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Quantum Field Theory and Critical Phenomena$
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Jean Zinn-Justin

Print publication date: 2002

Print ISBN-13: 9780198509233

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198509233.001.0001

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Field Theory in a Finite Geometry: Finite Size Scaling

Field Theory in a Finite Geometry: Finite Size Scaling

Quantum Field Theory and Critical Phenomena


Oxford University Press

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

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