Recording in progress very good. Okay This is the usual slide as you know, ah, let me Let's not make the same mistake as yesterday You can see the slide and also the car sir. No The department there whatever you want. We're fantastic. Okay, so good morning everybody. That's good afternoon for you It's morning for me This is the our last lecture and for the last lecture I've thought of something that's I Could say it's slightly more Advanced in the sense that it's a research topic. It's something that I have done quite recently last year But on the other end is very basic and we will see how and why it is very basic in in the following slides I believe this lecture should be quite short. So there will be room for a lot of questions But let's see how it goes as I unfold it Anyway, you can ask question at any moment during the lecture. I wanted to say that I would try to go to a Particularly slow pace today and this lecture has always to do with long range interacting system as usual And we are in the quantum limit The first lecture we have talked about that the equilibrium of strong long range interacting system So the equilibrium of quantum system with alpha Smaller than D and in particular smaller than one because you know that I like to make examples in one dimensions Which are my opinion simpler that which can often be generalized in any dimension And on the second lecture so the lecture of yesterday We have focused a little bit on the case of alpha larger than these all for larger than one and I've I've tried to show you how the the presence of a long range interaction term Modifies the scaling behavior of of a spin system, but also of a quantum gas Yielding critical exponents which are different from the expectation for short range interacting or nearest neighbor interacting systems and also try to show you how the different regimes for the critical exponents emerge and how can we They pick to this or can we can try to Determine this exponent with via a different approximation techniques So today we are Going back To the case of strong long range interacting system So do the case of alpha smaller than the dimension D But we are Focusing on the case of dynamical evolution so quantum dynamics of Long range interacting systems And as usual I'm going to use as an example the icing model that you know It's my favorite model and it's my favorite model because it's Connected with all of these experiments that you see here as I try to argue in the first lecture All of this experiment in certain regimes can be related or mapped into a quantumizing Hamiltonian So I go back to the quantumizing Hamiltonian So the icing with ferromagnetic long range interactions as I so vi j is power low decaying with alpha and Alpha now is so and now we see what happens as a function of alpha when we study the dynamics This I don't need to comment so much on this we already seen yesterday you remember there is the case of of weak long range interaction So alpha larger than the dimension in which we don't need any cat's rescaling and indeed this prefactor is just a constant It's just a number and instead there is also the case of strong long range interaction Now for smaller than one in which we need a cat's rescaling which depends on the system sides and is the number of spinning the chain and In order to have an extensive internal energy The dynamics that I want to study. It's a very simple one, but since I imagine that not many of you have Ever Have experience with quantum dynamics and it's especially many body quantum systems that I would try to go very slow and I So I am really asking you to ask questions to to tell me wherever you are not following So the idea is to study a phenomenon which is very Celebrated in long-range interacting system and which Stefano knows and probably talked to you about Which is the one of quasi stationary states? So quasi stationary states are dynamical states in the sense that they are not equilibrium states But their lifetime so once the system enters in such states It is stays in there for a very long time So basically this quasi stationary state is a concept that we use To to refer to the fact that long-range interacting system strong long-range interacting systems They do not like to equilibrate when we post them out of equilibrium. They stay out of equilibrium for very long time And normally this long time which is the lifetime of this quasi stationary states Grows with the system sides. So when we make the system larger and larger It tends to stay more and more time out of equilibrium How can we So this is a concept that is very well known in classical long-range physics Stefano has worked a lot on it many other people have worked on it But there is also the case that the analogous behavior It's also observed in quantum long-range system and I'm gonna make you a simple example The simple example was spelled out first by Michael Kassner in this in this physical review letter that you see here this reference But the simple it's so simple that I believe everybody can understand it It's it's a nice in model and the asking model is prepared at the equilibrium at in a state where at very high Magnetic field. Okay. So basically we start with our icing model Which has a very strong magnetic field in such a way that basically it doesn't feel any Contribution for the interim from the interaction term and all the spins are aligned with the transverse magnetic field spin Magnetic field so all the spins Basically are independent on each other The the state is a product state of independent spins and all the spins are aligned with the magnetic field That because they they want to align with it because it's very strong So basically the magnetization at the at the initial time of the dynamics of when T is going to zero you see here the The magnetization is basically one everybody's magnetizes the system is polarized along the magnetic field axis At this point we do what it's referred in the literature of Many body quantum system has a sudden quench so we abruptly change the Hamiltonian from the initial value Where the system was laying at the equilibrium to a new Hamiltonian where the magnetic field is completely off Okay, so this is what we call a sudden quench Is there anyone that doesn't know as ever has never heard about a sudden quench? Somebody's crazy. I haven't I had work good But was it clear what I said?  I think it Derives from what the blacksmith does with the sword now when you want to forge a sword You first have to heat the metal so the metal has to become very Flexible and that you can shape them all the and then when you when you are ready So when that when when you make you gave the metal the shape you want that you have to put it into cold water You have to cool it down very fast and this procedure I think in English or in in old English was called a quench as what we And it's for a by analogy We use this in physics to mean when we take a system and we switch very fast the Hamiltonian So it's basically the idea now of Lowering the temperature very fast by putting the hot metal in cold water is the same that happens here No, we take the system that is fully magnetized along the transverse magnetic field axis and Suddenly we switch the magnetic field off and the system has to find a new equilibrium Very well. So the system has to go and find a new equilibrium and this equilibrium Okay, the Equilibration of an isolated quantum system is a little bit of a tricky question But don't let's not talk about the system in general be let's talk about an observable Let's say let's talk about the magnetization observable. So this is something that should be easy enough As we say that the the magnetization starts at t equal zero Which is basically one because the system is fully polarized along the magnetic field axis and Then a time t equal to zero the magnetic field is suddenly switched off and the system has to evolve According to the new Hamiltonian the new quantum Hamiltonian the initial state was an iron state of the Initial Hamiltonian so no evolution was taking place But in the new Hamiltonian the initial state is not an iron state. It's just a product state and the evolution starts So the observable starts to evolve and it does what you will expect it to do So it decreases It has some bounce back This is a phenomenon that we will talk about in a minute and then there is some decoherence and the observable Equilibrates to its expectation value to the in the final state. No the final Hamiltonian as no magnetic field and then we expect it to show no Transverse magnetization and this is indeed what happened to the observable it equilibrates and it goes to It's expected equilibrium value As long as the system is finite This equilibration can never be perfect It exists a phenomenon that's a phenomenon that's also known in classical system Which is was the reason why a lot of people opposed Boltzmann Description of statistical mechanics in the in the beginning of the century. Well in the beginning of the last century And it will and it is that a closed system Always comes back to the initial Condition after a exponentially long time This phenomenon is called Poincaré recurrence time. Unfortunately, it's not visible in this Plot, but what you can imagine is that as long as the system size is finite There will be at some point at very large time. I'm a new Revival of the dynamics which goes up and then it goes down again This is the concept of a Poincaré recurrence time But what Boltzmann said to to oppose the criticism that was led by these many people Especially it's a male or other people is that this Poincaré recurrence time exists They are a thing but they they become exponentially large in the thermodynamic limit and as they become exponentially large We are allowed when we treat a statistically big system so system that has a navigator number of particles to this card this Poincaré recurrence time and to basically treat the system as it is as if it is ergodic as if it follows the hypothesis of the HTRM Did you know about this controversy between Boltzmann and Zermelo so Well, I guess you know it's about the story You know that the theory of Boltzmann was never was never really accepted by the people at this time No, they were opposing him and one of the reason of this opposition is is what I have described right right now Okay, so I think you I Hope you got this point. It's this is a feature that has nothing to do with long-range interaction So this plot in here. It's representing the evolution of a system with Short or weak long-range interaction so alpha larger than one knows So obviously as a function of alpha the situation would be a little bit different here and there But this picture here refers to the case alpha larger than one where the observable starts in the Initial value and then evolves towards the equilibrium value Quite fast and it basically stays at the equilibrium forever Apart from some exponentially large time which is washed away in the thermodynamic Now this is equilibration as we know it the in local systems And it's also worse in weak long-range interacting system. So alpha larger than one But we may now want to consider the case of long-range interaction system We're alpha is smaller than one or smaller than deep and see what happens to the Equilibration So we do they say the exact same procedure. So we start with the system fully polarized in the In the Full Magnetized state so where all the spins are aligned with the magnetic field and the state is a product state and Then we quench the Hamiltonian into a final Hamiltonian where the Magnetic field has been turned off and we look at the evolution and now this lower panel refers to the long-range interacting case To the case alpha smaller than one And you see that it's it's a very different from the from the case of Short-range interactions Because it starts at one and it stays there for for quite a long time No, you see here it was not saying at one at all So you don't even see one because my time axis starts from one and not from zero So this is one the one year would be somewhere here And then it's very fast in equilibrating here It persists in in the initial value for quite a long time and then it starts Going down and equilibrating towards the equilibrium value As we increase the system size as the system size got increased so you see this this This blue curve is 10 to the 7 spins the So as the shade of blue gets darker the number of spin grows All the curves were identically near so it's not that here I'm plotting just one curve with just at all these curves here in the short-range case They collapse on each other, but in the long-range case they do not collapse But actually as they increase the system size the magnetization persists for a longer and longer time on its initial value and I can tell you if you get it longer and longer the chain the longer the chain the longer you stay In the initial value and this is basically the the all marker the the basic evidence of Poise stationery states so a state which persists in its which persists and it's not basically Influenced by the dynamics for for very long time and this time grows as the system size grows As everybody clear the difference between the short-range physics and the long-range physics I think it's very stark, but please make some question on this point if you're missing anything. I Do not see questions and ecologic and Okay, so if you Are sure about it. Let's let's proceed And let's ask ourself. What is the mechanism that generate this question stationary states? No, so I told you there are this question I said I showed you one evidence You have to bear in mind that this evidence is quite unique because this is the only quench we can treat So we cannot do a quench between two finite values of h We cannot we can only do quenches from h equal to infinity to h equals zero This is the only case that we can treat analytically All the other things we have to treat in numerically so no way we can get a signature at such large sizes for For any quench only this peculiar quench It's special and can be solved analytically and the reason for that is that The initial state. It's a product state and the final Hamiltonian. It's a classical Hamiltonian You see the initial state. It's a product state because H was so big that we discarded interaction and so each spin the The state of which spin was not influenced by the neighbors Everybody was behaving on his own and aligning with the magnetic field And the final Hamiltonian is classical because if I turn off h I'm only left with an Hamiltonian, which is sigma x sigma x and so there is no non-commuting term in the Hamiltonian In this sense we say that the final Hamiltonian the one which we with h equal to zero is classical because when you remove this term This guy alone is there is no non-commuting term in the Hamiltonian And so it basically a classical Hamiltonian But we expect what the stationary states to be much more general and to be observed also in Different scenarios not just in this simple example But in order to show that in order to understand that we have to ask ourselves the question What is the mechanism of the root and the mechanism of at the root or at least The core mechanism there may be other ones, but the fundamental aspect of cross-stationary state Emerge if you look at the Fourier transform of long-range interactions in the lecture of yesterday if you remember I Mapped the ising model into a bosonic model. So this was the spin wave approximation And in the spin wave Approximation the dispersion of the spin wave. It's proportional to the Fourier transform of the coupling Vij, sorry here. I called it Vij and I called it jij later. So this guy but this jk is nothing but the Fourier transform of Vij and Yesterday you should have from the lecture of yesterday You should remember that the the Fourier transform of the Vij that unfortunately is now called jk It's the dispersion relation of the spin waves And if you remember yesterday, we talked about this dispersion relation and we say that this dispersion relation is non-analytic As long as alpha is it's larger than one But let's now try to do the calculation for alpha smaller than one and for alpha smaller than one We cannot forget that we have to introduce a scaling factor one over n to the alpha the cast scaling that they Depends on M and now something very peculiar appears This was this is really peculiar at least for me because I know a lot about tight binding amiltonians in condense matter And and what happens here is completely the opposite. So the presence of long-range interaction it makes it Very different from the standard case when you try to compute this Fourier transform you find out that this Fourier transform is discreet But it's not discreet because the system it's finite it remains discreet in the thermodynamic limit You see I am taking the limit n to infinity here the thermodynamic limit for this result We take I took the thermodynamic limit for this result and you see that as I take this thermodynamic limit The Fourier transform is discreet This is a stronger position with the case of Short-range interactions where the Fourier transform becomes continuous becomes a Fourier transforms I'm sorry where the Fourier series becomes Becomes continuous in the in the thermodynamic limit here. It does not Here the spectrum of this of the quasi particle the spectrum of the spin wave. It's discreet in the thermodynamic limit And it's not just that it's discreet as I because This will this may be surprising to say that's discreet because normally what we are used to to understand is that if we take a finite chain we do The mode of the composition We write the spin wave approximation and we count the modes We have a spin wave for each side not the number of degrees of freedom that The Fourier transform is a linear transformation. So the The number of modes should be the same And so in the thermodynamic limit, it's people normally understand that as you add the more and more Sides to your system. You are you are adding more and more modes to your Fourier to your momentum descriptions and these modes in the standard case of short-range interactions They go in between and they fill up the entire Spectrum until the spectrum it becomes continuous But in the case of long range interaction, that's not the case as We increase the system size we add more and more modes, but all these modes are added in the At the top of the band So sorry, I say the top and indicate the bottom, but this is because this is Jay, but my my Hamiltonian if you remember As a minus in here okay, so The actual spin wave energy is minus Jay and so this is the top of the band and this is the bottom of the bed So we add more and more modes to the system and the modes Do not go in the middle and fill up the band No, they go only at the top of the band and they accumulate at high energy So this shows you that somehow in the thermodynamic limit the summation over the modes is Dominated by these points. It becomes flatter and flatter. It's like if there is no dispersion relation in this in this model Basically the model any model with alpha smaller than one in the thermodynamic limit It looks like a fully connected model. It looks like a model that has only one model because All of the modes become degenerate in here and they know that generate more they are washed away in the thermodynamic limit We can understand the consequence of this if we do a simple very very simple example and It will be great if you already tried to do this calculation in the case of nearest neighbor interactions, but maybe you didn't so Let me try to outline us the calculation works We consider an example that's much simpler than the one of the ISV model that I've done before Is the example of a single particle? Hoping on a one-dimensional lattice. It's a single particle. No, there is no interaction whatsoever the single particle can hop on a chain and Each of the state each of the Sites in the chain. They have an orbital So there is an orbital that's fully localized on these sites and there is some hopping matrix That allows you to tunnel from one orbital to another. This is just a so-called tight binding emitter, you know And now let's imagine what happens when I put a particle at t equal to zero I say that the particle is fully localized in one of these Eigenstates is in one of these states. Let's say it's fully localized in the orbital zero Well, we know that the orbital zero is not an eigenstate of this Hamiltonian No, because the Hamiltonian capo's orbitals are different sites So what will happen is that the particle will evolve it will diffuse and If you ask yourself, what is the probability of finding the particle in Zero time t you may want to compute this quantity that I call the fidelity and also some other people call the fidelity, you know It's the initial state Evolved with the evolving operator, which is e to the minus ht in quantum mechanics and then sandwiched with the initial state No, you see this is there anybody that doesn't see that this number gives me the probability to find the particle at zero or in the state psi at any time Very good. It seems that everybody is getting this Now this is just Standard very basic quantum mechanics you start in the state psi you evolve With the evolution operator and you ask the question What is the overlap between the evolved state and the initial state? And this is what we call the fidelity at time t. Well, as I say the initial state is just Any orbital I choose zero for my example. So the particles started in here and then it evolves and if we want to compute The the evolution of the particle We need to use the eigenstate and the eigenstate of this Hamiltonian the Hamiltonian is translation I invariant the eigenstates are just plane waves. So they are just the Fourier transform of the Atomic orbitals not the block function, which are nothing but the Fourier transform of of the atomic orbitals So when we compute the evolution what we do is that we decompose the initial state which was just a single orbital into the basis of the block functions, which are Fourier compositions of the atomic orbitals and we compute the sum over the energies of the plane wave states So this key which is nothing but the Square root or what not the square root, but you see it's just this guy without the square We without the model square this key is just the sum over some time independent probability which tell me what is the overlap between my initial state What is the projection of my initial state on the plane wave eigenstate? So this pie is the projector This P n is the probability as I said that my the overlap between my initial state and the plane wave states and each of these p evolves with the energy of the state of the plane wave state of the corresponding plane wave state and so the key is the summation over n over all plane wave states of this evolution So now what do you physically? Expect that's gonna happen. So the particle starts in here This state the orbital state the single orbital state. It's not an eigen function of the Hoping Hamiltonian So it evolves it evolves and what happens to the particle? Well, the particle is start is gonna diffuse It's not a classical diffusion. It's a quantum diffusion The wave function is gonna get lost more and more. No, it's the overlap of the wave function with with people with the orbitals further and further away from the initial orbit that is gonna grow and The particle is gonna tunnel further and further away from The initial state from the initial orbital Now as it tunnels further away it can also tunnel back and in some sense this probability measures the probability that the particle is tunneling back so that That you find this is the probability that the particle that started here and started this quantum diffusion is back on the initial state as a time t as I go to the thermodynamic limit if the hoping is short-range you remember that yeah I'm talking about a nearest neighbor hopping. It can only hope to nearest neighbor sites as I go to the thermodynamic limit so as I add more and more states to the chain I Add more and more energy these energies and become dense and this some this submission becomes an integral over the energy and the And the chi goes to zero Because in the thermodynamic limit the spectrum of this plane wave state becomes continuous And as the spectrum becomes continuous The limit as t to infinity of chi becomes zero that means that the particle is completely lost No, there are so many sites in the chain that the particle is tunneling further and further away And it cannot come back and so the probability of finding it back in the initial orbital vanishes This is a mathematically exact statement So mathematically you can prove that no matter what are the e n as long as they become a continuous ensemble They this summation has to satisfy that this limit goes to zero and it's also satisfied some other properties Which are a bit more general but this I don't want to talk about to you now for the moment I want you just to grasp this basic point a particle hoping on lattice Stars localized in an in a certain state It diffuses because the the Hamiltonian is a hoping Hamiltonian a tight binding Hamiltonian its evolution depends on the spectrum of the Hamiltonian and the spectrum of the Hamiltonian Becomes continuous in the thermodynamic limit Due to the fact that the spectrum is continuous the probability for the particle to come back To the initial states at long time vanishes Is this clear for all of you is anyone that's this is not clear I will repeat it or we you can ask some question we can't verify And in color I have a question so maybe you can clarify so because you mentioned that it's a quantum diffusion Not classical diffusion in which sense In which sense so can you be more precise in the in the sense I'm just saying what I mean is that this is This dynamical evolution well this dynamic evolution is mediated by the Schrodinger equation so it's a completely conserving evolution and In some sense you can classically represent it as a particle which has a certain probability to be back and And in this case is completely clocked so you you are representing it as a classical system But from the quantum point of view The there is a wave function that is becoming flatter and flatter so what's what's really happening is that the The probability to find the particle in each of the site is become all equal for all of them Well, I guess that's That's just a difference in understanding you can map this You can do as usual the the trotter the composition and this will become exactly a classical diffusion problem Okay, thank you It's just a question of point of views No, so I mean when I say that it's a quantum diffusion just mean that It's it follows the evolving operator of the Schrodinger equation So the coherent evolution of the Schrodinger equation is not some Langevin or some Brownian motion, but you can map it on a brown and motion So if you do trotter the composition, you will see that the mapping it's on a brown and motion So in this perspective you can also say that it's the same thing Some other questions even more basic. Okay, so I hope that's clear The sample is very simple and I really in some sense So I encourage you to do this calculation. It's a simple calculation If you know base very basic quantum mechanics, it's very easy to do just Consider a chain of orbitals each site as an orbital centered on it And you say that this is the Hamiltonian and you compute what's the probability for a particle that has been placed in this orbital to come back a Time t and you would say that this probability goes to zero as you make the thermodynamic limit and the large time Now that's enough for nearest neighbor interaction Let's go to the Long-range case, let's go to the case in which the hopping is not just between nearest neighbors But I cannot vary far And this is what we have been talking about for the entire course somehow What happens when we generalize our knowledge of? Local system to a case in which the system is non-local And this is somehow an example super simple This is a super simple example that The the hopping is So it's not to be some people won't even call this long-range interaction because this is not an interaction effect It's a hopping effect So the particle does not interact with anyone. It's alone, but it It can tunnel and it can tunnel further and further away With respect to the case of nearest neighbor hopping where she where the particle could tunnel only between nearest neighbor sites So now it's the same problem as before exact same condition all of the same Just the tunneling coefficients are now not Vanishing for sites which are further than the one on each other. They are all Finite and they decay as a power law. You know it very well So the the tunneling coefficients decay as a power law and alpha is smaller than one The only thing that changes it's the spectrum No, this is what I've told you before Nothing has changed us of now Only the spectrum has changed all the calculation follows identically to the case of nearest neighbor interaction, but The energy of the spin waves is not becoming continuous anymore. It's it's staying discreet in the thermodynamic limit Exactly for the effect. I was saying before no, I told you that the Fourier transform of the V is Discreet it's not continuous in the thermodynamic limit And since it's not becoming continuous It's remaining discreet we cannot apply The Liriman Lebesgue lemma which was giving us this result We cannot apply this mathematically exact result We rather have to stay with a discrete spectrum and for a discrete spectrum When you do this summation the probability will not vanish at large time, but it will remain finite The function f t it's an almost periodic function and so it doesn't converge to any limit it will continue Equal it can also convert to some limit, but in most of the case it will continue oscillating and tunneling back and forth This is really a consequence of having a finite Poincare recurrence time. No, this phenomenon it was just the the simple the the simple statement that this closed system as Infinite Poincare recurrence time While this closed system as finite Poincare recurrence time But what's intuitively the reason for that the intuitively the reason for that is that as we say this is a tunneling process So it can be seen as a classical diffusion as we just said in which The diffusion is mediated by quantum fluctuations. So it's it's the Schrodinger dynamics that makes the tunneling happen the particle starts here it starts hoping and Get lost and if the chain is infinite It can never come back It gets lost and it can never come back. This is true only if the chain, sorry If you if there is time after your talk, I would like since we are in the last almost the last day of the week I would like to propose a discussion among the students about the Poincare recurrence time Absolutely, we agree that we might maybe Because I think it's What do you think Panteo? Yeah So because I think it's exactly what I had in mind. Okay. We are at the end of our lecture course. I can propose a more Yeah, yeah, I think I can be way of interacting with the students so to learn to know So what is wrong about the theorem of Poincare? So I said he proved it. He made a big effort and then you are saying He's wrong in some in some sense. Oh, no, it's just wrong in it's not wrong in the sense He's correct. It's just that the hypothesis do not extend the tool. Okay. Okay. Let's discuss these which other hypothesis and so on, okay so, but Okay, I think it's a very interesting Point that you are making with these These these paper by Nicolò is published I make a little bit of an Advertising since he's your last talk and you leave the effort of talking from Boston this paper of Nicolò is published in proceedings of the National Academy of Sciences and so it's Very and he published alone on this on this issue. So So it's a very important remark that he did so go on Nicolò don't feel too much pleased We will deal with with the Poincare So basically well basically somehow this is already we could already say That we have done and I wanted just to once again invite you For example, I myself also, sorry, I cannot find what I wanted to show you. Ah, this is what I wanted to show you So I myself I studied quantum mechanics quite a bit When I was in doing my master studies, but when I started working on this topic. I found myself that they I Needed some more background and I wanted to show you this paper. It's not as you see It's kind of all this. I don't think it's not even a paper. It's more of a chapter of a book It's a it's very mathematically style But I encourage any one of you that is interested in quantum theory that's interested in quantum mechanics to go and have a Look at this paper because it will show you it will make a summary of all the known results Exact mathematical results about quantum mechanical evolution and really show you how this if the spectrum is continuous So he said it. Well, he says it's a bit everywhere, but You see you basically any Hamiltonian any Hilbert space of an Hamiltonian you can decompose it into Absolute continuous singular continuous and pure point. So this is really a Statement which is generic about Hilbert spaces That you can decompose the Hilbert space of an Antonian into into sectors And there is continuous sector pure point spectrum. Well singular continuous as we were saying with seven or sometimes ago It's not so common When you do this the composition You can really see that what I told you it's somehow a bunch of exact results because the Riemann Lebeson Lemma is just a statement that any Fourier transform of the spectrum as As gonna go to zero as see the parameter of the Fourier transform goes to infinity This is what happens here. No This is just the Fourier transform of the spectrum because the Pn This Pn in here for a for a state that is fully localized is gonna be one everywhere not one one over n They are gonna just be identical for each plane wave So this Pn really doesn't enter in the computation. What this computation is is just making the Fourier transform of the spectrum of the spectrum in the Hilbert space and so You can really apply this bunch of mathematical Exact results not only to to prove what I said, but to prove a large number of Other theorem like this rage theorem which Involve also that the average of more complex operator that's just just the fidelity have to vanish So there is a lot that can be learned from just looking at the spectrum of this system and Whether this we can call this spectrum of the system pure point is still a bit to be seen but still you can Use a bunch of mathematical results and from this simple Things you can already understand a lot of why long range interactions are so different from the case of local systems Okay, very good. So basically If you allow me I would like to point out the property so of the short range versus long range of the example that you have made Okay, essentially in the example of a short range. Can you show? the matrix the matrix that you have to diagonalize is Three diagonal so there are elements only in on the diagonal and on the diagonals that are close to diagonal Okay, and of course, you know how to diagonalize such matrices. Okay, it's just the matrix is commutes with the translation operator so you the eigen the eigen functions of the translation operator are waves and Standard waves. Okay. So then you apply the matrix to the again functions that are standard waves And you get the eigen values and the eigen values are the ek that That Nicola is showing here. Okay. Now show the other matrix the other matrix is Is no more diagonal. Okay, but it shares the property of being diagonalizable in some cases because It is in some cases you if you ever so I don't know which example you were basically It is basically it's always a topolitz matrix So topolitz matrix are just a way that the mathematician used to say something which is translational invariant No, okay that the the entries of the matrix only depends on I minus J So if the entries of the matrix depends on I minus J They are always diagonalizable because it's like if the system is translational invariant You know that the iron states are plain waves. So this is this statements can be made mathematically So the the pile K that you see here are exactly the same again function. So the previous example What exactly what is different here is the values of the eigen of their eigen values are the eigen values and So so so that's the fortune that the lack of this example because you know the eigen functions So it's not It's possible to do the calculation because you know that you know the again functions and but this Difference you you have a model with the same eigen functions, but with different spectrum of eigen values Makes a big difference in the behavior of the fidelity in the retina to the to the initial states Okay, sorry, Nikola. I wanted to To make no, no, it's very good. The more we repeat I think it is This in some sense, it's advanced because it's a long-range interaction, but it's also very basic No, it entails a lot of concept which are from standard quantum mechanics So already getting until the year It's a lot of knowledge that you need from quantum mechanics But it's knowledge you will use a lot and this example as seven was saying it just violates the common law Because it's no local and the entries of this matrix. They are the matrix is not anymore be a tree diagonal or So normally matrix that we treat in quantum mechanics. They have known vanishing entries only close to the diagonal Well, this is different because it's a full matrix. I didn't find so many mathematical Results about it. I was a bit surprised, but maybe in the future. We will make some to really show that when the matrix is full you need to have some conditions on the decay of the of the entries of the matrix that Far from the diagonal in order to get a continuous spectrum Well, this is maybe too much advanced for the moment. You just remember full matrix discreet Discreet spectrum Sparse matrix or try diagonal matrix continuous back. Okay. I wanted to make just one more example, but Not gonna so I'm not gonna want to spend too much time on it It's a slightly more complicated because this is now not just one particle hopping on the lattice. It's many many fermions And they are not just hoping they are also have this non conserving term that we were talking about yesterday So this is again the model I was talking yesterday about it's a model of fermions hoping on the lattice with non Non conserving non density conserving operators. It's a model of superconductivity That can be also related to the icing model as I showed you yesterday this is not as much important for you I Will you don't need to know at this stage in your academic Trajectory how to solve was such model, which is a bit complicated. Well, it's not really complicated It's still just a Bogolubov transformation at its any ways of many body Hamiltonian what's it's interesting any way that even in this many body model or pseudo many body model The presence of a hopping matrix Which is long range which does not the key decay fast with the distance Basically cause the appearance of quasi stationary states and Since we want to discuss a little bit in two minutes I want to just prep it up But I want just to show you that if I take an operator which is just related to the density of the fermions You see this mx It's kind of the density of the fermions. No this measures how many fermions are at a certain point It's normalized. So if we consider just this operator, which is somehow related to the density of the fermions and we compute its evolution after a quench of The age which it's now the chemical potential of the fermions We find exactly the same signatures That I've shown you for the icing model and again this system Just this is just a consequence of the discrete spectrum so if we do such dynamics between a Large values between a large value of the chemical potential to a small value of the chemical potential will do the quench So we change the Hamiltonian suddenly from hi to hf That density operator will change this value will always start from one because I normalize it to be one at the beginning But if the interactions are short-range you see something very similar as to the icing model it goes fast Somewhere and then it starts oscillating and there is a day-facing for which it equilibrates at a Certain value and here you see the Poincare recurrence time here on this axis. We finally have An axis that is long enough to see the Poincare recurrence time. So, sorry, let me make it bigger so you can see it well, so You see when the system is small The guy does this kind of equilibration. You see the the greenish line here, but then at a certain point The dynamics starts back. This is a Poincare recurrence time, you know Tom and then it restarts oscillating and it goes it's flattens as You increase the system size the Poincare recurrence time occurs later and later and later until it goes away From the axis you see and This is really out what happens also in the thermodynamic limit as the more you increase the system size the more this Poincare recurrence time is pushed away to Infinite times basically you are you achieve actual equilibration and This is for alpha Basically infinity so nearest neighbor well for alpha. That's a bit smaller, but still larger than one So remember that this is the case alpha larger than one It You still see that if you increase alpha this equilibration it takes more time No, you see so if you decrease alpha you see this is alpha I think it's 1.5. So it's quite small, but still larger than one You see it takes more time, but the phenomenology is basically the same You have a Poincare recurrence if the system is small and as you increase the system size The Poincare recurrences are washed away and it just takes more time to reach some kind of equilibrium This is strongly opposed to what happens for alpha For alpha smaller than one for alpha smaller than one this is alpha. I think this is 0.9 or something 0.8 You see a finite size The the two curves are not so different you start from one It goes down and it then starts to oscillates around some kind of value that you may mistake for an equilibrium But the difference is that here as you increase the system size it doesn't Stay doesn't converge as more and more to the same value It's actually the opposite as you increase the system size Oscillations kind of continue, but they are pushed Closer and closer to the initial value and they are pushed closer to the initial value then the more The the system is long range. This is alpha equals 0.5 And so this is again the phenomenon of quasi stationary state There is not well-defined equilibrium here because there is no unique Value around which all the curve oscillate and converge radar as you increase the system size You get closer and closer to the initial value of the dynamics. You basically basically get a non-evolving system Okay, this is what somehow My point All of this is caused by the discreteness Versus continuous back through most of the statement can be proven exactly by mathematical theorems, I Do not want to tell you all of these because I would prefer to have some discussion. I Just make this conclusion Long range couplings long. This is not even talking as I said about interaction. This is just long range hopping couplings Have a spectrum which remains discreet to the thermodynamically and this is you can understand exactly that the way Stefano say this just a full matrix that now you have to diagonalize and this spectrum is different from the one that you will get from a Tridiagonal or a fast Empty matrix One can then study a system of particles with this discrete spectrum and this I say here It violates the kinematical cause hypothesis because basically what it what it's happening in here is That due to the discreteness of the spectrum, there is no De-phasing no Equilibration in the short-term case come from the fact that this guy becomes continuous And so you are mixing a lot of different frequencies That are all close to each other but all different and this they basically average to zero The same is not true here because there is not this de-phasing this de-phasing is called Kinematical cause hypothesis in some paper that I can send you if you wish And so this is violated by this long-range spin wave theory As a consequence of these violations Do you have a finite one career current time in the thermodynamic limit? This one career current time are not even finite, but they kind of tend to zero So they become very small not zero. Well, they they tend to become smaller and smaller as the system grows Another statement that you can make is that any system with from this perfect perspective any system with alpha between zero and one is Basically identical to the case of alpha equal to zero because the spectrum of the alpha equal to zero system will be just a single flat line The the different alpha they are not flat But as you go to the thermodynamic limit they basically become flat because you have Infinity many states on the flat line and you can basically discard the contribution for the few states which are finite Which do not belong to the flat line. Okay This I think close is what I wanted to say. I want to say thank you and We can have some discussion now if anybody is interested on What do you think we should ever break or we started this car? Obviously, we can have a break before Five and then are you are you available for discussion? Yeah, yeah, if I have And then