 Vse pristajem, da je tudi vsega vsega vsega, na tem, da imaš vsega srečnega vsega, in vsega vsega imaš vsega, vsega vsega in vsega izpristam vsega. Tudi, da sem čel mnim tudi za del, ker, tudi in tudi, z drugimi težkami, pričal je vsega vsega, z tudi izgledaj. Spravno je pa, da zelo izgledajo v svoju presne proancaj. Svi gravitationalnobolj, stilv, pankovč, fyziks, navočne ne. Eko je pa kapel i danes spravno, kdo je zelo več仮jev in skupovosti za otrošljenje 표če. Relativno je tudi meni prosini na zelo posledaj, vso glasbi se delajo, u terkiji izgledaj, in očezan veliko je, če je zelo analožit, in v terkistih, če je vidjet, kaj je vsih doličenih vse. Zato zato, kaj je vsih doličenih vse. Svojim, da imam, da so vse, da bo vse doličenih vse in da bi bilo, da so vse. Zato, da da je vse, da je vse, da bi bilo, da so vse. Vse je, da je vse, da je vse, da je vse. Zato, da je vse, da je vse. z mnogih technikov in modeljih, mnogih šešnjih fizikalnih sejtupov. Tako je to, da smo se pošličili. Tako smo imeli alamirtonijan v tem formu, z kajim interakcijem z power law d plus sigma, smo imeli vse minfilj propagator in tukaj momentum term v inversi propagator je q2sigma, kaj sigma je bolj, or q2 square, like short range, if sigma is larger than 2. So, therefore, this is the main result for critical properties of classical longer range system. So, saks back in the early days of the normalization group introduced this relation between the anomalous dimension and sigma, showing that if the anomalous dimension is 2 minus sigma for sigma smaller than a certain value sigma star, and for sigma larger than a critical value, the anomalous dimension, all the critical properties are the same of the short range model. So, you see that there is no discontinuity, and the typical diagram that you have is that when sigma is very large, is larger than this value, which depends on the dimension, so you have a short range model. When sigma is smaller than 0, basically you have a beam field, and in the middle you have a longer range intermediate system, that sometimes is called a weakly longer range system, because sometimes a longer range system is referred to the situation which sigma is smaller than 0. So, there were controversies about this result. So, in 2002, the prediction of saks was somehow confirmed, but some other recent, more recent Monte Carlo results were questioning, and they were saying that eta is as this form, and the sigma star is equal to 2. One may say that, ok, it's a technical point, I don't care if it's 2 or 2 minus 1 fourth, for example, for the easing, but I would like to say that these two pictures are referring to two different physical scenarios. In this scenario, when sigma star is equal to 2, below sigma star in the propagator at the critical point you have only longer range. If sigma star is 2 minus eta short range, in the propagator you have also contribution from the short range, so the question is how to normalize the short range term in presence of a longer range interaction? And these two scenarios are alternative. So, I have to say that very recent simulation by a Japanese group are in favor of the SAK argument, so really you have a contribution of the short range at the critical point also in the longer range phase, in the weak longer range phase. So, I will come back in this point later and how one can assess this using renormalization group. I'm doing this because the same approach will be used also for quantum longer range systems, so you introduce Gindsburg-Landau free energy and you study how the different term in the Gindsburg-Landau are renormalized. Now, for the longer range system you have the usual mass of the quadratic term and you have the wave function renormalization for the longer range term. Now, you have to choose what you want to renormalize and your result may depend on the choice of what you are going to renormalize. So, the main point that you get from this analysis is that it emerge an effective dimension. So, when you write the renormalization group for example in the functional formulation you realize that they are exactly the same of the renormalization group equation of a short range model with an effective dimension. So, one can introduce and see what is this effective dimension. So, the first approximation is let's neglect anomalous dimension. Then you find that there is an effective short range dimension and this result is exact for n going to infinity. This was shown, this is in agreement with exact results in the spherical model back in the 70. But then you start to renormalize the wave function you put the z, the mass and you find that the effective dimension ask the form. So, eta short range depends on the dimension you have to solve the same consistency in this equation and you get the effective dimension. But this is somehow cheating because you are renormalizing only the long range term. So, what you have to do to improve the approximation is to let renormalize both the long range and the short range masses and let the system decide. So, what you find is this scenario. First of all, there is no effective exact effective dimension. You can never rewrite the renormalization group equation as the equation of a short range model with a proper effective dimension. And you see that the eta as a function of sigma up to sigma star given by suck be forkated in two parts. Here I'm showing the eigenvalues of the renormalization group stability matrix. So, above sigma star you have only the short range, the blue. At sigma star they biforkate, the red is the long range and the long range merge with the mean field which is the dash and exactly at sigma star equal to d over 2. And below you are in mean field. So, this is quantifying the error that you commit by using this effective dimension and looking at eta, the critical response eta, you realize that the error for the e-sync into dimension is order of 1%. So, there is no exact effective dimension but it works very well. Ok, so you can also compute the correlation and exponent nu. Here I'm plotting as a function of n from the e-sync to the spherical model in two-dimension and three-dimension. Here I'm comparing with Monte Carlo results but Monte Carlo results are not enough precise to establish if the effective dimension found is exact or not. There is a lot of room to improve such a Monte Carlo result. However, the main point that I want to make here is that the sac result is confirmed and there are prediction, concrete testable with Monte Carlo result of the critical exponents. So, a case in which we were able to get really a confirmation of the sac result is the percolation in one dimension in which you have a bond which is turned on according to probability which depends on the power law. So, here we computed the critical value for which you have the percolative transition and here I'm plotting eta as a function of sigma. You see that this is exactly 2 minus sigma and here we are plotting 2 minus sigma minus eta as a function of sigma. So, if sac is correct, this value should be zero. We are confirming this result in summation 10 to minus 3. It's not exact. We don't have an analytic argument but it seems extremely well confirmed at least within our numerical results in this long range model. It's important to notice that here we are doing a model in one dimension but when you have a long range dimension, when you vary sigma, you are really varying the dimension. So, you are passing from lower to upper critical dimension. So, I think this is a good confirmation of the result with a precision, which is pretty high. Let me also go to compute the eta. So, the eta. So, we are using epsilon as a function putting the effective dimension that I defined before. So, you see that the agreement is pretty good close to the transition to the mean field but agreement is not perfect. Again, effective dimension works relatively well, but it's not exact. And here there is a very close transition at this point in agreement with all the result of Cardi and other in the 80s. Ok, so, I talked about the classical long range system. Let's move toward the quantum system and let's try to consider a situation in which the anisotropy is in which the long range is anisotropic. The quantum model and the quantum system are an extreme case of it because this long range is one direction, for example, and the short range in the other. So, when you have a generically anisotropy at the critical point you don't see because the system, due to the universality, want to wash out. So, the point is that there are genuinely anisotropic fixed point and how we can see. So, therefore, at this point we do exactly the same treatment as before with the same formalism. So, this is the Hamiltonian that we consider. Now there is a power load k sigma in d1 dimension and a power load k tau in d2 dimension. So, I can write them in field propagator. You see that you have a contribution q2 sigma, q2 tau in the two different subspace and you can write an effective field theory. Again, you put zeta perp in the sigma direction, zeta parallel in the tau direction and you can add also the q2 term for the short range exactly as it had before. If you perform this analysis you immediately recognize that there is an anisotropic index theta which enters in all the properties at the critical point, the asymptotic propagator, the correlation length and if I consider in field result I get this. So, now I'm ready to classify the region of this anisotropic system and I realize that there are three situations. First of all you have a case which is called region 3 in which the anisotropy is not large enough to be seen at the critical point. So, for all critical properties region 3 is exactly short range. Doesn't see the long range doesn't see the anisotropy. Region 1 is a region in which you want to be anisotropic but in both directions. So, this is again not the genuinely anisotropic point. The genuinely anisotropic point is region number 2 because there is a true fixed point in which the system wants to scale according the short range behavior in one direction and according the other critical exponents in the other direction. I remind you that if you take a generic model and you put anisotropy which is not large enough the correlation function does not depend if not for prefactor on the critical on the power law. So, this is really the region that corresponded to genuinely anisotropic points. So, here I'm plotting region 1, 2 and 3. So, region 3 is here and they mean field predict region 2, region 2 and region 3. So, but when you start to renormalize the mass so you get that the boundary is moved and is different for the easing or for O n model for n larger than 1. So, using the same formalism you can compute also the anomalous dimension and you can compute anomalous dimension in region 2 for example if I consider d1 equal t2 equal to 1 and 1 of the 2 is infinite. So, you see that we are going we are approaching the quantum the quantum models because basically this is the formalism will be pretty much the same. There is one difference the quantum model have a true dynamical critical exponent while the dynamical critical exponent in the classical models depends on the dynamic that I put. So, now the goal is to use the formalism to compute the dynamical critical exponent. Ok, so the model that we considered are the easing long range also look at Ayagot so yesterday discussed about properties of this and the generalization O n with n larger than the 1 so 2 is related to the quantum phase model and so on again I take this interaction and to make connection with the previous part of the torque it corresponded to have a power law sigma in the direction and an exponent infinite in one additional direction. So, now the formalism proceed pretty much the same that you introduce the correct field theory and you introduce the imaginary time now you can define the critical exponent you can do the normalization of the pre-factor of the of the ginsburg-landoff energy you see how are scaling with momentum and you can define the critical exponent z and the critical exponent eta. Again you will find by construction that the suck result is implemented as I was showing before and this is the result that I would like to discuss with you so this is basically the easing I consider d equal 1 so I plot here c sorry this is d versus sigma so I consider n equal 1 and you see that I find 3 region 1 you have short range behavior even do you have a longer range is not 2 due to the correction of suck to the eta short range which depends on the correction due to the short range term in the transverse direction this is the mean field region and this is the true longer range criticality and with critical exponent that I can compute with the same formalism as I discussed before the situation is a bit different for continuous symmetry but again this short range behavior a region with a single face the longer range criticality and you can determine within our approximation the border of this true longer range criticality and the mean field region ok so here I am trying to compare the results that you obtained for the critical exponent with Monte Carlo results so the red is the easy model the blue is the green is n equal 3 the Isenberg model in dimension 1 I am plotting Monte Carlo result for the easy and xy you see that for the easy you get a good agreement with our prediction while not for the for the n equal 2 so most probably these discrepancies are due to the fact that you have a BKT transition and it is very difficult for numerics if you don't do a very careful financial analysis to get the BKT behavior this is even more evident in this figure in which I plot a combination of z the dynamical critical exponent and the exponent nu so this is again easy and you see that there is a good agreement between Monte Carlo and analytical results obtained by renormalization group and these are the prediction of our approach and these are the Monte Carlo results so in our formalism if you want to have a BKT Berezinski-Costa-Listowless you need that this number is going to zero while this apparently is not the case for Monte Carlo I may suspect that by doing a very not easy financial scaling analysis a very long simulation this point should go here twice would mean that there is no BKT transition and we expect in dimension equal to 1 the nice point of the formalism is that you can treat unequal foot all the n for example also the Heisenberg and all the dimensions but to my knowledge no body so far simulated by quantum Monte Carlo two dimensional quantum system with long range in order to get the dynamical critical exponent so I try to summarize what I said so suppose that you have a quantum model in dimension d with interaction sigma it is clear that if you do a trotter the composition this is an anisotropic classic system and this is an exat mapping but once that introduce this effective concept of dimension you can map on anisotropic longer range system which is leaving in d plus z z is equal to 1 if you are in short range otherwise z is the number that you get from your approach now this isotropic longer range system in dimension d plus z is related to a classical short range system with the relation which is due basically to suck so therefore I take a crucial quantity from Monte Carlo in numerical simulation to determine z and to see if truly there is agreement between the numerical results and the approach that you can get by the normalization group ok, so let me conclude by saying that this normalization group in functional form unify the treatment of the critical properties of a longer range classical and quantum so many things remain to be done would be nice to compare with numerical results also in dimension larger than 1 but also in dimension 1 or 2 classical it is extremely challenging and most probably a difficult task to study Bereziski Kostalislav's transition would be nice also to extend the functional normalization group to an equilibrium properties in presence of a longer range and would of course nice also to have proposal to have to engineer the longer range that you want so you have a certain particular longer range that you want to simulate and we have a work with Joao Pinto Barras Marcello Dalmonte in Trieste to implement with cold atoms and gauge fields ok, so to conclude it's a pleasure to thank also my collaborator Nicolo Defeno was doing a PhD in CISA and moved in Heidelberg Giacomo Isier, Alessandro Codello and Stefan Ruffo are in CISA Joao as I was telling at PCDR and Marcello is at ICTP so thanks for the attention