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 not equal 0 fluctuations. Hence the exponent eta describing power-law decay of correlations at criticality is unambiguously eta = 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 chi 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.