 Yes, we can see. Thank you. Okay. Thank you very much for giving me a chance to this nice meeting. Today, I'd like to talk about construction of one function using crystal symmetry. I'm Takashi Koretsune from Tohoku University, Japan. Okay, this is the outline my talk. So first, I'd like to explain how to construct one function using symmetry. And the second topic is about symmetry adaptive one functions. Okay. As you know, the key quantity to calculate one new function is the overall matrix, which is defined like this. This is a K vector in the Fuglian zone and B is the vector that connect K point to its neighbors. So we have to calculate this quantity in all the K point and all the B vectors. So in the current implementation in the quantum express, we first calculate all the wave function in the Fuglian mesh and calculate overall matrix. And in some codes, for example, first calculate wave function in the Fuglian zone and then generate a function in the Fuglian mesh using symmetry and calculate this quantity. But it takes a lot of time and the file size of this matrix can be very large. So I'd like to explain how to calculate this matrix using symmetry. So to do this, we first define the wave function in the Fuglian zone, which can be written by symmetry operation and the wave function in the reduced Fuglian zone like this. But the point is that there is a several symmetry operation that moves K point in the Fuglian zone to this point. So yeah, this is K i and this is K f. So there are several symmetry operation for this. And we have to fix, determine which symmetry operation we use for each K point. And to calculate overall matrix, we have to fix this G zero for each K point. Okay, so now let's discuss the overall matrix in the Fuglian zone using the wave function in the reduced Fuglian zone. Then this matrix can be written like this and using the wave function in the reduced Fuglian zone and this becomes like this. And this G zero and G zero prime is symmetry operation. And in general, these symmetry operations are different. So we cannot rewrite this expression using this overall matrix in the reduced Fuglian zone. But we can decompose this G zero prime using G zero and H. Here, H is a symmetry operation in a little group of K. So this is a set of symmetry operation that doesn't change K point. And using this equation, we can rewrite this like this. And using this expression, and then we can rewrite this using the overall matrix in the reduced Fuglian zone and this matrix. This is called a representation matrix. And once we can obtain this quantity, then we can calculate overall matrix in the Fuglian zone. Okay. And next, let us consider the symmetry for projection matrix, which is defined like this. And this is an initial one in the orbital. And again, we consider the projection matrix in the Fuglian zone, which can be written like this and this. So we can rewrite this using the wave function in the reduced Fuglian zone. And this part, we know how this orbital transforms under the symmetry operation, which can be written like this. And this is a so-called rotation matrix. So using this equation, then we can rewrite this using the projection matrix in the reduced Fuglian zone and the rotation matrix. Okay, so that's what we did. So in the implementation, we first calculate wave function in the reduced Fuglian zone using the quantum express. And then we use modified PW2190 to calculate these matrices in the reduced Fuglian zone. And also we calculate representation matrix and rotation matrix. And combining these quantity and we calculate these matrices in the Fuglian zone. And for this, we implemented the new code. And using this Python code, we can generate this quantity. And then we can run 1A90 using this. This is what I did in this implementation. And this code works with any symmetry operation that quantum express can consider. So this works with symmetry symmetry operation and symmetry symmetry operation fractional calculations. And we can also use a symmetry of coupling and with the software and PW shoot potentials. And this code is implemented and uploaded here. So you can check how it works using this code. Okay, now let me show some example. And this case. This is a band structure of cobalt chandite with a spin of coupling. And this is calculated by original 1A90 compared with the DFT calculations. And this is a new calculation. So as you can see, both calculation agrees well with the DFT is one. And the total spread of one functions are very similar. This original one is this and the new calculation, the spread is like this. And in this calculation, we use about 500 k points. But in the calculation, we only calculated 65 k points from the symmetry. And so the computational time is about 10 times faster. And the MMN and the M file is the size is one tenth of the original ones. Okay, this is a first option. And let's move on to the next one, symmetry adaptive one function with frozen windows. Okay. Yeah, as pointed out by several talks in this meeting, symmetry of one function obtained by 1A90 is slightly broken. And this is convenient for analyzing the political properties. So there are two approaches. One is symmetrizing the Hamiltonian after one realization and the second approach is symmetry adaptive one. Okay, so I will explain this method. Okay. First, let me briefly summarize the previous approach. Okay, this is the definition of one a function, as you know, and this is new matrix, and we have to determine this new matrix. This new matrix can be written like this. This is a property function, and this is a free transform one function like this. And then we consider the symmetry relations for property function and one functions. Okay, so this is a relation for property function. And, yeah, when we apply symmetry operation for function, the function changes like this. We know all the function so we can calculate this quantity. And this is a relation for free transform of functions. And in this case, if we assume the symmetry adaptive function for this part, then we can calculate how this wave function transforms under the symmetry operation. Then we can calculate this D matrix. Yeah, in the current implementation, we can calculate this matrix matrices using symmetry adaptive mode and the calculated file. The data is written in DEMN file. But using the method in the first topic, we can also easily calculate this quantity. So actually this corresponds to the representation matrix and this corresponds to the rotation matrix. Okay. So these are three equation shown in the previous slide. And combining these relations equations which are easily obtain how this new matrix transforms under the symmetry operation. So this is the symmetric constraint for this new matrix. So once we obtain new matrix that satisfies this condition, then we can obtain the symmetry adaptive one functions. And to symmetrize this matrix, we first symmetrize UK in the real distribution zone using the little group of TK, and then expand UK to the whole brilliant zone like this. Okay, so this is our main idea of symmetry adaptive when you function approach. Now, let me move on to the entangled case. When you function can be written like this. And this, this is an optimal subspace for each K point defined like this. And this is a projection as you know. So, yeah, we have to determine this new matrix and this you opt. The relation between one a function and the property function is characterized by this you times you like this. Yeah, instead of you, we have to consider you talk and the symmetry constraint becomes like this. So there is only one constraint for the new matrix, you and you. And in the previous study, he assumed that you opt also obeys this symmetry constraint. This means this corresponds to the assumption that this web function also behaves like a free transform one function. So he added additional assumption and this assumption is not compatible with frozen window. So that's why we cannot use the frozen window in the previous approach. So we considered another implementation. In this approach, we first calculate you opt matrix without any symmetry constraint. And then we import the symmetry constraint when optimizing this matrix. So we first we calculate you opt and then you and then we can calculate you dot and then we symmetrize you dot and then we calculate you using this relation. And using repeating this process until we get the convergence. That's what we did in this new implementation. And the advantage of this method is that we can use the frozen window technique. But the problem is that this new matrix is this unity of this new matrix can be broken by symmetrization. Yeah, it depends on the choice of frozen window and mission of us. And anyway, so as a result, obtained Hamiltonian does not necessarily reproduce the original energy bands. So that's the problem in this approach. So let me explain this advantage and the problem in the example. Okay, this is an example. The case of my open. The comparison of the structure and one interpretive one. This is the original 90s result with frozen window and no symmetry constraint. And this one is original 90s symmetry adaptive mode with no frozen window. And this is new calculation. So as you can see, this calculation fails to reproduce the DFT results. But the first one, the third one will reproduce the result. DFT is real results. So, yeah, in this energy scale post works very well. So, let me check the detail of the boundary structure. So we focus on this area. And this is a magnified product of the boundary structure. On the gamma P line. And this black one is DFT is real result. And this one's a fast one. When you're 90 with frozen window and no symmetry. And the green one is a new calculation. DFT and 90 with frozen window will agree with each other. But this new calculation gives slightly different energy. But the difference is very small about 0.1 millib. Next, let me focus on this one. In this case, this band is double degenerate due to symmetry. I mean, the wave functions are characterized by two-dimensional representation. So we brought the energy difference of the W-degenerate band in this plot. And the DFT calculation, this band is completely degenerate. So the energy differences are zero like this. And the new calculation, we imposed symmetry, so the energy difference is also zero. But the original one in 90 with frozen window, there is no symmetry constraint. So this band is actually not degenerate. There is a, yeah, degeneracy is lifted slightly. Yeah, energy scale is very small, but there is a difference. So that's what we did. So, yeah, in this new approach, band structure is slightly different. But we can properly impose, yeah, symmetry is properly imposed. Okay, so that's what we did. So first, we explained how to calculate running function using the symmetry. And the second, I explained another implementation of symmetry-adapted approach used with frozen window. And this code is uploaded here. And yeah, I think it's very convenient if one in 90 can read this quantity and, yeah, regenerate MMM inside the plot. But it's a bit difficult to treat symmetry using photo and so, yeah, I don't know where to start. So I'm happy to discuss about this topic. Yeah, thank you very much for your attention. Okay, so thank you very much for the very nice talk, Takashi. So now we have time for questions. I see already one for one hand. Takashi, thank you very much for your really nice talk and really nice work. I think a lot of the people get very useful for lots of things. Because I think you told the problem. I'm going to do the right person. Maybe you already talked. Sorry. I'm sorry, but it's very, it's very difficult to hear. Yeah, because the microphone is just here. You can come closer to this. So, Takashi, I was just saying on the irreducible for one zone stuff. That's really nice work. Thanks very much for doing that and I think you're in touch with Marco for trying to get this into PWM 90 officially that's great. Can you give us a sense of the computational cost savings that you get because there's some there's some additional work you have to do with the D matrices right to expand it out but presumably that costs nothing almost. Sorry. What's the channel cost for what we're going from the irreducible belongs on to the full matrix presumably that costs very little doing the operation to expand out. I mean, I think it's not so heavy. Yeah, I use Python code but the calculation cost is not so much. And these D matrices are they the same D matrices in the symmetry of the photo list that the new matrices that need to be calculated. Yeah. In the previous implementation. We calculate the matrix in a full brilliant zone. In this case, this is only for irreducible brands on. Okay. Yeah, and then on the second part I had I had a question is, can you can you explain to us why the frozen window is is not compatible with the symmetry constraint is there an easy way to understand that. Okay. Yeah. Yeah, this is a. Okay. In the previous approach. I think that this way function behaves like one of three a transfer of one new function. So, this means this M correspond to the index of money orbitals. But on the other hand, in the frozen window approach, the Europe is determined like this. So, within the frozen window, the Europe is just a diagonal matrix and this band index is this index is just the same with this band index. So, yeah, this index is completely different in the, in the frozen window approach and this symmetry adaptive approach. So that's why we cannot use frozen window approach in this. Yeah, in the symmetry adaptive. Okay. I have one actually probably it's me and I must have missed this. Can you go there to be some slide when you show there is a difference in the interpolated fans to calculate them with a standard method and with a similar. There is a two million different. Sorry. It's difficult to hear. I'm sorry. I was asking if you can go back to the slide where you show that there is a small mismatch. Yeah, this one. Okay. Oh yeah. So what I want to this seem when you're, you're not, this is this are not the symmetry adaptive any function this is just the symmetry symmetrization you do at the very beginning of this overlap matrices and right. No, in this case, this red one symmetry adaptive one, the red line. The red one, red one is not that red one is original one. So there's no symmetry and the green one. Yeah, right. Okay, so so the difference comes from the symmetry adapted not from the, the, you know, the, the first method you show. Sorry. This is symmetry adapted many functions right. Yeah, see symmetry adapted so the discrepancy comes from the, the fact that you are, you know, it's a different kind of functions if you want right. It doesn't come from the symmetrization of the, you know, this unfolding of the and so that from the irreducible brains on to the full brain zone of the overall matrices and they're in the AM and the projection matrices. Yeah, it is unfolding. It is unfolding, but it is because of the symmetrization in the irreducible bridge and so on. Yes, due to the symmetrization. Okay. Interesting. Yeah, it is similar to the symmetry station after we get by new having fun. Right. But that should be, you know, identical it's just you're just, you know, yours, it's a let's say it's more convenient to do, you know, to use the symmetries but the values to be really, really identical right. Yeah, but even for tight binding symmetrizing the tight binding Hamiltonian, you're hoping slightly changes and the boundary slightly changes. Okay. So here in the second approach you opt slightly breaks the symmetry. Because we do not impose any symmetry constraint in the you opt in the in the second approach. Okay. Yeah. Maybe maybe we can discuss this, this later. Are there questions. Yes. Can you come here discuss the questions here. Yes. Hi, so is there a limit on which initial projections you can use with this symmetry do like initial projections have to observe the symmetry or can they, you know break the symmetry and then you kind of symmetrize them later. Yeah, yeah, initial projection is very important in both calculations. Yeah, in the first case, if the, yeah, when you function breaks this. Yeah, is not compatible with symmetry, then we cannot do. And of course, in the second approach. And yeah, if you use Yeah. Not good. In short, then we cannot get a good. Yeah, when you function. So for example for like transition metals we often put like s orbital in the interstitial site away from the atom. So that means that all would not work or Yeah, in the case of a couple iron. Yeah, we can put s orbital in the interstitial site and we can get very good one you functions. Oh, okay, it works. Okay. Yeah. Okay, thanks. Okay, so we have. Thank you very much.