High-temperature critical indices for the classical anisotropic Heisenberg model

The classical anisotropic Heisenberg model is studied by means of high-temperature expansions. The purpose of the work is to determine (in the context of the model) how many phase transitions (as characterized by the critical exponents) there are, and which features of the dynamics and kinematics of a given system determine the dritical exponents. A diagrammatic expansion for the Slpin-spin correlation function is derived and renormalized. The resulting form of the perturbation theory has been used to derive high-temperature series, for various lattices and anisotropies, through orderT (closepacked lattices) and T- 9 (loose-packed lattices). These series for the correlation functions are combined to form series for the zerofield susceptibility, second moment of correlations, and specific heat. The methods used to extract the critcal indices y (susceptibility), V (correlation range) and alpha (specific heat) from the series coefficients are discussed in detail. The results areconsistent with the. hypothesis that the critical indices only change when there is a change in the symmetry of the system, e.g. in interpolating between the Ising and isotropic Heisenberg models, indices remain Ising-like until the system is isotropic, at which point they appear to change discontinuously. The classical anisotropic planar Heisenberg model is also studied. The possibility of the isotropic limit being a lattice model of the A-transition of liquid He4 is discussed. The results for the 'model are compared with experiment. In view of recent interest in spinel structures and to present a graphic example of the dangers which lie in attempts to extrapolate too-short series, some results for that lattice are given.U of I OnlyThesi