 Thank you Lara. Thanks everybody and good morning. I'm going to speak about stuff which lots of people have talked about in different ways and I'm going to cut the problem probably slightly differently from what has been done to talk about some aspects which are not just common between these two materials which have become fashionable to discuss because there are lots of new experiments but to tell to talk about their connection with some old experiments and in this talk I will also use most of the stuff that I will talk about here in relation to these two materials is unpublished. I don't think it's unpublishable but we'll find out. I'm going to speak about, my talk will include some work done quite a while ago on quantum criticality and I will just barely mention a paper which I've worked very hard with the Hidemabashi in Tokyo which has just appeared on the archives which gives the exact solution of the Kubo equations for transport electrical conductivity, thermal conductivity, thermal power using what I believe are the fluctuations of the quantum XY model. My interest in twisted bilayer graphene arose after a meeting in Minneapolis and discussions there and waiting at the airport getting out of Minneapolis with Erisburg and then meeting Mike Zellertal at Berkeley and my interest in Tangsindai Seleneid was aroused by a discussion with Liang Fu. So I have organized my talk around the following question. Why are the quantum critical properties induced these two materials essentially identical to those in the cuprates, the heavy fermions and iron-based antifuror magnets even though the microscopic physics of these materials, each of these materials is quite different. The parameters in them vary by more than two orders of magnitude from one to the other. The order parameters are different but at quantum criticality most famously they have a resistivity which is linear in temperature or in magnetic field for magnetic field much larger than temperature. In the case of the cuprates and the heavy fermions and to some extent iron-based antifuror magnets there are a lot more experiments than in Tangsindai Seleneid and twisted bilayer graphene simply because of availability over the 30-year period of very high-quality crystals and armies of experimentalists and theorists to mislead them. In particular I want to stress that near quantum criticality in the case of the cuprates and the heavy fermions the specific heat is T log T and equivalently where thermal power is measured it is T log T. For example sometimes it's hard to get the specific heat because of superconductivity intervening and not being able to go low enough to the critical point but when measured the thermal power is T log T and I predict that although when thermal power is measured in twisted bilayer graphene and Tangsindai Seleneid in the regime where the resistivity is linear in T the thermal power is T log T. Thermal power unless I'm corrected by the experimentalist here is easier to measure than the specific heat. When known the superconductivity for example in the cuprates and the heavy fermions where it's very well known is definitely T wave superconductivity and trusting my experimental friend Ali Yazani the superconductivity in twisted bilayer graphene appears to be d-wave as well. I think as far as I know Tangsindai Seleneid twisted Tangsindai Seleneid has not yet appeared as a superconductor that's what Abhay Pashupati of Columbia tells me. So just to give you the final answer to this question the answer is that they all have quantum transitions which are described by this statistical mechanical model of quantum x5 model coupled in a particular way to fermions and by particular way I mean that is how the fluctuations of the quantum x5 model are obliged to couple the fermions so the important part is why are these quantum x5 models. So I'm not going to talk about cuprates or heavy fermions or iron compounds I will simply speak about Tangsindai Seleneid and twisted bilayer graphene. Fortunately the kind of order parameter that is proposed by them is not something that I propose the order I'll come to that in a moment but just to show you the quality of the data in the cuprates these are very recent measurements and crystals of lanthanum cuprate and you see this linear in p resistivity and if you reduce the transition temperature from about 40 Kelvin downwards by applying magnetic field you can go down to 2 Kelvin more or less not exactly the same slope but more or less and you see that this is a very high quality compound in the sense that the extrapolated resistivity to zero temperature is essentially zero the same thing when you apply magnetic field where here you have to apply magnetic field on the scale of 40 to 80 Tesla which you can do at the magnet lab and then if your temperature is high enough you find that for example here you're looking at the derivative of the resistivity with the spectrum magnetic field and that again shows you a linear behavior but very interestingly I'll come back to it the derivative of the resistivity with the spectrum magnetic field has a characteristic temperature dependence which I will talk about and contrast this with the experiments by Yao Yi an effort of this group in Barcelona and somewhere else on twisted bilayer graphene this is the phase diagram that they draw and at several specific doping for in particular the new equal to minus 1.1 minus 1.6 minus 2 minus 2.8 and 3.2 you see that the resistivity is from about 50 milli Kelvin to about 10 Kelvin it's linear in temperature I understand that not every group has gotten so many different four different doping at which the resistivity is linear in temperature but when they do get it the slope is similar when they do get it at one or the other so the results are consistent you also see that if you depart from their resistivity approaches t square here is you see this is not too far from from what you measure there it's here is for example this one is new equal to minus 3.7 and then this is 3.2 and you see that the resistivity does change and they have also done experiments in a magnetic field very interestingly the region where you find this resistivity is linear near there is a region in which there is a phase transition to some unknown symmetries and then below this region there is also superconductivity okay and same thing for twisted tungsten disilinear layer these are experiments from Pashupati's group at Columbia and here around some particular doping is the linear anti-resistivity so I'm not going to solve any model there are lots of people who do that sort of thing much better than me I'm just going to tell you why what for example Liangfu told me about tungsten disilinear indicates that it's described by quantum xy model okay and tungsten disilinear is easier to discuss than twisted bilayer graphene which actually will be very similar but even though the order parameter is quite different so this is a triangular lattice and in this triangular lattice there are these points k and k prime on the Brillouin zone and I'm told points k and k prime where there are nodes in the dispersion and I'm told that spin orbit scattering is so high that the k point has let's say spin pointing up and k prime has spin pointing down and spin orbit coupling is so strong that you can regard the z component of spin as a conserved quantity and if z component of spin is a conserved quantity obviously anything that's happening that's interesting fluctuating is in the plane and then I'm told that the most probable expected order is a three sub lattice order of these large semi classical spins lying in the plane and lying the order expected is at 120 degrees to each other okay so here are three semi classical spin vectors lying in the plane each of them associated with a lattice point so I will associate a b c with the three triangles with which this point is associated and similarly everywhere else and then the classical Hamiltonian is simply s i a plus s i b plus s i c squared let's say s is one so take out a number minus three and and the ground state is s i a plus s i b plus s i c equal to zero these three things are lying at 120 degrees to each other full symmetry of the lattice whereas if in the intermediate state you have a cooper fellow then the same word x will give you a dominant d wave attraction and that's very important that that's what hung me up in the case of cuprates because I knew that the self-energy was independent of momentum so I felt it could not be d wave superconductivity till I realized this coupling okay so now we have to talk a bit about the properties of the quantum x y model now after this the the quantum x y model with this particular coupling to fermions is as well solved as the classical uh costly stylist model okay and it's soluble it's it's a big surprise that there's quantum model which without doing any one over an expansion any disorder averaging any Gaussian approximation is exactly soluble and it's actually soluble for a simple and good reason so here is my uh Hamiltonian here are my rotors here are my generator of rotations this this is what I've just described and then I've also described to you this coupling to the fermions okay now it turns out that this model can be exactly transformed into a model of interacting topological defects just as uh costless and hollows did it for the classical model but in this case the uh the topological defects are not just uh topological objects in in space they also topological effects in time and the reason this model becomes soluble is that the topological defects in time and in space are orthogonal variables so you can solve them by uh doing the costless tricks twice okay I'm not going to show you the solution I'm practically going to show you an experiment which is that you can do quantum Monte Carlo on this model and that's just like doing an experiment okay and so this was done again by legion true and what we are going to what I'm going to show you is the correlation function e to the i theta x t x tau e to the i theta zero zero the x is the two-dimensional space variable and tau is the time variable and one of these i should have a plus sign okay uh which should be conjugated with each other and it turns out that this correlation function the fluctuation regime is identical to the and this can be analytically shown to the correlation function of the generator of rotations okay and here I'm going to show you tau times g theta so this is very extensive Monte Carlo the we had access to a computer in Shanghai gigantic computer and this was done on uh 200 by 200 by 300 uh in in and and my young friend legion true was extremely obsessive so we have results for all kinds of things to fourth decimal place okay you see you see this you're varying something here to get to the critical point so tau times g theta plotted as a function of tau to the half not tau as I vary these parameters you see that it is rapidly decaying and then it becomes constant you have arrived at the critical point and then it goes up and and and becomes constant at uh large tau and you can take all these curves and collapse them thereby getting the correlation length as a function of the parameters okay and that's shown here and with that what you discover is that this correlation function close to criticality let's if you just look at it it looks like one over tau times e to the minus tau over c tau to the half e to the minus x over cx now uh when one tries to do it analytically in the leading order rg at the level of uh costlets I I can only get uh an exponent without half but I believe this is a this is the correct result it's also correct result where it's a bit of a pain because you cannot do it's analytic continuation easier to do but at criticality we agree and this is a may I say it's an extraordinary result okay so you are talking about the correlation function which exactly at criticality are going as one over tau times log of space so first thing you observe is that the correlation function is of a product form in time and space okay this is not what happens when you do the hertz moria type models uh because of this everything further is calculable so the three remarkable features separable function of space and time and the other thing you notice is this one over tau hey I can do the analytic continuation of one over tau it is tangent hyperbolic omega over 2t and and uh that is the particular form that my colleagues and I had suggested in 1989 as the phenomenological secret behind the peculiar normal state properties of the cuprate near criticality okay so we finally have succeeded in deriving it okay when we wrote that paper in 1989 I got a quick note from Anderson saying would you make me a co-order okay and I have it in the historical record and I didn't reply to it yeah I suffered a lot for it now the spatial length scale that I've already shown you in the previous few graphs what happened to it this this spatial length scale you can prove is the log of the temporal length scale okay so so essentially the problem is effectively local the dynamical critical exponent is infinity now given that uh you can calculate all kinds of things uh I wasn't smart enough to calculate the conductivity exactly uh this has just been done by uh Hidai Maibashi you could actually this is the first instance where with any fluctuations the Kubo equation has been solved exactly yeah and uh what you discover is that umklab scattering appears as a necessary vertex correction which is not one I used to assume that when the spatial correlation length is essentially local the umklab scattering will give me a factor one so my previous results are modified by a factor which is in between 0.75 and 1 but at least now we have a precise solution of the Kubo equation so that one can have more confidence and what you find is that resistivity at low t over e s is given by appropriate angular average of the single particle scattering rate times the umklab factor the umklab factors between 0.75 and 1 we can put limits on it okay now uh let me quickly speak about the resistivity as a function of magnetic field uh because that will serve as a okay so so let's say that I'm doing the single particle self-energy right the single particle self-energy is now this correlation function which is effectively frequency independent it has some thermal occupation factor and then there's a factor which is the amplitude of the fluctuations okay now I already told you that there is a conjugate object to it and if I have to calculate properties which are related to what magnetic field does I have to find out what is the response now I have a vector field in the problem so immediately the idea arises the conjugate object is really a vortex and those things that I was calling s sub z if they are placed locally they are like vortices because I just can't put them there in the middle of the rest without having charges flow into it and out of it okay the magnetic field is a static object so the objects that I will generate will be proportional to the real part of the susceptibility if the imaginary part is going as hyperbolic tangent omega over t the real part at omega equal to zero goes as log of t log of omega c the cutoff in the problem divided by t and therefore I will get some scattering from these objects which will go as the real part and then just to cut along this question short what one can prove is that the slope as a function of temperature in the magnetic field divided by the slope as a function of temperature will be some factor which depends upon details of the problem times mu bh over kt times log omega c over t and there was some results on a type binding graphene which had already been published by esotov in his paper as a function of magnetic field and when I told them about this log mr yawyi re-plotted the thing as a log and at all those four compositions you see this at these four compositions where there was a linear in t resistivity is a linear in edge and the coefficient of the linear in edge is a log of t they have no point between two kelvin and 40 milli kelvin these are experimental issues you can they they do not have a continuous variation of temperature in a large magnetic field and so this last point is departing and I have various speculations on why that is there are only speculations but over this whole region you see this and then you see that the characteristic upper cutoff that you get is on the scale of 50 to 100 when you do the same thing for the cuprates the character cutoff is about 2500 and again you you fit the log okay so let me just run up and say that when you begin to get an experimentalist producing results like this where you are getting agreement of the logs you have reason to feel a little happy okay so I believe that this problem about quantum critical conductivity is a well solved problem this linear in t and linear in edge and here are my conclusions I have not talked much about I have not talked at all about superconductivity but let me just mention that superconductivity is related inevitably to fluctuations which scatter fermions and I've given you some physical argument of why that particular vertex gives you d-wave superconductivity d-wave in a general sense it will not depend upon how many pockets you have stuff like that and there's this prediction for twisted Braille graphene and for WSC2 that when they measure thermal power it will be t log t thank you okay thank you chandra is questions comment dissipation kernel it goes like theta square shouldn't it be periodic in theta sorry the dissipation kernel you mentioned is alpha omega times theta square why is not periodic in theta and a second question why is alpha at a critical point let me answer the first question because I haven't yet understood the second question so you are you are also allowed a function which is one minus cosine of theta okay which would be if the you you would think that only the vortex like objects are dissipating and not the spin wave part of the problem when and I cannot solve that problem analytically but we can do that with quantum Monte Carlo that was done published result and what to discover I wish I could give you the physical reason for that what you discover is that with the one minus cosine theta the properties remain that of the Galilean invariant 3d x4 of the 2dx quantum x sub model omega continues to go as as k okay so this even though if I put the the coefficient of that 10 times the coefficient of the theta square term the theta square term dominates okay what was your first second question why well there are three parameters in the problem right the the the coupling constant to the sort of like just in term the kinetic energy term and the alpha okay and the critical point is arrived at a particular linear combination of those okay the the actually the phase diagram is more interesting a little more complicated I've told you only of one one object it is the critical alpha that I have emphasized is going like that where this is the so called ordered phase and this is the disordered phase and we are doing criticality going in this direction of this direction so there's a whole line yes there is another question right one quick technical question and one more general question technical question is this you showed up bubble and you correctly said that if you have Keynes the vertices you have extra two powers of momentum uh when you calculated current current bubble but my question is uh internal part normally give you land-out damping which is omega over q why not extra one over q in no no the the the the answer is no no one land-out damping I talked to land-out about it only appears for conserved quantities we are really relaxing the current and and with with currents current is not a conserved quantity at infinitesimal temperature even in the purest system in that case there is no such thing as land-out damping it's a very popular belief that there should be land-out damping always but that's not what land-out said well okay more general question is this uh you said linear and t linear and h uh I'm talking about resist under conditions yeah do you have the full formula for the scaling function of t over h or it's just general statements that in both cases you should get linearity at the moment I have only limits on it okay except that I have I've that the linear and h asymptotically must have this coefficient log t if pressed I can try to calculate calculate the crossover function they are much harder to do and hey I had my 80th birthday in June it's enough it's one of the guys if one of you guys want to do it wonderful okay I know there are more questions from the audience otherwise there are a couple of questions from the chat let me go over there okay so the first question is can you explain the macroscopic origin of the coupling between x y spin fluctuation and the fermion that you propose in the case of twisted balear graphene where does this coupling come from macroscopically and also I wonder if there has been experimental evidence of the loop current flux state you mentioned twisted balear graphene many questions yeah first what is the origin of this macroscopic coupling between x y spin and fermion let me come to the second question but let me clarify is this question in relation to twisted balear graphene yes there is experimentally no evidence at the moment it is just that my very clever friends at Harvard and Berkeley have proposed this order if it is that order it is the quantum x y model if it's a quantum x y model I have the solution for the case of the cuprates it is loop current and we have plenty of experimental evidence for that for the case of heavy fermions it's always the sort of the spin order in the x y plane and for critical fluctuation if I can ignore the fourfold anisotropy and it turns out that for quantum fluctuations I can the fluctuations of the anti ferromagnet and of the ferromagnet are identical even classically if you are if you are an x y model okay the what was the first question was what is the macroscopic origin of the coupling to the x y spin and fermions in this in the case of twisted balear graphene so this comes to can I take a minute on this after minutes okay and as I said somebody fell into the trap and you guys have to pay the price okay so deriving the fermion collective mode coupling okay so so we are we are dealing with problems let's say I have an x y model which whether it's for the spins or for the currents I can represent it as some phase difference which is occurring on links okay and in the case of spin problems it would be a spin dependent phase different then you see my and here here's the important part it's a kind of gaze theory the dispersion of the fermions is in the presence of a static phi x or phi y change this fashion kx plus phi x okay but we are not interested in the static order we interest in the fluctuations so let me imagine that I am slowly varying this object in space delta phi x r delta phi y r let me further imagine that I have x y symmetry although I can do this calculation on the lattice as well what that means is that this object is independent of rotation in the kx ky plane if I if it kx and ky lie on a circle then the change of phase with space delta phi x delta phi y r is one over the model phase and minus phi y phi x times how much I have rotated phase okay you see the rotation is generating it's a it's a it's a generated rotations now my effect in Hamiltonian I can take this by differentiating this dispersion with relation with respect to delta theta so this gamma of k k plus q is the derivative of epsilon with respect to this and you get this and because it's a rotation object even I take the derivative in the k prime direction I have a minus sign with a phi y and a plus sign with a kx what that basically means is that I'm and and because of this property derivative with respect to flux is the same as derivative with respect to momentum which is a velocity which is k and therefore this coupling will immediately give me i k cross k prime okay that's magic this is very simple and I believe it's crucial in the whole physics okay so a second question is from pierce what does your theory predict for the temperature dependence of the whole coefficient at the moment at the moment I've not calculated the the whole coefficient and I plan to do that my young friend our credit sector has just measured the whole coefficient in the cuprates reliably I just learned about this last week at aspirin and I'm planning to to do that what you observe experimentally is that at high temperatures the whole coefficient tends to behave as if it's independent of h I have to see why that's happening I don't know the answer and Marco as a question maybe you can just switch on the marker from the ask it yeah yeah hi chandra nice to see you I have a question concerning the z equal infinity transition you find in some sense your situation is very close to an extreme case in which the psi the spatial part doesn't grow or grows slowly and then at the same time the time correlations are very long range so I was wondering whether this finding could be made even more radical in the presence of disorder in the sense that for instance you could have that psi doesn't grow at all in space and stays fine still you have a transition which is purely driven in the time axis with the psi tau which diverges although psi x still stays finite this could be tested with Monte Carlo did you ever think of this I thought a bit about disorder and being a being a practical guy who doesn't like to work much I look at the experiments first and I find that the slope of the linear anti-resistance in the case of cuprates is invariant when I change the residual resistivity by more than a factor of 10 that discourages me to work hard on disorder let me also say which I didn't point out that you know effort of results looked absolutely beautiful on plotted on a linear scale but he had carefully subtracted out the background resistance okay and in fact the background resistance in twisted bilayer graphene in effort of data which I believe is the best that has been produced on these problems the regime in which you see the linear anti-resistivity the variation is similar to the background resistivity in magnitude whereas the data that I was showing to you in cuprates the background resistivity was effectively zero so I so that's why I'm not worried so much about about disorder okay I think that time is running out so we can thank you again Chandra and the talk and even next talk will be hello