 Okay, welcome everyone to the second talk today. It's a pleasure to introduce our speaker York Schmalien, Professor York Schmalien from Karlsruhe Institute of Technology in Germany. So, can you help me or somebody else how to get these slides there so. Oh, wonderful. That looks better. Alex was fighting with the machine here. This is going backwards. And the pointer. Press and hold it. Press and hold. Oh, wonderful. Excellent. Now I'm well trained. It's great to be here. I already forgot what I have to do here. Good. The work I will be presenting was done with a number of wonderful collaborators. It started, in fact, with Ilya Estelist, who back then was still a graduate student and now goes on to get a faculty position in medicine. I'm from Lydon and students and a postdoc from Karlsruhe, who the work was done together with and essentially it was all because of Alexei Kamis, Kamis thought because Dima Bagratz, one of his collaborators came to Karlsruhe and talked about their work, the one he just presented. And I asked myself so I want to see if the branch that fermion that we've just seen becomes superconducting what happens if such a system becomes superconducting during this talk and then we started talking about this. So I will give a little bit of a motivation. What is the experimental background that drove some of our considerations. Then we will talk about a toy model that is as the as YK model just presented a version of it as ridiculous as the one that was presented as beautiful as the one that was presented. And we will try to solve it. We will do so with the clear conscience that this is a simple toy model, but we will then also see that making some progress along those lines will help us to understand actual critical family surfaces. And that is a question that was posed early on so that's however, talk about the motivation there are certain things about say really complicated superconductors such as the high TC superconductor that we actually understand. If you want to talk with a positive note we understand that they are for example d-wave superconductors. We understand that their face diagram is a mess, but we also understand that they form Boogaloo of quasi particles. So this is a photo emission results here, long ago taken by the argon group, where you see the dispersion and the backwards bending as you would expect it for Boogaloo of quasi particle in the BCS type of theory. And the ground state to some extent has to have some overlap with the BCS wave function and some well defined way. And if you then however look at this at the Boogaloo of quasi particle you notice also that it has its own subtleties and interesting aspects so this is again a spectrum photo emission in the spectrum taken again many, many years ago, where you see a very, very broad peak in the spectrum and in the superconducting state you see this coherent peak emerging which is our dispersing Boogaloo of quasi particle. And the thing that you should observe here is that its spectral weight is actually tiny. So whatever is done then the superconducting only grabs a small portion of the spectral weight of the system and changes it transforms it, which is at least something noticeable. And also make very nice connections on a phenomenological level with this, with this weight that we've just seen here as function of certain parameters in the chemistry of the system. And you see that the superfluid stiffness which is at least not the least important quantity of a superconductor actually is correlated to it in one way or another. In fact, there's even the condensation energy as say how it might be to define this sharply but obtained here from eat capacity extrapolation that so forth also seems to be correlating in some way with this coherent weight. So there's something to be said about the superconductors, which are forming out of a normal states that is very, very incoherent. And the message that I'm trying to bring across is one of the aspects that is unique is that they have a very small Boogaloo of co other particle weight. Another aspect of superconductivity, which doesn't have to be related to one I've just discussed is quantum criticality. It also happens to occur in a number of systems here. These are heavy formula materials serum palladium to silicon to a serum in doom three, we have a magnetic phase, the magnetic phase disappears here by pressurizing the material and at low temperatures, where the system seems to be disappearing in its order state superconductivity is taking place. We're also here in charge transfer assaults of the organic variety, where you have a density waste instability disappears and you have a superconducting instability. This is an iron based superconductor, you have a certain magnetically ordered state, it disappears here there's chemical composition changed. And again, wherever you are suppressing the order state you seem to have superconductivity that is even largest. There are some statements made about cup rates and quantum criticality. I am not entirely convinced that this is experimentally really the case is something interesting there. But clearly in these systems. I think these strong the argument in favor of quantum criticality does exist. This is to some extent surprising. Because if you recall. How do we get to superconductivity. One way to do so is by just assuming a family liquid and realizing that indeed superconductivity as many of us know is the completely natural ground state of family liquid. I have a single pole in my propagator. I can characterize this in a certain way with a self energy. I'm then summing up the diagrams that, at least in some well defined logarithmic sense of dominating in the system, and I'm looking for the susceptibility of pair excitations. And what I find is that the free gas of electrons has a logarithmic behavior and then of course, there will be an arbitrary small coupling constant sufficient in order to get me the superconductivity. This is what we know to happen in non interacting or weekly interacting systems, and it carries over not just to weekly interacting system is also carries over to family liquids. When the quantum numbers continue to be those of the free electron system, regardless how strongly the system the systems actually are interacting, but the quantum numbers stay intact. So, when I'm taking one of those carriers here, and another one, there might be a cloud around them that is made by many muddy excitations that are rather complicated but still the the analysis that we are seeing here can be carried over. What we however want to do is what happens when the systems are as ill defined as this illustration you suggest so for example we have a self energy that is characterized by a branch cut. So we have seen exponents, and, and Alex called them delta and I will call them gamma. And this is a pure homage to under two book often his collaborators who just use a different letter and I think one is just four times the other mind is one or one like this so they can be translated into one another very very easily. So if this exponent gamma is between zero and one which is the case I wish to study. This has a branch cut singularity, and I can now do be naive, and I do the exact same calculation that we've just seen. So if you're summing up the same diagrams. I have less justification to do so actually have no justification to do so but do it anyway. And if I do so, then I will find that the person stability because of this much less sharply defined for me on is less sharply pronounced, I don't get a cool point stability anymore. So this was true. And I could continue this and this line of reasoning as we just as we agree doing here then I would always need a critical strength of the pairing interaction in order to get a superconducting state. And this is just like the stoner instability for paramagnetism. There's nothing illegal about the stoner instability in the narrow sense or where when we seems to be some some some physical justification for it, but it's not a controlled calculation as such. And also, it wouldn't really naturally explain why the superconductivity is most pronounced where you have this quantum criticality it's a critical point. That's where that's an because that's where the quality particles are least defined. If this logic was true, of course, it's not I'm just trying to lead on to it. And, and this is why, why we this is at least something was well discussing. There's actually a paper from from Bardin, who discusses, why is sodium not superconducting because sodium should be superconducting it's actually not because there's some pulley screen coolant interaction in the game apparently. It should be probably superconducting eventually but not at these temperatures that are being controlled. The paper from Bardin. Why isn't sodium not ferromagnetic. Why because we have a stoner instability fulfilled and nobody's surprised that we don't. Right. So then, if this reasoning quantum critical superconductivity should be the exception rather than the rule. There are answers to this problem. And they these answers have been discussed. And there is a great deal of research going on. And what I want to do is I want to discuss two possible answers to you that appear to us completely different. And then at the end I want to show you that they're actually identical. So let's look at them. First of all, as being completely different. One is a theory that I think the first version of the theory goes back here to damn son was in the room who looked at soft blue and mediated color superconductivity. And I think independent understanding of the same physics by a bunch of two book open think it's time on spin fluctuation induced superconductivity due to anti-ferromagnetic or spin density wave system and a number of follow up. So of course. So what you do here is you have essentially a you cover type coupling, you have some both on I call it five and have for me on bilinear side dig up side. There's some coupling constantly some quantum numbers involved. So then you make you work hard. You look at some critical point. And if you try to make certain more or less unjustified approximation sometimes more sometimes less approximate you find such a branch cut at isolated quantum critical points. So there's lots of discussion of whether this is justified in each individual case and whether this is controlled and so forth. But let's be pragmatic about it for the time being. So then you get what we just get but the assumption of the of the of the analysis that we've just seen here was that we continued to use an instantaneous pairing interaction. So we still had a coupling constant that was just a number. So what we should be doing of courses. When you're in such a critical system to appreciate the fact that the interaction themselves are equally singular and power law behaved and strange, then, because we are in such a critical state. And if you do so, you find in a number of cases that the same exponent that just characterized the branch cut singularity of the fermions also determines the singularity of the pairing interaction. So if you have the same exponent, there's a chance of balancing the pain that you've just went through by having so well ill defined quality particles by having them interact more strongly. And this is balance that will give us the rise to something that you may call a generalized Cooper instability, taking place in a system that is quantum critical state. In fact, it turns out that it's more powerful. If you have a very well defined dial such as the coupling of as YK dots that Alex was just referring to. Good. This is one way to think about pairing in critical system. And there's a completely different one, which I have to very have to admit very openly clearly to have no idea I didn't understand a word of all of this. And it's called the holographic superconductivity and essentially is based and I will have to go into some more detail as we move on because I assume some of you at least know as little as I did. I have a point about it. So the idea is that we have a D plus one dimensional quantified theory and it lives. And it can be efficiently described by a D plus two dimensional gravity theory. And then you do what, let's just take this for granted, I have to say more words on this one for sure. But if it takes us for granted. I have to do in a superconductor. I do against for Glendale theory. Right. And I've write down my kids for Glendale action in that space. Of course, I have not yet justified why this should be done but let's assume it should be. So then I have a good for Glendale theory and then a little bit instability. So this is something that was put forward by the late Steve Gupta and by Sean from Hartnell Christopher Herzog and Horowitz Horowitz. Now in 2008 and has clearly stimulated a lot, how one could and should think about superconductivity in critical systems. So, again, my goal is to show you that actually this theory. And that you we are very, very closely related to one another and what we want to do is to use ESYK model as a tool from which we can build to actually describe critical Fermi surfaces to derive from a theory of this type. The gravity, we will start with a many body Hamiltonian and then step by step get a gravity theory out of it. And then there's no doubt it's obviously there. And then we have an understanding what is the extra dimension of the problem. Why is there a gravity formulation of that problem and so forth. So, what we are trying to do to the program that we are trying to do is to some extent similar to, even though we should be more humble but at least has an analogy to what left Petrovich Gorkov did. He started with the BCS theory and demonstrated to the world that you can derive from it to Ginsberg-Lando theory. Right. And there were non trivial implications of that current derivation. The first one was it made more convincing the statement that there's no real problem the BCS theory as far as Getsch invariance is concerned. And it told us what's the charge of the Ginsberg-Lando theories, namely it's two times E. You can read in the original paper what Ginsberg and Lando said about it and they said it's probably E. And with essay by Ginsberg on the subject who explains why they thought so, because Lando apparently realized that the mass can be different of this object in the kinetic energy term. It can be different for all kinds of reasons. And you look at free electron systems and put some in say some more complicated environment, and masses can also then be space dependent in principle. But to assume that the charge was space dependent was unacceptable to Lando because it meant that it would violate the gauge principle. And therefore what else do you do then it can change so there was E. But of course, the argument is not effective if I put an integer, or maybe some other topological number upfront the charge there right and therefore to E or 16 E are perfectly well allowed objects. So, good. So what we will be, and what's, and also what it clarified is what was the scholar field of the BC of the Ginsberg-Lando theory. Namely, it could be written in terms of the anomalous propagator, where you create or annihilate a pair of electrons here in the singlet spin channel opposite spin channel. So, where we have basically two different positions where we can do this in two different time points. And just to remind ourselves because it will become important. It is to some extent sensible to look at this object is depending on the center of gravity coordinate, which is the one that appears in fact here in the Ginsberg-Lando theory. And that describes in homogeneous states of vortex, for example. Then we can also look at the total time different of the system this is the one that appears in the Ginsberg-Lando theory if you make a time dependent. And that describes for example non equilibrium behavior to irradiate your sample electromagnetically and so forth you do time dependent Ginsberg-Lando. So, if you have a complicated this is maybe, then we have also the dependence between with regards to the relative coordinates here that doesn't appear in the Ginsberg-Lando theory it's actually being projected out. When we are going into the channel that is ultimately the superconducting one, but this important information, because it tells us what the internal structure of the pair is. It doesn't have to be a wave or a p wave, clearly, but also what its internal dynamics is. And here we have to be more careful when we do the holographic version at the critical when we look at the critical point doesn't have to anything with any of you in the narrow sense because we need to know something what's internal structure of the Cooper pair, if we are at a critical point, and it will be precisely this aspect that will lead us to the extra dimension of the gravitational description. So the internal dynamics of the Cooper pair. So what we will be there for doing here is that we will also start from the normal screen functions and then derive a field that lived in one extra dimension that very often is called zeta or Z or R or you or whatever right. So, this is our, our goal then we can also identify similarly, the origin of gravity, the physical interpretation of the dimension, and we will demonstrate that this will have some equivalence to critical both ones. Yes, are we. Absolutely correct. So this is a mystery, but of course we will try to resolve it. So, so let me, let me comment on what is a very, very good question. What I have here if I have the center of gravity and the relative coordinates. Then obviously the system in D dimension depends on not only D plus one but twice D plus one coordinates, but I only want to get to D plus two for D equals zero that's the same. I mean, only time and I get twice time it's the same than twice time, but then, but in five dimension it will not and we will have to see whether what really works out here, and whether what I'm saying is nonsense but we will have to get to that. It could be your right that in fact I get to a try to D plus one dimensional gravity theory, you won't, but you need to do it carefully then you will see that this is the case. Good. Now a few words on the holographic principle. So we are all on the same page and we know what we are talking about. Again, this goes back to truly pioneering papers written in the late 90s. So the partition function, for example, of some quantum field theory was mapped on to the gravity theory, and could be at C on, if you have some certain well defined large and limit and the gravity theory becomes classical and can be dealt with So one way to visualize this we have our quantum field theory here with space and time, and then we look at it from the site and add the extra dimension, and it is in this extra dimension where we have the usual interpretation, and that so-called radial direction of the problem, which in the usual interpretation of this community has somewhere the field theory living on on some boundary. And then you go and this scale has to do something with the renormalization group scale which you the probes the system on different energies. So, often, when you had finer temperatures we are very well familiar that renormalization group flow stops. And how would the holograph do this of course you put a black hole because then something seems to be changing at that scale. So that's how you do this here and to me this looked absolutely gorgeous. I have no idea what these people were talking about. And it's very interesting how the community works. I assume they all understand what they were doing but I could not understand why is there for no apparent reason an extra dimension appearing in the problem that is at least not manifest in the original formulation. And what this really means, and mostly some of the argumentation that I will be giving and calculation that we are doing, but driven by the urgent desire to understand this from say the pedestrian contents matter perspective. And then of course this is what I already mentioned to you you write down against for Glenda zero if you wish to study superconductivity, there's something quite interesting. If you look at such, which is related to this curved space story, which usually takes place in a so called anti to sit or space I have a picture of one in a second. So where you have the Mars in Ginsberg London theory usually at the mean field level, when this object changes sign that's when we are at the transition. In this case you curved space helps you a little bit, and you need to be more negative than a certain bond that was calculated long ago in the literature of and that is it. This is the example of bright and low non written. Good. So here, finally, we can see the Euclidean version of enter the center space to which is nothing else but a hyperboloid. I'm only drawing you half of the space the other half is not there is just a good old fashioned hyperboloid solution of this equation. So this is the ADS to if my time is Euclidean, which we do because we want to have a statistic mechanics description. And there's a very useful parameterization of these of this equation here, in terms of two variables which was sufficient. We have already called tau because it will be our imaginary time. And the other one is that because it will be that so the holographic version, and in this space, geodesics are these half circles, and the metric is given here. So basically the fact that this is curved, you can see by this coefficient you will was flat it would just be details squared and deseta squared. And in those coordinates you can also cover the, the age to but that doesn't look particularly beautiful because of course with this choice of variables we picked one coordinate to be different than naturally than the others it's not something you should make us nervous. So this is a nice way to look at this problem. So therefore we have this additional dimension here that is describing ADS. So this now we know everything we need to know about gravity because that's a symmetric. If I write this and I generalize this to us to a system that is this in the dimensions, therefore it goes to D plus two dimension problem then the metric would be the one of ADS D plus two again in imaginary time and this is how it would look like in real time that's just a minus And these are just for us to realize that these equations are perfectly straightforward for those who have never worked with them. And so if you, for example, write down now this scholar field in such a space, then we can calculate the determinant of the metric we can evaluate as the derivatives here and then you get an expression doesn't look that bad. There are four law coefficients up front here and then there's derivatives in the radar directional ready to fully transform in frequency in momentum. That's what it looks like. And there's something interesting also I can look at the saddle point of this equation, and that's the equation I would have to solve here. So this is an equation that is not too hard to solve. And we will recover it later when we look at Eliaschbach equations and I just wanted to re prepare to you because this is what what the client Gordon equation basically would be in this enter the center space if I assume that I can fully transform in momentum and frequency space and only have this addition dimension to really worry about. This is an equation that we should not have any fear. You can see already if you go to queue and on the guide to zero so we only have this part here. So this is so you know more genius and this additional dimension we can see that one immediate can diagonalizes in a few steps. And the transformation is the so-called maline transformation so basically a full transformation in logarithmic variables. And so you can write this action as just coupled harmonic oscillators. And here you see that the Mars has to be more negative than this value. This is our right and loan a frequent bound in order for this system to become unstable. Right. Otherwise this in these logarithmic variable this is how this would look like in this. Nothing all that complicated we have now accumulated sufficient knowledge to the kind of do do manipulations in this in this description. Yes. No it goes from minus infinity to plus infinity. Yeah, basically is the logarithm so if the scale goes from zero to infinity the logarithm goes from minus infinity to plus infinity that's why you have to do it like this. This is all this is only well defined above TC. Everything becomes more complicated when you're going below TC this is these are these Gaussian fluctuations of a non superconductor. That's what I'm trying to do here which is much easier than going into superconducting state. Yes, absolutely correct. So, there's more than a DSB plus two we need all of this because we need to make concrete calculation we will find results that require us to go a little bit beyond this. Because there's something special of course not too surprisingly if I look at this specific metric here at T to zero there's no black hole here we are not at T final temperatures. I would rescale my time and my length scale by the same scale factor, and I do this with this funny new dimension, since I don't know what it is I shouldn't lose much sweat over it what I need to do. But at least if I rescale time and space in the exact same way, then this line element here is invariant. What does that mean for the condense matter contact that's not precisely what we often see in the condense matter context what we rather see is that you need to rescale length scales, differently from time scales. And there's some so called dynamical scaling exponent Z. So this would not be something scales and time scales are scaled differently. So this would not be something naturally encompassed by this and that is it a space of course, and it will only be important once we even have space for the time being when we're talking about space why K we don't have much to say about space, but we will get to this. So then, it would be nice to generalize this and of course the community has worked on it. And here, for example, is something that goes into the name of lifted space time for the obvious reason because of an esoteric piece. So for those who don't know this when you have a usual five fourth theory and you have a gradient squared as usual in one direction, but for whatever reason the coefficient of the gradient span other direction is zero, then you take the next gradient which is force power in the system becomes an esotropic. Here we have an an esotropy not between different spatial directions but between time and see spatial directions. The name comes from and then you have essentially here a metric that just is an esotropic with regards to say the space and the temporal and also radial direction here. This goes under the name of the lift should space time, and there is in fact a very nice work that I at least managed to understand to some extent by Sean Hartnall and collaborator, where they could generalize something from a general relativity perspective. So what they did is they said well let me just add to my meta fields, the fluid that goes propagates to my system, and take an idea fluid description where the equation of states that relates density and pressure, that was taken from a Thomas family approach something straightforward and simple. And if you don't solve this problem you actually get whatever the value you want to have, depending on a parameter that determines the equation of state of the problem. This is not a particularly constructive statement maybe for us, but it tells us at least that if there's a back reaction of C critical fluid on the gravity, there can be more than just enter the center spaces. We need to still understand why we even talking about all this stuff. But I'm, it's a moment here to accumulate some knowledge so we can discuss this and more sophisticated basis. So, and we will have this of course has been called electron stars which is a beautiful name. And because it's precisely this you have some matter that affects the gravity doesn't necessarily form a black hole, and therefore exists there and changes the geometry. And of course also because it was motivated by a work from Oppenheimer on neutron stars, very much the same logic and so forth. Yes. Oh, this is there's nothing there's nothing there's also an equation of state for the current and the conductivity of the system that would then make it a charged object. What we need to do with this I just didn't write it down as there's another equation of state that relates the current and I think you can also get this without this. But the calculation was done you know maybe more. The calculation was done for a charged system that had to find out conductivity. And the star is it is a coefficient in the equation of state of the of the Thomas Fermi description is just a dimensionless parameter that characterizes the, the, the equation of state of the system all I want to tell you is, I don't need to write that they're very, very more complicated theories to get to these lifters gravity theories. This is at least something where can follow the philosophy of the of the physical motivation of it. Why do I do this because we will see that we get such geometries for specific critical system. In fact, an icing ferromagnetic quantum critical point in two dimension will behave just like this, we will see. So now we, however, go back to our S. Y. K. model now we've cleared the, the, the air and discussed some of the mathematics that we need to move forward, and now can go back to look at Hamiltonians and so forth, that we cherish, usually a bit more. So we're starting the discussion by looking at zero space dimension the problem that we've just seen in Alex Kamenev's talk, the S. Y. K. model. So, here's it again. As I wish to study superconductivity, I would be foolish in starting this description in terms of my Iran performance because we've just seen. I mean, it is already superconducting if you want so. Because it has not. I mean the particle number conservation is not is manifestly broken so therefore it's not a sensible question to ask. So therefore we will have charged ordinary complex for me and whatever you want to call them this is all to all interaction that goes to end. So I've just chosen that says one of the end stands up front here though I have an easier correlator and how to do this deal with this problem we've also seen. Instead of thinking in terms of the original fermions of the problem you think in terms of by local fields propagators and self energies are the fluctuating objects. So we've seen the tricks to get there to this result in the previous talk, and something like this can be done here as well as the large and infinity more precisely limit. This is the branch cut singularity that you might get. You can write down an analytic expression at least correct the lowest frequencies and then of course there's some at finite and some low scale where we see that this actually turns around. So that's one of the differences in the previous talk. I will mostly discuss this as my reference frame and look at small superconducting fluctuations relative to this state. And we will look at what how the superconducting fluctuations behave. We've also seen in the previous talk. You can do this reparameterization of time. And if you do so, you can do the reparameterization of time. If you ignore high energy behavior and physics that's related on the cut off because this is the derivative that was neglected in Alex previous talk. In addition, you have actually an infinity of solution. If you include however the derivatives the temple derivative of course you need to pay a price for such a reparameterization. And this is the price expressed in terms of the schwarzen derivative because it keeps the leftover movies transformation invariant that we've also seen. As we also heard, the gravitational theory of say a matter of free gravitational space in two dimensions and so called Jackie's title one geometry. Good. This is the subject I model and again. Our personal motivation was we want to see what happens to this beast, as it becomes superconducting. We want to have fun. No further justification for it. And to do so the need to actually have, or at least very very convenient to have a model that gives you at the large and mean field level superconducting solutions, mean field superconducting solutions at an infinity. And we therefore wrote down a slightly different model that goes now into the name of your cover is like a model here in this paper with Elias Bellis a couple of years ago now, where we have again our formula is just a chemical potential no kinetic energy. We now have both owns. Call them phonons. They have a conjugate momentum, and they have a finite frequencies are get these phonons are get some optical frequency everywhere the same. We can call them with a random or to all coupling constant see psi psi and five. So this is our model G are random couplings I have to say more about how we choose the distribution functions. This has, there's a essentially independent work by you shine one. We choose a slightly different versions of therefore he doesn't get superconductivity at the mean field level but otherwise the normal state is very much the same. It's actually an interesting experience I had we were sharing an office in Stanford when we did this work I didn't know yet that he was my office mate. He had done his calculations on the blackboard. And these were my equations and quote unquote, but with a different handwriting different written by somebody else so I was felt it was very very weird experience. So this, it was really done independently. And there are general relations to find out the mentioned that we will exploit later on. So how do we do the distribution functions. First of all, we need to realize that this Hamiltonian of course has to be a mission for the foot G J G I J must be G J I star. And then it can still however be complex as a real and imaginary part. And we take different distribution functions for the real and for the imaginary part which we parameterized by a value alpha. The proper distribution function for different indices the one that damson wanted for the J's, but maybe that's just too destructive, but important is that if for example this value alpha here is zero, then there is no imaginary part. So this is a purely real quantity then. So therefore we are, we are drawing for given K, the G I J's from Gaussian orthogonal ensemble. And if the distribution of the real and imaginary part are the same. So alpha would then be one, that's a half here and a half here, then we're drawing this from a Gaussian unitary ensemble. Now we will see that in the unitary ensemble we will not find superconductivity for the orthogonal ensemble however we do find superconductivity physically this means I have a random realization. We know the time reversal symmetry in one case and doesn't in the other and we know the time reversal symmetry can be poisoned for Cooper pair. And hence this result, and we introduce this parameter alpha here, basically to be able to interpolate between these two limits, which just is just convenient. Then you can see how things change, you can call alpha the pair breaking parameter. Yes, there is none. It's just electrons interacting with both zones. Yeah. So the procedure is now very much the same we want to calculate the partition some we usually have this for a given realization. There's some replica story and we've heard lots of discussions on the replica aspects already in Alex talk. This, this, this order problem here. And upon replication, there's a new replica indices I just basically square this up here. But this is really only the result that I get for the Gaussian unitary ensemble. So they're really imaginary parts of this coupling constant are the same. So basically the operator here twice and I've just rearranged them already that next to each other is the same index II JJ and LL that goes to N, or M, if you want to take a different number of both own flavors. And now of course, the introduction of these by local fields here as we've seen them here for both ones and fermions can be done, and you can write down the interaction energy as a trace and the appropriate way. So this would be the same steps that we've just seen for the from unique for from an interacting as like a model now done for the problem that we wish to study here. And so we get this term here we enforce these identities with large multiplier fields that gets in same letters and eventually it's a saddle point the same meaning as self energies. And then be integrated out, and we get an effective action only in terms of the by local fields that are so convenient to us. But before we do this, we need to see what happens when we have the generic assemble, not alpha equals one that we just looked, which was the one for the Gaussian unitary assemble. If we do this, all what changes is index orientation so we don't even want to look at this, but we have to for a second, because now we see that the same indices I and I that are so important at large and both come with creation operators, or with annihilation operators in pairs. We can now deal with this problem and do the just the same recipe for this problem again, we need to have Gorkov anomalous greens function as a fluctuating field. And the same analogy then introducing conjugated fields and so forth that enforce them would then give us an effective interaction that looks like this. We have our pair breaking parameter now we have the Gorkov fields here. And this are the usual grains function that we have that's all what changes but there's hope that we get solutions that will give us superconductivity. That's the technical steps to get there. Yes. By preparing this. Well, as I can always do them but then the contribution is swollen one of one. Yeah, I want to have. I want to tailor make my theory this is all engineering here. Right, we're not solving, we're not having nature in front of us and writing down nature's Hamiltonian and then solve nature's Hamiltonian no no we're cheating, we're writing down the Hamiltonian we can solve. But we're writing the Hamiltonian that gives us a physics we wish to see. So, let's just be completely honest about this. Yes. Yes, can see which central symmetry, central symmetry. I just don't know what it's meant by the time reversal symmetry is broken for realizations and of course restored on the average, but I don't know this the wrong way back. I don't know which is back. No. Oh, the bottom central symmetry. Maybe the person could just clarify what's meant by this I'm sure. So, good. How am I doing on time. Left excellent. So, so it's actually here. Oh no, if I point there this thing can go there right. So, if you. So this is this is the actual interaction with the randomness. Now g i j k i j l is, is a Gaussian distributed field. So the way you think about it is, is this here to clean the blackboard. You have i j k and some operator i j k, and you average over this one you need to do this of course with replications with some replica index a. And then you have g squared. Oh, I j a dagger a. Oh, I j k double. It's just a Gaussian integral because you have here some, you just do the integral the Gaussian integral over g. Good. So now integrating on fermions, I get a field theory in terms of these by local fields here written, all convolutions are somewhere hidden. I have a number structure now of course because I have all these anomalous propagators and of course when we look at this this looks like a lot of what functional. But of course it's not it's an actual action. And that's why it's so much fun to look into this, because we can look at the actual fluctuations of these by local fields. So if we want to work the first step is of course we take the saddle points, because I'm not doing the sophisticated things that Alex was referring to yet, or at least fluctuations beyond the first need to see what the saddle point looks like clearly. Right, of this problem. And is the number of chromions and is the number of both zones, or what matters is that they're both large and their ratio can then be arbitrary. So, what matters is, I get equations that are, we are very well familiar with. So recall, but Alex got when he did the saddle point was something that looked awfully like self consistent second order perturbation theory. In fact, we could have ignored all this stuff. So just self consistent second order perturbation theory of course, nobody has given us permission to do so in the system where the unperturbed system was zero. And the perturbation is infinitely larger than the perturbation than the bear part. Here we, the, the, the entire large n random and so forth coupling constant justifies to do something that we could have done otherwise. What we get here are basically a large pack equations. So we get, I mean the derivatives with respect to the respect to the self energies always give you the Dyson equation for both of the chromions. And the other ones now give me the self energies and this is just one loop self consistent one loop for the both on it for phone on it. And from me on it normal and the normal experience function, but this is F and this is G because the arrows and the diagram and so forth. So all what we have done is, we have given permission to solve a large box equations, and don't need to ask anymore and everyone who comes and tells oh what are what mitzvahs theory we say go away. I have done a large and calculation but that's all really what we have accomplished here so far on the saddle point. Right. So let's solve this problem, and we get for the greens function and for the, for both the boson, the, the fermion and the bosonic green something to get power law solutions. This is not trivial result, because our boson started out to be kept. This is an interaction of the system that ultimately and no matter what you do drives the system to a critical ground state. The renormalized boson mass always gets renormalized at T equals zero to zero, and it goes with the same power law basically that also governs the bosonic dynamics of the system. In a critical state. This is also governed by the exact same exponent here gamma, that is between zero and one if I choose and equals and I get whatever 0.68. So, important is that you get a critical state that you can tune this actually dimensionless constant in the problem which is, well our interaction and a certain power of the of the bare phone and frequency. It's dimensionless. And you didn't have a dimensionless coupling constant in the swk model because there was only one energy scale, and I think with zero it doesn't matter. Here we have a dimensionless parameter, and it depends a little bit with zero result doesn't depend on it but what happens at finite temperatures is affected by it, but at lowest temperatures we get this power law reason I'm prepared to do the analytical I don't need to get to this result on the blackboard, but I don't know that anybody cares to see the solution. You need to yell if you want so what you see is that you get these branch cut singularities you for the both fermionic and for the bosonic solution, and the final temperature gets softened. Good. So if you look at here at higher temperatures and the solution at large dimensionless coupling constant actually changes the boson gets extremely soft. And the fermion looks like it just sees impurities. So there's a wide intermediate regime here where the physics is somewhat different we have extremely incoherent fermions is not even a branch cut anymore it's just a blob. The fermions are perfectly sharp but extremely soft. That's just a thermally excited state all the bosons are firmly excited essentially. And therefore act as impurities for the electrons. And you get just a different state, you didn't have that state in this sense in the swik model at lowest temperatures however these states are the same yes please. Yes, there's a reprimandation invariance yes. Yes, exactly. So, if you go to the fixed point, and then try to, you find you have an equation at for Etsy at T equals zero have to be. And for those equations I can find a reprimandation invariance is a bit more subtle at finer temperatures because the bosons get massive. Yeah. Yeah. Is enslaved by the fermions by London damping. So, the bare initial dynamics of the boson is here completely relevant. All what matters is, which dynamics is imposed on to the phonons by London damping if you want so all by the self energy that is due to the fermionic system. Yeah, that's why they say in this. Now I need to think, sorry. That's why this, this is the exact same exponent here. Yeah. So, again, you need to decide do you want me to do the calculation. Nobody cares. It's good enough. Good. So, and you can also know the next this was all for the Gaussian unitary or something that wasn't superconducting. Now I make the system superconducting. The superconducting transition temperature is actually very simple you find the T squared with the dimension of the constant here it's more coupling, as opposed to exponentially small. So in that sense, TC is much higher than the one you would have gotten for a family liquid we get to this once we couple dots. Yes. TC, the transition temperature goes like this coupling constant to G squared is parabolic here. And this levels off because on the one hand, you get these very very soft both owns that is good, but your formula and so also pretty incoherent. And there's a balance and that they actually get a TC that goes to a finite value. Yes. Yeah. Yeah. No, it doesn't depend on the exponent is always G squared. And you can also see there's a Boogaloo of quite a part of these are numerical solutions here. This is very flat in the normal state, and we are here strong coupling in this regime somewhere. And then we get a very small coherent weight of the Boogaloo of part of that actually vanishes asymptotically. Why, because it's such a broad state. There's a way to be gained within the scale of the of the gap, if you want so, and that scales this universe power of this dimension is coupling constant so what is to be said here is here actually TC sets in at the exact same scale, where you would have gotten the system quantum critical. And here you're in the middle of this funny state where you get superconductivity, it therefore makes sense to dial our pair breaking parameter, because if I know going to a different direction here, and at this pair breaking then I can tune my transition temperature arbitrarily small. And then it vanishes here with this essential singularity type of behavior. And this will be no surprise at least to those who have thought about how you can break conformal symmetry such as damson. This is always while you would have to do this. And this is also how the differential singularities appear quite often. It is one universal value of Alpha C or one generic value of Alpha C independent on the details of the problem. And this is also how you would lose superconductivity in these ADS type models, incidentally, if you're going to this bright and loan of written bound for this funny exponential essential singularity type of behavior. And this is a superconducting state that we have now understood on the mean field level. Now we want to go back to understand it's a holographic relevance of the state. And for this we recall one more time, what we are doing is we really actually have to integrate all these fields and in particular we have to integrate over those pairing fields here. Those were the ones I recall that are relevant for these anomalous brains function so what I will do is now we will look at Gaussian fluctuations on top of this critical mean field state. Just to get a sense. How would they look like just superconducting Gaussian fluctuations the simplest calculation I could do to describe some aspect of superconductivity, just above TC. If we can bring TC to zero, I can also be just near this critical point that is the superconductivity might disappear. And in doing so, it is useful. Again, as I discussed earlier to go from the two time variables to the center of gravity time and we transform it to what I call Omega and the relative time and we transform it to what I call epsilon. Just two different ways of characterizing the same subjects. So I do so and I expand my Gaussian theory I get the results here this second part is known to us this is our effective interaction that we've just seen earlier already. All I do is since I'm expanding on the normal state I can insert here my solution from the normal state that we've just seen which is a power law. And I have some trace log to deal with, but I'm only doing this up to second order. So there are just something that has some diagrammatic interpretation, and it's essentially some particle particle propagator the trace log was for fine have integrated by the enormous self energy already. That's why it looks like this so this has interpretation again here. This is the power law behavior of the pair of the phone on propagator that was given to us from this critical normal state. And here we have the particle particle propagator of the normal state given by the chromions and characterized by its own exponents. The theory of Gaussian fluctuations I could stop you and say this is the theory of Gaussian fluctuations period. But what you can do is, you can look at this equation and for example at the stationary saddle point this is stationary saddle point of the problem this has to be equal to the linearized gap equation of the problem and of course it is equal. So all this object here by, and I go to omega equal to zero, I get an equation that many, many people have written down. I'm giving you some list. Actually not for the SPK model, but for systems in compressible Fermi surfaces with critical states in either ferromagnetic or anti ferromagnetic and some kind of systems where there's a different exponent we have our exponent here but you can actually change the self energy for the enormous self energy on the Fermi surface of the compressible system, and we will get to this more clearly as we move on, you can solve this equation. But here we already see that this gap equation is this is the, you know, self energy that we can see Cooper log and the singular pairing interaction that makes it stronger again and they both add basically to zero. That's what I mentioned earlier in this discussion when we said how can you avoid the Cooper, the death of the Cooper instability in singular systems. Good. So this is what we get from this as I came model as well. And we can now do a very, very nice analogy. And I have learned the trick from this paper here by them son because he sits in the first row I need to say this particular because he sits in the first row. You have this gap equation here, and you assume that this gamma is small, if it's smaller than this is a weekly varying function. And you can basically split the integral depending on whether epsilon prime is smaller larger than epsilon, and have two contributions, and then easily find a differential equation for the problem and this is the differential equivalent in this limit to this integral equation. And clearly, everybody likes differential equations more than integral equations because there's a command to dissolve. And then we know what to do. But what you can also do is that you take the second order differential equation and introduce a new variable of a little bit of experimentation, where zeta is just the inverse of the relative frequencies of the error energy. And if you do so, then you find this equation, which happens to be an equation I showed you earlier, namely the decline Gordon equation in enter to zeta space ADS to where the mass of this problem is given by the exponents and the coupling constant and so forth. And the correct law of treatment bond is actually one quarter. So depending on these two terms here we can see there's a critical coupling constant, where the system is in or outside the superconducting state. What we see from this analysis is that mathematically this is not an accident that there's an extra dimension, and that the physics of this extra dimension indeed is the internal dynamics of the Cooper pair that we need to pay to once a system is quantum critical. And that's why it's not enough to write down the order parameter theory of a D plus one dimensional problem in D plus one dimensional theories but better in D plus two dimension because we need to keep track of this additional internal dynamics of an order parameter. So if we then do something and go, even this was for a static solution we allow for time dependencies or should be a towel here. Then you can even give us a geometric interpretation. At first glance are a bunch of minus signs they don't work out the way we wanted to for the end of the space. And one has to do a little bit of a detour that cost us several months. In the end is not deep physics in it I believe what you need to do is that you need to not look at this original variables, but you need to average over geodesics of this problem which goes into the name of a rather than transformation I don't want to waste too much time on this because I couldn't increasingly believe it's not important. So what you do then is you take your a scholar field and average over well defined geodesics something you can learn from the holographic community. And in this variables you do find indeed a mapping between our Gorkov propagator and the scholar field, just with the meaning of the additional dimension is essentially again as I said earlier that the inverse frequency gives you this additional function and you get then indeed the metric of an ideas to problem in the sum so the superconductor has an action that looks like this is precisely the action we want with this mapping. This is hidden nothing else but our enormous greens function of the Gorkov problem, but keeping track of the fact that you have to pay attention to internal dynamics of the Cooper pair. There are two dimensions and not in one dimension where we had before just time. That's really all there is. And we can interpret the individual terms but I need to first know how much time I have because I want to get some more 20 minutes very good. I have enough time. So we can interpret all these different terms that occur here in this action because we have the, you know, the new dimension has a gradient. And it actually arises it's a non local term is, and it arises from this power law interaction. And it's actually the leading term of a gradient expansion of this power law interacting problem. So we can find out that this temple derivative here comes from the frequency dependent of this propagator here, and both terms contribute to the mass. There's a repulsive contribution from this term. This is our earlier result. But it's don't like superconductivity. And therefore, if there was only non familiar with physics and no attractive interaction that singular it would always get as a positive mass in the system never would want to superconduct. And then there's a negative contribution to the mass, coming also from the attractive interaction, which is actually by itself always more negative than the Brighton loan of pigment bond. And this would always love to be superconducting it's just being prevented from doing so, by the fact that these, but this is a non familiar with state. So, so there is this critical coupling constant that we saw that he vanishes is there because the system is not a family. We get. I think now a clear understanding in one very concrete, maybe over simplified example for why we can see gravitation of physics appearing in a critical state. So this is a set already of course you can take the equation the saddle point of this equation again and it's again equivalent to the electric equations. Why because it's the same theory. So it's instability better be the same it's not just somehow similar. It gives you the precise same condition for the onset of superconductivity number wise coefficient wise and so forth. So we can do a mapping a construct of mapping we know all the parameters of the theory, just like you know the parameters against the London theory. If you start from a BCS theory and follow go course recipe. So we can do source fields. I think I don't want to get into this and look at susceptibilities. You can go to finite temperatures. You can do that finite temperatures. The mapping gets more elaborate but actually step by step straightforward because they're mappings of the finite temperature metric that has a black hole at a certain horizon energy. So you can do to map this onto the zero temperature problem. And you can do the same with the, with the reparameterization invariance on the site of the, of the SPI came model you can link the map we've just found. And then you find indeed that the problem can be written as a scholar field in a metric that has a black hole, or what the black hole does is stop scaling for nothing to get overly nervous about. Our G stops at this temperature scale and here we see it's up to pi times T or whatever that means right. So that's very beautiful we can also, this is a D here. There's also actually a gauge field involved. So what this is gauge field is not an external gauge field of the problem. What it really is. It is an electrical field that is there, when the system deviates in its particle density from particle holds symmetry. So if you have an original problem, the SPI came model with a chemical potential that's not at zero, but you move it a little bit away from zero you get your break particle holds symmetry. And we do so the holographic version of the superconducting theory looks like it has an electrical field that simulates this broken particle holds symmetry. So if you care about broken particle or symmetry you want to look at, say the thermal power of that state or something then you will have to include in your description. So that's the actual this gauge field and the problem that's in some gauge, you can write like this. So therefore everything can be done. It's not just an accident at T equals zero, but it seems to be something much more natural that you can write this as a covariant final temperature theory in this gravitational space. This is the last one and the question that them song asked earlier, Fermi surfaces. So we want to, I skipped this because this is maybe more interesting. This was a discussion that I just skipped of coupled dots that you can look at the superconductivity there yes. It's in duality on the level of the action itself. Yeah. So now we look at again critical points but we want to look at finer dimension than the specific case I wish to study is an icing ferromagnet in two dimensions. There are other generalizations of the s y k logic to find a dimensions and giving some references here. And the idea is the following. We have some momentum state. That's periodic. The momentum states a spin. And I have my boson just before. And then there's a coupling here because I have an I think paramagnet a couple of my five field now to a Pauli matrix Sigma Z. Fine. I don't want charge because there's so much going on with acoustic phonons and solids. When charge fields condense that I rather think about spin but that's really the only reason the otherwise I could do the same. So now I have a random random coupling constant but the important thing is this random coupling constant is the same for all lettuce sites of my lettuce. So, basically, so it doesn't break translation in various it's not you know random here and random differently here and uncorrelated this infinitely correlated so the way you I think about this is, we have a model with a given constant G and we just average over the model. We have all kinds of different typical coupling constant values. It is a procedure to give us control calculations. And as many of those, you may question it's a physical sanity, but as such it is well defined so we have here now a coupling constant that's random, and it's random everywhere in the same fashion, I think the idea goes back here to this work with the bungeon and errors and and central and so forth. And then you average over these infinitely correlated systems in space infinitely and in time. If you do so, you buy local fields now become by local and coordinate and time and you do goes to the exact same recipe. So we need to introduce our normal as propagator if we have the right correlation functions and so forth if we have the correlation functions of the random couplings and we get the effective by local fields action and since I'm written in some abstract form. It looks the same they're not only traces in coordinate space or momentum space in addition to time, nothing else. Otherwise, we go to the saddle point equations. So that's the saddle point equations we reproduce the good old fashioned Elias back equations. In other words, this is yet again just a way to justify Elias back equations if you wish so, just like we did this before. We have solved them and they have been solved in the past, of course. So we have a both on propagator that at this critical point is characterized by London damping omega over Q, and we have a firm any propagator for this specific problem has an enormous dimension two thirds. We discussed already a normalist critical exponents, and it's always been irritating to me that this problem has different critical exponents dynamically critical exponents for the both and for the fermions right for rescale energies I need to rescale momentum by a third power here. And if I do it here because this is the momentum transfers to the family surface I need to do it by the powers rehabs. I'm in two dimensions at the moment yes. Because that's where you get these nice exponent otherwise you have to deal with logs and you get chest pain, but otherwise, it should go along the same so this is my critical points that I get. This is the analog of my critical solution of course now I'm only getting it at one point of my coupling constant, because the system is only critical at one quantum critical point has been studied also with numerics and so forth. I think this is understood. And that is always question from earlier. My, a normalist self energy at a system that has a superconducting and this superconducting state you will be a P wave triplet straightforward calculation to do so. The enormous self energy really only depends on the angle of the momentum, because the typical magnitude is always given just by the family momentum. And that means all the dependence on the relative momentum of the field is frozen in by the family surface. It's always stuck at the family momentum and it's only the relative energy that matters not the relative momentum. And if I do so again get a Gaussian theory now. Now of course everything depends on momentum, my pairing interaction here is still singular this exponent gamma was one third for the problem. I wasn't here to do this for generic gamma just to see what would change if if this were to change and I looked at a different critical point for which I can always write down just same type of generalized problem my particle particle probably can come out collided here as well. And you see it is very asymmetric in momentum and in frequency space. Just because we have a family surface that breaks for momentum very much be careful for finite momentum relative to the family service very much differently than it does so for energies. I can do the same. I can write down a mapping that is now a little bit different than what we had before. But again, I have the relative frequency here I call this my inverse holographic variable. And I have some power law upfront. And in terms of this object I can write down an action again that has the same looks than what we had before. But now of course, the momentum dependence and the frequency dependence is very asymmetric is a problem. That's not too surprising because we are looking at a kind of matter problem is a family surface of the gravitational point of view of course it's something what has to accommodate for. And if you do so you find that this is the metric of this problem is not trivial because these power laws all have to come up to the exact same power up front here. There has to be a geometric interpretation for my taste. But if you do so you find this metric, and this is nothing else but the lift ships quote unquote electron star metrics that we have seen earlier. And the dynamic it critically exponent is given by the power law gamma, and it's just one over one minus gamma. So to couple of limits here, what we have is of course one third and then the, the, this is one minus one thirds is apparently so two sorts of three halves. In other words, the form you're Nick, a normalist dimension of critical anomalous that dynamically scaling exponent that matters here for the holographic description not the bosonic one. Interestingly, I wouldn't know what to expect beforehand, you can go to gamma equals zero then you recover ideas. You can go to gamma equals one then you get an exponentially large power dynamically exponent or activated scaling and this goes is basically the metric of ideas to plus. This is what I said already the firm in momentum is the problem. And this is also what I was just about to say we can go to gamma to to these two limits and either we find ideas or we find the space that factor factor rises into ideas to and flat real space. This is also known in the holographic community to arise in the vicinity of a charge black hole, near its, its, its event horizon, but there. It is really a state that is perfectly local in space, and perfectly, not local in time, but our real situation is not like that, our real situation is this funny lift, it's geometry. And then ask, what is the critical instability of a superconductor to make some concrete statement and we find that say this state is rather robust against pairing. And this is extremely fragile and we immediately get basically that the superconductivity breaks down in this extremely local theories just to make one statement about this. And then we need to conclude here. What we've managed to do to accomplish is to give the wonderful principle that was put forward by a different community, a more concrete condense matter realization, having an interpretation with these variables mean. And if you wish so we have accomplished nothing, because we could have solved them just without doing this mapping. On the other hand, we all know that just having for example a BCS theory may not always be the ideal tool sometimes you do want to have a kids book London theory because you want to look at say in more genius states, and so forth and so on, that are just very difficult to analyze with those tools. And for example, non equilibrium dynamics of a superconductor is one perspective I can see where this formulation is just much more powerful than doing this you know brute force in terms of the greens function in terms of diagrams. In addition, we have seen that the superconductivity by critical both owns and the holographic superconductivity are really the same thing. At least one can be derived from the other. So we have a, at least for me, deeper understanding for either one of these two statements and maybe this will help us because both communities have worked on an arsenal of insights. That influence each other, and one can learn a little bit from one another. Thank you so much for your attention. I wonder what the zero temperature, the both of mass and that's like a problem. And, but then I would worry that if we go to a, maybe more physical phase diagram only describing a single line in this whole phase diagram, and how do I know that this effective action is even unstable for two days at all. But maybe it's just a critical description and actually, yeah, the physics is completely different. So, you're right in the swk model is a critical ground state. I didn't know there was a question mark or a voice raising at the end of this thing so I could say nothing at the moment but I'm saying something. So there was, yes, the swk problem is purely critical. This theory that I looked at the end that really only makes sense at one fine tuned critical point, which is what we usually do I think it was always clear to me that this critical phase is a caricature of something that is critical to us. And, at least, I would not have been able to do these calculations in finite dimension had I not known what to do from the zero dimension in case. So therefore, at least pedagogically it was useful but yeah, the line is actually a battle point. No, we know if we add hopping what happens, right. Yeah, yeah. So now for me really I came from a different perspective. I never wanted to know why this is a critical phase because it's a silly model so it behaves silly. So what it mostly taught me how to think about a certain problem, which I find much more interesting those systems with for example Fermi surface. Yes, there was also a question for a while. Yes. Oh, I mean, it's a phone on. Really. Right, it's exact same physics. It's attractive in because it's in the charge channel and it's a phone on. So we have triplet for the icing ferromagnetic case, because it would be repulsive in the charge channel. But so there is nothing really new compared to what we all know already how these pairing interactions come about in this specific case, because in the end of the day we just, if you just looking at the saddle point be seeing the electric equation, what might be new is the way to look at the fluctuations around those phases. Yeah. Oh, oh yeah you need to average. So yes. So what you do is, that's actually a tricky one to some extent. So you really have to expand. And there's actually index and here and this for each angular momentum. Is there a coupling. Actually this term at finite q couples different angular momentum. But if you do a gradient expansion, you can ignore this, because it's only coupled on anyway. But take the mode that gives you the most attractive channel. Now, the subtlety is which I didn't reveal to you is the decision I mean actually they're all infinitely degenerate on a sub leading level beyond the leading approximation. There is a term that would in this case you're just be some non local term that tells me that one is more attractive than the other. At Gaussian level I don't need to be losing too much sweat about it. But if I ever go beyond the Gaussian level I need to worry about how these modes are actually coupling. Yeah. And equals plus minus one. And because it was not allowed because it's a triplet it's a triplet and equals plus minus one it's a p wave is the most splitting is small. Yeah, and splitting is on a level that's on this action sub leading. It's funny. I mean, I'm not. Yeah. Oh, yes. Yeah. So my motivation is this one. I write down the theory that's perfectly stable as an action like this a plus k squared by K by minus K. So let me do a transformation where I'm writing phi of K as one plus a k squared at all K. So I just wait different momenta differently. I insert this in here actually there should be minus one I insert this here and expand to small momentum. I see that it looks to me like it's unstable. But it isn't, but Gaussian series by themselves naked have no meaning. You always need to have a source fields tells you how you should properly couple to the system. And therefore, I think it's just, you need to do this so called funny rather than transformation in the, in the other case. So it looks unstable but that's, it's not I think physically unstable because I can calculate the physical sustainability it's perfectly well defined. It is, because I'm looking at the system in inefficient variables, and you need to find the right variables. So for example, the future would have a change in sign of cave expanded for certain regime of parameters but that's just because I've been silly. And I need to need to fix that this is my current understanding of it I thought initially this was much deeper, but I think there's nothing deep about it. Yes. So, if I ever wanted to study the superconducting state. Right, deep in the superconducting state. Well, I mean, everything gets kept. So this in the gravitational language is a giant gravitational back reaction. My personal take on this is, well, I rather solve my electric equations. In this case, and I don't lose too much sweat on doing the gravitation theory for that. Maybe I end up being wrong here. But if I'm near the superconducting transition then I can do at least a small correction to the gravitational background and then I would have to solve Einstein's equation. And we know we have a gravitational theory. We have now the superconducting theory. And what we really need to do is we need to just do the variation with respect to both variables and couple them simultaneously. And then we will see what the answer is. This is doable, but it only makes sense in my view, near the transition temperature. Other deep in the superconducting state, I don't think personally, this is something you need to be doing, you just solve the equations that we have anyway. So again, the, you have a direction to form units, a creation and annihilation operate on a boson. And then we did several steps to solve it, but in the end of the day, that coupled integral equations that are being being controlled in certain crazy large end limits, where you get a self energy from a green function and a green function from a self energy. And you need to compute this and you should have raised your hand and say, I want to see you. I'm prepared to do this. Now it's too late on the blackboard. I did all of this. Remember all the exponents. And I'm happy to do this for you privately. Yeah. Yes. So if I'm deep in the superconducting state. I just have a gap state. I wouldn't expect much fun out of this calculation, but it may well be a relevant calculation for certain applications but if I'm just deep into superconducting state that's what I would expect. I deliberately didn't want this. I wanted the same critical fluctuations that would mean on family with behavior to be responsible for superconductivity not some extra source of pairing but in principle you could do this. You get a giant gap, if it's a strong coupling, and then not much fluctuates around it because it's all kept. So it is something you can solve and it could be relevant for many situations it's just not something that I felt much compelled to do so I have not much to say about it.