Dzyaloshinskii-Moriya interaction

DMI can be written in the form of the two-ion interaction as

  $$\hat{\mathcal{H}}_{DMI} = - \frac{1}{2} \sum_{nn^{\prime}... ..._{n}) \hat{\mathcal{I}}_{\alpha}^n \hat{\mathcal{I}}_{\beta}^{n^{\prime}}$$ (10)

where the operators are understood as $\hat{\mathcal{I}}_{1} \leftrightarrow \hat{J}_x$, $\hat{\mathcal{I}}_{2} \leftrightarrow \hat{J}_y$, $\hat{\mathcal{I}}_{3} \leftrightarrow \hat{J}_z$ and $\mathcal{J}_{\alpha\beta}(\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n})$ is a skew-symmetric matrix.