Magnetic exchange

Isotropic magnetic exchange interaction (e. g. Heisenberg exchange) is written as

  $$\hat{\mathcal{H}}_{X-J} = - \frac{1}{2} \sum_{nn^{\prime}... ..._{n}) \hat{\mathcal{I}}_{\alpha}^n \hat{\mathcal{I}}_{\beta}^{n^{\prime}}$$ (8)

where the operators are understood as $\hat{\mathcal{I}}_{1} \leftrightarrow \hat{J}_x$, $\hat{\mathcal{I}}_{2} \leftrightarrow \hat{J}_y$, $\hat{\mathcal{I}}_{3} \leftrightarrow \hat{J}_z$.