Eur. Phys. J. B 10, 687-703 (1999)
https://doi.org/10.1007/s100510050901

Violation of finite-size scaling in three dimensions

¹ Institut für Theoretische Physik, Technische Hochschule Aachen, 52056 Aachen, Germany
² Institute of Particle Physics, Hua-Zhong Normal University, Wuhan 430079, PR China

Corresponding author: ^a vdohm@physik.rwth-aachen.de

Received: 5 February 1999
Published online: 15 August 1999

Abstract

We reexamine the range of validity of finite-size scaling in the φ⁴ lattice model and the φ⁴field theory below four dimensions. We show that general renormalization-group arguments based on the renormalizability of the φ⁴ theory do not rule out the possibility of a violation of finite-size scaling due to a finite lattice constant and a finite cutoff. For a confined geometry of linear size L with periodic boundary conditions we analyze the approach towards bulk critical behavior as L->∞ at fixed ξ for T>T_c where ξ is the bulk correlation length. We show that for this analysis ordinary renormalized perturbation theory is sufficient. On the basis of one-loop results and of exact results in the spherical limit we find that finite-size scaling is violated for both the φ⁴ lattice model and the φ⁴ field theory in the region L>>ξ. The non-scaling effects in the field theory and in the lattice model differ significantly from each other.