Lattice distortions in one-dimensional Ising chains

Abstract

A model system consisting of a lattice of linear Ising chains with magnetic interactions only along the chains interacting with three-dimensional phonons has been investigated. It is shown that the thermodynamic properties of this model are equivalent to those of a three -dimensional pseudospin Ising model on a fixed lattice in an external magnetic field. This shows explicitly that one-dimensional systems can undergo phase transitions when coupled to three-dimensional phonons. Further, the mean-field approximation in which the phonon normal-mode coordinate is replaced by its thermal average is shown to be an arbitrarily bad approximation depending on the form of the phonon dispersion. When certain conditions are satisfied, the model exhibits a structural phase transition which increases the lattice periodicity. The relevant properties of the structural phase transition such as the lattice distortions, the soft-mode frequency, and the width of the central peak may be expressed exactly in terms of the properties of the pseudospin Ising model.