(3)-axis are then modelled as boson excitations, with a pseudospin that is entirely polarized along the (1)-axis represented by the vacuum state.

In this representation, the raising and lowering operators for site $i$ differ from the typical set by a cyclic permutation of the pseudospin axes. We then define [@holstein] $$\begin{aligned}
 \label{S+}
\hat{S}^{+}_{i} & = \hat{S}^{(2)}_{i} + i \hat{S}^{(3)}_{i} \\
& = \sqrt{2S} \left( 1 - \frac{1}{2S} \hat{a}^{\dag}_{i} \hat{a}_{i} \right)^{1/2} \ \hat{a}_{i},\end{aligned}$$ $$\begin{aligned}
 \label{S-}
\hat{S}^{-}_{i} & = \hat{S}^{(2)}_{i} - i \hat{S}^{(3)}_{i} \\ 
& = \sqrt{2S} \hat{a}^{\dag}_{i} \left( 1 - \frac{1}{2S} \hat{a}^{\dag}_{i} \hat{a}_{i} \right)^{1/2}.\end{aligned}$$ Since the polarization lies close to the (1)-axis in the paraelectric state, only low bosonic excitation states are relevant. In this case, $\hat{S}^{+}_{i} \approx \sqrt{2S} \hat{a}_{i}$ and $\hat{S}^{-}_{i} \approx \sqrt{2S} \hat{a}^{\dag}_{i}$. Additionally, the (1)-component of the pseudospin is defined as [@holstein] $$\label{S1}
\hat{S}^{(1)}_{i} = S - \hat{a}^{\dag}_{i} \hat{a}_{i},$$ and the (3)-component is $$\begin{aligned}
 \label{S3_a}
\hat{S}^{(3)}_{i} & = \frac{1}{2i} \left( \hat{S}^{+}_{i} - \hat{S}^{-}_{i} \right) \\
& = \frac{\sqrt{2S}}{2i} \left( \hat{a}_{i} - \hat{a}^{\dag}_{i} \right).\end{aligned}$$ Because $\hat{S}^{(3)}_{i}$ represents atomic displacements, $\hat{a}_i$ and $\hat{a}^{\dag}_i$ are therefore phonon operators.

Equations  and can now be substituted into equation . We transform to reciprocal space using $\hat{a}_{i}$ = $\frac{1}{\sqrt{N}}\sum_{{\mathbf{k}}}  e^{i{\mathbf{k}} \cdot {\mathbf{r}}_{i}} \hat{b}_{{\mathbf{k}}}$: $$\begin{aligned}
