08:31:10 So, as you see from the title I will be speaking about super diffusion and actually some very interesting aspects of super diffusion, which seems to be sort of something that you can little bit provocatively calls to play modality and try to explain what
08:31:28 I mean by that. So, since I have only 20 minutes I decided just to basically cover one paper, and that paper was already published last year so it's not very nice nice so I apologize for for experts, because they will not learn anything new.
08:31:41 But still I mean since this is a bit more let's say mathematical paper maybe it was not really noticed more broadly, I mean, is as I mean this this aspect which I mentioned the super diffusion, sorry the super users IDs diffusion of maybe you know it's
08:31:56 worth addressing.
08:31:58 So I mean, as you see me the buzzword here also is collaborative drunk universality so it seems that in either.
08:32:05 Another buzzword is interoperability and another one is not a billion cemeteries, so I should combine is going to be done on a billion cemeteries for the claim is that you get generically, keep this universality.
08:32:16 And for people who follow this, the story of KPZ for for for several decades now. I mean they know and couple of years ago people have connected KPZ knows it to me past I wouldn't I problems to on harmonic oscillators.
08:32:30 I would like to just make a disclaimer I mean this is a different aspect of KPMG morality because here we need to interoperability, so I mean it's kind of bizarre i mean you know for someone like lattices which are not interoperable, but they have three
08:32:42 kinds of quantities people have shown that you have a PC transport. But now, you have interoperability which means you have infinitely many kinds of quantities, but you need not a billion cemeteries and then you have a busy transport So, and this is just
08:32:55 I mean my talk is not about any groups, it's not about any any any physical mechanisms, is just about some simple, simple mathematical model which basically can reserve assimilated and very good data can be obtained to, to, to, to make conjectures let's
08:33:10 see.
08:33:11 So let me just before going into this model so that the class of models. If I just basically couple of us couple of slides for history. I mean, I think this was one of the first papers were this super diffusion internal systems was know that it was a
08:33:27 paper with wine drinker which, when we looked at a classical version of London. Basically a classical version of XXZ model which is this. So what Loughner lattice lambda licious model, which is the topic version of classical classical magnet.
08:33:42 So there is a parameter which you can tune and basically they are the motivation was different there we wanted to see whether it is a ballistic to diffuse if crossover similar tend to next xz.
08:33:51 So there is this little delta which is basically like a logarithm, quantum Delta. When there is positive, it's like easy access when there's negative is like is it blaming manager zero it's like as a topic.
08:34:02 So when we then we found that correlation functions well, medical blocks to this space diamond correlation functions if you want but the main plot is this one so the correlation function of spin spin.
08:34:17 I'm an integrated or space so time correlation function optimization, either sub plateaus for easy blame, or the case quite weak for easy access. And typically the case like Bala with this low which seems to be like two thirds I mean at that time we really
08:34:28 didn't even pay attention to that two thirds. I mean, We basically best fit was 0.65, but later.
08:34:48 is one over two thirds which is to two. And then I think we should also mentioned the paper by by Marco who looked actually even before that equation for the ex ex ex model that as a topic Heisenberg magnet and actually also found that the spit steady
08:35:02 state current decay is not like one of our nm is the system size, but in the case that one of them. And if one does some some scaling one sees that this is totally compatible with dynamic explanatory or something that was, I guess the first hint that
08:35:14 should be something like three or two that I'm an exponent, but maybe one. You know, I mean there is still some issues, whether steady state limb that is really a correct description or or let's say it captures all the, all the features of of demographics
08:35:31 of open systems of closed systems but let me that don't want to get into legislative discussion here so then I think this was the paper, actually, but this conjecture was kind of forced without really clear and loud, just our paper from 2019 where we
08:35:47 looked at really homogeneous crunch. Spin transport in xxx Heisenberg model, and it's described time into the body and we looked at the speed. Again, the gifts being correlation functions we found that I'm exposed to be able to go and also the profiles
08:36:05 will have those which were predicted for for KPMG universality I mean, she knows which are related, but the whole bunch point.
08:36:14 Okay, so, and there were many other papers.
08:36:18 I mean, after 2019 I think now there are much more papers that are listed here this is the first papers probably which came out, which basically tried to understand this this phenomenon they put out some some some some also some heuristic mechanisms.
08:36:33 I think we should probably mention papers by by by Roman a certain certain Gopalakrishnan who were the first to kind of derived. Then I've explored dynamic exploring three over to based on a particular picture, but I should also stress that is the only
08:36:47 applies to quantum models, and somehow classical models, even though you might think they are conceptually simpler. They may be are more difficult because it's not so clear what they're really correct was the particles.
08:36:58 I mean the internal beauty seems to be a bit more scope.
08:37:02 But anyway, so I mean at least what is understood now is more or less than damning exponentially three or two.
08:37:17 And also watch it also probably stretch the last day for the PR x often a surround can then company, who also derived this sobering reality in the sense that they pointed out that the three of us should be universal across more than a million signatures,
08:37:22 again, based on the quantum destruction of the, of the models.
08:37:28 And moreover I mean one should probably also stress very recent experiments.
08:37:32 Bye bye bye.
08:37:34 Quantum Gospel. The Gospel be where
08:37:39 people have actually been able to kind of very nicely demonstrate the standard respond to and take his escape.
08:37:47 So anyway, but this talk is not about physics of it I mean it's not about physical mechanisms for PPC transporting integral systems is just about simple and cute into the system we want to be in symmetry, which, maybe, you know, maybe the simplest, the
08:38:02 minimum model which has all that kind of one can tune also not only the, I mean one control actually that the Scimitar group because I mean that's why again now I'm trying to explain what I mean by super universality I mean that you have the number of
08:38:16 experiments which is insensitive to the senior group of the model.
08:38:19 So the model can have any Su and only one missing plastic USPS student so on any competition, it's a group of looking at the same. That's the claim, that's conjecture have the same, the same result.
08:38:31 You have the same, the same result. Okay, so since. Again, I just won't be getting a little more than it's more. So let me just give you three slides about the properties of the smaller the first of all the finished of this model.
08:38:42 So the model actually basically again falls into the category of this kind of circuit model so if you want to kind of sell an automaton models I mean time is discrete species discrete, so you only have to specify the fundamental fundamental dynamical
08:38:56 laws which apply let's say to to to neighboring sites. So now imagine that it each side of your lattice now that is windy, and also time is a lot this so you apply a map discreetly time.
08:39:10 So imagine that in each time step you apply a map to a to neighboring spins and this map is like a mapping, which actually two matrices. So the local degree of freedom is a matrix, I will specify more concretely what space of mattresses, what symmetric
08:39:23 space, I have to have to take it from, but let's just assume this is that the two mattresses for a moment. And now let me just, you know, throw from the air throw this this this irrational transformation of these two matrices and one of them to.
08:39:36 I'm sorry, there is a typo here on the left hand side there should be bribes. So this would be a mapping to that which maps and one of them to to Amazon Prime and enterprise.
08:39:46 So it's really a simple, you know, Russia transformation which is like the swap because here's what first you do the swap and then you conjugate contract with the, with the sum of matrices plus some items tower thousand.
08:40:01 Now this is how we play a lot of time of time step.
08:40:06 But anyways, so this map, this is now, kind of, I mean I will show you this map has nice properties for example is my business is letting.
08:40:13 So it defines kind of generalized Hamiltonian dynamics, so it has all the finalized properties of physics that you would like, okay, but also this map also has some other nice properties for example if you inverted them, just, you know, this map is parametric
08:40:27 independent right it depends on the external parameters.
08:40:30 So if inverted then it corresponds to minus top. Then it conserves, the property of matrices is this graph to one. Also the image matters is square one.
08:40:43 And if the madrasas our mission, then also the image message is our mission so these hints you, which direction you should look for compact space, the space.
08:40:53 And moreover I mean it also preserved. If you want to generalize speak. It was the sum of matrices.
08:41:03 I mean, you could already see if you have to add two by two matrix is with these properties that described one, and that their permission, that this would be like a classical space right this would be like a vector of boundaries. I mean, arbitrary arbitrary direction which is determined by a vector on the sphere. So then this would be like a classical spin, and this this
08:41:12 support property with black would be like a consideration of classical spring.
08:41:15 Under these two sides to site interactions right these are two sides interactions. Now what is the face base.
08:41:21 Now the face base in general is what is known as public squares Manion so it's part of my thrice not only by dimension of the metrics by by the rank of them.
08:41:30 But, but also by the rank of the metrics. Okay.
08:41:33 I mean this is not precisely rank but it is number of negative versus positive eigenvalues of this metrics, so these metrics is have to square to one.
08:41:44 So the eigenvalues only plus or minus one. So you can decide how many I can read his mind is one you want, and this gives you a rank of the first money because it was my name is basically the these metrics plus identity and then this becomes a project.
08:41:54 So, M square one. If you had identity becomes this vertical.
08:41:58 So these are projected.
08:42:01 Ok so now you can basically write the normal form of this matrix is which is given in terms of n Su Su and transformation. So think this is a diagonal.
08:42:15 These methods is similar to the plus and minus once this is what's the sometimes called signature. And then there is this GG which is basically. Yeah, but I'm a parametric you have a specific agenda SP classical instruments be okay it's a compact bodybuilders
08:42:33 Okay, so if you want dimensionality of the space space. I will use two times capital n minus k times k. So, the simplest example is when the metrics of two by two and rank is one, then this dimension is too.
08:42:45 So this is the measure of a sphere of a surface of a sphere.
08:42:49 Right. So this is a classical classical Spinoza quantum a sphere.
08:42:55 This is this is this is an SEM case as you to guess and then you can study su three common one is you for coming one comma to answer.
08:43:04 as you define the cork circuit, but now is not the quantum is a classical so it's basically a simplistic map that you basically built out of direct product if you want, or not there's a broader view but the direct product of these two by two legs.
08:43:27 Okay, and then you can define the space time dynamics this Mr has two indices lower index is a spatial optimism temporary index.
08:43:43 Discard the goals, I guess I should not go into much detail with definitions. I mean, I hope this is clear enough all right i mean you define now. The the mapping to the formatting, which I call capital, five, which is now the composition of even an odd
08:43:54 right, because now you see you still do it in a staggered way but it's applied to even there's 1234456 and then what bears 2345 and so on and then, you know, each square.
08:44:05 here is pluck out here is one application or each, each each each each red boxes one application of a two by two matrix, ma'am.
08:44:14 Okay.
08:44:24 Right. So now you do it in this way, and now the next statement is, I mean this is of course I mean, it's very simple but you know the first thing you can easily see from that and I could also reverse the discussion of course I could start from the abstract
08:44:28 notion of interoperability and to zero curvature then I could derive the map, but I didn't do that I just throw the, the map on the slide. And now what I will do next I will show that this map actually derives from part of the transport of zero curvature
08:44:41 condition, with respect to some super simple lacks operate.
08:44:45 So that's basically the language of visibility so if you can propose a laps operator and say that the map derives from a zero curvature condition for that labs operator, then basically have shown that this natural be integrated that these dynamics.
08:44:58 So in this case it's really simple.
08:45:00 So the labs operator is like, I mean you have basically you can you can imagine your have this space that lot is now you should basically Imagine you have some sort of ability transport problem.
08:45:11 And these lacks operators are actually active as property there's on this on this on this on this along this to the actions I took that to the actions that is to the animals that is this way and then is that way.
08:45:22 And once you want to do to claim is that, you know, that this to propagate those which are called a blast, which are these diagonals. and then minus which are these levels.
08:45:30 And first I will assume that there is a symmetry between these two the evidence to add plus and minus is the same as just depends on former parameter London the battle of the bands on metrics and so it's.
08:45:42 Yeah, so it's Alex metrics right so, and I would propose that it should be leaner inspector parameter, a leader in the body. So just lambda process.
08:45:50 And then there is this constraint which is one, which brings long yardage.
08:45:54 In his business which is asking the question square one.
08:45:58 And then do you think is now that you should ask that.
08:46:01 I mean, the zero curvature condition or faster property that is that if you go from here to here. This way, you should basically arrive to the same vector as you go that way, right.
08:46:12 So that means that these two products the flux operator should be the same with this problem of lack of operators but but these locks operators carry the metrics metrics variable.
08:46:20 So if you go that way, you carry the old valuables. If you go this way you carry the new variables. So this gives you the mapping between all the new buyers.
08:46:27 So if you do a little bit of algebra.
08:46:29 It's almost everything you need is on the slide. If you do a little bit of algebra you find that you get exactly the transformation that transformation which was on the first slide, this one.
08:46:42 But moreover I mean the general procedure is really for general so you can also assign to this mapping of what we call the twist.
08:46:48 Some transformation, so wanting vertigo submission and Sagittarius what the total transformation that to derive basically is a swap between the two mattresses and composition with with F times, product of some of matrices plus it, I think so.
08:47:04 I mean, why do we use with that we just tell you in terms of physical solutions Mrs Aqua net.
08:47:10 So if you take as you do that which would be just a medical editor.
08:47:14 Otherwise is some sort of general medical so you can play with it that will show some the metrics. I guess I should hurry up, to be able to show some American Journal, so much time do I have.
08:47:25 Well, same names we could be like five six. It's ok.
08:47:32 Okay, so now I will probably keep some of the slides, which is again sort of mathematical properly but you know it's just tell you that I mean if you are.
08:47:42 If you want to have a nice one with all the nice properties then you also want to have a bustling brackets right and here's the explicit expression for the porcelain porcelain break that you want to have environment measure.
08:47:52 So you want to know that this is really a known as a system with a with a synthetic structure and measured and.
08:48:00 And then after you have this you can actually prove it is not a synthetic respect to the special person record
08:48:21 and okay and then you know if you want to reclaim that ability what physicists would like to have is complete set of, or at least an extensive set of quantities Right, well first of all you want to have a transfer metrics and out of this you derive terms
08:48:23 of quantities. So let me just show in one slide that you can have that, first of all, you just take a product, you take a string of mitosis em to determine some point in time, right now, point in time is a series of mattresses because it's like a mini
08:48:34 body system, and tweet, tweet you assign a metrics and you put them on all the metrics, with depends on specter parameter. And this is a product of locks operators.
08:48:44 And these labs operators would have staggered set the parameters. So first one should have lambda but the second module is number four stall and then again London London.
08:48:52 London London style so there should be a staggering determined by the timestamp right i mean that's very similar to what people do in XXZ and make ensuring that it's still interesting one, it's this sort of staggering of transfer of parameters, but after
08:49:09 that you can actually show that if you take a trace of this model, then you get the first base function so I don't want to call it a transfer metrics it's a function of displacement variables, but it's easy to show this function is preserved preserved
08:49:20 exactly on the time evolution.
08:49:22 So it's it's basically like commuting with $100 a month, but there is no Hamiltonian is just conserved.
08:49:28 And moreover the definition of possible graduate ensures you that this, this way defined classroom up record transfer moto moto Moto X, this classroom is Muslim communities, plus one community itself for any parents with amateurs.
08:49:43 So these are basically is that your substitute of you're transferring magic to the generates consideration.
08:49:48 And if you take normal derivatives.
08:49:51 I mean, a logarithmic derivatives of that thing that you get that, then you obtain actually local considerations.
08:50:04 So when this projectors, which parameter is your face base crank one. So this is an interesting issue I don't want either have no time to get into it because you know if you have higher right, then I suspect that this constant motivation was only suitable,
08:50:11 These are not all that are not they're not always local The only local when your competence gas money as basic as and one.
08:50:18 Then I suspect that this constellation was only suitable, but maybe not even that, but we have at the moment no to to formalize that but it is they are not local.
08:50:23 Okay then, you know, you get also some sort of young master equation which actually did not meet. You don't know what is good for.
08:50:30 And then you have some space time with the policy which we also don't need this, we don't know how to use it.
08:50:35 And then you have continuous space continuous time versions of that model for example you can take continuous time version. You can finalize your twist like it to the Eiffel Tower beaver these some tricks, which is like a su empathetic field.
08:50:48 And then what you get at the end is this kind of Hamiltonian model, which is very similar to last longer Lucy's mom.
08:50:55 It's a generalization of London, London if it's model Atlantic standard if it's models Western.
08:51:03 Okay. And whatever I mean, then you can take this special case of as you do, then you get classical spins, which you could you could harmonize them by these capital S select ones which are normalized vectors are responsible sphere.
08:51:17 And then our rational transformation which gives us this electric to speed map is this explicit transformation which I only involved not product and.
08:51:26 So, it's probably the simplest rational function which is Su Su Su so resubmit.
08:51:34 Okay. And this is a full on that model we had a paper a couple of months before that was a graduate student, and then after What's with the Navy have somehow generalize it.
08:51:46 That is a special piece.
08:51:48 Okay, now let me give you a photo for the and some some metrics. So for example now just this is just now what you can do basically you can take charges right i mean screenshot this has been speaking speed components if you want, which are the skew alpha
08:52:02 qL follow take some example some traces madrasas now for su to this will be poly my thesis for su three the weaker man mattresses. So, I mean, in general, this will give you some, these will be some generalized among mattresses which will give you components
08:52:13 And it didn't can take correlation function of this Su and speed.
08:52:19 And if you plot it like for the Tech Data Collection function for the components.
08:52:26 Then again density which has this characteristic shapes will be the envelope which is scaling like Peter to say or do to the two sets of independent and.
08:52:38 So I mean, for example, I mean if you just look at the how the collection function the case on the on the on the on the show on this.
08:52:46 In the center this gives you actually that directly broke off that the exponent. Then you find that you get this is no metrics and on the line is the best fit in power law and it's just slopes like in two digits precision two thirds.
08:53:00 I mean this is all I mean, these are different graphs which corresponds to different business units with like 10 different ranks young Su Su three as you for one so before I don't know that the other cells are some simplistic form, I don't have no time
08:53:14 I don't have no time to go into detail but, but, I mean, the staple nice and, again, there's no groups by the conjecture is that one should have universality irrespective of the compact league.
08:53:30 Okay now. Yeah, this is now some fine detail.
08:53:35 Comparison of scaling of guts of curation functions that equal time scale to the none of the exponents or the size scale whatever it's excellent p to the Z, Z Stuart, and then compared to this graph of a spawn correlation function which is that Jane curve.
08:53:51 And finally have excellent the green one and the Radcliffe is the best fitting Gaussian which is just to convince you that it's not the Gaussian.
08:53:59 And now the last slide is just showing you the effect of magnetic field.
08:54:06 Progress let's keep that just a little bit. So, this, this, this, this, this, this magnetic field.
08:54:14 I mean of course this
08:54:16 consequence of that is that the speed is not a concern but there's a procession of the spin and.
08:54:22 And so you get this isolate this global oscillations in the correlation function as a consequence.
08:54:28 Okay, so here's my conclusion. So what I tried to give you some empirical evidence that
08:54:35 these metrics models which are defined for any longer symmetry exceeded to keep busy physics without responding to your to.
08:54:46 I mean there they have not yet tried all possible. Come back to the cemeteries, but only Unitarians electric. We should also define it for orthogonal groups but for that one has to modify our medical little bit.
08:55:01 So, and then of course this is the conjecture.
08:55:05 So, all the data shows now that are these kits based on models be the smoking circuits from to body synthetic unboxing maps, without having provided was taking medicine callback number yes metrics basis exhibit super diffusion of the PPC type in equilibrium
08:55:19 equilibrium states return for consumers. So what is important is that the initial state or the state in which complete corrections has another billion symmetry.
08:55:29 And that theory.
08:55:32 So that's, again, kind of funny because you know, KPZ was originally device that was it class. Here we find that the supplies on ethically them so it is a linear transport, giving you encouraged punches.
08:55:46 Okay, and then Bruce, we are still waiting for ideas.
08:55:51 Right, thank you very much for reading let's talk.
08:55:55 So we have time for questions.
08:56:00 Maybe well people I think I'll ask again.
08:56:03 First, I mean one thing we kind of found accidentally for diffusion but I guess it's applies to keep busy as well, that an integral systems once you reach the boundaries so you basically relaxed.
08:56:18 Dynamic starts to be very different. So like a non integral systems, basically exponentially decay for diffusion in time.
08:56:26 And it means spectral function is constant so you go to run the matrix.
08:56:30 But for integral model we always found the spectral function goes to zero so dynamic should be very, very different and sometimes I just wonder if this is something you ever thought about this is something you can test.
08:56:41 That's an interesting question but it's totally different. Different scaling different region right i mean here to what we want to do is take demographic limits so that we never see.
08:56:52 Yeah, it's different. So you just never thought about this, we think about this these issues in different contexts, but in terms of quantum gals and ADHD and all that but but not often this field so maybe we can discuss it later, but I think we had a
08:57:07 question.
08:57:09 Yeah, so, you know, please correct me if I'm wrong but it's my understanding that whether we understand the three halves, the exponent in various ways, or the theoretical understanding for the full cape is the universality glasses still not really fully
08:57:25 established or accepted.
08:57:28 Is that is that right and you know, in these classical models, or is there anything where there's actual where we really theoretically know why you get the key.
08:57:40 At the moment, no idea why this applies to this classical models.
08:57:44 We have understanding of three words to four quantum models based on quasi particle picture.
08:57:58 That doesn't cover the full cape is the universality class right it's just the rest. Okay, so we have no idea. I can be.
08:58:01 Okay,
08:58:05 additional question I think Marcus has a question.
08:58:09 Yeah.
08:58:11 So, I mean this integral role models right they have infinite concept quantities and of course they would have all these ballistic currents, right. So, do we understand, which objects will would exceed these super diffusion right so you have it for, for
08:58:26 the spin Quran, but can you, can you tell in general so I give you an integral role model.
08:58:33 You know the concept charges and so on so we are going to be the objects that will have super diffused.
08:58:41 That's it. That's a very good question, but I think, well that you see that this this this this these two classes of observers The so called even an auto observers.
08:58:51 I mean, and I believe that one should have on, I mean they believe they should be only for the so called observers observers which are kind of thought on this universal, that is Lena Lena and spin variables.
08:59:01 That is Lena Lena and spin variables. I mean in SDN models you have n squared minus one over to speed variables, or something like that. I mean, so all these number of greatness matrices, but this is about it I think you can't.
08:59:16 I mean, I mean, again, I mean, what we understand quite well now is the SU two case and an extra two case right i mean there you know that you have spin mobilization and then you have the transfer methods which is even under the under fleet of the screen.
08:59:33 So, the speed behaves very differently than and then the transfer metrics.
08:59:37 Right, so, so I mean I have to tell you honestly we have not really seriously, looked into anything which will be even. So, universal so I would guess if you asked me what my best guess is that the key is the only advice to auto sector.
08:59:54 Remember the last question by Leon yet and
09:00:00 so one of the compartments, of the variable is important for it is in your case, as well, I guess he has because we have to be able to define the averages.
09:00:13 Right. I mean, states expedition value of a state so we of course I mean I was very interested to see what can go beyond that, I mean, for example, at some point we were thinking what I'm not an expert so I didn't know how to go further, but you're thinking
09:00:28 maybe one can also do this for some sort of super mattresses or something I mean, just go beyond normal mattresses, or some some some other variables, but then as soon as you get some modern no compact variable that you don't know what to do it.
09:00:42 So, how did you find the energies.
09:00:46 Great question. So when you say
09:00:59 it's it can be turned on, then it basically can look at the correlation function at a given temperature, and it still may decay in a non trivial way, but when we're talking about the driven system.
09:01:03 Okay. What do I mean by Killebrew yeah well I mean that's kind of.
09:01:09 Yeah, of course it's not the prevailing that's sense because there is no Hamiltonian, I mean, what I mean is that this is these are states which our environment and the time and space translation so on the environment and the period of time.
09:01:24 Okay yeah that's kind of interesting.
09:01:31 Okay.
09:01:33 Transfer metrics and dial. Right then it should be this period in time so I mean right.
09:01:40 Right.
09:01:42 Okay, that's thank you so much again I think it's time to move to the next up on my Jenna dander from Esther was going to tell us about onset of Brownian motion and likeness.