 So, good morning everyone, my name is Pedro. I am working at, well let me publish myself properly, I'm working at Utrecht University as a postdoc and I'm here to give you your first lecture introducing you to the machinery behind many-body perturbation theory. I think there's a warning, no, okay, all right. So this is the schematic of what I'm going to be talking about. So first of all is a short introduction to the many-body problem within quantum mechanics. Then we will try to move into Green's function, its definition and properties, so how to extract information from the Green's function. Then we will talk about how to compute it using Eisen's equation and I will give you a short preview of what you'll be seeing during the school. All right, here it seems to be the best place. So the many-body problem within quantum mechanics. As Andrea showed you in the last lecture, the problem is that we need to diagonalize and obtain the eigenstates and eigenvalues for this Hamiltonian, which is, well, the basic of quantum mechanics. Now this is made of, we can describe it in two parts, so we can think of it as having a term that has all the single particle elements, either the kinetic energy or its interaction with the external potential, like an electromagnetic field, but then you have the rather complicated piece that deals with the particle interactions. And this is the problem that we will try to solve. Now if you can think of it as that the first many-body problem was actually the gravity, the introduction of the three-body problem, and you might think that with advancements in mathematics and physics, things became easier to solve, but in fact, no. And right now, even having no particles, so the electromagnetic vacuum, is something that is quite difficult to treat. However, there are tricks to go around this. So the first thing is that we have to consider what's going on. So the one particle or the single particle Hamiltonian, I quite like this picture and you can find in Mahtuk that you can see it as a horse, an isolated horse that runs through a field. But the field is not empty, it's not a horse in vacuum with a frictionless surface, otherwise the horse wouldn't run and it also would not breathe. You also need to consider what is going on. So the horse will move around and as it goes, it creates a cloud of dust around it due to its interaction with the field. And maybe some flies as well, things to smell. And so it's this effect that we will be discussing during the school, the concept of quasi-particle. So not just the electron itself or the hole itself, but also the effects of the surrounding electrons and holes that are in the material. And as in all cases, you have to consider that there are two limiting possibilities. One would be the ideal solution that you are fully able to solve your Hamiltonian. You have all the interactions. But this is often quite costly, especially for realistic materials. So it will be very hard to reach this point. But you can also not stay at the full isolated particle case. So we need to find a compromise, something that sits in between these two limiting situations. And you will have always to consider how feasible it is and how accurate, so how distant it is from the most easy to solve case or how long, how distant it is from the exact solution that you are very hard to ever reach. That's going to be very hard to ever reach. Now, some of you might be familiar with it. And a common approach is based on density functional theory. So all quantities are assumed to be expressed as a functional of the ground state density. And then you work trying to solve the quantum system of equations for a system of independent quasi electrons. And you have the fundamental object that describes the interaction as the quantum potential. However, as some of you might know, and probably the reason why you're at this school, DFT is quite good for ground state properties. But it's made for the ground state. It's not made to access excited state information or even non-equilibrium information. And fish is very good at swimming. But you cannot ask it to climb a tree. So you need something else. And one of the examples of the failure of DFT quite often is given as the gap that we can compute the band structure. And it's going to be quite nice reproducing the dispersion of the bands. But the gap is often incorrect. And we need something else to go beyond it. And it's often GW that you will see, I think, tomorrow, if I'm not wrong. Now, what happened in the previous slide? So why is DFT failing and why do we need to go beyond? Well, the gap is not exactly a single particle property. When you theoretically think of the gap, you are thinking about removing an electron from a system that has n particles and then putting it back into the conduction. And so within DFT, going into the approximation was too far. And we need to come back and be closer to the real-sick system and reintroduce some of the interactions between the particles that we are completely neglecting with DFT. Especially because if you think about it, the electronic gap and optical absorption are not just, as I said, a single electron phenomena. You need to consider what's going to happen, how the system will react once this electron is removed from the valence into conduction. So the electrons are not static, they're not frozen. They see each other, they interact with each other. And once an electron is removed, it leaves behind a hole which polarizes your material and all the other electrons are going to see this hole and they will try to react to it. And this is what DFT neglects quite often. Come on. So this is the general problem. So we have talked about how the issue that we have a lot of particles in the system and it's not often feasible to compute and to solve the Hamiltonian exactly. The DFT is quite good for the ground state, but we want to go beyond. We want to reach and compute accurate band gaps and often optical phenomena. Now a way to do this is based on the Green's function that I will talk about now. So consider this. If you have a solid and at the center point you will have an electron in RT and then it propagates and reaches the position r prime T prime, you can think, you can try to visualize the transition matrix element. So you are at the beginning at the ground state, then the electron is created at r, propagated from T to T prime reaching finally r prime and then you close the expression to obtain the matrix transition element. This brings is how you describe the movement of the electron going from here to here and you can think of it as the electron-electron Green's function. You can also think of a particle going the other way and this would be the whole which motivates also the expression for the whole's Green's function. But now instead of creating an electron you are destroying a state because the whole is the absence of an electron, so to say, that moves backwards in time. Now all of this motivates the expression for the time-ordered Green's function which will be the fundamental object for this lecture that is written in this way where here you have a time-ordering operator that allows you to replace or better to simplify this expression here involving the electron and the whole Green's function. Now what can you extract from the Green's function? Much like we said in DFT where you assume that all quantities are functional of the density and if you know how to write it and if you have the density of the ground state you can extract all the information that you need. Similar thing happens in the Green's function so you have a sort of general expression for the average value of an operator which in the case of the density is quite easy to write because it's just a diagonal part but there's more information hidden inside the structure of the Green's function. So for instance if we look at its poles and we do this by inserting a completeness relation between a system of n plus 1 or n minus 1 particles we can arrive at what's called the lemon representation of the Green's function. Once also you take the Fourier transform with respect to t minus t prime. Now what's hidden inside here though it looks rather complicated there's easy information to extract already. So first off you have the excitation energies here between the system of n minus 1 and n plus 1 and n minus 1 particles. They are separated by the chemical potential so the energy needed to either add a particle or remove a particle and here you have some matrix elements that will give you the amplitudes but once you plot this on the imaginary axis you see that from the poles of the Green's function you can access the excitation energies of your system. So not only you have information of the average value of operators you also have information on excitation energies of your system something that DFT was not providing immediately. Something I've told you works and it's true but there is a fundamental issue is that I'm telling you that once you know the solution so once you know the ground state you can get all the information about the ground state. The problem is then how do you actually get the ground state because it's not useful if I'm telling you that you need to know the solution in order to know the solution. It's somewhat topological and this is what we are going to discuss with Dyson's equation. So how to actually compute and evaluate the Green's function. So first of all we start from the time order expression for the Green's function and we have to assume that at some point there is a ground state a system that you can actually solve for which you know the solution and the interaction is given by this operator V that you are able to tune adiabatically. So at some point in the very far past you know where the system was you know the ground state and in the late future you also assume that it's going to go back to it but at the time that you are somewhat performing your measurement you are able to adiabatically bring the system to the fully interacting one and I have a new Hamiltonian that has a new state. Now how do we connect these two sets so the ground state Hamiltonian and the fully interacting Hamiltonian well at zero Kelvin so at zero temperature Green's function there's an actual theorem or a mathematical trick most accurately that can connect both. So it tells you that as long as this limit exists this will be an eigenstate of your fully interacting Hamiltonian. So you don't need to worry about evaluating these matrix elements here because as long as you're able to show and it's quite often possible that this element this sorry this limit exists then it's fine. You can connect the non-interacting system with fully interacting one. This however does not quite simplify your life because the expression for the Green's function becomes somewhat more complicated so you have an infinite sum of products of the interaction potential and we want to simplify this. So a way to or by the d-way and it's a very nice result from mathematics to sort out this time order product it's to use vix theorem that replaces the time ordering product as a sum of the normal ordering of all operators plus the normal ordering with one contraction between two operators normal ordering with two contractions of operators and so on and so forth until everything is contracted. That is a contraction so it's simply a difference between the time ordering and the normal ordering of two operators so it means that you have all annihilation operators to the right and all creation operators to the left if I'm not switching right and left which I often do and luckily the expectation value of the of a contraction is simply the expectation value of the time ordering operator. Now once you manipulate this very nasty looking expression here you're actually able to arrive at a simplified version of the Green's function that will give you all them diagrammatic expressions and all contributions to G. Now the fundamental idea that vix theorem tries to, oops moving too fast that vix theorem tries to do is that instead of dealing with this very complicated step you replace all your processes and this is what exactly the Feynman diagrams are doing you replace this one step complicated process with the sum over steps of much simpler processes and this is an analogy I made that well think of it as somewhat ideal or idealized experiment where you have a ball going through a black box and it's repeatedly bombarded with jets of ink and the only thing you know is that it comes out blue. Now instead of dealing with all the bombardments at the same time of ink you simply say okay let's deal with it as iterative steps the first thing that happens is that the ball which was white is just hit by blue ink and comes out as blue and then you can consider that it's first hit by red ink then blue ink and comes out as blue and more complicated processes and so on and so forth and if you sum all the probabilities of this hits by ink the idea between vix theorem is that the sum of all of this has to be equal to what's happening inside the black box that is very hard to describe and that you don't know how to deal with so instead of just doing the full complicated thing you just start dividing the problem first you do one transition then two transitions three four five six seven eight nine ten and ideally at the end the full sum would be equal to the entire process and so this is what you are trying to do when you're doing Dyson's equation or when you want to arrive at Dyson's equation more actually so here we have the first order contributions for the greens function with the Coulomb potential and you have all the all the six terms and now we can start trying to correlate them with diagrams so how does this work this is also an important part because for the rest of the school you're going to see lots of images with lines, wigglies, bubbles going on and so forth and you cannot just stare at it as if they were hieroglyphs you actually have to have a somewhat Rosetta stone so let's concentrate on expressions A and B if I can move the slides forward alright so first off you have an independent particle greens function going from one to two which are also the limits of your term for the fully interacting so it's basically free not interacting with anything and you have greens functions going from three to three and four to four that interact with the Coulomb potential so this is the equivalent expression the equivalent diagram of is a contribution for the B contribution it's almost the same so you have an isolated line but now you have two lines connecting the extrema of the potential and these two types of diagrams are what we call disconnected diagrams because well you have the you have a green you have two pieces that are not connected by any line now once we look at the other four contributions we will see that not all of them are different from each other so starting with contribution C we can understand that there is a greens function going from one to three three to four and four to two so you can draw it as a straight line already with two points that are then connected by the potential for term D however you have it going from one to three three to two so already reaching the final point of your integration but then there is a greens function that connects the twice to the one of the vertices of the potential now these are the two main contributions because if we look at expression E we see that instead of three we have the coordinate four and instead of four we have coordinate three so we have just flipped and you obtain a diagram that is equivalent to contribution C and the thing happens to F and D so even in your expression for the correction of the greens function not all diagrams are going to be different some of them are going to be topologically equivalent to each other and this is something you have to take into account so you are only going to need the diagrams that are topologically different when you're dealing with whatever system you're studying now as I mentioned before we have two types of diagrams two that are disconnected and two that are fully connected and thankfully you can actually prove that the greens function can be written in this way so you have the product of all connected diagrams with all the disconnected diagrams divided by all the disconnected diagrams so you can just cancel out and ignore completely the disconnected diagrams and you only need to consider the ones those that are connected and once you do it you arrive at this expression for the greens function now the full interacting one written in this way with the double line that starts to resemble something like this and you can notice that there are there's also another distinct distinction between the diagrams first is that you have diagrams that cannot be split by cutting through a greens function line and still leave you with a diagram for the greens function that are these three well these two you can cut through this greens function line here and still opt in a diagram so to the first kind we call irreducible diagrams because there is no way to cut them through a greens function line still opt in a greens function diagram while the later are called reducible because you can actually do it but not only this if even if you were to write your organize your sum as the sum of all irreducible diagrams as a building block and here i'm taking just this diagram as an example you will start to notice that the equation is somewhat repeating itself so this term is repeating this one here and then the next terms are also repeating the equation and this means that you can reorganize all your sum of diagrams cutting through this line here and take advantage of the repeating terms and write them in this way so that your fully interacting rings function is the non-interacting rings function times the non-interacting rings functions multiplied by the sum of all irreducible diagrams that don't repeat and again the same equation and this is the diagrammatic motivation for the Dyson equation as we're using the reducible self-energy that is written mathematically in this in this form now this is not enough unfortunately you because the self-energy is also an equation so you actually here you're going to have oops an integral differential equation because the self-energy is also made and function of the Green's function so you need to close the system of equations so we have one the equation for the Green's function but now we need the equation for the self-energy that deals with the interaction so this boson w here and the recommendation with the vertex you also need to be able to describe so all of this just a short note is made for the club potential you also need to be able to describe how the boson is computed so we in this case we do it by the screening you also are going to need to understand how electrons and holes are interacting or not in your system this is done by the polarizability and finally you have the last equation that closes everything which is the vertex and this is called hidden system of a equation so it's just together with the equation of motion for g is a system of five equations that if you are able to solve them they would give you the full solution for your problem it's usually represented in this way with the pentagram and the idea is that you start from one of one point and you then go on iterating through all of them until your your solution is converged now this is the full machinery or better the introduction to the full machinery what comes next and what are you going to see in the next lectures during the week so depending on your system and depending on what you're studying what you'll have to consider which diagrams are important which are the ones that you're going to have to include in your sum which are the ones that you can disregard and this will be based on the physics of your system so if you're studying interaction with electromagnetic fields or if you're studying phonons or if you're studying magnons plasmons whatever it is you you're going to have to adapt your your formulation of the problem to the exact interaction that you want to describe and this you need to understand the physics of the system and it's what's going to be explained to you in the next days so things like interaction range conserve quantities low high density limit of the of the electrons all of this will have to be taken into account when if you want to do a proper accurate description of your system so for instance one of the things that you're going to deal with in the next days is the accurate computation of the band of the electronic band gap so you'll move from tft into the correction of the band gap using gw but then once you reach optical absorption you'll see that gw is actually not good enough you need to go beyond and reach the level of the beta-sorbiter equation to actually accurately describe the interaction between the electrons and the electron in the whole pair that will give you the optical gap which is what you actually measure in absorption experiments and this is where I conclude if you want to know more especially since I glossed over a lot of details this is what we usually call the Fetrivaletska in 20 to 40 minutes lecture and this is a massive book with well with lots of theory and mathematical derivations you can also find a more didactic approach in Matuk and other ways to derive these equations using Strinatis with the functional derivative method and you also have some sort of let's call it Bible that encompasses all theories so not just what I've shown you which is the zero temperature formalism but also outlay up to the non-equilibrium formalism where the system is no longer in the ground state you're creating something that is already somewhat out of equilibrium and you need to deal with it with some special care using what I mentioned which is the Kalishkontor so thank you all for your attention and questions depends let's put it this way if I knew nothing and wanted a complete mathematical derivation of everything Fetter if I wanted something that at the start doesn't really have a lot of physical considerations but because the functional derivative technique is a mathematical trick then Strinati if you already know this too and you will want to learn a lot about physics you can read Abrikozov's book but this is written in Landauwisch so it's the the books you read after you know the subject so you understand once you understand what's going on you read them to have a deeper insight on physics hello I have a couple of questions so you said that we need this many body formalism to find the gap so suppose if you have an exact exchange correlation do we need to still use these many body methods to compute the gap because as you said you can't compute the gap with DFT what would you call the exact exchange correlation when suppose we have like a method which can give a good approximation good approximation for exchange correlation and the question is now does this work for all materials yes hypothetically so hypothetically this is the ideal case of DFT good luck finding it I mean like as I understood I mean you can't compute gap with a DFT I mean as far as I understood this is because we are using some approximations for example LDA the point is that you have to understand what is the theory developed for so in DFT even if I give you the exact potential you can calculate exactly only the density only the density and its authority of course because it's a functional of the density but in in DFT the conich charm equations are nothing more than a Lagrange multiplier's equations so you do functional derivative sorry a minimization as you do without refoc exactly the same with the with the exact energy functional and then you can prove exactly that you can rewrite the density in in a single particle representation exactly as it is at refoc but instead of at refoc we function you have conich charm where function but the energies as the Lagrange multipliers that ensure auto normalization of the wave function they have no physical meaning except for the lowest excitations the Koopman's theorem but this is not a story in general you cannot interpret the conich charm levels as gaps and in DFT this is pretty known actually it has been also demonstrated that even the exact conich charm get misses of appease is the the the derivative discontinuity terms it is due to the fact that if you do the theory instead that from the density from the total energies you say okay forget about densities I still accept exchange correlation potential and then you try to do the same story with the total energies you realize that when you add an electron the potential and the density change so abruptly that you have another piece that is completely missing conich charm so it's a different theory while many body in instead is is developed exactly to give the poles of the single part of this function and then the gap so with the exact suffrage instead you get the exact gap exact I mean it's not that we are not able to get the exact gap with DFT but this is somewhat trick we can use hybrid functionals DFT plus you and so on so on but it's not an it won't be an exact theory it won't be the exact functional ever and you can do it there it won't be general it won't apply to all systems and some hybrid functionals are extremely nasty to deal with computationally so the as I said the point is that DFT is not made for this not made to get the bandgap ever you can tweak it you can manipulate the functional as much as you want but it's it won't be a general framework thank you very much for this talk this many body formulation how many parameters are involved for instance that's the first question for instance if you have a system you have different systems and so the number of parameters and the number of parameters involved how they the same or many parameters for different systems yeah so that's just the first question and when because okay the competition of quantities okay we are talking about so a couple of things so we're talking about the competition of the exact bandgaps well I remember one time I was calculating one particular system which is it doesn't involve the calculation of exact bandgap but it involves the calculation of the difference in the bandgap I'm not interested in the exact but I'm interested in the the difference in the bandgap but in the question is do I need to go to these uh higher level calculation in order to to do my calculation but DFT is just enough okay so the number of parameters this is dependent on how you write it you can do type for instance type biting DFT which is going to be dependent on the series of parameters you can do some sort of semi empirical method which is also going to be dependent on lots of parameters you can do what we do which used to take the DFT as a starting point and then the only parameters you actually need are the chemical composition and the geometry of your system so it's what we call an issue the idea is that you just need to know what's there and how it's arranged and this from this you can extract all information but this is a philosophy I don't approach to to the problem so it you have to choose what will be the best so just let me stress what Pedro is saying is that ab initio many body perturbation theory is a parameter free simulation so you don't have to feed it without with any parameter this one point but then of course there are the parameters that you need to converge so there is the energy cut off the number of bands and in this sense there are many parameters that you need to converge so yes but those are not fitted to anything experimental these are the same thing as you do in numerical integration you have to converge your integrals for your second question I'm not sure I understood so you want your interest in the difference of gaps of systems what is this band alignment case well yeah you still need to go to to GW and because there is no guarantee that the difference of the gaps from the FT is going to is going to be accurate I mean I mean if it's for one particular system it might take it might happen it might happen but we cannot guarantee you that it will happen for all systems imagine that you have a let's say hbn and transition metal echocogenide the correction for the gap for the transition metal echocogenide it's of the order of the one electron volt which is enormous compared to the FT and this is similar of magnitude for hbn so if you are building an interface of this it means that your dft calculation difference between the gaps is not going to to be good I would say the perfect subject so I was wondering why you see that the difference in the eye gap does seem not that good compared to the exact but I mean it's clear it's clear I mean uh the the the it does absolutely clear also to us that the the the procedure I mean that the work you have to do from the money body to the funnel number is huge and I mean it is not the only way there are many other ways with the empirical methods intuition there is different ways to do physics millions of ways to do physics so absolutely our message is not that you have to be strict and formal from the beginning to the beginning to the end it is something that must be decided these are the basis of the project of the problem of the system there are so many ingredients but uh more you are aware of what you can use and more you can decide which one to use yeah you have to fit the solution to the problem you're studying and to the resources you have this is but this is the human and logistic limitation of it so I'm going to read a couple of questions from students online and the first question is uh how good is it to discard the vertex function so is it an issue that we discard the vertex function this your uh can I answer one by one so this is something uh I think you're going to see in the next lectures more often uh than not it's for all materials I've studied it's been quite good uh GW has an approximation I can't remember right now any system and any particular case where the the vertex was a fundamental part for the correction but no for the electronic gap it's the same version yes but a correct correction for the electronic gap to be the sub peter for optical absorption yeah but for the electronic gap going beyond GW I don't remember any system by heart where it has sorry okay okay so another question is so you made the point that DFT is underestimating the the gap no I made the point that DFT uh doesn't won't give you the correct gap yep and but the question here is but besides that is it the DFT band structure good enough for semiconductors I mean when you open the gap do you think the the band structure is okay you mean the dispersion of the curves not the exact values but how the shape well I can read the question it asks does it give a faithful band structure for example vines band maximum conduction band minimum position etc of semiconductors yes most cases that that's that's the case it won't give you the correct band gap and this is something I want to emphasize that I didn't say it will give you an underestimation of the gap because once I reach it good lord as I shown here this is actually a special case where the approximation taken to DFT actually gives a larger band gap than GW so it can be system dependent in most cases yes it's an underestimation but depending on how you actually treat DFT it can give you an overestimation okay maybe last question what is the difference between optical gap band gap and absorption coefficient so the electronic band gap is the difference between the Omo and Lumo so the highest occupied and lowest unoccupied state the optical gap is the lowest energy at which the system is going to absorb light this is going to be dependent on the interaction between electrons and holes because once you start to try to promote the electron up it leaves a hole behind they will interact see each other and the end the energy of this new state will give you the optical band gap the absorption coefficient do you want mathematical definition I think it's okay I mean it's part of the optical gap yeah it's part of it's will be related to the optical gap you're going to see it at the the BSE lectures later on so in the time ordering slide could you please go there the time-ordering operator yes yes so it just shifts all your operators from the earliest to the latest yeah so I have a question there yes so why do you have a negative sign I mean like because they are fermions but like as you said I mean yeah in the time-ordering inside the time-ordering yeah it shouldn't matter right no it does matter so the ordering of the operators matters because I'm dealing with fermions if they were bosons you'll have a plus sign they because this will have to respect the Fermi direct statistics I mean this is not an operator right this is what we do I mean this is a convention which we define this is not an operator right this is like something similar to a mathematical operator on the wave function I think the point he is making is that how we we define this time-ordering operator in your operator this time-ordering notation in a way so yeah it is defined in in a way that when you zoom yeah so it's not a quantum mechanical operator yeah that's true it's it's a convention it's a shortening convention you can call it in this way if you don't if you're not comfortable with calling it an operator but yeah it's not it's not a quantum it's so the weak the weak's theorem that is the theorem that allows you to expand the complexity product in elemental uh contractions and this requires in the case of fermions to have the correct statistics indeed in the case of bosons weak's theorem is more tricky to demonstrate okay I think it's time to move to the coffee break so we will resume in I don't know 15 15 20 minutes half an hour okay well okay let's say 25 minutes okay yeah you can go upstairs there will be something to drink and some food yeah so