[Pw_forum] energy-degenerate states with different irrep labels

[Silvia Bakalova]

Dear Gabriele,

Thank you for your help and really useful reply. I think that you are right
and this is the case also for D_3d (-3m) double group in my QE calculations,
as most of the states remain doubly degenerate, but with the same energy.
What I found is that this type of degeneracy is Kramers degeneracy and there
is a proof of the at least double degeneracy of the ground state when the
spin is included, based on the Kramers degeneracy theorem:

http://arxiv.org/abs/0809.4471v1

Thanks again. I just wanted to ask, because everywhere they prove this
degeneracy for odd number of electrons, and I have even number of electrons
in my case, is there any difference and could really double point group be
used to describe the electronic states?

Cheers,

Silvia

Silvia Bakalova wrote:
>* Thank you for the reply, Gabriele.
*>* Yes, this is @Gamma, the states are degenerate, but I wonder why they
*>* have different irreducible representations (G_5+ and G_6+)?
*
Because they actually belong to different representations of the double group...

Andrea and me some time ago kind of understood why in presence of
time-reversal there are
couples of bands which can be matched, in the sense that they are
degenerate not in the
usual sense, but in the following.
For each k, if there is an eigenvalue e_{k,v} belonging to a band v
of, say, G_5 symmetry,
there must be at -k an eigenvalue e_{-k,v'} with the same magnitude
belonging to the
matching band v' of, say, G_6 symmetry (and viceversa).

We checked this for the double group C_{2v}, which was the one of our
case study. Please
have a look at the band structure here
http://people.sissa.it/sclauzer/Data/COsu7PtFR_G3G4.pdf

You probably will understand better what I was saying above.
Since at Gamma k=-k, the bands must be degenerate there (in the usual
sense). I never
checked what happens with other double groups, but you may confirm (or
not) that the
situation is the same. The demonstration of such property of the bands
structure in
presence of time reversal can be found in this book (I don't remember
the page, I don't
have it at hand now):

Bassani F.; Pastori Parravicini G. (1975). Electronic states and
optical transitions
in solids. Pergamon Press, Oxford.

HTH

GS


On Fri, Oct 9, 2009 at 3:53 PM, Silvia Bakalova <silveto at gmail.com> wrote:

> Hi,
>
> I have one question: for spin-orbit calculations, the energy bands are
> labelled with double point group notation (D3d’ in my case).
>
> Some of the energy-degenerate states have different irrep labels and I
> wonder why…
>
> e.g. the valence band top:
>
>      e( 45 - 46) =      7.09176  eV     2   --> G_5+  L_4+
>
>      e( 45 - 46) =      7.09176  eV     2   --> G_6+  L_5+
>
> I would be grateful for your reply or some literature reference, as I am
> not familiar with the group theory.
>
> Many thanks,
>
> Silvia
>
>
> ----------------------
> Dr. Silvia Bakalova,
> Post Doctoral Researcher,
> HH Wills Physics Laboratory,
> Bristol, BS8 1TL, UK
> http://spectra.phy.bris.ac.uk/
>
>
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