 So this lecture will be about dynamical mean field theory. That's an approximate method to solve lattice, correlated lattice systems. And that's in any type of mean field theory. The basic idea is to map the lattice problem to a single site effective problem, which in this case is a so-called quantum impurity problem. So the idea is to map the lattice to a self-consistently defined quantum impurity problem. So let us take as an example for the lattice problem the Hubbard model. So we have the single band Hubbard model. So this model describes electrons which hop around some lattice. I should wear two. Oh, I see. OK. OK. So here we have the lattice system. Electrons hop around this lattice with some hopping amplitude t. And they interact on each of these lattice sites with some repulsion energy u. So the Hamiltonian for this model looks as follows. We have the kinetic term, which describes this hopping between nearest neighbor sites. And then we have a local term, which describes the interactions on each site and also a chemical potential term. And now in this dynamical mean field theory, this lattice problem is mapped onto a single site model, which consists of one correlated site in an uncorrelated medium. So that's a so-called Anderson impurity model. So we have one site with interaction u. In principle, infinite bath of uncorrelated sites with some energy epsilon of k. And then there is a hopping or hybridization v of k between this interacting site and the bath. So the Hamiltonian looks as follows. Hybridization term, which describes this electronic exchange between the impurity and the bath. That's this term. Then we have a term describing the non-interacting bath. Then we have also a local term, which describes this impurity with the interaction u, which is the same as in the original lattice problem. And we also have a chemical potential term. So these A operators are for the bath and the C operators for this impurity. And now we optimize these parameters of the bath, these hybridization amplitudes and the energy levels of the bath in a self-consistent way, such that the bath somehow mimics the lattice environment. So that kind of the hopping of electrons from the impurity into this bath and maybe after some retardation back onto the impurity mimics process where an electron hops from a correlated site into the lattice and then after some excursion through the lattice returns to the original site. So that's the idea. So that's the strategy. So these parameters v of k and epsilon of k, these bath parameters, they are kind of the mean field in this approximation. And now we have to define a self-consistent scheme, which allows us to optimize these bath parameters in such a way that somehow the electrons which are sitting on this correlated impurity field to some extent as if they were part of the original correlated lattice problem. And so in order to enable this mapping, we have to do an approximation. So basically the approximation which allows us to do this mapping is the assumption of a local or momentum dependent self-energy for the lattice problem. So the self-energy is sort of the object which describes how interaction effects change the propagation of electrons in the solid. And of course a single site model can only sort of produce a local self-energy. And so if we make this kind of assumption that in the lattice system we have a local self-energy, then we can sort of map the lattice problem to the quantum impurity problem as I shall describe. So the essential approximation is this approximation that the Harvard model self-energy, which is in principle non-local, can be approximated by the self-energy of an impurity system, which is local, but still time-dependent because we can mimic retardation effects by this bath construction. And so in practice it's often useful to work with action rather than Hamiltonian. And that allows us to integrate out the bath degrees of freedom which are non-interacting and then just represent those bath degrees of freedom through a so-called hybridization function in the action. We just have a correlated site and this kind of hybridization function which describes how electrons hop from the impurity into the bath and after some retardation hop back onto the impurity. So the action then has the following explicit form. So that's the hybridization term which describes this hopping into the bath and back. And then we also have the local term which describes the interaction and chemical potential contribution. So that's the impurity action. And in this formulation now the self-consistency condition fixes this hybridization function. So now the mean field becomes this hybridization function. And this hybridization function can be written in terms of these epsilon and v parameters of the Hamiltonian formulation as follows. Delta in Matsubara frequency space. You can be expressed as follows. Hybridization squared divided by i omega n minus epsilon. So it's really only this combination of hybridization and bath energies which matter. So it's this object which we want to optimize. Equivalently to this we can define the Green's function of the non-interacting impurity which is sometimes called the Weiss Green's function because it's like the Weiss field in the generalization of the Weiss field in the static mean field theory. So that's the Green's function of the non-interacting impurity problem. And that's related to the hybridization function by this formula. So the two are equivalent. We can either work with delta or we can work with this curly G. And depending on the method one uses, one or the other is more convenient. So anyhow, in the end we now have to define a procedure which fixes us in a self-consistent manner, say this Weiss Green's function, G0 which now plays the role of the dynamical mean field. So this mean field is fixed by the following self-consistency loop. So we start from some impurity problem with a given Weiss field or hybridization function. We have some G0. Or equivalently we have some hybridization function. Then we need some numerical procedure to solve this impurity problem, which means we want to compute now the interacting Green's function and the self-energy of the impurity problem. So this numerical procedure is called an impurity solver. We are going to discuss one powerful technique to do this just in a few minutes. So now let's assume we have this method and that gives us now the impurity Green's function and the self-energy. Now we somehow have to connect this impurity calculation to our lattice calculation. And this we do through this dynamical mean field approximation, which I have already mentioned. So we now pretend for the moment that this self-energy is a good approximation of our lattice self-energy. So in the end we want to optimize the bath in such a way that this is true. But let's assume it's the case. We can approximate our lattice self-energy by this impurity self-energy, which in particular means that we have to neglect the momentum dependence or the non-local contributions of this self-energy. And now we have some lattice self-energy, and with this we can compute the lattice Green's function as follows. So here in principle we would have the lattice self-energy but now we replace it by the impurity self-energy and that this gives us now a momentum dependent lattice Green's function, where the momentum dependence comes only from the non-interacting dispersion in this case. And given this momentum dependent lattice Green's function we can average our momentum to get the local lattice Green's function. We have the local lattice Green's function and we somehow have to go back to the impurity side and now we impose the self-consistency condition of dynamical mean field theory, which is that the local lattice Green's function and the impurity Green's function should be the same, which means that we now try to optimize the impurity system in such a way that the impurity Green's function gives us the local lattice Green's function. That's the goal of the whole self-consistent procedure. And this we can use now to define a new twice Green's function because we have a Dyson equation for the impurity problem and so we can use now this new estimate for the impurity Green's function, which we have obtained through this momentum averaging and the old self-energy for the impurity to get the new above Green's function just by this usual formula. And here in this calculation the self-energy comes from the previous step from here and the lattice Green's function has been updated here from this loop. And with this we can now continue. We have a new approximation for the vice field. We solve the impurity problem. We go through this loop a couple of times until this calculation converges. Yes, this? How I get it? This is... Well, first I compute this by some numerical procedure. That's not easy, so we will discuss how one can do this. And then we use... This is the non-interacting tree. This is the imp... No, no, no. This is sort of something equivalent to G zero. We have to compute it. This is to be computed as G zero minus one minus G, but that's sort of a non-trivial calculation how to get this. That's right, so that's what we will discuss next. So how to get the Green's function or any other type of observable for a given, say, hybridization function or vice Green's function. So there are many different techniques. I will not try to give an overview of all these different techniques, which exist. I will just describe one, which is very powerful, and that's Monte Carlo technique, based on a diagrammatic sort of expansion in the hybridization term. And that's called hybridization expansion. And because we will work in this method with the hybridization function, I will write the impurity model in terms of the hybridization function. And so the idea, which I just sketched, is to expand the partition function, which we can write as follows. We have some trace over the time-ordered exponential e to the minus s impurity. That's our partition function. This can be expanded in powers of the hybridization term, which is contained in this impurity function. And then we view the diagrams, which we obtain from this expansion as our Monte Carlo configurations and then implement the stochastic procedure to generate all these configurations. And once we know how to generate all these diagrammatic contributions, we just have to measure the contribution of each of these diagrams to the observable of interests, such as the Green's function or any type of observable. And in order to keep the discussion simple, we will first consider the spinless case because for this hybridization expansion, the spinless case is not the trivial and it allows us to understand the main ideas. So we have an action of a non-interacting impurity coupled to some hybridization, which looks as follows, and we have a chemical potential term. And so now we write our partition function and expand it in powers of delta. Now we expand this into a power series and are now writing the nth order contribution. So from the power series of the exponential, we get 1 over n factorial and then a trace of a time-ordered product of pairs of creation and annihilation operators from these terms. And then we have a product of hybridization functions. That's the expression we get after expanding the partition function in this, in powers of delta. Yes, so this, sorry, here it's also missing. The chemical potential term remains in the exponent and we expand only in the first term here. And so each term gives us two time integrals over tau and the tau prime that you see here. And the pair of creation and annihilation operators at these times, these are these pairs here. And then also a hybridization function whose time argument is the difference between tau and tau prime. So here we have n and such factors. And now we have basically what we need from Monte Carlo algorithm, namely here we have some kind of sum over configurations. So this is kind of a sum over configurations C where our configuration is just a collection of time points on the imaginary time interval. And here on the right we have the weight of this configuration. That's the weight, our weight WC. And so if we know the configuration space and the weight of the configurations we can use standard Monte Carlo techniques in order to generate all these configurations according to this weight. We'll have a bit closer look at these weights, what this is, how we can diagrammatically represent these weights. So let's look at the first auto diagram. No, sorry, zero auto diagram first. So this trace here, if we have no creation and annihilation operators, it has two contributions, one for the empty and one for the occupied impurity state. So we have sort of two diagrams which I can plot like this and like this. So this would be an imaginary time interval from zero to beta. This dashed line represents an empty state and this full line represents an occupied state. So here we have sort of zero electrons on the impurity and here we have one electron on the impurity. And so this gets as a weight, well, what is the weight? The weight is basically now coming from this chemical potential term here and if we don't have any electrons, then it's just one, so e to the zero is one. And here we have an electron on the impurity, so this gives us e to the nu times beta. So these are the weights of these two contributions and that's all we have at order zero in the hybridization. Now let's look at the order one diagram. So now we have one creation and one annihilation operator. We have a C dagger and a C operator and we have a hybridization line connecting these two, which I draw like this. Yes, so each hybridization function comes with a C and a C dagger operator. So it means that here an electron is annihilated on the impurity, so it hops into the bath, then it propagates in the bath and it comes back, it hops back onto the impurity at time tau prime and that's represented here. So basically now we have occupied impurity in this time interval, so basically the picture is that the electron hops in at this time, stays on the impurity up to this time and hops out. So that's more or less the meaning of this. And so what is the weight here? So if this is tau one and this is tau one prime, then this line here is e to the mu tau one prime minus tau one. So just the length of this segment here. And then we have a hybridization function which comes tau one minus tau one prime. So the total weight of this contribution is just the product of this exponential factor and this hybridization function. That's now one diagram in the order one space and of course we can now integrate all possible positions of these C dagger and C operators. So this is the C operator and these positions are integrated over. Here we have an integral and that will generate all the first order configurations. And then we can go to the second order configuration which we can graphically represent like a collection of two such segments. You can imagine we now insert two creation and annihilation operator pairs like this. They are somehow connected by hybridization functions that would be in our order delta squared contribution and so forth. Yes, you are right. But that corresponds to a different ordering of the C dagger and C. So in principle you have to integrate, I mean they can be at arbitrary places on this interval. So the other contribution which you mentioned that would correspond to one like this but of where the creation operator comes after the annihilation operator. But at this stage all possible positions are integrated over. Yes, now one point one can realize is that if we have more than one pair of operators there are different ways in which we can connect these operators by hybridization lines. I just showed one possible connection but there are other possibilities. We can connect creation and annihilation operators by hybridization lines. In total there are n factorial ways in which we can connect these and it turns out we can sum up all these n factorial contributions into some determinant. Maybe we should show this explicitly for the simple case of n equal 2. So here we have two possibilities. One is this diagram which I have already sketched. So say this is some length L1. This is length L2. And this is tau 1, tau 1 prime, tau 2, tau 2 prime. Then this diagram has the following weight. We have minus 1 squared coming just from the each hybridization function comes with minus sign. That's just this. And then we have a chemical potential contribution which is determined by the length of these segments. So e to the mu L1 plus L2. And then we have a product of two hybridization functions with the following time argument. That's the weight for this configuration. But there is a different second configuration which has exactly the same operator positions. That's d tau to the power of 4. So in principle this infinitesimals should also be considered part of the weight. In principle this, anyhow, it's not important for the following but in principle this is part of the weight of the configuration. So the second way we can connect this creation and annihilation operators is this. And also connect them like this. And that gives a different weight, namely the following, minus 1 to the power of 2 from the expansion at second order. And then we have a second minus 1 from this time ordering operator because now if the arrow goes into the opposite direction it means we have to somehow exchange a pair of creation and annihilation operators and that will give us a minus sign from the time ordering. So that is from time ordering. And then we have exactly the same contribution from the segments because the segments are the same. And then a different combination of hybridization functions. So now we have delta of tau 1 minus tau 2 prime times delta of tau 2 minus tau 1 prime. And now we can combine these two if we want as follows. And write this as minus 1 squared e to the mu L1 plus L2. And now determinant of a 2 by 2 matrix and the following matrix. So this contribution will give us the first term where we have the product delta tau 1 minus tau 1 prime of tau 2 minus tau 2 prime. And then we have another term which corresponds here now to the off diagonal elements delta tau 2 minus tau 1 prime delta tau 1 minus tau 2 prime. And now we have also the infinitesimal tau 2 to the 4. And I think you now already guess what is the general structure if we have n segments. We can just add all n factorial diagrams up into a determinant which generalizes this structure here. And now the algorithm becomes very nice because we can now forget about the hybridization functions. They are taken care of by this determinant and we can only focus on the time sequence of operators. So in other words, we can only draw these segments and then a collection of segments implicitly defines the determinant if we think of summing up all the different diagrams that we can obtain by connecting these segments by hybridization functions in all possible ways. So we really only have to think about the sequence of operators for this segment structure and the weight is then proportional to just this exponential factor with chemical potential times the total length of the segments, which I write as length of segments, and then a determinant of matrix which I call delta and that's sort of the hybridization matrix which is implicitly defined through the sequence of operators as I explained here for the two orbital case. So we can simply represent such a configuration by some arbitrary collection of these segments. Looks like this. And now we know the weight for such a configuration and you can easily guess how we sample such a configuration. We can now just think of a procedure of randomly inserting one of these segments, randomly removing another segment and that's how we can sample the entire space of such configurations according to this weight. This would then be tau one. OK, so this was the non-interacting case. Now, how do we treat interacting impurities? So that's very simple. Once we have this segment picture, we consider the Anderson impurity model with interacting spins. Now we have just one segment configuration for each spin and the hybridization determinant for each spin and the interaction contribution will be given by the overlaps of these segments for up and down spin. So let's make a picture. So we now have one time interval for up, one time interval for down and some segment contribution or configuration for up. Let's say this one and another one, for example like this, for down. So I mean these segments, they now indicate the time intervals where an up electron is on the impurity and where a down electron is on the impurity. That means the interaction happens when both up and down electrons are on the impurity so it corresponds to the time interval where we have overlaps between these segments. So it's this kind of time which met us for the interaction. So let's call this L overlap. So basically the interaction contribution will be interaction U times the length of this overlapping segment. And the chemical potential contribution will just be given by the length of the segment, so length for the up and length for the down segment. So we have that the weight is essentially given by this chemical potential contribution which is chemical potential times length for up segments, length for down segments, then minus the interaction contribution U times length of the overlaps and then a hybridization matrix determinant for the spin up and the hybridization determinant for spin down. So that's now the structure for the interacting problem. So it gives a very simple intuitive representation for what such configuration means. It really means an down electron hops in here, up electron hops in here, then we have two electrons on the side, they interact, then the down electron hops out, we have only singly occupied side and then the up electron hops out. And this also gives us very easy way to measure, for example, double occupation. It's just the average length of these overlaps in our configurations, but the occupation is just the average length of the segments and so forth. Green's functions are a little bit more complicated to measure. Maybe I'm not going to describe how we measure the Green's function but basically what one has to do is one has to remove one of the hybridization lines and then one has a C dagger and a C operator in the configuration and that's a Green's function configuration and one then measures those contributions. But that's a little bit technical so I will not go into this. But anyhow, so this is one technique which allows us to solve impurity problems numerically in a rather efficient way. So in the last half hour I want to discuss a little bit the problem of dynamical screening and what this interaction parameter in the Hubbard model really means and how we should properly define it if we want to simulate the material. Basically we have the problem of defining or calculating the Hubbard U in quotes for a low energy effective model for some complicated material. So we will consider a typical case where we have one narrow band, say a D band near the Fermi energy which is separated from other bands which we can assume to be more weakly correlated. So the situation is the following we have here our Fermi level we have some D band near the Fermi level and we have some other bands called screening bands near the high energies. So now we can partition energy space into a low energy window and a high energy window like this. We have here our low energy window which spans this or contains this single band near the Fermi energy and we have a high energy space which sort of contains the bands which are further away from the Fermi energy. So this is sort of the high energy subspace which I call H. So this is now considered high energy subspace because we are interested in how far the energy is from the Fermi energy and then we have the low energy subspace. So these are the occupied states and ultimately we want to now define a kind of Hubbard type model for this low energy subspace from a kind of up-in-issue calculation. Calculate this interaction we have to compute now the polarization of the system because the polarization tells us how the bare interaction is screened and gives us sort of the screened interaction which is relevant for the definition of this low energy model. Now the idea is that we separate the polarization into contributions which are associated only with the low energy subspace and into contributions which involve the high energy subspace. The contribution from the low energy subspace we call P L and the rest contribution from everything else which is not from the low energy subspace is called P H. Contributions involving somehow the high energy subspace so we have these two contributions from the low energy subspace and from everything else. Now the fully screened interaction is basically the bare interaction screened by this full polarization P. The fully screened interaction W kind of Dyson equation which relates the bare interaction V and the screened interaction W and the kind of self-energy in this equation is the polarization. And this we can re-express in the following form. So now what is P L and P H? Somehow we can represent this graphically in this picture. The low energy polarization has to do with transitions within the low energy subspace something like this. Whereas the high energy polarization is everything else so that includes transitions from occupied states in the high energy subspace into the low energy subspace like this or from the occupied states in the low energy subspace to empty states in the high energy subspace like this or of course also transitions from occupied high energy states to unoccupied high energy states. So this is all part of P H and this is part of P L. Now the point is that these polarization effects these kind of transitions they should be properly taken care of in our solution of the impurity model. If we believe that within dynamical mean field theory we can obtain a rather good description of the low energy physics so we should capture all these low energy polarization effects in our DMFT calculation. So it means that this polarization should not be considered in the definition of the Haber-Dieu otherwise we would sort of double count the polarization effects. So we have to remove this low energy polarization and only define our Haber-Dieu by considering the high energy polarization. So that's why we now introduce the so-called partially screened interaction. So we define this so-called partially screened interaction W H where we only screen the bear interaction V with the high energy polarization. I want to write it differently and you can do a little calculation using this relation and the definition of the fully screened interaction to show that this now implies that the fully screened interaction can be expressed as follows as 1 minus P minus W H P L minus 1 W H which means that the fully screened interaction is now obtained if we further screen the partially screened interaction by the low energy polarization and that is exactly what we want. We want to define our Haber-Dieu or our interaction in the low energy subspace in such a way that if we screen this interaction by the low energy polarization we get the fully screened interaction. So that's the proper definition of the interaction in our low energy model. Then we have our correct definition of the interaction in the continuum and now we want to define the interaction in the Haber-Dieu model and for this we have to construct now localized Vanier type orbitals and then evaluate the matrix elements of this interaction in these localized Vanier orbitals and that's now our correct definition of the Haber-Dieu. So let's call this basis of localized orbitals phi as sort of the lattice index and then we have our definition of the Haber-Dieu which is the following integral of the space I of r squared and this partially screened interaction depends on r r prime and frequency phi i of r prime squared. So that's our Haber-Dieu and the interesting or the important thing here is that it is frequency dependent. So that comes from the fact that the polarization is frequency dependent. All these transitions which contribute to the screening they have a characteristic energy given for example by the energies between different bands and so this introduces a energy dependence into the polarization and if the polarization is energy dependent then also this partially screened interaction becomes energy dependent. So v is not energy dependent, that's a constant but this is now frequency dependent so w is also frequency dependent and so also our Haber-Dieu is frequency dependent. So the proper definition of the Haber-Dieu in a low energy effective model like this is a frequency dependent object which ranges from a very large value basically the bare interaction in the solid is very large for transition metals something like 20 electron volts and then it is screened by all these high energy bands down to a value of just a few electron volts in the static limit. So if we sketch this typical structure of the interaction it looks as follows this is our frequency axis we start from a bare interaction which is relatively large and then, so this is the real part imaginary part would look something like this and there's usually a big peak corresponding to the so-called Plasmon frequency and so above this Plasmon frequency screening is not effective anymore and the interaction is approaching the bare interaction value but below this Plasmon frequency the interaction is strongly reduced and so that's our partially screened interaction where we have not taken into account the low energy screening from our transitions within the low energy subspace if we would also include this that would be down here somewhere a second sort of reduction of the screening and that would then be the fully screened interaction W but as I explained in the DMFT type calculation where we explicitly treat this low energy screening we don't have to take this polarization into account in the calculation of U and we should really use this partially screened interaction which removes the polarization which does not include the polarization of the low energy space so this W includes so now one further comment is that these formulas which I had on the board are the relation between screened interaction, bare interaction and polarization and so forth these are exact as long as we calculate the polarization exactly but in practice we are not able to calculate these polarizations exactly so we use some approximations typically random phase approximation to calculate the pH the high energy polarization and this scheme is called constrained random phase approximation CRPA that's one of the standard methods nowadays to compute these interaction effects and so you may wonder why is it okay to calculate the polarization in the random phase approximation so one argument which is typically given is precisely this that the low energy subspace is usually strongly correlated these are the D bands near the Fermi energy whereas the high energy subspace is more weakly correlated maybe S and P bands and so since we removed the low energy polarization anyhow we only sort of treat weakly correlated bands in the random phase approximations or maybe it's okay so that's sort of a general argument it's not so clear if this is actually true but it's basically in simple model setups to check if this kind of down folding is accurate and at least in these simple model setups which we checked it's not very good so there are some question marks and the last point I want to say is the following usually we know that in a metal for example the Coulomb interaction is short range because of low energy screening so we go from a long range Coulomb interaction to a short range maybe you cover type interaction thanks to the low energy screening which is given by this PL but now in this constrained random phase approximation we remove the low energy metallic screening so in principle the interaction again becomes quite long-ranged and so it's static that in practice we then simply use a Hubbard model description even though the interaction is not short-ranged and that's a major I think flaw in most calculations up to now that one really then just uses these parameters only the on-site terms whereas long-range couplings are not treated even though they are not small there may be a factor of too long but they are not negligible compared to the to the on-site terms and so tomorrow tomorrow I'm going to discuss two things first of all I will discuss how we can solve Hubbard type models with a frequency dependent interaction and I'm going to describe a formalism which is also based on dynamical mean field theory which can treat long-range Coulomb interactions that's a so-called extended dynamical mean field formalism and this is really then fundamentally dependent on our ability to treat these frequency dependent couplings because they are then somehow self-consistently updated if we treat these long-range screening from the from the long-range Coulomb interaction but that's what I'm going to discuss tomorrow so thank you for your attention