 His first talk is going to be about superconductivity via repulsive interaction Today's the day of lectures by organizers. Yes Tomorrow as well by the way Yeah, so There will be two lectures and They're both under this umbrella of superconductivity of strongly correlated electrons The first one will be a bit about concepts with funny pictures at least some of them and I'll try to go back and pretend that you didn't hear any lectures before on superconductivity and say something in general words and then In the second part of the lecture, I hope that I will have time I will basically be talking about how superconductivity a emerges out of repulsion So this will be the key subject of today and second how it competes with how it competes with Competing orders by which I mean magnetism orbital order charge density wave you name it and So this is what will be today Say today means before lunch And after lunch on the blackboard I will give you some ideas about what happens with superconductivity near a quantum critical point Where the issue of attraction versus repulsion will be just put by hand under the rug So interaction will be by construction attractive. So But there will be another issue superconductivity will be competing with non-formal liquid And so there will always be competition either with competing orders Which is more for me liquid like picture or with non-formal liquid which requires dynamics So this story will be story without dynamics. This story will be story with dynamics Let me start with something which is very crude Not exact, but I like it because it's really Simple concept what we need for superconductivity basically if you look at Standard through the theory for metals standard prediction from through the theory that can the resistivity should remain finite at t equal to 0 simply because Current is proportional to electric field with the coefficient, which is conductivity inverse resistivity And then if you calculate resistivity as long as you have finite lifetime of Excitation means fermions are moving so along just one direction scatter change their direction moving into other direction And if you include finite lifetime of excitation meaning finite time before between collisions you find out that resistivity is finite number and if you replace Hound by Musloff you will find the same and in his lectures. I guess it's the beginning But first of all, this is dissipative current. I mean that So sustained current you need always to borrow energy from the source of electric field Which means that you put electric field to zero and current stops But it has been known Actually before theory of superconductivity came in that if somehow The system of macroscopic number of particles is described by a single Macroscopic wave function meaning that there's infinite number of particles in the same quantum mechanical state Then which is called condensate Then this condensate wave function generally has amplitude and a phase and you use quantum mechanical formula for the current and you find that there is a current proportional to the gradient of the phase and This is not accompanied by any energy dissipation and generally would exist even if you are if Electric field is zero and if you have finite current and zero field, which means that this row is zero Which means that resistivity is zero So The outline of this simple reasoning again a little bit oversimplified of course is that once We have macroscopic condensate. We have superconductivity and at weak coupling. This is pretty much the story Then the next argument is even more simple that you open textbooks and you ask how to get condensate this was Einstein condensation So if you have a bosons, there is no problem bosons like to accommodate at the single level generally with K equal to zero and so you always get mark at zero temperature up to quantum fluctuations Bosons tend to sit at the same level with minimum energy because you can put all of them into this level There is no contradiction with quantum mechanics, but for electrons as we know is formula there is a power the expulsion principle and Two formulas cannot simply exist in one quantum mechanical state Which immediately brings second argument that however if you take two formulas and form bound state of two Formions even not necessary with zero momentum zero momentum is just for simplicity here Bound pairs of fermions of two objects with spin one half has been either zero or one So it behaves as a boson so you prepare and all prepare objects, which then contents again there is a separation between forming a condensate and Sorry forming bound pairs and condensation and this is story about the difference between BCS and BC Let's not go into that business here Let's just say that at the end of the day if we form bound pairs they will eventually condense Which brings of course? Simplifications that we need to pair fermions into bound state if we manage to find the way to pair them into bound state Then you follow this line of reasoning and you get eventually state with zero resistance Now This is the same phrase You go back in time you ask this wonderful Nobel Prize for I guess really probably the best theory in physics in 20th century and you find out that this BCS-Bardin-Couper-Schriefer theory consists of two parts one is Mostly attributed to Cooper in fact is a statement that you don't need Strong attraction any attraction will do the job meaning that infinitesimally small attraction small attraction is enough to create super to create bound pairs because We call it pairing susceptibility is logarithmically single or at small temperature We call it cooper logarithm, but in reality what it means is that in condensed matter physics as opposed to high energy physics we deal with the system with finite number of fermions they have Fermi surface and therefore states with zero energy are States close to this Fermi surface and if you take integration in any dimension in fact I put D3 here just to be in three dimensions then essentially up to a constant it translates into integration Transverse to the Fermi surface which is always one-dimensional if you don't have kf then D3k is D3k k2dk so you basically get out of case k2 or any in any dimension k to power of D minus 1 Very good. So this one part the second part is of course what actually accounts for attraction and second part of this theory was that if you consider exchange by phonons, which is quant of latest vibrations Yes, if you forget for a second about Coulomb interaction you get attraction because There are number of ways to explain this neither of this is exact But story goes like this you have one fermion that creates vibrate creates distortion of the latest Then after some time another fermion come into the same range Read the distortion of the latest and through distortion of the latest two fermions talk to each other and there is attractive They're just finished racing that yeah, go ahead. I cannot hear you. Sorry. Let's try to do it again Electron electron is attracting the positive ions But I'm talking about effective maybe I do not say you but I'm talking about effective interaction between Electrons mediated by vibration of a like it's so How can a single electron can attract the positive ions, which are very massive Only by little bit Yeah, it's let's move Okay, one way or another but This is the story that I don't want to go in details Exchange of phonons give rise to effective attraction between fermions, which for our purposes is this there is a foreign object With respect to electrons and this foreign object mediates interaction between electrons there is of course an issue about electron-electron repulsion with respect to this one and Yes, call it Prokofiev and his lecture probably will touch about what the interplay between coolant repulsion and electron phonon interaction Quite interesting that this problem, which is supposed to solve 50 years ago Is not solved so people are coming back in this year in fact and Discussing again how to properly Deal with interplay between coolant repulsion and electron phonon interaction But putting this aside for a second story is that if you consider this at this foreign object You have a possibility to get attraction between fermions The story in pictures goes like this Tc was superconductivity was discovered in 1913 then temperature was gradually going up and then went up Vertically almost overnight. It was basically a half a year when temperature went from 30s to over 100 Kelvin Probably this is the record Nobel Prize given next year after the discovery Which is unthinkable in other other fields and At this stage we are looking just as a picture Tc went up substantially There was new discovery new breakthrough in 2008 Superconductivity in iron-based compounds. I'm essentially repeating the lecture with Ian Galeas gave and Temperatures are not as high and the cup rates Although there were report about at least one materials which has Tc of close to 100 Kelvin But it's definitely record in terms of nature science communications and all those glossy journal papers that accompanies it So the question is is Tc relevant so far. I just play the game We get strong enhancement of Tc in the slide which I steal from Gersh Blumberg There is a mgb2 phonon superconductor as again Repeating the statement which doesn't belong to me apparently available by mail for 48 years But nobody bothered to measure although there were measurements that didn't go low enough in temperature So this material has Tc of about 40 Kelvin 40 Kelvin is quite comparable for iron at least iron-based Superconductors and first family of the cup rates. So in this respect when says this is high temperature superconductors It's a phonon superconductor I Hate asking people questions because questions means that I know the answer you don't but the record holder at least Before there was this publication Three weeks ago about new record holder which community is now digesting but before that the record holder material Has Tc of about 200 Kelvin under pressure and this is phonon based superconductor So in this respect if you ask which materials have high Tc These are ordinary phonon based superconductors with the caveats that ordinate still means that you need to Include all dynamics of electron phonon interaction and solve more accurately than just BCS theory So here's what I'm going to talk about the question is what is actually relevant and Then it turns out although there is no monolithic No Not everyone is a community share what I'm going to tell you about but if you ask if you take a survey I guess it will be 99% of the community That believes that if you consider cup rates iron pnechtites Routinates the material that was discussed here have a fermion materials organic superconductors The least can continue then electron phonon interaction probably is not responsible Probably means that yes There are people who are still saying that maybe it's electronic modified electron electron interaction And I don't want to present that the problem is solved completely But I'll just give you one example that From the group that worked on electron phonon interaction for years Even if you neglect coulomb interaction and just calculate what Tc would be in iron pnechtites You get one Kelvin you can Twist things change parameters a little bit and get it from 1 to 5, but you will never get 60 Kelvin So it's a huge difference and then on top of this there is also negative effect from coulomb interaction so if not phonons There is also symmetry reasons why phonons in some cases may not Be the case, but if not phonons if there is no foreign body that mediates interaction between fermions and what and The answer is there are not too many choices in this system. In fact, there is only one choice electron electron interaction So this is where I actually start how to get superconductivity out of cool in fact screen coulomb interaction, which is repulsive interaction and There were talks about as I said that electron phonon interaction is relevant Definitely relevant for some physics in this materials, but probably not a glue for superconductivity So how to get superconductivity out of repulsion is the subject of what I'm going to talk today And I go back in time I Will take this to her back to 60s for about 10 minutes and then Talk about what people do nowadays. So the story really started in early 60s of last century due to actually as history shows two independent works one was done by Landau and Pitevsky this is the work I know more about and There was also at about the same time the words by Phil Anderson and Morel pretty much saying the same and both groups said couple of things first Generally, it's a bunch of statements Statement number one that Firmance can form bound state with angular momentum the phones that we discuss the two formulas just attract each other We always assume that there's some constant attraction between fermions if constant attraction If there's a constant attraction then the only possibility to form bound state is with angular momentum zero Meaning no angular dependence whatsoever, but in principle interaction can depend on momentum transfer you can Look at the pairing quantum mechanical problem in fact of bound state of two fermions with arbitrary Angular momentum, which is not exactly superconductivity, but nevertheless for this reasoning. It's okay and What is most important? Is that pairing problem decouples between harmonics with different moment, which means you may have a repulsion in all channels with all momentum except for one and Strong repulsion and other channels will not affect pairing in a state with a given momentum So it's really factorization Which means that there is much less severe restriction on interaction. It doesn't have to be attractive at all Sorry, it doesn't have to be just simply attraction It can have just one attractive component and it will be enough and this is the most important part out of all this reasoning And there is a third piece that if you formally consider large value of angular momentum and ask from where this Interaction come from you find out that large distances are important again. It's a simple reasoning based on quantum mechanics you can calculate which relative distances contribute to interaction with different angular momentum and you find the Largest angular momentum the larger Which immediately brings me the issue that if you go to large distances Then what is screen coulomb interaction? We normally say was you cover potential so interaction gets screened and dies off Exponentially, but there is more at large distances. There is also oscillations and this is Interaction essentially become occasionally over screened and this oscillation are due to robustness of the Fermi surface And in fact these oscillations can off and off called Friedel oscillations and in fact There are people here in the audience who did very extensive studies of charge density wave and Friedel oscillations Have been used in studies of charge density wave. All we need for us is this minor things here That interaction which is generally repulsive positive has some range of distances Where it occasionally gets simply over screened and changes side and so the question then is this Is it enough? Can we just use the fact that in some ranges of distances between particles? There is occasional over screening and get attraction in some pairing channel Which brings me to the story which in my opinion revolutionized The field but was not really appreciated. Well at that time. It's a story by conlat in turn 1965 Who basically calculates this very accurately? What they did may look like just second-order perturbations theory, but they did much more They did much more accurate analysis in second-order perturbation What they said is let's suppose that all regular screening not taking to account Friedel Oscillations is already taken into account. So we have some regular potential u of r We have no idea what this u of r is but we assume that it's repulsive Repulsive means that all components of this interaction with any angular momentum are repulsive Then on top of this there is a bunch of diagrams which account for screening leading to Friedel oscillations and So you get effective interaction which includes this effect of Friedel oscillations You expand this interaction the screen fully screened interaction in angular harmonics and ask the question What happens and the answer is that interaction is definitely attractive at least for odd values of angular momentum With even values it can be attractive or repulsive depending what you put What are the interplay between u of 0 and u of 2 kf in momentum space? But for all four odd values of m. It's a minus full square. So it's definitely The numbers that they get in this paper were enormously small they get TC for d-wave Pairing being 10 to minus 17. So nobody took them seriously Although interestingly enough if they put the numbers in their papers for p-wave pairing They get one milli Kelvin, which is pretty much the same as Superfluid temperature for helium 3 and this was before superfluid in helium 3 has been discovered But from theory perspective, this was really big step forward because statement was if you have attractive interaction Yes, you get most like the s-wave superconductivity for interaction, which is repulsive because of Overscreening due to Friedel oscillations, which is a necessary effect You still get channels which are attractive and you do get superconductivity because pairing problem factorizes between different channels You don't care how strong your repulsive is in s-wave channel if you want to study channel with angular momentum 23 This was in my opinion first real example of what's called superconductivity from repulsive interaction You can take a simplified version of con-latenture. Yeah, go ahead No, no, wait a second Wait, thank you can these two things if you take fully screen interaction fully screen interaction You don't need to go to second order. You just take this fully screen interaction and expand the angular harmonics You get what I said what I said is that what they did they basically said Let's assume that all regular terms are included up to infinite order and give some attractive some repulsive interaction and then accurately considered effect singular effect due to Friedel oscillation you may ask What's the small parameter to do this value of m so their theory is rigorous for large value of m And then you find out that higher order Corrections due to Friedel oscillations are small in powers of 1 over m to 4th power So they took leading terms in the expansion in M but nowadays we break the rules and so more simple version suppose that instead of arbitrary interaction Actually answering to precisely the question that was asked instead of arbitrary interaction on all this rigorous physics, I will just take take Hubbard model and Just Hubbard model as you know it has it's attractive. It's repulsive interaction Sorry, but if it's a constant it only contributes to S wave channel simply because there's no angular dependence between particles on the form of interaction between particles on the form of surface so Just do take this you and go to second-order perturbation theory in you Well, if you go to second-order perturbation theory in you you find out interesting results that to gain to first order in you There was repulsion in S wave channel and nothing in all other channels you go to second-order in you You calculate standard second-order diagrams you find interesting results that all channels No matter even odd doesn't really matter. They all become attractive and the largest is p-wave so in some sense statement here is that if you Take Hubbard model and ask what superconductivity you get the answer will be p-wave If you get attractive in direction ask what superconductivity will be it will be S wave So it makes story even easier In fact, if you go back to Conlatinger and ask what they get in p-wave versus d-wave is they get the same They get attraction strongest attraction and p-wave channel. So it's the same story Okay, I want to talk about latest systems that people start to study study nowadays And then this concept doesn't work fully. There is no theorem in Lately systems that there is always will be superconductivity You may cook up a model that in principle will be repulsive in all channels the reason is you cannot Expand in angular momentum. There is also a discrete number of Artigonal one-dimensional two-dimensional representations and so we cannot say m equal to zero one two three four five You don't have this choice as a result of having no choice of having parameters that goes from zero to infinity There is less option and so I warn you that there is no statement that any system necessary will become Superconductor if it has a regionally screen coolant interaction at the same time At least I don't know any counter example in real systems that the system never becomes superconductor So it looks at any real system always find a way to find at least one attractive channel And so let's try to apply con-latent your reasoning to this Systems and see whether this will give a simple understanding of what's called non-phononic mechanism of superconductivity This is what I'm going to do. Let me start with cup rates. It will be really Level zero cup rates nothing particular compared to very detailed talks that you heard last week So let's just briefly look at the phase diagram Antifermidism is here superconductivity under either whole doping or electron doping See the gap which attracts so much attention over the last 15 years Let's go here or here. Let's go to high doping either Hold up or electron doped systems and just from Experiment this our system with large Fermi surface that satisfy a light integer count Which means that volume of the form a surface is proportional to the number of particles This what you expect if nothing particularly striking happens with the form illiquid So let's assume that these systems can be treated within in quote weak coupling Namely coupling may be of order of one but nothing drastic happens between small coupling and moderate coupling Just can look at this for me surfaces and simply ask where the action should happen So I will try to apply con light the same reasoning as con light integer did but to latest system And you ask simple question. Where's the action should happen? Well, I didn't tell you about but I'm sure that you all know That when you look for superconductivity to get dimensionless coupling constant you multiply interaction by density of states So you ask a question where the density of states is the highest and of course you find this corners as Region where the density of states is the largest? So let's look at fermions at this corners and play very simple game So we include two interactions. Let's take fermions in region one and in region two I will consider spin singlet pairing means that the gap that I have order parameters That I have here at more point K will be exactly the same as point minus K and the gap that I have here Will be exactly the same as here, so I don't need to include four I it's enough to include two points. It will be by symmetry the same as other So we take these two regions in not points regions, and I'll just introduce two interactions one will be within Patch on the Fermi surface. That's called g1 another will be between patches Let's call it g2 in standard notation one interaction that small momentum transfer while another interaction is at large momentum transfer So this is what I put here We need to solve force for the pairing so we do a standard BCS like analysis Solving for we introduce order parameters read them superconducting gaps here and here solve two by two problems This is pretty standard BCS problem with L is the Cooper logarithm notice minuses everywhere here means repulsion So if I forget about the second term for example and try to solve this equation I will not be able to solve this equation because I cancel out data and get one equal to minus something positive Times big positive logarithm, so there's no way I can solve for superconductivity if I don't include interaction between patches But this is two by two problem So everyone in ten seconds can write down what is eigenvalues and there are two eigenvalues for the problem One is the sum of the coupling another is the difference between coupling Convention is such that for the pairing you need lambda negative and you clearly see it here It will be great to change the sign of g1 to be make it negative Then you don't need this guy you solve this problem already So these are two couplings and we'll play the game with two couplings in this example and in the next two examples as well So let's do conlatinger story. What means conlatinger story? I'm not going to expand in angle harmonics I will take simplified version of conlatinger namely. I go to a station with Hubbard interaction and At first order in Hubbard interaction and this important because the same story holds in pnigtides Even if you take five orbital model and include all this fancy stuff about going from orbitals to bends What you find is that if you have online on-site repulsion you You will find that both interactions are you we don't care whether the interaction a small momentum transfer or a large momentum transfer In momentum space Hubbard interaction is a constant in real space is delta function So then both of them are the same and then you see Interesting situation in one channel you have strong repulsion Right, but then the other channel gets zero. You don't know what it is. It's neither repulsion nor attraction zero Very similar to the simple version of the conlatinger in an isotropic case when I take Hubbard U Remember there it was repulsion an S wave channel and zero in all other channels Now I don't have all other channels because of latiss But I still have in one channel strong repulsion and we are used it will be S wave case simple S wave But in the other we get zero So what we do what we now call conlatinger we do second-order diagrams Don't think that it's schematic all other diagrams are cancelling out So there's only one left out of second order So you calculate this coupling to second order and you get interesting result Let's not talk about how to calculate and how to get these results. You find out that this guy Interaction at large momentum transfer becomes larger than direction of small momentum transfer And it happens in all systems that people study no matter what Cooperate sneak tights top graphene that you will see Twisted graphene in this regard everything in all systems. It goes the same way you do perturbation theory You start if you start with Hubbard model interaction at small at large momentum transfer is the same you add second-order Corrections interaction at large momentum transfer gets larger than the one at small momentum transfer Which means that one of the couplings here becomes negative. This is what you need for pairing You see it's very similar to conlatinger original conlatinger story. You renormalize your interaction You screen it if you like and you get attraction in some other channel For the original one, which is S wave you started with repulsion. You cannot do anything. It's just continuous as repulsion You can ask what is the eigen function for this solution? Very easy to calculate eigen function changes sign between one and two Which is completely clear from actually this form. Remember that if G2 is nothing is Zero you don't get a solution how to get a solution make G make delta 2 of Opposite sign to delta 1 and make this guy larger than this one then effectively by changing sign of this guy You change sign of interaction so in some sense you get attraction between delta 1 and minus delta 2 And if this guy is larger you can get a solution, but we need opposite sign So sorry you get minus plus or plus-minus doesn't matter. This is just your eigen function, right? I told you it's been singlet. So what you have here is what you have here What you have here is what you have in this range Let's put it and you immediately recognize this is D wave Why because going over one segment of the Fermi surface you go from minus to plus So you ought to have zero in between we have four zeros and This is D wave So you get D wave right in the spot You don't need to do anything fancy to get D wave painting in this situation and out of all this infinite number of experimental result for Hide for corporate superconductors the ones that earn orders Buckley price is this one gap modulus of the gap in fact Extracted from our best measurements exactly around this arc of the Fermi surface and you see it goes from Maximum to maximum and I remember this modulus of the gap going through zero in between so it's exactly D wave Of course, it's only tip of the iceberg of Cooperate story. There is a bunch of other things this even least is incomplete. There is more physics that have feeling There is a pseudo gap for which every series has ideas sometimes to conflicting ideas from the same series There's a decay in coherence, which got of non-formal physics completely non-trivial spin dynamics, etc. etc. etc, but D wave can be understood simply. I mean that let's separate Different levels of difficulty. D wave I claim can really follow from completely basic stuff Of course, you need to calculate TC. You need to find out why TC is large But this is technical problem. Once you get a pairing symmetry, the rest is how to properly calculate TC for this particular pairing symmetry Let's continue for the next five minutes ten minutes with two other examples One I will just use because I will use it later when I will be talking about Competition between superconductivity and other ordered state. So iron pneigthites pretty much the same as cooperates some Deluded version of cooperates not that well Established mode physics and in view of some me included there's a little evidence for more physics and this materials but Still magnetism You dope either by electrons or by whole there is another axis here Which is called isovalent doping you get superconductivity either way pneumatic phase was subject of Discussions here in one lecture by Raphael Fernandez. I'm talking. I want to talk about Superconductivity and I just put a incomplete list of people who were talking on Iron-based superconductors of course well Raphael was not talking about iron-based superconductors, so I didn't forget He was talking about general pneumatic order so You look at fermiology of this system and of course it's a zoo There are many states are 5d orbitals of irons. There is also p orbitals of arsenic and It's really zoom on the level of say bit probably between minus 4 ev to 4 ev But if you zoom into low energies Then you have set of Fermi surfaces the ones that the plate with and cooperates there was very easy There was just one Fermi surface one d9 So one orbital there here is more than one orbital But out of this orbital physics in the bend language you get circular or nearly circular whole pockets in the middle You get electron pockets as the corners and this wonderful words whole pockets and electron pockets mean simple things Electron pocket means the dispersion is ordinary that you write when you ask to write any dispersion it goes up So chemical potential cuts somewhere and stays below chemical potential are filled Whole dispersion is opposite invert so it goes down So states you again cut by Fermi level states away Stands it low low energies are filled so states inside the pocket are empty So we have these two states and we don't care at this moment which one is electron which one is a hole What I want to do is simplify this story even further What I want to do is to say suppose that we have a place simplest possible game for pnektides Instead of all this orbital physics blah blah blah We just take one whole one pocket in the middle and one pocket at the corner This for just means the same because this is pi pi and this points are added just simply by symmetry So you really have to look into this part Very quickly, why don't we play the same game as before same game means we take two interactions One will be inside pocket and other will be between pockets. So what I did I just replace the word patch by the word pocket same story otherwise and Again, we have two repulsive interactions So we can play the same game solve two by two we get exactly the same story One cup one eigenvalue is the sum of the coupling another eigenvalue is the difference between couplings again Conventional such that we need negative lambda for superconductivity. Yeah, Yuri go ahead No, absolutely not now and in reality if you take real system g1 Inside whole pocket is not the same as that side electron pocket So no No, what you need the question was this if this is g1 and this is g1 star What we have? Well, you solve a little bit your expressions are a little bit more complex You get eventually the same result. Yes. Yes, you get eventually the same result Okay now Particularly right then look I can answer your question in a second suppose that we have hubbard interaction Then we don't care about anything Interaction at any momentum transfer is the same, right? What you really are saying that this size of this pocket is not exactly the same as the size of this pocket So in general interaction should be different, but if you take hubbard you don't care about this Yeah, so let's play the simplest game I really try to oversimplify the problem as much as I can we play exactly the same game as before We do consider hubbard you to first order again one of the couplings is zero another is positive Do second order same story in trap inter pocket repulsion exceeds inter pocket repulsion So then one of the couplings become negative and superconductivity can develop and you can say I'm telling you a completely boring story Because I didn't change anything with only one exception that now there are Fermi surfaces And now this plus minus sign change is not between patches on the same Fermi surface but between different Fermi surfaces and so I get on this Fermi surface plus on this Fermi surface minus This is the amplitude of the gap or if you like face of the order parameter between this Fermi surface And this Fermi surface changes by pi and You may ask is it still d-wave? No, it's not d-wave the reasoning is very simple because this is Fermi surface and circling zero So we take a circle around this Fermi surface no changes in the gap value So in terms of symmetry you make a circle around zero and you don't change anything with the gap Previously made a circle around zero we find four nodes. So that was d-wave now. It's not d. It's s There's nothing changing the only thing that changes is the sign from here to here So this is another s which is called which was termed s plus minus or sign changing s And so the question is what experiments tell you remember for Cooper's there was one experiment that clearly showed that there was no that 45 degrees right this is really an experiment for which buck the price was given now you look at Pnektides First ever experiment done on this angular dependence and you see it's actually almost constant much more sophisticated laser arches on this isovalent phosphorus dop material actually three whole pockets So it's a story not a one by one and you get the gap which is pretty constant So definitely not d-wave the gap does not change if you make a circle around this pocket Which is s wave how to prove that it's s plus minus not Ordinary s well you look at new trans scattering data and Neutron scattering data find very nice sharp resonance peak below twice the gap value in the Superconducting state and people remember story from the cooperates how to explain this peak basic message is very simple You don't need to do fancy physics all you do is a standard BCS physics with coherent factors You look at spin response of a superconductor and you find out that if gap between k-point and k plus pi at the point in momentum at which Measurements has been done if this is satisfied then there is a residual Attraction between particles in a superconductor in a manner similar to how Higgs mod get pushed below to Delta by charge density wave order residual attraction between fermions here push Resonance below to Delta in spin channel So this is nothing but spin response of a superconductor which has this property of a gap It doesn't say anything about pairing mechanism in reality. It doesn't say anything Except for that if you do a standard BCS theory with proper coherent factors you get this result But never the less if it's like this Theorists are saying that you are supposed to have a resonance peak below to Delta if it's not like this You get nothing you can judge by yourself. Yes, there is a resonance and to me this is that probably the best Indication for s plus minus not the only one there are other indications. They're wonderful Sdm data by Hanaguri Interpreted completely correctly as strong evidence for s plus minus for sign change of the gap, but never as historically I guess it was the first So looks like boring story But you get s plus minus and I have nick ties you get d-wave for the cup rates, right? Let's play one more game. It will give us a More interesting result. Let's suppose that we apply the same reasoning to dogged graphene Well graphene as Honeycomb latest made of carbon atoms It's famous for having direct points in the dispersion So we can write down what is nearest neighbor with dispersion if you take nearest neighbor hopping In fact, if you add near second nearest neighbor, nothing will change and story goes like this It all depends on what is the key where is the chemical potential if chemical potential is zero You have direct point right at the Fermi surface if chemical potential goes up or down You start creating Fermi pockets and if you look at this picture this case dirac point if you can see colors here You see a small Fermi surface in each initially appearing. We basically cut here. So we get small six disconnected pockets appearing near all this K points and then as chemical potential goes up up up this Disconnected parts gets larger and larger until finally at some points that they merge and the system Experience what's called one whole singularity and transition from six disconnected parts of the Fermi surface Into one large Fermi surface. You need pretty large chemical potential for this You need mu equal to plus minus t1 with this. So it's a huge scale scale of electron volt looks like science fiction, but By the way, yeah, if you have this particular point their points are six points in the brilliant zone where Different Fermi surfaces merge in one direction split in the other this called one whole point There was a small type of one of the KY's is K one of the K In the dispersion. Oh, okay. Yes, sure. Absolutely Thanks, what is stupid? Thank you. Ah, sorry about this Experiment in fact experiment photo emissions you put Callium and potassium under the suspended graphene, etc. Let's not go to details about which I don't know much But let's look at the figures So you start with the small pocket which you clearly see around Dirac point it grows it grows it grows it becomes more triangular side keeps growing and then you see one Triangle another triangle tends to merge and now you see merging of or almost merging of these two triangles exactly as it happens here So it's reality. You can't dope Question is of course what how good are your data, but you see it visible you see it would make time What's going on? So let's look what let's play our game in this situation So now we have a game play same game. Where is the largest density of states? Well, here no question in one of these points because basically these are one-half points and Density of states just diverges So what's shown here is in red is the Fermi surface right at the point where triangles match when we are going to go from Six disconnected pieces of the Fermi surface into one large Hermes surface in black is a brilliant zone boundary You may ask is there any meaning of this straight line here? The answer is no This is because we only included first and actually Yeah, sorry. There should be one of them through the cakes of course So It's only because we included nearest neighbor if we include certain neighbor you start seeing this portion here But the point survive so let's focus on physics around these points and play exactly the same game with we played What's the difference there is one difference that you can see right here how many how many points? Different points that we need to play Three right how many interactions well again three equivalent points So we have interaction inside patch and interaction between patches and they're completely equivalent So we have g1 and g2 two interactions Three points in which we solve for order parameter. So set of three equations with two interactions We get three coupling constants again the ones that is here is against any pairing The one that is in between patches well may help us if we've properly changed sign of the gap But it very much resembles triangular case right we have triangles So if you get plus minus between the two it's just like ising spins on triangle latiss the third one will be completely Unhappy because it doesn't know whether to get plus or minus. So let's see what the resolution will be so Two solutions, but notice this a b toward degenerate here Not surprisingly I should get three couplings. I have three by three So to our degenerate and there are special reasons why the general is not going to this so standard First order and you this is definitely positive Which means that this is conventional s wave plus plus plus will never realize because of repulsion this again zero and Go to second order and use same story always Interaction at large momentum transfer gets larger than interaction at small momentum transfer nothing else. So you get attraction in two channels immediately both are attractive Solutions doubly degenerate. Let's look. What are the solution? What are eigen functions now for this to degenerate eigen values? Let's look. This is one eigen function for this triangle zero Delta minus delta we consider spin singlet superconductivity. Sorry. This is the second one I'm sorry delta minus delta over two minus delta over two some of this v zero, but these are two different eigen functions By symmetry of course because it's been singlet what you have here. You should get here. So we get this Okay, let's play a game for 10 seconds. What symmetry of the pairing state? How many zeros when you go along this hexagonal? How many sign changes? That's how we check symmetry, right? We go along the Fermi surface and ask how many times our order parameter should change sign I showed this order parameter in the corners. I see that between corners is gradual. So the answer is D-wave 4 right look at this We change sign here. We change sign here. We change sign here and we change sign here four times here is zero one between this two second Third fourth. So both are D-waves. So we have two different D-wave states, right? They're degenerate call them There's a wrong words. They're originally called by us in fact DX y and DX square minus y square. This is a bad notations This is just by analogy with square lattice Don't call them this way call them D1 and D2 two different states Very good. But if I have the generacy between two different states I always ask a question what the system would prefer to develop below TC. Would it prefer to develop? This state this state or combination of the two well for this you just Calculate free energy. You need to calculate this coefficient. There is a standard technique In fact, it can be done really very simply you calculate this two coefficients And what is interesting is that if you take this order parameters and add phases This term depends on the relative phase of these two guys Then you do calculations you find out both coefficients are positive you minimize do 30-second calculations you found that magnitudes are the same on both which means both develop But the relative phase between the two is either plus by half or minus by half It's just because of proper sign of this K2 term Which means that you have two different states and the system develops one of them One state will be D plus ID Where the phase rotates by 4 pi along the furnace surface clockwise and others D minus ID when the first rotate by 4 pi minus clockwise This are the states that this are Spin singlet version of P plus IP superconductor You get it in five seconds You really get it by just doing simple con light energy analysis. You don't need anything fancy But the result is sort of fancy. You get a state which breaks time reversal symmetry. So you get this chiral Superconductor chiral because it's basically one is one component is symmetric with respect to k minus k and now there is anti-symmetric with respect to changing kx to minus kx or ky to minus ky So some superconductivity which breaks both time reversal and parity can be obtained by exactly the same mechanism as D-wave and the cooperates s plus minus and iron picnics We have some hope that in Twisted graphene where superconductivity was found relatively close to one whole point. Maybe this is a paving state But I have hope and there is I guess already three papers on the web saying exactly this that it's this state and That twisted graphene has this superconductivity. We'll see maybe not Now in the remaining 20 minutes, I will do something a little bit more sophisticated. Yes, please You are asking about numbers or you are asking about what happens if I go away from hubbard you and do different in direction Second it's this slide. What happens if I this story is too good to be true Absolutely because where I cheated you I cheated you just let me finish the frame I cheated myself and you in one aspect. I said hubbard model and by saying hubbard model I assumed one thing. I assumed that Interaction, let's put it this way that interaction at small momentum transfer and interaction at large momentum transfer to first order And you are the same so I started with zero my point of departure was zero either in Original conlattinger in all channels with non-zero angular momentum or in our case in d wave s plus minus d plus id You know the three channels. I started with zero then I could go to second order and Say okay to second order on top of zero. I get something negative great But if I ask any of you in realistic situation with screen Coulomb interaction Which one is larger? Then of course the interaction at small momentum transfer is larger think about 1 over q square weakly perturbed by by Screening same answering to your question if you consider you of q but not you which means if you like Interaction on-site second neighbor third neighbor. You will not get attraction will not get zero At the bare level you get repulsion at the bare level and then all the story apparently falls apart Because what I said we do perturbation theory on top of what on top of already repulsive interaction Well, we changed it a little bit, but we need to now overcome Something finite finite repulsion so the questions that I will address is the remaining whatever 15 minutes today and all lecture in The afternoon is Basically, how we think about how to overcome the repulsion in the meantime, of course, this is What we have right now and let's see there two Main theoretical approaches if we assume that nobody who is very strongly for more physics is in the room For people who very much like what physics I'm just not considering it here It's another possibility But two main approaches is this one approach is to say, okay Let's assume that we abandon perturbation theory. We say we don't do with perturbation theory We use some sort of phenomenology Experimentally motivated phenomenology to essentially justify that interaction at large momentum transfer by physical reasons It's already larger than interaction at small momentum transfer how to obtain this we say Well, I'll give you one example in the afternoon how to do it But neither of this example is theoretically justified So in reality is some sort abandoned perturbation theory takes something and pretend that it comes from experiment and use phenomenology to get it then the issue will be Essentially, this would be story about superconductivity near quantum critical point as I said the issue will be yes There is attraction I put it by hand, but then you will see that this attraction fights against non-formy liquid Which tends to destroy superconductivity? What we'll do is 15 minutes today. We'll set. Okay. How about we keep interaction weeks But see whether there is a possibility To go to enhance conlator meaning to go beyond second order and still be in a rigorous basis so in some sense we are searching for a ways to some infinite series of Corrections instead of going to second-order perturbation theory and This is what goes under the word parquet normalization group analysis I'll be mostly referred to what was done originally by Dilashinsky-Yakavenka and Sasha Finkelstein in this analysis But what I want to say is that park that RG exists with respect to the systems that I'm considering in two versions I'll be talking about analytical RG. There is also a large number of people in fact much larger group doing numerical Functional RG so far. There is no single case when this and this give different results, but we'll see What I mean by RG will come later as a word, but what I really mean is this so far Remember what we said we said we start in realistic system We start with Situation when interaction at small momentum transfer is larger than the one at large momentum transfer So we start with repulsion in both channels. Let's look iron-pnektide Simply because formulas look most easier in iron-pnektide case. So if we do this then The only thing we need to look at is this we can say are these the interactions that we need and The answer is no, these are terms in the Hamiltonian Terms in the Hamiltonian is what we call Bayer interaction and they're generally defined Before we integrate out high energy furnace. So these are in this respect Defined in the upper upper edge of the theory which is generally bandwidth However to solve for superconductivity What we need is interactions at the level comparable to for me energy or smaller than for me energy At least in this system If you don't take the zoo of states into consideration just look at the scales bend with few electron volts for me I'm just this is the upper boundary in some system for me. I'm just ten times smaller than this So there is a huge range in between what happens when you want to know what our interactions Here in this region you already need to integrate out Contributions to interaction from all this for months and this is what goes under the name Interaction flow when we go from higher energies to low energies flow means they acquire renormalizations when we progressively Integrate out high energy for months and want to come out with effective theory which only knows about What happens with the interaction between low energies? and This flow of interactions it happens in all channels Including particle particle channel and also particle whole channel. So all channels contribute here And you may say, huh, we are at weak coupling. So why don't we care about other channels? There is only one channel where we know there is logarithm But in fact if you look at iron pneigthites and go very quickly to this remember I told you there is one band dispersion down one band dispersion up this called electron bands Whole band. Let me now calculate simply particle whole not particle particle susceptibility at momentum transfer between these two channels and when you calculate this you find you get logarithm. I Guess this is was the first done by Kildesh and Kapayev many years ago in the context of excitonic insulator but All we need is basically logarithm here what it means It means that if we ask are there logarithmic color normalizations They are from two channels They are cooper logarithm from particle particle channel and there is another logarithm particle whole channel And this means that if we want to include logarithmic terms, we need to include both channels particle particle and particle whole And you may say well, what's the point of doing this with logarithms? Well, in fact presence of logarithmic perturbation theory is a blessing It's a blessing because we can go beyond the lowest order and perturbation theory and the way how it's done Basically, it's like BCS theory of superconductivity is done in BCS superconductivity. You say coupling is weak but Renormalizations are logarithmic and so if you want to solve order by order Realization of the pairing vertex this what's called Cooper logarithmic story Then you sum up terms Coupling next time will be coupling square times logarithm next time will be coupling cube times logarithmic square you neglect here terms Simply G square you neglect here terms G square times Sorry, you neglect here terms Say G cube times logarithm for example So you always neglect terms which are just simply small in coupling constant But there is a way to sum up all these terms namely, you know all this coefficients exactly and this give you a famous The result that tells you that if coupling is this is sign of multiplication. There's no star here If coupling is negative you get instability at BCS transition temperature The reason I mentioned this is that a to say that if you have logarithms You know the trick how to sum up infinite series of graphs and be that the same result exactly the same equation can be written as Differential equation. It's called RG equation, which is this respect synonym to the word differential equation when you consider this coupling as function of running energy Read it running temperature and you write the same set as Solution of differential equation solve this differential equation to get exactly the same results and then energy critical energy at which G blows up is the same as the critical temperature. So viewed like this you find out that interaction Flows from smaller value to larger value as temperature or energy goes down and blows up at some scale So this is one-to-one equivalence Some prefer this way some prefer this way. It's really Now we need to do this when we have two channels particle whole and particle particle when there are logarithms To do this what we need to do is to write down not only paving interactions This are g1 and g2 that we had before Interpocket and interpocket repulsive interaction if we stay only with this one We don't get anything because in fact what I did was equivalent to summing up logarithms only in a particle particle channel Then you cannot convert repulsion into attraction. It will stay repulsive interaction But we also need to include interaction in the particle whole channel because we know that this channel is also logarithmic There are two different interactions both in if you ask what they do they breathe the system close to density wave orders They are not contributing to particle particle channel per se Now what I wanted I want to write down a set of coupled differential equation for all these couplings It looks difficult, but in fact, it's not that difficult to write these equations You just accurately look at all the garrif mical Realizations in the problem solve them and they get this fantastic result. This is still under control my coupling is small and Only it's not small term is coupling times logarithms So what you see is this? This is really solution of this set of four equations This was a bad guy the one that was against superconductivity That was a good guy the one that gave us s plus minus in iron pnektides d wave D plus id whatever so this guy was originally smaller than the one This is statements that interaction at small momentum transfer is larger than direction large momentum transfer But you see what happens under our g at some point they cross and then the system self generates attraction It's really interesting situation. You start with the system with repulsive interaction You integrate start integrating high energy fermions and at some point repulsion converts into attraction Which means that upper H4 attraction is given within the system And as long as this upper H is still on the range when g times logarithms of order of one But G itself is small. You are at weak coupling. You can justify all this stuff And this is really merchants of con-latent your effect and the controllable Calculation because you have more than one logarithmic channel This is exactly how Morris rice and his collaborators in various papers treated cooperates close to one-half singularity This is how several groups both numerical and analytically treated System on a hexagonal lattice near the Read graphene near one-half point. It was always done the same way you start with repulsive interaction. You do Solve these equations number can be larger and you get this wonderful effect when repulsion eventually converts into attraction At the scale, which is still controllable within perturbations here Of course, you can say does it mean anything for iron pick tights now? Let me scare you a little bit There are few of states in iron pick tights I did our G with just one hole one electron pocket in reality if you look at the zoo There are five even if you neglect arsenic neglect oxygen There is a zoo of 5d orbitals So even if you look at low energy states, you actually have two two or in our times three whole pockets This is one iron zone two electron pockets different color is different orbital content of this pockets So you ask about low energy theory how many different couplings you have Previously we have four right two in pairing channel two in particle whole channel the total number is 44 So you need to write down 44 coupled equations if you forget about this guy You get 30 equations 30 actually can handle Claimers that you can also handle 44 But interestingly enough at least numerically what you get doesn't depend on how many couplings you have It's always the same story and then I have very few minutes left So let me briefly tell you what the result is the result is interesting because it has one more aspect than the story So previously the story that I told you before was a story like this you start with repulsive interaction If repulsion original it was zero great duper to second order perturbation you get attraction You don't need to worry much if it was repulsive first Then you need to solve this differential equation to see how all couplings flow you get attraction, right? What I didn't tell you is whether attraction will eventually win over other channels Because hidden in what I said was the reason why we get attraction and it turns out that Attraction was due to push from the channel which wants the system to develop spin density wave So this brings another question Superconductivity or spin density wave or something else. Let me quickly show you what the results are so The question is of course. What is the leading instability? We know that so far there will be attraction in a superconducting channel Question is is it the largest attraction or some something else will happen with the system? And for this we have various options We have magnetism as a possibility superconductivity as a possibility. There was a lot of talks about pneumatic Order, so let's put all of them into start meaning that we ask In which interaction we have in which channel we have the largest interaction and we ask who will come eventually to the finish And just very quickly to repeat initially what we have for spin density wave We don't need attraction repulsive interaction give rise to stoner criteria for spin density wave So repulsion is good for spin density wave in this respect converting words Interaction spin density wave channel is actually attractive meaning system wants to develop spin density wave interaction and superconducting and I didn't tell this pneumatic channels are repulsive in the process of our G That's we know Interaction and superconducting this we told and I didn't tell you also pneumatic channels for this you need to solve this 30 or 44 RG equations In fact, we also becomes attractive and both attraction are due to push from spin density wave channel This is the typical result at the end of the day what you really need to do so far We wrote RG for the cup rings then you need to go to second stage RG and write differential equation for the vortices From that you write differential equation for susceptibilities to make long story short. This is the result the Rule of the game as you calculate susceptibilities in different channels the channel in which susceptibility Diverges with the largest exponent is a winner. Why nobody really knows it just idea that if susceptibility diverges Strongest then probably the system will develop this order to be honest with you nobody checks this accurately But let's play this game. This is a magnetic channel. This is a superconducting channel and this also channel with orbital order I don't want to go into details except that the point gives this originally magnetism as a winner tendency toward superconductivity simply because we have repulsive interactions Superconducting and orbital channels are both repulsive Now let the system move meaning let's integrate out contribution from high-energy fermions this blue guy pushes red and also green up meaning this Interaction that leads to spin density wave pushes interaction between patches which is good for superconductivity and also good for pneumatic order and you see what happens at the end Red and then green actually wins. So depending on where you stop your RG It can be either Superconductivity or actually pneumatic order both possibilities are there But what is interesting and this is for this you need to really to go to two-stage RG that It happens quite often in politics That one interaction Attractive interaction creates two other attractions and these two are the attractions combined to create negative feedback effect on the source so basically what happens is that result of the source eat try to eliminate the source and This is exactly what happens here at the end of the day these two guys Superconducting and pneumatic channel when become attractive they give negative feedback on the channel that makes them attractive in the first place and As a result magnetic substitute doesn't diverge So why is it important? Well, because there is this wonderful material iron Selenium You probably have more than necessary about this material. There's a large number of talk about it Remember what happens in this material is here. There is a pneumatic order. There is superconducting order where it's magnetism no magnetism So I guess I'm done So let me very quickly flash conclusions conclusions so far is that a To get superconductivity if you start with Hubbard model is just and you get it You get it just doing second order perturbation theory and you seem to get right pairing channel for at least three classes of materials But if you want to do something more sophisticated You go and look into what happens look first of all why You get attraction because it turns out that in spin dense in interaction spin density channel In fact pushes Interactions at large mark pushes up interaction at large momentum transfer and at the end of the day give attraction to superconductivity but if you want to do more sophisticated way you find out that There is a feedback from Superconducting an orbital channel that tends to destroy the source destroy magnetism Which means simple thing that superconductivity or orbit or order happened before magnetism the system Before developing magnetism the system always develops either superconductivity or orbital order Okay, let's stop here and second lecture will be on the blackboard. Okay