Chapter

Statistical field theory

Hidetoshi Nishimori and Gerardo Ortiz

in Elements of Phase Transitions and Critical Phenomena

Published in print December 2010 | ISBN: 9780199577224
Published online January 2011 | e-ISBN: 9780191722943 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199577224.003.0005

Series: Oxford Graduate Texts

Statistical field theory

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Statistical-mechanical systems often involve discrete elementary degrees of freedom such as spins in the Ising model. Field theories, on the other hand, have continuous fields, defined over the whole space-time or part of it, as fundamental degrees of freedom. These two seemingly different descriptions of physical phenomena can be related close to the critical point. The present chapter summarizes how the description by continuous fields emerges from discrete degrees of freedom in a more systematic manner than in previous chapters. The phenomenological Landau-Ginzburg approach, based on the concept of order parameter, is expanded to generate effective field theories. The important roles of symmetry and topology are also elucidated in some detail. Also shown are some of the important consequences of having a broken-symmetry phase, such as long-range order, the emergence of Nambu-Goldstone modes when the symmetry involved is continuous, and topological defects.

Keywords: field theory; critical point; symmetry; order parameter; broken symmetry; topology; topological defects

Chapter.  11471 words.  Illustrated.

Subjects: Mathematical and Statistical Physics

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