 I'm fine. I'm just not seeing any more the room so I don't know if the people are in the room or not no I think yes Thank you very much. Yeah. Yes. Yes. Yes. We start again. I Couldn't see the room anymore. So I didn't know if the people were seated or not and I didn't know what was Very good So we can start back and Okay, so up to now I've been talking to you about how to relate how to relate the How to relate the The existence of the transition To the existence so the existence of the ferromagnetic transition of an icing model into the existence of a condensation transition in a Bose gas with Non-standard the non-analytic kinetic energy Now interestingly we cannot only relate the existence of the soul Basically, we say that that we can do the mapping between the spin and the bosons so we can Diagonalize the boson then we can take the continuum limit and from the continuum limit we get that they Ah That the transition always exists as long as this condition is fulfilled now We can do Something more we can also try to study the properties of the transition of the quantum critical point in terms of the bosons so and then the features of the quantum critical point that I'm interested in Are critical exponents Now I Don't know how many of you know about critical exponents in quantum phase transition I for the moment. I'm trying to give you some definitions and I ask you to make to stop me and make any questions If you have some doubt over these definitions the definitions are quite simple and kind of intuitive but But if you don't have much experience I some verification may be needed so we say that The the ising model can be related to this both a gas with this dispersion relation Now in order for the phase transition to a poor we need the the gas to be So we need the the chemical potential of the gas to attain a certain value, you know because in here you see this was the form of the Which was the form of the dispersion relation of the lattice and Quantum phase transition of course when the mode becomes soft it means when this omega Becomes basically zero Okay But in order for this omega to become zero It means that H has to become equal to vk equal to zero No, you see so Basically we at a tree at a quantum critical point this omega has to vanish and This vanishing omega it's related to H going close to the value of V zero of the V of zero momentum and The V of zero momentum well, you can compute it from here the V of zero momentum you can compute it from here and it's nothing but the Remands that a function or if you include the customization is not is just one so we can say That as H approaches the critical value, which is one so This omega becomes zero and now the critical exponents are defined by the scaling of the omega close to zero so This is the definition of the gap exponent. The gap exponent is called a zeb nu and And it is defined as the way in which the Zero momentum gap the zero momentum frequency vanishes as a function of age approaching the critical value and It's obvious from this computation that in our approximation this value is one half Because v of k equal to zero is one And this goes to an omega of k equal to zero goes to zero like the square root of h minus one And so this zeb nu is one half because there was a square root Another critical exponent is the dynamical exponent z and the dynamical exponent z is deduced by how the Frequency or the energy spectrum Scales at the critical point as a function of the momentum So what is the power of k with which this omega scales when h is equal to h critical? So once again, it's easy to do this in our approximation because we can just Take h equal to h critical and we just need to Take the expansion of vk as a function of k the expansion of vk is k to the sigma but if H is at the critical point we have the square root the square root of k sigma and so it means that this z is sigma over 2 So z nu is one half and z is sigma over 2 and then we can use These two values to also compute nu, which is the divergence of the correlation length But I don't want to define the correlation length in this moment. So let's just focus on these two definitions Very good So these are the definitions of the critical exponent as what what I have just said According to this definition in our Approximation, we have the following values z nu is equal to one alpha I've already told you and z is equal to sigma alpha and this is obviously valid as long as sigma is less or equal than 2 When sigma is larger than 2 this z is simply one because as you know Also as I already told you when sigma is larger than 2 this guy the k to the square wins And the lower energy theory is a standard boss a gas Okay, so Thanks to the mapping from of the spin to the bosons. We were able first of all to understand the possibility to have a transition the the yes them To understand the the condition for having a transition But we were also able to compute some critical exponents which are approximated but gives us a rough idea of What's going on in the system anyway, this is not the only approximation I can think of we can think of we can also think about an approximation in which the The spins of the original model are not mapped into bosons, but are mapped into fermions This approximation is quite more complicated I don't want to give you too many details about that because it will bring it will bring us too far outside. I Just want you to know that it exists It exists a such an approximation that maps the spins of the system into fermions and What is most interesting of this approximation is that this approximation is not a prox an approximation at all If the system is nearest neighbor if nearest if the system is nearest neighbor this Transformation that's called Jordan Wigner. It's not an approximation, but it's exact So up to now I have been showing you the study of the ising model in terms of bosons But I could also study the ising model in terms of fermionic excitation. So this C and C dagger The definition of this fermionic excitation is quite complicated, but the intuitive picture is very simple These fermionic excitations are domain wall No, are basically if you start from a fully paramagnetic state so a state which is fully aligned with a magnetic field the This the domain walls are just Kink in the in this magnetization that are core at certain sites And these are completely opposite to the spin waves because if you remember the spin waves That I've defined before they were completely non-local objects No These were spin waves which have low moment that they were fully non-local objects but these guys here are Very local things that they are just things. They are magnetization flips okay so I Can make I can map my ising model into a gas of King and This gas of King It's Also solvable analytically Because it's basically has the same property of the bosonic system It's just a system of fermions hoping on the lattice with some known with some anomalous pairing term So also in this case we have some thermal which does not conserve the number of particles. So it's a bit odd But you see the Hamiltonian it's Ressembles very closely the one of the fermions Basically of the bosons. It basically it's the same Hamiltonian. It's just the nature of the particles that has been changed and The solution is also comes in the same way It also comes by a bogolyub of transformation But by a bogolyub of transformation with fermions that it's also possible. It's still a two by two matrix Is there it's still a rotation in sq1 space? and When you do this transformation you find new quasi particles which are BCS so are the same quasi particle that you know from superconductive theory if you have done superconductive theory if you don't Have if you haven't done it is just the same the same procedure as before with some minus sign in the sense with some different With some different definition that it's all to the fact that these particles are fermions and When I diagonalize in the theory in terms of fermionic quasi particles, I get a different dispersion relation Okay, a different dispersion relation and this different dispersion relation is due to the fact that these are fermions and Now I don't only have the term H minus V as in the bosons But I only have a new term which is this delta which is the Fourier transform of the Interaction but made with the sign instead of the cosine So you remember with the boson you we had the cosine With bosons we had the cosine and now with fermions We have both the cosine in this epsilon and the sine in the delta Okay, but apart from that it's pretty standard It's just the same procedure as before But now I can compute Again the critical exponents and these are different critical exponents and You can try to do this yourself or take this definition Take this a question and reapply the same definition that I've given to you before so reapply the same definitions And when you do this procedure you find different critical exponents from the bosonic theory With one crucial difference That these critical exponents are now exact When sigma is very large so for when sigma is Larger than one when sorry when sigma is much larger than two We have that these critical exponents become exact because they are the correct critical exponent In the sigma infinity limit in the case of nearest neighbor interactions and so We get to our final picture This model it Can be mapped into bosons, which is the spin-wave theory and when it's mapped into bosons it basically Tells you true information The first information is that there is a threshold value as sigma equal to two Above which the model looks like a nearest neighbor model looks like a local gas with K-square kinetic there He gives you also a second information that I didn't have time to talk you about too much, but it is obtained by a Simple ginsburg-landau argument and this could be an exercise for you just take The bosonic theory that I showed you here and you are ginsburg-landau analysis to show That this theory also applies to the interacting case as long as a Certain condition d over sigma is to be satisfied And if you find do this analysis you will find that the bosonic theory becomes exact When sigma is below two-thirds Obviously, I'm talking about one dimension because I started talking about one dimension so dimension is one And so you see Immediately what's the situation from the bosonic theory from the spin-wave theory we have some critical exponents which should be Exact in the regime sigma going from zero to two-thirds, and I encourage you to verify it doing a ginsburg-landau argument in this region For sigma between two-thirds and two we have certain critical exponents coming from the bosonic theory But the ginsburg-landau argument tells us that this cannot be correct or may not be correct And for sigma larger than two we have that the bosonic theory is not useful anymore because for sigma larger than two The bosons will not condensate So in this regime the bosonic theory will tell you that there is no transition, but we actually know That there is a transition and we know that from the fermionic theory So the fermionic theory this is why this short range here is green Is because the fermionic theory predicts a transition For large sigma and we know that this treatment is exact for large sigma However at small sigma when sigma between Becomes more between two and one We don't know actually we know that we cannot trust the fermionic theory And the reason why we cannot trust the fermionic theory is that the The transformation that I have been doing here It's not quadratic anymore. So I have strong corrections I didn't have time to talk about this But I wanted you to at least get a glimpse of All that we can do with this long-range rising model with sigma larger than zero and the glimpse is as I said that The bosonic theory is good as small sigma The fermionic theory is good at large sigma They are both approximation at intermediate sigma. So in between This both this theory cannot give you the final answer the final answer It should be that the exact theory is mean field That means it can be described by the bosonic theory between zero and this value With in for sigma in this range and then in this range the theory becomes No mean field and it cannot be described neither in terms of boson neither in terms of fermions And then there is a threshold value which I call sigma star because I don't know which is this value It's not one and it's not two The boson say that the threshold value is two the fermion say that the threshold value is one But actually in the exact theory which I cannot study I can only guess that this sigma star is between two and one. It's some number in between these two in this At this sigma star The system the long range interactions are not relevant anymore are not important anymore at the transition And the model becomes back nears. It has the same critical exponents as the nearest neighbor model So basically this is the message. I wanted to give you In this lecture We can study the long range interacting guys in model not just for alpha smaller than d as I've done yesterday But for alpha larger than d and in the case everything turns out to depend on sigma Which is alpha minus d or in one dimension. It's alpha minus one And as a function of sigma, we have very different regimes And in each of these range in the in the boundary regimes We have two descriptions which are both exact for the critical exponents One is the spinway theory a small sigma which gives exact critical exponents in here One is the fermionic theory which gives exact critical exponents a large sigma But then we have an intermediate range Where the where the system it cannot be described Either in terms of bosonic quasi particles or in terms of fermionic quasi particles And in this middle regime It's a very strongly interacting regime and it needs to be treated with more advanced field theoretical arguments And this I think is too advanced for you. I will have some slides, but I don't want to talk to you about that I want you just to remember this crucial point As a function of sigma There are the regimes where the quasi particles are fermions regimes where the quasi particles are bosons and in the middle There is a regime which is strongly interacting and we don't know how to solve exactly Okay, I think I I have given you enough information for this lecture. Please make me some question I Would like to remind that I showed a slide in my lectures where This sort of Theory was a sketch and I what I promised that Nicola would have talked about theory of a regime for a positive Where the mean field exponents are still valid then an intermediate regime where we don't know much and then A regime for sigma large in which you go you go to the To the short range Okay, so this is exactly the type of analysis that That you can do to to make it more rigorous. Okay Okay, and you saw also this This line d over 2, um, ecolo, you are still there Yeah, I'm still here. Okay, so maybe you can make a comment on this D d over 2 sigma d over 2 I see Sigma d over 2 so one alpha. Yes In one dimension Sorry, so sigma sigma equal one alpha for from from the point of view of the theory that I have So you get to third in your theory. You don't you don't get Because I'm sorry. Just sorry because this is the quantum. Yes, obviously. Okay. Now. Now. I got what you mean so basically So I think Stefano showed you some plot which looks like this Basically, so there is the dimension on one axis and the sigma or the alpha in some other axis And you see that there is a mean field region above a certain value And then there is a strongly interacting region, which is the one I was talking about And then there is a short range behavior for large sigma. This is exactly the same kind of feature That justifies this. This is just the one-dimensional case. So this is just a cut Of this picture in one dimension or even here now for the What is the difference between the quantum and the classical case? Well This is a little bit maybe uh complicated, but You see There is a A zoology of different mappings that you can do. No And yesterday I told you that you can use shorter the composition to map A long range model a quantum model sorry into a classical model And yesterday we have done that for the fully connected case, but there is no reason to To not do it also for this for the Long range interactions, which are a little bit weaker, which are powerful And when you do this shorter the composition you see That the quantum model it's equivalent to a classical model, which is anisotropic So which has an additional dimension, which is the time dimension But that is a little that is an isotropic And so The quantum model somehow it's equivalent to a classical model and you see this is this this little part here It's equivalent to a classical model, but in a dimension which is much larger and And basically you see I told you That the long range quantum model Can be mapped onto a short range model in this effective dimension So I told you that the long range model can be related To the short range model in this effective dimension But it can also be related to a classical Short range model And if we relate it to the classical short range model the effective dimension is not only 2d over sigma, but it's 2d over sigma plus 1 That is telling you that the quantum In order to reduce the long range interaction to the Nearest neighbor interaction you have to use an effective fractional dimension So you have to say that d is equal to 2d over sigma But if you want to also map the quantum model into the classical one, we have you have to also add an additional plus 1 so you see there is a A zoology of transformation that allows you to understand How the critical behavior prolongs From quantum long range to quantum short range to classical short range And if you put all of this together you will see that If you redo the same argument in the class in the purely classical case Where there is no quantum nature in absence of any quantum nature this plus 1 will not be there So this plus 1 is only due to quantum fluctuations in the system If you redo the same argument I have done in the classicalizing model There will be no plus 1 and if there is no plus 1 It it it's immediate to show that the actual boundary In here is not two-thirds, but it's one half And what is this boundary that stephanow was mentioning this boundary is the boundary at which The spin wave approximation so that is the boundary at which There is no difference in studying the critical behavior of the spins or the critical behavior of the bosons and so it's the it's This is this of course also in nearest neighbor system, you know, but Probably this would be clear only for the people of you that are expert that know something about critical phenomena We know that the icing model the transition of the icing model becomes mean field When the dimension is larger than four And this is the case for the same case for the long grain system The long grain system becomes mean field where sigma is smaller than a certain value But this is nothing but what I told you before it's the effective dimension. So You can just say when is the effective dimension d2d over sigma becoming larger than four Well the effective dimension to the over sigma is becoming larger than four when sigma is larger than than the alpha And this is the mean field threshold exactly as stephanow has defined it to you But this is valid for a thermal phase transition for a phase transition like the one of the bose gas That appears as a function of the temperature For a quantum phase transition that appears at equal zero like the one of the spins Then One has to do a little bit of a different calculation and this goes to be two turns Okay, I think I said whatever I wanted to say about this Okay, so let's uh finish here So it's uh quite late. So Thanks, uh, nicola, you have the full day in front Yeah, okay, so and uh Let's meet tomorrow at 4 p.m. For the last talk Okay Bye. Bye