Self-avoiding walks and the Ising and Heisenberg models

Abstract

Analysis is made of the asymptotic form of the coefficients in high temperature series expansions for the Ising model of spin ¹/₂, the classical Heisenberg and planar classical Heisenberg models. For these models, if suitable expansion variables are chosen, only lattice constants of multiply connected graphs need be considered. Numerical investigations indicate that in the initial stages of each expansion the largest contribution is due to one simple basic graph. For the specific heat this graph is a polygon, and for the pair-correlation function and susceptibility it is a chain. If all other contributing graphs are ignored, critical indices are related to the geometrical properties of a self-avoiding walk on the lattice; hence this is termed the 'self- avoiding walk approximation'. Critical indices of higher derivatives with respect to magnetic field are then related to the virial coefficients of chains.