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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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(p.235) 10 Duality
Elements of Phase Transitions and Critical Phenomena

Hidetoshi Nishimori

Gerardo Ortiz

Oxford University Press

Exact solutions of model systems are the most reliable source of information. However, there are not many model systems that can be exactly solved and, consequently, many approximate schemes such as series expansions have been developed. It is sometimes possible, though, to extract exact information without directly solving those model systems. Arguments using duality transformations make it possible to derive the exact location of the phase transition point and the exact value of the energy at the transition point when the model is self dual. Duality not only determines the exact location of the transition point of the two-dimensional Ising model and related models but also is useful to rewrite the $XY$ model into a different form, the Villain model and its dual, which reveals new physical aspects of the system. Classical and quantum dualities can be related by certain mappings and Fourier transformation is at the root of such deep results.

Keywords:   classical duality, quantum duality, series expansions, Fourier transform, Villain model

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