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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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The O(2) Classical Spin Model In Two Dimensions

The O(2) Classical Spin Model In Two Dimensions

Quantum Field Theory and Critical Phenomena


Oxford University Press

Having established in Chapter 32 a few properties of two-dimensional models, this chapter now discusses the abelian O(2) spin model near and in two dimensions. As shown in Chapter 31, at low temperature its long distance properties can be described in terms of the O(2) non-linear σ-model. The O(2) case is special because the RG β-function reduces in the low temperature expansion to the dimensional term (d - 2)t and, therefore, the properties, from the RG point of view, are quite different. In particular, in two dimensions, the O(2) model is not asymptotically free. The origin of this difference can be found in the local structure of the manifold: for N = 2, the O (N) sphere reduces to a circle which is locally at manifold, that is, which cannot be distinguished from a straight line.

Keywords:   abelian spin model, low temperature expansion, two-dimensional models

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