 Okay, I would first of all like to thank the organizers for inviting me to this Distinguished workshop as this eminent scientific institution and I would like to speak about calculations for transition metal compounds and as as our chairman already mentioned density functional theory does not work very well for say 3d transition metal compounds a Frequently cited example is nickel oxide here You see the LDA band structure and this predicts the material to be a metal where you have here the Fermi surface which cuts into these bands Whereas an experiment this material is an insulator with a band gap of four electron volts and it is widely believed that the reason for this is the Strongly localized nature of the transition metal 3d shell which renders the Coulomb interaction within the D shell between Electrons in this D shell very strong so obviously we need to Deal better with this Coulomb interaction and One way to do this is the so-called cluster method which was pioneered by Fujimori and co-workers and here you see the angle integrated Valence band photo emission spectrum of nickel oxide and you can see essentially perfect agreement between experiment and cluster Diagonization so how does this method work? Since the problem is with the Coulomb interaction Obviously we have to find a more sophisticated way of treating this Now if you were asked to guess the energy of the end you would probably come up with something like this It's n times the orbital energy and then this is the number of electron pairs times something like the average Coulomb interaction per electron pair But if you have a multi orbital situation then Additional possibilities come into account So for example here you have your 5d orbitals which are labeled by the m quantum number and some electrons Distributed over these and now two of these electrons for example these two can scatter from each other by the Coulomb interaction and Be transferred to different orbitals so we have Coulomb scattering within the D shell Okay, so how do we describe this? We would of course calculate the matrix element for this and the matrix element then would look like this here We have the initial weight state wave functions and these are atomic wave functions That means they are the product of a radial wave function a spherical harmonic with L equals to and the spin wave function and if you and also Of course, we also have the final state wave functions And if you plug in this form of the wave functions and we also use the familiar multipole expansion of the Coulomb interaction Then after some rather easy calculations we arrive at an expression like this This is our matrix element and you see these delta functions. They express the conservation of spin and angular orbital momentum Orbital angular momentum and you see this sum here depends on the initial and final m values and These Ck here are integrals spherical integrals over three Spherical harmonics and they are called gaunt coefficients and they are so to speak pure geometry And you can find them tabulated in textbooks and these f of k are the so-called Slater-Contin parameters And if you look at this expression here, you see that the superscript k here is something like the multipole order of interaction So this f of k, which has the dimension of energy is something like a Hubbard u for k pole interaction And they can be calculated for example using atomic Hartree-Fogway functions And for a D shell only three of them matter f0 f2 and f4 This is why this sum here is zero two and four and these three numbers also can be expressed in terms of the three Raka parameters. That's just a change of notation more or less Okay Now once we know these fk We can then generate all such configurations of n electrons in ten orbitals And their number is this binomial coefficients, which is a couple of hundred So this index nu here goes from one to nc and then we can search out all possible scattering processes and Calculate the matrix element by the formula on the preceding slide And then we have the matrix elements of the Coulomb interaction in the basis of these configurations And then we can run this through a diagonalization routine and get the energies So here you see an example for d8 and the energies then you see they span a range of a couple of electron volts And when we have the wave functions, we can also calculate the spin and orbital angular momentum and they have of course to take these quantized values and we can then assign these standard Term symbols and you see also these energies are highly degenerate That's a good check for your computer program because if you have made the smallest buck You will never get these degeneracies and you can also see that the ground state is Compatible with the first two Hund's rules. It has the maximum spin and the maximum orbital angular momentum for that spin Okay, we can calculate this to experiment here. You see the more the energies of these multiplets and They are taken for you can get them from the optical Optical absorption spectra of nickel vapor for example, and if you convert convert the wave numbers into electron volts We can compare to our calculation and you can see that the calculation is quite good We get the correct sequence of the terms or multiplets and the energies agree to within 10% And many more examples can be found in Slater's famous textbook But now we are not so much interested about the nickel in vacuum But nickel in a solid and so what we do is we add a cage of nearest neighbors in the solid So for example in the case of nickel oxide we have nickel and an octahedron of oxygen to minus ions around it And now the crucial point is that we have to allow for charge transfer That means electrons are allowed to hop from oxygen onto nickel and vice versa and This of course then increases Considerably the dimension of the Hilbert space that means the NC from the previous slide But using the lunches algorithm We can still calculate the greens function the electron greens function and in this way we can get the photo emission spectrum Now here you see and the top one is always an experimental curve And these are angle integrated photo emission spectra of nickel oxide cobalt oxide and manganese oxide and below You can see the cluster calculation that means the photo emission spectrum Calculated from such an octahedron shaped cluster and you notice that in all cases the agreement is very good an interesting thing here is that all these compounds have rock salt structure and Nickel cobalt and manganese are neighbors in the periodic table So the LDA band structure for these compounds look almost indistinguishable Despite this you can see that the photo emission spectra are totally different And the reason is simply that what determines the photo emission spectrum is not so much the band structure But it's the multiple the multiplet structure of the transition metal ion Okay, you can also calculate x-ray absorption spectra within this formalism and here you see nickel You can always see a comparison between experiment and theory and in particular for cobalt oxide For example, you can see that there's really peak to peak correspondence between theory and experiment and in fact Even this tiny temperature dependence, which is seen in experiment is reproduced by the cluster calculation and finally you can also calculate core level photo emission spectra here you have a couple of Iron and nickel compounds and just by a first glance you can see that there's excellent agreement So to summarize this we have seen that we can describe many properties of transition metal compounds by considering a cluster of the material and thereby it's instrumental that the Coulomb repulsion within the transition metal 3d shell is treated exactly in the framework of Multiplet theory, but of course this is a band a cluster method So we have no access to band structures. We cannot discuss thermodynamics and of course according to no phase transitions However, I will now discuss how this cluster method can be extended to a real band structure method by use of the variational cluster approximation okay The basis of the variational cluster approximation is a very famous paper by Luttinger and Ward who have shown that the Grand canonical potential of an interacting fermion system can be expressed as a functional of itself energy okay, you see you have here a sum over Matsubara frequencies and You have the non-interacting greens functions, which is of course this and you have the self energy and Then here you have the so-called Luttinger Ward functional and just to mention briefly I'm of course in a multi band system the greens functions and the self energies also are matrices of dimension number of bands essentially Okay, now the Luttinger Ward functional This is derived from more precisely It's the functional the genre transform of a functional of the in fully interacting greens function Which you see here depicted as Feynman diagrams and this phi of G is the sum of all that means of course infinitely many closed linked skeleton diagrams where the non-interacting greens function is replaced by the full one So the prescription is you see you have here the red lines correspond to G which is the argument of the functional and The wiggly lines are the interaction lines in the Hamiltonian So the prescription to evaluate this would be each diagram can be converted into an integral by the Feynman rules Then evaluated we get the number and summing over all these infinitely many diagrams We would get the value of phi of G Now at this in this We see also a very crucial property or a property that will become crucial in a moment Namely you see that the functional except for G which is the argument this function involves only the interaction lines and Therefore if we have two systems which have the same interaction But maybe different hopping integrals different orbital energies or also a different number of non-interacting orbitals They will always have exactly the same Looting award functional Okay Now a looting and what also proved another key property Namely when viewed as a functional of the self-energy Omega is stationary at the exact self energy and this is the basis of the variational cluster approximation This is of course reminiscent to the variational property of the ground set wave function Which is put to use in the Ritz variational principle. So you could say well, we do something similar We devise some trial self-energy Which is maybe a sum of functions of k and omega with some expansion coefficients And we determine the expansion coefficients from plugging this in into the omega functional But this oh, how can I get back now? How can I scroll back? Actually, how can I scroll back? Thank you very much. Okay. Sorry. No Can you scroll back one more? Yes. Thank you Okay So we could devise some trial self-energy and plug it in but then we face the following problem because you see in our In our functional omega this innocently looking f of sigma that is really the Legendre transform of this phi of g and This phi of g was defined as a sum over infinitely many Feynman diagrams And so of course this is absolutely impossible to evaluate so we can never Evaluate this f of sigma for a given trial self-energy. So we could say okay so much about our nice variational principle, but now came Michael Pottoff with a Very brilliant idea and his idea was to introduce a so-called reference system So for example nickel oxide here you see the full nickel oxide crystal You have hybridization here between all orbitals So this is connected and the reference system then would be exactly the same nickel atoms But now hybridizing with a cage of so-called ligands Which are a kind of fictitious orbitals and the key point is that these Individual clusters here are completely disconnected. That's in indicated by this line here So there's no hybridization or interaction across these lines. Okay now If you make the approximation that we retain only the Coulomb interaction within the nickel 3d shell Which is reasonable because this is the real strong Coulomb interaction that we want to treat better than the intensity functional theory Then you see that these two systems have the same wiggly lines And just to remind you these wiggly lines they describe the scattering processes here and they correspond to these scorned coefficients and slater Integrals as outlined before so they have exactly the same wiggly lines And that means also the looting award function is precisely the same for these two systems And that will be instrumental in a moment Because now you see the reference system that contain that consisted of finite clusters and such a finite cluster We can always solve by exact diagonalization with the procedure that was outlined before and that means we can calculate all energies So we can evaluate the grand partition function and obtain omega for the cluster and you see till they means cluster quantity Similarly, we can also obtain the cluster greens function g alpha beta and alpha beta is a combined orbital and spin index So for example for a D shell that would be a 10 by 10 matrix and if you have no spin dependent Quantities then it would be decomposing to two five by five matrix And now we can invert this numerically and use the Dyson equation and extract the self energy the exact self energy of The cluster and now because we know omega from from the grand partition function we can use the looting award expression for omega the other way around and Calculate the exact numerical value of the looting award functional So the net result is that this procedure gives us a self energy and the exact numerical value of the looting award Functional and now we use this as a trial self energy for the lattice That means we just plug it into the expression for omega lattice where we have here the non-interacting greens function of the lattice So we have an approximation for omega But then the question is how do we be how do we perform the variation of this self energy and the answer is this You see that the requirement was only that the interaction lines are the same There is nothing that for example fixes the value of this hybridization integral or the energy of this ligand So the answer is we perform the variation of the self energy by varying the non-interacting Hamiltonian of the reference system namely, we vary this V or this epsilon ligand and then For each combination we evaluate this omega lattice by the procedure outlined before so that this becomes a function of V And epsilon ligand and then we just find those values where this is stationary and Obviously this procedure corresponds to seeking the best approximation to the self energy of the infinite system Amongst the cluster representable ones. Okay, that's that's the idea of the variational cluster approximation Okay, now some technical points We need of course a non-interacting H naught and this is just a tight binding fit to an LDA band structure Here you see this for cobalt oxide and you see they are hardly distinguishable so cobalt oxide for example really likes to be described by LCAO and Then we have these Slater integrals or Slater-Contin parameters and we evaluate them just by using Hartree-Fock Radial wave function however This raka parameter are a which is almost f0 and which is something like a Hubard U That is very strongly screened in the solid and Also the the 3d orbital has to be corrected by the Coulomb interaction That goes under the name double counting correction And what I did is just use this a and this Renormalized D level energy were adjusted so to get the correct Insulating gap and also the correct position of the satellite and the photo emission spectrum Okay, now this is just supposed to tell you that all these compounds that that means the Transition metal oxides with rock soil structure. They all undergo Enti-Pheromagnetic ordering, but I skip this. Okay now here you see the staggered and the uniform Susceptibility versus temperature for a nickel oxide and they are calculated just by differentiating omega and The staggered susceptibility diverges at the nail temperature. The uniform susceptibility has Curie-Weiss form as expected with this Curie-Weiss temperature You notice that the Curie-Weiss temperature is higher than the nail temperature But this is a consequence of the frustration in this type 2 Enti-Pheromagnetic structure and we can now go along the sequence of these oxides and If we readjust this later cost of parameter dp sigma, which is the most important one slightly by by this factor Then you can see that the nail temperature the calculated nail temperature and also the Curie-Weiss temperature agree quite well so we can describe The magnetic ordering reasonably well by this VCA now, let's look at Nickel oxide in particular So this is the grand potential versus temperature and here you see the paramagnetic solution and then at the nail temperature First anti-pheromagnetic solution branches off and then at lower temperature a second anti-pheromagnetic solution takes over And you see these two intersect with the finite slopes that would indicate a first-order phase transition Which is not good. So most likely my opinion is that there is maybe Rapid but continuous change of the state of the nickel ion around this temperature Okay, but now we focus on this solution here and What you see here is the angle integrated photo emission spectrum that means the k integrated spectral density For a nickel 3d and for oxygen 2p and that's compared to different experiments here Here you see hard x-ray photo emission which should show predominantly nickel So this should be compared to the red line and you see we can basically Identify peak by peak the second spectrum here has been taken with an x-ray energy Where the so-called satellite down here is resonantly enhanced and you see this corresponds quite well to this one And finally this one is XES and that should be compared to the blue line that shows mainly electron sorry Oxygen density of states so that should be compared to the blue line and that matches also reasonably well Now we go to the band structure What you see here is the dispersion along 100 by our best taken by our best And these are our perspective which are taken in different Experimental geometry and that means that due to matrix element effects. You can sometimes observe different states and this is an Intensity map of the spectral function from VCA and this is twice the same, okay? Now if you look at the experimental spectrum the most conspicuous feature are these flat bands here you have a multitude of almost dispersionless flat bands and Looking at the VCA you also find The reason name is just the multiplets of the nickel iron these multiplets are Generated by the strong coulomb interaction in the D shell and they obviously persist in the solid only they are split by a crystalline electric field effect and so these Multiplets in the when when nickel is embedded into a solid They give us these flat bands and as you can see they agree mostly quite well We come down below here You have bands which are derived from oxygen and they have a stronger dispersion and you can see for example This portion here you can see here maybe and for example if you go like this then this is perhaps this one and If you go here, then this part might correspond to this and you also have this which could be this So you see the the spectra the the artist data are a little bit scattered so one cannot make a very detailed Comparison but all in all I would say the agreement is not so bad And we can do the same for the 110 direction we also have our best and VCA and again you can see these dispersionless bands and then maybe this would correspond to this one and Perhaps you have another dispersionless band down here. So by and large The agreement is not so bad Okay, and this is still shows manganese oxide and cobaltis oxide and you see again for example this Which one this is hard x-ray which should be compared to the red curve and you see quite good agreement This is XES which should be compared to the blue line and again you can see quite good agreement And the same thing here for cobaltis oxide again This is hard x-ray where you can see this sequence of peaks that you also have here in the red spectrum And again, this is with the resonantly enhanced satellite which also is at the right position Of course, this was done by adjusting these two parameters. Okay, so this is maybe not that much of an achievement And lastly you see this is XES which should be compared to the blue line and again you can see Maybe quite reasonable agreement. Okay, this brings me to the end. I Hope that I have convinced you that the variational cluster approximation allows to implement the variational principle for the self-energy and solution of a reference system allows to extract allows more or less an exact treatment of the interaction and Maybe you will believe me that the results for nickel oxide cobalt oxide and manganese oxide are not so bad and Thank you for your attention