 All right, seem to be working check it. It's not sure. Okay. Thanks. All right, everybody. Good morning. Glad to see all of you and people on zoom. Can you hear me? Can you see my screen? Okay. If nobody objects will assume that it's okay. All right. Thanks, Yasha. Right. So Tigran basically said, almost all the, all things which need to be said, one more piece of information. There are a couple of small confidence rooms around here. So if you want to have a meeting and discuss physics undisturbed by many, many other people you ask Tigran for key. And you can have a separate apartment here. Okay. All right. Okay, so. I will give you today, sort of very basic introduction to a model. Probably most of you have seen it and know everything so I'll be moving relatively fast but if I'm moving too fast, then please don't hesitate to stop me and please ask as many questions as you, as you feel comfortable. All right. So let's see how it works. In some place. No, it doesn't work by some reason it doesn't work. Yes. Okay. So, so such defecative model is a is this beast so you think about interacting. So, in this case, it's my run a few months. I'll mention what it is in a sec. And the decisive feature is that you have your Hamiltonian which contains interaction if you wish for fear me on terms. Okay. That's not the way to intend it to be on sec. Okay, right. So it has four for fear me on terms. No, no quadratic term. So if you wish no kinetic energy whatsoever just just interaction energy. It's, it's sleeping. All right. So, now, now you have four of these objects again I'll tell you in a sec Huawei, and then there is this coupling constant, which, which is a tensor with four indices. And since we don't know to say much, nothing special about them, then let's just assume that it is a completely random, completely random means that it's up to symmetries. It's independent Gaussian variables with zero mean and the variance, which has this J square and J is the only unit of energy in the problem. It's convenient you will see why to normalize it to n cube and n is I forgot to mention it, and is just number of this species in the game. Okay, so now these guys are my runners in a sec I will mention, maybe it's a good thing to mention it even now. Yes, yes, you can equally well or almost equally well. Think about usual complex fear me on so you have see data see data CC. In this, in this case, you want to also, you may also want to have a chemical potential. So my runners are, and notice that if you want to think about complex ferments. So you want to think about n over two of them, because to my runners form one usual firmion. So you can pair them in some in some way. I was told that I can use blackboard. I can pair them and say let's say sky one plus I can do, and then see data is one minus I can do. So out of my runners, you can form. These two complex fear maps. And, and, and these two models are almost identical. But, but you need to have in mind that if you do this trick, then the this tie of Hamiltonian before my runners. Substitute this you will have terms like see digger see digger CC, but you will also have terms like see data seed, they are see they are one see. And you will also have terms with all see diggers. And whatnot. And that means that this model, as it is, it doesn't conserve number of particle of actual ferments. So that's a sort of generalization of bogalub of terms, whatever, superconducting terms to to interact to interact interactions. But other than particle number conservation or not conservation. These two models are practically equivalent. I may mention some differences as we go. So again, if you, you won't prefer to think about this model. You have this tensor or the four indices and it's again random with zero mean in and, and the variance which you parameterize as one number and field. Any questions so far. No, okay. So then. Yeah. You do. Yes, of course you do. So, so this one, for example, should be anti symmetric with respect to permutation of any. This J, I J KL time J, I prime J prime K prime. And then there are a lot of chronicle delta. You mean this one. Yes, exactly. Yes, you have a chronicle deltas up to all all symmetry which you need to include. So it should be symmetric with respect to I anti symmetric I and J permutation and you know. Yes. Thanks. Yes, there are quite a bit of sim symmetries apparent, hopefully apparent just from looking at it so if you, for example, if you permute I and J. Then it should change sign, because if you permute any, any of my runners that you, you should have a sign change. No right. Good. Good, good, good. Yeah, so a little bit of history. It was actually introduced much before such different and yeah. I'll show their paper paper in a sec. So it was suggested as a kind of random to body interaction model back in the early 70s. And what these people did they they investigate level statistics so I'll tell you is in a sec. Basically what what was known in in late 70s. So the early 70s. Something is wrong with translation between my computer and ICT P computer. So that's supposed to be paper of such different year from 93. And then Kitayev did gave a talk on it in 2015 and that sort of the start of a modern history of this subject. So, so the first thing you can do you can take this Hamiltonian as it is written on on my blackboard. And you can. So it's an over two complex fermions. So the Hilbert space has dimensionality two to the power and over two. Here it is. It's a basis of occupation numbers. So you have a string of an over two values and each one is zero or one. So it's two to the power and over two such strings and they form a basis of your Hilbert space. You can write this Hamiltonian or you can teach your computer to write this Hamiltonian as a matrix of the size to the power and over two times to the power and over two. Okay. And then you can ask your computer politely to to diagonalize this metrics. And if n is not too big, and you like it, then your computer may comply and will give you a spectrum. And that's an example of such spectrum so it's just a density of states is a histogram of how many energy levels have that given energy so the horizontal axis is energy. So you have a number of eigenvalues and and you got something like, like, like this. Okay, for an n equal 32. It's actually, if you go to bigger and you will see that it, it more look like a Gaussian, but you need to, to go to, or at least we believe that it will look like a Gaussian from the theory because we cannot diagonalize numerically. And you can write this which are much larger than 32. But, but, but that's what it is now. Most of the physics. So now, those are many body energy levels. Right. So the ground state is the very, very, very last or first, if you wish, energy level here. At a small temperature, the entire physics is sitting just in one single level here. And everybody else are completely red. If you have small temperature when physics is sitting in in this red rectangle and around their density of states looks something like this so there is a square root singularity like and dig not Dyson ensemble and that's of course not a coincidence. And then there's actually a theory, which I will mention as we go hopefully. And that's probably all what I want to say about it. Any questions. Yes, please. I believe this is for my run. But if you will do it for complex you will not be able to distinguish it at all. No, right. So the next game you can play is to, to take differences on between this energy levels and ask what statistics of the differences but what we usually do in random matrices. And that's what people in 70s actually did precisely base. They have smaller sizes of course because at that time computers were much less. So if you do that. So you, you, you form differences of energy levels and then you look for statistics of this delta epsilon ends, you will find perfect Wigner Dyson ensemble. Okay. It says perfect as you can imagine. And then you do to the very first energy difference if you will look for statistics between different realizations of your model, or the energy difference between ground state and the first excited state, you will still see perfect Wigner Dyson. That's an experimental fact. Nobody quite have a theory to explain it, but experimental fact that it is true. Maybe some of you will will find a good theory to that. All right. So now, let me mention time scales which are involved. So as I mentioned there is one single energy scale or time scale which is this capital G. So everything is met measured in units of capital G. I will sort of refer to it as being one, because there is nothing else in the problem, but nevertheless there is quite quite a few quite a few times scales or energy scales which are involved. So, one of them is heisenberg time and that's characteristic distance between this many body energy levels, which I just showed you in a previous slide. So the characteristic difference between them since there are exponentially many of them in a finite interval. So the distance between them is exponentially small or characteristic time is exponentially big. Okay, so that's the biggest time in a problem. It's called heisenberg time. And this is one, which is in this case UV cut off because that's my J. And that's a big energy scale. So that's a very small time scale. And then there are a couple of other times, one associated with logarithman. And one is associated with with capital M. So there is something else here you see there is a very big empty space in between. Again, nobody quite knows. So, what happens on this time scale so at the very longest time scale this Wigner Dyson statistics is established and that we know experimentally from computer experiment. Again, we don't have quite quite have a good fury yet. At a very short time, what we observe is exponential growth of certain correlation functions in particular this out of time order correlation function. And I'll probably not not mention it in this talk but if you have questions and I hopefully can answer. Now, from our point of view, from sort of technical point of view, what happens is that there is a mean field theory which I will tell you in a sec how it's built. And it works well up until that time scale, which is a capital N. So, it's sort of question is how do you take the limit. I don't think about n is being large, but question is how large is large. So you can take limit and as large but time is also large, such that the ratio of characteristic time and and and stays constant or you can take another limit there. You can take the limit of e to the power and over to and characteristic time stays constant and and so on so forth. So there are many scaling limits, which you can imagine how you take and to infinity and time to infinity of energy to zero. And depending how exactly you do you, you will recover very different type of physics. So, if you take and to infinity first you, you stay in in mean field, and it works nice and well, I'll tell you in a sec, what it does. If you take another scaling limit you you'll go to the region where so called three parameterization fluctuations plays the role. And they have a game. Nice theory for that. And it has to do with holographic theory and kind of gravitational part of the story and I understand that York are going to mention something about it. Okay, so I'll, I'll keep quiet about it. And if you go to that part, then we don't have a quite good theory yet. I like that it should be a sigma model since it is give a bigger dice and, but how do you build with sigma model. God knows. I mean, I exactly and says that he knows that I don't know right. So what you actually need for validity of most of what I'm going to say is this inequality that n is much less than e to the power and so this interval is, is, is a wide interval. And, and that's true for, I don't know an equal 10 certainly true. Maybe even less. All right. Okay, so now let me tell you a little bit how you build mean field. So you think about evolution operator. So you think about your Hamiltonian integrals. So HD D tau and tau runs from zero to beta. Let's think about finding temperature and kind of imaginary time. Evolution partition function if you wish. So that's my partition function. And now I what I'm doing here I'm averaging it over realizations of coupling constants j over ensemble of random coupling constants. And that's of course not a good idea, because we know that what should be averaged is actually logarithm of that. Partition function not partition function itself. So, so to do a good theory, reliable theory, you can play the replica trick, which is sort of written here. You, you take our capital R replicas of your system. And then you can build this replicas by by a and the late in late in indices. And then you can average over disorder z to the power air are. So you have this summation of replica going from one to capital R, and then you have your, your Hamiltonians. And they all have the same disorder. So now you have this capital J sitting in the exponent and capital J has Gaussian statistics. So that's, that's an easy Gaussian integral you can do it. So you will have now after you integrate over J's with the Gaussian date. You will have eight my run a fermions. Right. And then you can come with before to have an eraser. So you have this thing sitting in the exponent. That is it. So now you're integrated over the J and that's some Gaussian date. And Q. So something like this. I j l. And as done mentioned, should be some delta functions of course hidden here. So you have something like double integral detail detail prime and then eight my run affair. That goes at moment. And then another four, at the moment to prime something like this right. And this eight, you can arrange into four combinations. Somehow like this. And each combination like this you, you can call green function. So my, my, my green functions to my run a fermions. And since I have eight. I have G to the power for sitting here. Hopefully it's more or less clear. So this diagram supposed to represent it. So you have one vertex, the four fermions in the replica a and then another vertex in replica B server is this matrix in the replica space, which you call green function if you wish. And what you find is this this object to the power for it's not a matrix multiplication just each element of this matrix. It's not taken to the power for. But by the way, the reason why Sigma model is not easy to build is because this is not a rotational invariant object. So tricks, which we usually play deriving Sigma models do not quite work here. They find good. If you want to do, let's say, in replica space rotations. Right. So this is not matrix multiplication of G to the force. That's the rotation. No right. And then another trick which you play, which is also a simple trick you put a resolution of unity which tells you how to enforce this definition. This is written here under the delta function is nothing else but, but my definition of delta function so I just tell my theory that I want to, to employ such a definition, and I do it in this standard way so I put a delta function integrate over all cheese. So that's an identity, and then I put my delta function in the exponent with an auxiliary metrics field, which I call Sigma. Sigma is a good notation. So if you do that, you have a theory with two metrics fields, which are called G and Sigma. And now you can integrate out all your fermions, because they're now Gaussian. So let me, maybe too fast, but you see that the only place where fermions enter are is this in combination with Sigma and they enter quadratically. And since they're quadratic that now I can integrate them. That's, that's an easy thing. And I find determinant of this operator. It's called plus Sigma, and again Sigma is a matrix, both in a space of replicas and in a space of time indices. It's the non local object in time. So then I have my good friend G to the four and then I have this Lagrange multiplier, Sigma times G from from Delta resolution of a delta function. And the good thing about it is that if, if I have right scaling, like here in my legboard, this n cube. I choose n cube here. And that's the reason I, they did it. So then the only place capital N appear as a common pre factor in front of everything. And now N is explicitly sitting. If you wish as an inverse plan constant in coefficient in front of of an action. Okay, so at this moment you can say wow, that's very good because since I have a big, very small plan constant if N is big, I can go to semi classical limit of this theory. Instead of doing thinking about functional integral, I will just think about equations of motion classical equation of motion. That's good, because that also know how to derive. You just take a variation with respect to G and with respect to Sigma. Okay, and that's, again, not a big deal. So if you take variation with respect to Sigma you got inverse operator of that because of logarithm. And then you plus G, and that's equal to zero. If you take variation with respect to Sigma. Oh, yeah, that's what I just said, if you take variation with respect to Sigma you have inverse operator of that plus G, and if you take variation with respect to G. Then you have G cube equal to Sigma schematically. So that's what is written here. So that's equation, which is DSD Sigma. And the second equation is DSDJ. So the variation with respect to G, DG, so take it back, DSDG, so get you something like this. Now looking at that, you realize that this is nothing else, but self consistent Dyson equation. So, hopefully you know what's Dyson equation I don't need to explain you. So there is a self energy, which is a sum of all connected diagrams irreducible diagrams. And then the second statement is that the self energy is nothing else, but free green functions, but dressed green function, not the bear green function but dressed green function. So basically you say that you take the simplest possible self energy, and you take a self consistent approximation, but when you say that each line here is already dressed line. There are two coupling constant here and there. And the Gaussian random variables, so they are the range Gaussian averaging to connect these two together. Through this correlator of my Gaussian random variable. Okay, so this dotted line. That dotted line represent Gaussian averaging, which is written on my blackboard. Okay. Okay, very good. So you have now this to two equations. That's of course much easier than when the full quantum theory still not not an easy thing to to solve. So the next step, you say let me be brave, and let me disregard time derivative here. Okay, so for now I'm just doing it completely out of the blue. I'm just brave. I'll explain you in a sec, why it is not a crazy idea to do it. Okay, so if you do that, then your equations become sort of much more tractable, because now you have a Sigma times G is equal to one and Sigma is G Q. So notice that here it is a convolution so this equation you want to think an energy space. It looks nice and easy like in a textbook. This one is written in a time space. But nevertheless, you can again be brave and try to find the scaling solution. Namely you would say that your matrices G and Sigma depend on the on time difference. And it will depend on time difference in a power law in an algebraic way. So if you do that and you substitute your algebraic equations here, then just dimensional analysis will immediately tell you that the right power is T minus T prime to the power one half. Okay, so, so my green function. And that's indeed a green function as you see from my Dyson equation. You behave like one of our T minus T prime square root of that. This is very much not like in a Fermi liquid theory in a Fermi liquid it's like one of our time. Now if you do Fourier transform you find it that it's one of our square root of epsilon. In the same way you immediately know what is Sigma because Sigma is G to the third power so it's one of our T minus T prime to the power three half. You again do Fourier transform and you find that Sigma behave like square root of epsilon. And the good news is that square root of epsilon at small energy is much larger than epsilon. And since it that's is the case then neglecting with detail, which is nothing else but epsilon. Yeah, is not, not a bad idea, because indeed Sigma behave like square root of epsilon this behave like epsilon. And because of this inequality in a zero approximation. That's not bad. Okay. This semi mysterious statement that this form of a green function means that the scaling dimension of the fermions is is one quarter. And yeah, one second. I'll explain it in a few slides. Yes, please. Can you speak up. Can you hear me here. replica indices right. At what point did the fermion indices themselves disappeared. Oh, at what point the fermion disappeared. Yes, thanks that that's a very good question. So fermions disappeared at that point. So there is this term. Let's see it so go to. Yeah, so there is this piece of knowledge. And an extra piece of knowledge is that when I write my evolution equation. I also have a term like hi. Yeah, you have indices. Yeah, definitely it's important to do everything correctly. Right, very good. Yes, so what you what you will find is. So you will have Chi I detail plus Sigma Chi I. And then you will have product of all I, they become diagonal. So this expression become diagonal in in the synthesis. Because my definition of green function was manifest and self energy is diagonal in index I in color index if you wish. So this matrices G and Sigma they do not have color indices. That's an important point. So, and then this thing is a determinant of this operator. To the power and over to. All right. Very good. Very good. Okay, so, so what we are doing with time, still have one hour. Okay. So, so this is mean mean field, Yuri, which I mentioned in my discussion of time scales. Okay. So for relatively short time and you will see why it's only for relatively short time. But if you take you buy what I'm telling you, you discover that you have this thermionic model with correlation function with which behave in a quite unusual way. And finally, they have algebraic dependence on time or on energy, and the power law, the exponents of this algebra functions is is quite unusual. So, instead of having poles, it has a branch cut in energy space. So, people sometimes call call not non Fermi liquid another sort of whole mark of non Fermi liquidness is that self energy, both imaginary part and real part we both have the same. If you don't a little continuation from imaginary time to real time and you, you do all that which is not very easy but you can do it. You find that both imaginary and real part of self energy behave like square root of epsilon. So imaginary part of the self energy you attribute with the lifetime of your quasi particles. And you find that in this lifetime of your quasi particles, and you find that this lifetimes is much shorter than than the inverse energy of your quasi particle. And that's inequality by itself, if you understand it as an imaginary part of self energy, basically means that you do not have well defined quasi particles. All right, but you have a theory for this so you can sort of proceed and try to extract some consequences. So probably the most spectacular consequence come from thinking about the rate of such such defecative models. So you, you imagine a latest or whatever a graph, any way you want to think about it and in each node of this latest or a graph you, you put such defecative model. Sorry about that. And, and then you add some some kind of a tunneling. For example, quadratic and fermions that that's the simplest way to to connect between these dots. And then you have a model like this. So now, sorry, I flee switched from Kai to it, but that's the same, the same animal. Okay, that's still my my run the fermions. And now in this is a and the label, not replicas but yeah by the way I need to go back, but sides of my race. Again, not notations are not ideal. I was copying it from from different papers. So sorry for that. Yeah, by the way, I didn't tell you anything about replica. So maybe, maybe I should. So this solution, which I sort of equations as they are there's still metrics equation. Again, I was brave, and I looked for solution which is diagonal in in replica space. Okay, that's of course a very, very limited class of solutions, it is a solution but nobody promised that it is a unique solution. Finally, you can find solutions, which are have some structure in replica space and replica symmetry breaking and maybe one step replica symmetry breaking, infinite step replica symmetry breaking and all that. Quite a bit of time, investigating that, and the conclusion is sort of semi analytical semi numerical that none of these extra possible solutions is not is not relevant. Namely, what you can see in numerics can be perfectly explained with only replica diagonal saddle point. Now, if you want to wave your hands you can say that as I came model is maximally chaotic. It doesn't have a glass face. Level statistics is perfectly digner Dyson gain experimental fact. So you don't expect glass behavior and that's why you don't expect replica symmetry break. There are two words, and there is no good theory which would explain why only this saddle point should be taken. Moreover, we know that that's not the case at long time at long time when you have. If you want to describe the ignore Dyson seriously. You need to do something with replica, a little bit more sophisticated than that. It's relatively short time. It seems to be describing everything what we see in America. The statement is, after what you call the high and well, if you can be and not. Yeah, okay, so let me try to be as careful as I can. So it, it's definitely the case that after Heisenberg time replica should play some role because otherwise you will be able to derive the more dice. Okay, before that, nobody quite knows where it become important. Maybe at Heisenberg time, maybe at square root of Heisenberg time. At the mean field level. Yes. There's no vacation for negative. Nothing like that. Yes. The question is what happens between. Yeah. They're very good question so mean field is sort of brave approximation just assume and much larger than one. Okay. And you don't necessarily know up to what energy or time, you can trust your mean field. Okay, so to derive that you need to work out fluctuations around mean field. Okay, and that's what I will be doing soon. But at this stage we don't know mean mean field. Maybe it's good for any of us. No right. So let's continue. Right so if you play this game of as like a array, and you assume this coupling to be small, let's say, and so small and you do simplest perturbation theory you can imagine. And you can calculate conductivity. Okay, here you need to be careful if you want to calculate electrical conductivity you need to play with complex terms you need number conservation of particles, because otherwise, you don't have mass or charge current. So if you play with my arena, then we should discuss. So energy is still concerned so we should discuss thermal current energy current. But if you play with complex model, then we can discuss charge current and calculations are pretty much the same. So if you do that you calculate conductivity as a simplest loop. You know each of these two green functions you do integration as you need your substitute Fermi functions as as as needed and all that. And what you discover is that conductivity is proportional to one of our temperature. Just this simple calculation. This is sort of spectacular because that means that there is resistivity. Well, so you, you imagine that you have complicated atoms, and each of these atoms have many orbitals and orbitals degenerate. And they play somehow such defecative physics inside each atom, quote unquote, and you build a crystal out of these atoms, or unit cells. And you connect them for tunneling. And now you, you, you measure conductivity so it's pretty much, you know, look, people discuss how to build it out of graphene, the vortices and all that. I don't know how to experimentally build it. I mean, I don't want to go into it. But if you want to have a cartoon then I have some kind of a crystal where each unit cell has many degenerate orbitals by some chemistry reasons. Yes, please. Yes, in this model. They have independent realization of disorder. I don't think it's, it's very much important. But that makes the fury easier. Yes, good, good question. So you, if you do this calculation you you got the resistivity is linear in temperature, which make many people in strongly correlated community happy, because that's what is seen experimentally in in in many, in many compounds. However, you should be careful here, namely, you should be suspicious if this kind of perturbation theory is is a legitimate approach. Why the suspicion, the suspicion is because if you think about kind of RG scaling dimensions. And you think that s y k is your kind of starting point s like a fifth point is is the physics which, which is a bear. And that's the case because I take each green function as being just s y k green function. Now from that point of view, your perturbation, which has only two fermions is more relevant than my Hamiltonian, which was written on a blackboard. So this is a in a very naive level, this is a relevant perturbation. That means that if you go to the very small energy, your coupling constant. V is expected to grow, and you are not expected to trust this perturbation theory. And indeed, people believe that that's the case. Namely, there is a kind of crossover energy or crossover temperature, if you wish. So this is my coupling constant V tunneling between my s y k dots. And the temperature. So this result. T linear resistivity which is called strange metal is valid and indeed can be absorbed but if temperature is sufficiently large. If you go to very small temperature or very small energy, then that's become a relevant perturbation and you back to to Fermi liquid. So you're back to some kind of disorder Fermi liquid disorder because everything is random here. So you expect a constant resistivity. So overall prediction is that as a function of temperature, you have T linear resistivity at high temperature and then below certain crossover temperature it, it flatten out, and it is constant. That's a sort of mean field expectation for. For this model, which Misha is sort of justifiably skeptical about but that's what people are playing. Okay. No right so now I want to lead sort of go beyond mean field. Just a couple of steps. So if you have any pressing questions about mean field, then that's the time to ask. No right. So if not, then let's go. Let's go let's go. Okay. So, so here is, I'm back to my hermaphonic action. Yeah, that's precisely what I'm going to discuss in a sec. So if it's not clear in five minutes then ask again. So, right so you, you, you back to this action. Now what we did before we, we said that we want to forget about detail. Remember, when I was solving by mean field equation I said that I'm brave, and I forget about detail so essentially what I did I completely disregard that we still. So then my action is simply written in terms of fourth year months. And now conformal dimension is completely manifest. So you see that if you change your time, then fermions should should change as power one quarter of the time. If you want to formalize this conformal invariance, then the good way of thinking about it is the following. So we have initial time towel, which go in a circle from zero to beta. Everything is periodic or anti periodic with this time. So you may think about this time as being a sort of a clock, going with with constant speed around the circle. But you can allow yourself are changing time. So instead of tower you will use monotonically increasing function, but, but it, it sort of, it's something like this. You have a clock, which is going with variable speed, still going forward, still everything is periodic in in beta or in one hour in this picture, but but time is going sometimes faster sometimes slower. And you can sort of think about any deformation of my clock. I think Dali could could think about you, you can sort of parameterized by some function. H of tau. Okay. And mathematical way is saying is that H of tau belong to the group of defiomorphism of a circle. So it's a mapping of a circle on itself. It's an infinite parameter group which is mathematicians called deep as one. And then what you want to know you want to change your thermion. So instead of being a function of tau it's now function of new time, H of tau. And what I want to know I want to add this factor, sort of piece of Jacobian age prime to the power and power I want to, to pick one quarter. And that is that because now I have four fermions, each one of them come with this factor. H prime with power one quarter. So four of them give me just H prime and this detail. And this H prime to a power one. Simply give me DH. My action is written in terms of new time, precisely like it was written in terms of all time. Okay. So now everything is, is, if I, as long as I forget about this thing, everything is completely and manifestly invariant with respect to that transformation. Okay. Yeah, so we usually don't don't think about this kind of general relativity type of transformation, because for most of conventional Fermi liquid physics. This term is actually most important player. Right. But as long as you can allow it and somebody as like a allow you to to disregard this term all of a sudden you have new symmetry in a problem which you have to acknowledge. No, right. So let's, let's see what it does. So if, if you do this transformation then you need to transform your green functions correspondingly so you have two time object and it's now with Jacobian factors coming from here. And instead of tau, tau one minus tau two square root you have, you have this combination new time. So this two and this one quarter is precisely the square root which, which we used to have. There's a question from from online audience. Is this reparameterization is somehow a kind of homotopic transformation, a smooth and no folding transformation. Yes, at least that's the way I think about it rightly or only. All right. So you do this. Now I'm sure that this thing is still a solution of my mean field equations. Again, provided I forget about this detail. So, this guy is as good as a solution as the one which I showed you before. But the only thing is that it's not translation in there in time. So it doesn't depend on tau one minus tau two but but other than that it's, it's a possible solution of these two equations. And I have to sort of acknowledge it. What it mean is that although I had large and sitting in front of my action. I told you that I can sort of disregard on all quantum fluctuations and just look at the settle point solution equations. The equations are such that there is a completely flat or almost completely flat direction in the field space space of target space of your field theory. And deform your field G in this particular way, and your action is still exactly the same. Again, provided you disregard detail. So in this huge space of my fields G and fields Sigma, there are certain particular directions, their matrices so the big and complicated, but there are certain deformations, which we've now discovered of that form, which doesn't cost you any action, or almost doesn't cost you any action. So those are soft modes, if you wish, Goldstone fluctuations. This one should sort of explicitly take into account. Maybe there are some others, but actually if you play with complex fermions then there is also phase mode which is a soft soft fluctuation for my arena. Those believe to be the only soft mode. Okay, no right. So now what do you do with that. Oh, yeah, before we go. So this symmetry is even more subtle. What you discover. And I think it I was the first to appreciate that that that if you look carefully at this expression. And you take your age function as being fractional linear function of time. Do I have it. No, but notations. Right. So if you take your function age to be a tau plus be divided by C tau plus D. And you just substitute it here and you do three lines of algebra. You discover that your green function is still time translation in the end, and it's still precisely the same as it used to be before the transformation. That means that some of this reparameterizations actually are not a valid reparameterizations. You think you, you did a new time you think you sort of deformed your clock the clock, but in fact you didn't. Because your fields are oblivious to to this change. And what it means that that the actual group of soft modes manifold of soft modes is a is a closet space. So this is the form of the form this D first one divided by the group a sale to to our so this fractional linear transformation they are parameterized with ABC and D, and you see that overall multiplicative factor is doesn't matter. This ABC and D in a two by two metrics ABC D with unit determinant. And that's always matrices form group, which is called a sale to our. And then if you have a combinations of your reparameterization let's say you go from tau to age and now you take another reparameterization you take G of age. And you can take another and another. The sky is the limit. So then if you play with this fractional linear, which are called Möbius transformations. Then they're still Möbius transformations. But with the parameters which you can find by multiplying with small two by two matrices. And easy algebraic exercise each of you can do it in five minutes. You just play with this convolution, not convolution combinations of fractional linear transformations and you see that they work precisely as multiplication of two by two matrices. Yeah. Again, wait a second. No, no, of course not. Oh, say to our is that's. Yes. Yes. Yeah. It will be still the same. Right. So, so that mean that the actual soft modes are parameterized by orbits, if you wish, of a sale to our within a bigger group of mapping of a circle on itself. I mean, if you in in the medical site then algebra of deep as one is, is the r soror algebra. And the cell to our three generators minus one zero and one of, of, of a big for a soror algebra, and they form a subgroup, and you can sort of divide the r soror algebra by the sub sub algebra divide by the sub algebra. Okay, that that will take me too far away. So now you, you, you, what you need to do, you need to go back to this time derivative which damson just mentioned. This was my time derivative, yes. So, so far I completely neglected the time derivative. So now I, I take it back. So it's written on my blackboard as inside of a determinant. So I just consider it as being small, and I do perturbation theory in detail. I substitute my green function in in the form in this form. And I substituted and Sigma has a similar form of course and I substitute it into this determinant, and I expanded perturbatively in power some detail. So this way I want to derive action for my soft moments. Again, if I neglect it out then they are completely zero modes they don't cost me any action whatsoever. If I keep the towel, then I discover that yes they cost me a little bit. Okay. And I want to find what what this a little bit is is. So if you do this perturbation theory in detail. So what you discover is that your action in some of you. Assuming that age of tower sufficiently slow. You find your action is given by so called Schwartz and derivative. So this is an object which here is a definition of it. It is divided by first derivative and then second derivative divided by first to the power square. So it looks as a completely random and completely crazy combination of derivatives. And your first instinct that it cannot be true. Of course but but then you realize that this this object has a deep deep mathematical meaning. And finally, this is the simplest local differential operator which is invariant under this SL to our subgroup. So namely if you, as I said before, do a combination of two reparameterizations, and you consider, and you look what is a Schwartz and derivative of this thing. So if, if one of my reparameterization is fraction of your transformation belong to SL to our subgroup, then, then this thing is true. And if you, as Damson just again mentioned, if you take age belonging to to this, then it doesn't cost you anything. But if you have reparameterization which goes beyond the cell to our with Schwartz and derivative tells you how much you have to pay for this reparameterization. So soft modes. And we haven't. If you wish goldstone action which tells tells you how much. What is the factor in the exponent, which each parameter is reparameterization age of tau cost you. Parameter M here, which is proportional to capital and because our entire action was proportional to capital and from dimensional analysis you can easily find out that there is J should be Jane denominator, and then there is a numerical factor, which depends on regularizations and what's not which you need to work a little bit, maybe numerically to calculate. Yes. Yes, you should think about it as a gradient expansion and that's a sort of most relevant term in this gradient expansion. Yes, that's not the whole truth. You can invent more complicated. The operators which are invariant with respect to a sale to our but that's the simplest one. Well, well roughly speaking, if you think that your whole symmetry was deep as one, and you break it down to SL to our, then the goldstone modes are parameterized by this closet space of one divided by another. All right. So now, okay, I still have 15 minutes to torture you. So, okay, let's go ahead. Yeah, so, so, so this object actually makes heart of gravitational people to, to beat immediately, because what gravitational people know is that this object immediately related to two dimensional gravity. So I probably will sort of skip it. So, just to say a few words. What could I say. So, the smarts and derivative and a sale to our group are all marks of projective geometry, two dimensional projective geometry. So if you ever studied LaBachevsky geometry, then it's full of this object. So the LaBachevsky geometry is gravitational people call it a DS to it's it's some something like this. So it's a LaBachevsky plane. So if it's a complete LaBachevsky plane, and you calculate gravity Einstein action for the complete LaBachevsky plane as for any two dimensional plane, many space time. It's a constant this is a Gauss-Bonne theorem which tells you that if you integrate curvature, you got a constant. But then what we're saying, they say, let's cut a little bit out of the boundary. So this is a complete LaBachevsky plane, then you cut a little tiny piece around the boundary. And then you can deform this line with which you cut, and that's my dali clocks. This red line maybe parameterized by by the same reparameterization which I used to to reparameterize my dali clock. And then your action is not completely trivial, because now it's, it's not an integral over entire space of a negative curvature. It's an integral over part of the space and what remains is precisely with Schwarz and derivative of, of your red line, which you cut, cut out a piece of LaBachevsky plane. So that's in gravitational work is known as Jackie title bomb gravity, and hopefully York will will say more because I honestly cannot say much more I don't know much of it. Noray, but but back to, to my action and to the Schwarz and derivative. So now what you want to do, you want to calculate your green function, which is the following thing. It is expressed through possibility parameterizations. Each reparameterization come with an action which is with Schwarz and derivative. You want to integrate over all possible reparameterizations which span closet space defense one divided by SL to R. We were hard measure on on this symmetric space. But what your green function, apparently is, if you go beyond mean field, if you take into account the soft modes in your field. Now, good news is that although it looks sort of crazy, you need to know measure and you need to do functional integral over this beast. Actually, it's, it's not too complicated, you can even do it. So the first thing to notice is that the measure is relatively simple, and that was derived back in in eighties by none else but Edward beaten. So namely, if you remember my reparameterization someone at on the increasing time is always going forward. It means that age prime is always positive. And since it is positive why not to parameterize it as an exponent of some, if you wish, non compact phase. So if you do that, then in terms of this new field phi of tau, this measure is, is flat. That's a kind of mathematical miracle. Moreover, my crazy Schwartz and derivative in terms of this new field phi immediately start looking like something which we are familiar with just five dot square. Okay, which is also good news so now now we have a flat integral over defy here we have five dot square. Now you need a little bit of sort of algebra to massage this beast and you can do it. I will not go into much details. Basically everything is written here so this combination can be written as an integral simply using this definition. You can write it as an integral between tau one and tau two of either the power phi, and then you can elevate it into exponent. The fine mantrik and some auxiliary parameter alpha and and after you do that you you basically map to your partition, your green function you map to some kind of a quantum mechanical. You have an integral. So you have kinetic energy, which is five dot square, and then you have this potential, quote unquote, exponent of five which which came from, from this streak. So I realized that it's a little bit fast and you, you probably cannot follow the details, but you know either you just believe me or you can always find me and ask I'll explain you all, all the nitty gritty details. In the end of it, you realize that this is a certain quantum mechanics, which is known as a little bit quantum mechanics namely it's a quantum mechanics with a potential with exponential potential. There is a kinetic energy term and there is an exponential potential which looks like this. And then you can immediately sort of realize, see that the spectrum is from zero is continuous and it goes from zero up to infinity the energies, because your potential has, you know, exponent is flat here and then it. And the way function, they are about, you know, reflection of waves from this exponential potential. Okay, now the good news that you can even find it exactly. Basically, basically some kind of a basic functions. So you know precisely your spectrum, which is easier. And you know your way functions so you can go ahead and you can calculate everything about this quantum mechanics. So quarter appear only here. Well, you know, yes and no. So you can know in a sense that you can easily do calculation with any delta by the same money, but the results will be different. You can do a calculation. But, but you should work work a little bit to get the results. Yes. So, right, I, I think this integrals are this matrix element can be calculated with any delta. Because it just did this in functions. Okay, so once you do that, you find that your green function is actually slightly more complicated that when you used to think, namely, if your time is relatively short, and short means it's a combination so if time is, if you remember my one of my first slides dimensionless time. So if time is less than capital N, then you back to what we already know that the green function behave like one over square root of time. That's my mean field. That's what you expect. That's a good news. What is new here is that if time is larger than capital N, then behavior of your green function is actually different. It decay, instead of decaying as as T to the minus one half indicate as T to the minus three half. So that's a sort of small numerics, which show you that it indeed look like that so if time is relatively short. The slope is close to minus one half and if time is longer than the slope changes and with a loving eye, you can say that it approaches minus three half. Since capital N here was not too big then it's not definitive of course confirmation. Now that has important consequences, because now, instead of thinking of fermions as object with conformal dimension one quarter. You can think them as objects with conformal dimension of three quarter, because the ultimate low energy behavior of your green function changes it's not. It's not this but it's Robert one over T minus T prime to the power are three half and that means that each fermion brings three quarter of this of the power. And another news is that there is a new scale, which appeared in a problem and the scale is capital J divided by N. So at that scale mean fields stop to work. And with soft fluctuations take over and physics changes. Now whether it stays the same all the way up to exponentially long time over is other energy scales. In the middle, nobody quite knows. But, okay, that's all I can say. If you do it in energy space, you do Fourier transform then what you did discover is that if a large energy goes green function goes like one of our square root of epsilon. That was non Fermi liquid which we discussed before, then that energy which is less than this new energy scale. It changes behavior and it behave like square root of action. It's still non Fermi liquid still kind of a branch cut behavior, but very different. Namely, instead of a kind of non non Fermi liquid metal, you now have non Fermi liquid insulator, which it's a soft insulator it doesn't have a hard gap, but nevertheless density of states is suppressed down to zero. The single particle density of states which is a single particle green function so single particle density of states is suppressed at small energies, you can call it zero bias anomaly. Older people remember what zero bias anomaly was a big deal 40 years ago. So, yeah, I wanted to tell you that it's have some consequences for for the physics. Let me let me just flash. So if, if before the expected crossover between strange metal and Fermi liquid. If you take this reparameterizations into account, then the life is a little bit more interesting you have still have strange metal at high temperature. You still have a Fermi liquid, if coupling is big enough and this is a relevant perturbation as I explained, but, but now you have an insulator phase. So if coupling is less than a certain critical coupling, you have an insulator sitting sitting here in the left of your face diagram. And now you see that this strange metal, it, it sort of occupy quantum critical region of a certain quantum phase transition as function of coupling constant between my my grains. I have quantum phase transition between an insulator and and the Fermi liquid. And then strange metal come as a, as a quantum critical phenomenon. And so that's probably where I would stop because it's too much. Yeah, very good. Very good. So for zoom people, I will repeat the question. So Dima asked if this reparameterization story is only valid for replica diagonal settle point solution. You can work out reparameterizations for replica non diagonal settle point, namely you can allow for different reparameterizations in different replicas to play this game. And you can work out experiment sort of correlation functions with this assumption. And then you can go and compare it to numerics. And you see that numerics perfectly. Not perfectly, but as much as we can see numeric does agree with replica diagonal solution and does not agree with replica non diagonal solution. Why I don't know, but experimental fact. So you're talking about mean field or well. Yes, in in in mean field. It's sort of a big theorem, but it is understanding that conformal dimension of each firm is quarter not not one half as usually but quarter. Right. Well, not quite, I mean, let's say you have four fear mounds and you calculate for four different times relation function. So then it's a complicated object. And it may be written in terms of conformal ratios. So one minus two times two three minus four divided by insert and powers, and this powers follow from just kind of counting conformal dimensions of your fear mounds. I actually work out what what it is the three parameterizations as well it's sort of slightly more complicated, but don't remember, I want to say yes but don't, don't take my word for it. Pre Maya runs right. So, so the more general models come with the name as like a within the skew. Right. So in this case what I discussed it as like a four model. So you can imagine as like case with all the indices and there is something funny about them. Honestly, I never worked on them. So I don't remember but there is a literature on them and it is funny, funny objects. But that's as much as I can say. Thank you. Can we take any message about the narrative from most of them. Well, the honest question is, I don't know. But one observation is that if you believe that your self energy that quasi particles are not very well defined. So in some sense, self energy is much larger than detail. The epsilon itself right so then, but by whatever reason, maybe as I came in maybe some other reason but self energy is larger than detail right so then you say okay let me forget about detail and I'll be brave. So then maybe you immediately have to think about this reparameterization guys. They, they become soft, right. So the thing which make them not soft is detail. And as long as detail is less important than everything else may maybe you want to think about them. Speculative message is that maybe all non ferminic would have this kind of a soft reparameterization modes and holography associated with that as a generic feature. But again, it's just bullshitting. Yes. Can I defer it to York. No, that's a that's a very good question. Cool. So, if you do your reparameterization yes. So maybe that's a good transparency. So your green function looks like this and it is not time time translation invariance so you would think that energy is not concerned about that. But that's because you look for one given particular reparameterization. Once you average over all possible reparameterizations with proper weight measure blah, blah, blah. Then the result is still time translation and invariant. Okay, so in, in average, in the end of the day correlation functions are translation and invariant, of course, but for every given a reparameterization they are not. Yeah, yeah, yeah, yeah, yeah.