Quadruplet interaction

In Spirit, the quadruplet interaction term is written as

\[\mathcal{H}_{\rm quad} = - \sum_{ijkl} K_{ijkl} (\vec{n}_i \cdot \vec{n}_j)(\vec{n}_k \cdot \vec{n}_l)\]

First, notice, that \(i \ne j\) and \(k \ne l\), otherwise the Hamiltonian is constant. Therefore, only three cases are possible:

• \(i = k\) and \(j = l\) (or equivalently \(i = l\) and \(j = k\))

  This is a case of Biquadratic exchange.

• \(i = k\) and \(j \ne l\) (or equivalently \(i = l\) and \(j \ne k\) or \(j = k\) and \(i \ne l\) or \(j = l\) and \(i \ne k\))

  This is case 1. The Hamiltonian maps to the term of the general Hamiltonian with \(k = 4\), \(l = 4\) (\(m_{4,4} = 3\)).

  \[\begin{split}\mathcal{H}_{4, 4} = C_{4, 4} \sum_{\substack{\mu_1, \mu_2, \mu_3, \\ \alpha_1, \alpha_2, \alpha_3, \\ i_1, i_2, i_3, i_4}} V_{\mu_1, \mu_2, \mu_3; \alpha_1, \alpha_2, \alpha_3}^{i_1, i_2, i_3, i_4} \cdot X_{\mu_1; \alpha_1}^{i_1} \cdot X_{\mu_1; \alpha_1}^{i_2} \cdot X_{\mu_2; \alpha_2}^{i_3} \cdot X_{\mu_3; \alpha_3}^{i_4}\end{split}\]

• \(i \ne k\) and \(j \ne l\) and \(i \ne l\) and \(j \ne k\)

  This is case 2. The Hamiltonian maps to the term of the general Hamiltonian with \(k = 4\), \(l = 5\) (\(m_{4,5} = 4\)).

  \[\begin{split}\mathcal{H}_{4, 5} = C_{4, 5} \sum_{\substack{\mu_1, \mu_2, \mu_3, \mu_4, \\ \alpha_1, \alpha_2, \alpha_3, \alpha_4, \\ i_1, i_2, i_3, i_4}} V_{\mu_1, \mu_2, \mu_3, \mu_4; \alpha_1, \alpha_2, \alpha_3, \alpha_4}^{i_1, i_2, i_3, i_4} \cdot X_{\mu_1; \alpha_1}^{i_1} \cdot X_{\mu_2; \alpha_2}^{i_2} \cdot X_{\mu_3; \alpha_3}^{i_3} \cdot X_{\mu_4; \alpha_4}^{i_4}\end{split}\]

Quadruplet interaction (case 1)

By expanding the dot product, one gets

\[\begin{split}\mathcal{H}_{\rm quad}^{\rm case \, 1} = - \sum_{\substack{\mu_1, \mu_2, \mu_3,\\ \alpha_1, \alpha_2, \alpha_3, \\ i_1, i_2, i_3, i_4}} K_{ijl}^{i_1, i_2, i_3, i_4} n_{i}^{i_1} n_{i}^{i_2} n_{j}^{i_3} n_{l}^{i_4}\end{split}\]

where the tensor \(K_{ijl}^{i_1, i_2, i_3, i_4}\) is defined as

\[\begin{split}K_{ijil} &= K_{ijl}^{xxxx} = K_{ijl}^{yyyy} = K_{ijl}^{zzzz} \\ &= K_{ijl}^{xyxy} = K_{ijl}^{xzxz} = K_{ijl}^{yzyz} \\ &= K_{ijl}^{yxyx} = K_{ijl}^{zxzx} = K_{ijl}^{zyzy} \\\end{split}\]

and all other components are zero. Then one renames the indices as \(i \rightarrow (\mu_1, \alpha_1)\), \(j \rightarrow (\mu_2, \alpha_2)\), \(l \rightarrow (\mu_3, \alpha_3)\) and correspondence becomes clear

\[\begin{split}C_{4, 4} &= -1 \\ V_{\mu_1, \mu_2, \mu_3; \alpha_1, \alpha_2, \alpha_3}^{i_1, i_2, i_3, i_4} &\leftrightarrow K_{ijl}^{i_1, i_2, i_3, i_4} \\ X_{\mu_1; \alpha_1}^{i_1} &\leftrightarrow n_{i}^{i_1} \\ X_{\mu_2; \alpha_2}^{i_3} &\leftrightarrow n_{i}^{i_3} \\ X_{\mu_3; \alpha_3}^{i_4} &\leftrightarrow n_{j}^{i_4} \\\end{split}\]

Quadruplet interaction (case 2)

By expanding the dot product, one gets

\[\begin{split}\mathcal{H}_{\rm quad}^{\rm case \, 2} = - \sum_{\substack{\mu_1, \mu_2, \mu_3, \mu_4,\\ \alpha_1, \alpha_2, \alpha_3, \alpha_4, \\ i_1, i_2, i_3, i_4}} K_{ijkl}^{i_1, i_2, i_3, i_4} n_{i}^{i_1} n_{j}^{i_2} n_{k}^{i_3} n_{l}^{i_4}\end{split}\]

where the tensor \(K_{ijkl}^{i_1, i_2, i_3, i_4}\) is defined as

\[\begin{split}K_{ijkl} &= K_{ijkl}^{xxxx} = K_{ijkl}^{yyyy} = K_{ijkl}^{zzzz} \\ &= K_{ijkl}^{xxyy} = K_{ijkl}^{xxzz} = K_{ijkl}^{yyzz} \\ &= K_{ijkl}^{yyxx} = K_{ijkl}^{zzxx} = K_{ijkl}^{zzyy} \\\end{split}\]

and all other components are zero. Then one renames the indices as \(i \rightarrow (\mu_1, \alpha_1)\), \(j \rightarrow (\mu_2, \alpha_2)\), \(k \rightarrow (\mu_3, \alpha_3)\), \(l \rightarrow (\mu_4, \alpha_4)\) and correspondence becomes clear

\[\begin{split}C_{4, 5} &= -1 \\ V_{\mu_1, \mu_2, \mu_3, \mu_4; \alpha_1, \alpha_2, \alpha_3, \alpha_4}^{i_1, i_2, i_3, i_4} &\leftrightarrow K_{ijkl}^{i_1, i_2, i_3, i_4} \\ X_{\mu_1; \alpha_1}^{i_1} &\leftrightarrow n_{i}^{i_1} \\ X_{\mu_2; \alpha_2}^{i_2} &\leftrightarrow n_{j}^{i_2} \\ X_{\mu_3; \alpha_3}^{i_3} &\leftrightarrow n_{k}^{i_3} \\ X_{\mu_4; \alpha_4}^{i_4} &\leftrightarrow n_{l}^{i_4} \\\end{split}\]