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Z2 invariant calculation Date: 2022/10/14 04:38 Name: Alireza Baradaran   <alireza.baradaran@gmail.com> I want to compute the z2 invariant in 2D material by the Fukui-Hatsugai method based on the tight-binding approach not by the Z2FH code included in the OpenMX package. I saw the notes presented at a workshop at Kanazawa University by Dr. Sawahata and the OpenMX manual. I also study the references mentioned by Dr. Sawahata and his coworkers. I saw the Z2FH.C code and it seems any exp(iG.r) coefficients did not consider in the Kramars-pair or Time-reversal subroutines included in this code!; and any Translational symmetry is not included there, but the authors talked about three symmetry included: Translational symmetry, Time-reversal symmetry, and Kramar's degenerate. I know the wavefunctions in any k point derived from the Hamiltonian and then I can compute n-field, but my problem is about fixing gauge, especially about phase coeffect like exp(iG1.r) appears in the relation of symmetry gauges. I can't understand how I should implement these coefficients by a numerical calculation. gratitude to help me

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Re: Z2 invariant calculation ( No.1 ) Date: 2022/10/17 23:52 Name: Naoya Yamaguchi Hi, >I saw the Z2FH.C code and it seems any exp(iG.r) coefficients did not consider in the Kramars-pair or Time-reversal subroutines included in this code! All the treatments are addressed as appropriate, and the implementation of `Z2FH` is based on http://dx.doi.org/10.1016/j.cpc.2012.04.001 . Please refer to the above article to find which code corresponds to each treatment. Regards, Naoya Yamaguchi Re: Z2 invariant calculation ( No.2 ) Date: 2022/10/18 03:25 Name: Alireza Baradaran  <alireza.baradaran@gmail.com> Thank you Dr Yamaguchi Do you help me to know how the link variable defined by Dr. Fukui and Hatsugai derived? I can't find any reference. Re: Z2 invariant calculation ( No.3 ) Date: 2022/10/18 12:49 Name: Naoya Yamaguchi Dear Alireza Baradaran, In `Z2FH`, u, the periodic part of a Bloch orbital psi, does not appear explicitly, and instead, psi appears. Transformation from u to psi is necessary to compare the actual implementation with equations in the article. Regards, Naoya Yamaguchi Re: Z2 invariant calculation ( No.4 ) Date: 2022/10/18 20:23 Name: Alireza Baradaran  <alireza.baradaran@gmail.com> Dear Dr. Yamaguchi, This means that if we have diagonalized the Hamiltonian using the tight-binding method so that its output is eigenvectors, which are the periodic part of the wave function, we should include these eigenvectors in the definition of the link variable. This is right? Regards, Alireza Baradaran Re: Z2 invariant calculation ( No.5 ) Date: 2022/10/19 15:10 Name: Naoya Yamaguchi Dear Alireza Baradaran, What I mentioned is that the link variable can be obtained from u or psi. In the OpenMX code, the overlap matrix between u at two adjacent k-points is evaluated with Eq. 11 in http://www.openmx-square.org/tech_notes/tech11-1_1.pdf . Regards, Naoya Yamaguchi

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