[Pw_forum] phonon eigenvectors

[Stefano Baroni]

Dear Eric: I feel flattered (and a little guilty) for your taking me  
so seriously. Also, I may have had a role in starting the discussion  
taken up by Axel and Fernando which, to tell the truth, I do not  
fully understand. All in all, forgive my being (involuntarily!)  
paternalistic, and let's come to the real stuff!

> Question: What is the meaning of an eigenvector?
> Answer:  An eigenvector tells how the atoms are displaced in the  
> vibration.

OK

> Question: What is the imaginary part?
> Answer: The imaginary part is a phase factor.

Forgive my fussiness: its is *not* a phase factor. A non zero  
imaginary part is *due* to a non zero phase.

> If one atom has a significant imaginary part with respect to the  
> rest, then then it's displacement in the vibration will be phase  
> shifted with respect to the others.

OK - More precisely: a phase dfference between two different atomic  
components of a same eigenvectors imply that the motions of the two  
atoms are out of phase (i.e. the velocities of the two may vanish at  
different times)

> In the case of a completely imaginary eigenvector, the displacement  
> will be completely out of phase.

This statement, instead, is meaningless. Ou of phase WITH RESPECT TO  
WHAT? The only meaningful thing is the phase difference between the  
eigenvector components of two different atoms. An overall phase equal  
for all the atoms is simply equivalent to a shift of the origin of  
time. CAN YOU SEE THIS?

Can you see the analogies with the well know statement that quantum  
mechanical wavefunctions (a problem which is conceptually TOTALLY  
different) are defined to within an overall phase factor?

> Question:  If the imaginary part is a phase factor, then what does  
> it mean if all of the components are imaginary?
> Answer:  First answer: nothing.  If all of the atoms are "out of  
> phase", then they are "in phase" with respect to eachother,  
> therefore, having a completely imaginary or completely real  
> eigenvector should be equivalent.

VERY GOOD. YOU GOT IT!

> Question:  Great.  If this is the case, then why does this  
> eigenvector come up imaginary when all of the other vectors come up  
> real?
> Answer:  Hmmm.  That's a good one.  There must be some reason that  
> the code chooses an imaginary eigenvector for this mode...time to  
> get help on this one.  We are fresh out of answers.

Very good, Erich. You came to the point. The answer is deceivingly  
simple. If the the phase is *physically* irrelevant, how could a  
physically sound mathematical algorithm choose it? Answer (deeper  
than it may sound at first): AT RANDOM! Actual algorithms may not be  
really random, but it wouldn't harm if they were!

> And with this I come first to the pwscf discussion archive.  I  
> didn't find any discussion of imaginary eigenvectors with real  
> eigenvalues.

Hey, hey, hey! Slow down! Real eigenvalues are a consequence of the  
Hermiticity of the dynamical matrix and have nothing to do with the  
eigenvector being or not real. A Hermitean matrix may or may not have  
complex eigenvectors. If the matrix is real, the phases can alwayes  
(*not* "must": you see the difference?) in such a way that the  
eigenvectors are real. If the matrix is not real (still being  
Hermitean), I do not know: eigenvectors are in general complex (i.e.  
no overall phase can be chosen  so as to make all of their components  
real), but they may also be in some particular instance real ...

> I looked to the literature, with the same result.  There comes a  
> time when one asks himself questions to which he doesn't have the  
> answers, then it comes time to discuss with his peers.  With that I  
> come to you with my question:
>
> Why did the code "choose" to make this displacement imaginary.  Is  
> this simply an artifact of the matrix diagonalization,

YES

> or is there some physical implications to this?

NO

> Thank you for your time and patience.

Thank you for challenging them! ;-)
(I am not kidding: it's refreshing and instructive for us to help, no  
less that it is for you to be assisted - hopefully ;-)

Stefano

---
Stefano Baroni - SISSA  &  DEMOCRITOS National Simulation Center -  
Trieste
[+39] 040 3787 406 (tel) -528 (fax) / stefanobaroni (skype)

Please, if possible, don't  send me MS Word or PowerPoint attachments
Why? See:  http://www.gnu.org/philosophy/no-word-attachments.html



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