the modified model to ferroelectric thin films. In , we then apply the model to the LAO/STO interface, and show explicitly how the modification allows for the formation of the 2DEG.

Inhomogeneous Ferroelectrics {#sec:FE}
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We begin by describing a modified TIM () that contains an additional anisotropic interaction; depending on its sign, this interaction generates either a pseudospin easy axis or easy plane. We obtain mean-field equations for the pseudospin and susceptibility, and by comparison to the LGD theory () we show that the Landau parameters are under-determined by the conventional TIM. Essentially, the problem is that equation  contains three adjustable parameters ($\Omega$, $J_1$, and $\mu$), while the simplest LGD model requires four parameters to describe an inhomogeneous system. The additional interaction in the modified TIM fixes this discrepancy. In sections \[sec:fit\] and \[sec:J1\] we obtain fits to the model parameters for the case of STO. As a simple application, in we explore how the new term modifies the polarization distribution of a ferroelectric thin film.

The Modified TIM {#sec:TIM}
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The modified Hamiltonian for general pseudospin $S$ is $$\begin{aligned}
 \label{TIM_full}
\fl \hat{H} = - \Omega \sum_{i} \hat{S}^{(1)}_{i} - \frac{J_{1}}{2S} \sum_{\langle i,i' \rangle} \hat{S}^{(3)}_{i} \hat{S}^{(3)}_{i'} - \frac{J_\mathrm{an}}{2S} \sum_{i} \hat{S}^{(3)}_{i} 