New formalism for the study of lattice Hamiltonians

An expansion in 1/z is developed for a general class of lattice Hamiltonians, including ferromagnets and granular superconductors. This expansion technique, which overcomes many of the limitations of conventional mean-field theory and the spin-wave approximation, is especially useful to study the effect of quantum fluctuations on the transition temperature and specific heat. Any thermodynamic quantity can be expanded in powers of 1/z, where z is the number of nearest neighbors on the lattice. While the lowest-order term is the mean-field result, the first-order correction involves the coupling of fluctuations of the spin operator Si on neighboring sites. With the help of a recursion relation, the fluctuation corrections to the order parameter and free energy can be evaluated exactly. Three derivations of the 1/z expansion are provided. The first two, which are based on expansions of the order parameter and free energy, are entirely general. The third proof is an extension of the variational technique developed by Strieb, Callen, and Horwritz for the spin-1/2 Heisenberg model. This variational technique, also called the constant-coupling approximation, can be derived from the more general 1/z expansion when the operators in the Hamiltonian obey two sets of commutation relations.