[QE-developers] failing to calculating matrix representations of symmetry operators

[Gerson J. Ferreira]

Dear QE devs,

I'm not sure if I want to send this question directly to the mailing list,
as it might give away a big part of a code that I'm starting to develop.
But if you prefer I send it there, please let me know.

I'm sorry for the long email, but it is a difficult issue to explain. In
the first few paragraphs below I try to summarize and later I go into
details.

SUMMARY

I'm extracting the plane-wave coefficients and the total potential from QE
and planning to use it to calculate k.p matrix elements (Kane parameters).
So far, it works well without spin.

The problem is that the numerical eigenstates from QE are not necessarily
in the same representation that I would like to use to build my effective
H. So I have to apply a unitary transformation to adjust the basis. To do
this, I'm using the QE eigenstates to calculate the matrix elements of
symmetry operators.

I'm starting with a simple case: graphene. The symmetry
operators (generators) at the K point are C3z, My and Mz. Additionally,
it's useful to consider the composed symmetry TI = (time-reversal) x
(inversion), which is a magnetic symmetry.

The calculation of the matrix elements of C3z, My and Mz work well. The one
that is failing is the TI. So I wonder if the problem is that the composed
TI operation is not explicitly used in QE, since it is not listed in the
list of found symmetries. Therefore it might be that this TI symmetry is
broken in the numerical results, which explains why this is the only one
that does not work.

I know that QE can handle the magnetic symmetries, but it still has some
issues there. Is there a way that I can help QE identify this symmetry?

DETAILS

The figure below shows the absolute value of matrix elements of these
operators within 16 bands of graphene at the K point.

[image: image.png]

These seem ok. Most of it is set in 2x2 block diagonals, except at the
center, states [6,7,8,9], which is the Dirac point and it's nearly
degenerate. This degeneracy is why these states are allowed to mix into a
4x4 block.

What bothers me here is the eigenvalues of these matrices. For C3z, My, and
Mz I get the expected results within the 4x4 central block of states
[6,7,8,9]:

C3z = [-1. +0.j    -1. +0.j     0.5+0.866j  0.5-0.866j]
My = [-1. +0.j    -1. +0.j     0.5+0.866j  0.5-0.866j]
Mz = [-0.+1.j  0.-1.j  0.-1.j  0.+1.j]

But for the TI operator, the eigenvalues should be +1 or -1, but instead I
get

TI = [-0.969-0.247j -0.819-0.573j  0.819+0.573j  0.969+0.247j]

It's almost correct, but there's something slightly wrong.

Let me explain a bit how I calculate these matrix elements for the TI
operator. The inversion (I) does not do anything to spin, but changes K
into -K, while time-reversal (T) acts on spin as "i.sigma_y.θ" (where θ is
complex conjugation), and also takes K to -K. So the overall action of the
composed TI operator is to keep K fixed, and act only on spin as
"i.sigma_y.θ".

So I read the wfc...hdf5 file to get the plane-wave coefficients and
calculate the overlap integrals, which is almost the identity, so we can
neglect it. Then, to calculate the matrix elements of TI I use the same
plane-wave coefficients but flipping the spin up/down on the ket via the
action of i.sigma_y and take the complex conjugate.

I believe everything is correct in my code. So the only possible
explanation that I have at the moment is that the TI symmetry is broken,
and consequently I cannot calculate its matrix elements successfully.

Best regards,
--
Gerson J. Ferreira
Prof. Dr. @ InFis - UFU
----------------------------------------------
gjferreira.wordpress.com
Institute of Physics
Federal University of Uberlândia, Brazil
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