Spin-orbit coupling

The spin-orbit coupling (SOC) is implemented at atomic level,

$$\begin{equation} \hat{H}_{\text{SOC}}=\lambda\sum_{i}\vec{\mathbf{l}}_{i}\cdot\vec{\mathbf{s}}_{i}, \end{equation}$$

where, $\vec{\mathbf{l}}$ is orbital angular momentum, and $\vec{\mathbf{s}}$ is spin angular momentum. In second quantization form,

$$\begin{equation} \hat{H}_{\text{SOC}}=\lambda\sum_{\alpha\sigma,\beta\sigma^{\prime}}\left\langle\alpha\sigma\left|\vec{\mathbf{l}}\cdot\vec{\mathbf{s}}\right|\beta\sigma^{\prime}\right\rangle\hat{f}_{\alpha\sigma}^{\dagger}\hat{f}_{\beta\sigma^{\prime}}, \end{equation}$$

where, $\alpha$ is orbital index and $\sigma$ is spin index, and

$$\begin{eqnarray} \vec{\mathbf{l}}\cdot\vec{\mathbf{s}} & = & \frac{1}{2}\vec{\mathbf{l}}\cdot\vec{\mathbf{\mathbf{\sigma},}} \end{eqnarray}$$

where, $\vec{\mathbf{\sigma}}$ is Pauli operator.

Now the question is how to write down the matrix elements for $\vec{\mathbf{l}}\cdot\vec{\mathbf{\sigma}}$

$$\begin{eqnarray*} \vec{\mathbf{l}}\cdot\vec{\mathbf{\sigma}} & = & \hat{l}_{x}\hat{\sigma}_{x}+\hat{l}_{y}\hat{\sigma}_{y}+\hat{l}_{z}\hat{\sigma}_{z}\\ & = & \left[\begin{array}{cc} 0 & \hat{l}_{x}\\ \hat{l}_{x} & 0 \end{array}\right]+\left[\begin{array}{cc} 0 & -i\hat{l}_{y}\\ i\hat{l}_{y} & 0 \end{array}\right]+\left[\begin{array}{cc} \hat{l}_{z} & 0\\ 0 & -\hat{l}_{z} \end{array}\right]\\ & = & \left[\begin{array}{cc} \hat{l}_{z} & \hat{l}_{x}-i\hat{l}_{y}\\ \hat{l}_{x}+i\hat{l}_{y} & -\hat{l}_{z} \end{array}\right]\\ & = & \left[\begin{array}{cc} \hat{l}_{z} & \hat{l}_{-}\\ \hat{l}_{+} & -\hat{l}_{z} \end{array}\right] \end{eqnarray*}$$

where, $\hat{l}_{\pm}=\hat{l}_{x}\pm\hat{l}_{y}$, and

$$\begin{equation} \hat{l}_{\pm}Y_{l}^{m}=\sqrt{(l\mp m)(l\pm m+1)}Y_{l}^{m\pm1}. \end{equation}$$

We just write down $\vec{\mathbf{l}}\cdot\vec{\mathbf{\sigma}}$ in the complex shperical harmonics basis, the orbital order is: $$\begin{equation} Y_{l}^{-l}\uparrow,Y_{l}^{-l}\downarrow,Y_{l}^{-l+1}\uparrow,Y_{l}^{-l+1}\downarrow,\cdots,Y_{l}^{l}\uparrow,Y_{l}^{l}\downarrow. \end{equation}$$

For $p$ system, $$\begin{equation} \vec{\mathbf{l}}\cdot\vec{\mathbf{\sigma}}=\left[\begin{array}{cccccc} -1 & 0 & 0 & \sqrt{2} & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \sqrt{2}\\ \sqrt{2} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & \sqrt{2} & 0 & 0 & -1 \end{array}\right] \end{equation}$$

For $t_{2g}$ system, $$\begin{equation} \vec{\mathbf{l}}\cdot\vec{\mathbf{\sigma}}=-\left[\begin{array}{cccccc} -1 & 0 & 0 & \sqrt{2} & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \sqrt{2}\\ \sqrt{2} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & \sqrt{2} & 0 & 0 & -1 \end{array}\right] \end{equation}$$

For $d$ system, $$\begin{equation} \vec{\mathbf{l}}\cdot\vec{\mathbf{\sigma}}=\left[\begin{array}{cccccccccc} -2 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & \sqrt{6} & 0 & 0 & 0 & 0\\ 2 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{6} & 0 & 0\\ 0 & 0 & \sqrt{6} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 2\\ 0 & 0 & 0 & 0 & \sqrt{6} & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & -2 \end{array}\right] \end{equation}$$

For $f$ system, $$\begin{equation} \vec{\mathbf{l}}\cdot\vec{\mathbf{\sigma}}=\left[\begin{array}{cccccccccccccc} -3 & 0 & 0 & \sqrt{6} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & \sqrt{10} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \sqrt{6} & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & \sqrt{12} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \sqrt{10} & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{12} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & \sqrt{12} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & \sqrt{10} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{12} & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & \sqrt{6}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{10} & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{6} & 0 & 0 & -3 \end{array}\right] \end{equation}$$