SOC

One can just take the usual bands from a spin-independent Hamiltonian and say that the orbital \(\psi_{nk}\) describe two different bands (one with spin-up, one with spin-down).

Hamiltonian

SOC is always present. By taking the low-velocity limit of the Dirac equation one gets \[ \frac{[\nabla V\times \mathbf{p}]\cdot\mathbf{S}}{2m_e^2c^2}, \] where \(\mathbf{S}\) is the spin operator of the electron. This is still a relativistic effect.

In other words, \(H_{SO}\) represents a Zeeman coupling between the magnetic moment of the electron spin (in its reference frame) and \(\mathbf{B}_{\rm eff}\) (generated by other electrons moving relative to electron’s reference frame).

Bloch wavefunctions

Generally, these should be written using two different, real-space periodic wavefunctions \(u_{nk}\) and \(v_{nk}\) \[ \psi_{nk}(\mathbf{r}, \sigma) = e^{i\mathbf{k}\cdot \mathbf{r}}\left( u_{nk}\ket{\uparrow} + v_{nk}\ket{\downarrow} \right). \] Then, \([S_z, H_{SO}]\neq 0\) and \(S_z\) is no longer a good quantum number (not conserved). Spin-orbit-coupling breaks this degeneracy, unless both \(\Theta\) (time-reversal) and \(P\) (parity) symmetries are present.

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