 ... ... ... ... ... ... v Belizei, Georgia, nekaj sem bil nekaj fotograpov z tajem ljudev. Danes zelo smo zelo počkali na ACTP. Zelo počkali po nekaj 25 ljudev na Radgers. Tako da. Peres je tudi menej vrštih vrštih. Jedno je, da je vrštih vrštih vrštih vrštih. So much excited, that you always think that there are more than two peers around you if you excite him, so you feel that you are all surrounded by him. So today unfortunately I didn't work in strongly correlated systems for some years, so today Pearce will pay for his past sins, that he was interested many years ago together with me in finance zero point entropy at zelo, da je zelo, zelo temperatru. A nič ga se prišli, da so se počešal, da je zelo, da je dinamiz. Zelo, da se prišli, da je vse model simple. A vse, da se prišli, da se prišli, da se prišli, da se prišli, da se prišli, bo, da se moduč nekaj je večo artefizijan. Kaj je moduč? Moduč je zelo v glaskomunitivnih vživosti. Vse je kompletno vrednje klasičnje. Kompletno vrednje je, da je bilo nr. spin in spin in energič nekaj je kompletno vrednje konfiguracije spin. tk. teži se povede površenje in neko včasno naliv je zelo. Tukaj časno, klasikovosti model je izgledajo mnogo dveh njih latih v Daeridu in je še pričo, da je izgleda komunitete, da vzveš, da vzveš površenje vzvešenje koče vzvešenje, da je izgleda na vzvešenje vžave. Zelo se vzelo na dobro vse model. Zelo se tako zelo obrženja, da ne zelo, da je, jo, da imaš gašnji form, da je to energije, ki je vrst konstrativne energije, ki je odstupena na gašne, pa način je boljska faktor. minimizujte in vseš, da je domenjete z energijem, ki je, občas sem zelo, minus tudi energija, ki je proporšna na n, in tako je zelo, da neče inče inče vseče, ali potem vzelo, da vzelo, da je tudi energija vzela energija, ki so nekaj stvari. Zelo, zelo, nekaj konfiguracij, da je to na n. Zato, kako jih možete pričočiti, na nekaj energiju ne vedete nekaj stati všeč. Zato nekaj partičnja funkcija je dominačnja v zelo energiju, kaj je tudi nekaj stati, nekaj in tudi na dynamic limit, nekaj je dominačnja v zelo energiju, kaj je nekaj stati, in je tudi transiton in tudi režim. To je tudi glas transiton, kaj je tudi nekaj relovan, za moj diskussion. Nelj smo zelo však však v kvantom modu, v kateri, da imamo kvantom, sem zelo vzelo, očičje term, ko je, ko je, ko je zelo, ovo je sigma x. Ok, zelo... ... ... ... ... ... ... And before I discuss more quantum model, let's discuss a classical dynamics of this simplest model. Classical dynamics, you can think about some sort of Monte Carlo procedure in which you flip the spins with the probability, which is given by the Boltzmann factor. So in this case you can ask how long it takes me to explore the space. There is no structure in the space whatsoever so it takes naturally two to n steps because you don't know where you go. At each step you will find better or worse, but until you explore everything you do not know where you are. So as I said before, this is a big advantage from the point of view of solution, but also is a disadvantage because the model is quite unphysical. So now I would like to argue that this unphysical model from the viewpoint of normal physical system is in fact is quite relevant to some nonphysical applications. The application, which is parallel, it's not exactly, but it's very similar to the system which is well known in mathematical community, which is known as number partitioning problem. Not to confuse this partition function that I was talking about, they have nothing to do with whatever. So the problem is like that. Suppose you are given a set of numbers A1, A2, An and the numbers are known with very high accuracy. There are some mathematical works which tell you with what accuracy you need to know these numbers for this to work. And what you want to know is to minimize this strange energy, which is the absolute value of the sum of some numbers minus the sum of other numbers. You can also say that, look, I just distribute in optimal way plus and minus in the sum of all numbers, so to minimize this quantity. If you are physicist, then you say that you can also, the problem is equivalent to minimizing this sort of antiferomagnetic like Hamiltonian in which you have A as the given numbers and sigma, which are these plus minus is invariables. So, what is the common property of this and random energy model? If you just change, if you flip one spin in this problem, that is, you change one sign from plus to minus, immediately you go very, very far in energy, in this energy. The reason is qualitatively obvious, because you are sort of adjusted this sum to get the value, which is much, much smaller than the typical value in your energy. And just, if you just flip one sign, you immediately go very far. So, in this respect, this is also, this problem is very similar to random energy model, and what I am going to say, most likely applies to it as well. So, that is just one example of the sort of mathematical problems, which do not have any structured phase space, unlike normal glasses. Ok, so now, what we, I will discuss how far this model relaxes and so on in the main part of my talk. But before, I want to give you a very important reference point. How far, what is the fastest relaxation that can happen in such a problem? So, in very general terms, we are talking about completely unstructured search, sorry, we are talking about quantum search in unstructured space. If in unstructured space, the search can happen, we can use Grover algorithm, and it was shown by mathematicians that this is the best that one can do. What is this algorithm? The algorithm is works, the important thing is that algorithm works in the square root of the number of classical steps. That is, if you have a space like this, which has two to n elements, classically you just have to check all two to n elements. Quantum algorithm takes the square root of it that is two to the power of n over two. And the basic idea of it is that you create these two very special operators. One is done from this unknown state. Another is done from just a superposition of the old states that you have in your problem. And then you rotate your system basically in this two dimensional space. It's very important that this algorithm, the whole idea of this algorithm, if you look closely at it, is due to the fact that you limit your problem to rotation in this artificial two dimensional space. You reduce the phase space of the problem to only these two. And then you can go slowly. And once you realize it, you realize that mathematicians working on this field were struggling with for a long time that this algorithm is extremely sensitive to disorder in the sense that you nearly need to have to keep the phases of all components exactly equal. And vice versa, instead of this two dimensional rotation starts to deviates and goes God knows where. I did a little simulation just for the sake of this talk in which Green indicates how the algorithm works in a perfect system. And in black I introduced some perturbation and you see that it never reaches the ideal state. OK, so in our problem, which I am going, which is the random manager model, which can be thought as a search in this unstructured space, we expect because it's not fine tuned in any way that we cannot get very close to grow. And we should get better, we might be able to get better than classical, but never grow. So, OK, OK. I can skip this for the sake of time. And now I will talk about the actual properties. So, first of all, let's do some simple physical discussion, what we have. I will be talking exclusively about the regime of relatively small fields in the next direction in which because what I really want that the field in the next direction do not completely destroy the states that I have classically. And so then, what are the states that I have in this problem? If I forget about my classical disorder, then clearly I have the states which are characterized by the magnetization in the next direction. And the magnetization runs from minus n to n, so here are these states. So, now on the top of it, I have the classical states in the direction, the states corresponding to the potential in the direction, which are also Gaussian, but broader. So, each of these states in the next direction are highly degenerate, because, of course, if I fix magnetization in the next direction, there are many ways how I get this magnetization provided that this magnetization is not maximal. So, lowest energy level is, of course, non-degenerate, but all others are extremely degenerate. So, as shown here in the regime of small gamma, gamma less than one, my lowest energy polarized state is higher than the lowest energy classical state. So, clearly, if I am somewhere here, with energy somewhere here, my classical states are not affected much by the quantum processes, simply because in order to move from one classical state to another, I need to use these states in another sector and they are separated by the gap. So, also, it's clear that there are, depending on gamma, there are two regimes. As I said, I will be interested in the regime when gamma is smaller than this square root of log 2, which is shown here. But in principle, there is also another regime where gamma is large, and depending on what is larger, the lowest energy state that is zero, let's say zero energy state, is dominated by one or another, by polarized state or classical state. This strange picture that you have either dominance by polarized states or classical states result in this phase diagram, which is known for 25 years that people say that, look, at zero at small gamma, we have this classical Derrida glass, and we have a transition from classical paramagnet to classical glass, whereas at slow temperatures we have a transition as a function of gamma, as I explained you, whether we are dominated by classical states or polarized states. And that was sort of the whole story for many years. What I'm going to tell you now is that this picture is almost correct, but not correct, but not fully correct, because it is, even though it is correct, that the potential has very little effect on the polarized states at lower energies. However, at higher temperatures or high energies, some completely different thing happens, namely that you have two dynamical glass phases. And here I actually want to explain what are these two different dynamical glass phases. There is, of course, with these boring thermodynamic glass, which is not very interesting, where I just dominated by a single state, but then here you are dominated by many states in thermodynamics. However, if you ask what is dynamical properties, you see come full quantum localization. That is, if you start from one in one state, you remain there forever. And this is known as many body localization, of course. However, there is another phase transition here, above which you are still in a glass state in the sense that your entropy is still very low compared to what it should be, dynamical entropy. But you are not dominated by one state. So here you have localization, but you are still in a glass. And only finally at very, very high temperatures, which are beyond this plot, you go to a fully ergodic state. So, and that is the main message of my talk, that here we have completely different regimes. In the entropy is very dynamical entropy, that is the space, the phase volume that we cover by natural dynamics is completely different. Okay, so to explain very briefly how, why it's so, let's focus on some low energy states. And we'll see that there are two different regimes. So, the number of these states is simply given by the density. And then we can ask how many states we get if we flip a given number of spins in a starting from one state. We get this number of states, we get this number of new states. And you see that this function has a sharp maximum at d equal n over 2. What does it mean? That the number of states where we can go, if we start in perturbation theory to go step by step from our beloved state that we started with, grows very fast, more than exponentially. And that is the result, well, it's Gaussian actually, so as a result all quantum processes, even though the quantum tunneling amplitude typically decreases exponentially with distance, we're always dominated by larger distances because the number of states increases with distance very fast. So, that's a qualitative reason why we're always jumping very far. Now I imagine that you believe me and of course I will need to prove it more formally. You imagine that you believe me that it's true, that we're always dominated by large distances. Then in this problem is reduced to a very well-known problem in localization theory which is called Rosensweig-Porter model that was studied in a number of works, but most important for me will be the recent work by Volodya Kravtsov here. And this model is sort of simplest model for localization in a high-dimensional space. What do we have in this model? We have random energy and we have random hopping amplitudes and the hopping amplitudes scale as non-trivial power of the system size. So, in our, and this model, of course in our problem we need really to find what is this scaling, how this scaling exponent gamma is related to the parameters of our problem and that is a subject of some calculation. But what is important is that this is just a number which is a function of our parameters. Now if you just take this scaling and then use it to get the physical properties, you immediately see what happens that in the Roland-Sweig-Porter model there is a large intermediate phase in which the states are not localized because the sum of the matrix elements of the Hamiltonian is less than, is more than, the sum of the matrix elements diverges, but the sum of the squares is finite and therefore it has no, it doesn't have ergodic behavior. So, let me explain you these two properties. First of all, why this requirement for localization? It's very simple because if you have, if this is infinity, it means that you can always find another state in resonance with yours. If this is infinite as well, then you can always, you can find the, sorry, then the total scattering rate in your problem, sorry, the total decay rate is finite, it means that you go away very quickly. So, in the intermediate regime, if you put the definitions, you find that this sum, which appears here, is given by these equations and therefore in this intermediate regime when gamma is, sorry, what is this? When gamma is less than, more than one, but less than two, we have extended by non ergodic states. So, now, so what do we have there? So, it's a function of energy, how gamma, which is this parameter, increases naturally because simply gamma, you remember, contains this number, which comes from the fact that number of states increases in our system. So, it increases when we go to small energies and therefore, sorry, when it increases, when we go to high energies and it decreases and goes to gamma, to one at low energies. So, naturally, it crosses these two critical points, one and two. There is no way how it can avoid it. Exactly where it crosses depends on this factor, but as I said, this factor is not crucial. So, now, so we have therefore in our system and we have two regimes. One is this fully localized regime that I was talking about in which there is no life at all. We have intermediate regime in which the system relaxes, but slowly it goes, we can find some resonances into which you go, but it takes a lot of time. As a result, if you ask now what is the physical correlators in my problem, after all, I started saying that I am studying the spin glass, something like a glass. The spin correlators are very funny. They decrease exponentially, but the rate at which they decrease is anomalous power of the system size and they do not decrease fully, but there is a bit of leftover here and this leftover itself scales as a non-trivial power of the system size. All that shows that in this system there is a wide non-ergodic phase in which, of course, one naturally will associate with the quantum glass behavior. And in this respect here I compare these correlators with the correlate which were computed by direct numerics with the compute correlators in the Rosenzweig-Porter model in Kurtzhoff paper and you see that they are completely similar. Well, actually they can be mapped one to one exactly so this mapping to Rosenzweig-Porter model is probably correct, is almost certainly correct. So now the last thing that I might want to discuss is how to compute this matrix element that I told you that we need some this parameter alpha but it's just a computation and I should say that I computed simply by computing the green function of this problem in time and then taking the Fourier and then I can show that by Fourier transform in time I can get it in energy. So it's all quite simple, there is a non-trivali here which is the effect of the cross-terms in which that is effect of the interaction in z-channel, on the interaction on x-channel and backwards, which is not clear to us at the moment and so we did some a lot of numerics to check how what is the accuracy of this formula for alpha that we derived analytically here. So but even though it's very likely that we are that this formula is incorrect by 10 or 20%, it doesn't change the whole qualitative picture that I described before namely which is summarized on this slide that we have instead of very interesting dynamical glass phases in which there is, especially in this phase there is some life, there is non-zero entropy but a lot of entropy is hidden in the because the system does not dynamically cover all the phase space. Ok, so I think that that's my time, right?