 Okay, let me start by expressing my gratitude to the organizers for being here together in a beautiful Trieste. I'm really having a good time in this beautiful town. And what I want to present to you today are results done in collaboration with Igor Herbert and Vancouver on superconductivity and anisotropic non-familic with behavior in 3D luchines or semen metals. So I will explain all these notions during the talk, so don't worry. And you can also find them online in this publication, and we are also writing up the newer results. And I also want to say that I'm supported by the DFG, which is really great. Okay, so the quick outline will be, I will tell you a bit what quadratic bandtouching is, and then I will divide the talk into two main parts, which is about the superconductivity and the non-familic with behavior in an anisotropic system. Okay, and a currently very active field of condensed metal theory is semi-metallus with a Fermi point where two points would touch, and then you have complicated bandtouching like that, but the low-energy physics are described by these touching points, and if they turn out to be linear like here in graphene, you could employ Dirac theory to describe the materials, or even rather recently like Tandulum arsonite, and why semi-metals have been realized in experiment, all the signatures have been very side, so it's a really very impressive, very fast-paced field of solid state physics. And okay, well if you have the linear band crossing, so a natural point to look at would be a quadratic bandtouching, and for instance here I have the band dispersion of cretin, so it's a sort of tin at low temperatures, has dispensed dispersion, and if you zoom in here you see ah-ha, they touch power-boiligally, and what you would also for instance have it in a mercury tellerite, and the material I'm going to show you on the next slide, and so these are described by the Dirac Hamiltonian, or Weyl Hamiltonian, you call them Dirac and Weyl semi-metals, so we kind of suggest, since this is described by the Latinja Hamiltonian, to call them Latinja semi-metals. And as a word of warning, these are three-dimensional materials, so it has really nothing to do with a Latinja liquid in one day, it's just a name. Okay, so a very interesting class of material, or at least to me it's interesting, I can't tell about you, is pyrochloriridates. They have this structure, the formula, so you have a iridium sitting there, which is a 5D transition metal element, so it has a large number of nucleons, which means strong spin orbit coupling. So you have a strong spin orbit coupling, and then what can happen if you have your band structure, and then one of the bands gets inverted? This is what you need to get this lower band of the quadratic band touching point, and it has, so and then you have some rare earth atoms, and the pyrochloriridates is essentially this, so you take an FCC lattice, and on each side you put a tetrahedra, and on each of these tetrahedra points you put an iridium, then you get that, and well, it's sufficiently complicated, but it works, and then the rare earths, you know, these are those of the funny names, so you have proceudemium, neodymium, and then you can change this material, the other part always stays the same, but you change the rare earth ion, and then you can study how this material class evolves, and for instance for proceudemium 227, they are measured with arpers, and the occupation of states that this quadratic band touching point is really there, and you see it here, the temperature you also can kind of guess, the upper band has also been shown for the neodymium class, and there's also DFT calculations which kind of back up that these materials really have a quadratic band touching point, because of the strong spin orbit coupling, and also if you do a simple density of states calculation you see, short range of the actions are screened, but the colombian actions are not screened, and this is also an important aspect of these materials, and you also see the effective mass is six times the electron mass, so that's quite substantial, and what I find really amazing, a phase diagram is the one taken from this review by Leon Ballens and collaborators, so what is plotted here on the X, on the Y axis is the temperature, and here you have the ionic radius, and now they just change this rare earth metal, so it's really for me as a cold atom sky, it's just amazing to engineer these materials and look how the phase diagram changes, and you see that proceudemium, it is in a metallic phase, whereas all of these guys, they have a phase transition to a magnetic phase, and it's actually an all-in-all-out magnet, which is a strange configuration on these tetriles, so it's certainly an interesting class of materials, so you would like to understand a bit more about these quadratic band touching systems. Okay, so you see these are really real materials, I'm not just doing this as an academic exercise, so let me start with a little comparison, which might be helpful for some of you, so there's a system which has a similar dispersion, which is ultra-cold atoms at a Fesbach resonance, so it has a quadratic dispersion, so it's a non-relativistic theory, but somehow the upper branch is missing, so here this is the Latinscher semi-metal, two bands touching, chemical potential goes through, so the low energy physics is dead, and this would be ultra-cold atoms at vacuum, you have zero chemical potential, so it's not filled, other one is missing, so what is happening if I look at S-wave pairing in the particle-particle channels, so which would be the superconductor, the superfluid precursor, for the ultra-cold atom system, well, as has been found out by Sebastian Diel and Chris of Federation, also Satsdev and Nikolich, essentially at this vacuum problem, what you have to look at is the renormalization of the self-energy of the fermion, which is this fermion boson loop, and the self-energy of the bosons, which is the fermion fermion loop. Now, since this band is missing, this diagram, it is cyclic, so you can close the frequency integration in the half-plane of your choice, and it vanishes, so this diagram is strictly zero, which tells you that the fermion anomalous dimension is zero, and the dynamic critical exponent is strictly two, exact statement, and then you can actually show that also to all orders in epsilon, and also exactly, because it's a few-body problem in the end, the anomalous dimension is exactly one, so that is a cold atom case. Now, you see this argument that you can close this in the other half-plane, relied on the fact that there are no particles with negative energy. This now changes in the kinetic band-touching system, and in fact, you, of course, then the computation becomes more difficult, you have to employ approximations, like for instance, an epsilon expansion close to four dimensions, so epsilon will always be the difference to four dimensions, and in the end, we extrapolate to three spatial dimensions, so epsilon equals one, you know, the epsilon expansion, so we looked at this problem in this paper with Egor, and you do indeed find all the things you expect, so it looks roughly the same as the cold atom system, but of course, you have a non-fermiliquid behavior, so by non-fermiliquid, by the way, I mean anomalous fermion scaling, and a C, which is different from two, and of course, the bosons, it's kind of mimicking this system, but it's more intrigued, in principle, to go to really three dimensions, what would have to use, for instance, the FHG, so to resolve that issue, or higher order epsilon, or whatever you like. Okay, so that is this analogy, so now let me get a bit more to the actual things you compute, so the kinetic band-touching system in three dimensions is described by this first quantized four by four Lattinger Hamiltonian, so this H is a four by four matrix, and so how can you make a four by four matrix? Well, the unity matrix, this would be a good candidate, or you take these spin three-half matrices, they are four by four, and now you cook up something which is gotatic in the momentum, and Lattinger did this, and you can describe it like gallium arsenide, it has not the inverted band dispersion, or like here Cratin, it has an inverted band dispersion, and the Hamiltonian depends on three parameters, which are Lattinger parameters, alpha one, alpha two, and alpha three, you see these are the only ones, and you could look up in a book, or some tables, experimental tables, like for instance band-masses in the 111 direction, 110 direction, or stuff like that, then you could compute these numbers, or somebody measured it for you, so they are principally known, maybe not for all materials, and you see this is quadratic in P, this is quadratic in P, and this is certainly all quadratic in P, that's a funny term, it's Px squared, Jx squared plus Py squared, Jy squared, that's something sort of a bit out of the box, and in fact, if you look at the first two terms, they are rotation invariant with respect to spatial rotations in three-dimensional space, if you make a spatial rotation of a coordinate system, well P squared is a scalar, and P times J, because the operators get rotated as well, this is also a scalar under SO3, but that guy, depending on the difference between alpha three and alpha two, these numbers you look up in the catalogue, this guy is only invariant with respect to cubic rotations, which are sort of rotations by 90 degree, 180 degree, and in the end it comes down to permutations of the axis X, Y, C, so if you change X, Y, C, exchange X and Y, for instance, this term is certainly invariant, but it's not a scalar under the full rotation group, and whenever I say anisotropic, I mean this cubic invariant, you don't have a full rotation symmetry, you only have this cubic rotation symmetry, this will be a reoccurring theme now, in everything I do, so for quantum field theoretical treatment where you take this first quantum Hamiltonian, sandwich it with a four-component fermion, psi to the power of psi, perfect, and then you can actually rewrite it in terms of some Dirac matrices, gamma matrices which satisfy Clifford algebra, I don't want to go into detail, it just makes computations easy, I think you can, you can guess that it's very nice to have some Dirac matrices, and then the Hamiltonian has these structures with alpha one multiplying the p squared, this is constant, we just set to one later, and here you have this difference which is again the anisotropic term, and now I introduce two running couplings, so this guy I want to set to one in the end, so this X multiplying the p squared, this is the particle hole asymmetry, it tells you that the upper band is a bit more flatter than the other one, it will always be irrelevant for this talk, so just recognize X as particle hole asymmetry, you don't have to care too much about it, delta is more interesting, that is the difference between alpha two and alpha three, this is the parameter multiplying this anisotropic term, if you have a non-zero delta it means that sort of this material breaks full rotation invariance, okay, so now let's add some interactions, for instance in the superconducting channel you can show through a suitable fields transformation that if you just have an attractive contact interaction, so psi dagger psi squared, you generate a S wave superconductor or this gives you an instability to an S wave superconductor, so you have the kinetic term I've just shown you, complex boson, since it has linear frequency dependence, it's complex, so this is multiplied by a term Y, and here this is in fact an S wave term because you can show that this is the time reversal operator, so you're pairing a particle with its time reverse partner, so this is an S wave pairing, so you can show all that and you can read it there or just ask me, so this is the model and now we look at the Archive flow, so we did the simple this momentum shell Archive, so if you if you want the sharp cutoff, and then the first interesting finding is that this X, the particle hole asymmetry, and this Y, this frequency dependence of the boson, they both go to zero but in oscillatory manner, really in epsilon, so it's this this is really perturbatively exact if epsilon is small, so it's not some artifact of an extrapolation, so that's a quite good behavior and I think you don't have too many models where you find actually something like that in an epsilon expansion, even more interesting is the behavior of the anisotropy, it flows to zero but the flow is exceptionally small, so if you look at it close to the fixed point zero, it's minus 2 over 55 times epsilon delta, so it's in extremely flat directions, so for all practical purposes it's kind of constant, so it's exceptionally slow, typically people would say from loop indications you don't get a factor like that, but you really do, and so yeah okay, so keep that, I will later come back to this, so it's exceptionally slow, this flow, this flow, and yeah it's kind of a finding here, we did it in the Taylor expansion in delta, but it will now come back, where we did it for arbitrary delta, and this still persists, it's always an extremely slow direction, it's like a tuning parameter of the theory, which is very interesting I think, okay so this is the superconducting quantum critical point, nice little exercise, now let's go more into the direction of these pyrochloriridates, which have a magnetic transition, which means repulsion, and also I told you, well you have this long range repulsion, which is not screened because of the density of states going like the square root of energy, so the Hamiltonian or the Lagrangian you would like to look at is, well the stuff I've just shown you, coupled through some photons, so it's only the electrostatic photon in this case, and this is the photon propagator, and up the course of showed in a really ingenious paper that this is an extremely simple example for non-fermic liquid, because why is that? So this diagram, it is non-zero due to the same mechanism because you have these two bands, so you cannot close any loop, so it's non-zero, it's the fermion self energy, so you immediately get the fermion anomalous dimension, but since the photon is not frequency dependent, this guy does not renormalize the coefficient of the frequency terms, so C is just 2 minus eta, so it's also not equal to 2, and this guy is also not at all zero, the particle hole bubble, this gives you a renormalization of the photon wave function, which is a charge renormalization, so you get a fixed point at the finite charge, you get all the non-fermic liquid stuff, extremely simple, I challenge you to cook up something more simple than this up because of non-fermic liquid, so I think it's really nice example to do that, now you might think, okay I've shown you these materials, they have this quadratic band charging, Coulomb, so is it really a non-fermic liquid? Well actually it is not, as shown by Igor and Lukas Janssen whom you've seen on Monday, so what does it come about? So it's nothing wrong with these equations close to four dimensions for instance, well what happens is that you have these long range interactions, so photons hanging around in no short range interactions, but short range, so short range interactions are screened in the beginning, but they are generated during the Archie flow through this box diagram, so this is only you cover couplings to the photons, so this guy is certainly non-zero, but once you have it you also get this triangle diagram and this Fermi-Fermi coupling, so short range interactions, even if they are absent in the beginning, they get generated, now what happens in the plane of the short range couplings, here exemplified by G2, but there's actually two of them, so just G2 represents all the short range interactions and here's E squared, you have the Gaussian fixed point, you have apricots of the non-fermic fixed point which is completely attractive, you would always flow into that, and then you have here a quantum critical point, so it's not really interesting to you because it has a relevant direction too much and well it couldn't care less, but what happens is that as you lower the dimension from close to four dimension down to three dimension, these guys move closer to each other in the event, eventually collide and annihilate each other, so you see it's a very intriguing effect, you get rid of the apricots of scenario because just the fixed point is gone, it's a rather robust statement and then you can compute at which dimension this happens and then Igor and Lukas found out this happens above three dimensions, slightly above three dimensions, so this is the fate of the apricots of non-fermic fixed point for delta being zero, now you might ask what happens for finite delta, this unisotropy parameter and then you can write down a flow equation for this delta and it looks the following way, so it's one minus delta squared, so the way I've defined it, it can only be between minus one and one, so it's now too late to show you, but believe me, so it's between minus one and one, zero is the isotropic case, so this is the flow equation, here I've plotted it, you see three fixed points at minus one, zero and one, so here only the E2 doublet would contribute to the Hamiltonian, here only T3G, triplet, don't really have to care at this point, so you have a fixed point at vanishing anisotropy, you have a negative slope, so apricots have realized that the fixed point can only be stable at delta being zero, but look at the numbers on this plot, this is three percent, that's the maximum you get, it's an extremely slow flow, so it doesn't really matter where the fixed points are or whatever they are and where you start, delta is almost constant for all practical purposes, so what I will do in the following is just, I assume delta is a parameter of the theory, I don't know, a temperature or something like that, just a dimension or flavor number, it's just an expectation of the RG flow and I solve it for all values of delta, then you can see what happens to the beta functions and it's a nice deformation of this theory, I've just shown you of Ego and Lucas, okay, so for instance what happens to the charge fixed point, well you find the apricot of fixed point and also this anomalous dimension I've shown you, they survive for all values of delta, so at least in the epsilon expansion and you see that actually the value of E squared, the fixed point value, it actually gets smaller as you have more anisotropy, so it's approximately disparabular, so in fact the fixed point gets weakly coupled if you are more anisotropic, so you sort of lose dimension if you get more anisotropic, makes sort of sense to me, but maybe not really, so let's kind of, let's outcome of the calculation, okay, now for fermion interactions, now we have to see how many are there if you have an anisotropy, so let's first look at the isotropic case with delta being zero, so how many for fermion couplings can you write down which have a definite transformation property under rotations, so by a tensor of rank n I mean something that transforms as a tensor of rank n under spatial rotations, SO3 spatial rotations and then the unity matrix, this is this guy, this is a rank zero tensor, well okay, then the ji, these are the three spin matrices, they transform as a vector, now if I rotate my frame, they rotate as a vector, they have three components and I call them magnetic order, so all of them are four by four matrices, then we have actually five gamma matrices, so I've already shown them to you, they are actually the five components of rank two tensor, and then you as a total basis you need 16 operators, you have seven left and you can in fact show that these are the seven components of a rank three tensor, so this is a funny guy, so this is related for instance to this all-in, all-out order on these pyroclore lettuces, I won't go into this, in the details here you just have to believe me there's a rank three tensor, well it's a funny object, I suggest to call it magnetic order because it's sort of a magnet which doesn't point anywhere, it's just a magnet which singles out a permutation of the axis or kind of a direction of it, but it doesn't point anywhere, like in a magnetic order, and you can actually show that there's no rank four tensor as a result of the Cayley-Hamilton theorem, and rank five and so on, no question, so you get two independent couplings because you have two third-side entities, and so it looks like a mass but in the end you have two couplings, that's sort of treatable, now this was under rotation symmetry, you can classify them in these four groups, now if you break rotation symmetry you have this classification but a subdivision of these sectors, for instance this rank five components they split up into two plus three, seven components they split up into three plus three plus one, and you get another nasty one, so what you get for arbitrary delta are actually eight couplings, so you get the charge density wave, you get these EG and T2G doublet and triplet which are made of the gamma matrices, you still have the J vector, you get the W vector which are the first three components of the W, you get the W prime which are components four, five, six, you get W seven which is all in all out order, and you even get this quasi thing between J and W vector, so these are all the couplings you can generate and you will generate when you start with this up because of Lagrange and I wrote down, you start with Coulomb, you generate all of these guys, but now you have eight couplings but you also have five fields identities, so after all you have only three independent couplings, for instance take the first three of them, so that's kind of a nice thing, because in the beginning you would think just gets more complicated but three couplings, okay I can handle that, now you do just exactly the same program, you start with the Coulomb interactions, generate the short range interactions and see whether this fixed point collision is still there, and what you find is that the fixed point collision scenario is still happening all of this, so you get a bit more fixed points and stuff but they don't influence it, but you can lower the critical dimension, you know, so this is a plot of the critical dimension where this collision happens, so the non-familiquid fixed point is gone, as a function of delta which can be between minus one and one, so this would be strongly unisotropic, so it has sort of a maximum at delta being zero, it's remarkably symmetric this curve, for a reason I don't know, and then about 60% unisotropy you cross three dimensions in this calculation we did, and you can understand it sort of by this fixed point of, this apricot of fixed point being drawn to recoupling, so it's like you're reducing your dimension, so you sort of, if you have enough unisotropy it's like a large N, it's like a small epsilon, so you get it back, so this is kind of a nice picture, you get back this non-familiquid fixed point and then, well an experiment, you are close to three dimensions, so you can expect at least over some range to see this scaling, and of course one would have to go to higher loop orders to see whether this persists, but I think it's going to be roughly like that, okay, and soon it will be also online, okay, thank you for your attention.