Violation of Finite-Size Scaling in Three Dimensions

TL;DR

This paper investigates the limits of finite-size scaling in three-dimensional $\,\phi^4$ models, revealing violations in the scaling behavior for large system sizes relative to the correlation length, using perturbation theory and exact results.

Contribution

It demonstrates that finite-size scaling can be violated in 3D $\,\phi^4$ models due to finite lattice constants and cutoffs, challenging previous assumptions based on renormalizability.

Findings

Finite-size scaling is violated for $L \,\gg\, \xi$ in both lattice and field theory models.

Violations differ significantly between the lattice model and the continuum field theory.

One-loop perturbation theory and exact spherical limit results support the violation conclusion.

Abstract

We reexamine the range of validity of finite-size scaling in the $\phi^4$ lattice model and the $\phi^4$ field theory below four dimensions. We show that general renormalization-group arguments based on the renormalizability of the $\phi^4$ 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 $L \to \infty$ at fixed $\xi$ for $T > T_c$ where $\xi$ 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 $\phi^4$ lattice model and the $\phi^4$ field theory in the region $L \gg \xi$. The non-scaling effects in the field theory and in the lattice model differ significantly from each other.