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

P.H Lundow

doi:10.1016/j.nuclphysb.2021.115422

Abstract

High-dimensional (d≥5) Ising systems have mean-field critical exponents. However, at the critical temperature the finite-size scaling of the susceptibility χ depends on the boundary conditions. A system with periodic boundary conditions then has χ∝L5/2. Deleting the 5L4 boundary edges we receive a system with free boundary conditions and now χ∝L2. In the present work we find that deleting the L4 boundary edges along just one direction is enough to have the scaling χ∝L2. It also appears that deleting L3 boundary edges results in an intermediate scaling, here estimated to χ∝L2.275. We also study how the energy and magnetisation distributions change when deleting boundary edges.

Highlights

The 5-dimensional (5D) Ising model is well-known to have mean-field critical exponents so that, for example, α = 0, γ = 1 and ν = 1/2
Assuming the usual finite-size scaling (FSS) rules this would imply that the susceptibility scales as χ ∝ Lγ /ν = L2 near the critical point βc, where L is the linear order of the system
We have investigated several scenarios of deleting boundary edges from a periodic (cyclic) boundary conditions (PBC)-system, where an free boundary conditions (FBC)-system corresponds to deleting 5L4 edges

Summary

Introduction

[5] was it suggested on theoretical, if non-rigorous, grounds that for free boundary conditions (FBC) the rule χ ∝ L2 holds, for d ≥ 5. We start with a PBC-system and delete, for example, all boundary edges along one or more, say r, dimensions This means we delete rL4 edges and, as we will see, this is enough to change the scaling behaviour of χ to that typical of an FBC-system. Deleting rL3 edges seems to give a scaling behaviour between that of PBC and FBC, suggesting χ ∝ L2.275.

Definitions and details

Scaling of susceptibility

Magnetisation distribution

Scaling of energy quantities

Findings

Conclusion