 First of all, I want to thank the organizer for giving me the opportunity to present here my research. I will talk to you about anisotropic long-range interactions in spin systems. But first of all, I will give you a brief overview of what I mean for long-range interactions, and especially in spin systems. Then I will show you what are the traditional results. I will outline a controversy that emerged in the literature in recent years. I will show you some results on standard long-range interaction in spin systems, and then I will finally deal with the anisotropic case, showing you some scaling analysis, how the system can be divided in different regimes, and I don't want you to spoil the results. So, long-range interactions, as I were defined in the first lecture by Ibarre, are power-load decaying interactions with the distance, and we parameterized this power-load decaying with an exponent d plus sigma. And in my talk, maybe differently from the previous talks, I will focus on the case of positive sigma. So, when sigma is greater than zero, we have many interesting results. This kind of interaction are found in the correlation for DNA sequence, in the interaction between polymers. They have been studied as candidates for neural networks, and they can be responsible for some exotic behavior of materials. We decided to investigate these interactions in spin systems, because the spin systems are the testbed of statistical mechanics, and we have various Monte Carlo and perturbative results available. And our focus will be to determine the phase diagram, the critical exponent of spin systems with long-range interaction belonging to different symmetry groups. Then, if possible, I will also deal with, give some youth, some small information about Tiger order critical points. Okay, spin system with long-range interaction can be a lattice of spins. INJ are the labels of the lattice sites. The spin are bold because they are vectors of general n components. And the interaction is power-load decaying, as I said, with d plus sigma, and d is the dimension of the system. So if we are not, for example, cubic lattice, this is the dimension of the cubic lattice, and sigma is some real positive number, and j is always positive, so the interaction will be always ferromagnetic. We can calculate them in field propagator for this theory, and it is simply the inverse Fourier transform of the interaction matrix. The interaction matrix is this function here. We have two different behaviors. So we know that the universality class and so the critical exponent and all the behavior of the system at the critical point where the phase transition occurs can be derived only by the low momentum excitation. The low momentum excitation are obtained in the q to 0 limit of the propagator. In the case of long range interaction, we have two completely different behaviors for the propagator. When sigma is smaller than 2, the leading term in the small momentum propagator is going as q to the sigma, so with a non-analytic power as long as sigma is a real number. When sigma is greater than 2, the propagator is going as q to the square in the low momentum limit, and q to the square is the same behavior that the propagator will show in case of only short range interactions. So we are ready to outline some traditional results. So these results were derived by Fischer and other authors in the 70s, and they can divide the spin system with positive sigma in three regimes. Our first regime, when sigma is between 0 and the alpha, is a mean field region in the sense that the critical exponent and the universality class of the system can be retrieved using all the mean field approximation, and the results are given. The anomalous dimension is equal to 2 minus sigma, and the correlation length exponent is 1 over sigma. With these two exponents, you can reconstruct all the other exponents so you know all the information about the scaling laws at the critical point in this region. When sigma is between the alpha and 2, we have some peculiar long range exponent. We are in a correlated region as it happens for standard, short range, easing, or ON models, and we should determine our critical exponents numerically. Finally, they identified a region for sigma greater than 2 that if you understood this argument, it's obvious that for sigma greater than 2, we will recover the same exponents as in the short range case, because the low momentum propagator has exactly the same behavior as in the short range interacting case. So this was the picture retrieved by Fischer and others in the 70s, and they also calculated the critical exponent in this correlated region. They found a peculiar behavior for long range interactions that the anomalous dimension, even in the correlated region 2, is stuck to the mean field result. They cannot calculate any correction, even toward the epsilon to the third. They do also 1 over an expansion, and the result was the same, so they didn't find any correction to the mean field anomalous dimension, even in the correlated region, and they concluded that this result should be exact at all orders in the extra expansion, so it should be exact. This was somehow strange, because this would imply a discontinuity of the anomalous dimension as a function of sigma, because if this behavior should be valid till sigma equals 2, it will give us an anomalous dimension equal to 0, exactly as sigma equals 2. But for sigma greater than 2, we should have that the result should be the same as for a short range model that has a non-vanishing anomalous dimension. So this result, if combined with this one, would imply a discontinuity in the anomalous dimension as sigma equals 2. This traditional picture was then overcome by SAC that some years later did an improved renormalization group analysis, and they found that actually this behavior, the short range behavior of the system is recovered not at sigma equal to 2, but at some sigma star that should be determined as sigma star equal to 2 minus the dimension of the short range. So the discontinuity obviously disappeared. And so we get this diagram for the system regime. When sigma is greater than the half, we are in the mean field region. When sigma is below the half and less than this red boundary, that's sigma star, and it is, as you see, is depending on the dimension, because obviously the anomalous dimension of the short range system depends on the dimension. Here the system has peculiar long range exponents to be determined, and here it recovers short range behavior. This picture was later confirmed in 2002 by Monte Carlo Resalt by Luther Bloyton. But in recent years, in 2013, precisely, there were new Monte Carlo Resalt that found us like a different picture. Indeed, while the result of Luther Bloyton confirmed the result of Fisher for the anomalous dimension, so the anomalous dimension is equal to 2 minus sigma till some sigma star, which is exactly 2 minus eta short range, and then is straight equal to the short range value. The Monte Carlo Resalt found by Pico and Doders were claiming an anomalous dimension which has a known mean field correction, and so it is continuously moving till sigma star is now equal to 2. So we wanted to clarify this issue, and we decided to reanalyze the long range spin system using modern renormalization group technique. In particular, we used functional renormalization group technique. I don't want to go into the details of the calculation I did. If somebody is interested, I can discuss it later. The only thing I want to say is that we used... that the meter starts with Ginsburg-Landau. Here I report the standard Ginsburg-Landau for a spin system in the dimension with only short range interaction, which has a standard kinetic term that depends only on the... which is the laplacian of the field. In this case, it's the magnetization. And Ginsburg-Landau for a long range system that has this anomalous non-analytic kinetic term. Okay, it is important to note that the long range Ginsburg-Landau will contain, as I showed you before, also short range terms that are subleading for all sigma less than 2, but that becomes leading for sigma greater than 2. Okay, the first analysis we did, it was... we discard any renormalization for the kinetic term. And when we discard renormalization for the kinetic term, we find that the results obtained for the short range... for the long range system can be related to the... so in particular the critical exponent gamma, which is the divergence of the susceptibility, the critical exponent gamma of the long range system is the same as the one of a short range system at an effective fractional dimension. This effective fractional dimension is given by this result. This was obtained discarding an anomalous dimension effect, both in the short range and in the long range part. When we turn up to anomalous dimension corrections, we turn on this renormalization of the kinetic terms, and we get that the long range non-analytic term does not renormalize. So in agreement with the results of Fisher, we don't find any anomalous dimension correction to the long range power, and so we again find another relation for the critical exponent gamma, which now includes effects of the anomalous dimension. Okay, so we have a tower of approximation. The first one, as I said, gives you this result, and the second approximation in which we have anomalous dimension corrections gives this result. It is important to note that in an n equal to infinity limit, they both give an exact mapping. So in the n equal to infinity limit, n is the number of components of the spin, so n equal to infinity is a spherical model. We obtain the critical exponent of the long range system can be obtained exactly from the critical exponent of the short range one. Also, the first relation gives the correct ranges for sigma, but it gives us the result found by Fisher, while the second relation that is more complex and includes anomalous dimension corrections gives us the results found by Sack. Finally, we added short range corrections, so this term here. We added short range corrections to our theory, and we see that the picture is slightly more complicated. So this, the phase diagram of a theory with long range interaction has two fixed points. One where long range interaction are vanishing is reported in blue, and the other where long range interaction are non-vanishing is reported in red. For sigma greater than sigma star, only the short range interaction fixed point exists. For sigma star from the short range fixed point, it emerged a new fixed point, a long range one, where both zeta and zeta two are non-zero. This long range fixed point that are the red lines is leading with respect to the short range one. So in this region, as is seen by this plot, these are the stability exponents. In this region, the long range fixed point is more stable than the short range one, and so gives us the actual universality class of our system. However, at the point that is sigma star, that it is found to be exactly two minus zeta short range as it was found by SAC. The system present logarithmic corrections, which are probably responsible for the difficulties to pursue Monte Carlo simulations. Okay, we were also, sorry, we were also able to make some estimation of the error that one does when used this effective dimension correction. This estimation was done computing the critical exponent before for a long range system and then for a short range system at the effective dimension. The error, as you see, is very small. So at this approximation level, this effective dimension appears not to be exact, but it appears to be a very good estimation for the critical exponents as shown in this plot. And so we use the effective dimension relation to compute the critical exponent of spin systems with a generic number of components in dimension two and dimension three. Now, let me turn to the case of anisotropic interactions. Anisotropic interactions is when we can cut the system in two subspaces. One subspace of dimension D1, the spin that are sitting in this subspace, they interact with power law D1 plus sigma, where D1 is the dimension on the first subspace. In the second subspace, the spin interact with an interaction D2 plus tau. Tau and sigma are different, our two real numbers always bigger than zero. And D1 or D2 are the dimension of the two subspaces. So, this interaction has been meant to mimic an interaction that's anisotropic depending on the direction. It's the extreme case in which the spins interact only if they sit in the same subspace. And if they are not sitting on the same subspace, as is mean by the delta, they do not interact. Okay, we can compute the mean field propagator of the system and it appears to be anisotropic as expected with a momentum q parallel to the sigma and a momentum q perpendicular to the tau. And we can still derive against Burland out a theory called an effective field theory which represents the low energy behavior of the system I showed you. Okay, this system can seem a little exotic, but actually it has a simple realization. If we think to an easing spin chain or a quantum spin chain in general with long range interaction, we can map a quantum spin chain with long range interaction using classical correspondence, quantum to classical correspondence into a classical spin system with the anisotropic interaction I showed. So, if I take a quantum spin chain in dimension D with the power low decay sigma, it can be mapped on anisotropic classical spin system with dimension D1 equal to D. So, this D1 will be equal to D, sorry, the dimension of the classical, the dimension of the first subspace will be equal to the dimension of the quantum system. The dimension D2 of the second subspace will be equal to the dynamical critical exponent of the spin of the quantum spin system. Sigma will be the power low decay in the subspace 1 will be equal to the power low decay in the quantum system, while tau, the power low decay in the second subspace will be equal to 2. This means that the second subspace will always interact with only short range interactions. Okay, the quantum long range using model is obtained when you put Z equal to 1. So, this just to show you that this anisotropic spin system can be realized. In presence of anisotropy, we have four different critical exponents that are the anomalous dimension of the propagator in the two directions and the correlation length exponent in the two directions that diverge with two different exponents, one and two. Since we have only a critical point, even if the system is anisotropic, we have a relation between these four exponents and the total number of independent exponents is reduced to three. We can specify only one anomalous dimension, the corresponding correlation length exponent and we get the other two using this anisotropy index that should be computed. At mean field level, the result is that the anomalous dimension are vanishing in both the subspaces and the correlation length exponents are equal to one over sigma and one over tau. So, when we are at mean field level, the two subsystems don't see each other and the result is like if they were isolated. So, these systems can be divided in four different regions and this can be understood also by the results that I told you before. For sigma and tau less than two threshold values, sigma star and tau star, we are in the anisotropic long grain system here and both the two power load do not get any renormalization, even known at mean field level. Here, the blue shadow represents the mean field regions in two dimension and in the two plus one dimension and one plus one dimensions. Otherwise, when one of the two exponents, sigma or tau, are greater than certain threshold tau star or sigma star, we are in this that we call mixed regions. So, we are in a region where one subspace has a real power load decay while the other subspace is like a short range interacting one. These two regions, so two A and two B are the interesting one when we are treating long range quantum spin system. Then we have for sigma and tau greater than the thresholds sigma star and tau star, we recover an isotropic short range system because the long range interaction are irrelevant in both subspaces. Okay, we can pursue the same analysis we did before for a standard long grain system and we found that in region one, so here where the long range interaction are relevant in both subspaces, this anisotropic system can be mapped so again it has the same critical exponent gamma of a long range model in a dimension d1 plus theta d2 where theta is the anisotropy index and d1 and d2 are the dimensions of the two subspaces. This relation is again exact in the spherical model limit and gives us the following result for the correlation length exponents. We can validate our results in the spherical model limit comparing them with the one of the anisotropic next nearest neighbor model that's a model of spin, that's a spin system where we have next nearest neighbor interaction that are leading over the nearest neighbor interactions. These results are very well known in literature and if we do that, if we impose that that tau is equal to 2L as it should be for the case of ani model we get these two relations that are the one that one found in literature for the one over an expansion of the ani model. So we validate somehow our calculations we show that this anisotropic long range system are something meaningful because they can reproduce the results of a model that's a well-studied one and we can go on computing the anomalous dimension of the system. As I already told you in region one we have no anomalous dimension for the system both in both subspaces while in region two A or B where one of the subspace is only short range interacting we develop an anomalous dimension and the interesting thing is that the anomalous dimension depends on the value of the power law exponent in the other subspace and so when we are in region two A or B the anomalous dimension of the short range interacting sector depends on the power law of the long range interacting sector and we compute the dependence of these anomalous dimensions. We can again do a brief summary of what we have found in the region one we have an effective dimension that is exactly the N equal to infinite limit and gives correct ranges for the regions in region two we have again an effective dimension but this time it maps on a short range system and we can use it even if it has anomalous dimension corrections in region three and so these are the regions outside the mean field approximation region one the boundary are now corrected by the presence of the anomalous dimensions that is reported here so you see that the boundary are different for different spin components so the red is the N equal one so it's an easy model that has an unvanishing anomalous dimension also in D equal to so the boundary gets the curve are N equal to N three blue and green and since in two dimension this is a D one equal to one and D two equal to two they have a vanishing anomalous dimension also the boundary go back to the mean field one in this point and in this region we recover the exponent of a short range system okay this is the same plot but when we have D one equal to D two equal to one and so obviously it's not symmetric also in this case we can use the effective dimension correlation to compute the correlation length exponent we know that this will be not exact but there will be a very good approximation for the exact ones so in this way we were able to characterize all the phase diagram of anisotropic long grain systems in effective dimension relations that can be derived in the standard case with scaling arguments but which are much harder to derive in this anisotropic case and which we were able to use thanks to this functional aromization group technique thank you all for your attention