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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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Two-Dimensional Models and Bosonization Method

Two-Dimensional Models and Bosonization Method

Quantum Field Theory and Critical Phenomena


Oxford University Press

Chapter 31 discussed the generic O(N) non-linear σ-model. We have noticed that the abelian case N = 2 is special because the RG β-function vanishes in two dimensions. The corresponding O(2) invariant spin model is specially interesting: it provides an example of the celebrated Kosterlitz–Thouless phase transition and will be examined in Chapter 33. However, a thorough discussion of this model requires a new technique: we have to establish relations, special to two dimensions, between fermion and boson local field theories, by a method called bosonization. This chapter illustrates the derivation involves several steps with the help of various other two-dimensional models which are physically interesting in their own right. It first studies the free massless boson and fermion fields. It evaluates the determinant of the covariant fermion derivative in the presence of an external gauge field. The result is at the basis of the bosonization technique. The chapter then discusses the sine–Gordon (SG) model. It solves the Schwinger model, QED with massless fermions. Finally, it demonstrates the equivalence between the SG model and two fermion models with current-current interaction: the Thirring model and another model with two species of fermions.

Keywords:   free massless boson, covariant fermion derivative, sine–Gordon model, Schwinger model, Thirring model, two fermion model

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