Chapter

Field Theory in a Finite Geometry: Finite Size Scaling

JEAN ZINN-JUSTIN

in Quantum Field Theory and Critical Phenomena

Fourth edition

Published in print June 2002 | ISBN: 9780198509233
Published online January 2010 | e-ISBN: 9780191708732 | DOI: https://dx.doi.org/10.1093/acprof:oso/9780198509233.003.0037

Series: International Series of Monographs on Physics

Field Theory in a Finite Geometry: Finite Size Scaling

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Many numerical calculations, like Monte Carlo or transfer matrix calculations, are performed with systems in which the size in several or all dimensions is finite. To extrapolate the results to the in infnite system, it is thus necessary to have some idea about how the infinite size limit is reached. In particular, in a system in which the forces are short range no phase transition can occur in a finite volume, or in a geometry in which the size is infinite only in one dimension. This indicates that the infinite size extrapolation is somewhat non-trivial. This chapter presents an analysis of the problem in the case of second order phase transitions, in the framework of the N-vector model. It first establishes the existence of a finite size scaling, extending RG arguments to this new situation. It then distinguishes between the finite volume geometry and the cylindrical geometry in which the size is finite in all dimensions except one. It explains how to modify the methods used in the case of in infinite systems to calculate the new universal quantities appearing in finite size effects, for example, in d = 4 - ε or d = 2 + ε dimensions. Special properties of the commonly used periodic boundary conditions are emphasized. Finally, both static and dynamical finite size effects are described.

Keywords: transfer matrix calculations; finite volume geometry; Monte Carlo calculations; finite size effects

Chapter.  10483 words. 

Subjects: Mathematical and Statistical Physics

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