# How to write the second quantization form of spin-orbit coupling(Dzyaloshinskii-Moriya interaction)?

Spin orbit coupling is the single particle term, so the second quantization form can be written like:$\langle \alpha\sigma|s\cdot(\nabla V\times P)|\beta\sigma'\rangle c^{+}_{\alpha\sigma}c_{\beta\sigma'}$, but the matrix element seems hard to transform, from T.Moriya's paper, this matrix element can be transformed into angular momentums' matrix element. After writing the second quantization form of spin orbit coupling, there are a Dzyaloshinskii-Moriya term and an anisotropy exchange term. If we use Wannier basis, then we can see the spin orbit coupling can lead to hopping from one site to another, how to imagine this process? The core's field V make the electron hop?

• I have worked out the expressions, there are a Dzyaloshinskii-Moriya term , a $S(R_j)\cdot \Gamma_(ji)\cdot S(R_i)$ term and some wierd term like $S^+ (R_i)$,$S^- (R_j)$. It seems unnecessary to transform the spin-orbital coupling matrix element to angular momentum matrix element. – ZJX Mar 29 '16 at 3:39
• By the way, I have not found the detailed derivation of second quantization form of spin-orbit coupling. There is a derivation of DM interaction in Norberto Majlis 's The Quantum Theory of Magnetism [2nd ed] pg.107. – ZJX Mar 29 '16 at 3:47