A study of the (m,d,N)=(1,3,2) Lifshitz point and of the three- dimensional XY universality class by high-temperature bivariate series for the XY models with anisotropic competing interactions

High-temperature bivariate expansions have been derived for the two-spin correlation-function in a variety of classical lattice XY (planar rotator) models in which spatially isotropic interactions among first-neighbor spins compete with spatially isotropic or anisotropic (in particular uniaxial) interactions among next-to-nearest-neighbor spins. The expansions, calculated for cubic lattices of dimension d=1,2 and 3, are expressed in terms of the two variables K1=J1/kT and K2=J2/kT, where J1 and J2 are the nearest-neighbor and the next-to-nearest-neighbor exchange couplings, respectively. This report deals in particular with the properties of the d=3 uniaxial XY model (ANNNXY model) for which the bivariate expansions have been computed through the 18-th order, thus extending by 12 orders the results so far available and making a study of this model possible over a wide range of values of the competition parameter R=J2/J1.