 Miločno, da sem vprašim. Moj pa 있elして razmah solo, da osmah je bil, da sem njihajo vzivom, že teamento v tem dobrih z lapčного delov začne는je, kot nekaj tešen od ljubibera, vendiv kden je strm mjonek ljude ljubito, dve ljubiberi režim, z kaj bom izgledal vprašanleti reromajsacije, zpsegrandi reromajsacijo, in da, da je to je zelo skupil in zelo na FRG. Zato da smo izvokili... Ispešno? Ja. Tako, pa smo prišličniti v tečen, ki je na res tudi občin za dve piste. V prvisem piste so prišlično z vrteckim frekvencijnjem, zelo prišlično zvrteckimi definitivnih, of the one particle irreducible and the two particle irreducible vertexes, then I will give a diagrammatic understanding of the vertex structure that arises in frequency, and I will show how to decompose the vertex and how to use the information that comes from this, the composition to reconstruct the vertex in an efficient way for numerical calculation. The second part of the talk instead will be showing an application, I will introduce an application, namely the combination of dynamical mean field theory and functional randomization group that will be then discussed in the next talk, and for this application it will be essential to consider the frequency dependence of the vertex. So, let's start from something very basic. We know that if we consider the two particle greens function, it doesn't factorize as the product of the one particle greens function because we have interaction between the electrons, and these interactions are described by the one particle irreducible vertex, which is in some sense the equivalent of what the self-energy is at one particle level. What I want to stress now is that in functional randomization group, people have acquired over the years a big deal of experiences for understanding the momentum dependence of the vertex. So, we are getting familiar with this kind of plots in which we mainly consider the momentum dependence, and we can understand these plots, we can understand what are the leading instabilities if we look at the location of the structures in the momentum and we can compute the susceptibilities, but there is something which in the conventional functional randomization group has been always neglected, namely the frequency dependence of the vertex, which I want to address in this talk. So, my experience is always a bit confusing when we talk about the vertex denotation. So, I want to just fix a notation. We have an object which depends in principle on four arguments, but one can be fixed by frequency and momentum conservation, so we are left with three arguments, and we are free to choose these arguments in the way that is most convenient for our goals. So, I will choose always to show a vertex which depends on two fermionic four vectors, so which are the arguments of the incoming and outgoing lines, and the third argument will be rather a bosonic four vector which is a momentum transfer, is a combination of the arguments appearing at the outgoing and incoming arrows, and which combination depends on which channel we are considering. So, for example, if we consider the particle-particle channel, it will be appropriate to describe or to show the vertex in terms of the total incoming or outgoing frequency or momentum. Now, since I want to focus on the frequency dependence of the vertex, I will show results that are obtained for a model which only has frequency, which doesn't have any momentum dependence, namely the single impurity under some model in the wideband limit, and I will show results that are obtained in the means of functional randomization group. So, here I show a first plot of the vertex, and just for you to get familiar with this kind of objects, what we plot here is plotted at a fixed bosonic frequency, while the two fermionic frequencies here are left to change. What we can see is that this object from the color coding, we can see that this object is not flat, but it shows some diagonal structure which do not decay when the frequency is increased. So, they already suggest that there is a large frequency behavior that we need to take into account in our calculation. So, how to understand better the origin of this structure? Well, the first thing that we can do is to compose the vertex by using some diagrammatic classification, and it is well known that the vertex can be decomposed in the means of the parquet equation, and the f vertex is the one particle irreducible vertex, contains a collection of all the diagrams that do not fall apart by cutting one fermionic lines. It can be shown then that all these diagrams can be classified being either two particle irreducible or two particle reducible in only one channel. So, what doesn't mean two particle irreducible? It means that these diagrams do not fall apart by cutting two fermionic lines. In this context, let me stress that these are the diagrams. These diagrams cannot be computed by means of functionalization group at the one loop troncation level, so they are not part of our FRG calculation, and so this plot has been computed by means of exact diagonalization, and the lowest order of this diagram is the fourth order, so they are subleading at weak coupling, and also they show a decay in frequency, so they don't have this large frequency behavior. Next, we can focus on the two particle reducible, which classify as two particle reducible in the particle-particle channel, or in the particle-all or particle-hole cross channels. So, these diagrams can be instead computed in functionalization group by integrating each of the terms in the FRG differential equations in a separate way, and if we look at this object, they look like this if we have a color plot. No, I don't specify here if it is a particle-particle-particle-all or particle-all cross vertex, since the consideration that I'm going to make are rather general, they are not specific to a given channel, but apply to all of them, and in particular we want to understand what is the meaning or if there is some physical meaning into this structure that we see in this cross structure. So, the next thing that we have to do is to do this, we need to rely on a further assumption, namely, we will assume, as is usual for the Haber-like models, that the bear interaction is local in frequency and time, which will impose some constraints on the... which imposes some constraints of the diagrams. In fact, we can see that whenever we have diagrams in which two of the lines, two external lines are attached to the same bear vertex, the fermionic argument does not enter into this bubble or into these lines, which will only depend on the bosonic transfer in the given channel that we are considering. So, this bubble here, for example, would not depend on any of the fermionic arguments, k prime and k, but will depend only on the bosonic transfer in the particle-particle channel. This is a consideration, which is quite well known, that weak coupling has already been used several times, for example, for making a weak coupling the composition in the frequency or into the momenta or even for an homogeneous system. This is quite well known, but what we want to do is generalize the composition, if possible, to higher order. Well, we can use this consideration about the dependency of the diagrams to divide further the five functions into three classes of functions, kernel one, k2, k2 bar and rest, and the first class, k1, is characterized by the fact that all the external lines are attached to the same bear fermionic bear interaction vertex. The implication of this is that this diagram here will not depend on the fermionic argument, but will depend only on the bosonic argument. We have an object, which this k1 object will depend only on the bosonic argument, and what we can see by looking at the diagrammatic structure is that it includes exactly the same diagrams that we have included or that we have to include when we want to compute a physical susceptibility. If we look at our plots now, we see that since the plots are at fixed omega and the k1 does not depend on omega, this k1 appears here only into the background of this five function, and if we make a scan of the five function, we will see that the k1 object is rather sharp and center the zero frequency as it is consistent also with the susceptibility. Also, let me notice that if we look at the full vertex instead, we are plotting here again at one fixed bosonic transfer, which means that the other two bosonic transfer are left free as we change the frequency here and here, and this line here exactly corresponds to this structure since it corresponds to the line where omega1 and omega1 prime are equal and opposite, so they exactly correspond to a total incoming momentum of zero that we can see here into the plot of the k1. The next class of diagrams consists of diagrams of the class k2 in which two of the outgoing lines of the external lines are attached to their own vertex while the other two lines are attached to the same vertex. Also in this case, we will have a simplification of the frequency dependence of this object which will turn out to depend only on one fermionic and one bosonic argument. If we look at the color plot, we can see that we can obtain the structure of this k2 by making a scan at fixed omega frequency and keeping also the other frequency at fixed transfer frequency and keeping fixed also the other fermionic frequency and we will have a structure like this for our k2 as a function of the bosonic transfer frequency. Now, another thing that I want to notice about this is that this object also has a physical interpretation or can be connected to the fermion boson vertex as can be seen by considering the diagrams that contribute to the class k2. Finally, there is the rest function and for the rest function each of the external lines is attached to its own bare vertex and therefore the frequency dependence of this object does not simplify in any way but rather we have to consider a full frequency dependence on all the arguments. Let me notice that this is subleading at weak coupling since it is at least fourth order in the interaction and this is the only object for which we really need the full frequency dependence and numerically is the most expensive object to compute but it is also the most localized ones because it doesn't have any term which extends at high frequency so we can compute it on a relatively small box and the size of the box on which we need to compute it will depend on the strength of the interaction. It is also important to notice that the same consideration that apply for the frequency also apply for the momentum and the rest function is the one that is relevant or will contain the d-wave scattering since in the d-wave scattering for example we will have dependence on all the arguments and so it doesn't simplify as the previous cases. So now that we have understood the physical meaning and classification of the structures that appears in the vertex we want to use this information for our calculation to compute our phi functions in a finite box and this can be computed either by inverting the beta-salpeter equation if we have a method that gives us the one particle reducible vertex or they can be computed directly by FRG integrating each of the channel in a separate fashion and once we have computed these phi functions we can we can use them to extend the phi whenever we need to compute the vertex outside of this finite box so for example it is important that we consider also the high frequency asymptotics of the vertex when we are looking for the vertex on the right hand side here in the FRG flow equation and therefore we want to extend the size of the computed box so we first extract the function k1 the background and we extend it we do the same for the function k2 and k2 bar and we have also this this structure at high frequency and the last step is extracting the function r just to check that it is already decayed when we are at the border of the boxes and we don't need to because if it is not decayed we might need to increase the size of our calculation so with this I would like to conclude the first part of my talk and I have shown you that the interaction vertex shows a non-trivial frequency structure which we might need to include in our calculation and this vertex structure can be understood diagrammatically and by knowing by knowing these asymptotics of the vertex we can reduce our computational effort for function renormalization group in the second part of the talk instead I will discuss the dms2rg a method in which we want to combine dynamical mean field theory and function renormalization group and for this method the frequency dependence of the vertex is essential, that's my reason for showing now the method that will be then discussed further in the next talk so what is the goal? the goal is to combine the non-perturbative local physics that comes from the MFT with the non-local fluctuations that we can include by means of the function renormalization group so how does it work? one can show that in infinite dimensions or in the infinite dimensional limit the self-energy of the lattice the self-energy of an infinite dimensional lattice becomes exactly local and if we have a self-energy which is local this can allow us for an except mapping on an underson impurity model which we have to embed in a self-consistent frequency dependent buff so what we will have is a buff that will be mean field in space but due to its frequency dependence it will go beyond mean field in time so we will still have local fluctuations then the next step the infinite lattice on to the underson impurity model we can solve exactly the underson impurity model and by now there are several techniques that allow us to do that numerically like quantum Monte Carlo exact diagonalization and then we will take this underson impurity model this DMFT solution of the infinite dimensional lattice as a starting point for the FRG flow equation so conceptually we will start from we want to solve the lattice problem we will approximate this lattice problem with an infinite dimensional lattice which has the same density of states we will map the infinite dimensional lattice on a self-consistent impurity model and then after solving this impurity model we will flow to the original lattice so we can see this has approximated our lattice with an infinite dimensional one and then flowing away and then removing by means of the flow all the extra dimension that we have had for giving an approximate solution another way to see this is that instead of starting from an initial action that does not contain any interaction or any correlation we start from initial action which already contains a lot of correlation which already contains the local correlation and this is the initial action of the DMFT in the randomization group to be possible we require we need that the interaction part in the two actions in the initial and in the final action is the same while we can only act on the Gaussian propagator what we have is that the action of the DMFT has the vice field as a Gaussian propagator and this vice field is an important object that has all the information about the density of states of the original lattice so the vice field is the solution of the self-consistency equation which is the equation that tells us that the impurity problem has the same green function of the lattice once we have computed the green function of the lattice approximating the self-energy with the local self-energy of the impurity problem and we want to flow from this action in the initial action of the lattice so we have to recover in the end of the flow the Gaussian propagator of the lattice we are interested in and for doing this we just need to define a flow that allows us to go from this propagator to this propagator and the simplest choice is just an interpolation between the two propagators but let me also notice here that this is not at all the most general choice that can be done by including some regulating functions from which we only require some condition at the initial stage and at the final stage so for lambda initial and for lambda final the last thing that we need to change compared to the usual FRG are the initial conditions which already include the correlation of the MFT which means they already include the dynamical mean field theory energy and the dynamical mean field theory vertex which is strongly frequency dependent and for the effect of the frequency dependence of the for the effect of the inclusion of the MFT into FRG I will leave the word to the next speaker and I would like to conclude and thank you for your attention