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

Abstract

Nonlinear $σ$-model is an ubiquitous model. In this paper, the $O(N)$ model where the $N$-component spin is a unit vector, ${\bf S}^2=1$,is considered. The stability of this model with respect to gradient operators $(\partial_μ{\bf S}\cdot \partial_ν{\bf S})^s$, where the degree $s$ is 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 relevancy 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.