Critical exponents of the three-dimensional classical plane-rotator model on the sc lattice from a high-temperature-series analysis.

In three dimensions the two-component vector model is the simplest spin model in the universality class of the superfluid λ transition of He, and of the ferromagnetic transition of magnets with an easy magnetization plane[ [1]]. No high temperature (HT) series studies of this model have appeared in the last two decades in spite of remarkable experimental measurements of the critical parameters in superfluid He and intense theoretical activity in Renormalization Group calculations and by direct Monte Carlo simulations. In particular we should mention that the critical index ν, which describes the leading singularity of the superfluid fraction in He near the superfluid transition temperature, has been measured with high precision in a long series of experiments by G. Ahlers and his collaborators[ [2]]. As stressed by Ahlers, the superfluid fraction is the most accurately known singular parameter at a critical point, and correspondingly ν is the most accurately known critical index. The most recent experiments yield the value ν = 0.6705± 0.0006. Unfortunately the critical exponent γ cannot be measured in liquid He and, as far as magnetic systems are concerned, no precise measurements exist either for γ or for ν. A review of static critical properties of He can be found in Ref.[ [3]] and a general discussion of the interpretation of the measurements on He in connection with the problem of confluent singularities is given in Ref.[ [4]]. The Hamiltonian of the three-dimensional plane rotator (or XY) model is