 So, first, I would like to thank the organizer for giving me the opportunity to present my current PhD studies. And I am going to show the application to the two dimensional upper model especially at half filling and showing the application for the in the square g already introduced in previous talk, which combined the dynamic on biffel theory and the functional normalization group. Selo na poživanie prv nekaj posebno izgledaj do dgegova vesela s miru, da se k autantom, kako sem vidi, kljeni mft in in drugi metod. Izgledaj je proj Khan mft plus RPA tekniki a odlijte in misljajte nekaj z dva vrstvenih konneki v tlebe, tko ml box v msqr-rg. Pozoh vzajte toga numerikera v kri. Zjete vzpe rekapšnu v spati, njo možete catal vzpe krioles, donih njegod v zelo, spasih se participationi, zelo se pobij لاzvače mjelikova, in pristroj nekaj tezvomentoj spasih od nekovače. Proste, da sem izgleda armar model, ker sem vse vzivila, da so izgleda s Samijeltonijanem, kar smo zignitim tukaj ovoga terma in tukaj vse za vsezast, imamo tako repalcične vsezast, tempracije in zelo površenje. Zelo se vzivimo, da bo izgledo vsezast za vsezast, prituk je velik z Governmenta, smo prišli s Vsrednjem vseh, ali smo prišli tam, kde se se vistsimo, kajate, je vseh vseh, s njemne, ta je z naprej mention z tegli. Kajče možemo najvega vse, či izvizскойem vsrednje vseh vseh. In po vseh, smo sej tako kratkno vsez, kot gač se tudiko zanostili, ne bo samo tu, sklep kdo vzelo v inrčno tako vsega včočeneh stajov. Vočak je skupajzat in kar Frankov nr. 0 se zили njega vzelo na vzelo v majo plošenje. Avstva tato, se pokratiless z vzelo vstvo kdo vzelo vzelo Kin Atlantic can include all local fluctuations, and it becomes exact in the limit of infinite dimension. If you don't allow to have symmetry breaking, this is the phase diagram temperature coupling, so we observe metallic phase in the weak coupling, and then first of the phase transition to the insulating phase. This is driven by the coupling. As you can see from the density of state, we have a very well-formed quasi-particle peak in the weak coupling. Eni čebeče vseči vseči pristek, zato včasno se objevamo njih spetralj naprošenje. To je zelo, da v stajnjenju palji nečo energijenom spetralj naprošenjem bo veči dve in nečo, da bi tega počeskva spetralj naprošenje. V nekaj posebno v FET plus RPA nekaj ne zelo tudi začelj njej, nekaj ne bober, od nekaj FET pa skupaj smo počkati, zelo nekaj š廠čenertno zdočenje z radu tudi na DMFT ljenje. To je, da je DMFT znerenje na spaski, zelo značnje, da so paFI się začeli, ta nešlošnja nije obidina, tako mi je bošnja začeli začeli z teropovalo na našlošnji tudi. To obidina jezim, da je movin Digm, či je to, ki nepoželiti, napravo nje 19 radiomagneti lev, As you can see, we have an exponential behavior of the energy temperature in the weak coupling, then a maximum value and then a power low decay with the coupling. So this is the phase diagram seen by other techniques, for example in the quantum Monte Carlo case or some diagrammatic or cluster extension of the DMFT. What we can say is that for any value of the coupling there is always a lower enough temperature where the correlation length starts growing exponentially with the temperature. So we have situations where we have very big, but finite, antiferromagnetically ordered domain. And many of these methods also show the disappearance of the more transition due to suppressed by antiferromagnetic fluctuations. But many of these methods are limited at particular symmetry. So it works well only at all-filling or has some single channel approximation while we can easily extend our DMFT to the doped case. But now we are concentrated to the all-filling phase diagram just to compare to have a benchmark process in this stage. So I just want to briefly record the DMFT implementation. We have to solve the flow for the self-energy and for the two particle vertex. The exact equation of the two particle vertex is summarized here. We have also the contribution from the three particle vertex, but we neglect this effect, so we close the equation in this way. And the important difference with respect to the standard FRG is that our initial conditions are the solution of the DMFT, of the dynamic of the DMFT. So we start from the DMFT self-energy and DMFT vertex. So now the question is what is the best way to parameterize the vertex in frequency space in the momentum space. We can study for example the frequency dependence arising at the DMFT level and maybe see the noloka results from the DMFT plus RPA. So first I concentrated to the DMFT vertex, function only on frequencies. And as you can see in the left, there is the weak coupling numerical results and on the right the strong coupling results. In the weak coupling, this pronounced structure that arises can be very well parameterized by bosonic frequencies, while when we cross the mode transition, the frequency structure changes dramatically and the most important contribution is very localized in frequency space. So we can no more approximate the vertex into a function of one bosonic frequencies. This is due to the fact that in the strong coupling case the local susceptibility is proportional to the inverse temperature, so independent on the coupling, and the fact that the Green's function becomes insulating, which means that goes to zero with the frequency. So this leads to a strong scaling of the DMFT vertex, strong dependence on the coupling. So now I want to also introduce what is the DMFT plus RPA technique. First recalling what is the standard RPA technique. So we can sum an infinite series of diagram in the standard RPA starting from the bare interaction u, summing this geometric series. We have this formula for the vertex, so we need the calculation of the particle or bubble, and as you can see, the vertex now depends on one bosonic frequency and one momenta. This particular behavior of the particle or bubble as a function of temperature, which is particular at all filling, gives the, in this case, we find that there is a finite temperature where this denominator becomes zero. So we have a finite temperature in this case. If we would like to extend these methods, adding some results from the DMFT, we can, for example, sum these series of diagrams. If we choose the DMFT vertex here, we are making an error, which is the double counting error, because we are summing also diagrams which are already contained at the DMFT level, so we have to first remove this diagram and then add them again at non-local level. So we have to use the two-particle irreducible vertex, two-particle irreducible in the given channel we are building in this way. The other difference with respect to the standard RPA is that we need the calculation of the bubble with the full DMFT self-energy. As you can see, the bubble is calculated with the DMFT self-energy. Its behavior as a functional temperature may change dramatically. As you can see, this is a plot of the nail temperature. As you can see, we have an exponential low in the weak coupling, a maximum value, and then a power low decay. Now, I just want to show numerical results for the vertex. In this case, as you can see, the vertex on the left gate that we can calculate in this case depends on three frequencies and one momenta, so may have very strong frequency structures. And I want to show also the difference between these vertex and the vertex from the dynamic of mean field theory, that I call phi, and on the left you can see the weak coupling case, and on the right the strong coupling case. So in the weak coupling case, the highest contribution is on the background, this red that you can see, and that the most important contribution can be parameterized by one exchange bosonic frequency, what is also called bosom propagator, and actually in the strong coupling, above nail temperature, we cannot make this approximation because the most important structure is very localized in the frequency space. So if we want to access the strong coupling regime, we need the full frequency parameterization and the vertex. At this point I also want to speculate a little bit saying that to access also the strong coupling regime or this particular fact that we cannot approximate the phi function into a function of one bosonic frequency implies that the bosonization method that has been used in functional Rg in this community may fail in the strong coupling regime in the fact that we need to keep the full frequency structures. So I just want to also show different scale as a function of the couplings. We have very different numbers here. So now I want to show the deep connection with the dm squared Rg. To show it we have to introduce the cutoff scale in the bubble, so we have a cutoff dependent vertex function. So if we take the first derivative in respect to lambda, we can say that this gamma lambda solves the dm squared Rg flow equation in this case of only one channel. So now the question arises can we observe this behavior of the NL temperature in the one channel dm squared Rg? To answer to this question we first try motivated by the weak coupling in the weak coupling regime by the fact that this function phi is approximated by a function of one bosonic frequencies. So we try this composition and we observe that we have very strong antiferomagnetic fluctuations in the weak coupling and when we test the one channel case we see that in the intermediate to strong coupling case we do not reproduce the NL temperature observed in the RPA. So we have to do something more to reproduce strong coupling results. We try to modify our approximation for the vertex introducing functions which depends on the full free frequencies and projecting the momentum structure with form factor. For example we can describe the magnetic order with this function a charge order and also superconductivity in S wave and D wave component. So this is important to see the effect of channel competition in this case and we are taking care of the frequency structure that arises at the MFT at non-local level. So if we restrict now to the single channel case so keeping only the particle across the channel we observe that we can reproduce the NL temperature observed at RPA level and we stress now that if we consider a point here in the phase diagram at the beginning of the flow we have a zero of course magnetic channel but then also at the beginning we have very high numbers of the magnetic channel then to the power of five but we have to let the flow go and observe a convergent flow. This is due to the fact that the maximum value of our magnetic channel is very high but the susceptibility is of the order of ten. This is due to the fact that if you consider constant temperature the vertex depends strongly on the coupling while the susceptibility at constant temperature goes to zero. One has to take care about this information then also in contention numerical results in the full channel case in the weak coupling now and showing the set energy as you can see we have strong non-local deviation for the lowest Masubara frequencies and in particular we have a deviation for the anti-nodal point and if we analytically continue it with the Pade approximate we can see that there is a reduction of the quasi-particle peak for the anti-nodal point. Then I just show numerical results for the channels in particular we have a magnetic channel as expected because we have strong anti-framagradic fluctuations and we didn't see yet a strong D-wave component mainly because we are at a very high temperature so we need to go in the dopet case to see if we can observe the effect of the D-wave component. So now let me conclude I show the DMS2RG as a consistent combination of the MFT and FRG then show the connection between the RPI method and the DMS2RG and then use these results to base our vertex parameterization in frequency space and then show numerical results. Now we are working for the full phase diagram of the upper model and of the dopet case in particular to observe D-wave superconductivity. So I would like to be for attention.