On-site anisotropy (k = 4)¶
On-site anisotropy of the fourth order involves one site and four entities. Therefore, it can be expressed via the term \(\mathcal{H}_{4, 1}\).
\(k = 4\)
\(l = 1\)
\(m_{4,1} = 1\)
\[\mathcal{H}_{4,1}
=
C_{4, 1}
\sum_{\mu_1, \alpha_1, i_1, i_2, i_3, i_4}
V_{\mu_1; \alpha_1}^{i_1, i_2}
X_{\mu_1; \alpha_1}^{i_1}
X_{\mu_1; \alpha_1}^{i_2}
X_{\mu_1; \alpha_1}^{i_3}
X_{\mu_1; \alpha_1}^{i_4}\]
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 |
\(j \rightarrow (\mu_1, \alpha_1)\) |
\(C_{4,1}\) |
\(- \frac12\) |
\(V_{\mu_1; \alpha_1}^{i_1, i_2, i_3, i_4}\) |
\[V_{\mu_1; \alpha_1}^{xxxx}
=
V_{\mu_1; \alpha_1}^{yyyy}
=
V_{\mu_1; \alpha_1}^{zzzz}
=
K_{\mu_1, \alpha_1}\]
Other components are zero. |
\(X_{\mu_1; \alpha_1}^{i_1}\) |
\(n_{\mu_1; \alpha_1}^{i_1}\) |