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

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, S2 = 1, is considered. The stability of this model with respect to gradient operators (∂μS·∂νS)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.