Phase transitions in a simple model ferroelectric—comparison of exact and variational treatments of a molecular-field Hamiltonian

We pursue the qualitative features of ferroelectric phase transitions via a detailed examination of a simple model system, the Hamiltonian of which consists of a lattice of quartic anharmonic oscillators of mass $M$ interacting via a quadratic intercell interaction term. The intercell interaction term—assumed to be long range in character—is treated in a molecular-field approximation. The eigenvalue and eigenfunction spectrum of the resulting molecular-field Hamiltonian is obtained numerically exactly. This permits us to construct the density matrix associated with the lattice of coupled oscillators, and hence calculate the related statistical properties for different values of the intercell coupling $\chi$, the zero-point parameter $\lambda \equiv \frac{\hslash }{\surd M}$, and the temperature $T$. It is found that a second-order structural transition occurs at some finite temperature $T_c$ if $\chi >{\chi }_c^{\pm }\left(\lambda \right)$, where the superscripts refer to the two cases where the local particle potential possesses a single minimum (displacive) or a double minimum (order-disorder), respectively. The functional dependence of ${\chi }^{+}\left(\lambda \right)$ on $\lambda$ is qualitatively different from that of ${\chi }^{-}\left(\lambda \right)$; e.g., ${\chi }^{+}\mathrm{}\left(0\right)\mathrm{}$ is finite, whereas ${\chi }^{-}\mathrm{}\left(0\right)=0\mathrm{}$. A variational treatment of the molecular-field Hamiltonian employing a trial density matrix of the displaced-Gaussian form yields the prediction that if a transition occurs it may be either first or second order, depending on the values of the model parameters, as compared to the exact numerical treatment, where the transition is always second order. The implications of the first-order transition are discussed, with the changeover from second- to first-order behavior being examined.