[QE-users] symmetry traces in sym_band.f90

[Hongyi Zhao]

On Wed, Mar 9, 2022 at 9:04 PM Gerson J. Ferreira
<gersonjferreira at ufu.br> wrote:
>
> To write the formatted symbols I use the "Tex for Gmail" extension (only works in Chrome), which allows me to use Latex expressions in the email and it compiles using Unicode text or PNG images. Here I'm using the unicode option.
>
> Regarding your question, we can say that the numerical QE matrix is A, while B is the representation that I want to work with. So I know both and the only unknown is U. For an algorithm on how to find U, please check:
>
>> M. Mozrzymas, M. Studziński, and M. Horodecki
>> Explicit constructions of unitary transformations between equivalent irreducible representations
>> J. Phys. A. 47, 505203 (2014).
>
>
> My problem is that I want to use the QE wave-functions to do some calculation, but it needs to be in the B representation, so I need to find the similarity transformation (or basis transformation) U that takes A to B, so I can apply it to the QE wave-functions. As I said above, I have B (it's my choice of basis) and I need QE to calculate A, then my code finds U.

Taking your aforementioned 2-dimensional matrices as an example, based
on the tricks given here [1-2], I obtained the method as shown in the
attachment.


[1] https://mathematica.stackexchange.com/questions/98514/finding-a-unitary-matrix-in-mathematica
[2] https://math.stackexchange.com/questions/4199837/find-a-matrix-for-a-unitary-transform-between-matrices-or-prove-that-there-is-no/4199915

Regards,
Hongyi
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