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

We study the properties of a model system that exhibits a transition between ferromagnetic and helical order at a Lifshitz point, as interaction parameters R and S compete. Here R JJJ,, and S = J:/J,,, where J, and Jz denote interactions between nearest- neighbour and next-nearest-neighbour spin pairs respectively in the z direction, and J,, is a nearest-neighbour interaction between spin pairs in each xy plane. We calculate the high- temperature susceptibility series to order 8, 6, 5, and 35 respectively for the Ising, planar, Heisenberg, and spherical models (n = 1,2,3, and io). In order to verify our results, we derive rigorous results which provide strong checks on the series coefficients. Series analysis is focused on the ferromagnetic phase. In particular, we confirm scaling with respect to both parameters R and S. In addition, we find the critical region shrinks as the Lifshitz point is approached. This is indicated by analysing the spherical model series where asymptotic series behaviour is not evident, even at order 35. Finally, by exploiting simple geometric ideas about the dependence of the correlation length on R and S, we describe the full wavevector and temperature dependence of the structure factor.