PERTURBATION-THEORY PREDICTIONS AND MONTE-CARLO SIMULATIONS FOR THE 2D O(N) NONLINEAR SIGMA-MODELS

By using the results of a high-statistics (O(10(7)) measurements) Mont e Carlo simulation we test several predictions of perturbation theory on the O(n) non-linear sigma-model in 2 dimensions. We study the O(3) and O(8) models on large enough lattices to have a good control on fin ite-size effects. The magnetic susceptibility and three different defi nitions of the correlation length are measured. We check our results w ith large-n expansions as well as with standard formulae for asymptoti c freedom up to 4 loops in the standard and effective schemes. For thi s purpose the weak coupling expansions of the energy up to 4 loops for the standard action and up to 3 loops for the Symanzik action are cal culated. For the O(3) model we have used two different effective schem es and checked that they lead to compatible results. A great improveme nt in the results is obtained by using the effective scheme based on t he energy at 3 and 4 loops. We find that the O(8) model follows very n icely (within few per mille) the perturbative predictions. For the O(3 ) model an acceptable agreement (within few per cent) is found. (C) 19 97 Elsevier Science B.V.