 So this is the last day of the school. And our first speaker for today is Professor Yuzuke Nomura from Keio University. And he will give us an overview of symmetry-adapted Vanier functions, including both the theoretical sort of lecture and a hands-on tutorials. And so this is going to be a two-hours session with 30-40-minute talk plus the hands-on tutorials. Thank you a lot, Yuzuke, for accepting our invitation. And please, the floor is yours. OK, thank you very much for the interaction. So my name is Yuzuke Nomura from Japan. And the first, I'd like to thank the organizers for inviting me to this very nice conference. And also, I'm very sorry that I cannot attend in person. OK, let me start. So before going into the main topic, let me give just two advertisements. The first one is actually, we have just started a new group at Keio University. And if you are interested in, please visit my group website here. So the advertisement number two is QE-ELEPS. So it is an open-source program package for Contamnese Perso to compute irreducible representation of the growth wave functions. And the paper is, you can find the paper here and also the program from this GitHub page. And actually, we also have an interface to check topological math from the QE-ELEPS. So if you are interested in, please visit this GitHub page or this paper. OK, so let me start. So let me briefly explain the maximum localized warning functions. So basically, what we get from the DFT calculation is this kind of growth state. Then the transformation from the growth state to warning states are given by these equations. Then in this equation, the unitary matrix U is obtained by minimizing the total spread of the warning functions. So basically, this is the idea of the maximally localized warning functions. But sometimes, in the maximally localized warning functions, the symmetry of the warning function is broken. The famous example is given by Kappa. So in the case of Kappa, if we put the initial projection of the S and V orbitals at Kappa atoms, then after the maximally localized warning functions, then we obtain the not atom-centered S orbital, but the center of the S orbitals is located at some interstitial position. The reason why the symmetry of the warning function is broken is that the maximally localized procedure does not care about symmetry. But sometimes, we want to keep the symmetry of the warning function. Then Ray-Sapma proposes a way to combine the maximally localized procedure and the symmetry constraint. So basically, the symmetry-adapted warning functions obtained by the combination of the maximally localized procedure plus symmetry constraint. Then we can obtain the irreducible representation of subgroup of the full symmetry group or irreducible representation of the side symmetry group. And this functionality is now available in the 1A90, and it is available from version 3 of 1A90. Then basically, what we obtain is the irreducible representation of side symmetry group. So let me briefly explain what side symmetry group is. The side symmetry group is a subgroup of full symmetry group. Whose symmetry operations do not change the position of the side. Let me show an example using the two-dimensional square lattice. In this case, it is the C4V symmetry group. And the symmetry group consists of eight symmetry operations. So the symmetry group, symmetry operations, the C4 rotations, 90 degrees, 180 degrees, and 270 degrees, and 360 degrees. And also the reflection along the x-axis, and also along the y-axis, and the diagonal directions in this direction, and also these directions. And then if we consider the y-coff position of 0, 0, then at this 0, 0 point, all the symmetry group does not change the position of 0, 0. It means that the side symmetry group at 0, 0 consists of all the symmetry group. And the multiplicity of the side is just 1. But if we consider the y-coff position of 1, 0, then around this 1, 0, we do not have C4 symmetry about this point. So then in this case, the number of the side symmetry group consists of four symmetry operations without C4 rotational symmetry. Then the multiplicity of the side is 2. So if we apply the C4 rotational symmetry to this point, then this point moved to this point. So it means that the multiplicity is given by the number of full symmetry group divided by the number of symmetry operations in the side symmetry group. Then the concept of the symmetry-adapted one function is to put symmetry constraint to the maximum localize one-year functions. Then this is a schematic picture for the symmetry-adapted one-year functions. So in the case of maximally localize one-year functions, we try to find the global minimum of the total spread from all the unitary transformation space. But in this symmetry-adapted mode, we put constraint on this unitary matrix. So it means that the space for finding the minimum becomes smaller like this. Then sometimes the global minimum in this smaller subspace becomes local minimum within all the subspace. Of course, in many cases, these two maximally localize one-year functions become the same as the symmetry-adapted mode, symmetry-adapted one-year functions. But sometimes the symmetry-adapted one-year function becomes different from maximally localize one-year functions. And if they are different, the total spread of the symmetry-adapted one-year function should be bigger than the maximally localize one-year functions. Then here, let me explain a little bit more about how to put symmetry constraint on the unitary matrix. Because we are considering the symmetry, what we need to consider is the unitary matrix for the irreducible k-point. And then for other k-point, we can reproduce the unitary matrix from the symmetry operations. So what we need to care is just irreducible k-point. Then the U of k for this irreducible k-point, this unitary matrix should satisfy this equation. So here, we have two different matrix. One is d tilde and the other is capital D. And these two matrices are related to symmetry. And I will explain a little bit more later. And this g of k is the symmetry operations that not change the position of k-point. Then once u of k at irreducible k-point satisfy this symmetry constraint, then the unitary matrix for other k-point can be reproduced by the symmetry operations like this. Then the key quantities, these d tilde matrix and also these capital D matrices. Then let me explain what these matrices are. So first let me start from capital D. So capital D matrix shows the transformation of symmetry-adapted one-a-gauge both functions by the symmetry operations. So this both wave function is the one-a-gauge both function. Then we apply this symmetry operations g. Then these both functions are transformed by this formula. So this transformation is given by this capital D matrix. On the other hand, in the case of d tilde case, so it shows the transformation of the concham numerically obtained in the concham both wave function by the symmetry operations. So this is the concham wave function obtained in the DFT calculations. Then we apply these symmetry operations. Then these concham wave functions are transformed by this formula. And the information of these two matrices written in seed name dot dmn file in the case of symmetry-adapted mode. So this is the flow of the calculation. So basically the flow of the calculation is basically the same as the case of maximally localized one-a-functions. But we need additional input. But additional inputs are very simple. So here we assume that we already perform the NSCF calculations, and also we already created .mnkp file using the preprocessing of the one-a-90. Then what we do is first to learn the pw2090 to create the amn and also the mmn file in the case of maximally localized one-a-function. But in the case of symmetry-adapted mode, additionally, we need the information of this capital D and also this dq. Then if we put additional input of write dmn in the pw21.in, then this pw2090 create the additional input of the seed name.dmn file. And in this dmn file, the information of capital D and dq matrix is written in this file. Additionally, we also have the output of seed name.sin. So this is not used in the one-a-90. But in this file, the information of the symmetry operations employed in this pw2090 are written in this file. Then after that, we execute one-a-90. And for this input of one-a-90, we need additional input of size symmetry. Then with this additional input, the one-a-90 create the symmetry-adapted one-a-function. So basically, the calculation is very simple. Just we need additional inputs of write dmn and size symmetry. And then for the educational purpose, let me show you an example of the dmn file, capital D, using the calcium copper O2 case. So the calcium copper O2 is a member of high-TC cube rate. And the crystal structure is given by this. So it consists of calcium copper O2 layer and the calcium layer. And it is quasi-two-dimensional material. And the space group is p4mm. And we have 16 symmetry operations in total. Then let me consider to create the DP model for this compound. So if we perform the DFT band structure calculations, we obtain this band structure. So red band is the DFT band structure. And the green, sorry, the blue one is the winding band for the DP model, for the DP model. Then for creating the DP model, we put the projection of x square minus y square orbital for the copper side. And the px orbital for this oxygen and the py orbital for this oxygen side. Then the x square minus y square orbitals is a basis function for the one-dimensional irreducible representation for this size symmetry group of the copper. And the multiplicity of the side is one in this copper side. Next, we consider this oxygen-px orbital. It also forms one-dimensional irreducible representation. So here, about this side, we do not have C4 symmetry. Then the px and the py orbitals are not equivalent anymore. So the px orbital becomes a base function for the one-dimensional irreducible representation. And the multiplicity of the side is two. So mainly, we rotate this side by 90 degrees rotations. Then this side goes to this side. Then the px orbital centered at this side is transformed to py-like orbital centered at this oxygen side. Then this capital D matrix is a transformation of the symmetry-adapted one-dimensional functions by the symmetry operations. Then let me discuss the concrete example of this D matrix. So here, we do not discuss the D-tudor matrix because this D-tudor matrix shows how the constant wave functions are transformed by the symmetry operations. But numerically obtained, the constant wave function has a random phase. So this D-tudor matrix depends on this random phase. So here, we do not discuss this D-tudor. But we can discuss this capital D analytically. So let me consider the capital D matrix for the 90 degrees rotation at gamma point. So at the gamma point, the x square minus y square like a broad wave function looks like this. So at each copper side, we have x square minus y square orbital. And the phase of the x square minus y square orbital is the same among all the copper side. We also have the px-like broad wave functions. We also have py-like one-year gauge broad wave functions like this. Then capital D shows how these broad wave functions are transformed by the symmetry operations. And let me show an example for this 90 degrees rotation. So in the case of x square minus y square orbital case, so if we apply the 90 degrees rotations, then after the transformation, the broad wave function looks like this. Then we can see that we obtained the additional phase of minus 1. So here, this amplitude is plus. But here, it becomes minus here. Also, if we consider the transformation of px-like broad wave function, after the transformation, the broad wave function becomes like this. Then I transform to the py-like broad wave function. Similarly, the py-like orbitals are transformed to px-like broad wave functions. But here, we obtain the additional phase of minus 1. So in the case of gamma point and also the 90 degrees rotation case, the contents of the D matrix becomes like this. So the x square minus y square orbital obtain the additional phase of minus 1. And the px-like orbital is transformed to py-like orbital. And the py-like orbitals, it's transformed to px-like orbital with additional phase of minus 1. And you can clearly see that the matrix is brought down now for each irreducible representation. And the block size is given by the dimension of the irreducible representation times the multiplicity of the site. Then this is the obtained symmetry for the Gaussian kappa O2. So here, this is the x square minus y square orbitals for the kappa orbital, for the kappa site. And this is the px orbital for the oxygen site. And this is the py orbital for the oxygen site. So actually, in this case, the symmetry adopted one function is the same as the maximally localized one functions. But for example, in the case of kappa, as I already said, if we create the maximally localized one function for kappa, then the center of the s orbital deviates from the kappa site. But using this symmetry adopted mode, we can create the s orbital center of the kappa site. And if you want, we can also create the s-like orbital center at half by, half by, half by point if we use the information of the site symmetry group of half by, half by, half by. So finally, let me show an example for the H3S case. So for this H3S, you can also try the symmetry adopted one function in this tutorial. So the H3S is a member of hydride superconductors at high pressure. So if we apply extremely high pressure, then the H3S crystallizes into solid. Then its crystal structure is like a BCC-like structure. Then this solid shows high Tc superconductivity. Then if we perform the band structure calculation for this H3S compound, we obtain the band structure like this. So here, green, sorry. The black one is the DFT band structure. And the red one is the 1A band. Sorry, black one is DFT. And the red one is 1A band. Then in this case, we clearly see the difference from the maximally localized 1U function and the symmetry-adapted 1U functions. So this is the maximally localized 1U function. And this is the output of symmetry-adapted 1U functions. Oh, sorry. So here, we put the projection of SELFAR-S orbital, SELFAR-P orbital, SELFAR-D orbital, Hydrogen-S orbital. Then, for example, if we look at SELFAR-P block, we clearly see that the symmetry of Px and Py and Pz orbitals are broken. And also the center also deviated from the SELFAR site. SELFAR site is located at 0, 0, 0. And it clearly deviated from 0, 0, 0 site. But if we apply the symmetry-adapted mode, the SELFAR orbitals are located at 0, 0, 0 point. And also we have the symmetry among the Px and Py and Pz orbitals. The spread is the same. But as I said, because we put the symmetry constraint in the case of symmetry-adapted 1U functions, the total spread should be bigger than the maximally localized 1U functions. So here, the final spread is 10.99 something. But in the case of maximally localized 1U functions, it is 8.16. So clearly, the spread of the symmetry-adapted mode is bigger. But for example, if we look at the individual orbital, like hydrogen S orbital, here the spread is about 0.5, 2, or 0.51. But here it is smaller in the case of symmetry-adapted mode. So if we look at the individual orbital, the spread of the individual orbital can be smaller than the maximally localized 1U functions. But if we look at the total spread, it should always be bigger than the maximally localized 1U functions. Then now this symmetry-adapted mode is implemented in the version 3 of the 1U90. And you can also try the symmetry-adapted mode from the official example in the 1U90. So the symmetry-adapted mode are prepared by the example 21 and the example 22. So you can also try the symmetry-adapted mode from these examples. OK, finally, what we need to be careful is that sometimes there is a misunderstanding that we can always get 1U orbital whose centers are exactly at some atoms. But what we should be careful is that we cannot always get atom-centered 1U orbitals in the symmetry-adapted mode. So to get atom-centered 1U orbitals, actually we need proper symmetry. So let me show an example using the molybdenum disulphide case, monoreal case. So in the case of monoreal case, the saloper layer here is sandwiched by the vacuum layer and also the molybdenum layer. So it means that there is no symmetry between upside and downside. Then we do not have symmetry. Then if we put, for example, the PZ-like orbital projection to this saloper atom, then we can obtain the atomic-like 1U functions. But if we look at the center of the 1U orbitals in the standard output, then this center should deviate from the saloper atom. But in this case, the deviation is very small. And if we visualize, we do not see much difference. From the atom-centered, exactly atom-centered 1U orbitals. But numerically, the center should deviate from center because we do not have symmetry. So it means that we need proper symmetry to keep the center exactly at atoms. OK, to summarize, the symmetry-adapted 1U function is given by the combination of the maximum localization procedure and the symmetry constraint. Then we obtain the irreducible representation of the side symmetry group. Then for the future perspective, we are considering the frozen window. And for this topic, my colleague, Takashi Koretsune, will be talking about at the development of this meeting. Also, the extension to non-colonial case is also an interesting future perspective. OK, for the lecture part, that's it. Then now we can take questions. Thank you for your kind attention. Thank you very much for the very nice talk. So there are a few questions on the chat that I would like to ask first. So there were two questions that are essentially equivalent. And they ask, what about other Abinishia engines? Does this symmetry-adapted 1U function method also work with VASP? Has it been implemented only in Quantum Espresso? Yeah, for the moment, it is implemented only in the Quantum Espresso. But we can easily create an interface from VASP, or the Abinishia package. Because basically what we need is just the additional input of this seed name dot dmn. So in this seed name dot dmn file, we have the information of this capital D and this dtuda. So basically the interface from other program package to 1A19, they create AMN file and the MNN file. So if we can create dmn file from other Abinishia packages, we can use symmetry-adapted mode in the 1A19. But for the moment, I do not know how the code are implementing this dmn file. It should be not very difficult. OK. There was a question, but not really clear. But it's Lata-Belgina, if you can actually re-ask your question. Well, in the meantime, while we wait, are there any questions here in Trieste? Yes, there is one. Thank you very much. Very nice presentation. So I wanted to ask about the Atom Center at 1EF functions. So is it possible to constrain the 1EF functions to Atomic Centers? I could imagine some defect structure where only translation remains the symmetry operation, but I still would like to get 1EF functions centered at the Atoms. I don't think we can, from the symmetry-adapted mode, the answer is, I think no. Because as I already said, what we can do from the symmetry-adapted mode is put the symmetry constraint on the 1EF functions. And if we have defect or something other, if the symmetry is forward from, for example, defect or the slab structure, then if the symmetry is lost, then we cannot obtain the exact triatom center 1EF orders in that case. Thank you. So we always need some symmetry to do that. OK. So I see that Vlata has rephrased the question. If we could, what we can do is just we create the one-shot 1EF function without any minimization of the spread. Then in this case, in many cases, the center of the 1EF orbitals are almost same at the center of the symmetry is higher than the maximally localized 1EF functions. So it is a kind of practical solution. OK. So Vlata rephrased the question. And she asked, if we compare the hopings between the Hamiltonian constructed with symmetry-adapted 1EF functions and the one constructed with maximally-adapted 1EF function, will they be different? Yeah, if these maximally-adapted 1EF functions and symmetry-adapted 1EF functions are different, then it should be different. In many cases, if we create the maximally-adapted 1EF functions, they keep the original symmetry. Then in that case, the symmetry-adapted 1EF functions and the maximally-adapted 1EF function is the same. So the hopping should be the same. But for example, in the case of this H3S case, clearly, the symmetry is broken in the maximally-adapted 1EF function. Then the symmetry of the hopping should also be broken in this case. But basically, these two are just related by some unitary matrix. So if you plot the boundary structure, it is the same. But if we just look at the individual boundary of the hopping, then it should be different between the symmetry-adapted case and the maximally-operated case. OK. Are there any questions from the Adriatico? I have a question. Please go ahead, Stepan. Yeah, if you could. So thank you for the nice talk. My question is basically to the last slide about the frozen window. So just, of course, it's more a discussion for developers meeting. But briefly, if a problem in the formula is not just some small lack of implementation, what is the problem where it was not implemented from the beginning? Yeah, yeah. So in the original implementation of the frozen window, we just took the energy window for the frozen window. But the energy does not care about symmetry at all. So if we have some state that does not satisfy the symmetry in this frozen window, then we cannot create the symmetry-adapted mode. So to be compatible with the frozen window, the frozen window should also care about symmetry, not only the energy, but also the symmetry in the frozen window. That is the problem. OK. So if there are no other questions, I think it's time to move to the hands-on. Yeah, I think we have additional. There is one. OK, yeah, OK. We want to use this. Yeah, the question is whether it's better to use the symmetry-adapted, I think, or the usual maximalized to construct against a tight money model. Yeah, so if we want to analyze the symmetry property using the tight money model, the symmetry-adapted mode should be better. But if we, so as already Ibo answered on Wednesday, the physical quantity calculated by the one-year function, for example, the whole conductivity or optical conductivity, this should be the same. Basically, the symmetry-adapted mode and the maximal localized functions translated by the unitary transformation. And basically, the physical quantity computed by this model should be the same. But if you want to analyze the symmetry property, then the symmetry-adapted mode will help. OK, I guess we can move to the tutorial now. Yeah, please. So basically, you can find the tutorials in the day 5 AM 1, SAWF, Symmetry-Adapted Money Functions. Just copy this directly to what you want in the home directly or something like that. Then basically, we have three tutorials. One is the Gallium Arsenide and the others, Kappa and H3S. So what I suggest is that you start from Gallium Arsenide, then move on to the Kappa and H3S. Then you can follow the instruction of readme in each directory. And importantly, sometimes you need to modify the input file. Then if you get some error, most probably it is a mistake. There is a mistake in the input file. Then in this case, please compare the input. You can find the reference input file in the left directory. Then you can compare your input and the reference input. Then see the difference between them. Then you can avoid unnecessary error on the input. I think what we can do is that all the participants here, they just start to work on the tutorials. I have a microphone. So in case you have specific questions, you just raise your hand or call me and then I will bring the microphone and you can ask directly the question to Yuzuke.