Is the phase transition in the Heisenberg model described by the (2+epsilon) expansion of the non-linear sigma-model?

The non-linear sigma-model is a ubiquitous model. In this paper, the O(N) model where the N-component spin is a unit vector, S-2 = 1, is considered. The stability of this model with respect to gradient operators (partial derivative(mu)S . partial derivative(nu)S)(s), where the degree s is arbitrary, is discussed. Explicit two-loop calculations within the scheme of epsilon-expansion, where epsilon = (d - 2), leads to the surprising result that these operators are relevant. In fact, the relevance increases with the degree s. We argue that this phenomenon in the O(N) model actually reflects the failure of the perturbative analysis, that is, the (2 + epsilon) expansion. It is likely that it is necessary to take into account non-perturbative effects if one wants to describe the phase transition of the Heisenberg model: within the context of the non-linear sigma-model. Thus, uncritical use of the (2 + epsilon) expansion may be misreading, especially for those cases for which there are not many independent checks.