Finite-size scaling above the upper critical dimension

TL;DR

This paper provides a unified analysis of finite-size scaling above the upper critical dimension, clarifying the applicability of modified FSS and presenting simulation evidence from the 5D Ising model.

Contribution

It demonstrates that the modified FSS applies only to k=0 fluctuations and introduces a scaling variable T-T_L that unifies behavior at pseudocritical and critical temperatures.

Findings

Finite-size shift exceeds rounding with free boundary conditions.

Scaling collapse achieved using T-T_L as variable.

Susceptibility diverges as L^{d/2} at T_L and as L^2 at T_c.

Abstract

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 so there is only a single exponent eta describing power-law decay of correlations at criticality, in contrast to recent claims. 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 does collapse on to a scaling form which 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 5-dimensional Ising model.