method, and actually more about the method and later on about the veneration and this is a project as part of my PhD under the supervision of Professor Jeronkoly and Despoia. So the project was actually just some nice slides are not okay. So the project, just to give the outline of the talk, I'll be introducing you to basically the like the physical motivation to do to work with quasi to dimensional crystals and their optical properties then I'll talk about the motivation to develop our own code to calculate these properties some theory and examples as well as the final part which is related to the veneration. So basically in to the materials, the electric field is weekly screen and is highly non local which gives rise to many different and interesting properties, which are basically collective excitation modes within this type of materials and these are usually stacked layers of graphene analogous materials such as TMDs or hexagon bar nitride and similar. And what you have is that these modes, apart from existing within the material themselves can hybridize between layers, but they can also hybridize with an external electromagnetic field and create new hybrid modes which are polyerotonic modes, and depending on the material you will have different types of these collective modes, which will of course give very different types of dispersions. So once they hybridize with light, the basically the light line bends and you get these very interesting formations so the dispersion is basically like your fingerprint to observe these collective excitations. So basically going to be interested in excitons, which are strictly speaking like tightly bound electron hole pairs. And basically you have to do two things to describe excitons which hybridize with light, which are going to form external polyerotons. So you have to calculate an optical spectrum absorption spectrum at a very high level, and then we have to couple it with your photonic field to observe the paratonic exciton paratonic hybrid modes. And usually the first part people do it by showing a two particle Hamiltonian at a high level, but the second part is usually done in a microscopic model, and not really accounting for the quantum photonic field. So this is the thing that we are trying to address and do a proper gator approach to calculating these properties. So in the first part to calculate the optical spectrum, usually, like for example in packages like Yumbo people calculate the static screening they construct a basic kernel and then they diagonalize the two particles and then the Hamiltonian and eventually obtain the final dielectric function. But there are certain limitations to this approach. And one of the thing is that basically you have to solve the whole Hamiltonian and re diagonalize it each time that you want to change some little thing with it, you can you have the flexibility to separate different contributions like RPA or later contributions and recalculated just parts of it. Then again, the Hamiltonian method is also basically an eigenvalue problem you get a discrete eigen spectrum. It's a zero temperature model which requires some post processing to link to come to the finest temperature. And what is the biggest deal breaker is that you cannot directly describe the paratonic most because the photon photon retardation effects are neglected. So in the propagator approach. On the other hand, the spectrum is going to be continuous by design and fine temperature, and also we will include by using the photon propagators we are going to have retardation effects included. And also with some approximations in the Hamiltonian model you really have to add each atom in your system so if you want stacked layers you really have to include them in your crystal, while in the propagator approach with a bit of approximation you can easily do a very high level monolayer and then combine in a final equation these individual response functions and obtain like for a very large heterostructure your your basic response properties. Also if you want to include a substrate in the Hamiltonian model, you have the same same issue that you have to really include the atoms of the substrate while here we can just modify the photon propagator to include this type of scattering. So the methodology is basically can be divided in two steps you do the very classical thing or to obtain the grant state wave functions for example from quantum express and then you take these way functions and compute the bear and screen chrome interaction by GW but cause a particle corrections construct the BSE kernel and solve a quantum electrodynamic data so better equation. So how do we go about it. This Hamiltonian basically has free free parts which describes first the concham electron subsystem, then you have a part which describes the free photon subsystem, and you have the last term which contains your electron photon interaction, which is going to be important both to describe the electron photon interaction within the system but also with an external electromagnetic field. The last term contains the electron current operator which will be rated to the freemium field operator and contains your concham states, and you also will have a photon field operator. However, we will not actually be working with this Hamiltonian instead we recast the problem in terms of time ordered propagator for the electromagnetic field, and we perform like a perturbative expansion of this propagator. You can call connect diagrams, and you can extract basically a distance equation for the electromagnetic field in reciprocal space which will contain two major component. D zero which is your time order propagator for the free photon field and then a self energy photon self energy, which is going to describe the polarization within your positive dimensional crystal. You can also assume separable subsystems and that the operators are in the interaction picture. Okay, so if you draw out final diagrams for this contribution, so you have basically two steps that you have to do first is calculating the photon self energy, which will contain your different contributions for the RPA contribution then your self energy contribution basically just GW, and then vertex corrections which are essential to describe the binding of the electron whole pair. So this vertical. So this equation and then basically when you obtain the photon self energy plug it in in the Dyson's equation for the electrical magnetic field, which will finally give you also the hybrid like the hybridization with the external photons. Okay, so it's basically like calculating the optical absorption then obtaining the dispersion relations because these are the like things that we want to obtain in the end. So the photon self energy like we actually compute the two terms separate if the RPA is the really like a well known thing where you have the RPA self energy with the patient factors damping constant and energy differences. And in the end you multiply it with current vertices which will be related to the electron current within your crystal, and these current vertices will basically contain conchamble states and their derivatives. So when you go to the ladder approximation so this is this are the vertex corrections you have a bit of a different thing and the problem is that this diagram cannot be evaluated. It cannot be factorized so there are certain issues, but if you could evaluate it, you would again have current vertices, but this time in four places. You would have to evaluate the later later four point polarizability which contains a photonic kernel, and you have electron hole propagators which are going to be used to actually describe the the propagation of the electron and hold within the system and this in terms again contains greens functions, which in which you can plug in also your GW correction. But okay there's this issue that we cannot solve this so instead we have to do a certain approximation and we approximate the interaction between electron hole in the non-retardation limit and here we can replace the photon propagator here with a screen conval interaction. So we are basically assuming that the photon travels from one place to the another, instantaneously but while we still have the retardation effects within the crystals. When you do that you get, you can then evaluate the screen conval interaction which is going to contain just your background interaction and time order RPA polarizability. And when you do this whole thing you're able to also replace the current vertices with charge vertices which are much simpler to evaluate. So it's not that you no longer have current vertices but you got rid of them. In this point where the electron hole interact. So this simplifies your diagrams you have a letter four point present with polarizability where the photonic for Colonel is now your standard for Colonel. And then you just have to evaluating this you can track the fermionic lines and which is just a summation over bands and KK prime points. You can also try the current vertices and this is basically your for a ladder contribution to the photonic self energy. And then you plug this photonic self energy in the dice invasion as we said and this will give you the coupling. And if you just look at the photonic self energy we are able to describe very accurately the excitons and you get really sharp peaks. Well describing this is in force frame for example, and if you exclude the vertex correction as we would expect you get a completely. You don't even get the spectrum and the next on peak and you get a completely wrong shape of of the spectrum. This also the same thing can be applied to different materials and we get a consistently expected results so this is for hbn and often this fight. You can also see a significant difference between the optical gap and the band gap. And then when you saw the second part and obtain the hybrid modes by selling the Dyson's equation can see that for a single layer of hbn, the, the dispersion of light is almost unperturbed. This indicates you also the dispersion of sex it on, but you see where they cross there is a bit of a change so okay in a single layer hbn obviously the stabilization between the external part and photon is not strong. But if you make a bigger heterostructure and as you add layers, the this actually coupling strength increases significantly and you see the photon line actually bending significantly so you really get a very strong hybrid. So you get some parietonic modes, and yeah this this allows you to calculate these kind of properties for many different combinations of one of us heterostructures. Then finally, the question is, okay but why do you want to use the veneer bases at all. And the problem is also because the letter contribution is extremely difficult to compute it's a it's a potentially heavy thing. So you cannot really include many bands, and the thing is that your exotic properties, for example in hbn are going to be basically localized around a key point between these two bands, and you don't really need too many of the bands above. But the problem is that the contribution if you project on the atomic states the problem is that the symmetry of your band is not conserved. So you actually have to include these free electron bands and just because of the very small contribution at different places in the case space. But when you veneerize naturally the band symmetry is conserved. So essentially in the BSC calculation now we can just take two of these bands and get very accurate. So basically we have two properties without without having to do a very heavy computation with all the other bands. But what we do we have to do to actually do this and there are only two places we have change in the code so basically we have to rewrite the charge vertices and the current vertices in the veneer bases, which is kind of straightforward. These vertices can be expressed will also contain matrix elements of the of this form. So I didn't write it out because they're just, it's a very troublesome derivation but in the end you get like four terms similar to this. And to just change the basis of the charge vertices you just perform the veneration use the unitary matrices, and we want to stay in reciprocal space because that's easier. It's easier to evaluate the propagators in reciprocal space. So we actually work with the smooth block basis after the veneration. With the unitary transformation you can just keep using your, your veneerize charge vertices in the rest. What we were also looking about is could we maybe interpolate interpolate these matrix elements, but this turned out not to be so easy and this is something we're also looking maybe if somebody has some idea how to do it. So we tried, we realized that we have to free transport to real space and then back again, and the problem gets even worse with current vertices which we have four of these matrix elements which would basically negate any gains in speed. But it would be very useful for us because we really have to have a very dense K mesh and K plus Q mesh to obtain accurate, accurate axotonic properties. So if somebody has some idea, we are, we are looking forward to it. And the code is also going to be open sourced and soon available doesn't yet have a name but it is modular well documented and if somebody is interested in these kind of things we are also looking for collaborators. So just to conclude my talk, I would just like to repeat a few main points in this methodology and propagator approach works for both metals and semiconductors you can describe plasmas excitons and hybrids with polyethanes very well you get a fine you get a temperature model, and you can mix different levels of accuracy, without having to recompute everything from zero, like you would have to do in a Hamiltonian approach, and you have a really an ab initio approach which is like really taking it into account the quantities are quantized electromagnetic field. Yeah, and in the end, we can also notice that the linear functions might be a very good basis for this type of thing because the bands, and they can reduce the number of bands we need in the expensive parts of the computation. Thank you for your attention and thank you the organizers for giving me the opportunity to present this work. Thank you for the nice talk. Open for questions. Thank you for the nice talk. I could you repeat the last slide what was the why do you say that. Sorry, no, it's not efficient or it's difficult. Yeah, a calculating the vertices to find great interpolate to find great. Okay, what's the problem with that. So yeah, the problem is that the thing is that you cannot really eliminate. So when you look at this last line you cannot really eliminate the G dependence or so you really have this e to the i to the G times are in it and the matrix are between K and K plus Q. And the thing is the grid in Q has to be dense has to be contain really many points. So I don't see it's not really straightforward to do an interpolation so if you would make a sparse k grid, the thing is that the q one depends on the grid. So, I mean the thing is that we we can do some kind of interpolation, but not without actually going to the real space and working with veneer function in real space, and then applying a bigger size weight and things like that. And then we have to Fourier transform back, but this doesn't really work well because we have like for the current versatility vertices you have five I think five for your transformation back and forth. And this really negates any gains in speed from the from the interpolation. So you want some interpolation to a very dense grid but that's not obvious that's yeah it's not obvious how how we could evaluate this. Thank you. Anyone else related questions so here how many g vectors do you need to include to get good results. This also depends on the accuracy you want. So you could actually take a g equals zero approximation and this gives you some kind of really I mean it works for certain types of excitations, but usually you have to take about 130 g vectors. That's like the optimum in terms of accuracy and and speed. The feasible is this method for a very large systems. So obviously it's not feasible for very large systems if you want to do it really at the full level. But if you do some approximation like reducing the number of g vectors or you can also, the thing is that you can introduce an approximation let me just go back a bit. So you can introduce an approximation where you before this last step, and you compute like the irreducible, like the photonic self energy for individual monolayers. And then you can do a few tricks and partially for reactions form this equation in in a Z direction because most of your properties are going to be highly dependent on the Z direction, and not so much in plane because the photon wavelengths have a significant resolution in plane, they have significant mismatch to the photon basically sees just a plate, a conductive plate. So you can basically use some delta functions to fix in Z in the Z direction your, your photonic self energy contribution, and then, like a summation of these terms will give you like an approximate photonic self energy which you don't plug back into the dice is equation and then you can treat really very very large systems like I don't know like you can go to the bulk limit of Android. But this is an approximation. If you don't do any approximations then maybe we can treat like seven layers. Seven layers, when depends on how large your computer but you're just thinking more like more a system. No, no, no way, no, absolutely not. No, it's, it's not going to work like that. You have a question. So you mentioned that you do a GW approximation. Yeah, but this is not like the one, the standard one for electrons because you care about the photon, right. Yeah, I mean but it's, yeah you can do it for electrons but you could also do a G delta W correction which also includes the effects due to the photonic. Yeah, but what you do is, you know, you take an instruction of the phone, the phone, that's what you care about. Yes. Because if, and maybe Andrea can comment because he has more expertise than me but let's say in a GW calculation, the cheapest part is the what you call the, what you call dipoles so this matrix element that you mentioned the one they say that's the cheapest that you know takes like a fraction of the total cost. So, but here it seems that this is not the case. Okay, I mean the GW creation you can separate from the whole thing here I mean the charge vertices are not so bad as for example evaluating the fork kernel and you have some matrix inversions, which are actually very costly. But the GW correction you can, you can compute it in any way you like I mean we have our own methods also so. I mean, can you just be more specific about. I mean to make a parallel with what I know about, you know, just a bit for electrons and it seems that so many things are different life in your case, apparently you don't need so many empty states because you are okay with having just few linearized bands right that you this what you call charge vertices that I think is what we call dipoles and correct me if I'm wrong, it seems that they are sort of important part. Why for the electrons is, I mean, it's just like the you never care about them because you care about the response function. This is I think are also finite q right I think the dipoles in the amber just q going to zero. Yeah, here we have a finite. And also, yeah, we have to the GW corrections applied both to the election hall propagator. So you have to two propagators you have to correct. And yeah, also if you want to substrate. You also have to do this delta w because you have in the present if you add a substrate to the photonic propagator, then you also have an additional renormalization of the band gap. So this is also very costly type of computation. Yeah, I didn't talk much about the GW because it has like many repetitive equations but. Yeah, so. Did you find any any gauging issue in passing. I mean, matching with money found a 90 and the many body called the symmetries. Did you exploit symmetry by the way. Yes, as we exploit symmetry. I mean with the one year we don't but when we use the blog like once we're so as the base ground state and we actually exploit the symmetry and compute everything with the irreducible wedge. Yeah, but with one year I don't we just did like the simplest possible approach with just taking a full below in the home and work with that. Just make a comment so usually when people do this disentanglement. It's usually a problem that the band structure doesn't, you know, is not exactly, but for you are somehow you're helped by having this one band instead of having like all of these spaghetti bands right yeah yeah we are kind of interesting. So. I think the reason might I might get really okay bands which which really look nice is probably related to the fact that like you don't have a mixed system like a box system but you really have layers which are separate and they only weekly interact. So the band structure when you you calculate, for example, for graphene and HPN and combine them, they're basically unperturbed I mean a gap opens what so you get bands that closely produced the plane wave base. And then the other comment was, so I don't quite understand all this better so Peter but like if you compared your absorption with just regular BSE and this BSE maybe actually maybe I actually have a comparison to show. So this is some equation I don't remember for which system maybe maybe HBN, but for example if you do a GPOL calculation. Yeah, it's BSE and compared with our calculation the differences are not not very large. But then, but then where this helps is when you get these like bending of the. Yes, yes, you cannot describe the bending of the photon in in GPOL. So even with finite q there is BSE with finite q. Yeah, it's fine. Okay. And I mean there are also certain differences that yeah I mean they're mostly related to the approximations if you want to treat a larger system that here you just do some partial terms, eliminate some terms and then pre transform back, and you have a much simpler equation to solve so we can treat larger system and GPOL. And so when you say retardation effect that means you basically include like magnetic field also somewhere. Yeah, I mean the magnetic field could also be included but. So like physically what does it mean. Yeah, it is. Yeah, it is. But we don't actually look for magnetic moves which are also possible. And what if you just take like a bulk, you know, because we have a lot of calculations where we compute the, you know, optical property for bulk. Okay, and then for some metal right there have been a lot of talks but then people assume that you know when you cut this and make a, you know they use some kind of approximation to because an experiment you actually have a surface. So could you use this to also just apply it to like a normal metal, you know, gold or something with a surface and would you get anything different. So yes you can apply it on a metal like, like, I mean a layer of metal but you cannot really go to the bulk limit because we do certain tricks within the constructing the propagators to, you know, separate the XY and the Z components because the XY doesn't play a large role. It's really important because like a lot of optical properties we compute for like simple bulk metals that they're done in bulk. Yeah, I mean this method in principle can can reproduce that but the way that our code works you cannot just add a bulk system and expect everything to work. Antimo, what's. Yeah. So asking so how do you do, how do you do with the, how do you deal with empty states. I know I already asked this but I love it a bit more. So, is it everything done with a localized basis or not. I mean it works with any kind of basis because those empty states will basically have like a zero contribution to your response function eventually. It will not affect your, your pro like your final dispersion relations. Okay. Okay, but you know also, you know, also basic in the basic digital approximation you have no this empty states. Yeah, I mean we we also have some empty state states and if you add them. I mean in when you're yes we didn't. Okay, I just want to understand if they come if they come from the vanier functions or they are the plane waves. Yeah, so so the empty states come from the plane way. Okay, yeah, in one year. But as they have this very small like very small contributions which are also in the symmetry part, we can get rid of them. Coffee break now, but let's thank the speaker again. Thank you.