Low temperature asymptotic expansion for classical $O(N)$ vector models

TL;DR

This paper establishes that low-temperature expansions for classical $O(N)$ vector models in three or more dimensions are asymptotic series, providing explicit error estimates and extending previous methods to non-abelian symmetries.

Contribution

It proves the asymptotic nature of low-temperature expansions for $O(N)$ models and generalizes existing approaches to non-abelian symmetry cases.

Findings

Low-temperature expansions are asymptotic series with explicit error bounds.

The method applies to the spontaneous magnetization in the 3D Heisenberg model.

Extension of previous techniques to non-abelian symmetry and non-gradient observables.

Abstract

We consider classical $O(N)$ vector models in dimension three and higher and investigate the nature of the low-temperature expansions for their multipoint spin correlations. We prove that such expansions define asymptotic series, and derive explicit estimates on the error terms associated with their finite order truncations. The result applies, in particular, to the spontaneous magnetization of the 3D Heisenberg model. The proof combines a priori bounds on the moments of the local spin observables, following from reflection positivity and the infrared bound, with an integration-by-parts method applied systematically to a suitable integral representation of the correlation functions. Our method generalizes an approach, proposed originally by Bricmont and collaborators [6] in the context of the rotator model, to the case of non-abelian symmetry and non-gradient observables.