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

Abstract:

We study the large-distance behavior of the bulk order-parameter correlation function for T > T _c within the lattice version of the theory including lattice effects. We also study the large-L behavior of the susceptibility for T > T _c of the confined lattice system of linear size L with periodic boundary conditions. We find that the structure of the large-L behavior of of the confined system is closely related to the structure of the large-distance behavior of of the bulk system. Explicit results are derived in the spherical (large-n) limit and in one-loop order for general dimensions d > 2. For the lattice model with cubic symmetry we find that finite-size scaling must be formulated in terms of the anisotropic bulk correlation length (exponential correlation length) that governs the exponential decay of for large r rather than in terms of the ordinary isotropic bulk correlation length defined via the second moment of . We show that it is the exponential bulk correlation length in the direction of the cubic axes that determines the exponential finite-size scaling behavior of lattice systems in a rectangular geometry. This result modifies a recent interpretation concerning an apparent violation of finite-size scaling in terms of the second-moment correlation length . Exact results for the one-dimensional Ising model illustrate our conclusions. Furthermore we show for general d>2 that a description of finite-size effects for finite n in the entire region requires different perturbative approaches that are applicable either to the region or , respectively. In particular we show that the exponential finite-size behavior for above T _c is not captured by the standard perturbation approach that separates the homogeneous lowest mode from the inhomogeneous higher modes. Consequences for the theory of finite-size effects above four dimensions are discussed. We show that the two-variable finite-size scaling form predicts an exponential approach to the bulk critical behavior above T _c whereas the reduction to a single-variable scaling form implies a power-law approach .