On-site anisotropy (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 \(k\).

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

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

This term can be mapped to the general form in different ways.

Strategy #1

This strategy maps the term to the \(\mathcal{H}_3\) term of the general form.

\[\mathcal{H}_3 = C_3 \sum_{\mu_1, \mu_2, \mu_3, \alpha_1, \alpha_2, \alpha_3, i_1, i_2, i_3} V_{\mu_1, \mu_2, \mu_3; \alpha_1, \alpha_2, \alpha_3}^{i_1, i_2, i_3} X_{\mu_1, \alpha_1}^{i_1} X_{\mu_2, \alpha_2}^{i_2} X_{\mu_3, \alpha_3}^{i_3}\]

First rename the index \(j \rightarrow (\mu, \alpha)\), then define three entities

\[\begin{split}X_{\mu, \alpha}^{i_1} &= (1 - (\hat{K}_{\mu, \alpha}^{(1)}\cdot\vec{n}_{\mu, \alpha})^2)^{i_1} \\ X_{\mu, \alpha}^{i_2} &= (\hat{K}_{\mu, \alpha}^{(2)}\cdot\vec{n}_{\mu, \alpha})^{i_2} \\ X_{\mu, \alpha}^{i_3} &= ((\hat{K}_{\mu, \alpha}^{(1)} \times \hat{K}_{\mu, \alpha}^{(2)} ) \cdot\vec{n}_{\mu, \alpha})^{i_3}\end{split}\]

where component indices \(i_1, i_2, i_3\) assume the same integer values as \(n_1, n_2, n_3\) respectively. Then the interaction parameter can be defined as

\[V_{\mu_1, \mu_2, \mu_3; \alpha_1, \alpha_2, \alpha_3}^{i_1, i_2, i_3} = \delta_{\mu_1, \mu_2} \delta_{\mu_1, \mu_3} \delta_{\alpha_1, \alpha_2} \delta_{\alpha_1, \alpha_3} K_{\mu_1, \alpha_1}^{(i_1, i_2, i_3)}\]