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<p class="MsoNormal">Dear Gabriele,</p>
<p class="MsoNormal">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: </p>
<p class="MsoNormal"><a href="http://arxiv.org/abs/0809.4471v1">http://arxiv.org/abs/0809.4471v1</a></p>
<p class="MsoNormal">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?</p>
<p class="MsoNormal">Cheers,</p>
<p class="MsoNormal">Silvia</p>
<br><pre>Silvia Bakalova wrote:<br>><i> Thank you for the reply, Gabriele.<br></i>><i> Yes, this is @Gamma, the states are degenerate, but I wonder why they <br></i>><i> have different irreducible representations (G_5+ and G_6+)?<br>
</i><br>Because they actually belong to different representations of the double group...<br><br>Andrea and me some time ago kind of understood why in presence of time-reversal there are<br>couples of bands which can be matched, in the sense that they are degenerate not in the<br>
usual sense, but in the following.<br>For each k, if there is an eigenvalue e_{k,v} belonging to a band v of, say, G_5 symmetry, <br>there must be at -k an eigenvalue e_{-k,v'} with the same magnitude belonging to the <br>
matching band v' of, say, G_6 symmetry (and viceversa).<br><br>We checked this for the double group C_{2v}, which was the one of our case study. Please <br>have a look at the band structure here<br><a href="http://people.sissa.it/sclauzer/Data/COsu7PtFR_G3G4.pdf">http://people.sissa.it/sclauzer/Data/COsu7PtFR_G3G4.pdf</a><br>
<br>You probably will understand better what I was saying above.<br>Since at Gamma k=-k, the bands must be degenerate there (in the usual sense). I never <br>checked what happens with other double groups, but you may confirm (or not) that the <br>
situation is the same. The demonstration of such property of the bands structure in <br>presence of time reversal can be found in this book (I don't remember the page, I don't <br>have it at hand now):<br><br>Bassani F.; Pastori Parravicini G. (1975). Electronic states and optical transitions<br>
in solids. Pergamon Press, Oxford.<br><br>HTH<br><br>GS<br></pre><br><div class="gmail_quote">On Fri, Oct 9, 2009 at 3:53 PM, Silvia Bakalova <span dir="ltr"><<a href="mailto:silveto@gmail.com">silveto@gmail.com</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
<p class="MsoNormal">Hi,</p>
<p class="MsoNormal">I have one question: for spin-orbit calculations, the energy
bands are labelled with double point group notation (D3d’ in my case). </p>
<p class="MsoNormal">Some of the energy-degenerate states have different irrep
labels and I wonder why…</p>
<p class="MsoNormal">e.g. the valence band top:</p>
<p class="MsoNormal"><span style="font-size: 10pt; font-family: "Courier New";"><span> </span></span><span style="font-size: 10pt; font-family: "Courier New";" lang="ES">e( 45 - 46) =<span> </span>7.09176<span>
</span>eV<span> </span>2<span> </span>--> G_5+<span>
</span>L_4+<span> </span></span></p>
<p class="MsoNormal"><span style="font-size: 10pt; font-family: "Courier New";" lang="ES"><span> </span>e( 45 - 46) =<span> </span>7.09176<span>
</span>eV<span> </span>2<span> </span>--> G_6+<span>
</span>L_5+<span> </span></span></p>
<p class="MsoNormal">I would be grateful for your reply or some literature
reference, as I am not familiar with the group theory.</p>
<p class="MsoNormal">Many thanks,</p>
<p class="MsoNormal">Silvia</p>
<div class="gmail_quote"><p class="MsoNormal"><br></p><p class="MsoNormal">----------------------<br>
Dr. Silvia Bakalova,<br>Post Doctoral Researcher,<br>HH Wills Physics Laboratory,<br>Bristol, BS8 1TL, UK<br><a href="http://spectra.phy.bris.ac.uk/" target="_blank">http://spectra.phy.bris.ac.uk/</a><br><span style="font-size: 10pt; font-family: "Courier New"; color: black;"></span></p>
</div><br>
</blockquote></div><br>