Relation between bulk order-parameter correlation function and finite-size scaling

Abstract

We study the large-$r$ behavior of the bulk order-parameter correlation function $G(\bf{r})$ for $T>T_c$ within the lattice $ϕ^4$ theory. We also study the large-$L$ behavior of the susceptibility $χ$ of the confined lattice system of size $L$ with periodic boundary conditions. The large-$L$ behavior of $χ$ is closely related to the large-$r$ behavior of $G(\bf{r})$. Explicit results are derived for $d>2$. Finite-size scaling must be formulated in terms of the anisotropic exponential correlation length $ξ_1$ that governs the decay of $G(\bf{r})$ for large $r$ rather than in terms of the isotropic correlation length $ξ$ defined via the second moment of $G(\bf{r})$. This result modifies a recent interpretation concerning an apparent violation of finite-size scaling in terms of $ξ\neq ξ_1$. Exact results for the $d=1$ Ising model illustrate our conclusions. Furthermore, we show that the exponential finite-size behavior for $L/ξ\gg 1$ is not captured by the standard perturbation approach that separates the lowest mode from the higher modes. Consequences for the theory of finite-size effects for $d>4$ are discussed. The two-variable finite-size scaling form predicts an approach $\propto e^{-L/ξ_1}$ to the bulk critical behavior whereas a single-variable scaling form implies a power-law approach $\propto L^{-d}$.