 The first of all, there is a request from some of you to have the exam of the host next week on Tuesday afternoon. Sdaj se sredimo, da se se skupnimo v različenju v Wolpert. In bomo vzgledatiще na odličenju rufu. Iselo se vse vse? Stefano rufo, opravlj in različenju. V friadi sem se vse pripranj v Wolpert, oje? Kaj je to? Ok, ok. Zdaj je. Zdaj si počutite na slagšanju o Rufos kurs. Ok. Ok. Zdaj... Ok. Zdaj... Zdaj si počutite na 7. lektu z stefano Rufo Skorce, ki je zelo spesial. Tako. Včeš, če smo rekočati? Včeš, če smo rekočati? Tako, če smo rekočati, če smo rekočati? Tako, če smo rekočati, če smo rekočati? unfortunately we had about a invited Nicolas de Fenu, who is currently a postdoc at eth zureš, and is connected today from Boston to give the second part of the lecture course for Nicolas. It did his PhD at CISA, working on several aspects of long range interaction. I received his PhD at CISA and I sent him to of long-range interactions, so his first contributions were to understanding critical exponents of long-range interacting systems by devising some special innovation group approach for these systems. Then he went on and he has now collected a series of very interesting results on long-range interactions in the quantum domain. So I thought it was the most appropriate person to talk about this, to you. So I will leave the floor to Nicola for his presentation. So it's now 4-6, I think even if we make a short break, we could finish for 5.30, 5.40 with this presentation. And I will act, and also I will be helped by Mattel, if I understand well, some persons in the hall will also collect questions. Emanuele will collect the questions and I will also connect to Zoom and have a look at the chat to see if there are questions. So Nicola, I will act as your assistant in this lecture course. I see the faces of the students and so I can see if they open their eyes. So I cannot see unfortunately the rest of the face. So we are masked, but OK. So Nicola, I leave you the floor for the presentation. So the first talk will be about sort of all-to-all quantum systems and what happens if you add quantum mechanics to all quantum systems. OK, so good morning everybody. I hope the slides are visible, seems so. Yes, they are visible. Very good. So, well, first of all, let me thank Stefano for inviting me here, for giving this very nice introduction of my work and my research. I guess in this situation we are mostly focused on something which is pedagogical so that you can understand a little bit the basics of long range systems, quantum systems with long range interactions. You are probably much warmer than I am. So this is my first lecture that I will be teaching here while you already have six or maybe four, I don't know, six because it's a seventh. So you already have six lecture and so you are much warmer so we can dive directly in. Most of what I'm gonna say, you can find it in this archive preprint that you will find online. It's basically a review that has been written by myself and also Stefano is between the authors and there are also the collaborators of us. So most of what I will be saying during the lecture can be found in this review. The review is not completely pedagogical because it's a scientific review, but at least you can find references therein which are more pedagogical. So we are gonna roughly follow what's written in the review in a more pedagogical way and I can give you further references when you... Sorry? No, no, nothing, sorry. Okay. And by any way you are welcome to ask questions both regarding the physics and the references at any moment while I'm speaking. Okay, so the first part of this lecture is gonna be dealing with systems so how to realize long range interactions in quantum systems. And this I have to say that it's kind of a quite recent field. It has been... It has started developing around 10 years, maybe a little bit more and it's probably started with this kind of... So the example you see here in the top which is a platform of trap dions. And trap dions have been the first one where allowed us to realize long range interactions which have a tunable decay exponents. I don't need to define what is long range interaction for you because you have already had several lectures. It's just a two-body potential. So it's a system where the two-body interaction potential within the particles or the spins or any microscopic component of the system decays as a power to alpha. In the quantum regime as I was saying we have a few examples that can realize this kind of long range interaction potential and most of them are found in the case of atomic systems or atomic and optical systems where long range interactions are realized thanks to the mediation of some bosonic particle that can be phonon or photon. And these that I'm showing you here are the most celebrated examples of long range interacting quantum systems. The one here on the right is a system of quantum gases trapped in an optical cavity and the one on the top is, as I said, trapped ion system at low temperature. The one on the left is not an atomic system. The one on the left is a more traditional system. It's a condensed matter system, a film of a magnetic field. So this is a film I think of iron or copper where long range interactions arise somehow naturally. So they are not engineered like in the case of trapped ions or cavity systems. They are somehow naturally realized due to the fact that the atoms in the in the team for magnetic field have a dipolar interactions. And so you see the long range interactions in this condensed matter system realize this kind of striped behavior or lamelle which is very interesting, but unfortunately it's not what we are gonna be talking about. It's just to show that there are not just synthetic systems, synthetic quantum systems with long range interactions, but there are also natural ones. As I said, as I already said during these lectures, we are gonna mostly focus on the synthetic ones because the synthetic ones are a better playground for theory in some sense. They are systems where long range interactions can engineer in an environment which is highly tunable and controllable. And this makes it in such a way that we have more or less the possibility to realize our favorite theoretical situation in an environment where the phenomena are easily observable and controlled. The first case I'm gonna describe is the case of trapped ions. And trapped ions, as the names say, they are ions which are cooled at a very low temperature thanks to laser pooling. If you are not expert in AMO devices in cold atom, just take my word with a grain of salt. This is just to give you a glimpse, an intuitive understanding of what's gonna, so of why long range interactions are important and how we can use AMO devices to realize long range interactions. So basically, as I said, we are gonna talk about a system of trapped ions. And the ions are here trapped into an electron, what is called a pending trap. So you see basically the atoms are church particles. In this case, they are beryllium plus, but this is not so important. What is important is that they are positively church particles. And so they are confined in the plane by the fact that there is an electric field here. This plane is at a negative potential. So the ions like to lie in this plane. And since they are at very low temperature, they form a crystal. So ions are church particles at very low temperature. They form a crystal because they have to arrange a way to minimize their interaction potential, which is long range, is a cool interaction potential. And so they form a crystal. But the ions also have an internal degrees of freedom. So they basically have electronic degrees of freedom. And so we can have ions which are in a state zero or in a state one. So the ions have also a spin degrees of freedom on top of them. And this spin degree of freedom interact by some kind of by interaction, which are mediated by the crystal phonons. So basically if you, so if you see it here, the there is a coupling between the spin of the ion. So each ion is a spin, which occupies the place on this lattice. And the value of the spin is coupled to the position of the ion. So this Zi is the position of the ions on this plane. And so it's coupled to the position of the ion. So there is a coupling in the Hamiltonian between spin between the special and between the special degree of freedom and the spin degree of freedom. And this coupling can be tuned thanks to the presence of some electromagnetic field. In this case, are there some laser beams that you see entering here from the plates? And by tuning the shape of these laser fields, the angle in which the laser fields are somehow imprinted on the crystal, we can realize different kind of interaction potential between the spins. Because you see the spin is coupled to the emotional degree, to the special degree, but the special degree can be expanded in contribution from phonons, the phonons of the crystal, which are this P, I, and B, J. But the phonons of the crystal can then be integrated out, so they can be removed from the Hamiltonian because they are very fast. They have a line scale that is very short with respect to the one of the ions. And when the phonons are integrated out, we get a spin-spin interaction. So basically the spin can exchange one phonon, and so the interaction between the spin is mediated. And when the spin exchange such a phonon and the quadratic spin interaction is mediated, we can also tune the coupling between the spins, which is this J, I, J, by just tuning the property of the phonons. And the property of the phonons, as I said, they are tuned by these fields, by these laser fields that you see here. And this is basically a way to realize tuneable long range interaction. So we are realizing an amyltonian of spins. Here I just took the spin part, the interaction spin part in this amyltonian. And this interaction decays as a power law, and the power law also depends on now what is the angle of this imprinted laser fields. And you see that I can tune different kind of interactions. So this is a plot of the J, I, J as a function of the spin-spin separation. And you see that depending on the frequency and on the angle, one can find different interaction potential. We have alpha equal to zero. Sorry, this A, it means alpha. So it's the power law of the interaction decay. So alpha here is very small, basically zero. And then we can increase it by changing the frequency of this laser beam or actually the shift in frequency between the laser beams. And you see we can realize basically a kind of a large range of interactions, power. So basically the interaction that we can realize of alpha that goes from zero to basically three. And this is a very wide range of tuning parameters. And it's very good for realized model that they are interesting in theory. So this is a first example. So with trap dions, we can realize long range interactions. We can realize long range interaction that decay with a power law that we can tune basically almost as much as we want. I show you the two dimensional realization. There is also one dimensional realization in which one can tune the range of the interaction power law. Now we go to another kind of system, which is particularly interesting for us. It doesn't have the tuneability of the trap dions simulator. It's only allowed to realize strong long range interaction, which are basically flat. So alpha equal to zero, which I believe is the case that Stefanos treated the most in the previous lecture. So the case of very flat interactions. And in this kind of system, the interactions are mediated by light. So they are mediated by photon. So in summary, the example of before had tunable power law interactions, which were mediated by crystal phonons. Here we have a non-tunable, so a flat long range interaction, which is mediated by the photons. How do we realize this? We take a gas of particle of quantum gas. So even in this case, the gas, these are atoms which are cooled at very low temperature. So basically they behave quantum leaders, large superposition between the wave function of the atoms. And these atoms are trapped into an optical cavity. An optical cavity is just basically two mirrors, so no, two reflective mirrors with a very high finesse, which basically means that you can have, you can trap some radiation into the cavity because it's bounced forth and back by the mirrors. And when the atoms are trapped into the optical cavity, basically they are interacting with some nearest neighbor interaction, which you see here. This is the Hamiltonian of the atoms. It has some nearest neighbor interaction, which is this term here. And the atoms can move, so they have their momentum, but they are also coupled to a pump. So there is a laser field, which is shined on the atoms. And this laser field, it's very important that it's shined transversally. So the cavity, it's longitudinal, is on the x-axis, while the laser comes from the, it's directed towards the y-axis, so it comes transversally. So this is the setup that most people are dealing with. And what happens is that the cavity, so the pump sends photon into the cavity, as I said, transversally. When atom can scatter a photon from the pump, and it gets some momentum along the diagonal, and it also scatters this scattering effect as an effect, it produces a photon that goes into the cavity longitudinally. So you see, we are pumping transversally, but the scattering of the light with the atoms generates some photon to go longitudinally, so that they can be reflected from the mirrors. And then there can be another scattering from the photon to the atom, in such a way that the photon goes back into the pump, so it becomes longitudinal again, and the second atom is imprinted in the opposite momentum in the other direction. So you see, it's basically a two-atom process, and this atom process involves three photons, two photons from the pump, and one photon from the cavity. The contribution of the photon of the pump can be removed, so we can go to the rotating wave approximation and we can remove the contribution of the photon in the pump. And so basically we see this as scattering between the cavity photon and the atom. And this is what is described by this Hamiltonian here, which is the coupling between atom and cavity. And you see basically this is exactly, so psi is the atomic creation and annihilation operator, so is the atomic operator. And you see that this is the term that is responsible for the scattering of the photons. And this term is proportional both to the cavity mode, which tells you basically what is the distribution of the photon mode in the cavity, but is also proportional to the pump mode, which the operator has been eliminated, but it remains a contribution from the, like from how is the intensity profile of the pump field that you see here. Well, obviously there are also higher order effect with the coupling of the photon, but these are not so important. They are just a shift in the energy of the atoms and the cavity, which is not so important at this stage. So very good. This is the atom cavity, atom Hamiltonian and the cavity Hamiltonian. I guess you know how it works, because the cavity Hamiltonian is just, the cavity is a single mode cavity, so it has just one photonic mode with a certain frequency, which is called delta because it's in the rotating wave of the pump. So it's the shift of the cavity frequency from the pump frequency, nothing much. Very good. So the coupling between the photon and the atoms I've already explained and it's roughly proportional to the rabbi frequency and the rabbi frequency is calculated by the pump intensity, which is this omega and the delta, which is basically the line width of the atom. So even in this case we are talking about atoms, which have two states and so delta A is just the line width between the ground state and the excited state of the atoms. And as I said, the scattering process is 2-4. Nicola, there is a question from the hall. Prego, prego. I have a question. I don't understand the term with g lower case, half times psi dagger psi. In the second Hamiltonian. This one? Yeah, what does it represent? This is just the interaction between the atoms. Okay. So you see this is the atom annihilation operator, the atom creation operator, the atoms are bosons, okay? So it's a density-density interaction between the atoms, which is local. It's just to account for the fact that these atoms are repulsive bosons and sometimes when they meet, they just repel each other. Okay. Thank you. Okay. In some sense, this term is the one that is less important from the point of view of the description of the long range interacting system because we are interested in the interaction which is mediated by the photon. But one has to consider that these are anyway are bosons. So when they meet, they repel each other. Well, okay. So as I was saying, so there is this process, which scatter the atom, which is in the ground state. So remember that this is a quantum gas to start with. So the atoms are mostly in a condensate with zero momentum. And when they scatter a photon from the pump in the cavity and then back on the pump, they acquire a momentum which is proportional to the cavity momentum and the pump momentum. In the following, I'm gonna consider the fact that Kc is roughly equal to Kp. And so basically these photons are scattered, these atoms are scattered by the photons from a state, from a ground state to a state which has a diagonal momentum as it is represented here. Okay. Now I outlined you the phenomenology of the cavity. Now let's go to the theory basically. So you see the phenomenology is pretty complicated. There are a lot of cavity mode functions pump mode functions, a lot of little details, the local interaction. But from the theory perspective, things are pretty simple, at least as long we are in a dispersive cavity. And this is the definition of a dispersive cavity. It means that the K square, so which is, this is called the recoil momentum, the recoil energy, sorry, the recoil frequency, which is proposed, which means it's basically the energy that is imprinted of the atom by the scattering, by the scattering process. This energy, it's much smaller than the cavity line width. So the cavity line width is you see by this simple picture, each cavity, the cavity is never perfect. No, the mirror cannot be perfect. They never reflect 100% of the light. They always reflect only a portion of the light and there is a small portion that get lost. This is this K, this is the finesse of the cavity, how much the cavity dissipates and loses light. As long as this coupling is much larger than the momentum that the atoms acquire, we can basically say that the cavity is fast with respect to the atom. It means that the cavity dynamics, so the dynamics for which a photon goes into the cavity, bounces forth and back and then it's lost, it's much faster than the atoms dynamic. Now this is just a timescale reasonment. Photons do their job very fast. They enter the cavity, they are scattered, they bounce forth and back and then they are lost at a very high rate. A rate which is much faster than the rate at which the atoms evolve because the atoms are slow. They have a slow momentum, a small momentum. Within this assumption it makes sense to eliminate the cavity. Eliminate the cavity means we integrate over the cavity mode because the two timescales are very different. I would say, so as you are master students, I would say this is very similar in some sense of what you have, I think you have seen in your basic courses when we talked about the open-imer approximations for an atom. You normally consider the motion of the electron into the static field of the nucleus of the ion because the nucleus is very slow with respect to the electrons. And this is kind of the same thing. The atoms are small with respect to the photons that are faster and so we can integrate away the photons and the photons generate an interaction potential between the atoms. And this is what happens. So we can integrate away the atomic part of the wave function and we get effective Hamiltonian for the atom salon in which interactions between the atoms are mediated by the photons. And as I said in my analogy with the open-imer approximation it's like when you take you integrate away electrons from an atom and you only consider or from a crystal and you only consider the motion of the ions of the nuclei into the effective potential generated by the electrons. Very good. So this is basically what happens in this case is not a potential that's generated by the integration procedure. It's an interaction term in the sense that the photon interacts in the photon mediates interaction between the atoms. And this interaction potential is long range. It's given by a convolution as you see of basically the cavity mode and the pump mode. I am not going to give you the details how this is done. What is important is this is you see it's a check board potential in two dimension so if this is a two dimensional cavity as it often happens this basically generates an interaction potential which is a cosine. It's very similar to the one that you have already encountered in the Hamiltonian midfield model. And indeed this is really what happens so this system can be mapped into the Hamiltonian midfield model. And in order to do this one has to define a theta which is a order parameter which is basically the sum of the cosines of kxj. So k is the momentum of the cavity. The photon in the cavity has a certain momentum and this is the momentum of the photon in the cavity but it's also the momentum that get imprinted to the atoms by the scattering process. So k is just one over the cavity length so it does something like this. And xj are the position of each atom in the cavity. So you see basically that when this guy is maximum or minimum it means that all cosines are plus one or minus one and means that the atoms are occupying the maxima of this standing wave. So there is a standing wave in the cavity. This standing wave has a momentum k if the atom occupies either the maxima or the minimum of the standing wave you get the theta which is either plus one or minus one. So this order parameter theta it's finite and large well, large in the sense that it tends to one when there is a special order in the cavity where the atoms occupy these special points on the cavity standing wave. If the atoms are disordered so they occupy random places and they behave like a quantum gas and low temperature so they are basically a non-homogeneous condensate this theta is zero. And as a function of the cavity of the pump field so changing the pump field when the cavity when the intensity of the light of this pump light is small is lower in real value Are there any question Stefano? No, no, I don't think maybe it was background noise. Okay, sorry. So it's always present in cavity so don't worry. Exactly. In this case it was not fast enough that we didn't hear it. Okay. So what I was saying is that basically so basically when the intensity of the pump field is below some critical threshold it's like if the cavity weren't there. Basically the atoms are in a condensate they live in an homogeneous state mostly with zero momentum and there is just some small some small light intensity that impact on the atoms the intensity get immediately lost very fast because k is anyway somehow large. The intensity gets immediately lost from the cavity and so the pump just act as a little bit of a small temperature effect on the atom cloud. On the other end when omega grows above a certain threshold omega C then above the threshold it's when the situation becomes interesting because there is enough light scattered by the pump into the cavity that one generates a standing wave. So basically for small intensity there is no standing wave field in the cavity and there is no radiation detectable in the cavity only very small fluctuations. But when the pump overcomes this threshold a standing wave is generated this standing wave is the result somehow of the scattering of the atoms of photon from the pump to the cavity. You see this is the photon that goes it bounces towards an atom and it's reflected and it's scattered into the cavity and these photons that get scattered into the cavity are so many that basically the cavity starts to have a standing wave field inside it. So when you now want to measure the number of photons in the cavity you will find that the number of photons in the cavity it's finite. It means that there is a finite light intensity inside the cavity. But this light intensity has a standing wave mode function as we said so it's a cosine or a sine. And the atoms react to this electromagnetic field and they want to occupy either the maxima or the minima. And so they basically arrange themselves in order to be in these positions and they made a structural transition. This structural transition means that they arrange themselves into basically a lattice. And once they arrange themselves they also foster the scattering so when they are structurally arranged it's easier for them to scatter photons and so there is a feedback mechanism for which basically this becomes a laser in the sense that there is some light that comes out from the cavity which has a certain fixed intensity which is proportional to the intensity that you get inside the cavity. Obviously I call this a laser but there is no... So well, it's a form of laser because this is a coherent light there is a structure so there is a coherent light that goes out but obviously it's not an efficient one because the pump intensity is very high. It's just an interesting phenomenon from the theoretical point of view because it's a quantum phase transition. Well, yeah, it's a phase transition or on the quantum nature we can discuss about. It's a phase transition and this phase transition is somehow can be mapped, can be related to the phase transition that we have seen in the Hamiltonian field model. Because when you go and you boil down this system to the very minimal ingredients the very minimal ingredients are that there is a motion from the atom which is this p square and there is an interaction from the atom the interaction from the atom it's all-to-all so it's an all-to-all interaction and can be represented by this theta to the square which is the order parameter to the square. And when you take the order parameter to the square and you rearrange the terms you see that it looks like this and this is exactly this cos k xi minus xj is exactly the same that appears into the Hamiltonian B3 model that you know of which is here. There is also the cut scaling which is automatically implemented because this is like a standard results in cavity system the intensity of the light in the cavity decreases like one over the sides of the cavity itself. So basically the fact that the interaction is mediated by light it's automatically realizes cut scaling in this system. So you see the two systems so the system of atoms strapped into an optical cavity after integration over the cavity mode it looks really resembles the Hamiltonian of the Hamiltonian of the Hamiltonian mean field model. It has a cut scaling it has global interaction and it has the second order phase transition and the second order phase transition is basically the same it's in the same university class. The difference between these two systems it's that in the cavity there is also a breaking of translational invariance which is not existent in the Hamiltonian mean field model. So there is this cosine of xi plus xj term which leads to the breaking of translational invariance but anyway this is not relevant in the phase in the phase where there is structural order and it's not even relevant at the transition point so somehow the physics of the two model is very close to each other apart from this term and apart of course from the fact that this is a driven dissipative system so in this system there is noise there is yes there is noise there is an effect of dissipation which is this k which nothing of which is represented by the simplified Hamiltonian. But anyway the overall physics the qualitative physics is the same and this justify us as I mean theoreticians like me that work in the quantum field to focus on the study of fully connected spin system because this fully connected spin system represent well the overall qualitative feature of this transition which appears in cavity and also somehow of some ferromagnetic transition that can be simulated that has been simulated with trap diodes. Okay here I go finally to the theory so I think I talked long enough about the experiments it was just meant to give you a glimpse of what we can do with these experiments are there questions on the experimental part? Sorry, maybe I missed some point but where do you see that there is a long range in the reaction in the Hamiltonians that you showed us? Okay basically the long range interaction is seen here so okay this is not represented very well but let me try to let me try so first of all let me do a qualitative argument and then let me try to give you some formula even if not precise so the qualitative argument is the same is the following we said that the atoms want to have this structural transition they want to occupy the minima or the maxima of this standing wave field in the cavity now do you agree with me that this is affecting the sense atom always want to be either depending on the depending on some details of the atoms they don't always want to provide either the maxima or the minima of the of an electromagnetic field so this is this is the fact so now this electromagnetic field in the cavity is only there due to the fact that the atoms scattered the light okay so in some sense the existence of the field in the cavity depens from the fact that the that the atoms are efficient that the special distribution of the atoms it's efficient in scattering the light so basically it means that if I move one of the atoms outside of their favorite position it will be less efficient to scatter the light and the light intensity as to decrease a bit so this is something that the system doesn't want to happen because it's minimum it's equilibrium it's well it's not equilibrium because it's even dissipative it's it's stable configuration it's when the atoms occupy the minima or the maxima of this grid and so if the atoms moves a little bit out it gets pushed back by the field of the cavity it goes back into its it has some force so that who finds it in a certain place but this force but sorry I see, I hear some noise some strange noise do you hear it too? Yes, yes, yeah I can hear you just some background noise not in the I don't see okay, okay now it's anyone who they are all muted those in apart from you so okay very good so I can eliminate maybe the my so basically the atom feel the restoring force mediated by the cavity light but the intensity of the cavity light it's proportional to the number of atoms because the more atoms are in the cavity the most scattering there is so there is there is a proportionality coefficient between the light in the cavity and the number of atoms so it means that the light mediates an interaction between atoms because the intensity of the light itself is proportional to the number of atoms but the force that the atom feel to be that brings them to their position it's also caused by the cavity light so you see how the photon mediated interaction between the atoms do you agree with me on this? Yes, yes, I agree with you but perfect now let's go why is it long range? it is long range because the cavity it's a standing wave single mode cavity so as we say the intensity of the light it's just proportional to the number of atoms it does not depend on their position on their distance on their distribution it just globally depends on how many they are and if they are in the minima or if they are scattered so basically it means that if I move an atom here it feels a force that brings it back which is proportional to the number of atoms is not proportional to the distance and this you see from the fact that my theta my order parameter is just a sum over cosines no there is no distance between the atom this this theta is proportional to the position so this is this theta is the result of the sum over the position of the atoms with respect to the cavity standing to the cavity wave now in the amiltonian that I haven't the right for you but in the amiltonian here there is theta to the square and you see that this theta to the square it's an interaction between the atoms because it basically couples the position between the atoms but it couples them all to all it doesn't couple them proportional to the distance or anything it's just each atom feels the force of the cavity but the force of the cavity is proportional to the entire density of all the other atoms was this clear? at least intuitively? yes thank you now it's much clearer with the square thank you very good and then if you go into the review that I told you so this one this archive there are the details so now to go from this basically actually from this to this and you can try and you can really see that in the end you get this cosine of the distance but the cosine of the distance is just the result that these are the cosine of the position when you do cosine of the square you can do the prostiferency formula you get the cosine of the distance but there are some intermediate steps so one should be careful I don't want to bore you with the details the important point is that you get the overall picture everything is proportional to the density not to the density to the number of atoms very good someone else? no I don't see questions so you can go on very good so let's go to some theory and the theory is so in the sense that before was the theory of the experiment and if somebody of you has ever worked with experiment the theory of the experiment it's easier than the theory of the theory because and not because we are lazy theoretician but because in theory we want to have a model which is simple enough that we can really understand the basic phenomenon we want to understand the fundamental ingredients that lead to a certain phenomena we don't want to have to be bothered by all the tiny details like the the breaking of that symmetry or the the small dissipation which is not not relevant but it makes some contribution we want to focus on the necessary ingredient to study a certain phenomenon and the phenomenon that we have in mind here is the transition the structural transition between disorder both science and condensate in a cavity and disorder of both science and condensate in a cavity and structurally ordered atoms in the lattice with a standing wave light within them this transition is related sorry this transition is related to the Hamiltonian mean field model for the reason that we have discussed before and since it is related to the Hamiltonian mean field model let's focus on something which is somehow even simpler just a symbol called long range spin chain which has a quantum phase transition between a ferromagnetic and a paramagnetic state and the ferromagnetic state so and so I focus on this Hamiltonian where you have global all-to-all interaction you see this is to the square this is just the magnetization along the z direction the interactions are ferromagnetic it means that j is larger than 0 and we have the usual cut scaling to ensure that the energy is is extensive and these atoms these these spins these are one-half spin operators so you see here there are the Pauli matrices divided by one-half this is not so important these are one-half spin operators and they feel an external magnetic field which however act on the x component of the spin rather than on the longitude in our component this is what is called a transverse fieldizing model have you ever encountered this kind of system is there anybody that doesn't know anything about the transverse fieldizing model? okay, apparently everybody kind of knows it very good so this is nothing but the transverse fieldizing model as I said where interactions are global so each spin in the model interacts with each other's spin and we can in close analogy with what we have already done for classical system try to compute the partition function of the model by doing the trace of e-beta h so h is the Hamiltonian beta is the inverse temperature this is the Boltzmann weight the only difference is now instead of integrating over all classical state we have to make a trace over all quantum states but I think it's kind of the same the interesting part is that since this system it's global interacting it can be basically mapped into a single giant spin and we do this by defining the total magnetization operator which has this mz and y and max and when we define them in this way you immediately see that the I'm sorry here there is a mistake now it should be correct so you see now this is basically the Hamiltonian is first of all in this way it's explicit that is extensive because I have this overall n factor where n is the total number of spin but also you see that I map my Hamiltonian into a single giant spin which is self-interacting a giant fixed spin into a magnetic field but this giant spin now is as I say is the result of the sum of many a large number n of one-half spins and so by the usual formula of the addition between spin states you have that the spins can have a very large total a very large spin modulus so s square and also I have different magnetization state it is important to note that this Hamiltonian commutes with the modulus of the spin so each sector each total spin sector behaves on its own in the sense that if I if I initialize my system if I want to study the dynamic of my system and I initialize it in a certain spin state this spin so this modulus will be conserved the modulus of that spin state will be conserved by the dynamic now this is the standard you know this from basic quantum mechanics when the Hamiltonian commutes with some operator that operator is conserved so this is basically what happens to the total spin and I wanted just to remember for to you that when you have a large spin we basically use normally a basis of two quantum numbers which are the total models of the spin and the depolarization along the z-directions and so this I wanted just to remind to you okay very good now we want to solve this we want to calculate this partition function and we say that sorry Nikolaj, did you study the angular momentum in the course of quantum mechanics so these formulas are not odd to you so you know that okay anyway it's just a different view you can imagine of having as Nikolaj is saying a sort of giant spin made of all the spins of all the part sorry Nikolaj, I wanted to check the fact is that I don't see the faces of the people so it's good if sometimes you tell me what's going on so what I want so basically what Stefano was saying because if you studied the angular momentum rules this is just the rules of angular momentum oh yeah, you have a z you have a preferential component that we always call z you have an eigenvalue along that component and then there is another eigenvalue which is important to characterize the state which is the model of the total model square which is s so these are the standard to eigenvalue for angular momentum states in quantum mechanics in some sense they are not crucial for what follows but if you know them it's better so now we want to compute this this e to the minus beta so this partition function we want to compute it but we are faced with the fact that yes it is a single giant spin so it seems a pretty simple system but still in the Hamiltonian there are two operators that do not commute with each other and it does not commute with mz and it doesn't even commute with mz square ok so these two operators do not commute and since they do not commute it's not so easy to calculate the trace over all possible states because the states are many because remember this is a giant spin it has many values of s and it has many values of mz so basically mz goes from minus nalf and nalf and s can so there are a very large number of states which you have to sum over so it's better to make a trick and the trick you want to make is to use what is called the Suzuki Trotter transformation and this trick is very well known in quantum mechanics even if you maybe don't know it and it's a trick to map a quantum system into a classical one and this is why I wanted to do this for you not in all details, you can check all details in this reference for example or in reference therein, if you are interested we can discuss what is the best reference to study this kind of Trotter decomposition but I believe this long-range interacting system is one, a very good example of how this Trotter decomposition is applied so first of all the Suzuki Trotter transformation is based on the simple equivalence that is shown here so the exponential of a sum of operators remember that these are operators I didn't put the hat to keep it light but they are all operators so if you do not commute as you know you cannot use the composite in the product of two exponentials so but you can do that if you use this formula so it's true that the exponential of the sum is equal to the product of the exponential as long as this product is done in many slices you see so basically you divide each operator by ns and then you rise it the wall to ns so you repeat this ns time and when you send ns to infinity these two guys are equal so can you see the difference is there anybody that doesn't see the difference between these two okay so you see it there is a sharp difference this is the exponential of the sum this is the product of the exponentials but I have to repeat this product I have to reapply this product an infinite number of times this ns it means I'm reapplying and reapplying before this product is equal to the actual exponential okay so if you both this formula let's just apply it to the big to the calculation we want to do so to the trace to the partition function of the giant's pyramid and so instead of calculating the entire exponential I do the product of the exponentials but I have to repeat it ns times and in each of these ns terms in the so yes for each term in this product of exponential I put an identity operator sum over all possible states remember that the coherent spin states are a complete basis so I have a complete basis for my system I just insert the identity as the project on over all the states in the complete basis and I introduce this identity between each of these ns exponential products and I get this little bit of a messy formula let's try to decompose it so these are the two exponential operators that have been separated and they are sandwiched between two states and these states are just spin configurations so these are a configuration of microscopic spins so these states basically tell you that so they are just states in which each of the spin is polarized along z plus or minus one it's just a classical state of the spin this this kind of term is reproduced ns time this is the product ns times of these same terms with the difference that each basis is a different one because I added the identity ns times so these are ns terms in the product each one is done as this form and when you put two of them close to each other there is sigma alpha plus one, sigma alpha plus one in here and this is the identity and you can get back to the original form very good ok, this doesn't seem like we have done much progress now we have instead of exponential of the sum we have two exponentials but we have this product over all the trotter slices so each of this term in the product is called the trotter slice and we have also to sum over all possible states in the trotter slice the only interesting thing is that now we can put this operator outside the brackets because this mz we know basically these states are eigen states of mz so there is no tunneling in use between different states by mz and this can be put outside and we only have to put the tunneling in use by mx between different classical spin configurations ok, this in principle is doable but we can do something even easier and better because this mz term is squared and it's a little bit annoying because it couples all the spins in the system this is the reason why we say that this is a global spin in the sense we need to consider it as a global spin to solve it it will be somehow better if we could remove this square here and in such a way that we can only act on a single site and the Hamiltonian will be basically the Hamiltonian of a non-interacting system the interesting thing is that we can do that in the system and we can do that by making a simple trick and the trick is based on the fact that this guy here it's not anymore an operator even if it's not anymore an operator because it is diagonal in the state in this state here this state as I told you our classical state polarize along and z, so this guy now it's a classical number I can take it out, I didn't put the formula but you can imagine it, I just take it out and since I can take it out I can now do a trick which is to use delta potential which fixes my order parameter so this is why these kind of systems are called mean field models because we can define what I call them alpha which is nothing but the sum over the classical values of s, z as I said the sum of the sets of each site these are now classical values because as I told you, these states were the classical states and for each of the trotter slices we define a different order parameter and how we define this order parameter we just put a delta function of m minus this summation this equation we put this delta function which fixes the summation to m alpha and when we use the representation of the delta function as an exponential we get to this guy here you see, I haven't done much it's maybe a bit of passages that you can put to yourself at home if you're interested if you have an intuition for what's happening even better the intuition is simple I used the Suzuki trotter to decompose the exponential I took this guy outside because this guy it's classical, it's not an operator because the basis I chose it's the basis of the eigenvalues of m, z so this guy has classical values in this basis now I use the trick of a delta function to somehow define to make my integral in such a way that each of the integrals over that each of the integrals over the spin states it occurs at a fixed values of m alpha that's the point the point is that now the only part that depends on the on the sides being coupled is this part here but this part here I can take care of with a constraint that constraints some classical value to be the sum of the longitude there is a question, Nicolo, from the other please, please, tell me I have a question on the previous slide where we computed trace why are the eigenstates different sigma of alpha and then sigma of alpha plus one on the right hand side so, can you repeat the question sorry, my... we were computing the trace for the partition function but why we are doing the product of sigma of alpha on the left side and sigma of alpha plus one on the right side because basically remember, look at the Suzuki tractor the composition this guy which is the same as this guy repeated NS times no when I put it to the power I mean that I repeat it's a product of all the same guy NS times ok now, in here I can put it just as a power in here I do not have any power because in between of each terms of this product I have inserted a complete basis let me maybe I can do something maybe I can do something maybe it does not understand why it is labeled plus one simply a complete base in several points it's a way of numbering a complete base in several several license can you I will try to do something let's see if this works you see my screen no no ok, that's a problem sorry guys ok, now you see it yes ok let me sorry, I'm not the best at doing this I understand that it doesn't work ok, so finally I think I did it so now you see imagine that you have these two operators I'm trying to simplify the things no so this is AA1, AA2 to the square so this means that this is AA1 AA2 multiplied by AA1 AA2 AA2 ok, so this is a product between the two now I insert in here I insert the identity and the identity is the sum over all states so this sigma bar means over all possible configuration of the spins do you agree with me on this equation? do you understand it? sorry, who is the guy that made the question? yeah, yeah, it's ok it's ok, and also you see this is the identity now I insert the identity inside here I insert the identity inside there I insert the identity inside there and I get that there is the sum over sigma of AA1 AA1 AA2 sigma sigma AA1 AA2 but now remember that I have to compute the trace of this guy so I have to put another sandwich on the other side so this is the sum over sigma bar sigma prime bar sigma prime AA1 AA2 sigma sigma sigma AA1 AA2 sigma prime you see it now so this is sigma prime this is sigma sorry, so this is sigma prime and this is sigma prime ok so basically if we go back and unfortunately this I cannot share the two screams at the same time but if we go back to what I was showing before this is sigma and this is sigma prime and this product means that there is another guy coming which has the same shape but on the left it is sigma prime and on the right it is sigma second then there is sigma second again sigma third do you see it is the same calculation that I have done you on the black board or on the white board but ns times I have done it only two now this is ns times do you see it? yes, thank you fantastic ok, so it is not an easy calculation it needs some care so you can redo it at all the important thing is to understand for each factor in this infinite product of the trot at the composition I need to introduce an identity and then basically this is where it comes from the fact that on the left you have a sigma alpha then you have a sigma alpha plus one then there is another guy sigma alpha plus two sigma alpha plus three ok so this now I was going to another trick which is very common in field theory, quantum field theory statistical physics etc which is the one to restrict an integral via delta function and this is exactly what is happening here I want to use a delta function which I represent as an exponential of e i lambda in such a way that the values of m alpha which I have here which is also here is the sum over all s z and so basically it means somehow that I can simplify well, not simplify but I can rewrite this integral over all possible configurations which was a little bit unknown to be done explicitly this summation over all possible configuration now becomes an integral over all possible values of m alpha and an integral over d lambda alpha and d lambda alpha is not in lambda alpha is nothing but the Lagrange multiplier that fixes m alpha to be equal to sigma z and indeed you see that lambda alpha of course here in front of m alpha and here in front of sigma z because it tells you that this identity when doing the integration has to be fulfilled very good so basically thanks to using the properties of the delta function I have rewritten the integration in such a way that this is an integral over the value of the magnetization where the value of the magnetization is fixed to be what I want so to be its definition by a Lagrange multiplier alpha and I have also to integrate over the Lagrange multiplier alpha because this is the property of the delta function but now what I gained is that the quadratic guy is written in terms of the classical parameter m alpha and the term which is lambda alpha sigma z alpha I can put it back into the trace and now it is a single side term and now I have to do the trace here only over a single side operator ok this is a bit complicated I understand so I do not pretend that you press all of what I am saying you should just see the main point the main point is that well the Suzuki Trotter decomposition I guess we discussed it in details the main point is that in this representation this guy is a classical value so I can take it out but I am still a little bit bothered by the fact that this summation is over all possible configuration of an infinite giant speed so I need to compute the configuration it is possible that it is a bit annoying the multiplicity of each configuration that gives the same magnetization instead of doing it explicitly I can use a Lagrange multiplier to impose it and so introduce this delta function into the integral and simplify my integral in such a way that it is an integral over two parameters all the values of the magnetization and the values of this somehow Lagrange multiplier that fix this to be what I expected to be and when I do this trick I replace my integration over all the spin configuration into an integration over two variables and I need to compute only the trace of this operator that is back to be the Hamiltonian but this operator is now easy to compute because it is a single side operator very well I have hopefully given at least a glimpse of what is happening and now we come to the most important part the most important part is that all of this transformation that I have done I have always left n the sides of the system out you see this is the crucial part the crucial part is that there is always an n that lives in front of the exponential and since there is an n that lives in front of the exponential it means that in the thermodynamic limit I do not need to make this integration at all so I have done a lot of work to simplify this integral but actually I never need to do it because as I go to the thermodynamic limit it will be only the minimum or the extremum of this guy that contributes to absolutely the extremum this is a standard property a mathematical property that the exponent the integral over the exponential of something which is decreasing exponentially fast only receives contribution from the minimum of the exponent I guess this and so basically when I approach the thermodynamic limit I do not need to make an integral it is enough that I minimize it is near and to minimize the quantity it means basically that the free energy so this is the integral to be done as the partition function when I take the log of the partition function I get only this minimum only the minimum of this guy so the log of the partition function is the free energy the free energy is just the extremization with respect to lambda and the minimization with respect to m of this guy here and this guy here when you do the trace over the single side is this one ok I'm skipping a bit of passages as I said it will be very boring to do the entire computation so I leave it for you to do at home I'm trying just to give you a glimpse of what is the physics behind the physics behind is that this system is a giant spin and the global interaction and the global interaction are such that in the thermodynamic limit the integral for the partition function it receives contribution only from the saddle point and this is a very important part the very important part is that the integral receives contribution only from the saddle point and this is the typical thing that happens in long range interacting quantum system where there are strong long range interactions possibly like in this case flat fully connected interactions you see it won't be the same if this were a nearest neighbor coupling if this were a nearest neighbor coupling there would be no n here and this term will couple only nearest neighbor sides so there could be no possible trick of defining a global classical parameter especially there could be no possibility to take the saddle point as the result of the integral because there will be no n factor in here so this is very important in some sense then to the mean field nature of the of the interactions we can trade our integration the saddle point the saddle point is very easily done you just take the derivative with respect to lambda and m and you post them to zero and you immediately find that lambda is equal to 2jm and that m is equal to this and since lambda is lambda itself is m you see that this is a mean field condition it ties m with itself I guess you have also seen this condition but maybe not in this form but if you put h the external magnetic field to zero this is just the mean field condition of the classicalizing model because I think you know if you put h to zero these two lambas simplify each other and this is the m is the hyperbolic tangent of beta 2jm which is exactly what you get from the mean field treatment of the classicalizing model and here we found it in its quantum generalization where there is also the h the external magnetic field that plays a role and in this kind of system this treatment is exact because this saddle point that you do here it's exact so this is an exercise for you take transverse fieldizing model with nearest neighbor interaction the one that you know the standard case try to do the mean field approximation and check that you will get basically the same result with some coefficients which are different so since this is the mean field equation that we all know it's not surprising that there is a phase transition a second order phase transition basically shown here so as a function of the magnetization z sorry here I didn't put the mz but this is obviously mz mz as a function of j for different temperatures h has been fixed to 1 so I fix h to 1 and for two temperatures I give you the magnetization as a function of j and not surprisingly the larger is the temperature the stronger you need j to make the system magnetize and you can easily compute the phase diagram so the critical temperature as a function of j as usual this is j over h and this is tc over h so h has been set to 1 and if you do the computation you can do it easily you see that this is the shape of the critical temperature as a function of j and basically you see that the system is paramagnetic for t above this line and is paramagnetic for t below this line and you see another very important aspect of quantum phase transition which is the I believe this is the most studied case of quantum phase transition the quantum phase transition occurs at the end of a line of classical phase transition so when you cross this line you have a finite temperature phase transition that in critical phenomena it's always called a classical phase transition so even if the system hits quantum in itself when you cross this line it behaves as if it were a classical system and so we always call this a line of classical phase transition but as the temperature is reduced the line goes to this point and here there is a quantum critical point which occurs as a function of j at fixed h so remember line of classical phase transition which terminates into a quantum critical point and this is the standard the most known case of critical phenomena in quantum system do you know this concept of quantum critical phenomena in quantum system so that the quantum critical point may occur at the end of a line of classical transition is there somebody that doesn't know that everybody I think I think so but I think it's a very good illustration on a simple example exactly the cool thing is that in most of the book that you may find and study this diagram will be either obtained by intuitive arguments or by some kind of approximation in this model that we have here it is an exact result and it is an exact result because in this model the sides of the system N act as a control parameter basically as you increase the sides of the system N the partition function only receives contribution from the saddle from the saddle point and this it amounts to say that the system is exactly solved by its mean field approximation so while in most of the system the mean field approximation is just an approximation that has to be obtained with either intuitive or formal arguments but still an approximation in this case the mean field approximation emerges naturally from the computation and it emerges naturally just once again because N the sides of the system appears in front of the exponential and it acts as a control parameter ok and so basically in this system of fully connected spins we see that the phase diagram is exactly what you expect a line of thermal phase transition which terminates in a quantum critical point and this is the same that you will find on Wikipedia this is the plot that you find on Wikipedia of such a quantum phase transition which is exactly the same situation a quantum critical point which occurs at the termination of a line of classical phase transition the only point that you have to remember in here is that while in the standard case of nearest neighbor of local system one has a boundary so one has a region where there are strong fluctuations and so basically there is a region where classical statistical fluctuations are very strong and they make it very difficult to treat the system with mean field or any approximate technique in our case of strongly of long-range system this does not happen in the thermodynamic limit because in our case fluctuations are cut off by N and so when we approach the thermodynamic limit this region becomes thinner and thinner until it vanishes so this is why I wanted to share the comparison between this and this which are exactly the same plot with the difference that this shaded region here doesn't exist for us because there is no strong fluctuation regime in the thermodynamic limit I had some other thing to show you but we have gone a little bit slower which is good I think that it is enough because the last step to do the log of the trace of sigma x now we will take them some time I think to do it so they have to explicit the calculation of the log of the trace of absolutely and you can write me if any one of you has problems or questions this is the reference you should look for and in this reference there are other references the first part of the reference is basically a review so it is pedagogical but it contains a lot of textbook and other references inside so you can really use it this is this reference here and they don't have your email in the chat no, maybe I can write it on the blackboard as you wish yes, okay so you could add Nicola on Slack ah, yes I can also enter in Slack I don't have Slack on this computer you exchanged I think Matteo in email are there any questions on the part I have saved? questions I don't see in the chat this is the email okay, this is the analogous I would say of the Q revised model but made it one with an external field so it would be completely classical if h were zero so then if h is zero the model is in fact the Q revised classical model simple addition of h because h adds an operator which doesn't commute to the sigma z makes the transition quantum but only at zero only at t equal zero only at t equal zero so there is no classical phase transition in h because if I add an h in the Q revised I remove the phase transition classically because if I add an h in the Q revised instead of having a singularity in the magnetization I have a smooth behavior of the magnetization sorry Stefano, just to clarify the difference is that what Stefano is saying it will be exactly the same in the quantum case if you were adding a longitudinal model that multiplies an mz for a sigma z but since this is transverse so this is basically the quantum nature of the system you see from the fact that the interaction and the magnetic field they act on two different components of the magnetization that you don't have in the classical case because in the classical case the spin is just an arrow, it doesn't have components it goes just up or down is it clear ok I think it's a very nice example of a model that is solvable nevertheless it contains it's rich that it contains this concept of phase transition in an amyltonia which is made of two non-commuting operators so let's thank Stefano no I wanted to say my suggestion to really understand this little bit of intricacies is to try to take the amyltonian that is this one or this one as you prefer and study it either via this technique that I have shown that you find it in the reference or by just doing the mean field as you are used to be you can do it just in mean field just taking that the operator is just a classical values plus a small perturbation and you will see that you find the exactly same result but then in field tricks is kind of arbitrary because you make a decoupling that we know it's a little bit of arbitrary but with this way you really see that for the long range interacting system the fluctuations contribution to the partition function it's suppressed by M and then as Stefano said you're right I didn't show you in details how to do this trace but this is just a trace on an operator over a single site so it's easy to do it's just you have to diagonalize a 2 by 2 amyltonian basically and indeed you get a hyperbolic cosine you will see it when you do it and if you have problem you can write me and discuss but this is a very simple part is to understand how this n comes out ok, so tomorrow is at 2.15 your lecture 2.15, so 8.15 for me so it will be with coffee in Boston well it's even difficult to go to the barrier so you have to do coffee at home there is no everywhere ok, so and there is no in Boston so it's very cold I think super cold it's minus 4, minus 5 but it's also very humid so you feel it I feel it much more here than in Zurich ok bye bye ok guys are you sure you don't have any questions sure, I see that everybody wants to leave so it's late afternoon for you I understand ciao guys bye bye