[QE-users] symmetry traces in sym_band.f90

[Hongyi Zhao]

On Mon, Mar 7, 2022 at 8:58 PM Gerson J. Ferreira
<gersonjferreira at ufu.br> wrote:
>
> Easy answer first: I'm sorry for the formatted code. I've copy/pasted directly from the VSCode and it went like this. It was not on purpose, just a lack of attention to detail.
>
> Now regarding the irreps, notice that you are describing the characters of the representations, and not the matrices. I agree that if all we want is to identify the irreps that describe each band, it is sufficient to look at the characters / trace, and that's what the sym_band.f90 routine already does.
>
> These traces are invariant under unitary transformations (change of basis), but the full matrix representation DΓ(S) for S in G is not. In other words, let's call A a 2x2 matrix representation of the C3(z) operator under a certain basis set {ψ0, ψ1}, which transforms as some irrep Γ. If U is an unitary transformation that gives a new basis (within the same subspace) denoted {ϕ1, ϕ2} = U ⋅ {ψ0, ψ1}, then under this new basis the matrix representation of C3(z) becomes B = U.A.U†, which is also a rep of the same irrep Γ, same traces.
>
> I need the full matrices DΓ(S), and not just the characters / trace, because I need to make sure that numerical wave-functions from QE match the representations I'll use later in a separate python code. Is it clear now? So I'm editing this routine to get what I need.

Interestingly, you seem to have created some Python code for your
specific research. Where's the python code? Have you shared it on some
public code hosting platforms, such as GitHub?

Best,
Hongyi

> Regarding the bug, don't worry. One of the QE devs has already replied to me. It is indeed a bug in the D_3h classes that was unnoticed because both s_v and s_h classes have characters  equal to zero in the double group, so it does not affect the identification of the irreps. I'll try some suggestions he sent me to fix this.
>
> But if you want to reproduce this, simply run bands.x for graphene (full relativistic) or any other D_3h material and use the sym_band.f90 file attached here. But notice that in this code I'm printing a bunch of stuff to stdout for testing purposes. The relevant section for this bug is between lines 870--876.
>
> Best,
> --
> Gerson J. Ferreira
> Prof. Dr. @ InFis - UFU
> ----------------------------------------------
> gjferreira.wordpress.com
> Institute of Physics
> Federal University of Uberlândia, Brazil
> ----------------------------------------------