TB2J¶
Spin Hamiltonian¶
TB2J calculates parameters for the following bilinear spin Hamiltonian:
\[\mathcal{H}
=
-\sum_{i \neq j}
\mathbf{S}_i
\cdot
\mathcal{J}_{ij}
\cdot
\mathbf{S}_j
-
\sum_i
\mathbf{S}_i
\cdot A_i
\cdot \mathbf{S}_i\]
where \(\mathbf{S}_i\) is a classical spin vector at magnetic site \(i\) normalized to 1; \(\mathcal{J}_{ij}\) is a 3x3 exchange tensor, that can be decomposed into physical components
Isotropic Heisenberg Exchange
\[H_{\text{iso}} = -\sum_{i \neq j} J_{ij} (\mathbf{S}_i \cdot \mathbf{S}_j)\]Dzyaloshinskii-Moriya Interaction
\[H_{\text{DM}} = -\sum_{i \neq j} \mathbf{D}_{ij} \cdot (\mathbf{S}_i \times \mathbf{S}_j)\]Symmetric Anisotropic Exchange
\[H_{\text{sym-ani}} = -\sum_{i \neq j} \mathbf{S}_i \cdot \mathcal{J}^{\text{ANI}}_{ij} \cdot \mathbf{S}_j\]where \(\mathcal{J}^{\text{ANI}}_{ij}\) is the traceless symmetric part of \(\mathcal{J}_{ij}\).
Convention¶
Spin normalized |
yes |
Multiple counting |
yes |