 So, please, go ahead with the, yeah, here we go. Thank you very much. It's a great pleasure. Thank you for the invitation. It's a great pleasure to visit here at the PST and ICTP. And this talk contains some of the latest development of our recent efforts for improving the one-year algorithms for finding one-year functions. And this is pretty up-to-date, which has appeared on Archive yesterday. But there is a hidden trend in the talk, which is even more up-to-date. That was updated like 10 minutes ago. So, first of all, I don't think to this audience, I need to talk about much about the one-year functions. Now, the Mazari Vanderbilt construction for the MLWF has become the standard. And this is two examples, one from Silicon, one from the Graphene. And there is a lot of mathematical reasoning for the existence of exponentially localized one-year functions, at least for insulating systems, but more recently extending to the understanding of the existence of one-year functions for topological materials, which I won't discuss here today. And one-year functions are very useful. Here are only very incomplete lists for using one-year functions such as the analysis of chemical bonding, the band structure interpolation. You want to construct the basis functions for the occupied orbitals or the basis functions for excited state calculations, which is to represent the hard-mart product or the pair product of the orbitals. You can do it for strongly correlated systems, phonon calculation, et cetera, et cetera. So, it is a very useful thing. Mathematically, how do you construct this one-year functions? You first do your Kong-Shan calculation. You get your Kong-Shan orbitals called Psi. And you would like to find a gauge, which is obtained, which is called U. U is a unitary matrix. And it is called a gauge because irrelevant to the degree of freedom, it does not affect the density matrix and other physical observables. And you find this by minimizing the spread functional or the second moment of the localized orbitals. Here, I highlighted the word intuition here because later you will see that this is actually not taken for granted that this is always the bad thing you do. But nonetheless, the standard way to do it is to minimize the spread. And this, as you can see, is like a variance thing. This is the second-order moment. This is the first-order moment in the subtract. And you minimize this phi. But what you are really minimizing is with respect to U. And that is the gauge degrees of freedom. Once you have achieved the minimizer, and this is the corresponding spread, and this phi, what you obtain there, is the localized orbitals. So robustness issue has been consistently observed in this community for various kind of systems. Actually, robustness can be interpreted from two directions. And one is the so-called initialization problem. That is, the previous optimization problem is highly nonlinear and it is entirely possible to get stuck at the local minima if the initial starting point is not correct. And the second is the robustness with respect to the bandgap, that is, the case of the entangled band, which I'm going to talk about today. And if you directly do the localization for the entangled band in the system, it won't work directly, and you have to generalize these one-year functions. But the generalization is not unique. How do you do that? And in practice, people observe, in the case of entangled band, it becomes more challenging to tune these parameters. And both needs to be addressed for high throughput computation, because if you rely on the expert doing that for one or two systems, that's probably fine. But if you want to do it for screening with 10,000 materials, you want some methods that works more robustly. And this is a recent example, a pen just hours ago from Roberto's current student, Lucas Mucler. And we had some drawing work before. He also worked with Roberto on a new topological insulator called Tonsten.helleride. And it's a monolayer system, a lot of interesting properties, and this is a unit cell of that. The thing I want to show is he actually spent weeks tuning the parameters so that you can get the one-year functions for this system. In the end, he actually has to resort to the help of an expert in Professor Mazzari's group. And in the end, he obtained this begin projections thing. What it does is to exactly put the localized orbitals on some bonds, I mean along some directions. If this is what you need, you can see that this is not very friendly for the high throughput calculations. Now, if you run the new developer version of the one-year 90 code thanks to the development of Dr. Vitaly sitting in the audience, and all you need to do is this, and it becomes much easier to tune. And the talk contains two parts. And the first is the stuff we did in the past few years called the selected columns of the density matrix. And we're going to talk about that. And the second part, I'll talk about the recent variational formulation. So what is SCDM? The basic idea is that you have the Psi is a unitary matrix, and then P is equals to Psi Psi star, and this is the density matrix. It is a projection operator. Clearly, it is a gauge invariant in the sense that if you choose any U, and you can see that UU star exactly cancels, and you have the Phi Phi star. Remember, Phi is the localized orbital, which means that P is the sparse matrix. And at least it is close to a sparse matrix. And this is a simple example for a model 1D system. And the basic idea of the SCDM approach is, okay, it is kind of hard to find Phi directly on this U. However, this P, which is a gauge invariant, is already a sparse matrix. And can you select the localized orbital based on the information from some columns of this P? This is entirely possible, and actually it only uses one line of the MATLAB code, the simplest version, to do it. If this is a gamma point calculation, you take your Psi, coming out of, for example, from the output of a quantum espresso. You do this weird operation called a pivoted QR. You do a Psi, you transpose that, you do a QRCP factorization, and the previous Q factor is guaranteed to be a good gauge. It is only one line of the code, but the proof is actually it sends more than one line, and you're interested, you can read this paper. And it is very, clearly very easy to code and to paralyze. It is deterministic, no initial guess output whatsoever, and the permutation codes, the basic idea I talked about earlier, which encodes the selected columns of the density matrix implicitly. You can do this for the k-point, and the strategy is that you find one anchor k-point, usually you choose it to be the gamma point, and you select the columns of the density matrix, but now you have actually more than one density matrix, so you cannot do just one Q R decomposition to have all the gauge, and you do a few extra steps in alerting authorization procedure, and you will get the gauge for all the cases. And you might wonder whether this always works, although there is no direct proof, but our experience so far is if your insulating system, clearly this is still for insulating system, if the insulating system is topologically trivial, it really doesn't matter which point you choose, mostly, most likely the case, but if it is topologically non-trivial anyway, this won't work, so it is quite, well, in practice it becomes actually easy. So this is some examples of the SDM orbitals, the shape of that obtained from Kwon Davis-Brussel a couple of years ago from the gamma point calculation, and you can see that this is the shape of the SP hybridized orbitals located on the bound, and this is the localized orbitals localizing on the oxygen, pretty much agrees with the chemical intuition. Again, this is only for the valence band, and this is more recent result with the K point, and this is a system we obtained from Steve Lewis, former group member, Cynia Sarko, and he was working on the so-called chromium oxide example, which is quite challenging because it's been polarized, and you have the competition between the DO orbitals and the PU orbitals, so on and so forth, and you have to, if you directly run one year, it really depends on the initial guess. Here is an example. If you already know the orbitals you are interested in, which is the DXY, DYZ, DXZ, you start from here, that is the initial spread is about 70 nth from square, and after 30 iterations it converges, and if you start from a wrong guess, I'm just throwing some random thing there, let's say start from SP2, of course it's wrong, but the point is it should be robust. If you start from SP2 initial guess, you can see that it got stuck, and it never converges even if you run it for a thousand iterations, and if you do the SCDM, you can see that it's almost a flat line, but it's not exactly flat because SCDM is not guaranteed to minimize the spread of the one year function, but almost in the sense that the initial spread from the SCDM, that is run on the one line or two line of the code, is already 17.22 after the fully converged one year function is 16.98, so you're really almost there, and if you look from there, it's just a flat line. Okay, now I run very quickly through the case of the isolated band structure, but if you have questions about that, I'll be more than happy to explain after the talk. The reason why I rushed a bit because I want to talk about the entangled band, and in the case of entangled band, if you look at the density matrix, if there's no gap, which means that the density matrix as a projector along the energy spectrum is not a smooth function, but the decay of the one year functions is directly connected to the smoothness of the underlying matrix function, and if you don't have smoothness, you don't have the decay, and therefore for metallic system, directly do it, you don't have exponentially localized one year function. The idea is very simple, which is you use a quasi-density matrix, something that does have the smoothness property. And how do you do it? You can see this F plays the role of occupation number, and this is not something that is very ailing because in Combsham DFT calculations, you have a Fermi Dirac or Gaussian or this kind of things, you are smearing that anyway. However, you usually smear it to the extent of the physical temperature, that is about 100K, 300K, up to 1,000K, so on and so forth. What we are doing localization, which means that the energies we are looking at can span several hot trees, which means that here, in constructing the quasi-density matrix, it's only a mathematical tool, and we can actually afford to use a smearing that is about several EV width, and what it doesn't mean to have several EV, converting it to KBT, that's like tens of thousands of K. You can have pretty decent decay due to this artificial smearing. That's exactly the idea, and as I'm showing you here, for the case of isolated band, this is the HOMO, this is the LUMO, you have a gap, and therefore, although the spectral projector looks like a discontinuous function, but there's no difference between this function and something that is smoother in the gap, so you effectively have something that is smooth. For the entangled case one, this is obtained from the terminology from one of Professor Mazari's paper in 2007, and for the entangled case one, you are interested in localizing both the valence band and also part of the conduction band, and therefore, you choose the smearing function to be like this. You can have more than one way of doing this, and one way is to use this Earth-C function. It is given by two parameters, one's mu, where you roughly want to cut, and second is sigma, how large is the smearing? And these are the only two user parameters in the code. And the entangled case two is, for example, you have a copper, and you want to disentangle the D electrons from the rest of the S and P electrons, and therefore, you use a Gaussian-like function. Again, the shape of the function is not necessarily very important, but still, if you use a Gaussian, you have two parameters, one's mu and one's sigma, and you can tune this so that the resulting density matrix is smooth along the energy direction. So this kind of idea, if you want to try this idea, there are a few ways to do it, and one is we first put it on the MATLAB code, and recently it is on the version on Julia, but also with the variational formulation. And, okay, here is the hidden trailer that is obtained from Ivan that his paper appeared on archive this morning, and therefore, you have the most up-to-date reference for this, and also thanks to Dr. Vitaly's work, and this is also now in the developer, at least in the developer branch of 1N90. And as I said before, how would you use this instead of having the begin projections block? For the isolated band, there's no parameter whatsoever. All you need to do is the SCDM projection equals to true, and the SCDM entanglement, I think this equals to zero, means isolated band, and that's it. And for the entangled band, you need to select this one or two and give the mu and sigma parameters. You can see that it becomes a lot easier to tune than before, especially this mu. Usually you can safely put it to be the Fermi surface. So this is some examples. This is the band structure interpolation obtained solely by SCDM, which is this one trick. And you can see that it interpolates the band very well. This is benchmarked with the result obtained from quantum espresso. This is for the case one, for the full valence band, and also part of the conduction band. And this is case two for the copper. You can see that it's nicely selects out the d-orbitors in the middle of the spectrum. And as I said, spread is not everything. Here I'll give you the first example, which is quantitative. I'm using the 10 by 10 by 10 k points. And you start from six bands, and you like to reduce these to four bands. And we directly use SCDM and don't do this entanglement. And what you see here is it is very nicely interpolated up to the Fermi surface. Actually, even above the Fermi surface, it's not that bad either. And the SCDM orbitals, the average spread is 18 point, sorry, the total spread was 18.38 N squared. And if you turn on this entanglement, but the wrong, in a poor way. I'm not saying that one year couldn't do this system. Clearly one year can do it. But if you only focus on the spread, you can get some orbitals has a significantly smaller spread but the interpolation is just garbage. And this is to say that, I mean, you shouldn't, especially in the case of entangled band, you shouldn't only focus on the spread. And yeah, especially smaller spread does not necessarily mean that you have a better interpolation. Now with the rest of the 10 minutes or something, so I can talk about the recent thing that is the variational formulation for one year functions for the entangled systems. And as I said, it just appeared yesterday. And the standard approach, if I understand it correctly, in this community is called the disentanglement procedure proposed by Souza, Masari, and Vanderbilt in 2000, sorry, 2001. And the basic idea is the following. Because you have an entangled band, it is impossible to select n orbitals and find n one year functions and there's never going to be smooth, so it's not a good idea. Instead, you introduce something called the frozen orbitals, which is denoted by PF. This PF is just some, let's say, within a certain energy window, you say your one year functions should exactly reproduce all the states in this energy window, for the k-points you have selected. And this condition is mathematically realized by the following constraint, that is the projector corresponding to the one year functions projected to the frozen window, should be the frozen window. So that's it for all the k-points in the Berlin Zoo. And this is a subspace selection process with a frozen band constraint. And in practice, you need to work with the three numbers. One is an outer, that is how many states you have in your outer window. You can conceptually take it to be infinity, but practically you often reduce it to a more reasonable number. The second is the number of one year functions you are interested in. And third is the number of frozen bands for each k-point. I mean, these are three necessary parameters and you do need to work with more bands. Honestly speaking, at the beginning, we tried to understand this entanglement procedure. Thank you. We couldn't understand this very well because we don't know in the variational sense what it is minimizing. We really don't know. And now we think we have a better understanding because we now know the reason why we don't know why it is minimizing because this is not. It is actually better viewed as an incomplete minimization problem towards minimizing something. And it doesn't mean that it's not a good algorithm. It's still very good. Okay. So in order to enforce this constraint, and there are at least four equivalent ways, mathematically equivalent ways for enforcing the constraint, I don't have time to go through them. But the numerically most convenient one is this so-called XY representation. That is your gauge UK is given by this weird matrix. You have identity reflecting the frozen band. You have a YK gauge in the block diagonal form. And this YK is a unitary multiplied by another unitary matrix, which is SK. And this part corresponds to the gauge invariant part and this part corresponds to the gauge dependent part. So more specifically, when you look at the variational formulation, it looks like the following. It looks very much like the Masai revender built recipe. You still minimize the spread, but instead of minimizing U, you minimize these two things independently. And you have the X and the Y. I want to say this is a redundant representation and which means it is more difficult to minimize than the original problem. But you exactly put this answer in and you have the X and the Y to be unitary matrix, and that's it. You do minimize this. And after the work, we realized, I mean, as a euro when you write the introduction, you realize that this is equivalent, mathematically equivalent, to the so-called partly occupied one-year functions proposed by Theiger-Fern, Hans and Jacobsen in 2005, but it is what it is. And if you want to use this formulation, the preliminary version of the Julia code is available here. And the relation to the disentanglement procedure, as I said, is the following. In the disentanglement procedure, you split the functional into the gauge invariant part and the gauge dependent part. And you can do the so-called alternating minimization. You minimize the gauge invariant part, minimize the gauge dependent part, go back and forth and so on and so forth, and hopefully this will lead to a convergent algorithm. However, the actual implementation of the disentanglement procedure only does this in one step. You do the gauge invariant, gauge independent, sorry, gauge invariant, gauge dependent, then you are done. So it is actually a one-step realization of the alternating minimization for the variational problem, and therefore the total spread of the variational formulation is guaranteed to be no greater than the spread of the disentanglement procedure. So let me quickly run through a few examples. And one is the silicon system with eight bands. And as you can expect from chemical intuition, if you just have the valence band, the orbitals should look like the SP hybridized orbitals. But they have eight orbitals. They should really be the atom centered, not bond centered. It becomes an SP3 hybridized orbitals. So you can see the variational formulation indeed has a slightly smaller spread, and all the three methods give excellent band interpolation. And another interesting thing to see is although the SCDM orbitals have a significantly larger spread, you don't see the difference at all from the band interpolation. And a more interesting result can be seen if you resolve to the per orbital spread. You can see all the eight orbitals obtained from the variational formulation have exactly the same spread. And if you look at the one-year, they actually split into two groups, one the 3.16, one the 3.59. And if you look at the shape of the functions, they look a little bit different. And this one looks more like the SP3 hybridized orbitals, which agrees more with the chemical intuition. I mean, I'm not saying that the variational formulation should be expected to restore the symmetry always, but at least coincidentally here it restored the symmetry. Another thing is the SCDM does not recognize the symmetry. You can see that it is split into the two groups, mostly related to valence and conduction, respectively. And as I said in the abstract, we have actually a few other materials in the paper. You can read that, like aluminum, copper, and other things. But another very interesting thing is that at least in the literature, we're not aware of the report of the decay properties of this generalized one-year function for uniform electron gas. If you just think about the uniform electron gas, if you do the standard localization, you just have a 1 over r decay period. And if you do this generalized one-year function, thanks to the variational formulation, we can ask a definitive answer what is the decay property. I think that this thing definitely works. This is the SCDM initialized variational optimization. And you can see it exactly reproduces up to the cusp. And the second band, I don't constrain that, so it doesn't matter. And you can see the orbitals, they indeed decay, and you can measure that very precisely, a decay like 1 over r squared. You might wonder at this point, why does this thing decay like 1 over r squared instead of exponential decay? This is precisely because the one-year functions is minimizing the spread or the second-order moment, or mathematically speaking, the H1 norm in the Fourier space. What is H1 norm? It is only minimizing the first-order derivative and doesn't care at all about the second-order derivative. And the optimizer exactly respects the request. That is, you can see there's a kink. And therefore, this is really, you minimize something, has exactly compact support and zero outside. And therefore, you have precisely a 1 over r squared decay. And very recently, there's a mathematical understanding saying that if you do it correctly, you can actually at least get to super algebraic decay. We implemented the procedure, you can see that by relaxing, we don't minimize the spread, but ask the spread to be a little bit larger, this can be enhanced to super algebraic decay. And you can do this in 2D. You can see this very interesting effect of the, I mean, it's a non-trivial boundary shape due to the enforcement of the constraint, due to the further band constraint. I don't have time to talk into that, but you still have the 1 over r squared decay if you minimize the spread. So, let me conclude. And so, one-year functions localization now, we think it can be robustly initialized, for example, with the SCDM. And you can use it for high, hopefully, it will be useful for high throughput materials calculation. You can do this, do the variational optimization even for entangled band. Spread is not everything, hopefully. This is a take-home message. And at first, I put some future directions like symmetry restoration, topological materials, but I heard just yesterday during the poster session, Dr. Vitaly has already made very promising progress along this direction. Hopefully, we'll see that happen in the near future. Thank you very much.