Finite size scaling of the 5D Ising model with free boundary conditions

TL;DR

This study uses large-scale simulations to investigate the finite size scaling of the 5D Ising model with free boundaries, providing evidence supporting the standard $L^2$ scaling over alternative theories.

Contribution

The paper offers the largest simulation data to date for the 5D Ising model with free boundaries, clarifying the finite size scaling behavior and reconciling it with theoretical results from the random-cluster model.

Findings

Supports the standard $L^2$ scaling for susceptibility.

Large system simulations up to size 160.

Provides theoretical explanation for boundary condition effects.

Abstract

There has been a long running debate on the finite size scaling for the Ising model with free boundary conditions above the upper critical dimension, where the standard picture gives a $L^2$ scaling for the susceptibility and an alternative theory has promoted a $L^{5/2}$ scaling, as would be the case for cyclic boundary. In this paper we present results from simulation of the far largest systems used so far, up to side $L=160$ and find that this data clearly supports the standard scaling. Further we present a discussion of why rigorous results for the random-cluster model provides both supports the standard scaling picture and provides a clear explanation of why the scalings for free and cyclic boundary should be different.