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

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Abstract
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 Jz/Jxy and S identical to J'z/Jxy, where Jz and J'z denote interactions between nearest-neighbour and next-nearest-neighbour spin pairs respectively in the z direction, and Jxy 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.