Questions about the paper not the code

[Jaime Garrido]

Hi everyone!

I have been revisiting the original TB2J paper in order to try to try to fully understand the derivation of the Hamiltonian parameters. I thought about contacting the authors through email, but then I realized that this place might be a better option since the answer can be read later by the rest of the users.

I have two questions basically. Sorry if the answer is obvious :/ 

1.  The first one is about the paragraph between equations (7) and (8) where we can read: "In condensed form, the spin and orbital matrix for site i is now Pi = pi^0*I + \vec{p}i \cdot \vec{sigma}  ". I am getting confused with the dimensions of the matrices. Pi is 2Norb X 2Norb. and the Pauli matrices are just 2x2. Then, how is the equality possible? I guess there is something implicit that I am not following.
2. Before arriving to equation (13): "The change of H due to the rotation of the spin is \delta \vec{\phi} x \vec{p}...". Why is it exactly that quantity? I do not have any clue about where that relation is coming from. 

My ultimate goal is being able to write equation (13) from equation (12).

Thanks in advance!

Jaime

[Xu He]

Hello Jaime, 

Thanks. Indeed these questions are valuable to the community. 

1. This operation is for each orbital of one atom.   Pi is thus a 2x2 matrix, pi is a scalar, and \vec{p}_i is a 3-vector. 

2. \delta \vec{\phi} \cross \vec{p} is the viariation of \vec{p} when it is rotated along the axis \vec{\phi}.  

(I take this figure from google.) When the vector is rotated from A by \delta\theta to B, you can see easily B-A has the amplitude of |A  \delta \theta| , and the direction is perperdicular to both A and \delta \theta (the rotation axis can be seen as a vector). 

To go from Eq. 12 to Eq.13 is indeed not trivial. We did not put the details as it has been done in several pervious work and it would take a few pages to do so. 

I recommend this recent review paper which gives more detailed. (section V). 

Best regards, 

HeXu

[Jaime Garrido]

Thanks a lot for your answer.

Regarding the recent review you mentioned, I do not see any link or hyperlink on the message. Maybe you forgot to add it? I would love to take a look to the section you say.

Thanks again!

Jaime

[Xu He]

Sorry. Here is the link:

https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.95.035004

Best regards, 

HeXu