 Od行了 school I would like to,</ural organizer for inviting me for this meeting. Of course, I know Boris for a long time, but I never born in solid state physics and I think my talk will be the only one where in no experiments and even no proposal for experiments. But of course with cold atom one can do everything you want. v prinspeciju, da je posobil, da imetite nekaj modelov, kaj sem bilo diskussi. Tako, nekaj, da se diskussi in nekaj, da se diskussi, nekaj toj modelov nekaj spektrali statističnih. Toj modelov je, da je zašličnje v analitivnih kakulacijnih, ali nekaj strančnih, zato videli, da je to veliko in nekaj nezavršen behavior. Tako, nekaj, da vse pravimo, je zašličen. Zelo sem počučen, da sem počučen, da sem počučen. Zelo sem počučen, da sem počučen, da sem počučen, da sem počučen, da sem počučen, da sem počučen. je po težko težko prirodila, da sem jezobimila, zato je več karakteristika tukaj, ali vseč jaz ni najsimolji to imač, kako sem je mare iz vzelo obstuknje modela, ko mi ljube vseč ponenjo delno, in da tak samo naživamo vsečnje halliteti. Na daj model, ko je vzelo obstuknje map, pride ga važno izolit, ali sem ga da pokazala, da je ne musela dobro uživila vse delovosti vsega vsega vsega vsega vsega ini vsega metri, ko je mu zelo vsega laks metri. Vse sem pa tudi daj poštaj, na koncluziju. Vsega, dobro taj sreč, vsega vsega vsega, ne bo vsega ini vsega, ratistiki, ki so imate v mene, da barriersi, že tu imaš jeljavih translatev e1, e2, en plus 1. Zatenje tudi hospitalsiificación , energijskel, spetrem v tisliči 3D-ključjev tako ga najbolj napravite, nekaj zelo početno. Zelo početno n, kaj je n plus 1 minus en. E g. Zato početno zelo početno zelo početno. Zelo početno zelo početno e plus 1 minus e. Zelo početno zelo početno. 건 se, da je početno zelo početno zelo početno. Da je to bil doličivo disloc diagn맛, zelo početno. E dotatno visk. Plast je n gauge. Osve ownership tra nak plenty peh v sebe musiquewasser. câmera. Illajko, 85 dall 110. kako se pokazate. Če bo tukaj, da se razmah je kajne 7,8,4,4. Nesakaj ne je zelo, da sem tukaj, da je 6,8,4,4,4. Svoje to nekaj, da se zelo, da sem priliv, ki se jaz zelo, da je zelo, da je nekaj, da je zelo, da je nekaj, da je nekaj, da je nekaj, da je nekaj, da je nekaj, da je nekaj, ne nekaj, ki se občaj. In je to izgleda, da je tudi vse tako je. In, da je ovo to nekaj, izgleda je, da je početno vse z Bernoulli procesov, da je E-Piro-S. In za to, da je početno vse početno vse početno, da je početno vse početno. Početno početno početno vse početno, da je nekaj početno. But everybody believes that for intergable system, generic intergable system, spectral statistics, statistical energy level will be pro sonjo statics. On the other hand, if your billiard is chaotic, for example, that shape of the study room, and you do the same story, you will see that you'll get something else. You will get function, which is zero at the origin, it has exponential squared, squared table. Vzelo, Cathedral was the final function. It means that if you use random, standard random matrix, you can computeộibed with the resolution of the equations but it's completely the different behavior. My talk is about dynamical system,ils with neither Poisonian nor Karotic. Kaj je tukaj idej, od numerika, od odpravdu, je to ozvodno 3-dimensionala Andersen model. So, if you have a Andersen model, you have disorder and you have jump and disorder in some say box disorders, and you know very well that if disorder is very large, all states are localized. But if disorder is less than special value, pa se počekljili v nekaj poživnih posobilitvacjah. V centre delozirati spetrumov, v tem delozirati je. Nakaj, odrešal, in je valja, če je tudi metalozirati transicije, da je dobro tako ko se posobila. Zato, da je vse nekaj nekaj najznim za vzbim, in da je, ki se zelo, načešča, a imaš uživljana korakli. Tako, to ne so vzelo. To je ne vrvi numeriči. Vrvi numeriči ne zelo konvensite, ali, da ustavljate, na tudi, vzelo vzelo vzelo vzelo v zelo. Marko, da bomo počkati vzelo vredovati. A zelo vzelo vzelo vzelo v zelo, vzelo vzelo vzelo s 100, da imaš počkati, ali, da boš vzelo vzelo vzelo v zelo, vzelo vzelo vzelo vzelo, pa nekako sem bil, tako početne karakteristike. Ako je naredil, zelo je vseško pravno, da je posao, da ima nekaj nekaj behavior. Če pa je nekaj behavior karakterizena, pri lukovaj repušenosti, kako je, nekaj početne, na početnih vseh rejdenah, in tudi naživljenjijo delovih, If you compute number variance, what it means, you count how many labels exist in the interval of length l and you normalize it as I do it before, that mean value of labels corresponding to n of l, mean value equal l, and then you compute variance of this one. It's one more exercise objavljanje pernuli počkaj, neč prišlično. Nombu vrarnosti. To je nombu vrarnosti, včoč jazi nombu vrarnosti razbenju sev posnešljenih interval, kaj jazi nekaj režim, če se počkaj v zepetru in jazi nije nožno v random. V zepetru neč krati neč nekaj rešimi vrarnosti. občas, kaj je spetrelja kompresibilit, kot je 1 for Poisson, and 0 for random matrix. For all non-trivial Maeman-type statistics, you will see that K is some non-trivial number between 0 and 1. Another property of all this model, which I will discuss, it is that wave function are multi-fractal. But it means that if you compute psi squared, power to q, and you compute mean value, then it's scaled as a v is a volume size to the power q minus one dq in dq called fractal dimensions. It's easy to see that if you have localized state, you have only one element of c large or other small, so dq is equal to zero. If you have random matrix, they are all of the same order and due to normalizations, each of them is one over n. So, you do dq equal one for random matrix. For all model, which you will see, this quantity unknown trivial. So, what is known and what is well investigated? So, once more, if you have, what will be the random matrix model of this behavior? It's well known that if you have integral models, they are like diagonal matrices. Each may diagonal matrix, independent variables, and you get integral model. If you have random matrices, all elements roughly speaking of the same order, each e is interacting with all other. So, it was natural to consider matrices of the form e minus g power alpha. So, when alpha is small, it means that you can ignore that it looks like a constant. So, it should be close to random matrix theory. If alpha is large, it should be close to pass rule. And this also was, have been proved by Levitov, and actually Levitov, that when alpha equal one, you have a special type of statistics, the critical statistics. Standard model of critical statistics is what is called critical power low banded matrices. It's a random Gaussian variables, independent variables, but the variance is not a constant, but decaying like e minus g power one, if it's square root of h, each element is the decaying. So, when b equal infinity, it means that all of them are the same, and you get random matrix, if b equal to zero, you get both. What is known for this standard model? One can construct perturbation theory. For both, b equal much bigger than one, and b much less than one. For the large b, you use heavily sigma model, and I don't know how to do it without sigma model, but for small b, you do perturbation theory, and you get easily some confirmations that key is non-trivial, and fractal dimensions are non-trivial in the both limit. What is interesting symmetry relation, which has been derived, proposed by Mirlin in company, that if you compute, not dq, but if you compute this combination, then this combination is symmetric with respect to q to one minus q. So, main question, which I start, is as there are dynamical models with intermediate statistics. You remember that if system is integrable, statistics is Poissonian, if it's Karotik, it's random matrix. But there are another type of models, which are really mentioned. It's called Pseuda integrable models. Pseuda integrable, simplest example of pseudo integrable models is a plane billiard, where all angles are rational of pi. So, all angles of this polygon should be rational of pi in order that it will be pseudo integrable model. For example, you can have rectangular billiard with some part in it, or you can have triangular billiard with angle pi over n. For all this system, it is proved that trajectory are living, they are living on the higher genus surface. So, the genus of the surface should be compute by this formula. So, when all m e is one, it's the only case. Genus is wall, so it's integrable system. All integrable system trajectory are living on the torus, and torus is the surface of genus wall. For Karotik system, all trajectory are living or cova uniformly high dimensional energy surface, and so there is no two dimensional surfaces in the game. For example, if you use triangle pi over 4, pi over 4, pi over 2, or pi over 6, pi over 3, then it's right triangle that genus 1 is integrable, but if you do pi over 5, pi over 8, or something else, you will see that they are not because the genus don't corresponds to integrable model. So, what is the classical dynamics of this to the integrable model? It's also well established that it corresponds to what is called interval exchange map. For example, let me consider pi over 8 triangle. Instead of continuation, you will consider reflection in the sides, and you will have a hexagon, and you should identify parallel parts of this hexagon. So, all classical mechanics of trajectory in pi over 8 triangle corresponds to this, and you see immediately that it genus 2 surfaces, because you have four independent contours for this. But if you look more carefully, you will see that you can split it in four different parts, and all trajectories, say, from 0, 1 will go to 4, 5 region, from 1, 2 they go 5, 6, and so on. So, classical map for these trajectories. You start here, you go here, you come here, you reflect it, you go here, you go here, reflect it, and so forth. It's called interval exchange map, because you easily see that all information of trajectory are hidden in the permutation of interval. You have 0, 1, 1, 2, 3, 4 interval, and you just permuted it in the special man. It's all known that this system is neither integrable nor chaotic, and they have name through the integrable system. And it's even classical, as they are quite non-trivial, because you see that here classical mechanics lead to explicit discontinuity. So, if you iterate for a long time, you should be sure in what part of the discontinuity you are. It's a quite heavy talk, heavy method which permits you to do much higher numerics than usual. But what is about quantum mechanics? So, up to now, I take pi over n triangles, for example pi over 5 triangles, and somebody did calculation, and you see you have 10,000 levels, which you split in four equal parts. And if low energy part, here you have three curves. This curve is a random matrix prediction. I start it, it's not p of s, integrated p of s. It's much more sensitive to the small details, and for numerics it's much more useful, and much more sensitive to use, not p of s. P of s depends of the bin. N of s, they just count how many s is less than a fixed value. It's much more easy and much more stable to do n of s. So, it's random matrix prediction, it's Poisson prediction, and in between is something which is called semi Poisson, which I will discuss slightly later. It has label repulsion at small, and exponential tails in the large. And everything here, you can compute correlation function analytically. And you see that with increasing of energy, you might nicely go to the prediction to something else. What? You go to something else than it was before. And this is the same story, but for all triangles from 5 to 30, and there's a difference between semi Poisson, which is just a mark. And all triangles, we see clearly, this it will be a random matrix prediction. And all triangles are quite far, they depend of n. And here you have 20,000 labels, so it's quite a good statistics. So, what can be done analytically? Analytically, one can compute only the compressibility. Compressibility, if you compute it carefully through periodic orbits of these triangles, you will have some explicit formula of like this. So, the formula is very simple. It is heavy, mathematics is hidden in it, because to compute periodic orbits is possible only for what is called which triangle, where you have a which group in it. If somebody interesting, I can discuss, but if the triangle is pseudo integral, but not belong to this subgroup of all triangles, I don't know to do, because even the number of periodic orbits is not known explicitly. It's not only the bound, but it can oscillate it to all usines. It's just a consequence of very complicated character of classical mechanics for this simply looking system. You can do another system, and if you do the computation, you will see that for this value 5, 8, 10 and 12, k of zero is close to the value one half, which was predicted to Poisson. To predicted to semi Poisson, you can find small details, it's not important. This is a correlation function, correlation for a factor as numerically computed, and this value corresponds to this guy. Another model, which you also can belong to this same class, so mechanics is different, is a run of bomb flux line. You can put flux line in the rectangular billiard, and you will see that spectral statistics becomes from Poisson, becomes of intermediate tape. It has a level repulsion, and it has exponential tail numerically, and you can also compute through periodic orbits calculation some value for a factor of zero, just to confirm analytically strange character or intermediate character of this model. You can do of course calculation for circular billiard, whatever you wish, and I will not spend time for this. What is the difficulty of treating this model quantum mechanically? Look, if you have classical reflection of the half plane, for example, you have here the region, you have in this red point, you have two light, light coming, a light reflected. In this region you only one light, and here you have no classical light. So classical mechanics always for pseudo integral system is discontinuous. Quantum mechanics cannot have discontinuity, so it produces some complicated expression close to this line, which is called optical boundaries, and you also know that if you consider, say, a reflection of the wedge, of the electric wedge, then you have this formula and you see that exactly on this direction you don't see it, but probably you can believe exactly in the optical lines, it corresponds to some diffraction coefficient goes to infinity, which signifies that here the field is strongly perturbated and no perturbation theory can be applied in this region. The same story for the flux line, of course, diffraction coefficient of the flux line is well known. It's one more classical mechanics is continuous, but it appears phase. If you go in this part of flux line, you get phase e pi alpha, for this line you get phase e minus e pi alpha, and so on you see discontinuous phase. Here there is many interesting phenomena, which I don't have time to discuss because I see that I am too slow. In particular, one can prove that reflection from many lines like this corresponds in the dominant approximation, the mirror reflection. If angle is small, if angle is small, not small angle, but really the parameter should be square root of k, d, much less than 1, phi is the angle, but g is the distance from this, and if this criterion is fulfilled, the dominant effect of horizon reflection is reflection from the boundary, and you can easily construct approximate wave function leaving for this model. For example, this wave function corresponds to the barrier billiard, which I showed you before. You have barrier billiard, and then you split it in two parts. If it's equal, so you have this type of billiard. There is only a difference that here you impose anointment condition, and here you impose Dirichlet-Dirichlet condition. And all this pseudo-integral model leads to approximate semi-classical quantization of periodic obvious channel. I don't have time to discuss, but just look for this particular nice example. It's 10,000 levels. Everything is normalized. The distance between two levels is one, and you clearly see a regular structure for very high excited states, and no such structure appear in, say, rectangular billiard. And I have infinite many pictures of this, but it's pictures, but it leads to some kind of semi-classical quantization, which you see here. Approximate value is this, and exact value is this. The same story is here. It's long story of how to quantize surface of higher genus, and what I have in myons that you can do semi-classical quantization of surface of higher genus, at least of certain surface of higher genus. But these are the pictures, but I, like, more have formulas, so if you have expand this function to some kind of basis function. So this is a y, so you have zero, zero here, and here I put Lehmann condition here, and zero condition here. It's a full set of function in this rectangular. You can expand and you can plot. Here you will have m, here you have n, and you see that all coefficients for this particular function is small except one, or a few of them, which corresponds exactly to these propagating waves. Here if you do look carefully, it's much less, it goes to zero. It's a non-aligned picture, so the line means plus-minus, but amplitude here is approximately one, and here amplitude is very small. So it's like we call it super scar, because they survive, semi-classical limit goes to infinity. If you have numbers, you can compute participation ratio, and you can compute participation ratio for this numerics, which has been quite old one, and you know that participation ratio for higher moments should be scaled with n. Now n is k, because you have two-dimensional surface of conservation of energy, only the number of function proportional to k, and if you do computation, and you plot, say, this is from 1,000 levels to 800 levels here 4,400 days, 10,000, 10,000, 20, and then you compute, try to fit this curve, and you see that the fit with all this is old date, nowadays one can do, if you wish much more. You see that the fit, like for this point, for the stadium billiard, which is chaotic, and you see for the stadium billiard, fit is proportional to k, proportional to n, which means that d is equal to 1, but for barrier billiard, d is equal to 1,5 approximately. And here you have all labels still for 5,000, and you can fit it, and you can conclude that everything for the pseudo-integral models are fractal wave functions. OK, the model is model, but I like to more analytical results so I will simplify it. And also, instead of consider pseudo-integral model, I will consider quantum pseudo-integral map. Quantum pseudo-integral map even the most simplest, which interchange to integral. So this is the example of x plus x model of 1, so you have lines, discontinues and other lines and so on. It's also discontinued map, it's pseudo-integral, it will be pseudo-integral map if you look for these sequences of map. It's a reconcerting map because you have dynamical system and if alpha is rational, you have interval exchange map, if alpha is rational, you have only ergodic map but not pseudo-integral. And you can compute quantum map. What is quantum map is unitary matrix whose saddle points equal classical maps and it's not my invention, it was somebody else who do it for pure mathematical purposes. And so I use a slightly generalization of this map in the momentum space. If you do Fourier transformation of this map, you get that you can approximate this map like this. It is not approximation, exact, of course, Fourier transformation. Here you have a phase. In the dynamical model, it will be k squared and here I put just random phases and here you have unitary matrix. You can easily convince yourself that it's unitary, so it's not clearly seen and what I will consider will follow two different cases, non-symmetric matrices which is analog of GUE and symmetric matrices in minus k at the same. If alpha is irrational, I told you that I don't know, it's not pure interval, it's just ergodic map and you see that spectral statistic numerically very well described by standard random matrix example. This I cannot prove because I don't see how to, nobody knows how to prove that even for this simple matrix that all spectral characteristics do agree with standard example. But what I can prove is alpha is rational, say m over q is 2 comprime integer n is dimension of matrix equal plus minus 1, then spectral statistics agree with semi-pooson statistics with one parameter beta equal either k minus 1, either k minus 2 for non-symmetric symmetric. What is semi-pooson? It is exact result, numerics is just exact result but I don't have time to discuss. What is semi-pooson statistics? It's appear many years ago when we try to consider the simple possible random type logarithmic gas approximation. In random matrices you have all energy interacts with all energy. Here I put energy only nearby energies and then you do by some formulas and you get that all characteristics can be computed. In particular p of s equal s power beta the same beta is here and minus beta plus 1 s exponential tail k of 2 you can compute you can compute compressibility 1 beta plus 1 and the simplest case beta equal 1 it's a semi-pooson statistics which I discussed you before. So look, alpha can change a level repulsion as I wish. If you example if alpha 1 3, 1, 6, 1, 9 and you see p of s is s squared then you have s5, s8 the same for symmetric matrices it's shifted just to better so you have s1 half s3 half and 5 half this is exact result compared with some numerics. I have to go more quicker and if you do for example alpha 1 over 20 and n equal 8, 0, 1 you see the formula s 19 power e minus 20, but if you change a little n 8, 8 you see 8,000, 9 you see that curve is not far from random matrix result it's not exactly random matrix result because I can compute explicit formula for all for all k there are some explicit formula more complicated which is not important it is important but not immediately and correlation function for example you see as strange it is exponential of exponential is something it's very nicely fitted to numerics for this case n equal 8,001 if you compute numerically fractal dimensions you see that they are fractal dimensions very nicely follow it's for 1,3,1,5 by 9 but if you check this relation symmetry relation you see that they are not fulfilled there is no perturbation theory here you have one of them one delta another of them of another one it is not symmetric at all it is the same story for all alpha which we have investigation so eigen function of this strange quantum map multifractal but symmetry relation is not fulfilled but as I told you it is a very strange map because it appears somewhere else and we just recuperate it and to show its vector statistics it is nice to see something more generic consider for example this type of matrix random moment a random Q it's a diagonal matrix you have E so it's a Hermitian matrix and I will impose the measure what will be the measure the measure will be say trace L dagger L is natural and if I do only this I have this path AQ is not restricted because it's not here and I will put here beta Q squared it's very natural measure but if you look a little and you remember that you see this matrix before you will see that it's a lux matrix for collodgera-moser model L dot equal L exactly the same L you have M special which corresponds to as a consequence of this relation to collodgera-moser-gamiltonian rational collodgera-moser-gamiltonian but I don't have any dynamics but I do know who is the action in angle variables for collodgera model for example if you diagonalize this matrix this is a phases and this is there you can easily prove that this relation and what is important that double M is the angle variables double M angle value of lux matrix action variables it's important to me that this transformation between paying Q to WL and lambda are canonical it's very difficult to prove even by 2 by 2 matrices that it is canonical transformation you don't have any wrong scale so the only information I will use is the relation to some new variables so what is a GL is this one then I choose trace of LL dagger is lambda M squared Q squared is at Q dagger Q because it was eigenvalue of Q it is equal to this one so a random depended Q is the lambda double W and you get this factor for eigenvalue of lux matrices and you see that W integral you just throw it away and you get join distribution of eigenvalues without only for eigenvalues of lux matrices it is exact formula it's not approximated no approximation is done so all classical random matrix are based that you have big group of rotation which you can integrate rotated variables but lambda and then get join distribution from here you don't have explicit group invariance but you have hidden group invariance because you have an integral of motion which did not change eigenvalue of lambdas is a spectrum deformation and so you get by free expression for this join distribution so if you do it for something even more simpler than before you don't have time to discuss let consider even more simple matrix so you have here random variables and here you have k minus r without models but with e and you compute eigen distribution of p of s of this guy from this distribution like for the Wigner surmise you can convince yourself that the good Wigner type is the exponentially strong label repulsion p over s squared minus some quantity s so you have for example it's alpha g equal 2 it's a disk of you have exponentially big label repulsion and then you see how good is it for this model and this is a difference you can do it for all collodger model and you get a nice formula for all collodger model and the most important to me is that this strange looking model it's called Roussen-Artschneider model it has a name of relativistic generalization of collodger and never mind it's integrable model and the lux matrices after big simplification reduce exactly to the matrix which I have used for this interval exchange map but parameter r here is the arbitrary it's related with parameter r which is here and so on I used it before when r was equal alpha n now I can use it as r equal constant because I know how to compute it because I know spectral distribution for Roussen-Artschneider matrices exactly and this is some picture for hyperbolic models I don't have the time to discuss it but for example for this model for Roussen-Artschneider matrices so it's exactly the same matrix but now there is no n here and then you can check that spectral statistics depends of integer part of n for a between 0 and 1 spectral statistics is shifted for a zone so you don't have any eigenvalues in p of s before s is equal to a you have a gap is eigenvalues it's not even strong it's extremely strong rival repulsion and compressibility is also analytical formula for this case you can compute for some other values you can compute in principle for all necessary functions and you can easily check for example here a equal 1 half and you see there is no eigenvalues here and then exponential tail and everything for a equal 4 3 this was a curve which I showed you before and so on and so forth I will just two words that all these models are fractal you compute you compute fractal dimension and you see nice fractal formula and all of them are symmetric so symmetry relation of Merlin is working here and you can compute perturbation theory for all these models which I have in mind and they agree to the model of critical random matrix ensemble you can even compute perturbation theory for q less than 1 half which usually you don't need you cannot for normal model but here you can compute and you see that in the leading orders this is just formula and if you look for the formula which I showed you before I started slightly before so you will see that in all limit for small and large coupling constants there is this type of relation that key equals 1 minus d1 and d1 is just entropy of wave function is just one of the fractal dimensions related with the entropy but it's valid for small and large coupling constant in between if you do numerics even for critical bended random matrices you see one of them key in one of them is 1 over d and the point looking as a point for b to equal 1 for b to equal 2 here they should coincide but they are very nicely coincide in between for Russian-Artschneider ensemble one can compute told you exactly and this is a compressibility in 1 minus d1 which I compute numerically and you see that its points are on the point there is another model which I call ultrametric matrices once more this is a relation it's some kind of mystical relation it's valid only for the model which decays 1 minus as a first power and also it's related so in all possible cases which you have checked even for the cases where you don't have symmetry relation this conjecture works perfectly well so this is all what I would like to speak so first of all there are standard models which leads to intermediate statistics of eigenvalue what it is which was probably the first one where numerics have been performed it's pure the interval billiard for all of them you do have pure the intermediate statistics you can interval billiard with a flux line probably many of them what is the critical band and trend of matrices it has one large diagonal plus e minus g follow pure the interval models by definition diagonal or just shifted diagonals plus follow in small changes of matrices I did not show you numerics but you can see analytically numerically that small changes you think that it's small but it leads to big changes of spectral statistics because here everything is 1 over n when you have changes in the scale 1 over n you can change everything for the spectral statistics there is no universality you see that very strange models appear but this is exact and also interesting fact that lax matrices of all interval classical system give new soluble ensembles soluble means that I can write during distribution of eigenvalues not for not for all of them I can compute correlation function for Sennarchneider I do can but for all say scologer model I cannot but you can have very nice beginner type ansatz which agree very well with numerics for all models which I know eigenfunctions are multifractal there is no practically proof from any there is no I don't know any one model one can compute spectrum of fractal damage analytically and so is mostly use perturbation theory and numerics there is a symmetry relation it's valid for models with one large diagonal but it's a rule out at least numerically for the models both pseudo integral maps we put conjecture which we cannot need proof that key equal 1 minus d1 divided by dimension of the system and you can imagine that it's leads for some new developments of intermediate statistics it may be some new models physical models we do have this statistic thank you very much