 Okay, first of all, I, of course, have to thank the organizing for inviting me to this meeting. And I will talk about, in the main part of the talk, I will talk about the work that we actually have done with Boris and our postdoc, Monevel Pina. And at the very end, I will mention some very recent results that we've done together with Valeriy Kravtsov. But before I start to talk, of course, I have to, I want to, no, yes, I want to congratulate Boris on his birthday and recollect, as a previous speaker did some very old history, how I have first heard about Boris, which was probably the year of 79, if my memory is correct, when my supervisor, Tolya Larkin, came back from St. Petersburg, which was at this time Leningrad, of course, and said that he has seen a miracle. And the miracle was, as he described it, a boy who was, and he, as he described it, Aronov, who was, as you know, the supervisor of Boris, was drawing diagrams on the board and this miracle boy was writing, was writing the result in his head, which he computed in his head in the real time. So, and what I want to, what, was it like that? So, unfortunately, I couldn't find the photograph of the Boris at this time and this is the best, what you see is the best possible approximation to that. And although, as you know, Boris has changed a lot over time, I still, I'm sure that inside him still leaves this miracle boy of 79, and I wish that he continues to live there. Okay, so now let's go get to my talk. So my talk, as we were requested, as was requested, has two parts, has a sort of general introduction and then I will talk about the actual work that we have done with Boris. And then this general introduction, I would like to describe the situation, the theoretical and experimental situation with this old subject of Josephson junction arrays, which to me look very unsatisfactory. So, and what we are going to discuss, what I'm going to discuss in the new part of the talk will be a specific case of Josephson junction chains, which we expect to show very unusual behavior, but not directly related to what was observed, to what was observed experimentally in large arrays that were studied some years ago. So it's not, but it's not surprising because the regime is different. So these two parts of my talk are only vaguely connected. The main idea is that the main connection is that there are some strange intermediate phases which sometimes appear and the starting of these phases in many cases is pool. Okay, so let's start with this general introduction. So very naive model of Josephson junction array is that you have Josephson energy, which is cosine of the phase difference and which competes with the charging energy, which is proportional to the voltage squared and due to Josephson relation, voltage is the same as the time derivative of the phase. And thus, these two phase and the charging energy are represented by the canonically conjugated variables. And as a result, if you have large Josephson energy, it wins over the charging energy. Or if you have large charging energy, it wins over the Josephson energy. And you have basically two regimes. In one regime, you can call it insulator. That is, if you see it very large, you expect that low-temperature resistance should become infinite. You form an insulator at high-temperature at low-temperature in the opposite regime. You expect to find a superconductor. So that's a sort of naive and simplistic scheme. The reality, microscopic reality is very different. The reason for that is that on each island, there might be offset charge. Instead of this charging energy, which is simply quadratic in the charge, you have on each island the offset charge. Not only it is completely uncontrollable, but also it slowly changes with time on the timescales. Well, it depends on who did the sample and how patient was the person and how patient you are in waiting for these charges to relax. But they always, always move. So, well, this is a picture from the work of Aetnist and very similar by Zimmerman and very similar picture was observed, for instance, by Juro Pashkin, who is somewhere here in the audience. And you also see the same that if you study individual islands, the charge slowly drifts and some very patient people like this Zimmerman Aetnist told me that he waited for seven years and they didn't stop to drift. So, okay, so the one very important thing to have in mind when comparing, when thinking about realistic arrays is it completely uncontrollable and slowly, slowly changing charges here. So, let's compare and let's now discuss whether the simple picture that I gave you first is indeed observed in the arrays. So, in other words, what we, so the disagreement or some puzzle appears even at a very, very basic level of the face diagram that is observed, that is observed. Namely, in this simple picture, it's either the charging energy which wins or the superconducting energy it wins, which wins. And therefore, you either have superconductor here or insulator as a change disorder or you can also say that you change magnetic field. And therefore, you frustrate those on interactions and therefore you change from insulator to superconductor. However, what I'm going to show you that in many cases, but not very unpleasantly not in all, there is some intermediate regime which appears in between, which demands theoretical explanation which is absent. So, very first experiments after this picture was, after this radical picture was publicized, many experiments indeed show something which vaguely resembles either this, either the resistance curves up or down. So, and so, but also even at this time, there are all data in which resistance remained simply constant. So, my procrustus doesn't work. So, what I, so, so this little cartoon, which unfortunately doesn't work very well, shows the Greek hero as you know, procrustus who used to take people in his chamber and either cut their legs if they were too long or stretch them if they were too short. So, that's exactly what people try to do to fit the experimental data with this theory. And for instance, here you see very nicely superconducting regime. You see a single point here and then you see that this must be a quantum critical point between insulator and superconductor. But of course, if you remove this line, well, you see yourself. So, even worse here, in which you do see the same experiments at the nonzero magnetic field, you see that, well, naively you would expect if you look at this picture without these lines which guide your eye, supposedly. So, you will say that there is some intermediate regime and that's exactly what you see on these plots, but that's what is published. So, the situation actually became worse but was completely missed by community because for some non-scientific reasons, this work was basically abandoned with experimental work because these two people, Panitia and Siria, working in Grenoble and studying a race of some special topology, observed a huge regime of EGA over EC in which they see temperature-independent resistance. And this regime covered, well, more than one order of magnitude. So, very recently, a very similar picture was observed by Misha Gershengdon, who is somewhere here, maybe. Or maybe not, oh yeah, here, yes. Who observed that in some key, who observed that in a wide regime of the resistance is constant in a relatively wide regime of parameters, the race that he studied were also very a bit unusual. And what is important is that he measured, actually he represents his results in terms of the gap in this not real gap but sort of fit to this exponential activation or behavior. And this gap becomes exactly zero in some range of magnetic fields. So, all that, it's not always like that. Sometimes his arrays show this dependence, but some arrays show this strange bad metal behavior. He also, in keeping in tradition with Penetrientseret, never published his results. No, it depends whether it is, whether it depends, no, no, no, sorry, yeah, yeah, yeah, sorry, yeah, sorry. Okay, and finally, something very similar was observed by, very recently by Buschia for Superconducting Islands on Grafien, which is also a very good geoluson junction system. So, to summarize, and that was the end of my introduction, to summarize that many geoluson junction arrays show very wide intermediate metal state. And so, in this metal state, the resistance can be differ depending on the parameters on the magnetic field or on Ej over Ec by one or two orders of magnitude. And so, and featureless in the sense that there is no trace of some non-linear, some current voltage and the current voltage characteristics. So, it's really like a metal, but on the other hand, as your allies in this geoluson junction arrays, there are no normal electrons. So, it is, so, the proper way to think about it is that this state is some strange normal liquid of bosons or of wish, if you wish, of Cooper pairs. So, now, after that introduction and to the, under the rational review of the unsatisfactory situation in this, and our understanding of geoluson junction arrays, let me discuss geoluson junction chains, which you will think are even simpler object. So, and show and convince you that in this supposedly very simple object, there is still something very unexpected going on. So, the main point of my, the main point of my talk will be of this, that there is some non-trivial, non-ergodic, intermediate, non-ergodic phase, which appears in this geoluson junction arrays. And when I, when you hear the words non-ergodic, you usually think that non-ergodicity is something like a glass, but in the sense that it's due to large potential barriers, which forms spontaneously, and the four are the glass, which was made in the form of this beautiful Etruscan vase, or sorry, Egyptian vase remains in this form for thousands of years. But a non-ergodicity could also mean something different, that if you have strange kinematic rules, such as in this little child model, child-child play, in which you have allowed to move the cars only vertically and horizontally, you might get stuck, you know, like in a traffic jam. So, let's go to this, to my, to this subject of geoluson junction chain, and see where non-ergodicity happens. First of all, in this chain, if we discuss thermodynamics, oh sorry, so for a moment I will discuss just the ideal chain. Of course, if we, a bit later, I will talk about the effect of the disorder, which turned on this physics that I am describing, and this effect will turn out to be not qualitatively important, and that's very important. So, what we know about this geoluson junction chain? Well, first of all, we know what is its behavior at zero temperature, and at zero temperature the problem can, you can think about this chain as experiencing phase slips, which are exactly equivalent to the vortices in two-dimensional X-Y model, and therefore you expect Berezinski-Koslowski-Stowler's transition at some particular value of Ej over Ac. So, here you exactly at zero temperature on the real axis, you expect superconductor, here you expect insulator, and somewhere here you will say that, okay, that must be quantum critical point. Quantum critical point. So, you expect basically the same behavior that I was drawing before, that is a resistance, when you go down here, the resistance goes to zero, here it goes to infinity, and the same song and dance that I already performed. So, of course, we have to introduce these random charges, and the main claim of my talk is that this phase diagram is mostly wrong. In fact, what happens there is that there is indeed a region of good metal, which appears at large Ej over Ac. There is some region of insulator, which is here, but what happens at minimum increase temperature is completely different story. Namely, that there is a line which separates insulator formed at high temperatures from the metal, the metal which appears at low temperatures, and the phase in between is what you will call a bad metal, that is, it conducts somewhat, but it's non-ergodic, and bad in all respects. So, I will give you two types of arguments for this phase diagram. One argument is based on the classical, on the classical, or more precisely, quasi-classical limit here, and another argument is due to numerical simulations, which essentially probe the system along this line, and I will explain why numeric simulations are in some sense weird that they can probe the system only along these hyperboles. So, and also speaking about the effect of the charges, this behavior here and all quantitative values, for instance, what is Ej over Ac for, for the transition indeed depends strongly on the presence of random charges, but this transition, this part of the phase diagram, which is the most interesting for me, it does not. So, let's talk about the first, let's start with a quantum phase, with this transition first. So, the argument for that is that I look what happens at large Ej over Ac. At large, sorry, at large temperatures. At large temperatures, this term, which is cosine of the phase difference is always limited, so mostly the energy goes into the charging energy. When the energy is in this term, it means that the average value of the charge is also large, and when the average value of the charge is large, and because, and random, well, on different sites, the elementary process, which is generated by Josephson term, which corresponds to the transition of one pair between two neighboring sites, as shown here, has cost you a lot, a large energy. This energy can be positive or negative, it's not important, but the fact, but what is important is that in perturbation theory, you start to get large denominators, which are proportional to this difference of charge, but whereas your numerator is always fine, it is always fixed, it's just Josephson energy. Therefore, what happens is that this term, if you average it over the distribution of charges corresponding to some finite temperature, what you get is that typical value of this term is this Ej over square root of temperature times Cc. Square root of temperature appears simply because typical charges are of the square, are of the order of square root of, typical charges squared are of the order of temperature. Thus, you expect that the perturbation theory in this term converges if this term, if this typical value is smaller than one, and therefore, again, using the same arguments that were invented first by Anderson and then used for many body localization by Baskoal and Eralschuler, you see that what you get is that this, for when this parameter is much smaller than one, is smaller than one, we get many body insulating state. This argument, so in other words, what I have tried, argued that this line and this, when it happens is when T over Ej is larger than Ej over Ec as you see this, squaring this quantity, you get this condition. So, what I argued is that this line T over Ej is equal to Ej over Ec corresponds to real transition into many body localized state. So, when I write here insulator, it's really not just insulator, but many body localized insulator. Okay. So, well, one can do slightly better to convince oneself that this is indeed correct by studying the propagation of the wave function of the pair in the forward propagating approximation in which you average over the all processes in which the cooper pair moves some distance r, not necessarily that it moves just step by step, but it may move first here and then there and so you sum over all these processes, overall permutations of these steps and what you get is exactly the same result which you expected from this naive arguments and you can even discern the coefficient in this equation. So, now let's talk about what happens when in the phase, in this phase just before, just before full many body localization. So, classical system seems to be completely trivial because if, well, it has in the classical limit, you will think that, okay, I have charging energy and Josephson energy, these are two independent quantities so I can compute trivially the free energy. However, the equations of motion are not trivial and because, well, they're not exactly solvable. So, the question is whether these equations, what these equations of motion, whether they reproduce when if you follow the time evolution given by these equations of motion, if you give, get the same result as from the aerodynamic average. So, one can actually use for this classical arc for what, to understand what happens in the regime of very large temperatures and that will be a classical equivalent, what I described for the quant, for the many body localization transition. One can use the very similar arguments saying that I have now distribution of the velocities of these phases and then I ask how the noise which is created somewhere far propagates through my classical system and what I will get in this propagation is the exact analogy of the, of the denominator that appears in the quantum perturbation theory. Everything is completely equivalent and therefore the noise which propagates and therefore you, you conclude that the noise which propagates a large distance from the source contains the product of these factors and therefore averaging over this distribution of these velocities what you get is log normal distribution of the noise of the noise values. So, well, what you say is that typically noise decreases exponentially when you go away from the source and so, but then you should ask yourself where the noise is generated at all because that looks like that everything in the system at high temperatures dies, dies, dies, but it's not completely correct because in which you can see by studying more carefully the, what happens in the system if you study proper, if you include the fact that there are some low frequencies or to be more precise the frequencies which might be in resonance with the frequency of the noise that propagates and then I immediately see that there are some very small denominators and you have to be careful with these denominators. So, the qualitative argument is that in order to get the chaotic behavior in this system you need to have at least three islands because for two islands, for two islands with two close frequencies or two close charges if we translated back into the Josephson junction language you have still integrable equations but for three close frequencies you can generate, you can, you expect a chaotic behavior if their frequencies are close how often it happens. It happens if the, if these frequencies are close with the, have frequencies of the order of which differ by the, by unity and that happens with probability square root of two, square root of t for any pair of frequencies and therefore t for three frequencies. So, you, so the, therefore the following picture emerges that you have some rare triads of islands which happen to be in almost in resonance with each other. These resonances generate the noise as well we can easily see numerically and this, but these triads are rare and they talk to each other through this silent and dead regions of the, of the space. Okay, so, so then you, if you ask about, start to ask about physical properties then you should say that how the transport happens in my system. The transport happens like, like electrical current or conductivity happens in my system is, it happens in something which resembles very much mode transport in the conventional insulators that you have, that some charge which is, which moves from one chaotic system to another and then to another and each such motion is associated with the resistance which is distributed according to this log normal distribution that we discussed, that they discussed before. So, the sum of course of the chain is the sum of the resistances and that all translates into the, into the prediction that the resistance of this chain is, has the resistance of this chain, sorry, logarithm of the resistance of this chain scales as internal energy or temperature times logarithm of temperature. So, also this picture means that tells you that there are some distribution of relaxation times associated with the motion of the charge between these rare triads and therefore, and therefore the properties, the physical properties of the system has this long exponential, relatively long straight dependence, time dependence of the resist of the properties. So, well then we check this numerically and we saw both that the conductance, so that of course we saw trivial fact that at low energies the, at low, sorry, at, at, yes, at low temperatures and energies the conductance is large, we approach superconductivity, but that's of course whatever, what everyone would expect, but at high temperatures, at high energies, well not extremely high, you see the energies are still pretty decent here, the resistance is roughly exponential in temperature or the energy. So, but what is even more interesting and important, what we saw numerically and that is that the relaxation is very slow and if we study that thermodynamic quantities, such as fluctuation of this internal energy and extracted from the extrapolation of these curves to the infinite times, then we find that this extrapolation never leads us to the pure thermodynamic result, which is shown here by red line, whereas this, all these curves correspond to finite time to the measurement, to the measurement of this quantity at finite times and so if you extrapolate them, they go at most to this dashed line, so there is always some difference between the thermodynamic quantities and the extrapolated ones. In other words, this picture of these triads, which stay, talk to each other poorly, but also their position is more or less fixed. That is, of course, they move a little bit around their position, but they do not move very far, so the system, the whole face space of the system is broken into different parts corresponding to different position of these triads. At least that's my qualitative picture of this non-ergodic behavior. You can also, yeah, so here I can, well, I don't need all these song and dance here at this part, because I can really go to this very far long times, but if I, but for do, to do numerical, to do, well, these times become, to get, these times become so long that I really need to start to do some extrapolation. By, well, you see, it's not very good to talk about temperature, because I'm basically arguing that there is no temperature. What is the right control parameter is the average energy, which is exactly what I fix in the simulations. Yes, yes, yes. Okay, so let me now very quickly skip this to the, to this part of the, on the quantum simulations and just tell you that the reason that we did exact numerical diagonalization to study the behavior of the system in, with a fixed number of charging states, and instead of allowing the discharge to fluctuate from minus infinity to infinity, we allowed it to fluctuate from minus q to q, and therefore, and this is exactly equivalent to the probing the system along this line. And what we saw is exactly the same quantity that is entropy, which we define as a von Neumann entropy integrated over the half of the system is in some, it does not reach its thermodynamic limit in the large range of parameters. That is, we, in this system, we expect the transition to the fully localized state somewhere here, well, here, and there is a wide range of, roughly speaking, a factor of three, where the entropy remains remarkably less than its thermodynamic limit, which is logo, well, because we used just five, five charging states, while the, that's the, which is shown here by red line. Okay, so since I got somehow out of time, which I thought it was different, let me just go to the conclusions, which you can see here, and I'm ready for the questions. Thank you very much.