Violation of finite-size scaling in three dimensions

Abstract:

We reexamine the range of validity of finite-size scaling in the ${\varphi ^4}$ lattice model and the ${\varphi ^4}$ field theory below four dimensions. We show that general renormalization-group arguments based on the renormalizability of the ${\varphi ^4}$ 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_cwhere $\xi$ 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 ${\varphi ^4}$ lattice model and the ${\varphi ^4}$ 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.