Bilinear exchange¶
Bilinear exchange involves 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 |
|||||||
|---|---|---|---|---|---|---|---|
\(\mathcal{H}\) |
\[\frac{1}{2}
\sum_{i\neq j}
\boldsymbol{e}_{i}
\cdot
\boldsymbol{J}_{ij}
\cdot
\boldsymbol{e}_{j}\]
|
\[-
\sum_{i\ne j}
\mathbf{S}_{i}
\cdot
\boldsymbol{J}_{ij}
\cdot
\mathbf{S}_{j}\]
|
\[\sum_{i \neq j}
J_{ij}
\,
\mathbf{S}_i
\cdot
\mathbf{S}_j
+
\sum_{i \neq j}
\mathbf{D}_{ij}
\cdot
\left(
\mathbf{S}_i
\times
\mathbf{S}_j
\right)\]
|
\[-
\dfrac{1}{2}
\sum_{nn^{\prime}, \alpha\beta}
\mathcal{J}_{\alpha\beta}(\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n})
\hat{\mathcal{I}}_{\alpha}^n
\hat{\mathcal{I}}_{\beta}^{n^{\prime}}\]
|
\[\sum_{i \neq j}
\mathbf{S}_i J_{ij} \mathbf{S}_j\]
|
\[-\frac{1}{2}
\sum_{i\neq j}
J_{ij}^{\alpha\beta}
n_i^\alpha
n_j^\beta\]
|
\[\sum_{i<j}
\mathbf{S}_{j}
J_{ij}
\mathbf{S}_{j}\]
|
Indices renaming |
\(i \rightarrow (\mu_1, \alpha_1)\), \(j \rightarrow (\mu_2, \alpha_2)\) |
\(i \rightarrow (\mu_1, \alpha_1)\), \(j \rightarrow (\mu_2, \alpha_2)\) |
\(i \rightarrow (\mu_1, \alpha_1)\), \(j \rightarrow (\mu_2, \alpha_2)\) |
\(n \rightarrow (\mu_1, \alpha_1)\), \(n^{\prime} \rightarrow (\mu_2, \alpha_2)\), \(\alpha \rightarrow i_1\), \(\beta \rightarrow i_2\) |
\(i \rightarrow (\mu_1, \alpha_1)\), \(j \rightarrow (\mu_2, \alpha_2)\) |
\(i \rightarrow (\mu_1, \alpha_1)\), \(j \rightarrow (\mu_2, \alpha_2)\), \(\alpha \rightarrow i_1\), \(\beta \rightarrow i_2\) |
\(i \rightarrow (\mu_1, \alpha_1)\), \(j \rightarrow (\mu_2, \alpha_2)\) |
\(C_2\) |
\(\dfrac{1}{2}\) |
\(-1\) |
\(1\) |
\(-\dfrac{1}{2}\) |
\(1\) |
\(-\dfrac{1}{2}\) |
\(1\) |
\(V_{\mu_1, \mu_2; \alpha_1, \alpha_2}^{i_1, i_2}\) |
\(J_{\mu_1, \alpha_1; \mu_2, \alpha_2}^{i_1, i_2}\) |
\(J_{\mu_1, \alpha_1; \mu_2, \alpha_2}^{i_1, i_2}\) |
\(J_{\mu_1, \alpha_1; \mu_2, \alpha_2}^{i_1, i_2} + \sum_{i_3} D_{\mu_1, \alpha_1; \mu_2, \alpha_2}^{i_3} e_{i_1, i_2, i_3}\) |
\(\mathcal{J}^{i_1,i_2}(\mathbf{R}_{\mu_2, \alpha_2} - \mathbf{R}_{\mu_1, \alpha_1})\) |
\(J_{\mu _1, \alpha_1; \mu_2, \alpha_2}^{i_1, i_2}\) |
\(J_{\mu_1, \alpha_1; \mu_2, \alpha_2}^{i_1, i_2}\) |
\(J_{\mu_1, \alpha_1; \mu_2, \alpha_2}^{i_1, i_2}\) |
\(X_{\mu_1; \alpha_1}^{i_1}\) |
\(e_{\mu_1, \alpha_1}^{i_1}\) |
\(S_{\mu_1, \alpha_1}^{i_1}\) |
\(S_{\mu_1, \alpha_1}^{i_1}\) |
\(\hat{\mathcal{I}}^{\mu_1; \alpha_1}_{i_1}\) |
\(S_{\mu_1, \alpha_1}^{i_1}\) |
\(n_{\mu_1, \alpha_1}^{i_1}\) |
\(S_{\mu_1, \alpha_1}^{i_1}\) |
\(X_{\mu_2; \alpha_2}^{i_2}\) |
\(e_{\mu_2, \alpha_2}^{i_2}\) |
\(S_{\mu_2, \alpha_2}^{i_2}\) |
\(S_{\mu_2, \alpha_2}^{i_2}\) |
\(\hat{\mathcal{I}}^{\mu_2; \alpha_2}_{i_2}\) |
\(S_{\mu_2, \alpha_2}^{i_2}\) |
\(n_{\mu_2, \alpha_2}^{i_2}\) |
\(S_{\mu_2, \alpha_2}^{i_2}\) |
Note
-
\(e_{i_2, i_3, i_1}\) is the Levi-Civita symbol.
-
Indices \(i_1, i_2\) run either over three or six components, as the operators \(\hat{\mathcal{I}}_{\alpha}^n\) are interpreted either as
\[\begin{split}\begin{pmatrix} \hat{\mathcal{I}}_1^n \\ \hat{\mathcal{I}}_2^n \\ \hat{\mathcal{I}}_3^n \end{pmatrix} = \begin{pmatrix} \hat{J}_x^n \\ \hat{J}_y^n \\ \hat{J}_z^n \end{pmatrix}\end{split}\]or
\[\begin{split}\begin{pmatrix} \hat{\mathcal{I}}_1^n \\ \hat{\mathcal{I}}_2^n \\ \hat{\mathcal{I}}_3^n \\ \hat{\mathcal{I}}_4^n \\ \hat{\mathcal{I}}_5^n \\ \hat{\mathcal{I}}_6^n \end{pmatrix} = \begin{pmatrix} \hat{S}_x^n \\ \hat{S}_y^n \\ \hat{S}_z^n \\ \hat{L}_x^n \\ \hat{L}_y^n \\ \hat{L}_z^n \end{pmatrix}\end{split}\]