 One minute, five minutes. Yeah, thank you. Yeah, so thanks a lot, Andy, for the introduction. And thanks a lot to all the organizers for asking me to speak on this occasion, honoring peers. So like Eduardo, I did my PhD from Rutgers University. But unlike Eduardo, actually, so peers was not my PhD advisor formally. But fortunately for me, towards the end of my PhD, he decided to adopt me. So I am the adopted PhD student of peers. And so it was really from him that I learned about condo physics and heavy fermions, especially in the context of quantum phase transitions and to appreciate such problems. And so here are two papers that we had worked out together. But what I'm going to emphasize is more than the physics that he has, over all these years, been a fantastic mentor. And so somebody to whom I could go back and ask for help and advice whenever I needed. So in this context, actually, I would like to share something with you. So those of you who have gone through the training of trying to put up a presentation with peers, you know that he is going to tell you many, many times that a picture says things much better than words. So put a picture. So I'm sorry, peers, I don't have a picture. I have words. That's my Indian tradition, I suppose. So it's actually an email that I wrote to peers and his reply. So you have to read it, actually, from bottom to top. So this is what I'm writing. And the context is that this is three years after my PhD. I'm a postdoc. I don't have any active projects with him. But some anxiety during which I am writing this to him. And his reply to this is very characteristic. He's not even asking what is the matter or anything. So he's there to help. So this is something that is very characteristic. And anybody who has come to work with peers, I'm sure, has appreciated this generosity and this position in which he has been mentoring all of us. So really, thanks a lot, peers. So with that, I'm going to move to the scientific part of this presentation. And so I'm sure to be running out of time. So let me first acknowledge my collaborators. So Marcus Garst, who is a theorist at Technical University of Dresden, Dimitri Labad, who is doing his PhD with me, and Yan Gale, who had spoken on Wednesday and is probably currently absconding. So the subject is motivated by the iron base superconductors. So we have had several fantastic talks over this week on this topic, so I'm not going to get into it, but directly point out to the fact that if we are interested to study electronic nematic phase transitions, then the iron base systems are really a fantastic system to look at. So it is really in this context, in the context of this nematic phase transition, and in particular, about the fate of a metal close to a nematic quantum critical point that one can achieve by tuning some parameter that I'm going to address certain questions. So here is the outline of the talk. The first question that I'm going to address is what's the nature of the metallic state close to the nematic quantum critical point and is it a non-fermil liquid as we are used to think of, typically, for metals close to a quantum critical point? The second question which I would like to discuss is the following, is BCS pairing promoted by the fluctuations, by the soft fluctuations around this nematic quantum critical point? And you have heard earlier that these kind of questions are also relevant, not just in the context of the iron base systems, but if you look at also in the literature, they are also important in the context of the cube rates. The third question is somewhat technical, but if I come around to it, actually I'm going to explain that it's really not so, but let me define the question. So imagine you write down the susceptibility associated with the nematic variable. In general, this is a function of frequency and momentum. So you can go to 0 frequency and 0 momentum, but there are two ways of doing that. So one is the static limit where you first put frequency to 0 and then momentum to 0, and the other is the dynamic limit, which is vice versa of that. And the question is really whether these two limits coincide or not. Of course, if the variable would be a conserved quantity, then this is a trivial question, but it really is non-trivial because the nematic variable is not a conserved quantity. So I put up this set of three questions and you can ask why these particular questions. So there is really a motivation for that, which is connecting these three questions and it's really that the ultimate fate of these three questions, in order to determine them, one has to really go to physics beyond electron only models. So for example, in the case of the first question, what I will argue is that if we stay within electron only models, the answer to this question is yes. The state is indeed a non-formal liquid, but if you introduce electron acoustic phonon interaction, then the answer is going to be no. And similarly, for the same physics, what one can show is that one might think that close to a nematic quantum critical point, one would have a boost in the BCS pairing. But indeed, once one puts back the fact that there is electron phonon interaction, the situation is not so clear. And in all likelihood, the answer at least in the iron base systems is no. For the third question, actually, it's somewhat more delicate because if we stay within electron only models, we will find that the answer to this equality is a no, that the two limits actually do not commute. Now when you put back elastic scattering, it will turn out that the answer is yes. And then if you add on top of that electron phonon coupling, the answer is going to be a no. And there is actually important physics in these two answers in terms of what is being measured in the spectroscopic probes that are trying to probe the system close to a nematic phase transition. So I'm almost certainly not going to be able to go through all three of them, so I thought I will just put up the answers over here. And so if some of you are interested, you can ask me later in case I cannot go over them. So let me slowly go over the arguments for the first question. And so why electrons interacting with acoustic phonons preserve a Fermi liquid close to a quantum critical point. So it turns out that a nematic transition is also a structural transition. And this is simply because symmetry is somewhat like trust. You can break it only once. And so once you have broken it either on the electronic side or on the lattice side, the other one is going to fill it through their mutual coupling. So it's also a structural transition. And you can see that in the softening of this particular elastic medium. So at this point, what one can do is that forget about all the complications of an electronic system. But just look at and consider the acoustic instability of an elastic medium. So let's just think of just an elastic medium. So here I will remind you a few basic things about an elastic medium. So this is the free energy of an elastic medium written in terms of the various components of the strain tensor and the c's of the various elastic constants. And so if there is a tetragonal symmetry in the system, then there are six of those elastic constants which are independent. Now if, for example, the elastic constant C0 goes to 0, then there is a continuous phase transition from a rectangle, from a square to a rectangle, thinking in terms of two dimensions. Or if the C66 component goes to 0, then similarly there is a continuous transition from a tetragonal phase to a monoclinic phase. The next thing to keep in mind is that the same free energy actually describes the fluctuations of this elastic medium which are the acoustic phonons. So we know that the acoustic phonon frequency must be linear in its wave vector and the velocity is a square root of the elastic constant. But the very important point to keep in mind is that when a phonon is excited along generic directions or its polarization is along generic directions, then the excitation of such a phonon implies multiple elastic constants being excited or multiple strains being excited. So it's really when phonons move along certain high symmetry directions such as so, and its direction of propagation is also along a high symmetry direction that you have a pure strain mode which is being excited. So from this argument it follows that when there is a tetragonal to an orthorhombic phase transition, there is only one of the elastic constants which goes to 0, but all the remaining elastic constants are finite, which means that the signature of the second order phase transition which should happen with the disappearance of the phonon velocity is restricted to only these two high symmetry directions lines. So if you excite phonons along any other lines or along any generic polarization, then you will not notice that there is a second order phase transition going on because the phonon velocity is going to remain finite. So in other words, momentum space and isotropy is very crucial in order to feel this criticality. And it is restricted only to this high symmetry directions. So much about an elastic medium. And now let me bring back the electrons in the problem. And the only thing now to keep in mind is that because of electron phonon coupling, there is indeed a symmetry allowed what I'm calling a nematohelastic coupling between the strain and this phi, which is made up of electronic variables. So here I have written this phi as a Pomeranchuk variable, but any sort of nematic variable is going to do the job because this is a symmetry allowed coupling. So because of this linear coupling, actually what one can show is that the electron nematic modes, they now inherit this critical elastic properties. So just to keep things in perspective, so you remember Premi's talk on Monday where she was talking about where strain was coupling to square of order parameter. So this is really the Larkin-Pickin mechanism that she was talking about in her talk. And it is to be contrasted with what I am emphasizing now where strain is coupling to a linear order of the electronic order parameter. And so this I'm going to call the Larkin-Levanuk physics. And in some sense, they go in opposite directions. So let's see from the side of the electronic side how this critical elasticity is going to affect. So it turns out that the argument can be understood purely within the scenario of Landau's theory of phase transition. So where we are used to thinking that the free energy can be written down in terms of a mean field part and a fluctuating part. So exactly as in Premi's talk, it's actually very, very important in this case to separate out what is the exact q equal to 0 variable and the q not equal to 0 variables. So these are the fluctuations. Now you might think that here I have an A and a B over there. And it's a typo. And indeed in standard theories, these two quantities are the same. But one can ask the question, that do they need to be the same? Or more generally, whether B can be a function. So the only function that can make, so it can meaningfully be a function only of wave vector, the way this expansion is being done. Now already the q square term is over here. And if you want to maintain analyticity, then the only possibility over here is that B becomes a function of various brailleur zone angles. So B is a function not just of the modulus of wave vector, but only of angles associated with the wave vector. So I'm going to show that this is indeed the case. But let's continue just for a moment with the argument assuming that this is so. So the moment B becomes a function of angles in momentum space, you see that at the phase transition point where A goes to 0, there are only very special angles where B equal to A equation can be satisfied. And these are typically the high symmetry directions. So the moment you deviate out of the high symmetry directions, then one will encounter finite correlation length. In other words, loss of criticality. So let me very quickly show that indeed the nematohelastic coupling does exactly that. So here is the nematohelastic coupling. And then one can see that the total nematic susceptibility must have an electronic part and a correction coming from this coupling. And so here is the structure of that correction. And what you can see over here is that if you put frequency to 0, this numerator has two powers of momentum because derivatives are involved and it's squared. Whereas the denominator, it has information about the acoustic phonon dispersion. And so this also is q square. So in other words, we have generated a term which depends not on the modulus of wave vector but only on the angles. So then simply by analogy with the, do you know where is the panic button here? No, it's OK. So just from analogy with the elastic medium, now you can see that criticality is going to be restricted only to these isometric directions. So this, so now if you, let's say, move to a different coordinate which describes the critical directions. And I'm going to call them q1 and q2. And the z direction I'm going to call q3. And one can ask, how does the susceptibility look like in the vicinity of those critical lines? So if we move out of q equal to 0 along the critical line, so of course there will be a q1 square term over here. But the important question is what happens if we go in any transverse direction? So the moment we go in any transverse direction, such as q2, we will now discover this finite mass coming from the angles. And you can see purely from analyticity and the fact that the answer has to be consistent with tetragonal symmetry that the only possibility is this kind of corrections. So these are actually angles because they are ratios. So you see the structure of the critical susceptibility is very different from what we are used to in textbooks. So all of this can be shown visually in the following way that the volume of phase space where critical scaling holds is severely restricted to this yellow region as opposed to a typical case where it will be a sphere in momentum space. And one can show that this implies that the total effective dimension, space dimension goes to 5. In other words, the fluctuations become weak. And the theory flows to its Gaussian fixed point. So here is the result then. So in the absence of the electron phonon coupling, the phase transition would have happened at this parameter r equal to 0. And T0 would have been the transition temperature. And so because of this coupling, the transition temperature moves to Ts. And similarly on the 0 temperature axis, it happens at the value of r equal to r0. So if we define things correctly, then actually this r0 can be put to be a dimensionless parameter of the theory which measures the electron phonon coupling. And very roughly it is given by this difference in temperatures. So this ratio. And the most important conclusion is that just as there is a shift of the quantum critical point in the 0 temperature line, similarly as one comes down in temperature, there is a finite temperature scale below which a Fermi liquid is going to be established simply because the critical fluctuations are getting cut off by the shear modes. So I thought that I should point out that this is a very important theory constrained before one tries to explain any anomalous resistivity close to a nematic quantum critical point. And so it's important to keep that in mind while thinking of the wonderful data of Nigel Hussie that he had shown us earlier. So how much? Two minutes. So I'm going to totally skip the second and just say a few words about the third topic. So let me at least tell you why this kind of a question at all is interesting from an experimental point of view. So the whole point is that when we think of a phase transition, it is a divergence in a thermodynamic quantity. So the thermodynamic quantity is defined over here on the left, the static limit of the nematic susceptibility. So it is this quantity that we expect to diverge. So here I'm not putting the lattice. I'm thinking of an electron only problem. So this is going to diverge at some particular temperature. Now as you heard earlier in Gersh's talk that because the nematic variable is a tensorial quantity, it's very difficult to detect it. There is no simple probe that is going to measure this quantity. So one way is to do it through spectroscopic probes. So that's why in spectroscopic probes, what is being actually or what can ideally be measured is this quantity. And this is via the crimoschronic relations. So this quantity, the dynamical limit is related to the imaginary part of the susceptibility by causality. And so if one can at least in principle carry out this program of integration, then one has information of this quantity. But we are not sure that these two things are the same. So in other words, the question really is that can electron Raman spectroscopy because this is really what is measured in Raman spectroscopy. So it's a photon in and out process. There are two photons with two polarizations. And two vectors make a rank two tensor. So it can really couple to the nematic variable. But can electron Raman spectroscopy really detect nematic criticality? So that's the question which is hinging upon this equality. So once again, I'm going to skip certain arguments and really tell you that what really is crucial over here is to look at the property of the polarization associated with the nematic variable. And whether really the dynamical limit of that polarization goes to 0 or not. If the answer is yes, then the two limits commute. If the answer is no, then the two limits do not come out. And so this polarization can be the easiest way to look at it is diagrammatically. And one can do such calculations. But let me give some very quick heuristic argument over here. So when one tries to do such calculation, one realizes that actually, at least for low energy processes, the low energy nematic polarization and current polarization are pretty much the same objects. Modulate certain constants here and there. So this quantity can be written down like so. Here sigma is the complex conductivity. So to put it very simply, if this quantity is to remain finite at 0 frequency, the conductivity has to diverge. Either it's real or imaginary part. And the whole point is that the moment you introduce elastic scattering, this is not going to happen. Conductivity is a well we have quantity. And so once we realize that, then these two limits, we realize that these two limits in general in a real system has to be equal actually. And so one has to go through those arguments to really understand that the third dynamic instability at t0 then becomes accessible through spectroscopic probes. Modulate certain questions about experimental details. So let me skip why the answer then turns to be no. You can ask me later if somebody is interested. And so let me just summarize. I'll just let you read about it. I have run out of time. And I would like to say once again, happy birthday to your peers. And thanks a lot for your attention.