Biquadratic exchange¶
Bilinear exchange involves four entities (two entities per site). Therefore, it can be expressed via the term \(\mathcal{H}_{0, 4}\).
The summary of the mapping for each code is given in the table below.
Code |
||
|---|---|---|
\(\mathcal{H}\) |
\[\mathcal{H}
=
\sum_{i \neq j}
B (\mathbf{S}_i \cdot \mathbf{S}_j)^2\]
|
\[\mathcal{H}_{\rm quad}
=
- \sum_{ij} K_{ij} (\vec{n}_i \cdot \vec{n}_j)^2\]
|
Indices renaming |
\(i \rightarrow (\mu_1, \alpha_1)\), \(j \rightarrow (\mu_2, \alpha_2)\) |
\(i \rightarrow (\mu_1, \alpha_1)\), \(j \rightarrow (\mu_2, \alpha_2)\) |
\(C_{0, 4}\) |
1 |
-1 |
\(V_{\mu_1, \mu_2, \mu_3, \mu_4; \alpha_1, \alpha_2, \alpha_3, \alpha_4}^{i_1, i_2, i_3, i_4}\) |
\(B\cdot \delta_{\mu_1, \mu_2} \delta_{\mu_3,\mu_4} \delta_{\alpha_1, \alpha_2} \delta_{\alpha_3,\alpha_4}\) if \((i_1, i_2, i_3, i_4) \in \mathcal{K}\) and 0 otherwise. |
\(K_{\mu_1, \mu_2; \alpha_1, \alpha_2}\) if \((i_1, i_2, i_3, i_4) \in \mathcal{K}\) and 0 otherwise. |
\(X_{\mu_1; \alpha_1}^{i_1}\) |
\(S_{\mu_1; \alpha_1}^{i_1}\) |
\(n_{\mu_1; \alpha_1}^{i_1}\) |
\(X_{\mu_2; \alpha_2}^{i_2}\) |
\(S_{\mu_2; \alpha_2}^{i_2}\) |
\(n_{\mu_2; \alpha_2}^{i_2}\) |
\(X_{\mu_3; \alpha_3}^{i_3}\) |
\(S_{\mu_3; \alpha_3}^{i_3}\) |
\(n_{\mu_3; \alpha_3}^{i_3}\) |
\(X_{\mu_4; \alpha_4}^{i_4}\) |
\(S_{\mu_4; \alpha_4}^{i_4}\) |
\(n_{\mu_4; \alpha_4}^{i_4}\) |
Note
\(\mathcal{K} = \{(x,x,x,x), (y,y,y,y), (z,z,z,z), (x,y,x,y), (x,z,x,z), (y,z,y,z), (y,x,y,x), (z,x,z,x), (z,y,z,y)\}\)
-
Spirit includes more general Hamiltonian that is called "quadruplet interaction"
\[\mathcal{H}_{\rm quad} = - \sum_{ijkl} K_{ijkl} (\vec{n}_i \cdot \vec{n}_j)(\vec{n}_k \cdot \vec{n}_l)\]in the case \(i = k\) and \(j = l\) (or, equivalently, \(i = l\) and \(j = k\)) this interaction describe a biquadratic exchange. Other cases are considered in the page Quadruplet interaction.