[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.

It seems that the following is a similar tool for Firefox:

https://github.com/fred-wang/TeXZilla

But I'm still not sure, and here [1] I filed an issue to ask the developer.

[1] https://github.com/fred-wang/TeXZilla/issues/74

Best Regards,
Hongyi


> 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.
>
> Best,
> --
> Gerson J. Ferreira
> Prof. Dr. @ InFis - UFU
> ----------------------------------------------
> gjferreira.wordpress.com
> Institute of Physics
> Federal University of Uberlândia, Brazil
> ----------------------------------------------
>
>
> On Tue, Mar 8, 2022 at 10:28 PM Hongyi Zhao <hongyi.zhao at gmail.com> wrote:
>>
>> On Tue, Mar 8, 2022 at 10:00 PM Gerson J. Ferreira
>> <gersonjferreira at ufu.br> wrote:
>> >
>> > I'm sorry, but it works with SOC as well. This is just a basic change of basis in group theory notation, it does not matter if we are dealing with single group or double group. Two representations are equivalent if they are related by an unitary transformation as A = U.B.U†,
>>
>> How do you enter Unicode symbols such as "†" in email and keep their
>> formatted as they are, i.e., displayed as a superscript here?
>>
>> > where A are the representation matrices in one basis, and B the representation matrices in another basis, and the unitary transformation U is the same for all operators. So I just want to find the change of basis U.
>>
>> OK. Let me describe it further. Here, we have 3 matrices, A, B, and U.
>> You want to find U. So I want to clarify the following questions:
>>
>> 1. Is the QE numerical wave-function corresponding to A or B?
>> 2. In this unitary transformation, A = U.B.U†, suppose you only know
>> A, then it may have many possible (B, U) pairs which meet the
>> similarity transformation or conjugation condition [1].
>>
>> Maybe I don't really understand the essence of the problem, or I don't
>> really understand your meaning. Criticism and correction are welcome.
>>
>> > Maybe you are confused by my example where I started with the single group irreps for graphene and later I showed the 4x4 double group matrix. But notice that for graphene the double group irreps are simply a direct product of spin with the single group irreps. In other words, the SOC in graphene is of the sigma_z type, so the spin blocks are decoupled.
>>
>> A small supplement, the sigma_z is the 3rd Pauli matrix defined in Mathematica:
>>
>> PauliMatrix[3]
>> {{1, 0}, {0, -1}}
>>
>>
>> > Nevertheless, the graphene example is just a particular simple case. The change of basis and everything discussed here is valid for any single or double group.
>> >
>> > I'm thankful for the discussion and the tip about the sym_band_sub.f90 in the thermo_pw code. This will certainly help us finish our code and paper. I'll let you known once we have a first draft and make the code public.
>>
>> You're welcome. Wish you all the best!
>>
>>  [1] https://en.wikipedia.org/wiki/Matrix_similarity
>>
>> Yours,
>> Hongyi
>>


-- 
Assoc. Prof. Hongsheng Zhao <hongyi.zhao at gmail.com>
Theory and Simulation of Materials
Hebei Vocational University of Technology and Engineering
No. 473, Quannan West Street, Xindu District, Xingtai, Hebei province