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

TL;DR

This paper develops high-temperature bivariate series expansions for 3D XY models with competing interactions, enabling detailed analysis of the Lifshitz point and universality class across a broad range of interaction ratios.

Contribution

It extends the order of bivariate series expansions for the 3D uniaxial XY model by 12 orders, allowing comprehensive study of the Lifshitz point and universality class with anisotropic competing interactions.

Findings

Extended series to 18th order for the 3D XY model.

Enabled detailed analysis of the Lifshitz point.

Provided insights into the universality class with competing interactions.

Abstract

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.