Testing an Optimised Expansion on Z_2 Lattice Models

Abstract

We test an optimised hopping parameter expansion on various Z_2 lattice scalar field models: the Ising model, a spin-one model and lambda (phi)^4. We do this by studying the critical indices for a variety of optimisation criteria, in a range of dimensions and with various trial actions. We work up to seventh order, thus going well beyond previous studies. We demonstrate how to use numerical methods to generate the high order diagrams and their corresponding expressions. These are then used to calculate results numerically and, in the case of the Ising model, we obtain some analytic results. We highlight problems with several optimisation schemes and show for the best scheme that the critical exponents are consistent with mean field results to at least 8 significant figures. We conclude that in its present form, such optimised lattice expansions do not seem to be capturing the non-perturbative infra-red physics near the critical points of scalar models.