[QE-users] symmetry traces in sym_band.f90

[Hongyi Zhao]

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