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Elements of Phase Transitions and Critical Phenomena$
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Hidetoshi Nishimori and Gerardo Ortiz

Print publication date: 2010

Print ISBN-13: 9780199577224

Published to Oxford Scholarship Online: January 2011

DOI: 10.1093/acprof:oso/9780199577224.001.0001

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Renormalization group and scaling

Renormalization group and scaling

(p.52) 3 Renormalization group and scaling
Elements of Phase Transitions and Critical Phenomena

Hidetoshi Nishimori

Gerardo Ortiz

Oxford University Press

Mean-field theory is usually taken as a first step toward understanding critical phenomena, providing an overview that reveals qualitative behaviour of physical quantities. However, it is necessary to proceed beyond the mean-field theory to better understand the situation, both qualitatively and quantitatively, when fluctuations play vital roles leading to exponents that cannot be explained by dimensional analysis, thus introducing anomalous dimensions. The present chapter explains the basic concepts of the renormalization group and scaling theory, which allow us to analyze critical phenomena with fluctuations systematically taken into account. The essential step in a renormalization group calculation consists of establishing recursion relations between the parameters defining the Hamiltonian of the system. These recursion or renormalization group equations define a flow with well-defined fixed points. Details other than the values of the relevant operators have no influence on the critical exponents and this represents universality.

Keywords:   fluctuations, scaling, renormalization group, critical exponents, recursion relation, renormalization flow, fixed point, universality, relevant operators, anomalous dimension

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