On-site anisotropy (k = 2)¶
On-site anisotropy of the second order involves one site and two entities. Therefore, it can be expressed via the term \(\mathcal{H}_2\).
The summary of the mapping for each code is given in the table below.
Code |
||||
|---|---|---|---|---|
\(\mathcal{H}\) |
\[\mathcal{H}
=
\sum_{i}
\boldsymbol{e}_{i}
\cdot
\boldsymbol{K}_{i}
\cdot
\boldsymbol{e}_{i}\]
|
\[\mathcal{H}
=
\sum_{i}
\mathbf{S}_{i}
\cdot
\boldsymbol{A}_{i}
\cdot
\mathbf{S}_{i}\]
|
\[\mathcal{H}_{\rm uni}
=
\sum_j K_j (\hat{K}_j\cdot\vec{n}_j)^2\]
|
\[\mathcal{H}
=
-
\sum_{i}
\mathbf{S}_{i}
\cdot
\boldsymbol{A}_{i}
\cdot
\mathbf{S}_{i}\]
|
Index renaming |
\(i \rightarrow (\mu_1, \alpha_1)\) |
\(i \rightarrow (\mu_1, \alpha_1)\) |
\(j \rightarrow (\mu_1, \alpha_1)\) |
\(i \rightarrow (\mu_1, \alpha_1)\) |
\(C_2\) |
\(1\) |
\(1\) |
\(1\) |
\(-1\) |
\(V_{\mu_1, \mu_2; \alpha_1, \alpha_2}^{i_1, i_2}\) |
\(K_{\mu_1; \alpha_1}^{i_1, i_2}\delta_{\mu_1,\mu_2}\delta_{\alpha_1,\alpha_2}\) |
\(A_{\mu_1; \alpha_1}^{i_1, i_2}\delta_{\mu_1,\mu_2}\delta_{\alpha_1,\alpha_2}\) |
\[\begin{split}K_j^{i_1, i_2}
=
\delta_{\mu_1,\mu_2}\delta_{\alpha_1,\alpha_2}
\begin{pmatrix}
K_j \left(\hat{K}_{j}^{x}\right)^2 & K_j \hat{K}_{j}^{x} \hat{K}_{j}^{y} & K_j \hat{K}_{j}^{x} \hat{K}_{j}^{z} \\
K_j \hat{K}_{j}^{x} \hat{K}_{j}^{y} & K_j \left(\hat{K}_{j}^{y}\right)^2 & K_j \hat{K}_{j}^{y} \hat{K}_{j}^{z} \\
K_j \hat{K}_{j}^{x} \hat{K}_{j}^{z} & K_j \hat{K}_{j}^{y} \hat{K}_{j}^{z} & K_j \left(\hat{K}_{j}^{z}\right)^2
\end{pmatrix}\end{split}\]
|
\(A_{\mu_1; \alpha_1}^{i_1, i_2}\delta_{\mu_1,\mu_2}\delta_{\alpha_1,\alpha_2}\) |
\(X_{\mu_1; \alpha_1}^{i_1}\) |
\(e_{\mu_1; \alpha_1}^{i_1}\) |
\(S_{\mu_1; \alpha_1}^{i_1}\) |
\(n_{\mu_1; \alpha_1}^{i_1}\) |
\(S_{\mu_1; \alpha_1}^{i_1}\) |
\(X_{\mu_2; \alpha_2}^{i_2}\) |
\(e_{\mu_2; \alpha_2}^{i_2}\) |
\(S_{\mu_2; \alpha_2}^{i_2}\) |
\(n_{\mu_2; \alpha_2}^{i_2}\) |
\(S_{\mu_2; \alpha_2}^{i_2}\) |