 OK, I think we're ready to start the next session. This session has, as a common denominator, the 1990s. We have three speakers and one chair. The session starts with Eduardo Miranda, then moves to Indra Neal Paul, who are PhD students in the 90s, spanning the start to the end, and then Rutgers and Jimmy R.C. and myself, postdocs at Rutgers in the 90s as well. So, our first talk this morning in this session will be given by Eduardo Miranda from Campinas. He will tell us about emergent symmetry and transport in disordered chains. OK, thank you, Andy. And thanks to the organizers for this invitation to be here. It's a real pleasure to be here at this occasion. So I'm going to be talking about emergent symmetry in disordered chains, and if there is time at the end, maybe I'll talk a little bit about transport too. And of course, I'm going to start with a few words about peers. So, unfortunately, I couldn't find pictures that I had, that I own with peers from that time, because these are all printed photos and they are stuffed in boxes in my house and I'm very disorganized. I couldn't find them. Fortunately, Ide Takage was very nice and he sent me this photo from 2002 in Brasilia where I organized this conference. So we have many usual suspects here, Gabe Apley, Vlad Obrasavlevic, peers, myself and Ide. OK, so this is the only photo from around that time that I could find. So, however, in the absence of pictures, I decided to, since I'm probably the first student to talk, of course, we heard Rebecca speaking before, but today I'm the first student and I think one of the earliest students of peers. So I decided to tell an anecdote that I think shows a little bit about peers' personality and character. And so the thing is that just before the PhD defense, peers would warn us students that he would ask a question, a general question, completely unrelated to our work, completely unrelated to what we were doing and it could be anything. And so I was, of course, very nervous about this. I didn't know what was coming. And in my case, I don't know if he kept this tradition after me. And I don't know if he remembers what happened in my case, but in my case, he asked me about this, the potato clock. So this is a toy of his children and he brought the potato clock, he put it there, and he asked me to explain how it works. And of course, the potato clock is just a battery and I had no idea about the chemical elements inside the potato that made this run. So I gave him an answer. I can't remember what the answer I gave him. And I can't remember what his reaction was, but I think it was something like this. So the work I'm going to talk about, this work I have been doing with some students and a collaborator. So this is Victor Kito, he's the main name behind this work and also later work with Luis Faria, Pedro Lopes and a collaborator, José Aoyos, from Sao Paulo. So I'm going to talk about disordered chains. So disordered is a topic that has not been mentioned too much in this conference, so I'm going to take a different axis here and talk about these disordered spin chains. And the motivation is there are a few, not many, but there are a few quasi 1D compounds that realize these disordered spin chains. So here are some examples. Some of them are, most of them I would say are spin-a-half systems. There is however one which is a spin-one. And of course these are quasi one-dimensional systems and the particular feature of these systems is that for instance they don't seem to order to very low temperatures. So this is just the susceptibility of this system. And the theory I'm going to talk about is the theory that gives this dashed line prediction for the susceptibility. And I'll try to explain to you how this result is obtained. So, okay. There are some interest in this type of system because of cold atom systems that can be engineered to be one-dimensional and in many cases you can introduce disorder by hand with speckled disorder of laser. And there's also some interest in that direction. So what I'm going to describe now is the method that's used to analyze these systems. This is a very powerful method geared to describe disordered systems, especially strongly disordered systems. So of course you probably know that disordered is a very difficult aspect of condensed matter to deal with. And the advantage of the method that I'm going to talk about is that it gives you very reliable answers in certain cases regarding disordered systems. So let's take, as a first example, a disordered one-dimensional Heisenberg chain, spin-a-half. And so you have these coupling constants which are random variables. And I'm going to assume that these random variables are independently chosen from a certain distribution, PJ. This distribution has only positive values of just anti-ferromagnetic couplings and I'll assume it has a sharp cut-off omega at high energies. So the idea of the method is the following, was proposed by these people, is the following. Okay, this is a disordered system. What you do is the following. So it's a decimation procedure. You take the largest coupling constant in the chain, which I call omega here, so it should be close to the cut-off. If you look at the level structure of these two spins, it's going to be a ground singlet and an excited triplet separated by this very large energy scale omega. So the idea is the following. Just ignore that high energy triplet at low energy. It's probably not going to be very important. And besides you now calculate with perturbation theory, what the polarization of these middle singlet here does to the interaction between the neighboring spins, S1 and S4. So if you just do second-word perturbation theory, you can show that the polarization of this guy, which has been removed because it's just a singlet, introduces a direct exchange coupling between S1 and S4 that was not there before. And the coupling is given by this expression. No, it looks very much like second-word perturbation theory expression. And the new coupling, J tilde, you should note it's smaller than both J1 and J2, okay? So the tendency is for smaller couplings to be generated by this decimation procedure. So the net result is that S2 and S3 disappear, they form a singlet, and a new coupling appears between S1 and S4, okay? So you just iterate this procedure and what the work of these people showed was that the flow of this decimation procedure leads to universal distributions at low energies, okay? So this is a real flow of a speed-a-half system. So as you can see from the picture, because you generate these very low energy scale couplings, the distribution becomes extremely singular at low J. Even if you start with a uniform distribution, it becomes very singular as you iterate. So in fact you can show that they are all attracted by this fixed-point distribution, which is just a power law, okay? Very singular power law, and actually the exponent alpha here is going to zero very slowly with the log of the energy scale, and so this power law is becoming ever more singular as you iterate. Now the interesting thing about this fixed-point distribution is that if you calculate the width of the distribution compared to the mean, it diverges, okay? So the interpretation of this is that the effective disorder at low energies becomes increasingly larger and larger, and actually it becomes infinite at zero temperature, as you approach zero temperature or zero energy, okay? Now you remember that the procedure consists of doing perturbation theory, assuming that the central coupling is large, and because the distribution gets larger and larger, broader and broader, the error you make by doing a second order perturbation theory becomes smaller and smaller. So in fact the method is asymptotically exact, okay? So you can actually get exact results. Of course at the beginning you're making mistakes, so as long as the flow at the beginning is not terribly badly behaved, you should trust your results at the end, or another way of putting it, if you start with very strong disorder, the disorder gets even bigger and your results are probably reliable. So pictorially what you do is the following, you find the strongest coupling, you decimate it, you renormalize the adjacent spins, you do this progressively, you do this, you do that, and eventually you start decimating longer couplings, okay? So when you do that you generate new couplings like this, and eventually you have a larger coupling, and as you do that forever, you generate ever larger couplings and larger bonds between these spins. So the ground state that you get has this structure which sometimes called random singlet phase, very often called the random singlet phase, it's a sort of a valous bond glass, okay? So it's a sort of the glassy version of the valous bond solid. And the feature to remember here is that you have well separated spins of all sizes, spin pairs of all sizes, connected by these large bonds. Now what about excitations above this ground state? So the excitations are just the breaking of these bonds. So when you break a bond you get a triplet excitation, the localized triplet excitation. Now of course you can also show, I don't have time to do it here, but it's not too hard to show that the energies of these longer bonds, they are smaller and smaller because they are longer there, they happen later at the procedure, and they actually depend on the size through this sort of stretched exponential behavior with this characteristic exponent psi. So this is called activated dynamical scaling as opposed to the usual dynamical scaling of critical points, okay, where this is usually a power law. Now this can, if you analyze this, you can understand how low temperature properties are obtained. So for instance at finite t, all the bonds which are longer than a certain L0, a certain length, which is obtained just by inverting this expression, putting E equal to t and inverting this, so LT is given here. So all the bonds that are longer, this length scale are broken, okay? And so for instance at temperature t, they are going to be on the average separated by a distance one over LT, which is given by this. So for instance if you now look at the susceptibility of the system, you can with extremely good accuracy at low energies say that the susceptibility, the contribution to the susceptibility from the singlet is zero, and the contribution from the free spins is just curie-like. So in the end the susceptibility that you get is a curie expression with a law correction, okay? So because the distribution, once again the distribution is broader and broader at lower energies, this is better and better approximated by this expression, okay? In this sense it's asymptotically exact. So the expression I showed was shown at the beginning compared to the experimental data is this expression for the susceptibility. You can get other physical properties using similar arguments, but I'm going to just show this susceptibility argument. So what I want you to keep in mind is that at finite temperatures, the state of the system can be viewed as a collection of random singlets available on glass and free spins, okay? So what our consistent of was to look at the spin one system. A spin one version of the Heisenberg model has been studied extensively in the Heisenberg limit, but of course every spin one system can have also a bi-quadratic term like this. It's absent in spin a half, but it's in spin one, it's in principle there. And our goal is to understand sort of the phase diagram of this system. So it's convenient when you're discussing the system to talk instead of using these J and D variables to put them in a 2D plot and define radio variables and angular variables, which I call E and theta here. And I'm going to be looking at the case where the initial distribution is such that the theta is constant and the E's are disordered, okay? You can also analyze the case where theta is distributed, but I'm going to focus here because on this initial case where the theta is fixed, it's easier to understand the phase diagram in this case. So once again you can do the same thing that I showed before, you can try to decimate the system but now the interactions are different. And in the spin one case, the difference is that you have three possible level structures. So the ground state might be a singlet, a triplet, or a quintuplet, and the excited states also could be different. So the procedure consists of decimating these higher energy guys and looking at the effective Hamiltonian generated within the low energy multiplet, which now can be a singlet but can also be a spin one or a spin two, okay? So to understand when you have this case, that case, so that case it's useful to look at this 2D plot and the answer is the following. In this region here, you always have this situation here. So when J and D fall in this region, you always form a singlet, in this region here you form a spin one and in this outer region here you form a spin two. Now, I'm not going to talk about this region here. This region generates large spins. The physics is dominated by very large spins and the flow is completely different from the one that I'm going to talk about here. So I'm going to talk about just this portion of the phase diagram where you can either have that decimation or that one over there, okay? Now, once again, the analysis is... I can give you details of the analysis, but I don't have much time to do it here, so I'm going to give you just the answer and I'll try to explain why this happened. So it's not too surprising to say that in this whole region here, you get a phase which is very much like the phase that I explained before for the spin a half case, okay? Because you only form singlets, okay? So this is not too different from the Heisenberg case. Actually, the Heisenberg point is embedded there and once again you have at finite t a collection of random singlets and free spins, okay? So this is not very different from what I talked before. Activate the dynamical scaling, size equal to 1 half. This acceptability is given by that expression, okay? There's a small detail here which is that this phase ends at pi over four and here this goes to a smaller angle. So actually the random singlet phase extends a little bit more there, but these are details that I'm not going to go into. Now, the interesting phase we found is actually the one that lives in this wedge which I call one over there. Remember that is the edge where you can form spin ones when you decimate. So the nice thing about this case is that, as I said, you can decimate these two spin ones and get another spin one and this spin one can then decimate another spin one and generate a singlet. So this is a process by which you can form singlets out of three spins. Okay, so the picture of the ground state is still a random singlet phase but it's a random singlet trio phase, okay? Actually, you can also form sex steps, in other words, singlets made out of six or nine, any multiple of three, but they're less frequent than the trios. So I'm going to just talk about the trios here. Okay, so this is the structure now of the ground state. Now you can also work out the energy length scale relationship in this case. The exponent is different, it's one-third, okay? And anybody that guessed that these three he has to do with the number of spins in the trio is right. Okay, so this is three. And, but however, it's still a phase that's characterized by infinite disorder. You can still make exact claims at low energies and in particular you can also have this type of susceptibility and other properties like that. Okay, so now the building blocks, if I want to look at finite temperature, the building blocks are singlet trios and free spins, okay? Now, I want to talk about emergent SU3 symmetry in this system. Remember, this is an SU2 symmetric chain, okay? Just spins with scalar interactions. So what about SU3? Okay, so the SU3 comes in in the following way. We all know that spin one operators form a three dimensional representation of SU2. However, if you add to the usual spin operators, these so-called quadrupolar moments, bilinear operators in the spin operators, these eight guys here, they actually form a set of generators of the group SU3, okay? So actually, if you put these guys in an exponential with arbitrary constants, this is a general representation for the fundamental, actually for the fundamental representation of SU3, okay? SU3 is different from SU2 in the sense that there is also another one, another three dimensional representation, different from the other one, which is a so-called anti-fundamental representation, which is obtained if you just flip the signs of these five guys here, the five quadrupolar operators become, together with the usual spin operators, become generators of SU3. Okay, so you can actually rewrite the Hamiltonian which had only linear and bilinear spin operators in terms of these generators. It has this form that's given here. It's easy to show. And it's interesting that, for instance, in these two cases here, pi over four and minus pi over two, these are special points where this Hamiltonian is such that this coefficient here equals that coefficient and, or at least either they are the same or they have the same modules in different signs. And in that case, this Hamiltonian has exact SU3 symmetry. This is well known for the cling case for the pi over four point. It's actually an integrable point, very well understood. The minus pi over two is also SU3 symmetric. And so we actually solved the disorded case a few years back for, in the context of general disorded SUN chains. And there's a hat tip to Piers here, who I guess the apple never falls very far from the tree. So I, this is one of Piers obsessions, SUN group, large N expansion. So we decided to look at this disorded SUN chain, in particular to look, to check, since we can get the exact results, we wanted to check whether the SUN chain, the N equals infinity limit is characteristic of finite N chains. And the answer is no, but I'm not going to be able to discuss this here. So these points are special, but what I'm saying to you now is that the whole phase, one or two, actually have emergent SU3 symmetry. Remember that the Hamiltonians in this region here are not SU3 symmetric, okay? But at low energies, SU3 emerges as a symmetry. So the reasoning very quickly is the following. So these singlets here, which we all agree are SU2 singlets, once you put back all these SU3 generators, these are also SU3 singlets, okay? In other words, in technical terms, the eight SU3 generators, they annihilate this state of three or two spins. So these are both SU2 and SU3 singlets. So they both transform as the trivial representation of SU3. And you can actually look at this pair in terms of a quark, or a fundamental and an anti-quark representation of SU3. So this is like a meson, if you want to use high energy language. Whereas the trios are like baryons, they are bound states of three quarks, okay? This is exactly the same kind of thing that happens in hydrons, hydrons physics. And of course the free spins also transform as SU3 objects. So they also can be viewed both as SU2 and as SU3 objects. So since the building blocks of the system are the singlets and the free spins, and both of these transform as SU3, you should expect an emergent SU3 symmetry at low energies. And this is actually what happens, so we call this kind of phase where we have pairs, a mesonic phase and this trio phase, a baryonic phase because of this high energy analogy. So for instance, if you now couple an external field to all the eight generators and you calculate the susceptibility, the response of the system to this external magnetic field that's coupled to both dipolar and quadrupolar operators, they're all the same. They all behave with the same exponent, okay? So this is what we call an emergent SU3 for this system. The same thing happens, for instance, for the correlation function, which I'm not going to show here, but this is the sense in which we talk about an emergent SU3 in this system, okay? So I guess I'm not going to be able to talk about transport, but afterwards when we saw this, we said, well, let's just look at higher spins and we're probably going to find other cases of emergent symmetries at higher spins. And this can be done. It's a long analysis, but it's not too hard. Unfortunately, we do not find any other examples of these emergent symmetries at higher spins, okay? So we only get conventional psi equals one half phases. They actually have SU2S plus one symmetry, but they are conventional spin pair phases. We wanted to get these trios and quintuplets and so on. There are phases where psi is equal to one third, but it has absolutely no emergent symmetry. So we were not very happy with this and that's where things stood at this point. But then Victor said, well, there's another way of looking at this system, which is the following. SU spin one is also the fundamental representation of SU3, user rotations in three dimensions, okay? So maybe the way to go is not SU2 with spin one to SU2 with larger spins, but rather SU3 to SUN, okay? And this is actually the case. So if you write an SUN symmetric Hamiltonian, but with different coupling constants in these two sectors, so this has SUN symmetry, but no SUN symmetry because, but however, if this guy is equal to that guy, it would have SUN symmetry. So the same thing happens here. So when this guy is equal to that, you have explicit SUN symmetry, but when you do the disorder, so this can be viewed as a sort of anisotropic SUN model, but in the distorted case, when you flow through the same mechanism that happened in the spin one case, you also have phases where you have emergent SUN symmetry, emergent SUN baryons, emergent SUN mesons, and the baryons, you know, they form these objects where the number of spins in the singlet is given by N. So it could be, for instance, N equal to five. Five, for instance, in the case of SO5. Now, you may be asking yourselves, well, what is this good for? I mean, SON, what does it mean? Okay, but it turns out the following. Well, first of all, SO2 going to SU2, SU2 emerging out of SO2 is actually very old. It's just the XXZ chain that was analyzed early on by Fisher because, you know, XXZ has SO2 symmetry and actually Fisher showed that the correlation functions in the X-direction and the Z-direction are all the same exponent. SO3 going to SU3 is what we had seen before. And SO4, for instance, we know that SO4 is isomorphic to SU2 cross SU2, and this is just the Kugelkomsky model. So if you take the Kugelkomsky, where you have spin and orbital operators, the Kugelkomsky model, it is a realization of an SO4 Hamiltonian. When you do the disorder version, you have an emergent SU4 out of this problem. SO5 is isomorphic to SP4, symplatic group 4, but we didn't find anything interesting there. But SO6 is isomorphic to SU4. So it could have SU6 coming out of SU4, and it turns out that with cold atoms, it is possible to build systems in principle, nobody has ever done this, but in principle, it's possible to generate an SU4 chain, and if it's the disorder, you might see SU6 emergent symmetry coming out of that. So I guess I'm not going to talk about transports, so let me just skip to the conclusions. So what I just talked to you about is just emergent symmetry in very strongly disordered symmetries that are generally in this scheme SON to SUN. So I'll stop here and thank you very much. Thank you, Eduardo.