 At the end of the last lecture, I tried to motivate why in a realistic material context we have to deal with dynamically screened interactions. So today I would like to explain how we can do dynamical mean field simulations for models with dynamically screened interactions. And in the next lecture, we will then introduce some extension of the dynamical mean field formalism which also requires us to solve problems with dynamically screened interactions. So let us consider a Hubbard model which has a dynamically screened interaction. This is our model, like this model with hopping T and the interaction is frequency dependent. So in a Hamiltonian formulation this means we have a coupling to bosonic modes with characteristic frequency omega and this produces this frequency dependence in an action formulation. It just means we have a retarded Coulomb interaction. So if we map this now to the impurity problem in the interaction formulation, we just have an impurity system which has a retarded onsite interaction. So basically one correlated site in a non-interacting bath which is represented by this hybridization function and the interaction is now itself retarded. So it's a retarded interaction U of tau. U of tau is just the Fourier transform of U of omega. So the impurity action now has the following form, usual hybridization term, usual chemical potential term and then we have a retarded interaction which is of the following form, times U. So the screening does not distinguish between spins. So the long range interaction, the retarded Coulomb interaction also acts between the same spins. So we can write the interaction term in this form where the total density is just n up plus n down. And as I sketched yesterday, the typical interaction in the frequency space looks something like this. At high frequency we have a high value U bare and then at some frequency called the plasmon frequency there is a strong reduction of the interaction to some screened value U screened. So this is the real part of U of omega. And so we can separate this into a sort of static frequency independent contribution U bare and sort of screening contribution which is the rest. So we can write U of omega is U bare plus U tilde of omega. And this U tilde of omega now goes to zero at high frequency. So we basically shift everything down here by U bare and define this function U tilde which looks something like this. And this would be the real part of U tilde of omega. And this one goes to zero at high frequencies. And the static value of U tilde would then be U screened minus U bare. So U, that's just a shift. And then if we go to the time domain we will find that this constant value in frequency will give us a delta function contribution in time. So an instantaneous repulsive interaction corresponding to the bare value. And this will give us a retarded attractive contribution which now describes the screening effect. So U of tau is then of the following form, U bare delta tau plus U tilde of tau where this is sort of the repulsive bare Coulomb interaction. And this is an attractive retarded interaction which comes from the screening. So then the interaction we have to treat as the following form in the time domain. So it's a beta periodic function here zero minus beta plus beta. This is the time axis. So we have a delta function contribution with strength U bare. It's this one. And then we have an attractive other contribution like this which is U tilde of tau. And that is this contribution. And that should be compared to a simulation where we just take the screened static value. Very often up to now people have just said, okay maybe these are very high energies. We don't care what happens at these high energies. We just take the static value here. And then the description would be one which has the following interaction. It just has a delta function like interaction with the reduced screened interaction like this. So that would be the conventional sort of static description in terms of the screened interaction. But now we want to treat the full problem where we have a very strong instantaneous repulsion from the high frequency component or high frequency limit and then a reduction of this interaction from an attractive retarded component which describes the screening effect of the environment. And so in the following we will also use spectral representation of this retarded interaction. So this is a kind of bosonic function. It is beta periodic. So we can write it as follows. You can write U tilde of tau in terms of the imaginary part of the frequency dependent U as minus 1 over pi is infinity to infinity d omega. And then imaginary part of U tilde of omega. That's a spectral density and then this bosonic kernel e to the minus omega tau e to the minus beta omega plus 1, minus 1. Then we can also use the symmetry of imaginary part of U to split this integral into two parts. And then we can combine these two and just write integral from zero to infinity d omega. Imaginary part U tilde of omega and then cosine hyperbolic omega times beta half minus tau divided by sine hyperbolic beta omega beta half. So that's the relation between the imaginary part of U of omega and U of tau. And we will use this expression in the following. Yes? Here? Yes. You are wondering what this is? Yes. So if you would study a Holstein model where it just has one bosonic mode coupling to the electrons, your frequency dependent interaction would look as follows. You would have basically a pole at the boson frequency. And then if this is your U bare, it would have basically a pole structure like this. So that would be the effective retarding frequency dependent interaction from a boson with energy omega zero coupling to your electrons. And so now in reality you have sort of a continuum of bosonic modes in a certain energy range and then it smears out this pole and you get sort of a smeared out version of this pole structure. So for what you are saying in the following, it doesn't matter how... I mean, this function can be anything, just an input of the calculation. You put in this in a basically tabulated form and use it. So now I would like to explain how we can easily extend this algorithm on the color algorithm which I introduced yesterday to these types of retarded interactions. So I showed you these segment configurations and the only thing we now have to understand is how we can compute the interaction contribution for this type of frequency dependent or retarded interaction. So let me write the interaction term once again. It was this term with the total density and retarded U. So this we now have to evaluate for some given segment configuration where the total density is an up, plus and down. And retarded U has these two contributions, the bare one with the delta function and the retarded one like this. All right, and now we just plug this in and evaluate this formula. Then we get the following from this delta function. We get the term with the bare interaction like this. We also have contributions and up squared and then down squared. And then we use the thermionic property that n squared is n. And again, the delta function then gives us a second contribution U of half integral d tau n up plus n down. So basically these are the contributions from n up squared and then down squared. And then we have the contribution from the retarded part which looks as follows. We have the contributions from opposite spins and we have contributions from the same spins. Yes, sorry. So let's draw our segment configuration. This is the time interval. We have up and down. And suppose we have some segment here and here. Okay, this term corresponds to this overlap contribution which we introduced last time with the bare interaction. Okay, so this is this U bare times the length of the overlap. That's this first term here. This term here is basically a shift of the chemical potential by U bare half. So this is like a chemical potential contribution and this we can absorb into a shift of mu that takes care of this term. And now we have a retarded interaction between up and down electrons. So that would be something like this. So we have say tau 1 here interacts with some tau 2 here. That would be this term. And finally, we also have sort of intra-segment interactions of electrons in the same segment like this. So that's this contribution. Okay, so this one would be, for example, n up tau 1 U tilde tau 1 minus tau 2 and down. And now we have to integrate over all times in these segments. So let us split up this contribution into the contribution of all pairs of segments for all sort of intra-segment contributions for each segment. So to write this in a general way, we now introduce a name for each segment. We call it say the segment number i is denoted by i. Then we can, for this inter-segment contribution, we can just sum over all pairs of segments in our configurations. So this is the sum over all k1 different from k2. And then the integral over segment k1, the integral over segment k2. And then this retarded interaction. And then we have the kind of intra-segment contribution where the both integrals run over the same segment. And that is one half sum over all segments k. And then the integral of both times over the same segment like this. So this would, what we call the kind of inter-segment contribution, that would be the red guy here, would be the intra-segment contribution. And now we have to evaluate these double integrals. And that is easy if we know the twice integrated function u tilde. Then we can easily evaluate the double integral. So let's assume for the moment that we know it, and that it has specific symmetries, which we can assume by properly choosing the integration constants if we do this double integrals. So let us assume that a function h of tau is a bosonic function symmetric around beta half, which satisfies that the second derivative is the retarded interaction. This basically means that we want a function which has the same properties as the retarded interaction. It's beta periodic and symmetric. And the second derivative should be our retarded interaction. And then we can easily express these double integrals in terms of this function. For example, let's do this double integral here. I define the start and end point of segment number one. And so we want to evaluate this double integral. And with the help of our function h, we can express these as follows as minus h of tau one end minus tau two end plus h tau one end minus tau two end plus h tau one end minus tau two start plus h tau two tau one start minus tau two end minus h tau one start minus tau two start. So that's very elementary. And if we draw again our segment configuration, the one which I just erased, we now see that this contribution can be sort of considered as an interaction between the end points of these segments. For example, this would be tau one start, tau one end, tau two start, tau two end. So we have an interaction tau one end minus tau two end. That's something like this. And then tau one end minus tau two start. That's something like this. And we have also this kind of interaction. And this would be minus h of tau one end minus tau two end. So these are the input segment contributions. Now let's look at the intra-segment contribution. This we can also evaluate in a similar manner, tau just from tau start to tau end. And then the same tau start, tau end that we can then write as follows as minus two h of zero plus h of tau end minus tau start plus h of tau start minus tau end. And if we use the symmetry property of this function, we can write this as two times h of tau end minus tau start because h is even. And so then we obtain the following. We obtain h of tau end minus tau start minus one half h of zero minus one half h of zero. And this we can graphically represent as follows. This is tau end minus tau start. And then we also have sort of a loop here, which is minus one half h zero. And similarly for this segment, we would have an interaction line like this and a loop. So that's the evaluation of these double integrals. Then if we combine the contributions, basically these lines from all segments, we get the following interaction energy from the retarded interaction. And this follows minus n times h of zero minus now sum for all pairs of operators, i not bigger than j, s i s j h tau i minus tau j. So this s is a sine factor, s i is plus one, if the operator at position i is a creation operator and it's minus one if it's an annihilation operator. For example, here I probably just made a mistake because I don't know, it's okay. Yeah, for example here you see that this line connects a creation operator and an annihilation operator and it has the opposite sign from this line, which connects an annihilation operator to an annihilation operator. So depending on what type of operators we connect by these lines, we have a different sign here. So that's the interaction contribution and we can rewrite it a little bit if we want as follows as minus sum over all pairs and these sine factors and then h of tau i minus tau j minus h of zero. And you can check that basically because of cancellation effects from these signs, only each segment contributes in the end one factor h of zero and that gives them n times h of zero in the end. Good, so now we have sort of our weight contribution from the retarded interaction. We just have to figure out now what is h for some retarded interaction and so we have h, the second derivative of h is the retarded interaction u tilde and this we can represent or write in the spectral representation as this integral over omega from zero to infinity of the imaginary part of u tilde of omega and then this cosine hyperbolic function omega times tau minus beta half divided by sine hyperbolic omega beta half and now we just have to basically integrate this function twice and since it's written here as a kind of in Fourier space it just means divide twice by the frequency so each integral gives a division by omega so we basically have to now divide here by omega squared and that gives us the final formula which is that h tau minus h of zero which appears here in this weight is one over pi integral from zero to infinity d omega then imaginary part of u of omega divided by omega squared and then the difference between two cosine hyperbolic functions is like this so that's the explicit form of this function and this we can in principle just read in and tabulate in our program and then this determines these contributions to our weight which is then easily evaluated so in the end we can basically treat arbitrary frequency dependent interactions at almost the same computational cost because evaluating these lines is very cheap compared to for example manipulating the determinants of the hybridization matrices so it's basically for free we can treat these retardation effects so here I should say that this formula is valid for tau between zero and beta and then it is an even function of beta and the beta period it can even function and so this brings us to one slight subtlety which we still have to properly take into account namely this function h minus h of zero has a slope discontinuity at zero because of this symmetry so it looks like this here zero beta minus beta it's basically a function which looks like this and so the formula h double prime is u is only valid sort of between zero and beta but not at this point the first derivative has a discontinuity and the second derivative will give us a delta function because of this symmetry so at this point this relation is violated and now we have to correct for this violation of this relation so basically now the first derivative of h at this point has a jump and the second derivative has a delta function of zero whose weight we can compute from here the weight of this delta function is basically two times the first derivative at zero for two times h prime at zero plus that's the weight of the delta function and so let's calculate two times h prime of zero plus from this formula that will give us two times one over pi an integral from zero to infinity e omega imaginary part of u tilde omega over omega times minus one so that's the weight of this delta function and now if you look at this formula maybe it reminds you of the Kramos-Kronig formula which relates the real and imaginary parts of a complex function so then you can recognize that this is nothing else than the real part of u tilde at omega equals zero so the static part of u tilde that's basically by using the Kramos-Kronig formula so what this means is that if we use this h function we have to subtract the delta function contribution with weight real part of the static value of u tilde and so what was this static limit of u tilde so if we remember the picture so we had this shift by u bar so the static value is u screened minus u bar so that is minus u screened minus u bar delta of tau so the total instantaneous interaction therefore becomes the following so previously we have just the u bar term but now we have u bar plus this u tilde this real part of u tilde at omega equals zero and that if we use this relationship just gives us u screened times delta tau so basically what this does if we properly take into account this slope discontinuity contribution it just means that if we calculate the overlap here we have to calculate it with the screened interaction rather than with the bear one and that takes care of this so this now becomes u screened times the length of the overlap okay so then we are basically done and we know how to calculate the local interaction of a given segment configuration so the weight is exponential then we have the overlap contribution now with the screened interaction instead of the bear one then we have the shifted chemical potential also actually with the screened interaction times the length of the segments and then from the retarded part of the interaction we have this sum over all operator pairs the sign, a sign to each operator whether it's a creation or an annihilation operator and then this h function h of tau i minus tau j minus h of zero where we have the explicit expression for this h function this is basically the imaginary part of u divided by omega squared times this cosine hyperbolic function so the Monte Carlo sampling proceeds in exactly the same way as before except that we replace the bear interaction by the screened one shift the chemical potential and we have to add these sort of non-local interaction lines between all pairs of operators which we can explicitly evaluate with this formula and which is quite cheap so meaning if we insert the new segment here like this then basically whatever operators are around we have to compute all possible non-local interactions with these new operators like this but that's as I said is actually a cheap thing to do it's not really slowing down the simulation in any significant way so that's all I wanted to say about the algorithm so if you have questions about the algorithm you should ask it now how it would look like yes so if you want to represent this in a Hamiltonian form it would look like as follows so the local part of the Hamiltonian would be u times n up and down minus mu n up plus and down and then you would have a coupling to an infinite set of bosonic modes of the type say g lambda n up plus and down times v dagger lambda plus v lambda plus some lambda, omega lambda so this is an oscillator with frequency omega lambda and then you couple the electrons to these bosons via this type of coupling and basically the g lambda would be related to the imaginary part of u so this is more or less your imaginary part of u at the frequency omega lambda because now if you have such an electron boson system and you integrate out the bosons you will precisely get this kind of retarded or frequency dependent interaction but it's relatively complicated to think about it in terms of Hamiltonian so it's much nicer to work with the action where all these bosons are integrated out and you just have a retarded interaction in the imaginary time but physically this is really where it comes from I mean yesterday I sketched a little bit this constrained random phase approximation where you do the down folding and these bosons more or less the particle hole excitations of electrons into higher bands which produce these sort of bosonic modes which couple with a specific coupling strength and have associated energy which is related to the separation between the bands basically if you have this is your low energy theory which you are interested in now you have some high energy band and you can do this kind of transition that will produce some bosonic excitation with an energy say omega lambda like this that produces your screening of the Coulomb interaction of this particular energy the P-lambda the P-lambda should there be an extension of the electron of the band you are considering some modes you are not considering having? Yeah that sort of this separation into low energy and high energy screening modes which I discussed yesterday so in principle both right so for the fully screened interaction you would have also the low energy excitations within the band like this but in the if you want to compute for this low energy theory you should actually exclude these processes otherwise you double count these processes because in the solution of the impurity problem you explicitly create this kind of screening contributions but if you compute the fully screened interaction you also have to add these okay so then I would like to move on to the next topic which is how to define a dynamical mean field formalism for problems with long range Coulomb interactions yes or you want to stop now so maybe I can more or less get started and then we so the next topic is trying to devise a dynamical mean field formalism for models with long range interactions and as we will see is that this also leads to a formalism with dynamically screened interactions and the formalism is called extended DMFT and has been developed actually in various different contexts for spin systems for various types of systems by different groups and this is a DMFT formalism for models with especially non-local long-ranged interactions and then as an example we will discuss the so-called u v Hubbard model u is the onsite interaction v is whatever intersite interaction the Hamiltonian is the following we have the usual kinetic term the usual chemical potential term and then the onsite interaction as in the Hubbard model and now also an interaction between different sites where this n here is the total charge n i is n i up plus n i down and these angular brackets we denote the nearest neighbor pair so we take here just as an example the nearest neighbor interactions but this can in principle be anything so the new term is this non-local Coulomb interaction so it will be interesting to see how we can treat such a term in a DMFT approximation which is based on a local impurity problem but the first step is to rewrite this in an action formulation we will work with the action that's easier and first let's write the action for the lattice just this lattice model so we use a Grasman variables and then the action can be written as follows integral over the imaginary time of well some kinetic term that's the kinetic term written in the action formalism and then the interaction term plus u i n i up of tau and now i down and the interaction which we can write as a general v ij and n i tau nj of tau and now for this specific model with nearest neighbor now hopping and nearest neighbor interaction we would have that t ij is minus t times delta ij so only nearest neighbor hopping and this v ij would be v times delta ij so interaction v only between nearest neighbor sites but in principle it can be anything, it doesn't matter now the first thing is to slightly rewrite the interaction term so we write this term here as follows as u half sum over all sites and then n e up plus n e down squared minus u half sum over i n e up plus n e down so that is again using this property that n squared is n for fermions we can rewrite the interaction term like this and so this becomes again a shift of the chemical potential and here we have the total n this is n i squared and so we can now combine it with this long range part which is also written in terms of the total density on a given site so that's the reason why we rewrite it like this then we find the following expression we call mu tilde if we want this shifted chemical potential and here I have now written a v tilde and this v tilde in our case of the nearest neighbor interaction nearest neighbor hopping is just u times delta ij that's the on-site interaction plus v times delta for nearest neighbor interactions and so if we are on a square like this we can write down the Fourier transforms of the hopping and this interaction matrix easily so if we only have nearest neighbor hopings the Fourier transform of the hopping matrix gives us a epsilon of k which is minus 2t cos kx plus cos ky and similarly if we only have nearest neighbor interactions we can Fourier transform this v ij tilde and this gives us a v tilde of k which is u from the on-site interaction and then plus 2v also the same cosine okay now the interesting part starts and maybe we will then really go through this in detail after the break but now we want to map this to a single-site problem and in order to do this we have to first decouple somehow the long-range interactions and that we do by Hubbard-Stratonovich transformations and this transformation now replaces the inter-site coupling by the local coupling to the Hubbard-Stratonovich field and then if we have only local couplings we can sort of map the problem to a single-site effective model which has also a coupling to such a Hubbard-Stratonovich field and then we are back to a single-site description with coupling to a Hubbard-Stratonovich field and that is more or less a bosonic field and then we have a dynamically screened interaction and the screening now comes from this Hubbard-Stratonovich field and that's more or less the extended DMFT formalism and I think we probably make a break now and then after the break I will explain in detail how this decoupling works I don't know if I should repeat the break now Yes, okay, thank you