The R-S model for magnetic systems with competing interactions: series expansions and some rigorous results

The properties of a model system that exhibits a transition between ferromagnetic and helical order at a Lifshitz point is studied as interaction parameters R and S compete. Here R identical to J z /J xy and S identical to J' z /J xy , where J z and J' z denote interactions between nearest-neighbour and next-nearest-neighbour spin pairs respectively in the z direction, and J xy is a nearest-neighbour interaction between spin pairs in each xy plane. The high-temperature susceptibility series are calculated to order 8, 6, 5, and 35 respectively for the Ising, planar, Heisenberg, and spherical models (n=1,2,3, and infinity ). In order to verify the results, rigorous results are derived which provide strong checks on the series coefficients. Series analysis is focused on the ferromagnetic phase.