[Pw_forum] energy-degenerate states with different irrep labels

[Gabriele Sclauzero]

Dear 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


> 
> Silvia Bakalova 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…
> />/ 
> /
> At which k-point? I think that at Gamma they have to be degenerate if time reversal 
> symmetry holds (i.e. no magnetic field).
> 
> 
> GS
> 
>>/ 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


-- 


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| Gabriele Sclauzero, PhD Student                  |
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