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Quantum Field Theory and Critical Phenomena$
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Jean Zinn-Justin

Print publication date: 2002

Print ISBN-13: 9780198509233

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198509233.001.0001

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Phase Transitions Near Two Dimensions

Phase Transitions Near Two Dimensions

Quantum Field Theory and Critical Phenomena


Oxford University Press

This chapter discusses the generic O(N) non-linear σ-model. One property which plays an essential role is the UV asymptotic freedom of the σ-model in two dimensions. Some of its properties are expected to generalize to other asymptotically free models. In two dimensions, various fermion self-interacting models share this property. The symmetry which is then broken is the chiral symmetry which prevents explicit mass terms in the action. As in the case of the non-linear σ-model, IR divergences forbid the existence of a massless phase. One model of this kind can be defined in continuous dimensions, the Gross–Neveu (GN), a simpli cation of the Nambu–Jona-Lasinio model, which is studied in this chapter. It is renormalizable in two dimensions, and describes in perturbation theory only one phase, the phase with symmetry breaking. A model with the same symmetry can be identified, the Gross–Neveu–Yukawa (GNY) model which is renormalizable in four dimensions, and in which both phases can be reached already in the tree approximation. The study of these two models illustrates all the ideas and techniques developed in framework of the φ4 theory and the non-linear σ-models, that is RG equations near two and four dimensions, and large N expansion.

Keywords:   perturbation theory, non-linear model, massless phase, Gross–Neveu model, Gross–Neveu–Yukawa model, large N expansion

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