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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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Renormalization group: N-component fields

Renormalization group: N-component fields

(p.267) 11 Renormalization group: N-component fields
Phase Transitions and Renormalization Group

Jean Zinn-Justin

Oxford University Press

This chapter studies more general models with an N-component field (or order parameter), from the viewpoint of the renormalization group (RG). Indeed, one can find interesting physical systems for which the Hamiltonian does not have the O(N) orthogonal symmetry of the models studied in Section 10.5. A first family of such models is characterized by the presence of several independent correlation lengths. This happens typically when the quadratic part of the Hamiltonian involves several unrelated parameters. Generically, the different correlation lengths then diverge for different values of the temperature. The components of the fields that are not critical decouple and can be ignored in the study of the asymptotic large-distance behaviour (in the sense of the field integral, they can be integrated out). To classify the possible types of critical behaviour, one can thus restrict the discussion to models that, like the O(N) model, have only one correlation length in the disordered phase. Exercises are provided at the end of the chapter.

Keywords:   renormalization group, N-component field, Hamiltonian, gradient flow, correlation lengths

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