On Tue, Mar 8, 2022 at 10:00 PM Gerson J. Ferreira
<[email protected]> 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
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