Boundary effects on finite-size scaling for the 5-dimensional Ising model

TL;DR

This paper investigates how different boundary conditions affect the finite-size scaling of susceptibility in 5-dimensional Ising models, revealing that boundary modifications significantly alter critical scaling behavior.

Contribution

It demonstrates that deleting boundary edges along specific directions changes the susceptibility scaling exponent in high-dimensional Ising models.

Findings

Deleting boundary edges along one direction yields susceptibility scaling as L^2.

Removing all boundary edges results in susceptibility scaling as L^{5/2}.

Intermediate boundary deletions lead to intermediate scaling exponents around L^{2.275}.

Abstract

High-dimensional ($d\ge 5$) Ising systems have mean-field critical exponents. However, at the critical temperature the finite-size scaling of the susceptibility $\chi$ depends on the boundary conditions. A system with periodic boundary conditions then has $\chi\propto L^{5/2}$. Deleting the $5L^4$ boundary edges we receive a system with free boundary conditions and now $\chi\propto L^2$. In the present work we find that deleting the $L^4$ boundary edges along just one direction is enough to have the scaling $\chi\propto L^2$. It also appears that deleting $L^3$ boundary edges results in an intermediate scaling, here estimated to $\chi\propto L^{2.275}$. We also study how the energy and magnetisation distributions change when deleting boundary edges.