separately for each pseudospin (). [*[Inset]{}*]{}: The SrTiO$_3$ unit cell is illustrated, showing that the polarization is primarily due to the soft phonon mode (black arrows), in which the oxygen cage moves opposite to the titanium ions. The inset is re-published from [@atkinson17].[]{data-label="fig:STOX_compspin"}](figure1){width="0.7\linewidth"}

To show that the choice of $S$ has a small effect at the mean-field level, the uniform inverse dielectric susceptibility of STO is plotted for different values of $S$ in . From equation , the susceptibility for a weak uniform electric field $E$ is $$\label{X_gen}
\chi (T) = \left . \frac{1}{\epsilon_0} \frac{dP}{dE} \right|_{E=0} =\left . \frac{\mu \eta}{\epsilon_{0}} \frac{d S^{(3)}}{dE} \right|_{E=0},$$ where $dS^{(3)}/dE$ is obtained from equation  with ${\mathbf{h}}$ given by equation . shows results for $S=1$, $S=\frac 32$ and $S=2$. The fitting parameters $J_0$, $\mu$, and $\Omega$ depend on the value of $S$ and were determined by fitting to the experimental susceptibility, as described in below. (Note that $J_1$ is not explicitly used here because the calculations are for bulk STO.) The values of all these parameters are listed in .

Because the model was fitted to low- and high-temperature susceptibilities, the curves in are expected to be close in value at these limits. However, they also 