 Okay, first of course, let me also thank the organizers for giving me this possibility here to present our results and I will talk about a study that I've done in collaboration with Ritik Singh, Maxine Kudas and Andred Chubokov and Yeah, we investigated the interplay of different types of orders and iron-based superconductors so Let me start with a typical phase diagram of an iron-based superconductor here at the example of beryllium iron arsenide and Here we typically have a striped magnetic phase for zero doping which is sketched here Stripe means that the spins are anti pheromagnetically aligned in one direction and pheromagnetically in the other Then up in doping we find this superconducting dome with Cooper press and on top of the magnetic transition There's the so-called pneumatic transition where the four-fold rotation symmetry of the lettuce is broken down to a two-fold one and This can for example be explained by a so-called ising pneumatic order in this case the structure transition is due to the Stripe magnetism which not only breaks the spin rotation symmetry But also the lettuce symmetry due to these Different alignments in the both directions and it has been shown in many papers, but for example in this Reference that this transition might come ahead of the magnetic transition Okay, then here are two other phase diagrams of iron-based superconductors when one is for on the Left the one is for lithium iron arsenate and on the right for iron-silanite and This time it's a function of pressure, but I would like again concentrate on the parent compound For zero pressure and in lithium iron arsenate we find they are a superconducting transition so this is the critical temperature of the superconducting transition and no magnetism appears and Also for iron-silanite. There's a superconducting phase at zero pressure or zero doping and on top of this this orange phase is again the pneumatic phase the structural Transition which is here Appears for a very large region from I think eight to eighty five Kelvin and In this case, we don't have this striped magnetism which could explain this order and Then people have argued or assume that this structural Transition is due to orbital ordering where the different densities of the iron Orbitals are occupied differently Okay, and this leads me to the following question questions that I would like to address in my talk First of all, can we obtain superconductivity without the magnetism in the parent compound? Then what is the origin of this pneumatic order and there we have these two possibilities or mainly iso pneumatic order where this orbital order and Then We would like to address if there's a connection between all these different phase diagrams and Can you find kind of a universal description for the iron-based superconductors? Okay, at least one thing they all have in common which you can see here They're shown different compounds of iron-based superconductors And They all have in common this plane here where the red atoms are iron atoms and the yellow ones are nitrogen or kaigogen atoms and yeah, these iron atoms are arranged on a square lattice and then the Nitrogen or kaigogen atoms are arranged alternating on top and below this plane Okay, and we will also now it is assumed that this plane is responsible for the physics that we want to describe So we will also concentrate on this one Then one can have a look at the band structure Calculated from the orbiters or a tight binding district description for this plane and here you see on the left the Resulting Fermi surfaces in the brilliance and on the right the band structure Yeah, and here we have two whole pockets at gamma and M and two electron pockets and Yeah, we would like to have kind of a simplified description for this Which is why people has have often considered a so-called band model where they Restricted themselves to these Fermi surfaces and yeah, describe the bands to We produce these Fermi surfaces and On the left this would mean we restrict ourselves to this small area around the Fermi surfaces This we would also like to do but in addition we would like to account for the arbitrary degrees of freedom so that we can for example Investigate this orbital ordering and therefore As you see here, maybe there are only three colors and the colors encode the different orbitals So we will only include or we will not only but we will include three orbitals Which are important close to this for the slow energy model Yeah, and these orbitals are xz yz and xy iron 3d orbitals and Two of them xz and yz are connected by this four-fold rotation symmetry Okay, on top of this we then add interactions A starting point for our description They are I depicted a few of them here first one you won and We derive them Kind of are similar to a geology So we account for all symmetry allowed for Fermi couplings in this model and Yeah, these are two other examples you three you two and If we collect all of them in the full model we obtain 40 couplings Their starting point is given by the on-site interactions where you and you bar are the typical density-density interactions for Electrons in the same orbital and in different orbitals and then we have pair hopping and exchange interactions J and J prime Now of course one can ask the cave. I have 40 couplings. This is not really a simplified model But we will see that during the RG flow the relevant processes will be singled out and Only a few couplings will turn out to be important Yeah, let me again stress the advantage of this approach that we account for these orbital degrees of freedom They are as I said for example important to describe this ordering orbital ordering where the densities of Xc orbitals and Yc orbitals at the pre-horizon origin are occupied differently This has for example been observed in experiment in an AAPS experiment as example that I show here and It it is shown or as they splitting of the two bands Here which are degenerate for the four-fold rotation symmetry and then split due to these different occupancies and then also these circular Fermi surfaces Change to such ellipses Which they claim to see here Yeah, let me mention another type of order where this Is important it sketch here, but I don't want to go into detail They call it orbital antiphase s plus minus superconductivity Where the orbital degrees of freedom play a role for the superconducting gap structure and they they claim that This explains the superconducting gap in lithium iron arsenide Yeah, but so the message is orbital degrees of freedom are important Then with this setup we will perform a pocket RG study Which is kind of similar to the patching FIG scheme so We also integrate out high energy degrees of freedom. We start with our model as As model at the we then integrate down Thereby we include these one loop diagrams or particle particle particle whole diagrams Which also talk to each other in this way the the approaches unbiased and Yeah, they all diverge logarithmically and we will also only include these logarithmic contributions Yeah Then with this we can calculate the effect of couplings in the system And use them as inputs to calculate the susceptibilities for these different orders so that in the end we will decide With the help of the susceptibilities which order wins in the largest or the diverging susceptibility will make the game As I said it's similar to the FIG patching approach But we will the difference was mainly that we choose our couplings by symmetry and not by these patching schemes and Yeah We treat all these disabilities on equal footing and it's analytically feasible Then let me say a few words about the hierarchy in this of energy scales in this approach Yeah, here are two Yes, the energy scale is depicted and here this logarithmic energy scale, which will be our flow parameter And our equations are valid in this red Area we will start around the bandwidth or at a scale of the order of the bandwidth and then flow Go down to lower energies and the Fermi energy then cuts some diagrams and From here on the instabilities Evolve in an RPA like fashion so later series Should be this should describe the flow after the Fermi energy Okay, so this means if the instability there that we observe their ordering Could occur is behind the Fermi energy smaller than the Fermi Smaller than the Fermi energy, sorry larger than the Fermi energy Our description is valid and we can see Which were takes diverges the fastest This is the instability that we look for on the other hand if we had so this is depicted here Then the second case is that the Fermi energy appears before the scale of the of the Instabilities and from this scale on RPA physics takes over that means that probably the Vertex that has the largest Contribution which is largest at the Fermi energy will make the game in the end Which you can see here so the spin and the two wave vertex is larger at the Fermi energy and it will then win the competition and Then there's this third scale the Imperfect nesting scale If it's small, it's an irrelevant perturbation if it appears at any Before the instabilities it will cut the particle whole flow Okay, so We've performed this analysis and this is the result we observe a flow through strong coupling here depicted for do two different couplings you want and you want and and the signals and instability of our description the divergence of these couplings appears in an Universal way in the sense that Flow such that they have fixed ratios at the end of the flow And this is also called fixed trajectories if the couplings reach this regime to determine these fixed trajectories we reformulate our equations In terms of the ratios of the couplings these gammas then one of We sing it out one coupling which then diverges like this and Determine the flow equation for these ratios and since they become constant at the end of the flow We can then solve the fixed point equations for these ratios and also Determine if they are stable If we do so we find these results we find four different fixed trajectories and they are reached as function of the starting Interactions these on-site interactions and quasi particle masses here. I show the regimes of the Different fixed trajectories as function of on-site interactions between in the same orbital and between different orbitals and Yeah, many there are two different classes of fixed trajectories one Is depicted here in this case the flow is effectively to a three-pocket model There are all interactions with this Two whole pockets at the origin become sub-leading and only the interactions between these three pockets are relevant and the other case is a four-pocket fixed trajectory where the M pocket hole The M point hole pockets Become sub-leading So we can describe the system effectively with these four pockets Okay, so if you have determined these fixed trajectories we calculate the susceptibilities in terms of the couplings or the solution of the couplings that we've determined Yeah, here therefore we define Such vertices gamma for the different fermion bilinears here if sketch run for a spin density wave bilinear between and hole and an electron pocket and the Susceptibilities are then given by the square of these words vertices these bilinears and We find that the vertices scale like This critical scale minus our flowing scale to some exponent beta So they also diverge, but if we integrate then this equation for the susceptibility We find that it skates like this like one minus two beta and this only diverges as If beta is larger than one half if pictures are smaller than one half we will find that the susceptibility Will remain finite at this critical scale and this order won't develop Okay, this is what we then Calculated for the superconducting and magnetic instabilities These are the these exponents beta and we see that in both cases for four pocket and three pocket models The the number is smaller than one half And for the superconducting instability is larger than one half So this means that if our flow is well it down to low scale so the fermion energy does not cut the flow We will superconductivity will make the game you can see this here This is the susceptibility in both channels of spin density wave and superconducting channel and you see hopefully that The spin density wave susceptibility remains finite whereas the superconducting susceptibility diverges Yeah, then we can determine the Superconducting instability the gap structure and it's of s plus minus types. This means that The gap on whole pockets has a different sign than the one on the electron pockets and this orbital anti-phase Superconducting state that I mentioned earlier is only subtleties Okay, and furthermore Although we do not solve some kind of gap equation to determine the gaps more precisely We can anticipate due to this flow to an effective four or three pocket model The gap size on the whole pockets here or in the other case here is probably much smaller than the other gap sizes then We have as I said this orbital or pneumatic phase in the phase diagrams and Here the situation is different for the three and four pocket model If the instability occurs before the fermion energy is reached we find this In the three pocket model there is no orbital or ordering because these pockets fall out and In the four pocket model the orange line here as the promo run ship this orbital Instability and it makes so it wins the competition in the end on The other hand if we hit the fermion energy before these instabilities develop and RPA physics take over We see that in both cases the spin density wave is stability is the largest during most parts of the flow So it will develop in the end and then against for glano type analysis tells us in the three pocket case this magnetism is of order is of stripe type bare as in the Four pocket cases of checkerboard type so in the three pocket case We find ising pneumatic order and in the four pocket case. There's no Pneumatic phase But this brings me to the end Let me conclude I've shown you this pocket RG study for a full five pocket Low-energy model of iron-based superconductors There we find amizingly amazingly simple fixed trajectories namely We can describe them as effective three or four pocket models at in the infrared and Interestingly, we found that the same microscopic model provides two different Scenarios for these iron-based superconductors one is captures for example the physics of the real your iron arsenide They are the Fermi energies are Probably larger than this instability energies and in this case Stripe spin density wave occurs for zero doping in the parent compounds and Superconductivity then develops up-and-doping if nesting cuts this particular whole instability and Due to the stripe race we have ising pneumatic order Then the other scenario likely applies to lithium iron arsenide and iron salinide their Fermi energies are pretty small and in this case superconductivity can occur without magnetism already in the parent compounds and Orbital order is possible. Okay, so and then hopefully these results will be Appears soon and also I would like to Say I Mention these preprints This one is the basis for this calculation which did a simplified Calculation of this model and we will also soon Put in on the internet a study of these weekly Unserver fixed trajectories which might also explain some important physics. So thank you for your attention