 So our next speaker is Sophie Beck from the Flatiron Institute and her talk is about correlation effects on realistic materials modeling with DFT plus DMFT. Thank you for the introduction. Can you give me a ride or should I speak closer? Welcome everybody. Good morning. Thanks for coming. Also welcome to those on Zoom. My name is Sophie Beck. I'm a postdoc at the Center for Computational Quantum Physics at the Flatiron Institute in New York. And yeah, I will talk about correlation effects and dynamical mean field theory. So this is just to give you a broad overview of what DFT plus DMFT means. This is essentially, as you can see from this image, okay, my mouse doesn't work. DFT plus DMFT kind of stands on three pillars, one of them being density functional theory, then downfolded Hamiltonian, which we use Vani94 and then the DMFT part. So I also want to just thank all the developers and kind of inventors of Vani90 already for their work because without it, this kind of method would not work as it does right now. Right. So we've heard in the past few days a lot about how to construct a localized kind of downfolded Hamiltonian. So I will focus mainly on dynamical mean field theory. And then in the second part, I will talk about the Trix software project that we're going to see also in the hands-on session later. And I also want to thank the organizers for inviting me. This is a really, really great workshop so far. So, right, as I said, I'm going to do an introduction to DMFT and I'll talk briefly about charge self-consistency with quantum espresso Vani90 and Trix. And then the second part is going to be about the Trix software package, the basic building blocks, which is the Trix library. And then I'm going to focus on two applications, Trix solid DMFT and Trix FermiC, which my colleague, Alex Sampe, who's in the audience, and I have spent a lot of time working on and also with other people. And then we'll go to the tutorial in the next hour. So I'm going to start with this Nature article by Tokura Kawasaki Nagahosa, where they talk about strongly correlated electrons, materials as the next generation electronics. And you see they identify four key aspects that are interesting about strongly correlated materials. And we've heard in the past few days also about topological properties. But today I'm going to focus on electronics. And you see here, the key concept, they identify as electron correlation. And then there's also a target industry for high energy efficient electronics or energy harvesting. So this is a really kind of interesting field. And what makes strongly correlated materials so interesting is that they have a complex interplay between letters, orbitals, spin and charge degrees of freedom. And what this means is that they're very sensitive to small changes in external parameters, such as changing temperature, pressure or doping. And this gives rise to a kind of what we call rich phase diagrams, where they host a couple of different kind of phases, such as the famous high TZ superconductivity, colossal magneto-resistance, multi-phoric phases or kind of mott physics, which I'll focus on today. The materials that we consider strongly correlated are mostly those with open D or open F shells. So those that are kind of marked here. I've also marked ruthenium. You're going to see or hear me mentioning ruthenium a couple more times with its kind of seven electrons in the D shell. So these are the transition metal D shells or the rare earths or actinides that we consider strongly correlated. And just in terms of this kind of next generation electronics, what makes them also very special is that due to like tremendous progress in thin film growth techniques, experimentalists can nowadays really kind of grow these materials atom by atom. So very similar to how we kind of drag and drop atoms in Vesta. And they can exfoliate them, grow them really in atomic layers. And this gives rise to this kind of materials by design principle, where we would engineer materials to be in a specific state that we could use for electronic applications. And here's just one example of a prototype of a mott transistor based on thin films of rarethniculate that supposedly switches much faster and is more energy efficient or this kind of prototype for a mott solar cell based on the mott insulator lanthanum vanadate. That would have a higher energy efficiency than current semiconductor solar cells. Now let me define a little bit more closely what strong correlations means. I will actually define the opposite. This is something that we're mostly like everybody here is very familiar with what I consider weakly correlated systems. So these are systems where we can use an effective single particle picture and this means that we can construct the many body wave function from a product ansatz where we assume that the particles are independent, then we anti-symmetrize them and we add a slater determinant or a linear combination of slater determinants. And of course this is the kind of the typical band picture where you have for each momenta, for each band you have a concrete like a single eigen energy and this gives these bands. And if you think about this in real space you can imagine an electron moving through a lattice potential and just interacting through an effective potential. Now strongly correlated systems are defined by exactly the breakdown of the single particle picture where you don't have bands anymore but now you have a spectral function and you see in the spectral function they're kind of these washed out areas but then there's also areas of very high density and we're going to define what this is in a bit. And in the real space picture now you imagine kind of electrons moving through the lattice but now you really have to think about the time dynamics of what these electrons do on the lattice how they interact with each other. And this is kind of where dynamical from dynamical mean field theory comes in. And right so these strongly correlated systems the kind of the what causes the strong correlations is the local coolant interaction that we've also heard in the previous talk. And these systems are often between ionic localization and itinerant behavior what makes them very interesting. Now the spectral function that I've just showed I want to briefly define what that is. So the spectral function is more or less this commutator of the usual annihilation and I forgot the other word creation operators. So we're going to create a particle at location zero at time zero and we're going to measure the correlation when we take out another particle at later at a different location at a different time. And this is essentially what the Green's function is and then we can so this is in real space and time we can Fourier transform it and then we get the lattice Green's function as function of momentum and frequency. And if we take the imaginary part then we get the spectral function. And the spectral function is exactly what experimental colleagues would measure in photo photo emission spectroscopy in particular Arpus except that they would only probe the occupied side of the spectrum. Now in the non-interacting case we've already talked about this in a non-interacting case so this is kind of a band diagram with k in plane and the frequency you have specific eigen states for each momenta and this forms your bands. And in this case the Green's function has this typical shape so this is the non-interacting Green's function which is one over frequency minus your cone sham or tight binding Hamiltonian plus a small broadening otherwise this thing blows up. And the spectral function the corresponding one is just a sum of like a series of delta peaks which is exactly what you know what we what we'd expect. Now in the interacting case the picture changes a bit and the change is mostly that instead of like having this delta peaks you have quasi particle peaks which now have a different can have a different height they can have a different widths and so they kind of get a lifetime to them. And of course the kind of the integral has to be conserved so the spectral weight that decreases at the at the Fermi level is going to move to to the Hubbard bands which are kind of more atomic like excitations at higher and lower energies. And in terms of theory this can this can be described if we add a self-energy so this is what we call this big sigma. Sigma is a complex quantity so it has a real part and an imaginary part and is frequency dependent most importantly. And the corresponding spectral function now you can see that the eigen energies the previous delta peaks are now shifted by the real part of the self-energy so this is a kind of a quasi particle of renormalization that can change your bandwidth and then you have a the imaginary part that acts as a scattering and kind of washes out the effects. Now we're going to talk a little bit more about what we can learn from this spectral function if we do some approximations we can talk about quasi particles. So this is the the image that I showed before the spectral function and the self-energy that corresponds to this this is by the way strontum ruthenate I want you about this. So this is a self-energy that corresponds to this spectral function you have the real part as function of frequency and the imaginary part as function of frequency. And if you want to define quasi particles we can if you look at this picture we can still find kind of maxima of this of this intensity plot and define this as quasi particles and this is exactly what you see in this plot now these blue dots are now the kind of the maxima of this previous intensity plots and the red red lines are the is the tight binding vany Hamiltonian and you see exactly this kind of band renormalization so you change the bandwidth a little bit and if you if you want to if you do this you can kind of factorize or you can split up your greens function into something that we call coherent part which are the quasi particles and the incoherent part and for the coherent part you now have a quasi particle renormalization set which is one minus the real part of the self-energy and the inverse of that you're going to have renormalized quasi particle dispersion which is just the eigenvalue equation if we take the the conchamp states minus the real part and then you're going to have a scattering which is the inverse lifetime of the particles which come from the imaginary part okay so this works as long as the imaginary part of the self-energy is not too large but we can actually break it down a little more if you really just focus on the low frequency regime where you know most of the interesting physics takes place so if you look at the self-energy in this regime you can see that you can approximate you can you can expand your self-energy as a Taylor series and you're going to have a linear term in the real part of the self-energy and a quadratic term in the imaginary part of the self-energy and this gives you an analytic form for the self-energy that is very interesting to study kind of this is the liquid the Fermi liquid regime where things simplify a little bit more so z is now a frequency independent constant that you get from the slope of the real part of the self-energy so you just take the slope do one minus and the inverse of that and this is also the kind of yeah as I said the quasi particle renormalization so this is related to the mass renormalization of the electrons and then the scattering scattering rate is just the imaginary part of the self-energy at zero frequency time set okay so if you feel still a little bit uncomfortable with the concept of the self-energy let me remind you that there's one particular type of self-energy that you're most likely very familiar with which is that of dft plus u calculation so in dft plus u you would have zero scattering which means that the quasi particles still have infinite lifetime and you also have zero frequency dependence in the real part of the self-energy and a zero zero slope which means that your quasi particles are not renormalized and you can you know from dft plus u you can get you can the only while you can get a frequent an orbital dependent kind of shift in the real in the in the quasi particle in the in the eigen states but you won't get an insulating state unless you break translational symmetry and this is really the kind of fundamental difference to dft okay so let's talk a little bit about the mott metal insulator transition so kind of mott physics historically typical control parameters are kind of bandwidth control where you change the ratio of the Coulomb repulsion over the bandwidth and this can be achieved by applying pressure either kind of like pulling your pulling your sample out or like together or by chemical pressure then there's filling control which you can achieve by electrostatic or chemical doping and then there's a kind of a trivial mott transition which is by temperature and then there's also dimensionality control but in some ways they're also related so in practice what this looks like is that these materials of course they have complicated face diagrams as I introduced so here's you have temperature versus actually I'm gonna take this axis temperature versus pressure so this is an example for bandwidth control this is vanadium 203 so essentially you can change the hopping parameters or the bandwidth by applying applying pressure which actually you get via doping in this case and for vanadium 203 in the ground state you're in a paramagnetic metal phase but if you decrease or if you apply negative pressure you go into a paramagnetic insulating state and of course there's more phases down here but that's not so important right now another examples are rarest nicolates where you can change the bandwidth by like by a tilting angle by exchanging the east side cation or a nickel sulfide selenide where you can change the bandwidth via effective hybridization to the to the ligand states now the other thing filling control that's the kind of the typical example this is lanthanum 2 copper oxide so here you have temperature versus hole concentration and this is really kind of a phase diagram of the parent compound where you have an anti fermenting insulator and then if you apply a while hole doping you go into all sorts of weird phases the pseudogap phase then the famous superconducting phase but also the metallic phase okay and we're going to actually do in the tutorial we're going to look at this this material all right so physics and in theory this was already wonderfully introduced by the by the previous lecture professor ita so the most fundamental or most basic kind of model that we can use to describe a metal insulator transition is the so-called hubbert model where you have two terms one is the hopping term where you know the kind of the amplitude with which electrons move around from one side to another and then you have the local onset repulsion that electrons experience if they are located at the same side and this model looks deceivingly simple but i can assure it's not at least in more than one dimension and i also want to remind you that you might be might have become you know more familiar with the hubbert model in the past two years you know with this kind of internal competition between wanting to leave the house and move around and not being too close to somebody who's coughing um i keep waiting to take this slide out but i think we're getting there slowly um right but let's go back to the to the hubbert model and we look at the limits so if t is much larger than u and we take the half filling case then what we can do is we can just take single particle picture and describe electrons moving around in in these letters which is you know exactly what dft is about and the system is metallic so that's that's a simple case if we take the other case where t is much large and much much smaller than u um the system is essentially just a bunch of of isolated atoms we can do exact diagonalization is also not a problem the problem is really the intermediate regime as usual and this is however the interesting regime so if we had a wishlist for a theoretical method then we would kind of wish for it to be able to to handle the competition between itinerant and localized states uh we want to describe both quasi particles but also atomic multiplats um we have to constantly switch between reciprocal space and real space we want to describe all sorts of frequency regimes low frequency high frequency and we want this to work on kind of models but also from from and as you can imagine the solution kind of is dft so in dft uh we start kind of again from a from a lattice model which is the Hubbard model and we can map this lattice model to an effective impurity model so we take an atom that is kind of embedded in a bath of non-interacting uh of non-interacting states and uh the coupling is described by a hybridization function that has a frequency dependence so again this is the dynamical part of that dft so that we can really capture the time dynamics of what happens at this specific site and this impurity bath coupling uh this is to be determined self consistently and if you want to read more about this uh of course the kind of review articles are uh are great recommendation now in order to understand this a little bit more um we can we can start from from classical meanfield theory well usually you have a lattice model uh you have a local observable and then because you cannot really compute the lattice model you're going to construct an effective local model and uh describe uh the interaction of of your I don't know your local observable with uh with the rest of the of the particles via an effective medium that we call vice meanfield and you have to solve this uh self consistently and so for the easing model what this means is you have this Hamiltonian of interacting spins uh you're interested in computing the magnetization your effective local model is a spin and an effective uh in an effective medium and this kind of effective medium or vice meanfield is just uh the uh the product of your j your your exchange coupling and the number of of nearest neighbors and uh after doing some algebra you can get to the self consistency uh uh condition and you can actually solve this graphically in this case now you can already guess uh in for dmft we cannot solve it graphically um okay so dmft is a little bit more complicated so the first uh first line you've already seen you're very familiar with the Hubbard model by now then the local observable that we're interested in is the impurities greens function uh which we compute which has this kind of shape of um uh an iteration and creation operator that we've seen before time ordered and the effective local model is the Anderson impurity model which has an impurity Hamiltonian that is just the crystal field levels uh plus an interaction Hamiltonian then we coupled this to an effective bath of non-interacting fermions and this coupling is described so the coupling so this has you know creation annihilation operators a and the impurity has c and so here you have a bilinear coupling described by this coupling constants v and so the vice mean field in this case is what we call curly g and this is mostly described by this coupling v and the the eigen energies of the non-interacting path and so this second part of the curly vice the curly g is the delta is the hybridization function and we can also relate this curly g with the impurity greens function or the local greens function and the self-energy via the so-called Dyson equation and so for the self-consistency what we'll have to achieve is we have to kind of equal or we have to yeah we have to get the local uh lattice greens function to uh agree with the impurity greens function and so the local lattice greens function means that I just do a sum over k where I kind of uh average over over the momentum degrees of freedom and once the impurity greens function and the local greens function match then I've achieved then I have self-consistent solution however there's one problem so here you see kind of this self-energy coming up again the problem is that we don't have a momentum resolved self self-energy and so the fundamental dmft kind of approximation is we have a local self-energy and we can take this local self-energy from the impurity and put it in the lattice greens function and this approximation becomes exact in the limit of infinite connectivity of your lattice so that's kind of the fundamental outline of dmft and we have to solve this numerically and by far the heaviest part of this work is solving this impurity problem quantum impurity problem and there's a whole range of impurity solvers out there I don't have time to talk about this today but yeah there's a lot of lectures out there so I know this is a handful kind of a lot of equations here's kind of like a schematic overview again that is a little bit simpler so we start from from your non-interacting density of states or non-interacting greens function in this case it would be the vani-hematonian you construct your local greens function which is just a sum over k from this you can construct your curly viscimene field which is in the first iteration is identical the local greens function you solve the impurity greens function under kind of the action of this viscimene field and then you from this impurity greens function and the viscimene field you can you can compute the self-energy and you plug this into your lattice greens function and then you iterate until you converge there's one example where this is particularly simple so dmft people love the the beta lattice it has this structure so you can see it's not a crystallographic lattice it's an infinite Cayley tree but what is interesting about this is that if you go to the limit of infinite connectivity so here each side has three neighbors but if you do this for infinite kind of neighbors you can get an analytic expression for the density of states and this is just a semi-circular kind of semi-alyptic density of states which of course is not too far from from you know what we know in our in our materials even though it's not a crystallographic lattice and in this case since you have this analytic expression your dmft self-consistently simplifies drastically because you can like you can automatically plug your impurity greens function back into your curly g and just iterate over this thing and I'm going to I've showed you this because I will show you later how you can kind of code this self-consistency in tricks and I counted less than 12 lines but of course in a in a more general case if you want to do a real kind of real like material calculations it's it's a little more complex than that and so this is success of dmft can be shown by looking at this kind of face diagram of the Hubbard model where you have temperature versus coolant repulsion and you have even though this is a you know the details of this face diagram are going to depend critically on or not critically but like going to depend quite a bit on the specific material that you want to study the overall kind of faces they're actually quite you know quite well represented by the Hubbard model and fit many materials so on the left hand side for low interaction you have a Fermi liquid so a good metal and then on the right hand side you go to a paramagnetic insulator at low temperature of course you're going to enter an ordered phase magnetic or orbital order you have a first order transition where you have kind of you lose the metal at this kind of transition line and you lose the insulator at the left one then you have a second order critical point and above that you have something that we call a bad metal or a bad insulator not a band insulator and if you kind of move horizontally from the left from zero interaction to the right you get this kind of very famous picture where you start from a just a normal non non-interacting density of states you increase kind of the correlation you get this kind of infamous three peak structure and at some point you open a gap and this is exactly what this mod metal insulator transition is okay so this was kind of the general overview over dynamical mean field theory so how do we combine this with density functional theory or kind of ab initio methods and we've already had a great overview over this in the previous talk but i'll just briefly walk you through kind of the ingredients what you need for this calculation which is mostly kind of we call them projector functions but in in the vani language they're just kind of the u u mutt i guess the the u matrices the unitaries which is the overlap between your vani functions and the kohn-schamp states then you need of course the the hopping hopping terms hopping integrals and you need an interaction term and then you can run your dmft and after that you can get kind of post-processing steps you can get spectral functions and if you want to know more about this i recommend in particular the the amazing collection of lecture notes of the julich autumn school on correlated electrons and i also will briefly talk about a recent addition which is if you want to achieve charge self-consistency you of course have to kind of update you also have to iterate over the of a charge density going back to dft and i also want to mention that the whole kind of workflow is implemented in in solid dmft which you're going to use in the tutorial later okay so as for the ingredients we kind of start with the target bands so this is a slide for you kind of to relax you've seen this now several times in the past few days so you kind of have your band structure and you create maximally localized vani functions for what we call target bands which you know you identify as being not being or being strongly correlated and that need additional kind of treatment so you start from your kundram states and you create the with a you know the usual vani functions at let aside i and orbital index alpha i'm going to use a slightly different notation and then you have your hopping elements if you evaluate your kundram Hamiltonian in terms of these funny functions and integrate over over a real space then a slightly subtler thing that i have not mentioned so far is are these projector functions so um kind of in this dichotomy of constantly changing between the reciprocal space and the lattice and your local space and the impurity space uh we we do something that we call down folding or up folding where we kind of sum over uh sum over k and you get a local quantity and then for the up folding you do kind of an embedding and this is handled by these projector functions um which are most most of the times unitaries um but yeah so this requires a little bit of thinking um but uh you can get these from from vani 90 and then you do the reverse once you do uh once you embed the self-energy in in in your led screens function as for the interaction Hamiltonian i've showed a slightly simpler form before so this is a kind of a quadrat uh like a four-body kind of Hamiltonian in the most general case um this is usually a very complicated object uh we've we've seen in the previous lecture by by Professor Rita how we can use constrained random phase approximation to compute uh these Coulomb tensors so in the kind of simplest case uh if we have a just kind of the bare Coulomb repulsion this is just like the the Coulomb operator evaluated in the in the vani functions but this is this would only be valid for kind of like an atomic picture and of course you have these screening effects uh which was were also discussed in a previous lecture uh so this is a little bit of a complex business but um aside from that this form usually simplifies drastically if we have kind of a crystallographic kind of cubic matrix where we embed the atoms in this case many of the terms actually um are zero by symmetry and so then we have uh what we call the the usual kind of Hubbard kind of moory interaction so this is now just five terms and you have like the first line our density density terms where you have um you know um two spins of the same orbital uh two spins at different orbitals and then uh of the same like uh of the same orientation at different orbitals we have now to take into account the exchange interaction and then you have um also spin flip and pair hopping terms okay and so for the next steps you're gonna have to use an impurity solver for this I recommend the lecture by Olivier Parculé from the Arnold Sommerfeld School in 2017 and once you've solved your DMFT you can analyze spectral properties most of the impurity solvers actually work on the imaginary frequency axis so then you have to kind of you have to use an analytic continuation to transform this back to the real frequency axis and once you have that you can compute something like these Fermi slices so this is data by a colleague of mine Xiaodong for Strontium Wuthonate again where you see the kind of the the red uh data is calculated and the blue shade shaded area in the back is ARPIS measurements by Anna Tamay from University of Geneva and you see that in this case this is a material with spin orbit coupling you see that of course DFT kind of without spin orbit coupling is not is not a good a good way to to approximate the system DMFT without spin orbit coupling is also not great once you add spin orbit coupling you kind of improve the agreement but only if you add kind of what we call correlation induced kind of effective spin orbit coupling you actually get excellent agreement and then there's also a couple more post-processing steps that you can do you can compute optical or thermal conductivity uh you can compute susceptibilities or kind of two particle correlation functions policy big coefficients resistivity and so on and uh because of uh this yeah I briefly want to talk about optical conductivity uh because I this is also kind of a you know one of the wonderful things about the bunny 90 community so I um last year so prior to to last year this was only calculated in tricks from v2k and uh I wanted to update this uh to include uh start starting from bunny 90 and then I stumbled over the the great papers by by a lot of people who were present here David and and evil and Jonathan and I actually asked in the in the user forum and I got within two days or so got a wonderful response by evil and and sepan and this is really kind of one is one of my favorite discoveries of the last year I think so um in right for optical conductivity we started from kind of in a linear response regime we start with a kubo formula where you have just the um the current current operators and in a single particle picture you get the kubo greenwood formula which is actually I just copied from the bunny 90 input from the bunny 90 um user guide but for the many body analog so in this kind of single particle picture you just have a velocity matrix elements in a many body analog you have something called transport distribution and now not only do you have this velocity matrix elements but you also have the spectral function so this is slightly more complicated and as I said prior to to kind of last year we would have to go through this right of like evaluating the velocity matrix elements explicitly from a dft code and so on and uh now I uh kind of uh we might use of of bunny 90 and the bunny berry tool to really uh call this within our kind of dft routine and do this very quickly so bunny interpolation to the kind of berry connection um and this is a really great thing and to show you why this is uh kind of the many body analog is important so here's um the conductivity again for strontium ruthenate the experimental lines are kind of these drastically oscillating uh lines and this is just uh the uh the druide part so typically for for optical conductivity the druide part has a kind of lorenzian shape but you see that in this material you have kind of like a second uh um uh I think they call it a non druide foot kind of a second foot that is kind of um very typical for non for Fermi liquids and you can you can uh describe this with in the kubu formalism and so this I repeated the calculation with this kind of new implementation and this gives excellent agreement with uh with the prior implementation okay so now uh I want to briefly talk about the aspect of of charge for self-consistency this is a recent implementation um uh together with my colleague Alex Hampel uh Olivier Parcolet Claude Edera from ETH Zurich and Antoine Georges um right so you've seen this kind of outline over dft plus dft now several times the important part of the charge self-consistency is really going back from dmft to dft to quantum espresso and for this you have to compute kind of charge density updates and uh right I just want to mention that um as for the code additions there were some parts necessary for dmft of course and some parts uh in quantum espresso and this is all handled at the level of hd5 archives so there's no kind of reading in or files or writing files other than that and there was actually from from other things that were necessary for this project money 90 was an excellent shape there was nothing to be added this was really this is really a pleasure to work with right so I'm just going to give you a brief glimpse of what what is uh you know what is what needs to be done for this charge self-consistency so we can describe the interacting charge density if we kind of sum over the lattice greens function we take the trace sum over k sum over the matzubar frequencies and evaluate this in real space and we can think about this interacting charge density as the cone charm charge density plus a correction right and uh here the cone charm charge density is kind of the usual uh square root over the over the cone charm states and so the idea is really to compute this delta row and we you know we can just subtract the lattice greens function the interacting lattice greens function we can subtract the cone charm non-interacting one and this defines the delta n right so we just have to evaluate this and kind of feed it back to quantum espresso to recompute the you know to construct the new charge density recompute the potential and do the whole kind of whole thing again construct new value functions and iterate the whole thing so there's not only the dmft self-consistency there's now also that kind of the full charge self-consistency and so uh here's uh our benchmarks this is the orbital polarization and calcium vanadate so this is a pair of skyte and if you apply tensile potential strain uh you get for a single it has a single d electron so you kind of favor an insulating state because you apply a crystal field splitting that kind of favors this this orbital polarization and so on the right you see the occupation of the three t2g states as a function of onsite repulsion and you also see the the spectral weight at the Fermi level which is metallic at low u and then it drops to zero meaning that the material is now insulating and this goes in hand with uh a dramatic kind of increase in the orbital polarization and this is particularly strong for the one-shot case but is quite a bit reduced for the charge self-consistency so this is one way charge self-consistency can affect your results and we've benchmarked this against other implementations so this has shows excellent agreement with our prior calculations that were done with vasp and you can also look at this in you know in real space here's the difference between the charge densities plotted so the left hand side is the difference between the inter kind of the non-interacting representation of the full charge self-consistent charge density minus the initial conchamp state and on the right hand side is the one-shot version and you see that kind of the blue shaded areas are an increase in density which are kind of the I think the dxy orbital and this is uh overestimated for the one-shot case and slightly lower for the charge self-consistent case and then there's another example which is serum two or three where we did total energy calculations um so we apply strain for minus six to six percent and you see that for the for pbe the lattice constant is slightly underestimated in comparison to the experimental value then if we do one-shot calculations it gets overestimated but if we do charge self-consistency we actually get extremely well very good agreement with the experiment and one of the new aspects of this charge self-consistent implementation I should mention this has already been used for v2k and vasp but one of the what they use kind of internal projections in the dft code and this is usually very different like difficult to control because all you see is a bunch of numbers and you don't really know what your downfolded Hamiltonian actually kind of looks like or what it does and one of the new aspects is that thanks to the vanu90 interface we can kind of have excellent control we can at each step look at the band structure again see if our downfolded Hamiltonian changed drastically or anything else and we can you know track this as function of of iterations so just to summarize this part the major benefit of this implementation is really the vanu90 kind of ecosystem as was described in the in the prior days yes so this is a kind of a great addition for us to have more control over the downfolding procedure and and every other aspect of the of the construction of the local Hamiltonian all all parts of this implementation of really open source and MPI and K parallelized and this is integrated in quantum capacitance versus 7.0 and recently else in tricks 3.1 and the workflow is implemented in solid dmft okay so right so this uh summarizes the kind of the introduction to dmft part so maybe i'm gonna take like five minutes of questions here and then before i switch to the software part thanks for the great talk uh are there any questions here in presence one there so thank you for the talk it's been really nice i have a question about the slide that you have shown about the Fermi liquid theory so i guess that if you think of you know a scenario for example in which you have a non-interacting material and you start increasing the Howard interaction term then it's gonna if there's a more transition in which the material becomes insulator there's gonna be a breakdown of this picture i guess so how can we uh diagnose this breakdown and in particular to the more what happens with the gamma term that is related to the lifetime when you approach this more transition that's a good question let me think about this um right so i mean okay so you start from an from an infinite lifetime right um so you would have no scattering and then um how would this appear i'm not sure if there's anything uh it's super exciting happening with the scattering i would assume it just uh increases i'm not sure if i have anything smarter to say about this good question okay any other questions raise your hand okay this is a question on the chat that is my mouse sorry how do you construct a veneer function on each step of self-consistent calculations by hand or automatically that's a good question um so i think initially there were some concerns that um you know when for vani 90 for the initial vani functions you you know usually i look very much by hand what the correct energy window is did i did i capture the disentanglement procedure correctly and so on and of course there's a fear that once you introduce this charge density update things could shift around drastically but what you have to remember is that your dft band structure is not going to there's not going to be a gap opening or anything drastic right you're going to do very subtle kind of changes to the charge density and it turns out that uh so far we hadn't didn't have a single case where uh things would kind of if we keep the same parameters things would screw up drastically so in in practice you can just keep the same parameters but just make sure by kind of checking your vani functions each iteration that things haven't gone out of control i have a follow-up questions that is also a follow-up of yesterday's discussion we had so um you mentioned it's very important to have the symmetries right in the vani functions you know for for your workflow what is it exactly that this you know in the steps that you showed what is what is that this appears clearly because you know you know if for the eye of a non-expert like me it could be you know i could just throw in some decent vani functions and it should work but actually this is not true that's an excellent question um so i um you you kind of caught me here i completely ignored this this aspect so what we usually do is um let's go to the example of calcium vanadate um right so calcium vanadate is a perovskites uh in the perovskite orthorhombic perovskite structure which means that it has kind of four vanadium sites which are by symmetry equivalent and so it would be really stupid to compute like four impurity problems right so we just do a single one and then we take this self energy and apply the rotations to map them to that different sites and so if these symmetries are not captured correctly uh we might run into trouble so in practice we have to like we have to think about this is also actually encoded in these projector functions um let me go back right so these projector functions will take care of kind of folding the self energy that you get from the single impurity to all the ones that are symmetry equivalent it was okay a question by david hi that's a very nice talk i'm just trying to understand you know i um lived down the hall from christian howler who does uh you know embedded dmft so it's dmft and a kind of a density functional context self consistent so i'm trying to understand what is the relation and and what your method buys you that for example that style doesn't buy you well i i also still try to understand that um i would say this is a very difficult difficult difficult question for me to answer um right so as far as i understand christians method is kind of more from the from the functional approach which is really very appealing and in many ways uh i think he also has a real space projector functions which is also superior to our case actually i'm not sure if our method buys anything extra uh i would probably yeah um um yeah so i'm not sure i can i can give a conclusive response to that i think well yeah i mean i would have to probably get my hands on and code the other route in order to understand this better it was another question down there i would just curious if your setup for including correlations in the optical conductivity also works with spin orbit coupling so would you have access to the for example magnet optical part of the response in magnetic materials um uh we did not consider that yet i think magnetic materials um i'm i'm also not sure if i can answer that um yeah so this is a kind of a recent development so we've only looked at just looks very promising yeah okay okay any other questions we have actually a minute so so if you want i actually kind of so the second part of sorry maybe i was unclear i'm still going to use part of this lecture for the softer part ah okay okay so then you can just go on okay right so now for the the software part um right so here i should mention the main maintainers and kind of inventors of the trick software project niels wenzel who's a data scientist at ccq then my colleague alex sample who's in the audience and who's going to help with the tutorial later too he actually constructed most of the tutorial so i should give all the credit to him and he's also a data scientist then we have michel ferrero who's a who's a group in in paris and then olivier parcoulis so what is tricks tricks stands for a toolbox on for research on interacting quantum systems it contains both the tricks library which is kind of the fundamental building blocks of the software and then it contains a series of applications that are based on this tricks library it's fully open source uh it's high level interfaces in python three python three right uh low level is modern c plus plus so this makes it a very kind of very efficient implementation of course there's a lot of people involved but i'm not going to go through this here um so kind of you can think about tricks or try to summarize it in a slide as well if dmft is kind of your hammer tricks is your toolbox and then you can you can you know apply to the materials and the fundamental building blocks are kind of greens implementation of greens function objects they're really kind of the basis of every every sort of every calculation that you want to it has hd5 support so everything works via hd5 uh there's this kind of c plus plus to python layer it's fully open source fully npi paralyzed to speed things up we have our own statistics package for the qmc um the quantum Monte Carlo analysis we have integration tests and a bunch of tutorials if you're interested and here's just kind of an excerpt of what this looks like so in python you would uh import from tricks greens function you can import a mesh an imaginary frequency mesh or real frequency and the greens function you can define your mesh as a fermionic mesh at a specific temperature and a specific number of matzabara frequencies and then you can kind of feed this to create the greens function and then for the greens function you can just put your vani Hamiltonian in and that's as simple as that and then you can also do uh you know you can you have access to the many body operators like the uh like n or or the annihilation and creation operators and you can do exact diagonalization and so on so this is just a fundamental building box which is already um a lot so tricks uh the packaging we have uh you can install it via anaconda uh we have deviant packages uh binder docker images singularity and and easy build you can ask alex if you if you're interested in in uh downloading it we have a bunch of tutorials we actually also have a summer school coming up in in canada which is uh held every two years um right and i promised you to show this example of the beta letters so uh again we have this uh exact or kind of uh analytic expression for a density of states you import all the important things from from tricks here's a bunch of definitions like bandwidths or chemical potential then you initialize the solver uh you construct your uh greens function from a semicircular density of states you're gonna do um i think we set five iterations five dmft loops where you do like a spin averaging of the up and down channel and then you feed this into the into the curly g where you just take a t square so half bandwidth over two uh to the square times your greens function you plug this into your solver by and also defining um an interaction Hamiltonian and that's it for the beta letters it is that simple of course for real materials it's not that simple uh then you can look at at your greens function as function of of iterations so this would be a metallic state and we actually have an own kind of mudplot lip implementation that makes it easier so it's called oplot that makes it easier to look at greens function objects okay so this is the fundamental building blocks there's a couple of applications i'm just going to talk about three of these there's kind of the dft tool side which is really the glue to all the ab initio codes there's solid dmft which is kind of an optimist way to run the whole kind of call the whole routine there's max end which does the analytic continuation and then there's a whole range of impurity solvers where you really take one for each each problem so this summarizes this a bit more i'm not going to talk about the cluster extension of vertex methods but focus on the dft plus dmft part so we have kind of interfaces to all known or popular dft codes and then the whole range of impurity solvers and solid dmft which runs this so this is dft tools um historically this was interfaced with into k but has been interfaced with vani 90 for a very long time there's also like internal interface with wasp and quantum responses only mainly via vani 90 a recent implementation with elk and our colleague olivier jangras working on ab init and what dft tools does is really kind of these basic basic basic functions that i've talked about in during during the talk like constructing lettuce greens function or extracting the local greens function which is just really the sum over k um down folding and up folding meaning this kind of uh protector functions calculating the chemical potential or the transport distribution then as for the impurity solvers i'm not going to go into details here uh the most important one is ct hype which is a qmc solver i think also the one that christian howley uses the most um we have ct seg ct ind and so on the main important aspect is that we have mapped out most of the like difficult spaces of materials but there are still challenges um one of them being strong off diagonal hybridization spin orbit coupling and low temperature uh there's recent kind of next generation solvers which is uh fork tensor product states that's actually the self that produced the self energy that i've shown in the in the slides there's inchworm and and then there's a recent addition which is kind of a hargy fox solver that was developed by jonathan karp under the supervision of alex so he can he can respond to questions about that and this makes it kind of really exciting because this is really a kind of a dft plus u solver but using vani functions right uh then i'll briefly talk about solid dmft which is a project with max merkel and albert ukata from eth suric um under again the supervision of alex hamper um so this is uh you know this you can look at the documentation but the workflow i've you know i've shown you this several times kind of just handles all the uh all the parts that are necessary for the dft plus dmft calculation this can actually be handled like a a bit like a dft code so you have a a config file where you specify all the necessary things for dmft you call the executable and you write all the green's functions are getting written into an hdr5 archive but you also have an observable file to track the convergence like tracking the occupation or the chemical potential and we also have convergence metrics where you see the change in the impurity green's function and so on um and so this can be installed like any other tricks application and is uh scalable with scriptable config files what we really kind of put emphasis on is the reproducibility um so it has versioning control hdr5 storage we have convergence metrics and we have a very flexible impugy solver choice that are developed at uh at the flood iron institute uh that i've also mentioned city hub city sec and so on and of course online documentation and tutorials so then i'm going to use the last two minutes to uh uh introduce our latest addition this is uh tricks pharmacy which is a web app that you're also going to see in the tutorial um so this is a project again with alex nears olivian entran and is based on input of vani 90 and tricks um and is based on uh plotly dash on the on the software side and if you want to break down this code into a single line what it does is compute the spectral function that you've seen several times and the only input that you need is the vani Hamiltonian and a self-energy and the self-energy can either be computed within codes like solid dmft or you can also do uh kind of analytical approaches where you give like a Fermi liquid parameters and so this uh web is online you can test it now but i prefer if you do it in the tutorial um and i'm just going to finish by showing a quick kind of clip what this looks like so you can upload a full h5 uh config i'm going to skip the tight binding parts but you can recompute it i'm stopping here for a second you can recompute it on a different uh you know different k path and so on change you can compute the chemical potential automatically and now we're going to enter a self-energy you can either upload it or enter manually in terms of quasi particle renormalization for the three different orbitals for example and also a correlation induced crystal field splitting and you have to enter a broadening and then you compute the spectral function this takes a second in the background and there you have the spectral function uh we also have an edc which is an energy distribution curve which is just a vertical line through the spectral function and you have an mdc momentum distribution curve which is kind of a horizontal line and you can uh you can just click uh and get the corresponding data uh you can show you can also plot the quasi particle dispersion you can change the color scheme and so on we also have the filmy surface and hopefully soon optical spectroscopy takes a second again because now we of course we need to we use vani interpolation to get a much nicer kind of representation and this just inherits everything from the prior slide there we go okay all right so with this the software part is finished in my talk as well thank you very much for your attention