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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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Conformal field theory

Conformal field theory

(p.128) 6 Conformal field theory
Elements of Phase Transitions and Critical Phenomena

Hidetoshi Nishimori

Gerardo Ortiz

Oxford University Press

The present chapter is an introductory account of the basic concepts and important consequences of conformal symmetry, i.e. the invariance under local scale transformations, in field theories characterizing critical behaviour. The goal is to catalogue universality classes as a list of possible values of critical exponents and to find restrictions on the functional forms of correlation functions, which satisfy conformal Ward identities. From a mathematics standpoint, conformal symmetry applies to continuum theories, and therefore its obvious application to critical phenomena is formulated in the language of field theory. The energy-momentum tensor plays a fundamental role in defining the conformal generators that satisfy the Virasoro algebra, and any conformal field theory is characterized by the central charge a number that is important to classify critical field theories. One of the most remarkable applications of conformal field theory is found in the analysis of finite-size effects.

Keywords:   local scale transformation, conformal invariance, universality class, correlation function, primary operators, Ward identity, Virasoro algebra, central charge

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