 All right, so we will now decouple this long-range Coulomb interaction by Habat-Stratonovich. And there are various types of Habat-Stratonovich transformations for fermions, bosons, real periodic fields. And so the density is real and better periodic. So we will use the corresponding transformation, which is the following. This is some kind of general interaction written with some matrix A. That can be decoupled as follows, one over square root of a determinant, and then the integral over all the Habat-Stratonovich fields. So capital N is the number of sites in the lattice. And we have as many Habat-Stratonovich fields as lattice sites. Now we have a term which is quadratic in these fields. And we have a coupling between the density and the Habat-Stratonovich field, where in this notation somehow the summation is implied by a kind of matrix vector notation. And these fields, phi, they are also real and beta periodic. And here this A is, of course, just the non-local interaction v tilde. So if we apply this transformation to our lattice action, we obtain the following transformed action. So the hopping part, now from the Habat-Stratonovich transformation, quadratic part in the Habat-Stratonovich field phi i tau v tilde r minus 1 ij, ij of tau, plus then a local coupling between the Habat-Stratonovich fields and the electrons. And here in these curly brackets, we recognize the non-interacting or the inverse of the non-interacting Green's function of the lattice. So this we can write as G0 minus 1 component ij, where G0 is the non-interacting lattice Green's function. And if we use this notation, then we see sort of an analogy now between the fermionic part, which has this quadratic term with the non-interacting lattice Green's function, and the bosonic part, which has a quadratic term with this inverse Coulomb interaction. And so this then motivates us to map this system in the extended DMFT approximation to an impurity problem, where we have a fermionic vice field, which describes this hopping of electrons into the bath and back. And in analogy to this fermionic vice field, also bosonic vice field, which sort of describes the retarded interaction between these Habat-Stratonovich fields. So then if we just let's just write down the form of this action. So we map it now to the following impurity action in this e DMFT formalism, namely a quadratic term with the fermionic vice field and the retarded sort of hopping of the electrons. That's the usual term, which we have in DMFT, where this is the inverse of this non-interacting impurity Green's function. And then in analogy to this, we have a similar retarded term for the Habat-Stratonovich fields with some non-local interaction, which I call curly u inverse, which now couples these fields at different times tau and tau prime. And that is sort of now the bosonic vice field. And this is the fermionic vice field. And then we still have the local interaction between this bosonic field and the electrons like this. So this thing here would then be our fermionic vice field. And this curly u would be our bosonic vice. And these need to be fixed by a self-consistency condition in such a way that this impurity action somehow reproduces the properties of this lattice model, at least the local properties. So the fermionic vice field is just kind of Green's function. So this has sort of the properties of a Green's function in imaginary time. It's beta-antipariotic, for example. It looks like this. This is g of tau, where this is beta, 0. Whereas this curly u of tau is more or less of the form of the dynamical interaction, which we have discussed before. It's sort of a beta-teriotic function. It looks like this, this kind of general structure. And both of these functions now have to be fixed in a self-consistent way. Now, it's somehow not very convenient to work with these Haberstra-Donovich fields. But once we have done this mapping onto the signicide problem of this form, we notice that it's quadratic action in the phi fields. So we can sort of undo our Haberstra-Donovich transformation now for the impurity problem and integrate out, again, the phi fields and go back to a purely electronic action. And that will now become an action with a retarded interaction between the electrons. So in the following, we can, after sort of undoing our Haberstra-Donovich transformation, we find the following action impurity system. The usual fermionic part, which is unchanged. Well, this quadratic phi term, after the Haberstra-Donovich transformation, is replaced by a quadratic term in the density. And with the u, instead of the u inverse, and this part here disappears. So we have one half integral 0 beta theta theta prime. And now n of tau currently u tau minus tau prime, n of tau prime. And also from these determinants, which appear in the Haberstra-Donovich formula, we get another term, which is minus logarithm of square root of determinant of u. Now you see that we have end up with an action, which is exactly of the form which we have sort of studied in the last lecture. Some impurity system with a general retarded interaction. But now this interaction has to be computed in a self-consistent way by a self-consistency condition, which I still need to explain. And so this term with the logarithm of the square root of the determinant, we can rewrite a little bit as, for example, one half logarithm of the determinant of u. And then we use a formula which says that the determinant of u can be written as the exponential of the trace of logarithm of u. So this is exponential of trace log u. And then the exponential cancels with this logarithm. And then we find this is one half trace log u. So this is the action, which we really want to use then in the calculations. This is the action with the Haberstra-Donovich field, which was useful to derive the single side action. Now we need to derive two self-consistency equations, one for the thermionic-wise field and one for the bosonic-wise field. Basically, this will be, again, based on the identification of the local laggis-greens function and the impurity-greens functions. And both the thermionic and the bosonic-greens functions. So now we have to compute the thermionic and the bosonic-greens function of the impurity or problem. So how are they defined? Thermionic-greens function is just, as usual, the expectation value, or maybe with minus sign of C, C dagger, that you should be familiar with. That's the impurity-greens function, where this expectation value means now the expectation value evaluated with this impurity action. And the bosonic-greens function, we call w, is the corresponding correlation function for the phi field. So that is the expectation value, phi tau phi of 0. So now we want to work with this action in which we have integrated out the phi field. So it's a little bit non-trivial how we can get this phi-phi correlator from this action. So we need to do a little manipulation to see how we can get this from such a purely electronic action. And so first we start with this action here, which I call star. So from this action with this Harbott-Stratonovich field, we can easily see that this correlation function is simply the derivative of the logarithm of the partition function with respect to u minus 1. So if we take the derivative with respect to this bosonic-wise field here, we get the phi-phi correlation function. Bosonic-greens function can be written as 2 times d ln z d u minus 1. So that follows immediately from this expression here. And now we can manipulate this a little bit and see that this is the same as 2 times d ln z d u times d u d u minus 1. And d u d u minus 1, this is the same as minus u squared. So this we can write as minus 2 u d ln z d u times u. And now we go to this action here, with the phi field integrated out here. You see we have the retarded interaction u in the action. So we can easily take the derivative of the logarithm of the partition function with respect to u. Now we evaluate this term from this action here. And we find d ln z d u is nothing else than, well, from this term we get the density-density correlation function. If we take the derivative with respect to u, so 1 1⁄2 times n of tau every time ordering. And now if it matters n of tau, 10 of 0. And now we have to be a little careful. So we have another factor of u here, 1⁄2 log u. And so if we take the derivative of this one, we get another contribution, which is minus 1⁄2 1 over u. Now we plug this into here. We find the following expression that the bosonic impurity grains function can be expressed as u minus u chi local u with chi local density-density correlation function. And this is something which can be very easily measured with the algorithm that I explained in the previous lecture. So we calculate this with say Monte Carlo. And we know u. And then we can compute the screen interaction. So that's how we can do it. Maybe a little practical remark. Usually it's better to define the interaction term with respect to the density fluctuations. So we subtract the average density from these densities here. And then this chi local becomes sort of the connected charge correlation function. So OK. Now I think we have everything to set up these two self-consistency loops. First loop is the fermionic self-consistency loop, which is completely identical to the one for usual dynamical mean field theory. Let's rewrite it as follows. We have calculated our impurity grains function by the Monte Carlo technique, which I described. Then we compute next the impurity self-energy by using the impurity Dyson equation. And then we say that the lattice self-energy, the fermionic one, should be this impurity self-energy. So we approximate sigma of k i omega n by sigma impurity of i omega n. Where here i omega n are the fermionic Matsubara frequency. And then we use this approximate self-energy to compute the lattice grains function. So we write gk i omega n, which I mean the exact expression from the Dyson equation would be the non-interacting lattice grains function inverse minus sigma k i omega n. And this whole thing inverse, that's the exact expression. And this we approximate now with the impurity self-energy here. We have a lattice grains function. And next we compute the local lattice grains function by averaging over momentum. And now we use the dynamical mean field self-consistency and say this should be the same as the impurity grains function. So this is the DMFT self-consistency. If we do this, then we can use, again, the impurity Dyson equation and the impurity self-energy to obtain our new Weiss field, g inverse nu, which is now g local, inverse plus the old impurity self-energy. And then we use our Monte Carlo or whatever impurity solver to solve the impurity problem with this new Weiss field. And that gives us the grains function. So that's the fermionic. Now in complete analogy to this fermionic loop, we have a second loop for the bosonic. G grains function and the bosonic Weiss field. So let's assume we have computed our bosonic impurity grains function. Then we define the self-energy, bosonic self-energy, which is nothing else than the impurity polarisation function, which I call here pi. And that is defined as the bosonic Weiss field minus the inverse of the impurity grains function. Next step is that we approximate the bosonic lattice self-energy by this impurity self-energy. So we approximate the p of k and i nu n. So nu n is now a bosonic Matsubara frequency. We approximate by this impurity polarisation. And then we compute with this approximate self-energy or polarisation the screened interaction in the lattice. And so the exact expression would be the following, that w of k and Matsubara frequency i nu n would be, well, the bare interaction, the inverse of the bare interaction, 1 half v tilde minus 1 minus this momentum dependent polarisation function. And this inverse. But this we now approximate by this impurity polarisation. So we find 1 half v minus 1 minus impurity polarisation. That's now our approximation for the lattice grains function, bosonic one. And now we sum this over k, or average over k, to get the local grains function or the local screen interaction, supposed to be the same thing as our impurity fully screened interaction. That's the bosonic self-consistency condition. And if we use this, we can now use the impurity Dyson equation again for the bosonic version of the Dyson equation to extract a new bosonic vice field curly u. And as follows, we get the new curly u minus 1, which is from the bosonic Dyson equation, this w local minus 1 plus the impurity i. Now, in principle, we can solve our impurity problem to get new impurity bosonic grains function, or screened interaction. But we need this additional step. So the solver basically gives us, first, the chi local, this density-density correlation function. And from the density-density correlation function, we can then, and this curly u, we can get the impurity w. So that's the bosonic loop. And so now you see the complete analogy between these two loops, which has to sort of replace the vice field chi by u, curly u, the self-energy by the polarization pi, and the grains function by the sort of screened interaction w. And then there are two completely analogous loops, which fix both the hybridization function, or this curly chi, and the retarded interaction u in a self-consistent way. And of course, to solve this, we have to employ a method which can create arbitrary retarded u interactions. I think you do it at the same time. I mean, you compute w and chi in the same simulation, and then you go through both loops at the same time. You update both the curly u and the curly g. So it's sort of simultaneously the same. You update both. And the solution of the impurity problem gives you both the w and the g. Yeah, it's one loop. Yes, yeah, yeah, it's one loop. Yeah, in principle, each step contains both. You do it simultaneously. So then maybe as a last topic, I can say a few words about combining the so-called GW method with this extended dynamical mean field formalism. Of course, up to now, I mean, what is very nice about this formalism is that we now self-consistently compute the interaction u. So we sort of create the screening effects in the system in a self-consistent way. And that gives us the dynamical u. But we still have a completely local self-energy. So this pi and this sigma is completely local. And in a real system, of course, we have some momentum dependence in these self-energies, sigma and pi. So one idea to build in some momentum dependence into this scheme is to combine this DMFT or extended DMFT self-energy with the momentum dependent self-energy of some weak coupling perturbation theory. And one obvious weak coupling perturbation theory is the GW method. So GW method is sort of the lowest order self-energy approximation in a systematic expansion in the screen interaction w. What does the GW method? So in the GW method, we approximate the self-energy, sigma, by the product of the interacting Green's function G, and sort of the fully screened interaction w. That's why it's called GW. And we approximate the polarization pi simply by the bubble of interacting Green's function like this. But this is now momentum dependent. So there can be different sites, i and j, here. So these are non-local interactions. And so one idea, which is actively sort of explored in this Abinishio and DMFT community, is to apply this method, which is widely used sort of in the Abinishio world with dynamical mean field theory. And the idea is just to basically take this perturbation theory and replace the local contribution from these diagrams by the DMFT analog, which sort of corresponds then to a different subset of diagrams. So DMFTs believe sort of supposed to sum up all the local diagrams. And now in addition to these local diagrams, we can add the non-local GW diagrams. So that's the idea. So written as a formula, we now construct a self-energy sigma GW plus EDMFT, which depends on momentum and frequency. As follows, we take the EDMFT self-energy, which is only frequency dependent. We add to it the momentum dependent GW self-energy. Now to avoid the double counting between these two, we subtract all the local GW self-energy diagrams. So we subtract minus k sum G sigma GW k i omega n. And the same thing for the polarization. So we can define a polarization i in GW plus EDMFT, which depends on momentum and on sonic frequencies. That is the polarization function from EDMFT. We add to it this non-local polarization bubble from GW. And now to avoid double counting, we have to subtract all the local GW diagrams. So that's how we can construct some momentum dependent self-energy. And so how does the scheme look like now in this version? Well, we start, for example, from a converged EDMFT calculation. So this converged EDMFT calculation. This gives us the impurity self-energy and the impurity polarization. Now we construct a momentum dependent Green's function and polarization in the usual extended DMFT spirit. So we compute the G of i omega n, which is approximated as G0 minus 1 minus sigma impurity minus 1. And w a i nu n approximated as 1 half v tilde minus 1 minus polarization of the impurity. And then we compute the local Green's functions and from this device field. So that's all the same as before. So we compute G local i omega n by summing over k. And then we define the nu vice field using the impurity Dyson equation as G local minus 1 plus impurity self-energy. And the analogous thing, we have our impurity problem defined. And so we solve this impurity problem. So this gives us a nu impurity Green's function and a nu impurity w. And then we can define a nu self-energy. So the sigma impurity is the vice field minus the inverse of the impurity Green's function and the high impurity, the same with the bosonic vice field and the bosonic Green's function. OK. So up to now, everything is as before. And now comes basically this GW plus DMFT step. Do we? Yeah? OK. This one? Yes. Yeah, in principle, yeah. I mean, that's just a GW self-energy, which is non-local. I mean, in this diagram, the Green's function and W are non-local quantities. So it's, in principle, a k-dependent object. Yeah, all the local GW diagrams. So if we write this out, say a local GW diagram would mean something like this from i to i. And now the interaction sort of contains all kinds of doubles like this. And here, in principle, we can have i, k, l, i. So basically, all diagrams which go from i back to i will be eliminated. And all the diagrams which go from i to j different from i, we keep. Yeah? Yeah, that's the idea of this double counting. First, remove all local GW diagrams. Yeah, that's right. So that's what I'm just going to write again. So now we do the GW calculation. Now, as we evaluate the GW self-energy, so pi of GW. So I write it here now in tau space is sum over q. Green's function, q of tau. Green's function, q minus k of minus tau. So that's what I drew as a kind of bubble. And the self-energy, the only one, is the product of the Green's function and the screen interaction of this diagram. So that's the calculation of the GW diagrams. And then we extract a non-local part of these diagrams by subtracting the local contribution. So we calculate pi GW non-local. That's a usual GW diagram minus all the local GW diagrams. This really is just subtracting all the local polarization bubbles. And similarly for the self-energy, we combine this non-local part with these local impurity self-energies. Right now, the momentum-dependent polarization has the sum of the impurity polarization plus this non-local part of the GW polarization. And we write the self-energy fermionic one in the same way. This is impurity like this. And then we go back to this step two until convergence is reached. So now we have a new estimate of a momentum-dependent self-energy. With this, we can compute the local Green's functions and new wise fields. And we write this a couple of times until a converged solution is found. And so in principle, this is nice because if we now want to combine this with our initial calculations, where if we start from this GW, GW band structure, then we know exactly what are in a diplomatic form, what are the correlation effects included in this GW band structure. And then we can properly do the double counting if we add the DMFT self-energy. And that's not the case if we start from a LDA band structure where we don't know what the correlation effects are in a sort of diagrammatic language. So this is now an area of active sort of research to implement this scheme really in a material context. Good. So now, in principle, I have prepared some exercises. But I think we are running a bit late to start exercises. So maybe I'll just discuss one of the interesting things, which was also part of the exercises on the board here. Yeah? Yeah. If you want to do it for materials, yes. That's right. You get sort of, yeah, you get the orbitals to define your kind of hubbub model for the correlated orbitals. And a single band model. Few band models, yeah. That's in practice, you can of course only treat a few correlated bands. So then you define some, again, some low energy subspace for just a few orbitals. And you get the interaction from GW for these orbitals. And then you start this loop within this low energy subspace, and you recompute the interaction in a self-consistent way. The GW, you also, but yeah, I mean, yeah. In principle, there would be a feedback even from this low energy subspace to the high energy bands. And I think this, at the moment, is not realistic to do this. But what we can do and already do now is if you stay, if you do one GW calculation, then you restrict yourself to the low energy space. Then you can update the GW and the DMFT calculation self-consistently within this small subspace. And I hope that on a slightly longer time scale of maybe one year or so, we can maybe define an intermediate energy window where we really feedback the DMFT self-energy sort of into the GW part and self-consistently update the GW in a larger window of maybe 20 electron volts or something, but not the 100 electron or 200 electron volts for which the initial GW calculation is done. I think that's not realistic. Because nowadays, I mean, the GW calculation for this very large energy window is much more expensive than the DMFT calculation. So yeah, I think we can only really self-consistently do it within a small space. But that's what we do now. And it will be interesting to see if it's really an improvement over previous schemes. But. Yeah, this has already been done. Basically, you do GW, and then you do DMFT. Maybe you just combine the results at the end. Yeah, I mean, that you can do. And it's not that clear whether the self-consistent procedure is giving you better results. I mean, usually in the GW community, it's known that one shouldn't do GW self-consistently because the results are just not good. And the hope was that if you combine it with DMFT, if you have sort of a local vertex from the DMFT, that you can really do it self-consistently. Because this kind of one-shot philosophy of GW, that's really a bit.hoc and unsatisfactory. And maybe if one combines GW with the local part of the self-composure from DMFT, maybe one can really do a meaningful self-consistent calculation. That's the hope. At least on the model level, it works. And we now are trying to see what it gives for materials. So then maybe in the last 15 minutes, we can just touch a sort of interesting little topic, which is the relationship between this hybridization expansion, which I have described, and a very simple sort of semi-analytical scheme, which is called a non-crossing approximation. So that's maybe some people in the audience may be familiar with non-crossing approximation. And so maybe let's briefly discuss the relationship. And so let's look again at the simple case of a spinless fermion model. The partition function is the one we had on the board yesterday, trace time ordered product of e to chemical potential term u and the hybridization term. That's our non-interacting electron, which is hybridized with some non-interacting bath. And now let's expand this partition function in powers of the hybridization function. Then we get our partition function expressed in terms of these segment configurations, which we had on the board yesterday. We have these right simplicity like this. So here we have a C dagger on C, also the other way around like this, C, C dagger, and so forth, and many, many more. So in second order, we have this kind of diagram, of course, also this kind, and also this one, and also this one. And now finally, at third order, we find a diagram which has crossing hybridization lines. The first one with crossing hybridization lines looks like this, and many, many more, OK? What's the weight of these diagrams? So basically here we had 1. Here we had e to the beta mu. Here we had e to the mu times tau 2 minus tau 1, some times delta, tau 2 minus tau 1, and so forth. Now the point is that we can generate the whole subset of diagrams without crossing hybridization lines in an elegant way with some kind of Dyson equation. So all diagrams, except for this one on this board, we can generate in a sort of iterative manner by defining so-called pseudoparticle Green's functions. So let me explain what this is. Important point, those which do not have crossing hybridization lines, we can generate by introducing so-called pseudoparticle propagators. We have two in this system. We have two states. We have the empty state, and we have the full state. And for each of these two states, we now introduce such a propagator. For the empty state, we call it G0. And this sort of bare propagator has the following form. It's minus e to the minus energy of the empty state times tau, the energy of the empty state in this problem. I mean, I'm talking now about the empty atomic state. It's just 0. So if you have no electron on the impurity, the energy is 0. Similarly, we can define sort of a bare pseudoparticle propagator for the singly occupied state minus e to the minus energy of the singly occupied state, which in this model is just mu. Singly occupied electrons have energy mu. And you can see that in these diagrams, we basically always have these atomic energies appearing in the weight factor. But we can really sort of produce these weights by considering such pseudoparticle propagators which have the atomic energies, eigen energies, in the exponential here. And so how do we do this? We now define the following pseudoparticle Dyson equations, the Dyson equation for the empty state as follows. So we have an interacting pseudoparticle Green's function for the empty state. This is the bare pseudoparticle Green's function for the empty state, which is defined here, plus the particle Green's function for the empty state. And now comes a kind of self-energy insertion, which is of the following form, the interacting Green's function for the singly occupied state times the hybridization function. Then comes the bare and the full pseudoparticle Green's function for the empty state again. So this is a usual kind of Dyson equation where this is our self-energy. So this thing here is now like a self-energy, this pseudoparticle Green's function. So we have a sigma 0 in this non-crossing approximation, which is of the form interacting with the pseudoparticle Green's function 1 times hybridization function beta. This would be delta of minus tau, and this would be the self-energy tau. And in the same way, we can write down Dyson equation for our singly occupied state. That's the bare pseudoparticle Green's function for the singly occupied state plus bare times some other self-energy for now the singly occupied state, which has the form interacting Green's function for the empty state times the hybridization function, which goes in the opposite direction like this, and then G1. And so here we have another self-energy now for the singly occupied state, which is sigma 1 NCA equals G0 and delta. There may be some sign from the arrow which goes to the right. If you think about it for a little while, you will find that if you iterate, these are now two coupled equations which you can iterate starting from these pair propagators. You will generate exactly the subset of diagrams without these crossing hybridization lines. Because in the first iteration, you will have one hybridization function like this. Then you replace this interacting Green's function here by the expression here that will generate you further insertions of hybridization functions that you will never have a crossing between hybridization functions if you iterate this. And so it's a very simple calculation. So if we write down these Dyson equations explicitly, look as follows. So on the time imaginary time axis, these two Dyson equations can be written like this, from 0 now to tau, from 0 to tau 2. So that's the first one. And similarly for the G1, where you have two coupled differential equations. Here, you have the definition of G0 and G1. And basically, you have to solve this with the initial condition that G alpha of 0 is minus 1. Similar to this, their friends functions for alpha of 1 and 2. And then you get these interacting pseudo-particle Green's functions. And that's a very, very simple problem. Then you can easily see that the partition function is nothing else than the sum of these interacting 0-particle Green's function evaluated at beta and the interacting 1-particle Green's function also evaluated at beta. And why is this the case? This is because this G0 evaluated at beta is the sum of all diagrams without crossing hybridization lines, which start in an empty state and end in an empty state. And this expression here is nothing else than the sum of all diagrams which start in an occupied state and end in an occupied state. So these two give you all the diagrams without crossing hybridization lines from 0 to beta. So this is basically the contribution of the empty state, basically this contribution to the trace. And this is the contribution of the occupied state. And similarly, you can easily see, so just one minute, I have to maybe say how we get the Green's function. Now, the physical one, these are sort of auxiliary type Green's functions. So we need to say how we get the physical Green's function. And so what is the physical Green's function? Well, the physical Green's function is this. Within the NCA approximation, now the physical Green's function is 1. It's basically 1 over the partition function in NCA, which we have just written down here. And then graphically, say we have here a C dagger. So here's our time interval. Here's a C dagger. And here's a C operator. And this is time 0. And this is time beta. So we want to measure the Green's function where we create a particle at 0. And we annihilate it at time tau. Now we can obtain all the diagrams consistent with C dagger here and C here by just taking here the modified Pseudoparticle Green's function, G1, because here we start with occupied state. So it's a G1. And then here we annihilate an electron. So here we are in the empty state. So we continue with G0. That gives us all the diagrams, which are consistent with a creation operator here and an annihilation operator here. And that we can write as 1 over C NCA times a bubble of Pseudoparticle Green's functions like this. Now this is G of 1. And this is G of 0. So we basically take the product of G1 of tau and G0 of minus tau. And that gives us the physical Green's function. So that's a very simple calculation. And a few years ago, when I gave this as a kind of exercise to implement this NCA calculation, and one student basically solved this in less than 10 minutes. So he programmed the whole thing. And so it's very, very simple and fast impurity solver for the MFT, which works at least qualitatively in the strongly interacting regime. Basically multi-insulating regime. This is a good method. In the metallic regime, it's not very good, because this restriction to non-crossing diagrams is not really a good approximation in the metallic regime. But if you have time, and are interested in these methods, maybe it's fun to implement these coupled equations and solve them and see what it is. Yes? Something like these diagrams with cross denials. Well, it's somehow the result is always too insulating. Basically, the hybridization function in a metal is decaying slowly. So in an insulator, your G of tau is going like exponential, decaying exponentially with time. Whereas in a metal, your G of tau is basically going like 1 over tau. So it's decaying slowly at long times. And the hybridization function is behaving in the same way as the Green's function. So the hybridization function in an insulator is decaying very fast. The hybridization function in a metal is decaying slowly. So that's how one can understand why in an insulator maybe crossing hybridization functions are not so relevant. But in a metal where your hybridization function is not decaying fast, of course the crossing diagrams are important. And that's why this approximation is not working well in the metallic phase. It even becomes unphysical. If you do this approximation at low energies, at low temperature in the metallic phase, you'll get unphysical results, like causality violations in the spectral function and things like this. But in the mod insulator, where electrons don't hop very much, basically this approximation works rather well. OK, I guess if there are no further questions, we can stop here. Thank you for your attention.