 Okay, so welcome back everyone. So our next speaker today is Professor Linlin from the University of California at Berkeley, and it will talk about selected columns of the density matrix for many functions with entangled energy bands. So thank you a lot, Professor Lin for accepting our invitation and the floor is yours. Thank you very much. First, I'd like to thank the organizers for giving me the opportunity to give this talk. It's really a wonderful activity and sorry that I cannot be there in person. I'm organizing a semester long program at IPAM UCLA right now. So, interesting if there are quite a number of people here interested in one year or two. So, this one year definitely has a lot of impact in even in the applied math community. So this is a joint work with Neil Damley, who's currently assistant professor at Cornell, Antoine Lebed at INREA and Lexine E. Stamford. This is an expert audience and they have been a lot of wonderful talks in the past two days. So I'll directly start from the optimization problem. So as we know, the maximally localized one year function, it optimizes a functional called a spread functional denoted by this omega. So it is a function of the one year orbitals called phi or w and this phi is a unitary rotation of the bands called psi. The geometric intuition is that if you want to localize it, you can minimize a second moment. So this is, if you have n orbitals, you add up the square of the position operator and minus is center. And this one, if you sum over all of them, this minimizes the total spread. And this is exactly what localization wants you to do. Here this U is a code of gauge degree of freedom. It doesn't change anything physically such as the instimatrix. However, this U determines the local extent that this orbitals can be localized here I'm presenting the real space or the molecular version for simplicity for the contents matter or periodic systems. This X needs to be carefully defined but these are well documented in the literature. So there. It has been very successful. And we have seen a lot in the past two days. And numerically there are some robust in this issue. One is it suffers from the so called initialization problem, which means that it's a quartic functional and with a non trivial constraint. So the, the non linear op. It's a non linear optimization and it's quite possible to get stuck at a local minimum. So example of it. Then for the entangled fan structure, what we need to do is to perform localization in the absence of a band gap. This is quite different from the Euro scenario of one year localization that can be established theoretically. That says, if the energy plants are isolated, then, and the turn number is is non trivial. And then you can have localization and otherwise, otherwise you don't but when you really talk about entangled that structure is everything's always, there is no gap, or either up or below. So if you do localization is actually not entirely obvious. There are a number of ways one can do it. And both problems needs to be carefully addressed, especially if you want to do high throughput computation one around one year for many materials to do some screening, and the robust issue robustness issue is really the key. I'm going to say that other than the original paper by Mazar in the manabill and there are several alternative methods developed by many groups in the past two decades. This is by no means exhaustive list of methods. For example, Cyrus and talk and had this a partly occupied a one year function. This is somewhat related to the entanglement entangled band that I'm going to refer to later. And that's why Gigi has a work using the so called a cosine sine decomposition that is more robust, especially it allows you to identify all the orbitals to the extent that you can localize, if certain orbitals cannot be localized, you just leave them to be localized. So, in chemistry literature, Nizia is to my knowledge, one of the relatively recent works that has generated impact, at least within a potential chemistry. So the IO method, this intrinsic atomic orbital. It was, if I understand correctly, it was originally presented for the molecular case but it has been increasingly used for the condensed matter systems as well. So, I think our co organizers must have a co. So they and Steve Louis, they developed a method to improve the robustness by selecting some initial projected orbitals, and then do a more chemistry like approach to improve the robustness of the issues of finding localized one year functions in 2015. Then there are two math groups. So, also interested in the numerical aspects of this topic. They also both groups have developed a number of things along the theoretical lines as well in the past few years. So if you're interested along this direction, you can check some of the authors and find the works. The PRB paper, Kansas, Leavitt, Panetti, and Soles, and they presented a very interesting method, kind of like a generalizing the famous construction of a one year functions in 1D. And then this was further extended by Stubbs, Walsh, and Lou last year, and generalizing again this TXG type of constructions. What we did in the past few years since about 2015 is a very simple method called the selected columns of the dense matrix at CDM, which is in the talk. And tomorrow, Valero also did a very nice tutorial in terms of how to use that in one year. It's a, it's can be used in one year through Quan-Mistrasso such as PW to one year subroutine. So I'll first explain the idea behind the CDM for the molecular case and then explain how that works for the periodic case. The basic idea behind the CDM is rather straightforward, which is, our goal is to find some orbitals that are localized. But so what kind of thing is naturally or localized, even without any choice? The answer is very simple, which is the dense matrix. So the dense matrix is for a DFT system is defined to be the outer product of the Concham orbitals. The nice thing of the dense matrix is that it is a projection operator. It is so-called gauge invariant in the sense that you put any unitary rotation U and the answer doesn't change. So as to this, you can see that if I really apply U star transformation, U is a unitary to this side, then this becomes Phi U star U and this is exactly canceled and it becomes Phi Phi star. So from this argument, you can also immediately see that if the system is localizable, then the density matrix must be a sparse matrix, why? Because if this Psi, it only has very narrow support, then you can see that from the matrix level, those zeros will extend to the full matrix. And this is a picture of it. So all the eigenfunctions, they are completely delo localized in the system, but if you really consider P, this is actually a very narrowly bounded matrix. And this is a 1D system, but also it works with the same argument for 3D2. Another observation is that, okay, so you have this low rank and sparse matrix is almost to the thing you want. The question is whether you can use the already localized density matrix P to help you to find linear functions. You can say that, okay, it is a low rank of rank n, why don't you just arbitrarily pick n columns of this density matrix P and call them localized orbitals. Yes, so these are localized. If you need these n orbitals, then they are mathematically linearly independent of each other. And so the its linear combination is indeed the same as the span, the span of all the original functions, but there's a problem. If you really pick just arbitrarily n columns, if the two columns are too close to each other, then they are extremely linearly dependent. So the one-year functions also needs to be orthogonal. Right, so you have n very localized orbitals, but when you authorize them, this can significantly introduce a lot of extra tails. So this is what you don't want. So you not only want to select n localized columns, but this should also be very much linearly independent. So how do you solve this problem? The solution is actually extremely simple. So the pseudocode just takes two lines of mallocode. The theoretical justification is more complicated and which I'm not going through it here, but this is what you do. So let's say that this psi, which is a matrix of ng, ng, g means number of grid, you see that the sparsity is a super coordinate dependent thing. It's not like I change the representation of the pxy to pgg prime in the Fourier domain, is there still sparsity? No. You have to really represent the orbitals in the real space. So you can, that's why where this pw to one-year comes in, and you dump out the orbitals in the real space grid, and here ng is a number of electrons or a number of bends. Well, for simplicity, I assume the system is a gap. Then what you do here is you have this tall skinny matrix, you do a transpose of that, and you run a QR zero. This zero means something called a pivoted QR. What the pivoted QR means is literally do what I said. You pick the columns so that they're using a greedy-ish algorithm. It picks a set of NE columns that are maximally linearly independent of each other by heuristically maximizing some volume in the high dimensional space. So this will return you to three matrices. The Q or U is a unitary matrix. R is an upper triangular matrix, so we won't use them. And a set of permutations, which tells you which columns are to be selected. That's why it's called the selected columns of the insta matrix. You also see that I'm not directly running this to the insta matrix, which is of size ng by ng. That's enormously large and you cannot even fit that in the memory. You can prove that the SDM for the insta matrix, the selected columns of the insta matrix, it can be equivalently implemented using this much more economic guy. So once you have this U, which is of size ng by ng, you just literally multiply this band by this U and boom, you're done. So these are the SDM generated localized function. Those are different from the optimized one-year orbitals because, well, throughout this procedure, these two lines, I didn't like optimize with respect to any given functional, but the difference can be very, very small, as you will see later. It's very easy to code and to paralyze, right? And it's deterministic. There's no initial guess. You don't need to select, hey, this band needs to be of a feature sp3, the other is sp2. No, it will find it automatically. And as I said, the sperm vector, you don't need to use them, but in codes, the idea of the SDM. So these are two pictures showing you that it gives you the sensible thing you expect. You get these pictures by literally wrong, the two lines. So you do a supercell calculation, and for the silicon, you get the U, you transpose, you plot one of them, you find that naturally has a sigma bond. I didn't put any initial orbitals saying the center of the one-year orbital is not on the atom, but on the bond, but it will find automatically if it's on the bond and for the water. And again, it will find this nice orbitals automatically. So it's very paralyzable. There's not that much you need to do to make it a parallel. So if the SDM becomes more expensive and you can just run parallel QR and QRCP, QR with the column pivoting, and the paralyze as well. This is a number of years ago with the Scala pack, but recently they have been increasing interest in accelerating this algorithms with a randomized numerically natural packages. I believe the RAND QRCP performs even better, both sequentially and in the parallel setup. So let me, before going to the SDMK, let me talk about two other things to make it faster. One is to make it faster, the other is a way to treat periodic system. To make it faster, the bottleneck is really the QRCP procedure. The next one is just a matrix multiplication, you call it a gem. It's a general matrix matrix multiplication, which is very efficient. So in this paper, we thought whether this QRCP, which can be upon computational bottleneck can be accelerated for large systems of interest, without suffering the quality. So this is a procedure that we developed called the approximate localization refinement, what it does is something that is very straightforward. So quite intuitive. The intuitive idea is that you first do some random sampling, let's say of the electron density, you want to find any columns, but you pick a few more. Theoretically, it's bounded by any times log and you call the candidate points. So those should be sampled according to the electron density, which is just a vector that's very efficient. So with that point, you can run the QRCP that gives you some approximate localized orbitals. It sounds like this should be okay because you select a few points and we should be, you have some buffer, you run the QRCP and should give you pretty localized orbitals. But it turns out, it turns out that this is not good enough. So what I'm showing here is like a simple example, a small molecule is a dissociation of a beach three inch three. So, the idea is that at the beginning, and when at the equilibrium. There should be some one year orbitals like that is shared between the boron nitrogen and us is stretches apart and this orbitals increasingly goes on nitrogen and the dissociates from the board. This is a nice test to see the robustness of the algorithm, and the y axis and showing the percentage of the non zeros. And you can see that the randomized algorithm almost got it, but compared to the SDM, it seems to be a little bit more delocalized, and this is expected because So to overcome this issue, what we do is a refinement step, which is, you already have some approximately localized orbitals, and that allows you to locally group some of the columns. So you can. So these are the approximately localized orbitals, then you can do a simple detection algorithm to test like to do the QRCP locally. And let's say for the molecule, if it is dissociated, then I can at least run these two groups, local QRCP, and that gives you a set of points that then you do a small global refinement with a very tiny QRCP. So this is a two step procedure code localization and refinement. We can apply this thing to larger systems of interest and see whether it indeed outperforms the original QRCP. So if this is a box of water with 256 water molecules, if and because of the relatively large e cut and the the app if you run the QRCP sequentially that still takes like 4,000 second. So, how do you speed speed this up, you do the randomized algorithm, and that is extremely fast and like only takes like 15 second. And so the final step, which is matrix matrix multiplication that's also much shorter than the QRCP. So the randomization step that is very fast. And if you look at the, like y axis, which is the number of orbitals corresponding to a certain percentage of non zeros you can also merit by the spread. You find that, like, there are a bunch of orbitals that are more delocalized that compared to the original SCDM. So you do the refinement steps, and the total cost is like, like more than one order magnitude, like a 30 times smaller than the SCDM. But if you really look at the statistics and the difference, the difference with original SCDM becomes negligible. However, I would like to say that. Again, if you read the mirror in terms of the spread functional, the for this particular problem and the one years will can further localize it because it literally the goal is to minimize the spread. So, okay, so the, now let me just say something on the SCDM for the periodic systems. So the, there are actually two works of a SCDM for periodic systems, which can be a little bit confusing. So, the original SCDM K with a K points was actually we designed this method in this paper without the sufficient amount of knowledge what one year 90 does, and therefore it was became a little bit overly complicated. In other words, you can also say that it is generous, even more localized orbitals than what one in 90 generators. The reasons of the following. So, if you really look at the supercell perspective, there's no reason why you cannot allow a gauge matrix, gauge matrix, mixing the bands from all the but you immediately run into a problem, because the authorization of so many orbitals alone, it seems to be a very daunting task. And therefore, in one year 90, you can think that it chooses a mix of the life much easier by asking the gauge matrix to be K local. It doesn't necessarily generate the most localized one year functions, but it's easy to work with. So, when we first wrote this paper, we didn't have this knowledge in mind, and therefore we just thought, okay, how do you really allow the most general gauge mixing matrix mixing then from all key points, and naive implementation. Like it gives you all the n cube skating and k cube skating number of key points. And when the number of key points is 10 to the five. I mean, this is impossible. But it turns out that with some relatively intricate usage of a fast Fourier transforms, you can reduce the entire cost of all the NK log NK. And this method. Yeah, and maybe confusing a bit confusing hindsight was originally called SDM K, but it really works. It's, this is for model problems, you can see the shape of the orbitals for the, like, as it. This is the before authorization this after authorization, they look really like localized orbitals, this is zooming version of the authorized one year orbitals. And this thing scales. Indeed, like this is NK, this is NK log NK, like all the way to when you have even have more than 100,000 key points. So, just to say that this more elaborate way of mixing the gauge, it is a possibility. And then after SDM K, the reason is that if this is the version that's implementing the one in 90, thanks to a valid real work, and is the following. You only allows the gauge to mix orbitals within each K block. So that allows you to implement some relatively simple strategy to make this work at least for topologically non trivial system. And that I haven't talked about in tango then so far. So, the idea is that, okay, I already know how to run an SDM for the gamma point. So I just literally take those like points using once anchor K point such as gamma, and to find the columns, and they use the same set of columns for all the case. The idea is that you will rotate the gauge so that the rotating bands, they are smooth in K, so that when you just apply the regular one year, which is a Fourier transform, the orbitals becomes localized. So let me explain this procedure is sounds pretty complicated but implementation wise this is actually very simple. First, take one, the orbitals from one set of bands, quote, sorry, the bands corresponding to one K point, such as gamma, you do a transpose and do a sort of q RCP. This is the queue and are these are the selected columns. Remember that before I said that this side times q are the one year but now you need to do something more elaborate because it's a periodic system. So that is columns using this information in pie. The pie is usually stored as a commutation permutation matrix, or just a vector permutation vector. You pick the first and be number of bands per K point columns, and you select the columns of this for the of the band for each K. Okay, this is cannot as anchor point. This is like for each or the other case. And then you can prove that the, the gauge that you want, very close to the one year optimized one is literally the lobbying authorization of this small matrix. So the justification is kind of like, so these are the conceptually the rotated bands with a smooth gauging K, and you multiply the gauge matrix to the bands for all the case. And you, sorry, this one goes crop. So you can see that the selected column, you need to work a few lines and see that the selected columns of the density matrix for each K is a literally this guy, and this extra factor is correspond to the circle of the quantization. Let me, there are a few number of formulas here, but I let me just emphasize operationally what you need to do is to pick one set of bands and do a QRCP. And then you generate to this matrix, and you do this. You compute this small matrix for each K, you multiply to depend orbitals you're done. So this is a very simple implementation. This is how it works. So this is a pretty challenging premium oxide example that we learned from Sinisa a number of years ago. So there are, there are like a bit issue with respect to initialization for this particular problem, because if you don't initialize it correctly. So if you want to initialize the shape of the orbitals, let's say using sp2 orbitals instead of using the D orbitals, you will find that the one year 98 will get stuck. So it won't like localized to the right thing, only if you know exactly the nature of the orbitals and initialize properly, and then you see the spread beautifully optimizes and within converge within 30 iterations. And the point is that the, if you use SCDM, you can see that has converged from iteration one. So the reason is that if you look at like the value of the spread from the SCDM just by running this algorithm has nothing to do has no knowledge of the nature of the orbitals whatsoever. So you find that the initial spread is already like a 17 and the localized the spread is like 16.98. So from this picture, you can see that the span is like on all the hundreds, and this difference is completely negligible. Now let me talk about the entangled bands. So, with all the preparations, we can try to generalize the idea of the isolated bands to the entanglement band, and the idea is to really try to construct orbitals that has a smooth age with respect to the K index. So, this is not possible, if you have a metallic system or entangled bands structure, but you can engineer to have like entangled smooth gate idea is to look at something called a quasi density matrix. And you don't have the psi k psi k bra, but you add some like a profile F so that entire is called a smooth smearing function, so that this thing is actually smoothing with respect to K, it doesn't have the sharp edges. So, particularly in localization, you can easily afford a smearing on the order of EVs, and that is not possible if you really think about, like, smearing the DFT calculation, which is really on the order of MEVs, for example. So, the procedures following for the isolated band, you choose this profile F to be a step function, where the step is between the energy gap. Then this just picks the occupied bands, this summation sums over all the bands. If you choose a step function, this just picks the occupied bands. For the entangled band, however, these are not necessarily the optimal choices, it's just the two examples, you can definitely come up with better functions yourself. So, if I want to localize below a certain energy level, I can use our C function. And that is a smooth, like across, you can call it a Fermi surface or like whatever energy level that you're trunking. Or, let's say for the copper or copper oxide or something, you might just want to get the one-year functions corresponding to the D orbitals, and you can use a bump function to penalize the contribution from both high energy bands and low energy bands. And this is just a, let's say, a Gaussian. So, these are two examples, give you, see, like, how this works for a model system in 1D. So, the eigenfunctions that are completely delocalized, if the system has a gap, and you just run the one-year localization for the isolated bands, then you see this nice functions, each color is a function itself, so just a clean bump. If you do entanglement, entangle the case one, which is this RFC, you can see that it has more wiggles, but still, it is very much localized. And for the entangle case two, there will be more wiggles, but nonetheless, you get localized orbitals. So this is the idea. So, this SCDMK turns out, I think, maybe, yeah, to be useful for a number of things, and originally they have been some math lab or Julia code written by Anil and Antoine, respectively. And for this kind of meal, Aburonny and Kainousi, a couple of years ago, had this very nice paper illustrating the usage of a SCDM, not a K version. And for the for the, for localization and and to accelerate hybrid hybrid functional calculations. And also thanks a lot to Valerio for his very nice work in these two papers. One is the updated one year paper and the other is more automated high throughput calculation and I believe he's going to give a more detailed talk on this and the usage of SCDM in one year and 90 tomorrow. So let me show you how this thing works for the band structure interpolation and for the silicon and you can try to find not only localized to one year interpolation for the occupied band but if you set the mu C the energy level to be around here then you can also get a very nice interpolation of the conduction bands to and this not only works for the RC but also for the bump function for the copper you can also get all the d bands pretty well. So this is a more stringent test which is the graphene and because especially around the Dirac point that you care about whether this SCDM procedure is good enough to really get the like band crossing at for example the Dirac point and you can see that indeed it interprets very well even though there's a situation. So what is kind of interesting that we found from this study is that it has been long known anecdotally that when you do the disentanglement and it is not necessarily to optimize the spread that you're after but we also independently found this from a concrete example that is you have a you want to interpret the band structure of aluminum you start from six eigenfunctions and you reduce this to four and you find that if you really let the one year totally adjust the focus on optimizing the spread the resulting interpolation quality is very bad but really if you want to direct SCDM without further optimizing the spread function of the interpolation is much better although the spread is much so which means that at least when you deal with the entanglement band you shouldn't just focus on the smaller spread so the convergence of the one year interpolation procedure also works pretty well so here we compare the mean and the max of the error of the one year interpolation procedure for the SCDM and the one year orbitals they're pretty much like one is the tensed matrix the other is all formalized version and they're pretty much along parallel to each other and the converges as k increases both for semiconductors and for metallic systems okay so in the remaining ten minutes let me or so let me just talk about this variational formulation for the one year functions for the entanglement bands because before the procedure sounds like just a one shot calculation although it can be very good one shot but we wonder whether it is possible to combine like the variational aspects of one year together with this SCDM procedure it turns out to be possible so the so this for this we need to introduce the idea of frozen band if you look at the problem of the aluminum earlier the one problem was that we didn't freeze any band but you should really if you're you care about the quality of the interpolation within certain energy window you should really try to freeze it and this is the well-known this entanglement procedure like proposed by Sosa-Mazzara Reveneville in 2001 so the idea is that you select certain for each k point you say that a certain bands called NF must be fixed and how do you impose this condition it's like the projector corresponding to the one-year functions applied to the frozen band should also give you the frozen band so this is why it is frozen so this problem is much more complicated than the even more complicated than the one year 90 sorry that the one year localization thing we talked about earlier because it also involves the subspace selection process so you essentially have three parameters you have an outer which is the number of eigen functions or maybe all eigen functions you can work with and then you need to sub-select a set of one-year functions and this subset is larger than the number of frozen bands if you set number of one year to the number of frozen then you're back to the isolated case and this has often has obstructions you need to work with more bands so how do we enforce this constraint from an optimization perspective it turns out that there are a few equivalent ways of writing down this pretty awkward constraint pwpf equals to pf so the idea is that if you freeze certain bands and the gauge matrix is a correspondingly partition into the frozen part and the remaining part then there are a few ways of I like expressing this I don't have time to explain all of them but let me explain this fourth thing which is the most useful in practice called xy representation it is very intuitive the xy representation says the following for the for the frozen orbitals you keep them assist which is the identity the rest of them you do a sub-selection for the all of them together you have nw by nw gauge matrix and rotate okay so the y matrix does the job of a selection and one together with the identity here it creates a subspace and once the subspace is fixed and you can do a localization so there's no reason why you have to do one after another you can actually put them in one single optimization problem so this is xy we call it xy variational formulation so you minimize with respect to not you but the x and the y this x and the y together define the unitary gauge rotation so that it satisfies the authentic conditions by itself this one is the formulation is different but it's equivalent to what we found afterwards or after submitting the paper we found that it is equivalent to this partly occupied one-year function procedure so implemented in a different code so there's also a natural relation with this entanglement procedure so in the paper by so some other variable you can split the spread functional into a gate invariant part and the gate dependent part so the gate invariant part exactly optimizes this y matrix and you split this like a minimization single minimization into two consecutive steps one is to minimize this gauge invariant part to get this like to the subspace selection and then you minimize this gate dependent part which is the x to find the to further minimize the spread functional and by definition because this is like one step after another the variational procedure by considering them both gives you a slightly smaller spread so these are some numerical results and first is still the silicon and you can see that going from the SCDM for this particular example for this particular example the the initial spread of SCDM is still not bad because if you don't do I mean the spread is very high and you do the with some frozen bands and you do the around the one-year 90 and this gives you the spread is like 27 but the variational algorithm gives you even a slightly smaller spread and the quality of the interpolation is it's very good so actually if you look at the quality of the interpolation for SCDM which is blue it is almost like a pretty good almost everywhere but there are still a few points that are not so good and the variational formulation corrects that so another interesting feature is that it maybe coincidentally restores some symmetry so we know that all the the or the nature of the orbitals and so because once you if you don't include the conduction band it's a sigma bond but if you include the connection band and s should be sp nature and so indeed you get the the shape of the orbitals and you see this the value of the spread for all the localized orbitals is exactly the same which is not true for the SCDM which seems to separate them into two groups but not true when you run the two-step optimization one year either it is still separate into the two groups but although the difference is is much smaller so using this a much more stringent test of this procedure is to apply this to the uniform electron gas so seems I'm running out of time let me just quickly say this so the for the 1D you can try to interpolate you can freeze the lower band and you have two bands and let it interpolate it would because there are band crossings here and here it wouldn't be able to interpret everywhere as well but if you freeze one band it does a very good job and these are the shape of the one-year functions what is very interesting is that so the resulting one-year function it it oscillates but has a one over r square decay of the envelope which means is the resulting thing is definitely not exponentially localized it's algebraically localized but but nonetheless it's like it has a localized feature and looks something like this in the Fourier space so so the algebraic localization is related is related to the behavior of the one-year function in the Fourier space and there is a so-called gauge smoothing technique that you can apply to enhance this algebraic decay to the so-called super algebraic decay which is numerically can be seen as a almost exponential decay and the theory of this procedure is established in this paper published in annals of 100 Poincare in 2019 and we in our paper we provided the implementation of a possible gauge smoothing technique and which fixes some of the smoothness issue in the Fourier space and indeed in the real space you see super algebraic decay you can apply this thing to 2d as well these are the shape of the one-year functions pretty weird looking I think there are a lot of mathematical questions you can ask even though the systems like uniform natural gas like you in the Fourier space these are the shape of the 2d one-year like the one-year localized orbitals for uniform natural gas like in the real space again if you don't apply any gauge smoothing you get one raw square decay so let me conclude that we think that SCDM is a useful technique for one-year 90 and because the current procedure of using it is that one SCDM provides you an initial set of gauge matrix I think in one-year 90 called AMN and you don't any require any input especially not like input of where the orbitals are and where the centers are so on and so forth so SCDM should provide you in a black box fashion and this is and then you can run one-year or other like variational tools to further improve the quality of the orbitals this is really useful for some high throughput calculations and again MeloRail will talk about that tomorrow and so the variational optimization further helps especially when you have entangled band but I think the understanding of that is still at the beginning I hope I have conveyed to that the spread is not necessarily everything you need to consider a task so a very interesting thing is about symmetry I think this workshop and also the summer school there are other hands-on talks on that I think the symmetry is a very important especially for topological materials and we're also thinking along this direction as well and finally one particular system we're currently interested in are our mori systems and which has quite intricate natures very interesting looking one-year orbitals so with this I'd like to thank you very much for your attention. Thank you very much Lin for the great talk now we open it up for questions please raise your hand if you have a question Stepan will go through the desk and give you the microphone and while we do this if there are participants online who have questions please type them in the chat or raise your hand and we will unmute you. Thank you for the interesting talk. I have a question about your lab part based on what do you think the kind of strategy will be actually useful to achieve superoperative creating integral things? Sorry could you let me know the slide number and I didn't hear the question very clearly. You can come here and ask it here too. I'm not sure it's the volume thing yeah it's just I didn't hear it. Okay yeah so do you think this gauge smoothing technique will be practically useful to achieve like to generate more localized minor functions? Oh thank you yeah so it is an interesting technique but I'm not sure whether it's practically useful it's more of an interesting observation the procedure is described in our paper I can simply yeah so the the high-level idea is that you see these kinks and you want to smooth them out and as long as the thing is smooth in the Fourier domain then you can mathematically prove that this is going to have decay in the real space and this paper says that for some general metallic systems you can do similar things but it also this I'm sure it also distorts like some other like physical properties because as I said a spread is not the only thing and another thing you should pay attention to is that although the decay is nice the pre-constant may not be because you know I mean this one is initially the one-year thing decays pretty fast but this one initially is not necessarily decayed I mean maybe similar but not necessarily faster especially if you truncate the tail to this range. So yeah short answer to your questions I don't know but for specific systems that you know where the kinks are you can certainly experiment with those. Okay are there any other questions here raise your hand anyone on zoom well there is a question on zoom from Valerio so let Valerio just give me a second and I will unmute you. Hello can you hear me? Yes hi hi so yeah I was thinking about these issue with the symmetry and I was wondering so when you do a CDM and you want to vanyarize a set of bands that let's say have the same center and then you want to do one shorter CDM how important is the grid the real space grid whether it has to include that particular center or a week of position or something like that how important is that the grid the real space grid that has that point that is the center of the vanyar function. Oh yeah it's a great question thanks so the short answer is I don't have a direct experience of this for the symmetry I'm aware of like two works one is the thing symmetry adaptive work that's already going to be presented at length in this workshop the other is a more recent one I forgot sorry I forgot the authors like published the last year and which is like again projected to some other orbitals that you can work with the symmetry groups both are very nice approaches and I don't have a good idea whether the CDM will be a more useful approach in that context but indeed as you said the if you want to use CDM and make it symmetry adapted the most straightforward idea is to select the points to satisfy the corresponding symmetry property and to be able to do this you might have some constraints on the set of grids you use for example if you want to implement a c3 symmetry I mean it's going to be pretty annoying in the context of like a plane wave grid right the dual is not likely going to satisfy the c3 symmetry so therefore I mean I think it will be nice to see the performance for simpler symmetry c4 or c2 or something like that and see whether that already does the job but more generally I my current thinking against just the thinking I still like hasn't been implemented is to work with the ceiling matrices and like to impose some sort of algebraic conditions in the very within the variational framework is probably the most straightforward idea but again I don't have a like a working knowledge on this topic but if you have a or other people in the audience have insights to share I'll be very interested to hear. Thanks and I have also another question what's interesting to you about my racism? Oh yeah yeah so there are two things and one is I don't know you or whether you or other people in the audience are maybe are experts on this so if you consider the twisted body of graphing for example for other things like TMDC and the things are different and has been reported back in 2018 that if you try to localize within the two flat bands per spin in the valley and you get this kind of like a like a fidget spinner type of like three chloral leaf kind of one-year functions and yeah which is on a very vast scale and it's kind of interesting and it is accompanied with the so-called fragile topology which is kind of like interesting we want to understand this a little bit better but there has also is related to symmetry has been arguments from the physics side saying that this one-year functions although they localize but if you start truncating them they don't generate as nice like physical observables they want to detect later such as C2T symmetry breaking because the choice of the one-year already breaks some sort of the symmetry that's all this kind of arguments is always a source of mystery to me and yeah we also have a little bit I mean I but we do have a project on this twisted body as a matter of fact the reason why I cannot be there in trusty is because this week there is this Mori workshop like in the ATAC and so I look forward to hearing experts like McDonald or other people like on this on this topic okay hope I can get more in the future yeah thanks okay that's great are the any other questions here if not I think we can thank again thank you okay stop the recording now