 Thank you very much, Matteo. It's a pleasure to be here and give these lectures to you. I should thank the ICTP for giving me this opportunity to begin with. My name is Shaheen, maybe a little unusual for you to remember, so I write it down. What I'm going to do is to actually give you in nine lectures a very broad view. So it will be a little bit harder to be introduced to all these topics, so I'm going to start from a critical phenomenon, go into conformal invariance, shramble of net evolution, applications of it to various statistical mechanical models. And the central view is bringing in some analysis from conformal field theory to analyze curves which appear at the critical point of the model. So it's what I have called them critical curves. Actually, many people call them critical curves. These are curves which appear at the critical point in the sense of domain walls or interfaces between two phases of a system. So I will try to explain many topics, so I will not go very deep into them because I have to explain many topics. I have set a number of exercises for the course. I have talked to the directors. I have agreed to give five credit points for the five marks to the exercises. If you do them and you show them to the tutors and me and if they are correct and you haven't copied anybody else, you've done it yourself, you get five points of the exam. Fifteen points will be for the final examination. These exercises, some of them are analytical, some of them are computer exercises, which would be very nice to do because you will get some sense of the problems as we talk about. I have put them up on the portal so you can download the exercises from there and I suggest you start doing them and as you go through them hand-dominated, but they don't have to be in sync with the lectures. You can be ahead of the lectures. It doesn't matter. It covers the whole lot. Now, let's start with this. The first lecture will be on critical phenomena. Oh, I better give this overview of this course. I will cover... Shahid, before you start, I think all of you forgot something. Ah, ah, ah, ah, yes. Everybody forgot. Remember, otherwise... Yes. So, this is going to be what we try to cover in this course. I will give you some examples of critical curves and geometric approach. This is actually creating a geometrical approach to a thermal problem. Critical phenomena or the critical point is a thermal phenomenon that systems go as you tune their temperature to a critical value. They go critical and these curves appear. Now, the question is, can I analyze this event from a geometrical point of view? So, I would have to, for instance, derive the critical exponents based on geometrical properties of the system, which we will do eventually. To cover critical phenomena, we have to know certain things such as realization group, conformal field theory, fractals, et cetera. Then, I will talk about Shamlovna revolution, which is a method in stochastic processes which is used to characterize conformally invariant traces. So, these are stochastic processes which are conformally invariant. Then I will show you how you can do calculations with SLE. Then we go on to talk about surface growth. And surface growth is an important topic in physics of its own, but from our point of view, if you take a grown surface and you take a level cut through it, you will see many loops. You will see level sets of a surface. These level sets then can be analyzed from a statistical point of view. If you take a section of them, they are SLEs. If you take the whole loop, which you can do, a much more difficult mathematical tool is needed, which is a mathematical methods for analyzing level sets, loop models, and eventually conformal loop ensems. So, this is the whole list of topics which I have to cover. And this I will put on the interface. There are some general references which you can look at for critical phenomena, which you are probably familiar with, but anyway, conformal field theory, normalization group, SLE, surface growth, and loop models, and conformal loop ensems. These are some standard references which you can look up for the topics that you need to learn for this course. Okay, today what I'm going to talk about is start with scale-free random paths, talk about critical phenomena, order parameter symmetry breaking, et cetera, talk about ergodicity and breaking of it, universality and the realization group. So, hopefully today's lecture is completely familiar. So, let's start with this. This is a simulation of an Ising model in which blue is, say, a spin up and white is a spin down. So, you see a number of clusters which are formed by blue and a number of clusters which are formed by white. What I'm going to be interested in is the interface between the two groups, so which would be, I don't know how to get the laser beam out of this. So, you see that I have tried to trace out the boundary of one of these clusters and this is now a curve or a domain wall and a curve in 2D. As the Ising model moves towards this critical point, this curve will become critical or it becomes scale free. So, these are the things that we are interested in. We want to note, oh, in the side. Okay, yeah, so these curves are what we are interested in and the proposal is that if I can analyze this curve, I have a lot of information about the critical phenomena which is happening here and this is a geometrical approach to critical phenomena. That means I study something geometrical to explain something thermal and this is probably the interesting point about this work. So here, let's look at the problem in a little bit more detail. This is Ising model on a triangular lattice. And the way it is organized is that a path starts from A goes to C then to B. A spin up is on the left side of the path and a spin down is on the right side of the path. So this is just a domain wall between the two phases of up and down. How can I characterize this path? This is the basic question is how do I separate, characterize such a path? This is obviously some kind of a stochastic path because as you move towards criticality, the probability of seeing two spins will be equal. So this path may waver anywhere on the lattice. And there must be some sort of Markov property as well. This is what we will eventually see that we can divide the problem into going from A to C, a path which connects A to C and then a path which connects C to B. Somehow this must have a Markov property because we know the Ising model has a Markov property. So going from A to C should be, or going from C to B, I should say this way. C to B is independent of going from A to C. Here is an interface from a much larger simulation. And it points out it's been, if you look at it carefully, you see that the boundary conditions are set such that these are all a spin up and this is all a spin down. And so this path is forced to come in at this corner and go out at the other corner. So we have a separation between the two spins. Obviously, islands of the other spin is appear in the opposite spin. But here we have a complete interface and this interface actually looks like a fractal. I'm glad that in the previous lecture, fractals were discussed because I will give a fuller discussion of fractal for my purpose, but this is clearly a fractal. So we are going to have to deal with these paths which are fractals. Why is it obvious that this is going to be a fractal because at the critical point, the physical system is scale free. So I should get the same path if I change the scale. As we make the scale bigger or smaller, you should not be able to tell at which scale I am. Now, for this particular problem, I think model, this is what we know. If the temperature is below Tc, so one of the spins is dominant, the other spin is happening in a rare state, essentially nothing interesting happens because the shortest distance between two points is the line and I don't have a fractal. When it approaches T equal to Tc, now I get a fractal. And this fractal is of dimension 11 eighths which we will see how this 11 eighths will be derived later. However, when T goes above Tc, which the system is not critical anymore, it will have islands of one spin inside the other. But however, what we have is a cluster boundary which is still a random path and it is scale free because we can interpret that as a percolation, a boundary of site percolation. And therefore that has a fractal dimension of seven quarters because that comes out of site percolation. This is the taste of sort of ideas and calculations we will do in this course. There are other scale free paths which we will have to deal with, not just critical points. For example, we can create a random walk. A random walk is created on a square lattice by a standing at any point and you drop, you choose a random number. Probably one quarter, you may go right, left, up or down. So a path like this is created, which we call, which is a random walk. This is also a scale free path, but it is not a sort of path which I'm interested in because it crosses itself. I expect critical interfaces not to cross each other itself because I have to be able to interpret one side of it as one spin, the other side as the other spin. So I have to do something more complex. I can either create a self-avoiding random walk which is this guy, that means you come to a point, you come to another point, but here you don't have a quarter for the choice because you cannot go back and you cannot go forward. You then have, you only toss a coin for one half probability to go up or down. And in this way you create a self-avoiding walk which it always avoids walking onto itself. Alternatively, what you can do is to remove these loops. You create a random walk and then you walk along the walk and every time you come to a loop, you remove it. So if you remove the loops, you end up with this path which is a loop-raised random walk in the sense that all the loops have been cleaned out. This gives me another random walk which doesn't have loops so it comes into sort of thing that I have to study. This is also a scale-free path. So somehow my methods must have an answer for this situation as well. If you've never seen a random walk, this is a computer simulation of 10,000 steps for a random walk. It crosses itself many times. It's a fractal, not in the exact sense that fractals are things which look exactly like themselves when you take a smaller scale. But here if I take a smaller scale, it doesn't look exactly like itself but almost like itself. So in this sense, we also have fractals which are not identical to themselves when you change the scale but almost similar to themselves. Right, so this is the setting, my motivation and I have to start from critical phenomena because that's where things start. I also am very receptive and happy to answer questions as I talk if you see things which you disagree with or you don't understand, just raise your hand and ask your question. This is a photograph which I've taken in Firenze and I thought I'm sitting exactly in the same place as Enfest. But okay, it doesn't have to do with, physics doesn't have to do with fountains. This is not how you become a great physicist. But it was good to have this photograph. So poor Enfest proposed the first idea of what a phase transition happens and he classified them by their order because at this point, at the point of phase transition, the free energy becomes singular in some sense. In some sense means that one of its derivatives can be divergent. And he said that maybe if we set the problem like this that if the nth derivative becomes singular, we call it the nth order transition. This turned out to be wrong because actually physics is not organized like that. Physics is organized by having first order transitions or if you like to call them discontinuous transition as opposed to continuous transition. So we essentially have two major classes and then continuous transitions are classified by their exponents. So here is a phase diagram of water or a solid liquid vapor system. But this is very similar to the water phase diagram. You have a solid phase in a pressure, temperature graph. You have vapor and then you have liquid. So this would be the gas phase, this would be the fluid phase and the solid phase. And these are lines which separate these phases. If you move along the line like this, then you see a sharp change in volume or density and you see as you go from vapor to liquid and we call this line a line of phase transition and then there is this point. At this point, beyond or beyond this point, you no longer have this separation between vapor and liquid. So something as strange is happening at that point. These phases come to a similar state or you can start from vapor, go and turn around and come back here and you end up with a liquid. You have a continuous phase along which things change and then eventually you end up in a different phase of matter without going through any singularity which you would see through this line. So this point is playing a very important role and this was how phase transitions were discovered in the 19th century. What this point was observed experimentally and at that point you get an effect which is called critical opalescence and that is water becoming completely opaque and it's no longer transparent to light. It happens because at this point, all fluctuations become important. You have a divergence of fluctuations and because all fluctuations are present, you no longer get a, you can absorb light at all frequencies and it becomes opaque. And from that observation, the study of phase transitions and critical phenomena started. Another way of looking at it is to look at a magnetic system. In a magnetic system, I have a second, so H is really playing the role of pressure. It's the external field and I have non-zero magnetization below its critical temperature and I have a second-order transition at this point. The first-order transition and sorry, the second-order transition at this line and I have non-zero magnetization here. So this critical point is exactly playing the same role as what you saw in water. So here is the general picture. At low temperature, I have more order and less symmetry. At high temperature, I have less order and more symmetry. So this is usually called an order-disorder transition which is a little bit of a dilemma because we have the second law of thermodynamics which says that entropy is always on the increase. There is no process by which you can decrease entropy. So this doesn't make sense that a system can change its order, it's the level of its order because it has to always increase its entropy so it must always go towards less order. But what actually is true, the truth is that in the free energy, you have two terms. At high temperature, you can ignore this so that the free energy is decreased by increasing temperature. So this gives you the disorder phase. However, at low temperature, this term can be ignored and you can go to the minimums of the internal energy and typically atoms of molecules come and sit down in a very ordered fashion next to each other and it decreases their entropy. But this can only be done if the extra energy can be dissipated. So this order-disorder transition can only exist if you have an open system. It doesn't make sense for a closed system and the second law applies to closed systems. At this, so as we make this change, we have less symmetry in the ordered phase which is a little bit of contradiction. Always I have to explain to people that increasing order reduces symmetry. Conceptually it may be strange but what actually happens is that because atoms or molecules sit down in a special places along the ellatis, you lose a lot of symmetry. You get much more symmetry if they are allowed to move around and they have random positions. So some kind of symmetry breaking is happening as you go from high temperature to low temperature and you make this transition. At this very specific point, when T is equal to TC, I also have a scale invariance. So whatever symmetry I had here, a scale invariance is added to it. So at this point, I have even larger amount of symmetry. Once I pass through it, I have less symmetry than here, symmetry drastically reduces. So here I must have a symmetry breaking for this to happen and symmetry breaking gives me a scale invariance and a scale invariance, I am going to present to you an argument that it leads you to a much larger symmetry which we call conformal symmetry. And this is a tool which will be very useful in analyzing this phenomena as I have a much larger symmetry group or in other words, the conformal symmetry group. So I have three seemingly different concepts which come together. And I'm going to explain each one in a little bit more detail. I told you that spontaneous symmetry breaking happens. This has to mean ergodicity breaking and it also has something to do with renormalization group flow. Okay, I'm again at a natural stop. Any questions? Yes. If I have a, yes. If I have a closed system, this closed system has an entropy and delta S has to always increase. So if I go, if I have a, in this system as I change the temperature, delta S can never decrease. So I can never go to higher order because delta S always has to be positive. If you ignore U, this is okay because as S increases, F decreases and I go to the minimum of free energy, there is no problem. However, if T is a small, you have to ignore this term and the minimization of free energy is the minimization of internal energy. Minimization of internal energy increases, increases the order, but decreases the entropy. This can happen, a decrease of entropy can happen if you are able to dissipate some energy. So some energy must come out of this system for this to happen. So you cannot really say any of this about a closed system. You know, it's a contradiction that we have to worry about because we somehow think that we are always dealing with a closed system, we are not. In critical phenomena, you are not dealing with a closed system. If the system is closed, you can never have an order disorder transition. Could you start from the beginning, the question again? No, it is not mathematically the same, but it's physically the same. So here is the point. Every physical system, I shouldn't say every physical system, and a standard physical system usually has invariance under translations plus rotations. So these two invariances are guaranteed for us. As we approach the critical point, to this we add a scaling variance. So this is a transformation which x goes to Rx, where R is a rotation. This is, sorry, wrong. These are translations where x goes to x plus some numbers. A vector is added to every coordinate. Rotations is where x goes to Rx. R is a rotation matrix, so you take the x. A scaling variance is another space time transformation which changes x by some number. This transformation is a very special transformation. This is not the kind of invariance you usually have in physical environments. Like here, a room is usually organized around the scale of a meter. So that's because humans are meter-long entities and they live in rooms which are 20, 30 meters square. And physics usually has a scale. So either we talk about megaparsecs when we go up to high into cosmology, or we talk about micrometers or nanometers as we go down. Physics always has a scale. A kind of physics which does not have a scale will have this symmetry. And it is difficult to imagine a kind of physics which doesn't have a scale. It's so natural and intuitively clear for everything to have a scale. Freedom from a scale only happens near a critical point because all fluctuations become equally relevant. Yes, every distance will be equally the same. So you can multiply distances by a scale or not. Now, the point is that this forms a group which we know, for example, this can be the Galilean group or the Poincare group depending on whether you accept a special relativity or not. Now, when you add a scale to it, this is a single generator. It's just one operation. This guy, G plus one, this itself is a group. Now, so mathematically, this is a complete set of transformations but the amazing thing is that it doesn't stop here. Some other transformations are induced. So there are induced extra symmetries. And this happens not because of mathematics but because of physics. Because from a mathematical point of view, this would have been enough. These guys get induced. So we get another set of generators which let's call them K which when added to that, it will give me a much bigger group which is the conformal group. So the induction of this extra symmetry is not mathematics, it's physics and it can only happen in physical systems. The proof says that if you have conservation of energy and momentum, if you have unitarity and if you have local operators, then this will be induced. If any of these do not hold, then it's possible that this jump will not happen. However, unfortunately, this is a sufficient but not necessary condition. So there are situations in which unitarity doesn't hold and still we have this enlargement. So that part of the story is a little gray and a lot of work has been done to clarify it but it's ongoing research. Okay, okay, I'm going to tell you why these three concepts meet together to explain critical phenomena. SS dot spontaneous symmetry breaking. Symmetry is defined as an operator which commutes with the Hamiltonian. If you have an operator which commutes with your Hamiltonian that is the symmetry of your theory. The problem is with the vacuum. You usually have that. You start with that. You have a commutation with Hamiltonian but the vacuum may or may not be invariant under this transformation. If it is invariant so that this operator annihilates it then this is a symmetry of the system and I have no problem, fine. But if it is not that is the action of the symmetry on the vacuum gives you another state which obviously is also a vacuum because this means that all the vectors of S will have the same energy. This means that I have another vacuum and this is a contradiction in quantum physics. We cannot have degeneracy in the vacuum state. So something must go wrong when this happens and what goes wrong is called spontaneous symmetry breaking. You, this symmetry actually stops holding and it happens by a mechanism which is very, very interesting and exciting. It is through spontaneous symmetry breaking that the Higgs particle was invented and eventually observed a few years ago. So this is a simulation of an event in the CMS experiment where a Higgs particle is produced. And it's through observation of this event that we observed that actually the standard model of particle physics is correct in the sense that it has a Higgs particle. As you know it is not exactly correct because it doesn't explain certain other phenomena. I didn't explain all of it. See the problem is that if this happens so vacuum is invariant under S, everything is okay. So this is fine. But the problem appears when we have the second situation. So if I have two states O and O prime, they all, they both will have zero energy. So they're both vacuum. And assumption is that they are vacuum, you know what I mean? Meaning that there is no state with lower energy. Now it's actually easy to show that this state, this wave function can have an energy lower than zero if I choose alphabet properly. So there is a contradiction here, which I didn't expect. If you have more than one vacuum, I can always construct a state with lower energy. And therefore there is something contradictory happening. Either this has to be your ground state or there is something wrong with what you are saying. The physical example of it is, for example, the NH3 ammonia molecule in which the nitrogen can sit either on this side or the other side. Both states have the same energy, same configuration. Neither of them can be the ground state. Because by this construction, if you take them ground state, I can make a lower energy ground state, which is the symmetric sum of these two states. So the Z2 symmetry in that case has to break because of the lower energy argument. However, in something like molecular physics, which where you have just discrete symmetry like that, these two states can tunnel to each other. But if you have a very large system, then the probability of tunneling vanishes and you have to accept a spontaneous symmetry break. So this is a model which people use typically to explain a spontaneous symmetry break. For T positive, this potential has one minimum. And therefore one vacuum at phi equal to zero and there is no problem. But if the parameter small t becomes negative, then you see that you end up with a double well potential. With a double well potential, you have two minima, which we can call phi minus and phi plus. Now I have two minima, which they both can act as the vacuum. So you would have to either create your Hilbert space on top of this guy or on top of this. Now if you have just the quantum mechanical problem, then tunneling will create a mixing between these two states and the lower ground state will appear in this form. And that will be the new vacuum. But that's not called the vacuum as a ground state. If you look at this as a field theory, then this has to happen at every degree of freedom and there are infinite degrees of freedom and therefore this tunneling will never happen. System has to break. It either chooses phi plus or the phi minus as its minimum and builds the tower states on top of that. And hence the symmetry is broken. So this is essentially what happens. If you have a, if you have t bigger than tc, you just have one vacuum. As t goes below tc, the system has to choose to go either this side or the other. If the system goes to this side, it can no longer tunnel to the other side because this potential barrier for an infinite degrees of freedom will need an infinite energy and you can never go back to this side. Which is good because we believe that the present state that I'm walking in and you're sitting in it is the ground state of a standard model and it's in one of these vacuos. If a tunneling to the other vacuum would be possible, we would all vaporize and never reappear. Which was what people worried about. I don't know if you remember that when LHC was being built, some people said that doomsday is happening and this transition happens and they even made a movie about it. And they called it ground zero somewhere where everything disappeared down a black hole. But the same physicists were assuring people that the probability of that happening is zero. Although it can happen, but the probability of it is zero. And but it had very little effect in the sense of politicians. Now this has an effect on magnetization and the critical phenomena. In the sense that I define the order parameter for the 2D Ising model as this guy. And now, if I have a field interpretation of it, this has to be either the positive vacuum or the negative vacuum. And in this sense, M can be non-zero. Otherwise, if there is total symmetry here, phi is allowed to become minus phi and M will be zero. I explained that again. So M is related to expectation value of the spin. That sum is usually written to make it correct, but essentially the expectation of SIGMA is independent of the site so that sum gives you a factor N and N goes out. You just need to look at this. However, this equation, this can never be non-zero because you know that the Ising model is invariant under SIGMA going to minus SIGMA. This is the invariance of the Ising model. Therefore, if this is correct, this means that M is equal to minus M and hence zero. It will never be non-zero to observe a critical phenomenon. So I need to have a spontaneous symmetry breaking in the sense that Z2, this is no longer a symmetry. So, a spontaneous symmetry breaking is necessary. However, it also means that the ergodic principle has to break. Ergodic theorem plays a central role in statistical mechanics because we can never take ensemble averages. We always need, in the lab, take time averages. Hence, we need a theorem which tells us that a time average is equal to the ensemble average. This happens only if a time average visits the entirety of the phase space. So, if I have a phase space, the dynamic path will go through the phase space and it has to cover the entire phase space so that the time average will be equal to the phase volume average. However, a path will never cover an entire space. The solution is that I divide my phase space into cells and I accept the cell as visited if the path passes through it. Now this makes sense. I call this coarse-grainning and this coarse-grainning is actually necessary for many arguments based on the ergodic theorem. One of them is the second law of thermodynamics. You know that entropy equals the volume of phase, log of volume of phase space and because of Liouville's theorem, this guy is always independent of time, so the SDT vanishes. But this is in contradiction to the second law. How do I understand that? I also understand this in the sense of grainy phase space, that if you have a grainy phase space, then the Liouville's theorem doesn't apply to it anymore and you can have inch actual delta S bigger than time as Hamilton-Jacobi dynamics is slowly fills out the phase space. This is called the ergodic theorem, the fact that we expect a path to fill out the phase space. Now, yes? It's much bigger than that, yes. Eventually, if you reduce coarse-grainning, suppose you say, okay, I want to make my system very exact and I reduce the coarse-grainning to the ultimate limit, you cannot reach a point because quantum mechanics stops you. But what I'm dealing with here is a much bigger coarse-grainning. So because this is here delta Q delta P for this square is much, much greater than H bar, thousands of folds greater, yes. But yes, you are right. It's in fact has its roots there. Now, the ergodic theorem, so these are things which I have already told you, the ergodic theorem has to break because of symmetry breaking because of this argument here that not all points in the phase space are now acceptable. In fact, in the Ising model, if I have the entire phase space to add, this guy is always zero because I can always take sigma to minus sigma. This is invariant and that vanishes. The only way that this can be stopped is that if I divide my phase space into two parts, one part is positive and one part is negative so that you cannot sum all on all of it. So it's basically breaking means that the ergodic theorem has to break as well. Now, to explain the ergodic theorem, there is a very interesting mathematics. The mathematical problem is that can you somehow prove the ergodic theorem? That is, you take a time path in the phase space as I have here. It's randomly chosen initial position and over time this position fills the entire phase space. Mathematical proofs are very hard to get. So in fact, only in certain circumstances proofs exist. One of them is this Bunimovich stadium. It's a very special stadium in this shape that there are two semicircles and two straight lines connecting them. So it's not a complete rectangle. It's not a complete void. And if this thing works, it won't work because I have a PDF. This is, you can find this on the internet. There is a very nice simulation where you set your particles in the center of the stadium and then they fill out the whole stadium. So there is a, this is just an aside to tell you that the mathematical study of ergodic theorem is very interesting. Now, the third concept which comes in which is the renormalization group. How does that come in? We observed that critical points are marked by critical exponents. That is to say, every, at the near the critical point every observable such as the order parameter itself is related to, sorry, this is wrong. It doesn't mean the minus, the way I have defined. Yes, it does. So M is zero for T positive and it is non-zero for T negative. This is by requirement a good order parameter. It tells you that you are in a different phase, paramagnetic phase and the ferromagnetic phase. But the way it behaves near zero is by a power. And these guys are called critical exponents. There are many of them as you look at many different quantities near the critical point. Many of these critical exponents appear. One of the, another one is the specific heat which diverges near the critical point but with an exponent alpha and there are many others. Why is that the case? And that is the case because we have a scale events. Here is a complete list of the critical exponents for a system with two, for a system with two parameters, T and H, the reduced temperature and the reduced magnetization, external magnetic field. Why should that be the case? Why should we get only divergences of power law structure? That is a question. The other question which surrounds critical points is universality. What universality tells you is that critical exponents of many seemingly different physical systems are the same. So what has happened in this graph is that a re-parametrization in terms of density and so on has happened. And then you see that the order parameter care for all of these very different systems coincide. These systems in terms of interactions at the atomic level are different, very different. But they all show the same critical exponent. Why is that? And it is in fact this effect which allows us to classify critical points. We give the critical exponents and an infinite number of systems fall into the same class. And again, the question is why does that happen? Why doesn't the details of interaction play a role? Now, all this has to do with a scale invariance and generalization group equations. What do I mean by a scale invariance? There are certain physical quantities in this case correlation function which calculates the correlation of a spin fluctuation as position R and a spin fluctuation at position zero. We expect that to be physically a decreasing quantity as R increases and it is, it falls exponentially. However, the correlation length which you observe here diverges as you go towards the critical point. With exit diverging, what you have is that at the critical point, the correlation length has a parallel behavior. In, this is the dimension of the space and therefore at two in two dimensions, you just have R to the power minus eta or you have other physical quantities such as the specific heat which have a parallel behavior. This parallel behavior means that I have a scale covariance if you like because this lambda comes out. So I have quantities in physics in which a change in a scale produces a factor on the outside. This kind of power law dependence therefore is a byproduct of a scaling variance. And other scaling relations exist for the order parameter or susceptibility where you get these power laws. And a little bit of argument shows you that the free energy also has to have this form for it to have, for it to support these guys because all of them have to come by differentiations from this form. And then this form of the free energy because you actually have managed to formulate the free energy in terms of just two exponents alpha and delta means that all this zoo of critical exponents cannot be independent because all these quantities come out by differentiation of the same form. And for the example which you have here you see that these three exponents will have to be related which we call the scaling relations. This is a table for the test of scaling relations. So you have two different materials and various scaling exponents are listed here and these relations are tested here. As you can see they're almost true. Some discrepancy can be observed here but probably this is result of difficulty of experiments. This happens because of the renormalization group equations. So now I will take a few minutes to explain what the renormalization group is. Maybe that will help understand the situation a little bit. In general we describe a physical system by a set of operators which I call OM. And then these OM come into the Hamiltonian with certain coefficients which these we call these coupling constants. The RG mentality claims that these coupling constants are not independent of a scale. That I have to give you the scale before you can tell me what the coupling constants are. So for example what does this mean? It means that for example if you take a quantum system the coupling constants which go into it which are measurable parameters of physics for example the final structure constant or the mass of the electron. They are not numbers which are independent of a scale. They are numbers which are determined at a certain scale. You do an experiment to determine the final structure constant and this then goes into here. So CNs will change with lambda. And lambda is actually the number which determines my course-graining scale here. So remember I told you these boxes come here in the phase space. These guys are determined by lambda. How big I take them has to do with lambda. And as I change the course-graining the coefficients change. What this means is that at some scale you might have a Hamiltonian for example for Fermi interaction and then as you increase the scale then you will have a more fundamental theory at a lower scale which explains the Weinberg-Salon model or the standard model. And there may even be a smaller scale which explains a more fundamental theory which we don't yet know. And you can also go the other way around. There must be a scale higher than Fermi theory which is maybe quantum mechanics. It's not a story which is completely consistent in the sense that I cannot go from quantum mechanics to classical mechanics by increasing the scale but my belief is that you should be able to. However, it's not doable. Hence the dynamics, the fundamental dynamics of physics is not just given by the Dirac equation which we certainly is true but you must also give me the dynamics of this constants. How do they change under a change of a scale? And these follow the RG equations. So physics tells me that you need a pair of equations like this which tells you how your constants or parameters of physics change as well as how the dynamics of system follows. It is, however, the case that these things which we call the beta functions, we have to calculate them for each specific Hamiltonian separately. It is we don't yet have a universal beta function for everything but I have to solve a pair of equations not just one. As it happens, this is a much easier equation to solve than that because this gives you dynamics on the Hilbert space but this gives you some sort of classical flow over a parameter space. So these are called the RG equations. Now what happens is that what if the right-hand side vanishes and it can happen that for certain set of beta n's the right-hand side vanishes. So I might have beta n C star equal to zero. We call this a fixed point of these equations and this can happen that for some certain C star this point is obtained. That means that at this point, at this combination the these guys are no longer lambda dependent and they become constant. This means that they are independent of a scale or I have a scaling variance. It should happen for all n's, yes. They should all vanish at the same value of C star. I said that too quickly but I think it's correct. Now why is this now an important point? The reason is that if you have this situation then you have a critical point. So here is a typical RG flow. This point you have a zero of the beta functions at the Gaussian point. The Gaussian point is where both C's are vanishing. And so at C1 equal to C2 equal to zero you have a zero of beta function. And then a Wilson Fisher point is a non-trivial fixed point which is it is happening at a non-trivial value of the coupling constants. Now the other thing which is interesting about Wilson Fisher point, we will come to see the importance of it is that it's actually a saddle point. So if you go outside of the Wilson Fisher point in one direction it is an attractive point. In another direction it is a repulsive point. So a little fluctuation around WF will take you away along this line. So these black lines are in fact indicating that that if you're a little bit out of Wilson Fisher you will actually go up that way. Or parts which are not exactly on the move to WF point they come in, get close to it and then go away. This gives you a very interesting fixed point. Now these are details which I have to come to now. Before I explain this slide, let us explain that this point has to be a critical point in the physical sense. When I say in the physical sense is that I always have some mathematics and but this mathematics implies something physical. I always have to be able to translate what I mean from the mathematical point of view in the physical sense. So I'm claiming that when this happens you have a critical point. Why is that? The reason is that I have something called a correlation length which you saw in the correlation two point function. At the scale three point, lambda zeta has to equal zeta because zeta is typically length and when I change a scale all length scales change and therefore zeta changes. So you have a point in which lambda zeta and zeta are equal if this holds. This has only two possible solutions. Either there is no correlation length or that zeta is divergent. Just that aside to sort of make you a little, a little bit of fun is always good. In this very room I was participating in a meeting where physicists and mathematicians were sitting next to each other. And Lohler, who's a very good mathematician from Chicago, is strongly objected to the way physicists behave. He said you are never, ever allowed to write something like that. Nothing ever becomes divergent, infinity. It becomes divergent, it grows without bounds but it never becomes infinity. And there was a lot of discussion about changes between physics or mathematics. As I wrote that, I was reminded of Lohler's objection. But we are very sloppy in physics. I hope you forgive me for being sloppy. So this means, the fixed point means that I have to have divergent correlation length. Divergent correlation length means that I am at the critical point because we saw in the correlation function that zeta diverges, I'm at the critical. So this property, lack of dependence of Cn and lambda means a scale invariance and it means critical point and it means zeta is divergent. Now why does it bring in exponents? There is a very simple calculation. If I expand around the fixed point, so we take a small distance away from the fixed point and I expand around it by a Taylor expansion. What is left from the better functions is this term which is the first term in the Taylor expansion. So I have an Mn, sorry, I have my matrix M and the matrix M gives me how I actually move away from the fixed point and the calculation I have done actually shows that the matrix is the derivative of the better function. So now I can diagonalize M and if I diagonalize M it will give me the directions in which this expansion is happening. So the eigenvalues of M will be of this form and these will show me how each direction in the Cs space behaves. If my N is positive, this means that around the fixed point or around the critical point, there is a direction in which I have a growing coefficient. This means that the operator which is multiplying this coefficient, this operator is relevant, it is important. Whereas if my N is negative, I have a decreasing value, so decreasing C, so this operator, the corresponding operator is irrelevant. There is a middle point as well where my N is zero and I have a marginal. For the moment, ignoring this possibility, we observe that near the fixed point, there are a set of operators which are relevant and they are the ones which determine the physical interaction of the system at that fixed point. This explains universality. This means that it doesn't matter what physical system you take. The details of the interactions are not important. Details of the interactions are the irrelevant operators. The relevant operators, whichever they are, they are the determining parameters in your physical system. So you may have an infinite number of physical systems which share the same number of relevant operators, so they all fall into the same universality class, which was the graph by Guggenheim which I showed you. It also explains the exponents. It shows you that all the dependence of your parameters near the fixed point are power loss. So you can actually rewrite the free energy in terms of these eigenvalues, the device of the system near a fixed point. And if you can do that, then every exponent, every exponent in your physical system will depend on y t and y h. And of course, the dimension of the space time d on which the system is explained. In this case, I have only two relevant operators. If I have more than two relevant operators, of course I have a more complex situation and free energy will be more complex and this and very interesting things can happen. But this simple system is in fact, applicable to many things that we study for our course, including the Ising model. This also gives you a very definitive explanation as to what I mean by a scale invariance, which I discussed 20 minutes ago, that a scale invariance near the critical point has been added to my invariances. A direct differentiation, such as the specific heat at h equals to zero gives you this behavior and gives you alpha in terms of the space time dimensions and this y t exponent of the Rg equation. And of course, as I explained, this scaling form of the energy, free energy, implies the scaling relations, which were things like this, connecting the various exponents. Hence, this also goes back to the fact that I have a fixed point. And here is the full set of scaling relations for which Kenneth Wilson won his Nobel Prize. Too easy to win a Nobel Prize for, just algebra. So for all the systems which share the same Rg fixed point will be in the same universality class, they will have the same set of exponents. And for each universality class, there will exist an infinite many physical systems which differ in their irrelevant operators. These are all the things which we have. So two couple of questions arise. One is that what are the physical properties shared between systems which share the same fixed point? If I give you two physical systems on the table, say water and glass, can you tell me that they share a fixed point or not? Before you write down the Rg equations and stuff in there, are the fundamental physical properties that they share if they share the same fixed point? This is a question worth asking. It's a difficult question. I think in my opinion it is an unsolved problem. I haven't seen a reasonable response to that. But there is another question which is this. That's not enough. If you like, I explained it and I said I wanted to ask the question in another way. As if they share the same fixed point, they are explained by the same conformal field theory. What is it about them which makes the same conformal field theory explain them? For example, it could be the symmetry group. It could be the spacetime dimension. But from the work I have done, this is not enough. Maybe you have an angle on that. Why? When you go from a physical theory to a corresponding conformal field theory, you have now explained everything about this critical behavior. But how do you make this connection? And why? What are the properties that two theories must have, must be sharing to give you the same conformal field theory. But I can set you another problem which you can do. I'm not sure if it is in the exercises or not. That is, if you can diagonalize M like this, which you have to do to do all these calculations, then M has to be a symmetric matrix. Otherwise these guys would come out to be complex and you can no longer do that. Why is M always symmetric? M cannot be symmetric because I remind you M in M is defined like that. It's the nth derivative of the mth beta function. Clearly not symmetric under exchange of N and M. How come its exponents are always real? This is what you can answer so you can think about it. Okay. I think I'm at the end of my first lecture. I should have no more next lecture. Goes into conformal field theory, fractals, and self-similarity. So before I start that off, do you have any questions? Yes? Last question? Okay. You have to prove the derivative with respect to M of beta M is equal to the derivative with respect to M of beta M. For all physical systems. That is like Cauchy conditions. It's like a question. So if you find a potential with respect to which the betas are the derivatives. Exactly. So it means, at least, that you... Solve M times N... These are equations. That there is something... So this is not always true. Yeah? So if I write from... Let me explain it right from beginning to cover both questions. Thank you. So you have this equations. C N dot is equal to beta N of C. And this has a fixed point at C star. And I define some fluctuations around C star. Okay? Now, a little bit of Taylor expansion around C star gives me this relationship. Where M and M are derivatives of beta functions. Okay? So these steps are understood. So now I claim various things. First of all, that the eigenvalues of M are always in this form. Luler will be happy. Thank you. So first thing, show that the eigenvalues are always of this form. This is first thing to show. Second thing is to show that yk are always real. Okay? This is the second thing to show. For them to always be real, as pointed out correctly, M has to be symmetric. So this means that I have to have this relationship which implies that this guy has to be true. Which means that beta N have to be derivable from some potential. These are like Riemann Cauchy conditions and the solution to them is something like that. And the question is why is that always true? Why can you always get the beta functions from a potential? To help you a little, this is rather easy to show in two dimensions, but really difficult in four dimensions. Right. Okay. So next topic which I want to cover is conformal filtering, self-similarity and fractals. In fact, Professor Dar's lecture has already helped me in explaining fractals. But why do we want conformal symmetry? What happens as I explained in my previous presentation is that critical point we now know has a scaling variance because of these arguments. Now the problem, the point is that we know that a scaling variance plus physical theory, I mean physical theory by a theory which has unitarity, locality, et cetera, leads to conformal invariance. At least we know that this is true in two dimensions. It has some chance of being true in higher dimensions, but not complete proof doesn't exist. I think the proof in two dimensions is complete and good. But four dimensions is another question. Now, if the scaling limit of a lattice model holds, then we can assign to it a conformal filtering. What the scaling limit means is that the lattice model goes to a continuum model without problem. So I have a model defined on the lattice, et cetera. In some limit, this must go to some field theory. And the limit is when you let the lattice spacing 10 to zero. In other words, what tends to zero is a over zeta. The correlation function tends to zero, which is this is dimensional parameter. A dimensionless parameter that tends to zero is this combination. So correlation length is much bigger than the lattice spacing. So the details of the lattice doesn't bother me and I can have a continuum theory representing the same lattice theory. This is called the scaling limit. And it's difficult to prove because of obvious reasons. You have various things can go wrong in this limit and proofs are very hard to come by. But for example, it does exist for the Ising model. This proof does exist in 2D for the Ising model. In 3D it has some obstruction. It probably is not true in 3D. In 4D a proof exists, but as we say in person, a hair has been found in the yogurt. The yogurt is okay, except that there is a hair in it. That hair has to be, if it is removed, the proof is complete. So almost everything has been done in 4D, except the very last step. And I think in higher dimensions, nothing has happened. So in odd dimensions you have an obstruction, so the next dimension to think about is six and nothing has happened in six. But the proof in four is very, very young. It's something five years ago. Proof in 2D is old. Goes back to the 80s. I will give you a very simple version of the proof in 2D, you can see what goes into it. Not a real proof, but a physicist's proof I will give you in 2D. This has a very, so if this holds, this scaling limit exists, this means that every critical point corresponds to a conformal field theory. So I've made a jump, of course, if I have accepted the proof that a scale invariance plus physical properties leads to conformal invariance, and if I have a conformally invariant field theory, then I have a conformal field theory, which is a quantum field theory with conformal invariance. And every critical point then will be described by a conformal field theory, and there is then a recipe, very rather simple recipe that I can extract from this conformal field theory, all my critical exponents. And the classification of conformal field theories then will give me a classification of critical phenomena. And this has happened to a certain extent, it's not complete. So I'm interested in conformal invariance. And because I'm interested in conformal invariance, there was a question, yeah? No, I think I have said something which has created that concept. I need this so that from here I can construct a field theory corresponding to my statistical mechanics model. But this correspondence implies that the lattice spacing is going to zero, or the ratio of lattice spacing to correlation length is tending to zero. So I can make this correspondence usually only near the critical point. It's not always the case as in these lectures I will give you examples that you are not at the critical point but you have a corresponding field theory. But usually is the case that I need this to go to infinity. So the correspondence between field theory and lattice models is usually at the critical point. So then that is a conformal field theory. No, it's not necessary to have exact zero, you are right. All you need is to have what is called the hydrodynamic limit to hold. For that means derivatives can be approximated by finite differences and so on. Hence conformal symmetry is important to me and I want to spend some time on defining and talking about conformal symmetry. So let's talk about this statement first. That is why does a scale invariance lead to conformal invariance? There is a lot of literature and this literature goes under the title of scale versus conformal invariance. You can find many papers including this one which is called a scale invariance versus conformal invariance. It's a physics report by Nakayama. Pretty good physics reports which covers all the aspects of this correspondence. There is a proof in two dimensions that a scale invariance implies conformal invariance for a physical system. If this physical system has conservation of energy momentum, it has unitarity, it has a locality and this extra condition is the spectrum in scaling dimensions. These are scaling dimensions, the vise, which I have taken out. This means that you have a discreet spectrum of critical exponents, not a continuum of them. And the proof goes something like this. In two dimensions, I can rewrite the energy momentum tensor as a complex entity. It's a complex entity, the energy momentum tensor. It has three components. It's symmetric, so it has three components. A z, z component, a z bar, z bar component and a trace. Three independent components go into the energy momentum tensor. And the equations of conservation of energy momentum become that. This is all algebra. It's very easy to do. Now, a scale invariance means that the trace of the energy momentum tensor vanishes, which is this theta vanishing. Hence that equation of energy momentum tensor conservation means that the derivative of t with respect to the complex z is zero. I'll just remind you that we had this equation. If you set theta equal to zero, you immediately get this expression. What does this mean? It means that the complexified energy momentum tensor is only a function of z. If it is only a function of z, it means that it's an analytic function over the complex plane. And all analytic functions over the complex plane carry representations of the conformal transformations. This means that any transformation that you make over the complex plane, which takes z to omega of z, will leave this equation invariant. It also means that the two dimensional conformal field theory is fully integrable. Integrrable, for it to be integrable, it has to have an infinite number of conserved currents. So if you have this equation, which means that t is only a function of z, then you can take any power of it. Any power of t will also be a function of z, and hence its t bar will vanish, d bar will vanish, and these are an infinite number of conserved quantities. Hence I have a field theory, which has an infinite number of conserved quantities, therefore integrable. It's the king of integrable systems. Other integrable systems also have an infinite number of conserved quantities, but they're not so easy to write down. Very easy to derive. So we are able to correspond to every critical point, if those conditions above hold a conformal field theory. These conditions, although they are sufficient, they are not necessary. That means that if the conditions hold, you arrive at conformal invariance from a scale invariance, but if they don't hold, sometimes you do arrive. And that is a strange thing. And then there are some counter examples, where you can show that the conditions don't hold, and you do not arrive at conformal invariance. The enlargement from a scale to conformal invariance does not happen. So here we have a gray area. If they don't hold, then we have to test them one by one. We don't know what happens in general. Here is an example of a scaling limit. I will try to give a derivation of that another day, that this field theory is in fact a model for the Ising model. It's the field theory of a fermion in two dimensions, the two component fermion field with mass, but it's free, there is no other interaction. And in fact, mass in this theory plays the role of temperature. So if temperature tends to zero, you lose this term, and if you lose that term, what is left behind is conformally invariant. If not, this is still integrable, but not conformally invariant. I'm going to stop here and continue tomorrow, inshallah. Thank you.