Expansion in 1/zfor the transition temperature and specific heat of ferromagnets

An expansion in powers of 1/z is used to study the effect of spin fluctuations on the transition temperature and specific heat of the Ising and Heisenberg models of ferromagnetism, for general spin s. The 1/z corrections to the Curie temperature $T_C$ agree with the cluster-expansion technique, which yields the same free energy and order parameter to first order in 1/z. For the Heisenberg model, the fluctuation correction to the specific heat contains a peak below $T_C$. This peak is caused by the quantum fluctuations induced by the transverse degrees of freedom, which are absent from the Ising model. Specific-heat measurements in some materials may have found such a peak. As the spin s increases, the peak becomes better defined and decreases in temperature. In the classical limit s→∞, the peak disappears and the fluctuation contributions in the Heisenberg and Ising models become identical below $T_C$. The features of the Heisenberg model are very similar to those of a granular superconductor, where quantum fluctuations are induced by the charging energy and the classical limit is reached when the grain capacitance is infinite.