differ only slightly in between, indicating that STO is well-described by the simplest case shown, $S=1$, when using mean-field theory. In particular, the model accurately captures both Curie-Weiss behaviour at high temperature, and the saturation of the susceptibility at low temperature (where the ferroelectric transition is suppressed by quantum fluctuations).

[c c c c]{} & Spin-1 & Spin-3/2 & Spin-2\
$\Omega$ (meV) & 4.41 & 3.53 & 2.94\
$J_{0}$ (meV) & 3.88 & 3.10 & 2.58\
$J_{1}$ (meV) & 30-130 & 40-160 & 50-200\
$\mu$ ($e$Å) & 1.88 & 1.37 & 1.09\

\[tab:pars\]

Comparison to the Landau-Ginzburg-Devonshire Expansion {#sec:LGD}
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While equation  is the fundamental self-consistent equation for $S^{(3)}_{i}$, the role each parameter plays in determining the pseudospin is not transparent. For example, it is not immediately evident from this expression why the conventional TIM (with $J_\mathrm{an}=0$) is unable to describe inhomogeneous systems. To explore this point, we expand equation  in powers of $h_{i}^{(3)}$ and compare the coefficients to those in a typical LGD expansion. We show that the transition temperature and correlation length cannot be set independently unless $J_\mathrm{an}$ is nonzero.

The typical LGD free energy with order parameter $S^{(3)}({\mathbf{r}})$ has the form $$\begin{aligned}
 \label{F}
\mathcal{F} &=& \eta \int 