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Phase Transitions and Renormalization Group$
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Jean Zinn-Justin

Print publication date: 2007

Print ISBN-13: 9780199227198

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780199227198.001.0001

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The non-linear s-model

The non-linear s-model

(p.353) 15The non-linear s-model
Phase Transitions and Renormalization Group

Jean Zinn-Justin

Oxford University Press

This chapter discusses the non-linear s-model, a field theory characterized by an orthogonal O(N) symmetry acting non-linearly on the fields. The study has several motivations. From the viewpoint of statistical physics, the model appears in the study of the large-distance properties, in the ordered phase at low temperature, of lattice spin models with O(N) symmetry and short-range interactions. Indeed, in the case of continuous symmetries, the whole low-temperature phase has a non-trivial large-distance physics due to the presence of Goldstone modes with vanishing mass or infinite correlation length. Moreover, the model possesses, in two dimensions, the property of asymptotic freedom (the Gaussian fixed point is marginally stable for the large-momentum or short-distance behaviour) and the spectrum is non-perturbative. These properties are shared, in dimension 4, by quantum chromodynamics (QCD), a non-Abelian gauge theory and a piece of the Standard Model of fundamental interactions describing physics at the microscopic scale.

Keywords:   non-linear s-model, lattice spin models, renormalization group, RGE, correlation functions

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