[QE-users] Expectation values of total angular momentum in SOC case

[Thomas Brumme]

Dear Guido,

no problem! Input is welcome and I also wanted to know if my reasoning 
is wrong and in this case more people can help better - it's like the 
"Ask the Audience" joker in "Who wants to be a millionaire" :D

The spin expectation values can be calculated for each k point like in a 
text book. The expectation value of the Pauli matrices with the spinor 
wave functions.
And I use this to get a Hamiltonian consisting of the Pauli matrices and 
with k-dependent prefactors. But the length of the spin vector (i.e., 
the vector with the 3 expectation values Sx, Sy, Sz) is not 1/2 but 
0.468 in the TMD heterostructure at and close to the conduction-band 
minimum. I expected 1/2 like in the first system I calculated.

That's why I was thinking if only the length of the spin vector for the 
whole band is 1/2... How to do the averaging in this case? Or is it due 
to SOC and only the total angular momentum makes sense? How to define 
this for a band and not an atom?

Concerning the actual calculation: this is done in 
PP/src/compute_sigma_avg.f90 and seems to be correct... Or is there 
something missing in the PAW case which is not important for US-PP?

Thomas

On 1/23/20 3:37 PM, Guido Menichetti wrote:
> Dear Thomas,
>
> sorry if I intrude on the conversation.
>
> How do you evaluate the DFT expectation values for Sx, Sy, Sz from QE?
> Could the discrepancy arise from the way it is calculated?
>
> Regards,
> G.
>
> Il giorno gio 23 gen 2020 alle ore 15:22 Thomas Brumme 
> <thomas.brumme at uni-leipzig.de <mailto:thomas.brumme at uni-leipzig.de>> 
> ha scritto:
>
>     Hey Lorenzo,
>
>     the "problem" is actually more complex and it is not a real
>     problem but
>     something I thought about and maybe I'm just missing something.
>
>     I calculate the band structure for some 2D systems including SOC and
>     want to fit a model to the spin state such that I can extract SOC
>     parameters. First order would be Rashba-type SOC but 2nd and 3rd
>     order
>     is something else which also depends on the local symmetry. For one
>     system this works without problems. Then I wanted to transfer the
>     ideas
>     and my "code" to a heterobilayer of TMDs and there it sort of
>     works but
>     there is one problem:
>
>     In order to fit the model, I first fit a generic Pauli Hamiltonian
>     (to
>     which the model is fitted) - in this way the code can be easily
>     adapted
>     to other local symmetries because only the 2nd stage needs to be
>     changed. Anyways, in the Pauli Hamiltonian I assume that the spin
>     is 1/2
>     - an electron or hole. Yet, the DFT expectation values for Sx, Sy,
>     Sz do
>     not result in a spin of 1/2 (for the TMD heterostructure) but a
>     little
>     bit less, 0.468, and this value is too different from 1/2 to say
>     it is
>     numerical noise. And then I thought that, well, spin is not a good
>     quantum number and I would need the total angular momentum. Or do
>     I need
>     to calculate the spin expectation values for the whole BZ and then a
>     single band would add up to 1/2? Is it OK to just, lets say, use
>     S^2 =
>     0.468 instead of 1/2 and say that this is due to SOC?
>
>     Regards
>
>     Thomas
>
>     On 1/23/20 12:36 PM, Lorenzo Paulatto wrote:
>     > Hello Thomas,
>     > if I remember correctly, the fact that the spin does not commute
>     with
>     > the Hamiltonian mean that the spin can be:
>     > 1. k-point dependent, you do not have spin-up and spin-down bands
>     > which can be separated
>     > 2. aligned along any direction, instead of just Z
>     >
>     > I think, but am not 100% sure, that if J is a good quantum
>     number for
>     > isolated atoms with mean-field interacting electrons, this is
>     not true
>     > for bulk crystals (what is L in the bulk?)
>     >
>     > With the options of bands.x setting lsigma=.true. you can plot the
>     > spin projected over x y and z and do some kind of color-codes
>     plot of
>     > the bands
>     >
>     > cheers
>     >
>     >
>     >
>     > On 22/01/2020 16:57, Thomas Brumme wrote:
>     >> Dear all,
>     >>
>     >> I tried to find something in the archive but was not successful.
>     >>
>     >> In noncollinear calculations I can plot the spin expectation
>     values
>     >> using bands.x.
>     >> Those are calculated using the standard Pauli matrices. Yet,
>     spin is
>     >> not a good
>     >> quantum number anymore once I have SOC. Thus, I actually have
>     to look
>     >> at the
>     >> total angular momentum, J. Is it possible to get the expectation
>     >> values of J?
>     >> Does it make sense at all to think about implementing it?
>     >>
>     >> Regards
>     >>
>     >> Thomas
>     >>
>     >
>
>     -- 
>     Dr. rer. nat. Thomas Brumme
>     Wilhelm-Ostwald-Institute for Physical and Theoretical Chemistry
>     Leipzig University
>     Phillipp-Rosenthal-Strasse 31
>     04103 Leipzig
>
>     Tel:  +49 (0)341 97 36456
>
>     email: thomas.brumme at uni-leipzig.de
>     <mailto:thomas.brumme at uni-leipzig.de>
>
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>
> -- 
> ***************************************
>
> Guido Menichetti
> Post-Doc researcher in Condensed matter physics
> Istituto Italiano di Tecnologia
> Theory and technology of 2D materials
> Address: Via Morego, 30, 16163 Genova
> Email: guido.menichetti at iit.it <mailto:guido.menichetti at iit.it>
> guido.menichetti at df.unipi.it <mailto:guido.menichetti at df.unipi.it>
> menichetti.guido at gmail.com <mailto:menichetti.guido at gmail.com>
>
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-- 
Dr. rer. nat. Thomas Brumme
Wilhelm-Ostwald-Institute for Physical and Theoretical Chemistry
Leipzig University
Phillipp-Rosenthal-Strasse 31
04103 Leipzig

Tel:  +49 (0)341 97 36456

email: thomas.brumme at uni-leipzig.de

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