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

Abstract The non-linear σ-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 ( ∂ μ S · ∂ ν S ) s , where the degree s is the arbitrary, is discussed. Explicit two-loop calculations within the scheme of ϵ-expansion, where ϵ = ( 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 + ϵ ) 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 σ-model. Thus, uncritical use of the (2 + ϵ ) expansion may be misleading, especially for those cases for which there are not many independent checks.