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

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

1996-05-14

arxiv.org/abs/cond-mat/9605088v1

Nucl.Phys. B485 (1997) 613-645

Physics

Condensed Matter

RevTex, 33 pages, figures embedded

Scientific paper

10.1016/S0550-3213(96)00617-7

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.

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