 Perfetto, ok. Sì, quindi da me ne vogliamo questo, ma the enter full screen, basta se ci sono. The view, enter full screen, fit. Da come mai? View. Diamo così, comat F. Eccolo te. Non ci sono riusciti ad arrivare, ci ho provato. Ok. Ok. Thank you. I'm happy of being here, of course. You can see my affiliation. Actually I had an office to put floors off, but given the mandatory retirement age of Europe, I'm no longer, this is my main institution now. I wanted to comment on chemistry, what you said before. Many, many, almost all useful concept in chemistry don't have a first principle definition like unicity, covalency, aromaticity, whatever. Unicity, for instance, hopefully all unicity scales are monotonic also, but there are many, many different scales. Ok. Now, what is this about? I will pick up many, many things. I will just jump on what. I'm happy that his presentation by Nicola was very good because I will speak of many, many things he has already introduced very, very well. Ok. Now, this is the, I will start with the main concept about polarization. As you have learned, right now, position operator is not defined if you adopt periodic boundary condition. It is not an operator in maps a function which fulfills periodic boundary condition, bond for carbon, periodic boundary condition into some function which does not. And that was really the main bottleneck which hindered the development of polarization theory. While, of course, in a molecule there is no problem because if you adopt what is usually called the open boundary condition, that is, you use a square integral wave function, position is a very simple, elementary, multiplicative operator. Now, the change of paradigm occurred exactly 30 years ago, about by the day, because there was a workshop in the United States and David Vanderbitt and his students, Dominic Kismit, had a poster. I was there, I looked at the poster and that was really the ultimate solution of the polarization problem, at least in a band insulator. At least at the independent particle level. Now, nowadays, the polarization, routinely computed, according to the modern theory, but modern in quotes, because sometime ago a referee told me, why do you call it still modern? It is not modern anymore. It is 30 years. Ok, by now it is implemented in most quotes, you may know that, but very few textbooks have taken notice of the paradigm. And to look back at some textbooks, this is the Feynman lecture of physics, it says a complex crystal lattice can have a permanent intrinsic polarization. The book suggests that it is the dipole of the cell. But of course you also have learned that the choice of the cell is completely arbitrary. So if you choose this cell, you find this dipole. If you choose this cell, you find the opposite dipole. And then you may ask, which dipole is the right one? Neither one is the good, the polarization is something else. Other books are the Kittel, he says a ferroelectric crystal exhibits an electric dipole moment even in the absence of an electric field. And in a ferroelectric state, the center of positive charge does not coincide with the center of negative charge. But it is clear that the center of positive charge and the center of negative charge from the previous slide is ill-defined if you use periodic boundary condition, of course. If you cut the system in some way, with some surface, of course it is. But otherwise, if you use periodic boundary condition, that means you rub this side on this side, you get… What about this? Ok. Now, the astromermin… Ok, no, it's fine. Now, the astromermin says that the crystal whose natural primitive cell has an advantage in dipole moment are called pyroelectric, but it's clear also that there is no such thing as a natural primitive cell, as I've shown also earlier. So in the obsolete textbook, the focus is invariably on the charge density of periodic solid. And instead, slowly, let's say by the end of the 80s, early 90s, we can to the idea that the charge density is not all… There must be something else. And in fact, the density obtains from the square modulus of the conchamp block orbitals, and after the so-called modern theory was established, it is pretty clear that instead the only entries for the polarization theory are the phase factor of the conchamp orbitals as a function of k, more precisely the u orbitals as a function of k. And as the magnetic formulation yields the polarization in terms of the charge density of the 1A function. So I will start adopting this alternative formulation, of course because it is a school on 1A function, but also because here it is very, very intuitive. Now, one important point is that polarization is well-defined only if you have a charge-neutral system, exactly like the dipole of a molecule. Only if the molecule is neutral, you have a dipole which is uniquely defined. It does not depend where you set the origin or coordinate space. Now, suppose that you have a nuclei of charge EZL at some sites in a given unit cell. You choose a given unit cell, and then the nuclear, you can write the nuclear term in the polarization just as the dipole of the cell is here. But of course this is arbitrary modulus. Since I said a given unit cell, a given... What happens here? Why I've lost the... I've done something with this wonderful engine, I've done something wrong. But I said a given unit cell, and in fact the problem is that you might choose a different cell. If you choose a different cell, you find a different quantum, a different polarization which differs by the previous one, by this factor which usually is called the quantum of polarization. Next, if you write in terms of any function, then you need the electronic term. The electronic term is very simple, as shown by Nicola earlier. You take, again, you take a function in a given cell, which we call the central cell. You define the various centers, and then the electronic term has a very, very simple expression. It's just the dipole of the given cell times two because they are twice occupied. Here, again, it is arbitrary modulus of the quantum, and it is also arbitrary modulus of what I call the Marseille-Vanderville banter summation. That is just the capital U of K matrix, which was shown before. So no matter how you mix the occupied bands, you always get the same quantity. Again, modulus of the quantum. So this is the polarization theory. And you may ask, for an unbounded periodic solid, the total polarization is the sum as it must be of the nuclear term and the electronic term, the finite modulus of the quantum, and invariant by translation of the origin. Now, if you have a bounded sample, a very large bounded sample, a microscopic bounded sample, then you may ask, a polarization there is not a multivalent observable. It has a well-precise value, and this is fixed by nature. By nature, because nature wants to minimize the energy. So depending on what you attach at the end of the solid, you may have a different value of the quantum. Instead, if you are doing the theory with periodic boundary condition, either block state or guany function, you are by definition an unbounded solid without any surface. Insofar as you want to deal with a solid which is infinite, periodic, fulfills all of the space with block states and the like, you are not allowed to fix that quantum. The quantum must remain there. It is intrinsic in the observable. Even recently, we had some objections by knowledge of colleagues and we discussed with them. It is multivalent, but it is uniquely defined. It is uniquely defined and provided you use a primitive set. Because of course, if you take a superset, the quantum looks like different. But if you go back to Kittel, Kittel tells you that the lattice is an abstraction. In some sense, whenever a material is crystalline, by definition you may attach a concept which is a lattice, which is an abstraction. You may have quantum nuclei, finite temperature, solid alloys, chemical mix and the like. Whenever you have this abstraction, there is uniquely defined minimum volume of the cell. So you have to use the cell which has the minimum volume. In this way, it is sure that you have a multivalent, but uniquely defined observable. I tend to insist because this is a method which does not pass easily, even with two knowledgeable colleagues. That would not be good. I see it was, and it is smiling. Now, a simple example. Let's take alternating polyacetylene. Polyacetylene here in one dimension, the polarization is a dipole per unit length. That means as the dimension of the pure charge. So the quantum equals the unit charge. We focus only on the pi electrons. There are two pi electron per cell. So there is only one band, which is doubly occupied. There is only one relevant one function, which sits at the center of the double bond. And it is a very simple exercise to prove that in this particular one-dimensional system the polarization must be zero, model one, model one unit or charge. And you may ask, is this an artifact or is this something which occurs in nature? What happens if I have a bounded sample? So we made a pedagogical simulation several years ago with Roberto Carr and a postdoc in Princeton, a very cost-accuding. As an example, we took two polyacetylene chains. We terminated them in a different way. And both termination were polar. So the molecule has a dipole. Both molecules must have a dipole because they are asymmetric. And we computed just without using any very phase, any one function, just using the Gaussian code, which is the reference code for quantum chemistry. We calculated the dipole of this as a function of the number of monomers. And the result is shown here in the bottom figure that you see that insofar as the chain is very short, both have a dipole, which is non zero. But as the chain becomes longer and longer, you get that they converge to a value. This is dipole per monomer. If you divide by the cell A, you find just zero or one. So these are two possible manifestations. In the periodic system, you have zero, module one. You can find a bounded system where it is zero, another bounded system which is one. For a three-dimensional system, there are complications which I prefer not to speak here. It is the fact that if you have a bounded sample, you have the polarizing fields while the theory, it was almost implicit that the polarization theory only holds for zero electric field. So when you have a function, you have zero macroscopic field. And bounded samples which are spontaneously polarized in particular, if an electric material has a polarizing field. So there are complications. But the one-dimensional system is free from those complications. That's the reason why we have chosen a one-dimensional system. Now, in order to go on speaking of better phases and the like, I want to introduce quantum geometry. What is quantum geometry? The geometry, first of all, you have to define a distance to make geometry. And I define the distance in this way. Two state vectors, which belong to the same Hilbert space, have a distance which is minus the logarithm of the scalar product to the square. This is clearly gauging variant, of course, and is zero if the two states coincide apart for a phase. The states are normalized, of course, and it is infinite if the states are orthogonal. If you go back to the definitions and to axioms of a distance in the calculus book, this is not really completely co-share. There is an axiom which is not fulfilled, but it is not a problem. I think it is better to use in this way. And then, and so this is the distance. Now, the distance, you can, it is an identity. You can, the logarithm of the square is equal, of the square modulus is equal to the logarithm of the scalar product times the complex conjugate, the logarithm of the complex conjugate. Each of the two terms is not gauging variant. Each of the two terms may have an imaginary part. The real part keeps you the distance, but the imaginary part, what is the meaning of the imaginary part? The meaning of the imaginary part is very simple. If you write the scalar product as the modulus of the scalar product times a phase factor, the imaginary part is just the phase. And so this is the key quantity in what comes on because the connection fixes the phase difference between the two. It is clearly arbitrary. You can choose, you know pretty well that you can choose an arbitrary phase factor for Psi 1, another arbitrary phase factor for Psi 2, so this cannot have any physical meaning. And now you can say, given that it is arbitrary, why do you bother about this phase? And in fact, nobody bothered until, almost nobody until the early 80s when Michael Berry came out and said, phases are not so irrelevant, even if individually they are gauge dependent, they can be used, must be used to build quantities which are gauging invariant, and therefore are measurable. So it is famous, if I have, at that time he knew only a couple of phenomena which had the characteristics of what we nowadays we call a Berry phase. That was the Aron of Bohm experiment and that was the conical intersection in molecule. There was nothing else. But after that, for many years, Berry phases came out in every field of physics, including our one, electronic structure and polarization. If I have summarized the main message of this paper is that, first of all, we have learned since we were very young in order to have a measurable quantity you need an operator, you have to invent an operator such that the observable is the expectation value or the eigenvalue of the operator. And the paper by Berry says there are observable which are not related to any operator. They are just gauging invariant phases of the wave function, so completely revolutionary kinds observable. And the second message is whenever in quantum mechanics you may write a gauging invariant quantity, this quantity is, in principle, something which is measurable. And I will come back to this at the very end of this talk. That's just to anticipate. So whenever you have a gauging invariant quantity in quantum mechanics, in principle this can be measured. Now, if we want to do geometry in block states, with block states, you cannot do it with psi k, because psi k and different k are orthogonal. In a sense they are not in the same Hilbert space, and that you do instead in the u. The u are in the same Hilbert space, the u functions are all periodic, all obey the periodic condition on the cell, on the cell, not on the other. So they have a finite distance. And by definition the matrix is the infinitesimal distance, the infinitesimal distance between k and delta k. And the quantum, if you take the two derivatives of what is up there, of the logarithm, then you find what is in the bottom line. That is that I've done something wrong. I didn't want to do that. I don't know. Okay, I will not use the pointer. That's fine. I don't need the pointer. So in the last line you have the quantum metric tensor for one month, which is the infinitesimal distance in case space. And it is gauging invariant. It's not clear from the expression, but you can prove it is gauging invariant. Then you have also the phase, of course. Each of these two terms I told are not gauging invariant. The infinitesimal phase difference is by definition the connection. So the infinitesimal distance between the u states at k and k plus delta k is linear in delta k. You need the first derivative. And A is what is called the bery connection. The bery connection is gauge dependent because you can have an arbitrary phase factor in the u which contributes to the derivative. It is a real vector field because the scalar product, the product is purely imaginary. So you have an I in front in order to make it real. And if you take the curl of this bery curvature, you get something which is gauging invariant, which sometimes here in three dimension can be written also as a pseudo vector, of course, as a pseudo vector, but you can write as an anti-symmetric matrix, a pseudo scalar in three dimension, if you want to go beyond, it is an anti-symmetric matrix. Now, if there are many bands, the bery connection is that the sum on the occupied bands, of course, you are dealing with an insulator so the number of states is the same at all different case. And you can define a metric curvature tensor of the occupied manifold, such that the quantum matrix is just a real part of this tensor and the bery curvature is just the imaginary part, a part for a factor minus 2. At the end of this talk, I will also address very briefly methods and so in methods, I take the curvature of the occupied manifold, just summing on the occupied state with a Fermi function, which cuts the states. Now, bery phase, the famous theory, this is the very, very famous formula by Van der Mielenkismit, which came out, as I told, published in 1993, but showed the first time to the world in a poster in May 1992. And it is equivalent to the Van der Mielenkismit This has been shown by Nicola that was already known since 1962 so the equivalence between this formula of course in 62, bery phase was not existing yet, bery phase came 20 years afterwards, nobody called that integral bery phase, of course, but by now we recognize, actually Joshua Zuck recognized that it is a bery phase in 89 and so the transformation was known since the 1960s, what was not known was the physical meaning of this because when a function were a very formal, very formal topic used just for demonstrating theorems and in fact I would say the Marzari van der Bee paper have shown that any function are not just a formal topic as a very useful practical topic that I think this is also what changed with electronic structure in the late 90s. So the equivalence was known, this is the formula and it is just the the Brilman zone integral or even the reciprocal cell integral of the bery connection. The periodic gauge is mandatory so you have arbitrary as Nicola has shown you have an arbitraryness in choosing the phase of different states a different case you need to choose the same phase whenever you translate k by a reciprocal vector that means that the u are the translated u means a plane wave of course. So in this way you have a closed loop in reciprocal space and so the integral actually the integral over a closed path. The figure was drawn by David in one of the few papers very few paper we wrote together so one of the few papers called David and me and I've drawn this picture was drawn by him and came out from there. Now the key features of this this is invariant by translation of the coordinate origin both terms both terms of course depend on the origin both the electronic and the nuclear but not the sum of the two this is gauge invariant as I've shown before this is gauge invariant by the u the charge of the branch now you have a multi-valid observable if you want to use this as usually is done to evaluate born effective charges or piezoelectricity the choice of the branch is obvious you have to choose the same part even for spontaneous polarization is a little bit more delicate so this is used in that way now suppose I have a cubic lattice in three dimensions you have only one band the integral on the reciprocal cell is this triple integral but where is the very phase the very phase is the inner integral the inner integral the red one is a gamma for any fixed ky kz you have the integral or one dimensional very connection and now the problem is how to discretize because when you when you solve find the ground state by numerical diagonalization you have a diagonalization routine which gives you the eigenstate and the u they are not differentiable so the u at a different k have a completely random phase a completely random phase so you cannot do a numerical derivative as you would do in the standard way in other words the integral in ky in kz rather trivial you take a mesh and the like but the integral in kx that integral in kx is done on a function which is which is which has an erratic phase so it's not differentiable you cannot resolve by finite difference and the solution is like this we have to compute the integral the very connection that could they do and then you have to remember that as I've shown before the very connection is the infinitesimal phase difference and but we had the formula for what was the finite phase difference was just minus the imaginary part of the logarithm if delta k is along the x direction and now we discretize with n points at constant ky and kz we have to compute the integral of each of those lines the integral of those lines is computed using discretization shown there that means sum of the logarithms is equal to the logarithms product and now it is clear that the erratic phase factors are irrelevant are irrelevant if you remember that the last factor there the last factor there you don't need to diagonalize n plus 1 points in this way you say you have k0, k1 k2 up to kn that would be n plus 1 points but you want to close a group you want to diagonalize only on n points and in fact the last one must use the periodic gauge condition so if you do with periodic gauge condition and now this is politically clear that whenever you have even if the guy who coded the diagonalization routine inserted a random number generator and gives you by construction a random phase factor completely random, the random phase factor cancel out in the bra and the ket each bra is compensated by a counterpart in a ket and this goes away so the most erratic factor you can imagine is irrelevant harmless for this kind of formula if you have many bands you have n by n connection matrix the number of occupied bands and the formula is minus the imaginary part of the logarithm of the product of these matrices even in this case it is clear that since the determinant of a matrix is invariant by any unitary transformation here again the Mazzari Vanderbilt capital U matrix doesn't make any harm and you have something which is very well defined and this is the way actually how it is implemented in many many density functional codes so this is the way the pelagician theory works practically in all of the codes which have it there is a rather delicate point how to choose the quantum when you are on different lines when you have on different lines a different on different y and z I would say ok now let me let me after we understand that polarization I think I wanted to say the first one is the since both these observable have to do with conductivity I have to say one slide for fundamentals linear conductivity is a tensor of course is a tensor the current as a function of the electric field it has a symmetric part and an anti-symmetric part anti-symmetric part gives rise to a current which is always orthogonal to the field so it is a whole current or the transverse current that one is longitudinal of course if the material is an isotropic the current that will not be proportional to the field but of course since a tensor symmetric tensor can always be diagonalized there are at least three principal directions where the current is in the same direction as a field now transverse conductivity requires breaking of time reversal symmetry so for instance a ferroelectric material or a macroscopic external field macroscopic external field is a problem for one structure theory I will say a word at the end in general so we only speak of the so-called anomalous sole effect that means anomalous is something which happens because material by itself breaks time reversal symmetry for instance because of ferromagnetism if you have a pristine metal the electrons undergo free acceleration so if you have the homelow is only due to extrinsic effect if you imagine a sole without any extrinsic effect then the the system responds only at frequency omega equals zero there are no other response so the conductivity is a delta to omega equals zero the real part you have a real and imaginary part because of the Kramer-Krenig sum rule because of causality whenever you have a real part you also need an imaginary part in this formula you see that the real part has a delta to omega and then has a regular pair the delta is the finger print of free acceleration and the coefficient in front called the drood weight tells you the inverse inertia of the many electron system typically is a density over a mass for free electrons is simply the density over the mass otherwise you can think of it as an effective density over mass ratio for a given metal so in a given metal you always have an acceleration if we deal with insulator for time being in insulator the drood weight is zero and also the conductivity is gapped that means it's zero until omega reaches the main gap and now back to the Susa Wilkes-Martin sum rule this was in the in the PhD thesis of Ivo Sousa who is here and Martin was a supervisor let's go back to omega i omega i has been shown by Nicola what it is it is a lower bound for it is gauge invariant as I said but it's not, in general it's not a minimum except in one dimension in a two dimensional system in general it's not a minimum it's a lower bound but it is it is gauge invariant e non depend on the capital U matrix e now going back to the to the very main message whenever you have something gauge invariant is measurable and the question is can you ideally measure omega i how can you measure which kind of experiment do you have to perform ideally to measure omega i omega i is a ground state property clearly it cannot be measured by probing the ground state itself but it can be measured with some rule for the conductivity so Sousa Wilkes and Martin in 2000, the year 2000 proposed is integral just as a discriminant between insulators and methods all insulators and all methods not only mean field insulators and methods but also correlated methods disordered methods whenever you have something which is insulating or or metallic this integrates either divergent or finite in the special band case it is clear that in a metal this is divergent because you have a delta of omega over omega and this clearly is something which badly diverges instead if you have a band insulator as I said before the conductivity is 0 until the gap the integral is no longer from 0 to infinity but it is from the gap to to infinity and you have this integral and now you see the relationship so ideally if you want to measure omega i omega i non è una quantità intensa si tratta di una quantità intensa diventata da il volume del sen e questo è una parte di un constanto è solo uguale a Sousa, Winkens e Martin integrati il prossimo il prossimo è una conduttività anomalosa una conduttività anomalosa come ho detto è qualcosa che solo occorre in materiali di riferimento in materiali di riferimento una conduttività anomalosa ha da fare con il corbatore pericolato pericolato di un manifold occupato per un metallo o per un insulator questa è la funzione che è smooth in insulator e è piecewise continuous in metallo che è chiaro che posso sempre trovare una smooth gauge sempre una smooth gauge in questo senso nel caso topologico ci sono sabili su smooth gauge ma non per questa quantità e adesso dalla formula cuba la conduttività di scrivere come sempre lo troviamo in una teoria di risponsi lineari sulla formula cuba ma la formula cuba può essere trasformata Questa trasformazione della formula cuba in proprietà del grado è un grande risultato di Taules Comoto, TKNN, Taules Comoto, Deniz, e io credo che Deniz, il Nobel prize per Taules nel 2016, è quello che fa la quantizzazione, ma la formula è la stessa. La formula è la stessa, è stata soltanto... E rispettando il lavoro da Taules da 1982, il fatto che, anche in metalli, c'è una contribuzione di solito, che ha avuto solo in l'anno 2000. Ora, la formula... All'inizio, da la formula cuba, e per fare la trasformazione per Taules, potete scrivere la formula cuba per la conduttività transversa di DC, in qualsiasi insulatari e metalli, in qualsiasi dimensioni o tre dimensioni, nel modo in cui ho scritto, e per le due dimensioni, il caso è molto speciale, perché potete notare che c'è un H bar, in la prima formula c'è solo H, in la seconda formula. In un insulato, che è in grado è 1 over 2 pi times a dimensionless integral. Il corbatore ha la dimension of the squared length, quindi hai 1 over 2 pi times a dimensionless integral. Questo è un'integra. Questo è chiamato la CERN, quindi la CERN theorem in the 40s. Well before, this was useful in physics. Mathematicians have dealt with those quantities and got very interesting results. This shows that the conductivity is quantized and transverse resistivity of course is also quantized and by now H over E squared is a standard of resistance, universal standard of resistance, the inverse of what is there is called a one-clitzing. So one-clitzing, so the conductivity is just an inverse-clitzing times an integer. Let's go to what happens in metals, insulators and other. In ferroelectric metals, in ferromagnetic metals, sorry ferroelectric metals, this was discovered already by Hall himself, so Hall discovered one year or two years earlier the regular Hall effect and then discovered the anomalous Hall effect in materials which break time reversal, symmetry, spontanergy instead. The theory has been very, very controversial until the early 2000, as I said. There are for sure intrinsic contributions, like collids in different ways. There are two different... Difficult to separate experimentally, but they exist two different contributions, Cuse, Carter and Sejan. But it was universally recognized, as I said, about 20 years ago, there is an intrinsic geometrical contribution, which is exactly the same as for the quantum Hall effect, except it is not quantized. If you instead want what is now called quantum anomalous Hall effect in two-dimensional time reversal brick insulators, the theory came in 1988 in a very, very important paper by Duncan Alley, who also got the Nobel Prize in 2016 for this work and also for another important work, I would say. But it was very, very difficult to synthesize materials who may have this... where you may see this quantization. This was finally achieved in 2013 in China and then a couple of years afterwards in California and there is a very, very good career. In this case, of course, contrary to the metallic case, there are no intrinsic contribution. That is, a basic tenet of topology is that whenever you have some property which is topological, then you can do... This is very robust. You can do whatever you want to the system. The conductivity cannot change insofar as a system remain insulator. So only if the system passes through a metallic state, then you can switch the chair number. Suppose you have material where the chair number is 1. The conductivity remains 1, even in disordered case, even whatever you do, except when material becomes metallic. Instead, as I said before, in time reversal brick and metal, you must have extrinsic contribution because, of course, you would not otherwise have... Since all law is a fact of experimental, a fact of life, that means that impurities may also... must also contribute to the transverse. And finally, going back to the experiment, famous experiment by Fonklitz in Dortmund Pepper, the theory came out by TKNN, as I said before, but in this case, K cannot be a block vector for a very good reason. Whenever you have a macroscopic field, there is no way of writing a vector potential which gives you a periodic Hamiltonian. So the Hamiltonian is non periodic. You have to do something else. There are very exotic things, like the density of states changes abruptly. The problem is non-analytical B equals 0. You have a Landau level, so everything is very delicate. But in this respect, you can... the fact that you need the real reason why you have block states actually is translational invariance of the Hamiltonian, you may find a more complex symmetry which is the magnetic translation. You translate and you do something else, and so all of the algebra goes in the same way, and so you have the same thing. Next, I show only some reference, so the main reference of what I've been saying here in the first talk, and this one I would say is the book by David Landau, it came out in 2018, I guess, around four years ago by now. And also there are my lecture notes, which you find online. The address is a bit complicated, but if you google Raffaella Resta, you find either me or an hairdresser, a woman hairdresser in your room, and it's easy to choose, and you find those notes. And with that, thank you for your attention. Thanks a lot Raffaella for the very nice talk. So we are a little bit late on schedule, but I think there is definitely time for one, two questions. Sure. So thank you for the talk, first of all. I was just wondering how do these representations of polarization, so as a Berry phase or as a Vanier function, how do they differ, if at all, from the Dyson equation representation of polarization, for example, in the Eden's equations? That's Dyson equation, now it depends, Dyson equation is not Hermitian in a sense. So if you think about the GW representation, then in that case, I would say there is no problem, because they make a very big simplification. So the important point is that one body density matrix in the one body density matrix is a projector in our business, and it is not if you really want to do Dyson equation, but in GW, the approximation is that in general is a projector, so you can do, you can take, let's say the Dyson orbitals are not orthonormal in general, the true ones, but within GW you approximate it orthonormal, so in that case it can be done, so the GW polarization can be done. Instead I take this occasion for telling you something, which I forgot to say, good to know, if the chair number is non zero, in that case, you cannot have a periodic gauge and you don't have any functions. So the polarization in a topological non trivial material where chair number is non zero is a tricky business, something can be done, there is a very pretty paper by David and co-worker, but the theory as it is cannot work if the chair number is non zero. Thank you. Okay, so are there any other questions? Oh, thanks Raffaele, maybe just a little bit of a wild one, since at the end, electric fields and magnetic fields are a quadrivector in a fully relativistic treatment, all mixed together, shouldn't there be a fully relativistic sort of expression for all these quantities? No, because there is a profound mathematical difference, that is, the polarization came from the one form, there is only one derivative or one differential or whatever and the conductivity and manifestation also came from the two form. And this is enormous difference, in fact you cannot, you can write, you can write all of the servers coming from two form like transverse conductivity or magnetization in terms of the ground state projector. You cannot write polarization in terms of ground state projector. The only way would be to diagonalize the ground state projector and then compute the very phase. Any questions on Zoom? Please raise your hand or write something in the chat. Yeah, there is one. Okay, question why we can use polarization to compute the electric constant or can we do it? This was, I think I'm sure that there was already in one, in my slide, in fact cannot, not in the, not in its original form. There is a variant, which is also due to paper by a couple of papers, one of them by Ivo Sousa and David Vanderbitt, but not with the standard because as I've seen, as I said somewhere here in any case, whenever you have an electric field you don't have block states, you don't have one year function in their standard form, so you can compute derivatives of polarization by numerical differentiation. You can take with any perturbation, like piezoelectricity, ferroelectricity or born effective charges, even pyroelectricity, things like that, but not the electric field. This is the only one because what is implicit in all of this theory in its standard form is that the field is zero. Okay, so if there are no more questions I think we can thank Rafael again.