About cookies on this site Our websites require some cookies to function properly (required). In addition, other cookies may be used with your consent to analyze site usage, improve the user experience and for advertising. For more information, please review your options. By visiting our website, you agree to our processing of information as described in IBM’sprivacy statement. To provide a smooth navigation, your cookie preferences will be shared across the IBM web domains listed here.
Paper
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.