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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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Classical statistical physics: One dimension

Classical statistical physics: One dimension

Chapter:
(p.79) 4 Classical statistical physics: One dimension
Source:
Phase Transitions and Renormalization Group
Author(s):

Jean Zinn-Justin

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199227198.003.0004

This chapter, within the framework of classical statistical mechanics, discusses a family of models defined on one-dimensional lattices. It studies the simplest local examples: models that involve only interactions between nearest neighbours on the lattice. For such models, correlation functions can be calculated by a transfer matrix formalism. The chapter first describes some general properties of transfer matrices in one-dimensional models. This formalism is used to establish various properties of correlation functions, like the thermodynamic or infinite volume limit, the large-distance behaviour of the two-point correlation function, and introduces the very important concept of correlation length. Connected correlation functions, cumulants of the distribution, play a particularly important role. Indeed, these functions satisfy the cluster property, which characterizes their decay at large distance. The transfer matrix formalism is applied to the example of a Gaussian Boltzmann weight, which is studied in detail. The chapter calculates the partition function and correlation functions explicitly, and observes that , the correlation length diverges, making it possible to define a continuum limit. It shows that results of the continuum limit can be reproduced directly by solving a partial differential equation in which all traces of the initial lattice structure have disappeared. Finally, it exhibits a slightly more general class of models which share the same properties: divergent correlation length and continuum limit. Exercises are provided at the end of the chapter.

Keywords:   one-dimensional lattices, transfer matrix, correlation functions, thermodynamic limit, Gaussian model, continuum limit

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