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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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Exact solutions and related topics

Exact solutions and related topics

(p.204) 9 Exact solutions and related topics
Elements of Phase Transitions and Critical Phenomena

Hidetoshi Nishimori

Gerardo Ortiz

Oxford University Press

Only a limited number of models of phase transitions and critical phenomena can be solved exactly. These examples nevertheless play important roles in many aspects including the verification of the accuracy of approximation theories such as the mean-field theory and renormalization group. Mathematical methods to solve such examples are interesting in their own right and constitute an important subfield of mathematical physics. In particular the exact solution of the two-dimensional Ising model occupies an outstanding status as one of the founding studies of the modern theory of phase transitions and critical phenomena. The present chapter shows simple but typical examples of exact solutions of classical spin systems such as the one-dimensional Ising model with various boundary conditions, the n-vector model, the spherical model, the one-dimensional quantum $XY$ model, and the two-dimensional Ising model. An account on the Yang-Lee theory will also be given as a set of basic rigorous results on phase transitions.

Keywords:   mathematical physics, Ising model, boundary condition, spherical model

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