Second quantization derivation of spin-orbit coupling.

[todemax]

Can anyone recommend books/reviews that derives the spin-orbit coupling in second quantization. I am working on a tightbinding model and I should be able to convert the spin-orbit hamiltonian from k-space to atomic representation using Warnier states, but I can't figure out some of the aspects of deriving the hamiltonian, even though I feel I should be able to.
My derivation so far is at a stand still at the following equation

$\sum_{n,n'} c_n^+c_{n'} \langle n | (E \times k) \cdot \sigma |n'\rangle$

Thank your for any help in advance.

 

[eljose79]

I would recommend checking out the book "Quantum Theory of the Electron Liquid" by Giulio Fano and Franco Ortolani. Chapter 7 specifically covers the spin-orbit coupling in second quantization and provides a detailed derivation using Warnier states. Another helpful resource is the review article "Spin-Orbit Coupling in Quantum Materials" by Piers Coleman, which also walks through the derivation and includes additional references for further reading.

In general, the key to deriving the spin-orbit coupling in second quantization is to first express the spin-orbit interaction in terms of the electron's position and momentum operators, and then rewrite these operators in terms of creation and annihilation operators using the appropriate basis states. This will result in an expression similar to the one you have provided, with the addition of the spin-orbit coupling constant and the appropriate spin matrices.

I hope this helps and good luck with your research!