 So indeed, my name is Thierry Jean-Marie and I'm from the University of Geneva, so we continue with the Swiss session this morning. Now I'm sure many of you know the University of Geneva for my colleagues, Tasmianov and Ego Diminil Copa. I'm in the physics department, not in the math department. So you might wonder what a physicist is doing in this nice conference. And I must admit that sometimes I'm wondering also. But I first would like to thank very warmly Frank, the organizing committee, the scientific committee for having invited me. It's a pleasure to be here. The reason is of course the same reason than all of us. Frank, I must say that I have to correct the first speaker of the conference. He was not the one who knew Frank from the longest period of time. I'm probably having this honor because we met first in 1980 during the preparatory school. We were not in the same class, but in the same cours de recréation. No, we were in members essentially. And then we met again because both of us, Frank in mathematics and myself in physics, went to the economy in Paris, where we take also some course together in table soccer and Francois Gaudes, who is here, and testify that we were very, very good students in this field. So it's a real pleasure, of course, to be there for this conference in honor of Frank. Frank, I have to show you two photographs. Don't worry. I've not been generated by deep faith. And just as a puzzle for you. Okay, here is Frank. And the second photograph, you notice this very nice costume. So, Frank, as a puzzle for you during the talk, you have to find where and when these two photographs were taken. This one is easier. Yeah, this one is more easier. Okay, I'll whisper the answer in your ear at the end of the talk. And of course, happy birthday. We celebrated the birthday a little bit before the conference and of course, many, many happy returns. Now, of course, the first moment of pleasure passed for physicists to be in a mathematician conference. You can imagine the sheer terror. What am I going to tell? Because Frank is a specialist of blow up in particular. It's not at all close to my own research. And so I have to find, quote, nice story to tell you. And that's what I will try to do. But of course, I will tell it from the point of view of a physicist, which means that first we have a problem, which is the same then for those who saw the movie here, communication. We don't have the same language. So please don't hesitate to interrupt me. Don't hesitate to ask any question during the talk. Don't wait till the end of the talk. Because if I am saying things that you don't understand, just stop me. It doesn't matter how far I'm going in my story. Let me tell you a part of it, but hopefully a part that you can enjoy and that you can understand. I'm also realizing that I'm the only thing standing between you and a lunch. So I'll try to stay on time. So what I'm going to tell you is I needed to find a wave equation. And of course for a physicist, Schrodinger's equation is the equation. And I needed to put some nonlinear terms. And there are many ways one can put nonlinear terms in physics. You will hear other talks I'm sure on this from Ginsberg-Landau equations and so on. But I wanted to tell something on which I have at least some experience. And so I will put disorder in the Schrodinger equation. And what I will try to show you is that this is a very, very beautiful problem. A good part of this problem has been analyzed and beaten to death. But a part of it is very much alive. And I would say for this, the physics community certainly needs the help of the mathematician community. Because there are some questions which are extremely tough for us. And I'll try to point out these questions. So let me first mention why disorder. Maybe I can push myself here. Of course you all know that if you take classical particles, disorder is ubiquitous in nature. It's very hard to avoid having some dirt in any piece of material that you look at. And we all know since essentially Einstein and Sutherland that if you have impurities, if you have dirt into classical equations, it leads to diffusion. Instead of particle moving ballistically with a distance which is proportional to the time, they move with a sort of mean distance which is only growing as the square root of the time. And now one question you could wonder is we know that in reality because of quantum mechanics, particles are waves. And what does it change? And this is something which you could imagine saying, well it changes a lot of things I'll explain this line because if I have an obstacle, classically I cannot pass the obstacle if my energy is less than the height of the potential. But if I am in quantum mechanics, I can tunnel through the obstacle. So you would imagine that in quantum mechanics, moving in a disordered environment is more easy than it is for classical particles. And the first answer that came in the mid-20th century was very disappointing. It's not that it's more easy or more difficult or whatever, it doesn't change anything. If you compute the conductivity which is the relation between the current and the force, if you want the electric field, you find a very classical answer with the lifetime of the particle. This is the so-called Drude formula. So it seems that the wave nature of particles is not really important to compute the transport in a random environment. And I'm sure you all went through these very nice problems where you have an electron which is considered as a little ball, which bumps into an impurity, goes randomly in all directions, bumps into another impurity, and so on and so forth. And then you compute the current. Actually, this is not completely true. So rather than write on the board, I wouldn't dare to do that. It's too nice a board. So I wrote on my tablet and I took a copy of this. So it's a compromise. So be gentle with me. Again, don't hesitate to ask any question if something I'm saying is not clear. So we have the Schrodinger equation. So again, I have my system which is described by an Hamiltonian. I have my eigenstates here, so I have to obey the Schrodinger equation, which I've written here using the so-called Dirac notation. H psi is equal to the eigenenergy times psi. And the typical Hamiltonian for one particle is p squared over 2m, the momentum of the particle. And then I can put a local potential, which is dependent on space. For the moment, let me just ignore this. I set this to zero. And of course, if I just have momenta, this is trivial to diagonalize. And we know the eigenstates and the eigenvalues. The eigenstates are just plane waves. Essentially, the particle is everywhere with an homogeneous density in the system. Now we know already that there are very interesting effects. Most of the time, I'll be in one dimension, both because it's easier to draw, but also because most of what I will tell you is particularly important in one dimension. If I move to other dimensions, I will explicitly mention it. If you have any doubts, again, don't hesitate to ask. We know that potentials here, when we add them to this equation, change drastically the physics. And this is obvious if you put a periodic potential. This is something which is very easy to solve. And what happens is that every time you get one period of the potential, part of the wave which is going here is backscattered by the potential. And this creates interferences which makes that not all the eigenenergies are possible, which was the case for the plane waves. The energies are k squared over 2m. So if you change the momentum, you change the energy in a continuous way. But if I have a periodic potential, there are some energies that are forbidden, completely forbidden, and this is what people call the energy bands in solids or the energy gaps in the spectrum. So we already see that there is something weird, because this will happen even if the potential is very, very small here and in particular much smaller than the kinetic energy of the particles. Of course, the gap itself shrinks as the potential is going to zero. And so the question is what happens if now, instead of having a periodic potential, we are having a disordered one. What happens if I have something which is random and I'll try to explain what I mean by random in a moment. And again, there is something which is physically obvious. Imagine the potential is super high. It's the Mont Blanc. It's something super huge. Well, I cannot go through the Mont Blanc, so I'll be localized. I cannot go through the potential. But on the other hand, if the energy of the particle of the eigenstate is supposed to be very high compared to the random potential, one would imagine that the particle will surf over the random potential. Even a classical particle should surf over the random potential. And so, more or less give or take, we keep plane waves or we go to something which is diffusive or something simple is happening. Okay, so let's see whether this is true or not true. First, let me, because again, I have to be precise, so let me define what I call disorder. I remember a conference in Santa Barbara where I was writing Grad U and we spent half an hour deciding with the audience what Grad U meant. I swore I would never do the same mistake again, so okay, I'll try to explain what is the disorder. So the best disorder probably is to say that you get impurities. So each impurity is a potential which is put at a random position. So you put a disorder where the strength of the potential is, let's say, the same just for simplicity, and you have random positions for the impurities, and you spread, you throw balls, let's say, on a 2D surface at random. This is what physicists call the Poissonian disorder. It has the advantage that you can control the amplitude very well of this disorder. Now it's not very convenient to work with this, so people have worked with other distributions. You take a V of R, which is a random variable drawn with a Gaussian distribution, or a box distribution between a maximum and a minimum. So we've used all of this disorder, oops, sorry, I didn't go back. We used all of these disorders. Of course, if I am in the continuum, I have to suitably regularize these distributions to avoid problems. Let me just mention another point, which we will forget immediately, but I have to mention it, if you do an experiment, you have one realization of the disorder. You have your sample, and that's it. But it's big. So if it's big, you could say that this part sees a certain disorder, this part sees a certain disorder, and in a way, when you look at the whole sample, again, dimensions, distance between atoms are very small, angstrom, 10 to the minus 10. We are talking about samples which are centimeter long, let's say, 10 to the minus 2. So it makes a lot of atoms and a lot of impurities, and we can consider that various parts of the system kind of provide for you the averaging of a disorder. So theorists like me prefer, of course, rather than dealing with one realization and a big sample to do explicitly the averaging of a disorder. So you draw several realizations of the disorder with a certain probability distribution, and then you do the averaging. This is totally mandatory if you do numerical calculations because there you don't have 10 to the 23 particle, but you have 100. And then the fact that there are several realizations in several parts of the system is not so true anymore. Okay, I just want to introduce to you in case you don't know it, I'm sure many people in the audience know it, a discrete version of what I showed, so Schrodinger equation plus random potential, but let me discretize in a way the Schrodinger equation. This is what is known as the tight binding model. This is a model which is very well suited to describe solids where you say that you get a discrete set of states which, let's say, symbolize a particle which is around an atom, a fixed atom, and then you have a matrix element which can take the particle from this site and put it to the next site or the previous one. I should have written the other part which I didn't. Of course, this is Schrodinger equation, so this has to be done coherently. And then I can put on each one of this site a random potential. So my Hamiltonian is very simple. It has random elements on the diagonal which is my random potential and it has two elements away from the main diagonal which are just minus t. t is not the time. Here is the tunneling. Minus t which is fixed and the problem consists in solving, diagonalizing this matrix. So today it's trivial. On a computer, you can easily do 10,000 by 10,000 matrix like this and in a couple of microseconds you get the eigenstates and you get the eigenvalues. But there was a time when it was not so easy and this was in 1957 because computers were not what they are where a physicist of the name of Phil Anderson who is one of the greatest physicists of the 20th century in Conance Matter, a Nobel Prize of course discovered something which was totally unexpected which is that if you put even a very small amount of disorder on the diagonal the eigenstates change completely. The eigenvalues not so much they sort of go through practically the same set of numbers than the eigenvalues of the undisordered system but the eigenstates are completely different. Instead of plane waves which are everywhere you get exponentially localized wave functions. So here is Phil Anderson. This is a phenomenon which is known as Anderson localization and it's very interesting to read what Anderson himself said during his Nobel Prize lecture about the phenomenon which says very few believe localization at the time and even fewer so its importance among those who failed to fully understand it at first was certainly its author. It has yet to receive adequate mathematical treatment still true and one has to resort to the indignity of numerical simulations to settle even the simplest question about it. Not true for one particle because now as I said you can do it on a computer like this but you will see when one puts interactions things become a little bit more fun. So I think the statement which was done in 77 for the adequate mathematical treatment is still true very precious few is known rigorously on this and I will try to point out what I know is known but again don't hesitate to contradict me if you know things which have been discovered. Here is just for, I won't comment on this here is some references that you can have fun reading if you want to know more. These ones are essentially physicist references that were done for the 50 years of Anderson localization this is a school in Boulder on essentially Anderson localization and there is here something which is maybe closer to your heart which was a little day organized by Simon's collaboration on wave and disorder whose PI is Svetlana Maibodora and this was an interesting day because the day was split in two in the morning there was a physicist who was supposed to explain to the mathematicians in the collaboration what Anderson localization was I was the victim and in the afternoon there was a mathematician that was supposed to explain to the physicist of the collaboration what Anderson localization was and this was a very very nice day because okay let's say I think each part learned a lot so the videos are recorded if you want to see both the math and the physics aspect I recommend that you go to this to this one okay so let me discuss a little bit the consequences of having disorder and now I'll concentrate on the 1D solution I'll come to a higher dimension later question what is the nature of the spectrum what are the eigenvalues what is the density of state which is sort of the number of states which have an energy between E and E plus DE okay what is the level statistics what is the correlation between having an eigenvalue at energy E and having an eigenvalue at energy E plus U is it changed compared to a plane wave where it's stupidly equally spaced so these are legitimate questions and of course they can be answered numerically you take again a 10,000 by 10,000 matrix you diagonalize it very easy to do you get your eigenstates and then you do this thing okay there are some theorems on this but not much what is the nature of the eigenstates as I said the big surprise was that even an infinitesimal disorder even a small disorder was completely changing the nature of the eigenstates and from plane wave you were going to states which are exponentially localized with a characteristic length here which gives you the exponential decay which people call the localization length which is of course a function of the energy of the eigenstates and what is remarkable in Wendy is that all eigenstates are localized so you cannot find any single eigenstate which is extended which has a uniform probability not uniform but extended probability to exist everywhere the fact that this is true for strong disorder has been proven mathematically as far as I know the fact that the critical value in Wendy is zero I don't think it has been proven but I might be wrong on that one and in two and three dimensions only the strong disorder result is known so there is room for improvement there and I'll show you what happened in 2D and 3D for physicists yep, sorry there was a question for physicists there is another quantity which is super important after all that was the one that was targeted initially which is the transport imagine you have a piece of junk like this with impurities you put contacts here, contact there which means you put a force which will push the particles going through the system and you want to know what is the current which is going through the system in response to this force this amounts in taking the Schrodinger equation let's say on the discrete system that I showed you and put a linearly decreasing potential like this from site 1 to site n and this will induce a current that will pass through the system and out of this there are formulas that allows you to compute the linear response to this potential something called the Kubo formula I'll show it a little bit later in the talk and of course one quantity you can extract is the so-called conductivity which relates the current in the system with the force or the electric field that has been exerted and of course what you want to know is what is the conductivity of the system as a function for example of its size and if it's localized you would expect that it exponentially decreases with the size that's one question yep ok there are many questions of course that people were asking I'll show you that even very famous people can say wrong things but ok it was still 1968 so this is someone named Sir Neville Mott who is again one of the greatest condensed matter theorist of the 20th century he invented a metal insulator transition of his own which is called the Mott transition but here is discussing about Anderson localization and he says ok let's see the metal non-metal transition so the first question is is there a transition if you change the strength of disorder will you go from a situation where you get plane waves essentially and the system is delocalized to a situation where everybody will be localized in 1D the answer is no everything is localized but maybe in 2D or maybe in 3D ok so what Mott was saying is the transition is first order meaning there is a jump chock if the disorder is weak I have a good conductivity and suddenly I pass a critical value of the disorder and boom it jumps to zero and then it's zero it's wrong we know now that it's wrong but in 68 that sounded like a good idea ok I'll skip point B and C which are more interesting for physicists but I just want to point out the question which is is there a metal insulator it calls it non-metal but let's call it insulator transition as a function of the strength of the disorder as I said in 2D and 3D I don't think this question has been answered mathematically physicists have their own answer I will show you the answer but I don't think this question has been answered in a proof so the answer to this question has been given by various people and this goes under the name scaling theory which I will not explain at all but maybe I should go to the other side I'm blocking your view ok so it goes under the name of scaling theory and it's based on the work of this gentleman David Thaules who got the Nobel Prize in physics for other works there is something called the Berezinski-Kostalitz-Thaules transition but he did absolutely seminal work on the field of Anderson localization and the thing he invented was reused by a gang of four it's the official name, the gang of four which is Elioa Brahms, P.W. Anderson, Lysiardello and Ramakrishnan here is the photo of this gentleman I'm sorry I don't have a photo of Lysiardello it seems that he disappeared in the continuum and they sort of went trying to build a very simple theory of what is the conductivity or rather the so-called conductance which is the total transport of a piece of metal of size L and I won't go into detail of how to do it if you ask me I can explain something that can be explained in five minutes what they found is that if you're below the equal to essentially there is no value of the disorder so let's say this is the conductance itself so if you want this is a kind of measure of how good metal or insulator the system is so in a way it's a measure of the strength of the disorder so if you're below the equal to there is no value of the disorder which can give you a finite conductance so the system just always plunges to insulator if you're in D equal to it's marginal but it's based on this theory it seems that it's always the same fate so in D equal to there is also not localization, delocalization, transition but the states are much less localized than they are in 1D in 1D if you want if you take let's say the mean free path the distance between impurities if you want the localization length is about the mean free path said brutally you have a particle moving it bumps on an impurity, goes back bumps on to another impurity, goes back and that's it, it's localized twice or whatever in two dimensions this length here is exponentially large in the mean free path which means you can easily reach the Earth-Moon distance so you can call it localized but that's a view of let's say it's in an idealized theoretical world but nevertheless the claim is that there is no transition in two dimensions and in three dimensions there is a transition so if the disorder is smaller than a certain quantity then the system is metallic it has a finite transport and if the disorder is larger than a certain quantity which people call the mobility edge if you want then the system is insulated again to the best of my knowledge this is consequences of this theory and of course the many sophisticated improvement that have been made on the top of this and the many numerical simulations that have been made on the top of this but there is no mathematical proof to the best of my knowledge of this and there is no mathematical analysis of what happens close to the transition because you can imagine that the wave functions the eigenstates will be very weird at this critical point for example here is a numerical solution of the wave function at the critical point by Sacha Mirlin at Karlsruhe and it seems the wave function shows a multifractal structure again these are essentially numerical evidence very hard to do a theory there are theories which have been done in physics but they are extremely difficult to do just to show you that this is reality, it exists I cannot resist showing you an experiment you know this was done in cold atomic gases so there are gases which have been trapped by using lasers I'm sure all of you know this gentleman because he got the Nobel Prize this year it's Anna Aspe and this is Massimo Inguscio I would say he's homolog in Florence, in Italy and both of them studied using these gases the localization due to disorder by slightly different tricks so here is the image this is not a drawing from this is the image of the wave functions of a particle in the random potential which has been created by light and you see here the exponential tails that correspond to the wave function decay in the system Massimo Inguscio doesn't use random potential but use what people call quasi-periodics which is a sum of two cosine with incommensurate wave vectors and what you see is that it produces essentially the same effect which is an exponential localization of the wave function this class of potential has been less studied it has been studied a lot by Serge Aubry it's called the Aubry-André model and there are a certain number of things that can be proved relatively rigorously but I would say it's also something for which we certainly lack information now let me move to what I would say is probably the heart of the matter at the moment in the physics community which is the case of interacting particles as I said the case of non-interacting particle it's difficult, it needs to be proven there are a lot of very fine quantities like the structure of the wave function and the transition and so on that we don't know but essentially you take a 10,000 by 10,000 matrix you diagonalize it on a computer you can get a lot of information but all this finishes as soon as you put more than one particle and you have interactions between them so that's what I would like to describe and that becomes more difficult for those who have seen the movie okay so let me show you two models which physicists love to deal with one is the so-called Gross-Betayevsky model the other is the Liebleninger model I remember a conversation with Frank we were discussing mathematical physicists and I was telling Frank oh there is this guy you might know as a mathematical physicist he's really great, he did a lot of things he's a Lyotlib Lyotlib, what do you mean? Lyotlib is a mathematician so you know I think it's a good sign when two communities want to get you so the other is the so-called Liebleninger model that I will try to explain so this model which was introduced by Petayevsky here is essentially a sort of very equivalent for those of you who know of the Ginsberg-Landau model and this is a model which is supposed to describe a bunch of bosons, many many bosons who interact and have a Bosenstein condensation so they are in a coherent state they are in a superfluid state and this superfluid state is described by this wave function psi which is the wave function which is a collective wave function which depends on the spatial coordinates so three coordinates if you're in 3D and eventually the time if you want and you get essentially the equivalent of a non-linear so you get the linear term here in the Hamiltonian which when you write the equation is very much like the Schrodinger equation but then you get this non-linear term which is describing the fact that the particle interacts and of course it gives you a cubic term in the equation here now without the randomness this is a model that has been intensively studied with the randomness this becomes a very nasty model and we know again mostly numerical things about it and I'm sure there are a certain number of things that one could tackle by methods which are not the usual methods of a physicist a variant on this model is the so-called Lieblinger model it's the same ID you want to describe n bosons and now of course your wave function has to contain n coordinates I'm in 1D so I have a wave function which is the wave function describing my n bosons of one of the bosons x2 the coordinate of another one x3 and so on if you're wondering why I'm not saying x1 is the coordinate of the first boson x2 the second and whatever is because bosons are indiscernible particles so the wave function has to be totally invariant by permutation 2 by 2 of each pair of particles so you don't really know which boson is which and that's a constraint you have to implement on the wave function and then the Hamiltonian is what you would naively expect the kinetic energy this is the p2 over 2m that's just the derivative with respect to the coordinate of each one of the particles but then there is an interaction which tells you that you have an extra energy cost c or 2c when two particles are at the same point in space when they are on the top of each other you pay a penalty the penalty is not necessarily infinite it's a finite energy cost so this is a nasty model by itself and it was solved by Lieben Leninger using a technique which is known as better on that which consists in writing essentially explicitly the wave function but which is let's say it's OK to write explicitly the wave function it's much more difficult to extract physical observables out of this study and the technique was invented by Hans Better in 1934 if I remember correctly and it's only at the beginning of the 21st century that people could compute the first physical correlation functions out of the better on that very often when people tell you a model is exactly solvable by better on that the only thing that they know is that it's solvable they don't know anything else but the solution ah because if you put Coulomb it's not solvable by better on that so of course you could put Coulomb you could put 1 over r square so 1 over r square is also solvable was solved by Haldane and Chastry any other potential is possible you could put a v of x1 minus x2 and of course there are approximate techniques field theory bosonization and so on where you can deal with these potentials but if you want to use the exact solution you have to have a delta function potential so of course you're totally correct I must say if I take the cold atoms that I showed you before cold atoms are neutral so they don't see each other very far and the delta function is actually a pretty good approximation so it's not this is a model that is realized experimentally actually although it would seem like a kind of crazy model but it's a very experimentally relevant model now not in condensed matter but in cold atomic systems ok if you put the disorder on the top of it ok again this is something extremely difficult let me give you a discrete version again as I was doing for the simple transport and that's something called the xxz spin chain I choose this one because it's easy to visualize you say on each side I have a system which is can be in two states either let's say down or up and therefore on each side I have a Hilbert space which is described by the Pauli matrices if you want a 2 by 2 matrix and my energy is essentially what you would expect for a magnetic interaction so it's sxsx plus sy sy ok here I've put a different coefficient for zz and then you put on each side a random magnetic field which favors either spin up or spin down on each side why I'm saying that this system is very much like the ones before if you think in your head that the spin down is like not putting a particle on the site and the spin up is like putting a particle when you do something like this you have the particle jumping to the next site and when there are two particles which are nearby up and up they don't have the same interaction on this term then if you have one particle and a hole here on this site and so this is very much like the Lieblinger model plus disorder except it's on a discrete it's on a discrete lattice ok what are the questions we want to work just a remark the Hilbert space now is 2 to the n which means you can do 40 sites on the best computers compute 2 to the power 40 and then you say I have a matrix 2 to the power 40 times 2 to the power 40 and you will see this blows up your computer and if you add one site you double the Hilbert space which means even if your computer can do it the best you can do is add one site not all the size of the system just add one site so brute force calculation you're dead if you want to deal with this kind of technique and that's the kind of thing that we need to solve to describe experiments so what are the questions same questions if there is no temperature so if we do quantum mechanics what is the ground state what is the nature of the ground state localized not localized what are the physical observables and essentially what you would like to know is do the interaction help the localization or do they kill it after all I tell you bosons are super freed a super freed is something that can flow without being blocked by dirt so you would imagine if I put a little bit of disorder on the top of a super freed it shouldn't be localized it should zoom through the disorder on the other hand if I have a lot of disorder I shouldn't have the super freed so one immediately realize that there must be a competition between disorder and interactions which will change the deal compared to the non-interacting system so that's the kind of question that one wants to know and since unfortunately experiments are done usually at finite temperature we cannot just compute things in average in the ground state but we have to compute the usual displacement trace where we have to average over all the states with a distribution which is given by exponential of minus beta H where beta is the inverse temperature which makes the calculation of course even more complicated maybe I will skip this but just to show you that actually using dirty technique one can essentially predict transition a super freed state and localized state which people call the Bose glass and this is confirmed by some numerical calculations let me go very fast there some numerical calculation here is a phase diagram that was obtained by people in Munich where you see very clearly but look this is numerics the transition between super freed and the Bose glass just to show you look these are numerical works which span 90 to 2009 and it continues of course today so again it would be nice to have definite answers on this type of issues because there are a lot of questions let me maybe skip the quasi periodic because I would like to keep a little bit of time for questions just to show you people can realize this in experiment here is very similar experiments than the one I showed you before they are done in the group of Massimo Inguscio in Florence this is again putting two lasers with two different wave vectors cosine q1x plus cosine q2x here is what they measure just without going into details this quantity here tells you how good or bad the super freed the system is when this is blue it's a good super freed when it's red it's a bad super freed now you're wondering what a good and a bad super freed is ask me the question if you really want to know but I won't go in detail but you see here that there is a region where you would say it's super freed and there is here another region where you would say it's not super freed so this goes well with the idea of having a phase transition between a localized and a super freed system one question for you on which the physics community cannot decide and maybe where the math community could answer what is the transport so what is the conductivity and I'll explain what is the conductivity as a function of the temperature of the problem of two sort of schools one is to say okay if I am at t equals 0 I'm localized so let's say I'm in 1d I'm totally localized so my conductivity is 0 when I put a small temperature I can jump from one localized state to the other and if I put an electric field particle will drift from one place to the next so I have a very small conductivity here which is one formula that has been put forward is exponential of minus 1 over the temperature to some power it has been invented by Mott it's known as Mott variable range hopping I won't go how this is derived but this is a very plausible argument where particles jump from one exponentially localized state to the next and then more recently there was the first paper was a paper by a liner at Schuller-Basco which was saying no this is only true if the system can exchange energy with the outside world but if the system is totally isolated actually the conductivity will stay 0 up to a critical temperature 0 0 0 and it's only at a finite temperature that the system can start moving and of course if you want to distinguish between these two situations numerically it's a nightmare because 0 is 0 here but here it's exponentially small so before you see that in numerics and you distinguish in numerics so that's very difficult so maybe a bound could answer this question so I'm putting this forward in front of you as maybe a challenge take it as a challenge it's a very simple object you have this current operator that I defined in a previous slide which is relatively simple you make it evolve with time using the standard Eisenberg representation in quantum mechanics this is not trivial to compute but this is trivial to define and then the conductivity is just the integral from time 0 to time infinity forget even the frequency of the commutator of j at time t and j at time 0 computed in average as I defined before with a trace of exponential of minus b times so this is a well defined object perfectly well defined if the Hamiltonian is known and which is supposed to have one of these two behavior or third one that nobody has guessed yet sorry, a couple questions on the slide so first of all what sounds of the system isolated because there's a t there so there's a thermal bath that's actually a very good question when you use this formula here that is known as the Kubo formula which was established to do linear response in a perturbation what you do is the following you have time here you assume that at time minus infinity your state is distributed with the distribution exponential of minus h at time minus infinity but then the thermostat is removed so each one of these states now evolves as it wants but with the isolated system and when you compute this commutator and whatever you compute it for a totally isolated system it's just at time equal minus infinity that you decided that you had thermal distribution of state it's not the same thing then saying that during the whole time of evolution your system is in contact with a big outside world and is able to receive or give energy and why it's not innocent is because if you try to do a move between two localized states which are not at the same energy if you have such a thermostat you can give the difference of energy to the thermostat and then the move can be accepted then another move will take the energy or whatever so in average you don't exchange energy but for each one of the move you can if you are totally isolated then this move is forbidden it doesn't conserve energy so by law of quantum mechanics it's forbidden therefore an isolated system has a much harder time jumping from one localized state to the next actually for the non-interacting system we know the answer the conductivity is zero at any temperature because there is no way all states are localized and there is no way you can transport current if you want it to be non-zero you need to put the bath and the second question is the kubo formula so it's well defined it's not clear that this limit when omega goes to zero is actually well defined it's far from obvious I agree with disorder actually I cheated a little bit but I can give you the clean formula because there is usually a little regulator which is written here with an exponential of zero plus time t but I totally agree with you it's not obvious that this limit is well defined there are systems for which there could be a delta function at omega equals zero but usually with disorder the limit omega tending to zero is reasonably well defined you have to take the limit of the number of particles going to infinity first there is a very well defined order of limit that you should respect first the size of the system and the number of particles go to infinity and then you're allowed to take the limit omega goes to zero otherwise you get discrete levels and then again you're sensitive to the distance between eigenstates absolutely but in the localized state this is okay because in the localized state the conductivity is supposed to go to zero so sigma of omega and let's say t equals zero is doing something like this where this is going to zero you say it's totally correct in a superconductor what you will get is a delta function here and then you get a regular part it doesn't matter what it is but this delta function means you have a superconductor if you have dirt let's say normally you should avoid that if you're in the localized phase so your point is well taken and it's totally correct for which there is a delta function here but this is usually in the superconducting phase or in the superfree phase that I was mentioning for the bosons but not on the localized side but that's why I put it that way because I think that's where we can discuss and try to formulate it as cleanly as possible so that then a mathematician can work on it because for physicists we are used to hop only I think it's close to being full proof but I totally agree with the points that are made and so one has to be very careful in the, there are questions which are hidden in this Kuboff formula that physicists have lived with for the last whatever 1950s or so 70 years and which we sort of swallow immediately without even worrying about it which need to be defined and indeed if I take a clean superconductor the limit omega goes to 0 as is more complicated than that point perfectly well taken are there other questions on this? Yes, would you have a physical explanation of what happens at t star because it's from mathematical standpoint makes sense that the function has no reason to be smooth at t star but from a physical standpoint what is happening for that all of a sudden you get this rush? Nobody knows the calculation that is in this paper is a particular actually it was already in the Anderson paper in 1958 57 and the idea is the following that if you have this problem of not conserving energy and you have no interaction this is dead you cannot do anything but if you have interactions you put the energy to other localized states because you interact with them so the system could act as a bath for itself and when you write a kind of mean field theory of this you find the transition point whether this exists or not we know mean field theory giving you transition for systems for which there is no transition take the 1D easing model it has no phase transition but you do mean field you find one so whether this point is correct or not whether it's an artifact of this way of doing things nobody knows and nobody has been really able to compute essentially what happens around this point the only thing I can say is that the scale is matching the one that for example I derived with 30 methods for which the disorder kind of blow up so it's the scale at which the disorder effect becomes really strong and that's the scale at which if you have a thermostat you would sort of start in the mod variable range hopping but again whether there is this transition or not I think it's a genuinely open problem are there other questions on this I have only one more slide so don't hesitate to ask questions ok one last question in the same set of questions in the same way then there is this question on transport at low temperature you could sort of move to very high temperature say I don't care about temperature I just do an average over all the states I go to infinite temperature all states are equally probable essentially if I am localized maybe I don't explore the whole Hilbert space I am confined to a subset of the Hilbert space and if the disorder is weak clearly I can explore the whole Hilbert space so maybe there is an ergodicity transition that occurs as a function of the strength of the disorder because I am going from a system which is localized even at finite energy density or if you prefer infinite temperature and a system which would be kind of delocalized so this is a little bit the equivalent of this transition point that I showed you for the low temperature phase I won't go into details in this I just point a very nice review that exists in review of modern physics that you can look at but there are clearly attempt to break this problem coming from the math community here is a paper by Embry where he attacks this problem to the best of my knowledge this is not a proof I heard the qualification quasi-proof I wouldn't there tell you what is a proof or quasi-proof but just to show that there is issues that one can try to attack and again these are issues that are difficult to address by the standard tools that the physicists have at their disposal okay I think that's a good point to conclude I leave you with the conclusions essentially what I tried to show you is that Anderson localization so Schrodinger equation plus disorder is a fantastically rich phenomenon even today it is still very rich for non-interacting particles but of course as soon as you put interactions the Hilbert space just explodes and this becomes very difficult an open problem but one which I think could still be reachable by rigorous tools that are the tools of mathematics so I think it makes for a nice maybe convergence point between the two communities and on that I'll stop thank you for your attention and of course wishes Frank again happy birthday so there are all these results by Erdos and Nyao about the case with very weak potential so that's the Schrodinger equation Schrodinger equation and I think it shows that you have diffusion in this very weak potential you mean in 1D or in more than 1D then at least in 3D there should be the answer that I thought was not there because that's my memory of what was said at this meeting but maybe I... there are very different methods because they are not respectful methods but they are just like so then it would be nice to see what is the result in 2D because in 3D people expect a mobility edge but not in 2D so if the result gives a delocalized phase in 3D that's perfect at weak disorder but if you go to 2D the same method should either fail and show absence of diffusion and certainly in 1D so in 3D I'm not sure that you get the linearized Boltzmann equation and then for last time 2D I don't remember but it would be interesting to see because then it answers the question that this side is delocalized localized in 2D and localized in 1D and I think that's the Poissonian case that you mentioned I mean if the disorder is weak Poissonian gives Gaussian anytime so at least for physicists it will not matter I don't know for mathematicians because now because you get a very large number of impurity per sort of zone which is concerned by the system and then you have central limit theorem which reduces the disorder to a Gaussian disorder so I don't think it matters let me say that way I would be extremely surprised if there is an answer which is given for weak Poissonian disorder strong Poissonian disorder is a different story because then you can be pinned by one very strong impurity and not collectively but for weak Poissonian disorder I don't think it matters so if it's there for Poissonian we should tell me what is the reference just an age question okay I see you start that's a good picture nobody has tried a large equation it would be easier in some sense that's interesting question so people have tried of course there is a paper by Efetov who is a very very well known physicist and in D equal to infinity he found very weird results and people don't believe very much that these results were correct but again if you go too high in dimension the system will become a little bit trivial unless you manage to scale the strength of disorder yeah so you have to find a way to scale up this point properly as you as you move up in dimension at least the only work I know moving up in dimension is this work by Efetov where he found some weird results and also went on to the Cayley tree and but I would say here the excitement also for physics reason is around D equal to but your point is well taken I cannot give you other reference I'm sure other people have tried to look at but I cannot comment on those it's a good point more questions more questions