Perturbative renormalization group, exact results and high temperature series to order 21 for the N-vector spin models on the square lattice

Abstract

High temperature expansions for the susceptibility and the second correlation moment of the classical N-vector model (also known as the O(N) symmetric Heisenberg classical spin model or the as the lattice O(N) nonlinear sigma model) on the square lattice are extended from order beta^{14} to beta^{21} for arbitrary N. For the second field derivative of the susceptibility the series expansion is extended from order beta^{14} to beta^{17}. For -2 < N < 2, a numerical analysis of the series is performed in order to compare the critical exponents gamma(N), nu(N) and Delta(N) to exact (though nonrigorous) formulas and to compute the &#34;dimensionless four point coupling constant&#34; g_r(N). For N > 2, we present a study of the analiticity properties of chi, xi etc. in the complex beta-plane and describe a method to estimate the parameters which characterize their low-temperature behaviors. We compare our series estimates to the predictions of the perturbative renormalization group theory, to exact (but nonrigorous or conjectured) formulas and to the results of the 1/N expansion, always finding a good agreement.