Finite-size scaling above the upper critical dimension.

We present a unified view of finite-size scaling (FSS) in dimension d above the upper critical dimension, for both free and periodic boundary conditions. We find that the modified FSS proposed some time ago to allow for violation of hyperscaling due to a dangerous irrelevant variable applies only to k=0 fluctuations, and "standard" FSS applies to k≠0 fluctuations. Hence the exponent η describing power-law decay of correlations at criticality is unambiguously η=0. With free boundary conditions, the finite-size "shift" is greater than the rounding. Nonetheless, using T-T(L), where T(L) is the finite-size pseudocritical temperature, rather than T-T(c), as the scaling variable, the data do collapse onto a scaling form that includes the behavior both at T(L), where the susceptibility χ diverges like L(d/2), and at the bulk T(c), where it diverges like L(2). These claims are supported by large-scale simulations on the five-dimensional Ising model.