Finite size scaling of the correlation length above the upper critical dimension

TL;DR

This paper demonstrates that in five-dimensional Ising models, the correlation length at criticality scales with system size as L^{5/4}, confirming a modified finite size scaling theory for dimensions above the upper critical dimension.

Contribution

The study provides numerical evidence that finite size scaling in dimensions above four requires replacing L with L^{d/4}, refining the understanding of critical phenomena in high dimensions.

Findings

Correlation length scales as L^{5/4} in 5D Ising model

FSS expressions in d>4 involve L^{d/4} instead of L

Logarithmic corrections to FSS are analyzed in 4D

Abstract

We show numerically that correlation length at the critical point in the five-dimensional Ising model varies with system size L as L^{5/4}, rather than proportional to L as in standard finite size scaling (FSS) theory. Our results confirm a hypothesis that FSS expressions in dimension d greater than the upper critical dimension of 4 should have L replaced by L^{d/4} for cubic samples with periodic boundary conditions. We also investigate numerically the logarithmic corrections to FSS in d = 4.