 Hi everybody. Good afternoon, good evening, good morning, whatever is more appropriate where you are at the moment. So you see that there's been a change of setting. So today's lecture will, I will be given from tablet and I will try to save the notes and send it to you. In the meanwhile, some of you might have noticed that on the on the matrix channel, I've already sent you all the references, including both last lecture material and today's material. And at the very end of this lecture, I will also send you a temporary PDF file, where you will find all the equations that we have been discussing and all the concepts or most of the concept that we have covered. You will find additional references. So today I will be discussing few things. I mean, the first is that it's connected what we to what we discussed last time. And what we had, we had this concept of partition function. What we want, what we started doing was to define a data structure corresponding to this partition function, which entail two questions. The first one is to find a proper embedding space and we already discussed that. And the second one was instead related to the characterization of a feature space that we have not discussed and will be the starting point of today's lecture. Remember, everything we will be doing here will be done in the context of this very simple three side is more because it's the one where we can write down everything explicitly in a single. Okay, so this will be the first part of the lecture. And we will do, we will move to the second part and we will discuss how we generate this corresponding feature spaces with stochastic methods. And these will be doing in two parts. First, I will give you a very, very short reminder about Markov chains that is at the heart of computational map. What is a feature space is a feature space of your features is the space that contains the characteristics that you, you want to try to probe. You will see that in the example. I think the example is self explanatory in this respect. So, and we will see the Markov chains and we will connect them to one of the most used methods to probe stochastic properties of partition functions which are not the only methods that one can use for our purposes, especially if one moves away from the classical case and start discussing the quantum case. There are different techniques that can be used to do exactly the same stuff that we have discussed they will be discussing today. Variational methods based on the network or even exact methods, but we will be forgetting about them for the sake of time and simplicity. Okay, so, once we will have defined Markov chains in the context of Monte Carlo, we will illustrate what is the main challenge in defining meaningful stochastic representations of partition function classical partition functions and these main changes is so-called critical slowing down critical and then we will see that it is actually possible to solve this problem in a specific set of problems out of which you can imagine the 2D easy model belongs to and the methods that solve critical slowing down essentially are called cluster algorithms. So this is, these are the three topic will be discussing today. So, let us then start game characterization of partition functions. Here it is, sorry. Oops, for some reason page is not moving. Let me try one sec. You like this? I will place everything. So partition functions. So yesterday we have seen, so what was the embedding space of these three cities in model? Just a quick reminder, we said, okay, every configuration is just labeled by the three possible values that this thing can have. We can label our configuration sigma one, two, three. And in order to describe the switcher space, we just use a cube. Okay, that was the most natural thing to do. I'm so sorry. Yeah, please. Is it possible to say again, what is embedding space? Yes, the embedding space is the space that contains our data. Okay. So remember what our partition function is. Okay. You wrote an example of possible partition function. This will have to be some. E, and then you will have a sum of possible configurations of this object. And you can decide that the configuration that appeared in your model are just some of the configurations that are possible. For instance, you can add configuration with a coefficient of 0.2 plus the configuration, which is up, down, down, efficient 0.5 plus configuration, which is down, up, down with a coefficient 1.2. Okay, this will be reasonable partition function. Maybe not very reasonable, but tool partition function, then the embedding space will just be the space on the door, which you picture your partition function. Okay. In this case, this will be represented by numbers that corresponds to typical configuration. So each configuration lives in a space, which is, if you want, in the case of this model, nothing but the configuration space. Now you can think about it as the configuration space. Cheers. The reason why I did not use the word configuration space from the very beginning is that for instance, if you think about extending some of the concept that we discussed today to the quantum mechanical problem, then one has to be more careful. And it's a long story, but then it's not configuration space to consider anymore. So our embedding space is made out of these vectors. And now if we want to represent a given state, we can decide for instance that the state on the bottom left of our cube. Sorry, somebody, I will have to mute somebody. So the state on the bottom left is nothing but the state where all the spins are down. And you can imagine that if you move one on the right then the third, the first thing changes. If you move again. But now in the y direction you are flipping the second spin. And if you move now in the C direction you're flipping the third spin. In this simple representation, you see moving in this direction as we have the first spin moving on this direction, the second spin and moving vertically as we have the third spin. So that's our embedding space and that is very simple. In principle now, for example, we want to depict where the state down down up is where it is okay the first thing is down so as to lie on the bottom. The second spin is down so as to lie on the y equal to zero line. But the third thing is up so this will be the state. So the point here corresponds to a state, but then some of you asked already last time is asked the following question I mean now if we have a partition function. In general, I mean, the weights in front of this difference it will be different so how do we now describe the partition function inside this embedding space and this is exactly what we will be discussing now. So there could be third spin. There is a third spin in the sense that this is this guy. Is it clear or what do you mean by third spin or third value of the spin like in a pot's model. So something which is up down at zero. Was this your question. I got it or you got it. That's fine. I just want to mention that whatever we do now is spin one off indeed. But it can be extended to arbitrary spins and even to continuous variables. The point is when we will do that. We will lose the simple intuition that we get from the easy model and actually in one of the reference that I sent you there is easy model three state post model. XY models. So all the different types of possible data sets that come out of spins spin. So now we want to define our future space. And to do that, I have to completely raise this feature feature space. And the idea of this feature space is because of this specific coefficients that we have in our partition function. What we want to do now is instead of having something that fully characterize it, we will characterize it on a statistical. So we will give up the full knowledge of the partition functions or give up and replace it with a stochastic representation. How do we do this in practice for our simple three site easy model. Okay, reference point. So there is the following. So we define a number of states that we plan to sample. States that we sample states here you can replace with configurations. Okay. And this number in the note I've wrote it. Who has M. No, the visual space has nothing to do with face space. No, not in our context at least. You will see it now. Cheers. Cheers. So we sample states. And what we do, we collect several, we collect several such a space equal to configuration space. No, let me finish. Okay, but they're there, of course, very closely connected. But they're not the same thing. So now remember what we have. Our embedding space is made out of this strange squares that form this cube. So now what we do we, we, we take a finite number M of samples for in our case we can take M equal to three. You don't have to do it typically and is actually very large, but we do it for our. We sample our partition function, but it's not that we just take three random states. With probability distribution, which is the one defined by our partition function. It's minus beta. This has to be normalized so that it's a probability distribution. So what does he, what does it imply? Okay. Now, suppose that we work at temperature, which is much smaller than the corresponding critical temperature, or in our units, this is much more than one, which is the coupling strength. Now, what are the states that we will sample most of the time in this regime. If the temperature is really low, you will mostly sample the states which have the lowest temperature. So this implies that our state with only three samples is likely it in this guy here and this guy here, which are the state up up up and down down down most of the times. Okay. The other states exactly will spin up or spin down. Correct. So most of the time we will not be exploiting the fact that the configuration space that benefits is three dimensional, we will only see two points. Okay, now you can tell me three points out of it is not a great is not a great set to define a geometry. But with a bit of wishful thinking, you can understand that two points define a line. So most of the time we just sample lines. Of course, there will be some specific cases where this is not happening. We are also simply something outside. So out of the state we sample, we will get mostly configurations up up up down down down. And we will just repeat it several times. And this will define our future space. So you will see there is a connection of course between the future space and the configuration space the configure the future space is embedded. Okay. So this is not a configuration space in the sense that future space is a collection of configurations. If you want, but this is not a random collection of configurations is a collection of configurations that are selected by our sampling that is dictated by the probability distribution that is the very same one that defines the partition function e to the minus beta. So we imagine to apply the very same procedure to every other whether defined probability distribution is just that in our case energy is the most significant one. Now, what we're interested when we do data mining of the many body problem, we are interested in properties of the future space. What we want to study properties. I have a bit of a problem with the letter u. I'm sorry for that. We are in these two states in the thermodynamic limit to logically. We are not talking about them on the clinic limit yet. Okay, the money on the clinic limit will come in the last lecture. Now we are only seeing a simple three set example to the best of my knowledge. This is not correspond to any many bodies, but it's just to give you a physical interpretation of what is going on. Here for low temperature case you have chosen and equal to two. And you caught me okay in principle I said okay there are only two points here, maybe one of them is repeated if I sample more times. Okay, you can imagine. Indeed, this is actually tricky. Okay. Repetitions will be a problem. But I'm hiding this under the carpet. I'm writing this like check the repetitions and the reason why it will be a problem. Oops, a will be discussed by Alex. I don't think next week, but next to next. Okay, but it's a very good point here I've chosen and equal to three and I sample one of these states maybe twice so I forgot that I sample it twice, but it will be important later on to be careful about not sampling points more than twice. You will see that if I have a small system you always sample points more than twice. So this is a bit problematic. However, if you start looking at the many body system and we will be looking at easy models where you have 100,000 to million spins, the probability of checking getting the same configuration twice unless you really go to zero temperature to temperature which is minus 14. It's not there. And so at the many body level, this repetition problem will not appear. Markov chain Monte Carlo in coming to play in the construction is very good. Okay. One of you has already understood where we are going. We will utilize I mean now we can sample our configuration almost by hand because we can write down everything, but at some point we will not be able to do that. It's essential to have a method that allows to allow us to stochastically sample a system and then we will have to use Markov chain Monte Carlo. It's exactly what we will need to generate our feature space. Now you can also be forward thinking and say why the hell do you have to do Markov chain Monte Carlo. Let's just do an experiment and collect snapshots of a partition function. Well, indeed, this is actually true. Some of the things that I will be telling you are not only applicable to numerical experiments, but they can also be applied to real experiments. That is, however, one fishy point in doing this is that when we do metrics, we have all errors under control and we do Markov chain Monte Carlo specifically we are all errors under control. Thanks to the fact that we are just evaluating statistically. So we know how this series are generated. So apart from memory that will be a problem that we will be discussing everything is mathematically extremely well defined. When you have experienced what can happen is that there are systematic errors that you are not aware of. Because you are collecting samples and maybe that day the metro has been running a bit too fast and the magnetic field in the underground was a bit higher and then your atoms don't behave nicely or you are doing a bio experiment and there is a reactant that is not perfectly, I don't know. Systematics and systematics when we discuss about when we think about their roles in a many body problem. In particular we've seen in our context here they can create outliers and outliers can be problematic because they can give them the interpretation of the features we are interested in because we are not able to decrypt them anymore. So that's why we will mostly focus on Monte Carlo. Okay, so now I'll show you the feature space for the for a low temperature space we can try to do the same trick now for high temperature. Now it's a very simple exercise. Oops, let me raise this and we start looking at the high temperature case. If we take T much larger than Tc. So, and we reconstruct our very nice cube. So what will happen in practice is that whenever I have a fan number of samples in my first sample gets this guy, this one, and this one. Then I take another sample, and my next sample to get this, take this again, and take this, then I get another sample, and this other sample will get this, this, and this, and so on and so forth so we are spanning the full cube. Because there is no energy selection anymore at very high temperatures, all the weights are of the same order which is one divided by a number of configurations, all the probabilities. In this case will be one over eight for us. So all states are equally probable. So that you see now that there is rather fundamental difference in the feature space of high temperature partition functions of these three settings is immoral and the low temperature phase. So, what is one, what is one difference that strikes us. Okay, and it's important to understand that because we will not be looking at very fine human effects we want to study many body effects so we want to have features that are visible. Well, there is one that it is clear, I mean remember what I told you before I mean when we were looking at very low temperature, we added all our points where just sitting on a line. Okay, so T much smaller than TC, what the ad is real line. And now, if we have T much larger than TC essentially what we're doing. Let me try to use this. We are covering the full 3D structure right. So what we in details, and that is a cube. And what is the most striking difference between a cuba line. You see that is that dimensionality. And indeed, what we will be looking at is a specific quantity which is called intrinsic dimension of a data set. Oops, apologies. And we will be calling this quantity ID. And what I've tried to show you now in this simple example is that ID is equal to one 40 much more than TC, and I see is equal to three 40 much larger. So ID is a simple proxy to distinguish phases, which are fundamentally different. Okay, already in this three sites model, we're talking about phases. It is really not something we can do as was questioned already. So I had a question. We will have to use a random walk on this space to collect. Yes, it is absolutely reasonable to use random walk in that case is actually equivalent to an equal weight. But you will have to use a lot of work with a very, very large. If you do many body, you will have to use a lot of work with a very, very large working time because otherwise you start adding memory effects and memory effects are not good to create statistically uncorrelated. So we have to find a feature space. And so now I would like to take a very short break, like two minutes also because we need some water. And during this break, feel free to add questions to the zoom chat, and we will start the next part of the lecture by discussing your question. So we come back in three minutes. Okay, so three minutes. Okay. That's now. So let me start by reading some of the questions that you guys have asked and comment on that. So the first question is, which is the ID on TC critical temperature. Well, let me make two comments here. First, we have a, we have a finite small system. So there is a really not critical temperature. It's not this transition, but this will be the question that we will be asking when we stop many body. And then it will be I don't really allow to understand what happened to TC at the critical point and I will show you at the next section. So this you have to wait a bit. Can you explain ID again? Yes. Even though it could be remind yourself that I'm not giving you a rigorous full-fledged explanation of what the idea of a data space is I'm just giving you simple intuition, what it is. So essentially ID is the, you can see it as follows you can see ID basic dimension. It has the minimum number of variables that you need to describe your space. Of course, they have to be independent to describe your space. Now, if I have a line, I only need one variable with the position along the line. If I have a cube, in order to tell you where you are on the cube, I cannot specify a single point. You can specify three points. Direction X, Y and Z, or if I continue distance from the center and two angles. So you can see that as ideas, really the minimum number of variables that are needed to describe the set, independent variables, of course. I hope that is clear. We can get the TC in this space. No, this is again, good question. You will see the last lecture. What is equal to TC? Again, you will see there. It's good that all of you asked this because it means that you will be interested in the last lecture. Okay. In TC, why all configurations have the same probability? Well, the reason is the following is that they don't have mathematically the same probability. But what you can do, you can at very high temperature, you can expand this explanation. Very high temperature implies beta, very small. So this will be approximately one minus beta E plus. So you see that the leading order in this correction is energy independent, which implies that the dominant contribution to probability of the states will be the same for all states. Professor, can you repeat the part why we have to give up full knowledge to create the official space? Okay, the reality is that we don't have to give up. The reality is that it is, there is no conceptual reason why to give up. But there is a very simple practical reason that that typically when we deal with a problem which is very complicated, you will not know the fuel, the full feature space of the partition function to start with. Because computing all the coefficients in front of the, that make up the partition function will not be possible is exponentially. So I think it's more of a practical reason why we need the knowledge rather than fundamental. There are also, there is also fundamental reason that I would forget about. There is another question here. Line for T much more than Tc has only two points and the quads are as it, then why is the dimension one three, since we can only sample finite property. It's just to classify. Yes, it is just to classify. You're correct, of course. But it's just to give you an understanding that in this simple space, one can have this simple structures. And if you want to do an example, where you can see a true line and a true and a true cube in a simple three site model, you can you can take the three side x y model, not not really, of course, it's too complicated. Because there you have continued that continuous variable so the configuration space is really are three. So then you can really see lines and cubes. And actually this is what we do in one of the archive reference that I put on the elements channel. Is there some relation between the national with the fact when there is, of course, you're there's a very good point. I don't want to elaborate in elaborate on it further. Since I'm not so sure. Most of you are familiar with the fact that Alex might be discussing this, but there is a very strong relation. And what is ID at T much more than TC for anybody problem. This we will be discussing in lecture seven. So this is really one of the questions we will be answering. What happens. So it is useful to use stochastic representations for many spins. Yes, it is, it will be super useful with fundamental actually because you cannot handle anything. There are very, very few things exactly. So you will have to rely on a stochastic representation for instance automatically perfect. Okay, so now let me know we will have to go a bit faster on the next point but I don't care to be honest because I prefer to answer your questions and if you want to study the cluster algorithm I will send you the notes and you can study that in detail. But now we have to discuss a bit Markov chains. Let me try to use the same notation. Now, let us briefly review a few definitions. I mean, suppose that you have a set of configurations. We call them x1 x2 blah blah blah xm we denote a series of these configurations of Markov chain probability distribution of the full series can be written as the probability of each single one of those in our my connection is unstable. Sorry, I hope he's back. So what does this condition tell us in terms of the sampling that we discussed. Well, it is telling us that the probability of observing a given event in our Markov chain is independent on its history probability history of the series. So of course this concept has huge implications. Think about master equation. The quantum systems and so on and so forth, but it's also very important for us. Now, because we have given a definition of our future space in our future space we said okay look, we want to collect configurations. We want to be statistically independent. Okay, because it has to be only as to be only a sample of the probability distribution so it looks like the configuration that we are searching for are exactly this Markov chains. Okay, it's exactly matching. Perfect match feature space definition questions. Just a second. Maybe more, maybe with example what he depends. Yes, I will explain it in the next slide. What's the future space. No, no, it's just a collection. In our case it will be the future space but for a marketing is just a collection of configurations, but or that this is very nice for my definition. This is how we generate a Markov chain in something like the easy model we are thinking of. Here I need to have a bit of a larger many body example so let me take a square lattice will define I will define as this mean future space is memories. Is that the face is generated by a process which is memory less. The concept of memory in future space does not really make sense. Okay, but the way it is generated. It is with the process which is memory less. Is this clear. Cheers. Now I want to write down my easy model. And I can put spins. Suppose I have a configuration of this type and I'm working at very, very high. Now, yes, I did before at very, very high temperature. All configurations roughly at the same weight, the same probability. So they can be generated by just flipping single spins. And the idea if I give a configuration this I call this will be my x one, then my next configuration that I will call x two. I can just obtain by taking a random spin and flipping. For instance, I can take this guy and then my next step three. I can do the same with another spin and flip it. And I can go on so on and so forth by just having to flip single spins. So this for those of you that are familiar with this so called single spin update in Monte Carlo methods and in Metropolis, which is good. Because he's telling us that if we want to generate statistically meaningful samples for very, very high temperatures. So in the easy model, we can just take configuration and randomly spin flip with a probability essentially one. One has to write down the corresponding balance equations and forth but just to just the degree of the physics is not difficult to generate essentially statistically meaningful samples for easy model T larger than TC. So let us do the same exercise, but not in the high temperature case, but in the low. So we now take T much smaller than TC, redraw my spins, even of who cares. I mean, yeah, no, maybe that me. Let me raise this completely because then it's not clear what they're doing. There is minus minus minus. What you meant. Let us draw our picture again but now low temperature. So what will happen at low temperature is that most of the spins will be parallel to each other. Not all of them, of course, they will be from time to time spins that one can further flip. So if you look at this plot, and if we propose to flip the spin here, we will likely not flip it in a Monte Carlo method because this will actually increase the energy. So this is unlikely, even though it is still possible. But if we now propose to flip this spin, this is a spin up. So if I flip it, I get a lot of energy because I'm actually favoring the firm magnetic state. So this will likely be free thing can happen. X2 can happen recursively. So you will be able to create a local excitations that violate the firm magnetic state. But at the same time, lower the energy by favoring large for magnetic states. So again, also in the T much more than TC, single spin flip are fine. So the simple Monte Carlo method with simple spin flip will allow us to generate very large data sets in particular in the form of Markov chains at both T much larger than TC and T much smaller than TC. Now the question is very clear. What happens close to TC? And close to TC, okay, we are very short on time. What happens is that there is some of this single spin flip will not be efficient at all. Okay. And the reason why we will not be efficient at all is because what happens close to TC. Okay. At very small and very low and very high temperature, the correlation length is over a lot of fuel at this side is very short. At T equal to TC or TC, the correlation length. This implies that if you modify only locally spins, you will take a huge amount of time to actually generate two statistical and independent configuration. So what we instead need to do, we have to change our configurations at very large scales, which looks impossible. So this implies need to configurations scales. I wanted to show you a video of why it is so. Okay. Because I think it is more. Let me retrieve it. Sorry. This clip from YouTube. Very beautiful. It's one of the things that you should not be watching before going to sleep because otherwise, it stays here. Okay. Let me show it to you. Just a sec. Computer. Share screen. Share it here. It's possible to give us the name. Sorry. Is it possible? Is it possible to? I cannot hear you. Can you please repeat? Sorry. Okay. Is it possible to give us the link of that video? Sure. Sure. Sure. I will send to you. Thank you. So can you see the video? Yes. So this is a person that has decided to put a set of snapshots of a Monte Carlo configuration of the easy model as a function of Monte Carlo time. Now, the critical temperature for this model is 2.269 is this stuff here on the bottom on the top right is below the critical temperature because before below the critical temperature the spins are either mostly up or mostly done. Okay. You can see that. Here you will have mostly red or mostly green. Or everything green, everything red, two green, four green, five green, one red. And so this is the federal phase. Either you change all the spins or you change one or two, no more. Instead, if you go above TC, for instance, in this case, 2.8, it's sufficiently above TC, you will see that the spins are a bit random. And then they can change you, some change here, some change here, some change here, so on and so forth. They don't change by a lot. One by one. Now, let us look at what happens at TC. Okay. If we run TC, you see that there are clusters. There's a clusters of red, cluster of red, cluster of red, they are not single ones. They're really clusters. We move and the rest is green clusters, clusters, clusters, cluster of green now cluster of green. So this implies that the physics of the is more at the transition point, and also in its vicinity, of course, it is not governed by the behavior of single spins, like these in the very high and very low temperatures. But rather the opposite. And this implies that if we try to have a meaningful mark of chain, or there is more if we try to break it down, there is no that we can do that with single spin flips. The formal reason for those of you that are in the known is because something which is called autocorrelation time diverges out of the correlation time of the algorithm of single spin, it diverges at the critical. It's huge. It's a good system size. Because it is not capturing the correct degrees of freedom. Right. So in order to fight this, I mean, we would like to study phase transition. So we have to fight this problem. And okay, I ultimately didn't have time to explain to you how we do it. You will read it in the notes. The idea is that we will need specific type of new algorithms that generate mark of chains, not by flipping single spins, but now they flip the new degrees of freedom which are clusters. So they will be able to flip huge amount of spins that are connected one to each other, one to the other, at the same time with a single move. So that's exactly what was done in the simple YouTube video. And you can write a code of this type with 80 lines in Python is very simple. You can find also some in the Internet, of course. And the idea is that these, these algorithms, not only they were fundamental in the 80s and the 90s to actually simulate the easy model. I mean, the model at criticality was not really efficiently simulated until 1989. They are also fundamental nowadays, because if we want to generate reasonable and physically interpretable feature spaces of a partition function, we really rely on these algorithms. Questions, will we see any numerical example about cluster or just the technical, you will always see the technical explanation. But if you wish, I can point you out. I mean, what I've included in the notes is a video of banner crowd lecture notes at ICTP where it discusses also cluster algorithms. And I mean, it's discussed also in my notes. Actually, you can write down the algorithm out of my notes. It is very easy. And if some of you is interested, I can also give more information. Of course, it's no problem. So let's run now. I want to know is meta dynamics. So it's doing something actually different. You can also use meta dynamics to sample efficiently, the easy model in 2D, but he's a different thing. But thank you for the question. So with this my lecture today I think has also been out of time and I want to take anything so maybe there are more questions do cluster algorithms rely on knowledge of your parameter. What is the microscopic degrees of what are the collective degrees of freedom. They do not immediately rely on the order parameter. Cheers. Okay, so if there are no other questions. Then. Thank you very much, Marcelo. And we look forward for the next lecture. Thank you and everybody. If you have more questions, just post them on the element channel. I've just opened it for questions. Question related to the course and I will be happy to answer that element. Sorry, not all of us has has invitation for element yet. I mean, some of us don't have it. Once you enter, you will be able or you can drop me an email if you have a question drop me an email. I would like to be part of the element if, if you could like make it public or something so that we can join. Yes, sure. Right. The secretaries and they will, they will let you in. Okay. Okay. Okay, so thank you very much. So we take a short break and we reconvene at quarter past 15 central European time with the Dominica. Thank you very much.