was developed by deGennes in 1963 to describe the ferroelectric transition in hydrogen-bonded materials like potassium dihydrogen phosphate (KDP) [@degennes63]. As suggested by its name, the model formally describes a system of magnetic Ising moments in a transverse magnetic field [@stinchcombe73], and since its discovery it has become significant because it is one of the simplest models to exhibit a quantum phase transition [@Sachdev:2011]. The focus of this work is more practical; we explore the use of the TIM to describe the dielectric properties of SrTiO$_3$. Indeed, the TIM has been used widely to model the low-energy physics of systems in which local degrees of freedom can be represented by pseudospins [@stinchcombe73]. In KDP, for example, the $S=\frac{1}{2}$ Ising spin states represent the two degenerate positions available to each hydrogen atom, while the transverse field represents the quantum mechanical tunneling between the states.

Because the TIM starts from a picture of fluctuating local dipole moments, it naturally describes materials, like KDP, with order-disorder transitions. However, the model has also been applied to materials like SrTiO$_3$, which are close to a displacive ferroelectric transition. While there are some clear discrepancies between the model and experiments [@Muller:1979wa], the mean-field TIM nonetheless gives 