On-site anisotropy (k)

On-site anisotropy terms involve one site and \(k\) "entities". It should map to the terms of the Hamiltonian that can be written as

\[\mathcal{H}_{on-site} = \sum_{k=1}^{\infty} C_{k,p(k)} \sum_{\mu_1, \alpha_1, i_1, \ldots, i_k} V_{\mu_1, \alpha_1}^{i_1, \ldots, i_k} \cdot X_{\mu_1; \alpha_1}^{i_1} \cdot \ldots \cdot X_{\mu_1; \alpha_1}^{i_k}\]

Contributions to the term with \(k = 1\) are discussed in Zeeman interaction, with \(k = 2\) in On-site anisotropy (k = 2), with \(k = 4\) in On-site anisotropy (k = 4).

In this page we discuss the term with arbitrary \(k\).

Spirit

In Spirit, the on-site anisotropy term is written as

\[\begin{split}\begin{alignedat}{1} \mathcal{H}_{\rm biaxial} & = \sum_{j} \sum_{n_1,n_2,n_3} K_j^{(n_1, n_2, n_3)} (1 - (\hat{K}_j^{(1)}\cdot\vec{n}_j)^2)^{n_1} \cdot (\hat{K}_j^{(2)}\cdot\vec{n}_j)^{n_2} \cdot ((\hat{K}_j^{(1)} \times \hat{K}_j^{(2)} ) \cdot\vec{n}_j)^{n_3} \\ & = \sum_{j} \sum_{n_1,n_2,n_3} K_j^{(n_1, n_2, n_3)} \cdot [\sin(\theta_j)]^{2n_1} \cdot [\cos(\varphi_j)\sin(\theta_j)]^{n_2} \cdot [\sin(\varphi_j)\sin(\theta_j)]^{n_3} \\ & = \sum_{j} \sum_{n_1,n_2,n_3} K_j^{(n_1, n_2, n_3)} \cdot [\sin(\theta_j)]^{2n_1 + n_2 + n_3} \cdot [\cos(\varphi_j)]^{n_2} \cdot [\sin(\varphi_j)]^{n_3}, \\ \end{alignedat}\end{split}\]

#TODO (mapping is possible, but the interpretation of the original Hamiltonian is unclear at the moment)