Is the phase transition in the Heisenberg model described by the $(2+\epsilon)$-expansion of the nonlinear $\sigma$-model?

Abstract

Nonlinear $\sigma$-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 $(\partial_{\mu}{\bf S}\cdot \partial_{\nu}{\bf S})^s$, 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 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 $(2+\epsilon)$-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 misleading, especially for those cases for which there are not many independent checks.