[QE-developers] Phase of wave-function

[Stefano de Gironcoli]

On 08/04/20 21:38, Duy Le wrote:
> Dear developers,
>
> After reading many threads about the subject, and lot of testing, I 
> still not very confident in handling the phases of wave functions. I 
> would appreciate if you could help me with the following:
>
> 1. Do all wave-functions (of all bands) of a given k-point have the 
> same phase? In other words, is psi(k, ib1) psi(k, ib2)* gauge invariant?

no. the phase of an eigenvector is determined by the diagonalization 
routine. The safe assumption is that it's basically random, and for 
degenerate eigenvalues you can assume a random (unitary) rotation inside 
the degenerate manifold.  this is why in molecular dynamics simulations 
when you want to extrapolate the starting wavfunctions of a given time 
step from previous ones, before performing the extrapolation, you need 
to align the two (or more) wfc subspaces.

basically you compute W=<psi(t')|psi(t)> and built the unitary rotation 
that best align the two wfc sets from SVD of W ->  U  then build  
|psi'(t')> = |psi(t')> U as the set at t' best aligned to the wfc at t.

check routine   extrapolate_wfcs   in file update_pot.f90 in PW/src

> 2. I want also to calculate psi(k, ib1) psi(k2, ib2)*. How should it 
> be done so the random phases of wfc do not affect the result?

similarly random phases should be assumed ( in addition to the exp(+ikr) 
phase from the Bloch vector) so you better work with the periodic part 
of the wavefuncitons u(k)

you can compute  the matrix of the overlap of the periodic part of the 
wfcs at the two kpoints  W = < psi(k1)| 
exp(i(k1-k2)r)|psi(k2)>=<u(k1)|u(k2)>,  extract the best unitary 
transformation from that and play with it in order to align wfcs at 
different k's. this is one of the basic operation that Wannier 90 code 
peforms.

> 3. Could you please tell me if there is any part of the PWSCF code 
> that deals with phase of wave function so I can learn how it is done?

see above

best

stefano

>
> Thank you very much. Be safe!
>
> Duy Le
> University of Central Florida
>
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