Thermodynamic and magnetic properties of a cooperative spin-phonon coupled system

A systematic perturbation technique is developed for calculating the thermodynamic and magnetic properties of a system undergoing the cooperative Jahn-Teller effect. The leading-order results are calculated within the limits of a molecular-field theory, and the connection with the Ising or transverse Ising models is established. The higher-order corrections are obtained by a diagrammatic method. The results are applicable at any arbitrary temperature to within the first order in $\frac{1}{z}$, where $z$ is the number of nearest neighbors at each site. Low-temperature properties, particularly the specific heat, are discussed in some detail.