On-site anisotropy (k = 4)¶
On-site anisotropy of the fourth order involves four entities. Therefore, it can be expressed via the term \(\mathcal{H}_{0, 4}\).
\[\begin{split}\mathcal{H}_{0, 4}
=
C_{0, 4}
\sum_{\substack{\mu_1, \mu_2, \mu_3, \mu_4 \\ \alpha_1, \alpha_2, \alpha_3, \alpha_4 \\ i_1, i_2, i_3, i_4}}
V_{\mu_1, \mu_2, \mu_3, \mu_4; \alpha_1, \alpha_2, \alpha_3, \alpha_4}^{i_1, i_2, i_3, i_4}
X_{\mu_1; \alpha_1}^{i_1}
X_{\mu_2; \alpha_2}^{i_2}
X_{\mu_3; \alpha_3}^{i_3}
X_{\mu_4; \alpha_4}^{i_4}\end{split}\]
The summary of the mapping for each code is given in the table below.
Code |
|
|---|---|
\(\mathcal{H}\) |
\[\mathcal{H}_{\rm cubic}
=
- \frac12 \sum_j K_j ([\vec{n}_j]_x^4 + [\vec{n}_j]_y^4 + [\vec{n}_j]_z^4)\]
|
Index renaming |
\(j \rightarrow (\mu_1, \alpha_1)\) |
\(C_{0, 4}\) |
\(- \frac12\) |
\(V_{\mu_1, \mu_2, \mu_3, \mu_4; \alpha_1, \alpha_2, \alpha_3, \alpha_4}^{i_1, i_2, i_3, i_4}\) |
\[V_{...}^{xxxx}
=
V_{...}^{yyyy}
=
V_{...}^{zzzz}
=
K_{\mu_1, \alpha_1}
\delta_{\mu_1,\mu_2}\delta_{\mu_1,\mu_3}\delta_{\mu_1,\mu_4}
\delta_{\alpha_1,\alpha_2}\delta_{\alpha_1,\alpha_3}\delta_{\alpha_1,\alpha_4}\]
Other components are zero. |
\(X_{\mu_1; \alpha_1}^{i_1}\) |
\(n_{\mu_1; \alpha_1}^{i_1}\) |
\(X_{\mu_2; \alpha_2}^{i_2}\) |
\(n_{\mu_2; \alpha_2}^{i_2}\) |
\(X_{\mu_3; \alpha_3}^{i_3}\) |
\(n_{\mu_3; \alpha_3}^{i_3}\) |
\(X_{\mu_4; \alpha_4}^{i_4}\) |
\(n_{\mu_4; \alpha_4}^{i_4}\) |