[Pw_forum] energy-degenerate states with different irrep labels

[Gabriele Sclauzero]

Dear Silvia,

Silvia Bakalova wrote:
> 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

I think you are right on this point


> 
> 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?

I'm not sure of this, but since KS equations are single particle equations, then you 
should treat this case as a single electron one (your KS eigenstates are one electron 
wavefunctions, not many-body wavefunctions).

Cheers

Gabriele


> 
> 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 
> <mailto: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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| Gabriele Sclauzero, PhD Student                  |
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