constant. The pseudospin is usually taken to be $S=\frac 12$, and $S^{(3)}_{i}$ is the third component of the corresponding three-dimensional pseudospin vector ${\mathbf{S}}_{i}$. The other two components, $S^{(1)}_{i}$ and $S^{(2)}_{i}$, are fictitious degrees of freedom, with only the projection of ${\bf S}_i$ onto the $(3)$-axis corresponding to the physical polarization. (The unpolarized state is therefore described by the pseudospin lying entirely in the $(1)$-$(2)$ plane.) In a quantum model, $S^{(3)}_{i}$ is the expectation value of the operator $\hat{S}^{(3)}_{i}$, which is identical to the spin matrix $\hat{S}^{z}$ but which acts within pseudospin space.

The simplest version of the $S = \frac 12$ TIM is [@hemberger96] $$\label{TIM_orig}
\hat{H} = - \Omega \sum_{i} \hat{S}^{(1)}_{i} - J_{1} \sum_{\langle i, i' \rangle} \hat{S}^{(3)}_{i} \hat{S}^{(3)}_{i'} - \mu \sum_{i} E_{i} \hat{S}^{(3)}_{i},$$ where $\Omega$ plays the role of a transverse magnetic field that flips the Ising spins, $J_1$ is a nearest-neighbour coupling constant with $\langle i,i' \rangle$ indicating nearest-neighbour sites, and $E_i$ is the electric field in unit cell $i$. For $J_1>0$, the model tends towards a ferroelectric state at low temperatures; however, this is limited by $\Omega$, which disorders the ferroelectric state. Under mean-field theory the model predicts a ferroelectric phase transition only if $\Omega < Z 