 So I'm going to speak English for the benefit of some young postdocs and students who are here in the room. And I think also Boris is going to speak English and Marjorie is going to speak English. So the scientific part of the meeting is going to be in English. So I'm Slava Rychkov, I'm professor of physics here at H.S. So I'm too young to have met Louis Michel, but I have read many of his papers, so I know him through his papers. And I also know him through speaking to his collaborators, to Pierre Toledano, to Edard Brezen, who unfortunately both they couldn't be here today. So I, because of this I got to know Louis and I got to respect his work. So for this meeting I decided to present a small, very small part of his work, the work that he did around 1980-1985. So it's the work related to the theory of phase transitions, critical phenomena and the realization group. Simply because this is something that is related to my own research and something that I found useful for my own work and tried to carry a little bit further. So let me just start with some very basic definitions and facts. So I'm going to talk about phase transitions and so what is a phase? I have in mind some substances, some materials and when we have such a material, we can study how its properties change when we vary thermodynamic parameters like pressure or temperature. And when we do this, then the space of parameters gets divided into several regions, which are called phases. Within each region the properties of the material they change analytically. So they change smoothly and analytically with temperature and with pressure. And when you cross the boundary between these two regions then there are some non-analytistic, some discontinuities. And so the main thermodynamic quantity is free energy. So free energy can change continuously in this space, in this plane. But when you cross these boundaries then you can have typically either discontinuities in the first derivatives of the free energy and this is called first-order phase transitions. Or sometimes you may have discontinuities in the second derivative. So the first derivative is continuous. So this is called second-order phase transition. And the example of the first-order transition is the liquid solid transition. And the example of the second-order transition is the Curie point in magnet. It's a function of temperature. And so for this talk the important, I'm going to focus on second-order transitions. And so the first successful theory of the second-order transitions it was built by Landau in 1937. So Landau he understood that these second-order transitions they are most often related to the change in the symmetry. Of the substance. With the reduction of symmetry. So when you vary some parameter, for example temperature, you will have a more symmetric phase and a less symmetric phase. It's also called broken phase. And less symmetric or more symmetric is formulated mathematically in terms of group theory. So there is a group G which is associated with a more symmetric phase. And there is some subgroup, some proper subgroup of this group H which is associated with the less symmetric phase. And so what do those groups act on? Well, Landau introduced the concept of the order parameter. So the order parameter is a vector phi. It's an n-dimensional vector. And this n-dimensional space carries a representation, a reducible representation of the group G. Of the more symmetric group G. So it's an orthogonal representation. So the group G thus becomes a subgroup of the group O M. So this is the basic setting. And you see here the appearance of groups and Louis Michel was the master of group theory. So clearly here is something that he could have played with, as we will see. And so another important quantity that Landau introduced is the free energy functional. So the free energy functional f of phi or function, it's a function of phi, so this order parameter. So it has to be actually minimized over phi to get the true free energy. And this function, it has the main property of this function is that it has to be invariant under the action of the group. So it's a gene variant function. And so it has, for simplicity, we can limit ourselves to two terms in this function. The first term is the quadratic term. So it's just some parameter mu times the length squared of phi, so phi phi. So this is the quadratic term. And then there is a quartic term, p4 of phi, which is some homogeneous quartic polynomial of phi, which is an invariant polynomial of the group G. So every group G in a reducible representation will have a certain number of invariant polynomials. And so this p4 of phi should be some linear combination of these linearly independent gene variant polynomials. So it's a problem of group theory how to classify these polynomials, which I'm not going to discuss here. And in the Lando theory, the only property of this p4 polynomial is that it should be growing at infinity, so that at large values of phi, this free energy function goes to infinity, and at small values of phi, something interesting may happen. And so what happens at small values of phi depends on the values of this parameter mu. It can take positive values or negative values. If it takes a positive value, then we get a symmetric phase. Why is that? Well, because this function f of phi has then this sort of shape, so it has a minimum at phi equals zero. So this is the... And the value phi equals zero is symmetric with respect to action of the whole of group G. On the other hand, if we take negative value, then you get phi equals zero is an unstable point. It's an extreme, it's a local maximum. And so there's going to be some minimum at a non-zero value of phi. And that's the broken phase. And what is the value of the... what is the group H? This is the broken symmetric group. It's the isotropy subgroup of phi zero. So it's a subgroup of G, which leaves phi zero invariant. So this is how Lando said things up. And it's been a very successful theory. And actually one more important condition, which group G has to satisfy. So I talked about second order invariant, phi squared. I talked about quartic invariance of the group G. And the more very important condition is that the group G should not have any cubic invariant. And if it does have this cubic invariant, then the prediction of Lando theory is the tradition should be first order. Because then we would have to add a cubic term into this Lando potential f of phi. And this cubic term, it would change the picture that I described in such a way that the parameter phi changes not continuously, but discontinuously. Okay, so we are in 37. And a very important addition to this Lando theory was made by Kenneth Wilson in 1971. So when he created the theory of the termization group and the Bayes theory of the second order phase transitions. So the theory of Wilson, it builds upon Lando's theory. So it basically takes over all the things that I said so far. And it adds to them one extra requirement. So in Lando's theory, as I said, this polynomial p4 of phi, it could be arbitrary invariant polynomial of the group G. And in Wilson's theory, this polynomial should satisfy just one extra constraint. It's not arbitrary, but should satisfy one extra constraint. So let me describe this constraint. So to describe this constraint, I write this polynomial p4 of phi. I write it as a linear combination of monomials. I introduce here this tensor lambda, which is a symmetric gene variant for tensor lambda. And I define a certain function on all four tensors, a function called A. So this function is a real function for tensors, and it takes the following form. So it's a cubic function. It has two terms. One term, which is a negative lambda squared. And when I write lambda squared, I mean a contraction of lambda with itself over all indices. What I write here. That's the first term. And the second term is lambda cubed. So it's a contraction of the product of three lambdas over indices. Now each lambda has four indices. And now I split them into two pairs of two by two. And I contract the indices accordingly. So this is what I have here. We have a term where I just take four indices of lambda and contract it with four indices of other lambda. It's the first term. And this is the second term where I take two indices of lambda and contract them with this lambda, with this lambda, with this lambda, like this. So that's a possible function of lambda to consider. And the most important property of this function is that this function is oan invariant. So if I take the tensor lambda, I rotate it by an n transformation, then a doesn't change. Now, I just want to go back forward. Okay. So the main rule of Wilson's theory is that we are only supposed to consider tensors lambda, such that they are local minima of this function a of lambda. Not arbitrary, but only local minima of this function a of lambda. Okay, function a of lambda is defined on all tensors of lambda, but there is also this group G. So we only have to consider this function a of lambda restricted to the set of tensors which are invariant under group G. And find local minima of this function. So here I have to make two remarks. So where is the remunization group? There's no remunization group here. Well, for mathematical purposes, I formulated everything in terms of this function a of lambda. But we can view this function as a potential of a flow, which is going to correspond to the gradient of this function a. This is the potential flow. And this flow is the remunization group flow in some approximation. So this flow tells us that the tensor lambda doesn't stay constant, but it changes as we vary from short distances to long distances. And the fact that we are only supposed to consider lambdas which are local minima of this function a means that we are only interested in lambdas which are stable fixed points of this remunization group flow. So that's the meaning of this rule. So instead of talking of local minima, I can also call them stable fixed points. Always within this subspace of tensors which are invariant under the group G. And the second remark that I would like to make is that the existence of this local minima of stable fixed points is not at all guaranteed. So our function a is a cubic function. It's not bounded from below. At infinity in some directions it goes to minus infinity, plus infinity. So it might be that this function doesn't have any local minima. It might be that in any direction you will just escape to minus infinity. So what happens then if you don't have any local minima? Well then Wilson's theory predicts that the phase transition should be first order. So that's an interesting new type of first order predictions which are called fluctuation driven first order phase transitions. So now I finally can formulate the problem on which Louis-Michel worked. The problem is this. I have here number N which is the dimension of the order parameter. I have a group G which is a subgroup of O-N. So this group has to satisfy already on basis of London theory. It has to satisfy two requirements. It has to be an irreducible subgroup of O-N. And it has to satisfy the London condition, no cubic invariance. So the problem is to determine all such groups G for which there exists a local minimum on the subset of tensors invariant under the group G. Are you talking about Lee subgroups or discrete subgroups? Could be Lee subgroups, could be discrete subgroups. Both cases are interesting. The Lee case is easier because there is a small number of Lee subgroups but there is a huge number of discrete subgroups. And Louis-Michel knew the theory of groups very well. He knew some complicated cases of group theory like associated with crystallographic groups. So he was well prepared to think about this problem and he thought about this problem for five or six years from 1980 to 1985. And in fact, so he arrived to think about this problem. As I said, Wilson introduced this history in 1971. And so Michel started to work on this only about ten years later. So in these ten years, a lot of results have already been obtained by people who didn't know as much group theory as Louis-Michel but who just proceeded an intuitive level and then maybe by trial and error and they obtained many results. And so the contribution of Louis-Michel was twofold. First, he was able to prove a couple of structural results which clarified a lot this picture which was before obtained just by trial and error. So it explains some things that people have seen but didn't really understand why they were the case. And he also, as a second contribution, he solved one particular case of this problem completely. But in general, this problem is still far from being fully solved. So let me then describe a couple of theorems that Louis-Michel proved in his work. So the first theorem is very interesting because it has a corollary that if a stable fixed point, if a local minimum exists, then it's necessarily unique. And so this theorem goes as follows. So let's suppose that there are two, well, here's a misprint. So let's suppose that lambda 1 and lambda 2 are two, not local minima, this is a misprint, are two extremal points, two critical points of the function a of lambda. So let us consider, let us compare the values of this a function on these two points, lambda 1 and lambda 2. So I'm going to compare a of lambda 1 versus a of lambda 2. And so there are two possibilities, like one possibility is that one of these critical points, so lambda 1 and lambda 2 are critical points, not necessarily local minima, we don't know what they are. So if a of lambda 1 is less than a of lambda 2, then the theorem says that lambda 2 is necessarily unstable, cannot be local minima. That's easy, I'm going to explain how it follows. And the second part of the theorem, which is less trivial, it says that suppose that a of lambda 1 is equal to a of lambda 2, well, then you could say, well, perhaps they are both stable local minima, no. In this case, the theorem says that actually both of these points are unstable, cannot be a local minima. So, well, the proof of this theorem is really very, very easy. Let me just give it for you. So the key point is that this function a is a cubic function. It's minus lambda lambda plus lambda lambda lambda. So if I have two points here, lambda 1 and lambda 2, let me consider what this function does on the line which contains the two points, lambda 1 and lambda 2. Well, it's a cubic function and at one point, these are both extremal points. So we have extremal point at lambda 2, extremal point at lambda 1 and they are not equal to each other. Well, it just means that this function does something like that. So this is lambda 1, this is lambda 2 and this is lambda 1. And since it does something like that, it means that at the point lambda 2, it's not a local minima. End of the proof of the first part. The second part is more interesting. So we have equality here. And so in this case, actually, if you consider the value of the function on this line connecting lambda 1 and lambda 2, then on this whole line, because it's a cubic function, it has to be a constant. So we have a whole line where this function is a constant. And then the idea of Louis Michel was to consider what this function does in a plane, in the whole plane which contains lambda 1 and lambda 2. We have this line on which this function is a constant, but also recall that this function is a non-invariant function. So it's a rotation invariant. And so it's also constant on every radial line, radial curve, not radial circle around the origin. It is constant. So in particular, it is constant on this curve. And so if you look now at the point lambda 1, we see that there are two directions in which the function is constant. It means that the second derivative of the function vanishes in two different directions. And there is one direction, namely the radial direction in which the derivative is positive. Well, a quadratic form which has one positive direction and two zero directions necessarily has at least one negative direction. So, end of the proof. So that's the theorem. And this theorem is very powerful because it says indeed that if the local minimum exists, then it's unique. And that explained what people have seen before Louis Michel in many, many explicit calculations. And by explicit calculations, I mean take a group G, compute its set of quartic invariants, write down this function A, look for local minima, and you always found that the local minima, if it existed, was always unique. Well, here's the reason. Now, let me explain another theorem of Louis Michel so that this theorem explained something else that people have seen in many, many calculations before Louis Michel. So, they observed that, okay, I talked about this group G. The group G should not be very small group. It should be sufficiently large. Why is that? Well, because if the group is small, then first of all, it's not going to be irreducible. So, of course, irreducible subgroups of A and A should be sufficiently large. Also, if the group is small, it will have some cubic invariants. And as we said, there should be no cubic invariants. But even if you impose a disability and the Lando condition, there are still many possibilities for these subgroups G, some of them large and some of them small. And people observed that in the cases where this subspace of quartic invariants is large, then typically you never find a stable fixed point. You find some fixed points. You find some critical points of this function A, but they're all unstable. Why is that? So, Louis Michel proved the theorem, which explains why this is the case. It was in collaboration with his student Jean-Claude Toledano. So, the theorem has a group theoretical criterion. So, it says, take this critical point lambda star. So, it has its own symmetry G star. And this critical point, it lives in this space of quartic tensors lambda 4G. With this space of quartic tensors, you can associate another subgroup of OAN, which is the normalizer subgroup. It's a subgroup consisting of all elements of OAN, which leave this space T4G invariant. Now, let us look at this normalizer subgroup and let us compare it to the symmetry of the fixed point lambda star, to the subgroup G star. If this space T4G is large, then the normalizer subgroup is also going to be large. And it may happen that this normalizer subgroup is strictly larger than G star. So, this is going to typically happen. If the space is large, the normalizer subgroup is going to be very large, and you will have that this normalizer subgroup is larger than G star. Well, the theorem says that in this case lambda star cannot be local minimum. You can conclude it without doing any computations. And this theorem has an even simpler proof than this first theorem. It's actually an easy corollary of the first theorem. So, here we have a critical point lambda star. Let us suppose that the normalizer subgroup is larger than the symmetry group of lambda star. Well, then exists some element of the normalizer subgroup, which acting on lambda star gives us some other tensor, let me call it lambda star prime, which lies in the same space T4G. And if lambda star is a critical point of A, then lambda star prime is also the critical point of A. And since one was obtained from the other by acting by ion transformation, they have exactly the same value of A. And so, according to theorem 1, if you have two critical points which have exactly the same value of A, then they are both unstable. End of proof. So, to conclude, let me come back to this problem that I stated. Given n, the dimension of the order parameter, it is of importance for physics to classify all these subgroups of n, which give rise to a stable fixed point. Because by classifying these subgroups, we are going to be able to classify all possible second-order phase transitions, which are of interest to physics. And unfortunately, this problem has been so far only partly solved. So, so far, this problem has been solved only for n equal 2, 3, 4 and 5. For other values of n, there are only partial results. So, let me say what is known. For n equal 2 and 3, these are simple cases. You can solve this problem basically by inspection. And you find that the only interesting fixed point is the fully ion invariant fixed point. So, there is no fixed point which breaks symmetry. For n equal 4, the next case, the problem is from the group theory point of view, the problem becomes extremely complicated. Because of 4, well, of 4 is a big group. It has many, many, well, it has infinitely many subgroups. And in 1985, Louis Michel with his students Jean-Claude and Pierre Toledano, and with Edouard Brise, they solved the classification problem of stable fixed points for this setting. So, actually, the paper is extremely impressive. So, I think it was well beyond, well ahead of its time. And it goes through, like, it goes through the full list of the subgroups of 4, very long lists. So, then, they do necessary group theory to select out of these groups irreducible subgroups, to select the subgroups which satisfy the Landau condition, then they build group-subgroup relations between these groups. For each group, they work out the set of invariant polynomials that was well ahead of, you know, at the time there was no Mathematica or no software, so they may, they must have done everything by hand. And in the end, they make a full list of stable fixed points which exist in this dimension. And so, until recently, this was the state of the art. And so, this year, Jun Chen-Rong, who is our postdoc, he realized that this problem is doable also in Anical 5. And so, now I'm happy to say that we have solved this problem that Louis Michel solved 40 years ago. Only now we managed to solve the Anical 5 case. And so, Louis Michel lives on. When you present the functional A, you have the quadratic term to be termed, but the coefficient is one. Is it because you rescale the Landau? Yeah, for this, normally, if you know, it's done in four minus epsilon dimensions and there is a coefficient epsilon in front of lambda and there are also higher nonlinear terms. But for this question of stability, you can neglect the higher nonlinear terms and then you rescale the first term to have coefficient one. So, that's what I did for the purposes of this talk. Thank you again.