 Yeah, just up and down, and then you can, you can, one of you. What title is Bariology Using Vani Interpolation? That was the question. Right. Great. Yes. So, exactly, I'll be talking about Bariology Using Vani Interpolation. And what I thought I would do is divide the talking two parts. So in the first part, I will give you a feeling for some of the physical effects where Bari phase type quantities appear. And I will focus on optical effects and transport effects. And in the second part, then I will try to tell you a little bit about how in practice we use Vani functions in the sense that David was just mentioning as a tool to calculate such physical responses for physical, for real materials. Okay. I guess you already saw my little joke there, but so this, this was an idea of Stepp and Zirkin. He told me, why don't you go to a search engine and type Bariology to see what comes out. So I did that. And this is what appeared. So something to do with the berries and Mediterranean diet. So clearly we are in the right country to talk about Bariology, so that's great. But yeah, we'll talk about different types of berries. And I will choose two to focus on this talk, which is the berry connection, which is for, so in David's talk, the berry connection was defined mostly by taking a cell periodic block state and taking dinner products of that block state in Bend N with its gradient and again Bend N. I'll do a slight generalization where the brand, the cat have different Bend indices and so it's now a matrix in the Bend indices at every K point. And I will argue that this quantity appears very naturally in the description of optical properties. And then the berry curvature is defined by taking the Bend diagonal elements of these berry connection matrix and then taking another derivative, so taking its curl and so it's a property of a single band and again as a function of K. And I will show you, actually David already did it, but a bit more about how this quantity appears in the description of transport. Okay, so let's start with optics and let's remember the basics that we learn in undergraded quantum mechanics in atomic physics. So we start with a two level system and let's focus on two levels which I label N and L with energy E N and E L and we come with the light whose frequency is precisely the difference between these energies. So we're going to optically excite an electron from the lower to the upper state and let's say that the light has polarization along some Cartesian direction A. So one of the basic quantities that we learn about is the oscillator strength that tells you the strength of this transition and what the formula looks like is there is some pre-factor containing the energy difference omega L N and then there is a matrix element of the position operator R along the polarization of light between the two states and then you take the magnitude squared of that. Now you remember from David's talk that the velocity operator can be defined as the commutator of the coordinate operator with the Hamiltonian and if you do that you can rewrite the oscillator strength as shown here in the second line. So it's actually easier to go from the second to the first by writing V is the commutator of R and H. What did I do? Are we still on zoom? It says connecting on my screen. I don't know if I maybe might have if you do multiple fingers you can switch to a different screen. Did I do just one finger? Yes. Okay, sorry about that. Okay so the point of this little derivation was just to tell you that to calculate optical transitions one way of doing it is to evaluate matrix elements of the velocity operator. So that was for atoms, now I cannot scroll. Now it works, okay. I'm sorry. Okay so now if you are in a crystal of course the energy levels are organizing two bands but we have pretty much the same situation where now your state your initial state is a state in a valence band N and your final state is in some connection band L and you want to evaluate the oscillator strength and of course the states are block states so a plane wave with a cell periodic part and I would like to spend a little time in this derivation just to show you how the very connection appears. So we take the matrix element of the velocity operator between the block states L and N and we're going to write the velocity as the commutator of R and H that's the first line. Now to go to the second line what we do is we plug this formula for psi as a plane wave times U and I'm going to absorb these phase factors into the definition of H of K. Okay so what we have is still a commutator of R with this H of K and you can easily convince yourselves that if we differentiate this formula here with respect to K so my notation is that dA is just a shorthand for d or a partial derivative with respect to the A component of the wave vector so if you just differentiate this expression here you get the commutator right. Okay so that's to go from the second to the third line and the final step to go from the third to the fourth is another little trick which is you take these so these are block eigenstates so they are eigenstates of the Hamiltonian so this matrix has this form and we're just going to take the derivative with respect to KA of this entire equation. So you're going to get a bunch of terms but one of them is the one that appears in the third line which is the one where the derivative acts here right then you're going to have another term where it acts here another term where it acts there and then also it acts over there. Anyway you collect the terms in the right form you solve for the one you want in the third line and what you get is that you get two terms one contains the derivative of the energy eigenvalue and then it has a delta function so it vanishes if the bands are different that's the interband velocity so the band gradient which we all know about and then conversely if L is even from n that term vanishes but then we have the other term which actually contains the very connection and the way you see that is that for example when you act with the derivative on this cat here then you're still left with the Hamiltonian that can act on the bra and it pulls out an energy eigenvalue and then conversely when you act with the derivative here you get to the energy eigenvalue of the other state and you organize them into a difference yes right so the interband part is basically like the Helm and Feynman theorem I'm basically deriving if you want the Helm and Feynman theorem yeah well I don't think we made any assumptions apart from the state being eigenstates of the Hamiltonian so this is just straight forward manipulations but that's sort of how you derive the Helm and Feynman theorem yeah so if you go back to our diagram we were interested in interband transitions so this first term which is interband doesn't really play any role but you see that to evaluate the velocity matrix elements that we wanted they basically involve the berry connection so that was sort of what I wanted to motivate that it's a very natural quantity that appears in the description of optical transitions and by the way it appears multiplied by this omega ln so when you plug it there since it's squared the omega ln will go to the numerator so actually the berry connection matrix itself is basically the matrix elements of the position operator between different block states in different bands so it's kind of like an interband dipole matrix element okay so when you write something an optical response like the dielectric function and you focus on the interband part it's then very natural that the formulas that you obtain involved off diagonal elements of the berry connection as shown here for the dielectric function so there is here an F and L which is just the difference between the Fermi occupation factors of the two bands and so this is the kind of quantity that we want to evaluate to calculate within band theory the dielectric function of some material I will flash very quickly a slightly more complicated example you don't have to understand everything but it's just to give you the feeling that you want to go to for example nonlinear optical responses so this is now a current that is second order in a frequency dependent electric field so things get a bit more complicated of course you have more indices the farmers get a bit longer but the basic building blocks are still the same so you need energy eigenvalues to get the energy conservation you get the berry connection for the oscillator strength and in this nonlinear response you get the second berry connection that appeared here in the linear response in the nonlinear response becomes kind of a K derivative of the berry connection matrix it's actually what's called a covariant K derivative what that means is that if you only had the first term it would not be well behaved under gauge transformations but when you add these difference between band diagonal berry connections the entire object is now well behaved and and then the physical response is gauge invariant which should be to be physically measurable but so basically it's again all about berry connection type quantities okay so that was the examples I wanted to give about optics let's now switch to transport and and see how the berry curvature appears so David mentioned in responding to one of the questions this me classical picture of Chiang news group where you create a wave packet in some band so by the way here I'm focusing on a single band so I remove the band index so we have some band that is well localized in case space but also somewhat localized in real space we can define its average position in real space are an average position K in reciprocal space and we apply electric and magnetic fields so the total force electric force and Lawrence force gives you the change in K the rate of change in K and the total velocity in real space of the wave packet has two terms one is the band velocity and the second one is K dot cross the berry curvature and that terms looks very similar to the Lawrence force you replace our dot by K dot and replace the magnetic field by the berry curvature so the berry curvature is a kind of a magnetic field in reciprocal space if you want so for the rest of this talk I will set to zero the real magnetic field so I will talk about transport in response to just electric fields so we cancel out that term and we replace the expression that remains for K dot in the first line here and we get this expression for the velocity so there is the band velocity and then there is a second term which is known as the anomalous velocity by the way here the berry curvature was a vector and here I've rewritten it as a second rank tensor so in three dimensions a vector has three components but an anti-symmetric tensor has three independent components so we can repackage the vector as an anti-symmetric tensor so okay so that's one ingredient we need to calculate the current so to get the total current in the crystal what we do is we add up the contributions from the wave packets everywhere in the band but the band may be partially occupied so we need to only add up up to the Fermi level so we need to multiple supply by a Fermi occupation factor so to get the current we plug this expression here for r dot into this formula here and we still need an expression for the distribution function and solving the Boltzmann transport equation to linear order in the electric field we get this expression here so f that f prime is V there is the band velocity so the band gradient so we insert r dot here and f there and by the way these are just leading terms in an expansion in the electric field but we stay at low order so we collect terms to linear order in the field and that gives us the linear conductivity and we collect terms to second order in the field and that gets us the first nonlinear conductivity that has three indices so the linear conductivity has two terms one is we take the band velocity and perturbed by the electric field and we multiply it by the distribution function at first order in the fields so that gives us a product of two band velocities and f prime and then there is another first order contribution which is you take the velocity corrected to first order in the field that's the anomalous velocity and we multiply it by the unperturbed distribution function and that gives us this second term here so notice that the first term is symmetric in the Cartesian indices A and B so that's the usual textbook ohmic conductivity the second one as I told you before the very curvature written as a second rank tensor was anti-symmetric so that gives you an anti-symmetric or whole conductivity so that is the linear conductivity to second order in the electric field in the expression I wrote there is only one term which is the anomalous velocity that picks up one here and the first order distribution function that picks another e over there so we get basically the very curvature multiplied by the velocity and the distribution function f prime I'm going to do a little manipulation here which is to say that I'm going to write Vc f0 prime as one zero that's what the prime means so there is basically a chain rule here I can write that as one of a bar df0 dk and then I can do an integration by parts and transfer the derivative to the very curvature and I get the expression here in the second line so to summarize focusing now just on the whole parts of the responses highlighted in yellow we have a linear whole effect that is given by the integral of the very curvature over the occupied states and at second order in electric field we have a nonlinear whole effect that is given by again summing up over the occupied states not the very curvature itself but it's gradient and these are called anomalous whole effects in the sense that David explained that they occur at zero magnetic field so remember early on I crossed out the magnetic fields everywhere so they are anomalous because the usual whole effect of course requires a magnetic field okay let me talk a bit about symmetry to understand under which conditions the linear and the nonlinear anomalous whole effect can occur so if you have time reversal symmetry the energy bands are the same at k and minus k and the very curvature is equal and opposite at k and minus k so that means that when you do this integral you're going to pick up equal and opposite contribution from k and minus k that cancel out and so the net integral is going to be zero because of course if a state is occupied at point k since the energy is the same but minus k it will also be occupied but the very curvature will be opposite regarding the nonlinear whole effect the situation is the following so let's consider the effect of inversion symmetry again the energy bands are even under k but now the very curvature is also even so if the very curvature is an even function of k its gradient is going to be an odd function and so again there's going to be a cancellation between k and minus k when you do this integral with inversion symmetry so the conclusion is that to get a linear anomalous whole effect we need a system that breaks time reversal symmetry and to get a nonlinear anomalous whole effect the system should break inversion symmetry to guarantee that the required integrals are nonzero by the way if the system breaks both symmetries but it has the combined symmetry of time reversal and inversion the combination of these two conditions means that the very curvature vanishes everywhere in the Bruno's zone and in that case both the linear and the nonlinear anomalous whole effects vanish okay so here are some pictures I'm going to focus on a two-dimensional system on the left hand side it's a magnetic system with magnetization out of the plane and the very curvature is now a scalar or a pseudo scalar that also points out of the plane and in a system that breaks time reversal symmetry the very curvature has this kind of profile so in this case it's mostly negative but it's it has basically the same sign let's say in most of the Bruno's zone in such a way that when we integrate over the occupied states we get a nonzero value so this the experimental setup is that you apply a voltage along x for example and because of this very curvature vector pointing out of the sample along z you're going to have a transverse current along y in the case of the nonlinear anomalous whole effect now on the right side the system is non-magnetic but it breaks in inversion symmetry so because it's non-magnetic the net very curvature in some bands is zero so there is equal amount of blue and red so blue is negative red is positive but because it breaks inversion symmetry locally at a given point in case space the very curvature is nonzero and so in this case it's negative on the left positive on the right so there is going to be a kind of a dipole indicated by this pink arrow which is precisely this gradient quantity that I wrote before that describes the nonlinear anomalous whole effect so the geometry here is that if you apply an electric field along the direction of these pink arrow let's say along x you're going to have a whole current a second order in the field along y okay so that is the survey I wanted to do about the effects that we are interested in and to summarize for optics we need energy eigenvalues and off diagonal elements of the very connection and for transport we need the eigenvalues in their gradients the band velocities and also the very curvature and when you go to nonlinear responses then you may need further derivatives of such quantities such as for example the gradient of the very curvature and what I would like to explain in the second half of the talk is how vanier interpolation is a very convenient tool for calculating such quantities so the setting we have in mind here is what David was mentioning in response to the last question we have some system with several bands and we're going to vanier rise like you did in the tutorial yesterday a few of those bands that that where the action is taking place so if it's an interband transition we want to include the initial and final band within our energy window if it's transport we want to make sure that the Fermi level is in this inner window so that the Fermi surface is very well described and then we want to evaluate these kinds of quantities so we have our vanier functions I'm using this notation where capital R is the cell the lattice vector of the cell and J labels the vanier functions in the cell and then we're going to define a block basis function by doing this kind of Fourier sum and by the way I'm going to use a slightly different phase convention to the one that you've seen in all the talks so far there was not this tau J here in the similar formulas you've seen before in previous talks this is just a pure matter of convention what this tau J is is the center of the vanier function J in the home unit cell zero and the reason I want to do this will be will become clear later on but bear in mind that this is just a choice that you can make so vanier 90 does not include this tau J in the Fourier sums Python type binding does for example but okay I'll come back to that later so we have our block basis functions that span the vanier eyes bands and we set up at any given K the matrix elements of the Hamiltonian between different block basis states and we get this Fourier sum expression where on the right hand side we have the matrix elements of the Hamiltonian between the finding functions okay so we do this Fourier sum at a given K point and we get a small matrix n by n where n is the number of vanier functions in per unit cell and we diagonalize that matrix with some unitary matrix you and the eigenvalues are the interpolated energy bands so when we plotted yesterday in the tutorial some energy bands that's all that the code did very simple and remember these are small matrices because we only vanier eyes a relatively small number of bands so this can all be done very quickly across the brilliance on okay so this is nothing but orthogonal type binding using the funny functions as the basis orbitals and once you've tabulated these real space matrix elements of the Hamiltonian in the vanier basis so they are the onsite energies and the hopping so we can reuse them at every K point to evaluate the energy bands okay so the basic workflow is that we perform the I've initial calculation on the course I've initial mesh that was the NSCF step in yesterday's tutorial let's say for example a 4 by 4 by 4 mesh we use that as an input for Vani 90 to get our vanier functions then we set up this Hamiltonian matrix elements we start them and then we Fourier transform back to a dense mesh of K points to do the things we want like the energy bands and I'll show you later how to do the more complicated quantities like the very connection okay let me however before doing that show you just an example this is a work we did recently with you and the bank is asperos and for them to the one so this is a 2d material is kind of like graphene but some of the chains are pure carbon chains and then they are interleaved with bottom nitride chains and what I'm showing you here in this figure is the hopping matrix elements of the Hamiltonian in Vani basis so what we did here we we chose a Vani basis where there is one PZ orbital on each atom and what you see is that because Vani functions are exponentially localized these hopping matrix elements are very short range so they are only large between Vani functions that are near to each other and as you move far further apart then they very quickly drop down to zero because of the exponential localization so that means that in those Fourier sums you only need to keep just a few near neighbors hoppings to to get accurate results for the interpolated energy bands showed shown in red here so that's also related to what Jonathan Yates was showing yesterday that the accuracy of the interpolation was exponential in the number of k points because the number of k points in that initial mesh tells you how many Vani functions you have in your supercell so how many of these neighbors you can keep so in the end you don't need that many because when you are here you can include them but they don't contribute anything so the accuracy just converges exponentially fast okay so as I said before this is basically tight binding in the Vani basis and these matrices you that diagonalize the Hamiltonian I want to think of the columns of those matrices you as the tight binding eigenvectors and the next thing I want to do is starting from the block basis states which were this size with the W superscript I want to rotate them using the same matrix U that diagonalize the Hamiltonian and those will be my Vani interpolated block eigenstates and the reason I want to do that is that later on we want to calculate things like very curvatures and very connections which are properties of the wave functions so we need to have the Vani interpolated wave functions themselves to calculate these kinds of wave function derived quantities but before doing that I just want to do a couple of steps of algebra that I will need for what will come up next so bear with me for a little bit so this is the same equation I wrote before the one that we used to interpolate the energy bands but now I want to think of these columns of the U matrix as states so the tight binding eigenstates so in that notation I can rewrite this equation in a kind of a DRock Fanny DRock notation with double lines which I used to distinguish from the true DRock notation for the true block eigenstates so this is just a regular looking matrix element of the tight binding Hamiltonian between two tight binding states and since they are eigenstates this is a diagonal matrix and what I want to do next is precisely what I did here I did before for the true eigenstates I want to repeat the same algebra for the tight binding states and I want to do that just for off diagonal matrix elements and I will go very quickly here because it's the same that I did before but I just want to introduce a little bit of notation that will be used in what follows I will introduce a matrix D beggar as a Cartesian index a U beggar K drift of U so this is a matrix so it has two band indices L and N like so and in my condensed tight binding cat notation this would be just L so you see that this looks a lot like the very connection it's only missing an I but it's kind of a very connection for the tight binding states as opposed to the very connection for the real ab initio or Vanny interpolated block eigenstates and okay and by the way so what we need here to evaluate these D matrix is just sandwich between you dagger and you which we already have the gradient of the tight binding Hamiltonian age of W and that's very simple because remember that age of W was just as free as some and we want to take a K derivative but K only appears in the exponential so we only take the exponent down when we take the derivative so this is very simple to evaluate and again these are all small matrices okay so to define things like very connection and very curvatures we want the cell periodic block eigenstates so they are defined in the same way the Vanny interpolated ones just by rotating the the ones with the W sub superscripts with the U matrix and those are given here and so now we can proceed and evaluate things like the very connection and very curvature so let me start with the very connection so that was the definition and we want to take the gradient in K of the cat right which has the band index and so remember from the previous slide that the state was given by this first equation here so it has it is rotated by this U matrix so now now we want to take the K derivative of that so the derivative is going to act on two places going to get on this state oops right and then it's going to act on the U but remember the formula I wrote here U dagger U grad K U now I want I need the grad K U so I need it now because I want when I act with the gradient on this equation one one of the terms is going to be the gradient of U so anyway I do that and I get this expression here for the gradient of the Vanny interpolated block state it has two terms because of the two terms over there and I plug that into the expression for the very connection matrix and with a few steps I get this formula here so there are two terms one is and I've removed the band indices just to and clutter things so these are all matrices one is ID so now I put an I here there and it kind of looks like a very connection but in terms of the so this is the tight binding better connection if you want and then there is an additional term where you did sandwich between you dagger and you you have a very connection again defined in terms of the block basis states not the block eigenstates so the ones with the W superscript and you remember what those were they were given here in this formula here so I can just very easily obtain this expression for those matrix elements so this is very similar to what David showed in the previous talk when he was talking about hybrid Vanny functions that you needed matrix elements of the position operator between different hybrid Vanny functions so here it's the same but it's between real Vanny functions and here comes the point where it is useful to use this face convention that includes so these little towels there the reason is the following is that remember that when we were talking about the Hamiltonian I talked about on-site energies and hoppings so the on-site energies were the diagonal elements of the Hamiltonian in Vania basis and the hoppings were the off diagonal ones so we can use a similar terminology for position matrix elements so there are also on-site matrix elements which are which are just the Vania centers in the home cell and then there are off diagonal matrix elements which I can call our hoppings for example and which are given by this expression and that's precisely what appears here in this equation okay so again I'm reading on writing on top the expression for the very connection matrix with those two terms and the point I want to make is that a very common approximation in tight binding is to discard these are hopping matrix elements and so if you do that these matrix a of w is zero and so the very connection the Vanny interpolated very connection matrix reduces to the tight binding one okay but when we do Vanny interpolation we don't really need to make this approximation but for example when Sinisha will talk in a couple of days about Python tight binding that's the approximation that is done there so it's kind of nice to see how Vanny functions can be used to bridge the tight binding model worlds that some people are familiar with and I've initial world and you can somehow systematically go from one to the other by dropping certain terms so that's kind of what we're doing here okay so to summarize in Vanny interpolation you always work with small matrices and you do Fourier transforms back and forth in the case of the energy bands the matrix elements you need are those of the Hamiltonian in the case of optical matrix elements they are Hamiltonian matrix elements and also position matrix elements and both can be divided into on-site and hopping terms and sometimes people neglect the hopping terms for the position matrix elements and we were actually curious to see how good of an approximation that is for real materials so recently we did a calculation on that material I showed before BC to N and here's the result so remember before I showed you how the hopping matrix elements of the Hamiltonian decade with distance and I'm showing you something very similar here but these are now the hopping matrix elements of the position operator so these are on the left panel they are the X component and on the right panel they are the Y component this is a two-dimensional system and you see that eventually they also drop off very quickly but actually it's kind of interesting that the first nearest neighbor hopings are kind of small and it goes up and then it goes down again and so we can calculate the dielectric function that I showed the formula very early on in the talk and the shift current so they all depended just on very connections and those derivatives and we can evaluate them first by completely disregarding these hopping matrix elements of R so doing what would be a tight binding calculation and then we can progressively bring back the first second third nearest neighbor hopings and see how the calculated optical spectrum changes what you find is that for the linear response so the linear dielectric function already the tight binding the bare tight binding result is basically converged so when you add corrections from hopings of these are operator the shape and the curve basically doesn't change at all for the nonlinear response is actually a little bit more interesting it doesn't change a lot but here in the band edge region I think you you know you make a kind of a signal you know not so small error if you want to be quantitative and the interesting thing is that when you add so the solid line is the tight binding result when you add the first near neighbors it stays basically the same but when you add the second nearest neighbors then you get a decisable change so to get a quantitative kind of converged results you should go up to second nearest neighbors in these are hopings that are beyond the bare tight binding kind of formula that model people use so this is kind of a cautionary tale that there is a little bit more beyond just the simple tight binding description okay how much time do I have five minutes perfect okay so I'll go quickly through the procedure so this was all about optics and I'll talk a little bit about how to do the same strategy to interplay the Berry curvature for anomalous hall effects and really it's all very similar but let me first tell you so remember that Berry curvature was the curl of the diagonal elements of the Berry connection and if you do a bit of algebra you arrive at this expression here so you have so the Berry connection already had one K derivative you take a second one so in the end you get a formula with two K derivatives one on the cat and one on the bra so when people started evaluating the anomalous hall conductivity of ferromagnets back in 2003 2004 what they did is they massaged this formula using K dot P preservation theory I guess so we call these the K derivatives formula because it is this K derivatives here but you can write it as a Kubo or some overstates formula in this form here and so in the early days that's what people did so they implemented within their I've initial codes these some overstates formula where in principle this sum goes over an infinite number of bands which of course in practice you always truncate and actually you don't need that many because there is an energy denominator squared so the Berry curvature of some band N is going to be dominated by the coupling of that band to the nearby bands because that coupling is recently enhanced by these energy denominator squared but anyway so people would evaluate with non-self-consistent calculations this Berry curvature over many K points and then evaluate these matrix elements and some over some number of bands and until they got conversions so what we did is so the reason we got interested in this problem is that in those early calculations what people found I will show that later I should have that slide here is that to convert those calculations people needed to evaluate that integral of the Berry curvature they really really many many K points for a metal like iron on the order of millions and that starts to be quite expensive because it's in the end you know you do a plane wave calculation even if it's non-self-consistent over a million K points that's not so easy and then evaluating all those matrix elements so our strategy was the following we don't we revert from the kubo some of our state's formula we go back to the K derivative formula so I've written it here again and we apply the same Vani interpolation expression that I derived before to evaluate the Berry curvature and I just plug it into places so for the Berry curvature I just plug it on the cat there was only one derivative here we do the same and we just plug it on the bra as well and we get some formula I will not write all the terms but again there's going to be two types of terms one just involves these D matrix here and the other so that's kind of the tight binding formula and the other one has some terms which I'm not writing down but the point is that if you set to zero those off diagonal matrix elements of R like tight binding people do that term disappears so in Python tight binding for example what you would implement is the first term there but again in post W90 or in Vani Berry when you do of any interpolation of the Berry curvature like in the tutorial later there is this first term and then the other terms I didn't write down but so the first term is basically like a cubo formula and I'm calling it the tight binding cubo formula because D if you remember it has this expression I wrote a few slides ago and it looks basically like the cubo formula because these U dagger U's are the tight binding eigen vectors and this gradient of K of the Hamiltonian matrix the is the velocity matrix element essentially and we have two D's and so we're going to get an energy denominator squared so the point is that when you do Vani interpolation the sum of our bands is only a sum over the finite number of bands that you have vanishing but the funny thing is that you're not making any truncation error on the bands because somehow it all magically works out that the terms that you left out from the bands that you didn't vanu rise they magically show up like in the R hopping terms but so there is really no bent truncation in these Vani interpolation expressions which is kind of satisfying. Okay well maybe I will skip this it just to say that to calculate the anomalous conductivity you sum the Berry curvature over the occupied states and then that total Berry curvature only reacts strongly to the difference in energy between the occupied and the empty states and it does not react strongly to small energy denominators or band crossings among the occupied states and so I'm showing you here a plot this is now iron magnetized along Z so we need the Berry curvature summed over the occupied states the Z component and this is the band structure of iron along some high symmetry lines the top panel is the bands here's the Fermi level and what you see is that when like for example here when the Fermi level cuts through a pair of bands so one of them is just below and the other is just above the Fermi level then when that happens you get a huge spike in the Berry curvature because of those energy dominators squareds and so that's why this is such a tricky quantity to calculate you need to somehow zoom in on these regions of k space and in fact these spikes are not even all along high symmetry lines here is a heat map on a plane and you see that on these regions where two Fermi contours approach each other there is like a hot spot of large and positive Berry curvature that's the red color and here again two Fermi lines are very close to each other and there is a hot spot of negative Berry curvature so you have to capture all this stuff and so it's a perfect problem for vinyl interpolation and that was really the motivation back in 2006 2007 with David Jonathan E8 and David Cedens Shinji Wang that got us thinking about using vinyl functions to do to make basically k-points cheap and this was released in 1990 eventually as part of the post-1990 module and more recently many other properties have been this kind of methodology has been applied to many other Berry type quantities I've listed a few here including orbital magnetization that David mentioned and okay I'll just leave these slides for the end where kind of to to hand over to Steppen and the tutorial because Steppen wrote this wonderful code Vani Berry that has many very clever optimizations and I list here some of the features that he has implemented and that you will see in today's tutorial so thank you very much any questions maybe online thanks a lot for the presentation I'm myself also in like between this definition type ending world so while working on this I always like came up with some like questions about the if I'm doing things right and I always see or tend to see that you usually do like one-shot procedure in in when you are doing the bannerization and I get that you are getting a really good physical insight but you have some pre-established or you have some physical intuition behind this and I wanted to ask them like if I have maybe not really good physical intuition about the material and then I do a bannerization by disentanglement and process a procedure and then a bannerization how are these results good or how would you like apply your what you do to these results do which extent is better to do a one-shot procedure rather to do a full bannerization and as a last question also you think is it possible to do like some kind of symmetrization of this Hamiltonian after doing this bannerization yeah thank you yes so let me start to the second question I think Steppen will may mention that because I think that's a recent feature in Vania Berry so yes definitely you can do that you can either symmetrize the Hamiltonian a posteriori or maybe even better I think you can do some symmetrization before setting up the the matrix elements but I think maybe Steppen will do you want to add something or there are actually three ways at which stage you can do it one is that you keep symmetries during your variabilization which is the course all the symmetry adopted when your functions which also there is a tutorial later all right okay yeah second when you constructed your money of functions but they are slightly asymmetric if you did a one-shot then you can really restore the symmetry so you symmetrize your Hamiltonian and all for this position element matrix is and whatever you have and you get a now a symmetric model and so do you can just calculate with whatever you have and like when you integrate that in the case space and you get your tensors or vectors whatever quantities like an almost whole conductivities and others and you just symmetrize the result in tensor just a posteriori so the second and third are implemented in linear Berry so in the first is in one year ninety well with some limitations but we will know it later so maybe I can comment a little bit on your first question so the philosophy here is that we use many functions as a basis so if the basis is good in some sense the results should be for the physical observable should be independent of the basis that you choose the way it works out in money interpolation is that you remember I wrote these formulas that contain several terms and what happens is that if you go from a set of any functions for example the projected ones the one-shot projection to the makes me localized many functions there is going to be a change basically in book keeping so the values of the individual terms will change but the net result should be basically the same as long as the funny functions are somehow good which in practice means well localized so the message is as long as they are well localized it shouldn't really matter of course if you want to build models you probably want to keep the atomic picture and that's why many times in you know people you do the one-shot projection depending on what they want to do later but for the purpose of funny interpolation it shouldn't really matter as long as they are fairly well localized maybe I can say something of this there is one special case that is timer versus symmetry so sometimes if you know when you study it with spin orbit a system that has timer versus symmetry and and the one-year functions that you construct often they are not perfectly timer versus symmetric the interpolation can be tricky you know the bands that should be degenerate cannot be degenerate right and this comes back to the other question which is symmetrizing the results because right so you can it's a bit annoying when you have these slight symmetry breakings in your band yeah okay any questions from zoom yes is it possible to calculate the berry curvature in real space yes yes yes no you should answer but the answer is yes okay do you want so you mean like the these things about non-clinic spin textures is it possible to calculate the berry curvature in real space so yes certainly all I've talked about for the purpose of this presentation was the case space berry curvature if I understand correctly the question I think indeed one can define berry curvatures in other spaces including in real space and in fact they also contribute to an anomalous effect in systems with non-clinic spins but to be honest I've never really thought about how you would go about using many functions so maybe I'm isn't the questions because I thought you know it was asking if you can calculate yes so this they say the intrinsic anomalous conductive it in a in a metal in real space and this is something that we did Rafael and even before a fellow Bianco did the work for you know the old the chair number in real space so you know if they you I don't know if the participant meant that but if that was the question the answer is yes you're right both for metals in insulators