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

Alessandro Giuliani and Sébastien Ott

Vol. 6 (2025), No. 2, 439–477
DOI: 10.2140/pmp.2025.6.439
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 (1981) in the context of the rotator model, to the case of nonabelian symmetry and nongradient observables.

Keywords
asymptotic expansion, statistical mechanics, lattice spin models, low temperature, continuous symmetry, nonabelian symmetry, Gaussian, Heisenberg model
Mathematical Subject Classification
Primary: 82B20
Milestones
Received: 8 March 2023
Revised: 16 July 2024
Accepted: 23 November 2024
Published: 12 March 2025
Authors
Alessandro Giuliani
Dipartimento di Matematica e Fisica
Università degli Studi Roma Tre
Roma
Italy
Sébastien Ott
Institute of Mathematics
École Polytechnique Fédérale de Lausanne
Lausanne
Switzerland