For a 1D system, there is a way to calculate the Zak phase in the discrete form. Suppose C is some closed path in k-space (a 1D BZ). If we suppose the path is discretized into (not necessarily equidistant) ki steps with i=1,…,N and kN+1≡k1, the end result is:
(the formula exists in the appendix)
You can view this as the product (i.e. phase summation) of N small rotations of the eigenvector's phase as it's transported along C; the Im(log)-part merely picks out the phase.
If C is a non-contractible path in the BZ along a reciprocal lattice vector G, it is desirable to enforce a periodic gauge, in which case one would take:
(the formula exists in the appendix)
How we can implement the periodic gauge in numerical calculations? We know it is necessary when we talk about the berry phase. I mean how I can do a numerical calculation for considering periodic gauge mentioned above.
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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