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

[Thomas Brumme]

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