# Phase transitions in a model of interacting anharmonic oscillators

## Abstract

As a prototype of a ferroelectric crystal we consider a system of interacting anharmonic oscillators. This model is treated via a simple Hamiltonian which consists of a sum of single-particle quartic-anharmonic- oscillator Hamiltonians together with a quadratic intercell interaction term. The interaction term typically long range in nature is treated in a molecular-field approximation, yielding an effective lattice Hamiltonian which can be scaled in terms of two parameters only an effective-inverse-mass parameter and an effective coupling strength. For computational simplicity we treat the inverse-mass parameter as an effective temperature, with zero-point fluctuations playing the role of thermal fluctuations. The effective lattice Hamiltonian is solved numerically exactly to determine whether the lattice of coupled oscillators will, at some temperature, undergo a transition to a state in which the average value of the particle displacement is nonzero. From the properties of the exact solution it is shown that one can have a second-order transition or no transition, depending on the magnitude of the intercell coupling. If the anharmonic potential in which each particle moves possesses a double-well character, a second-order transition will occur for any value of the coupling strength greater than zero. On the other hand, if the anharmonic potential exhibits only a single minimum, then a transition will occur only if the coupling strength exceeds a critical value. These and other exact results establish a basis for ascertaining the range of validity of certain approximate treatments of the molecular-field Hamiltonian. In particular, we discuss in detail (i) variational treatments in which a set of trial displaced-oscillator wave functions are introduced as solutions to the molecular-field Hamiltonian and (ii) a so-called "two-level" approximation which is analogous to the de Gennes pseudospin model of hydrogen-bonded ferroelectrics. Finally, we discuss the collective properties of the system of coupled oscillators within the context of both the exact and approximate treatments. © 1973 The American Physical Society.